mm-251 === Subject: Re: Rationals are Uncountable>If there is no rational number that can be c then we> will find some n such that the open interval (a_n,b_n)> contains no rational.>Don't be ridiculous.>In particular, do not confuse the following two statements:>(1) For every rational c, there is an n such that c is not> in (a_n, b_n).>(2) There is an n such that for every rational c, c is> not in (a_n, b_n).>Statement (1) is true. Statement (2) is false. The order of the>> quantifiers> matters.OK.>> If (1) is true the rationals are not dense.Wrong. If (2) is true, then the rationals are not dense. There is no>> conflict with (1).If (2) is false there exists a rational c in every interval.Correct. That's what it means for the rationals to be dense.>>Proof:>> Let X equal the intersection of all (a_n,b_n).>> If X is empty then one of the intervals was emptyNo, that does not follow. You have been presen with examples.>Let A be a set and B a proper subset of A.> Take the intersection of A and B.> AxB = B> Is this a correct statement?> Does it matter if A and/or B are infinite?> If this is correct, the only way the intersection> can be empty is if B is empty.That holds for a finite collection of nes intervals, but not for aninfinite collection.-- Dave SeamanJudge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. Let X equal the intersection of all (a_n,b_n).> If X is empty then one of the intervals was empty> proving the rationals are not dense.Incorrect. A set of intervals can certainly have an emptyintersection without one of the internals in question === Discussion, linux)> Now I want readers to see what I've been talking about, and why I find> posters like Nora Baron so, unsettling.> This poster is now clearly engaged in more than just critiquing the> idea.> Clearly Nora Baron is about more than just trying to set me straight> mathematically or question this or that mathematical position.> This poster is out to GET ME, and it's something that I find, odd.> Why?> I put Nora Baron in quotes because the name is apparently a> pseudonym, and it's not even clear that Nora Baron is even female.> Who is this poster? Anybody know? I'm curious to know if anyone will> even post to say that they know who this poster is, without revealing> the name.> I just wonder if *any* of you know who is the person behind the name.It's just as you suspect. Nora Baron is the pen-name of Andrew Wiles.I mean, duh.-- If you *still* believe that [my proof is wrong], then I have to thinkthat your mind is limi [...], and it may be the case that noteveryone *can* achieve that, as the mental wiring may not be there forthe task. -- , on faculties needed to === Triplets linux)>> There must be something SERIOUSLY wrong with CORE MATHEMATICS!!> You really need to get a life.> I had an idea, tossed it out, and it didn't fly. No big deal.No, you didn't just toss out an idea. You proclaimed that yourmethods proved a theorem which was unprovable with contemporarymathematics and that this was evidence of the superiority of yourmathematics.That's rather different than casually tossing out an idea and seeingthat it didn't fly.> I wonder about some of you.Yeah?-- Jesse HughesBy definition m is a variable. By definition all then (sic) numbersrepresen by letters are variables--that's algebra[,] Magidin. -- shows deep === T89ZdQZX8eBeEJjsWrdNaChTIh4CCCgS+QcOVQyS-IvFj5b3S1Xz6ZHallo,Is this calculation (& approach) correct?(switch to Courier New.):###Let |K be a body(?). Let |V be a |K^3-vector space. Let /2 /3 /5 B = L( |7 |,|11|,|13| ) 17/ 19/ 23/and /0 /1 /4 C = L( |6 |,|8 |,|9 | ) 10/ 12/ 14/be bases of |V.If I want to get T_(B, C) (read: T from B(up) to C(down); this is thetransformation matrix), I must express each vector from B through C,and have to put the coefficients of this representation into thetransformation-matrix, correct?(You don't have to check the following if you don't want to. I just wanto try it out.)det(matrix(C)) =|0 1 4 | |0 1 4| |0 1 4| |0 1 4| |0 1 4| |2 0| |2 3||6 8 9 | = |6 6 1| = |2 2 -4| = |2 3 0| = |2 3 0| =- + 4|10 12 14| |4 4 5| |4 4 5| |4 4 5| |2 0 1| |2 1| |2 0|= -2 + 4*(-6) = -26/3 |11| = C19/ |0 3 4 | |0 3 4| |0 3 4| |0 3 4| |3 4|=> |6 11 9 | = |2 3 4| = |2 0 0| = |2 0 0| = -2 = 2*17 = 34 |10 19 14| |4 8 5| |4 8 5| |0 5 1| |5 1|b_C = -34/26 = -17/13; ( b_C from |K. )-17/13 + 4c_C = 3 <=> 4c_C = (39+17)/13 = 56/13 <=> c_C = 14/13 (c_C from|K)6a_C - 17*8/13 +9*14/13 = 11 <=> 6a_C - 2(17*4 - 9*7)/13 = 11<=> 6a_C - 10/13 = 11 <=> 6a_C = (11*13+10)/13 = 11+10/13<=> a_C = 11/6 + 5/39 = 51/26 (a_C from |K)/5 |13| = C23/|0 1 5 | |0 1 5 | |0 1 5| |0 1 5| |1 5||6 8 13| = |2 4 3 | = |2 4 3| = |0 5 1| = 2 = 2(1-5^2) = 2(1-5)(1+5)|10 12 23| |4 4 10| |2 0 7| |2 0 7| |5 1|= -2*4*6 = -6*8 = -48 => c_0_C = 48/26 = 24/13 (c_0_C from |K)b_0_C + 4*24/13 = 5*13/13 <=> b_0_C = (5*13-4*24)/13 = -31/13 (b_0_C from|K)=> 6a_0_C - 31*8/13 + 9*24/13 = 13 <=> 6a_0_C - (31*8-9*24)/13 = 13<=> 6a_0_C - 8(31-27)/13 = 13 <=> 6a_0_C - 8*4/13 = 13<=> a_0_C = (13^2+8*4)/(6*13) = (169+32)/78 = 201/78 = 67/26 (a_0_C from |K)/2 |7 | = C17/|0 1 2 | |0 1 2 | |0 1 2| |0 1 2| |0 1 2| |1 2||6 8 7 | = |6 8 7 | = |2 4 -3| = |2 4 -3| = |0 4 -16| = 2 = 2*-24|10 12 17| |4 4 10| |4 4 10| |2 0 13| |2 0 13| |4 -16|= -48=> c_1_c = c_0_c = 24/13 (c_1_c from |K) => b_1_c + 4*24/13 = 2<=> b_1_c = (26-96)/13 = -70/13 (b_1_c from |K)=> 6a_1_C-8*70/13+9*24/13 = 7 <=> 6a_1_C - (8*70 - 9*24)/13 = 7<=> 6a_1_C - 8*43/13 = 7 <=> a_1_C = (7*13+8*43)/78 = 145/26 (a_1_C from |K)So the transformation matrix is: / 145/26 51/26 67/26 T_B_C = | -70/13 -17/13 -31/13 |. 24/13 14/13 24/13 /Is this correct?Best === calculation?)> Hallo,> Is this calculation (& approach) correct?> (switch to Courier New.):> ###> Let |K be a body(?). Let |V be a |K^3-vector space. Letfield> /2 /3 /5 > B = L( |7 |,|11|,|13| )> 17/ 19/ 23/> and> /0 /1 /4 > C = L( |6 |,|8 |,|9 | )> 10/ 12/ 14/> be bases of |V.> If I want to get T_(B, C) (read: T from B(up) to C(down); this is the> transformation matrix), I must express each vector from B through C,> and have to put the coefficients of this representation into the> transformation-matrix, correct?Yes. So you calculate T_(each basis vector of B) and express it in terms ofthe basis vectors of C.> (You don't have to check the following if you don't want to. I just wan> to try it out.)Well, I won't. Too many numbetoo === TlOuT-ZEQeS6HIJr6f0jJ5arpniyy0sTUnohWYCklLT6cRLujCsM44> If I want to get T_(B, C) (read: T from B(up) to C(down); this is thetransformation matrix), I must express each vector from B through C,> and have to put the coefficients of this representation into the> transformation-matrix, correct?> Yes. So you calculate T_(each basis vector of B) and express it in terms> of> the basis vectors of C.Ok, but I think, what I have done wouldn't be a typical task in an exam. :-((I did this only for my understanding of this issue.)I still don't know what you can do with these matrices (except the factthat you hold the coefficients of the bases-transformation in them).Thus I don't know how a typical task for this issue could look like.(For example, we learned some relations for these t.-matrices:like: T_B_D = T_C_D*T_B_C // In which type of task can I use this? Why do // I need this relation? Or in other words: // In which situation will I need this one?same problems with: T_B_C = (T_C_B)^(-1))We also have the following relation:given: M_B_B(f). B is the canonical basis of |R^mrelation: M_C_C(f) = M_B_C(Id_(|R^m))*M_B_B(f)*M_C_B(Id_(|R^m)).(I know this seems to be the application of the 1st rule above.But I just transform some t.-matrices into other t.-matrices. Forwhich purpose. Why is this useful?)My problem is, that everything is so vague to me about this issue. @{I seem to miss the starting point.(And I would lie, if I say that I understand this good enough tosolve abstract tasks (like prove this and that ) ).It would be great, if somebody of you could give me some common === eulers>I have the following problem:>I have an object that has an orientation described with Euler angles and>on which I can calculate the forces exer, thus finding the torques and>angular accelerations wrt the object's axis. The problem is how to>compute the new angular position of the object.. >In my case the euler angles describe rotations in the order X Y Z (world>axis). I tried to convert these angles into a quaternion representation.>The angular acceleration allows me in the object's frame gives me the>angular velocity in the object's frame using simple Euler integration:>w'(t+dt) ~= w'(t) + w''(t)*dt>If there are no forces exer on the object then w' is constant, i.e. it>should correspond to a rotation around a stationary axis. It should be>simple to transform this axis from the object's to the world's frame of>reference... but I cannot seem to be able to do it..>Maybe the problem lies in the fact that the euler angles describe a>series of rotations.. and thus I perhaps miss something when converting>from Euler to quaternion. What do you do when you convert them? You should multiply thematrices in the correct order, for example Rz*Ry*Rx for the orderx-axis, y-axis then z-axis (if you multiply column vectors on theright side), then convert from this matrix to quaternion form.Otherwise you'll === rotations w quaternions and eulersmy mistake. if the Euler thing is the samedirection cosines, then it's overdeterminedby one parameter (except to fix the octantof the unitsphere). a sort-of homogeneity. > What do you do when you convert them? You should multiply the> matrices in the correct order, for example Rz*Ry*Rx for the order> x-axis, y-axis then z-axis (if you multiply column vectors on the> right side), then convert from this matrix to quaternion form.> Otherwise you'll have to give more details.--Give the Gift of Dick Cheeny -- out of office, at last!http://www.wlym.com/pages/music.htmlhttp://www.rand.org/ publications/randreview/issues/rr.12.00/http:// === w quaternions and eulersin other words, using direction cosines,the three angles that make them are notof rotations on three, particular axes;the angles are just *away* from the X, Y znd Z axes,not around them. that is to say, it beats me! take-away message:don't wait for the election, to route Cheeny andhis fellow Leo-cons (re-open the Senate investigation --and hirry !-)> my mistake. if the Euler thing is the same> direction cosines, then it's overdetermined> by one parameter (except to fix the octant> of the unitsphere). a sort-of homogeneity.--Give the Gift of Dick Cheeny -- out of office, at last!http://www.wlym.com/pages/music.htmlhttp://www.rand.org/ publications/randreview/issues/rr.12.00/http:// === w quaternions and eulersas any two rotations are equivalent to one,you can work the formula for that, twice! > Maybe the problem lies in the fact that the euler angles describe a> series of rotations.. and thus I perhaps miss something when converting> from Euler to quaternion. as you know, if you've read anyhting that I typed,there was a conspiracy of states that DID vote Gore -- andthe Supremes sealed that conspiracy on March 27, 2000,by refusing to hear the appeal in LaRouche v. Fowler.(Don Fowler was the DNC Chair in '96;Sentelle's 3-judge panel made the Voting Rights Act unconsitutitonal.but, hey; it's up for re-auhtorization in '07 !-) I am frankly scared of the touchscreen mentality. just likethe Supremes' abrogation of the USA and Florida constitutions fomentsa pop-culture hatred of the electoral collegeon the part of some rabid Democrats. ah, so;imagine if North Dakota ... rather, imagine ifWyoming had less than one electoral college vote for president. anyway,every one of us in this debate knowsabout the Texas cirterium for chad --it ain't just missing confetti! thus saith:but, at the national level,will you be able to maintain the DNC's Any One But George --unless it's Lyndon! media glut?... are you on the boardof GOPpers For Howie Troisieme? http://www.wlym.com/pdf/iclc/communism.pdf--Give the Gift of Dick Cheeny -- out of office, at last!http://www.tarpley.net/bush25.htm (Thyroid Storm ch.)http://www.rand.org/publications/randreview/issues/rr === Re: What is the meaning of imaginary numbers?> I can easily understand how the motion of a pendulum (for example) can be> described by something like this:> x(t) = cos(wt)> but what kind of a real world oscillation can be described by: > x(t) = e^(iwt) = cos(wt)+isin(wt) ???Well, at that point I invite anybody to falsify some musing of an engineer in a new thread: 'Fundamental basics of unilaterality'So far I only faced fine English politeness.I see the complex function of time an indication for being within the fictive so called frequency applies to the real world and to causality, so far.The very problem might not be to swallow imaginary numbers but to follow physics into arbitrary and more or less slippery choices. Perhaps the second arbitrary decision is the choice of an origin as a consequence of choosing R instead of R+. A chain of further arbitrariness is an inevitable === meaning of imaginary numbers?> Apologies if this question has been answered before...> I am trying to improve my perception of imaginary numbers. I already have> done an extensive search in the web but I keep finding information about how> to manipulate them (add, multiply etc), about their representation on a> plane with two axes and other things like that, that I am already familiar> with.> Can anybody point me to any resource describing their TRUE physical meaning,> how the need for their introduction into mathematics rose and generally the> whole concept behind them?> I just can't believe that mathematicians came up with them just because help.Ma'at, karma, yin yang and complex numbers. Two opposite === $$$ Im at a loss for words THIS WORKS $$$In sci.math, M.B (ml):> MAKE MONEY!!!MAKE MONEY!!!> MAKE THOUSANDS!!![rest snipped]Charles Ponzi died peniless in a charity hospital.-- #191, ewill3@earthlink.netIt's still === words THIS WORKS $$$> In sci.math, M.B (ml)> :> MAKE MONEY!!!MAKE MONEY!!!>MAKE THOUSANDS!!!