mm-228 === Subject: Re: Sex and Mathmike_deeth@yahoo.com (Mike Deeth) made some kool advert, but Cantor isall about Bijections, (or lack of them), So I tnk we need to showa bijection between the two most prominent features of s pic.Hence I remove the coverings. (Here I use Taylor's === can't a conjugate of a subgroup be a proper subset of === books?I'd recommend Schaum's Outline of Complex Variables, by Spiegel.Best,Kerry Soileau> The subject says it -- however, I'd like to clarify> one detail: I'm looking for a book on *analysis*, as> opposed to Calculus (i.e., that covers rigorously> the concepts and proofs on Complex numbers and> Complex variables functions). However, I'm just a hobbyist, so I'm not looking for> the ultimate, advanced reference book (i.e., I'm not> a mathematician or even a student in Mathematics; I'm> an engineer, who already knows (at least *knew* very> well :-)) about complex numbers, but I'm beginning> to appreciate and enjoy the rigorous side of maths,> and I find that complex numbers do have a great> === ideasI>> In sci.physics, >> saw(c_1 x + 7)(c_2 x + 7)( c_3 x + 1) = >>> 49(x^3 + 5x^2 + 3x + 1)with the c's algebraic integers, I tnk few of you would have a>> problem realizing that only two of the c's have 7 as a factor.Oh, look, he's changed polys on us! But OK, let's check>> ts one.Recall that, if P(x) = 49 * (x - x_1) * (x - x_2) * (x - x_3),>> then P(x) = (c_1 * x + 7) * (c_2 * x + 7) * ( c_3 * x + 1)>> implies that c_1 = -7 / x_1, c_2 = -7 / x_2, c_3 = -1 / x_3,>> for some permutation of the x_i. Again, c_1 * c_2 * c_3 = 49,>> as required, since x_1 * x_2 * x_3 = -1.Therefore, c_1 and c_2 both satisfy the equationc^3 * P(-7 / c)/49 = c^3 - 21*c^2 + 245*c - 343and c_3 of course satisfiesc^3 * P(-1 / c)/49 = c^3 - 3*c^2 + 5*c - 1so it turns out that ts time, James, you got it more>> or less right. :-) (And c_3 is even a unit, to boot.>> Come to tnk of it, so are the x_i, wch is probably>> one big reason why ts particular case actually works.)Also, it is obvious that c_1 and c_2 have factors of 7,>> as well, in the ring of algebraic integers.>> But, of course, you're looking at *functions* of x, as you have f_1(x) = c_1 x, f_2(x) = c_2 x, and f_3(x) = c_3 x, so I could also write it as>> (f_1(x) + 7)(f_2(x) + 7)( f_3(x) + 1) = 49(x^3 + 5 x^2 + 3x + 1).Notice that dividing both sides by 49 gives (f_1(x)/7 + 1)(f_2(x)/7 + 1)( f_3(x) + 1) = x^3 + 5 x^2 + 3x + 1as long as you're in a ring where 7 is not a factor of 22.OK, stupid question. Where did the 22 come from?Oh, I've been used to arguing with people about(5 a_1(x) + 7)(5 a_2(x)+ 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22)where the a's are roots ofa^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x)and b_3(x) = a_3(x) - 3, wch is a substitution made because a_3(0) => 3, so that I can isolate constant terms.My point is that certain rules apply as even with my example above, I> just have rather basic functions of x.Great. Try your logic on ts polynomial.(c_1 x + 7)(c_2 x + 7)( c_3 x + 8) = 49 * (x^3 + 8).If you don't like that one, try ts one:(c_1 x + 7)(c_2 x + 7)( c_3 x + 2) = 49 * (x^3 + 2).>> But yes, for ts particular example, your odd math actually does>> work, as you are dealing with x_i wch are in fact units.So you believe that math is quirky?Your math sure === of a set of pointsI've got a list of points (in x,y form) that are the corners of a> polygon, in order. It's not necessarily a convex polygon. Is there a> standard algorithm for determining whether a given point is inside> or outside the polygon? All ts takes place in the plane... For> practical purposes, an algorithm is ok if it can handle up to 7 points> decently. Many -Adrian> Ts is probably in the faq at comp.programming.games.algorithms, or > sometng like that. Google will probably also help.I know it's in the comp.grapcs.algorithms FAQ.http://www.faqs.org/faqs/grapcs/algorithms-faq/Check out === It doesn't seem as if your personality disorder is> curable and that strikes me as very sad.How many psychotherapists does it take to change a lightbulb?Only one, but the bulb _really_ has === down?C.Stevens scribbled the following:>> If you allow finite numbers constructed from a series of mathematical>> formulae, I tnk Graham's number takes the cake, by far.> FYI,Harvey Friedman's lower bound for n(4) in s block subsequence theorem is> much larger than Graham's number- it involves the Ackermann numbers A(n,n).> let A(n) equal A(n,n)> A(A(A........(A(1)..) where there are A(187,196) A'sAren't Ackermann's functions sometng like A(1,1)=1+1, A(2,2)=2*2,A(3,3)=3^3, A(4,4)=(((4^4)^4)^4)^4, and so on? In that case A(187, 196)must be pretty huge.I read that webpage about writing big numbers on a piece of paper. IfI am allowed to refer to Ackermann's function only by name, I tnk Ican write a finite natural number that is much larger than HarveyFriedman's number.-- /-- Joona Palaste (palaste@cc.helsinki.fi) - Finland ---------- http://www.helsinki.fi/~palaste --------- rules! --------/My === Cardinality of 2^n numbers?W. Dale Hall scribbled the following:> Ts is simply not so. If K is the set of powers of 2, it is just not> the case that 2^K is the set of all natural numbers. Since you're the> one making ts assertion, I'll ask you to tell me how one associates> a natural number with an arbitrary subset of K, so that every natural> number is matched with a subset, and so that no two different subsets> are matched to the same natural number. According to Cantor's theorem> it cannot be done. Given any set X, its collection of all subsets has> more elements than X.OTOH, isn't N bijective with the set of the *sums* of the *finite*subsets of the powers of 2?-- /-- Joona Palaste (palaste@cc.helsinki.fi) - Finland ---------- http://www.helsinki.fi/~palaste --------- rules! --------/Life without ostriches is like coffee with milk. - Mika P. === conjugate of a subgroup be a proper subset of it?Surely it can (consider the subgroup generated by the matrix (Mathematica notation) {{1,1},{0,1}} inside GL_2(Q)). Why do you === differentiable...problem...> if f is differentialbe on (0, infinite)and lim [f(x) +f'(x)] = L (x->infinite)show that lim f(x) = L (x->infinite) and lim f'(x) = 0 (x->infinite)> Everyone responds with L'hospital's rule, but that hardly helps onesee *why* it is true. Here's how I look at it (although ts is NOTrigorous so dont use it in your answer).Let O be an extremely huge number and o an extremely tiny one. Precisely, let O be so large that f(w)+f'(w) is witn a given, tinyreal epsilon of L for all w>O (wch we know we can do by hypothesis). Then we should have sometng likef(O) + f'(O) = L +/- oIn other words, f(O) + f'(O) ~= L (~= means approximately equal to)We can approximately rewrite f(O) as L-x and approximately rewritef'(O) as x. Here x is conjured out of tn air by the fact that(L-x)+x=L. What we are asked to prove is basically that x is verysmall. So suppose that, on the contrary, x is not small at all. Butx is approximately f'(O), so if it is not small, we are forced toconclude that the rate of change of f at O is not small either. Butwe stipulated that for all w>O, f(w)+f'(w) is witn epsilon of L. Sowe must conclude that the rate of change very quickly balances outso as not to violate ts stipulation. But how quickly must ts beaccomplished? Since epsilon can be made arbitrarily small, no matterhow quickly the rate of change balances itself out, it just ain'tquick enough... so we're forced to conclude that x is, in fact, small,wch is to say that f'(O) is approximately 0 and f(O) isapproximately L.(I must reemphasize ts is completely nonrigorous and very handwavey. The point is to make it clear *why* the proposition is true- L'hospital's rule is calculus's version of induction, it gets the jobdone but doesn't explain much)P.S. I find it somewhat amusing how in your postings you try topresent yourself as some sort of stereotypical ghschool-agedhot-girl. I'm guessing you are really a 40 year old male truckdriver from Texas? === of a subgroup be a proper subset of it?Because if H is a subgroup of G, h in H and g in G with gHg' asubset of H, then g'Hg is also a subset of H. What does thatimply about the === recommendation I am a senior and I will be needing to fill out applications for graduateschool. My problem is that I am on the quarter system and the quarter is aboutto end. We will not resume until the beginning of January. I'm pretty shy andhave not asked any of my professors for letters. I'm just not sure how to askthem and now that I basically have no time left, I was just wondering how Ishould go up to them and ask. I have had the same professors for a lot ofclasses in some cases but I'm not sure whether or not lower level classes wouldgive a professor enough to say anytng about me. Is there any standard way togo about ts? Most schools seem to want three and for me === map IN -> ZZ^3 (for computation in MAGMA)?Your help would be === core issues>>I've been tnking about my problems with getting any kind of>>admission that my math arguments showing the core error in mathematics>>are correct, so I've gone to marketing books.You know, in case you're curious, ts sounds really really stupid.>> If your results were correct you'd be able to convince people of>> them by explaining the proofs carefully. But in fact they're wrong,>> people continually explain what the errors are, and tactics from>> marketing books are not going to change that.If s results were correct, the same people who point out errors to m >> with saintly patience would instead have provided all these explanations.>Having spent some time now reading marketing tactics, I can clearly>see that both Ullrich and Bau are selling a viewpoint to the>readersp.>Here the assertion is that if I were right then people would>necessarily agree with me!!!