A122 === Subject: Zoomable ListPlot I made a small function to produce a zoomable listplot, hold alt down and draw a rect to zoom, shift click to return. I'm not very familiar with the eventhandlers, maybe someone else can do this much better, without redrawing the plot as many times? I would also like to be able to turn data series on and off by clicking the legend, but I have no idea on how to do that... PListPlot[fun_, opts : OptionsPattern[{Options[ListPlot], Legend -> {}, Frame -> True, ColorScheme -> 1}]] := DynamicModule[{p1, p2, pr = Automatic, pt = {{0, 0}, {0, 0}}, pl = {}, mylegend, is, style}, mylegend[data_] := Framed[Grid[ MapIndexed[{Item[ Graphics[{ColorData[OptionValue[ColorScheme], First@#2], Thick, Line[{{0, 0}, {1, 0}}]}, AspectRatio -> .1, ImageSize -> 30], Alignment -> {Center, Center}], Item[Style[#1, FontSize -> 16, FontFamily -> Helvetica], Alignment -> {Left, Bottom}]} &, data]], Background -> RGBColor[1, 1, 0.99999]]; is[x_] := {x, x/GoldenRatio}; EventHandler[ Dynamic[ ListPlot[fun, PlotRange -> pr, Epilog -> {pl, Inset[mylegend[OptionValue[Legend]], ImageScaled[{1 - 0.02, 1 - 0.02}], {Right, Top}]}, Evaluate@ FilterRules[{opts}, Options[ListPlot]], MaxPlotPoints -> ControlActive[1000, Infinity]], SynchronousUpdating -> Automatic], MouseDown :> (If[CurrentValue[AltKey], p1 = MousePosition[Graphics]]; If[CurrentValue[ShiftKey], pr = Automatic]; ), MouseUp :> (If[CurrentValue[AltKey], p2 = MousePosition[Graphics]; pt[[1, 1]] = p1[[1]]; pt[[1, 2]] = p2[[1]]; pt[[2, 1]] = p1[[2]]; pt[[2, 2]] = p2[[2]]; pr = pt; pl = {}; ] ), MouseDragged :> (If[CurrentValue[AltKey], p2 = MousePosition[Graphics]; pt[[1, 1]] = p1[[1]]; pt[[1, 2]] = p2[[1]]; pt[[2, 1]] = p1[[2]]; pt[[2, 2]] = p2[[2]]; pl = {Line[{p1, {p2[[1]], p1[[2]]}, p2, {p1[[1]], p2[[2]]}, p1}]}; ]; ) , PassEventsDown -> True ] ] === Subject: Text-based interface: Editing line input I have been enjoying mathematica, for a bit. While some might consider me a minimalist, I enjoy the text-based interface over the notebook style usage. However I severely miss the ability to move the cursor left and right for editing a single line. The documentation itself suggests pasting lines, but most of my pasting requires a little editing too. So instead of pasting whole lines of input, I resort to partial line paste, adding different text, then pasting the remaining relevant portion of a line. Is there any way to be able to freely move the cursor on a line? I have obviously tried the arrow keys, and also have tried emacs style movements as well with no luck (ctrl-b ctrl-f for backwards and forewards). Any suggestions? === Subject: ListContourPlot3D, no output I am trying to use ListContourPlot3D on a table of the form {{x,y,z,f},...}. Execution does not give any error number and I get only a system of axis, with all the ranges {0,1}. I made sure that the table is properly formatted. I am really puzzled. Adi === Subject: Alternating sums of large numbers I'm using a Mathematica 6, and I've faced with the following problem. Here is a copy of my Mathematica notebook: -------------------------------------------------------- NN = 69.5; n = 7; coeff = (Gamma[2*NN - n + 1]*(2*NN - 2*n))/n!; f[z_] := Sum[((-1)^([Mu] + [Nu])*Binomial[n, [Mu]]*Binomial[n, [Nu]]*Gamma[2*NN - 2*n + [Mu] + [Nu], z])/(Gamma[2*NN - 2*n + [Mu] + 1]*Gamma[2*NN - 2*n + [Nu] + 1]), {[Mu],0, n}, {[Nu], 0, n}]; Plot[coeff*f[z], {z, 0, 100}] -------------------------------------------------------- As you can see, I want to calculate a double alternating sum, consisting of large terms (products of Gamma-functions and binomial coefficients). Then I want to plot the result, in dependence on parameter z, which takes part in the summation as an argument of the incomplete Gamma-function, Gamma[2*NN - 2*n + [Mu] + [Nu], z]. Apart from this, I have another parameter, n, which is an upper limit for both of sums, and also takes part in Gamma functions. When this parameter grows, the expression next to summation also increases. At some point, Mathematica begins to show very strange results - and my question is actually about this. For instance, if the parameter n=5, everything is O.K., the plot shows a smooth curve. When we set n=6, there appears a little noise at 60 Hello everybody, > I have a list of {x,y} data and from a theoretical calculation I kno= w > that the data should be fitted by an equation, for example: > y=ax+bx+c; where a, b, c are constant parameters. > The question is how i can find the best fit for my data finding the > value of this parameters. > Do mathematica has a specific function for that? Try searching the documentation for fitting ... the first result, Fit, > does exactly what you describe here. when I try to do the FindFit command, the parameters have to be positive, so i was searching for the optimal values of the parameter that fit the data. How I can modify my code in order to find the best parameters fitting the data?I tried to do the Norm between the value of the data and the value of the equation but i cant do more. Remove[Global`*] data = {{1, 1}, {28, 0.719188377}, {54, 0.35746493}, {81, 0.182114228}, {117, 0.166082164}, {260, 0.132765531}}; express = (1 - k*x)*(1 - k*x/q)*(1 - p*k*x/q) this is the equation with which i want to fit the data (1 - k x) (1 - (k x)/q) (1 - (k p x)/q) {xx, yy} = data[Transpose] {{1, 28, 54, 81, 117, 260}, {1, 0.719188, 0.357465, 0.182114, 0.166082, 0.132766}} Try1 f1 = FindFit[ data, (1 - k*x)*(1 - k*x/q)*(1 - p*k*x/q), {{k, 0.01}, {p, 1.5}, {q, 1}}, x, MaxIterations -> 200] {k -> 0.00586032, p -> 2.86841, q -> 2.86841} here we have the parameters fitting the curve expr1 = express /. f1 (1 - 0.00586032 x) (1 - 0.00586032 x) (1 - 0.00204306 x) yyfit = expr1 /. x -> xx {0.986295, 0.658775, 0.415684, 0.230288, 0.0751917, 0.128567} yy - yyfit {0.0137055, 0.0604133, -0.0582186, -0.0481737, 0.0908904, 0.00419858} Norm[%] 0.133516 Norm[yy - ((express /. f1) /. x -> xx)] 0.133516 f11 = {k, p, q} /. Table[FindFit[ data, (1 - k*x)*(1 - k*x/q)*(1 - p*k*x/q), {{k, k0}, {p, 1.5}, {q, 1}}, x, MaxIterations -> 200], {k0, 0.01, 1, 0.1}] // TableForm f11 = Table[ FindFit[data, (1 - k*x)*(1 - k*x/q)*(1 - p*k*x/q), {{k, k0}, {p, 1.5}, {q, here i am trying with k from 0.01 to 1 1}}, x, MaxIterations -> 200], {k0, 0.01, 1, 0.1}] Norm[yy - ((express /. #) /. x -> xx)] & /@ f11 {0.133516, 0.133516, 0.133516, 0.133516, 0.162362, 0.133516, 0.162362, 0.133516, 0.133516, 0.133516} % // Min 0.133516 Here i find the minimum for Norm and find the position at which i have the best value of the fitting Position[%%, %] {{4}, {6}, {8}} % // Union {0.133516, 0.133516, 0.133516, 0.133516, 0.162362} === Subject: Transformations in expressions Hi all, How can I to transform a specific expression to other by substitution of a part of it. For example, I want to transform the expression Log[x^2+y^3]/ Sin[x^2+y^3] in Log[z]/Sin[z] by substitution x^2+y^3 --> z. And other question: Is the new expression dependent of z. I see it is, but how can I to programar it. === Subject: Re: Debugger > Welcome to the club! > I gave up on debugger a long time ago. > WRI has developed a (supposedly) good functionality, but they did not > bother to document it well. > Why should we, the users, waste time trying to figure out how it > works? > You will find some explanations (by WRI people) on this group about > Debug, but not much. > I suggest you use ON[] and Off[] and sprinkle the code you are testing > with Print commands to see the value of variables. > This way you clearly see the behind-the-scenes actions of your code. > Trace gives you the same information, but the output is not so well > formatted and is difficult to read. > Workbench is supposed to be the Integrated development environment > (IDE) for Mathematica, but well-known mathgroupers have told me that - at the > moment - it is more an headhache, rather than an Aspirin. > I too gave up on the debugger a long time ago for all the reasons debugger based on TraceScan, and described it at a developer conference. Because it had to execute many Mathematica steps for every step of the program, it was unfortunately rather slow, but it had more features than the built-in offering, and was used by quite a few people. Among other extra features: 1) It recognised that certain 'steps' should be skipped. For example, rather than stopping on a complicated CompoundExpression construct, it skipped that step and went to the first part of the CE. This made it much more practical to use. 2) It kept a step count, which it would print out in various circumstances. The idea was, that if you performed one debug session and got too far, you could note the step count, and debug the program again up to, say count-100. Again, this made it feasible to debug far more complicated examples. 3) There were various tools to let the user hide certain parts of the code from the debugger. For example, you could specify that the debugger should not step into code that was not in Global` context. I hope WRI make the notebook debugger into a real feature for the next release. I don't really want to use the Workbench just to get a better version of the debugger that is meant to be present in the notebook interface! David Bailey http://www.dbaileyconsultancy.co.uk === Subject: Problems with Frame in ListPlots with Legends I have a problem with using Frame in a ListPlot if it has a legend. Needs[PlotLegends] a = ListPlot[lowData, Joined -> True, PlotRange -> {{1, 3}, {.5, 1}}, PlotStyle -> {Hue[0], Hue[1/3], Hue[2/3]} ] b = ListPlot[lowData, Joined -> True, Frame -> {{True, False}, {True, False}}, FrameLabel -> {X, Y}, PlotRange -> {{1, 3}, {.5, 1}}, PlotStyle -> {Hue[0], Hue[1/3], Hue[2/3]} ] Plot a and Plot b both work. Plot b has a frame and frame labels. I can add a legend to Plot a. Adding a legend to Plot b works, but gives errors. l = {{{Hue[0], 1}, {Hue[1/3], 2}, {Hue[2/3], 3}}, LegendPosition -> {1, 0}, LegendSize -> {0.5, 0.5}}; ShowLegend[a, l] (* This works *) ShowLegend[a, l] (* Makes a plot with a legend, but gives errors *) The errors are a set of Axes::axes: {{True,False},{True,False}} is not a valid axis specification. I can't work out what is going on. It works with Frame->True, but I only want axes on the left and at the bottom, and I need frame labels. It is a shame that the PlotLegend option does not seem to work with ListPlot, although the documentation says it should. Any help much appreciated. Neil. === Subject: Simulate a finite-state markov process I've seen a message in comp.soft-sys.math.mathematica : I found it very interesting and tried to develop a similar program using the same idea. The problem is that I'm not that good with programming... Suppose I have the following Emission matrix: {{0.470949,0.260239,0.268812},{0.529412,0.470588,0},{0.625,0,0.375}, {0.722222,0,0.277778},{0.446667,0.24,0.313333},{0.5,0.5,0}, {0.481301,0.252846,0.265854},{0.471374,0.257634,0.270992}, {0.47293,0.269108,0.257962},{0.52027,0.239865,0.239865}, {0.595238,0.214286,0.190476},{0.444444,0.296296,0.259259}, {0.478528,0.263804,0.257669},{0.507874,0.233071,0.259055}, {0.434251,0.308869,0.256881},{0.504274,0.262108,0.233618}, {0.419753,0.320988,0.259259},{0.473684,0.315789,0.210526}, {0.512012,0.231231,0.256757},{0.5,0.19469,0.30531}, {0.390244,0.195122,0.414634},{0.510949,0.19708,0.291971}, {0.464286,0.25,0.285714},{0.285714,0,0.714286}} The alphabet is {A,B,C} And the following Transition Matrix: {{0.470949,0,0,0,0,0,0.260239,0,0,0,0,0,0,0.268812,0,0,0,0,0,0,0,0,0,0}, {0.529412,0,0,0,0,0,0,0.470588,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0.625,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0.375,0,0,0,0,0,0,0,0}, {0.722222,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0.277778,0,0,0,0,0,0}, {0.446667,0,0,0.24,0,0,0,0,0,0,0,0,0,0,0,0.313333,0,0,0,0,0,0,0,0}, {0.5,0,0,0,0.5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0.481301,0,0,0,0,0,0,0,0.252846,0,0,0,0,0,0.265854,0,0,0,0,0,0,0,0,0}, {0.471374,0,0,0,0,0,0,0,0,0.257634,0,0,0,0,0,0.270992,0,0,0,0,0,0,0,0}, {0.47293,0,0,0,0,0,0,0,0,0,0,0,0.269108,0,0,0,0.257962,0,0,0,0,0,0,0}, {0.52027,0,0,0,0,0,0,0,0,0,0,0,0.239865,0,0,0,0,0.239865,0,0,0,0,0,0}, {0.595238,0,0,0,0.214286,0.190476,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0.444444,0.259259,0,0,0,0,0,0,0.296296,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}, {0.478528,0,0,0,0,0,0,0,0,0,0.257669,0,0.263804,0,0,0,0,0,0,0,0,0,0,0}, {0.507874,0,0,0,0,0,0,0.233071,0,0,0,0,0,0,0,0,0,0,0.259055,0,0,0,0,0}, {0.434251,0,0,0,0,0,0,0.308869,0,0,0,0,0,0,0,0,0,0,0,0.256881,0,0,0,0}, {0.504274,0,0,0,0,0,0,0.262108,0,0,0,0,0,0,0,0,0,0,0,0,0.233618,0,0,0}, {0.419753,0,0,0,0.320988,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0.259259,0,0,0,0}, {0.473684,0,0,0,0.315789,0,0,0,0,0,0,0,0,0,0,0,0,0,0.210526,0,0,0,0,0}, {0.512012,0,0,0,0,0,0,0.231231,0,0,0,0,0,0,0,0,0,0,0,0,0,0.256757,0,0}, {0.5,0,0,0,0.19469,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0.30531,0,0}, {0.390244,0,0.195122,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0.414634,0,0}, {0.510949,0,0,0,0,0,0,0,0,0,0,0.19708,0,0,0,0,0,0,0,0,0,0,0.291971,0}, {0.464286,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0.285714,0.25}, {0.285714,0.714286,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}} How can I develop a program to generate x sequences of a specified length? By the way, does anyone know how to calculate the probability of a specific sequence to occur within n generated sequences?? Edson Ferreira === Subject: Re: How to simplify ArcCos[x/Sqrt[x^2+y^2]] to Pi/2-ArcTan[x/Abs[y]]? >> I don't think there is a way to simplify one of these expressions >> into the >> other, but one can use Mathematica as an aid in proving that they >> are equal >> (for real x and y). One way to do this is: If this is case, I therefore might have to rely on hand calculation. > But I would think Mathematica should be improved so that mathematical > expressions can be manipulated in more versatile ways. Peng I always agree that Mathematica should be improved (although perhaps I put the limit the point where it would start putting human mathematicians out of their jobs). However, that is easy to say, the hard thing is to find the right algorithms and to implement them. Algorithmic simplification of expressions is a very complex problem and only for certain types of expressions (like polynomials, rational functions, exponentials, logs and a few others) sufficiently general methods exist (and this ignores the not exactly trivial problem of defining when one expression is to be considered simpler than another). But the situation is much worse when you wish to convert from one form to another an expression, when the conversion is only valid under certain assumptions. If you can suggest a concrete general algorithm that you think should be implemented in Mathematica and is not implemented now I am sure WRI will be happy to oblige but just saying it should be improved isn't very helpful. You can be sure that there are people who are trying to do that all the time. Andrzej Kozlowski === Subject: Re: How to simplify ArcCos[x/Sqrt[x^2+y^2]] to Pi/2-ArcTan[x/Abs[y]]? I don't think there is a way to simplify one of these expressions into the other, but one can use Mathematica as an aid in proving that they are equal (for real x and y). One way to do this is: expres = {ArcCos[x/Sqrt[x^2 + y^2]], Pi/2 - ArcTan[x/Abs[y]]}; Subtract @@ Assuming[Element[x | y, Reals], Simplify[Sin /@ expres]] 0 Subtract @@ Assuming[Element[x | y, Reals], Simplify[Cos /@ expres]] 0 So for real x and y, the two expressions have equal sines and cosines. That means that they must differ by an integer multiple of 2Pi. However, since the difference is a continuous function of x and y, it has to be constant. Now, putting (for example) x=0, y=1, we see that the constant must be zero, hence they are equal. Andrzej Kozlowski ArcCos[x/Sqrt[x^2+y^2]] and Pi/2-ArcTan[x/Abs[y]] are the same. But I can not get the first expression be simplified to the second > one. And the following command can not be simplified to zero in > mathematica as well. FullSimplify[ArcCos[x/Sqrt[x^2 + y^2]] - ( Pi/2 - ArcTan[x/Abs[y]]), > Element[x, Reals] && Element[y, Reals]] I'm wondering if there is any walkaround to do this simplification. Peng > === Subject: Re: Problem with replacement rules > I use Solve[expression(x,y),x]. Then, x is obtained as a one-to-one function of y. I want to define the function x(y) using the output of Solve. How should I do? In[1]:= (* Solve the expression for x and save the result in sol *) sol = Solve[a * x + y == 7, x] (* Define x as a function of y *) x[y_] = x /. sol[[1]] (* Use the function *) x[10] Out[1]= {{x -> (7 - y)/a}} Out[2]= (7 - y)/a Out[3]= -(3/a) -- Jean-Marc === Subject: Re: Problem with replacement rules makeFunction[expr_, x_] := Module[{var}, var = Complement[Variables[expr], {x}]; Function @@ {var, x /. Solve[expr == 0, x][[1]]} ] and f = makeFunction[2 + x - y^2, x]; Plot[f[z], {z, -2, 2}] ?? > I use Solve[expression(x,y),x]. Then, x is obtained as a one-to-one function of y. I want to define the function x(y) using the output of Solve. How should I do? > === Subject: Re: How can I get list form like this expr = Part[f, #] & /@ Range[5] // Quiet % /. f -> {a, b, c, d, e} {f[[1]], f[[2]], f[[3]], f[[4]], f[[5]]} {a, b, c, d, e} -- David Park djmpark@comcast.net http://home.comcast.net/~djmpark/ As we know f/@Range[n] gives a list {f[1],f[2],f[3],=85=85}; > Here what I want is a list form like this{f[[1]],f[[2]],=85=85}(*It's too > tedious to type one by one*) > It means the 1st,2nd,=85=85part of list f. > Looks similar to each other, however I can't figure it out. > How to do this ? === Subject: Re: How can I get list form like this > As we know f/@Range[n] gives a list {f[1],f[2],f[3],=85=85}; > Here what I want is a list form like this{f[[1]],f[[2]],=85=85}(*It's too > tedious to type one by one*) > It means the 1st,2nd,=85=85part of list f. > Looks similar to each other, however I can't figure it out. > How to do this ? Map a pure function such as f[[#]]&. For instance, depending on whether the expression holds some values before the mapping or not, In[1]:= f = {1.1, 1.2, 1.3}; (f[[#1]] & ) /@ Range[3] Out[2]= {1.1, 1.2, 1.3} In[3]:= (HoldForm[g[[#1]]] & ) /@ Range[3] g = {1, 2, 3}; ReleaseHold[%%] Out[3]= {HoldForm[g[[1]]], HoldForm[g[[2]]], HoldForm[g[[3]]]} Out[5]= {1, 2, 3} -- Jean-Marc === Subject: Re: How can I get list form like this Hold[Part[##]] & @@@ Table[{f, i}, {i, 1, 5}] As we know f/@Range[n] gives a list {f[1],f[2],f[3],=85=85}; > Here what I want is a list form like this{f[[1]],f[[2]],=85=85}(*It's too > tedious to type one by one*) > It means the 1st,2nd,=85=85part of list f. > Looks similar to each other, however I can't figure it out. > How to do this ? === Subject: Re: How can I get list form like this Hi. The answer is Quiet[f[[#]] & /@ Range[10]] but will it really works as you want? === Subject: Re: How can I get list form like this Part[f, #] & /@ Range[85] returns a list in the form you requested, but produces warnings because f is not defined. If you have f defined, you will get the parts. As we know f/@Range[n] gives a list {f[1],f[2],f[3],=85=85}; > Here what I want is a list form like this{f[[1]],f[[2]],=85=85}(*It's= too > tedious to type one by one*) > It means the 1st,2nd,=85=85part of list f. > Looks similar to each other, however I can't figure it out. > How to do this ? === Subject: Re: How can I get list form like this As we know f/@Range[n] gives a list {f[1],f[2],f[3],=85=85}; > Here what I want is a list form like this{f[[1]],f[[2]],=85=85}(*It's too > tedious to type one by one*) > It means the 1st,2nd,=85=85part of list f. > Looks similar to each other, however I can't figure it out. > How to do this ? > You _could_ write f[[#]]& /@ Range[n], but Take[f,n] is shorter and most propably faster. Peter === Subject: Re: How can I get list form like this As we know f/@Range[n] gives a list {f[1],f[2],f[3],=85=85}; > Here what I want is a list form like this{f[[1]],f[[2]],=85=85}(*It's too > tedious to type one by one*) > It means the 1st,2nd,=85=85part of list f. > Looks similar to each other, however I can't figure it out. > How to do this ? > If you need to slice lists, use one of the following: f[[ Range[n] ]] f[[ 1;;n ]] f[[ ;;n ]] (The last two will only work in Mathematica 6) === Subject: Re: How can I get list form like this Use a simple pure function: f[[##]]&/@Range[n] Look at the command Function in the help file for more detail about this construction. Januk As we know f/@Range[n] gives a list {f[1],f[2],f[3],=85=85}; > Here what I want is a list form like this{f[[1]],f[[2]],=85=85}(*It's too > tedious to type one by one*) > It means the 1st,2nd,=85=85part of list f. > Looks similar to each other, however I can't figure it out. > How to do this ? === Subject: Re: Comparison of coefficients Coefficients[]. === Subject: Re: Problem with replacement rules Clear[x]; x[y_] = x /. Solve[y == x/a + b, x][[1]] -a (b-y) x[2] -a (b-2) I use Solve[expression(x,y),x]. Then, x is obtained as a one-to-one function of y. I want to define the function x(y) using the output of Solve. How should I do? -- === Subject: Re: How can I get list form like this f = a + b + c + d + e; f[[#]] & /@ Range[Length[f]] {a,b,c,d,e} or more simply List @@ f {a,b,c,d,e} Or, if as you state, f is a List, then a list of its parts is just f f = {a, b, c, d, e} {a,b,c,d,e} (f[[#]] & /@ Range[Length[f]]) == List @@ f == f True As we know f/@Range[n] gives a list {f[1],f[2],f[3],=85=85}; Here what I want is a list form like this{f[[1]],f[[2]],=85=85}(*It's too tedious to type one by one*) It means the 1st,2nd,=85=85part of list f. Looks similar to each other, however I can't figure it out. How to do this ? === Subject: Re: Trying to speed up a function Hi Abhishek, no mistery here. By definition Table[expr,{n}] completely evaluates expr n times. Therefore, it is a good idea to take out redundant calculations. Note also that Table has the attribute HoldAll and does therefore not evaluate its arguments. Daniel > Say I have a previously defined matrix 'vecs' and I want to calculate > certain sums of columns. I find it easier to write in terms of the > Transpose but it seems that when I put Transpose[vecs] in the > definition of a table, mathematica evaluates the transpose over and > over to get at every element of the matrix. Which takes much longer > than doing it in 2 steps (2nd method). I only found this out by trial > and error so I am curious about the explanation for this behaviour and > how to avoid it in general. BTW I am aware that the expression can be written in 1 line without > using Transpose also, but I'd still like to know how to tell that if a > sub-expression in a function is going to be evaluated repeatedly. > Abhishek In[15]:= Dimensions[vecs] Out[15]= {1620, 1620} In[23]:= Table[Total[Transpose[vecs][[n]][[811 ;; 1620]]], {n, 1, > 1620}]; // Timing Out[23]= {27.844, Null} In[24]:= vecst = Transpose[vecs]; > Table[Total[vecst[[n]][[811 ;; 1620]]], {n, 1, 1620}]; // Timing Out[25]= {0.016, Null} Internet: === Subject: Re: Error in file / load entire notebook into memory mathematica files are XML I think; i.e. they're text with extra kind-of flags in it. So the file will open using a good text editor. It should be possible to identify what is input and what is output. So what I'd do is eliminate the output then try to fix the input so that it looks right compared with another valid mathematica file. Then save and run the file. I've done this before with a student's exam file that got corrupted - it works. You're still left with the versions 5/6 issue however. good luck Peter 2008/9/9 Adam Weyhaupt : > I am teaching Calc I this semester and am having my students do > Mathematica labs. The labs are written in Mathematica 6, but we have both > Mathematica 6 and Mathematica 5.2 on campus, so I can't guarantee that the labs were never opened in 5.2. On a few (about 3 files out of 60), I am > getting the following error. This occurs in a pop up dialog box > after scrolling down through the file. There appears to be an error in the file named below. You should > close it without saving, then check the Load entire notebook into > memory checkbox in the Notebook Options dialog. File name: /Users/ > adam/Desktop/downloads/lab1_exercises_1_-2.nb. On two of these labs, the file seemed to be badly damaged --- I > couldn't read most of it. On a third, I can see no visual or > computational problem with the file. Do you have any thoughts about what might be causing these errors? > Have you seen this before? I can't find the load entire notebook > into memory option; do you have any suggestions about where to find it? I'm using Mathematica 6.0.2, Mac OS X 10.4.11, 2.2 GHz Intel Core 2 > Duo. (The files were also opened and worked on on Win XP, > Mathematica 6.0.0, I think. There could have been other machines > involved.) > http://www.siue.edu/~aweyhau/ > Science Building 1316 (618) 650-2220 === Subject: Re: About NDSolve k = .3; Clear[g]; > g[b_?NumberQ] := > g[b] = {x[t], v[t], b} /. First[NDSolve[{x'[t] == v[t] , > v'[t] == -x[t]^3 - k*v[t] + b*Cos[t], x[0] == 1, v[0] == 0}, {x, > v}, {t, 0, 2000}, MaxSteps -> Infinity]] > pts = Table[ > Evaluate[Table[g[b], {b, 0.2, 0.6, .01}]], {t, 1950, 2000}]; If I use h[s_]:= s^3 and replace x[t]^3 by h[x[t]], NDSolve in pts complains. Why can't I do this? > Piecewise or Which (e.g., Which[x[t] < -1, -1, x[t] > 1, 1, True, x[t]]) on the place of x[t]^3 also gives problems. > Any suggestions? > P_ter > I tried this, but didn't get any errors. === Subject: Re: How I can fit data with a parametric equation? > Hello everybody, > I have a list of {x,y} data and from a theoretical > calculation I know > that the data should be fitted by an equation, for > example: y=ax+bx+c; where a, b, c are constant parameters. The question is how i can find the best fit for my > data finding the > value of this parameters. Do mathematica has a specific function for that? > Dino > Given A your collection of {x,y} data, just : FindFit[A, a x + b, {a, b}, x] (careful, y = ax + bx + c should just be y = (a + b)x + c = dx + c !) Luca P. === Subject: Re: How can I do a grep -v equivalent in Import[]? > Valid point: > This will likely line wrap: --snip-- > CPU NFS CIFS HTTP Total Net kB/s Disk kB/s Tape kB/s Cache > Cache CP CP Disk FCP iSCSI FCP kB/s > in out read write read write age > hit time ty util in out > 24% 0 741 0 767 9381 6144 88815 0 0 81011 9 > 98% 0% - 31% 0 26 0 0 > --snip-- I'm trying to avoid importing through a pipe and I'm trying to avoid > preprocessing the data (for no real reason other than trying to get a handle > on data processing in Mathematica). I've done the following: importTemp = Import[ToFileName[Directory[], input.txt], {Lines}]; > regexMatch = ^d{2}.*; > sysstat = StringCases[importTemp, RegularExpression[regexMatch]]; which gets me ALMOST what I'm looking for.... I end up with the the > non-matched lines being empty data sets which Part:partw doesn't like. e.g. output is like this: {{}, {}, {24}, {29}, ... -jbl > I am still not 100% sure I know what you want to do, but assuming you want just the first two digits of lines that start with 2 digits, I am still not 100% sure I know what you want to do, but assuming you want just the first two digits of lines that start with 2 digits, tmp2 = Select[importTemp, StringMatchQ[#, DigitCharacter ~~ DigitCharacter ~~ __] &]; Map[StringTake[#, 2] &, tmp2] Alternatively, you could clean up the output that you have obtained already: {{}, {}, {24}, {29}} /. {} :> Sequence @@ {} With all the new string matching functionality, it is rarely necessary to resort to C preprocessing, unless speed is critical (i.e. your input file is very big!). David Bailey http://www.dbaileyconsultancy.co.uk === Subject: Problem with replacement rules I use Solve[expression(x,y),x]. Then, x is obtained as a one-to-one function of y. I want to define the function x(y) using the output of Solve. How should I do? === Subject: Rules from TreeForm Hi all, I am wanting a simple way to get the ordered list of rules that are internally generated using TreeForm, given the user has input an expression to that function. TreeForm usually generates a GraphPlot (or LayeredGraphPlot?) image. I am after the rules underpinning that graph visualisation. (BTW I searched this group but could not find a matching solution to this problem. Apologies if I have missed something obvious.) === Subject: Re: How I can fit data with a parametric equation? >Hello everybody, I have a list of {x,y} data and from a theoretical >calculation I know that the data should be fitted by an equation, >for example: >y=ax+bx+c; where a, b, c are constant parameters. >The question is how i can find the best fit for my data finding the >value of this parameters. >Do mathematica has a specific function for that? Yes, FindFit. Assume data is a list of {x,y} pairs. Then FindFit[data, a x + b, {a,b}, x] will find values for a and b that represent the best fit (least squares) line. FindFit supports non-linear models as well === Subject: Re: How can I do a grep -v equivalent in Import[]? >I've done the following: >importTemp = Import[ToFileName[Directory[], input.txt], >{Lines}]; regexMatch = ^d{2}.*; sysstat = >StringCases[importTemp, RegularExpression[regexMatch]]; >which gets me ALMOST what I'm looking for.... I end up with the the >non-matched lines being empty data sets which Part:partw doesn't >like. >e.g. output is like this: >{{}, {}, {24}, {29}, ... This last is easily dealt with using DeleteCases That is if StringCases[importTemp, RegularExpression[regexMatch]] produces the output you showed then DeleteCases[StringCases[importTemp, RegularExpression[regexMatch]],{}] will produce {{24}, {29}, ... === Subject: Re: How to simplify ArcCos[x/Sqrt[x^2+y^2]] to Pi/2-ArcTan[x/Abs[y]]? >> ArcCos[x/Sqrt[x^2+y^2]] >> and >> Pi/2-ArcTan[x/Abs[y]] >> are the same. >> But I can not get the first expression be simplified to the second >> one. And the following command can not be simplified to zero in >> mathematica as well. >> FullSimplify[ArcCos[x/Sqrt[x^2 + y^2]] - ( Pi/2 - ArcTan[x/Abs[y]]), >> Element[x, Reals] && Element[y, Reals]] >> I'm wondering if there is any walkaround to do this simplification. Try Assuming[x > 0 && y > 0, > FullSimplify[ > TrigToExp[ArcCos[x/Sqrt[x^2 + y^2]] - (Pi/2 - ArcTan[x/Abs[y]])]]] (1/2)*I*(Log[1/(I*x + y)] + Log[I*x + y]) > Obviously Mathematica isn't making a guess to which leaf of the complex > Log your algebraic complex travel history brought even if you arrived on > the positive real line. What I want is to simplify ArcCos[x/Sqrt[x^2 + y^2]] as Pi/2 - ArcTan[x/Abs[y]]. The two expressions are the same for any real x and y (maybe also true for complex x and y?). The use of such manipulation is useful when I want to convert the expression to CForm and eventually integrated in a C++ program, as the second expression is less likely to under or overflow than the first one. But I just don't find a way to do such manipulation in Mathematica. The more general question is how to manipulate an arithmetic expression to make it less likely to over or underflow. Another example is that the abs of a complex number is never computed as sqrt(x*x+y*y), it is always computed as |x| *sqrt(1+std::pow(x/y,2)) if |x| >= |y|. or |y| *sqrt(1+std::pow(y/x,2)) if |x| < |y|. I doubt Mathematica could solve this problem in general. But can I make it solve at least those particular cases? === Subject: How can I get list form like this As we know f/@Range[n] gives a list {f[1],f[2],f[3],=85=85}; Here what I want is a list form like this{f[[1]],f[[2]],=85=85}(*It's too tedious to type one by one*) It means the 1st,2nd,=85=85part of list f. Looks similar to each other, however I can't figure it out. How to do this ? === Subject: Re: Real and Complex Roots presented in a single plot > z[x_] = 1.3 Sin[1.7* x] + 0.6 Sin[4* x] ; > Plot[z[x], {x, 0, 18}] In the above plot we can see all the real roots of z very > approximately at {2.1,3.4, 5.5, 7.6, 9, 11, 13.1, 14.5, 16.5}, as the > curve crosses x-axis. We can also recognize and see the real parts of all the > complex roots of z where the curve is nearest to x -axis at {1.4, > 4.2, 6.6, 9.6, 12.3, 15.3, 17.7}. They are near to x-values where the > local maxima/minima occur.But we cannot 'see' their complex parts, as > they need to be computed. After computation of all real and complex roots I would like to > represent all roots, real and complex roots alike, on an ( x - y ) > Argand diagram (in a 2D plot) with the real part on x - axis and > imaginary part on y - axis. How to do this ? Narasimham This computes all roots with nonnegative imaginary part. You can flip across the real axis to get the conjugates. ListPlot[ Map[{Re[#], Im[#]} &, x /. N[{ToRules[ Reduce[{z[x] == 0, -100 <= Re[x] <= 100, 0 <= Im[x] <= 1/2}, x]]}]]] I capped the imaginary part at 1/2 because FindInstance tells me I can do so. In[27]:= FindInstance[{z[x] == 0, Im[x] > 1/2}, x] Out[27]= {} You can also get a hint of this from plotting the absolute value, letting the imaginary part of x go from 0 to 1. Plot3D[Abs[z[x + I*y]], {x, 0, 18}, {y, 0, 1}] Daniel Lichtblau Wolfram Research === Subject: Re: Real and Complex Roots presented in a single plot > z[x_] = 1.3 Sin[1.7* x] + 0.6 Sin[4* x] ; > Plot[z[x], {x, 0, 18}] In the above plot we can see all the real roots of z very > approximately at {2.1,3.4, 5.5, 7.6, 9, 11, 13.1, 14.5, 16.5}, as the > curve crosses x-axis. We can also recognize and see the real parts of all the > complex roots of z where the curve is nearest to x -axis at {1.4, > 4.2, 6.6, 9.6, 12.3, 15.3, 17.7}. They are near to x-values where the > local maxima/minima occur.But we cannot 'see' their complex parts, as > they need to be computed. After computation of all real and complex roots I would like to > represent all roots, real and complex roots alike, on an ( x - y ) > Argand diagram (in a 2D plot) with the real part on x - axis and > imaginary part on y - axis. How to do this ? Assuming I have correctly understood what you are looking for, visualizing real as well as complex roots can be achieve by formatting the list of solutions as a list of pairs {Re, Im} that can be displayed with ListPlot. For instance, z[x_] = 1.3 Sin[1.7*x] + 0.6 Sin[4*x] g = Plot[z[x], {x, 0, 18}] sols = Reduce[Rationalize[z[x], 0] == 0, x] // N Select[List[sols /. C[1] -> 0 // ToRules], (0 <= Re[x /. #] <= 18 &)] pts = Through[{Re, Im}[x /. %]] // Transpose p = ListPlot[pts, PlotStyle -> {Orange, PointSize[Large]}] Show[g, p] HTH, -- Jean-Marc === Subject: Re: Real and Complex Roots presented in a single plot Here's some suggestions. First, since the roots of the function include complex numbers, it's better to call the input variable z, and then use a different name for the function. Second, you might have better luck if you rationalize the coefficients, solve exactly, then take the numerical values of the exact roots found. That way you won't have to do lots of FindRoot instances. Third, it seems to me that plotting the real and imaginary parts of each root separately, on the x- and y-axes, respectively, will not be very enlightening unless all you're interested in is those real and imaginary parts: you won't be able to tell which real part goes with which imaginary part. So I also suggest that you plot the complex points themselves. If you want, in addition, to plot the real and imaginary parts of each on an axis, that should now be simple to add to the plot. Here's one way I would do it; undoubtedly there's a simpler way than I show to obtain the {x,y} pairs from the real and non-real roots. f[z_] := (13/10) Sin[(17/10) z] + (6/10) Sin[4 z] (* find the roots *) roots = z /. Solve[f[z] == 0, z] nRoots = N /@ roots (* massage roots to get {x,y} coordinates for each *) coords = nRoots /. Complex[a_, b_] -> {a, b} xyPairs = Replace[coords, x_?(Length[#] == 0 &) -> {x, 0}, 1] (* plot the roots *) ListPlot[xyPairs, PlotStyle -> PointSize[Medium]] With David Park's Presentation package, this is MUCH easier, as no massaging of the roots is needed: << Presentations`Master` Draw2D[{PointSize[Medium], ComplexPoint /@ roots}, Axes -> True, AspectRatio -> 0.5] > z[x_] = 1.3 Sin[1.7* x] + 0.6 Sin[4* x] ; > Plot[z[x], {x, 0, 18}] In the above plot we can see all the real roots of z very > approximately at {2.1,3.4, 5.5, 7.6, 9, 11, 13.1, 14.5, 16.5}, as the > curve crosses x-axis. We can also recognize and see the real parts of all the > complex roots of z where the curve is nearest to x -axis at {1.4, > 4.2, 6.6, 9.6, 12.3, 15.3, 17.7}. They are near to x-values where the > local maxima/minima occur.But we cannot 'see' their complex parts, as > they need to be computed. After computation of all real and complex roots I would like to > represent all roots, real and complex roots alike, on an ( x - y ) > Argand diagram (in a 2D plot) with the real part on x - axis and > imaginary part on y - axis. How to do this ? Narasimham === Subject: Re: Real and Complex Roots presented in a single plot Narasimham, I'll do your problem using the Presentations package. Needs[Presentations`Master`] First I find your notation just a little jarring. z is usually used for the complex variable, x is usually the real part of a complex number and I'll use f for the function. f[z_] := 1.3 Sin[1.7 z] + 0.6 Sin[4 z] The easiest way to see the zeros is to make a contour plot of the modulus of the function. All the zeros occur in a narrow strip about the x-axis. With[ {zmin = -I, zmax = 18 + I, contourlist = {0, 0.1, 0.2, .5, 1, 2}}, Draw2D[ {ComplexCartesianContour[f[z], {z, zmin, zmax}, Abs, Contours -> contourlist, ColorFunctionScaling -> False, ColorFunction -> (ContourColors[contourlist, ColorData[SolarColors]][#] &), PlotPoints -> {50, 15}, MaxRecursion -> 3, PlotRange -> {0, 4}]}, AspectRatio -> Automatic, Frame -> True, FrameLabel -> {Re, Im}, RotateLabel -> False, PlotLabel -> Row[{Modulus of , f[z]}], ImageSize -> 900] ] The following gives a closeup of the region that contains the first seven zeros. With[ {zmin = -.5 - I, zmax = 5 + I, contourlist = {0, 0.1, 0.2, .5, 1, 2, 10}}, Draw2D[ {ComplexCartesianContour[f[z], {z, zmin, zmax}, Abs, Contours -> contourlist, ColorFunctionScaling -> False, ColorFunction -> (ContourColors[contourlist, ColorData[SolarColors]][#] &), PlotPoints -> {30, 15}, MaxRecursion -> 3, PlotRange -> {0, 11}]}, AspectRatio -> Automatic, PlotRange -> {{-.5, 5}, {-1, 1}}, Frame -> True, FrameLabel -> {Re, Im}, RotateLabel -> False, PlotLabel -> Row[{Modulus of , f[z]}], ImageSize -> 900] ] I might be happy with those diagrams, but if we want numerical root values we can click the root locations off the diagram, using DrawingTools, and then refine the values with FindRoot. roots = z /. FindRoot[ f[z], {z, ToComplex@#}] & /@ {{-0.005546, -0.02102}, {1.285, 0.3771}, {1.279, -0.3678}, {2.081, 0.01109}, {3.462, -0.008173}, {4.194, -0.3485}, {4.213, 0.345}} // Chop {0, 1.27946+ 0.374308 I, 1.27946- 0.374308 I, 2.09155, 3.43663, 4.19786- 0.355809 I, 4.19786+ 0.355809 I} The following then plots just the roots in the Argand plane. Draw2D[ {ComplexCirclePoint[#, 3, Black, Red] & /@ roots}, AspectRatio -> Automatic, PlotRange -> {{-.5, 5}, {-1, 1}}, Frame -> True, FrameLabel -> {Re, Im}, RotateLabel -> False, PlotLabel -> Row[{Some Roots of , f[z]}], BaseStyle -> {FontSize -> 12}, ImageSize -> 900] -- David Park djmpark@comcast.net http://home.comcast.net/~djmpark/ > z[x_] = 1.3 Sin[1.7* x] + 0.6 Sin[4* x] ; > Plot[z[x], {x, 0, 18}] In the above plot we can see all the real roots of z very > approximately at {2.1,3.4, 5.5, 7.6, 9, 11, 13.1, 14.5, 16.5}, as the > curve crosses x-axis. We can also recognize and see the real parts of all the > complex roots of z where the curve is nearest to x -axis at {1.4, > 4.2, 6.6, 9.6, 12.3, 15.3, 17.7}. They are near to x-values where the > local maxima/minima occur.But we cannot 'see' their complex parts, as > they need to be computed. After computation of all real and complex roots I would like to > represent all roots, real and complex roots alike, on an ( x - y ) > Argand diagram (in a 2D plot) with the real part on x - axis and > imaginary part on y - axis. How to do this ? Narasimham > === Subject: Re: How I can fit data with a parametric equation? > Hello everybody, > I have a list of {x,y} data and from a theoretical calculation I know > that the data should be fitted by an equation, for example: y=ax+bx+c; where a, b, c are constant parameters. The question is how i can find the best fit for my data finding the > value of this parameters. Do mathematica has a specific function for that? Try searching the documentation for fitting ... the first result, Fit, does exactly what you describe here. === Subject: Re: How I can fit data with a parametric equation? > I have a list of {x,y} data and from a theoretical calculation I know > that the data should be fitted by an equation, for example: y=ax+bx+c; where a, b, c are constant parameters. The question is how i can find the best fit for my data finding the > value of this parameters. Do mathematica has a specific function for that? http://reference.wolfram.com/mathematica/ref/FindFit.html http://reference.wolfram.com/mathematica/guide/CurveFittingAndApproximateFun ctions.html -- Jean-Marc === Subject: Re: How I can fit data with a parametric equation? Fit fits well! !(Fit[yourlist, {1, x, x^2}, x]) === Subject: Re: How I can fit data with a parametric equation? Look up Fit and FindFit. Hello everybody, I have a list of {x,y} data and from a theoretical calculation I know that the data should be fitted by an equation, for example: y=ax+bx+c; where a, b, c are constant parameters. The question is how i can find the best fit for my data finding the value of this parameters. Do mathematica has a specific function for that? === Subject: Re: How to change the FrontEnd Encoding? > I'm using Mathematica 6 on Linux and I would want to know if there is > the net and only found a way to change the Kernel encoding. Most modern linux distributions use UTF-8 as its base encoding but the > FrontEnd is fixed in ISO8859-1. Therefore I'm having some problems > when I open files in directories that have characters with tildes or > something, like /home/f=F3=F3. In the open dialog the FrontEnd does recognize these directories. But > the window title is not displayed correctly and if I try to save the > file with Save as, the open dialog throws me a lot of errors saying > Could not read directory /home/f~~ I am not hundred percent sure that this is what you are looking for and if the option is available on Linux (I use Mac OS X), but open the option inspector (menu Format -> Option Inspector) and under Global Preferences, look for Formatting Options -> Font Options -> CharacterEncoding. In the drop down menu select the encoding scheme you wish. === Subject: Re: Computing Euler Rotation angles(?) If you have Mathematica it has a command, RotationTransform, that will provide the transformation function to rotate one vector into the direction of a second vector, which is what I think you are looking for. If you are looking for Euler angles the Mathematica Presentations package from my web site has an EulerAngles routine that will calculate both sets of Euler angles for all possible xyz rotation sequences. There is also a routine that will extract the rotation direction and rotation angle from any 3D rotation matrix. I'm sure there are many other pieces of software that will do the same things. http://home.comcast.net/~djmpark/ > Hello - I would like to compute the Euler rotation angles (rotX, rotY, rotZ) > required to rotate one vector A to another B. Actually what I'm trying > to do is to aim a newly created directional light -- that points in > the direction A = (0,0,-1) by default -- in the same direction as the > camera view direction B. === Subject: Re: HELP: Compounding equation that sum all the FUTURE VALUES. > Can you convert this to an excel formula where NP is dynamic? :) I'm sure that Excel has a built-in function to do what you want... It's the accumulated value of an annuity, isn't it? Check out FV or FVAL.. === Subject: Spiral problem The following appeared on one of my relative's math tests which was left on the kitchen table. There are 25 multiple choice (A,B,C,D or E) problems. The problems are supposed to be quickly solvable by a 17 year old and I can do most of them. To show the level, one of the questions says the reciprocal of 3/10 is 1+1/x what is x? I can do that in a few seconds. But the following problem is giving me brain ache. The question shows a grid of numbers arranged in a spiral with 1 in the centre square. 2 is to the right of 1. 3 is above 2. 4 is above 1. 5 is above 6. 6 is to the left of 1. 7 is below 6. 8 is below 1. 9 is below 2. 10 is to the right of 9. 11 is above 10. Numbers up to 17 are shown with a down arrow showing how to continue. And here is the question. What is the sum of the number that appears directly above 2007 and the number that appears directly below 2007. A) 4014 B) 4016 C) 4018 D) 4020 E) 4022 Maybe I'm missing the obvious but I can't see how to be sure of the answer quickly. If 2007 is in a vertical column with n+1 and n-1 above and below then the answer is A but is that a correct assumption? If I'd been doing the test I'd have answered A and moved on. === Subject: Re: Spiral problem >The following appeared on one of my relative's math tests which was left on >the kitchen table. There are 25 multiple choice (A,B,C,D or E) problems. >The problems are supposed to be quickly solvable by a 17 year old and I can >do most of them. To show the level, one of the questions says the reciprocal >of 3/10 is 1+1/x what is x? I can do that in a few seconds. >But the following problem is giving me brain ache. >The question shows a grid of numbers arranged in a spiral with 1 in the >centre square. 2 is to the right of 1. 3 is above 2. 4 is above 1. 5 is >above 6. 6 is to the left of 1. 7 is below 6. 8 is below 1. 9 is below 2. 10 >is to the right of 9. 11 is above 10. Numbers up to 17 are shown with a down >arrow showing how to continue. >And here is the question. What is the sum of the number that appears >directly above 2007 and the number that appears directly below 2007. >A) 4014 ÊB) 4016 ÊC) 4018 ÊD) 4020 ÊE) 4022 >Maybe I'm missing the obvious but I can't see how to be sure of the answer >quickly. >If 2007 is in a vertical column with n+1 and n-1 above and below then the >answer is A but is that a correct assumption? If I'd been doing the test I'd >have answered A and moved on. I get E, arguing that 2007 = 45^2 - 18, therefore the numbers > directly above and below it are 43^2 - 17 and 47^2 - 19, with > sum: (45 - 2)^2 + (45 + 2)^2 - 36 Why didn't you just add 43^2 - 17 and 47^2 - 19? > = 2*(45^2) + 2*(2^2) - 36 = > 4050 - 28 = 4022. To the OP: in such a spiral, the even squares form the upper left corners of the spiral rings and the odd squares form the lower right corners of the spiral ring. 2007 falls between 44^2 (1936) and 45^2 (2025), 1936 | | | | |---------------2025 and 2007 is more than halfway between 1936 and 2025, so it's past the lower left corner, so it's on a horizontal part of the spiral ring. 1936 | | | | |----2007----------2025 which means the numbers above and below it are on different rings, eliminating (n+1 plus n-1) as a solution. Where Angus got his numbers is from realizing that the numbers kitty-corner to 45^2 are 43^2 and 47^2 1936 | | | | 1849 |----2007----------2025 2209 with offsets 1 less and 1 greater, respectively. 1936 | | | | 1832------1849 |----2007----------2025 2190--------------2209 1832 + 2190 = 4022 -- > Angus Rodgers > Contains mild peril === Subject: Re: More with Quadratic Diophantine Theorem posting-account=Yn5cwwoAAADntcMuRwk-EwLg-DMZ_hXN Gecko/20070509 Camino/1.5,gzip(gfe),gzip(gfe) > i think you would be surprised at gauss' elementariness Yeah, hero worship is fun an easy, especially with a dead guy. He's > safe, right? I think you would be surprised if he were alive to consider what you > and your society do with current mathematical results of an elementary > nature, like in my original two posts in this thread. Do you really think Gauss would accept you as part of math society? For what? Because you'll say nice things about him? Do you think Gauss would like you or something? gauss was a paranoid asshole who screwed his kids over and spent so much effort inflating himself crying me me me that many of his contemporaries disliked him that said he was a damn fine mathematician no hero though i figured that if you spent some time reading him you too could be that paranoid asshole inflating yourself with me me me and finally be a fair mathematician to boot -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- === Subject: Re: More with Quadratic Diophantine Theorem > i think you would be surprised at gauss' elementariness > Yeah, hero worship is fun an easy, especially with a dead guy. ÊHe's > safe, right? > I think you would be surprised if he were alive to consider what you > and your society do with current mathematical results of an elementary > nature, like in my original two posts in this thread. > Do you really think Gauss would accept you as part of math society? > For what? ÊBecause you'll say nice things about him? > Do you think Gauss would like you or something? gauss was a paranoid asshole > Ê who screwed his kids over > Ê and spent so much effort inflating himself > Ê Ê crying me me me > that many of his contemporaries disliked him that said > Ê he was a damn fine mathematician no hero though i figured that if you spent some time reading him > Ê you too could be that paranoid asshole > Ê inflating yourself with me me me > and finally be a fair mathematician to boot I'm contemplating this awesome result linking all quadratic Diophantine equations in 2 variables and wondering how far it goes. You're mumbling. History will remember me. But what about you? Oh, you don't care, right? Hundreds of years from now, some people may be wondering about who I am and what kind of person I was, where one may hero worship, and another may mouth off about my many failings. But who will even know about you? James Harris === Subject: Re: More with Quadratic Diophantine Theorem [...] > > I'm contemplating this awesome result linking all quadratic > Diophantine equations in 2 variables and wondering how far it goes. There's the Perfect cuboid problem which no one has solved. Any Perfect cuboid (if there is one) is an Euler brick. http://mathworld.wolfram.com/PerfectCuboid.html I wonder if a perfect cuboid exists... What are the chances? David Bernier > You're mumbling. > > History will remember me. But what about you? > > Oh, you don't care, right? > > Hundreds of years from now, some people may be wondering about who I > am and what kind of person I was, where one may hero worship, and > another may mouth off about my many failings. > > But who will even know about you? > > > James Harris === Subject: Re: More with Quadratic Diophantine Theorem posting-account=Yn5cwwoAAADntcMuRwk-EwLg-DMZ_hXN Gecko/20070509 Camino/1.5,gzip(gfe),gzip(gfe) >(for an example of what i came to dislike >http://r6.ca/blog/20051210T202900Z.html >) How Dedekind Screwed Up a Hundred Years of Mathematics [...] My friend suggested that for most of the 3600 year history > of mathematics, mathematics was done constructively. I've been suspecting something of the kind, and am hoping to sneak > up on the philosophical problem of constructivism via the history > of mathematics (as my mind remains blank on the philosophy). [...] But Dedekind wasnÕt happy with the equivalence relations > in the theory and he created the set-theoretic formulation that > we suffer with today. I hear the sound of an axe being ground, and ... [...] Once again, Dedekind wasnÕt happy with this, and in 1872, he > proposed his own notion of real numbers based on set-theory by > Dedekind cuts of the rational numbers. A bit of sneaky rhetoric there. I doubt that the objective issues > are reducible to Dedekind's subjective happiness or unhappiness, > although undoubtedly his subjectivity would have given him (and, > via his writings, others) a handle on the objective issues. But > I must reserve judgement until I understand those issues better. Also, there /was/ no set theory on which to base the theory of cuts > (or, for that matter, ideals); rather, surely it was these convincing > applications of a still non-existent theory which helped to bring the > theory into being? Whether Dedekind himself would recognise ZFC, for > example, as his child is another question. of course you are right about there being no set theory and the overreaching imagination of his desires and the axe-to-grind style and i don't agree with the idea that it was somehow dedekind's fault mathematics was screwed up for a hundred years (though i think it was screwed up) it's a mentality that gets annoyed with questions of fundaments and moves to belittlement or a mentality that seeks certainty but doesn't want to obvious mythology so substitutes other unobservables like bivalence or ... there's a hundred and one reasons why it was screwed up and dedekind was just one of the authorities that made authority appeal easier i linked to the text mostly to point to the issues that dedekind created and didn't intend all of the rhetoric (just some of it :) ) [as an interesting aside dedekind's construction of the reals is not constructively equivalent to cauchy's i actually suspect the initial acceptance of equivalence may have helped confuse constructive foundations enough to delay it beyond the classical formalisation] -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- === Subject: Re: More with Quadratic Diophantine Theorem just because the ideals were not made to solve Fermat's Last Theorem, they don't work?... there's a good case-book that uses the golden section as the exemplar. I agree, though, it is best to avoid them til you might need them; I certainly don't need'em. > [as an interesting aside > Ê Êdedekind's construction of the reals > Ê Êis not constructively equivalent to cauchy's Êi actually suspect the initial acceptance of equivalence > Êmay have helped confuse constructive foundations > Ê Êenough to delay it beyond the classical formalisation] thus: an aether simply provides a prealistical model of the phenomenon of waves; one need not deal with photons at all, since they are just formally dual to waves, not at all pardoxical, as shown by Dirac. far too much of theory rests upon Einstein, saying that's impossible, which is the reality behind Michelson-Morley and those who carried the examination further -- but, herr doktor-professor Albert said, Nein!, when he was once in his dysused office at Caltech. let us remember him for those things which he was really good at, not the Big Bang Constant and its infernal klingons (fridge magnets) ?! > I don't need any aether to stick a magnet to a fridge, light up an thus: so, to ask an ignorant question of personal interest, how much of this could be used toward a new or old proof of quadratic reciprocity? > That Ac and -Dc must be quadratic residues, mod D and A, respectively, > is a trivial result: > Let Ax^2 - Dy^2 = c. [1] > Take this mod A: > -Dy^2 = c mod A > (Dy)^2 = -Dc mod A > Therefore, -Dc is a quadratic residue mod A. > Similarly, taking [1] mod D: > Ax^2 = c mod D > (Ax)^2 = Ac mod D > Therefore, Ac is a quadratic residue mod D. thus: ah, sort of a solopsistic monad. I am reminded of hte Many Words Interpretation of the Copenhagenskool: to be or not to be that is the shape of the box or the bubble -- inside or out? > Q. What is a tautological space? > A. A tautological space. thus: speaking of Young's anhialation of Newton's photons, here is the earlier elaboration on light by Fermat: http://www.wlym.com/~seattle/dynamis/issues/august08-fermat.pdf > as we know from Newton's iron-poor corpuscles. --ROTC, your summer vacation in the Sahara Desert ( S u d a n ) ; presage the Draft for your middleschool class of '12 -- brought to you by Allstate (tm) and Oxford U. Press! http://larouchepub.com/pr/2008/080813moloch brown.html http://wlym.com === Subject: Re: More with Quadratic Diophantine Theorem days. My association with the Department is that of an alumnus. [...] >What I don't see is how a proof of non-existence fits into the >constructive/non-constructive classification. For instance, consider >the classic proof that sqrt(2) is irrational: Assume it is rational, say a/b, with at least one of a, b odd. 2 = a^2/b^2 > 2 b^2 = a^2 > Therefore, a must be even, say a = 2r > 2 b^2 = 4 r^2 > b^2 = 2 r^2 > Therefore, b must be even. > This is a contradiction, therefore sqrt(2) is irrational. Would a constructionist allow that proof? For most meanings of constructionist (they may vary a bit), the implication is acceptable. It is the implication not(for all x (P(x))) ==> exists x (not(P(x))) that is considered unacceptable in the absence of a construction of x. (Constructionists also reject not(not(P)) --> P, so you also have to be a bit careful there). -- === Subject: Re: What binary operation lambda quantifier corresponds to? I think Tegiri Nenashi is on to something here. > Observation: all quantuifiers correspond to some binary algebraic > operation. I would say instead all binding constructs. Lambda and limit are not quantifiers, but they are binding constructs. > Consider: > 1. Sigma summation and integral are iterative forms of binary plus. > 2. Pi-capital product is iterative form of multiplication. > 3. Lattice supremum is iterative form binary meet. > 5. Lattice infinum is iterative form binary join. > 6. Existential and universal quatifiers are lattice supremum and > infinum. > Now, there must be a binary operation that lambda is iterative form as > well? I agree. Let {e1 -> e2} be the partial function that does nothing except mapping e1 to e2. If f1 and f2 are partial functions with disjoint domains, let f1 /+/ f2 be the union of the partial functions; i.e. the partial function whose graph is the union of the graph of f1 and the graph of f2. Then lambda x . e is indeed {e1 -> e[x/e1]} /+/ {e2 -> e[x/e2]} /+/ {e3 -> e[x/e3]} /+/ ... for some enumeration e1, e2, e3, ... of all the closed terms. I also think Tegiri was on the mark with the comment about limit and the second projection function. Let (a /$/ b) be b, and let /$/ be left-associative. Then limit (x -> infty) e is indeed (in some sense that can probably be made more precise) e[x/0] /$/ e[x/1] /$/ e[x/2] /$/ ... assuming we are talking about x as an integer. If x is a real number, then there is probably some interesting generalization. --Jamie. (efil4dreN) andrews .uwo } Merge these two lines to obtain my e-mail address. @csd .ca } (Unsolicited bulk e-mail costs everyone.) === Subject: Re: What binary operation lambda quantifier corresponds to? [F'up again to clf] >> Now, there must be a binary operation that lambda is iterative form as >> well? > I agree. Let {e1 -> e2} be the partial function that does > nothing except mapping e1 to e2. If f1 and f2 are partial > functions with disjoint domains, let f1 /+/ f2 be the union of > the partial functions; i.e. the partial function whose graph is > the union of the graph of f1 and the graph of f2. Then > > lambda x . e > > is indeed > > {e1 -> e[x/e1]} /+/ {e2 -> e[x/e2]} /+/ {e3 -> e[x/e3]} /+/ ... > > for some enumeration e1, e2, e3, ... of all the closed terms. You have only said that a function is the union of all single-point mappings. That's pretty trivial, and such a scheme will work for every construct that's binding a variable. And it's quite different from what happens with (finite) sums, products, and lattice operations. These are all non-trivial folds. If I'm inventive enough, I can of course find some binary operation everywhere. For example, every natural number is connected to the binary operation (+), because I can e.g. write 3 = 1 + 1 + 1, or 5 = 1 + 1 + 1 + 1 + 1. It's also connected to the binary operation (*), because it can be written as a product of prime powers. Nevertheless, we normally think of numbers as a whole. As we do of functions. > I also think Tegiri was on the mark with the comment about > limit and the second projection function. Let (a /$/ b) be b, > and let /$/ be left-associative. Then > > limit (x -> infty) e > > is indeed (in some sense that can probably be made more precise) > > e[x/0] /$/ e[x/1] /$/ e[x/2] /$/ ... > > assuming we are talking about x as an integer. The trick is of course that you have to made it more precise, because then you see that it doesn't work. The limit is *not* somehow the last element of a sequence. Not even for integers. Sorry. The whole point of the limit is that you can make some statement about what infinite sequence is approximating. So you need at least some notion of proximity. Which you won't get with this operation. And dealing with the infinite can be subtle, so you *must* make the step going from finite to infinite explicit. Just writing ... is not enough. - Dirk === Subject: Formula for quarter of a circle/or half of a parabola midpoint 0,0 and radius 5. Specifically, I'm looking for the quarter in the first quadrant. Alternatively, I also attempting to derive the formula for a half a parabola positioned in the first quadrant with a y-intercept of 5 and an x-intercept of 5. I don't have a particular focal point in mind. Keith === Subject: Re: Formula for quarter of a circle/or half of a parabola > midpoint 0,0 and radius 5. Specifically, I'm looking for the quarter > in the first quadrant. y = sqrt( 25 - sqrt(x)^4 ) > Alternatively, I also attempting to derive the formula for a half a > parabola positioned in the first quadrant with a y-intercept of 5 and > an x-intercept of 5. I don't have a particular focal point in mind. y = sqrt( 5 - sqrt(x)^4 / 5 )^2 Dave === Subject: Re: Formula for quarter of a circle/or half of a parabola > midpoint 0,0 and radius 5. Specifically, I'm looking for the quarter > in the first quadrant. y = sqrt( 25 - sqrt(x)^4 ) ******** What's with sqrt(x)^4 instead of x^2 , and y = 25 - x^2 ??? > Alternatively, I also attempting to derive the formula for a half a > parabola positioned in the first quadrant with a y-intercept of 5 and > an x-intercept of 5. I don't have a particular focal point in mind. y = sqrt( 5 - sqrt(x)^4 / 5 )^2 ********* Ahh, of course, there is only one solution in this case as well, because all parabolas look the same except for scale, as do circles. Only one free variable. But why now the redundant squaring and square root, instead of simply y= 5 - x^2 /5 === Subject: Re: Formula for quarter of a circle/or half of a parabola > midpoint 0,0 and radius 5. Specifically, I'm looking for the quarter > in the first quadrant. y = sqrt( 25 - sqrt(x)^4 ) ******** What's with sqrt(x)^4 instead of x^2 , and y = 25 - x^2 Ê ??? Because 25 - x^2 is defined for -5 <= x <= 5, so it gives the top half of the circle, but sqrt(x)^4 is defined only for 0 <= x <= 5, giving only the top half of the circle. With y = 25 - x^2, you have to add the domain restriction 0 <= x <= 5, but that is implicit in my definition. > Alternatively, I also attempting to derive the formula for a half a > parabola positioned in the first quadrant with a y-intercept of 5 and > an x-intercept of 5. I don't have a particular focal point in mind. y = sqrt( 5 - sqrt(x)^4 / 5 )^2 ********* Ahh, of course, there is only one solution in this case as well, > because all parabolas look the same except for scale, as do circles. Only > one free variable. But why now the redundant squaring and square root, > instead of simply y= 5 - x^2 /5 The inner square root makes sure that x >= 0, and the outer one makes sure that x <= 5. Again, it avoids domain restrictions, because the function is defined and real only in the first quadrant. Dave === Subject: Re: Formula for quarter of a circle/or half of a parabola > midpoint 0,0 and radius 5. Specifically, I'm looking for the quarter > in the first quadrant. Alternatively, I also attempting to derive the formula for a half a > parabola positioned in the first quadrant with a y-intercept of 5 and > an x-intercept of 5. I don't have a particular focal point in mind. For the quarter circle, how about: y = Sqrt[25 - x^2] where {x, 0, 5} === Subject: Re: Formula for quarter of a circle/or half of a parabola > midpoint 0,0 and radius 5. Specifically, I'm looking for the quarter > in the first quadrant. y = sqrt(25-x^2) for 0<=x<=5 Alternatively, I also attempting to derive the formula for a half a > parabola positioned in the first quadrant with a y-intercept of 5 and > an x-intercept of 5. I don't have a particular focal point in mind. There are a lot of possibilities. Would y = 5 - x^2/5 for for 0<=x<=5 do? > Keith === Subject: Re: Set Theory Notation Question >I am translating a soma puzzle solution into laymans terms and am >not familiar with all of the notation used here: >http://www.cs.dartmouth.edu/~mckeeman/references/soma/sld013.htm Which notation are you unfamiliar with? As far as I can tell, it is > all reasonably standard. The first line says, given u, and given s a subset of u, a Gamma, and > a P 0. The second line says let P be the union of all Gamma(p), with p > running over all elements of P 0 (that is, P consists exactly of all > elements x that lie in some Gamma(p), with p in P 0). The third line says and if we let R be the set that conists of all V > which are: > Ê Ê (i) subsets of P; > Ê Ê(ii) the union of V is an element of Gamma(s) > Ê Ê Ê Ê [that is, take all elements y of V, and then consider the > Ê Ê Ê Ê collection of all Êx that are in at least one of those y; this > Ê Ê Ê Ê should be an element of Gamma(s), where s is the specified > Ê Ê Ê Ê subset of u given in the first line]; and > Ê (iii) the union of V is disjoint from V itself (that is, there is no > Ê Ê Ê Ê element y of V which is also an element of an element of V). And the final line says that S (the set of solutions) consists > exactly of those Gamma(V) for which V is an element of the R defined > in the third line. If the problem is the meaning of the Gamma(p) and so on, then the > question is not about set theory notation, but about the meanings > given to the symbols in the context of space fillings. and I guess I had missed the subset notation notes. === Subject: Re: Set Theory Notation Question days. My association with the Department is that of an alumnus. [...] >and I guess I had missed the subset notation notes. Do yourself two favors: First, get yourself an actual book on set theory that will introduce you to the necessary particulars. My recommendation would be Paul Halmos's Naive Set Theory. Second: realize taht wikipedia may (emphasis on but it may not) be a good place to start, but it should never be the place to end your investigation. It is not always reliable, and it is always incomplete. I would not trust an encylopedia which underwent real, serious vetting as the final word, even if just published. Why give that kind of final word to wikipedia? -- === Subject: Re: Set Theory Notation Question >and I guess I had missed the subset notation notes. Do yourself two favors: First, get yourself an actual book on set theory that will introduce > you to the necessary particulars. My recommendation would be Paul > Halmos's Naive Set Theory. Second: realize taht wikipedia may (emphasis on but it may not) be a > good place to start, but it should never be the place to end your > investigation. It is not always reliable, and it is always > incomplete. I would not trust an encylopedia which underwent real, > serious vetting as the final word, even if just published. Why give > that kind of final word to wikipedia? === Subject: Re: traceless matrix unitarily similar to matrix with 0 diagonals? > Yes, there is a 2x2 matrix with trace 0 but the two columns are not > orthogonal. For example [ 1 1 ; 0 -1]. > A is any complex matrix. The way to solution is to find a unit vector > v with following property : v* A v = 0 > Rest of the problem is induction. > Where * is hermitian transpose. > But how do we guarantee existence of such v ? (Maybe I am late with this, but who is supposed to wade through unrelated > stuff?) Also, it may be a big tool for a small problem, but why not. The set of all (v* A v) over all unit vectors v is the numerical range, > which is known to contain all eigenvalues of A, and to be convex in the > complex plane. If zero is the trace of A then the sum of the eigenvalues of A is 0, and > so is their average, which is in the convex hull of these eigenvalues. > In short, 0 is in the numerical range of A. Hence the existence of a unit vector v such that (v* A v) = 0. > If A is a real matrix, and the unitary is required to be real (i.e. orthogonal), one has to look at the real parts of the eigenvalues when A is brought (by orthogonal similarity) to a real Schur form first, that is, block diagonal with at most 2-by-2 blocks corresponding to non-real complex conjugate pairs of eigenvalues. It's a little messier. Anyone with a shortcut? === Subject: Re: an interesting sum <10027060.1220971993506.JavaMail.jakarta@nitrogen.mathforum.org>, > Well, calculation of several hundred roots and elementary calculus give a > head-start. Really? Exactly how did you calculate that 4 is a root? I didn't check my work very carefully, but when I calculate S(4) I get approximately 0.0181922290846 . (Anyone else have a numeric value for S(4)?) > Using induction and derivatives we have peaks +/- at every prime. > That's sort of built into the sum I think. -- === Subject: Re: an interesting sum You are correct. I looked harder at large values because I thought the function was more likely go awry there. Some of those values are off on the order of 10^-20 but off they are. It's an interesting function. The part about a zero between successive primes (the part that one can prove using derivatives)is apparently true. And it mimics the conjectured property in a pretty audacious way. I did look at hundreds of zeros, just ones so close I took the difference to be computational error. === Subject: Re: an interesting sum > You are correct. I looked harder at large values because I thought the function was more likely go awry there. Some of those values are off on the order of 10^-20 but off they are. It's an interesting function. The part about a zero between successive primes (the part that Êone can prove using derivatives)is apparently true. And it mimics the conjectured property in a pretty audacious way. Ê I did look at hundreds of zeros, just ones so close I took the difference to be computational error. > Just a note that when you reply to posts and remove the original message - the readers have no clue to which message you are replying. You are welcome! ~Amzoti === Subject: Re: an interesting sum Well, before I throw this away, I did take the trouble to take derivatives and so on, and I have that S'(x)=0 when (x-prime[k])=0 for all k. Taken together with the sign changes, can we safely say the function has max/min at each prime and a zero between successive primes? > Just a note that when you reply to posts and remove > the original > message - the readers have no clue to which message > you are replying. > > You are welcome! > > ~Amzoti === Subject: Re: an interesting sum posting-account=HaopWgoAAADs72-s8RQYwP_-ruRUuNzX Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > <10027060.1220971993506.JavaMail.jaka...@nitrogen.mathforum.org>, > Well, calculation of several hundred roots and elementary calculus give a > head-start. Really? Exactly how did you calculate that 4 is a root? > I didn't check my work very carefully, but when I calculate > S(4) I get approximately 0.0181922290846 . (Anyone else have a numeric value for S(4)?) > Using induction and derivatives we have peaks +/- at every prime. > That's sort of built into the sum I think. -- > Here are some exact values - (you have correct magnitude - but missing negative sign). e = exponential k = 1 {s(1), s(2), s(3), s(4)} = {-1/e, -1, -1/e, -1/e^4} k = 2 {s(1), s(2), s(3), s(4)} = {1/e^4, 1/e, 1, 1/e} k = 3 {s(1), s(2), s(3), s(4)} = {-1/e^16, -1/e^9, -1/e^4, -1/e} k=4 {s(1), s(2), s(3), s(4)} = {1/e^36, 1/e^25, 1/e^16, 1/e^9} . . . k=100 {s(1), s(2), s(3), s(4)} = {1/e^291600, 1/e^290521, 1/e^289444, 1/ e^288369} HTH ~A === Subject: Re: an interesting sum >> Well, calculation of several hundred roots and elementary calculus give a >> head-start. >> Really? Exactly how did you calculate that 4 is a root? >> I didn't check my work very carefully, but when I calculate >> S(4) I get approximately 0.0181922290846 . >> (Anyone else have a numeric value for S(4)?) >> Using induction and derivatives we have peaks +/- at every prime. >> That's sort of built into the sum I think. >> -- >> >Here are some exact values - (you have correct magnitude - but missing >negative sign). >e = exponential >k = 1 >{s(1), s(2), s(3), s(4)} = {-1/e, -1, -1/e, -1/e^4} Huh? S(x) is a function of one variable - there's no > such thing as S(1) for k = 1 and S(1) for k = 2. Maybe your s(x) is not supposed to be the same as S(x); > if so I don't know what you mean by s(x). The definition > of S(x) was this: S(x) = Sum [(-1)^k * Exp[-(x-prime[k])^2], We're summing over k there. >k = 2 >{s(1), s(2), s(3), s(4)} = {1/e^4, 1/e, 1, 1/e} >k = 3 >{s(1), s(2), s(3), s(4)} = {-1/e^16, -1/e^9, -1/e^4, -1/e} >k=4 >{s(1), s(2), s(3), s(4)} = {1/e^36, 1/e^25, 1/e^16, 1/e^9} >. >. >. >k=100 >{s(1), s(2), s(3), s(4)} = {1/e^291600, 1/e^290521, 1/e^289444, 1/ >e^288369} >HTH ~A Understanding Godel isn't about following his formal proof. > That would make a mockery of everything Godel was up to. > (John Jones, My talk about Godel to the post-grads. > in sci.logic.) Can you please provide an algorithm for finding Prime[k] for k ranging from 1 to infinity over that sum? How did you find your value of S(4)? (Given that we got the same result for it.) === Subject: Re: an interesting sum > Can you please provide an algorithm for finding Prime[k] for k ranging > from 1 to infinity over that sum? If we had an algorithm for finding prime[k] for all k, we'd be a very happy bunch of people, here, believe me :-) I don't know how David did it, but I am guessing that he used an approximation of the original function with k running from 1 to some large value, say 1000 or something. > How did you find your value of S(4)? (Given that we got the same > result for it.) -- I.N. Galidakis === Subject: Re: an interesting sum If you look at Amzoti's calculations, as k increases the value of S(4) decreases quickly. At k=100 he's at 1/e^288369. k doesn't stop there, does it?? === Subject: Re: an interesting sum <10669042.1220999228176.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=HaopWgoAAADs72-s8RQYwP_-ruRUuNzX Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > If you look at Amzoti's calculations, as k increases the value of S(4) decreases quickly. At k=100 he's at 1/e^288369. k doesn't stop there, does it?? Certainly not - want more - here you go. k = 100000 {s(1), s(2), s(3), s(4)} = {1/e^1689240885264, 1/e^1689238285849, 1/ e^1689235686436, 1/e^1689233087025} . . . k = 1000000 {s(1), s(2), s(3), s(4)} = {1/e^239811921883044, 1/e^239811890911321, 1/e^239811859939600, 1/e^239811828967881} Enough? HTH ~A === Subject: Re: an interesting sum === Subject: Re: an interesting sum <10669042.1220999228176.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=HaopWgoAAADs72-s8RQYwP_-ruRUuNzX Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > If you look at Amzoti's calculations, as k increases the value of S(4) decreases quickly. At k=100 he's at 1/e^288369. k doesn't stop there, does it?? Certainly not - want more - here you go. k = 100000 {s(1), s(2), s(3), s(4)} = {1/e^1689240885264, 1/e^1689238285849, 1/ > e^1689235686436, 1/e^1689233087025} . > . > . k = 1000000 {s(1), s(2), s(3), s(4)} = {1/e^239811921883044, 1/e^239811890911321, > 1/e^239811859939600, > Ê Ê 1/e^239811828967881} Enough? HTH ~A Okay - last one. k = 10^8 {s(1), s(2), s(3), s(4)} = {1/e^4153748653978366564, 1/ e^4153748649902217081, 1/e^4153748645826067600, 1/ e^4153748641749918121} === Subject: Re: minimum cycle length a^k mod n As examples, consider the minumum cycle length of 3^k mod 10^i, for 1 <= i <= 8; according to my computations, these are, respectively, 4, 20, 100, 500, 5000, 50000, 500000, 5000000. I don't see anything posted so far that explains all of these values. In the last five cases (i = 4, 5, 6, 7, 8) the minimum cycle length is 10^i/20 -- how would one determine that fact other than by brute force? Is the formula 10^i/20 valid for all higher moduli 10^i? Not coincidentally, this relates to some properties of power towers mentioned in the section Decimal digits of Graham's number at http://en.wikipedia.org/wiki/Graham's_number (and the discussion page). === Subject: Re: Longevity of Math Books (was Re: Quadratic Diophantine Theorem) <6996c4lha799djvtrktcfnld533gdkbnl0@4ax.com> posting-account=jPnQ2goAAAA461y3QD0lbyw0oKeThma1 AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.2 Safari/525.20.1,gzip(gfe),gzip(gfe) Apostol begins the course with integration, as per Liebniz; there's a special name for that, per Liebniz: primitive equation or prime equation? this was the subject ot the keynote speaker at the Ninth Annual Nonlinear Science Conference at UCLA, although I no longer blame Newton exclusively for the controversy with Liebniz. anyway, apparently, Hooke did not do the math in a formalistic way; after all, he was the Royal Society's experimenter. the other part of the joke is that the '99 2-pound coin is inscribed, On the shoulders of giants. on the edge; at least, Newton was effective in his office-of-reward. > stole the solution from Hooke, to boot -- on the shoulders of a giant, > who was physically a dwarf; hence, the joke. Wrong again, but excusably, since that meme has spread widely. Read the > shoulders bit in context, and you can see it was not an insult. thus: ah, Lord Berty's thing was called the Unity of Sciences; well, here's a reference: http://www.larouchepub.com/other/2002/2949moonification.html The two operations of Wells and Russell from which Moon sprung are these: The Moral Re-Armament Movement, founded at a 1921 meeting between a wacky Lutheran preacher from Philadelphia and two British delegates to the Washington Disarmament Conference, Lord Arthur Balfour and H.G. Wells. Moral Re-Armament became the mass organizational vehicle for implementation of Wells' 1928 call in The Open Conspiracy, for a worldwide movement for draft resistance. The environment of Moon's Korean ministry was under control of Moral Re-Armament when he was picked up as an intelligence asset during the Korean War. The Unity of the Sciences movement. Founded in 1935 under the supervision of Lord Bertrand Russell and John Dewey, it brought together Trotskyite academics Albert Wohlstetter (mentor of current Defense Policy Board Chairman Richard Perle), Sidney Hook, and Ernest Nagel, with members of the radical-positivist Vienna Circle. Merging with Robert M. Hutchins at the University of Chicago in the 1950s, this operation took over the teaching of science in the United States. Thomas Kuhn's widely read fraud, The Structure of Scientific Revolutions, was published as the second volume of their Encyclopedia of Unified Sciences. In 1972, the Moonies were given the Unity of Sciences franchise, sponsoring the first of their still-ongoing International Conferences of the Unity of Sciences. Their early sessions featured such notables as Manhattan Project physicist Eugene Wigner, the lifelong ally of that truly mad scientist Leo Szilard (the model for Dr. Strangelove, in Stanley Kubrick's film of that name), and environmental fascists Alexander King and Aurelio Peccei, founders of the no-growth Club of Rome. Before looking back to the history of these projects, let us first briefly dispense with the person of Rev. Sun Myung Moon. Moon as a personality is of very little importance, in himself. The real Reverend Moon is a pathetic, if nonetheless nasty, victim of Japanese internment and North Korean torture sessions. He is what the.... thus: an aether simply provides a prealistical model of the phenomenon of waves; one need not deal with photons at all, since they are just formally dual to waves, not at all pardoxical, as shown by Dirac. far too much of theory rests upon Einstein, saying that's impossible, which is the reality behind Michelson-Morley and those who carried the examination further -- but, herr doktor-professor Albert said, Nein!, when he was once in his dysused office at Caltech. let us remember him for those things which he was really good at, not the Big Bang Constant and its infernal klingons (fridge magnets) ?! > I don't need any aether to stick a magnet to a fridge, light up an thus: so, to ask an ignorant question of personal interest, how much of this could be used toward a new or old proof of quadratic reciprocity? > That Ac and -Dc must be quadratic residues, mod D and A, respectively, > is a trivial result: > Let Ax^2 - Dy^2 = c. [1] > Take this mod A: > -Dy^2 = c mod A > (Dy)^2 = -Dc mod A > Therefore, -Dc is a quadratic residue mod A. > Similarly, taking [1] mod D: > Ax^2 = c mod D > (Ax)^2 = Ac mod D > Therefore, Ac is a quadratic residue mod D. thus: ah, sort of a solopsistic monad. I am reminded of hte Many Words Interpretation of the Copenhagenskool: to be or not to be that is the shape of the box or the bubble -- inside or out? > Q. What is a tautological space? > A. A tautological space. thus: speaking of Young's anhialation of Newton's photons, here is the earlier elaboration on light by Fermat: http://www.wlym.com/~seattle/dynamis/issues/august08-fermat.pdf > as we know from Newton's iron-poor corpuscles. --ROTC, your summer vacation in the Sahara Desert ( S u d a n ) ; presage the Draft for your middleschool class of '12 -- brought to you by Allstate (tm) and Oxford U. Press! http://larouchepub.com/pr/2008/080813moloch_brown.html http://wlym.com === so, to ask an ignorant question of personal interest, how much of this could be used toward a new or old proof of quadratic reciprocity? > That Ac and -Dc must be quadratic residues, mod D and A, respectively, > is a trivial result: > Let Ax^2 - Dy^2 = c. [1] > Take this mod A: > -Dy^2 = c mod A > (Dy)^2 = -Dc mod A > Therefore, -Dc is a quadratic residue mod A. > Similarly, taking [1] mod D: > Ax^2 = c mod D > (Ax)^2 = Ac mod D > Therefore, Ac is a quadratic residue mod D. thus: ah, sort of a solopsistic monad. I am reminded of hte Many Words Interpretation of the Copenhagenskool: to be or not to be that is the shape of the box or the bubble -- inside or out? > Q. What is a tautological space? > A. A tautological space. thus: speaking of Young's anhialation of Newton's photons, here is the earlier elaboration on light by Fermat: http://www.wlym.com/~seattle/dynamis/issues/august08-fermat.pdf > as we know from Newton's iron-poor corpuscles. --ROTC, your summer vacation in the Sahara Desert ( S u d a n ) ; presage the Draft for your middleschool class of '12 -- brought to you by Allstate (tm) and Oxford U. Press! http://larouchepub.com/pr/2008/080813moloch_brown.html http://wlym.com === Subject: Why does MySpace condone child rape and pedophiles? They keep withholding vital evidence from the District Attorneys and continue to let sexual offenders pour into and operate on Myspace. http://www.youtube.com/watch?v=8PV9FrNLjto === Subject: probability question I have a probablity question: Suppose we throw a dice 100 times. What is the probability that at least 1 number appears exactly once. I would be grateful for your help, You can do it by inclusion-exclusion. Let E_k be the event that k appears only once in 100 rolls. You want P(E_1 U ... U E_6), which equals P(E_1) + ... + P(E_6) - sum P(E_j n E_k) + sum P(E_i n E_j n E_k) - sum P(E_i n E_j n E_k n E_l) + P(E_i n E_j n E_k n E_l n E_m) This equals [ 6*100*5^99 - 15*100*99*4^98 + 20*100*99*98*3^97 - 15*100*99*98*97*2^96 + 6*100*99*98*97*96 ]/6^100 I'm getting 0.00000144896 in Excel. === Subject: Re: probability question Have you learned about the binomial distribution yet? === Subject: Re: probability question > > I have a probablity question: > > Suppose we throw a dice 100 times. What is the probability that at > least 1 number appears exactly once. Very, very close to zero. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: probability question posting-account=eFTX7goAAACJBRoHHOnDC9IuTZeZT1_H 1.1.4322),gzip(gfe),gzip(gfe) I have a probablity question: Suppose we throw a dice 100 times. What is the probability that at > least 1 number appears exactly once. I would be grateful for your help, > eugene I'll guess For a particular die face (doesn't matter when it happens) it would have to not happen 99 times, thus (5/6)^99 for one die face (i.e.number) But there are six die faces (same odds for each and they're independent)... 6 ( 5/6)^99 =~ 8.4 * 10^-8 Am I right? What do I win? === I'll guess For a particular die face (doesn't matter when it happens) it would > have to not happen 99 times, thus (5/6)^99 for one die face (i.e.number) But there are six die faces (same odds for each and they're > independent)... 6 ( 5/6)^99 =~ 8.4 * 10^-8 Am I right? ÊWhat do I win? Nope, I'm not right - I'm high by a factor of 6.... (See why?) so the answer is .....1.4*E-8 === Subject: Re: probability question > > I have a probablity question: > Suppose we throw a dice 100 times. What is the probability that at > least 1 number appears exactly once. > I would be grateful for your help, > > eugene > I'll guess > For a particular die face (doesn't matter when it happens) it would > have to not happen 99 times, thus > (5/6)^99 for one die face (i.e.number) > But there are six die faces (same odds for each and they're > independent)... > 6 ( 5/6)^99 =~ 8.4 * 10^-8 > Am I right? ÊWhat do I win? > > Nope, I'm not right - I'm high by a factor of 6.... (See why?) No, in fact you're low. Think about just the probability p_1 of getting exactly one 1 in 100 rolls. The 1 can go into 100 slots, each time with probability 1/6. Once the 1 is placed, the other 99 slots are randomly chosen from {2, ..., 6}. So p_1 = 100*(1/6)*(5/6)^99. > so the answer is .....1.4*E-8 === Subject: Re: probability question > I have a probablity question: > Suppose we throw a dice 100 times. What is the probability that at > least 1 number appears exactly once. > I would be grateful for your help, > I'll guess > For a particular die face (doesn't matter when it happens) it would > have to not happen 99 times, thus > (5/6)^99 for one die face (i.e.number) > But there are six die faces (same odds for each and they're > independent)... > 6 ( 5/6)^99 =~ 8.4 * 10^-8 > Am I right? ÊWhat do I win? > Nope, I'm not right - I'm high by a factor of 6.... (See why?) No, in fact you're low. Think about just the probability p 1 of > getting exactly one 1 in 100 rolls. The 1 can go into 100 slots, each > time with probability 1/6. Once the 1 is placed, the other 99 slots > are randomly chosen from {2, ..., 6}. So p 1 = 100*(1/6)*(5/6)^99. > so the answer is .....1.4*E-8- Hide quoted text - - Show quoted text -- Hide quoted text - - Show quoted text - Yes,..the factor of 100 is needed. 1.4E-6 There are a hundred ways ...so 100 (1/6)(5/6)^99 The OP asked about *at least* one number appears only once, so I think we've covered all the basis with this answer. How the 99 are distributed among the 5 remaining numbers, we could care less. === Subject: Re: probability question > On Sep 9, 9:07Êpm, The World Wide Wade I have a probablity question: > Suppose we throw a dice 100 times. What is the probability that at > least 1 number appears exactly once. > I would be grateful for your help, > > eugene > I'll guess > For a particular die face (doesn't matter when it happens) it would > have to not happen 99 times, thus > (5/6)^99 for one die face (i.e.number) > But there are six die faces (same odds for each and they're > independent)... > 6 ( 5/6)^99 =~ 8.4 * 10^-8 > Am I right? ÊWhat do I win? > Nope, I'm not right - I'm high by a factor of 6.... (See why?) > No, in fact you're low. Think about just the probability p_1 of > getting exactly one 1 in 100 rolls. The 1 can go into 100 slots, each > time with probability 1/6. Once the 1 is placed, the other 99 slots > are randomly chosen from {2, ..., 6}. So p_1 = 100*(1/6)*(5/6)^99. > > so the answer is .....1.4*E-8- Hide quoted text - > - Show quoted text -- Hide quoted text - > - Show quoted text - > > Yes,..the factor of 100 is needed. 1.4E-6 There are a hundred > ways ...so 100 (1/6)(5/6)^99 > > The OP asked about *at least* one number appears only once, so I think > we've covered all the basis with this answer. How the 99 are > distributed among the 5 remaining numbers, we could care less. No certainly not. What about a 2 on the first roll and all 1's thereafter? See my other post in this thread. === Subject: Re: probability question > I have a probablity question: > Suppose we throw a dice 100 times. What is the probability that at > least 1 number appears exactly once. > I would be grateful for your help, > eugene > I'll guess > For a particular die face (doesn't matter when it happens) it would > have to not happen 99 times, thus > (5/6)^99 for one die face (i.e.number) > But there are six die faces (same odds for each and they're > independent)... > 6 ( 5/6)^99 =~ 8.4 * 10^-8 > Am I right? ÊWhat do I win? > Nope, I'm not right - I'm high by a factor of 6.... (See why?) > No, in fact you're low. Think about just the probability p 1 of > getting exactly one 1 in 100 rolls. The 1 can go into 100 slots, each > time with probability 1/6. Once the 1 is placed, the other 99 slots > are randomly chosen from {2, ..., 6}. So p 1 = 100*(1/6)*(5/6)^99. > so the answer is .....1.4*E-8- Hide quoted text - > - Show quoted text -- Hide quoted text - > - Show quoted text - > Yes,..the factor of 100 is needed. Ê 1.4E-6 ÊThere are a hundred > ways ...so 100 (1/6)(5/6)^99 > The OP asked about *at least* one number appears only once, so I think > we've covered all the basis with this answer. ÊHow the 99 are > distributed among the 5 remaining numbers, we could care less. No certainly not. What about a 2 on the first roll and all 1's > thereafter? See my other post in this thread.- Hide quoted text - - Show quoted text - I don't understand. That case is covered by the 100*(1/6)*(5/6)^99 calculation. Once we have calculated the probability of having a particular number appearing only once, the multiplicity of the rest of the 99 doesn't matter. If the OP had asked for the probability of a number appearing only once and then gave conditions on the rest of the 99 then we would further work to do. But the OP didn't say that each number had to appear at least once. === Subject: Re: probability question > I have a probablity question: > Suppose we throw a dice 100 times. What is the probability that at > least 1 number appears exactly once. > I would be grateful for your help, > I'll guess > For a particular die face (doesn't matter when it happens) it would > have to not happen 99 times, thus > (5/6)^99 for one die face (i.e.number) > But there are six die faces (same odds for each and they're > independent)... > 6 ( 5/6)^99 =~ 8.4 * 10^-8 > Am I right? ÊWhat do I win? Nope, I'm not right - I'm high by a factor of 6.... (See why?) so the answer is .....1.4*E-8- Hide quoted text - - Show quoted text - Raphanus, you are still not right. Let us consider rolling a 1 precisely once in 100 tosses. The 1 could be rolled either on the first roll, the second roll, the third roll, etc. The probability of any of these events occuring is, of course, (1/6) * (5/6)^99, but it can happen 100 different (disjoint events) ways, so the probability of rolling precisely one 1 in 100 tosses is 100 * (1/6) *(5/6)^99. Unfortunately the events, rolling precisely one 1, rolling precisely one 2, ..., rolling precisely one 6, are not disjoint events. Therefore, you really have to work harder. I leave it to you to contain working on it because I am much too lazy and have other things to do right at the moment. Achava === Subject: Re: probability question > > I have a probablity question: > Suppose we throw a dice 100 times. What is the probability that at > least 1 number appears exactly once. > I would be grateful for your help, > > eugene > I'll guess > For a particular die face (doesn't matter when it happens) it would > have to not happen 99 times, thus > (5/6)^99 for one die face (i.e.number) > But there are six die faces (same odds for each and they're > independent)... > 6 ( 5/6)^99 =3D~ 8.4 * 10^-8 > Am I right? =A0What do I win? > Nope, I'm not right - I'm high by a factor of 6.... (See why?) > so the answer is .....1.4*E-8- Hide quoted text - > - Show quoted text - > > Raphanus, you are still not right. Let us consider rolling a 1 > precisely once in 100 tosses. The 1 could be rolled either on the > first roll, the second roll, the third roll, etc. The probability of > any of these events occuring is, of course, (1/6) * (5/6)^99, but it > can happen 100 different (disjoint events) ways, so the probability of > rolling precisely one 1 in 100 tosses is 100 * (1/6) *(5/6)^99. > Unfortunately the events, rolling precisely one 1, rolling precisely > one 2, ..., rolling precisely one 6, are not disjoint events. > Therefore, you really have to work harder. I leave it to you to > contain working on it because I am much too lazy and have other things > to do right at the moment. For each of the six faces, the probability of that face coming up exactly once is a_1 = 100 * (1/6) * (5/6)^99 For each of the (6 choose 2) = 15 pairs, the probability of those coming up exactly once each is a_2 = 100 * 99 * (1/6)^2 * (4/6)^98 For each of the (6 choose 3) = 20 triples, the probability of those coming up exactly once each is a_3 = 100 * 99 * 98 * (1/6)^3 * (3/6)^97 For each of the (6 choose 4) = 15 quadruples, the probability of those coming up exactly once each is a_4 = 100 * 99 * 98 * 97 * (1/6)^4 * (2/6)^96 For each of the 6 quintuples, the probability of those coming up exactly once each is a_5 = 100 * 99 * 98 * 97 * 96 * (1/6)^5 * (1/6)^95 It is impossible for all 6 to come up exactly once each. By the inclusion-exclusion principle, the probability of at least one face coming up exactly once is 6 a_1 - 15 a_2 + 20 a_3 - 15 a_4 + 6 a_5 = 39443044639617518482737726181871382116510270399912518309933101857486225/ 2722160931250295442069542779825240918904765460303978631429466526539571422822 4 or approximately 1.4489607938609723460 * 10^(-6). -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: probability question posting-account=eFTX7goAAACJBRoHHOnDC9IuTZeZT1_H 1.1.4322),gzip(gfe),gzip(gfe) On Sep 10, 12:13Êam, Robert Israel > I have a probablity question: > Suppose we throw a dice 100 times. What is the probability that at > least 1 number appears exactly once. > I would be grateful for your help, > eugene > I'll guess > For a particular die face (doesn't matter when it happens) it would > have to not happen 99 times, thus > (5/6)^99 for one die face (i.e.number) > But there are six die faces (same odds for each and they're > independent)... > 6 ( 5/6)^99 =3D~ 8.4 * 10^-8 > Am I right? =A0What do I win? > Nope, I'm not right - I'm high by a factor of 6.... (See why?) > so the answer is .....1.4*E-8- Hide quoted text - > - Show quoted text - > Raphanus, you are still not right. ÊLet us consider rolling a 1 > precisely once in 100 tosses. ÊThe 1 could be rolled either on the > first roll, the second roll, the third roll, etc. ÊThe probability of > any of these events occuring is, of course, (1/6) * (5/6)^99, but it > can happen 100 different (disjoint events) ways, so the probability of > rolling precisely one 1 in 100 tosses is 100 * (1/6) *(5/6)^99. > Unfortunately the events, rolling precisely one 1, rolling precisely > one 2, ..., rolling precisely one 6, are not disjoint events. > Therefore, you really have to work harder. ÊI leave it to you to > contain working on it because I am much too lazy and have other things > to do right at the moment. For each of the six faces, the probability of that face coming up exactly > once is a 1 = 100 * (1/6) * (5/6)^99 Ê > For each of the (6 choose 2) = 15 pairs, the probability of those coming up > exactly once each is a 2 = 100 * 99 * (1/6)^2 * (4/6)^98 > For each of the (6 choose 3) = 20 triples, the probability of those coming up > exactly once each is a 3 = 100 * 99 * 98 * (1/6)^3 * (3/6)^97 > For each of the (6 choose 4) = 15 quadruples, the probability of those coming > up exactly once each is a 4 = 100 * 99 * 98 * 97 * (1/6)^4 * (2/6)^96 > For each of the 6 quintuples, the probability of those coming up exactly once > each is a 5 = 100 * 99 * 98 * 97 * 96 * (1/6)^5 * (1/6)^95 > It is impossible for all 6 to come up exactly once each. > By the inclusion-exclusion principle, the probability of at least one face > coming up exactly once is > 6 a 1 - 15 a 2 + 20 a 3 - 15 a 4 + 6 a 5 > = Ê394430446396175184827377261818713821165102703999125183099331018 57486225/ > 272216093125029544206954277982524091890476546030397863142946652653957142282[ CapitalEth]24 > or approximately 1.4489607938609723460 * 10^(-6). Ê > -- > Robert Israel Ê Ê Ê Ê Ê Ê Êisr...@math.MyUniversitysInitials.ca > Department of Mathematics Ê Ê Ê Êhttp://www.math.ubc.ca/~israel > University of British Columbia Ê Ê Ê Ê Ê ÊVancouver, BC, Canada- Hide quoted text - - Show quoted text - Yeah,...what Robert said. === Subject: Re: probability question posting-account=plkOvwoAAABPOKn1e8t3zJKYVRn7nmuq Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) On Sep 10, 6:13 am, Robert Israel > I have a probablity question: > Suppose we throw a dice 100 times. What is the probability that at > least 1 number appears exactly once. > I would be grateful for your help, > eugene > I'll guess > For a particular die face (doesn't matter when it happens) it would > have to not happen 99 times, thus > (5/6)^99 for one die face (i.e.number) > But there are six die faces (same odds for each and they're > independent)... > 6 ( 5/6)^99 =3D~ 8.4 * 10^-8 > Am I right? =A0What do I win? > Nope, I'm not right - I'm high by a factor of 6.... (See why?) > so the answer is .....1.4*E-8- Hide quoted text - > - Show quoted text - > Raphanus, you are still not right. Let us consider rolling a 1 > precisely once in 100 tosses. The 1 could be rolled either on the > first roll, the second roll, the third roll, etc. The probability of > any of these events occuring is, of course, (1/6) * (5/6)^99, but it > can happen 100 different (disjoint events) ways, so the probability of > rolling precisely one 1 in 100 tosses is 100 * (1/6) *(5/6)^99. > Unfortunately the events, rolling precisely one 1, rolling precisely > one 2, ..., rolling precisely one 6, are not disjoint events. > Therefore, you really have to work harder. I leave it to you to > contain working on it because I am much too lazy and have other things > to do right at the moment. For each of the six faces, the probability of that face coming up exactly > once is a_1 = 100 * (1/6) * (5/6)^99 > For each of the (6 choose 2) = 15 pairs, the probability of those coming up > exactly once each is a_2 = 100 * 99 * (1/6)^2 * (4/6)^98 > For each of the (6 choose 3) = 20 triples, the probability of those coming up > exactly once each is a_3 = 100 * 99 * 98 * (1/6)^3 * (3/6)^97 > For each of the (6 choose 4) = 15 quadruples, the probability of those coming > up exactly once each is a_4 = 100 * 99 * 98 * 97 * (1/6)^4 * (2/6)^96 > For each of the 6 quintuples, the probability of those coming up exactly once > each is a_5 = 100 * 99 * 98 * 97 * 96 * (1/6)^5 * (1/6)^95 > It is impossible for all 6 to come up exactly once each. > By the inclusion-exclusion principle, the probability of at least one face > coming up exactly once is > 6 a_1 - 15 a_2 + 20 a_3 - 15 a_4 + 6 a_5 > = 39443044639617518482737726181871382116510270399912518309933101857486225/ > 27221609312502954420695427798252409189047654603039786314294665265395714228224 > or approximately 1.4489607938609723460 * 10^(-6). > -- > Department of Mathematics http://www.math.ubc.ca/~israel === Subject: Re: Quadratic Diophantine Theorem 7. Summary Starting with Euler in [Eul1773], many authors have come up with essentially the same idea: solving the Pell equation X2 - dY 2 = 1 (29) becomes easier by looking at certain auxiliary equations. The first step is writing down Legendre's equations rx2 - sy2 = 1, 2 for d = rs. (30) There is one nontrivial equation among these with a solution, and the smallest solution will have about half as many digits as the smallest solution of (29). The second step is to look at equations whose solutions give rise to a solution of one of the equations (30); but this second step has never been completed in full generality. What we have are specific equations applicable only in special situations; these were first discovered by Euler, rediscovered by Hart, Sylvester, G¬unther, G«erardin, Hardy & Williams, and Arteha, and slightly generalized by Bapoungu«e. HIGHER DESCENT ON PELL CONICS 11 References http://www.fen.bilkent.edu.tr/~franz/publ/pell-b.pdf Tuesday, July 27, 1638 Sir, The receipt of your letter in which you do me the honor of promising your friendship, gave me no less joy than if I had received it from a mistress whose good graces I passionately desired. And your other writings which preceded it reminded me of the Bradamante of our poets, who would take no one as a servant who had not previously proved himself against her in combat. It is not every day that I compare myself to that Roger who was the only one in the world capable of standing against her in combat; but such as I am, I assure you that I honor your merit considerably. And seeing the most recent method that you used to find tangents to curved lines [in your last letter], I have nothing to say in response, other than that it is very good, and that, if you had explained it that way in the beginning, I would have had no disagreement. The remainder of the letter deals specifically with geometry, and has not been included here. http://www.wlym.com/~seattle/dynamis/issues/august08-fermat.pdf thus: AP, you nut.... actually, it is a commonplace hearsay that the founding parental units were atheists, usually called Deists -- I'm the Higher Power of the 13th Step -- ME ?!? ... but it isn't true. if you read LaRouche, you'd know that, but there is some question about Shakespeare as a Third Language. thus: Apostol begins the course with integration, as per Liebniz; there's a special name for that, per Liebniz: primitive equation or prime equation? thus: this was the subject ot the keynote speaker at the Ninth Annual Nonlinear Science Conference at UCLA, although I no longer blame Newton exclusively for the controversy with Liebniz. anyway, apparently, Hooke did not do the math in a formalistic way; after all, he was the Royal Society's experimenter. the other part of the joke is that the '99 2-pound coin is inscribed, On the shoulders of giants. on the edge; at least, Newton was effective in his office-of-reward. > stole the solution from Hooke, to boot -- on the shoulders of a giant, > who was physically a dwarf; hence, the joke. thus: ah, Lord Berty's thing was called the Unity of Sciences; reference: http://www.larouchepub.com/other/2002/2949moonification.html The two operations of Wells and Russell from which Moon sprung are these: The Moral Re-Armament Movement, founded at a 1921 meeting between a wacky Lutheran preacher from Philadelphia and two British delegates to the Washington Disarmament Conference, Lord Arthur Balfour and H.G. Wells. Moral Re-Armament became the mass organizational vehicle for implementation of Wells' 1928 call in The Open Conspiracy, for a worldwide movement for draft resistance. The environment of Moon's Korean ministry was under control of Moral Re-Armament when he was picked up as an intelligence asset during the Korean War. The Unity of the Sciences movement. Founded in 1935 under the supervision of Lord Bertrand Russell and John Dewey, it brought together Trotskyite academics Albert Wohlstetter (mentor of current Defense Policy Board Chairman Richard Perle), Sidney Hook, and Ernest Nagel, with members of the radical-positivist Vienna Circle. Merging with Robert M. Hutchins at the University of Chicago in the 1950s, this operation took over the teaching of science in the United States. Thomas Kuhn's widely read fraud, The Structure of Scientific Revolutions, was published as the second volume of their Encyclopedia of Unified Sciences. In 1972, the Moonies were given the Unity of Sciences franchise, sponsoring the first of their still-ongoing International Conferences of the Unity of Sciences. Their early sessions featured such notables as Manhattan Project physicist Eugene Wigner, the lifelong ally of that truly mad scientist Leo Szilard (the model for Dr. Strangelove, in Stanley Kubrick's film of that name), and environmental fascists Alexander King and Aurelio Peccei, founders of the no-growth Club of Rome. Before looking back to the history of these projects, let us first briefly dispense with the person of Rev. Sun Myung Moon. Moon as a personality is of very little importance, in himself. The real Reverend Moon is a pathetic, if nonetheless nasty, victim of Japanese internment and North Korean torture sessions. He is what the.... thus: an aether simply provides a prealistical model of the phenomenon of waves; one need not deal with photons at all, since they are just formally dual to waves, not at all pardoxical, as shown by Dirac. far too much of theory rests upon Einstein, saying that's impossible, which is the reality behind Michelson-Morley and those who carried the examination further -- but, herr doktor-professor Albert said, Nein!, when he was once in his dysused office at Caltech. let us remember him for those things which he was really good at, not the Big Bang Constant and its infernal klingons (fridge magnets) ?! > I don't need any aether to stick a magnet to a fridge, light up an thus: so, to ask an ignorant question of personal interest, how much of this could be used toward a new or old proof of quadratic reciprocity? > That Ac and -Dc must be quadratic residues, mod D and A, respectively, > is a trivial result: > Let Ax^2 - Dy^2 = c. [1] > Take this mod A: > -Dy^2 = c mod A > (Dy)^2 = -Dc mod A > Therefore, -Dc is a quadratic residue mod A. > Similarly, taking [1] mod D: > Ax^2 = c mod D > (Ax)^2 = Ac mod D > Therefore, Ac is a quadratic residue mod D. thus: ah, sort of a solopsistic monad. I am reminded of hte Many Words Interpretation of the Copenhagenskool: to be or not to be that is the shape of the box or the bubble -- inside or out? > Q. What is a tautological space? > A. A tautological space. thus: speaking of Young's anhialation of Newton's photons, here is the earlier elaboration on light by Fermat: http://www.wlym.com/~seattle/dynamis/issues/august08-fermat.pdf > as we know from Newton's iron-poor corpuscles. --ROTC, your summer vacation in the Sahara Desert ( S u d a n ) ; presage the Draft for your middleschool class of '12 -- brought to you by Allstate (tm) and Oxford U. Press! http://larouchepub.com/pr/2008/080813moloch brown.html http://wlym.com === Subject: Re: Short Mars travel times at high speed. posting-account=jPnQ2goAAAA461y3QD0lbyw0oKeThma1 AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.2 Safari/525.20.1,gzip(gfe),gzip(gfe) cannot tack into the solar wind? well, you could always burn the sail for reaction-mass! 7. Summary Starting with Euler in [Eul1773], many authors have come up with essentially the same idea: solving the Pell equation X2 - dY 2 = 1 (29) becomes easier by looking at certain auxiliary equations. The first step is writing down Legendre's equations rx2 - sy2 = 1, 2 for d = rs. (30) There is one nontrivial equation among these with a solution, and the smallest solution will have about half as many digits as the smallest solution of (29). The second step is to look at equations whose solutions give rise to a solution of one of the equations (30); but this second step has never been completed in full generality. What we have are specific equations applicable only in special situations; these were first discovered by Euler, rediscovered by Hart, Sylvester, G¬unther, G«erardin, Hardy & Williams, and Arteha, and slightly generalized by Bapoungu«e. HIGHER DESCENT ON PELL CONICS 11 References http://www.fen.bilkent.edu.tr/~franz/publ/pell-b.pdf Tuesday, July 27, 1638 Sir, The receipt of your letter in which you do me the honor of promising your friendship, gave me no less joy than if I had received it from a mistress whose good graces I passionately desired. And your other writings which preceded it reminded me of the Bradamante of our poets, who would take no one as a servant who had not previously proved himself against her in combat. It is not every day that I compare myself to that Roger who was the only one in the world capable of standing against her in combat; but such as I am, I assure you that I honor your merit considerably. And seeing the most recent method that you used to find tangents to curved lines [in your last letter], I have nothing to say in response, other than that it is very good, and that, if you had explained it that way in the beginning, I would have had no disagreement. The remainder of the letter deals specifically with geometry, and has not been included here. http://www.wlym.com/~seattle/dynamis/issues/august08-fermat.pdf thus: AP, you nut.... actually, it is a commonplace hearsay that the founding parental units were atheists, usually called Deists -- I'm the Higher Power of the 13th Step -- ME ?!? ... but it isn't true. if you read LaRouche, you'd know that, but there is some question about Shakespeare as a Third Language. thus: Apostol begins the course with integration, as per Liebniz; there's a special name for that, per Liebniz: primitive equation or prime equation? thus: this was the subject ot the keynote speaker at the Ninth Annual Nonlinear Science Conference at UCLA, although I no longer blame Newton exclusively for the controversy with Liebniz. anyway, apparently, Hooke did not do the math in a formalistic way; after all, he was the Royal Society's experimenter. the other part of the joke is that the '99 2-pound coin is inscribed, On the shoulders of giants. on the edge; at least, Newton was effective in his office-of-reward. > stole the solution from Hooke, to boot -- on the shoulders of a giant, > who was physically a dwarf; hence, the joke. thus: ah, Lord Berty's thing was called the Unity of Sciences; reference: http://www.larouchepub.com/other/2002/2949moonification.html The two operations of Wells and Russell from which Moon sprung are these: The Moral Re-Armament Movement, founded at a 1921 meeting between a wacky Lutheran preacher from Philadelphia and two British delegates to the Washington Disarmament Conference, Lord Arthur Balfour and H.G. Wells. Moral Re-Armament became the mass organizational vehicle for implementation of Wells' 1928 call in The Open Conspiracy, for a worldwide movement for draft resistance. The environment of Moon's Korean ministry was under control of Moral Re-Armament when he was picked up as an intelligence asset during the Korean War. The Unity of the Sciences movement. Founded in 1935 under the supervision of Lord Bertrand Russell and John Dewey, it brought together Trotskyite academics Albert Wohlstetter (mentor of current Defense Policy Board Chairman Richard Perle), Sidney Hook, and Ernest Nagel, with members of the radical-positivist Vienna Circle. Merging with Robert M. Hutchins at the University of Chicago in the 1950s, this operation took over the teaching of science in the United States. Thomas Kuhn's widely read fraud, The Structure of Scientific Revolutions, was published as the second volume of their Encyclopedia of Unified Sciences. In 1972, the Moonies were given the Unity of Sciences franchise, sponsoring the first of their still-ongoing International Conferences of the Unity of Sciences. Their early sessions featured such notables as Manhattan Project physicist Eugene Wigner, the lifelong ally of that truly mad scientist Leo Szilard (the model for Dr. Strangelove, in Stanley Kubrick's film of that name), and environmental fascists Alexander King and Aurelio Peccei, founders of the no-growth Club of Rome. Before looking back to the history of these projects, let us first briefly dispense with the person of Rev. Sun Myung Moon. Moon as a personality is of very little importance, in himself. The real Reverend Moon is a pathetic, if nonetheless nasty, victim of Japanese internment and North Korean torture sessions. He is what the.... thus: an aether simply provides a prealistical model of the phenomenon of waves; one need not deal with photons at all, since they are just formally dual to waves, not at all pardoxical, as shown by Dirac. far too much of theory rests upon Einstein, saying that's impossible, which is the reality behind Michelson-Morley and those who carried the examination further -- but, herr doktor-professor Albert said, Nein!, when he was once in his dysused office at Caltech. let us remember him for those things which he was really good at, not the Big Bang Constant and its infernal klingons (fridge magnets) ?! > I don't need any aether to stick a magnet to a fridge, light up an thus: so, to ask an ignorant question of personal interest, how much of this could be used toward a new or old proof of quadratic reciprocity? > That Ac and -Dc must be quadratic residues, mod D and A, respectively, > is a trivial result: > Let Ax^2 - Dy^2 = c. [1] > Take this mod A: > -Dy^2 = c mod A > (Dy)^2 = -Dc mod A > Therefore, -Dc is a quadratic residue mod A. > Similarly, taking [1] mod D: > Ax^2 = c mod D > (Ax)^2 = Ac mod D > Therefore, Ac is a quadratic residue mod D. thus: ah, sort of a solopsistic monad. I am reminded of hte Many Words Interpretation of the Copenhagenskool: to be or not to be that is the shape of the box or the bubble -- inside or out? > Q. What is a tautological space? > A. A tautological space. thus: speaking of Young's anhialation of Newton's photons, here is the earlier elaboration on light by Fermat: http://www.wlym.com/~seattle/dynamis/issues/august08-fermat.pdf > as we know from Newton's iron-poor corpuscles. --ROTC, your summer vacation in the Sahara Desert ( S u d a n ) ; presage the Draft for your middleschool class of '12 -- brought to you by Allstate (tm) and Oxford U. Press! http://larouchepub.com/pr/2008/080813moloch brown.html http://wlym.com === Subject: Re: Short Mars travel times at high speed. http://www.allentwood.com/essays/lordofflies.html My path to the Unification Church began in 1966 when I matriculated at the University of the South in Sewanee, Tennessee. I entered college with little concrete notion of what I wanted to do in life. I had rejected the establishment rewards that my secondary education promised. To me the academic challenge at Sewanee was meaningless. Absorbed in a search for values by which I could live, the Peace steps of the burned out ROTC building at the University of the South in the spring of 1968, I was the first student to speak publicly against the War at Sewanee. That summer in New Jersey I worked for the Presidential campaign of Senator Eugene J. McCarthy. In August I went to The National Democratic convention in Chicago as a messenger for the New Jersey delegation. The ferocity of the police riots during the convention destroyed what lingering faith I had in the political process, and I returned to college in the fall bewildered by our nati onal leaders' refusal to respond to the people's wish to end the war in Vietnam. Unable to concentrate on my studies, I began to explore Eastern religions, specifically the religious synthesism of RamaKrishna. In the spring of 1969 I dropped out of the University of the South and hitchhiked to California, a stop on my way to India. In May I arrived in Berkeley, with a suitcase and a navy duffel bag and a copy of the Gospel According to Rama Krishna. I didn't know it, but it was the end of the line. === Subject: Re: Short Mars travel times at high speed. posting-account=BEQ9kgkAAAA_-zTZLFIXg8hu7ZTTi4Q8 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > http://www.allentwood.com/essays/lordofflies.html My path to the Unification Church began in 1966 when I matriculated at > the University of the South in Sewanee, Tennessee. I entered college > with little concrete notion of what I wanted to do in life. I had > rejected the establishment rewards that my secondary education > promised. To me the academic challenge at Sewanee was meaningless. > Absorbed in a search for values by which I could live, the Peace > steps of the burned out ROTC building at the University of the South > in the spring of 1968, I was the first student to speak publicly > against the War at Sewanee. That summer in New Jersey I worked for the > Presidential campaign of Senator Eugene J. McCarthy. In August I went > to The National Democratic convention in Chicago as a messenger for > the New Jersey delegation. The ferocity of the police riots during the > convention destroyed what lingering faith I had in the political > process, and I returned to college in the fall bewildered by our nati > onal leaders' refusal to respond to the people's wish to end the war > in Vietnam. Unable to concentrate on my studies, I began to explore Eastern > religions, specifically the religious synthesism of RamaKrishna. In > the spring of 1969 I dropped out of the University of the South and > hitchhiked to California, a stop on my way to India. In May I arrived > in Berkeley, with a suitcase and a navy duffel bag and a copy of the > Gospel According to Rama Krishna. I didn't know it, but it was the end > of the line. Okay, when the effects of the LSD wore off, what happened next? === Subject: Re: Short Mars travel times at high speed. posting-account=jPnQ2goAAAA461y3QD0lbyw0oKeThma1 AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.2 Safari/525.20.1,gzip(gfe),gzip(gfe) wow, maybe you could advise me. so, which is better, LSD or mild ergot poisoning -- will the latter flashforward to cure flashbacks from the former? most of which I did not read & that I just noticed, is, that little statement about synthesizm; no synesthezia required? also, Moog just came-out with a guitar (saw the ad; at 6500US, it's time to roll one's own .