> [rest snipped]> Charles Ponzi died peniless in a charity hospital.On the other hand, the evidence indicates that Charles Ponzi really believein what he was doing, which was allowing the little guy to get into thebig money game. He just wasn't competent and therefore used today'sdeposits to pay today's withdrawals and didn't actually invest the money.If fact, that's probably why he failed. If he'd been a crook, he'd havejust taken the money and run. (Instead, he was an incompetent who broke thelaw.)Jon === coincidence or is anyone familiar with why this happens? Inthe four columns below, 1st column is the fibonacci sequence, 2ndcolumn is position within the sequence; it also represents the decimalposition in PI PHI and e which are lined up in the 3rd 4th and 5thcolumns respectively (example: if you read the PI column downward itis 3.1415926, 3 being in position 0 and 6 in position 7 in thesequence). The 12th position is the first occurrence where fibonacciis exactly divisible by its position (144/12), and this is also thefirst occurrence where the decimals for PI, PHI, and e are equal(= 9). What's interesting is that the decimals for PI PHI e are not equalagain until you carry this to the 99th position! Lots of 9's here.I left out the lines 13 through 98 to save room. Also, my spreadsheetwill not handle the accuracy for the Fibonacci sequence. Has anyonecarried this further to see what the next postion is, where thedecimals for PI PHI and e are equal; and, if there continues to be anyrelation to the fibonacci number?Fib Pos. PI PHI eNo.0 0 3 1 21 1 1 6 71 2 4 1 12 3 1 8 83 4 5 0 25 5 9 3 88 6 2 3 113 7 6 9 821 8 5 8 234 9 3 8 855 10 5 7 489 11 8 4 5144 12 9 9 9 ***...218922995834555000000 99 7 7 7 ***Bob === decimal positions out to 10,000 digits that are all the same inPi, E and Phi are:{12, 99, 169, 395, 499, 595, 606, 693, 824, 827, 840, 940, 1282, 1291,1384,1594, 1705, 1742, 1905, 2020, 2060, 2153, 2257, 2302, 2359, 2367,2507, 2546,2557, 2710, 2724, 2791, 2832, 2857, 3036, 3051, 3280, 3309, 3429,3497, 3518,3591, 3651, 3709, 3867, 4210, 4292, 4390, 4493, 4719, 4826, 4859,4862, 4892,4934, 4940, 5087, 5315, 5427, 5480, 5488, 5653, 5699, 6155, 6426,6617, 6838,6854, 7113, 7155, 7202, 7358, 7390, 7659, 7685, 7721, 7761, 7816,7833, 7867,7923, 8417, 8570, 8611, 8653, 8731, 8914, 9051, 9077, 9133, 9283,9286, 9310,9704, 9717, 9724, 9805, === coincidence or is anyone familiar with why this happens? In> the four columns below, 1st column is the fibonacci sequence, 2nd> column is position within the sequence; it also represents the decimal> position in PI PHI and e which are lined up in the 3rd 4th and 5th> columns respectively (example: if you read the PI column downward it> is 3.1415926, 3 being in position 0 and 6 in position 7 in the> sequence). The 12th position is the first occurrence where fibonacci> is exactly divisible by its position (144/12), and this is also the> first occurrence where the decimals for PI, PHI, and e are equal(= 9).> What's interesting is that the decimals for PI PHI e are not equal> again until you carry this to the 99th position! Lots of 9's here.> I left out the lines 13 through 98 to save room. Also, my spreadsheet> will not handle the accuracy for the Fibonacci sequence. Has anyone> carried this further to see what the next postion is, where the> decimals for PI PHI and e are equal; and, if there continues to be any> relation to the fibonacci number?> Fib Pos. PI PHI e> No.> 0 0 3 1 2> 1 1 1 6 7> 1 2 4 1 1> 2 3 1 8 8> 3 4 5 0 2> 5 5 9 3 8> 8 6 2 3 1> 13 7 6 9 8> 21 8 5 8 2> 34 9 3 8 8> 55 10 5 7 4> 89 11 8 4 5> 144 12 9 9 9 ***> .> .> .> 218922995834555000000 99 7 7 7 ***> Bob CarlsonIt is conjectured that pi, phi, and e are all normal numbers.Going further to guess that they are jointly normal or whateverthe term should be, we expect that the three digits will coincideon the average once in 100 lines. So line 99 is about right, butslightly sooner than we would expect.-- === Fibonacci PHI, PI, e Relation> It is conjectured that pi, phi, and e are all normal numbers.> Going further to guess that they are jointly normal or whatever> the term should be, we expect that the three digits will coincide> on the average once in 100 lines. So line 99 is about right, but> slightly sooner than we would expect.I agree the average is once every 100 lines, but to have the threedigits coincide at the special 12th position where the fibonaccinumber divided by its position is a whole number seems more than acoincidence. I was wondering if there were a similar relationship atthe 99th position. I am looking up published lists of the Fibsequence now to check, since I don't have the required accuracy in === Fibonacci PHI, PI, e Relation> Is this a coincidence or is anyone familiar with why this happens? In> the four columns below, 1st column is the fibonacci sequence, 2nd> column is position within the sequence; it also represents the decimal> position in PI PHI and e which are lined up in the 3rd 4th and 5th> columns respectively (example: if you read the PI column downward it> is 3.1415926, 3 being in position 0 and 6 in position 7 in the> sequence). The 12th position is the first occurrence where fibonacci> is exactly divisible by its position (144/12), and this is also the> first occurrence where the decimals for PI, PHI, and e are equal(= 9).> What's interesting is that the decimals for PI PHI e are not equal> again until you carry this to the 99th position! Lots of 9's here.> Fib Pos. PI PHI e> No.> 0 0 3 1 2> 1 1 1 6 7> 1 2 4 1 1> 2 3 1 8 8> 3 4 5 0 2> 5 5 9 3 8> 8 6 2 3 1> 13 7 6 9 8> 21 8 5 8 2> 34 9 3 8 8> 55 10 5 7 4> 89 11 8 4 5> 144 12 9 9 9 ***> .> .> .> 218922995834555000000 99 7 7 7 ***> It is conjectured that pi, phi, and e are all normal numbers.> Going further to guess that they are jointly normal or whatever> the term should be, we expect that the three digits will coincide> on the average once in 100 lines. So line 99 is about right, but> slightly sooner than we would expect.But the 99th Fibonacci number won't be divisible by 99 - in fact, it's not divisible by 3; you can work out that a Fibonacci number is a multiple of 3 if and only if its position is a multiple of 4, which 99 isn't. Also, the 12th position is *not* the first time the Fibonacci number is exactly divisible by its position; this happens at 0, 1, and 5 before it happens at === in resolving the followingquestion, which is a Number Theory problem. I've tried and tried butcan't figure out the right way to solve it.The problem is:Let q(x,y) be a primitive (that is, with (a,b,c)=1) quadratic formax^2+bxy+cy^2,and n a positive integer. Show the existence of two integers s,t , === problemI hope you can give me some advice in resolving the following> question, which is a Number Theory problem. I've tried and tried but> can't figure out the right way to solve it.> The problem is:> Let q(x,y) be a primitive (that is, with (a,b,c)=1) quadratic form> ax^2+bxy+cy^2,> and n a positive integer.> Show the existence of two integers s,t , with (s,t)=1, such that> (q(s,t),n)=1.Use Chinese remainder theorem to reduce to case n = p^r, p prime.Notice that is equivalent to === Re: theorem vs. proposition> lemma:> a significant part of the proof of a theorem (and as such less> important than the theorem itself)... or a *very* important theorem which, besides its importance initself, can be used to prove other theorems, e.g. Zorn's Lemma.Nobody is entitled to call his theorems lemmas in this sense, this hasto be done by the subsequent generation of mathematicians.Helmut === for that clarification. The entry in Wikipedia doescite Zorn's and Gauss' Lemmas in this category. (Not that I knoweither of them, not being a mathematician and only recently havingresumed my studies in logic)Love and respectChris> lemma:> a significant part of the proof of a theorem (and as such less> important than the theorem itself)> ... or a *very* important theorem which, besides its importance in> itself, can be used to prove other theorems, e.g. Zorn's Lemma.> Nobody is entitled to call his theorems lemmas in this sense, this has> to be done by the subsequent generation of mathematicians.> === a *very* important theorem which, besides its importance in> itself, can be used to prove other theorems, e.g. Zorn's Lemma.> Nobody is entitled to call his theorems lemmas in this sense, this has> to be done by the subsequent generation of mathematicians.historical questions...What was Zorn's Lemma a lemma for?What was Fatou's Lemma a lemma for?...other named lemmata, same question...-- === vs. proposition> What is the difference between a theorem and a proposition?qualifications/disclaimers/comments:1) I mean these in the technical mathematical sense2) I feel like I've seen (mathematical) papers/books which> used both terms but I could not determine the distinguishing> characteristics.3) I suspect that their primary content is identical, just> their usage depends on context (like a corollary is a> kind of theorem, but a theorem is just a corollary of the> 2nd to last item in a proof).4) Heath's commentary on Euclid discusses lots of terminological> questions but doesn't seem to touch on this one. Did I miss it?5) Is this a problem of modern usage? how old are these two terms?> In Heath, he explains that porism is a synonym of corollary,> whereas (I thought) the modern usage is a theorem schema (a theorem> whose construction produces many possible constructions/solutions> (e.g. steiner's porism))5) Is there such a pair in other languages (in the original Greek> or modern languages (in German it seems Satz is the common> translation for both)-- > Mitch Harris> (remove q to reply)>> imo,> All statements that are true or false are propositions.> All theorems are provably true propositions. They are analytic and true.> All propositions that are false or provably false are not theorems.I agree with this characterization fairly closely. I prefer to go evenstronger, though.In a given language which can assign truth values exclusively of trueor false to certain sentences of the language, to those sentenceswhich are assumed to be strictly either true or false are calledpropositions. The truth value of the proposition may be unknown butnot provably unknowable (undecidable), since that would violate itsdefinition.A theorem is a statement in a language that can assign truth values tostatements (complica sentences with possibly multiplecontingencies) such that its truth assignment is either true or falseor provably unknowable. Thus, the theorem concept is weaker, butbroader, than the proposition concept. Example: prior to Fermat's LastTheorem being proved, it had the status of being possibly true orfalse or provably undecidable. In a language strong enough to makestatements whose truthfulness is undecidable, it may still be possibleto make propositions, at a minimum, on any decidable subset of thelanguage, such as statements made about well-defined finite subsets ofthe things of the language.As to whether or not to exclude false theorems is a matter semanticsbased on preference. I see no problem with allowing this case, sinceone can only use in mathematical proofs true theorems anyway. Atheorem is true if it has a true proof.Just for completeness, a principle is a statement in a mathematicallanguage which is either proved true or assumed to be true. Inphysics, the term is a bit weaker, for it includes all statementswhich one has === As a newcomer I wasparticularly interes to note the use of the trichotomy 'true' 'false'and 'undecidable'.Love and respectChris> I agree with this characterization fairly closely. I prefer to go even> stronger, though.> In a given language which can assign truth values exclusively of true> or false to certain sentences of the language, to those sentences> which are assumed to be strictly either true or false are called> propositions. The truth value of the proposition may be unknown but> not provably unknowable (undecidable), since that would violate its> definition.> A theorem is a statement in a language that can assign truth values to> statements (complica sentences with possibly multiple> contingencies) such that its truth assignment is either true or false> or provably unknowable. Thus, the theorem concept is weaker, but> broader, than the proposition concept. Example: prior to Fermat's Last> Theorem being proved, it had the status of being possibly true or> false or provably undecidable. In a language strong enough to make> statements whose truthfulness is undecidable, it may still be possible> to make propositions, at a minimum, on any decidable subset of the> language, such as statements made about well-defined finite subsets of> the things of the language.> As to whether or not to exclude false theorems is a matter semantics> based on preference. I see no problem with allowing this case, since> one can only use in mathematical proofs true theorems anyway. A> theorem is true if it has a true proof.> Just for completeness, a principle is a statement in a mathematical> language which is either proved true or assumed to be true. In> physics, the term is a bit weaker, for it includes all statements> which one has strong confidence in, such as === proposition>>What is the difference between a theorem and a proposition?>>For a good definition of 'proposition' as used in osophy and logic>studied from a osophical viewpoint see:>>http://en.wikipedia.org/wiki/Proposition>>Ah. didn't think of checking there (and I don't have a physical copy of>>a mathematical dictionary nearby)>>OK. Their 2nd paragraph talks about it with respect to Aristotelian>>logic, and that explanation seems tantalizingly close to ...>For what looks like a better definition of 'proposition' as used in>mathematics see:>>http://en.wikipedia.org/wiki/Theorem>>Ah nice...mostly fits well except for...>>proposition: a result not associa with any particular theorem.>>This really doesn't do much for me. (I've pulled it out of context,>>but the context didn't help me either)> I sympathise Mitch - it didn't do much for me either at first reading. I> think it makes more sense in context, which seems to say to me> that the distinction is only one of degree, based on how interesting or> imporant the result is. In summary I think the Wikipedia entry means:> theorem:> a statement which can be proven true within some logical> framework and which is interesting or important in some way> lemma:> a significant part of the proof of a theorem (and as such less> important than the theorem itself)> corollary:> also significant but very easily deduced from a theorem (and as> such less important than the theorem)> proposition:> proven true but not significant enough to be called a theorem or,> even, a lemma or a corollaryOK. That 1) makes more sense and 2) corroborates the not yet grown upcomment by Virgil. It doesn't correspond to my inner feeling about the word; I'll have to find some examples to check (by the way, all the facts proved in Heath's Euclid are titled proposition...)