>Is that true in your experience?It's certainly true in _my_ experience that when I'm right about> sometng and have a valid proof that I'm right then competent> mathematicians agree I'm right, after I've explained the proof,> yes. That includes cases where they were certain at first I was> wrong, by the way.Don't you agree, , that from C1-C4--indeed, from C3,C4 alone--any> *competent* mathematician could have deduced Ex~(x=x)?C1 AxAy[x=y -> Az(z in x <-> z in y)]> C2 AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)] > C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in> A)Classification> C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}] (Weak> Extensionality)someone will point out the error.?Did your Homies correct you? Or did they *defend* your mistake--> and you for having made it? --fuffyThe police and so forth only exist insofar as they can demonstratetheir authority. They say they're here to preserve order, but in factthey'd go absolutely mad if all the criminals of the world went onstrike for only a month. They'd be on their knees waiting for acrime. That's the only existence they have.William S. Burroughs (American writer) === supplemental text to aid in my learning of ODE andPDE next semester. Currently my undergraduate text is ElememntaryDifferential Equations with Boundry Value Problems by Derrick andGrossman. Ideally, I would appreciate an text wch emphsizestheoretical foundation over applied examples. Any === Math recently came up with the excellent idea of using modern> advertising techniques to get s theories through to the public. I> have decided that ts is an excellent way for me to promote my> anti-Cantorian doctrine to the public. Now, it is well known that sex> sells. Therefor, without further ado (please maximize your window): > I hope that you enjoyed ts complimentary ASCII babe, brought to you> free as a gift from your friends at the mathematical advertising> society.Your dear friend,> Nathaniel Deeth> Age 11> fuffyThe police and so forth only exist insofar as they can demonstratetheir authority. They say they're here to preserve order, but in factthey'd go absolutely mad if all the criminals of the world went onstrike for only a month. They'd be on their knees waiting for acrime. That's the only existence they have.William S. Burroughs (American writer) === convergence>!>It was noted several times in Google Groups (by Ronald Bruck), then>uniqueness of Banach limit of a sequence is equivalent to almost>convergence of ts sequence. Can you give me a nt where to find a>proof of ts result. (Web resources are prefered, but also book or>paper in a journal would help.)Have you tried searcng the _web_ using Google? If you typeBanach limit almost convergenceinto the search box you get a few ts, one of wch says ts> result was proved by Lorentz in A contribution to the theory> of divergent sequences, Acta Math 80(1948).in advance!> Sleziak> The police and so forth only exist insofar as they can demonstratetheir authority. They say they're here to preserve order, but in factthey'd go absolutely mad if all the criminals of the world went onstrike for only a month. They'd be on their knees waiting for acrime. That's the only existence they have.William S. Burroughs === convergence & differentiation >In many analysis-type texts I've found statements like since the series>converges uniformly, we can apply term-by-term differentiation and uniform>convergence allows us to differentiate term-by-term, but I've never>actually seen a proof or explanation of WHY you need uniform convergence to>legitimize term-by-term differentiation. Can someone explain ts? First we need to clarify exactly what result we're talking about! Your> post _sounds_ like you're saying you've read ts:(1) If the sum f = sum(f_n) converges uniformly then f' = sum(f_n').I hope you've never read that, that statement is simply false.What's true is ts:(2) If f = sum(f_n) and if sum(f_n') converges uniformly then > f' = sum(f_n').Yes, uniform convergence is needed in (2). Some people have pointed> out that uniform convergence is not a necessary condition; it's> not - that's not what it means to say the condition is needed,> what I mean by that is that if we simply omit the condition> (2) becomes false. That is, (3) is false:> (3) If f = sum(f_n) and if sum(f_n') converges (pointwise) then > f' = sum(f_n').A simple counterexample to (3): Let I_n be the ooen interval> (1/(n+1), 1/n). Choose a differentiable function f_n such that> f(n)x = 0 for all x not in I_n, but such that f_n = 1 at the> center of I_n. Then the two series f(x) = sum(f_n(x)) and > sum(f_n'(x)) both converge for _all_ x, but f is not even> continuous at 0, hence not differentiable at 0.> The police and so forth only exist insofar as they can demonstratetheir authority. They say they're here to preserve order, but in factthey'd go absolutely mad if all the criminals of the world went onstrike for only a month. They'd be on their knees waiting for acrime. That's the only existence they have.William S. Burroughs === Set] (1/n)>I recall seeing a theorem saying> Sum [n in Set] (1/n) converges iff (some criterion on Set)> There is also a conjecture of Erdos wch I tnk states that> the sum diverges iff Set contains arbitrarily long arithmetic> progressions.>That should be => , not iff . Obviously we could have arbitrarily>long arithmetic progressions and still have convergence, e.g. if>Set = { 10^k + m , 0 < m < k }. I have a notes saying Erdos offered>$3000 for a proof of => . You can still collect.>>In the future, you can post or mail; doing both is unnecessary.>Maybe, but one of these days I'll learn that what IS necessary is to>remember to flag my email so that it doesn't look like it was (also) a>newsgroup post ...Well, I for one was pleased to see s reply to your email (although> puzzled where the post from you was) because I didn't know about the> Erdos tng...>dave> The police and so forth only exist insofar as they can demonstratetheir authority. They say they're here to preserve order, but in factthey'd go absolutely mad if all the criminals of the world went onstrike for only a month. They'd be on their knees waiting for acrime. That's the only existence they have.William S. Burroughs (American writer) === needed?The police and so forth only exist insofar as they can demonstratetheir authority. They say they're here to preserve order, but in factthey'd go absolutely mad if all the criminals of the world went onstrike for only a month. They'd be on their knees waiting for acrime. That's the only existence they have.William S. Burroughs (American writer) === functionThe police and so forth only exist insofar as they can demonstratetheir authority. They say they're here to preserve order, but in factthey'd go absolutely mad if all the criminals of the world went onstrike for only a month. They'd be on their knees waiting for acrime. That's the only existence they have.William S. Burroughs (American writer) === functionThe police and so forth only exist insofar as they can demonstratetheir authority. They say they're here to preserve order, but in factthey'd go absolutely mad if all the criminals of the world went onstrike for only a month. They'd be on their knees waiting for acrime. That's the only existence they have.William S. Burroughs (American writer) === functionThe police and so forth only exist insofar as they can demonstratetheir authority. They say they're here to preserve order, but in factthey'd go absolutely mad if all the criminals of the world went onstrike for only a month. They'd be on their knees waiting for acrime. That's the only existence they have.William S. Burroughs (American writer) === convergenceThe police and so forth only exist insofar as they can demonstratetheir authority. They say they're here to preserve order, but in factthey'd go absolutely mad if all the criminals of the world went onstrike for only a month. They'd be on their knees waiting for acrime. That's the only existence they have.William S. Burroughs (American writer) === circumference of a circle is proportional to it's radius?The police and so forth only exist insofar as they can demonstratetheir authority. They say they're here to preserve order, but in factthey'd go absolutely mad if all the criminals of the world went onstrike for only a month. They'd be on their knees waiting for acrime. That's the only existence they have.William S. Burroughs (American writer) === sha1:HK6y10cGWFXc8XfymnCirFJxIUU=> The concept of free speech encompasses the right of people to say stupid > and hateful tngs. The Supreme Court said you can burn the American > flag or claim that Falwell had sex with s mother in an outhouse, > and that those tngs are protected speech. I for one defend your right > to say whatever the hell you want.You can burn the flag, but you have no right to say that Falwellhad sex with s mother in an outhouse (unless it's true). There arelibel laws.-- Jesse Hughes If you really tnk there's a bug you should report a bug. Maybeyou're not using it properly... It turns out Luddites don't know howto use software properly, === There are no other solutions... how to prove it? how can I prove that the ONLY solutions of the ODE y'' = -y are the ones given by A cos(x) + B sin(x) ?Easy: Wronskian W(cos,sin) = cos^2 + sin^2 = 1 != 0, soVariation of Parameters (2) specialized to _homogeneous_ case shows that any other solution is a linear combination of cos, sin with _constant_ coef's. Below are easy proofs,wch generalizes to gher-order LODEs (and recurrences).The proof is slicker in matrix form, e.g. Lemma 4.1 in (3).THEOREM If f,g,h are solutions on an interval I of the LODE y'' = p y' + q y, p,q continuous on Iand gh'-g'h != 0 for all x in I, then there exist constants c,dsuch that f = c g + d h on IPROOF: The below equations [0],[1] have a unique solution (c,d)since det = W(g,h) = gh'- g'h != 0 on I.[0] f = c g + d h[1] f' = c g' + d h'[2] qf+pf' = f = c g + d h via q[0]+p[1], g=qg+pg', h=qh+ph'[3] 0 = c'g + d'h via [0]'-[1][4] 0 = c'g' + d'h' via [1]'-[2] (1b)The above equations [3],[4] have unique solution (c',d') = (0,0)since det = gh'-g'h = W(g,h) != 0 on I. Thus c,d are constants.-------THEOREM If f,g,h are solutions of the recurrence y'' = p y' + q y, where y'(n) := y(n+1)with Wronskian W = gh'-g'h != 0 then there exist constants c,dsuch that f = c g + d hPROOF: [0],[1] below have unique solution (c,d) via det = W != 0[0] f = c g + d h[1] f' = c g' + d h' Now q[0] + p[1] yields:[2] qf+pf' = f = c g + d h via qg+pg' = g, qh+ph' = h[3] 0 = (c'-c)g' + (d'-d)h' via [0]'-[1][4] 0 = (c'-c)g + (d'-d)h via [1]'-[2]The above equations [3],[4] have solution (c'-c,d'-d) = (0,0),unique via det = W' = g'h-gh' != 0. So c,d are constants,since c' = c means c(n+1) = c(n).-Bill Dubuque(1) L. E. Pursell: A simple uniqueness theory for ordinary linearhomogeneous differential equations, Amer. Math. Monthly, 74, 1967, 47-50 http://links.jstor.org/sici?sici=0002-9890(196701)74:1%3C47%3E (2) Variation of Parametershttp://planetmath.org/encyclopedia/ VariationOfParameters.htmlhttp://ltcconline.net/greenl/courses /204/appsgherOrder/variationgher.htm(3) Marius van der Put: Symbolic analysis of differential equationshttp://msri.org/activities/programs/9899/focm/soggy/ MSRIintro/van_der_Put2.psd:Symbolic analysis of differential === distributed digits, does 1/x?> If so, how do you prove its normality?He can't. That's why he said it was s guess, not s theorem. Well at least it would be nice to get an idea why such a reciprocalwould be normal.