-) >http://www.allentwood.com/essays/lordofflies.html > religions, specifically the religious synthesism of RamaKrishna. In > Okay, when the effects of the LSD wore off, what happened next? thus quoth: Fannie Mae was created by President Franklin D. Roosevelt in 1938, as a government agency to buy mortgages from lenders, as a way of funding the purchase of homes in the Great Depression. In more recent years, Fannie Mae and its sibling Freddie Mac, were taken over by what FDR attacked as the economic royalists, and turned into vehicles for derivatives speculation. Under the great Greenspan bubble, Fannie and Freddie were turned into money machines to feed the run-up in real estate values to provide assetsÑin the form of mortgage debtÑas fuel to the derivatives markets. This scheme was bound to fail, as it spectacularly has, leaving Fannie and Freddie, and the U.S. banking system, utterly bankrupt. However, Fannie and Freddie are at the heart of Treasury Secretary Henry Paulson's and Federal Reserve chairman Ben Bernanke's insane scheme to bail out the banks by dumping all their bad mortgage paper into the two government-sponsored enterprises, effectively transferring the banks' losses to the government, and ultimately to the taxpayer. The government is not really bailing out Fannie and Freddie, but merely funding their conversion into the largest toxic waste dumps in history. Far from being saved, Fannie Mae and Freddie Mac are being destroyed. http://larouchepub.com/other/editorials/2008/3536tantamount treason.html === Subject: Re: Analysis with integral before.. >> |1 log(x) >> | ------ dx >> | 0 1-x >> |oo t e^{-t} >> = - | -------- dt >> | 0 1-e^{-t} >> |oo -t -2t -3t -4t >> = - | t ( e + e + e + e + ... ) dt >> | 0 >> oo >> --- 1 |oo -t >> = - > --- | t e dt --------(*****) >> --- k^2 | 0 >> k=1 >> For that the substitution u = kt was used. >> Of course, this only explains why is it that >> oo >> --- |oo -kt >> - > | t e dt (*) >> --- | 0 >> k=1 >> is equal to >> oo >> --- 1 |oo -t >> - > --- | t e dt >> --- k^2 | 0 >> k=1 >> But it doesn't explain why (*) is also equal to: >> |oo -t -2t -3t -4t >> - | t ( e + e + e + e + ... ) dt >> | 0 > Hm... it's not easy. > S(x) = Sum{n=1 to oo} x.e^{-nx} is not uniformly convergence on [0, oo). > Because, > If x = 0 , then S(x) = 0 > If x =/= 0 , then S(x) = x / (e^x - 1) --> 1 as x --> 0. > so, not continuous. > so, not uniformly convergence. > But > int{0 to 1} log[x] / (1-x) dx = -(pi)^2 / 6 is true by mathematica. > > Ok, I already showed that int{0 to 1} log(x) / (1-x) dx exists. > > so, int{0 to 1} log(x) / (1-x) dx > = lim{b->1-} int{0 to b} log(x) / (1-x) dx > Let x = e^{-t}. > = lim{b->1-} (-)int{-log(b) to oo} {t.e^(-t)}/{1-e^(-t)} dt > = lim{b->1-} (-)int{-log(b) to oo} t[e^(-t) + e^(-2t) + e^(-3t) + ...] dt > = lim{b->1-} (-)int{-log(b) to oo} sum{n=1 to oo} t.e^(-nt) dt > Of course, -log(b) > 0. > > Sum{n=1 to oo} x.e^(-nx) is uniformly convergence on [-log(b), oo) > by Weierstrass M-test. > so, int <==> sum (exchangeable) False. Uniform convergence on an infinite interval does not imply int <==> sum can be interchanged. > so, we can calculate it as Rob Johnson. > > ---------------------------------------------------------------------- > OR) > int{0 to 1} log(x) / (1-x) dx > = lim{a->0+, b->1-} int{a to b} log(x) / (1-x) dx > = lim{a->0+, b->1-} int{a to b} sum{k=0 to oo} (x^k).log(x) dx > by log(x) / (1-x) = sum{k=0 to oo} (x^k).log(x), > > Sum{k=0 to oo} (x^k).log(x) is uniformly convergence on [a, b] > by Weierstrass M-test. > > so, int <==> sum (exchangeable) > > Calculate the int{a to b} (x^k).log(x) dx. > Integrating by parts, > = [{x^(k+1)/(k+1)}*log(x)]_{a to b} - int{a to b} (x^k)/(k+1) dx > = -1/(k+1)^2 as a -> 0 , b -> 1. > > Consequently, > answer is -sum{k=1 to oo} 1/k^2 No, you still have to show sum and limit can be interchanged. A nice way to see int{0 to 1} log(x) /(1-x) dx = sum{k=0 to oo} int{0 to 1} x^k log(x) dx is to use the dominated convergence theorem. === Subject: Re: Analysis with integral before.. >> |1 log(x) >> | ------ dx >> | 0 1-x >> |oo t e^{-t} >> = - | -------- dt >> | 0 1-e^{-t} >> |oo -t -2t -3t -4t >> = - | t ( e + e + e + e + ... ) dt >> | 0 >> oo >> --- 1 |oo -t >> = - > --- | t e dt --------(*****) >> --- k^2 | 0 >> k=1 >> For that the substitution u = kt was used. >> Of course, this only explains why is it that >> oo >> --- |oo -kt >> - > | t e dt (*) >> --- | 0 >> k=1 >> is equal to >> oo >> --- 1 |oo -t >> - > --- | t e dt >> --- k^2 | 0 >> k=1 >> But it doesn't explain why (*) is also equal to: >> |oo -t -2t -3t -4t >> - | t ( e + e + e + e + ... ) dt >> | 0 > Hm... it's not easy. > S(x) = Sum{n=1 to oo} x.e^{-nx} is not uniformly convergence on [0, oo). > Because, > If x = 0 , then S(x) = 0 > If x =/= 0 , then S(x) = x / (e^x - 1) --> 1 as x --> 0. > so, not continuous. > so, not uniformly convergence. > But > int{0 to 1} log[x] / (1-x) dx = -(pi)^2 / 6 is true by mathematica. > > Ok, I already showed that int{0 to 1} log(x) / (1-x) dx exists. > > so, int{0 to 1} log(x) / (1-x) dx > = lim{b->1-} int{0 to b} log(x) / (1-x) dx > Let x = e^{-t}. > = lim{b->1-} (-)int{-log(b) to oo} {t.e^(-t)}/{1-e^(-t)} dt > = lim{b->1-} (-)int{-log(b) to oo} t[e^(-t) + e^(-2t) + e^(-3t) + ...] dt > = lim{b->1-} (-)int{-log(b) to oo} sum{n=1 to oo} t.e^(-nt) dt > Of course, -log(b) > 0. > > Sum{n=1 to oo} x.e^(-nx) is uniformly convergence on [-log(b), oo) > by Weierstrass M-test. > so, int <==> sum (exchangeable) False. Uniform convergence on an infinite interval does not imply int <==> sum can be interchanged. > so, we can calculate it as Rob Johnson. > > ---------------------------------------------------------------------- > OR) > int{0 to 1} log(x) / (1-x) dx > = lim{a->0+, b->1-} int{a to b} log(x) / (1-x) dx > = lim{a->0+, b->1-} int{a to b} sum{k=0 to oo} (x^k).log(x) dx > by log(x) / (1-x) = sum{k=0 to oo} (x^k).log(x), > > Sum{k=0 to oo} (x^k).log(x) is uniformly convergence on [a, b] > by Weierstrass M-test. > > so, int <==> sum (exchangeable) > > Calculate the int{a to b} (x^k).log(x) dx. > Integrating by parts, > = [{x^(k+1)/(k+1)}*log(x)]_{a to b} - int{a to b} (x^k)/(k+1) dx > = -1/(k+1)^2 as a -> 0 , b -> 1. > > Consequently, > answer is -sum{k=1 to oo} 1/k^2 No, you still have to show sum and limit can be interchanged. A nice way to see int{0 to 1} log(x) /(1-x) dx = sum{k=0 to oo} int{0 to 1} x^k log(x) dx is to use the dominated convergence theorem. === Subject: Re: Analysis with integral before.. > |1 log(x) > | ------ dx > | 0 1-x >> |oo t e^{-t} > = - | -------- dt > | 0 1-e^{-t} >> |oo -t -2t -3t -4t > = - | t ( e + e + e + e + ... ) dt > | 0 > oo > --- 1 |oo -t > = - > --- | t e dt --------(*****) > --- k^2 | 0 > k=1 >> For that the substitution u = kt was used. >> Of course, this only explains why is it that >> oo > --- |oo -kt > - > | t e dt (*) > --- | 0 > k=1 >> is equal to >> oo > --- 1 |oo -t > - > --- | t e dt > --- k^2 | 0 > k=1 >> But it doesn't explain why (*) is also equal to: >> |oo -t -2t -3t -4t > - | t ( e + e + e + e + ... ) dt > | 0 >> Hm... it's not easy. >> S(x) = Sum{n=1 to oo} x.e^{-nx} is not uniformly convergence on [0, >> oo). >> Because, >> If x = 0 , then S(x) = 0 >> If x =/= 0 , then S(x) = x / (e^x - 1) --> 1 as x --> 0. >> so, not continuous. >> so, not uniformly convergence. >> But >> int{0 to 1} log[x] / (1-x) dx = -(pi)^2 / 6 is true by mathematica. >> Ok, I already showed that int{0 to 1} log(x) / (1-x) dx exists. >> so, int{0 to 1} log(x) / (1-x) dx >> = lim{b->1-} int{0 to b} log(x) / (1-x) dx >> Let x = e^{-t}. >> = lim{b->1-} (-)int{-log(b) to oo} {t.e^(-t)}/{1-e^(-t)} dt >> = lim{b->1-} (-)int{-log(b) to oo} t[e^(-t) + e^(-2t) + e^(-3t) + ...] dt >> = lim{b->1-} (-)int{-log(b) to oo} sum{n=1 to oo} t.e^(-nt) dt >> Of course, -log(b) > 0. >> Sum{n=1 to oo} x.e^(-nx) is uniformly convergence on [-log(b), oo) >> by Weierstrass M-test. >> so, int <==> sum (exchangeable) False. Uniform convergence on an infinite interval does not imply int > <==> sum can be interchanged. Yes.. int{0 to 1} log(x) / (1-x) dx = lim{a->0+, b->1-} int{a to b} log(x) / (1-x) dx Let x = e^{-t}. = lim{a->0+, b->1-} (-)int{-log(b) to -log(a)} {t.e^(-t)}/{1-e^(-t)} dt = lim{a->0+, b->1-} (-)int{-log(b) to -log(a)} t[e^(-t) + e^(-2t) + e^(-3t) + ...] dt = lim{a->0+, b->1-} (-)int{-log(b) to -log(a)} sum{n=1 to oo} t.e^(-nt) dt >> so, we can calculate it as Rob Johnson. >> ---------------------------------------------------------------------- >> OR) >> int{0 to 1} log(x) / (1-x) dx >> = lim{a->0+, b->1-} int{a to b} log(x) / (1-x) dx >> = lim{a->0+, b->1-} int{a to b} sum{k=0 to oo} (x^k).log(x) dx >> by log(x) / (1-x) = sum{k=0 to oo} (x^k).log(x), >> Sum{k=0 to oo} (x^k).log(x) is uniformly convergence on [a, b] >> by Weierstrass M-test. >> so, int <==> sum (exchangeable) >> Calculate the int{a to b} (x^k).log(x) dx. >> Integrating by parts, >> = [{x^(k+1)/(k+1)}*log(x)]_{a to b} - int{a to b} (x^k)/(k+1) dx >> = -1/(k+1)^2 as a -> 0 , b -> 1. >> Consequently, >> answer is -sum{k=1 to oo} 1/k^2 No, you still have to show sum and limit can be interchanged. I already used the Weierstrass M-test by |(x^k).log(x)| <= (a^k).log(a) for 0 A nice way to see int{0 to 1} log(x) /(1-x) dx = sum{k=0 to oo} int{0 > to 1} x^k log(x) dx is to use the dominated convergence theorem. Yes, By dominated convergence theorem, int{0 to 1} lim{n->oo} sum{k=0 to n} x^k.log(x) dx = lim{n->oo} int{0 to 1} sum{k=0 to n} x^k.log(x) dx Because, Sum{k=0 to n} |x^k.log(x)| <= Sum{k=0 to n} |log(x)| on [0, 1] and Sum{k=0 to n} |log(x)| = n.|log(x)| and int{0 to 1} n.|log(x)| is integrable. === Subject: Re: Difference between P(a|b) and P(b|a) > Hi all > I m confuse about a probability quistion. I know there is a Difference between P(a|b) and P(b|a). But i cant understand. For example, > There is a probability that shooter can miss the target is 10%(event A). Also the probability that gun m.93ss the target (due to fault) is 15%(event B). answer is not same for P(a|b) and P(b|a). But when speaking both will give same meaning. Can someone clear my confusion. A picture often helps. Draw a circle, and put a cross inside it, offset from the centre of the circle. Call the area to the left of the vertical line A, and the area to the right A', meaning not A. Call the area above the horizontal line B, and the area below B' (not B). Note that every point of the circle (representing all possible events) is either in A or in A', and also either in B or in B'. The area of overlap between A and B is AB (both A and B are true). The probability of A given B, P(A|B), is equal to the fraction of the area B that the area AB takes up: P(A|B) = P(AB)/P(B) Similarly, the probability of B given A is the fraction of area A that the area AB takes up: P(B|A) = P(AB)/P(A) === Subject: Re: Difference between P(a|b) and P(b|a) Now i've understand. Here is an example. And i m sure u people will reply. Let p(a)=0.6 be the probability that the rocket will destroy during its flight. And p(b)0.02 be the probability that the engine will not start.What will the probability that the rocket will reach into space successfuly. === Subject: Re: Difference between P(a|b) and P(b|a) > Try another example ... Consider one roll of a die. Say A is: the result is 2, and B is: the result is even. So: P(A|B) = 1/3, but P(B|A)=1. That is a nice clear example. === Subject: Re: Algorithm collection >> I created a collection of algorithms which is available under >> http://seed7.sourceforge.net/algorith/algorith.htm > > People who use recursion to calculate Fibonacci numbers shouldn't be allowed > on sci.math. Or the Internet, for that matter. > (In case you don't know why, try your algorithm to calculate the 100th > Fibonacci number.) I can happily use recursion to calculate the 1000000th Fibonacci number. People who are unable to efficiently calculate Fibonacci numbers using a recursive algorithm shouldn't be allowed on comp.theory. Phil -- The fact that a believer is happier than a sceptic is no more to the point than the fact that a drunken man is happier than a sober one. The happiness of credulity is a cheap and dangerous quality. -- George Bernard Shaw (1856-1950), Preface to Androcles and the Lion === Subject: Re: Why does everyone do it? <177a7$48be4212$82a1e228$14394@news1.tudelft.nl> <87y729ihzh.fsf@phiwumbda.org> <8fd68$48be8174$82a1e228$7232@news1.tudelft.nl> <87od35maro.fsf@phiwumbda.org> <2891e$48bfc562$82a1e228$12161@news1.tudelft.nl> <8bd8a$48c4f6d9$82a1e228$23154@news1.tudelft.nl> <1hufc1lqfnx0q$.1us46cizd7gel$.dlg@40tude.net> <87abeiku0r.fsf@phiwumbda.org> <19ue9liygdw2a.5lx04p1nb73p$.dlg@40tude.net> posting-account=yxbZkgkAAABQBvyYeebYQ-PAvi0uT3tG Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) > So from > Ax(x = x) > we may not _conclude_ (using the rule of derivation UE) > George W. Bush = George W. Bush. There's an existential time component that is implied there. In one logical framework, GWB in 1988 is the same (person) as GWB in 2008 (where person has a specific meaning and implies a specific set of properties). In another framework, they are not the same person (where person implies, say, the collection of molecules, thoughts, memories, etc. of the person in question). The same can also be said about electrons. Electron A and B are identical if one of the properties they share is exists at location (x,y,z) at time t. Otherwise, they're not the same electron. === Subject: Re: Why does everyone do it? > >>So from >> Ax(x = x) >>we may not _conclude_ (using the rule of derivation UE) >> George W. Bush = George W. Bush. > > > There's an existential time component that is implied there. > > In one logical framework, GWB in 1988 is the same (person) > as GWB in 2008 (where person has a specific meaning and > implies a specific set of properties). > > In another framework, they are not the same person (where > person implies, say, the collection of molecules, thoughts, > memories, etc. of the person in question). > > The same can also be said about electrons. > Electron A and B are identical if one of the properties > they share is exists at location (x,y,z) at time t. > Otherwise, they're not the same electron. As I understand it, when available states are counted in solid state physics, you have to assume that electrons are identical for the numbers to come out right. If you mentally swap electrons between orbitals and count them as two two different states, then the predictions of various thermodynamic properties will be wildly wrong -- so this is not a mere philosophical quibble. On the other hand, some people have proposed that all the electrons (and positrons) in the universe are actually /one/ electron-positron moving backward and forward in time. (That I do consider mere philosophical quibbling, unless there is some physical consequence of this one-electron theory that I haven't heard about.) That would preserve the identity of indiscernibles, in this case, at least. If there are exceptions to the usual laws of identity (especially if the exceptions are confined to the Quantum Realm), I would find this very interesting but not the end of the world. It seems to me analogous to the classical addition of velocities working very well in almost all situations we'd be likely to use it, but needing correction at very high speeds. Jim Burns === Subject: Re: Why does everyone do it? <87od35maro.fsf@phiwumbda.org> <2891e$48bfc562$82a1e228$12161@news1.tudelft.nl> <8bd8a$48c4f6d9$82a1e228$23154@news1.tudelft.nl> <1hufc1lqfnx0q$.1us46cizd7gel$.dlg@40tude.net> <87abeiku0r.fsf@phiwumbda.org> <48C7F71B.7010009@osu.edu> posting-account=Yn5cwwoAAADntcMuRwk-EwLg-DMZ_hXN Gecko/20070509 Camino/1.5,gzip(gfe),gzip(gfe) >>So from >> Ax(x = x) >>we may not _conclude_ (using the rule of derivation UE) >> George W. Bush = George W. Bush. > There's an existential time component that is implied there. > In one logical framework, GWB in 1988 is the same (person) > as GWB in 2008 (where person has a specific meaning and > implies a specific set of properties). > In another framework, they are not the same person (where > person implies, say, the collection of molecules, thoughts, > memories, etc. of the person in question). > The same can also be said about electrons. > Electron A and B are identical if one of the properties > they share is exists at location (x,y,z) at time t. > Otherwise, they're not the same electron. As I understand it, when available states are counted in solid > state physics, you have to assume that electrons are > identical for the numbers to come out right. If you mentally > swap electrons between orbitals and count them as two > two different states, then the predictions of various > thermodynamic properties will be wildly wrong -- so this is > not a mere philosophical quibble. On the other hand, some people have proposed that all the > electrons (and positrons) in the universe are actually > /one/ electron-positron moving backward and forward in time. > (That I do consider mere philosophical quibbling, unless > there is some physical consequence of this one-electron > theory that I haven't heard about.) That would preserve > the identity of indiscernibles, in this case, at least. If there are exceptions to the usual laws of identity > (especially if the exceptions are confined to the > Quantum Realm), I would find this very interesting but > not the end of the world. It seems to me analogous > to the classical addition of velocities working very > well in almost all situations we'd be likely to use it, > but needing correction at very high speeds. everything in modern theories is the Quantum Realm (tm) this is certainly a mathematical question it all depends on your metamathematical theory of truth if you adhere to the correspondence theory of truth then identity is the least of your problems in a standard quantum theory the true or correct logic is not classical it is an orthomodular lattice which has nondistribution between joins and meets and suffers noncommutativity and with this logical change not only is identity affected (differently for bosons and fermions) but the concept of number itself completely changes (it becomes a fock space operator) and no feynman's back and forth does not restore identity even ontologically (let alone epistemically) we can recover some of our classical understanding taking the operational resolution but that only gives a constructive (heyting algebra) base -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- === Subject: Re: Why does everyone do it? > > 2) when are two objects a and b the _same_, i.e. a = b ? > No _two_ objects are _ever_ the same object (i.e. identical). Roughly speaking: to say of two things that they are identical is nonsense [...]. (L. Wittgenstein, TLP 5.5303) B. === Subject: Re: Why does everyone do it? Just a question: does any of this still have relevance for the op's > question? > Or has it become a oudemannenhuisdiscussie? You don't really expect a thread about Cantor cranks to stay on topic? -- Jesse F. Hughes It's your choice though, if you do not believe in mathematics, in the importance of its healthiness and correctness, then you can just walk away now. -- James S Harris, on the Pythagorean Oath === Subject: Re: Why does everyone do it? > >> Just a question: does any of this still have relevance for the op's >> question? >> Or has it become a oudemannenhuisdiscussie? > > You don't really expect a thread about Cantor cranks to stay on topic? Well, we've got to give credit to WM: now that he's away, the crackpot threads float much more to all different directions. With WM, the focus always kept on the original topic, no matter how long the threads lasted. -- Herman Jurjus === Subject: Re: Why does everyone do it? I claim we could develop mathematics without that mythology, and the > resulting mathematics would not in any way be deficient. We could > still do [this and that]. > So *please* don't hesitate to develop such mathematics. :-) Meanwhile we may stick to the following observation by Amir D. Aczel: Unproven statements carry little weight in the world of mathematics. B. === Subject: Re: Why does everyone do it? > >> I claim we could develop mathematics without that mythology, and the >> resulting mathematics would not in any way be deficient. We could >> still do [this and that]. > So *please* don't hesitate to develop such mathematics. :-) > > Meanwhile we may stick to the following observation by Amir D. Aczel: > > Unproven statements carry little weight in the world of mathematics. > > > B. There was a quite interesting thread in December 1999 to Jan. 2000: ``Continuum Hypotheisis : axioms of set theory, truth, etc. David Bernier === Subject: Re: Why does everyone do it? <177a7$48be4212$82a1e228$14394@news1.tudelft.nl> <87y729ihzh.fsf@phiwumbda.org> <8fd68$48be8174$82a1e228$7232@news1.tudelft.nl> <87od35maro.fsf@phiwumbda.org> <2891e$48bfc562$82a1e228$12161@news1.tudelft.nl> <8bd8a$48c4f6d9$82a1e228$23154@news1.tudelft.nl> <87bpyyril4.fsf@phiwumbda.org> <87r67ulfft.fsf@phiwumbda.org> <6wvr87fzhxao$.18em0a7t64vy1$.dlg@40tude.net> <7beff$48c66b73$82a1e228$19553@news1.tudelft.nl> <873ak9bfoy.fsf@phiwumbda.org> posting-account=Yn5cwwoAAADntcMuRwk-EwLg-DMZ_hXN Gecko/20070509 Camino/1.5,gzip(gfe),gzip(gfe) so all noncomputable reals are equal? -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- === Subject: Re: Why does everyone do it? > so all noncomputable reals are equal? > > -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- > Depends on how uncomputable they are. An uncomputable that is computably negative shouldn't ever equal one which is computably positive. === Subject: Re: Why does everyone do it? <87y729ihzh.fsf@phiwumbda.org> <8fd68$48be8174$82a1e228$7232@news1.tudelft.nl> <87od35maro.fsf@phiwumbda.org> <2891e$48bfc562$82a1e228$12161@news1.tudelft.nl> <8bd8a$48c4f6d9$82a1e228$23154@news1.tudelft.nl> <87bpyyril4.fsf@phiwumbda.org> <87r67ulfft.fsf@phiwumbda.org> <6wvr87fzhxao$.18em0a7t64vy1$.dlg@40tude.net> <7beff$48c66b73$82a1e228$19553@news1.tudelft.nl> <873ak9bfoy.fsf@phiwumbda.org> posting-account=Yn5cwwoAAADntcMuRwk-EwLg-DMZ_hXN Gecko/20070509 Camino/1.5,gzip(gfe),gzip(gfe) > so all noncomputable reals are equal? > -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- > Depends on how uncomputable they are. An uncomputable that is computably negative shouldn't ever equal one > which is computably positive. you're right except that since there is no finite specification available with which to extract such computable properties the collection of such examples is null now switch the quantifiers around using contrapositive does this say that there are only countably many reals that can be distinguished? (think of all the discussions on diagonalisation here...) -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- === Subject: Re: Why does everyone do it? > so all noncomputable reals are equal? > -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- > > Depends on how uncomputable they are. > An uncomputable that is computably negative shouldn't ever equal one > which is computably positive. > > you're right > except that since there is no finite specification available > with which to extract such computable properties > the collection of such examples is null I do not see that at all. That a number is not computable does not mean that one can know nothing at all about it, but only that it cannot be approximated to an ARBITRARY degree of precision. There is nothing theoretically impossible about knowing that a number can be approximated to within some fixed positive epsilon but no further, as when say, one is given a number between 1 and 2, but with no other properties. === Subject: Re: Why does everyone do it? <2891e$48bfc562$82a1e228$12161@news1.tudelft.nl> <8bd8a$48c4f6d9$82a1e228$23154@news1.tudelft.nl> <87bpyyril4.fsf@phiwumbda.org> <87r67ulfft.fsf@phiwumbda.org> <6wvr87fzhxao$.18em0a7t64vy1$.dlg@40tude.net> <7beff$48c66b73$82a1e228$19553@news1.tudelft.nl> <873ak9bfoy.fsf@phiwumbda.org> posting-account=Yn5cwwoAAADntcMuRwk-EwLg-DMZ_hXN Gecko/20070509 Camino/1.5,gzip(gfe),gzip(gfe) > so all noncomputable reals are equal? > -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- > > Depends on how uncomputable they are. > An uncomputable that is computably negative shouldn't ever equal one > which is computably positive. > you're right > except that since there is no finite specification available > with which to extract such computable properties > the collection of such examples is null I do not see that at all. > That a number is not computable does not mean that one can know nothing > at all about it, but only that it cannot be approximated to an > ARBITRARY degree of precision. There is nothing theoretically impossible > about knowing that a number can be approximated to within some fixed > positive epsilon but no further, as when say, one is given a number > between 1 and 2, but with no other properties. can you give me an example of such a number? -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- === Subject: Re: Why does everyone do it? > > so all noncomputable reals are equal? > -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- > > Depends on how uncomputable they are. > An uncomputable that is computably negative shouldn't ever equal one > which is computably positive. > you're right > except that since there is no finite specification available > with which to extract such computable properties > the collection of such examples is null > I do not see that at all. > That a number is not computable does not mean that one can know nothing > at all about it, but only that it cannot be approximated to an > ARBITRARY degree of precision. There is nothing theoretically impossible > about knowing that a number can be approximated to within some fixed > positive epsilon but no further, as when say, one is given a number > between 1 and 2, but with no other properties. > > can you give me an example of such a number? There are lots of them, pick one for yourself. === Subject: Re: Why does everyone do it? > so all noncomputable reals are equal? > -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- > > Depends on how uncomputable they are. > An uncomputable that is computably negative shouldn't ever equal one > which is computably positive. > you're right > except that since there is no finite specification available > with which to extract such computable properties > the collection of such examples is null > I do not see that at all. > That a number is not computable does not mean that one can know nothing > at all about it, but only that it cannot be approximated to an > ARBITRARY degree of precision. There is nothing theoretically impossible > about knowing that a number can be approximated to within some fixed > positive epsilon but no further, as when say, one is given a number > between 1 and 2, but with no other properties. > can you give me an example of such a number? There are lots of them, pick one for yourself. how would i know that i've picked one of these noncomputable numbers i am talking about and not e/2 or pi/3 or whatever? just given a number chosen from [1, 2] doesn't guarantee that -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- === Subject: Re: classical positions on natural numbers [...] >> at others on positivism, skepticism and realism. >> I sort of wonder to what degree of precision one >> can aspire in philosophical discussions. >> I think I read that with more context, the meaning >> of a sentence is easier to apprehend. > > there are times when i wonder > why phenomenalism isn't presented in a standard course > on the foundations of metamathematics > > there is a genealogy starting with brentano > > his student husserl gave some of the most explicit descriptions > of the foundations of meaning in experience > > and his student twardowski went on to bring > these phenomenalist foundations into the field of logic > establishing the lvov-warsaw school > > here grew the great polish school > ajdukiewicz > lesniewski > lukasiewicz > kotarbinski > > eventually from these strong teachers came tarski > whose contributions to metamathematics > semantics > and the foundations of model theory > laid the foundations for much of the modern research here > (he did tons of other things > a great mathematical polyglot) > > model theory was much the husserlian answer to logicism > and husserl's work was strongly pursued by the polish school > > i say this only to stress > there is philosophy that has strong formal consequences > when given the proper rigor > >> So I have this test question which relates to >> vagueness in language, and whether categories exist >> or are defined (e.g. there are black swans >> in Australia. But kangaroos aren't mammals, >> and a platypus isn't a bird, etc.). >> The bioinformatics specialists Hobolth, >> Christensen, Mailund and Schierup published >> a paper in an on-line genetics journal on >> the genealogical tree of humans, chimps and >> orangutans. >> In the abstract, they write: >> << We find a very recent speciation time >> of human[CapitalEth]chimp (4.1 ± 0.4 million years)[...] >> What was it like on the day of human[CapitalEth]chimp speciation? > > :) > > the same questions arise with life in general > but might be starkest with the notion of multicellular life > > your skin cells die off and are replaced every 2 weeks > > as a great swans song goes > where does a body end? > > this past weekend i (finally) saw the amazing > noriko's dinner table > where > along with it's predecessor > the great topological question is asked > are you connected to yourself? There are different kinds of speech/writing, depending on the intent. For example, a book of recipes is already pretty good if it's practical and one can learn how to cook from it. On the other hand, a book on Japanese history will only cover so much, and more could be said, maybe a better presentation used, and so on. So we have (a) recipe books (the proof of the pudding is in the eating) (b) history books (narration of events that are now history) Then I suppose (c) philosophy books (about philosophy). At least for recipe books, where the say a pinch of salt, leaving some things vague is fine; it's procedural knowledge. So that leaves (b), (c) and the rest. David Bernier > although immediately relevant to the suicides investigated > it eventually becomes clear they are talking about > the generation gap and the nature of parenthood > > as with all such questions though > when there seems to be a conceptual separation > there is often some attractor space that can be identified > which helps provide definition to the fuzzy boundaries > > with species > the real question is often the ability to interbreed > or more operationally > whether interbreeding has occurred > > to clarify the vagueness that arises in the collective > this concept should be applied to individuals separately > > so > bringing this ramble back to math > species are more clearly defined relative to individuals > > so species(x) could operationally be seen as > the collection of individuals who have bred with x > > but that's just mates(x) > which is depend on time > > in fact > many vague properties can be made more operationally clear > by providing specific temporal redefinitions > and it is often this case that vagueness arises > from atemporal abstractions that are not operational > > husserl talked about this quite a bit > > :) > > -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- > === Subject: Re: most of the USA Founding Fathers were science-athiests #20 (sic) Plutocracy (Science Council) will in the future replace our Democracy posting-account=yxbZkgkAAABQBvyYeebYQ-PAvi0uT3tG Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) >> I do not know whether McCain or Biden or Obama are Creationists? IF >> they are, they should be disqualified. > You present an interesting interpretation of Article VI of the U.S. >> Constitution. >> And of course that disqualifies every president since Washington. Well, this would be a valuable book for historians. That the USA > Founding Fathers, in large part were Athiests. To name a few: Ben Franklin > Thomas Jefferson > John Adams > Alexander Hamilton > James Madison > George Washington Most of them were deists, not atheists. === Subject: Re: Separation of church & state makes Founding Fathers athiests #22 Plutocracy (Science Council) will in the future replace our Democracy posting-account=jPnQ2goAAAA461y3QD0lbyw0oKeThma1 AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.2 Safari/525.20.1,gzip(gfe),gzip(gfe) there's no seperation of church & state clause in the Constitution, even though No Child Left Behind, Come the Rapture, is illegal. and, maybe you *should* join the Moonies. thus: yeah, I usually try to be charitable towards HSJ, but the question of whether he is a conscientious troll is unanswerable; if he is, he never lets the mask down, that I've noticed. so, I usually really give the typical acerbic, no-math critique of his grandiloquent form & theory of mathematical plausibility. maybe, it only *seems* bogus, and he is really the 99th Monkey. not ferferring to the typing-Shakespeare kind, and he doesn't seem to quite be the 100th, after which all monkeykind will mysteriously do the math that he won't try. if that sounds racist, just forget it; genetically, we're almost the same as chimps, any way. thus: wow, maybe you could advise me. so, which is better, LSD or mild ergot poisoning -- will the latter flashforward to cure flashbacks from the former?... most of which I did not read & that I just noticed, is, that little statement about synthesizm; no synesthezia required?... also, Moog just came-out with a guitar (saw the ad; at 6500US, it's time to roll one's own .-) >http://www.allentwood.com/essays/lordofflies.html > religions, specifically the religious synthesism of RamaKrishna. In > Okay, when the effects of the LSD wore off, what happened next? thus quoth: Fannie Mae was created by President Franklin D. Roosevelt in 1938, as a government agency to buy mortgages from lenders, as a way of funding the purchase of homes in the Great Depression. In more recent years, Fannie Mae and its sibling Freddie Mac, were taken over by what FDR attacked as the economic royalists, and turned into vehicles for derivatives speculation. Under the great Greenspan bubble, Fannie and Freddie were turned into money machines to feed the run-up in real estate values to provide assetsÑin the form of mortgage debtÑas fuel to the derivatives markets. This scheme was bound to fail, as it spectacularly has, leaving Fannie and Freddie, and the U.S. banking system, utterly bankrupt. However, Fannie and Freddie are at the heart of Treasury Secretary Henry Paulson's and Federal Reserve chairman Ben Bernanke's insane scheme to bail out the banks by dumping all their bad mortgage paper into the two government-sponsored enterprises, effectively transferring the banks' losses to the government, and ultimately to the taxpayer. The government is not really bailing out Fannie and Freddie, but merely funding their conversion into the largest toxic waste dumps in history. Far from being saved, Fannie Mae and Freddie Mac are being destroyed. http://larouchepub.com/other/editorials/2008/3536tantamount treason.html === Subject: Re: Maximum gap between primes? <9Qoxk.179209$oT7.48941@newsfe10.ams2> posting-account=D3mnJwoAAAAcfg0CE9jgFeeLgEgVn6Li Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Consider the natural powers of two, 2,4,8,16, 32, and so on. > There is no largest such power of two but also there is no largest > difference between successive such powers of two. > Why should primes be different? > Because the series of primes is irregular? ÊThe OP's questions seemed to doubt that Êany Êinfinite sequence of > naturals behaving like the powers of two could exist. An example shows > him otherwise. That doesw not prove that the set of differences between successive > primes MUST be unbounded, only that it might be. > I suppose it was doubt, though it was implicit doubt, not something I thought of prior to posting. I confess that I didn't even think of examining analogies with other sequences. The example of the perfect squares helped a lot with that. But I think Steven's point is a good one. It was less obvious to me that prime gaps can have no maximum (twin primes keep appearing) until the explanations here helped me see that even though prime gaps grow and shrink, there will eventually be a bigger gap as the primes involved get bigger. I guess the only thing I am still unclear on is the difference between infinite (as in the natural number sequence 1, 2, 3, ...) and unbounded (as in the sequence of prime gap values 1, 2, 3, ...) as the term unbounded has been used in this thread. It seems to me that one can express the sequence of prime gaps in exactly the same form as one can express the sequence of natural numbers, assuming that one removes duplicates from the prime gap sequence. I'm defining gap as the Prime Pages page below does (I love this site though a lot of it is over my head): For every prime p let g(p) be the number of composites between p and the next prime. http://primes.utm.edu/notes/gaps.html By that definition, it seems that g(2) = 0, since there are no composites between 2 and the next prime, 3. And g(19) = 3. I don't think the specific results matter to my question, but I would like to know whether I am plugging in the numbers correctly. :-) As far as I can tell, g(p) can equal 1 or 2 or 3 and so on, depending on the two primes involved. And since g(p) can grow arbitrarily large, I can see how g(p) tends toward infinity (meaning no maximum gap). But I don't see how finiteness applies to g(p). Also, has it been proven that there is a one-to-one correspondence between natural numbers and g(p) - in other words, that there really is a prime gap of every length starting with 1 and growing by 1? That somewhere in the sequence, g(p) is, say, 7,312? Sorry if my questions are a messy jumble of ignorance and misused math terms. I'm fascinated by primes and just want to understand this last part of prime gaps, because I now understand the answer to my first question about whether a max prime gap can exist in the first place. Shepherdmoon === Subject: Re: Maximum gap between primes? <9Qoxk.179209$oT7.48941@newsfe10.ams2> the next prime. > http://primes.utm.edu/notes/gaps.html By that definition, it seems that g(2) = 0, since there are no > composites between 2 and the next prime, 3. And g(19) = 3. I don't > think the specific results matter to my question, but I would like to > know whether I am plugging in the numbers correctly. :-) > Let p_n be the n-th prime. Then g(p_n) = p_(n+1) - p_n - 1. No, it does not equal p_(n+1) - p_n. For example 0 = g(2) = 3 - 2 - 1 and not 3 - 2. > As far as I can tell, g(p) can equal 1 or 2 or 3 and so on, depending > on the two primes involved. And since g(p) can grow arbitrarily large, > I can see how g(p) tends toward infinity (meaning no maximum gap). But > I don't see how finiteness applies to g(p). g(2) = 0 is even. Other that than for any prime p > 2 g(p) is the difference of two odd numbers minus one which is odd. 2j + 1 + 2k + 1 - 1 = 2(j + k) + 1 is odd. Thus for all prime p > 2, g(p) is odd. How many primes p are there for which g(p) = 1? > Also, has it been proven that there is a one-to-one correspondence > between natural numbers and g(p) - in other words, that there really > is a prime gap of every length starting with 1 and growing by 1? That > somewhere in the sequence, g(p) is, say, 7,312? No, except for g(2), gaps are only odd and they can be as big as one wants (see my other post in this thread). ---- === Subject: Re: Maximum gap between primes? > I guess the only thing I am still unclear on is the difference between > infinite (as in the natural number sequence 1, 2, 3, ...) and > unbounded (as in the sequence of prime gap values 1, 2, 3, ...) as the > term unbounded has been used in this thread. The members of the set of naturals are unbounded (have no maximal element within the set) while the set of natural numbers is infinite ( at least in the Dedekind sense of being injectable to a proper subset). Note that if a non-empty set has an ordering under which its members are unbounded then that set is infinite, though no member of it need be infinite. === Subject: Re: Maximum gap between primes? > Also, has it been proven that there is a one-to-one correspondence > between natural numbers and g(p) - in other words, that there really > is a prime gap of every length starting with 1 and growing by 1? That > somewhere in the sequence, g(p) is, say, 7,312? You are very confused. First of all, there is no proof that for every even number there is a gap between consecutive primes equal exactly to that number. It is conjectured that this is true, and no one seriously doubts that it is true - in fact, it is conjectured that for every even number k there are infinitely many primes p such that the next prime is p + k - but as of yet there is no proof. What is proved is that there is a gap of length AT LEAST k (and, it follows, infinitely many such gaps). But that has nothing to do with one-to-one correspondence, at least not in the way that that phrase is always used in mathematics. Two sets are said to be in one-to-one correspondence if there is any way of pairing off the two sets so nothing is left out in either one. So for instance there is a one-to-one correspondence between the natural numbers and the prime numbers, even though the prime numbers don't give all the natural numbers; simply pair 1 with 2, 2 with 3, 3 with 5, 4 with 7, 5 with 11, and so on, and each natural will be paired with its own personal prime, with nothing left out in either set. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Maximum gap between primes? <9Qoxk.179209$oT7.48941@newsfe10.ams2> posting-account=OKTeIQkAAAAZk6JK1hK7-grwpoUDNy98 CLR 1.1.4322; .NET CLR 2.0.50727; InfoPath.1),gzip(gfe),gzip(gfe) > Consider the natural powers of two, 2,4,8,16, 32, and so on. > There is no largest such power of two but also there is no largest > difference between successive such powers of two. > Why should primes be different? > Because the series of primes is irregular? > ÊThe OP's questions seemed to doubt that Êany Êinfinite sequence of > naturals behaving like the powers of two could exist. An example shows > him otherwise. > That doesw not prove that the set of differences between successive > primes MUST be unbounded, only that it might be. I suppose it was doubt, though it was implicit doubt, not something I > thought of prior to posting. I confess that I didn't even think of > examining analogies with other sequences. The example of the perfect > squares helped a lot with that. But I think Steven's point is a good one. It was less obvious to me > that prime gaps can have no maximum (twin primes keep appearing) until > the explanations here helped me see that even though prime gaps grow > and shrink, there will eventually be a bigger gap as the primes > involved get bigger. I guess the only thing I am still unclear on is the difference between > infinite (as in the natural number sequence 1, 2, 3, ...) and > unbounded (as in the sequence of prime gap values 1, 2, 3, ...) as the > term unbounded has been used in this thread. It seems to me that one > can express the sequence of prime gaps in exactly the same form as one > can express the sequence of natural numbers, assuming that one removes > duplicates from the prime gap sequence. I'm defining gap as the > Prime Pages page below does (I love this site though a lot of it is > over my head): For every prime p let g(p) be the number of composites between p and > the next prime.http://primes.utm.edu/notes/gaps.html By that definition, it seems that g(2) = 0, since there are no > composites between 2 and the next prime, 3. And g(19) = 3. I don't > think the specific results matter to my question, but I would like to > know whether I am plugging in the numbers correctly. :-) As far as I can tell, g(p) can equal 1 or 2 or 3 and so on, depending > on the two primes involved. And since g(p) can grow arbitrarily large, > I can see how g(p) tends toward infinity (meaning no maximum gap). But > I don't see how finiteness applies to g(p). Also, has it been proven that there is a one-to-one correspondence > between natural numbers and g(p) - in other words, that there really > is a prime gap of every length starting with 1 and growing by 1? Growing by 1? I believe that once you are past 2,3 all prime gaps are even since all primes are odd from that point on. > That > somewhere in the sequence, g(p) is, say, 7,312? Sure. As long as it's even. But it's not divisible by 6, so it might be harder to find. Sorry if my questions are a messy jumble of ignorance and misused math > terms. I'm fascinated by primes and just want to understand this last > part of prime gaps, because I now understand the answer to my first > question about whether a max prime gap can exist in the first place. Shepherdmoon === Subject: Re: Maximum gap between primes? <9Qoxk.179209$oT7.48941@newsfe10.ams2> posting-account=D3mnJwoAAAAcfg0CE9jgFeeLgEgVn6Li Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Consider the natural powers of two, 2,4,8,16, 32, and so on. > There is no largest such power of two but also there is no largest > difference between successive such powers of two. > Why should primes be different? > Because the series of primes is irregular? > ÊThe OP's questions seemed to doubt that Êany Êinfinite sequence of > naturals behaving like the powers of two could exist. An example shows > him otherwise. > That doesw not prove that the set of differences between successive > primes MUST be unbounded, only that it might be. > I suppose it was doubt, though it was implicit doubt, not something I > thought of prior to posting. I confess that I didn't even think of > examining analogies with other sequences. The example of the perfect > squares helped a lot with that. > But I think Steven's point is a good one. It was less obvious to me > thatprimegaps can have no maximum (twin primes keep appearing) until > the explanations here helped me see that even thoughprimegaps grow > and shrink, there will eventually be a bigger gap as the primes > involved get bigger. > I guess the only thing I am still unclear on is the difference between > infinite (as in the natural number sequence 1, 2, 3, ...) and > unbounded (as in the sequence ofprimegap values 1, 2, 3, ...) as the > term unbounded has been used in this thread. It seems to me that one > can express the sequence ofprimegaps in exactly the same form as one > can express the sequence of natural numbers, assuming that one removes > duplicates from theprimegap sequence. I'm defining gap as the >PrimePages page below does (I love this site though a lot of it is > over my head): > For everyprimep let g(p) be the number of composites between p and > the nextprime.http://primes.utm.edu/notes/gaps.html > By that definition, it seems that g(2) = 0, since there are no > composites between 2 and the nextprime, 3. And g(19) = 3. I don't > think the specific results matter to my question, but I would like to > know whether I am plugging in the numbers correctly. :-) > As far as I can tell, g(p) can equal 1 or 2 or 3 and so on, depending > on the two primes involved. And since g(p) can grow arbitrarily large, > I can see how g(p) tends toward infinity (meaning no maximum gap). But > I don't see how finiteness applies to g(p). > Also, has it been proven that there is a one-to-one correspondence > between natural numbers and g(p) - in other words, that there really > is aprimegap of every length starting with 1 and growing by 1? Growing by 1? I believe that once you are past 2,3 allprimegaps are > even since all primes are odd from that point on. > That > somewhere in the sequence, g(p) is, say, 7,312? Sure. As long as it's even. But it's not divisible by 6, > so it might be harder to find. that the prime gaps have to be even for all prime gaps past (3,5)! Fascinating. So if the prime gaps are multiples of 2, can they still be mapped to a 1, 2, 3, ... sequence? Also, shouldn't g(2) be 0 given the definition of gap as the number of composites between p and the next prime? Are there different definitions of a prime gap? Shepherdmoon === Subject: Re: Maximum gap between primes? <9Qoxk.179209$oT7.48941@newsfe10.ams2> posting-account=OKTeIQkAAAAZk6JK1hK7-grwpoUDNy98 4334.34; Windows NT 5.1; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) spider-mtc-th09.proxy.aol.com[400C70E9] (Prism/1.2.1), HTTP/1.1 cache-mtc-ad05.proxy.aol.com[400C74C7] (Traffic-Server/6.1.5 [uScM]) > Consider the natural powers of two, 2,4,8,16, 32, and so on. > There is no largest such power of two but also there is no largest > difference between successive such powers of two. > Why should primes be different? > Because the series of primes is irregular? > ?The OP's questions seemed to doubt that ?any ?infinite sequence of > naturals behaving like the powers of two could exist. An example shows > him otherwise. > That doesw not prove that the set of differences between successive > primes MUST be unbounded, only that it might be. > I suppose it was doubt, though it was implicit doubt, not something I > thought of prior to posting. I confess that I didn't even think of > examining analogies with other sequences. The example of the perfect > squares helped a lot with that. > But I think Steven's point is a good one. It was less obvious to me > thatprimegaps can have no maximum (twin primes keep appearing) until > the explanations here helped me see that even thoughprimegaps grow > and shrink, there will eventually be a bigger gap as the primes > involved get bigger. > I guess the only thing I am still unclear on is the difference between > infinite (as in the natural number sequence 1, 2, 3, ...) and > unbounded (as in the sequence ofprimegap values 1, 2, 3, ...) as the > term unbounded has been used in this thread. It seems to me that one > can express the sequence ofprimegaps in exactly the same form as one > can express the sequence of natural numbers, assuming that one removes > duplicates from theprimegap sequence. I'm defining gap as the >PrimePages page below does (I love this site though a lot of it is > over my head): > For everyprimep let g(p) be the number of composites between p and > the nextprime.http://primes.utm.edu/notes/gaps.html > By that definition, it seems that g(2) = 0, since there are no > composites between 2 and the nextprime, 3. And g(19) = 3. I don't > think the specific results matter to my question, but I would like to > know whether I am plugging in the numbers correctly. :-) > As far as I can tell, g(p) can equal 1 or 2 or 3 and so on, depending > on the two primes involved. And since g(p) can grow arbitrarily large, > I can see how g(p) tends toward infinity (meaning no maximum gap). But > I don't see how finiteness applies to g(p). > Also, has it been proven that there is a one-to-one correspondence > between natural numbers and g(p) - in other words, that there really > is aprimegap of every length starting with 1 and growing by 1? > Growing by 1? I believe that once you are past 2,3 allprimegaps are > even since all primes are odd from that point on. > That > somewhere in the sequence, g(p) is, say, 7,312? > Sure. As long as it's even. But it's not divisible by 6, > so it might be harder to find. that the prime gaps have to be even for all prime gaps past (3,5)! > Fascinating. Have a look at this (I apologize if I said something stupid there, but my official excuse is that that page is almost 10 years old and I was a lot stupider back then. I didn't even know that all primes >3 are 1 mod 6 or -1 mod 6.) be mapped to a 1, 2, 3, ... sequence? Why not? You can map the Naturals to the Evens, can't you? 1 -> 2 2 -> 4 3 -> 6 4 -> 8 etc. Also, shouldn't g(2) be 0 given the definition of gap as the number > of composites between p and the next prime? I was thinking the distance between primes, p n+1 - p n, such as 7 - 5 = 2, 17 - 13 = 4 or 29 - 23 = 6. > Are there different definitions of a prime gap? Well, I don't use number of composites, so there obviously is more than one way informally. But maybe that's just me. Shepherdmoon === Subject: Algebra question I'm working through a chapter on Relativity, and I can't follow the logic behind this transformation: a^2 - b^2 = a^2(1 - (b^2/a^2)). -- Craig Franck craig.franck@verizon.net Cortland, NY === Subject: Re: Algebra question posting-account=3WPJYgoAAAA55VjhzK9i07RN8h8u8eEs 1.7; .NET CLR 1.0.3705; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30),gzip(gfe),gzip(gfe) > I'm working through a chapter on Relativity, and I can't follow > the logic behind this transformation: a^2 - b^2 = a^2(1 - (b^2/a^2)). -- > Craig Franck > craig.fra...@verizon.net > Cortland, NY You're simply taking out a factor of a^2 from both terms. If you're not convinced, just plug in numbers: let a=2,b=3. Is (4-9) = 4 * (1 - 9/4)? M === Subject: Re: Algebra question > I'm working through a chapter on Relativity, and I can't follow >the logic behind this transformation: a^2 - b^2 = a^2(1 - (b^2/a^2)). >You're simply taking out a factor of a^2 from both terms. >If you're not convinced, just plug in numbers: let a=2,b=3. >Is >(4-9) = 4 * (1 - 9/4)? Yes, I am convinced that it works, I was wondering what the intermediate steps were. I see how a^2 - b^2 = (a + b)(a - b) = a^2 + ab - ab - b^2 works step by step, but don't recall coming across that other factoring in an algebra book before. It's obvious b^2/a^2 takes the a^2 out b^2, but the other part isn't clear. -- Craig Franck craig.franck@verizon.net Cortland, NY === Subject: Re: Algebra question > > > I'm working through a chapter on Relativity, and I can't follow >the logic behind this transformation: a^2 - b^2 = a^2(1 - (b^2/a^2)). > >You're simply taking out a factor of a^2 from both terms. >If you're not convinced, just plug in numbers: let a=2,b=3. >Is >(4-9) = 4 * (1 - 9/4)? > > Yes, I am convinced that it works, I was wondering what > the intermediate steps were. I see how > > a^2 - b^2 = (a + b)(a - b) = a^2 + ab - ab - b^2 > > works step by step, but don't recall coming across that other > factoring in an algebra book before. It's obvious b^2/a^2 takes > the a^2 out b^2, but the other part isn't clear. You can use the distributive law (in reverse) to pull out a common factor. The distributive law says that a(b + c) = ab + ac. By reverse I mean that if you have ab + ac, then you can rewrite that as a(b + c). But now the trick is to *recognize* what you can choose to be 'a' in this case. So, with a^2 - b^2 you can see that there is a factor of a^2 in the first term 'a^2'. But there is also a factor of a^2 in the second term 'b^2'. In fact, b^2 = a^2 * (b^2/a^2) so you can factor an a^2 out of the b^2 term too. Even though we can't explicitly see it in the equation, it's still a factor. It's exactly the same as saying that 5 = 3 * (5/3). It's the exact same principle. And the only thing you have to be sure about is that you are working in a number system that allows arbitrary (nonzero) division. And if it's a text on relativity, it's a pretty good bet they're working in the real numbers, or possibly the complex numbers. In both the real and the complex numbers, you can always divide any number by any nonzero number. So we can do that factorization. Now if you were reading a text on number theory, you could not factor out a^2. Because in number theory we're generally working with the natural numbers, in which arbitrary division isn't allowed! So this is a case where the factorization actually depends on the context in which we're working. That's what makes it harder to learn to see right away. To see that a^2 is a factor of a^2, we only have to look at the symbols themselves. To see that a^2 is also a factor of b^2, we have to reason about the meaning of the symbols. Philosophically, and also apparently in brain processing, this is a significant difference. === Subject: Re: Algebra question > I'm working through a chapter on Relativity, and I can't follow > the logic behind this transformation: a^2 - b^2 = a^2(1 - > (b^2/a^2)). > You're simply taking out a factor of a^2 from both terms. >> If you're not convinced, just plug in numbers: let a=2,b=3. >> Is >> (4-9) = 4 * (1 - 9/4)? Yes, I am convinced that it works, I was wondering what > the intermediate steps were. I see how a^2 - b^2 = (a + b)(a - b) = a^2 + ab - ab - b^2 works step by step, but don't recall coming across that other > factoring in an algebra book before. It's obvious b^2/a^2 takes > the a^2 out b^2, but the other part isn't clear. There's no need to factor a^2 + b^2 into (a + b)*(a - b). You simply divide both terms by a^2, then multiply the whole thing by a^2 (effectively multiplying by one, since a^2/a^2 = 1. Thus: (a^2 - b^2) = (a^2/a^2 - b^2/a^2)*a^2 = (1 - b^2/a^2)*a^2 which is the same as your original RHS. === Subject: Re: Algebra question posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/20070530 Fedora/1.5.0.12-1.fc5 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) > I'm working through a chapter on Relativity, and I can't follow >the logic behind this transformation: a^2 - b^2 = a^2(1 - (b^2/a^2)). >You're simply taking out a factor of a^2 from both terms. >If you're not convinced, just plug in numbers: let a=2,b=3. >Is >(4-9) = 4 * (1 - 9/4)? Yes, I am convinced that it works, I was wondering what > the intermediate steps were. I see how a^2 - b^2 = (a + b)(a - b) = a^2 + ab - ab - b^2 works step by step, but don't recall coming across that other > factoring in an algebra book before. It's obvious b^2/a^2 takes > the a^2 out b^2, but the other part isn't clear. a^2 - b^2 = a^2 1 - a^2 (b^2 / a^2) = a^2 ( 1 - (b^2 / a^2) ) -- m === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation posting-account=wVv_VwoAAAAVTfUuyxLzug5SzYWCgHj1 Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) This is truly masterful trollwork, a veritable clinic in trollery at its best. I will point out a few examples in the following - > Now that I have a general theory Troll-technique #1: Start off immediately with exaggerated bragging. This is really guaranteed to get under the skin of real mathematicians, who know that what Harris has accomplished here is minor high- school level manipulation of simple equations. There is no 'general theory'. > for all 2 variable quadratic > Diophantine equations it's worth coming back to note again the weird Troll-technique #2: Liberal use of the word 'weird' - pretending there is deep mystery where there is only shallow symbol-pushing. Again very efficiently designed to enrage the real mathematicians. > connection I found between certain Pythagorean Triplets and Pell's > Equation in the form x^2 - Dy^2 = 1 when D-1 is a perfect square. For instance for D=2, I have that for > every solution of Pell's Equation you have a Pythagorean Triplet! > Troll-technique #3: Hyped excitement! Wow! Trollery here is much more a matter of marketing than of genuinely interesting math. But it does effectively rankle the reader and provoke responses [this one included]. > But the triplets are special in that with u^2 + v^2 = w^2, v = u+1. > The connection is that w is x+y from Pell's Equation. The more general result is that u = sqrt(D-1)j, and v = j+1, while w > still equals x+y. Intriguingly Troll-technique #2 again, but with 'intriguingly' rather than 'weird'. It's NOT intriguing, it's vapid. Did Gauss EVER say 'intriguingly'? Galois? Riemann? Probably not even in personal correspondence. When JSH says 'intriguingly' or 'fascinating', he is describing exactly one person's reaction. Guess whose. > that means that proof that there are an infinite number > of solutions for certain Pell's Equations is proof that there are an > infinity of Pythagorean Triplets of a certain form! > Troll-technique #3 again: hyped excitement!!! > An easy example with D=2, is x=17, y=12, where notice you are paired > with the triplet 20, 21, 29. That is just some low-hanging fruit Troll-technique #4: Use of the trendy buzz-phrase. As if this is just a wee taste of the deep wonders to come [except this is as deep as it will ever get]. And it makes the real mathematician see red. > that I thought I'd mention. Troll-technique #5: Just an offhand little remark, delivered with the obligatory sneer. > Kind > of been a whirlwind of results flowing from playing with my > Diophantine Quadratic Theorem. > Troll-technique #6: 'Whirlwind'? Good. Similar alternatives: 'avalanche', or 'flood', or 'supernova' or 'shitload'. More like: 'trickle' or 'driblet' or 'wetspot' or 'flyspeck'. And 'playing with' - false modesty laid on to really get the professional's goat. Every Harris post is a FASCINATING [note buzzword!] lesson in trollery - starting-out trolls should begin taking notes - this is the all- time Master. Marcus. > James Harris === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation posting-account=wVv_VwoAAAAVTfUuyxLzug5SzYWCgHj1 Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > This is truly masterful trollwork, a veritable clinic in trollery >at its best. I will point out a few examples in the following - >[...] LOL! Very well dissected. But do you really think James is a > conscious troll? I've come across many trolls, and he never > strikes me as being one of their number. Doesn't he believe > his own hype, and isn't the trolling effect merely incidental? > better stop there. :-) -- > Angus Rodgers > Contains mild peril Right, he's not a troll. He has deluded himself into believing his own propaganda. But he is the international grand- master of troll METHODS. No one does it better. Of course, that's somewhat like saying, no one produces a more beautifully formed turd. Real trolls, whose goals are to get attention and provoke responses, perhaps preferentially hostile ones, could take lessons from him for years to come. The Leonardo da Vinci of the troll art! If he were as good at math as he is at trollery, he would rival Euler. Marcus. === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) Ê This is truly masterful trollwork, a veritable clinic in trollery > at its best. I will point out a few examples in the following - > Now that I have a general theory Ê Troll-technique #1: Start off immediately with exaggerated > bragging. > Ê This is really guaranteed to get under the skin of real > mathematicians, > Ê who know that what Harris has accomplished here is minor high- > school > Ê level manipulation of simple equations. ÊThere is no 'general > theory'. Um, well, yeah, there is, now. I put a link to the paper on my math blog. Infinite chains of Diophantine quadratics. And a remarkably simple way to check for whether or not integer solutions exist. Oh, and best yet! Proof that people like you either deliberately lie or deliberately refuse to accept mathematical proof, when you don't like the discoverer. I proved Fermat's Last Theorem using tautological spaces but there was a problem!!! I was proving a negative. Now I've turned them to studying an area where solutions exist and look at you. Denial is about weakness. You refuse to accept mathematical truth so you cannot be a true mathematician. But you can be a pretender who got away for so long, so long, and hurt humanity in the process. The future of the human race is not yet lost. Despite those like you. The future is not yet decided against this species. Despite all those who have fought so desperately to condemn it. Yes, you have not yet totally failed. But there is finally a light at the end of the tunnel. There is finally, hope. James Harris === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation <2s5fc49fa6egfsdgpl1cp6a3uageg92v6u@4ax.com> posting-account=jPnQ2goAAAA461y3QD0lbyw0oKeThma1 AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.2 Safari/525.20.1,gzip(gfe),gzip(gfe) yeah, I usually try to be charitable towards HSJ, but the question of whether he is a conscientious troll is unanswerable; if he is, he never lets the mask down, that I've noticed. so, I usually really give the typical acerbic, no-math critique of his grandiloquent form & theory of mathematical plausibility. maybe, it only *seems* bogus, and he is really the 99th Monkey. not ferferring to the typing-Shakespeare kind, and he doesn't seem to quite be the 100th, after which all monkeykind will mysteriously do the math that he won't try. if that sounds racist, just forget it; genetically, we're almost the same as chimps, any way. > Your strictures apply in some measure to all of us (so I feel) thus: wow, maybe you could advise me. so, which is better, LSD or mild ergot poisoning -- will the latter flashforward to cure flashbacks from the former? most of which I did not read & that I just noticed, is, that little statement about synthesizm; no synesthezia required? also, Moog just came-out with a guitar (saw the ad; at 6500US, it's time to roll one's own .-) >http://www.allentwood.com/essays/lordofflies.html > religions, specifically the religious synthesism of RamaKrishna. In > Okay, when the effects of the LSD wore off, what happened next? thus quoth: Fannie Mae was created by President Franklin D. Roosevelt in 1938, as a government agency to buy mortgages from lenders, as a way of funding the purchase of homes in the Great Depression. In more recent years, Fannie Mae and its sibling Freddie Mac, were taken over by what FDR attacked as the economic royalists, and turned into vehicles for derivatives speculation. Under the great Greenspan bubble, Fannie and Freddie were turned into money machines to feed the run-up in real estate values to provide assetsÑin the form of mortgage debtÑas fuel to the derivatives markets. This scheme was bound to fail, as it spectacularly has, leaving Fannie and Freddie, and the U.S. banking system, utterly bankrupt. However, Fannie and Freddie are at the heart of Treasury Secretary Henry Paulson's and Federal Reserve chairman Ben Bernanke's insane scheme to bail out the banks by dumping all their bad mortgage paper into the two government-sponsored enterprises, effectively transferring the banks' losses to the government, and ultimately to the taxpayer. The government is not really bailing out Fannie and Freddie, but merely funding their conversion into the largest toxic waste dumps in history. Far from being saved, Fannie Mae and Freddie Mac are being destroyed. http://larouchepub.com/other/editorials/2008/3536tantamount treason.html === Subject: Re: Solution manual to a First Course in Differential Equations with Modeling Applications (7th ed.) > I am a solutions manual collector, I offer solutions manual and ebook > services > Note: > all solutions manual in soft copy > that mean in Adobe Acrobat Reader (PDF ) format. if you want any book > not just solutions just contact with us.81B to get the solution manual > you want .81Cplease send message to > sharesolut...