-- Mitch === proposition> What is the difference between a theorem and a proposition?The difference is explained in most elementary texts of logic.A proposition is a sentence which makes an assertion which might be true orfalse but not both.A propositional schemata has the same properties as a proposition ie it maytake up the values true or false.The obvious difference is that a proposition as a binary variable cannot beassigned a value anymore than the 'n' or 'x'of algebra can without further information. The judgement of whether aparticular atomic proposition is true or false can only be made by assigningit a meaning. There are 'analytic' propositions but these are tautologies ortheir contraries. eg All red things are red. There are also falsepropositions like All propositions are true Theorems are true bydefinition, but their truth can only be known for sure in the context of aproof structure. This proof structure is not a proposition or apropositional schemata but a metalogical structure.A proposition is a logical object which can only take up two values (inbinary logic).A theorem of logic is about the relationships between distinct propositionsin a propositional schemata.A theorem of algebra is about the mathematical relationships betweennumerical variables.In both cases, the theorems could be described as molecular propositions,because they can take up the values true or false.However, it is worth keeping in mind the difference between the propositionas an atom and the proposition as schemata.In the predicate calculus, the proposition is no longer atomic but the sameprinciple applies.Tony Thomas> qualifications/disclaimers/comments:> 1) I mean these in the technical mathematical sense> 2) I feel like I've seen (mathematical) papers/books which> used both terms but I could not determine the distinguishing> characteristics.> 3) I suspect that their primary content is identical, just> their usage depends on context (like a corollary is a> kind of theorem, but a theorem is just a corollary of the> 2nd to last item in a proof).> 4) Heath's commentary on Euclid discusses lots of terminological> questions but doesn't seem to touch on this one. Did I miss it?> 5) Is this a problem of modern usage? how old are these two terms?> In Heath, he explains that porism is a synonym of corollary,> whereas (I thought) the modern usage is a theorem schema (a theorem> whose construction produces many possible constructions/solutions> (e.g. steiner's porism))> 5) Is there such a pair in other languages (in the original Greek> or modern languages (in German it seems Satz is the common> translation for both)> --> Mitch Harris> === proposition>>What is the difference between a theorem and a proposition?> The difference is explained in most elementary texts of logic.> A proposition is a sentence which makes an assertion which might be true or> false but not both.Oh. Yes. That particular usage slipped by me. That is a distinct technical usage of the term in logic. Instead, I am specifically referring to the other usage, that of general mathematicians in presenting their results. -- Mitch === if we assume (S1 undecidable), we are led to the inescapable> conclusion that (S undecidable), which is precisely the statement> (S1 true). So if S1 is undecidable, then it is the case that S1 is> true; which looks an awful lot like a contradiction!Yeah, I think you've got it.> Caveat: The contradiction depends on our being able to express the> above argument entirely in the system under discussion. It's quite> possible to have true, undecidable, statements. I'm pretty sure that> Godel proved that we *must* have such statements in *any* system.> But that doesn't automatically mean that this proof is wrong. :)Actualy Godel's proof requires that the system contains Peanoarithmetic. (It also avoids the sort of non-constructible reasoningthat the intuitionists dislike.) But IIRC Peano arithmetic is all youneed to reason about === Undecidability> Yeah, I think you've got it. Why?> Actualy Godel's proof requires that the system contains Peano> === to Pres. Bush I was laid off as a F-T community college math> instructor in August 2002 and have been unable to find another college> position since.> could you elaborate?> While I continue looking for another teaching job I was> wondering what other job options you nice folks can think of.> go back to school and get another degree. math/math-ed degrees are> essentially worthless in begetting jobs outside academia.Bah! and Humbug! They are liberal arts degrees. The two things businessneeds are communication skills and problem-solving ability. (You can learnthe jargon and the established process as you go along.)If you have a math degree, you should know how to read and write (how tocommunicate), and how to solve problems. (The two procedures that work bestseem to be logic and lucky guesses. Keep in mind that even business majorshave trouble with delegation as a problem-solving technique, even though itcan be quite effective. It turns out that hiring a consultant solves thepolitical problem of how to sell your solution internally, and not of howto solve the original problem.) So the problem of getting a job becomesone of learning marketing techniques very quickly. You can get help inthis. A good place to start might be _What Color is Your Parachute?_. Ormight not.In other words, your *degree* doesn't get a job. *You* get hired -- andhope to create a career out of it.Also, don't expect to do mathematics (unless you have a Ph.D.). You mayor may not use a mathematical approach to some problems in your career, butyou will do business. If that bothers you, what about the time you spendcooking, or cleaning house, or bathing? Shouldn't you be doingmathematics? Why didn't you get a Ph.D. in the first place? Or if youdid, why didn't you publish enough to be gran tenure?Time spent getting another degree (unless you are genuinely interes) istime was. Unless you are getting a professional degree, of course. Buteven then, it's not a panacea (engineers get laid off, there are still morearchitects gradua every year than currently practicing in the US, somelawyers make very little money, etc., and there are some people with mathdegrees working as engineers and analysts of various stripes). As nearly asI can tell, the best road to riches (on average, in the US) is still tobecome a doctor. But 85% report that they wouldn't do it again.The key to marketing yourself is to start talking to people that dosomething. Assess your abilities and interests and (surely) you knowsomeone who does something along those lines. Talk to them (1/2 hour, haveat least three questions that indicate you know something about what you'retalking about, and then end with Can you give me the names of two otherpeople that you know that I could talk to to get more information.) Then,when you call the next two, start with, X referred me to you, I'minteres in . . .. Because of the personal connection, you will gettalked to. If you've been honest about your abilities and interests,eventually (since 2^n grows pretty fast), you'll eventuall meet someone in abusiness with an interest and a need similar to your interests andabilities.Of course, you can use the more traditional means (resume floods, job fairs,newspaper ads, etc.) as well.Jon === but Python does include eval() and> exec(), allowing one to execute code that comes from data> instead of source files. (And traditional advice is to avoid> these functions - the reasons people give for this advice> is why I ask the question above.)In Perl (and I presume Python if it works the same way) eval() is alsovery useful as a simple way (although not the only one I grant you) forrunning code internal to the program, such that the eval() code levelstays in control if the code run by the eval() aborts, for example afatal error trying to execute an erroneous database statement.So for example if one sets up a script to run unattended as a backgroundmonitoring process which must have high availability, and the ability toalert someone if it runs into trouble itself, it is advisable to encloseas much code as possible, all of it basically, in an eval() followed onlyby a simple low-risk piece of reporting code such as an alert emailer.That's what I always do (not that the emailer is ever needed of course ;)Incidently, my understanding is that the Parrot project (Perl v6) aimsto produce a common backend or 'core' which will === thing, but Python does include eval() and>> exec(), allowing one to execute code that comes from data>> instead of source files. (And traditional advice is to avoid>> these functions - the reasons people give for this advice>> is why I ask the question above.)>In Perl (and I presume Python if it works the same way) eval() is also>very useful as a simple way (although not the only one I grant you) for>running code internal to the program, such that the eval() code level>stays in control if the code run by the eval() aborts, for example a>fatal error trying to execute an erroneous database statement.>So for example if one sets up a script to run unattended as a background>monitoring process which must have high availability, and the ability to>alert someone if it runs into trouble itself, it is advisable to enclose>as much code as possible, all of it basically, in an eval() followed only>by a simple low-risk piece of reporting code such as an alert emailer.>That's what I always do (not that the emailer is ever needed of course ;)If I'm following this, the right thing to do in Python would be toenclose the iffy code in a try-except block. That would be the thingto do in any language with structured exception handling (whichI gather Perl doesn't have?)try: something that might cause an errorexcept: do whatever to report the error and/or recover>Incidently, my understanding is that the Parrot project (Perl v6) aims>to produce a common backend or 'core' which will be usable by === Java>> [...]>>Python is hard to>>parse; people have to be taught the syntax. (Those two go together,>>of course.) >> This one I don't get at all. Are you claiming that people know how>> to write valid Lisp without having to learn how to do so???>No, I'm saying that the syntax is taught in about five minutes, and it>is then all known. And that's for people who don't know *any*>existing syntaxes. Python is easy IF you already know Algol-like>languages. Lisp syntax is easy right from the get go. It's as easy>as this:>* a number looks like a number (3, 3.1415, -156, etc.)>* a variable is a set of characters that doesn't look like a number,> and doesn't include parens or spaces; also a single period isn't a> variable.>* a list is a pair of parens with things in them (numbevariables,> other lists)>That's it. There are schools which use Scheme as their intro CS>class, and it's a commonplace that their students are learning about>programming techniques while the students in the school down the>street are still struggling over Pascal syntax I find that hard to believe. (Which is not to say it's not so - ifI had any _evidence_ that it's not so I'd say so.)>(that was in the 80s, I>suppose now the struggle over Java syntax or whatever).Here it's Java. Which has two things in common with C andC++: (i) although I don't actually _know_ the language, Ifeel I have a fairly good idea how it works on the basis ofreading things about it (ii) it strikes me as an abysmalchoice for a introduction to programming. The introlanguage should allow the student to concentrate onalgorithms, etc, without worrying about declaringvariables and such. Could well be that the onlyreason functional languages seem to tricky to meis what you suggest, that my brain was corrup bylearning procedural languages first - _could_ bethat Scheme is a good choice. Could be that Pythonwould be an excellent choice; that's certainly theopinion of a lot of Python people.>> Core language issues such as variable scoping rules are>>subject to frequent change in the language. (Indeed, the designers of>>Python initially thought that variable scoping issues were>>*unimportant*.) >> I don't know that the scoping rules change frequently - they>> do try to avoid major changes unless they have a very good>> reason. The new scoping rules are regarded as an improvement.>Sure. But why did they screw up in the first place? Don't know. Probably Guido is just stupid.Lisp _is_ a lot older than Python, right? Were the first fewversions of Lisp as wonderful as the current versions?>We're now on the>third or fourth set of Python scoping rules, and *none* of the issues>here are new. === Java permission for an emailed response.X-Zippy-Says: Hey, waiter! I want a NEW SHIRT and a PONY TAIL with lemon sauce!> Lisp _is_ a lot older than Python, right? Were the first few> versions of Lisp as wonderful as the current versions?Surely === emailed response.X-Tom-Swiftie: Don't forget to shut down the computer, Tom said haltingly> I find that hard to believe. (Which is not to say it's not so - if> I had any _evidence_ that it's not so I'd say so.)The two cases I know of are Oberlin and MIT. There are more; just askMIT Press which schools put in orders for SICP and then see whichclasses they === why a compiler written in the language itself should>> be a requirement, unless we're requiring that _one_ language>> should do everything - requiring that doesn't seem like a>> good idea to me, having high-level languages that do>> subtle things very easily and low-level languages that do>> simple things efficiently seems like a more reasonable>> plan.>Not a good idea to require it. Fine.>But perhaps there are languages which>dominate Python--that do what it does, better, and also have more>features? Regardless, the self-hosting comment was one explanation of>what a large system *is*. Seems more accurate to me to say that it's one _example_ ofa large system (which you gave as part of an explanation ofwhat a large system is.)The point being to make it clear that there are _other_examples of large systems.>If you're now saying that Python is only>intended for small systems, and not large ones, then that rather>grants the point I was making.But in granting that perhaps Python is no good for writingan optimizing Python compiler I'm _not_ agreeing that it'sno good for large systems, I'm (tentatively) agreeing thatit's no good for that particular example of a large system.>> (i) How hard is it to write Python code to do symbolic>> differentiation? Turns out to be very easy - you don't>> have to parse expressions or anything like that:>Right. Doing this in Scheme is *even easier*! Has it ever occured to>you there may be a reason that the Macsyma project used Lisp?No, that could not possibly have occured to me, since I had noidea it used Lisp. If I'd known that then yes, it would haveoccured to me...