>No one knows how to prove pi is normal, but the smart money >is all riding on that guess. Same for the suggested number.Smart money? Did I miss the JSH === advice on letters of recommendation> I am a senior and I will be needing to fill out applications for graduate> school. My problem is that I am on the quarter system and the quarter is > about> to end. We will not resume until the beginning of January. I'm pretty shy > and> have not asked any of my professors for letters. I'm just not sure how to > ask> them and now that I basically have no time left, I was just wondering how I> should go up to them and ask. I have had the same professors for a lot of> classes in some cases but I'm not sure whether or not lower level classes > would> give a professor enough to say anytng about me. Is there any standard way > to> go about ts? Most schools seem to want three and for me that is a lot.> What worked for me many years ago was to ask the professors of the upper-division classes where I had done well. I agree it can feel a little embarrassing to feel like you have to hustle like ts, but the profs are quite accustomed to it, it's part of their jobs. Just pick the three profs who have seen you do your best work, and go to them during office hours and simply ask for a letter. They've done === Newsgroup survey: Math and personality assessment proof is an argument wch beginning with some true statement> proceeds by logical steps to a conclusion wch then must be true.James, ts is not the definition of a proof. Not all proofs beginwith a true statement (read: with an axiom, I suppose). Very manyproofs begin with an assumption -- an assumption that may be provenfalse in the end.I know you're stupid. You can't even grasp the use of commas whenaddressing your correspondents. Nonetheless, try to grasp ts plain,basic fact. It is not the case that a proof is an argument beginningwith some true statement.You pretend to be a logical man. In order to make the pretensionslightly more plausible, you should drop ts baby definition of aproof. If you'll actually read a proof or two, you'll realize howstupidly inadequate your characterization is.-- Jesse HughesBasically there are two angry groups. I am a harsh force ofone. Against me is a society of mathematicians. So far it's been adraw. -- === NEED HELP BADLY (sorry, maths not psych)Expires: 28 days>> If fields acted instantaneously, we would be> able to use them for instantaneous communication.>>Please explain how you can use the fact that a force is>>a static electric field, for instantaneous communication.>>Paul, puzzled>> I'm sure some bright QMian would find a way.Backing out again, Henry?>You cannot defend your assertion,>but you will repeat it, won't you?>And you will back out once more when>asked to defend your repeated assertion, won't you?That's Henry Wilson's eternal circle of fleeing>restatements of fled assertions.Paul, not surprised>Paul, it should be obvious that if the effect of a field is instantaneousthroughout the universe, then switcng it on an off in an intelligent mannercan be used for instantaneous communication.Henri Wilson. See the Stupidity of === Need advice on letters of recommendation I have another (probably stupid) question. If you're applying tomultiple schools and they all want a letter of recommendation, does theprofessor have to print out a lot and sign them all or can === question) folks,Okay, ts is going to be some very stupid question to most of you.The axiom of foundation says that: ForAll B ~= emptyset, ThereExists y in B such that (y intersect B)=emptyset.Now suppose B = {lime, orange, lemon}. What is such y among ts setof citruses? Suppose y = lime. How can lime be a set? (myunderstanding is that only sets can intersect).I told you, it is a stupid question. But I want to === I have another (probably stupid) question. If you're applying to> multiple schools and they all want a letter of recommendation, does the> professor have to print out a lot and sign them all or can I xerox them?Ask someone in the math department office, for example the office manager. As I recall, they didn't give me my letters, in fact I never saw them. You just tell the prof what schools and the rest gets === stupid question)Jared> The axiom of foundation says that: ForAll B ~= emptyset, ThereExists y in B such that (y intersect B)=> emptyset. Now suppose B = {lime, orange, lemon}. What is such y among ts set> of citruses? Suppose y = lime. How can lime be a set? (my> understanding is that only sets can intersect).lime intersect B = emptysetMaybe the confusion is due tolime <> {lime}and indeed{lime} intersect B === stupid question)> folks,Okay, ts is going to be some very stupid question to most of you.The axiom of foundation says that: ForAll B ~= emptyset, ThereExists y in B such that (y intersect B)=> emptyset.Now suppose B = {lime, orange, lemon}. What is such y among ts set> of citruses? Suppose y = lime. How can lime be a set? (my> understanding is that only sets can intersect).I told you, it is a stupid question. But I want to know.It's not stupid, just based on a bit of a confusion between formal and intuitive set theory.The tng about formal set theory (or at least ZFC formal set theory - I tnk there are versions where ts isn't true) is that *everytng* is a set. For example 0 = empty set, 1 = {0}, 2 = {0, 1}, etc. You define the natural numbers in terms of sets, and then build up from there. When tngs get complicated enough tngs stop *looking* like sets - for example writing down the real number 1 as a set would be ghly non-trivial (and very dependent on how you constructed the real numbers), but the point remains that at some fundamental level they are still sets.So you couldn't form a set like B = {lime, orange, lemon}, because limes, oranges and lemons aren't sets (well, you could give the name 'lime' to a set, but it would still be a set).Hope that === permission for an emailed response.X-Tom-Swiftie: You've put too much peanut butter on the sandwich, Tom said tckly> I have another (probably stupid) question. If you're applying to> multiple schools and they all want a letter of recommendation, does the> professor have to print out a lot and sign them all or can I xerox them?When I applied to grad school (not in math, but the procedure isbasically the same regardless), each school had a separaterecommendation form.I provided each professor with a labelled envelope, the form for eachschool to wch I was applying. I had filled out the applicantportion of each form and as much of the recommender portion as madesense, and then gave each professor their stack.Most of the schools wanted the recommenders to mail theirrecommendations directly to the school; for those, the labelledenvelope was completely addressed and stamped. Other schools wantedthe recommenders to give sealed signed-across-the-flap envelopes tothe applicant, and for that purpose I gave the professor a largeenvelope addressed to me to put them all in and drop in the mail tome.I made it my responsibility to do as much as I could to minimize thehassle on the part of the recommenders, wle being careful topreserve the confidentiality of the recommending process. === into a flame war, but coming from sci.math> I have to object to what seems to me to be some false> theorems in your reasoning, namely that> politically liberal => PETA> Sierra Club => environmental terrorist> environmentalist => every wacko animal rights sentimentI would describe myself as a political liberal and an> environmentalist (to the point of bicyling across> Pladelpa to work on occasion). But I don't> identify with anytng on the right hand side.I refuse to get into an argument about my positions,> or yours. I just want to point out that you have> dismissed the other side of the political spectrum> as not being a spectrum. Randy, dear boy, if I might jump in here, it's rather quite axiomatic> that liberals never identify themselves as being liberal in their> politicsReally? Liberals never call themselves liberal? Areyou sure? How do you read the sentence beginningI would describe myself...? Just curious, if you'regoing to throw words like axiomatic around.> the presumption being that liberals by definition (their> definition, BTW) are open to all views they see as reasonable (again,> their definition of what's reasonable).So you would say all liberals consider themselves opento, say, Ann Coulter or Rush Limbaugh? Where are yougetting ts from?> As for your above equations, they ignore the larger reality that most> PETA freaks are overwhelmingly liberal in their politics.No they don't. Again, I'm not going to get into a politicalwar, just commenting on the error in logic. Your statementis that A => B, where A is person X is a PETA freakand B is person Y is a liberal.That does not in any way mean that B => A.> So, too, is> the wte wine and brie crowd who by and large make up the Sierra Club> constituency. As for the environmentalists, as a group, if anytng> they're even worse than liberals in their politics, the vast majority> of them being Bolsheviks.Bolsheviks? Are you a fan of Tom Potter? He's the onlyother person I know to use that term to describe peoplepost-1920 or so.> No, dear man, these people can be called a> lot of tngs, but conservative or republican aren't numbered among> them.Nor did I imply that it should be.Consider the phrase political spectrum. The politicalspectrum is actually a multivariate space, and there's lotsof room in the quadrant called liberal to separateenvironmentalist from eco-terrorist or PETA freak.Again you seem to be making the same logical error(s)that cause you to be unable to recognize that people,even liberal people, have multiple attributes thatdistinguish their political views.By the way, I'll also point out that democrat/republicanis not equivalent to liberal/conservative, thoughthere is certainly a gh degree of correlation. Up tilla couple of years ago, I lived in a liberal congressionaldistrict in a liberal state with a well-loved Republicancongressional representative.In fact, prior to the early 80s, I identified myself asa liberal === help.I'm having trouble setting up these 3 integrals. Let S be the integralsymbol.1) Evaluate the integral SS (over R) dA/(1 + X^2 + Y^2)^2 over the regionenclosed by one loop of the lemniscate (X^2 + Y^2)^2 - (X^2 -Y^2) =0.2) Find the surface area of the portion of the sphere X^2 + Y^2 + Z^2 = 16between the planes Z = 1 and Z = 2.3)Find the volume cut from the solid sphere X^2 + Y^2 + Z^2 = a^2 by thecylinder r = a * sin === the fastest algorithm for computing factorial, for very large > numbers (e.g. 10000!)