@msn.com .81Csharesolution(at)msn.com.81C replace (at) to > @ ,please email to me . > This is my part of solutions manual list ,If you want any other > solutions manual which is not in my solutions list, don't give > up .please email to sharesolution(at)msn.com Solutions manual to Vector Mechanics Dynamics Beer 8th Edition > Solutions manual to Fundamentals of Applied Electromagnetics 5th > edition by Fawwaz T. 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John Finnemore, Joseph B Franzin > Solutions manual to Derivatives Markets, 2nd , by Robert L. > McDonald (Solutions by Yufeng Guo) > Solutions manual to Options, Futures and Other Derivatives, 4th, > Solutions manual to a First Course in Differential Equations with > Modeling Applications (7th ed.) and Zill & Cullen's Diferential > Equations with Boundary-Value Problems (5th ed.) By Peter Tye hello, I am looking for , solution manual, for the book.A first course in differential equations, by zill, 7th, 8th 0r 9th edition. === Subject: Re: Solution manual to a First Course in Differential Equations with Modeling Applications (7th ed.) HI , I Need solution manual of first course in differential equations by Denis Zill . 5th edition that will be great if u have it. i don't know if that is going to be for free .. please be kind to reply as soon as possible === Subject: --- --- --- Reference for number theory question Cc: deepkdeb@yahoo.com posting-account=iJfBPwgAAACaBAH7DreA6VfAOYvL5VDz 1.0.3705; .NET CLR 1.1.4322; Media Center PC 4.0; IEMB3; IEMB3),gzip(gfe),gzip(gfe) Consider the following equation (1) under the given conditions. x^k + y^k = z^2 (1) Conditions: x, y, z are relatively prime integers, k is a prime > 5, z is even such that k|z It is known that (1) has no integer solutions [1]. But the general proof is very complex and applies the same technique as was used for the solutions of FLT. But for some special case a simple proof may exist. Any reference about the solutions of (1) under the given conditions will be highly appreciated. [1] Darmon,H., Merel,L.,Winding quotients and some variants of Fermat's Last Theorem, J.Reine Angew.Math 490(1997) === Subject: Re: Model and straight line posting-account=8wyvFgoAAAAJYWyfLHRzREe3lxFCHRTd MathPlayer 2.10b; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30),gzip(gfe),gzip(gfe) On Sep 8, 6:30Êam, Dominijanni Simone Hi. I know that SL (straight line) is a elementary object and that > there is a bijection > from R (real number) to SL. > In school when the professor speak about of the SL says that > MODEL: SL is an infinite number of points in a row > This model makes the idea of a SL without holes. > If this is a right model of the SL, then for every point of the SL > exists the successor > point. By this argument follows that if I identify a point of SL whit > the real number 0 > then, considering that exists the successor point of 0 (that I > denote with 0'), > 0' = 0. infinite 0 and as last digit 1 (0.0..01 , .. = endless > zeros ). > But I know that 0' isn't a real number because between two real > numbers there are > infinite real numbers. But, considering that 0' isn't a real number > and the bijection > from R to SL, doesn't exist the successor point in the SL of point > identified with 0. > But through this reasoning I deduce that in the SL there are infinite > holes. > Both the sets of real numbers and natural numbers are totally ordered for > the relation less or equal than, meaning that for any pair of numbers in > the set you can say if one is less or equal than the other. > This may lead to expecting that you can place all the numbers of the set in > order (which you can) and go from one number to the next, which you can for > natural numbers, but not for reals. Between any pair {a, a+e} there's an > infinite number of other reals, like a+e/2. You'll always find numbers > between a number and a candidate successor. So no immediate successor > exists. But that doesn't mean there are gaps. > Note that the set {1, 2, 3, 4, 10, 11, 12, 13} with the relation less than > or equal doesn't have gaps either! The successor of 3 is 4, and the > successor of 4 is 10, not 5. As far as the set is concerned there is no such > thing as 5. > Steven- Nascondi testo citato > - Mostra testo citato - The principle problem that I have, it consists in the affirmation: > there's a bijection from R to SL I think the SL as > CONCRETE AND REAL MODEL:a infinite wire stretched without holes. > In the example of Steven: > the set A = {1, 2, 3, 4, 10, 11, 12, 13} with the relation less > than > or equal doesn't have gaps > there isn't CONCRETE AND REAL MODEL as for SL whit which I can > represent the set A without holes. Then is the CONCRETE AND REAL MODEL thought (the wire), that is the > standard model on the basic geometry's books,another thing respect to the real number or > not? If the answer is not , how can I prove (or at least justified) that in > the real numbers there aren't holes if I identify those with the SL (that I think by > CONCRETE AND REAL MODEL?) Isn't the bijective map between R and SL what one imagines through the > multiples of the units of measure but, maybe, very more complicated? (Does it exist by > well-ordering theorem <--> AC as porky pig jr says?) - Show quoted text - Mathematics (derived from set theory and logic) can't prove that your wire has no holes -its an atomic physics problem .R is a mathematical object (a set with further structure which can be constructed to from set theory in such a way as to have no holes (assuming the positive integers exist (a set with a successor function satisfying th Peano axioms) .To talk about the bijiction between R and and the wire you really have to model the wire in set theory .That is done as part of foundations of Euclidean Geometry (See Forder's book -Dover ) and there it is specifically assumed in the theory (after 100 pages ) that the model wire has no holes .If you were doing atomic Physics it is probably wrong but for usual uses of wires the assumption is useful and accepted (like in studying music on a voilin. I am responding because I noticed that no one really saw what your question was .i hope this helps. smn === Subject: Re: Model and straight line smn schrieb im Newsbeitrag >> Hi. I know that SL (straight line) is a elementary object and that >> there is a bijection from R (real number) to SL. This is only true if real numbers are overly acurate rational numbers as set theory demands. >> In school when the professor speak about of the SL says that >> MODEL: SL is an infinite number of points in a row >> This model makes the idea of a SL without holes. The notion of infinity in set theory deviates from the original and clearly understandable meaning. >> If this is a right model of the SL, then for every point of the SL >> exists the successor point. Every point of the SL? What arrogant ignorance! I do not deny that there are reasonable statements using the expression every number, for instance: The successor of an even number is definitely an odd one. However, in order to leave the world of genuine (non-Hausdorff) continuum and enter the quite different world of numbers, one has to either deliberately or tacitly neglect what Cantor himself called an abyss. Uncountables are losing their feature when they get approximated by countables. To those teachers who are having problems to answer frequently asked questions like yours, I recommend to be honest and understandable: - Several mathematicians introduced approximative notions of continuity and infinity. - This arithmetisation is still considered inevitable in order to use numbers in case of non-linear functions. - Admittedly some tenets that are obviously wrong from the original, i.e., the pre-Dedekind point of view, are easily to swallow even for intelligent people if one explains them in a winking manner. Infinity of modern mathematics can be easily understood if one considers it like a huge infinitely expandable bag. [Just a comment to the teacher: It does not matter that Cantor denied this bag-model. Well, he was able to think as if he was schizophantastic, eating the cake and still have it. Tragically he himself believed in his transfinite whole numbers.] - Get calm. The best recommendation on can give to students is to learn pertaining analysis like a text from bible. The axiomatic method detracts from some otherwise obvious discrepancies between the immediately understandable original notions and sophisticated, arbitrarily twisted mathematical counterparts. - As a rule, it is not wrong to treat variables that denote continuous and infinite quantities as if they were really exact, i.e., (un)real and at a time also rational. - However do not believe that the belief of the mathematicians in what Hilbert called gewisse Zusammenhaenge (= certain tenets) always fits to more comprehensive and in particular physical relations. Return to you: You are quite right: Without an end of the cue there is no successor. Forget AC. If one is able and ready to think consequently, then one gets aware that the thinking by Dedekind as well as Cantor and their fellows is based on admittedly missing or incorrect definitions and evidences. Even Ebbinghaus did not only reiterate the naive theory, he also did not comment on his quotation of Lessing which referred to an obvious but valuable error. I do not share the opinion that set theory is valuable. I feel it an inconsequent or maybe dishonest cover of the impossibility to subordinate the world of the ideal mathematical continuum to the likewise ideal world of discrete numbers. What would not work without the alephs? Ask physicists whether they are using reals or rationals. They will not agree on that. When I asked mathematicians how to deal with zero when splitting IR into IR+ and IR- I got as much answers as possiblities. None was free of arbitrariness. I had to find a unique and compelling answer myself. >> then, considering that exists the successor point of 0 (that I >> denote with 0'), >> 0' = 0. infinite 0 and as last digit 1 (0.0..01 , .. = endless >> zeros ). How to distinguish it? >> But I know that 0' isn't a real number because between two real >> numbers there are infinite real numbers. >> But, considering that 0' isn't a real number and the bijection from R >> to SL, >> doesn't exist the successor point in the SL of point identified with >> 0. >> But through this reasoning I deduce that in the SL there are infinite >> holes. Stop mocking. >> Both the sets of real numbers and natural numbers are totally ordered >> for >> the relation less or equal than, meaning that for any pair of numbers >> in >> the set you can say if one is less or equal than the other. You can say and believe it. For the numbers of a genuine continuum, this is all you can do. Trichotomy ceases to be valid across the abyss. >> This may lead to expecting that you can place all the numbers of the >> set in >> order (which you can) and go from one number to the next, which you can >> for >> natural numbers, but not for reals. Between any pair {a, a+e} there's >> an >> infinite number of other reals, like a+e/2. You'll always find numbers >> between a number and a candidate successor. So no immediate successor >> exists. But that doesn't mean there are gaps. Without gap no successor and vice versa, discrete or continuous, tertium non datur. >> Note that the set {1, 2, 3, 4, 10, 11, 12, 13} with the relation less >> than >> or equal doesn't have gaps either! In the context of a line there are gaps between all elements, even 1 and two. >> The principle problem that I have, it consists in the affirmation: >> there's a bijection from R to SL If R is merely quasi-continuous, i.e., continuous in the sense of set theory, why not twisting the notion of a line accordingly? > Mathematics (derived from set theory and logic) can't prove that your > wire has no holes -its an atomic physics problem .R is a mathematical > object (a set with further structure which can be constructed to from > set theory in such a way as to have no holes (assuming the positive > integers exist (a set with a successor function satisfying th Peano > axioms) .To talk about the bijiction between R and and the wire you > really have to model the wire in set theory .That is done as part of > foundations of Euclidean Geometry (See Forder's book -Dover ) and > there it is specifically assumed in the theory (after 100 pages ) that > the model wire has no holes .If you were doing atomic Physics it is > probably wrong but for usual uses of wires the assumption is useful > and accepted (like in studying music on a voilin. > I am responding because I noticed that no one really saw what your > question was .i hope this helps. smn Could you please also offer help to J. v. Neumann who in 1932 introduced Hilbert space and already in 1935 admitted he did no longer believe in Hilbert space? Salviati: ... in ultima conclusione, gli attributi di eguale maggiore e minore non aver luogo ne gl'infiniti, ma solo nelle quantit.88 terminate. IR>|>IR+>|>IR === Subject: a rather pedestrian sum Am posting this as a separate topic because it sort of is. Regarding the sum S(x) = Sum [(-1)^k * Exp[-(x-prime[k])^2],k running from 1 to Infinity, prime[k] being the kth prime, I have by taking derivatives that there are alternating max and mins at successive primes, and therefore zeros between successive primes. Is this much true? === Subject: Re: a rather pedestrian sum posting-account=HaopWgoAAADs72-s8RQYwP_-ruRUuNzX .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; InfoPath.1),gzip(gfe),gzip(gfe) > Am posting this as a separate topic because it sort of is. Regarding the sum S(x) = Sum [(-1)^k * Exp[-(x-prime[k])^2],k running from 1 to Infinity, prime[k] being the kth prime, I have by taking derivatives that there are alternating max and mins at successive primes, and therefore zeros between successive primes. Is this much true? > Please post some example. === Subject: Re: What if: the Church had NOT condemned Galileo > >Modern scientists tend to misinterpret the recent rehabilitation of >Galileo Galilei as indicating that Church admits that they were wrong >to prosecute him, at the time. This is most certainly not the the >case. All the Church is saying is that Galileo was not a bad person, >and that his writings, even his satires of the Church, no longer pose >any social threat. >http://www.catholicnews.com/data/stories/cns/0801299.htm >... >Jesuit Father Sabino Maffeo, the Vatican Observatory's vice director >for administration, told CNS that Galileo ran into trouble with the >Holy Office because he did not have proof for his claims. >Not having proof ... (the Holy Office) was forced to hold on to the >centuries-old concept that saw Earth as the center of the cosmos, he >said. >If he had had proof, which did not come for another 100 years with >discoveries made by Isaac Newton, Galileo's fate could have been much >different, Father Maffeo said. He added that Italian Cardinal Robert >Bellarmine, who was part of the 17th-century Vatican commission that >admonished Galileo not to hold or defend the Copernican theory, had >told Galileo the day in which you bring a demonstration then we will >have to look at how sacred Scripture gets interpreted differently, >but >as long as there is no proof, we will continue to interpret >(Scripture) literally as we have all along. >... >What would have happened if the Church had not prosecuted or censured >Galileo? Would Newton have had the same incentive to develop his >comprehensive Copernican explanation of the Universe? Would society >have been destabilized by lack of faith in the Church and conventional >social order? >What implications does this example have for modern Church criticisms >of scientific theories such as Evolution and the Big Bang? Does the >Church, or other social institutions have some role in integating >scientific concepts into the broader social perspective? Was Galileo >a kind of idiot-savant, not understanding how to fully develop his own >ideas? Is this a common problem among scientists in general, who are >not holistic thinkers, however skilled they may be in their own >speciality? >>Though I thought they [Galileo's views on the solar system] were >>based on very certain and evident proofs, I would not wish, for >>anything in the world, to maintain them against the authority of the >>Church. - Rene Descartes. >>Descartes fled to the Protestant part of Europe after Galileo's trial.- Hide quoted text - >>- Show quoted text - >Nevertheless, the scientific method is a Christian concept, and was >nurtured systematically by the Church. The notion that scientists >should always go unchallenged is not necessarily a constructive one. >The one extreme is Church suppression. The other is academic self- >indulgence and incompetence. There may be a happy medium that should >be striven for. >>You apparently have no idea how science works. The point of science is >>to look for the truth which is, of course, testable. Scientists test >>things that other scientists do. They challenge each other all the time >>both for professional and ego reasons. >>Openness gets you truth. The church is not interested in that.- Hide quoted text - >>- Show quoted text - >I know exactly how science works, Doug. Professional bureaucrats play >political games to drum anyone out of the field who threatens their >own position. >>First of all, scientists are not bureaucrats. You are showing your >>ignorance. >> Scientists lie systematically to advance their own >agenda at the expense of their opponents, >>You are thinking about religion. Science is testable. Lies are not. >>You need to study something about science. >> and don't give a damn about >anything except their own power. Professional science is exclusively >concerned with power. Truth is a matter of total indifference to the >professional scientific community. >>No, you are confusing this with religion. See above. >>You have no clue about science. You have hate and bias. Learn something >>and you will be embarrassed. I have been in this business for over 40 >>years and you are wrong.- Hide quoted text - >>- Show quoted text - >You're a crook, Doug. Admit it. >>This is humorous but before you go out to read your first book about >>science, first go buy a dictionary (you know the one with all the >>words but not much of a plot) and look up crook. You clearly do not >>understand the word. In science, you are not allowed to randomly >>redefine words the way you do in philosophy. >>I do not know whether to pity you for your ignorance or to be >>disgusted with your reveling in it.- Hide quoted text - >>- Show quoted text - >Doug, you're a moron. >>When you get your dictionary, look up the word moron. You may be >>surprised to see what it means. >As, indeed, are quite a few scientists. By glorifying stupidity, >like yours, >>Science glorifies facts and truth. >>and calling questioning it ignorance, you help to >prevent progress, and maintain the status quo. >>Yes we have stopped the internet, stopped all the industrial >>development, stopped all medical research etc. What world do >>you actually live in? >>Resources are witheld> from useful work, in preference to bells and whistles technology of >the type Corporate Giants like Microsoft favor. Don't change >anything, that might undermine the current power base. We don't want >progress, we want stability. Just like the mediaeval Catholic >Church. Actually, they might have been more progressive than the >current military-industrial-scientific bureaucracy that you favor! >>I finally realized that your issue is jealousy. You feel that someone >>else got something you deserved. This is a very childish behaviour >>and I cannot help you with your problems. It must be a very sad >>life for you though. >>- Hide quoted text - >>- Show quoted text -- Hide quoted text - >>- Show quoted text - > > > I see I'm dealing with a member of the military-industrial-scientific > bureaucracy here. Anyone who doesn't want to kill and steal for a > living is jealous of those who do. Yeah, right. My goodness, your jealousy is really getting to you. You certainly live in a sad world. Why are you looking to make yourself so unhappy? This does explain why you hate Galileo and were happy to have him get persecuted because of the stupidity of the church. > > > === Subject: Re: What if: the Church had NOT condemned Galileo posting-account=5PxZDwoAAABv9p271mvbXvIpZRL9I54K CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) >Modern scientists tend to misinterpret the recent rehabilitation of >Galileo Galilei as indicating that Church admits that they were wrong >to prosecute him, at the time. ÊThis is most certainly not the the >case. ÊAll the Church is saying is that Galileo was not a bad person, >and that his writings, even his satires of the Church, Êno longer pose >any social threat. >http://www.catholicnews.com/data/stories/cns/0801299.htm >... >Jesuit Father Sabino Maffeo, the Vatican Observatory's vice director >for administration, told CNS that Galileo ran into trouble with the >Holy Office because he did not have proof for his claims. >Not having proof ... (the Holy Office) was forced to hold on to the >centuries-old concept that saw Earth as the center of the cosmos, he >said. >If he had had proof, which did not come for another 100 years with >discoveries made by Isaac Newton, Galileo's fate could have been much >different, Father Maffeo said. He added that Italian Cardinal Robert >Bellarmine, who was part of the 17th-century Vatican commission that >admonished Galileo not to hold or defend the Copernican theory, had >told Galileo the day in which you bring a demonstration then we will >have to look at how sacred Scripture gets interpreted differently, >but >as long as there is no proof, we will continue to interpret >(Scripture) literally as we have all along. >... >What would have happened if the Church had not prosecuted or censured >Galileo? ÊWould Newton have had the same incentive to develop his >comprehensive Copernican explanation of the Universe? ÊWould society >have been destabilized by lack of faith in the Church and conventional >social order? >What implications does this example have for modern Church criticisms >of scientific theories such as Evolution and the Big Bang? ÊDoes the >Church, or other social institutions have some role in integating >scientific concepts into the broader social perspective? ÊWas Galileo >a kind of idiot-savant, not understanding how to fully develop his own >ideas? ÊIs this a common problem among scientists in general, who are >not holistic thinkers, however skilled they may be in their own >speciality? >>Though I thought they [Galileo's views on the solar system] were >>based on very certain and evident proofs, I would not wish, for >>anything in the world, to maintain them against the authority of the >>Church. - Rene Descartes. >>Descartes fled to the Protestant part of Europe after Galileo's trial.- Hide quoted text - >>- Show quoted text - >Nevertheless, the scientific method is a Christian concept, and was >nurtured systematically by the Church. ÊThe notion that scientists >should always go unchallenged is not necessarily a constructive one. >The one extreme is Church suppression. ÊThe other is academic self- >indulgence and incompetence. ÊThere may be a happy medium that should >be striven for. >>You apparently have no idea how science works. ÊThe point of science is >>to look for the truth which is, of course, testable. Scientists test >>things that other scientists do. They challenge each other all the time >>both for professional and ego reasons. >>Openness gets you truth. ÊThe church is not interested in that.- Hide quoted text - >>- Show quoted text - >I know exactly how science works, Doug. ÊProfessional bureaucrats play >political games to drum anyone out of the field who threatens their >own position. >>First of all, scientists are not bureaucrats. You are showing your >>ignorance. >> ÊScientists lie systematically to advance their own >agenda at the expense of their opponents, >>You are thinking about religion. Science is testable. ÊLies are not. >>You need to study something about science. >> and don't give a damn about >anything except their own power. Ê Professional science is exclusively >concerned with power. ÊTruth is a matter of total indifference to the >professional scientific community. >>No, you are confusing this with religion. ÊSee above. >>You have no clue about science. You have hate and bias. ÊLearn something >>and you will be embarrassed. I have been in this business for over 40 >>years and you are wrong.- Hide quoted text - >>- Show quoted text - >You're a crook, Doug. ÊAdmit it. >>This is humorous but before you go out to read your first book about >>science, first go buy a dictionary (you know the one with all the >>words but not much of a plot) and look up crook. You clearly do not >>understand the word. ÊIn science, you are not allowed to randomly >>redefine words the way you do in philosophy. >>I do not know whether to pity you for your ignorance or to be >>disgusted with your reveling in it.- Hide quoted text - >>- Show quoted text - >Doug, you're a moron. >>When you get your dictionary, look up the word moron. You may be >>surprised to see what it means. >As, indeed, are quite a few scientists. ÊBy glorifying stupidity, >like yours, >>Science glorifies facts and truth. >>and calling questioning it ignorance, you help to >prevent progress, and maintain the status quo. Ê >>Yes we have stopped the internet, stopped all the industrial >>development, stopped all medical research etc. What world do >>you actually live in? >>Resources are witheld> from useful work, in preference to bells and whistles technology of >the type Corporate Giants like Microsoft favor. Ê Don't change >anything, that might undermine the current power base. ÊWe don't want >progress, we want stability. ÊJust like the mediaeval Catholic >Church. ÊActually, they might have been more progressive than the >current military-industrial-scientific bureaucracy that you favor! >>I finally realized that your issue is jealousy. ÊYou feel that someone >>else got something you deserved. ÊThis is a very childish behaviour >>and I cannot help you with your problems. ÊIt must be a very sad >>life for you though. >>- Hide quoted text - >>- Show quoted text -- Hide quoted text - >>- Show quoted text - > I see I'm dealing with a member of the military-industrial-scientific > bureaucracy here. ÊAnyone who doesn't want to kill and steal for a > living is jealous of those who do. Ê Yeah, right. My goodness, your jealousy is really getting to you. You certainly live > in a sad world. ÊWhy are you looking to make yourself so unhappy? ÊThis > does explain why you hate Galileo and were happy to have him get > persecuted because of the stupidity of the church. - Hide quoted text - - Show quoted text -- Hide quoted text - - Show quoted text - Doug, you're a moron. Keep talking. You help to prove my point. === Subject: Re: What if: the Church had NOT condemned Galileo >Oh, certainly, I agree the Church's objections were psychological -- >>or, shall we say, socio-psychological. Ø ØThey feared social >>disruption from crticism of the current conception of things without >>a >>comprehensive alternative being presented. ØAs for Newton being just >>a >>link in the chain, Newton's Principia is a pretty big link! ØAnd, >>probably the final one. ØA truly comprenhensive system of things to >>be >>put alongside Thomas Aquinas' work. ØI doubt Galileo was capable of >>work of this type, I suspect he was more of an engineer than a >>scientist. ØAnd, he got out of his depth, to his cost. >-- >Michael Press >Galileo's work with the inclined plane alone >qualifies him as a great scientist. >> Galileo may have been a great scientist. ØHe was not a great >> theoretician. Ø Which may have been what was bothering the Church >> authorities. >> This has got to be one of the all time unbelievably silly statements >> even for usenet. >> Ø He was not providing a comprehensive theoretical >> alternative to Aristotle's ideas, which had been integrated with >> Catholic theology by Thomas Aquinas. >> He was presenting an idea that the church did not like because >> it threatened their control. >> Ø ØNewton's work did. ØGalileo was >> ridiculing the existing theoretical conception while providing no >> alternative to it. >> Presenting facts is not ridiculing. It is called science. >> - Hide quoted text - >> - Show quoted text -- Hide quoted text - >> - Show quoted text - >> Presenting facts is not ridiculing. Calling the Pope a fool in print >> is. This is exactly what Galileo did. This is what got him into >> trouble. Umm, would you be interested in replying to any of the posts in which I > have noted that this is false? And maybe support your argument, as I > supported mine? We could even have a useful debate. > I? we? You have 'asserted' that it is false - your 'noting' of a position is not strictly 'falsification' - notwithatanding what yhur brother may once have thought. === Subject: Re: What if: the Church had NOT condemned Galileo posting-account=fNhuGwoAAAAcu_mWZdkQrPnHIPR_gQVH CLR 1.0.3705; .NET CLR 1.1.4322; Media Center PC 4.0),gzip(gfe),gzip(gfe) >>Modern scientists tend to misinterpret the recent rehabilitation of >>Galileo Galilei as indicating that Church admits that they were wrong >>to prosecute him, at the time. ÊThis is most certainly not the the >>case. ÊAll the Church is saying is that Galileo was not a bad person, >>and that his writings, even his satires of the Church, Êno longer pose >>any social threat. >>http://www.catholicnews.com/data/stories/cns/0801299.htm >>... >>Jesuit Father Sabino Maffeo, the Vatican Observatory's vice director >>for administration, told CNS that Galileo ran into trouble with the >>Holy Office because he did not have proof for his claims. >>Not having proof ... (the Holy Office) was forced to hold on to the >>centuries-old concept that saw Earth as the center of the cosmos, he >>said. >>If he had had proof, which did not come for another 100 years with >>discoveries made by Isaac Newton, Galileo's fate could have been much >>different, Father Maffeo said. He added that Italian Cardinal Robert >>Bellarmine, who was part of the 17th-century Vatican commission that >>admonished Galileo not to hold or defend the Copernican theory, had >>told Galileo the day in which you bring a demonstration then we will >>have to look at how sacred Scripture gets interpreted differently, >>but >>as long as there is no proof, we will continue to interpret >>(Scripture) literally as we have all along. >>... >>What would have happened if the Church had not prosecuted or censured >>Galileo? ÊWould Newton have had the same incentive to develop his >>comprehensive Copernican explanation of the Universe? ÊWould society >>have been destabilized by lack of faith in the Church and conventional >>social order? >>What implications does this example have for modern Church criticisms >>of scientific theories such as Evolution and the Big Bang? ÊDoes the >>Church, or other social institutions have some role in integating >>scientific concepts into the broader social perspective? ÊWas Galileo >>a kind of idiot-savant, not understanding how to fully develop his own >>ideas? ÊIs this a common problem among scientists in general, who are >>not holistic thinkers, however skilled they may be in their own >>speciality? > If I understand Jerry correctly, he is just saying that the Church use > to put a high penalty on exposing a scientific theory with inadequate > proof. > Standards have certainly declined since the 17th Century. ÊThese days, > a scientist would get nothing more than a rejection letter for his > paper if proof was substandard. > Contrast this with the treatment of Giordano Bruno. ÊUnlike Galileo, > Bruno refused to fully recant on his support of Copernicus' > Heliocentric Theory and he claimed that the Sun was just another star > in infinite space. ÊAfter his conviction at the Inquisition in Rome, > his jaw was clamped shut with a iron gag, his tongue was pierced with > an iron spike, and another was driven into his palate. ÊThen he was > burned at the stake. > According to Jerry, this probably because he misspelled something and > that, of course, justified his treatment and he should have been > happy that they treated him so leniently.- Hide quoted text - > - Show quoted text - You should study some theology and philosophy, Doug. ÊAlso, read some > good literature. ÊYou might learn how to write a coherent sentence. > You might, also, get some comprehension of reality beyond what you've > read in science textbooks. ÊWhich is, unfortunately, a pretty small > and systematically distorted slice of reality. ÊWhich, of course, most > people are well aware of.- Tekst uit oorspronkelijk bericht niet weergeven - - Tekst uit oorspronkelijk bericht weergeven - who the fuck cares asbout history === Subject: Re: What if: the Church had NOT condemned Galileo posting-account=jPnQ2goAAAA461y3QD0lbyw0oKeThma1 AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.2 Safari/525.20.1,gzip(gfe),gzip(gfe) presumably, you can cite a letter from Galileo to Kepler. > Q. What is a tautological space? > A. A tautological space. thus: speaking of Young's anhialation of Newton's photons, here is the earlier elaboration on light by Fermat: http://www.wlym.com/~seattle/dynamis/issues/august08-fermat.pdf > as we know from Newton's iron-poor corpuscles. --ROTC, your summer vacation in the Sahara Desert ( S u d a n ) ; presage the Draft for your middleschool class of '12 -- brought to you by Allstate (tm) and Oxford U. Press! http://larouchepub.com/pr/2008/080813moloch_brown.html http://wlym.com === Subject: Re: What if: the Church had NOT condemned Galileo posting-account=jPnQ2goAAAA461y3QD0lbyw0oKeThma1 AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.2 Safari/525.20.1,gzip(gfe),gzip(gfe) Tuesday, July 27, 1638 Sir, The receipt of your letter in which you do me the honor of promising your friendship, gave me no less joy than if I had received it from a mistress whose good graces I passionately desired. And your other writings which preceded it reminded me of the Bradamante of our poets, who would take no one as a servant who had not previously proved himself against her in combat. It is not every day that I compare myself to that Roger who was the only one in the world capable of standing against her in combat;1 but such as I am, I assure you that I honor your merit considerably. And seeing the most recent method that you used to find tangents to curved lines [in your last letter], I have nothing to say in response, other than that it is very good, and that, if you had explained it that way in the beginning, I would have had no disagreement. The remainder of the letter deals specifically with geometry, and has not been included here. thus: AP, you nut.... actually, it is a commonplace hearsay that the founding parental units were atheists, usually called Deists -- I'm the Higher Power of the 13th Step -- ME ?!? ... but it isn't true. if you read LaRouche, you'd know that, but there is some question about Shakespeare as a Second Language. thus: Apostol begins the course with integration, as per Liebniz; there's a special name for that, per Liebniz: primitive equation or prime equation? thus: ah, sort of a solopsistic monad. I am reminded of hte Many Words Interpretation of the Copenhagenskool: to be or not to be that is the shape of the box or the bubble -- inside or out? > Q. What is a tautological space? > A. A tautological space. thus: speaking of Young's anhialation of Newton's photons, here is the earlier elaboration on light by Fermat: http://www.wlym.com/~seattle/dynamis/issues/august08-fermat.pdf > as we know from Newton's iron-poor corpuscles. --ROTC, your summer vacation in the Sahara Desert ( S u d a n ) ; presage the Draft for your middleschool class of '12 -- brought to you by Allstate (tm) and Oxford U. Press! http://larouchepub.com/pr/2008/080813moloch_brown.html http://wlym.com === Subject: Re: What if: the Church had NOT condemned Galileo posting-account=jPnQ2goAAAA461y3QD0lbyw0oKeThma1 AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.2 Safari/525.20.1,gzip(gfe),gzip(gfe) this was the subject ot the keynote speaker at the Ninth Annual Nonlinear Science Conference at UCLA, although I no longer blame Newton exclusively for the controversy with Liebniz. anyway, apparently, Hooke did not do the math in a formalistic way; after all, he was the Royal Society's experimenter. the other part of the joke is that the '99 2-pound coin is inscribed, On the shoulders of giants. on the edge; at least, Newton was effective in his office-of-reward. > stole the solution from Hooke, to boot -- on the shoulders of a giant, > who was physically a dwarf; hence, the joke. Wrong again, but excusably, since that meme has spread widely. Read the > shoulders bit in context, and you can see it was not an insult. thus: ah, Lord Berty's thing was called the Unity of Sciences; well, here's a reference: http://www.larouchepub.com/other/2002/2949moonification.html The two operations of Wells and Russell from which Moon sprung are these: The Moral Re-Armament Movement, founded at a 1921 meeting between a wacky Lutheran preacher from Philadelphia and two British delegates to the Washington Disarmament Conference, Lord Arthur Balfour and H.G. Wells. Moral Re-Armament became the mass organizational vehicle for implementation of Wells' 1928 call in The Open Conspiracy, for a worldwide movement for draft resistance. The environment of Moon's Korean ministry was under control of Moral Re-Armament when he was picked up as an intelligence asset during the Korean War. The Unity of the Sciences movement. Founded in 1935 under the supervision of Lord Bertrand Russell and John Dewey, it brought together Trotskyite academics Albert Wohlstetter (mentor of current Defense Policy Board Chairman Richard Perle), Sidney Hook, and Ernest Nagel, with members of the radical-positivist Vienna Circle. Merging with Robert M. Hutchins at the University of Chicago in the 1950s, this operation took over the teaching of science in the United States. Thomas Kuhn's widely read fraud, The Structure of Scientific Revolutions, was published as the second volume of their Encyclopedia of Unified Sciences. In 1972, the Moonies were given the Unity of Sciences franchise, sponsoring the first of their still-ongoing International Conferences of the Unity of Sciences. Their early sessions featured such notables as Manhattan Project physicist Eugene Wigner, the lifelong ally of that truly mad scientist Leo Szilard (the model for Dr. Strangelove, in Stanley Kubrick's film of that name), and environmental fascists Alexander King and Aurelio Peccei, founders of the no-growth Club of Rome. Before looking back to the history of these projects, let us first briefly dispense with the person of Rev. Sun Myung Moon. Moon as a personality is of very little importance, in himself. The real Reverend Moon is a pathetic, if nonetheless nasty, victim of Japanese internment and North Korean torture sessions. He is what the.... thus: an aether simply provides a prealistical model of the phenomenon of waves; one need not deal with photons at all, since they are just formally dual to waves, not at all pardoxical, as shown by Dirac. far too much of theory rests upon Einstein, saying that's impossible, which is the reality behind Michelson-Morley and those who carried the examination further -- but, herr doktor-professor Albert said, Nein!, when he was once in his dysused office at Caltech. let us remember him for those things which he was really good at, not the Big Bang Constant and its infernal klingons (fridge magnets) ?! > I don't need any aether to stick a magnet to a fridge, light up an thus: so, to ask an ignorant question of personal interest, how much of this could be used toward a new or old proof of quadratic reciprocity? > That Ac and -Dc must be quadratic residues, mod D and A, respectively, > is a trivial result: > Let Ax^2 - Dy^2 = c. [1] > Take this mod A: > -Dy^2 = c mod A > (Dy)^2 = -Dc mod A > Therefore, -Dc is a quadratic residue mod A. > Similarly, taking [1] mod D: > Ax^2 = c mod D > (Ax)^2 = Ac mod D > Therefore, Ac is a quadratic residue mod D. thus: ah, sort of a solopsistic monad. I am reminded of hte Many Words Interpretation of the Copenhagenskool: to be or not to be that is the shape of the box or the bubble -- inside or out? > Q. What is a tautological space? > A. A tautological space. thus: speaking of Young's anhialation of Newton's photons, here is the earlier elaboration on light by Fermat: http://www.wlym.com/~seattle/dynamis/issues/august08-fermat.pdf > as we know from Newton's iron-poor corpuscles. --ROTC, your summer vacation in the Sahara Desert ( S u d a n ) ; presage the Draft for your middleschool class of '12 -- brought to you by Allstate (tm) and Oxford U. Press! http://larouchepub.com/pr/2008/080813moloch_brown.html http://wlym.com === Subject: Re: roots to 4th, 5th, 6th & 7th degree polynomials I share your doubts, but if mathematics doesn't forsake me, it has to be right. When a curve f(x) crosses the x-axis at point t, for small distances close to t, the curve approximates a straight line, right? NO, IT DOESN'T. That's one flaw in my method. If I take the derivative of f(x) at t, the tangent line is a straight line. One area is between f(x) and the x-axis from t to (t + delta t), and the other area is between f(x) and the x-axis from (t - delta t) to t. I simply say that when f(x) crosses the x-axis at f(x)=0, THE TWO AREAS CANCEL TO ZERO. A better strategy is to use the areas above and below the tangent line when f(x) crosses the x-axis. g(x) = f '(x) x + b equation of tangent line INT (f '(x) x + b) dx from (t - delta t) to (t + delta t) = 0 For instance, using the example suggested by Fran.8dois and letting delta t = c, f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1, f '(x) = 4x^3 - 12x^2 + 12x - 4 INT (4x^4 - 12x^3 + 12x^2 - 4x + bx) from (t - c) to (t + c) = 0 = [(4/5)x^5 - 3x^4 + 4x^3 - 2x^2 + (b/2)x^2] from (t-c) to (t+c) = 0 = [(4/5)x^3 - 3x^2 + 4x - 2 + (b/2)] from (t-c) to (t+c) = 0 h(t)= +(4/5)[(t+c)^3 - (t-c)^3] -3[(t+c)^2 - (t-c)^2] +4[(t+c) - (t-c)] - 2 + b/2 =0 f '(t)= 4t^3 - 12t^2 + 12t - 4= -b/t= [f(t+c) - f(t-c)]/(2c) b = -t f '(t) h(t)= +(4/5)[(t+c)^3 - (t-c)^3] -3[(t+c)^2 - (t-c)^2] +4[(t+c) - (t-c)] - 2 - (1/2)t f '(t) =0 Express h(t) as a Maclaurin series, take the limit as c approaches zero and WA LA! There's the solution to the quartic in terms of the known solution to the cubic. Notice, however, in the original equation, f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1, The lonely 1 has been entirely omitted out of the solution. This is a heinous injustice to the lonlely 1. Maybe one of you can FIX IT. -- Jon G. jon8338@peoplepc.com http://mypeoplepc.com/members/jon8338/math/index.html > roots to 4th, 5th, 6th & 7th degree polynomials http://mypeoplepc.com/members/jon8338/math/id19.html The best I can promise is all denominators are nonzero, and dimensional > analysis passes. See if you can understand the concept. If it works, it can find the roots > to any degree polynomial. If it doesn't, then at best it may give a rough > estimate. -- > Jon Giffen > jon8...@peoplepc.com Have you tried the formulae for a variety of polynomials. Your cubic might be right; the quartic is probably wrong since a resolvent cubic is needed and you do not have one; for higher degrees, Abel's theorem says Nope! === Subject: Re: roots to 4th, 5th, 6th & 7th degree polynomials <2tGdnazT3KdLv1rVnZ2dnUVZ_u-dnZ2d@earthlink.com> posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2008071615 Fedora/3.0.1-1.fc9 Firefox/3.0.1,gzip(gfe),gzip(gfe) > I share your doubts, but if mathematics doesn't forsake me, it has to be > right. The difference between mathematics and the natural sciences is that in math one does not simply state something and wait for a rebuttal: one is expected to actually *prove* that what one says is true. -- m === Subject: Re: roots to 4th, 5th, 6th & 7th degree polynomials > ... > A better strategy is to use the areas above and below the tangent line when > f(x) crosses the x-axis. > As mentionned: if all works as you planed, you only find roots where the sign changes. But ok, you could look at f'(x) to find the roots where f(x) has a root of even multipicity. > g(x) = f '(x) x + b equation of tangent line > Mh, the equation of tangent line at x0 is g(x) = f '(x0)(x-x0) + f(x0) isn't it? -- Dr. Detlef M.9fller, http://www.mathe-doktor.de oder http://mathe-doktor.de === Subject: Re: roots to 4th, 5th, 6th & 7th degree polynomials posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2008071615 Fedora/3.0.1-1.fc9 Firefox/3.0.1,gzip(gfe),gzip(gfe) > roots to 4th, 5th, 6th & 7th degree polynomials http://mypeoplepc.com/members/jon8338/math/id19.html The best I can promise is all denominators are nonzero, and dimensional > analysis passes. See if you can understand the concept. ÊIf it works, it can find the roots > to any degree polynomial. ÊIf it doesn't, then at best it may give a rough > estimate. -- > Jon Giffen > jon8...@peoplepc.com So, what are the roots of X^5 + 20 X + 16 = 0 ? -- m === Subject: Re: Mathematics: how to start again posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022; Tablet PC 2.0),gzip(gfe),gzip(gfe) > Give an example of a topological space that is both connected and totally > disconnected. Spacetime. === Subject: Re: Mathematics: how to start again posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022; Tablet PC 2.0),gzip(gfe),gzip(gfe) > Give an example of a topological space that is both connected and totally > disconnected. If you consider randomness as if it were a fluid embedded in dimension, then the resulting space should satisfy the question. I do not think that it is really provable, but I also dont see that as much of a problem. === Subject: Fundamentals of Machine Component Design posting-account=NwTOtgoAAACNXr76jlZ-zCkqjWd861uC 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) Do you have this solutions manual available?.... R.C. Juvinall and K.M. Marshek, Fundamentals of Machine Component Design, 4th edition, John Wiley and Sons, Inc. (2005) If so, how do I go about obtaining a PDF solutions manual? === Subject: ChartPro Ver. 1.8 posting-account=HZNw1AoAAACj5O59E7bG9-eTCyZA24mT .NET CLR 1.1.4322; .NET CLR 2.0.50727; Zango 10.0.370.0),gzip(gfe),gzip(gfe) spider-ntc-tc01.proxy.aol.com[CFC87041] (Prism/1.2.1), HTTP/1.1 cache-ntc-ac02.proxy.aol.com[CFC87483] (Traffic-Server/6.1.5 [uScM]) ChartPro Ver. 1.8 12 Chart Formats, 7 Two Dimensional, 5 Three Dimensional Control Key 3D Chart Rotation in any position. 50 Rows Max, 180 Columns Max. Programmable user data inputs. UnLimited Different Charts (Any name saved) AutoSave Functions : chart type, charting data. rows, columns, legends, chart loading type. http://www.sharewaresoftware.org/chartpro/chartpro.htm === Subject: combinatorics question I've been trying to figure out what's happening in the Clay Institute's P v. NP Problem description. It talks about the difficulty of generating the answers to the following scenario: you have 400 students, and you want to figure out how many ways you can put them into a dormitory that only holds 100 students. Also, there are certain pairs of students who can't be in the dorm together. I think that the number of combinations of students overall would be 400!/(300!)*(100!). If this is correct, a question: what difference would it make to the problem if instead of just considering the 100 students as an undifferentiated mass, you had to generate the number of *pairs* of students that would fit in the dorm at once? Does that even make sense? I haven't the slightest idea how you would go about eliminating prohibited pairs of students without generating the whole list and striking off those pairs. === Subject: Re: combinatorics question > I've been trying to figure out what's happening in the Clay Institute's > P v. NP Problem description. It talks about the difficulty of generating > the answers to the following scenario: you have 400 students, and you > want to figure out how many ways you can put them into a dormitory that > only holds 100 students. Also, there are certain pairs of students who > can't be in the dorm together. I think that the number of combinations > of students overall would be 400!/(300!)*(100!). If this is correct, a > question: what difference would it make to the problem if instead of > just considering the 100 students as an undifferentiated mass, you had > to generate the number of *pairs* of students that would fit in the dorm > at once? Does that even make sense? I haven't the slightest idea how > you would go about eliminating prohibited pairs of students without > generating the whole list and striking off those pairs. This problem sounds like an extension of bipartite matching (quadpartite, I guess). Bipartite and tripartite matching problems can be solved via network flow, but the structure of the problem makes this solution impossible for quadpartite matching. That said, one could easily represent this problem as a boolean satisfiability problem (or SAT) [1]. P_m_n represents whether or not the person number m is rooming in room n; the clauses are in two sets: statement prohibitions (P_m_n prohibits P_m_k for any k != n, and there must be four people to a room) and the given constraints. This is a naive encoding, yes, but it's late and I'm not in the mood to find more ingenious ones. The entire essence of NP-complete problems is that they are the ones for which all known correct algorithms are essentially try every possible combination, with the best-known correct algorithms merely making it easier to quickly identify solutions which cannot be correct. SAT is one of the more extensively studied problems; I cannot give a discourse on the best algorithms, though. [1] Well, any NP problem can be represented in SAT by virtue of Cook's Theorem, if you want to be technical. In fact, any NP problem can be represented any NP-complete problem by definition [2]. SAT was merely the first encoding I could come up with off the top of my head. [2] Okay, while we're on the topics of technicalities, `NP' here is used to refer to decision problems. I believe, however, that Cook's Theorem represents a form of reduction that allows the non-decision variants to be represented in the NP-hard version of SAT, by merely knowing where on the tape the final answer will be encoded and querying values. === Subject: Re: Simple Soln for Complex Root to System of Eqns? > Consider a system of N polynomials. I take it these are polynomials in one variable. > Each polynomial has degree 2N and has known complex coefficients. > I want to find a solution of this system of complex polynomials that > lie on the unit circle. By a solution of a system I take it you mean a complex number where all the polynomials vanish. But this is absurd. Even if you only have 2 polynomials, there is no reason to think there will be a complex number that will be a root of both of them. And even if you have only one polynomial, there is no reason to think it will have a solution on the unit circle. What do you really mean? -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Simple Soln for Complex Root to System of Eqns? posting-account=3WPJYgoAAAA55VjhzK9i07RN8h8u8eEs 1.7; .NET CLR 1.0.3705; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30),gzip(gfe),gzip(gfe) Here is a simple example. Consider the system of 2 eqns (e1) C11*z^4 + C12*z^3 + C13*z^2 + C14*z + C15 = 0 (e2) C21*z^4 + C22*z^3 + C23*z^2 + C24*z + C25 = 0 Where all Cij are complex and z is the complex variable of interest. Now the condition that z lies on the unit circle implies that z also satisfies: (e3) (C15^*)*z^4 + (C14^*)*z^3 + (C13^*)*z^2 + (C12^*)*z + (C11^*) = 0 (e2) (C25^*)*z^4 + (C24^*)*z^3 + (C23^*)*z^2 + (C22^*)*z + (C21^*) = 0 (where ^* denotes complex conjugation) The idea is that each eqn in this case has 4 solutions, but there is one unique solution that satisfies all of them. In this case, we have 4 linear eqns (in the z^n) in 4 unknowns, so it seems that there should be a single unique solution. However, the X_k for z^k, then I will get some solution, but how can I be guaranteed that the solution obtained will satisfy X_k = z^k for all k=1,...N? Right now, I'm going off of the fact that this problem arises in the context of a physical application that I expect to have a solution, so I'm inferring that the existence of a solution means that the one I get is the one I am after, but I'd like to be on firmer mathematical ground in making such a claim. TIA, Matt === Subject: Re: Simple Soln for Complex Root to System of Eqns? > Here is a simple example. Consider the system of 2 eqns > > (e1) C11*z^4 + C12*z^3 + C13*z^2 + C14*z + C15 = 0 > (e2) C21*z^4 + C22*z^3 + C23*z^2 + C24*z + C25 = 0 > > Where all Cij are complex and z is the complex variable of interest. > > Now the condition that z lies on the unit circle implies that z also > satisfies: > > > (e3) (C15^*)*z^4 + (C14^*)*z^3 + (C13^*)*z^2 + (C12^*)*z + (C11^*) = > 0 > (e2) (C25^*)*z^4 + (C24^*)*z^3 + (C23^*)*z^2 + (C22^*)*z + (C21^*) = > 0 > > (where ^* denotes complex conjugation) > > The idea is that each eqn in this case has 4 solutions, but there is > one unique solution that satisfies all of them. > In this case, we have 4 linear eqns (in the z^n) in 4 unknowns, You don't have 4 unknowns, you have 1 unknown, z. Once you know z, you know z^2, z^3, and z^4. Your (e1) has 4 solutions. Your (e2) also has 4 solutions, and there is no reason whatsoever to think that any of them will be among the 4 solutions of (e1). For example: what are you going to do if (e1) is z^4 - 14 z^3 + 81 z^2 - 154 z + 120 = 0 and (e2) is z^4 + 14 z^3 + 81 z^2 + 154 z + 120 = 0 If I've done my sums right, (e1) has the solutions 2, 3, 4, and 5, and (e2) has solutions -2, -3, -4, -5, so they have no common solutions, and neither one has a solution on the unit circle. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Simple Soln for Complex Root to System of Eqns? posting-account=3WPJYgoAAAA55VjhzK9i07RN8h8u8eEs 1.7; .NET CLR 1.0.3705; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30),gzip(gfe),gzip(gfe) > Your (e1) has 4 solutions. Your (e2) also has 4 solutions, > and there is no reason whatsoever to think that any of them will be > among the 4 solutions of (e1). > In general, that would be correct. I know, however, based on physical grounds (the eqns are derived from a problem I'm working on involving wide-band radar) that a solution does exist. The question then is, if I know a solution exists, is this solution unique? Intuition tells me it should be if all N of the equations are distinct since letting Xn = z^n for n=1,...N, I can obtain a unique solution for [X1,...XN]. Since I know one z satisfies these system of eqns, then this z must yield Xn=z^n for all n=1,...,N. Does this make sense? Matt === Subject: Re: Simple Soln for Complex Root to System of Eqns? posting-account=3WPJYgoAAAA55VjhzK9i07RN8h8u8eEs 1.7; .NET CLR 1.0.3705; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30),gzip(gfe),gzip(gfe) The last equation: (e2) Ê(C25^*)*z^4 + (C24^*)*z^3 + (C23^*)*z^2 + (C22^*)*z + (C21^*) = 0 should have been labeled (e4) Sorry, M === Subject: A wise decision? The reason for Alfred Nobel to decide against a Nobel price for mathematics was Leffler-Mittag. Who can tell more? Maybe, Nobel disliked or even distrusted Cantor's set theory? At least, Nobel would be correct in that transfinite numbers did not at all contribute to useful application in science and technology within more than one hundred years. Salviati: ... in ultima conclusione, gli attributi di eguale maggiore e minore non aver luogo ne gl'infiniti, ma solo nelle quantit.88 terminate. IR>|>IR+>|>IR === Subject: Re: A wise decision? > The reason for Alfred Nobel to decide against a Nobel price for mathematics > was Leffler-Mittag. Who can tell more? Maybe, Nobel disliked or even > distrusted Cantor's set theory? At least, Nobel would be correct in that > transfinite numbers did not at all contribute to useful application in > science and technology within more than one hundred years. By Hardy's criterion, that makes it the best mathematics of the past hundred years. === Subject: Re: A wise decision? > The reason for Alfred Nobel to decide against a Nobel price for >> mathematics >> was Leffler-Mittag. Who can tell more? Maybe, Nobel disliked or even >> distrusted Cantor's set theory? At least, Nobel would be correct in that >> transfinite numbers did not at all contribute to useful application in >> science and technology within more than one hundred years. By Hardy's criterion, that makes it the best mathematics of the past > hundred years. Really? I am not familiar with Godfred Harold H. However, aren't transfinite numbers and the infinitum creatum sive transfinitum unfounded and extremely ugly rather than beautiful and understandable? While I ascribe rather too much ambition to Georg (not Moritz) Cantor, was it a sign of Cantor' professional pride when he dealt with Shakespeare instead of doing serious work? Why did Cantor claim having received the CH immediately from god? Does intellectual curiosity mean just cheeky ignoring of what Archimedes, Galilei, Spinoza, Leibniz, Gauss, and many others found out? Is mathematics really harmless? Do not forget that Archimedes and many European mathematicians were killed, killed themselves like Hausdorff or were forced to emigrate. Salviati: ... in ultima conclusione, gli attributi di eguale maggiore e minore non aver luogo ne gl'infiniti, ma solo nelle quantit.88 terminate. IR>|>IR+>|>IR === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation <9a6dc49orob4ssug6gn6nrj9horhnp6kjs@4ax.com> posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) >Please point out on that page where Pell's Equation is related to >Pythagorean Tripes and specifically where, for instance, you can find >that there must exist a Pythagorean Triple of the form, >u^2 + (u+1)^2 = w^2 >for every integer solution to x^2 - 2y^2 = 1, where w = x+y. Anyone who has studied Pythagorean tripes - sorry, triples! - > can derive this result (or something very like it - see comment > at the end) without effort, using the generating formula given > at the very start of that Wikipedia web page. Yeah, anyone could have, but who did? Hindsight is 20-20. You seem very interested in dismissing results that you didn't realize before I told you them first. That Pell's Equation can be directly connected to Pythagorean Triplets may not be news but I don't think it's not news because YOU say so, because, who are you? Who do you think you are? I suggest you behave with a little more humility when talking about a mathematical subject that is 2000 years old. Cite. James Harris === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation > Who do you think you are? I suggest you behave with a little more > humility when talking about a mathematical subject that is 2000 years > old. I nominate this post for Projection Post of 2008. -- Jesse F. Hughes Mama: Do you need help? Sir Quincy: Nay, I'm a big knight. Knights don't need help with pee-pee. === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation [...] note again the weird connection I found between certain > Pythagorean Triplets and Pell's Equation in the form x^2 - Dy^2 = 1 > when D-1 is a perfect square. For instance for D=2, I have that for > every solution of Pell's Equation you have a Pythagorean Triplet! > > But the triplets are special in that with u^2 + v^2 = w^2, v = u+1. > The connection is that w is x+y from Pell's Equation. The more general > result is that u = sqrt(D-1)j, and v = j+1, while w still equals x+y. This is a special case of classical results going back to Euler, if not further. For some history and more modern interpretations I HIGHLY RECOMMEND the following exposition: Franz Lemmermeyer. Higher descent on Pell conics. II. Two centuries of missed opportunities http://www.fen.bilkent.edu.tr/~franz/publ/pell-b.pdf http://arxiv.org/abs/math.NT/0311296 See also parts I and III at http://www.fen.bilkent.edu.tr/~franz/publ.html --Bill Dubuque === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > [...] note again the weird connection I found between certain > Pythagorean Triplets and Pell's Equation in the form Ê x^2 - Dy^2 = 1 > when D-1 is a perfect square. ÊFor instance for D=2, I have that for > every solution of Pell's Equation you have a Pythagorean Triplet! > But the triplets are special in that with u^2 + v^2 = w^2, v = u+1. > The connection is that w is x+y from Pell's Equation. The more general > result is that u = sqrt(D-1)j, and v = j+1, while w still equals x+y. This is a special case of classical results going back to Euler, > if not further. For some history and more modern interpretations > I HIGHLY RECOMMEND the following exposition: Franz Lemmermeyer. Higher descent on Pell conics. > II. Two centuries of missed opportunitieshttp://www.fen.bilkent.edu.tr/~franz/publ/pell-b.pdfhttp://arxiv .org/abs/math.NT/0311296 > --Bill Dubuque You're doing better than the others. So then, of course you already knew that every Pell's Equation of the form x^2 - 2y^2 = 1 connected with a Pythagorean Triplet, right? Yes? Or no? James Harris === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation >> [...] note again the weird connection I found between certain > Pythagorean Triplets and Pell's Equation in the form x^2 - Dy^2 = 1 > when D-1 is a perfect square. For instance for D=2, I have that for > every solution of Pell's Equation you have a Pythagorean Triplet! > But the triplets are special in that with u^2 + v^2 = w^2, v = u+1. > The connection is that w is x+y from Pell's Equation. The more general > result is that u = sqrt(D-1)j, and v = j+1, while w still equals x+y. >> This is a special case of classical results going back to Euler, >> if not further. For some history and more modern interpretations >> I HIGHLY RECOMMEND the following exposition: >> Franz Lemmermeyer. Higher descent on Pell conics. >> II. Two centuries of missed opportunities >> [1] http://www.fen.bilkent.edu.tr/~franz/publ/pell-b.pdf >> http://arxiv.org/abs/math.NT/0311296 >> See also parts I and III at >> http://www.fen.bilkent.edu.tr/~franz/publ.html > > So then, of course you already knew that every Pell's Equation of > the form x^2 - 2y^2 = 1 connected with a Pythagorean Triplet, right? > Yes? Or no? Yes. Anyone familiar with the history of Pell's equations would be quite familiar with results of this nature since they were among some of the first techniques applied, e.g. see p. 2 of [1]. You can find many historical references in the papers I cited above. Furthermore, in Franz's papers you can find a very beautiful modern treatment of the arithmetic of conics in close analogy with the arithmetic of elliptic curves - something you will not easily find anywhere else. E.g see also esp. Franz Lemmermeyer. Conics - a poor man's elliptic curves http://www.fen.bilkent.edu.tr/~franz/publ/pell-sum.pdf http://arxiv.org/abs/math.NT/0311306 --Bill Dubuque === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > [...] note again the weird connection I found between certain > Pythagorean Triplets and Pell's Equation in the form Êx^2 - Dy^2 = 1 > when D-1 is a perfect square. For instance for D=2, I have that for > every solution of Pell's Equation you have a Pythagorean Triplet! > But the triplets are special in that with u^2 + v^2 = w^2, v = u+1. > The connection is that w is x+y from Pell's Equation. The more general > result is that u = sqrt(D-1)j, and v = j+1, while w still equals x+y. >> This is a special case of classical results going back to Euler, >> if not further. For some history and more modern interpretations >> I HIGHLY RECOMMEND the following exposition: >> Franz Lemmermeyer. Higher descent on Pell conics. >> II. Two centuries of missed opportunities >> [1]http://www.fen.bilkent.edu.tr/~franz/publ/pell-b.pdf >> Ê Êhttp://arxiv.org/abs/math.NT/0311296 >> See also parts I and III at >>http://www.fen.bilkent.edu.tr/~franz/publ.html > So then, of course you already knew that every Pell's Equation of > the form x^2 - 2y^2 = 1 connected with a Pythagorean Triplet, right? > Yes? Or no? Yes. Anyone familiar with the history of Pell's equations would > be quite familiar with results of this nature since they were > among some of the first techniques applied, e.g. see p. 2 of [1]. > You can find many historical references in the papers I cited above. Cool. I went over the same territory and found the results in less than 4 days. > Furthermore, in Franz's papers you can find a very beautiful modern > treatment of the arithmetic of conics in close analogy with the > arithmetic of elliptic curves - something you will not easily find > anywhere else. E.g see also esp. But I can re-do and simplify faster than anyone else in the world. > Franz Lemmermeyer. Conics - a poor man's elliptic curveshttp://www.fen.bilkent.edu.tr/~franz/publ/pell-sum.pdfhttp://arxiv.org/ abs/math.NT/0311306 --Bill Dubuque Um, dude. I couldn't care less about the subject before Friday. Yet today I'm pondering a complete theory for 2 variable quadratic Diophantines that I finished out yesterday: 4 days to complete a theory that encompasses over 2000 years of mathematical history But not a surprise if you know that I used tautological spaces 6 years ago to prove Fermat's Last Theorem. Proving a negative is not as easy to get across when the experts are fighting you, so I went to something else. I'm not terribly interested in the subject, and am less interested now that I've completed it, but I can and will use it ad infinitum against people like you. My point is that you are not worth being paid to do what you do with public funds. If you wish to continue this battle to see what I do next, what other field I wrap up completely, then continue. I have the time as I know what happens after--you lose everything. The real point as we move forward is that people like you cannot be intelligent as I keep telling you that at the end of this story you lose. And then I find things that make it clear that if you have an ounce of intellect and even a modicum of a sense of self-preservation you'd stop fighting mathematical proof now. Because I do not want you crying for mercy later. That's why. To answer the question you were thinking. The ONLY point of the time here is to remove plausible deniability. I have to remove any and all defenses. ANY and ALL defenses. So if any of you wondered why? Why the delay? Why do you seem to win against humanity's latest major discoverer? THAT is why. My analysis of the danger of the situation has indicated that to defeat you and your kind I must have absolute and total victory, which requires that I remove any and all defenses that you might present to a public that may wish to believe as you are their sons and daughters. And they will not want to believe that you tried to destroy the human race. James Harris === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation >> So then, of course you already knew that every Pell's Equation of > the form x^2 - 2y^2 = 1 connected with a Pythagorean Triplet, right? > Yes? Or no? >> Yes. Anyone familiar with the history of Pell equations would >> be quite familiar with results of this nature since they were >> among some of the first techniques applied, e.g. see p. 2 of [1]. >> You can find many historical references in the papers I cited above. > > Cool. I went over the same territory and found the results in less > than 4 days. The result in question is ancient - in essence going all the way back to Euclid's time. It is quite trivial - one of the first things one might discover when naively experimenting with Pell equations. >> Furthermore, in Franz's papers you can find a very beautiful modern >> treatment of the arithmetic of conics in close analogy with the >> arithmetic of elliptic curves - something you will not easily find >> anywhere else. E.g see also esp. >> Franz Lemmermeyer. Conics - a poor man's elliptic curves >> http://www.fen.bilkent.edu.tr/~franz/publ/pell-sum.pdf >> http://arxiv.org/abs/math.NT/0311306 > > But I can re-do and simplify faster than anyone else in the world. Ignoring modern mathematics and restarting from scratch certainly is not a particularly intelligent way to go about learning math. Not even the brightest minds of today would get very far in such a foolish endeavor. To get nontrivial results in mathematics it is essential to stand on the shoulders of mathematical giants. --Bill Dubuque === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif AppleWebKit/525.13 (KHTML, like Gecko) Chrome/0.2.149.29 Safari/525.13,gzip(gfe),gzip(gfe) > So then, of course you already knew that every Pell's Equation of > the form x^2 - 2y^2 = 1 connected with a Pythagorean Triplet, right? > Yes? Or no? >> Yes. Anyone familiar with the history of Pell equations would >> be quite familiar with results of this nature since they were >> among some of the first techniques applied, e.g. see p. 2 of [1]. >> You can find many historical references in the papers I cited above. > Cool. ÊI went over the same territory and found the results in less > than 4 days. The result in question is ancient - in essence going all the way back > to Euclid's time. It is quite trivial - one of the first things one > might discover when naively experimenting with Pell equations. For those not knowing what the poster is calling trivial it is the result that every Pell's Equation of the form x^2 - Dy^2 = 1 is directly linked to either a Pythagorean Triplet or to an integer solution of (D-1)u^2 + v^2 = w^2 when D-1 is not square. So it's a result showing an intimate relationship between Pell's Equation and ellipses. The full result though is more interesting as from my Quadratic Diophantine Theorem I have S^2 - D(x+y)^2 = - D + 1 where again x^2 - Dy^2 = 1, so I have a second Pell's Equation of a more general form, where solutions *must* exist for the second equation for every case they exist for the first. So with D=2, you have that S^2 - 2(x+y)^2 = - 1 and the result that for every solution to x^2 - 2xy^2 = 1, you are linked to a solution to S^2 - 2(x+y)^2 = -1. So with a solution for one you can immediately get a solution to the other. Um, do you claim that result goes back to Euclid as well? I'm curious. >> Furthermore, in Franz's papers you can find a very beautiful modern >> treatment of the arithmetic of conics in close analogy with the >> arithmetic of elliptic curves - something you will not easily find >> anywhere else. E.g see also esp. >> Franz Lemmermeyer. Conics - a poor man's elliptic curves >>http://www.fen.bilkent.edu.tr/~franz/publ/pell-sum.pdf >>http://arxiv.org/abs/math.NT/0311306 > But I can re-do and simplify faster than anyone else in the world. Ignoring modern mathematics and restarting from scratch certainly > is not a particularly intelligent way to go about learning math. > Not even the brightest minds of today would get very far in such > a foolish endeavor. To get nontrivial results in mathematics it > is essential to stand on the shoulders of mathematical giants. --Bill Dubuque I think math people repeat that so much in order to believe it, and maybe to be able to ignore results put right in front of you? The Quadratic Diophantine Theorem links ANY 2 variable Diophantine quadratic to another Diophantine quadratic of the general form u^2 + Ay^2 = B allowing you to simplify studying any 2 variable Diophantine quadratic. That trivially follows from use of the theorem as I've explained at my math blog and even put in a paper accessible from that site. And I started from scratch, learning most of what I know about prior research AFTER I got my own theorem and started looking for ways to use it. The result you trivialize above was one of the first I got, so I paralleled mathematicians 2000 years ago, if what you said above is correct. But within 3 more days I had completed the full theory for 2 variable Diophantine quadratic equations, meaning I traversed 2000 years of mathematical research history in 4 days, with a simpler and more complete theory!!! That is the power of tautological spaces which previously I used to prove Fermat's Last Theorem and posters like you argued against that research, and look at you now, when I went to a positive case versus proving a negative. James Harris === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation posting-account=33KaEgkAAAA9tz8WICNABjrkyMKXFbGS Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) > But I can re-do and simplify faster than anyone else in the world. > **************************************************************************** **** > Ignoring modern mathematics and restarting from scratch certainly > is not a particularly intelligent way to go about learning math. > Not even the brightest minds of today would get very far in such > a foolish endeavor. To get nontrivial results in mathematics it > is essential to stand on the shoulders of mathematical giants. **************************************************************************** **** > --Bill Dubuque Sagaciously said Bill, indeed your words resound with 'Nihilo Nihil Fit ' or 'Re-inventing the Wheel' . Even if obvious, with so much relevance they could even be cut out in stone and placed at the entrance of a modern Mathematics Academy,.. somewhat like Plato's portal engraving making it out of bounds for those ignorant of geometry. .. It's quite appealing. Narasimham === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation posting-account=HaopWgoAAADs72-s8RQYwP_-ruRUuNzX .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; InfoPath.1),gzip(gfe),gzip(gfe) > [...] note again the weird connection I found between certain > Pythagorean Triplets and Pell's Equation in the form Êx^2 - Dy^2 = 1 > when D-1 is a perfect square. For instance for D=2, I have that for > every solution of Pell's Equation you have a Pythagorean Triplet! > But the triplets are special in that with u^2 + v^2 = w^2, v = u+1. > The connection is that w is x+y from Pell's Equation. The more general > result is that u = sqrt(D-1)j, and v = j+1, while w still equals x+y. >> This is a special case of classical results going back to Euler, >> if not further. For some history and more modern interpretations >> I HIGHLY RECOMMEND the following exposition: >> Franz Lemmermeyer. Higher descent on Pell conics. >> II. Two centuries of missed opportunities >> [1]http://www.fen.bilkent.edu.tr/~franz/publ/pell-b.pdf >> Ê Êhttp://arxiv.org/abs/math.NT/0311296 >> See also parts I and III at >>http://www.fen.bilkent.edu.tr/~franz/publ.html > So then, of course you already knew that every Pell's Equation of > the form x^2 - 2y^2 = 1 connected with a Pythagorean Triplet, right? > Yes? Or no? > Yes. Anyone familiar with the history of Pell's equations would > be quite familiar with results of this nature since they were > among some of the first techniques applied, e.g. see p. 2 of [1]. > You can find many historical references in the papers I cited above. Cool. ÊI went over the same territory and found the results in less > than 4 days. > Furthermore, in Franz's papers you can find a very beautiful modern > treatment of the arithmetic of conics in close analogy with the > arithmetic of elliptic curves - something you will not easily find > anywhere else. E.g see also esp. But I can re-do and simplify faster than anyone else in the world. > Franz Lemmermeyer. Conics - a poor man's elliptic curveshttp://www.fen.bilkent.edu.tr/~franz/publ/pell-sum.pdfhttp://arxiv.or.. . > --Bill Dubuque Um, dude. ÊI couldn't care less about the subject before Friday. ÊYet > today I'm pondering a complete theory for 2 variable quadratic > Diophantines that I finished out yesterday: 4 days to complete a theory that encompasses over 2000 years of > mathematical history But not a surprise if you know that I used tautological spaces 6 years > ago to prove Fermat's Last Theorem. Proving a negative is not as easy to get across when the experts are > fighting you, so I went to something else. I'm not terribly interested in the subject, and am less interested now > that I've completed it, but I can and will use it ad infinitum against > people like you. My point is that you are not worth being paid to do what you do with > public funds. If you wish to continue this battle to see what I do next, what other > field I wrap up completely, then continue. I have the time as I know what happens after--you lose everything. The real point as we move forward is that people like you cannot be > intelligent as I keep telling you that at the end of this story you > lose. And then I find things that make it clear that if you have an ounce of > intellect and even a modicum of a sense of self-preservation you'd > stop fighting mathematical proof now. Because I do not want you crying for mercy later. ÊThat's why. ÊTo > answer the question you were thinking. The ONLY point of the time here is to remove plausible deniability. I have to remove any and all defenses. ÊANY and ALL defenses. So if any of you wondered why? ÊWhy the delay? ÊWhy do you seem to win > against humanity's latest major discoverer? THAT is why. My analysis of the danger of the situation has indicated that to > defeat you and your kind I must have absolute and total victory, which > requires that I remove any and all defenses that you might present to > a public that may wish to believe as you are their sons and daughters. And they will not want to believe that you tried to destroy the human > race. James Harris- Hide quoted text - - Show quoted text - Kiss my ass you delusional twit! You are so full of shit. Someone had to point out something you should have discovered if you gave a shit about anything. You are a pathetic joke and always will be. You will be remembered as a crank and super troll with NPD. Go see a doctor - loser! === Subject: Statistical Challenges and Advances in Brain Science posting-account=QrG9mwoAAAD4swwif73Cg69MYQStuBea 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648; .NET CLR 1.1.4322; InfoPath.2),gzip(gfe),gzip(gfe) Challenges and Advances in Brain Scinece, with invited editorials by Professors Karl Friston and Nicholas Lange. Please see the attached readers, we will take orders of this single issue. The hard copy via surface mail is US$13, and airmail, US$20. Please email me (karen@stat.sinica.edu.tw) your mailing address and indicate the number of copies you need before Oct. 10, 2008. Karen Li, on behalf of the editorial office PS. Checks in US currency must be payable to: Statistica Sinica Institute of Statistical Science, Academia Sinica Taipei 11529, TAIWAN Statistical Challenges and Advances in Brain Science A supervised singular value decomposition for independent component analysis of fMRI Ping Bai, Haipeng Shen, Xuemei Huang and Young Truong A bootstrap test to investigate changes in brain connectivity for functional MRI Pierre Bellec, Guillaume Marrelec and Habib Benali Encoding cortical surface by spherical harmonics Moo K. Chung, Richard Hartley, Kim Dalton and Richard J. Davidson Continuous-time filters for state estimation from point process models of neural data Uri T. Eden and Emery N. Brown An overview of modeling techniques for hybrid brain data Apratim Guha and Atanu Biswas Mixture modeling for dynamic PET data Huiping Jiang and R. Todd Ogden Dealing with spatial normalization errors in fMRI group inference using hierarchical modeling Merlin Keller, Alexis Roche, Alan Tucholka and Bertrand Thirion Nonparametric bootstrap for mixed-effects models for repeated spatial proximity measures (K-functions) with applications in neuropathology Sabine Landau and Ian P. Everall The acquisition and statistical analysis of rapid 3D fMRI data Martin A. Lindquist, Cun-Hui Zhang, Gary Glover and Lawrence Shepp Residual analysis for detecting mis-modeling in fMRI J. M. Loh, M. A. Lindquist and T. D. Wager Penalized PARAFAC analysis of spontaneous EEG recordings Eduardo Martinez-Montes, Jose M. Sanchez-Bornot and Pedro A. Valdes- Sosa Spatio-spectral analysis of brain signals Hernando Ombao, Xiaofeng Shao, Elena Rykhlevskaia, Monica Fabiani and Gabriele Gratton Inferring individual differences in fMRI: Finding brain regions with significant within subject correlation Sumitra Purkayastha, Tor D. Wager and Thomas E. Nichols Uncovering sparse brain effective connectivity: A voxel-based approach using penalized regression Jose M. Sanchez-Bornot, Eduardo Martinez-Montes, Agust?Ân Lage- Castellanos, Mayrim Vega-Hernandez and Pedro A. Valdes-Soda Scope of resampling-based tests in fNIRS neuroimaging data analysis Archana K. Singh, Lester Clowney, Masako Okamoto, James B. Cole and Ippeita Dan Penalized least squares methods for solving the EEG inverse problem Mayrim Vega-Hernandez, Eduardo Martinez-Montes, Jose M. Sanchez- Bornot, Agustin Lage-Castellanos and Pedro A. Valdes-Sosa Hybrid permutation test with application to surface shape analysis Chunxiao Zhou and Yongmei Michelle Wang Statistical modelling of brain morphological measures within family pedigrees Hongtu Zhu, Yimei Li, Niansheng Tang, Ravi Bansal, Xuejun Hao, Myrna M. Weissman and Bradley G. Peterson === Subject: Re: Total quotient ring and integral closures > Exercise: If A is a zero-dimensional commutative unitary ring. Then > it is reduced > > No: Z/p^2Z. I think that you need some assumption on localizations here. > > and all quotients over prime are fields. > > This is true. :) ... Sorry for that :) - Let's change zero-dimensional to regular In fact let's see if we can prove this characterization.. Let A be commutative unitary ring. Then A is regular iff A is reduced and each quotient over primes is a field. Proof: => Let A be not reduced, then theres a nontrivial nilpotent element (reduced=semiprime in commutative unitary rings) say a in A. A is regular, so a must have a quasi-inverse b (i.e. ba^2=a). Morever there is without los of generality a positive integer n such that a^2n = 0 .. but look.. a^2n b^n = a^2 b = a (ab is an idempotent) a contradiction. Let P be a prime that is not maximal, there is a maximal ideal M that contains P. Suppose a is in M not in P.. a has a quasi-inverse b.. either ab is in P or ab - 1 is in P (because ab(ab-1) = 0) .. if ab is in P then a=a^2b is in P .. a contradiction. If ab-1 is in P.. then ab is in M and so ab-1 is NOT in M.. yet it is in P.. a contradiction. <= Let a be in A, if it's a unit or 0 we are done (quasi-inverse is the inverse or 0). If it's not a unit, then it's contained in at least one maximal prime ideal of A (because A is reduced!)... hmm.. I want to do this with as little sheaf theory/algebraic geometry as possible.. I guess I will only show a sketch.. cause my train is coming and I need to catch it to go to work -_-;; Let D(a)={ P in spec A : a is not in P} .. this is clopen now consider the ideal I=I(D(a))={a' in A : a' is in P for all P in D(a)}. Consider A/I .. this has the same properties as A .. i.e. it is reduced, and all quotient over primes is a field. a mod I is in NO prime ideal of A/I .. thus it has an inverse in A/I say b mod I .. for some b in A.. Exercise: b is a quasi-inverse of a (Sorry the last part is messy.. and Im a bit late for my train.. =) .. there will be minor errors around ^^; ) Jose Capco === Subject: Re: Total quotient ring and integral closures > Exercise: If A is a zero-dimensional commutative unitary ring. > Then it is reduced >> No: Z/p^2Z. I think that you need some assumption on localizations >> here. >> > and all quotients over prime are fields. >> This is true. > > :) ... Sorry for that :) - Let's change zero-dimensional to regular > > In fact let's see if we can prove this characterization.. > > Let A be commutative unitary ring. Then A is regular iff A is reduced > and each quotient over primes is a field. > > Proof: => Let A be not reduced, then theres a nontrivial nilpotent > element (reduced=semiprime in commutative unitary rings) say a in A. > A is regular, so a must have a quasi-inverse b (i.e. ba^2=a). Morever > there is without los of generality a positive integer n such that > > a^2n = 0 .. but look.. a^2n b^n = a^2 b = a (ab is an idempotent) a > contradiction. > > Let P be a prime that is not maximal, there is a maximal ideal M that > contains P. Suppose a is in M not in P.. a has a quasi-inverse b.. > either ab is in P or ab - 1 is in P (because ab(ab-1) = 0) .. if ab > is in P then a=a^2b is in P .. a contradiction. If ab-1 is in P.. > then ab is in M and so ab-1 is NOT in M.. yet it is in P.. a > contradiction. > > <= Let a be in A, if it's a unit or 0 we are done (quasi-inverse is > the inverse or 0). If it's not a unit, then it's contained in at > least one maximal prime ideal of A (because A is reduced!)... hmm.. I > want to do this with as little sheaf theory/algebraic geometry as > possible.. > > I guess I will only show a sketch.. cause my train is coming and I > need to catch it to go to work -_-;; > > Let D(a)={ P in spec A : a is not in P} .. this is clopen now > consider the ideal I=I(D(a))={a' in A : a' is in P for all P in > D(a)}. Consider A/I .. this has the same properties as A .. i.e. it > is reduced, and all quotient over primes is a field. a mod I is in NO > prime ideal of A/I .. thus it has an inverse in A/I say b mod I .. > for some b in A.. > > Exercise: b is a quasi-inverse of a > > (Sorry the last part is messy.. and Im a bit late for my train.. =) > .. there will be minor errors around ^^; ) > No doubt that this is true since I was working with the characterization reduced and dim = 0. -- Best wishes, J. === Subject: Re: Hilbert vs Brouwer.. <6cad9$48c4f41d$82a1e228$23154@news1.tudelft.nl> posting-account=Yn5cwwoAAADntcMuRwk-EwLg-DMZ_hXN Gecko/20070509 Camino/1.5,gzip(gfe),gzip(gfe) >>While I agree generally with the tenor of Keith's remarks, >>I think he has been a bit glib at one point. >Hilbertthought he was battling for whose >mathematics would win out, and that he had to >win, to make sure that Brouwer didn't RUIN math. >>This is more or less correct, I think, and perhaps >>Hilbertwas being a bit panicky, even paranoid, >>but he DID have a point. It's worth noticing, >>for example, thatHilberthad no real qualms >>about Kronecker, Poincare, etc, who had also >>shown constructivist tendencies; but Brouwer >>was quite a different kettle of fish altogether. >>Brouwer was very combative, and worse, like >>the more-or-less contemporary Bohr in physics, >>was taking a very METAPHYSICAL, even MYSTICAL >>line toward math. He seemed to be espousing >>an almost post-modernist (hence repugnant) line >>that, if that's YOUR way of doing it, then it's >>as valid as anyone else's. SUBJECTIVIST math. >>I think it was this aspect of Brouwer's approach >>that alarmedHilbert, and even seemed to be making >>a little headway (Weil), and thatHilbertfelt >>(correctly IMHO) could well RUIN math, at least >>in part, if left to rage unchecked. >>Hence his very dismissive remarks, and Annalen battle. >There's no reason to believe >that he was interested in any kind of >cooperative coexistence, in which both kinds >of mathematics would be done. >>He wasn't, because Brouwer had made it seem >>that constructivism HAD TO involve subjectivism. >>It doesn't;Hilbertwas misled; and it was largely >>Brouwer's fault that these two became conflated. >>Hilbertwas very right to battle against subjectivism! >He wanted >to get Brouwer off the board, so that he would >not be perpetuating intuitionist mathematics. >>And now we see why. IMHO the fault was Brouwer's. >Hilbert'sanger at Brouwer was such that he was >beyond being logically reasoned with. >>Hilbert, OC, saw it in reverse! >>Modern constructivism, of the Russian, or Bishop >>styles, is fully objective, andHilbertwould have >>had no problem with it, I'm sure. >>Hilbertwas (rightly) obsessed with mathematical >>*hygiene*, in the form of logic and formalism; >>and Brouwer's anti-formalist ideas were, and are, anathematical. >>IMHO. > this is a very strange look at brouwer > subjectivism has very little to do > with brouwer's view of a kantian a priori categorisation > it's possiblehilbertheld this view > buthilbertwas a very poor metamathematician > (not discounting his [many] strengths) > -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- > No,galathaea, you are wrong and Bill Taylor is right. You should only > read Brouwer's dissertation to see it with your own eyes: Over de Grondslagen van de wiskunde. Dissertatie > - Universiteit van Amsterdam (19 februari 1907) http://genealogy.math.ndsu.nodak.edu/id.php?id=19467 > http://www.math.uu.nl/people/hogend/brouwer2.pdf Unfortunately, any reference is in Dutch, and I can't > provide an English version of it. Don't even know if > some of it has been translated anyhow. Individualism and solipsism are quite predominant > with intuitionism. Hilbert is right for one time. psychologism is not necessarily subjectivism subjectivism requires no possible connection either through non existence of other or the impossibility of comparison (often through analysis of comparison) something brouwer did not believe i'm not defending brouwer's position but there has been a whole lot of misunderstanding here even by constructivists in his dissertation he goes into a long discussion of the ur-intuition not of himself but of mathematics he denies logicism because it fails to describe the (one) mathematics but only attempts to describe it's communication he certainly believes the psychological foundations of truth (he speaks of truth to his ego as the only relevant truth) already though his strong kantian influence is apparent and he makes clear his feelings about absolute notions of correctness he even states in this dissertation mathematics is independent of logic practical and theoretical logic are applications of various parts of mathematics and after his dissertation if this was not clear he went on to clarify his position over the years (and yes i think it can be argued his position changed but i do not think these positions ever concerned subjectivism) his attacks on formalism and the consistency program were that consistency didn't matter if the system itself was incorrect in fact brouwer like heidegger gave the kantian response to subjectivism the strongest statement to this effect i am aware of is: whether the emotional representation evoked by the same word in various individuals is quite different (or rather incomparable as comparison presupposes mathematics) it does not obstruct the fact that on the basis of the similarity of the relations that is of the mathematical systems the acts of the hearer can be moved with sufficient accuracy in the direction wished by the speaker even though the representations connected to the act in different individuals do not resemble each other at all so despite incomparability considered he describes the connections inherent to ideas he does toy with solipsism but as an explanation for this connection and the absolute fundamentality of mathematical process again this is not to argue his position it's my view that his position is ill-formed and that his mysticism like heidegger's incorrectly leads him to the a priori stand it's just that his stand like kant's is in opposition to berkeley's which could never argue a notion of correctness -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- === Subject: Vector fields and differential forms? posting-account=hKYFlQkAAABAbYdS4rm5V2f0Hfgp0AVC .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) spider-ntc-te07.proxy.aol.com[CFC87087] (Prism/1.2.1), HTTP/1.1 cache-ntc-ac02.proxy.aol.com[CFC87483] (Traffic-Server/6.1.5 [uScM]) I'm trying to learn some differential geometry so I can understand general relativity, but mathematical physics texts seem to give conflicting definitions of vector fields and differential forms. Let $C^infty(M)$ denote the set of smooth functions on a manifold $M $. Some texts define a vector field to be a mapping from $C^infty(M)$ to $C^infty(M)$ which is linear over the real numbers and satisfies Leibniz's product rule. Let $Vect(M)$ denote the set of vector fields on $M$. These same texts define a 1-form to be a mapping from $Vect(M) $ to $C^infty(M)$ which is linear over $C^infty(M)$. On the other hand, you can define a vector field to be a section of a tangent bundle, and you can define a 1-form to be a section of a cotangent bundle. These definitions are different from the ones above because the domain of a section is the manifold itself, while the domain of a vector field as defined above is $C^infty(M)$ and the domain of a 1-form as defined above is $Vect(M)$. So what are vector fields and 1-forms? === Subject: Re: Vector fields and differential forms? > I'm trying to learn some differential geometry so I can understand > general relativity, but mathematical physics texts seem to give > conflicting definitions of vector fields and differential forms. > > Let $C^infty(M)$ denote the set of smooth functions on a manifold $M > $. Some texts define a vector field to be a mapping from $C^infty(M)$ > to $C^infty(M)$ which is linear over the real numbers and satisfies > Leibniz's product rule. Let $Vect(M)$ denote the set of vector fields > on $M$. These same texts define a 1-form to be a mapping from $Vect(M) > $ to $C^infty(M)$ which is linear over $C^infty(M)$. > > On the other hand, you can define a vector field to be a section of a > tangent bundle, and you can define a 1-form to be a section of a > cotangent bundle. These definitions are different from the ones above > because the domain of a section is the manifold itself, while the > domain of a vector field as defined above is $C^infty(M)$ and the > domain of a 1-form as defined above is $Vect(M)$. > > So what are vector fields and 1-forms? You might start from here http://en.wikipedia.org/wiki/Tangent_space#Definition_via_derivations to see that the two notions are indeed equivalent. -- Best wishes, J. === Subject: Re: Vector fields and differential forms? posting-account=hKYFlQkAAABAbYdS4rm5V2f0Hfgp0AVC .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) spider-ntc-te05.proxy.aol.com[CFC87085] (Prism/1.2.1), HTTP/1.1 cache-ntc-ab03.proxy.aol.com[CFC87443] (Traffic-Server/6.1.5 [uScM]) On Sep 10, 2:45?am, Jannick Asmus I'm trying to learn some differential geometry so I can understand > general relativity, but mathematical physics texts seem to give > conflicting definitions of vector fields and differential forms. > Let $C^infty(M)$ denote the set of smooth functions on a manifold $M > $. Some texts define a vector field to be a mapping from $C^infty(M)$ > to $C^infty(M)$ which is linear over the real numbers and satisfies > Leibniz's product rule. Let $Vect(M)$ denote the set of vector fields > on $M$. These same texts define a 1-form to be a mapping from $Vect(M) > $ to $C^infty(M)$ which is linear over $C^infty(M)$. > On the other hand, you can define a vector field to be a section of a > tangent bundle, and you can define a 1-form to be a section of a > cotangent bundle. These definitions are different from the ones above > because the domain of a section is the manifold itself, while the > domain of a vector field as defined above is $C^infty(M)$ and the > domain of a 1-form as defined above is $Vect(M)$. > So what are vector fields and 1-forms? You might start from herehttp://en.wikipedia.org/wiki/Tangent space#Definition via derivationsto > see that the two notions are indeed equivalent. -- > Best wishes, > J.- Hide quoted text - - Show quoted text - Oh, sorry. I didn't look closely at the link. === Subject: Re: Vector fields and differential forms? posting-account=hKYFlQkAAABAbYdS4rm5V2f0Hfgp0AVC .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) spider-ntc-te05.proxy.aol.com[CFC87085] (Prism/1.2.1), HTTP/1.1 cache-ntc-ab03.proxy.aol.com[CFC87443] (Traffic-Server/6.1.5 [uScM]) On Sep 10, 2:45?am, Jannick Asmus I'm trying to learn some differential geometry so I can understand > general relativity, but mathematical physics texts seem to give > conflicting definitions of vector fields and differential forms. > Let $C^infty(M)$ denote the set of smooth functions on a manifold $M > $. Some texts define a vector field to be a mapping from $C^infty(M)$ > to $C^infty(M)$ which is linear over the real numbers and satisfies > Leibniz's product rule. Let $Vect(M)$ denote the set of vector fields > on $M$. These same texts define a 1-form to be a mapping from $Vect(M) > $ to $C^infty(M)$ which is linear over $C^infty(M)$. > On the other hand, you can define a vector field to be a section of a > tangent bundle, and you can define a 1-form to be a section of a > cotangent bundle. These definitions are different from the ones above > because the domain of a section is the manifold itself, while the > domain of a vector field as defined above is $C^infty(M)$ and the > domain of a 1-form as defined above is $Vect(M)$. > So what are vector fields and 1-forms? You might start from herehttp://en.wikipedia.org/wiki/Tangent space#Definition via derivationsto > see that the two notions are indeed equivalent. -- > Best wishes, > J.- Hide quoted text - - Show quoted text - I understand that you can use the first definition to define tangent and cotangent vectors and then consider sections of the tangent and cotangent bundles. I'm just confused because the two notions are technically not equivalent. The first definition says that 1-forms act on functions called vector fields (which are not associated to individual points of the manifold). The second definition says that a 1-form is actually a *function* of the manifold. === Subject: Primitive r-th root of identity matrix posting-account=M-YpHwkAAADVpyTlN__wac51o3MO1nxE .NET CLR 2.0.50727; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) What are primitive r-th roots of an N by N identity matrix? I want to find all A such that A^r=I_N, where I_N is an N by N identity matrix. I search on Google and cannot find any related references. Chris === Subject: Re: Primitive r-th root of identity matrix > What are primitive r-th roots of an N by N identity matrix? > I want to find all A such that A^r=I_N, where I_N is an N by N > identity matrix. > I search on Google and cannot find any related references. Chris > I suppose a matrix A is sought such that A^r=I but A^k is different from I for all k such that 01. (You were not as specific in your post!) ----------- Recipe, complex case: Let z be any primitive r-th root of 1, and set up a diagonal matrix D such that its (1,1) entry is z, and the others are any r-th roots of 1. Then the most general A is T * D * inv(T) (T invertible). (Consider Jordan normal form to see it.) ----------- Recipe, real case: you need A to be N-by-N with N>1. Pick z as above, so z = cos(u) + i*sin(u). Set up a block diagonal matrix D whose top 2-by-2 block is [ cos(u) -sin(u) ] [ sin(u) cos(u) ] and fill in the rest of the 1-by-1 or 2-by-2 blocks with similarly formed blocks which are r-th roots of the appropriate identities, not necessarily primitive. As above, A = T * D * inv(T) will be the most general answer (T real invertible this time), ----------- Counting the equivalence classes (by similarity) of such matrices would be an exercise in combinatorics (maybe not too hard). ZVK(Slavek). === Subject: Re: Primitive r-th root of identity matrix > > > > What are primitive r-th roots of an N by N identity matrix? > I want to find all A such that A^r=I_N, where I_N is an N by N > identity matrix. > I search on Google and cannot find any related references. > Chris > > > I suppose a matrix A is sought such that A^r=I but A^k is different from I > for all k such that 01. > (You were not as specific in your post!) > > ----------- > > Recipe, complex case: Let z be any primitive r-th root of 1, and set up a > diagonal matrix D such that its (1,1) entry is z, and the others are any > r-th roots of 1. > > Then the most general A is T * D * inv(T) > (T invertible). (Consider Jordan normal form to see it.) Not quite. The diagonal entries must be r-th roots of 1, but none of them needs to be primitive. All that's required is that there is no positive integer s < r such that they are all s-th roots of 1. e.g. for r=6 the diagonal entries could be -1 and exp(2 pi i/3) (primitive 2nd and 3rd roots of 1). -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Primitive r-th root of identity matrix posting-account=Yn5cwwoAAADntcMuRwk-EwLg-DMZ_hXN Gecko/20070509 Camino/1.5,gzip(gfe),gzip(gfe) > What are primitive r-th roots of an N by N identity matrix? > I want to find all A such that A^r=I_N, where I_N is an N by N > identity matrix. > I search on Google and cannot find any related references. a nice example can be given when r = n in that cases matrices like | 0 1 0 0 0 ... 0 | | 0 0 1 0 0 ... 0 | | 0 0 0 1 0 ... 0 | | . . . . . .. . | | . . . . . . . . | | 0 0 0 0 0 ... 1 | | 1 0 0 0 0 ... 0 | are roots of unity it's easy to take the idea to a number of other circulant and permutation matrices when r =/= n these examples are more complex but when n is even and r = n/2 you can use the shift matrix above squared and so on for all divisors of n in general i would look at the characteristic polynomial and apply cayley-hamilton -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- === Subject: Re: Primitive r-th root of identity matrix > What are primitive r-th roots of an N by N identity matrix? > I want to find all A such that A^r=I_N, where I_N is an N by N > identity matrix. Every matrix over the complex numbers of finite order is diagonalizable, because the minimal polynomial is separable. Best wishes, J. === When the Argus Leader of Sioux Falls South Dakota did a story on Archimedes Plutonium in late June, 2008, shortly thereafter nor the photo pictures can been seen in the Google News Archive. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Plutonium posting-account=JpxxPAgAAAAgwzQIYqn4j6syK-YhOmcF Gecko/20070309 Firefox/2.0.0.3,gzip(gfe),gzip(gfe) On Sep 10, 5:04 am, Archimedes Plutonium on Archimedes Plutonium in late June, 2008, shortly thereafter > nor the photo pictures can been seen in the Google News > Archive. Archimedes Plutoniumwww.iw.net/~a_plutonium > whole entire Universe is just one big atom > where dots of the electron-dot-cloud are galaxies Hadron Collider just started functioning. Might be one of the side- effects. === Plutonium posting-account=HaopWgoAAADs72-s8RQYwP_-ruRUuNzX Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) On Sep 10, 10:22Êam, porky pig...@my-deja.com On Sep 10, 5:04 am, Archimedes Plutonium When the Argus Leader of Sioux Falls South Dakota did a story > on Archimedes Plutonium in late June, 2008, shortly thereafter > nor the photo pictures can been seen in the Google News > Archive. > Archimedes Plutoniumwww.iw.net/~a plutonium > whole entire Universe is just one big atom > where dots of the electron-dot-cloud are galaxies Hadron Collider just started functioning. Might be one of the side- > effects. ;-) Great point! We have seen our first black hole and it swallowed the electron-dot- cloud and universal atom! ;-) === Plutonium posting-account=HaopWgoAAADs72-s8RQYwP_-ruRUuNzX Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) On Sep 10, 2:04Êam, Archimedes Plutonium on Archimedes Plutonium in late June, 2008, shortly thereafter > nor the photo pictures can been seen in the Google News > Archive. Archimedes Plutoniumwww.iw.net/~a plutonium > whole entire Universe is just one big atom > where dots of the electron-dot-cloud are galaxies Is that necessarily a bad thing? Heck - most folks here probably consider it a good thing - I know I do! You are delusional! === Subject: Re: Newsweek: The Large Hadron Collider > The Large Hadron Collider is a symptom of America's decline in > > http://www.newsweek.com/id/157514 > > Ah well... best wishes to the Europeans For what it's worth: The first single bunch of about 5e9 protons at the injection energy of 450GeV has made three complete turns in the LHC this morning. That's a good start. Jeroen Belleman === Subject: Re: Newsweek: The Large Hadron Collider reply-type=response >> The Large Hadron Collider is a symptom of America's decline in >> http://www.newsweek.com/id/157514 >> Ah well... best wishes to the Europeans For what it's worth: The first single bunch of about 5e9 protons > at the injection energy of 450GeV has made three complete turns > in the LHC this morning. That's a good start. Jeroen Belleman === Subject: Re: Newsweek: The Large Hadron Collider > > The Large Hadron Collider is a symptom of America's decline in >> http://www.newsweek.com/id/157514 >> Ah well... best wishes to the Europeans >> For what it's worth: The first single bunch of about 5e9 protons >> at the injection energy of 450GeV has made three complete turns >> in the LHC this morning. That's a good start. >> Jeroen Belleman > > > > The Large Hadron Collider is a symptom of America's decline in >> http://www.newsweek.com/id/157514 >> Ah well... best wishes to the Europeans >> For what it's worth: The first single bunch of about 5e9 protons > at the injection energy of 450GeV has made three complete turns > in the LHC this morning. That's a good start. >> >> Yeah, right. But the media are all buzzing and many seem to > believe we're going to gobble up the world. Maybe Gabriel's Trumpet will sound. ;-) Rich === Subject: Re: Newsweek: The Large Hadron Collider > The Large Hadron Collider is a symptom of America's decline in >> http://www.newsweek.com/id/157514 >> Ah well... best wishes to the Europeans >> For what it's worth: The first single bunch of about 5e9 protons >> at the injection energy of 450GeV has made three complete turns >> in the LHC this morning. That's a good start. > Yeah, right. But the media are all buzzing and many seem to >> believe we're going to gobble up the world. > > Maybe Gabriel's Trumpet will sound. ;-) > There is a Gabriel among the control room crew. He doesn't play the trumpet, to my knowledge. ;-) Jeroen Belleman === Subject: Re: Newsweek: The Large Hadron Collider reply-type=response >> The Large Hadron Collider is a symptom of America's decline in >> http://www.newsweek.com/id/157514 >> Ah well... best wishes to the Europeans > For what it's worth: The first single bunch of about 5e9 protons > at the injection energy of 450GeV has made three complete turns > in the LHC this morning. That's a good start. >> > Yeah, right. But the media are all buzzing and many seem to > believe we're going to gobble up the world. >> Maybe Gabriel's Trumpet will sound. ;-) > > There is a Gabriel among the control room crew. He doesn't play the > trumpet, to my knowledge. ;-) Does he look like this ? http://imagecache2.allposters.com/images/pic/CLASS/130-297~Gabby-Hayes-Poster s.jpg === Subject: Re: Newsweek: The Large Hadron Collider reply-type=response >> The Large Hadron Collider is a symptom of America's decline in >> http://www.newsweek.com/id/157514 >> Ah well... best wishes to the Europeans >> For what it's worth: The first single bunch of about 5e9 protons > at the injection energy of 450GeV has made three complete turns > in the LHC this morning. That's a good start. >> Jeroen Belleman >> Yeah, right. But the media are all buzzing and many seem to > believe we're going to gobble up the world. Jeroen Belleman The media have always done that, it sells newspapers... will the cretins ever learn? No, of course not. We'll be hearing about these mythical black holes and Higgs bosons for months to come until the sales drop off and the whole shebang fades into obscurity. Didn't that Jesus bloke wander off into the desert to have his divine inspirations, just like Higgs wandering around the Scottish Highlands? Oh, we are all going to witness the Big Bang all over again and make physics a fuckin' spiritual experience. I hallucinate, therefore it is. What a fuckin' crock! White elephants have got nothing on CERN, bright green flying elephants are much better value for money. === Subject: Re: Newsweek: The Large Hadron Collider > > The media have always done that, it sells newspapers... will > the cretins ever learn? No, of course not. We'll be hearing > about these mythical black holes and Higgs bosons for months > to come until the sales drop off and the whole shebang fades > into obscurity. Didn't that Jesus bloke wander off into the > desert to have his divine inspirations, just like Higgs wandering > around the Scottish Highlands? Oh, we are all going to witness > the Big Bang all over again and make physics a fuckin' spiritual > experience. I hallucinate, therefore it is. What a fuckin' crock! > White elephants have got nothing on CERN, bright green flying > elephants are much better value for money. Do you ever have a coherent thought? -- http://improve-usenet.org/index.html aioe.org, Goggle Groups, and Web TV users must request to be white listed, or I will not see your messages. If you have broadband, your ISP may have a NNTP news server included in your account: http://www.usenettools.net/ISP.htm There are two kinds of people on this earth: The crazy, and the insane. The first sign of insanity is denying that you're crazy. === Subject: Re: Newsweek: The Large Hadron Collider > Of what use is the ISS? >>I don't know - what did Queen Victoria expect Columbus to find? >> >> I doubt she ever met him. > > Especially since Columbus had been dead a couple hundred years when > Victoria was born. > > Well, whoever the hell it was that funded him. Isabella? Just 'cause I > got the wrong queen doesn't alter the fact that _SOMEBODY_ paid for his > expedition - whoever it was must have expected _SOMETHING_ back. Columbus went to the Queen of Spain To ask for ships and cargo. He promised to kiss the Royal Arse And bring her back Chicago. -- Michael Press === Subject: Re: Newsweek: The Large Hadron Collider > On Sep 8, 1:00 pm, Phil Hobbs The Large Hadron Collider is a symptom of America's decline in > http://www.newsweek.com/id/157514 > Ah well... best wishes to the Europeans >> Tee hee. >> single technologically useful result it's generated since about >> Fermi's day--or is there something I've missed? It's primarily an >> aesthetic activity. >> There are lots better ways to spend a billion or two dollars on >> science than building ever larger colliders. If we really have to >> find the Higgs boson, maybe support work on wake field accelerators >> or other new methods, because keeping on turning the same old crank >> has got us way, way into the diminishing returns region. >> The argument seems to be that we have to keep dumping some obscenely >> large fraction of our limited physical science budgets into >> accelerators to avoid falling behind in a field that doesn't produce >> anything useful...or to avoid losing the expertise of the field >> because there's no new data...come again? There's really no sense in >> encouraging that many smart people to pour their careers down a rat >> hole like that, data or no data. >> I'm all for the search for origins and so on, but it's only worth >> spending so many billion bucks on such a brute force approach to a >> primarily aesthetic goal, particularly when it's in competition with >> much more valuable work that could train lots more people for lots >> less money. >> Phil Hobbs > Phil, I totally agree! There's way too much hype for the refurbished > LHC. The only thing I'm expecting out of this hugh machine is the > mass of the Higgs boson. It harldy seems worth it. I expect more > physics community has a great PR machine though... > George >> So....the higgs bosen was invented to explain mass. So..how come it >> has mass itself. Seems a bit like darkness to me... >> Kevin Aylward >> kevin@kevinaylward.co.uk >> www.kevinaylward.co.uk > > > > smitherinos I have those for breakfast. Mmmmmmmmmm. -- Michael Press === Subject: Re: Newsweek: The Large Hadron Collider posting-account=5GUrzQkAAADun29oaK3p_W_saUVxxHUF Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > The Large Hadron Collider is a symptom of America's decline in http://www.newsweek.com/id/157514 Ah well... best wishes to the Europeans ---------------------------- No Higgs and No hoggs and no schmigs !!!! look for the Circlon ATB Y.Porat ----------------------------------- === Subject: Re: Newsweek: The Large Hadron Collider >> The Large Hadron Collider is a symptom of America's decline in >> http://www.newsweek.com/id/157514 >> Ah well... best wishes to the Europeans >> I can't recall a single technologically useful result it's > generated since about Fermi's day > The argument seems to be that we have to keep dumping > our limited physical science budgets into accelerators > to avoid falling behind in a field that doesn't produce anything > useful >> >> Before radios, telegraphs, electric motors, etc, etc, Queen Victoria >> once asked British physicist Michael Faraday of what use his >> studies in electricity and magnetism were. He famously replied, >> Madam, of what use is a baby? > > has had big bucks poured into it for what, 60 years now? (Not counting > the Manhattan Project--nuclear has produced useful things.) When the > Faraday effect was 60 years old, it was powering half the world. > Somebody pointed out long ago that the further the energy scale gets > from kT, the fewer the useful results. Well, if you're looking for practical results... http://www.foxnews.com/story/0,2933,419404,00.html Coincidentally, the LHC is scheduled to go online in just a few hours, at 3:30am EDT Wednesday, 9/10/08, possibly creating black holes that will destroy the Earth... Someone will spot a light ray coming out of the Indian Ocean during the night and no one will be able to explain it, retired Professor Otto Roessler told London's Mail. Very soon the whole planet will be eaten in a magnificent scenario if you could watch it from the moon. A Biblical Armageddon. Even cloud and fire will form, as it says in the Bible. And we even have... A pair of Russian scientists even think the LHC will be the world's first time machine, and that we should expect visitors from the future to arrive soon after it goes into operation. And, hey, don't tell me these guys are kooks until you double-check what newsgroup you're posting to. -- John Forkosh ( mailto: j@f.com where j=john and f=forkosh ) === Subject: Re: Newsweek: The Large Hadron Collider > The Large Hadron Collider is a symptom of America's decline in > http://www.newsweek.com/id/157514 > Ah well... best wishes to the Europeans >> I can't recall a single technologically useful result it's >> generated since about Fermi's day >> The argument seems to be that we have to keep dumping >> our limited physical science budgets into accelerators >> to avoid falling behind in a field that doesn't produce anything >> useful > > Before radios, telegraphs, electric motors, etc, etc, Queen Victoria > once asked British physicist Michael Faraday of what use his > studies in electricity and magnetism were. He famously replied, > Madam, of what use is a baby? >> >> has had big bucks poured into it for what, 60 years now? (Not counting >> the Manhattan Project--nuclear has produced useful things.) When the >> Faraday effect was 60 years old, it was powering half the world. >> Somebody pointed out long ago that the further the energy scale gets >> from kT, the fewer the useful results. Well, if you're looking for practical results... > http://www.foxnews.com/story/0,2933,419404,00.html >Coincidentally, the LHC is scheduled to go online in just >a few hours, at 3:30am EDT Wednesday, 9/10/08, possibly >creating black holes that will destroy the Earth... > Someone will spot a light ray coming out of the Indian > Ocean during the night and no one will be able to explain it, > retired Professor Otto Roessler told London's Mail. > Very soon the whole planet will be eaten in a magnificent > scenario if you could watch it from the moon. A Biblical > Armageddon. Even cloud and fire will form, as it says in > the Bible. >And we even have... > A pair of Russian scientists even think the LHC will be > the world's first time machine, and that we should expect > visitors from the future to arrive soon after it goes into > operation. >And, hey, don't tell me these guys are kooks until >you double-check what newsgroup you're posting to. A hot cosmic ray can have the kinetic energy of a well-pitched baseball, far more than the LHC can manage. Certainly in the age of the Earth two such cosmic rays have collided nearby. We're still here. John === Subject: Re: Newsweek: The Large Hadron Collider jjlarkin@highNOTlandTHIStechnologyPART.com says... > >Of what use is the ISS? >>http://www.theregister.co.uk/2008/09/08/griffin_leaked_email/ >>John >Couldn't they stop the war for a day or two, that would give NASA >quite a lot of money? >martin > > > Not a day or two. NASA's budget is around $18 billion; over half is on > useless and dangerous manned spaceflight. There is nothing wrong with dangerous. The useless part makes the dangerous part a stupid undertaking. -- Keith === Subject: Re: Newsweek: The Large Hadron Collider > jjlarkin@highNOTlandTHIStechnologyPART.com says... >> Of what use is the ISS? >> http://www.theregister.co.uk/2008/09/08/griffin_leaked_email/ >> John > Couldn't they stop the war for a day or two, that would give NASA > quite a lot of money? >> martin >> Not a day or two. NASA's budget is around $18 billion; over half is on >> useless and dangerous manned spaceflight. > > There is nothing wrong with dangerous. Uh-oh. You are not being politically correct. > The useless part makes > the dangerous part a stupid undertaking. /BAH === Subject: Re: Newsweek: The Large Hadron Collider > > Uh-oh. You are not being politically correct. 'Politically correct' is for cowards and idiots. -- http://improve-usenet.org/index.html aioe.org, Goggle Groups, and Web TV users must request to be white listed, or I will not see your messages. If you have broadband, your ISP may have a NNTP news server included in your account: http://www.usenettools.net/ISP.htm There are two kinds of people on this earth: The crazy, and the insane. The first sign of insanity is denying that you're crazy. === Subject: Re: Newsweek: The Large Hadron Collider <48C56A35.A9B43B9D@hovnanian.com> posting-account=21qK2woAAADgDXzdjprGEPNP9dUjWxRr 1.1.4322; .NET CLR 2.0.40607),gzip(gfe),gzip(gfe) > Of what use is the ISS? Zero. Spam in a can, looking for something to do, which is mostly to > stay alive. Sadly I agree with you. The only thing the ISS provides is a relatively safe destination for the shuttle to fly to for damage inspection. > The new moon rocket thing is even sillier, since we've > been there already and nothing's going on. It would be interesting to see what the water ice detected near the lunar poles might be like. We have only looked at a few small spots on the lunar surface. And it would be fun to bring back some Apollo era momentos, together with some lunar dust and rub the fake lunar landings nutters noses in it. I expect China will be next on the moon. For the price of the ISS and the death-trap shuttles, we could fund > hundreds of world-class telescopes, launch scores of unmanned space > and planetary probes, or defend the Earth from the next comet/asteroid > strike. Manned space flight is still useful for some purposes although I do largely agree with you that robotic missions are the way forward unless and until we run into something that our robots cannot handle. Flesh and bone is too fragile in interplanetary space. Even in near Earth orbit like the ISS they get a fair suntan from the solar wind. Martin Brown === Subject: Re: Newsweek: The Large Hadron Collider <48C56A35.A9B43B9D@hovnanian.com> posting-account=eUaIfQoAAABt8kzNLoSYY_P3tYw0Y68G Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > It would be interesting to see what the water ice detected near the > lunar poles might be like. We have only looked at a few small spots on > the lunar surface. And it would be fun to bring back some Apollo era > momentos, together with some lunar dust and rub the fake lunar > landings nutters noses in it. Don't forget the only art piece on the Moon (of course, it's best placed there): http://en.wikipedia.org/wiki/Fallen Astronaut Tim === Subject: Re: Newsweek: The Large Hadron Collider >> >> And the funding was tiny compared to what NASA has spent. >>Which amount ($17.6 Billion) is approx. 4 (four) percent of what >they're spending ($439.3 billion) on war and weapons of mass >destruction. >> >> We've been paying gigabucks per year for the manned spaceflight >> program for four decades. Net return is zero, or negative if you count >> the deaths. All those resources, engineering talent, scientists, >> workers, energy, may as well have been used to dig holes and then fill >> holes. >I guess poetic crap like Expand the scope of human knowledge and >Because it's there doesn't mean much to a mundane like yourself. The robotic probes have delivered excellent, amazing science. The manned stuff, zero. So, don't go!! But don't put restrictions on what other people want to do. I would if I could! I'd make a pretty good emperor. John === Subject: Re: Newsweek: The Large Hadron Collider >> > And the funding was tiny compared to what NASA has spent. >>Which amount ($17.6 Billion) is approx. 