>All your examples are demonstrations of how a certain feature of>Python works wonders--and they are nice examples. The _second_ example, the one that I said was my main point,was also supposed to be an example of a large system wherePython works very well. (Or a system that is surely not largeby your definition, but which seems to me large enough andwhich seems to work well enough that it convinces me thatthere's no reason Python could not work very well in asystem that would be large by your definition.)>Now it turns out>that what in Python is a special feature, is a natural and quite>automatic thing in Scheme; it's not a special feature, because the>structure of the language makes it inherently easy. It falls out of>more general facilities, that are not even thought of as facilities,>they are just *there* like the air you breathe.>So having seen what good things a facility like Python's is in these>areas (and it is, certainly), now what about a language where that>kind of thing is *everywhere* and is so === Push for Java permission for an emailed response.> The point being to make it clear that there are _other_> examples of large systems.Sure. The reason one asks about self-hosting is because one assumesthat the implementors of a language are fans of it--perhaps thegreatest fans, as well as those who know more about it than anyoneelse. So if it were susceptible to self-hosting, wouldn't they do so?When they don't, an initial assumption with some merit (butrebuttable, yes) is that the language's greatest fans and the peoplewho know more about it than anyone else don't think it's suitable forsuch a === bother asking what percentage of programmers in general>> _do_ have experience in lisp-like languages, or the same question>> for Algol-like languages...>Of course many more have experience in Algol-like languages. But>that's a property of society, not a property of the languages. Yes, it is a property of society, not a property of the languages.It's a property of society that has a lot of relevance to the questionof how often it happens that people learning Lisp find themselvessolving actual problems a few hours after their first exposure to thelanguage, versus how often that happens with === Java permission for an emailed response.> Yes, it is a property of society, not a property of the languages.> It's a property of society that has a lot of relevance to the question> of how often it happens that people learning Lisp find themselves> solving actual problems a few hours after their first exposure to the> language, versus how often that happens with Python.Yes, but my point is that this is really irrelevant to how good thelanguage is in itself. If everyone know COBOL backwards and forwards,and nobody knew C, Lisp, or Python, that may well mean that knowingCOBOL is a more marketable skill, but it doesn't make COBOL a === <878yk9m4yp.fsf@becket.becket.net> <874quvyfjt.fsf@becket.becket.net> <87ektu8leo.fsf@becket.becket.net> <87isj655pt.fsf@becket.becket.net> <87r7xsdh48.fsf@becket.becket.net> Discussion, linux)> No, I'm saying that the syntax is taught in about five minutes, and it> is then all known. And that's for people who don't know *any*> existing syntaxes. Python is easy IF you already know Algol-like> languages. Lisp syntax is easy right from the get go. It's as easy> as this:> * a number looks like a number (3, 3.1415, -156, etc.)> * a variable is a set of characters that doesn't look like a number,> and doesn't include parens or spaces; also a single period isn't a> variable.> * a list is a pair of parens with things in them (numbevariables,> other lists)> That's it. There are schools which use Scheme as their intro CS> class, and it's a commonplace that their students are learning about> programming techniques while the students in the school down the> street are still struggling over Pascal syntax (that was in the 80s, I> suppose now the struggle over Java syntax or whatever).I admit that my first introduction to programming was BASIC, thenPascal, C, etc. and so I learned functional languages like Lisp afterI had already learned the other dominant type (say, what *are* theycalled? I forgot).I've always thought that functional languages are harder to learn thanlanguages like C, just because they're written backwards from the wayI think. I think in terms of: First do X and then do Y. Thus,writing things as X; Y seems natural to me. Writing (Y (X)) is lessnatural. (My pseudocode is pretty bad here, since the second appliesY to the return value of X and the first doesn't appear to do anythinglike that, but I hope that the point is more or less clear.)Now, I've always thought in those terms, despite the fact that Iprefer to write the composition of mathematical functions as g o frather than f;g. I suppose that my own biases may be attributable to the way I learnedprogramming, but I just don't think so. I wonder, then, if there areany studies about the ease with which students learn functionallanguages as opposed to that other kind. I don't mean syntax. I meanhow quickly they learn to put programs together.I don't have a bias against functional languages, honest. I think theresulting programs often look slicker'n owl on a brass doorknoband I always like it when I can put things together in a clever way ina Lisp program. But it's been difficult for me to learn that skilljust because the order in which we write commands in a functionallanguage is backwards from the order in which we write basic recipesfor doing tasks.So, I'm just not too sure that learning to program in Lisp is just asstraightforward and simple as you claim. It would be nice to see ifthere's any real evidence that students that learn a functionallanguage first start writing programs just as quickly and easily asthose who learn some other language (where the programs solve the sameproblems for each).-- [Lancelot] sighed, defea. 'It is as practical to hurry an acorntoward treeness as to urge a damsel when her mind is set.' -- John Steinbeck, /The Acts of King === Java <878yk9m4yp.fsf@becket.becket.net> <874quvyfjt.fsf@becket.becket.net> <87ektu8leo.fsf@becket.becket.net> <87isj655pt.fsf@becket.becket.net> <87r7xsdh48.fsf@becket.becket.net> Discussion, linux)>> This one I don't get at all. Are you claiming that people know how>> to write valid Lisp without having to learn how to do so???> No, I'm saying that the syntax is taught in about five minutes, and it> is then all known. And that's for people who don't know *any*> existing syntaxes.Evidently I am slow. I seem to recall that lisp has all sortsa funnysyntax involving ticks, backticks and whatnot that I just nevergrokked completely. I always have to look at examples forconstructions involving those buggers.But maybe I just use funny versions of Lisp.(Common Lisp? Someone's sick and twis joke. How can a group ofinterpreters all implement something called Common Lisp and all beincompatible? What does *Common* mean?)Nonetheless, I *like* Lisp. Really I do.-- Conservative, n: A statesman who is enamored of existing evils, as distinguished from the Liberal who wishes to replace them with === permission for an emailed response.> Evidently I am slow. I seem to recall that lisp has all sortsa funny> syntax involving ticks, backticks and whatnot that I just never> grokked completely. I always have to look at examples for> constructions involving those buggers.Semantics, not syntax, but I'll admit that I didn't list them in myoriginal post, because they can always be dropped. However:'foo is an abbreviation for (quote foo)`foo is an abbreviation for (quasiquote foo),foo is an abbreviation for (unquote foo),@foo === Re: JSH: Push for Java > I don't see why a compiler written in the language itself should > be a requirement,... > Not a good idea to require it. But perhaps there are languages which > dominate Python--that do what it does, better, and also have more > features? Regardless, the self-hosting comment was one explanation of > what a large system *is*.I have given two examples of systems written in Python. On of thoseyou have commen on, the other has apparently failed to attract yourattention.(1) A multimedia authoring system, including a GUI, incorporating pictures, videos, and whatever you want. You did not comment on that. small-scale. I will write about it a bit more. It consists of a number of programs running on different machines. The first is the smallest, it is a two-way multiplexer between TCP/IP and four RS232 ports. It sends commands received by TCP/IP to the appropriate port, listens to the four ports and sends answers back. This all in real-time. This is cosily running on a 80286. The other part runs on another machine; it takes a mathematical function in 3-D, translates the subsequent places of the object in time to coordinates in 3-D, translates that again to the length of cords on which the object would be suspended from four different points, allowing for the fact the none of the cords should be lose. Transmittes that through TCP/IP to the other machine where the four RS232 ports control the four motors controlling the cords. This is in no way a small system. It would have been a nightmare to do it in C (or C++). I do not think that it would have been much better than a nightmare in CL or Scheme. I have done this with a colleague in reasonable time. Most time was spent understanding how the Basic commands === Push for Java permission for an emailed response.> (1) A multimedia authoring system, including a GUI, incorporating> pictures, videos, and whatever you want. You did not comment on> that.I haven't seen it; I've seen things that fit this description whichare fabulous tools, and others which are front ends for web sites (andalas, the latter get described in fabulous tool language). Sounless you can say more, I can't comment on it.As for the other system, can you say why it would have been anightmare to implement in Scheme or === authoring system, including a GUI, incorporating > pictures, videos, and whatever you want. You did not comment on > that. > I haven't seen it; I've seen things that fit this description which > are fabulous tools, and others which are front ends for web sites (and > alas, the latter get described in fabulous tool language). So > unless you can say more, I can't comment on it.See which describes a multimediaauthoring model. The implementation referred to in that thesis (CMIFed)was written in Python. > As for the other system, can you say why it would have been a > nightmare to implement in Scheme or Lisp?I said that it *would* have been a nightmare in C and C++, and that Isuspec it would have been similar in Scheme or Lisp. But I do notknow enough about the latter two to be sure about it. Anyhow, themultiplexer is === for Java permission for an emailed response. What a COINCIDENCE! I'm an authorized ``SNOOTS OF THE STARS'' dealer!!> I said that it *would* have been a nightmare in C and C++, and that I> suspec it would have been similar in Scheme or Lisp. But I do not> know enough about the latter two to be sure about it. Anyhow, the> multiplexer is about 300 lines of Python code.Good grief, that's not even a small === be good things, _if_ they're conduc> rationally, with people _supporting_ their statements. So far> the only reason you've given for Python being bad is that it> does not work well for large projects, and in spite of repea> requests, including one in the post you're replying to, you> have declined to explain exactly what aspect of Python makes> it unsuitable for large projects.If I had to give a reason why Python may not be good for large projects itwould have to be its dynamic type system. While dynamic typing is a goodidea in some contexts for large projects it can cause problems if the codeis not extensively tes (which rarely happens). With a statically typedlanguage a lot of checking (of this sort) can be done at compile time (ie.checking if two types support a reques operation) whereas in a languagelike Python it is left to runtime.I have not used Python for large projects but this would be one of my mainconcerns (I am a professional software engineer and use === can be good things, _if_ they're conduc>> rationally, with people _supporting_ their statements. So far>> the only reason you've given for Python being bad is that it>> does not work well for large projects, and in spite of repea>> requests, including one in the post you're replying to, you>> have declined to explain exactly what aspect of Python makes>> it unsuitable for large projects.>>If I had to give a reason why Python may not be good for large projects it>would have to be its dynamic type system. While dynamic typing is a good>idea in some contexts for large projects it can cause problems if the code>is not extensively tes (which rarely happens). With a statically typed>language a lot of checking (of this sort) can be done at compile time (ie.>checking if two types support a reques operation) whereas in a language>like Python it is left to runtime.Yes, that was exactly the aspect of Python that I conjectured (somewhere in this thread) could be regarded as a problemusing Python with large projects.Otoh in yet another place in this thread you see a description ofa somewhat-large project that depends crucially on this aspectof Python.(One might suggest that a large project where the code is notextensively tes is in trouble regardless of what language isused. Yes, the dynamic nature of Python means you need toinclude tests that would not be needed in a strongly typedlanguage. But whether it usually happens or not, surelyregardless of the language every module _should_ beaccompanied by a test suite that checks all the functionalityin the module...)>I have not used Python for large projects but this would be one of my main>concerns (I am a professional software engineer and use === Java permission for an emailed response.X-Tom-Swiftie: My clothes are coming apart, Tom said orgasmically> If I had to give a reason why Python may not be good for large projects it> would have to be its dynamic type system. While dynamic typing is a good> idea in some contexts for large projects it can cause problems if the code> is not extensively tes (which rarely happens). With a statically typed> language a lot of checking (of this sort) can be done at compile time (ie.