?Normally ts would take n-2 multiplications, by multiplying out each > term n(n-1)(n-2)(n-3)...3.2. Is a better way known?Just do the multiplications carefully in the right order. Multiplying k-digit numbers takes O(k log k) time using the best known algorithms, and obviously requires at least O (k) time. If you start with 1, multiply by 2, multiply by 3, and so on, the numbers involved get larger and larger, so the time will be O (n^2 log^2 n). Instead you should use a binary tree for the multiplications: To find the product of k numbers for k > 2, divide the numbers into two sets of equal size, calculate the product of the numbers in the first set, the product of the numbers in the second set, and multiply the products. That way, you should be able to calculate n! in O (n log^3 n). (Actually, I wouldn't count 10000 as a very large === police and so forth only exist insofar as they can demonstrate> their authority. They say they're here to preserve order, but in fact> they'd go absolutely mad if all the criminals of the world went on> strike for only a month. They'd be on their knees waiting for a> crime. That's the only existence they have.William S. Burroughs (American writer)> Guardian, 1966(off-topic)When I heard that Bill Burroughs had died, I immediately askedhow could they tell?.(apologies to === issues>>I've been tnking about my problems with getting any kind of>>admission that my math arguments showing the core error in mathematics>>are correct, so I've gone to marketing books.You know, in case you're curious, ts sounds really really stupid.>> If your results were correct you'd be able to convince people of>> them by explaining the proofs carefully. But in fact they're wrong,>> people continually explain what the errors are, and tactics from>> marketing books are not going to change that.If s results were correct, the same people who point out errors to m >> with saintly patience would instead have provided all these explanations.>Having spent some time now reading marketing tactics, I can clearly>see that both Ullrich and Bau are selling a viewpoint to the>readersp.>Here the assertion is that if I were right then people would>necessarily agree with me!!!>Is that true in your experience?It's certainly true in _my_ experience that when I'm right about> sometng and have a valid proof that I'm right then competent> mathematicians agree I'm right, after I've explained the proof,> yes. That includes cases where they were certain at first I was> wrong, by the way.Don't you agree, , that from C1-C4--indeed, from C3,C4 alone--any*competent* mathematician could have deduced Ex~(x=x)?C1 AxAy[x=y -> Az(z in x <-> z in y)]C2 AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)] C3 EyAx[x in y <-> Et(x in t) & A] (with y not free inA)ClassificationC4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}] (WeakExtensionality)someone will point out the error.?Did your Homies correct you? Or did they *defend* your mistake--and you for having made it? === anyone know if an errata exists for ts text? I wanted to checkbefore going through the drudgery of contacting the publisher. As faras I can tell there is no errata available on === advanced conceptsIn sci.physics, <3c65f87.0311120919.271d4d70@ I've changedthe equations slightly for various reasons, so am not usingthe customary chevrons. However, as far as I can tell,James is of the opinion that the equation doesn't matter;it should work for equations. (I do not share tsopinion, mind you, but am wondering what the issues are.)My comments are indented in (), witn the emulated text.[begin emulation/check]If one sees(c_1 x + 7)(c_2 x + 7)( c_3 x + 15) = 49(x^3 + 9*x^2 + 23*x + 15),with the c's algebraic integers, I tnk few of you would havea problem realizing that only two of the c's have 7 as a factor. (Note that: (x^3 + 9*x^2 + 23*x + 15) = (x + 1) * (x + 3) * (x + 5). Since I happen to know the roots (by construction) of ts particular polynomial, let's see how far we get with ts logic. It is not clear whether JSH is arguing witn the space of all degree 3 polynomials in Z[x] with a_3 = 1, or the space of all degree 3 polynomials in Z[x] with a_3 = 1 and a_0 = 1, or even the rather esoteric space of all degree 3 polynomials in A[x] with a_3 a unit, where A is the ring of all algebraic integers.)But of course, you're looking at *functions* of x, as you havef_1(x) = c_1 x, f_2(x) = c_2 x, f_3(x) = c_3 x,so I could also write it as(f_1(x) + 7)(f_2(x) + 7)( f_3(x) + 15) = 49(x^3 + 9*x^2 + 23*x + 15) (Yeah, so?)Notice that dividing both sides by 49 gives(f_1(x)/7 + 1)(f_2(x)/7 + 1)( f_3(x) + 1) = x^3 + 9*x^2 + 23*x + 15as long as you're in a ring where 7 is not a factor of 1. (That is one representation of result, yes. Other representations are possible, such as (f_1(x)/49 + 1/7)(f_2(x) + 7)( f_3(x) + 1) = x^3 + 9*x^2 + 23*x + 15. Also, see below.)That's an important point and represents an issue over wch I'vefound a lot of people willing to argue, and it may seem vague to moveto functions, even though in the previous example they were actuallyfunctions but they weren't being *called* functions.So some rules need to be outlined in generalizing from the basicpolynomial factors with their simple functions like c_1 x, where c_1is constant, to more complicated ones that we might not even haveimagined yet.One tng that's clearly important is that the functions *must* equal0 when x=0, as then you have factors of the constant term oppositethem. (Not a problem here so far.)For instance(f_1(x) + 7) versus (f_3(x) + 15), where at x=0, both functions equal0, and 15 and 7 are both factors of the constant term. (Wch they are.)Next, the factorization must multiply out correctly, wch just meansthat(f_1(x) + 7)(f_2(x) + 7)( f_3(x) + 15)multiplies out to give49(x^3 + 9*x^2 + 23*x + 15).so f_1(x) f_2(x) f_3(x) = 49, for instance. (Actually, 49 * x^3. And even that may not be the case; one could, for example, set f_1(x) = 49(x^3 + 9*x^2 + 23*x + 15) - 7, f_2(x) = -6, f_3(x) = -14. The product of these would be a multiple of 49 * 12, not 49 * x^3. I'm not even certain one can conclude that their product is divisible by 49; consider f_1(x) = -6, f_2(x) = -6, f_3(x) = 49(x^3 + 9*x^2 + 23*x + 15) - 14, The product there is divisible by 7, that much I can tell you.)I'm abstracting and generalizing to functions because I've facedarguments with a much more complicated example, where the basicprinciples are the same:(5 a_1(x)+ 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22)where the a's are roots ofa^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) (I've left the above two equations intact. Unlike my example above the polynomial in () is irreducible in Z[x] -- and for that matter in Q[x].)so they are functions of x, and since one of the roots equals 3 atx=0, I haveb_3(x) = a_3(x) - 3, so that I can see all the constant term factors.What you have just seen is a major advance in mathematical tnkingwhere I've used a rather simple abstraction and a special polynomialto analyze the *roots* of another *different* polynomial. (Looks more sloppy than advanced to me; JSH is leaping to some strange conclusions. For example, given that (c_1(x) + 7)*(c_2(x) + 7)*(c_3(x) + a_0) = 49(a_3*x^3 + a_2*x^2 + a_1*x^1 + a_0) I can conclude very little about the c_i() functions. If one postulates that they are witn Q[x] of degree 0 or 1, one might fare slightly better -- but that's a postulate JSH has currently not made.)[end of === working in the study of some crystals with vacancies. Crystal arestudied like an orderen arrangement of atoms that forms a net. In one dimension ts can be represented like a sequence of points at theseme distance:(i would represent the unit cell, ts part of the secuence isrepetead infynit times.). . . . . . . . . . . . . . Now, if we have put some vacancies (faults of atoms) The secuence canbe. . . . . . . . . . . I want to know if there is a definition of uniformity in the folowingform:The most uniform distributions of the vacances in such a sequence iswhen the vacancies are as far as we can from each other. (There is asmal trick that the secuence is repetead so, the end and begining ofthe secuence are conected)TIAZunbeltz IzaolaUniversity of the Basque Country-- Remove XXX from === explicit bijective map IN -> ZZ^3 >(for computation in MAGMA)?>Start withan explicit bijection f between N and Z (you can code the sequence 0,1,-1,2,-2,3,-3, ... using (-1)^n judiciously)an explicit bijection g between N and N^2, e.g., n -> (g1(n),g2(n)), where n = 2^(g1(n)) * (2*g2(n)+1)then n -> (g1(n),g2(n)) -> (g1(n),g1(g2(n)),g2(g2(n))) -> (f(g1(n)),f(g1(g2(n))),f(g2(g2(n))))shows how to compose these into the desired bijection.Note: g1(n)=floor(2log(n)) and g2(n)=(n*2^(-g1(n))-1)/2KP-- E-MAIL: K.P.Hart@EWI,TUDelft,NL PAPER: Faculty EWIPHONE: +31-15-2784572 TU DelftFAX: +31-15-2786178 Postbus 5031URL: http://aw.twi.tudelft.nl/~hart === Complex Analysis books?> The subject says it -- however, I'd like to clarify> one detail: I'm looking for a book on *analysis*, as> opposed to Calculus (i.e., that covers rigorously> the concepts and proofs on Complex numbers and> Complex variables functions). However, I'm just a hobbyist, so I'm not looking for> the ultimate, advanced reference book (i.e., I'm not> a mathematician or even a student in Mathematics; I'm> an engineer, who already knows (at least *knew* very> well :-)) about complex numbers, but I'm beginning> to appreciate and enjoy the rigorous side of maths,> and I find that complex numbers do have a great> appeal. Others have recommended Churcll, Brown, & Verhey; if you have not studied complex variables at all, I would reommend that as a *start.* Afterwards (or if you already know the equivalent of most of CBV), take a look at Conway, Functions of One Complex Variable. -- Stephen J. Herschkorn === proof> Show with combinatorial reasoning thatC(3n,3) = 3*C(n,3)+6n*C(n,2)+n^3, where C(n,k) is n choose k.Am I supposed to show that both sides of the equation counts the tng? The text where you found ts exercise should probably explain what they mean by combinatorial reasoning. You should attempt to use their definition. It also might help to try a simpler one:C(2n, 2) = 2*C(n, 2) + n^2> Does it qualify as combinatorial reasoning if I use combinatorial > identities or prove it by induction?In my definition of combinatorial reasoning or proof, sometimes identities or arithmetic or induction can be used, but the -main- intent is to interpret the symbols as statements about combinatorial objects (sets, subsets, tuples, trees, etc) and the manner in wch they are counted. To show an equality (as above) you take one mathematical situation (most likely simply described by one side of the equation), and interpret it another way (corresponding to the other side of the equation).So I would say -first- use combinatorial ideas, and only then for small lemmas might you bother with identities/induction/arithmetic.Of course, the above identity is -very- trivial with arithmetic, just replace C(n,k) with the appropriate factorials and simplify. (No, I can't do that in my head, but I can do the combinatorics very easily in my head.) Trivial means mindlessly, just applying the rules you have without thought.