4 (four) percent of what >>they're spending ($439.3 billion) on war and weapons of mass >>destruction. > > We've been paying gigabucks per year for the manned spaceflight > program for four decades. Net return is zero, or negative if you count > the deaths. All those resources, engineering talent, scientists, > workers, energy, may as well have been used to dig holes and then fill > holes. >I guess poetic crap like Expand the scope of human knowledge and >>Because it's there doesn't mean much to a mundane like yourself. The robotic probes have delivered excellent, amazing science. The >manned stuff, zero. > I loved the repair of one of the mars rovers,(huey, dewey?), with the priority inversion problem when they sent up the source code and got it to compile it and reeboot. That problem solving excersise was not given enough air time on the TV progs, Real Belt and Braces Engineering martin === Subject: Re: Newsweek: The Large Hadron Collider > On Sep 8, 7:19 pm, Phil Hobbs > There is often a lag of five to ten decades before fundamental > research surfaces in practical applications. > This was true in Newton's day, in Lavoisiere's day, in Kelvin's day. > PD >> results, over 60 years. > > As I believe I just said, it is not unusual at all for five to ten > decades to pass before useful applications arise from fundamental > research. > If you find that an abomination, that's fine -- lobby your congressman > to stop funding fundamental research. > As for myself, I consider it an unwise display of impatience. > You pulled those numbers out of thin air. Do you have a recent example of something that sat around for a century unused and then made a big splash? I can't think of one. Note that we aren't talking about Leonardo's helicopter, here--he didn't know how to make one fly. We're talking about Faraday's dynamoelectric effect, or the transistor, or....? Phil Hobbs === Subject: Re: Newsweek: The Large Hadron Collider > On Sep 9, 2:37 pm, Phil Hobbs > On Sep 8, 7:19 pm, Phil Hobbs > There is often a lag of five to ten decades before fundamental > research surfaces in practical applications. > This was true in Newton's day, in Lavoisiere's day, in Kelvin's day. > PD >> results, over 60 years. > As I believe I just said, it is not unusual at all for five to ten > decades to pass before useful applications arise from fundamental > research. > If you find that an abomination, that's fine -- lobby your congressman > to stop funding fundamental research. > As for myself, I consider it an unwise display of impatience. >> You pulled those numbers out of thin air. >> Do you have a recent example of something that sat around for a century >> unused and then made a big splash? I can't think of one. Note that we >> aren't talking about Leonardo's helicopter, here--he didn't know how to >> make one fly. We're talking about Faraday's dynamoelectric effect, or >> the transistor, or....? >> Phil Hobbs > > Sure. > > Newtonian gravity -- proposed 1687. First practical application, if > you want to call it that: discovery of Neptune 1846 (16 decades). > First big splash: planned trajectories of launched satellites 1950s > (26 decades). > > Mendelian genetics -- proposed 1866. Ignored until 1918 (5 decades). > First big splash: industrial agricultural breeding for harvest crops > 1950's (7 decades). > > Boyle's law -- proposed 1662. First practical application: Savery > engine 1712 (5 decades). First big splash: Watt engine 1775 (11 > decades). > > Coulomb's law of charge redistribution in conductors - 1785. First > application: Faraday cage 1836 (5 decades). First big splash: > Electrodynamic revolution 1890's (11 decades). So you mean, 'Sure' in the sense of 'Not hardly'. The only arguably recent claim you make is the one about Mendel, and Mendelian genetics was used in cross-breeding animals well before the 1950s. === Subject: Re: Newsweek: The Large Hadron Collider On Sep 8, 7:19 pm, Phil Hobbs There is often a lag of five to ten decades before fundamental >> research surfaces in practical applications. >> This was true in Newton's day, in Lavoisiere's day, in Kelvin's day. >> PD > physics--as > results, over 60 years. >> As I believe I just said, it is not unusual at all for five to ten >> decades to pass before useful applications arise from fundamental >> research. >> If you find that an abomination, that's fine -- lobby your congressman >> to stop funding fundamental research. >> As for myself, I consider it an unwise display of impatience. > You pulled those numbers out of thin air. >> Do you have a recent example of something that sat around for a century > unused and then made a big splash? I can't think of one. Note that we > aren't talking about Leonardo's helicopter, here--he didn't know how to > make one fly. We're talking about Faraday's dynamoelectric effect, or > the transistor, or....? > Phil Hobbs >> Sure. >> Newtonian gravity -- proposed 1687. First practical application, if >> you want to call it that: discovery of Neptune 1846 (16 decades). >> First big splash: planned trajectories of launched satellites 1950s >> (26 decades). >> Mendelian genetics -- proposed 1866. Ignored until 1918 (5 decades). >> First big splash: industrial agricultural breeding for harvest crops >> 1950's (7 decades). >> Boyle's law -- proposed 1662. First practical application: Savery >> engine 1712 (5 decades). First big splash: Watt engine 1775 (11 >> decades). >> Coulomb's law of charge redistribution in conductors - 1785. First >> application: Faraday cage 1836 (5 decades). First big splash: >> Electrodynamic revolution 1890's (11 decades). > > So you mean, 'Sure' in the sense of 'Not hardly'. > > The only arguably recent claim you make is the one about Mendel, and > Mendelian genetics was used in cross-breeding animals well before the > 1950s. > > > > Phil Hobbs Einstein, 1917, Stimulated Emission of Radiation. -- Dirk http://www.transcendence.me.uk/ - Transcendence UK http://www.theconsensus.org/ - A UK political party http://www.onetribe.me.uk/wordpress/?cat=5 - Our podcasts on weird stuff === Subject: Re: Newsweek: The Large Hadron Collider > On Sep 9, 2:37 pm, Phil Hobbs > On Sep 8, 7:19 pm, Phil Hobbs > There is often a lag of five to ten decades before fundamental > research surfaces in practical applications. > This was true in Newton's day, in Lavoisiere's day, in Kelvin's day. > PD >> physics--as >> results, over 60 years. > As I believe I just said, it is not unusual at all for five to ten > decades to pass before useful applications arise from fundamental > research. > If you find that an abomination, that's fine -- lobby your congressman > to stop funding fundamental research. > As for myself, I consider it an unwise display of impatience. >> You pulled those numbers out of thin air. >> Do you have a recent example of something that sat around for a century >> unused and then made a big splash? I can't think of one. Note that we >> aren't talking about Leonardo's helicopter, here--he didn't know how to >> make one fly. We're talking about Faraday's dynamoelectric effect, or >> the transistor, or....? >> Phil Hobbs >> Sure. >> Newtonian gravity -- proposed 1687. First practical application, if > you want to call it that: discovery of Neptune 1846 (16 decades). > First big splash: planned trajectories of launched satellites 1950s > (26 decades). >> Mendelian genetics -- proposed 1866. Ignored until 1918 (5 decades). > First big splash: industrial agricultural breeding for harvest crops > 1950's (7 decades). >> Boyle's law -- proposed 1662. First practical application: Savery > engine 1712 (5 decades). First big splash: Watt engine 1775 (11 > decades). >> Coulomb's law of charge redistribution in conductors - 1785. First > application: Faraday cage 1836 (5 decades). First big splash: > Electrodynamic revolution 1890's (11 decades). >> So you mean, 'Sure' in the sense of 'Not hardly'. >> The only arguably recent claim you make is the one about Mendel, and >> Mendelian genetics was used in cross-breeding animals well before the >> 1950s. >> Phil Hobbs > > Einstein, 1917, Stimulated Emission of Radiation. > The maser, 1952. Optical pumping, also early 1950s. Pretty far from a century. Phil Hobbs === Subject: Re: Newsweek: The Large Hadron Collider > On Sep 9, 8:40 pm, Phil Hobbs > On Sep 9, 2:37 pm, Phil Hobbs > On Sep 8, 7:19 pm, Phil Hobbs > There is often a lag of five to ten decades before fundamental > research surfaces in practical applications. > This was true in Newton's day, in Lavoisiere's day, in Kelvin's day. > PD >> physics--as >> results, over 60 years. > As I believe I just said, it is not unusual at all for five to ten > decades to pass before useful applications arise from fundamental > research. > If you find that an abomination, that's fine -- lobby your congressman > to stop funding fundamental research. > As for myself, I consider it an unwise display of impatience. >> You pulled those numbers out of thin air. >> Do you have a recent example of something that sat around for a century >> unused and then made a big splash? I can't think of one. Note that we >> aren't talking about Leonardo's helicopter, here--he didn't know how to >> make one fly. We're talking about Faraday's dynamoelectric effect, or >> the transistor, or....? >> Phil Hobbs > Sure. > Newtonian gravity -- proposed 1687. First practical application, if > you want to call it that: discovery of Neptune 1846 (16 decades). > First big splash: planned trajectories of launched satellites 1950s > (26 decades). > Mendelian genetics -- proposed 1866. Ignored until 1918 (5 decades). > First big splash: industrial agricultural breeding for harvest crops > 1950's (7 decades). > Boyle's law -- proposed 1662. First practical application: Savery > engine 1712 (5 decades). First big splash: Watt engine 1775 (11 > decades). > Coulomb's law of charge redistribution in conductors - 1785. First > application: Faraday cage 1836 (5 decades). First big splash: > Electrodynamic revolution 1890's (11 decades). >> So you mean, 'Sure' in the sense of 'Not hardly'. >> The only arguably recent claim you make is the one about Mendel, and >> Mendelian genetics was used in cross-breeding animals well before the >> 1950s. >> Phil Hobbs > Einstein, 1917, Stimulated Emission of Radiation. >> The maser, 1952. Optical pumping, also early 1950s. Pretty far from a >> century. > > And the laser did not make it into mainstream applications until the > 1980's -- seven decades. Riiight. Revolutionizing spectroscopy in the first couple of years isn't good enough for you, it has to be 'mainstream'. We'll be seeing TeV sources in DVD players Real Soon Now. > > And you'll note that I said it was not unusual for fundamental > research to take five to ten decades to percolate into signficant > consumer impact, ten decades being an extremum. > > You asked for ONE case, because you could not think of a single one. > > I gave you several, which is certainly not an exhaustive list. You > chose to reject several as being not recent enough for your tastes, > but allowed that one of the four I provided does demonstrate the case. > Yet you whiiiiine that it's still wrong. You moved the goal posts, because you couldn't think of one either. > Phil Hobbs === Subject: Re: Newsweek: The Large Hadron Collider posting-account=_kLX7QoAAAChyHRjRDyv2ieHnlQ_-li2 CLR 1.1.4322) w:PACBHO60,gzip(gfe),gzip(gfe) > On Sep 9, 2:37Êpm, Phil Hobbs On Sep 8, 7:19 pm, Phil Hobbs There is often a lag of five to ten decades before fundamental > research surfaces in practical applications. > This was true in Newton's day, in Lavoisiere's day, in Kelvin's day. > PD >> results, over 60 years. > As I believe I just said, it is not unusual at all for five to ten > decades to pass before useful applications arise from fundamental > research. > If you find that an abomination, that's fine -- lobby your congressman > to stop funding fundamental research. > As for myself, I consider it an unwise display of impatience. > You pulled those numbers out of thin air. > Do you have a recent example of something that sat around for a century > unused and then made a big splash? ÊI can't think of one. ÊNote that we > aren't talking about Leonardo's helicopter, here--he didn't know how to > make one fly. ÊWe're talking about Faraday's dynamoelectric effect, or > the transistor, or....? > Phil Hobbs Sure. Newtonian gravity -- proposed 1687. First practical application, if > you want to call it that: discovery of Neptune 1846 (16 decades). > First big splash: planned trajectories of launched satellites 1950s > (26 decades). Mendelian genetics -- proposed 1866. Ignored until 1918 (5 decades). > First big splash: industrial agricultural breeding for harvest crops > 1950's (7 decades). Boyle's law -- proposed 1662. First practical application: Savery > engine 1712 (5 decades). First big splash: Watt engine 1775 (11 > decades). Coulomb's law of charge redistribution in conductors - 1785. First > application: Faraday cage 1836 (5 decades). First big splash: > Electrodynamic revolution 1890's (11 decades).- Hide quoted text - - Show quoted text - PD, Are any of these from the last century... discounting gravity which made a big spalsh before 1950. I think the trend of the last 25 years is to exploit any discovery as quickly as possible... before the other guy does it. In the past there just wasn't the competition of today. You could what till the end of your career and then publish some of thie things you did. Today if there's an aplication or advance it's used right away. The people at the LHC are not stupid. If they had found some new thing that could make them some money... or get them tenure you cna bet it wouldn't just sit there for 50 years. George === Subject: Re: Newsweek: The Large Hadron Collider posting-account=_kLX7QoAAAChyHRjRDyv2ieHnlQ_-li2 .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > On Sep 9, 2:37Êpm, Phil Hobbs On Sep 8, 7:19 pm, Phil Hobbs There is often a lag of five to ten decades before fundamental > research surfaces in practical applications. > This was true in Newton's day, in Lavoisiere's day, in Kelvin's day. > PD >> results, over 60 years. > As I believe I just said, it is not unusual at all for five to ten > decades to pass before useful applications arise from fundamental > research. > If you find that an abomination, that's fine -- lobby your congressman > to stop funding fundamental research. > As for myself, I consider it an unwise display of impatience. > You pulled those numbers out of thin air. > Do you have a recent example of something that sat around for a century > unused and then made a big splash? ÊI can't think of one. ÊNote that we > aren't talking about Leonardo's helicopter, here--he didn't know how to > make one fly. ÊWe're talking about Faraday's dynamoelectric effect, or > the transistor, or....? > Phil Hobbs > Sure. > Newtonian gravity -- proposed 1687. First practical application, if > you want to call it that: discovery of Neptune 1846 (16 decades). > First big splash: planned trajectories of launched satellites 1950s > (26 decades). > Mendelian genetics -- proposed 1866. Ignored until 1918 (5 decades). > First big splash: industrial agricultural breeding for harvest crops > 1950's (7 decades). > Boyle's law -- proposed 1662. First practical application: Savery > engine 1712 (5 decades). First big splash: Watt engine 1775 (11 > decades). > Coulomb's law of charge redistribution in conductors - 1785. First > application: Faraday cage 1836 (5 decades). First big splash: > Electrodynamic revolution 1890's (11 decades).- Hide quoted text - > - Show quoted text - > PD, ÊAre any of these from the last century... discounting gravity > which made a big spalsh before 1950. Well, I think that's the point. > Quantum mechanics, originated in the 1930's, has started to make > significant splash some 50 years later. > But weak interaction physics, originated in the 1940's, hasn't really > started to make a splash yet, but could very well in the next two > decades. Controlled fusion power is the most likely candidate. > Strong interaction physics, originated in the 1960's, probably won't > make a splash for another three decades. > Gravitational physics, originated in 1915 and then ignored for three > decades, probably won't make a splash for another four decades or so. > I think the trend of the last 25 years is to exploit any discovery as > quickly as possible... before the other guy does it. That's not so for fundamental research. > ÊIn the past > there just wasn't the competition of today. ÊYou could what till the > end of your career and then publish some of thie things you did. > Today if there's an aplication or advance it's used right away. ÊThe > people at the LHC are not stupid. ÊIf they had found some new thing > that could make them some money... or get them tenure Tenure doesn't buy you anything except for continued servitude. PD, > But weak interaction physics, originated in the 1940's, hasn't really > started to make a splash yet, but could very well in the next two > decades. Controlled fusion power is the most likely candidate. Wow, are you really expecting some electro-weak application. OK I guess you are taking the long view! Since I expect nothing from this in my lifetime (another 30 years if I'm lucky), or even my kids lifetime. But I do have an example for your view. How about General Relativity. Without which we would not be able to synchronize all our clocks in orbit. George === Subject: Re: Quadratic equations : need help > Problem : the roots of ax^2 + bx + c are B and nB where both of them > are real. Prove that (n+1)^2 * ac = nb^2 Let r = B. Handle the case r = 0 separately. Recall the formula for the sum of the roots and the product of the roots. r + nr = -b/a; nr^2 = c/a (1 + n)r = -b/a ac(1 + n)^2 = cb^2 / ar^2 = nb^2 ---- ---- === Subject: Re: A consideration concerning the diagonal argument of G. Cantor <87zlmrzxtn.fsf@phiwumbda.org> <1d79a$48bce687$82a1e228$604@news1.tudelft.nl> <6b855$48be3911$82a1e228$14221@news1.tudelft.nl> <%cywk.251241$gc5.174605@pd7urf2no> posting-account=6xUtvgkAAAD_jypmLa2oo2HnrV0e8X9q 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648),gzip(gfe),gzip(gfe) >> Exactly so. Did I ever suggest that physics (especially astronomy) is >> free from dogmas, oh well, let's say: questionable extrapolations ? >> You have clearly suggested that it is the sole source of all truth. > I've only suggested that mathematics could borrow some truth from that > source called physics. >> The trouble is that the objects that mathematics deals with >> (numbers, sets, etc.) are not physically observable phenomena. >> That's where any similarity between mathematics and physics >> breaks down. > The natural numbers are as observable phenomena as mass, gravity, > colour, form, etc. > > Really? About the natural numbers, has anybody ever observed the truth > of GC, or of the statement there are infinite even natural numbers that > don't satisfy GC? Do someone knows the truth of mass? maybe at the LHC we may learn something more. But, I'm sure, there arises more questions as questions will be answerd. -- > To discover the proper approach to mathematical logic, > we must therefore examine the methods of the mathematician. > (Shoenfield, Mathematical Logic) === Subject: Re: A consideration concerning the diagonal argument of G. Cantor <87zlmrzxtn.fsf@phiwumbda.org> <1d79a$48bce687$82a1e228$604@news1.tudelft.nl> <6b855$48be3911$82a1e228$14221@news1.tudelft.nl> posting-account=6xUtvgkAAAD_jypmLa2oo2HnrV0e8X9q 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648),gzip(gfe),gzip(gfe) >> I've only suggested that mathematics could borrow some truth from that >> source called physics. > >> The trouble is that the objects that mathematics deals with >> (numbers, sets, etc.) are not physically observable phenomena. >> That's where any similarity between mathematics and physics >> breaks down. > > The natural numbers are as observable phenomena as mass, gravity, > colour, form, etc. Have you observed one lately? What did it look like? If you are owner of two hands, put them together. What do you feel? Maybe two hands? === Subject: Re: A consideration concerning the diagonal argument of G. Cantor >> I've only suggested that mathematics could borrow some truth from that >> source called physics. > >> The trouble is that the objects that mathematics deals with >> (numbers, sets, etc.) are not physically observable phenomena. >> That's where any similarity between mathematics and physics >> breaks down. > > The natural numbers are as observable phenomena as mass, gravity, > colour, form, etc. > Have you observed one lately? What did it look like? > > If you are owner of two hands, put them together. What do you feel? > Maybe two hands? Maybe ten fingers (including thumbs). So which natural number(s) exist in one's hands? Is ten not represented as well as two. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor <87zlmrzxtn.fsf@phiwumbda.org> <1d79a$48bce687$82a1e228$604@news1.tudelft.nl> <6b855$48be3911$82a1e228$14221@news1.tudelft.nl> posting-account=6xUtvgkAAAD_jypmLa2oo2HnrV0e8X9q 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.648),gzip(gfe),gzip(gfe) >> Exactly so. Did I ever suggest that physics (especially astronomy) is >> free from dogmas, oh well, let's say: questionable extrapolations ? >> You have clearly suggested that it is the sole source of all truth. > I've only suggested that mathematics could borrow some truth from that > source called physics. > The trouble is that the objects that mathematics deals with > (numbers, sets, etc.) are not physically observable phenomena. > That's where any similarity between mathematics and physics > breaks down. > The natural numbers are as observable phenomena as mass, gravity, > colour, form, etc. All of which are artificialities which we impose on reality and by which > we interpret reality. Ah, very interesting. And, please, might you tell me: What is reality? You are unable to make a difference between artificialities and not- artificialities, are you? So, what we are talking about? Things like mass, gravity, colour, form, natural numbers, ..., which are artificialities and things like ??? which are not artificialities? Please give an example for ???. Your standpoint is very weak, I think. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor >> Exactly so. Did I ever suggest that physics (especially astronomy) is >> free from dogmas, oh well, let's say: questionable extrapolations ? >> You have clearly suggested that it is the sole source of all truth. > I've only suggested that mathematics could borrow some truth from that > source called physics. > The trouble is that the objects that mathematics deals with > (numbers, sets, etc.) are not physically observable phenomena. > That's where any similarity between mathematics and physics > breaks down. > The natural numbers are as observable phenomena as mass, gravity, > colour, form, etc. > All of which are artificialities which we impose on reality and by which > we interpret reality. > > Ah, very interesting. And, please, might you tell me: What is reality? That is a problem for physicists, of which I am not one. Humans are pattern seekers, and tend to impose their own order on whatever they perceive. Mathematicians deal with some of those patterns without worrying, at least as mathematicians, about what lies beneath them. > You are unable to make a difference between artificialities and not- > artificialities, are you? So, what we are talking about? Things like > mass, gravity, colour, form, natural numbers, ..., which are > artificialities and things like ??? which are not artificialities? > Please give an example for ???. Not my problem. > > Your standpoint is very weak, I think. Just limited. === === Subject: Simple doubt on continuity of multivarible function Hello All, I have a one simple doubt. Suppose f: R^n -> R function is such that: fix any (n-1) variables, then it is continuous function of a single variable. (e.g. suppose f is function of x,y. Then for any fixed x, f is continuous in y and for any fixed y, f is continuous in x) Will it mean that f is continuous function? --- Sujit Gujar. IISc Bangalore. Web: http:/people.csa.iisc.ernet.in/sujit === Subject: Re: Simple doubt on continuity of multivarible function posting-account=AdyLXQoAAABgRay99CKv1O8Y_7jjivwq InfoPath.1),gzip(gfe),gzip(gfe) > Suppose f: R^n -> R function is such that: > Ê fix any (n-1) variables, then it is continuous function > of a single variable. (e.g. suppose f is function of x,y. > Then for any fixed x, f is continuous in y and for any > fixed y, f is continuous in x) Will it mean that f is continuous function? William Elliot has given you a counterexample. If you're interested in more about this topic, see the following post: Continuity in each variable vs. joint continuity Dave L. Renfro === Subject: Re: Simple doubt on continuity of multivarible function > >> Suppose f: R^n -> R function is such that: >> fix any (n-1) variables, then it is continuous function >> of a single variable. (e.g. suppose f is function of x,y. >> Then for any fixed x, f is continuous in y and for any >> fixed y, f is continuous in x) >> >> Will it mean that f is continuous function? > > William Elliot has given you a counterexample. If you're > interested in more about this topic, see the following post: > > Continuity in each variable vs. joint continuity > > Dave L. Renfro Multivariate Limits and Continuity: A Survey of Calculus Textbooks http://eric.ed.gov/ERICDocs/data/ericdocs2sql/content_storage_01/0000019b/80/ 1c/6b/5c.pdf Dirk Vdm === Subject: Re: Simple doubt on continuity of multivarible function > (e.g. suppose f is function of x,y. Then for any fixed x, f is > continuous in y and for any fixed y, f is continuous in x) Will it mean that f is continuous function? > No. f(x,y) = xy/(x^2 + y^2), when x,y /= 0 f(0,0) = 0 is continuous in x and in y. Now let (x,y) -> (0,0) along y = ax, a /= 0. Then f(x,y) = ax^2 / (1 + a^2)x^2 = a/(1 + a^2) does not approach 0. === Subject: New SMS Concept - Enjoy Unlimited Income posting-account=3cMYmAoAAADj7pZIB_VD9V4ha4qQwlxk InfoPath.2),gzip(gfe),gzip(gfe) Just wanted to share some good information with fellow friends :) As the indian economy is booming, the concept is now slowly coming to india it seems gud.... it has started recently and suppose to be not fake. somewhat new concept...u receive sms and earn 10 to 20 paise for each sms... so go ahead and register in order to earn something extra without applying much effort :) http://www.mginger.com/index.jsp?inviteId=33568 adjust and put the above in the web browser link and register urself. === Subject: Re: -- Wrong limits do not commute > >Synopsis: All this time Han claimed to understand limits better than >the rest of us, it never occurred to them that (1) they could be >nested and (2) limits don't necessarily commute. >>Hence (?) the mathematicians are wrong. >>Let's consider again what a (double) limit really is. >> lim lim f(x,y) = F >> x->oo y->oo >>For any real number epsilon > 0 there exists real numbers M , N such >>that if x > L and y > M then | f(x,y) - F | < epsilon . Right ? > > Wrong. The above says that for every epsilon > 0 there exists > N such that |F - lim_{y->oo} f(x,y)} < epsilon for every x > N. > > What you give is the definition of > > lim_{(x,y) -> (oo, oo)} f(x,y) = F. > > That's something _different_. If you think the two > are the same then of course you're going to deduce > that there are problems somewhere. But that's > not a problem with mathematics, it's just a problem > with your misunderstanding of mathematical notation. Han de Bruijn === Subject: Re: -- Wrong limits do not commute > So, among mathematicians, there is distinction between > > lim_{x->a} lim_{y->b} f(x) > and > lim_{y->b} lim_{x->a} f(x) > and > lim_{(x,y)->(a,b)} f(x), > > particularly in the case when the last of these does not exist but one > or both of first two do exist. Okay. Thus, if I write > For any real number epsilon > 0 there exists real numbers M , N such > that if x > L and y > M then | f(x,y) - F | < epsilon . then the correct notation for this is: lim_{(x,y)->(a,b)} f(x,y) = F . Am I right this time? Hand de Bruijn === Subject: Re: -- Wrong limits do not commute > > So, among mathematicians, there is distinction between > > lim_{x->a} lim_{y->b} f(x) > and > lim_{y->b} lim_{x->a} f(x) > and > lim_{(x,y)->(a,b)} f(x), > > particularly in the case when the last of these does not exist but one > or both of first two do exist. > > Okay. Thus, if I write > > For any real number epsilon > 0 there exists real numbers M , N such > that if x > L and y > M then | f(x,y) - F | < epsilon . > > then the correct notation for this is: lim_{(x,y)->(a,b)} f(x,y) = F . > > Am I right this time? > > Hand de Bruijn Not quite. Try lim_{(x,y)->(oo,oo)} f(x,y) = F === Subject: Re: -- Wrong limits do not commute <87myiilf5a.fsf@phiwumbda.org> <60c3$48c68189$82a1e228$19985@news1.tudelft.nl> <9ea0a$48c77903$82a1e228$31708@news1.tudelft.nl> posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2008071615 Fedora/3.0.1-1.fc9 Firefox/3.0.1,gzip(gfe),gzip(gfe) > So, among mathematicians, there is Êdistinction between > lim {x->a} lim {y->b} f(x) > and > lim {y->b} lim {x->a} f(x) > and > lim {(x,y)->(a,b)} f(x), > particularly in the case when the last of these does not exist but one > or both of first two do exist. > Okay. Thus, if I write > For any real number Êepsilon > 0 Êthere exists real numbers ÊM , N Êsuch > that if Êx > L Êand Êy > M Êthen Ê| f(x,y) - F | < epsilon . > then the correct notation for this is: Êlim {(x,y)->(a,b)} f(x,y) = F . > Am I right this time? > Hand de Bruijn Not quite. Try lim {(x,y)->(oo,oo)} f(x,y) = F While this notation is probably correct, it is rather unusual, I think. The idea of limit that Han is entertaining is the one corresponding to the convergence of nets phi : R x R --> R where R x R, the cartesian square of the real numbers, is ordered with the product order of the usual orders in the factors. This kind of limit rarely shows up in calculus, and while your notation is quite indcative of the specific kind of limit, I do not think it is standard. -- m === Subject: Re: -- Wrong limits do not commute > >So, among mathematicians, there is distinction between >>lim_{x->a} lim_{y->b} f(x) >and >lim_{y->b} lim_{x->a} f(x) >and >lim_{(x,y)->(a,b)} f(x), >>particularly in the case when the last of these does not exist but one >or both of first two do exist. >>Okay. Thus, if I write >For any real number epsilon > 0 there exists real numbers M , N such >that if x > L and y > M then | f(x,y) - F | < epsilon . Huh ? Did I say this ? Must be: For any real number epsilon > 0 there exists real numbers L , M such that if x > L and y > M then | f(x,y) - F | < epsilon . >>then the correct notation for this is: lim_{(x,y)->(a,b)} f(x,y) = F . >>Am I right this time? > > Not quite. Try lim_{(x,y)->(oo,oo)} f(x,y) = F Sorry. That's what I meant, of course. The other one is: For any real number epsilon > 0 there exists real numbers L , M such that if | x - a | < L and | y - b | < M then | f(x,y) - F | < epsilon Han de Bruijn === Subject: Re: -- Wrong limits do not commute <87myiilf5a.fsf@phiwumbda.org> <60c3$48c68189$82a1e228$19985@news1.tudelft.nl> <9ea0a$48c77903$82a1e228$31708@news1.tudelft.nl> <222af$48c77ce4$82a1e228$31944@news1.tudelft.nl> posting-account=suWj4AkAAADE1IvGmj55Nmq3f98qb17e InfoPath.1; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) On Sep 10, 10:53Êam, Han de Bruijn So, among mathematicians, there is Êdistinction between >lim {x->a} lim {y->b} f(x) >and >lim {y->b} lim {x->a} f(x) >and >lim {(x,y)->(a,b)} f(x), >particularly in the case when the last of these does not exist but one >or both of first two do exist. >>Okay. Thus, if I write >For any real number Êepsilon > 0 Êthere exists real numbers ÊM , N Êsuch >that if Êx > L Êand Êy > M Êthen Ê| f(x,y) - F | < epsilon . Huh ? Did I say this ? Must be: For any real number Êepsilon > 0 Êthere exists real numbers ÊL , M Êsuch > that if Êx > L Êand Êy > M Êthen Ê| f(x,y) - F | < epsilon . >>then the correct notation for this is: Êlim {(x,y)->(a,b)} f(x,y) = F . >>Am I right this time? > Not quite. Try lim {(x,y)->(oo,oo)} f(x,y) = F Sorry. That's what I meant, of course. The other one is: For any real number Êepsilon > 0 Êthere exists real numbers ÊL , M Êsuch > that if | x - a | < L Êand | y - b | < M Êthen | f(x,y) - F | < epsilon Han de Bruijn- ************************************************************* A waste of time...again... **Sigh** Han, buddy: why do you insist over and over again to show to everybody in this forum that you suck, and big time, in pretty basic, elementary stuff in mathematics? Why this craving for notoriety in being wrong in mathematics in a mathematics forum, and even discussing and arguing nonsenses over and over again...with mathematicians!? This time is not only Cantor and set theory: now you're pissed off about limits, how they're defined (something that seems to be way above your comprehension), how to work with them, etc. So, in some rather amazing, surprising way, you seem to be trying to expand your field of cranking from set theory to other realms of mathematics...why?? Why won't you first master thoroughly in depth all the crankhood details in set theory and THEN pass over to crank around in calculus, geometric algebra, number theory, etc? Hey, who knows: perhaps after some time you'll be able, just as JHS has, to achieve heights like proving FLT with the unique and only aid of an abacus and prayers to Zoroaster, or even showing RH to be true by using matches and dancing the cherokee dance rain. Stay in what you're good, Han: Set Theory and Cantor cranking. Tonio === Subject: Re: -- Wrong limits do not commute > On Sep 10, 10:53 am, Han de Bruijn So, among mathematicians, there is distinction between >lim_{x->a} lim_{y->b} f(x) >and >lim_{y->b} lim_{x->a} f(x) >and >lim_{(x,y)->(a,b)} f(x), >particularly in the case when the last of these does not exist but one >or both of first two do exist. >>Okay. Thus, if I write >For any real number epsilon > 0 there exists real numbers M , N such >that if x > L and y > M then | f(x,y) - F | < epsilon . >>Huh ? Did I say this ? Must be: >>For any real number epsilon > 0 there exists real numbers L , M such >>that if x > L and y > M then | f(x,y) - F | < epsilon . >>then the correct notation for this is: lim_{(x,y)->(a,b)} f(x,y) = F . >>Am I right this time? >Not quite. Try lim_{(x,y)->(oo,oo)} f(x,y) = F >>Sorry. That's what I meant, of course. The other one is: >>For any real number epsilon > 0 there exists real numbers L , M such >>that if | x - a | < L and | y - b | < M then | f(x,y) - F | < epsilon > > ************************************************************* > > A waste of time...again... **Sigh** > > Han, buddy: why do you insist over and over again to show to everybody > in this forum that you suck, and big time, in pretty basic, elementary > stuff in mathematics? That pretty basic, elementary stuff in mathematics is pretty basic in our debates about _foundational_ issues. Look up e.g. in the WM threads (WM = Wolfgang Mueckenheim) how often the counter argument is used (by meanstreamers) that WM's limits do not commute, while WM thinks they do. > Why this craving for notoriety in being wrong in mathematics in a > mathematics forum, and even discussing and arguing nonsenses over and > over again...with mathematicians!? Uhm, if you find it disturbing, please leave this Internet coffeeshop and go to the censored 'sci.math.research', where you certainly can meet people of your own taste. (BTW. It's so incredibly boring there .. ) > This time is not only Cantor and set theory: now you're pissed off > about limits, how they're defined (something that seems to be way > above your comprehension), how to work with them, etc. > > So, in some rather amazing, surprising way, you seem to be trying to > expand your field of cranking from set theory to other realms of > mathematics...why?? Why won't you first master thoroughly in depth all > the crankhood details in set theory and THEN pass over to crank around > in calculus, geometric algebra, number theory, etc? > > Hey, who knows: perhaps after some time you'll be able, just as JHS > has, to achieve heights like proving FLT with the unique and only aid > of an abacus and prayers to Zoroaster, or even showing RH to be true > by using matches and dancing the cherokee dance rain. > > Stay in what you're good, Han: Set Theory and Cantor cranking. I'm not going to (im)prove on Fermat's Last Theorem, not going to prove Goldbach Conjecture or prove the Riemann Hypothesis. Maybe I'm a crank, but not _that_ much. (Actually, I've done research on these topics, but never published it, for good reasons. Oh well, everybody did an attempt on these issues .. Not ? ) Han de Bruijn === Subject: Re: -- Wrong limits do not commute <87myiilf5a.fsf@phiwumbda.org> <60c3$48c68189$82a1e228$19985@news1.tudelft.nl> <9ea0a$48c77903$82a1e228$31708@news1.tudelft.nl> <222af$48c77ce4$82a1e228$31944@news1.tudelft.nl> <678fd$48c7a989$82a1e228$32631@news1.tudelft.nl> posting-account=IBUqVwoAAADepmzxVr9iEYD5Z0A483SY rv:1.9.0.1) Gecko/2008070206 Firefox/3.0.1,gzip(gfe),gzip(gfe) That pretty basic, elementary stuff in mathematics is pretty basic in > our debates about _foundational_ issues. Look up e.g. in the WM threads > (WM = Wolfgang Mueckenheim) how often the counter argument is used (by > meanstreamers) that WM's limits do not commute, while WM thinks they do. He's an eejit, just like you. Victor Meldrew I don't believe it! === Subject: Re: -- Wrong limits do not commute Tonico a .8ecrit : > On Sep 10, 10:53 am, Han de Bruijn > So, among mathematicians, there is distinction between > lim_{x->a} lim_{y->b} f(x) > and > lim_{y->b} lim_{x->a} f(x) > and > lim_{(x,y)->(a,b)} f(x), > particularly in the case when the last of these does not exist but one > or both of first two do exist. >> Okay. Thus, if I write > For any real number epsilon > 0 there exists real numbers M , N such > that if x > L and y > M then | f(x,y) - F | < epsilon . >> Huh ? Did I say this ? Must be: >> For any real number epsilon > 0 there exists real numbers L , M such >> that if x > L and y > M then | f(x,y) - F | < epsilon . >> then the correct notation for this is: lim_{(x,y)->(a,b)} f(x,y) = F . >> Am I right this time? > Not quite. Try lim_{(x,y)->(oo,oo)} f(x,y) = F >> Sorry. That's what I meant, of course. The other one is: >> For any real number epsilon > 0 there exists real numbers L , M such >> that if | x - a | < L and | y - b | < M then | f(x,y) - F | < epsilon >> Han de Bruijn- > > ************************************************************* > > A waste of time...again... **Sigh** > > Han, buddy: why do you insist over and over again to show to everybody > in this forum that you suck, and big time, in pretty basic, elementary > stuff in mathematics? > > Why this craving for notoriety in being wrong in mathematics in a > mathematics forum, and even discussing and arguing nonsenses over and > over again...with mathematicians!? > > This time is not only Cantor and set theory: now you're pissed off > about limits, how they're defined (something that seems to be way > above your comprehension), how to work with them, etc. > > So, in some rather amazing, surprising way, you seem to be trying to > expand your field of cranking from set theory to other realms of > mathematics...why?? Why won't you first master thoroughly in depth all > the crankhood details in set theory and THEN pass over to crank around > in calculus, geometric algebra, number theory, etc? > > Hey, who knows: perhaps after some time you'll be able, just as JHS > has, to achieve heights like proving FLT with the unique and only aid > of an abacus and prayers to Zoroaster, or even showing RH to be true > by using matches and dancing the cherokee dance rain. > > Stay in what you're good, Han: Set Theory and Cantor cranking. No, youre wrong. He is not such a good crank. Otoh, he is quite good at trolling... > > Tonio > === Subject: Re: -- Wrong limits do not commute > No, youre wrong. He is not such a good crank. Otoh, he is quite good at > trolling... He's a *great* crank. You take that back. -- I believe that my job is to go out and explain to the people what's on my mind. That's why I'm having this press conference, see? I'm telling you what's on my mind. And what's on my mind is winning the war on terror. --- George W. Bush === Subject: Re: -- Wrong limits do not commute Jesse F. Hughes a .8ecrit : > >> No, youre wrong. He is not such a good crank. Otoh, he is quite good at >> trolling... > > He's a *great* crank. You take that back. > Never. That would implies he really believes what he says (most of the time). You wanna bet ? (but I admit he is a *great* troll, if you like) === Subject: Re: -- Wrong limits do not commute > Jesse F. Hughes a .8ecrit : >> > No, youre wrong. He is not such a good crank. Otoh, he is quite good at > trolling... >> >> He's a *great* crank. You take that back. >> > Never. That would implies he really believes what he says (most of the > time). You wanna bet ? (but I admit he is a *great* troll, if you like) > I think I'd take that bet, if we restrict it to his mathematical claims. But how will we settle it? -- We want a single platform. We're trying to get there using the carrot, or blackmail, or rewards, or whatever you call it. -- Madison, WI, superintendent Rainwater grasps subtlety in the operating system wars. === Subject: Re: -- Wrong limits do not commute > >Synopsis: All this time Han claimed to understand limits better than >the rest of us, it never occurred to them that (1) they could be >nested and (2) limits don't necessarily commute. >Hence (?) the mathematicians are wrong. >>Let's consider again what a (double) limit really is. >> lim lim f(x,y) = F >> x->oo y->oo >>For any real number epsilon > 0 there exists real numbers M , N such >>that if x > L and y > M then | f(x,y) - F | < epsilon . Right ? > > Not really. > > Consider the trivial example: it should be obvious that > > lim_{x -> +infty} lim_{y -> +infty} x/y = 0 > > yet the function x/y is not even bounded on any > set of the form { (x,y) : x > M, y > N }. So you find my definition: not right. Instead, what you do is first take the (single) limit lim_{y -> +infty} x/y (keeping x fixed) resulting in 0 and then you take the (single) limit lim_{x -> +infty} 0 giving (of course) 0 . That's what you did, right? Han de Bruijn === Subject: Re: -- Wrong limits do not commute <87myiilf5a.fsf@phiwumbda.org> <60c3$48c68189$82a1e228$19985@news1.tudelft.nl> posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I > Consider the trivial example: it should be obvious that > lim {x -> +infty} lim {y -> +infty} x/y = 0 > yet the function x/y is not even bounded on any > set of the form { (x,y) : x > M, y > N }. So you find my definition: not right. Instead, what you do is first take > the (single) limit lim {y -> +infty} x/y (keeping x fixed) resulting > in 0 and then you take the (single) limit lim {x -> +infty} 0 giving > (of course) 0 . That's what you did, right? Since that is what the notation means, yes. Do not confuse the iterated limits A: lim lim f(x,y) x->oo y->oo (first vary y , then vary x) With the double limit B: lim f(x,y) (x,y)->(oo,oo) (vary x and y in either order or both together) The two are related, (e.g. if the limit in B exists then the limits in A commute), however even if the limit in B does not exist the limits lim f(x,y) and lim f(x,y) are not wrong. - William Hughes === Subject: Re: -- Wrong limits do not commute > > >Consider the trivial example: it should be obvious that > lim_{x -> +infty} lim_{y -> +infty} x/y = 0 >yet the function x/y is not even bounded on any >set of the form { (x,y) : x > M, y > N }. >>So you find my definition: not right. Instead, what you do is first take >>the (single) limit lim_{y -> +infty} x/y (keeping x fixed) resulting >>in 0 and then you take the (single) limit lim_{x -> +infty} 0 giving >>(of course) 0 . That's what you did, right? > > Since that is what the notation means, yes. > > Do not confuse the iterated limits > > A: lim lim f(x,y) > x->oo y->oo > > (first vary y , then vary x) > > With the double limit > > B: lim f(x,y) > (x,y)->(oo,oo) > > (vary x and y in either order or both together) > > The two are related, (e.g. if the limit in B exists then > the limits in A commute), however even if the limit in B does > not exist the limits > > lim f(x,y) and lim f(x,y) > x->oo y->oo > > are not wrong. In a sense that they are not wrong in _common_ mathematics. But I think you can know quite well, meanwhile, what my objective is: the _meaning_ behind all this, in a physical sense - (or if you prefer: _geometrical_ sense; the latter is advantageous in that you stay within mathematics). I can see quite well how - what's the name of it - the following limit works out in this sense: B: lim f(x,y) (x,y)->(oo,oo) I can imagine that one devises a path, and travelling along that path leads to infinity. It may happen that the limit is independent of the path and then we have this limit. As e.g with: f(x,y) = 1/(x^2 + y^2) But I still find the meaning of the following limit at least confusing: A: lim lim f(x,y) x->oo y->oo x.y When applied to e.g. --------- = sin(2.phi)/2 with x = r.cos(phi) x^2 + y^2 y = r.sin(phi) Then with limit A we first go to infinity along a line parallel to the y-axis, giving 0 because the angle phi approaches Pi/2 . Right ? But then we have to shift that parallel line towards infinity and _that_ stumps me, because now I can't see anymore what's happening. (This is merely a consequence of _both_ my brain halves working. I can understand it with the formal half but not with the visual half). Consequently, I still find that the limit A is indeed wrong. Flame shield ... well, think I can take it. Do we have to live with counter intuition or can we refuse and let the other brain half speak ? Han de Bruijn === Subject: Re: -- Wrong limits do not commute > > > >Consider the trivial example: it should be obvious that > lim_{x -> +infty} lim_{y -> +infty} x/y = 0 >yet the function x/y is not even bounded on any >set of the form { (x,y) : x > M, y > N }. >>So you find my definition: not right. Instead, what you do is first take >>the (single) limit lim_{y -> +infty} x/y (keeping x fixed) resulting >>in 0 and then you take the (single) limit lim_{x -> +infty} 0 giving >>(of course) 0 . That's what you did, right? > > Since that is what the notation means, yes. > > Do not confuse the iterated limits > > A: lim lim f(x,y) > x->oo y->oo > > (first vary y , then vary x) > > With the double limit > > B: lim f(x,y) > (x,y)->(oo,oo) > > (vary x and y in either order or both together) > > The two are related, (e.g. if the limit in B exists then > the limits in A commute), however even if the limit in B does > not exist the limits > > lim f(x,y) and lim f(x,y) > x->oo y->oo > > are not wrong. > > In a sense that they are not wrong in _common_ mathematics. But I think > you can know quite well, meanwhile, what my objective is: the _meaning_ > behind all this, in a physical sense - (or if you prefer: _geometrical_ > sense; the latter is advantageous in that you stay within mathematics). > > I can see quite well how - what's the name of it - the following limit > works out in this sense: > > B: lim f(x,y) > (x,y)->(oo,oo) > > I can imagine that one devises a path, and travelling along that path > leads to infinity. It may happen that the limit is independent of the > path and then we have this limit. As e.g with: f(x,y) = 1/(x^2 + y^2) > > But I still find the meaning of the following limit at least confusing: > > A: lim lim f(x,y) > x->oo y->oo > x.y > When applied to e.g. --------- = sin(2.phi)/2 with x = r.cos(phi) > x^2 + y^2 y = r.sin(phi) > > Then with limit A we first go to infinity along a line parallel to the > y-axis, giving 0 because the angle phi approaches Pi/2 . Right ? > > But then we have to shift that parallel line towards infinity and _that_ > stumps me, because now I can't see anymore what's happening. (This is > merely a consequence of _both_ my brain halves working. I can understand > it with the formal half but not with the visual half). Consequently, > I still find that the limit A is indeed wrong. Flame shield ... well, > think I can take it. Do we have to live with counter intuition or can > we refuse and let the other brain half speak ? > > Han de Bruijn Note that, in mathematics, lim_{(x,y)->(oo,oo)} x*y/(x^2 + y^2) does not exist, since, for example, along the path y = m*x for m > 0 the limit would be m/(1+m^2), which depends on m. However both iterated limits lim_{x->oo} lim_{y->oo} x*y/(x^2 + y^2) and lim_{y->oo} lim_{x->oo} x*y/(x^2 + y^2) exist and are zero. If physicists chose to restrict their attentions to functions for which this sort of behavior does not occur, that does not mean that there are no such functions. === Subject: Re: -- Wrong limits do not commute <87myiilf5a.fsf@phiwumbda.org> <60c3$48c68189$82a1e228$19985@news1.tudelft.nl> posting-account=suWj4AkAAADE1IvGmj55Nmq3f98qb17e SIMBAR Enabled; SIMBAR={70306B22-CB8C-4d52-BFF4-18424E217075}; MathPlayer 2.10b; FunWebProducts; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > But I still find the meaning of the following limit at least confusing: Ê ÊA: lim Ê Êlim Ê f(x,y) > Ê Ê Ê x->oo Êy->oo > Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê x.y > When applied to e.g. Ê--------- = sin(2.phi)/2 Êwith Êx = r.cos(phi) > Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Êx^2 + y^2 Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê y = r.sin(phi) Then with limit A we first go to infinity along a line parallel to the > y-axis, giving Ê0 Êbecause the angle Êphi Êapproaches ÊPi/2 . Right ? But then we have to shift that parallel line towards infinity and that > stumps me, because now I can't see anymore what's happening. (This is > merely a consequence of both my brain halves working. I can understand > it with the formal half but not with the visual half). Consequently, > I still find that the limit A is indeed wrong. Flame shield ... well, > think I can take it. Do we have to live with counter intuition or can > we refuse and let the other brain half speak ? Han de Bruijn- ************************************************************ So you can't see anymore what's happening, and instead of putting the blame on your lack of mathematical education and/or on your own stupidity, you'd rather put the blame on mathematics....aha. Some rather comfortable way to put the blame on others rather that in you. And Han: your several brain halves (how many do we already have there: six, seven halves...?) speaking to you might be a sign for you to search for some professional help: you know, those little voices in your head...tsk,tsk,tsk. Not to mention that those voices seem to be as lost in basic mathematics as you are. Tonio === Subject: Re: -- Wrong limits do not commute <87myiilf5a.fsf@phiwumbda.org> <60c3$48c68189$82a1e228$19985@news1.tudelft.nl> posting-account=IBUqVwoAAADepmzxVr9iEYD5Z0A483SY rv:1.9.0.1) Gecko/2008070206 Firefox/3.0.1,gzip(gfe),gzip(gfe) > But then we have to shift that parallel line towards infinity and _that_ > stumps me, because now I can't see anymore what's happening. (This is > merely a consequence of _both_ my brain halves working. Or of _neither_ of your brain halves working? Victor Meldrew I don't believe it! === Subject: Re: -- Wrong limits do not commute > But then we have to shift that parallel line towards infinity and _that_ > stumps me, because now I can't see anymore what's happening. (This is > merely a consequence of _both_ my brain halves working. I can understand > it with the formal half but not with the visual half). Consequently, > I still find that the limit A is indeed wrong. Flame shield ... well, > think I can take it. Do we have to live with counter intuition or can > we refuse and let the other brain half speak ? I don't know why you have problems with it. Take a function f(x,y). A graph of that function is a surface. If we consider the limit as x -> oo of f(x,y), we get a curve: a function g(y). Now, you can visualize what it means to take lim_{y->oo} g(y), so there you go. -- Jesse F. Hughes I get to make things move just by saying a few things. When I post now the math world has to tremble, even if it does so quietly, hoping that no one else notices. -- James S. Harris has the power. === Subject: Re: -- Wrong limits do not commute <87myiilf5a.fsf@phiwumbda.org> <60c3$48c68189$82a1e228$19985@news1.tudelft.nl> posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I >Consider the trivial example: it should be obvious that > lim {x -> +infty} lim {y -> +infty} x/y = 0 >yet the function x/y is not even bounded on any >set of the form { (x,y) : x > M, y > N }. >>So you find my definition: not right. Instead, what you do is first take >>the (single) limit lim {y -> +infty} x/y (keeping x fixed) resulting >>in 0 and then you take the (single) limit lim {x -> +infty} 0 giving >>(of course) 0 . That's what you did, right? > Since that is what the notation means, yes. > Do not confuse the iterated limits > A: lim lim f(x,y) > x->oo y->oo > (first vary y , then vary x) > With the double limit > B: lim f(x,y) > (x,y)->(oo,oo) > (vary x and y in either order or both together) > The two are related, (e.g. if the limit in B exists then > the limits in A commute), however even if the limit in B does > not exist the limits > lim f(x,y) and lim f(x,y) > x->oo y->oo > are not wrong. In a sense that they are not wrong in common mathematics. But I think > you can know quite well, meanwhile, what my objective is: the meaning > behind all this, in a physical sense Even in a physical sense, meaningful limits may or may not commute. (Hint, check posts that you ignored for an example) So limits that do not commute are not wrong in in either the mathematical or the physical sense. - William Hughes === Subject: Re: -- Wrong limits do not commute > >Consider the trivial example: it should be obvious that > lim_{x -> +infty} lim_{y -> +infty} x/y = 0 >yet the function x/y is not even bounded on any >set of the form { (x,y) : x > M, y > N }. >>So you find my definition: not right. Instead, what you do is first take >>the (single) limit lim_{y -> +infty} x/y (keeping x fixed) resulting >>in 0 and then you take the (single) limit lim_{x -> +infty} 0 giving >>(of course) 0 . That's what you did, right? >Since that is what the notation means, yes. >Do not confuse the iterated limits >A: lim lim f(x,y) > x->oo y->oo >(first vary y , then vary x) >With the double limit >B: lim f(x,y) > (x,y)->(oo,oo) >(vary x and y in either order or both together) >The two are related, (e.g. if the limit in B exists then >the limits in A commute), however even if the limit in B does >not exist the limits >lim f(x,y) and lim f(x,y) >x->oo y->oo >are not wrong. >>In a sense that they are not wrong in _common_ mathematics. But I think >>you can know quite well, meanwhile, what my objective is: the _meaning_ >>behind all this, in a physical sense > > Even in a physical sense, meaningful limits may or may not commute. > (Hint, check posts that you ignored for an example) > So limits that do not commute are not wrong in > in either the mathematical or the physical sense. I did not ignore it, but rather thought it was a confirmation instead of a refutation. (Suppose you mean the story about transistors and chips) Han de Bruijn === Subject: Re: -- Wrong limits do not commute <87myiilf5a.fsf@phiwumbda.org> <60c3$48c68189$82a1e228$19985@news1.tudelft.nl> <1ef5$48c7d31b$82a1e228$18396@news1.tudelft.nl> posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I On Sep 10, 10:00 am, Han de Bruijn Even in a physical sense, meaningful limits may or may not commute. > (Hint, check posts that you ignored for an example) > So limits that do not commute are not wrong in > in either the mathematical or the physical sense. I did not ignore it, but rather thought it was a confirmation instead of > a refutation. Nope. Two meaningful limits that do not commute. A clear refutation to the idea that limits which do not commute are wrong. - William Hughes === Subject: Re: -- Wrong limits do not commute > > >Synopsis: All this time Han claimed to understand limits better than >the rest of us, it never occurred to them that (1) they could be >nested and (2) limits don't necessarily commute. >Hence (?) the mathematicians are wrong. >>Let's consider again what a (double) limit really is. >> lim lim f(x,y) = F >> x->oo y->oo >>For any real number epsilon > 0 there exists real numbers M , N such >>that if x > L and y > M then | f(x,y) - F | < epsilon . Right ? > > Not really. > > Consider the trivial example: it should be obvious that > > lim_{x -> +infty} lim_{y -> +infty} x/y = 0 > > yet the function x/y is not even bounded on any > set of the form { (x,y) : x > M, y > N }. > > So you find my definition: not right. Instead, what you do is first take > the (single) limit lim_{y -> +infty} x/y (keeping x fixed) resulting > in 0 and then you take the (single) limit lim_{x -> +infty} 0 giving > (of course) 0 . That's what you did, right? > > Han de Bruijn Han is again wrong mathematically , to define lim lim f(x,y) = F x->oo y->oo to mean the same as lim f(x,y) = F (x,y)->(oo,oo) Mathematically, at least, they are different, as the example lim_{x -> +infty} lim_{y -> +infty} x/y = 0 quite properly shows. === Subject: Re: -- Wrong limits do not commute > >>LOL. Look, the word fixed is there to help you; if you find it >>confusing, then just abandon it. For each x in [0,1] you have a >>sequence of numbers (g_n(x)), and for each n in N, you have a function >>g_n on [0,1]. That's all that is being said. > > By the way, did you run screaming out of calculus class when the > professor said fix x and take the derivative with respect to y? Are > partial derivatives also wrong limits? > > Please read my response to . There is no problem at all > when taking only _one_ limit with a function of two variables. IMO there > is only a problem when taking a _double_ limit with such functions. Uhm, > you know about the rule d^2f/(dx.dy) = d^2f/(dy.dx) , for those partial > derivatives, which is always valid in physics, but not in mathematics. > > Han de Bruijn So how come int_[a,b] int_[c,d] f(x,y) dx dy = int_[c,d] int_[c,d] f(x,y) dy dx for any continuous f on [a,b] x [c,d]? Aren't those double limits? Are they wrong limits? === Subject: Re: -- Wrong limits do not commute >>LOL. Look, the word fixed is there to help you; if you find it >>confusing, then just abandon it. For each x in [0,1] you have a >>sequence of numbers (g_n(x)), and for each n in N, you have a function >>g_n on [0,1]. That's all that is being said. > > By the way, did you run screaming out of calculus class when the > professor said fix x and take the derivative with respect to y? Are > partial derivatives also wrong limits? > > Please read my response to . There is no problem at all > when taking only _one_ limit with a function of two variables. IMO there > is only a problem when taking a _double_ limit with such functions. Uhm, > you know about the rule d^2f/(dx.dy) = d^2f/(dy.dx) , for those partial > derivatives, which is always valid in physics, but not in mathematics. > > Han de Bruijn > > So how come > > > int_[a,b] int_[c,d] f(x,y) dx dy > > = int_[c,d] int_[c,d] f(x,y) dy dx > > for any continuous f on [a,b] x [c,d]? Aren't those double limits? > Are they wrong limits? I can think of lots of situations in which int_[a,b] int_[c,d] f(x,y) dx dy = int_[c,d] int_[a,b] f(x,y) dy dx But,unless a = c and b = d, I don't really expect that int_[a,b] int_[c,d] f(x,y) dx dy = int_[c,d] int_[c,d] f(x,y) dy dx very often. === === Subject: Re: Palin's religious craziness posting-account=WtNqtQoAAADXIopd0ISJV1G4rXJHaFyY Gecko/20070914 Firefox/2.0.0.7,gzip(gfe),gzip(gfe) > Experiments on the power of prayer have always showed statistically > significant results. You leftie atheists always deny such data. no, the data shows those prayed for do worse, but that's just a coincidence.:) > Should we do any and > all experiments even if there is a good chance they'll turn the earth > into a cinder? What about if it's only a fair chance, or a minor > chance. What if it's just a very very slim chance. Tell us just where > your ego would draw the line. and you tell us where your ego draws the line on wars for oil or for protecting the almighty us economy from crashing and burning or for religious dominance... > You don't know how lucky you are that > there are beings monitoring what you are yours are up to so that you > can't turn the place into a cinder. what beings are those? > So much for your evolution by > chance being fact. yes, it's not evolution by chance, look it up. We want to > stop violent crime. So the way to do it is to disarm all potential > victims while arming all criminals. Sure, that will work well. NOT. who is proposing arming all criminals? === Subject: Re: Palin's religious craziness Distribution: Global says... > >> Got that right. But who said this was God's fault? > Everything is his fault. He created the bloody mess in the > first place, didn't he? > > No, She didn't. But I agree that She should have kept an eye on that > singularity on the shelf. > > > There is no God. It's that simple. Just like all of you arguing over that Republicunt. === Subject: Re: Palin's religious craziness MPG.233188873574ed5989891@nntp.motzarella.org > says... >> >> Got that right. But who said this was God's fault? > > Everything is his fault. He created the bloody mess in the > first place, didn't he? >> >> No, She didn't. But I agree that She should have kept an eye on that >> singularity on the shelf. >> >> >> > > There is no God. It's that simple. Just like all of you arguing over that > Republicunt. Right. Too bad that Voting America is just ready for her. Prepare for the worst. Dirk Vdm === Subject: Re: Palin's religious craziness posting-account=WtNqtQoAAADXIopd0ISJV1G4rXJHaFyY Gecko/20070914 Firefox/2.0.0.7,gzip(gfe),gzip(gfe) > To a leftist, Atheism is the STATE religion. There is NO > separation of church and state! Atheism is forced as the state > religion just as in the USSR. Nothing else is permitted! really? you are not allowed to go to your church and pray in the us? > And that > means that we all have to believe that all of life started and > developed by CHANCE. do you think the theory of evolution is based on chance? > It's all survival of the fittest. Thus, to an > atheist, morality or ethics doesn't exist. survival of the fittest certainly is the way of nature. do you think morality and ethics are tied to nature? > If you get away with it, > there is no crime! > The 10 commandments are fairytales for suckers! If > you want something you STEAL it. If someone gets in your way, you KILL > them. If you can fool someone, you LIE to them. There is no punishment > for these acts! It is the RESULTS that count. Survival of the > fittest! THAT is the 'religion of the left! that all sounds like bush & the repubs to me...you're getting a little worked up here, don't you think? > The amount of > ridicule and lies being heaped by the Left on Palin is virtually > without precedent! why virtually? === Subject: I need a solutions manual posting-account=-3B5sQoAAAA1XGcT_mQY5BE0LuPAtasr rv:1.9.0.1) Gecko/2008070206 Firefox/3.0.1,gzip(gfe),gzip(gfe) I am looking for a solutions manual for Elements of Chemical Reaction Engineering (4th edition) by H. Scott Fogler. === Subject: LHC a touchstone for mathematics? In Switzerland right now (today) a most expensive experiment has begun which is designed to check purely mathematical constructs. Several physicist doubt that the hypothetical Higgs bosons will be found which would confirm the standard model of quantum theory. You might argue that mathematics is independent of physics. Heaviside uttered that mathematics is an empirical science too. Maybe, set theory is not appropriate in some cases. See the thread Very basic mistakes 03.09.08. Salviati: ... in ultima conclusione, gli attributi di eguale maggiore e minore non aver luogo ne gl'infiniti, ma solo nelle quantit.88 terminate. IR>|>IR+>|>IR === Subject: LHC no touchstone for mathematics posting-account=IBUqVwoAAADepmzxVr9iEYD5Z0A483SY rv:1.9.0.1) Gecko/2008070206 Firefox/3.0.1,gzip(gfe),gzip(gfe) > In Switzerland right now (today) a most expensive experiment Very expensive. > has begun which > is designed to check purely mathematical constructs. Will it solve the Riemann hypothesis? Will it refute Wiles's proof of FLT? Will it help mathematics in any way? > Heaviside > uttered that mathematics is an empirical science too. If Heaviside said that, it must be right, mustn't it ?! Victor Meldrew I don't believe it! === Subject: path of maximum length I am currently faced with the following problem: Given are N s-dimensional points x_1,...,x_N. Find a subset of k points for which the length of the path along these k points is as large as possible. Somehow, this sounds to me like a well-known problem... If somebody could point me to relevant literature or algorithms on how to solve this problem, this would be very helpful. Bart -- === Subject: Re: path of maximum length ... > Given are N s-dimensional points x_1,...,x_N. Find a subset of k points > for which the length of the path along these k points is as large as > possible. Your phrase length of the path along these k points seems unclear. Suppose that by path you mean an ordered list of k distinct points, and path length is the sum of the k-1 point-to-point successive distances. If k is less than N, one can make a longer path by adding another point to the path; mentioning subsets of points is superfluous. > Somehow, this sounds to me like a well-known problem... If somebody > could point me to relevant literature or algorithms on how to solve this > problem, this would be very helpful. For two points x, y in the set of points S, let Dxy = distance(x,y). Let M = max{x,y in S} Dxy, and define Gxy = M-Dxy. Let closed path P' solve the traveling-salesman problem over points S with distance function G. (A closed path being a path with an implied closing link from last point to first.) Break P' into two parts P0, P1 by removing its longest Gxy link, and form path Q by appending P0 to P1, of course treating edge cases appropriately. Then Q is a shortest path on S with distance function G, and a longest path on S with distance function D. -- jiw === Subject: Is change of coordinates an open mapping? posting-account=ZUczkQoAAABCznvSjYJPkJbPEzZyPyBU rv:1.8.1.16) Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) Let (x,z) in R^{n+m}, where x is n dimensional and z is m dimensional and R is the reals. I have a continuously differentiable function F(x,z) that maps (x,z) into R^m (as a side note, in my setting, I know something much stronger, which is that F also happens to be such that each component function F^i is real analytic, which I believe implies that F is a C^infinity function). Now I want to consider the change of coordinate (x,z) mapsto (x,F(x,z)). Let us denote this change of coordinate mapping as T. What I am trying to understand is if I can put conditions on F(x,z) to ensure that T is an open mapping, i.e., T maps open sets to open sets. My conjecture was that if for every x in R^n, the derivative matrix of F(x,z) with respect z has full rank m for every z in R^m, then the result goes through. I was wondering if anyone could confirm whether this is correct and offer any guidance on a proof. === Subject: Re: Is change of coordinates an open mapping? > Let (x,z) in R^{n+m}, where x is n dimensional and z is m dimensional > and R is the reals. I have a continuously differentiable function > F(x,z) that maps (x,z) into R^m (as a side note, in my setting, I know > something much stronger, which is that F also happens to be such that > each component function F^i is real analytic, which I believe implies > that F is a C^infinity function). each F^i analytic => each F^i C^oo <=> F C^oo > Now I want to consider the change of coordinate (x,z) mapsto > (x,F(x,z)). Let us denote this change of coordinate mapping as T. What > I am trying to understand is if I can put conditions on F(x,z) to > ensure that T is an open mapping, i.e., T maps open sets to open > sets. > > My conjecture was that if for every x in R^n, the derivative matrix > of F(x,z) with respect z has full rank m for every z in R^m, then the > result goes through. I was wondering if anyone could confirm whether > this is correct and offer any guidance on a proof. This follows from open mapping theorem. Jose Carlos Santos === Subject: Re: Is change of coordinates an open mapping? > Let (x,z) in R^{n+m}, where x is n dimensional and z is m dimensional > and R is the reals. I have a continuously differentiable function > F(x,z) that maps (x,z) into R^m (as a side note, in my setting, I know > something much stronger, which is that F also happens to be such that > each component function F^i is real analytic, which I believe implies > that F is a C^infinity function). >> each F^i analytic => each F^i C^oo <=> F C^oo > Now I want to consider the change of coordinate (x,z) mapsto > (x,F(x,z)). Let us denote this change of coordinate mapping as T. What > I am trying to understand is if I can put conditions on F(x,z) to > ensure that T is an open mapping, i.e., T maps open sets to open > sets. >> My conjecture was that if for every x in R^n, the derivative matrix > of F(x,z) with respect z has full rank m for every z in R^m, then the > result goes through. I was wondering if anyone could confirm whether > this is correct and offer any guidance on a proof. >> This follows from open mapping theorem. > > Is that by a more direct argument than I gave, such as that the > matrix of partial derivatives of T has the form: > > / I 0 > * A / > > where I is the n x n identity matrix, and A is the nonsingular > m x m matrix of partial derivatives of F(x,z) with respect to z? Yes, together with the fact that if a matrix has maximal rank and more columns than rows, then the square submatrix obtained from it deleting the first columns is nonsingular. (Or, for that matter, *any* square submatrix obtained from it deleting some columns.) Jose Carlos Santos === Subject: Solutions manual to Engineering Mechanics, dynamics 6th by J. L. Meriam, L. G. Kraige, posting-account=y7Z6OAoAAAD9FX_IL8yyi-5ioDwYBttu CLR 2.0.50727),gzip(gfe),gzip(gfe) solutions manual (To search click in keyboard Ctrl+F) Solutions Manuals in Electronic (PDF)Format! Just contact with , solutionpayfee (at) hotmail.com (my email address), these are parts of our solutions, if the solution you want is on the list, please email to me. NOTICE: if the solutions manual that in my list ,please note it in your email . 