> checking if two types support a reques operation) whereas in a language> like Python it is left to runtime.Quite the opposite is true. Dynamic typing is actually a great boon.Indeed, C++, your preferred language, doesn't have GC. I think thedisasters due to memory errors far outweigh the occasional typingmistakes in dynamically typed languages. Experience has shown thatdynamic typing, far from decreasing reliability, increases itconsiderably, especially in polymorphic languages like CL or Python.Scheme, which is not polymorphic, achieves the same benefits by theuse of a much more highly functional style than is usual in Lisp;again, the dynamic aspects of the typing thus === Java>> If I had to give a reason why Python may not be good for large projects it>> would have to be its dynamic type system. While dynamic typing is a good>> idea in some contexts for large projects it can cause problems if the code>> is not extensively tes (which rarely happens). With a statically typed>> language a lot of checking (of this sort) can be done at compile time (ie.>> checking if two types support a reques operation) whereas in a language>> like Python it is left to runtime.>Quite the opposite is true. Dynamic typing is actually a great boon.>Indeed, C++, your preferred language, doesn't have GC. I think the>disasters due to memory errors far outweigh the occasional typing>mistakes in dynamically typed languages. Experience has shown that>dynamic typing, far from decreasing reliability, increases it>considerably, especially in polymorphic languages like CL or Python.Huh. Of course dynamic typing is a great boon for many reasons,in particular it's crucial in the sort of things that I find fascinating (long description of an example elsewhere), but I'msurprised that you say this - I thought the standard wisdomwas that !Q's point about what the Python guys call bondage anddiscipline languages was exactly correct. Could you give ahypothetical example of how dynamic typing can increasereliability?(Or was the point about memory errors your main point here?If so, is there some reason there could not exist a stronglytyped language with garbage collection, theoreticallyimmune from memory errors? Hmm, Java is such a language,is it not?)Note these are actual questions, not assertions disguisedwith question marks...>Scheme, which is not polymorphic, achieves the same benefits by the>use of a much more highly functional style than is usual in Lisp;>again, the dynamic aspects of the typing thus introduced === JSH: Push for Java permission for an emailed response.X-Zippy-Says: I'm encased in the lining of a pure pork sausage!!> Huh. Of course dynamic typing is a great boon for many reasons,> in particular it's crucial in the sort of things that I find > fascinating (long description of an example elsewhere), but I'm> surprised that you say this - I thought the standard wisdom> was that !Q's point about what the Python guys call bondage and> discipline languages was exactly correct. Could you give a> hypothetical example of how dynamic typing can increase> reliability?The B&D languages remark is actually one that the Lisp community didfirst too. :)> If so, is there some reason there could not exist a strongly> typed language with garbage collection, theoretically> immune from memory errors? Hmm, Java is such a language,> is it not?Sure; they are more or less orthogonal === Scheme, and CL.> Yes you did - he wan't paying attention.Yes, === Re: JSH: Push for Java> You have claimed that Python is not suitable for> large systems, but have produced neither reasons nor> evidence.> Evidence: the lack of any.Well, I would contend that such systems do exist (includinge.g. the complex image processing system written inPython/C that I am part of a team working on).Elsethread it would appear that your comments aredirec more at the type of system than the size.Classify your large system critisism:evidence very weak, reasons none> You imply that Python has other weaknesses but do> not list them.> Python maintains the data/program distinction. Python is hard to> parse; people have to be taught the syntax. (Those two go together,> of course.) Core language issues such as variable scoping rules are> subject to frequent change in the language. (Indeed, the designers of> Python initially thought that variable scoping issues were> *unimportant*.) It is not possible to capture continuations in> Python. It is not possible to create lexical closures in Python.parse/syntax This is in direct opposition to my experience and that of many others. I find the Python syntax natural and was able to learn enough in a few hours that I was able to do serious programming.scoping We have found language stability issues to be an irritant but nothing worse. We have never had any problems with changes in scoping rules.Overall your list of weaknesses sounds like a whine thatGuido van Rossum doesn't worship at the One True Church ofFunctional Programming. > You claim that there are tools which dominate Python> but for some reason are being coy as to their identity.> I lis two: Scheme, and CL.In a post that didn't arrive on my news server (Google) until afterI pos (and in this post though you imply that you favourthese languages, you do not claim they dominate Python).In the post in which you state There are languages which are extremely fabulous for the areas in which Python is good, *and* which are also fabulous for programming large systems.you mention Scheme but not CL (and Scheme is mentioned in a differentcontext (operator overloading). I stand by the === permission for an emailed response. CONGRATULATIONS! Now should I make thinly veiled comments about DIGNITY, self-esteem and finding TRUE FUN in your RIGHT VENTRICLE??> Well, I would contend that such systems do exist (including> e.g. the complex image processing system written in> Python/C that I am part of a team working on).> Elsethread it would appear that your comments are> direc more at the type of system than the size.Hrm, working on is different from comple. Regardless, manylanguages which are not suitable for large systems nontheless are usedfor them. Examples: C, which is suitable for some things, but notlarge systems. Bliss, which isn't suitable for anything, but they> parse/syntax This is in direct opposition to my experience> and that of many others. I find the Python syntax> natural and was able to learn enough in a few hours> that I was able to do serious programming.Sure, because you were already familiar with Algol-like languages. Bycontrast, the syntax of Lisp can be explained in about five minutes tosomeone with no programming background at all.Parsing, you're just wrong; Python is a pain to parse (just as C orPascal are a pain). Python is a fair bit easier than C, and a fairbit harder to parse than Pascal, and a bajillion times harder to parsethan Lisp. > scoping We have found language stability issues to be an> irritant but nothing worse. We have never had any> problems with changes in scoping rules.Presumably because you have stayed away from anything === JSH: Push for Java>[...]>Parsing, you're just wrong; Python is a pain to parse (just as C or>Pascal are a pain). Python is a fair bit easier than C, and a fair>bit harder to parse than Pascal, and a bajillion times harder to parse>than Lisp. Well, the only thing I've ever written a parser for is XML, or rathera subset of it called QML (and the reason I was able to do thathas something to do with the fact that XML is sort of self-parsing...)So what I thought about this is probably totally wrong, and I'mnot going to be able to ask the question very efficiently: Therecertainly exist standard automatic parsethat accept a formaldescription of a grammar. Some programming languages donot admit description in these standard grammar-definitionlanguages, but my impression was that Python does. Yes?>> scoping We have found language stability issues to be an>> irritant but nothing worse. We have never had any>> problems with changes in scoping rules.>Presumably because you have stayed away from anything that would get>near them. Not a good sign.This strikes me as, um, having a hard time finding the rightadjective. If someone hasn't had a problem with a supposedflaw in a language some people would take that as evidencethat the flaw was not that serious, not as evidence that theperson who hadn't had any problems was missing === JSH: Push for Java permission for an emailed response.> So what I thought about this is probably totally wrong, and I'm> not going to be able to ask the question very efficiently: There> certainly exist standard automatic parsethat accept a formal> description of a grammar. Some programming languages do> not admit description in these standard grammar-definition> languages, but my impression was that Python does. Yes?Yes. C does too. GCC's parser for C is amazingly complex, as isfew thousand lines long. A parser for Scheme can be written in abouta hundred === contend that such systems do exist (including> e.g. the complex image processing system written in> Python/C that I am part of a team working on).> Elsethread it would appear that your comments are> direc more at the type of system than the size.> Hrm, working on is different from comple. Regardless, many> languages which are not suitable for large systems nontheless are used> for them. Examples: C, which is suitable for some things, but not> large systems.I repeat my question. Have you been taking gnomic lessons from Harris?We now know that you think it is possible to come up with usefulcriteria for why a programming language would be suitable for largesystems. We know you don't consider C or Python suitable for largesystems. You have not actually given an example of a language thatyou do consider suitable for large systems, although I assume that Scheme fits into this catagory. However, we are still completelyin the dark as to what criteria you are using. I hope its somethingmore than === Re: JSH: Push for Java permission for an emailed response.X-Tom-Swiftie: I've finished counting the horses, Tom said summarily> I repeat my question. Have you been taking gnomic lessons from Harris?> We now know that you think it is possible to come up with useful> criteria for why a programming language would be suitable for large> systems. We know you don't consider C or Python suitable for large> systems. You have not actually given an example of a language that> you do consider suitable for large systems, although I assume that > Scheme fits into this catagory. However, we are still completely> in the dark as to what criteria you are using. I hope its something> more than functional good, procedural bad.I think Scheme is good, Common Lisp is as well. I could imaginePython evolving into something that would be good.I have already identified some specific features that I think aremissing in Python, and other factors that are quite === trig algebra> I thought the rule was don't divide through by something that may be zero> or you may lose some solutions. Is there a simple valid rule I canstate?Sure. Take two cases: something = 0 (and therefore . . .) or something !=0 (and therefore you can divide by it).So in your (dele) example of sin(x) = 5cos(x), either cos(x) = 0, inwhich case sin(x) = 0 in which case x = ?, or cos(x) != 0, in which === Understanding trig algebra by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id William + Robin too).>I'm still troubled by this though...>> hm no...>> sin(x)=5cos(x)>> therefore>> sin(x) - 5cos(x) = 0>> therefore>> cos(x)*sin(x)/cos(x) - 5cos(x) = 0 unless cos(x) = 0 !>> [ because you are not allowed to write this when cos(x) = 0 ]>> therefore>> cos(x)(tan(x)-5)=0 unless cos(x) = 0>> therefore>> tan(x)-5 = 0 unless cos(x) = 0>> etc...>I'm not convinced, because using this logic, can't you incorrectly prove>x^2-x=0 has only the one solution? E.g.>x^2-x=0>therefore x(x^2/x-1)=0 provided x != 0>therefore (x=0 or x=1) provided x != 0>therefore x=1>So I now think the mistake is as follows:>sin(x) = 5cos(x) CORRECT>sin(x) - 5cos(x) = 0 CORRECT>cos(x)(tan(x)-5)=0 CORRECT>therefore cos(x)=0 or tan(x)=5 INCORRECT>This last line is wrong because a.b=0 implies a=0 or b=0 fails when a=0>implies b=infinity which is what we have in cos(x)(tan(x)-5)=0 since>cos(x)=0 implies (tan(x)-5)=infinity>So this means logic a.b=0 implies a=0 or b=0 always needs treating>carefully.>It works fine on x(x^2/x-1) since here x=0 does not imply>(x^2/x-1)=infinity.>Do you think I have this understood careful. Starting from sin(x)= 5cos(x) you can say AS LONG AS cos(x) IS NOT 0, we can divide by cos(x) to get tan(x)= 5 so x= tan^(-1)(5). Now check what happens if cos(x)= 0: IF cos(x)= 0, then sin(x)= 1 or -1 so that does not give a solution. The only solutions are values of x for which tan(x)= 6. Your note that cos(x)=0 implies (tan(x)-5)=infinity simply shows that x such that cos(x)= 0 does not satisfy the equation so you don't have to worry about that. You CAN divide by some quantity involving the unknown, x, as long as you note that solution is assuming the quantity is not 0 and are ALSO careful to see what happens to your equation IF that quantity === supercalifragilisticexpialidocious day, after dancing about singing Bibbety bobbety boo!, John Griffin ishkabibbled:(ridiculous attempt at flame war snipped)I think the name of this thread should be (for your sake )Use Cyanide And Perish!And tell Chris Reeve he's lame TO HIS FACE YOURSELF, chicken who has nothing better to do than insult the handicapped.Plonk!-- The Queen of DXeas well asQueen of the Commonwealth of Virginia, as well asThe Ruler of A.D.P., as well asSaint Debbe, as well asOur Lady of the Black Hole Exploratory Input Services as OhFishAlly Appoin by the Psychedelic Pope, a/k/a Saint Isidore of SevilleAn Oin Minister of the Universal Life ChurchReverant of the Church of the SubGenius, UnOrthodoxSuperior Mutha Superior of the Little Sistahs of the Politically IncorrectWorshipper of Eris, === Re: Logs, bases and Casio calculators> for my calculations is not allowed for an exam I am sitting shortly.> Therefore I have been forced to borrow a less complica Casio FX-83WA.I'm familiar with converting from base 10 logs to other bases, but does> anyone know wether the Casio family of characters has a button or syntax> for specifying the base? Surely I don't have to perform manual> calculations each time I want a base other than 10.> For heaven's sake.... Real Mathematicians never use base 10 logs.Bah! Henry Briggs was a Real Mathematician.Real Mathematicians never use base 10 logs *any more*!Jon === === calculators>Subject: Re: Logs, bases and Casio calculators>For heaven's sake.... Real Mathematicians never use base 10 logs. So what bases do real mathematicians use? e of course. Logs to base 2>> are>> useful in many cases. Any others that come up in real math? What's this math that you and others talk about? Anyway Real Mathematicians use logarithms, but never bases.>> All your base are belong to us.