> Any good nts, by the way?I will refrain from commments about the other nt ;).My nts are these:+ corresponds to disjoint union of sets.* corresponds to ...what? ... you can tnk of it as repeated summation, or as first doing one tng, then doing another (i.e. 3*C(n,3) means do sometng in three possible 3 ways first, then do sometng taking C(n,3) ways).n choose k is not only the -number- of subsets of size k out of n.^ corresponds to ... what?... repeated multiplication or tuples or functions from x to y (wch is counterintuitively counted by y^x)Combinatorial reasoning just assigns meaning to the parts of the === compactnessI have a small question wch I have been unable to solve on my own. I>know that if f: M->N is a homeomorpsm, thenK compact in M <=> f(K) compact in Nand I'd like to know if the opposite way holds, that is, if f is>bijective andK compact in M <=> f(K) compact in Nis f then a homeomorpsm? I have proven ts to be true for locally>compact spaces, but I would like to know whether the result holds>generally.>Here is a counterexample:X = N times N plus one extra point *the points in N times N are isolated, the (basic) neighbourhoods of * are of the formU(n,f) = {*} union {(k,l): k>=n, l>=f(k)}where nb runs through N and f through all functions from N to NIn the resulting space all compact subsets are finite, so the identity map betweenX and === initialsJesse F. Hughes> fishfry> The concept of free speech encompasses the right of people to say stupid> and hateful tngs. The Supreme Court said you can burn the American> flag or claim that Falwell had sex with s mother in an outhouse,> and that those tngs are protected speech. I for one defend your right> to say whatever the hell you want. You can burn the flag, but you have no right to say that Falwell> had sex with s mother in an outhouse (unless it's true). There are> libel laws.> Jesse HughesWell, I wouldn't worry about libel from JSH. s manifest bad faith, not tosay incompetence, render m incapable of harming anyone with s words.Moreover, as an ancient legal saying goes, The abuse of a humble tongue ischeap.> If money is your motivation, you'd do better selling penis enlargementpills on the Internet.A good point, not often made in Harrisland. People make money in all mannerof fraud: astrology, formulas to beat the lottery, age-reversing drugs, andso on without limit. But they make money by indulging the _buyer's_ vanityand ambition, not their own! Harris is a narcissist and narcissists wantattention, not money per se. But get famous in mathematics? Grosslyillogical. Ask the man in the street to name one living mathematician. Inall likelihood you'll get Nash (because of the illness and the movie)or else no answer at all.There is another tng that makes math such an illogical area in wch toattempt fraud: The standards of evidence are gher in ts field than inany other. I theorise that someone, maybe a teacher or teachers, once toldHarris that he was good at math; when s narcissism later kicked in, mathwas therefore the area in wch he assumed he must be Superman.One more bit about narcissism. In JSH's mind, these are two incontrovertibleaxioms:1) JSH is right.2) JSH is great.Ts at once explains JSH's inability to learn, or indeed === integral help.> I'm having trouble setting up these 3 integrals. Let S be the integral> symbol.> 1) Evaluate the integral SS (over R) dA/(1 + X^2 + Y^2)^2 over the region> enclosed by one loop of the lemniscate (X^2 + Y^2)^2 - (X^2 -Y^2) =0.Use plane polars (x = r cos(p), y = r sin(p), dA = r dr d(p) )[nt: substitute u = r^2, take the limits of the theta integral as-(pi)/4, pi/4] 2) Find the surface area of the portion of the sphere X^2 + Y^2 + Z^2 = 16> between the planes Z = 1 and Z = 2.Use cylindrical polars (x = r cos(p), y = r sin(p), z = z,dA = r d(p) dz ) 3)Find the volume cut from the solid sphere X^2 + Y^2 + Z^2 = a^2 by the> cylinder r = a * sin (theta).Use spherical polars ( x = r cos(p) sin(theta),y = r sin(p) sin(theta), z = r cos(theta),dx dy dz = r^2 sin(theta) dr d(theta) d(p) )(I'm not sure r = a *sin(theta) is a cylinder; I tnk the cylinder isgiven by a = r *sin(theta) [ x^2 + y^2 = a^2, -oo < z < oo ] )-- P.A.C. SmithThe vast majority of Iraqis want to live in a peaceful, free world.And we will find these people and we will bring them to === came up with the excellent idea of using modern> advertising techniques to get s theories through to the public. I> have decided that ts is an excellent way for me to promote my> anti-Cantorian doctrine to the public. Now, it is well known that sex> sells. Therefor, without further ado (please maximize your window):Brilliant. You've convinced me that you're right, although I've beentemporarily distracted from what the question was.Also, an 11-year-old shouldn't have access to such explicit images. Ihope someone tells your momma.-- I've been tnking about my problems with getting any kind ofadmission that my math arguments showing the core error in mathematicsare correct, so I've gone to marketing books. -- === Re: Newsgroup survey: Math and personality assessment police and so forth only exist insofar as they can demonstrate> their authority. They say they're here to preserve order, but in fact> they'd go absolutely mad if all the criminals of the world went on> strike for only a month. They'd be on their knees waiting for a> crime. That's the only existence they have. William S. Burroughs (American writer)> Guardian, 1966 huffyWhere huffy == Correy (and so is self-identical)?In another post by huffy, you signed (and clearly spoke as Correy). Why the (additional) pseudonym?-- We want a single platform. We're trying to get there using thecarrot, or blackmail, or rewards, or whatever you call it. -- Madison, WI, superintendent Rainwater grasps subtlety in === (sorry, maths not psych) >>> If fields acted instantaneously, we would be> able to use them for instantaneous communication.>>>Please explain how you can use the fact that a force is>>a static electric field, for instantaneous communication.>>>Paul, puzzled>>> I'm sure some bright QMian would find a way.>Backing out again, Henry?>You cannot defend your assertion,>but you will repeat it, won't you?>And you will back out once more when>asked to defend your repeated assertion, won't you?>That's Henry Wilson's eternal circle of fleeing>restatements of fled assertions.>Paul, not surpriseda static electric field in an accelerator, the force doesn't actthe same instant as it enters the field, ts could beused in instant communication.> Paul, it should be obvious that if the effect of a field is instantaneous> throughout the universe, then switcng it on an off in an intelligent manner> can be used for instantaneous communication.Funny, eh? :-)Paul, enjoying the acrobatic === of a subgroup be a proper subset of it?It can!>Because if H is a subgroup of G, h in H and g in G with gHg' a>subset of H, then g'Hg is also a subset of H.That's not true. Take H cyclic subgroup of GL(2,Q) generated byh = ( 1 1 ), ( 0 1 )and let g = ( 2 0 ) ( 0 1 )ghg^-1 = h^2 but g^-1hg = ( 1 1/2 ) ( 0 1 )Derek Holt.> What does that>imply about the equation === <87znf0wyzo.fsf@pwumbda.org> <95Isb.44352$jy.37753@clgrps13> sha1:JW2UgH+Flw0op2Jp6b9KeGaCEMo=> Jesse F. Hughes>> fishfry>> The concept of free speech encompasses the right of people to say stupid>> and hateful tngs. The Supreme Court said you can burn the American>> flag or claim that Falwell had sex with s mother in an outhouse,>> and that those tngs are protected speech. I for one defend your right>> to say whatever the hell you want.>> You can burn the flag, but you have no right to say that Falwell>> had sex with s mother in an outhouse (unless it's true). There are>> libel laws.>> Jesse Hughes> Well, I wouldn't worry about libel from JSH. s manifest bad faith, not to> say incompetence, render m incapable of harming anyone with s words.> Moreover, as an ancient legal saying goes, The abuse of a humble tongue is> cheap.No one was talking about whether JSH would sue for libel. -- Jesse HughesYou see 300 of sometng, anytng, and you go `[Man], that's a lot ofstuff.' -- === Re: Turing macne question>> A Turing macne question:Can you build a Turing macne if - instead of a pen and an>> eraser to mark on the tape you have a *finite* number of paper>> clips, wch can be clipped on and off the tape?You could certainly build such a macne, but its power, as,>> say, a language recognizer, would be severly limited. There>> would certainly be useful tngs such a macne could do-->> unary addition is just one example.The power would not be severely limited unless you mean merely that it > would be slower. Two-counter macnes are capable of simulating > arbitrary Turing macnes. [...] - ts and Minsky macnes proved to be useful search terms.-- __________ |im |yler http://timtyler.org/ === png Magic Square via a free web applicationNow you can create your PNG Magic Square for free using the webapplication you can find at http://www.pivari.com/squaremaker.htmlIt is based on our perl module Math::MagicSquare you can find athttp://search.cpan.org/ , exactly athttp://search.cpan.org/~fpivari/Math-MagicSquare-2.02/If you want to give a similar service inside your web site (internetor intranet) to your students, surfers, ... we can sell you the cgiapplication (a simple executable file that doesn't requirecustomization) for Windows, Linux, Solaris, AIX, HP-UX, FreeBSD,Tru64, SGI, ...Contact us directly at mailto: === does not seem to out of place and I hope that you are able tohelp me. I am about to complete my second year of my maths degree. I am going to bedoing a major in mathematics next year and I a am tnking about doinghonours the year after next. I have been looking into some areas of maths that I might be interested indoing more study in and one of them is that of complex (chaotic) systems.Well pretty much anytng to do with Differential Equations I could beinterested in I just need to find out what they are really about. What I am after is just some good books on the area so that I can get a bitof a taste of it before I decide it is what I want to do further study in.Any of the other areas of applied maths that are rather interesting or newand exciting that would be helpful as well. I mean really anytng as Ireally just want to see what is out === Looking for NTS on proving a limitRonald Bruck Ca.96estro escribi.97 en that limit without> use (e^x)' = e^x?