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Martin(chapter 1 to chapter15) solution manual for Probability and Statistical Inference ( 7th edition by Hogg & Tanis) solution manual for Fundamentals of Communication Systems by John G. Proakis ,Masoud Salehi === Subject: Re: Group homomorphism > > does a surjective group homomorphism of the symmetric groups S_5 -- > S_4 exist? I know such a homomorphism for S_4 --> S_3. Does the > existence have perhaps something to do with resolvability? > You mean, whether the simplicity of A_5 plays a role? > Yes: consider the kernel K of such a surjective group homomorphism. > Its intersection with A_5 is normal in A_5, and therefore either > K includes A_5 Ê or Ê ÊK intersects A_5 trivially. > Both cases are impossible because of |K| = 5. > Marc > > Why is the case K intersects A_5 trivially in contradiction to |K| > = 5? > > Check the cardinality of the subgroup H = < K , A_5 > of S_5 > generated by K and A_5: > because both H and A_5 are normal subgroups, we have H = KA_5. ______________^^^__ Sorry, this should be K . > But this forces |H| |K intersect A_5| = |K||A_5| = 5*60 > which in turn rules out the possibility of |K intersect A_5| = 1. === Subject: Re: The Computable Reals (beta) posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > The Computable Reals (beta) > --------------------------- > Julio Di Egidio (LV) > julio at diegidio dot name > (c)2008 LV on behalf of sci.math, sci.logic, comp.theory > All right reserved. I have published a revised and working code here: http://julio.diegidio.name/Sandbox/ComputableReals.beta.htm It is fairly undocumented and very essential, but should be very clear > to those who have followed the discussion and know what to look for. At the moment, it shows first a bunch of random strings with their > corresponding index. Then it goes through few iterations of the > inductive construction for the list of binary expansions, using the > very list entries to build sample strings whose index is given aside. > Hit reload to generate another series of the random samples. The > source code is available, just do a view-source: all should be there, > but please feel free to ask for clarifications if needed. I have extended the above page a little bit: now it also shows the sequence of indexes for two uncomputable numbers, one of them being the Champernowne constant in binary, and the other being another example that had come out in this thread. All such sequences provably converge over the implied metric space (hope this is the correct term). http://julio.diegidio.name/Sandbox/ComputableReals.beta.htm -LV === Subject: Re: The Computable Reals (beta) > The Computable Reals (beta) > --------------------------- > Julio Di Egidio (LV) > julio at diegidio dot name > (c)2008 LV on behalf of sci.math, sci.logic, comp.theory > All right reserved. > I have published a revised and working code here: > http://julio.diegidio.name/Sandbox/ComputableReals.beta.htm > It is fairly undocumented and very essential, but should be very clear > to those who have followed the discussion and know what to look for. > At the moment, it shows first a bunch of random strings with their > corresponding index. Then it goes through few iterations of the > inductive construction for the list of binary expansions, using the > very list entries to build sample strings whose index is given aside. > Hit reload to generate another series of the random samples. The > source code is available, just do a view-source: all should be there, > but please feel free to ask for clarifications if needed. > > I have extended the above page a little bit: now it also shows the > sequence of indexes for two uncomputable numbers, one of them being > the Champernowne constant in binary, and the other being another > example that had come out in this thread. All such sequences provably > converge over the implied metric space (hope this is the correct > term). > > http://julio.diegidio.name/Sandbox/ComputableReals.beta.htm > > -LV What julio seems to be carefully ignoring is that a set having a sequence of numbers converging to some value does not mean that the set contains that value. The ennumerable set of rationals contains sequences converging to every real, but most reals are not rational. So, for example, a sequence of rationals converging to the Champernowne number does not itself contain the Champernowne number. === Subject: Re: The Computable Reals (beta) > I have published a revised and working code here: >http://julio.diegidio.name/Sandbox/ComputableReals.beta.htm > It is fairly undocumented and very essential, but should be very clear > to those who have followed the discussion and know what to look for. > At the moment, it shows first a bunch of random strings with their > corresponding index. Then it goes through few iterations of the > inductive construction for the list of binary expansions, using the > very list entries to build sample strings whose index is given aside. > Hit reload to generate another series of the random samples. The > source code is available, just do a view-source: all should be there, > but please feel free to ask for clarifications if needed. > I have extended the above page a little bit: now it also shows the > sequence of indexes for two uncomputable numbers, one of them being > the Champernowne constant in binary, and the other being another > example that had come out in this thread. All such sequences provably > converge over the implied metric space (hope this is the correct > term). >http://julio.diegidio.name/Sandbox/ComputableReals.beta.htm What julio seems to be carefully ignoring is that a set having a > sequence of numbers converging to some value does not mean that the set > contains that value. I didn't say that. It's you who have troubles getting the relevance of it. http://en.wikipedia.org/wiki/Computable number the computable numbers, also known as the recursive numbers or the computable reals, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm And that's what I have. -LV === Subject: Re: The Computable Reals (beta) > The Computable Reals (beta) > --------------------------- > Julio Di Egidio (LV) > julio at diegidio dot name > (c)2008 LV on behalf of sci.math, sci.logic, comp.theory > All right reserved. > I have published a revised and working code here: >http://julio.diegidio.name/Sandbox/ComputableReals.beta.htm > It is fairly undocumented and very essential, but should be very clear > to those who have followed the discussion and know what to look for. > At the moment, it shows first a bunch of random strings with their > corresponding index. Then it goes through few iterations of the > inductive construction for the list of binary expansions, using the > very list entries to build sample strings whose index is given aside. > Hit reload to generate another series of the random samples. The > source code is available, just do a view-source: all should be there, > but please feel free to ask for clarifications if needed. > I have extended the above page a little bit: now it also shows the > sequence of indexes for two uncomputable numbers, one of them being > the Champernowne constant in binary, and the other being another > example that had come out in this thread. All such sequences provably > converge over the implied metric space (hope this is the correct > term). >http://julio.diegidio.name/Sandbox/ComputableReals.beta.htm > What julio seems to be carefully ignoring is that a set having a > sequence of numbers converging to some value does not mean that the set > contains that value. > > I didn't say that. It's you who have troubles getting the relevance of > it. > > http://en.wikipedia.org/wiki/Computable_number > the computable numbers, also known as the recursive numbers or the > computable reals, are the real numbers that can be computed to within > any desired precision by a finite, terminating algorithm > > And that's what I have. > > -LV You do not have anything like all of them yet, and even if you did have all of them that you can ennumerate, your ennumeration itself would provide the basis for computing a number which you have omitted. === Subject: Re: The Computable Reals (beta) posting-account=Y877ggoAAAB-75kOgSLDnLSoS2_jn90U Gecko/20071126 Fedora/1.5.0.12-7.fc6 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) > The Computable Reals (beta) > --------------------------- > Julio Di Egidio (LV) > julio at diegidio dot name > (c)2008 LV on behalf of sci.math, sci.logic, comp.theory > All right reserved. > I have published a revised and working code here: >http://julio.diegidio.name/Sandbox/ComputableReals.beta.htm > It is fairly undocumented and very essential, but should be very clear > to those who have followed the discussion and know what to look for. > At the moment, it shows first a bunch of random strings with their > corresponding index. Then it goes through few iterations of the > inductive construction for the list of binary expansions, using the > very list entries to build sample strings whose index is given aside. > Hit reload to generate another series of the random samples. The > source code is available, just do a view-source: all should be there, > but please feel free to ask for clarifications if needed. > I have extended the above page a little bit: now it also shows the > sequence of indexes for two uncomputable numbers, one of them being > the Champernowne constant in binary, and the other being another > example that had come out in this thread. All such sequences provably > converge over the implied metric space (hope this is the correct > term). >http://julio.diegidio.name/Sandbox/ComputableReals.beta.htm > What julio seems to be carefully ignoring is that a set having a > sequence of numbers converging to some value does not mean that the set > contains that value. I didn't say that. It's you who have troubles getting the relevance of > it. http://en.wikipedia.org/wiki/Computable_number > the computable numbers, also known as the recursive numbers or the > computable reals, are the real numbers that can be computed to within > any desired precision by a finite, terminating algorithm And that's what I have. -LV That you can approximate a real arbitrarily well, whether you call it computable or not, does not mean that it is in the enumerable set. I commented on this in an earlier post, haven't you been reading the replies? > [...] > Let me clarify something, albeit I'm not sure if this is necessary. > That a computable real is in a list does not mean that there are > arbitrarily close numbers to that real, the real itself has to be in > the list. That a real is a computable real means that there is a > finite model able to approximate it. Simply because you deal only with > computable reals does not mean that you can, in an analogous sense, > reinterpret statements concerning reals. Reformulations are however > welcome. So are you claiming that your enumeration contains every real, or that it computes every real? === Subject: Re: The Computable Reals (beta) posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > The Computable Reals (beta) > --------------------------- > Julio Di Egidio (LV) > julio at diegidio dot name > (c)2008 LV on behalf of sci.math, sci.logic, comp.theory > All right reserved. > I have published a revised and working code here: >http://julio.diegidio.name/Sandbox/ComputableReals.beta.htm > It is fairly undocumented and very essential, but should be very clear > to those who have followed the discussion and know what to look for. > At the moment, it shows first a bunch of random strings with their > corresponding index. Then it goes through few iterations of the > inductive construction for the list of binary expansions, using the > very list entries to build sample strings whose index is given aside. > Hit reload to generate another series of the random samples. The > source code is available, just do a view-source: all should be there, > but please feel free to ask for clarifications if needed. > I have extended the above page a little bit: now it also shows the > sequence of indexes for two uncomputable numbers, one of them being > the Champernowne constant in binary, and the other being another > example that had come out in this thread. All such sequences provably > converge over the implied metric space (hope this is the correct > term). >http://julio.diegidio.name/Sandbox/ComputableReals.beta.htm > What julio seems to be carefully ignoring is that a set having a > sequence of numbers converging to some value does not mean that the set > contains that value. > I didn't say that. It's you who have troubles getting the relevance of > it. >http://en.wikipedia.org/wiki/Computable number > the computable numbers, also known as the recursive numbers or the > computable reals, are the real numbers that can be computed to within > any desired precision by a finite, terminating algorithm > And that's what I have. Ê That you can approximate a real arbitrarily well, whether you call > it computable or not, does not mean that it is in the enumerable set. I do have the list of all the binary sequences over finite induction. That is enough to (inductively) define the computable reals. The rest is the usual noise. -LV === Subject: Re: The Computable Reals (beta) > The Computable Reals (beta) > I have published a revised and working code here: >http://julio.diegidio.name/Sandbox/ComputableReals.beta.htm > It is fairly undocumented and very essential, but should be very > clear > to those who have followed the discussion and know what to look > for. > At the moment, it shows first a bunch of random strings with their > corresponding index. Then it goes through few iterations of the > inductive construction for the list of binary expansions, using the > very list entries to build sample strings whose index is given > aside. > Hit reload to generate another series of the random samples. The > source code is available, just do a view-source: all should be > there, > but please feel free to ask for clarifications if needed. > I have extended the above page a little bit: now it also shows the > sequence of indexes for two uncomputable numbers, one of them being > the Champernowne constant in binary, and the other being another > example that had come out in this thread. All such sequences provably > converge over the implied metric space (hope this is the correct > term). >http://julio.diegidio.name/Sandbox/ComputableReals.beta.htm > What julio seems to be carefully ignoring is that a set having a > sequence of numbers converging to some value does not mean that the set > contains that value. > I didn't say that. It's you who have troubles getting the relevance of > it. >http://en.wikipedia.org/wiki/Computable_number > the computable numbers, also known as the recursive numbers or the > computable reals, are the real numbers that can be computed to within > any desired precision by a finite, terminating algorithm > And that's what I have. > Ê That you can approximate a real arbitrarily well, whether you call > it computable or not, does not mean that it is in the enumerable set. > > I do have the list of all the binary sequences over finite induction. > That is enough to (inductively) define the computable reals. It is not enough to LIST the computable reals, so is irrelevant to properties of such listings. > The rest > is the usual noise. The chief noisemaker here is julio himself. Much ado about his nothings. === Subject: Re: The Computable Reals (beta) posting-account=Y877ggoAAAB-75kOgSLDnLSoS2_jn90U Gecko/20071126 Fedora/1.5.0.12-7.fc6 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) > The Computable Reals (beta) > --------------------------- > Julio Di Egidio (LV) > julio at diegidio dot name > (c)2008 LV on behalf of sci.math, sci.logic, comp.theory > All right reserved. > I have published a revised and working code here: >http://julio.diegidio.name/Sandbox/ComputableReals.beta.htm > It is fairly undocumented and very essential, but should be very clear > to those who have followed the discussion and know what to look for. > At the moment, it shows first a bunch of random strings with their > corresponding index. Then it goes through few iterations of the > inductive construction for the list of binary expansions, using the > very list entries to build sample strings whose index is given aside. > Hit reload to generate another series of the random samples. The > source code is available, just do a view-source: all should be there, > but please feel free to ask for clarifications if needed. > I have extended the above page a little bit: now it also shows the > sequence of indexes for two uncomputable numbers, one of them being > the Champernowne constant in binary, and the other being another > example that had come out in this thread. All such sequences provably > converge over the implied metric space (hope this is the correct > term). >http://julio.diegidio.name/Sandbox/ComputableReals.beta.htm > What julio seems to be carefully ignoring is that a set having a > sequence of numbers converging to some value does not mean that the set > contains that value. > I didn't say that. It's you who have troubles getting the relevance of > it. >http://en.wikipedia.org/wiki/Computable_number > the computable numbers, also known as the recursive numbers or the > computable reals, are the real numbers that can be computed to within > any desired precision by a finite, terminating algorithm > And that's what I have. > That you can approximate a real arbitrarily well, whether you call > it computable or not, does not mean that it is in the enumerable set. I do have the list of all the binary sequences over finite induction. > That is enough to (inductively) define the computable reals. The rest > is the usual noise. -LV Does this entail that you also claim that your enumerable set contains every infinite binary sequence, or just the finite ones? === Subject: Re: The Computable Reals (beta) > I have extended the above page a little bit: now it also shows the > sequence of indexes for two uncomputable numbers, one of them being > the Champernowne constant in binary, and the other being another > example that had come out in this thread. All such sequences provably > converge over the implied metric space (hope this is the correct > term). However, the Chapernowne constant *is* a computable number, on any normal account of what computable numbers are; we can write a program that will for a given n, return the first n digits of the number. > -LV -- Alan Smaill === Subject: Re: The Computable Reals (beta) posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > I have extended the above page a little bit: now it also shows the > sequence of indexes for two uncomputable numbers, one of them being > the Champernowne constant in binary, and the other being another > example that had come out in this thread. All such sequences provably > converge over the implied metric space (hope this is the correct > term). However, the Chapernowne constant *is* a computable number, > on any normal account of what computable numbers are; > we can write a program that will for a given n, return the > first n digits of the number. Right, sorry for the confusion: those are two irrationals, the Chapernowne constant being transcendental (the other one I wouldn't know). -LV === Subject: Will rocket reach on target Can some one help me to solve this quistion. Let p(a)=0.6 be the probability that the rocket will destroy during its flight. And p(b)0.02 be the probability that the engine will not start.What will the probability that the rocket will reach into space successfuly. I've an idea to solve it. P'(a) intersection P(b) + P(a) intersection P'(b) and P'(a) intersection P'(b)?? === Subject: Re: Will rocket reach on target 23805696.1221058603043.JavaMail.jakarta@nitrogen.mathforum.org > Can some one help me to solve this quistion. > Let p(a)=0.6 be the probability that the rocket will destroy during its flight. > And p(b)=0.02 be the probability that the engine will not start. > What will the probability that the rocket will reach into space successfuly. > I've an idea to solve it. > P'(a) intersection P(b) + > P(a) intersection P'(b) and > P'(a) intersection P'(b)?? Your notation is a bit messy here. Intersection is an operator for sets. Probabilities are numbers, operated on with addition and multiplication. The rocket must start AND it must not destroy. What is the probability that it starts? What is the probability that it does not destroy once it started? How do you combine an AND of two possibilities? Dirk Vdm === Subject: Re: Will rocket reach on target The probability that it start is 1-p(b). The probability that the rocket it doesnt destroy once it is started is started p(b'|a')= [P'(b) intersection P'(a)]/P'(a) i hurrily typed and instead of + oops. Now again check the statement. === Subject: Re: Will rocket reach on target 25974868.1221066483505.JavaMail.jakarta@nitrogen.mathforum.org > The probability that it start is 1-p(b). > The probability that the rocket it doesnt destroy once it is started is started p(b'|a')= > [P'(b) intersection P'(a)]/P'(a) As I said, intersection is an operator for sets, not for numbers. Dirk Vdm > i hurrily typed and instead of + oops. > Now again check the statement. === Subject: Re: Will rocket reach on target posting-account=eFTX7goAAACJBRoHHOnDC9IuTZeZT1_H 1.1.4322),gzip(gfe),gzip(gfe) > Can some one help me to solve this quistion. > Let p(a)=0.6 be the probability that the rocket will destroy during its flight. And p(b)0.02 be the probability that the engine will not start.What will the probability that the rocket will reach into space successfuly. > I've an idea to solve it. > P'(a) intersection P(b) + > P(a) intersection P'(b) and > P'(a) intersection P'(b)?? Probablity the engine starts = (1.-.02) Probability the flight is successful = (1.-.6) (1.-.02)*(1.-.6)= .392 === Subject: Re: Will rocket reach on target > Can some one help me to solve this quistion. Let p(a)=0.6 be the > probability that the rocket will destroy during its flight. And > p(b)0.02 be the probability that the engine will not start.What will > the probability that the rocket will reach into space successfuly. > I've an idea to solve it. P'(a) intersection P(b) + P(a) intersection > P'(b) and P'(a) intersection P'(b)?? P(not (a or b)) = 1 - P(a or b) = 1 - [ P(a) + P(b) - P(a and b) ]. Since I am assuming that it can't be the case that the engine doesn't start and the rocket is destroyed, P(a and b) = 0. Therefore, the answer is: 1 - P(a) - P(b) = 1 - 0.6 - 0.02 = 1 - 0.62 = 0.38. === Subject: Re: Will rocket reach on target > Can some one help me to solve this quistion. Let p(a)=0.6 be the > probability that the rocket will destroy during its flight. And > p(b)0.02 be the probability that the engine will not start.What will > the probability that the rocket will reach into space successfuly. > I've an idea to solve it. P'(a) intersection P(b) + P(a) intersection > P'(b) and P'(a) intersection P'(b)?? P(not (a or b)) = 1 - P(a or b) = 1 - [ P(a) + P(b) - P(a and b) ]. Since I am assuming that it can't be the case that the engine doesn't > start and the rocket is destroyed, P(a and b) = 0. Therefore, the answer is: 1 - P(a) - P(b) = 1 - 0.6 - 0.02 = 1 - 0.62 = 0.38. Hmmm...take an extreme to see if your calculation works. Suppose the probability of the rocket engine not starting was .7. Then your calculation would give 1 -.6-.7 =-.3 which doesn't make sense. But (1.-.7)(1.-.6) = .12 would. === Subject: Re: Will rocket reach on target Ok let move ahead a bit more. What this statement shows. P(a) int. P(b) + P'(a) int. P(b) + P(a) int. P'(b) + P'(a) int. P'(b). It must be equal to one. Now move further. P(a) int. P'(b). I think it is impossible. Dont u think? What the remaining part tell us. Now i undestand that my required answeq is P'(a) int. P'(b). Right? === Subject: Re: Will rocket reach on target reply-type=response ga8o0c$95i$1@aioe.org >> Can some one help me to solve this quistion. Let p(a)=0.6 be the >> probability that the rocket will destroy during its flight. And >> p(b)0.02 be the probability that the engine will not start.What will >> the probability that the rocket will reach into space successfuly. >> I've an idea to solve it. P'(a) intersection P(b) + P(a) intersection >> P'(b) and P'(a) intersection P'(b)?? > > > P(not (a or b)) = 1 - P(a or b) = 1 - [ P(a) + P(b) - P(a and b) ]. > > Since I am assuming that it can't be the case that the engine doesn't > start and the rocket is destroyed, P(a and b) = 0. Therefore, the answer is: > > 1 - P(a) - P(b) = 1 - 0.6 - 0.02 = 1 - 0.62 = 0.38. What if P(a) = 0.7 and P(b) = 0.8 ? ;-) Dirk Vdm === Subject: Re: Will rocket reach on target > Since I am assuming that it can't be the case that the engine > doesn't start and the rocket is destroyed, P(a and b) = > 0. Therefore, the answer is: > 1 - P(a) - P(b) = 1 - 0.6 - 0.02 = 1 - 0.62 = 0.38. > > What if P(a) = 0.7 and P(b) = 0.8 ? ;-) Then there is a rather large positive probability that a rocket is destroyed in space while it still stands on earth. === Subject: Re: Will rocket reach on target qoty720f1qt.fsf@ruuvi.it.helsinki.fi > Since I am assuming that it can't be the case that the engine > doesn't start and the rocket is destroyed, P(a and b) = > 0. Therefore, the answer is: > 1 - P(a) - P(b) = 1 - 0.6 - 0.02 = 1 - 0.62 = 0.38. >> >> What if P(a) = 0.7 and P(b) = 0.8 ? ;-) > > Then there is a rather large positive probability that a rocket is > destroyed in space while it still stands on earth. If you are joking, then :-) Otherwise, quoting the OP's words (!) and applying to my values: p(b) = 0.8 is the probability that the engine will not start, which means that the rocket will probably remain on the ground. p(a) = 0.7 is the probability that the rocket will destroy during its flight, which means it's not a good rocket and that, once it gets started, it will probably be destroyed. Keywords: during its flight. Dirk Vdm === Subject: Re: Will rocket reach on target >> Since I am assuming that it can't be the case that the engine >> doesn't start and the rocket is destroyed, P(a and b) = 0. >> Therefore, the answer is: >> 1 - P(a) - P(b) = 1 - 0.6 - 0.02 = 1 - 0.62 = 0.38. > What if P(a) = 0.7 and P(b) = 0.8 ? ;-) >> Then there is a rather large positive probability that a rocket is >> destroyed in space while it still stands on earth. > > If you are joking, then :-) I thought you were joking, and that I just interpreted your joke. > Otherwise, quoting the OP's words (!) and applying to my values: > > p(b) = 0.8 is the probability that the engine will not start, > which means that the rocket will probably remain on the ground. > > p(a) = 0.7 is the probability that the rocket will destroy during > its flight, which means it's not a good rocket and that, once it > gets started, it will probably be destroyed. Keywords: during its > flight. I admit I'm still interpreting these as referring to one attempt with the rocket, as I did with the bullet in the earlier thread, and I admit the OP did already tell me (*) that I do not understand the statement, so maybe your probability assignment makes some kind of sense to him. (* I think he meant me. He actually followed up to his own message without quoting anything.) I would say the probability of the destruction of the rocket _given it got started_ is P(a|not b), and the probability of a succesful launch _and_ destruction during the flight is P(a and not b). The latter is just P(a)P(not b|a) = P(a) because I think destruction during the flight _implies_ a succesful launch, so that P(not b|a) = 1. It makes no sense, to me, to say that a rocket is destroyed during a flight that never started, so I think P(a) + P(b) <= 1. But this is rocket science, so maybe I just misread it all. === Subject: Re: Will rocket reach on target qotsks8eyu7.fsf@ruuvi.it.helsinki.fi > Since I am assuming that it can't be the case that the engine > doesn't start and the rocket is destroyed, P(a and b) = 0. > Therefore, the answer is: > 1 - P(a) - P(b) = 1 - 0.6 - 0.02 = 1 - 0.62 = 0.38. >> What if P(a) = 0.7 and P(b) = 0.8 ? ;-) > Then there is a rather large positive probability that a rocket is > destroyed in space while it still stands on earth. >> >> If you are joking, then :-) > > I thought you were joking, and that I just interpreted your joke. > >> Otherwise, quoting the OP's words (!) and applying to my values: >> >> p(b) = 0.8 is the probability that the engine will not start, >> which means that the rocket will probably remain on the ground. >> >> p(a) = 0.7 is the probability that the rocket will destroy during >> its flight, which means it's not a good rocket and that, once it >> gets started, it will probably be destroyed. Keywords: during its >> flight. > > I admit I'm still interpreting these as referring to one attempt with > the rocket, as I did with the bullet in the earlier thread, and I > admit the OP did already tell me (*) that I do not understand the > statement, so maybe your probability assignment makes some kind of > sense to him. (* I think he meant me. He actually followed up to his > own message without quoting anything.) > > I would say the probability of the destruction of the rocket _given it > got started_ is P(a|not b), and the probability of a succesful launch > _and_ destruction during the flight is P(a and not b). The latter is > just P(a)P(not b|a) = P(a) because I think destruction during the > flight _implies_ a succesful launch, so that P(not b|a) = 1. > > It makes no sense, to me, to say that a rocket is destroyed during a > flight that never started, so I think P(a) + P(b) <= 1. Yes, that is where we differ, but it's no big deal. The OP has everything he needs now. After all, *he* is the ultimate judge about what *he* actually meant, so to speak. > > But this is rocket science, so maybe I just misread it all. Dirk Vdm === Subject: Re: ultrafinitism and ultraformalism Originator: tchow@lebesgue.mit.edu.mit.edu (Timothy Chow) >> Can't you see that someone else is equally justified in balking >> at *your* dogmas? No. [...] >I don't take it for granted. I see that they are obviously true. It's >not a philosophical stance, it's knowledge. They are so basic, if you >can't (having learned the meanings of the symbols) see that they are >true, there's no helping you. All right, so where is your cutoff? Which of the following do you regard as proven mathematical theorems, i.e., either obviously true, or proven to follow from things that are obviously true? 1. Brouwer's fixed-point theorem. 2. Robertson and Seymour's graph minor theorem. 3. Martin's Borel-determinacy theorem. 4. Every commutative ring with unit has a prime ideal. 5. The product of compact topological spaces is compact. 6. ZFC is consistent. 7. Projective determinacy. This is not an arbitrarily chosen list. (1) is provable in WKL_0 but not RCA_0. (2) is provable in Pi_1^1-CA_0 but not ATR_0 (and hence, by some definitions of predicativism, is impredicative). (3) is provable in ZFC (I think even in ZF) but not in Z. (4) is provable in ZFC but not in ZF but weaker than full AC. (5) is equivalent to AC (over ZF). (6) is not provable in ZFC but is provable by any large cardinal axiom. (7) is not provable in ZFC but is seriously proposed by Woodin as a new basic axiom. All of them have been challenged by some people and also regarded as obviously true (or following from obviously true statements) by others. Where do you draw the line between so basic that there's no helping you and incoherent? Why do you draw the line there and refuse to see that other people draw the line elsewhere? -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === Subject: Re: ultrafinitism and ultraformalism >> Can't you see that someone else is equally justified in balking >> at *your* dogmas? >No. > [...] >I don't take it for granted. I see that they are obviously true. It's >not a philosophical stance, it's knowledge. They are so basic, if you >can't (having learned the meanings of the symbols) see that they are >true, there's no helping you. All right, so where is your cutoff? Which of the following do you regard as > proven mathematical theorems, i.e., either obviously true, or proven to > follow from things that are obviously true? 1. Brouwer's fixed-point theorem. > 2. Robertson and Seymour's graph minor theorem. > 3. Martin's Borel-determinacy theorem. > 4. Every commutative ring with unit has a prime ideal. > 5. The product of compact topological spaces is compact. > 6. ZFC is consistent. > 7. Projective determinacy. This is not an arbitrarily chosen list. (1) is provable in WKL_0 but not > RCA_0. (2) is provable in Pi_1^1-CA_0 but not ATR_0 (and hence, by some > definitions of predicativism, is impredicative). (3) is provable in > ZFC (I think even in ZF) but not in Z. (4) is provable in ZFC but not in > ZF but weaker than full AC. (5) is equivalent to AC (over ZF). (6) is > not provable in ZFC but is provable by any large cardinal axiom. (7) is > not provable in ZFC but is seriously proposed by Woodin as a new basic axiom. All of them have been challenged by some people and also regarded as > obviously true (or following from obviously true statements) by others. Where do you draw the line between so basic that there's no helping you > and incoherent? Why do you draw the line there and refuse to see that > other people draw the line elsewhere? i'm glad you mention woodin i've been meaning to mention that his approach to consistency in projective completeness has a lot of similarities to the predicativist positions concerning the consistency of PA some of the similarity is purely arms-length inspection they're both looking at consistency in certain limiting processes in opposition to commonly accepted theories and believe the commonly accepted theories are in some sense inconsistent but there are some formal similarities when one views the iterative (projective) completion recursively built from the axioms of ZF in the same light as predicative derivations recursively built from atoms of a predicative theory ZFC + projective determinancy =?= contradiction PA + projective predicativeness =?= contradiction for some time i've had this vague expectation of some woodin-nelson connection sitting latent out there but that's all speculation... -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- === Subject: Re: ultrafinitism and ultraformalism > Can't you see that someone else is equally justified in balking > at *your* dogmas? >> No. > [...] >> I don't take it for granted. I see that they are obviously true. It's >> not a philosophical stance, it's knowledge. They are so basic, if you >> can't (having learned the meanings of the symbols) see that they are >> true, there's no helping you. > > All right, so where is your cutoff? Which of the following do you regard as > proven mathematical theorems, i.e., either obviously true, or proven to > follow from things that are obviously true? > > 1. Brouwer's fixed-point theorem. > 2. Robertson and Seymour's graph minor theorem. > 3. Martin's Borel-determinacy theorem. > 4. Every commutative ring with unit has a prime ideal. > 5. The product of compact topological spaces is compact. > 6. ZFC is consistent. > 7. Projective determinacy. > > This is not an arbitrarily chosen list. (1) is provable in WKL_0 but not > RCA_0. (2) is provable in Pi_1^1-CA_0 but not ATR_0 (and hence, by some > definitions of predicativism, is impredicative). (3) is provable in > ZFC (I think even in ZF) but not in Z. (4) is provable in ZFC but not in > ZF but weaker than full AC. (5) is equivalent to AC (over ZF). (6) is > not provable in ZFC but is provable by any large cardinal axiom. (7) is > not provable in ZFC but is seriously proposed by Woodin as a new basic axiom. I think it's an interesting list. I'm rather curious about things that can be stated in ZC, can't be proven in ZC, and are provable in ZFC. It seems to me that (3) can be stated in ZC [we need to say what a Borel set is, and what a strategy is]. Does ZC + Borel Determinacy imply anything like the existence of cardinals larger than beth_n (all n in omega) ? David Bernier > All of them have been challenged by some people and also regarded as > obviously true (or following from obviously true statements) by others. > > Where do you draw the line between so basic that there's no helping you > and incoherent? Why do you draw the line there and refuse to see that > other people draw the line elsewhere? === Subject: Re: ultrafinitism and ultraformalism Originator: tchow@lebesgue.mit.edu.mit.edu (Timothy Chow) >It seems to me that (3) can be stated in ZC [we need to say >what a Borel set is, and what a strategy is]. >Does ZC + Borel Determinacy imply anything like >the existence of cardinals larger than beth_n >(all n in omega) ? Yes, I think so, though this is not an area I know that much about. Friedman showed that Borel determinacy requires uncountably many infinite cardinals, and I think that you need beth_alpha to prove determinacy for a set in level alpha of the Borel hierarchy. The fundamental reference is Higher set theory and mathematical practice by Friedman, Ann. Math. Logic 2 (1970/71), no. 3, 325-357. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === Subject: Re: ultrafinitism and ultraformalism Originator: tchow@lebesgue.mit.edu.mit.edu (Timothy Chow) >2. Robertson and Seymour's graph minor theorem. [...] >(2) is provable in Pi_1^1-CA_0 but not ATR_0 (and hence, by some >definitions of predicativism, is impredicative). Correction here...(2) is not provable in Pi_1^1-CA_0. It's certainly provable in ZFC, and Friedman has some conjectures about what is really needed to prove it, but I don't think anyone knows of any axiom-like equivalent of the graph minor theorem. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === Subject: Re: ultrafinitism and ultraformalism > > I am not sure exactly what you mean by first order logic, but > I am fairly sure whatever you mean, it does not make sense to me. If you do not know the meaning of first-order logic then you should not be making such strong statements about foundational/philosophical topics in mathematics. --Bill Dubuque === Subject: Re: ultrafinitism and ultraformalism > > I am not sure exactly what you mean by first order logic, but > I am fairly sure whatever you mean, it does not make sense to me. > > If you do not know the meaning of first-order logic > then you should not be making such strong statements > about foundational/philosophical topics in mathematics. > > --Bill Dubuque His is quite within his rights to say that it does not make sense to him. Everyone has a right to declare his own ignorance. It is sometimes a first step to wisdom! === Subject: Re: The Computable Reals (alpha version) posting-account=Y877ggoAAAB-75kOgSLDnLSoS2_jn90U Gecko/20071126 Fedora/1.5.0.12-7.fc6 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) > It occurs to me that limits are used often in calculus, and people > maintain (at least in some cases) that the limit is a specific number > and IS the solution. However when recent postings postulated limits in lists, the idea was > roundly dismissed. Perhaps this analogy is (some of) the cause of the confusion. Could someone explain why limits prove the existence of an actual > number in one case, but are invalid arguments in the other case? -paul- Limits generally prove a property of a particular number, not the existence of one in a set. If I may reply with a basic, but hopefully not insulting, comparison. Take the definition of left-continuous functions as an example, f is left-continuous at point t iff lim_{x->t} f(x) = f(t). We note that there are natural examples of functions for which this is true (e.g. the constant function f(x) = 0), and natural examples for which it isn't (e.g. the step function f(x) = [ x >= 1 ] = 1 if x >= 1 and 0 otherwise). Now, if f(x) = 3x, and we consider lim_{x->1} f(x) = 3, then it doesn't follow that there is an x in [0,1) such that f(x) = 3. Because the function is continuous and monotone, we can however, through a simple and convenient application of the limit, conclude that the function's range for this set is f[[0,1)] = [0,3). Letting x approach 1 may seem somewhat arbitrary, so we may instead look at g(x) = 3(1 - e^-x). For this function, lim_{x->inf} g(x) = 3, even though there is no real x such that g(x) = 3. That the limit of a sequence (or a function) in the sequence (in the considered range) is therefore a non-trivial property and such an argument must be well- motivated. For the problem that is being discussed, we can provably show that you cannot provide such a motivation as it stands. === Subject: Re: The Computable Reals (alpha version) >> I translate that as I don't know. > No, it's just that I had already been asked that question (as as all > others) in all possible shapes, and I had already tried with no > success to answer in any possible way, actually asking for a support > that has never come. > In fact, if you say so, I must take that you rather didn't bother to > read the thread before asking, including the fact that at that very > moment I was surely not in my best mood (and for obviuos reasons). > I've read almost every post in this thread, and so far it > appears that your computable reals comprise only a > proper subset of the rationals. ÊSpecifically, all of them > end in a repeating sequence of 1's or 0's exclusively. > If your definition differs from this, I must have missed > it. ÊIf so, please explain. > The construction as per version beta already provides all the binary > strings over finite induction. I have not been able to provide an > extension to transfinite induction. > Note that my presentation might be far from satisfactory from a > mathematical standpoint (or at least so I have to suppose given the > feedback), still a mathematician should be able to fill in the gaps as > to the simple fact that what I am saying trivially holds over -again- > finite induction. > Maybe I should ask: what would you think I am really missing here? (I > mean, apart from the cantorian dogma, which is quite clear.) > -LV What set are you referring to as all binary sequences? It is not simply a set. It is the list of all the binary sequences over finite induction. This point is indeed delicate in that it is here that Cantor's list is not a list, at least in no constructive sense. > You certainly > cover all finite binary sequences, Finite but unbounded. Finite induction. > and you could easily extend to > cover every eventually periodic sequence. No, there is no notion of period involved in the definition of such a list: it is just the list of all (and unique, i.e. without repetitions) binary sequences. Then you can use it -- and rather a compound form of it -- to index the rationals, but that's a further step. > If you wish to cover the > entire set of, not necessarily finite, binary sequences, then you need > to explain why these are included - I do not see any way to extend > your algorithm so that it isn't easily proven that it doesn't cover > them. What I am saying trivially holds over finite induction. I have not been able to extend to transfinite induction, although even to me it seems very easy to at least come out with a notion of limit to state the extended (lambda) case. -LV === Subject: Re: The Computable Reals (alpha version) posting-account=Y877ggoAAAB-75kOgSLDnLSoS2_jn90U Gecko/20071126 Fedora/1.5.0.12-7.fc6 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) >> I translate that as I don't know. > No, it's just that I had already been asked that question (as as all > others) in all possible shapes, and I had already tried with no > success to answer in any possible way, actually asking for a support > that has never come. > In fact, if you say so, I must take that you rather didn't bother to > read the thread before asking, including the fact that at that very > moment I was surely not in my best mood (and for obviuos reasons). > I've read almost every post in this thread, and so far it > appears that your computable reals comprise only a > proper subset of the rationals. Specifically, all of them > end in a repeating sequence of 1's or 0's exclusively. > If your definition differs from this, I must have missed > it. If so, please explain. > The construction as per version beta already provides all the binary > strings over finite induction. I have not been able to provide an > extension to transfinite induction. > Note that my presentation might be far from satisfactory from a > mathematical standpoint (or at least so I have to suppose given the > feedback), still a mathematician should be able to fill in the gaps as > to the simple fact that what I am saying trivially holds over -again- > finite induction. > Maybe I should ask: what would you think I am really missing here? (I > mean, apart from the cantorian dogma, which is quite clear.) > -LV > What set are you referring to as all binary sequences? It is not simply a set. It is the _list_ of all the binary sequences > over finite induction. This point is indeed delicate in that it is > here that Cantor's list is not a list, at least in no constructive > sense. Whenever I've called it a list, you've complained that you do not have *one* list. Now I will simply refer to it as an enumerable set, even though one implies the other's existence. > You certainly > cover all finite binary sequences, Finite but unbounded. Finite induction. As Virgil earlier pointed out, what about the infinite sequences? > and you could easily extend to > cover every eventually periodic sequence. No, there is no notion of period involved in the definition of such a > list: it is just the list of _all_ (and unique, i.e. without > repetitions) binary sequences. Then you can use it -- and rather a > compound form of it -- to index the rationals, but that's a further > step. Alright, I'm sorry. I was still uncertain about whether or not you wanted to use the periods in your output. Even though you told me earlier that you don't, my poor interpretation of your subsequent statements seemed to imply that you had either changed your mind or misinterpreted in the first place, mostly because this extension would imply that you cover all the rationals. Now, only the dyadic rationals are in your set, but every real can be approximated arbitrarily well by numbers in it. > If you wish to cover the > entire set of, not necessarily finite, binary sequences, then you need > to explain why these are included - I do not see any way to extend > your algorithm so that it isn't easily proven that it doesn't cover > them. What I am saying trivially holds over finite induction. I have not > been able to extend to transfinite induction, although even to me it > seems very easy to at least come out with a notion of limit to state > the extended (lambda) case. May I ask what it is you wish to show with transfinite induction? -LV === Subject: Re: The Computable Reals (alpha version) > ... Cantor's list is not a list ... Cantor does not give a list. Rather he says, you give me a list, and I can prove that things are missing. So you cannot legitimately base an argument against Cantor's proof on Cantor's list. The above is an imprecise statement of the proof. The proof has everything (like what he means by the thing I refer to as list above) very precisely defined. -paul- === Subject: Re: The Computable Reals (alpha version) > ... Cantor's list is not a list ... Cantor does not give a list. Rather he says, you give me a list, and > I can prove that things are missing. Right, and that proof is invalid. Not only: my list is a counter example. -LV === Subject: Re: The Computable Reals (alpha version) > ... Cantor's list is not a list ... > Cantor does not give a list. Rather he says, you give me a list, and > I can prove that things are missing. > > Right, and that proof is invalid. Your claim of its being invalid is what is invalid. > > Not only: my list is a counter example. You have provided no list (in the sense of an ennumeration, a function from the set of naturals to the set of numbers you want to list) which contains all reals, not even one which contains all computable reals, so the only thing your list counters is your own claims. === Subject: Re: The Computable Reals (alpha version) >> I translate that as I don't know. > No, it's just that I had already been asked that question (as as all > others) in all possible shapes, and I had already tried with no > success to answer in any possible way, actually asking for a support > that has never come. > In fact, if you say so, I must take that you rather didn't bother to > read the thread before asking, including the fact that at that very > moment I was surely not in my best mood (and for obviuos reasons). > I've read almost every post in this thread, and so far it > appears that your computable reals comprise only a > proper subset of the rationals. ÊSpecifically, all of them > end in a repeating sequence of 1's or 0's exclusively. > If your definition differs from this, I must have missed > it. ÊIf so, please explain. > The construction as per version beta already provides all the binary > strings over finite induction. I have not been able to provide an > extension to transfinite induction. > Note that my presentation might be far from satisfactory from a > mathematical standpoint (or at least so I have to suppose given the > feedback), still a mathematician should be able to fill in the gaps as > to the simple fact that what I am saying trivially holds over -again- > finite induction. > Maybe I should ask: what would you think I am really missing here? (I > mean, apart from the cantorian dogma, which is quite clear.) > -LV > What set are you referring to as all binary sequences? > > It is not simply a set. It is the _list_ of all the binary sequences > over finite induction. This point is indeed delicate in that it is > here that Cantor's list is not a list, at least in no constructive > sense. Cantor places no limitations on what binary sequences may be presented in a list so Cantor will accept any list of such sequences that julio, or anyone else, chooses to present, whether over finite induction or otherwise, and compute for THAT list a binary sequence not listed in it. Why julio thinks that placing additional conditions on a list will allow it to include binary sequences otherwise excluded is not clear. > > You certainly > cover all finite binary sequences, > > Finite but unbounded. Finite induction. No single sequence then is anything but finite, although the set of all such sequences can not be finite. > > and you could easily extend to > cover every eventually periodic sequence. > > No, there is no notion of period involved in the definition of such a > list: it is just the list of _all_ (and unique, i.e. without > repetitions) binary sequences. No it is not ALL of them, since it specifically excludes endless sequences, some of which, at least, are computable. > Then you can use it -- and rather a > compound form of it -- to index the rationals, but that's a further > step. Until it is a completed step, it may not be relied upon. > > If you wish to cover the > entire set of, not necessarily finite, binary sequences, then you need > to explain why these are included - I do not see any way to extend > your algorithm so that it isn't easily proven that it doesn't cover > them. > > What I am saying trivially holds over finite induction. If it is so trivial, why do you try to make it seem so important. You are saying no more than that a certain set of finite binary sequences is no more than countably infinite, which was well known and universally granted well before you began saying anything. > I have not > been able to extend to transfinite induction, although even to me it > seems very easy to at least come out with a notion of limit to state > the extended (lambda) case. If it is so easy why haven't you been able to do it? Could it be that there is more to this than meets your eye? === Subject: Re: The Computable Reals (alpha version) > It occurs to me that limits are used often in calculus, and people > maintain (at least in some cases) that the limit is a specific number > and IS the solution. > > However when recent postings postulated limits in lists, the idea was > roundly dismissed. > > Perhaps this analogy is (some of) the cause of the confusion. > > Could someone explain why limits prove the existence of an actual > number in one case, but are invalid arguments in the other case? There's a well-defined notion of convergence for a sequence of rational numbers converging to a unique real number. There is no such notion for a sequence of essentially arbitrary digit strings converging to a unique something. -- --------------------------- | BBB b Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum viditur. | BBB aa a r bbb | ----------------------------- === Subject: Re: The Computable Reals (alpha version) > It occurs to me that limits are used often in calculus, and people > maintain (at least in some cases) that the limit is a specific number > and IS the solution. > However when recent postings postulated limits in lists, the idea was > roundly dismissed. > Perhaps this analogy is (some of) the cause of the confusion. > Could someone explain why limits prove the existence of an actual > number in one case, but are invalid arguments in the other case? There's a well-defined notion of convergence for a sequence of rational > numbers converging to a unique real number. ÊThere is no such notion for > a sequence of essentially arbitrary