>One leg is both the same.Now if that was one log is both the same, the thread would be back === calculators>For heaven's sake.... Real Mathematicians never use base 10 logs. So what bases do real mathematicians use? e of course. Logs to base 2>> are>> useful in many cases. Any others that come up in real math? What's this math that you and others talk about? Anyway Real Mathematicians use logarithms, but never bases.> All your base are belong to us.> Que?It's an example of Engrish that's become a popular Internet === Subject: Re: wooooooooooosssssssssshhhhhhhhhhh.............On a supercalifragilisticexpialidocious day, after dancing about singing Bibbety bobbety boo!, Shaun Rimmer ishkabibbled:^^>^> All centuries are the same length, bimbo.^^Heheheh! Ya hear that Queenie? You're a bimbo now! LOL, FFS!^^> There are no short^> ones. Don't take my word for it. Ask your mommy.^>^> And I was born with a plastic spoon in my mouth. My parents^> couldn't afford silver till much later.^>^> You can take it out now.^^Silly little idiot!^^Actually, Shaun, that pertickular hillbilly has been killfiled.And I never said I WASN'T a bimbo!!! Just a remarkably educa one.-- The Queen of DXeas well asQueen of the Commonwealth of Virginia, as well asThe Ruler of A.D.P., as well asSaint Debbe, as well asOur Lady of the Black Hole Exploratory Input Services as OhFishAlly Appoin by the Psychedelic Pope, a/k/a Saint Isidore of SevilleAn Oin Minister of the Universal Life ChurchReverant of the Church of the SubGenius, UnOrthodoxSuperior Mutha Superior of the Little Sistahs of the Politically IncorrectWorshipper of Eris, Goddess of DiscordI WON'T grow === wooooooooooosssssssssshhhhhhhhhhh.............X-Who-Cares: Who cares?> On a supercalifragilisticexpialidocious day, after dancing> about singing Bibbety bobbety boo!, Shaun Rimmer> ishkabibbled: ^> message > ^^> All centuries are the same length, bimbo.> ^> ^Heheheh! Ya hear that Queenie? You're a bimbo now! LOL,> FFS! ^> ^> There are no short> ^> ones. Don't take my word for it. Ask your mommy.> ^^> And I was born with a plastic spoon in my mouth. My> parents ^> couldn't afford silver till much later.> ^^> You can take it out now.> ^> ^Silly little idiot!> ^> ^> Actually, Shaun, that pertickular hillbilly has been> killfiled. The faggot says he's a coward, besides being a bimbo.> And I never said I WASN'T a bimbo!!! Just a remarkably> educa one. Why do you take such pains to conceal that while crowing about it, === wooooooooooosssssssssshhhhhhhhhhh.............On a supercalifragilisticexpialidocious day, after dancing about singing Bibbety bobbety boo!, Shaun Rimmer ishkabibbled:^^> On a supercalifragilisticexpialidocious day, after dancing about singing^> Bibbety bobbety boo!, Shaun Rimmer ishkabibbled:^> object in my hand and communicate, was something that, as^little^> ^>> as 200 years ago, would have had one burned at the stake. Now it's^> ^>> commonplace and easily understood.^> ^>^> ^>^> ^>Could you explain what you mean?^> ^>^> ^> i think pianowow means a cellphone...either that or a magic spoon.^> ^^> ^I think he was definitely referring to cutlery.^> ^^> ^^> ^^> ^Shaun aRe, born with a magic spoon in his mouth.^> ^^> ^^> ^^> ^^> Actually, I think that was MY quote, and I was referring to^> technological advances that would have been seen as witchcraft a few^> short centuries ago.^^Sorry for propagating the misatribution there! Oh, knew excatly what you^were referring too though ',;~}~^^> And I was born with a plastic spoon in my mouth. My parents couldn't^> afford silver till much later.^^They shoulda saved up soon as they knew you were on the way!^^^Shaun aRe^^^Actually, my mom gave me her Mickey Mouse cereal spoon she'd gotten as a kid out of a Kellogg's box. That beats a silly ole silver spoon ANY day!!!-- The Queen of DXeas well asQueen of the Commonwealth of Virginia, as well asThe Ruler of A.D.P., as well asSaint Debbe, as well asOur Lady of the Black Hole Exploratory Input Services as OhFishAlly Appoin by the Psychedelic Pope, a/k/a Saint Isidore of SevilleAn Oin Minister of the Universal Life ChurchReverant of the Church of the SubGenius, UnOrthodoxSuperior Mutha Superior of the Little Sistahs of the Politically IncorrectWorshipper of Eris, Goddess of DiscordI === algebraic geometry texts> Anyone know of any good ones?Although I am totally ignorant of the subject matter, I will say thatwith a search engine you can use the compound termalgebraic geometry lecture notesand come up with free materials that may be worth your consideration.david === support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0LDsh817363;>The geometry of === Nothing squared plus one equals zero (0).> Because zero (0) is a number in Calculus, every algebraic equation is> now solved. For example:> Consider the equation: x - 3 = 0 > The solution for x - 3 is the number 0 A equation is solved for a variable (in this case, x), manipulatingboth terms fo the equation until only x remains at one side, and aexpression not containing x in the other side. So, the solution forthe equation x - 3 = 0 is 3.x - 3 is not an equation, thus can't be solved; it's just anexpression.Replies in newsgroup only, please.Duran Castore <4e63857.0401211028.13dc1f18@ Garry Denke> Is this useful?If you're into solving equations> What do you think?Other than being depressed last night and all day about the Eagles? > (x^2 + 1)/0 = (0)/0, Socrates' Nothing is Everything. Denke, 1655OK, time for the engineering hat.An equation represents, say, the stress on a bridge. An architect> has to decide whether to use concrete (rocks), paper, or scissors --> erm, I mean, iron -- for a certain critical subspar. He solves> an equation and gets the answer 0/0. Which does he use, and why? Paper, because his paper covered rock (limestone) cut by iron.>> Uh huh.>> Please show me a (non-model) bridge made out of paper.> http://www.bennington.com/chamber/Bridges/images/paper_ mill.jpg> PAPER MILL BRIDGE> and follow to first street on left, Murphy Road (.5 miles) Bridge may> be seen from Rt. 67. This bridge, originally built in 1889 by Charles> F. Seawas completely reconstruc in 2000.> DIMENSIONS: 125.5 feet long, 14.25 feet wide, 8.67 feet high at truss,> 11.17 feet high at center. This bridge was built by the son of the> Silk Bridge builder.> Want to see a Silk Bridge?Arrgh. What I meant was one using paper in its construction as theprimary structural element (as opposed to wooden beams, steel girders,concrete, etc.)-- #191, ewill3@earthlink.netIt's still legal to go === GtFpSuZOgeLMu3yi+U0NQcAikkLagIbEZFge+8VFxQjN6AFw-y+MwMHallo Jase,> I have a couple more boolean algebra expressions that need minimized.> I have the answers but I don't know exactly === Boolean Algebra> I have a couple more boolean algebra expressions that need minimized.> I have the answers but I don't know exactly how to get to them:> 1) X + Y' + X'.Y + (X+Y').X'.Y> answer: 1> 2) ((P'.Q)'.(P.R)')'> answer: P'.Q + P.R> Can someone show me the steps to get these answers please.more or less blindly applying the rules of propositional logic doesn't work for you?as an old prof of mine would put it:proof of the above: by === couple more boolean algebra expressions that need minimized.> I have the answers but I don't know exactly how to get to them:> 1) X + Y' + X'.Y + (X+Y').X'.YShow (x+y)x'y = 0 and prove useful theorema + a'b = a + b = b + b'aThis theorem was also used previous threadso you're seeing it's repea usefulness.Thus put it in your notes to have as needed.> answer: 1> 2) ((P'.Q)'.(P.R)')'> answer: P'.Q + P.R> Can someone show me the steps to get these answers please.Ah, the === Algebra> I have a couple more boolean algebra expressions that need minimized.> I have the answers but I don't know exactly how to get to them:> 1) X + Y' + X'.Y + (X+Y').X'.Y> answer: 1Well, you just do ordinary algebraic manipulations using the followingrules:1: (A+B)' = A'.B'2: A.A' = 03: A+A' = 14: 0.A = 05: 1+A = 16: A'' = ASo, in the above expression, use rule 1 and rule 6 to rearrange the thirdterm. Then combine the result with the first and second term using rule 3.Finally it turns out that you can forget about the fourth term because ofrule 5.It's much easier than you think, kind of like riding a bike. Once you getthe hang of it, it all seems so === table2) deMorgan's laws, distributive laws|| I have a couple more boolean algebra expressions that need minimized.|| I have the answers but I don't know exactly how to get to them:|| 1) X + Y' + X'.Y + (X+Y').X'.Y|| answer: 1|| 2) ((P'.Q)'.(P.R)')'|| answer: P'.Q + P.R|| Can someone show me the steps to get === AlgebraThis isn't logic, it's algebra; so how do truth tables apply?top postingleaves questionto be rememberedorto be seenlater> 1) truth table> | 1) X + Y' + X'.Y + (X+Y').X'.Y> === algebra; so how do truth tables apply?Well, you're using algebra to try to prove an identity. Normally, youcanot prove an identity by example, since you cannot try every valueof each variable. You can disprove an identity by counter-example,but not prove it.However, in Boolean algebra, you CAN try every possible combination ofvalues, since each variable has only two values, 1 or 0, True orFalse, On or Off, etc.An identity with four variables in it would only involve 16 possiblecombinations of values. This is certainly possible in a === logic, it's algebra; so how do truth tables apply?> However, in Boolean algebra, you CAN try every possible combination of> values, since each variable has only two values, 1 or 0, True or> False, On or Off, etc.> An identity with four variables in it would only involve 16 possible> combinations of values. This is certainly possible in a truth> table...Can you? Consider the Boolean algebra of subsets of integers with* + ' as === FNKKCLHLDLDCIBIBLOBLPLNJFLKDECOBBNEMEMIKIn physics impenetrability is the inability of two portions of matter tooccupy the same space at the same time: Or the property of matter thatprevents two bodies from occupying the same space at the same time. That is any attempt to move any two bodies into the same place will cause themto mutually thrust against each other; and force them out of each other'sway: Changing each other's previous speed and or direction, while the world_continuously_ keeps on rolling along.That is thrusts aren't instantaneous: They consist of the force _and_ itsduration. They are 'impulses', which for just a light touch, may be brief,or sustained; as is the force of a body's weight: Where bodies resting onEarth's surface are being continuously displaced from falling further; at arate of about 16 feet per second, per second, by the surface on which theyrest.For a body on Earth's surface then, its inertia, and the quantity of matterthat it contains is the ratio of the product of its weight-force [w] and theduration [t] during which it is exer on it by the ground; that ispreventing further displacement [s]; at a rate of s/t = gt/2 = about16'/sec^2. Where [g] is its gravitational acceleration.For any body, anywhere then, its inertia, and the quantity of matter that itcontains is the ratio of the _net_ mechanical-impulse [ft= F-uwt] exer onand/or by it to the rate of forced displacement [s/t] that it causes:ft/(s/t) = ft^2/s; is a Constant, and the ratio [s/t = at/2]; where a is itsinertial acceleration. ----- Pos via NewsOne.Net: Free (anonymous) Usenet News via the Web ----- http://newsone.net/ -- Free reading and anonymous posting to 60,000+ groups other postsmade through NewsOne.Net violate posting === and Displacemrnt> In physics impenetrability is the inability of two portions of matter to> occupy the same space at the sand. Really pack itin and tamp it down. Now, add salad oil.Hey Dumb Donny Head, crystallize a 1-A zeolite crystal. Now addup to 20 wt-% water. It will not be wet at the end, nor will itsvolume have changed. Where did all the water go?Hey Dumb Donny Head, compress cryogenic solid hydrogen in adiamond anvil press. It doesn't leak out, so where does it go when itdisappears?Hey Dumb Donny Head, mix exactly 50 milliliters of distilled waterat ambient temperature with exactly 50 milliliters of anhydrousethanol at ambient temperature. Measure the volume of the mixture atthe same ambient temperature. It will be 95 milliliters. How can (50 ml) + (50 ml) = (95 ml)Dumb Donny Head?Hey dumb Donny Head, if you take a cubic centimeter of purelithium metal and force in 862 cm^3 of hydrogen gas, you get a nicewhite lump of volume 0.8 cm^3. How can(1 cm^3) + (862 cm^3) = (0.8 cm^3)?Dumb Donny Head?Hey Dumb Donny Head, don't pull your head out of your ass, but docease listening for echoes.http://www.mazepath.com/uncleal/sunshine.jpg--Uncle Alhttp://www.mazepath.com/uncleal/qz.pdfhttp:// www.mazepath.com/uncleal/eotvos.htm (Do something naughty to === mathematician, so may be for the puremathematicians my question will be too stupid. Sorry for this, butI'll be kindly apprecia for your assistance.Is it possible to say something about the derivative of such functionsof the complex variable as:exp(i*Arg z)orz/|z|where z belongs to C excepting (0,0).It seems that according to the Cauchy-Riemann conditions thesefunctions are not differentiable, isn't it?If so, how this corresponds with the fact that exp(z) is analytic onC?Another question is: what is possible to say about thedifferentiability of the functions |z| and Arg(z), where z iscomplex?I will kindly appreciate your kind === variable function exp(i*Arg z)>Is it possible to say something about the derivative of such functions>of the complex variable as:>exp(i*Arg z)>or>z/|z|>where z belongs to C excepting (0,0).>It seems that according to the Cauchy-Riemann conditions these>functions are not differentiable, isn't it?Correct.>If so, how this corresponds with the fact that exp(z) is analytic on>C?So what? The problem isn't the exp, it's the Arg.>Another question is: what is possible to say about the>differentiability of the functions |z| and Arg(z), where z is>complex?They aren't differentiable anywhere. A real-valued function of a complex variable is never differentiable unless the derivative is 0.This is easy to see from Cauchy-Riemann.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of === Recommend a Book?Hi all,I am a graduate - well 4 years ago anyway. I havnt been using maths to anyreal standard and I don't want to forget the knowledge.I wonder if anyone could recommend a good reference book - I would like thebook to contain details/tutorials etc on topics such as calculus, Fourier,ODE/PDE - matrix ops... I know these topics are wide ranging - so I wonderdoes a good book covering these topics === all,> I am a graduate - well 4 years ago anyway. I havnt been using maths toany> real standard and I don't want to forget the knowledge.