>> I.e., it is possible to prove directly, using only the definition of> derivative, that (e^x)' = e^x?>> Well, Jos.8e H. Nieto give me a satisfactory answer in es.ciencia>> matematicas.>> Lim((1 + 1/x)^x, x, inf) = e>> let x = 1/t,>> Lim((1 + t)^(1/t), t, 0) = e ==> Log(Lim((1 + t)^(1/t), t, 0)) = Lim(Log((1 + t)^(1/t)), t, 0) = 1>> Lim(log(1 + t)/t, t, 0) = 1>> Then, let 1)/y, y, 0) = 1>> With that is possible study the derivative of exponential function>> before than the derivative of the logarithmic function and the>> composite or inverse function. I tnk there's more than a little circularity here. What are you starting from? That lim_{x to infty} (1+1/x)^x = e?> And the x-th power of 1+1/x is...what? Or do you mean to let x only> range through integers? In wch case let x = 1/t has t approacng> zero only through...reciprocals of integers? And what is Log? And why is it continuous? And why is Log (b^x) = x> Log b? And why is Log(1+t) surjective? (So that you can just casually say> Then, let y = Log(1+t)?) IF YOU WANT TO KNOW HOW TO DO TS, READ THE EXERCISES AT THE END OF> CHAPTER 1 OF RUDIN'S PRINCIPLES OF MATHEMATICAL ANALYSIS. There is> a development there of one (non-integral) approach to logarithms and> exponentials. (I don't remember whether the differentiability is> addressed in these exercises or not.) You persist in dribbling bits and pieces, without ever addressing a> clear sequence of ideas or of strategy. The problem of developing> exp,> log, sin and cos without power series, and without using the inverse> function theorem, isn't DIFFICULT, but it requires several steps.> There is no panacea. (Except by power series, where everytng becomes easy.) --Ron BruckTs questions are said in the previous post:The definition of the function f(x) = a^x, a real, a > 0, that I alwaysuse, since secondary school to now, is the function whose value at x isthe number a raised to the power x. And the definition of the number a^xisa^x = Lim(a^(x(n)), n, inf)where Lim(x(n), n, inf) = x, and the x(n) are rationals.e = Lim((1 + 1/n)^n, n, inf)As I said before in that thread. The existence of the limit is no toodifficult to show, viewing that (1 + 1/n)^n is increasing with n andbounded.All that, as the logarithmic function, is previous in the curriculum to theexposition of derivatives. Power series are much later in the curriculum (inspanish secondary school, at least). I was looking for a method to exposethe derivative of f(x) = e^x without the derivative of the inverse functionor the derivative of the composite function (chain's rule). And I found ityet.,Ignacio Larrosa Ca.96estroA Coru.96a === asking for your opinionMy opinion is that if your Differential Equations teacher had been a f***ingmindkilling brainslayer with a psycho-like uncomprehendibility and optionalTotal-Encephalon-Annilation ModeT like my teacher was, then you wouldn'tbe willing to study that field. Luckily yours, it seems, wasn't. Excuse mefor the rhetorics and for the unusefulness of my post, but you know... onone hand, people must know that being into maths doesn't mean that youcannot have a good command of the language, on the other hand, sometimespeople need to feel luckier than someone else to be happy ;-).Stephen ha scritto nel messaggio> I hope ts does not seem to out of place and I hope that you are able to> help me. I am about to complete my second year of my maths degree. I am going tobe> doing a major in mathematics next year and I a am tnking about doing> honours the year after next. I have been looking into some areas of maths that I might be interestedin> doing more study in and one of them is that of complex (chaotic) systems.> Well pretty much anytng to do with Differential Equations I could be> interested in I just need to find out what they are really about. What I am after is just some good books on the area so that I can get abit> of a taste of it before I decide it is what I want to do further study in. Any of the other areas of applied maths that are rather interesting or new> and exciting that would be helpful as well. I mean really anytng as I> really just want to see what is out there just so I have === subgroupsWhy can't a conjugate of a subgroup be a proper subset of it?Surely it can (consider the subgroup generated by the matrix > (Mathematica notation) {{1,1},{0,1}} inside FinlandYes, I realized the result was false shortly after posting. Ts meansthere is an error in Spanier's Algebraic Topology, p. 74. There heclaims that two subgroups of a group are equal because they areconjugate and one is a subset of the other, but actually they areequal for a different === conjugate of a subgroup be a proper subset of it?Because if H is a subgroup of G, h in H and g in G with gHg' a> subset of H, then g'Hg is also a subset of H. You sure? I would have thought a superset, not a subset.> What does that> imply about === will be sure to tell my DE's teacher that > My opinion is that if your Differential Equations teacher had been af***ing> mindkilling brainslayer with a psycho-like uncomprehendibility andoptional> Total-Encephalon-Annilation ModeT like my teacher was, then youwouldn't> be willing to study that field. Luckily yours, it seems, wasn't. Excuse me> for the rhetorics and for the unusefulness of my post, but you know... on> one hand, people must know that being into maths doesn't mean that you> cannot have a good command of the language, on the other hand, sometimes> people need to feel luckier than someone else to be happy ;-).> Stephen ha scritto nel messaggio> I hope ts does not seem to out of place and I hope that you are ableto> help me.> I am about to complete my second year of my maths degree. I am going to> be> doing a major in mathematics next year and I a am tnking about doing> honours the year after next.> I have been looking into some areas of maths that I might be interested> in> doing more study in and one of them is that of complex (chaotic)systems.> Well pretty much anytng to do with Differential Equations I could be> interested in I just need to find out what they are really about.> What I am after is just some good books on the area so that I can get a> bit> of a taste of it before I decide it is what I want to do further studyin.> Any of the other areas of applied maths that are rather interesting ornew> and exciting that would be helpful as well. I mean really anytng as I> really just want to see what is out there === conjugate subgroups > Why can't a conjugate of a subgroup be a proper subset of it?Yes, I realized the result was false shortly after posting. Ts means> there is an error in Spanier's Algebraic Topology, p. 74. There he> claims that two subgroups of a group are equal because they are> conjugate and one is a subset of the other, but actually they are> equal for a different reason.Sorry, but wch chapter, section, etc is that? I only have the Tata-McGraw llpaperback edition available, and there p.74 is at the beginning of section 4of chapter 2 (section titled The Lifting Problem). I couldn't immediatelylocate the claim you mentioned on that page or any === === differentiable...problem...>> if f is differentiable on (0, infinite)>> and lim [f(x) + f'(x)] = L (x->infinite)>> show that lim f(x) = L (x->infinite) and lim f'(x) = 0 (x->infinite)>| f e^x (f + f') e^x>| lim f + f' = L => lim f = lim ----- = lim = L>| x->oo x->oo x->oo e^x x->oo e^x>> Not even the existence of lim(x->oo) f(x) ? >L'Hospital's rule for the form lim f/g, lim g = oo >needs no hypotheses on existence or value of lim f>> [1] A. E. Taylor, L'Hospital's Rule>> Amer. Math. Monthly, Vol. 59, No. 1 (Jan., 1952), pp. 20-24..>> [2] A. M. Ostrowski, Note on the Bernoulli-L'Hospital Rule>> Amer. Math. Monthly, Vol. 83, No. 4 (Apr., 1976), pp. 239-242.. >Jstor requires a subscription (e.g. most major universities). >Alternatively, many public libraries subscribe to the Monthly.Oh, I'd have to ask for an inter-library copy be mailed.For math papers, four pages might not be short and simple.How demanding are they? Witn the scope of 2nd year calculus?All the l'Hopital's rules I know come of Cauchy's mean value theorem.If f',g' exist on [a,b], g(a) /= g(b), then some z between a,b with f'(z)/g'(z) = (f(b)-f(a))/(g(b)-g(a))Does Taylor or Ostrowski have a similar approach?Is one to be recommended over the other?> However when f e^x -> k; then> f = fe^x e^-x -> 0> f e^-x = f e^x e^-2x -> 0>> L = lim f+f' - lim 2f = lim f'-f> 0 = lim f = lim f e^-x / e^-x = (f' - f)e^-x / -e^-x = lim f-f' = -L> 0 = lim f+f' - lim f = lim f'>> So if lim f = k and lim f' exists, then lim(x->oo) f+f' = k + lim f'> k = lim(x->oo) f = k + lim f'; lim f' = 0>> If lim f' doesn't exist, then for n >= 2> f_n(x) = sin(x^n)/x -> 0> (f_n)'(x) = nx^(n-1) cos (x^n)/x - (sin x^n)/x^2 -> === oscillation----Subject: Re: Homeomorpsms and === M->N is a homeomorpsm, then > K compact in M <=> f(K) compact in N >if f is bijective and > K compact in M <=> f(K) compact in N >is f then a homeomorpsm?f:X -> Y proper when for all compact K, f^-1(K) compactYou ask are bi-proper bijections, bi-continuous?Arturo showed counterexample: finite discrete M, indiscrete NHere's T1 counterexample: let K be the cofinite integers. K and the disjoint sum K+K aren't homeomorpc.Yet, as all subsets of K and K+K are compact, all bijections between the two are bi-proper. >I have proven ts to be true for locally compact spaces, >but I would like to know whether the result holds generally.For compact Hausdorff M,N, it is === immediate.How you do it for locally compact?----Subject: Re: === independence/square roots/primes >>How to prove that the square roots of the >>primes are linearly independent over the rationals? >Let p1, p2, ..., p_i, ... be a sequence of different primes. >INDUCTION HYPOTHESIS: sqrt(p_n) does not belong to the field > E_(n-1)=Q[sqrt p1, sqrt p2, ..., sqrt p_(n-1)] >Proof n-1 => n: By ind. hyp., E_(n-1) has 2^(n-1) automorpsms >(sqrt p_i -> +- sqrt p_i). Suppose sqrt (p_n) belongs to E_(n-1). >Apply all automorpsms, and you see that in the Q-linear >decomposition of p_n there are AT MOST two nonzero summands. >More exactly, >(1) sqrt p_n = r+s*sqrt(q_1*q_2*...