digit strings converging to a unique > something. Cool! That indeed is a very clear way to tell why -- on a side -- I do have the computable reals, and why -- on the contrary -- Cantor's argument is invalid. -LV === Subject: Re: The Computable Reals (alpha version) posting-account=Y877ggoAAAB-75kOgSLDnLSoS2_jn90U Gecko/20071126 Fedora/1.5.0.12-7.fc6 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) > [Restored cross-posts.] > It occurs to me that limits are used often in calculus, and people > maintain (at least in some cases) that the limit is a specific number > and IS the solution. > However when recent postings postulated limits in lists, the idea was > roundly dismissed. > Perhaps this analogy is (some of) the cause of the confusion. > Could someone explain why limits prove the existence of an actual > number in one case, but are invalid arguments in the other case? > There's a well-defined notion of convergence for a sequence of rational > numbers converging to a unique real number. There is no such notion for > a sequence of essentially arbitrary digit strings converging to a unique > something. Cool! That indeed is a very clear way to tell why -- on a side -- I do > have the computable reals, and why -- on the contrary -- Cantor's > argument is invalid. -LV Limits of sequences are well-defined w.r.t. given metrics. I do not think that Cantor's diagonalization argument for binary strings ever considered the limit however as it was enough simply to show that there is no element in the (possibly infinite) sequence with the property in question. === Subject: Re: The Computable Reals (alpha version) posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > [Restored cross-posts.] > It occurs to me that limits are used often in calculus, and people > maintain (at least in some cases) that the limit is a specific number > and IS the solution. > However when recent postings postulated limits in lists, the idea was > roundly dismissed. > Perhaps this analogy is (some of) the cause of the confusion. > Could someone explain why limits prove the existence of an actual > number in one case, but are invalid arguments in the other case? > There's a well-defined notion of convergence for a sequence of rational > numbers converging to a unique real number. ÊThere is no such notion for > a sequence of essentially arbitrary digit strings converging to a unique > something. > Cool! That indeed is a very clear way to tell why -- on a side -- I do > have the computable reals, and why -- on the contrary -- Cantor's > argument is invalid. Limits of sequences are well-defined w.r.t. given metrics. I do not > think that Cantor's diagonalization argument for binary strings ever > considered the limit however as it was enough simply to show that > there is no element in the (possibly infinite) sequence with the > property in question. That's another way to tell how the diagonal argument fails: no metric possible with a list that is actually just a set. -LV === Subject: Re: The Computable Reals (alpha version) > [Restored cross-posts.] > It occurs to me that limits are used often in calculus, and people > maintain (at least in some cases) that the limit is a specific number > and IS the solution. > However when recent postings postulated limits in lists, the idea was > roundly dismissed. > Perhaps this analogy is (some of) the cause of the confusion. > Could someone explain why limits prove the existence of an actual > number in one case, but are invalid arguments in the other case? > There's a well-defined notion of convergence for a sequence of rational > numbers converging to a unique real number. ÊThere is no such notion for > a sequence of essentially arbitrary digit strings converging to a unique > something. > Cool! That indeed is a very clear way to tell why -- on a side -- I do > have the computable reals, and why -- on the contrary -- Cantor's > argument is invalid. > Limits of sequences are well-defined w.r.t. given metrics. I do not > think that Cantor's diagonalization argument for binary strings ever > considered the limit however as it was enough simply to show that > there is no element in the (possibly infinite) sequence with the > property in question. > > That's another way to tell how the diagonal argument fails: no metric > possible with a list that is actually just a set. Metrics are irrelevant to the completeness of lists, as any positive distance, however small, between a listed number and a test number is enough to separate them. That there is a sequence of numbers listed does not necessitate that any limit point of that sequence is listed. And julio repeatedly assumes that contrary to fact property to get his list. === Subject: Re: The Computable Reals (alpha version) posting-account=Y877ggoAAAB-75kOgSLDnLSoS2_jn90U Gecko/20071126 Fedora/1.5.0.12-7.fc6 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) > [Restored cross-posts.] > It occurs to me that limits are used often in calculus, and people > maintain (at least in some cases) that the limit is a specific number > and IS the solution. > However when recent postings postulated limits in lists, the idea was > roundly dismissed. > Perhaps this analogy is (some of) the cause of the confusion. > Could someone explain why limits prove the existence of an actual > number in one case, but are invalid arguments in the other case? > There's a well-defined notion of convergence for a sequence of rational > numbers converging to a unique real number. There is no such notion for > a sequence of essentially arbitrary digit strings converging to a unique > something. > Cool! That indeed is a very clear way to tell why -- on a side -- I do > have the computable reals, and why -- on the contrary -- Cantor's > argument is invalid. > Limits of sequences are well-defined w.r.t. given metrics. I do not > think that Cantor's diagonalization argument for binary strings ever > considered the limit however as it was enough simply to show that > there is no element in the (possibly infinite) sequence with the > property in question. That's another way to tell how the diagonal argument fails: no metric > possible with a list that is actually just a set. -LV I could say that the fact that your (infinite) set is, by construction, enumerable means that we can produce an infinite number of lists which exactly agrees in members (and the number of times each appears). However, I ask instead, what is the formal definition of a list which you are using in these contexts? What is the realtion to sequences? To a bijection from N, or a subset thereof? To an enumeration? === Subject: Re: The Computable Reals (alpha version) > [Restored cross-posts.] > > It occurs to me that limits are used often in calculus, and people > maintain (at least in some cases) that the limit is a specific number > and IS the solution. > However when recent postings postulated limits in lists, the idea was > roundly dismissed. > Perhaps this analogy is (some of) the cause of the confusion. > Could someone explain why limits prove the existence of an actual > number in one case, but are invalid arguments in the other case? > There's a well-defined notion of convergence for a sequence of rational > numbers converging to a unique real number. ÊThere is no such notion for > a sequence of essentially arbitrary digit strings converging to a unique > something. > > Cool! That indeed is a very clear way to tell why -- on a side -- I do > have the computable reals, and why -- on the contrary -- Cantor's > argument is invalid. Julio manages not to see what is there to be seen while managing to see what is not there eat all. The only way that julio will ever convince anyone of his dogma is by becoming more of a logician and mathematician than his present performance indicates is possible. === Subject: Breadline schmuck's basic mistakes <48c30b31$0$3550$9b4e6d93@newsspool3.arcor-online.net> posting-account=IBUqVwoAAADepmzxVr9iEYD5Z0A483SY rv:1.9.0.1) Gecko/2008070206 Firefox/3.0.1,gzip(gfe),gzip(gfe) >Then that is a matter for physics, and a physics newsgroup >is more appropriate for such discussion. The course of the last 60 years at least has been one in which > mathematicians are kept out of physics and in which the > standards of mathematics applied to physics are inadequate > to deal with the fundamental problems facing physics. Then physicists need to put their house in order: a matter for physics. > restricted to Heisenberg's complex criterium but seems to be a more criterium? > My name here is Salviati. Your name is Eckard Blumschien. >> Hilbert intended to also create an axiomatic physics and metaphysics. >Then discuss Hilbert's ideas on physics in sci.physics (or worse). There is little reason for that. Save the sensible practice of discussion in appropriate forums. Is = an inequality sign? Are you irrelevant? >Examples of totally ordered sets >include N, Z, Q and R with the standard ordering. In what differs R from Q if both are totally ordered? Learn some elementary mathematics --- there are many totally ordered sets. Your question presumes that all totally ordered sets should be the same --- that is stupidity. > If one did consequently distinguish R from Q, > then every piece of it would consist of an uncountable > amount of fictitious elements. Each interval in R is an uncountable set. > So far mathematics is still far from admitting this. fictitous is not a mathematical term. > Well, that's why 1,000,000.00 Mark were paid in 1925 Was that about 2 shillings then? > Having experienced bad things, Did you see something nasty in the woodshed? > Cantor's thinking and Hilbert's wording often correspond to the > hollow pathos of the German empire. Is this Godwin's law? >In order to fully reveal all > nonsense you should even be familiar with the bible. I'll eschew nonsense, thank you. > My main topic is IR+. However you ignored what I am claiming. You have neither defined IR+ (that's not a standard notation) nor have isolated any mistakes in mathematics. > The mistakes are more or less related to each other. > However, denial of the categorical difference between Q and R > seems to be a standard ritual in modern mathematics. Schroedinger's kitty seems to indicate that it harms. Again, you can't help it: that's physics not mathematics. Let's face it, your uncontrollable will-to-power demands you to subjugate mathematics to physics. But all it has achieved is to make you a laughing-stock here. > - A number is a number is a number. A rose is a rose is a rose. > - Dedekind: One can split the entity of ALL rational numbers: >, =, or <. > Ê (I argue: The entity of ALL rational numbers is identical with the reals.) Liar: you may assert that, but you haven't argued it (even unsuccessfully). > - IR+ is not worth to be taken seriously. What is IR^+? > Enough for today. Enough of your vacuous posturing forever. Victor Meldrew I don't believe it! === Subject: Re: Triangle Grid Of Coprimality: Game By the way. Are people in Usenet-land seeing ALL the games I have posted? Or are most newsreaders blocking posts made from Google Groups? I suspect that my posts become visible to most people ONLY when someone posts a reply to them from somewhere other than Google Groups. Is this correct? Leroy Quet === Subject: Re: Triangle Grid Of Coprimality: Game > By the way. Are people in Usenet-land seeing ALL the games I have > posted? Or are most newsreaders blocking posts made from Google > Groups? > > I suspect that my posts become visible to most people ONLY when > someone posts a reply to them from somewhere other than Google Groups. > Is this correct? > > Leroy Quet > > I see them just fine. I don't know if I'm seeing all of them, though. I've seen at least four or five in the last week to ten days. === Subject: Re: Triangle Grid Of Coprimality: Game > By the way. Are people in Usenet-land seeing ALL the games I have > posted? Or are most newsreaders blocking posts made from Google Groups? I don't know of any newsreader software that as-written specifically blocks posts from Google Groups, although a number of users have set up filters to block such posts. I think the number of such users is fairly small and that most of them also set up whitelist filters to avoid filtering out desired posts. I haven't previously seen [in sci.math] any posts in thread Triangle Grid Of Coprimality: Game > I suspect that my posts become visible to most people ONLY when someone > posts a reply to them from somewhere other than Google Groups. Is this > correct? -- jiw === Subject: Re: Triangle Grid Of Coprimality: Game James Waldby said: > >> By the way. Are people in Usenet-land seeing ALL the games I have >> posted? Or are most newsreaders blocking posts made from Google Groups? > > I don't know of any newsreader software that as-written specifically > blocks posts from Google Groups, although a number of users have > set up filters to block such posts. I think the number of such users > is fairly small Perhaps, but I'm certainly one of them. > and that most of them also set up whitelist filters > to avoid filtering out desired posts. to compensate. (On the other hand, I don't always have filters turned on.) -- Richard Heathfield Email: -http://www. +rjh@ Google users: Usenet is a strange place - dmr 29 July 1999 === === === Subject: What is the anti-derivative of |f(x)|? posting-account=nBdEXgoAAAB34FFmV1nYZfZ4VIw_WVgB Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) Hi there, I'm struggling with the following problem which is hopefully clear for others: Let f(x) be an integrable function and let F(x) be its anti- derivative. Also, we define g(x) := |f(x)| and let G(x) be the anti-derivative of g(x). Now, the question is: what is the anti-derivative of |f(x)|? In other words: what is int |f(t)| dt = G(x)? Clearly, if f(x) is positive for all x, then |f(x)| = f(x) and we have G(x) = F(x). Similarly, if f(x) is negative for all x, then G(x) = -F(x). But what happens if f is negative for some intervalls and positive for others? What is G(x) in this case? Sorry if the question is too easy I've already spent all day on it and not sure whether my answer is correct or not. I have just worked out the case where f(x) = x. Then G(x) = |F(x)|. Is this correct and does this hold in general? Miranda === Subject: Re: What is the anti-derivative of |f(x)|? > Hi there, > > I'm struggling with the following problem which is hopefully > clear for others: > > Let f(x) be an integrable function and let F(x) be its anti- > derivative. > Also, we define g(x) := |f(x)| and let G(x) be the anti-derivative > of g(x). > > Now, the question is: > > what is the anti-derivative of |f(x)|? > > In other words: what is int |f(t)| dt = G(x)? > > Clearly, if f(x) is positive for all x, then |f(x)| = f(x) and we > have G(x) = F(x). > > Similarly, if f(x) is negative for all x, then G(x) = -F(x). > > But what happens if f is negative for some intervalls and positive > for others? What is G(x) in this case? In general, G(x) = signum(f(x)) F(x) + C(x) where C(x) is constant on each interval where f(x) has constant sign but will usually change from one interval to another, in such a way that G(x) is continuous. -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: What is the anti-derivative of |f(x)|? posting-account=JpxxPAgAAAAgwzQIYqn4j6syK-YhOmcF Gecko/20070309 Firefox/2.0.0.3,gzip(gfe),gzip(gfe) > But what happens if f is negative for some intervalls and positive > for others? What is G(x) in this case? > Due to the great variety of integrable functions (I assume you are working with Riemann integral), the most we can say (in general) is that | int f | le int |f|. That is, in general we *can't* express one as some function of the other. For the trivial examples of strict inequality, consider f(x) = -x on [-1,1], or f(x) = sin x on [0, 2 pi]. === Subject: Re: What is the anti-derivative of |f(x)|? posting-account=nBdEXgoAAAB34FFmV1nYZfZ4VIw_WVgB Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) On 10 Sep, 18:21, porky pig...@my-deja.com But what happens if f is negative for some intervalls and positive > for others? What is G(x) in this case? Due to the great variety of integrable functions (I assume you are > working with Riemann integral), the most we can say (in general) is > that | int f | Êle int |f|. That is, in general we *can't* express > one as some function of the other. Yes, let's f is Riemann integrable. Another thought: I guess we can write |f(x)| = sgn(f(x))*f(x) where sgn(x) denotes the signum function. Then int^x |f(t)| dt = sgn(f(x))*F(x) where F(x) is the anti- derivative of f(x). If so, what is int a^b |f(t)| dt =: I(a,b)? Would we have I(a,b) = sgn(f(b))*F(b) - sgn(f(a))*F(a)? === Subject: Re: What is the anti-derivative of |f(x)|? > On 10 Sep, 18:21, porky_pig...@my-deja.com But what happens if f is negative for some intervalls and positive > for others? What is G(x) in this case? > Due to the great variety of integrable functions (I assume you are > working with Riemann integral), the most we can say (in general) is > that | int f | =A0le int |f|. That is, in general we *can't* express > one as some function of the other. Yes, let's f is Riemann integrable. > Another thought: I guess we can write |f(x)| =3D sgn(f(x))*f(x) where sgn(x) denotes the > signum function. Then int^x |f(t)| dt =3D sgn(f(x))*F(x) where F(x) is the anti- > derivative > of f(x). If so, what is int_a^b |f(t)| dt =3D: I(a,b)? > Would we have I(a,b) =3D sgn(f(b))*F(b) - sgn(f(a))*F(a)? No. Your idea gives a piecewise antiderivative, which is not normally continuous. Consider, for example, f(x) = sin(x). Your idea would give - sgn(x) cos(x) as an antiderivative, but that function has jump discontinuities at integer multiples of pi. A correct antiderivative of |sin(x)| on R would be, for example, - sgn(x) cos(x) + 2 floor(x/pi) + 1. David === Subject: Extending a group's operation beyond the group Distribution: world Keywords: Algebra I'd like to define the extension of a binary operation defined on a group, G, to a superset of G. If the superset is a subset of a group, H, that has G as a sub-group, the extension is simple enough, since the operation's already defined on H. However, that's not quite general enough for my purposes. The particular group that's motivating this is the multiplicative group of mxm non- singular matrices, and the superset that I'd like to extend to is the set of mxm non-zero matrices. Obviously, that's not a group, since some elements don't have inverses. The axioms that I'm considering are: 1. G is a group with group operation *. 2. G is a subset of H. 3. For any a, b in H, a#b is in H. (closure) 4. For any a, b in G, a#b = a*b. (isomorphism) 5. For any a, b, c in H, (a#b)#c = a#(b#c) (associativity) 6. If e is the identity of G, a#e = e#a = a for any a in H. (identity) I would assume that such a concept already exists. What's it called? In the very unlikely event that I'm the first person to ever have this idea, can anybody see any obvious flaws (such as the possibility of generating contradictory statements) in what I've come up with? -- Michael F. Stemper #include Nostalgia just ain't what it used to be. === Subject: Re: Extending a group's operation beyond the group Keywords: Algebra days. My association with the Department is that of an alumnus. >I'd like to define the extension of a binary operation defined on a group, >G, to a superset of G. If the superset is a subset of a group, H, that has G as a sub-group, the >extension is simple enough, since the operation's already defined on H. However, that's not quite general enough for my purposes. The particular >group that's motivating this is the multiplicative group of mxm non- >singular matrices, and the superset that I'd like to extend to is the >set of mxm non-zero matrices. Obviously, that's not a group, since >some elements don't have inverses. The axioms that I'm considering are: 1. G is a group with group operation *. >2. G is a subset of H. >3. For any a, b in H, a#b is in H. (closure) >4. For any a, b in G, a#b = a*b. (isomorphism) >5. For any a, b, c in H, (a#b)#c = a#(b#c) (associativity) >6. If e is the identity of G, a#e = e#a = a for any a in H. (identity) I would assume that such a concept already exists. What's it called? Your H will be a monoid, with G a subgroup (a submonoid which is also a group). Other nomenclatures are possible (for example, semigroup with identity). There is an entire hierarchy of sets-with-a-single-binary-operation. If all you have is a binary operation, and you do not even have a warrant for associativity, you get a magma. Quasigroups are magmas in which division is possible. Loops are quasigroups with identity elements. Semigroups are magmas in which the operation is associative. Groups correspond to associative loops (the intersection between loops and semigroups). You have abelian varieties of each of them, etc. === Subject: Probability: well-known result? If we have N entities, the ith generating an independent random value u[i] uniformly in [0,1], then if we assign positive weight w[i] to the ith, and let w be the sum of the w[i], then choosing the entity with maximum value of u[i]^(1/w[i]) chooses entity i with probability w[i]/ w. (Using ^ for exponentiation.) This result is used in work by Rao (no, not that one) and Stoica (An Overlay MAC Layer for 802.11 Networks) and obviously regarded by them as sufficiently novel that they prove it. But that raises for me the questions: (a) Is it well known? If so, does it have a name? An accessible reference? (b) Is it obvious? Rao and Stoica's proof is integrating PDFs and stuff like that. Quite acceptable as a prrof, but it doesn't make the result obvious to me. Is there a way of looking at it that makes it obvious. (Yes, that's highly subjective, but if it's obvious to you, it might be to me.) (c) Any useful generalisations or related results? === Subject: Re: Probability: well-known result? > If we have N entities, the ith generating an independent random value > u[i] uniformly in [0,1], then if we assign positive weight w[i] to the > ith, and let w be the sum of the w[i], then choosing the entity with > maximum value of u[i]^(1/w[i]) chooses entity i with probability w[i]/ > w. (Using ^ for exponentiation.) > > This result is used in work by Rao (no, not that one) and Stoica (An > Overlay MAC Layer for 802.11 Networks) and obviously regarded by them > as sufficiently novel that they prove it. But that raises for me the > questions: > > (a) Is it well known? If so, does it have a name? An accessible > reference? > (b) Is it obvious? Rao and Stoica's proof is integrating PDFs and > stuff like that. Quite acceptable as a prrof, but it doesn't make the > result obvious to me. Is there a way of looking at it that makes it > obvious. (Yes, that's highly subjective, but if it's obvious to you, > it might be to me.) > (c) Any useful generalisations or related results? If U is uniform in [0,1], -ln(u^(1/w)) = -1/w ln(u) is exponential with rate w. It is indeed well-known that if X_i are independent exponential random variables with rates w_i, P(X_i = min_j X_j) = w_i/sum_j w_j. See e.g. Ross, Introduction to Probability Models, sec. 5.2.3. One way to make the result obvious is this. Consider independent Poisson processes N_i(t) with rates w_i, and let X_i be the waiting time for the first occurrence of process #i. Now one way to get these n independent Poisson processes is to start out with one Poisson process of rate w = sum_i w_i and assign each occurrence independently to one of the processes, taking process #i with probability w_i/w. When looked at this way, it's clear that the probability that the first occurrence is in process #i (corresponding to X_i = min_j X_j) is w_i/w. -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: binary integer non-linear programming excel solver premium posting-account=hrbKuAoAAABB-pktWwb5o12FCbdbJca0 Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) Howdy, I am working a homework problem involving non-linear programming for an operations research class. The problem is in three parts. The objective is to minimize the total distance traveled between a facility and thirty different towns each year by optimally locating the facility amongst the towns. There are a fixed number of trips between each facility and each town each year. The second and third parts of the problem require locating two and three facilities. I believe IÕve got the first part correct. ExcelÕs solver give a consistent and reasonable answer. Of course, two and three facilities seems to produce local minima. IÕve downloaded and IÕm trying to use Frontline systems Premium solver add-in. Other than randomly sorting through different starting points, how do I configure the premium advice or help. === Subject: Re: Andrew Wiles > (The real test would be to see if he says math or maths...) In Simon Singh's really excellent documentary Fermat's Last TheoremWiles > usually says mathematics, except once when he says math. > He also says zee squared (rather than zed). I'm afraid Sir Andrew has gone native. Victor Meldrew I don't believe it! === Subject: Card Probability The question here is that we played a game with 7 cards,Two cards had the same meaning we dealt it, played then gave the cards for a reshuffle.Now the thing here is Joe got two of those cards 7 times in a row. Now what i ask is does the probability stay the same becuse every time we deal its a new case,which means the probability stays the same, or does the probability for him to get those two cards 7 times in a row is smaller than just getting one of them once ? === Subject: Differential equation question hi guys, i found a problem in an old math book im reviewing that i dont even know how to approach. here it is: suppose y=f(x) is a differentiable function a)show that x-intercept of the tangent line at a point P ( x0, y0) on the graph of y = f(x) is equal to x0 - f(x0)/f ' (x0) b) show that the distance between any point P (x0, y0) on the graph of y = f(x) and the x-intercept of the tangent line to y = f(x) at P (x0, y0) is equal to |f (x0) square root of 1/ [ |f '(x0)]^2 + 1| any comments are appreciated === Subject: Re: Differential equation question days. My association with the Department is that of an alumnus. >hi guys, i found a problem in an old math book im reviewing that i >dont even know how to approach. here it is: suppose y=f(x) is a differentiable function a)show that x-intercept of the tangent line at a point P ( x0, y0) on >the graph of y = f(x) is equal to x0 - f(x0)/f ' (x0) Can you find the equation of the tangent line? HINT: It's related to the derivative at the point. If you are given the equation of a nonvertical line, y = mx + b, do you know how to figure out the x-intercept (if one exists)? Put the two together. >b) show that the distance between any point P (x0, y0) on the graph of >y = f(x) and the x-intercept of the tangent line to y = f(x) at P (x0, >y0) is equal to |f (x0) square root of 1/ [ |f '(x0)]^2 + 1| Your equation is a mess: you have three absolute value bars, when the number should be even. You have a left bracket followed by an opening absolute value bar, and you close the bracket before closing the absolute value. You'll need to be more careful. In any case: if you take (a) for granted, that means you know where the point is, and you know where the x intercept is. Do you know how to compute the distance between two points on the plane? Then do it, and compare it to the expression you are given. >any comments are appreciated These are not questions about differential equations, but about basic calculus/geometry. A differential equation is an equation in which the unknown is a function, and the equation involves derivative (or partial derivatives) of the unknown. Mislabeling the kind of problem you are considering will impact the quality and number of responses you receive. -- === Subject: Re: More with Quadratic Diophantine Theorem > i think you would be surprised at gauss' elementariness Yeah, hero worship is fun an easy, especially with a dead guy. He's > safe, right? I think you would be surprised if he were alive to consider what you > and your society do with current mathematical results of an elementary > nature, like in my original two posts in this thread. Do you really think Gauss would accept you as part of math society? For what? Because you'll say nice things about him? Do you think Gauss would like you or something? gauss was a paranoid asshole who screwed his kids over and spent so much effort inflating himself crying me me me that many of his contemporaries disliked him that said he was a damn fine mathematician no hero though i figured that if you spent some time reading him you too could be that paranoid asshole inflating yourself with me me me and finally be a fair mathematician to boot -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- === Subject: Re: More with Quadratic Diophantine Theorem posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > i think you would be surprised at gauss' elementariness > Yeah, hero worship is fun an easy, especially with a dead guy. ÊHe's > safe, right? > I think you would be surprised if he were alive to consider what you > and your society do with current mathematical results of an elementary > nature, like in my original two posts in this thread. > Do you really think Gauss would accept you as part of math society? > For what? ÊBecause you'll say nice things about him? > Do you think Gauss would like you or something? gauss was a paranoid asshole > Ê who screwed his kids over > Ê and spent so much effort inflating himself > Ê Ê crying me me me > that many of his contemporaries disliked him that said > Ê he was a damn fine mathematician no hero though i figured that if you spent some time reading him > Ê you too could be that paranoid asshole > Ê inflating yourself with me me me > and finally be a fair mathematician to boot I'm contemplating this awesome result linking all quadratic Diophantine equations in 2 variables and wondering how far it goes. You're mumbling. History will remember me. But what about you? Oh, you don't care, right? Hundreds of years from now, some people may be wondering about who I am and what kind of person I was, where one may hero worship, and another may mouth off about my many failings. But who will even know about you? James Harris === Subject: Re: More with Quadratic Diophantine Theorem [...] > > I'm contemplating this awesome result linking all quadratic > Diophantine equations in 2 variables and wondering how far it goes. There's the Perfect cuboid problem which no one has solved. Any Perfect cuboid (if there is one) is an Euler brick. http://mathworld.wolfram.com/PerfectCuboid.html I wonder if a perfect cuboid exists... What are the chances? David Bernier > You're mumbling. > > History will remember me. But what about you? > > Oh, you don't care, right? > > Hundreds of years from now, some people may be wondering about who I > am and what kind of person I was, where one may hero worship, and > another may mouth off about my many failings. > > But who will even know about you? > > > James Harris === Subject: Re: More with Quadratic Diophantine Theorem How Dedekind Screwed Up a Hundred Years of Mathematics [...] My friend suggested that for most of the 3600 year history > of mathematics, mathematics was done constructively. I've been suspecting something of the kind, and am hoping to sneak > up on the philosophical problem of constructivism via the history > of mathematics (as my mind remains blank on the philosophy). [...] But Dedekind wasnÕt happy with the equivalence relations > in the theory and he created the set-theoretic formulation that > we suffer with today. I hear the sound of an axe being ground, and ... [...] Once again, Dedekind wasnÕt happy with this, and in 1872, he > proposed his own notion of real numbers based on set-theory by > Dedekind cuts of the rational numbers. A bit of sneaky rhetoric there. I doubt that the objective issues > are reducible to Dedekind's subjective happiness or unhappiness, > although undoubtedly his subjectivity would have given him (and, > via his writings, others) a handle on the objective issues. But > I must reserve judgement until I understand those issues better. Also, there /was/ no set theory on which to base the theory of cuts > (or, for that matter, ideals); rather, surely it was these convincing > applications of a still non-existent theory which helped to bring the > theory into being? Whether Dedekind himself would recognise ZFC, for > example, as his child is another question. of course you are right about there being no set theory and the overreaching imagination of his desires and the axe-to-grind style and i don't agree with the idea that it was somehow dedekind's fault mathematics was screwed up for a hundred years (though i think it was screwed up) it's a mentality that gets annoyed with questions of fundaments and moves to belittlement or a mentality that seeks certainty but doesn't want to obvious mythology so substitutes other unobservables like bivalence or ... there's a hundred and one reasons why it was screwed up and dedekind was just one of the authorities that made authority appeal easier i linked to the text mostly to point to the issues that dedekind created and didn't intend all of the rhetoric (just some of it :) ) [as an interesting aside dedekind's construction of the reals is not constructively equivalent to cauchy's i actually suspect the initial acceptance of equivalence may have helped confuse constructive foundations enough to delay it beyond the classical formalisation] -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- === Subject: Re: More with Quadratic Diophantine Theorem just because the ideals were not made to solve Fermat's Last Theorem, they don't work?... there's a good case-book that uses the golden section as the exemplar. I agree, though, it is best to avoid them til you might need them; I certainly don't need'em. > [as an interesting aside > Ê Êdedekind's construction of the reals > Ê Êis not constructively equivalent to cauchy's Êi actually suspect the initial acceptance of equivalence > Êmay have helped confuse constructive foundations > Ê Êenough to delay it beyond the classical formalisation] thus: an aether simply provides a prealistical model of the phenomenon of waves; one need not deal with photons at all, since they are just formally dual to waves, not at all pardoxical, as shown by Dirac. far too much of theory rests upon Einstein, saying that's impossible, which is the reality behind Michelson-Morley and those who carried the examination further -- but, herr doktor-professor Albert said, Nein!, when he was once in his dysused office at Caltech. let us remember him for those things which he was really good at, not the Big Bang Constant and its infernal klingons (fridge magnets) ?! > I don't need any aether to stick a magnet to a fridge, light up an thus: so, to ask an ignorant question of personal interest, how much of this could be used toward a new or old proof of quadratic reciprocity? > That Ac and -Dc must be quadratic residues, mod D and A, respectively, > is a trivial result: > Let Ax^2 - Dy^2 = c. [1] > Take this mod A: > -Dy^2 = c mod A > (Dy)^2 = -Dc mod A > Therefore, -Dc is a quadratic residue mod A. > Similarly, taking [1] mod D: > Ax^2 = c mod D > (Ax)^2 = Ac mod D > Therefore, Ac is a quadratic residue mod D. thus: ah, sort of a solopsistic monad. I am reminded of hte Many Words Interpretation of the Copenhagenskool: to be or not to be that is the shape of the box or the bubble -- inside or out? > Q. What is a tautological space? > A. A tautological space. thus: speaking of Young's anhialation of Newton's photons, here is the earlier elaboration on light by Fermat: http://www.wlym.com/~seattle/dynamis/issues/august08-fermat.pdf > as we know from Newton's iron-poor corpuscles. --ROTC, your summer vacation in the Sahara Desert ( S u d a n ) ; presage the Draft for your middleschool class of '12 -- brought to you by Allstate (tm) and Oxford U. Press! http://larouchepub.com/pr/2008/080813moloch brown.html http://wlym.com === Subject: Re: More with Quadratic Diophantine Theorem > >(for an example of what i came to dislike > http://r6.ca/blog/20051210T202900Z.html >) > > How Dedekind Screwed Up a Hundred Years of Mathematics > > [...] My friend suggested that for most of the 3600 year history > of mathematics, mathematics was done constructively. OK, I think I understand the difference between a constructive proof and a non-constructive proof for the existence of a given thing. E.g., if you wanted to prove that there exists irrational a and b such that a^b is rational, this would be a non-constructive proof: Consider sqrt(2)^sqrt(2). If that is rational, then a=sqrt(2), b=sqrt(2) are irrational a, b such that a^b is rational. If, however sqrt(2)^sqrt(2) is irrational, then let a = sqrt(2)^sqrt(2) b = sqrt(2) Then a^b = 2. That's non-constructive because it doesn't actually give us a and b that works. It just narrows it down to two possibilities, one of which must work. The result could be provided constructively by giving specific, known irrational, values for a and b that work, such as a = e, b = log 2, after you first prove that e and log 2 are irrational. What I don't see is how a proof of non-existence fits into the constructive/non-constructive classification. For instance, consider the classic proof that sqrt(2) is irrational: Assume it is rational, say a/b, with at least one of a, b odd. 2 = a^2/b^2 2 b^2 = a^2 Therefore, a must be even, say a = 2r 2 b^2 = 4 r^2 b^2 = 2 r^2 Therefore, b must be even. This is a contradiction, therefore sqrt(2) is irrational. Would a constructionist allow that proof? If not, how would a constructionist show that sqrt(2) is irrational? -- --Tim Smith === Subject: Re: More with Quadratic Diophantine Theorem days. My association with the Department is that of an alumnus. [...] >What I don't see is how a proof of non-existence fits into the >constructive/non-constructive classification. For instance, consider >the classic proof that sqrt(2) is irrational: Assume it is rational, say a/b, with at least one of a, b odd. 2 = a^2/b^2 > 2 b^2 = a^2 > Therefore, a must be even, say a = 2r > 2 b^2 = 4 r^2 > b^2 = 2 r^2 > Therefore, b must be even. > This is a contradiction, therefore sqrt(2) is irrational. Would a constructionist allow that proof? For most meanings of constructionist (they may vary a bit), the implication is acceptable. It is the implication not(for all x (P(x))) ==> exists x (not(P(x))) that is considered unacceptable in the absence of a construction of x. (Constructionists also reject not(not(P)) --> P, so you also have to be a bit careful there). -- === Subject: Re: Longevity of Math Books (was Re: Quadratic Diophantine Theorem) <6996c4lha799djvtrktcfnld533gdkbnl0@4ax.com> posting-account=jPnQ2goAAAA461y3QD0lbyw0oKeThma1 AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.2 Safari/525.20.1,gzip(gfe),gzip(gfe) Apostol begins the course with integration, as per Liebniz; there's a special name for that, per Liebniz: primitive equation or prime equation? this was the subject ot the keynote speaker at the Ninth Annual Nonlinear Science Conference at UCLA, although I no longer blame Newton exclusively for the controversy with Liebniz. anyway, apparently, Hooke did not do the math in a formalistic way; after all, he was the Royal Society's experimenter. the other part of the joke is that the '99 2-pound coin is inscribed, On the shoulders of giants. on the edge; at least, Newton was effective in his office-of-reward. > stole the solution from Hooke, to boot -- on the shoulders of a giant, > who was physically a dwarf; hence, the joke. Wrong again, but excusably, since that meme has spread widely. Read the > shoulders bit in context, and you can see it was not an insult. thus: ah, Lord Berty's thing was called the Unity of Sciences; well, here's a reference: http://www.larouchepub.com/other/2002/2949moonification.html The two operations of Wells and Russell from which Moon sprung are these: The Moral Re-Armament Movement, founded at a 1921 meeting between a wacky Lutheran preacher from Philadelphia and two British delegates to the Washington Disarmament Conference, Lord Arthur Balfour and H.G. Wells. Moral Re-Armament became the mass organizational vehicle for implementation of Wells' 1928 call in The Open Conspiracy, for a worldwide movement for draft resistance. The environment of Moon's Korean ministry was under control of Moral Re-Armament when he was picked up as an intelligence asset during the Korean War. The Unity of the Sciences movement. Founded in 1935 under the supervision of Lord Bertrand Russell and John Dewey, it brought together Trotskyite academics Albert Wohlstetter (mentor of current Defense Policy Board Chairman Richard Perle), Sidney Hook, and Ernest Nagel, with members of the radical-positivist Vienna Circle. Merging with Robert M. Hutchins at the University of Chicago in the 1950s, this operation took over the teaching of science in the United States. Thomas Kuhn's widely read fraud, The Structure of Scientific Revolutions, was published as the second volume of their Encyclopedia of Unified Sciences. In 1972, the Moonies were given the Unity of Sciences franchise, sponsoring the first of their still-ongoing International Conferences of the Unity of Sciences. Their early sessions featured such notables as Manhattan Project physicist Eugene Wigner, the lifelong ally of that truly mad scientist Leo Szilard (the model for Dr. Strangelove, in Stanley Kubrick's film of that name), and environmental fascists Alexander King and Aurelio Peccei, founders of the no-growth Club of Rome. Before looking back to the history of these projects, let us first briefly dispense with the person of Rev. Sun Myung Moon. Moon as a personality is of very little importance, in himself. The real Reverend Moon is a pathetic, if nonetheless nasty, victim of Japanese internment and North Korean torture sessions. He is what the.... thus: an aether simply provides a prealistical model of the phenomenon of waves; one need not deal with photons at all, since they are just formally dual to waves, not at all pardoxical, as shown by Dirac. far too much of theory rests upon Einstein, saying that's impossible, which is the reality behind Michelson-Morley and those who carried the examination further -- but, herr doktor-professor Albert said, Nein!, when he was once in his dysused office at Caltech. let us remember him for those things which he was really good at, not the Big Bang Constant and its infernal klingons (fridge magnets) ?! > I don't need any aether to stick a magnet to a fridge, light up an thus: so, to ask an ignorant question of personal interest, how much of this could be used toward a new or old proof of quadratic reciprocity? > That Ac and -Dc must be quadratic residues, mod D and A, respectively, > is a trivial result: > Let Ax^2 - Dy^2 = c. [1] > Take this mod A: > -Dy^2 = c mod A > (Dy)^2 = -Dc mod A > Therefore, -Dc is a quadratic residue mod A. > Similarly, taking [1] mod D: > Ax^2 = c mod D > (Ax)^2 = Ac mod D > Therefore, Ac is a quadratic residue mod D. thus: ah, sort of a solopsistic monad. I am reminded of hte Many Words Interpretation of the Copenhagenskool: to be or not to be that is the shape of the box or the bubble -- inside or out? > Q. What is a tautological space? > A. A tautological space. thus: speaking of Young's anhialation of Newton's photons, here is the earlier elaboration on light by Fermat: http://www.wlym.com/~seattle/dynamis/issues/august08-fermat.pdf > as we know from Newton's iron-poor corpuscles. --ROTC, your summer vacation in the Sahara Desert ( S u d a n ) ; presage the Draft for your middleschool class of '12 -- brought to you by Allstate (tm) and Oxford U. Press! http://larouchepub.com/pr/2008/080813moloch_brown.html http://wlym.com === Subject: Re: Quadratic Diophantine Theorem 7. Summary Starting with Euler in [Eul1773], many authors have come up with essentially the same idea: solving the Pell equation X2 - dY 2 = 1 (29) becomes easier by looking at certain auxiliary equations. The first step is writing down Legendre's equations rx2 - sy2 = 1, 2 for d = rs. (30) There is one nontrivial equation among these with a solution, and the smallest solution will have about half as many digits as the smallest solution of (29). The second step is to look at equations whose solutions give rise to a solution of one of the equations (30); but this second step has never been completed in full generality. What we have are specific equations applicable only in special situations; these were first discovered by Euler, rediscovered by Hart, Sylvester, G¬unther, G«erardin, Hardy & Williams, and Arteha, and slightly generalized by Bapoungu«e. HIGHER DESCENT ON PELL CONICS 11 References http://www.fen.bilkent.edu.tr/~franz/publ/pell-b.pdf Tuesday, July 27, 1638 Sir, The receipt of your letter in which you do me the honor of promising your friendship, gave me no less joy than if I had received it from a mistress whose good graces I passionately desired. And your other writings which preceded it reminded me of the Bradamante of our poets, who would take no one as a servant who had not previously proved himself against her in combat. It is not every day that I compare myself to that Roger who was the only one in the world capable of standing against her in combat; but such as I am, I assure you that I honor your merit considerably. And seeing the most recent method that you used to find tangents to curved lines [in your last letter], I have nothing to say in response, other than that it is very good, and that, if you had explained it that way in the beginning, I would have had no disagreement. The remainder of the letter deals specifically with geometry, and has not been included here. http://www.wlym.com/~seattle/dynamis/issues/august08-fermat.pdf thus: AP, you nut.... actually, it is a commonplace hearsay that the founding parental units were atheists, usually called Deists -- I'm the Higher Power of the 13th Step -- ME ?!? ... but it isn't true. if you read LaRouche, you'd know that, but there is some question about Shakespeare as a Third Language. thus: Apostol begins the course with integration, as per Liebniz; there's a special name for that, per Liebniz: primitive equation or prime equation? thus: this was the subject ot the keynote speaker at the Ninth Annual Nonlinear Science Conference at UCLA, although I no longer blame Newton exclusively for the controversy with Liebniz. anyway, apparently, Hooke did not do the math in a formalistic way; after all, he was the Royal Society's experimenter. the other part of the joke is that the '99 2-pound coin is inscribed, On the shoulders of giants. on the edge; at least, Newton was effective in his office-of-reward. > stole the solution from Hooke, to boot -- on the shoulders of a giant, > who was physically a dwarf; hence, the joke. thus: ah, Lord Berty's thing was called the Unity of Sciences; reference: http://www.larouchepub.com/other/2002/2949moonification.html The two operations of Wells and Russell from which Moon sprung are these: The Moral Re-Armament Movement, founded at a 1921 meeting between a wacky Lutheran preacher from Philadelphia and two British delegates to the Washington Disarmament Conference, Lord Arthur Balfour and H.G. Wells. Moral Re-Armament became the mass organizational vehicle for implementation of Wells' 1928 call in The Open Conspiracy, for a worldwide movement for draft resistance. The environment of Moon's Korean ministry was under control of Moral Re-Armament when he was picked up as an intelligence asset during the Korean War. The Unity of the Sciences movement. Founded in 1935 under the supervision of Lord Bertrand Russell and John Dewey, it brought together Trotskyite academics Albert Wohlstetter (mentor of current Defense Policy Board Chairman Richard Perle), Sidney Hook, and Ernest Nagel, with members of the radical-positivist Vienna Circle. Merging with Robert M. Hutchins at the University of Chicago in the 1950s, this operation took over the teaching of science in the United States. Thomas Kuhn's widely read fraud, The Structure of Scientific Revolutions, was published as the second volume of their Encyclopedia of Unified Sciences. In 1972, the Moonies were given the Unity of Sciences franchise, sponsoring the first of their still-ongoing International Conferences of the Unity of Sciences. Their early sessions featured such notables as Manhattan Project physicist Eugene Wigner, the lifelong ally of that truly mad scientist Leo Szilard (the model for Dr. Strangelove, in Stanley Kubrick's film of that name), and environmental fascists Alexander King and Aurelio Peccei, founders of the no-growth Club of Rome. Before looking back to the history of these projects, let us first briefly dispense with the person of Rev. Sun Myung Moon. Moon as a personality is of very little importance, in himself. The real Reverend Moon is a pathetic, if nonetheless nasty, victim of Japanese internment and North Korean torture sessions. He is what the.... thus: an aether simply provides a prealistical model of the phenomenon of waves; one need not deal with photons at all, since they are just formally dual to waves, not at all pardoxical, as shown by Dirac. far too much of theory rests upon Einstein, saying that's impossible, which is the reality behind Michelson-Morley and those who carried the examination further -- but, herr doktor-professor Albert said, Nein!, when he was once in his dysused office at Caltech. let us remember him for those things which he was really good at, not the Big Bang Constant and its infernal klingons (fridge magnets) ?! > I don't need any aether to stick a magnet to a fridge, light up an thus: so, to ask an ignorant question of personal interest, how much of this could be used toward a new or old proof of quadratic reciprocity? > That Ac and -Dc must be quadratic residues, mod D and A, respectively, > is a trivial result: > Let Ax^2 - Dy^2 = c. [1] > Take this mod A: > -Dy^2 = c mod A > (Dy)^2 = -Dc mod A > Therefore, -Dc is a quadratic residue mod A. > Similarly, taking [1] mod D: > Ax^2 = c mod D > (Ax)^2 = Ac mod D > Therefore, Ac is a quadratic residue mod D. thus: ah, sort of a solopsistic monad. I am reminded of hte Many Words Interpretation of the Copenhagenskool: to be or not to be that is the shape of the box or the bubble -- inside or out? > Q. What is a tautological space? > A. A tautological space. thus: speaking of Young's anhialation of Newton's photons, here is the earlier elaboration on light by Fermat: http://www.wlym.com/~seattle/dynamis/issues/august08-fermat.pdf > as we know from Newton's iron-poor corpuscles. --ROTC, your summer vacation in the Sahara Desert ( S u d a n ) ; presage the Draft for your middleschool class of '12 -- brought to you by Allstate (tm) and Oxford U. Press! http://larouchepub.com/pr/2008/080813moloch brown.html http://wlym.com === Subject: Re: Quadratic Diophantine Theorem posting-account=aLpfCwoAAACh4BOs3HOlQBCoxUpEgyxc Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) > Ê Ê[...] >> THEOREM. Let d be a positive squarefree integer, greater than 1. >In fact I don't see anything below which uses the fact that d is >squarefree, rather than the weaker fact that d is not a perfect >square. Well, the equations are equivalent. If d = m^2*n with n squarefree and > greater than 1, then solutions to Ê Ê Êx^2 - dy^2 = k yield solutions to Ê Ê x^2 - nY^2 = k Of course, silly me. > [...] That said, I believe the reason that LeVeque restricts to squarefree d > is that he works in part with elements of Z[sqrt(d)], and number > theorists traditionally restrict to squarefree d in this case so that > Z[sqrt(d)] = Z[sqrt(e)] if and only if d = e. Sorry if I'm being thick here, but isn't this true even without this restriction? Suppose that d and e are not perfect squares, and let d = n^2*r, e = m^2*s with r, s squarefree and greater than 1, and n, m positive, and suppose that Z[sqrt(d)] = Z[sqrt(e)]. Then since n*sqrt(r) is in Z[sqrt(d)] we have, for some x and y n*sqrt(r) = x + y*m*sqrt(s), so (1) n^2*r = x^2 + y^2*m^2*s + 2*x*y*m*sqrt(s), therefore x*y = 0. The case y = 0 is eliminated by the irrationality of sqrt(r) and we are left with n^2*r = y^2*m^2*s (2) Let p be any prime dividing r, and let i be the largest power of p dividing the LHS of (2); then i must be odd. Therefore p also divides s, since otherwise the exponent of p in the RHS of (2) would be even. The same argument shows that any prime dividing s divides r, and since every exponent in the prime factorisation of r or s is 1 we have r = s. Going back to (1) now gives n = y*m so m divides n; repeating this argument with d and e reversed shows that n divides m so n = m. === Subject: Re: Quadratic Diophantine Theorem days. My association with the Department is that of an alumnus. [...] >> That said, I believe the reason that LeVeque restricts to squarefree d >> is that he works in part with elements of Z[sqrt(d)], and number >> theorists traditionally restrict to squarefree d in this case so that >> Z[sqrt(d)] =3D Z[sqrt(e)] if and only if d =3D e. Sorry if I'm being thick here, but isn't this true even without this >restriction? My mistake. Should have been the fields rather than the rings. There is a one-to-one correspondence between quadratic extensions of Q and squarefree integers d different from 1; with this restriction, the ring of integers of Q(sqrt(d)) is Z[sqrt(d)] if and only if d is not congruent to 1 modulo 4 (if d is congruent to 1 modulo 4, then the ring of integers is Z[(1+sqrt(d))/2]. If you simply take d to be non-perfect squares, the conditions refer to the square-free part of d, making them more annoying. In any case: the point is that number theorists are used to restricting to d being squarefree because it makes life easier, and since in this situation there is no loss in generality once you have all the result for squarefree d that include how to generate every solution, it is probably more comfortable to work with that restriction. -- === Subject: Re: Quadratic Diophantine Theorem >So that's your real name? Yep, 'fraid so. >And you're an old geezer too. Quite old, yes; geezer, never! >Knowledge is valuable Mr. Rodgers, even when you find it inconvenient. >Your age may explain though your puzzling lack of interest in the >reality of the hugeness of the find. >I take it you are an established mathematician? Far from it. (I'll refrain from going into the excruciatingly > embarrassing and painful details.) Well, you're younger than Kurt Heegner was when he solved Gauss' class number 1 problem... http://en.wikipedia.org/wiki/Kurt Heegner === Subject: Re: Introduction to Heat Transfer (5th Ed., Incropera, DeWitt) Is there any chance you can forward this solutions manual to me also: subyspeed00@aol.com === Subject: Re: A couple of engineering solution manuals... > I have the following solution manuals: Water-Resources Engineering by David A. Chin 2nd edition Structural Analysis by R.C. Hibbeler 6th edition Mechanics of Materials by R.C. Hibbeler 7th edition Essentials of Fluid Mechanics: Fundamentals and Applications by > Cimbala and Yengel Electrical Engineering Principles and Applications by Hambley 4th > edition Engineering Mechanics Statics by Bedford 4th edition (almost the same > as the 5th edition) If you are interested in any of these, please send me an e-mail at > viktor912(at)hotmail.com I now also have the solutions manual for: Principles of Geotechnical Engineering 6th edition by Das. $15 for any of the solution manuals. Please e-mail me at viktor912(at)hotmail.com === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation posting-account=wVv_VwoAAAAVTfUuyxLzug5SzYWCgHj1 Gecko/20080702 Firefox/2.0.0.16,gzip(gfe),gzip(gfe) This is truly masterful trollwork, a veritable clinic in trollery at its best. I will point out a few examples in the following - > Now that I have a general theory Troll-technique #1: Start off immediately with exaggerated bragging. This is really guaranteed to get under the skin of real mathematicians, who know that what Harris has accomplished here is minor high- school level manipulation of simple equations. There is no 'general theory'. > for all 2 variable quadratic > Diophantine equations it's worth coming back to note again the weird Troll-technique #2: Liberal use of the word 'weird' - pretending there is deep mystery where there is only shallow symbol-pushing. Again very efficiently designed to enrage the real mathematicians. > connection I found between certain Pythagorean Triplets and Pell's > Equation in the form x^2 - Dy^2 = 1 when D-1 is a perfect square. For instance for D=2, I have that for > every solution of Pell's Equation you have a Pythagorean Triplet! > Troll-technique #3: Hyped excitement! Wow! Trollery here is much more a matter of marketing than of genuinely interesting math. But it does effectively rankle the reader and provoke responses [this one included]. > But the triplets are special in that with u^2 + v^2 = w^2, v = u+1. > The connection is that w is x+y from Pell's Equation. The more general result is that u = sqrt(D-1)j, and v = j+1, while w > still equals x+y. Intriguingly Troll-technique #2 again, but with 'intriguingly' rather than 'weird'. It's NOT intriguing, it's vapid. Did Gauss EVER say 'intriguingly'? Galois? Riemann? Probably not even in personal correspondence. When JSH says 'intriguingly' or 'fascinating', he is describing exactly one person's reaction. Guess whose. > that means that proof that there are an infinite number > of solutions for certain Pell's Equations is proof that there are an > infinity of Pythagorean Triplets of a certain form! > Troll-technique #3 again: hyped excitement!!! > An easy example with D=2, is x=17, y=12, where notice you are paired > with the triplet 20, 21, 29. That is just some low-hanging fruit Troll-technique #4: Use of the trendy buzz-phrase. As if this is just a wee taste of the deep wonders to come [except this is as deep as it will ever get]. And it makes the real mathematician see red. > that I thought I'd mention. Troll-technique #5: Just an offhand little remark, delivered with the obligatory sneer. > Kind > of been a whirlwind of results flowing from playing with my > Diophantine Quadratic Theorem. > Troll-technique #6: 'Whirlwind'? Good. Similar alternatives: 'avalanche', or 'flood', or 'supernova' or 'shitload'. More like: 'trickle' or 'driblet' or 'wetspot' or 'flyspeck'. And 'playing with' - false modesty laid on to really get the professional's goat. Every Harris post is a FASCINATING [note buzzword!] lesson in trollery - starting-out trolls should begin taking notes - this is the all- time Master. Marcus. > James Harris === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008070208 Firefox/3.0.1,gzip(gfe),gzip(gfe) Ê This is truly masterful trollwork, a veritable clinic in trollery > at its best. I will point out a few examples in the following - > Now that I have a general theory Ê Troll-technique #1: Start off immediately with exaggerated > bragging. > Ê This is really guaranteed to get under the skin of real > mathematicians, > Ê who know that what Harris has accomplished here is minor high- > school > Ê level manipulation of simple equations. ÊThere is no 'general > theory'. Um, well, yeah, there is, now. I put a link to the paper on my math blog. Infinite chains of Diophantine quadratics. And a remarkably simple way to check for whether or not integer solutions exist. Oh, and best yet! Proof that people like you either deliberately lie or deliberately refuse to accept mathematical proof, when you don't like the discoverer. I proved Fermat's Last Theorem using tautological spaces but there was a problem!!! I was proving a negative. Now I've turned them to studying an area where solutions exist and look at you. Denial is about weakness. You refuse to accept mathematical truth so you cannot be a true mathematician. But you can be a pretender who got away for so long, so long, and hurt humanity in the process. The future of the human race is not yet lost. Despite those like you. The future is not yet decided against this species. Despite all those who have fought so desperately to condemn it. Yes, you have not yet totally failed. But there is finally a light at the end of the tunnel. There is finally, hope. James Harris === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation >You refuse to accept mathematical truth so you cannot be a true >mathematician. But you can be a pretender who got away for so long, so long, and hurt >humanity in the process. The future of the human race is not yet lost. Despite those like you. The future is not yet decided against this species. Despite all those >who have fought so desperately to condemn it. Not ... aliens? I knew it! >Yes, you have not yet totally failed. But there is finally a light at the end of the tunnel. There is finally, hope. You're funny, but your struggle is a serious one. (Never mind Hamlet, how about Don Quixote?) I mean, you project a fantastic amount, but that's one way of dealing with demons, and it reflects the truth that the demons aren't simply in your mind. (Bwahahahahah!) Your strictures apply in some measure to all of us (so I feel) - and that includes you, I'm afraid. === Subject: Re: JSH: Pythagorean Triplets and Pell's Equation yeah, I usually try to be charitable towards HSJ, but the question of whether he is a conscientious troll is unanswerable; if he is, he never lets the mask down, that I've noticed. so, I usu