> I wonder if anyone could recommend a good reference book - I would likethe> book to contain details/tutorials etc on topics such as calculus, Fourier,> ODE/PDE - matrix ops... I know these topics are wide ranging - so I wonder> does a good book covering these topics decently Stewart or Howard Anton. Both are good,but I prefer === know: - origin of circle - line x,y - line gradientI would like to know whether the line and circle intersect and how fardown the line the intersection occurs.Someone sugges that there is a simultaneous equation that could solvethis. However, as this is forming part of a computer program, asimultaneous equation is not ideal.Any help would be === circle> I know:> - origin of circle> - line x,y> - line gradient> I would like to know whether the line and circle intersect and how far> down the line the intersection occurs.> Someone sugges that there is a simultaneous equation that could solve> this. However, as this is forming part of a computer program, a> simultaneous equation is not ideal.> Any help would be apprecia.Here is something based on solution of triangles:This can be the intersection of a distance with a direction.For example, a radius point with a N,E of 20,20 , a radius of 11, and a linewith N,E endpoints of 10,40 and 30,23 .The direction of the line develops as InvTan((23-40)/(30-10)) or 40.3645degrees. Since (23-40) is negative that is W and since (30-10) is positivethat is N . NW represents the fourth quadrant and so that is a linedirection of 319.6355 .Now a baseline from an endpoint of the line to the radius is developed asInvTan((20-40)/(20-10)) or 63.4349 degrees in the fourth quadrant for abaseline direction of 296.5651 . The baseline distance is theSqrt((20-40)^2 + (20-10)^2) or 22.36 .So the interior angle off the baseline, at the line endpoint, and up theline is (319.6355-296.5651) or 23.0704 .Now side-side-angle of a triangle are known. So by the law of sines Angle B= InvSin((Sin(23.0704) * 22.36) / 11) or 52.8017 degrees.The remaining interior angle in the triangle is180 - (23.0704 + 52.8017) or 104.1279 degrees.Well, now consider the direction of the baseline as 116.5651 out instead of296.5651 in and the direction to the intersection point is(116.5651-104.1279) or 12.4372 degrees in the first quadrant.So the N coordinate of the intersection point is 20 (as the N coordinate ofthe radius) + (Cos(12.4372) * 11) and the E coordinate of the intersectionpoint is 20 (as the E coordinate of the radius) + (Sin(12.4372) * 11) foran intersection N,E of 30.74 , 22.37 .The second intersection depends on development of a baseline from the radiusto the === between a line and circle> I know:> - origin of circle> - line x,y> - line gradient> I would like to know whether the line and circle intersect and how far> down the line the intersection occurs.> Someone sugges that there is a simultaneous equation that could solve> this. However, as this is forming part of a computer program, a> simultaneous equation is not ideal.> Any help would be apprecia.Given (x0,y0) as the center of the circle, and m and b respectively as theslope and y-intercept of the line, there is an intersection iff thefollowing equation has real roots:x^2 (m^2 + 1) + 2x(bm - my0 - x0) + === Intersection between a line and circleTo try and clarify the problem, i've drawn a diagram:http://www.dur.ac.uk/grey.jcr/question.jpgSo, I know:- === know:> - origin of circle> - line x,y> - line gradient> I would like to know whether the line and circle intersect and how far> down the line the intersection occurs.> Someone sugges that there is a simultaneous equation that could solve> this. However, as this is forming part of a computer program, a> simultaneous equation is not ideal.> Any help would be apprecia.Ok, let's assume that the center of the circle is at the origin (translatingit shouldn't be too diffifcult). If this is the case, the equation of thecircle is x^2+y^2=r^2. Now we have a line of the form y=ax+b.x^2+(ax+b)^2=r^2x^2+a^2x^2+2abx+b^2=r^2(a^2+1)x^2+ 2abx-r^2+b^2=0x=(-2ab+/-sqrt(4a^2b^2-4(a^2+1)(-r^2-b^2)))/(2a^ 2+2)x=(-2ab+/-sqrt((2ab)^2-(4a+4)(-r^2-b^2)))/2a^2+2x=(-2ab+/- sqrt((2ab)^2-(-4ar^2-4ab^2-4r^2)))/(2a^2+2)And then y=a((-2ab+/-sqrt((2ab)^2+4ar^2+4ab^2+4r^2)))/(2a^2+2))+bI === Intersection between a line and circleError correction below> I know:> - origin of circle> - line x,y> - line gradientI would like to know whether the line and circle intersect and how far> down the line the intersection occurs.Someone sugges that there is a simultaneous equation that could solve> this. However, as this is forming part of a computer program, a> simultaneous equation is not ideal.Any help would be apprecia.>> Ok, let's assume that the center of the circle is at the origin(translating> it shouldn't be too diffifcult). If this is the case, the equation of the> circle is x^2+y^2=r^2. Now we have a line of the form y=ax+b.> x^2+(ax+b)^2=r^2> x^2+a^2x^2+2abx+b^2=r^2> (a^2+1)x^2+2abx-r^2+b^2=0> x=(-2ab+/-sqrt(4a^2b^2-4(a^2+1)(-r^2-b^2)))/(2a^2+2)> x=(-2ab+/-sqrt((2ab)^2-(4a^2+4)(-r^2-b^2)))/2a^2+2> x=(-2ab+/-sqrt((2ab)^2-(-4a^2r^2-4a^2b^2-4r^2-4b^2)))/(2a^2+2) > And then y=a((-2ab+/-sqrt((2ab)^2+4a^2r^2+4a^2b^2+4r^2+4b^2)))/(2a^2+2) )+b> I don't know if this will work, but it might.> === know:> - origin of circle> - line x,y> - line gradientI would like to know whether the line and circle intersect and how far> down the line the intersection occurs.Someone sugges that there is a simultaneous equation that could solve> this. However, as this is forming part of a computer program, a> simultaneous equation is not ideal.Any help would be apprecia.>> Ok, let's assume that the center of the circle is at the origin(translating> it shouldn't be too diffifcult). If this is the case, the equation of the> circle is x^2+y^2=r^2. Now we have a line of the form y=ax+b.> x^2+(ax+b)^2=r^2> x^2+a^2x^2+2abx+b^2=r^2> (a^2+1)x^2+2abx-r^2+b^2=0> x=(-2ab+/-sqrt(4a^2b^2-4(a^2+1)(-r^2-b^2)))/(2a^2+2)> x=(-2ab+/-sqrt((2ab)^2-(4a+4)(-r^2-b^2)))/2a^2+2> x=(-2ab+/-sqrt((2ab)^2-(-4ar^2-4ab^2-4r^2)))/(2a^2+2)> And then y=a((-2ab+/-sqrt((2ab)^2+4ar^2+4ab^2+4r^2)))/(2a^2+2))+b> I don't know if this will work, but it might.> The above will find the intersection (if it exists), but then to find thedistance from the origin, if I read the problem right, will === and circle>I know:> - origin of circle> - line x,y> - line gradient>I would like to know whether the line and circle intersect and how far>down the line the intersection occurs.>Someone sugges that there is a simultaneous equation that could solve>this. However, as this is forming part of a computer program, a>simultaneous equation is not ideal.>Any help would be apprecia.By line x,y I assume you mean a point (x0,y0) on the lineBy line gradient I assume you mean a direction vector (Vx, Vy) of theline.The parametric equation of line is:x=x0 + t*Vxy=y0 + t*VyThe circle equation is (x-xc)^2 + (y-yc)^2=radius^2 where (xc,yc) isthe center.Replace the x and y of the line equation into the circle equation, youget a quadratic polynomial in the parameter t. Solve it with thequadratic formula.If there is 0 real solution for t then no intersection.If there is 1 real solution then the line is tangent to the circle.If there are 2 real solutions then the line line cuts the circle. You simply substitute the values of t into the line equation above tofind === between a line and circle> I know:> - origin of circle> - line x,y> - line gradient> I would like to know whether the line and circle intersect and how far> down the line the intersection occurs.> Someone sugges that there is a simultaneous equation that could solve> this. However, as this is forming part of a computer program, a> simultaneous equation is not ideal.> Any help would be apprecia.Circle-Line Intersection === Subject: Re: Intersection between a line and circle> Circle-Line Intersection> http://mathworld.wolfram.com/ Circle-LineIntersection.htmlAnother item which might be helpful:Intersection of circle or circular arc with line, ray, or segment in 2Dat David === circle> I know:> - origin of circle> - line x,y> - line gradient> I would like to know whether the line and circle intersect and how far> down the line the intersection occurs.> Someone sugges that there is a simultaneous equation that could solve> this. However, as this is forming part of a computer program, a> simultaneous equation is not ideal.> Any help would be apprecia. (0) Translate the circle's origin to (0,0), and the line's equationy=mx + b accordingly. Your option. 1) Calculate the minimum distance of the line from the circle'sorigin. Standard high school geometry problem - erecting aperpendicular to a line through a given point. 2) Compare distance in (1) with the circle's radius.-- Uncle Alhttp://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals)Quis custodiet ipsos custodes? The === I know:> - origin of circle> - line x,y> - line gradient> I would like to know whether the line and circle intersect and how far> down the line the intersection occurs....> .... this is forming part of a computer program .... There are formulae which will give you that information just by plugging numbers into them, but writing those formulae requires more facts than you've given. (Also, your origin of circle and line x,y aren't very clear mathematical language, although I can guess what they may mean.) How are your data specified? Do you have the circle's equation, or its radius and the coordinates of its centre? Do you have the line's equation in the form lx + my + n = 0, or in the form y = mx + c, or some other facts to tie === between a line and circle> I know:> - origin of circle> - line x,y> - line gradient> I would like to know whether the line and circle intersect and how far> down the line the intersection occurs.> Someone sugges that there is a simultaneous equation that could solve> this. However, as this is forming part of a computer program, a> simultaneous equation is not ideal.> Any help would be apprecia.1. Translate the line x, y origins so they coincide with the origin of thecircle. This will require only addition and subtraction.2. Using geometric means, determine the closest approach distance of theline to the now-common origin.3. Compare that distance to the circle === I know:> - origin of circle> - line x,y> - line gradient> I would like to know whether the line and circle intersect and how far> down the line the intersection occurs.> Someone sugges that there is a simultaneous equation that could solve> this. However, as this is forming part of a computer program, a> simultaneous equation is not ideal.> Any help would be apprecia.What's the radius of the circle?--There are two things you must never attempt to prove: the unprovable -- === Intersection between a line and circle>> I know:>> - origin of circle>> - line x,y>> - line gradientoops, the radius is also === I know:>> - origin of circle>> - line x,y>> - line gradient> oops, the radius is also know.If there is a solution to the two equations (the equation of the circle andthat of the line) then they intersect.For example, if the circle has center at (x0,y0) and a radius r, then itsequation is:(x - x0)^2 + (y - y0)^2 = r^2If the line passes through the point x1, y1 and has a slope m, its equation(point-slope form) is:(y - y1)/(x - x1) = mIf there is one (or at most two) solution x,y that satisfies both equations,than x.y is a point of intersection === of the two lines.Tom DavidsonRichmond, VASubject: Assessing is normal and I am not sure if Ishould be looking at the entire spread or if I should split it up bytreatment group. Everywhere I look it says to make sure your data isnormal but it does not necessarily say how. It just says totransform if it is not.----== Pos via Newsfeed.Com - Unlimi-Uncensored-Secure Usenet News==----http://www.newsfeed.com The #1 Newsgroup Service in the World! >100,000 Newsgroups---= 19 East/West-Coast Specialized Servers - Total Privacy via Encryption === the sample mean and variance. Then calculate higher ordersample moments and see how they compare to corresponding normaldistribution moments for the same mean and viariance.----== Pos via Newsfeed.Com - Unlimi-Uncensored-Secure Usenet News==----http://www.newsfeed.com The #1 Newsgroup Service in the World! >100,000 Newsgroups---= 19 East/West-Coast Specialized Servers - Total Privacy via Encryption === we have a positive REAL, possibly irrational, x,>then we can find a sequence (many many sequences..., I bet) of>positive integers>{n(k)}>such that>sum{k=1 to M} 1/n(k) = x,>where M is infinity if x is irrational.>I also wonder about alternative EF-expansions, both with all positive>terms and with the possibility of having negative terms.One class of alternative EF-expansions for 00 increment i and goto 2 otherwise stop.It's easy to see that x=k/a[0]+k/(a[0]*a[1])+k/(a[0]*a[1]*a[2])+...If k=1 (or k=2 and a[0] is even, or ?) then x=sum(1/n[i]), as desired. For agiven x, different methods of choosing a[i] in step 2 usually will result in adifferent expansion of x. So, yes, there are many possible expansions. Somesimple choices for a[i] (with k=1) that I have played around with (and posto sci.math) over the years are discussed below. To begin, let m[i] be the smallest possible choice of a[i] and M[i] be largestpossible choice in step 2. If a[i]=M[i] then 2<=a[i]<=5 for i>0. For a randomx it has been shown (not by me) that prob(a[i]=2)=206/337. Also, some pairs ofconsecutive digits never occur. Another interesting choice is a[i]=max(m[i],M[i]-1) applied to x=1/p, p prime. IIRC, if p is prime then the sequence is eventually constant. Morestatements of the form if p is prime then (...) are suprisingly easy to find. [Somewhat off topic is the generalization:x=0/a[0]+1/(a[0]*a[1])+2/(a[0]*a[1]*a[2])+... Here if x=1/p, a[i]=M[i], youget (it seems): p is prime iff a[i] is small. A simple, useless, and*unproven* observation about the 'digits' of the prime numbers.] My favorite choice for a[i] is: if rnd<.5 then a[i]=m[i] else a[i]=M[i]. Applyit to a rational x, plot the choices for various trials (i.e. until x[i+1]=0),and you get a binary tree that represents x. Prune off the banches that arelonger than average and you have a nice, framable, picture of x. One can even formulate a two person game around these representations asfollows: let x=4/n, where n is some (possibly unknown?) positive integer. P1and P2 alternately pick an a[i]. The first player to pick an a[i] so thatx[i+1]=0 wins. It would be kind of cool to find a familiar, irrational, x and a simple way ofchoosing a[i] so that is provably well behaved. The only (lame)progress I've made is to note that the first 100,000 digits (with a[i]=M[i]) oflog 2 are not distribu as I expec them to be. I've since changed myexpectations. So, yes, there are *lots* of possible EF-expansions and many of them are quiteinteresting. Interesting enough to periodically arouse the curiousity of thisamateur, at least.Richps: As for your comment on not finding anything in the OEIS, I wonder what theoverall rate (as a percent, to the nearest integer, by day) === Course***Advanced R/Splus Programming***Boston, January 29-30, San Francisco & DCHappy New Year XSolutions Corp (www.xlsolutions-corp.com) is proud to announcea 2-day Advanced R/Splus programming taught by R DevelopmentCore Team Guru. *********San Francisco ---------- TBD (email for current status)*********Washington DC ---------- TBD (email for current status) Reserve your seat Now (payment due after the class)Registration:www.xlsolutions-corp.com/training.htmEmail Sue Turner: sue@xlsolutions-corp.comPhone: 206-686-1578Course outline:- Overview of R/S fundamentals: Syntax and Semantics- Class and Inheritance in R/S-Plus- Concepts, Construction and good use of language objects- Coercion and efficiency- Object-orien programming in R and S-Plus- Advanced manipulation tools: Parse, Deparse, Substitute, etc.