*q_i) , r,s from Q. where q_j >are some different primes from the sequence p1, ..., p_(n-1).How's that === compact definition....book-topology def)A collection B of subsets of a space X is said to cover X,or to be a covering of X, if the union of the elements of B is equal to X.it is called an open covering of X if its elements are open subsets of X.def)A space X is said to be compactif every open covering B of X contains a finite subcollection that alsocovers X--------i tnk that meaning of compact is X = U(G_i_n) by upper definition.but in the other book,i saw that definition of compact is X C U(G_i_n) {C : inclusion sign}wch === Re: asking for your opinion> I hope ts does not seem to out of place and I hope that you are able to> help me. I am about to complete my second year of my maths degree. I am going to be> doing a major in mathematics next year and I a am tnking about doing> honours the year after next. I have been looking into some areas of maths that I might be interested in> doing more study in and one of them is that of complex (chaotic) systems.> Well pretty much anytng to do with Differential Equations I could be> interested in I just need to find out what they are really about.> What I am after is just some good books on the area so that I can get a bit> of a taste of it before I decide it is what I want to do further study in.(Table of contents at http://www.jesus.ox.ac.uk/~dacheson/calcon.html)Glendinning, P. Stability, Instability and Chaos: An Introduction to theTheory of Nonlinear Differential Equations, Cambridge University Press1994.Essentially the lecture notes of a trd year Cambridge undergraduatecourse.> Any of the other areas of applied maths that are rather interesting or new> and exciting that would be helpful as well.Acheson, D.J. Elementary Fluid Dynamics, Oxford University Press 1990(http://www.jesus.ox.ac.uk/~dacheson/efdcon.html)And, in the interests of equal time,Batchelor, G.K. An Introduction to Fluid Dynamics, Cambridge UniversityPress 1967 (reprinted 2000)Ts, however, is far more dense and less readable than Acheson.-- P.A.C. SmithThe vast majority of Iraqis want to live in a peaceful, free world.And we will find these people === sft, core issuesAt least now I won't have to feel guilty if I decide to use tactics,>>as I've given fair warning.And if ts doesn't work, you can start pestering Congress to get your >theorem passed into law.He'll have to kick their cots to wake === Re: Mike Turner in NY Times Science 11/11/03>> I tnk we are so confused that we should keep an open mind to>> tinkering with gravity, said Dr. Michael Turner, a cosmologist at>> the University of Ccago. previouslyQ to Ed Witten: How can the cosmological constant be so close to>> zero but not zero?DL> Close to zero? WTF does close mean? Is there a maximum valueDL> we can compare it to to see how close it is?The current estimates are that the energy density contributed by thecosmological constant (or dark energy) is 0.7, in units of the closurephysics that make predictions about the cosmological constant. Thetypical value predicted is 10^50 or so. -- Lt. Lazio, HTML police | e-mail: jlazio@patriot.netNo means no, stop rape. | http://patriot.net/%7Ejlazio/sci.astro === Re: is Sadistic .>> > By the time I saw your response to my other post, I have already>>replied to ts post of yours. I would not have otherwsie.>>Put it in your tnk head. You cannot demand; you can only request.HUH?!>>Did you have to use the word 'f***ing'? >>You could have asked What countries were they from?And you still haven't answered the damned^W question.>>I am not obligated to answer, am I?Gawd, I hate cute. Nope, you aren't obliged to answer,>> especially when you used the datum as the basis>> of your argument. > I said Africans. You said that Etheopia used to send their brighterst; I replied, they>were not Etheopians.I waited to say the countries name because I was forgetting where 2>were from (I do remmeber that it wasn't Etheopia) and I decided to>tnk about it a bit before saying anytng since I knew that I would>come to remember, at least one of that two.Then say that you don't remember or you don't know or you will getback to the question when you can remember. Instead you screw around.> When asked for a clarification of that>> datum, you were rude Honey, you haven't seen rude.> you deliberately ignore the question,You were rude and so I ignored to answer.I see. That's why you answered the post. I thought you saidyou were going to quit wch is why I stopped posting.> add all kinds>> of red lines to divert the lack of an answer. What lack of answer? > Ergo, you>> don't feel very strongly about the subject but do want>> to get a free ride based on perceived discrimination.Note: No amount of your accusation will make me tell you the names of>those countries. Live with it.So you really don't know. You assume, based on their looks, thatthese people came from Africa. These people could have come fromBrazil or Jamaica or the USA or England./BAHSubtract a === .>Apparently you don't have anytng at all to say that's>> relevant to any of the three newsgroups to wch you>> continue to post. How *do* you justify your existence?She seems to be looking for a Muslim scientist to t>on her so she's advertising in sci newsgroups and for>bait is trying to appear intelligent, passionate, and>desirable.It didn't sound like that. I have a different hypothesis./BAHSubtract a hundred and === personality assessment>> The police and so forth only exist insofar as they can demonstrate>> their authority. They say they're here to preserve order, but in fact>> they'd go absolutely mad if all the criminals of the world went on>> strike for only a month. They'd be on their knees waiting for a>> crime. That's the only existence they have.>> William S. Burroughs (American writer)>> Guardian, 1966>> huffyWhere huffy == Correy (and so is self-identical)?In another post by huffy, you signed (and clearly spoke as> Correy).Presumably it wants to avoid killfiling. I have no idea why ittnks that the opinion of ts late novelist, heroin addictand wife-murderer is at all relevant though. These huffy postsare considerably more === tedious than the rantings of D S Kabatoff.-- Subject: Re: ability for solving differential equations, shemainly works through itteraative processes. Chaos theory is about analsingthe possible outcomes of those itterative processes.It also links in with the biggest debate in science: Einsein's God does notplay dice. Einstien's world was deterministic obeying the laws of Newton(with relativitstic corrections). Quantum mechanics took the view thateverytng works by chance. Chaos theory shows that complex systems work ina psudorandom fason.My guess is that 21st century appied maths is going to be about applyingchaos theory to classical physics to propery Harveybruce@bearsoft.co.ukThe Alternative Physics Sitehttp://users.powernet.co.uk/bearsoft> I hope ts does not seem to out of place and I hope that you are able to> help me. I am about to complete my second year of my maths degree. I am going tobe> doing a major in mathematics next year and I a am tnking about doing> honours the year after next. I have been looking into some areas of maths that I might be interestedin> doing more study in and one of them is that of complex (chaotic) systems.> Well pretty much anytng to do with Differential Equations I could be> interested in I just need to find out what they are really about. What I am after is just some good books on the area so that I can get abit> of a taste of it before I decide it is what I want to do further study in. Any of the other areas of applied maths that are rather interesting or new> and exciting that would be helpful as well. I mean really anytng as I> really just want to see what is out there just so I have plenty of options === anytng at all to say that's> relevant to any of the three newsgroups to wch you> continue to post. How *do* you justify your existence?> Feeling hurt..still? > Can you come up with even one post that on-topic> somewhere and avoid the otherwise inevitable trip to> everyone's killfile?Do it INSTAED OF wnning about it. I didn't expect that there would be people like you to take tngs out> of context and attack me. Moreover, I realized that it was a mistake> for me to even bother to reply to people like.Like? > Now ... put me in your killfile instead of talkign about it. You will> be helping me greatly.It'd pobably help more if you didn't drink wle posting. Either thator pay more attention to your spell === 2^n numbers?> OTOH, isn't N bijective with the set of the *sums* of the *finite*> subsets of the powers of 2?Well N u {0} *is* the set of sums of the finite sets of powers of 2.I suspect ts is whatssorhername's conceptual problem. He/she/ithas forgotten that infinite sets have lots of === initialshttp://www.giganews.com/info/dmca.html>Some of you may have realized that I have remarkable power to draw>attention on several newsgroups to the extent that I even have my own>dedicated repliers, like or Ullrich, who like to>obsessively insult me! Like when I called you a ing piece of dogst. Poor torturedJames.No, wait, actually it was you who called me a ing piece ofdogst. Never mind...>Anyone have a good term for people who just>follow around a popular poster insulting m? Well I call them critic>trolls.In any event, so like I said I have ts drawing power, and lots of>people besides and Ullrich get upset with me over>strange tngs, and there's little doubt there are people out there>bothered by my use of my initials JSH in subject lines, as if it's>some kind of arrogant tng, so here's the story as to how I happened>to start using my initials in subject lines.Let me take you back a bit.I started looking for simple yet profound math discoveries back in>1995. My tnking was that if I could look in places that others>thought were well-worked and find some spectacular, but basic result,>I could make money.Ok, so yeah, I'm in it for the money.Indication number 37 of your utter cluelessness regarding almosteverytng.>In any event, at first I talked directly to mathematicians, primarily>at math journals, and found that when I made mistakes with my>mathematics, it was embarrassing and time consuming to find someone>else to talk ro about new ideas, and then in 1996, I discovered>Usenet.It seemed like the perfect place! I could talk about mathematics on>the newsgroup sci.math where other people were talking about math. >Maybe I'd even get some people who would be sympathetic to my idea.What I found was a lot of hostility.Apparently certain posters see the sci.math newsgroup as their>personal territory, and have a territorial reaction to people posting>there. They feel they can control the newsgroup content, by>controlling posters by, you guessed it, insulting them.Ts bit about controlling posters needs to be answered. In fact_nobody_ has _ever_ taken any steps to try to prevent you fromposting to sci.math. The _only_ person in all ts who _has_ triedto control posters is _you_: You've complained to my _employer_about various tngs, and later admitted explicitly that thereason you made those complaints was to try to get me to stopreplying to your posts.>Well I ended up in lots of arguments, but meanwle kept posting my>ideas, as I realized that maybe I'd signed on to a really big task,>looking for a spectacular but simple proof, and I sfted more to idea>generation mode, you know--brainstorming.Well that *infuriated* the territorial posters who became more>energetic in their insults, as like I said, they'd try to insult>posters like me into shutting up.And I didn't exactly like all of those attacks, but hey, I kept at>coming up with ideas and posting them, but found myself first angry,>then intrigued when one poster--angry at *accidentally* reading my>posts--suggested that I give some identifier so that people could know>when I was the one who started a thread.So at first I replied haughtily about my freedom of speech, and right>to post whatever subject line I wanted. But then I thought, hey,>that's not a bad idea!!!So I added my initials to the subject line to show that it was me>posting, as requested.Now when I see a need, like now, to identify who started a thread, I>add JSH to the front, but of course, Usenet sparks imitators, so>sometimes other posters will stick *my* initials on a thread THEY>started, wch is just one of those tngs.In any event, in case you were wondering, no, I didn't tnk of>tossing my initals on subject lines--I was *ordered* to do it by a>poster angry at not always being able to tell threads that I>started!!!Isn't Usenet a wacky, wacky world?>My math discoveries, found for === Tbl9NaZ1we++y9QKom6kuOERHmzG5xwqLk5ZS+wfxmx9HlTWqmzpYuWhat is the fastest algorithm for computing factorial, for very large> numbers (e.g. 10000!)?>(...)(Actually, I wouldn't count 10000 as a very large number)Yes, but what about the OP's 10000! ;-) === H#SM:U1U-/6#NN83s6?Die557~]Dfifz~-|V:wSKGL6T-|!qk{U4/M7+k5Py!- {q=2Q/%0@ E29yc_kQC&^What is the fastest algorithm for computing factorial, for very large> numbers (e.g. 10000!)?> (...)(Actually, I wouldn't count 10000 as a very large number)Yes, but what about the OP's 10000! ;-) That's large itself, but it doesn't take long tocompute. computing 10000!! isn't feasible whatever algorithmyou use.-- J.97n Fairbairn === counterpart theory, and contingent identity> IIVrb.16233$Rah1.1560@twister01.bloor.is.net.cable.rogers.com> ...> ixFx = ixFx> is necessarily true, wle> FixFx is contingently true.> Not so, for Russell's description theory.> > 1. G(ix:Fx) <-> Ey(Ax(x=y <-> Fx) & Gy), see PM *14.1> 2. E!(ix:Fx) <-> Ey(Ax(x=y <-> Fx)), see *14.2> 3. E!(ix:Fx) <-> F(ix:Fx)> 4. E!(ix:Fx) <-> (ix:Fx)=(ix:Fx)> 5. E!(ix:Fx) <-> EG(G(ix:Fx))> 6. F(ix:Fx) <-> (ix:Fx)=(ix:Fx)> 7. [](F(ix:Fx)) <-> []((ix:Fx)=(ix:Fx))> By: 6, |-p -> |-[]p, and, [](p <-> q) -> ([]p <-> []q).> 8. []E!(ix:Fx) <-> []((ix:Fx)=(ix:Fx)).> Youre not wrong insofar as there really seems to be a problem for the> NFL-logician, for whom (1) ixFx = ixFx -> E!ixFx is impeccable. If the normal rules of modal logic (particularly the rule of> necessitation and the rule of box distribution), are applied to (1),> we indeed get the following result: [](ixFx = ixFx) -> []E!ixFx But once again, were being confronted with the notorious ambiguity of> modal formulas. Read de dicto we have: If 'ixFx = ixFx' is necessarily true, then 'E!ixFx' is necessarily> true.By convention, imo, [](ixFx = ixFx) -> [](E!(ixFx)) is not ambigious.It is read: If 'ixFx = ixFx' is necessarily true, then 'E!(ixFx)' isnecessarily true.(ixFx) []= (ixFx) -> ([]E!)(ixFx), is also not ambiguious, it is readIf 'ixFx is necessarily equal to ixFx' then 'ixFx is necessarily existent'.The 'de dicto' and 'de re' distinctions are needed and made specific forexpressionscontaining described objects, independent of modal predications.D1. f(G(ixFx)) defined f(Ey(Ax(x=y <-> Fx) & Gy)) {de dicto}D2. (fG)(ixFx) defined Ey(Ax(x=y <-> Fx) & f(Gy)) {de re}Ts distinction is required even for non-modal truth functions.For example: ~(G(ixFx)) <-> ~(Ey(Ax(x=y <-> Fx) & Gy)), {de dicto}and (~G)(ixFx) <-> Ey(Ax(x=y <-> Fx) & ~(Gy)), {de re}.E!x -> {f(Gx) <-> (fG)x}E!(ixFx) -> {f(G(ixFx) <-> (fG)(ixFx)} The point is that in negative free logic ixFx = ixFx is not a> necessary truth, since it is possibly false (in case ~E!ixFx). There> is a possible world in wch ixFx = ixFx is false because ixFx does> not exist in that world; and if there is a possible world in wch> ixFx doesnt exist, E!ixFx cannot be a necessary truth either, since> necessary truth is defined as truth in all worlds! Even though (ixFx = ixFx) -> E!ixFx is a necessary NFL-truth, both> ixFx = ixFx and E!ixFx are no necessary NFL-truths!Certainly, eg. (the present king of France)=(the present king of France) isfalse and,E!(the present king of France) is also false.It seems to me that ts version of NFL is simply description logic,contained witn classical logic.ixFx = ixFx, is invalid. i.e. ~[](ixFx = ixFx).E!x <-> ixFx = ixFx, is valid. i.e. [](E!x <-> ixFx = ixFx). But, luckily, that circumstance does not render the following> implication untrue: [](ixFx = ixFx -> E!ixFx) -> ([](ixFx = ixFx) -> []E!ixFx) The antecedent is true but neither the antecedent nor the consequent> of the consequent are true in NFL! So we have 1 -> (0 -> 0)> 1 -> 1> 1 ! By the way, the rule of necessitation can also be unproblematically> applied to the following NFL-theorem: Ax(x = x -> E!x)> []Ax(x = x -> E!x)> [](Ax(x = x) -> AxE!x)> []Ax(x = x) -> []AxE!x Since in NFL both Ax(x = x) and AxE!x are axioms, everytngs fine> here!These are theorems of classical modal logic, aren't they?> What if we interpreted (1) *de re*? In NFL E!x <-> Ey(y = x) holds (by definition), and so we can write (1*) ixFx = ixFx -> Ey(y = ixFx)(1*) ixFx = ixFx -> Ey(y = ixFx), is valid in description logic but,it is not the case that (ixFx) is a value of the individual variable unlessE!(ixFx). i.e. it is not an instance of x=x -> Ey(y=x).|- Ey(y = ixFx) <-> Ey(Ax(x=y <-> Fx)), because, y = ixFx <-> Ax(x=y <->Fx).|- E!(ixFx) <-> Ey(y = ixFx)> and [](ixFx = ixFx) -> Ey[](y = ixFx)I don't tnk so.[](Ey(y = ixFx)) -> Ey([](y = ixFx)), is invalid.[](E!(ixFx)) -> Ey([](y = ixFx)), is invalid.[](E![ixFx]) -> Ey([](y = ixFx)), is valid.[ixFx] defined (iy: [](Ax(x=y <-> Fx))).Witt If ixFx is necessarily self-identical, then there is sometng such> that it is necessarily identical with ixFx. (It doesnt mean ... there necessarily is sometng such that> ...!!!) Ts de re interpretation is impeccable, for now both the antecedent> and === sft, core issueshttp://www.giganews.com/info/dmca.html>>I've been tnking about my problems with getting any kind of>admission that my math arguments showing the core error in mathematics>are correct, so I've gone to marketing books.You know, in case you're curious, ts sounds really really stupid.> If your results were correct you'd be able to convince people of> them by explaining the proofs carefully. But in fact they're wrong,> people continually explain what the errors are, and tactics from> marketing books are not going to change that.If s results were correct, the same people who point out errors to m > with saintly patience would instead have provided all these explanations.>>Having spent some time now reading marketing tactics, I can clearly>>see that both Ullrich and Bau are selling a viewpoint to the>>readersp.>>Here the assertion is that if I were right then people would>>necessarily agree with me!!!>>Is that true in your experience?It's certainly true in _my_ experience that when I'm right about>> sometng and have a valid proof that I'm right then competent>> mathematicians agree I'm right, after I've explained the proof,>> yes. That includes cases where they were certain at first I was>> wrong, by the way.Don't you agree, , that from C1-C4--indeed, from C3,C4 alone--any>*competent* mathematician could have deduced Ex~(x=x)?Two comments, the second for anyone who missed your first few hundredrepetitions of ts question:(i) your question has no relevance here - you seem to be confusinga statement and its converse.(ii) the original claim was that Ex~(x=x) followed _in_ standardset theory. Since Ax(x=x) is a theorem of standard set theory, ifts follows then it also follows that C1-C4 are simply inconsistentwith standard set theory.>C1 AxAy[x=y -> Az(z in x <-> z in y)]>C2 AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)] >C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in>A)Classification>C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}] (Weak>Extensionality)>someone will point out the error.?Did your Homies correct you? Or did they *defend* your mistake-->and you for having made it? A few people recently have pointed out that to conclude that whatI said was erroneous you need to take it out of context. Wch isof course true.>--Giggle: When you set up a new yahoo account with a silly namelike pzuffy the idea is to avoid using your real name in postsyou make under the silly name.(I'm assuming the point to pzuffy was to make it appear thatthere's someone other than you out there who agress with thetngs you say. If the point was sometng other than thatthen never mind. Although it's hard to see what other pointthere could have been - if the point was to set up a newaccount because the old one was than ding youridentity, it's hard to see why you'd choose a === software wanted.> My school gets a lot of kids that are bend their nominal peer group> in math skills.> I'm looking for a self-contained math remediation system for junior> and senior gh.See if ALEKShttp://www.aleks.com/will meet most of those needs. They offer UNLIMITED numbers of 48-hour freetrials, so you'll have time to check it out. The 48-hour free trials don'tallow CONVENIENT tracking of a particular student's progress over time,wch is how the company gets people to sign up for the real service.I don't work for the company; I've just used the free trial a few timeswle checking my son's progress in other math courses.Hope ts helps!-- Karl M. Bunday Christ has set us free. Galatians 5:1Learn in Freedom (TM) http://learninfreedom.org/kmbunday AT earthlink DOT net (preferred email address)-- submissions: post to k12.ed.math or e-mail to k12math@k12groups.orgprivate e-mail to the k12.ed.math moderator: kem-moderator@k12groups.orgnewsgroup website: http://www.tnkspot.net/k12math/newsgroup charter: === and MathIn sci.math, Bill Taylor< mike_deeth@yahoo.com (Mike Deeth) made some kool advert, but Cantor is> all about Bijections, (or lack of them), So I tnk we need to show> a bijection between the two most prominent features of s pic.Hence I remove the coverings. (Here I use Taylor's uncovering lemma!)[Taylor's uncovering lemma used too many times :-) ]> Hmmmm... there's a thought.... I wonder what another application > of the uncovering lemma would reveal??>