- How to fully take advantage of Vectorization- Generic and Method Functions; S4 (S-Plus 6)- Search path, databases and frames Visibility- Working with large objects- Handling Properly Recursion and iterative calculations- Managing loops; For (S-Plus) and for() loops- Consequences of Lazy Evaluation- Efficient Code practices for large computations- Memory management and Resource monitoring- Writing R/S-Plus functions to call compiled code- Writing and debugging compiled code for R/S-Plus system- Connecting R/S-Plus to External Data Sources- Understanding the structure of model fitting functions in R/S-Plus- Designing and Packaging efficiently Early-bird group research fee: $995!This course will also deal with lots of S-Plus efficiency issues andany special topics from participants is welcome.Please let us know if you and your colleagues are interes in thisclass to take advantage of group discount. Over half of the seats inthese classes are currently reserved. Register now to secure your seat in this course! Elvis Miller, PhDManager Training.XLSolutions Corporation206 === 686 1578www.xlsolutions-corp.comSubject: Re: Matrix inverse inversions.> Can someone give me exaples of matrix 2x2 3x3 and 4x4 - how it should look > after inverse (best - some easy examples - with small numbers)> Or maybe program for linux that will do such thing?Check out winmat at http://math.exeter.edu/rparris/winmat.htmlIt can do inverses, so you can plug a matrix into it and your program to compare results.-- Will Twentymanemail: wtwentyman at copper dot === small program for array inversions.> Can someone give me exaples of matrix 2x2 3x3 and 4x4 - how it should look > after inverse (best - some easy examples - with small numbers)> Or maybe program for linux that will do such thing?The inverse of the 2x2 matrix - - | 1 -v | | 0 1 | - -is - - | 1 v | | 0 1 | - -formed by interchanging the diagonal elements, taking the negatives ofthe off diagonal elements, and then dividing the elements of thatintermediate matrix by the determinant of the === Isoperimetric Zepp> Consider the problem of maximizing the integral> of some function F(x,y) over a simply connec> region whose boundary C is allowed to vary> subject to these two constraints:> 1) C contains a fixed point (x_0,y_0), and> 2) C has fixed length L.l(Caution: the last letters in the URL are html, not htm):~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ We have 100 meters of fencing, which we use to enclosea pasture. Our fence must begin and end at the oak tree.Ground south of the oak tree, or less than 20 metersnorth of the oak tree, is worth $100 per square meter;but ground more than 20 meters north of the oak treeis worth $200 per square meter. What shape maximizes thetotal value of the enclosed pasture, and what is thisvalue? Clarifications: The oak tree is a single point.20 meters north refers to an east-west line passing20 meters north of the tree. Begin and end at the oaktree means that both ends of the fence touch this point.All of this takes place on a flat Earth.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~The situation is just the one described by Jim Ferry:The oak is standing at the origin (-20,0). Length of C isL = 100. Function F(x,y) gives the price per squaremeter: F(x,y) = 100 for x <= 20 F(x,y) = 200 for x > 20Two principles (too nice to be false help finding thesolution. Since the circle has maximal area for givencircumference, the boundary C has to consist of circle arcs: (1) the lower the price the flatter the arc, or: F * r = const. (where r the radius of the arc) (2) no corners except for the one at the oak, or: arcs meet with common tangents.I proudly present my solution below, which describes C andgives the maximal value of 114256.5483 Dollars.> Rainer coined the name Zepp for this shape.Not really: just for the function F(x,y) = x, please!There are as many shapes as there are Functions F(x,y),or nearly as many: The shape in the puzzle is definedby the ratio cheap/expensive = 100/200. The length Lis not relevant for the shape!Ingmar Rubin proposed a price function F(x,y), whichvaried continuously with the difference of x. He putthe oak at (0,0) and proposed F(x,y)=x Dollars persquare meter. Without the knowledge of the calculus ofvariations I eagerly struggled around with this niceproblem, making use of the principles (1) and (2) above.Klaus Nagel gave an algorithm for pointwise constructionof an approximative solution and later produced a niceJava Applet (maximaum value is 15468.50 here):http://home.t-online.de/home/nagel.klaus/zaundir/ zaun.htmWolfgang Kirschenhofer succeeded with an explicit solutionin terms of elliptic integrals. It is given in a pdf-filewritten in German. He kindly allowed me to send it topeople interes in this paper.I learned a lot about the fascinating calculus of variationsin the course of the discussions and say thank you to allthe people involved (for example to Jim Ferry, who was happyto re-read his 7 year old notices regarding this problem).One shape quite similiar to the Zepp is the lemniscate.I was very happy when I found out that the lemniscate is thesolution of the given problem of maximizing the enclosedvalue, if the price is given by F(x,y) = sqrt(x^2 + y^2)i.e. where the price varies with the distance. The apex angleof the lemniscate is 90 degrees, the one of the Zepp is slightlylarger than 80 degrees.> Surely such a wonderful shape cannot be new, and> indeed Rainer has traced it back to Euler. I'll let> him tell you more about that.Michael Lonhard from Munich sent me an e-mail some days ago,and he gave the lovely references to Euler and the success.Further links were given by our mentor of de.sci.mathematik,Hermann Kremer (give him a big hand!):1) The source of Eulers calculus of variations with appendix 1,regarding 9 classes of curves, where the Zepp is the fifth:http://posner.library.cmu.edu/Posner/books/book.cgi?call =517.4_E88M_1744(See the pages 262 and 263 and the figures past pages 320.)It is phantastic to have these informations online, THANKS! towhoever.2) Jakob Bernoulli, Acta eruditorium, Leipzig, Juni 1691 (p. 451):http://gallica.bnf.fr/scripts/ConsultationTout.exe?E=0&O= n050009.htm> Maybe he will also explain why the two characteri-> zations give the same shape.Formula (1) ... My Latin is quite bad, but good enough to feelthe enthusiasm of Euler in presenting his findings. He is quiteexci and praises the Bernoullis Daniel and Jacob quite often.When I detec (1) in the midst of many messy numbers in mysimulation Excel file - I was very exci too. Funny enough Icalled the shape Euler-Zepp in honor of Euler:~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~Und einen Namen habe ich bei der Gelegenheit heute fr.9fh auchschon gefunden: Das Euler-Zepp (TM (C) Rainer Rosenthal)Darin verbirgt sich der Zeppelin, der hier am Bodensee vielWerbung macht und der mit seiner Form ja entfernt an unsereZielfigur erinnert. Und der Name Zepp ist kurz und peppig,einsilbig wie Kreis und auch im Englischen gut aussprechbar.Na ja und den Euler tue ich halb aus Verehrung und halbaus Wichtigtuerei dazu~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ two pointsG and H, where the fence crosses the border between the two areas.The figure we are looking for, does contain this triangle and maybe looked at as a blown up version of the triangle.Due to the optimality, curves have to be parts of a circle.The optimal figure's fence goes from the oak tree E on a circleline to the border point G, passes thru the high-priced area viasymmetrical to the starting line back to the oak tree E.Here are two principles from which the solution can be compu:Principle 1:The blowing up force is proportional to the price in the respectivearea, whereas the radius of the respective circle is inverseproportional to the price: r_cheap = 2 * r_expensive.Principle 2:There is no corner at the meeting point of cheap and expensive: sothe centers M of a cheap_circle and Z of the expensive_circle arein line with the corresponding border point G.I found these principles and presen them in the ci thread.I admit I didn't find them the easy way but only after heavy trialand error and numerical approximations with polygons. The more I amhappy to be able to present them as if being self evident.The rest of the computation is best explained with the followingsketch (fixed size font necessary): | Cheap G 100 $/m2 |. | Expensive | . 200 $/m2 | | . | -- E --------------O---Z--------------- -> North . | . | . . | | . r_cheap = |MG| ' | r_expensive = |ZG| | . Basic ideas: H * ' 1. r_cheap = 2*r_expensive | M 2. G,M,Z collinear |I put everything into cartesian coordinates.Let O be the crossing of border and the line from E to north.We have E = -20. Approximation shows that Z is on the right.Z = (E2-3G2)/(4E) computable from |ME|=|MG|Let alpha = <)OEM so that alpha = atan(G/(2Z-E))Let beta = <)OGZ so that beta = atan(Z/G)Then we have <)EMG = Pi/2 - alpha - beta.The circumference U of the final figure can be compu as U(G) = (3*Pi - 4*alpha - 2*beta) * sqrt(G2+Z2)by adding up the arc lengths.Solving for G in U(G) = 100 gives G = 14.0310005: G Z alpha beta U---------------------------------------------------------- 14,0000 2,35 0,515651153 0,166306754 99,79066214,0300 2,3815338 0,515475493 0,168143099 99,99323714,0310 2,3825860 0,515469612 0,168204236 99,99999714,0310005 2,3825866 0,515469609 0,168204267 100,000000The value W of the enclosed area is the sum of the values of thesingle areas A1 = triangle EGH, A2 = segment EG, A3 = segment EH(A3 = A2 by symmetry) and A4 = segment GH.The formula for that is W=100(A1+2*A2)+200*A4: A1 A2 A3 A4 W--------------------------------------------------------- 280.62001 45.31558133 45.31558133 385.6571549 114256.5483 findings andinteresting encounters. Thamks to all the readers who cameup to this end of my long posting!Rainer === questionLet H be a Hilbert space, let x in H, and let {v_n} be an orthonormalsequence in H. BUT, let {v_n} be uncountable. Prove there are at mostcountably many of the v_n such that (v_n,x) = 0.My first thought was to try to suppose otherwise (for contradiction) and endup with something likeSum (v_n,x)^2 < infty,which must be false since the left-hand side is an uncountable series ofpositive terms. I tried to say, well, if H is a Hilbert space, then thenorm|| Sum (v_n,x)v_n||must be finite. This then meansSum (v_n,x)^2 < infty, contradiction.BUT, then I realized that I couldn't make the assumption that|| Sum (v_n,x)v_n|| must be < infty, since I haven't shownSum (v_n,x)v_nto even be an element of H.Any === Hilbert space, let x in H, and let {v_n} be an orthonormal> sequence in H. BUT, let {v_n} be uncountable.It can't be a sequence and be uncountable. I think you want to say let {v_n} be an orthonormal set in H that is uncountable.> Prove there are at most> countably many of the v_n such that (v_n,x) = 0.Lemma: If A is uncountable and t_a > 0 for each a in A, then sup_F sum_{a in F} t_a = oo,where the sup is taken over all finite subsets F of A.Apply the lemma and Bessel's inequality === hilbert space question> Let H be a Hilbert space, let x in H, and let {v_n} be an orthonormal> sequence in H. BUT, let {v_n} be uncountable. Prove there are at most> countably many of the v_n such that (v_n,x) = 0.> My first thought was to try to suppose otherwise (for contradiction) and end> up with something like> Sum (v_n,x)^2 < infty,> which must be false since the left-hand side is an uncountable series of> positive terms. I tried to say, well, if H is a Hilbert space, then the> norm>|| Sum (v_n,x)v_n||> must be finite. This then means> Sum (v_n,x)^2 < infty, contradiction.> BUT, then I realized that I couldn't make the assumption that>|| Sum (v_n,x)v_n|| must be < infty, since I haven't shown> Sum (v_n,x)v_n> to even be an element of H.> Any suggestions?Parseval's theorem?-- Dave SeamanJudge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.> Hello sci.math people.I'm presently attempting to produce a model of a system as it evolves>> over time. It involves the interaction of four components. The>> derivatives are:dydt[0] = B * y[0] * (A *y[3] - N_1) + seed * y[3] * gamma;>> dydt[1] = B * y[1] * (A * y[3] - N_1) + seed * y[3] * gamma;>> dydt[2] = C * y[2] * (y[3] - N_0);>> dydt[3] = -B * (y[0] + y[1]) * (A * y[3] - N_1) ->> C * y[2] * (y[3] - N_0) - y[3] * D * gamma;where A and D are ~1. B and C are ~ 10^-12. N_0 and N_1 are>> ~10-^26. Gamma is a decay rate (~10^3 per sec) and seed is a small>> number (I've used 10^-8). If you're interes, this is supposed to>> be the interaction of two lasers (y[0] and y[1]) being amplified by>> some gain medium (y[3]), pumped by another laser (y[2]). The above is>> for the interaction at a single position - I repeat this many times>> I've worked through my program and triedscaling the components. The flaw seems to be elsewhere, however, sincewhen I reduced the complexity of my model to just the above step itseemed to behave properly. I shall have to spend more time === example 2> I made a post about a quadratic example that Rick Decker, a professor> at Hamilton (I said University before but it might be College), gave> in a recent post. I haven't seen any replies to that yet in Google> Groups, so I'll leave it and make another thread to consider Decker's> example in more detail, again here are some headers to allow you to> find his === original post:> Subject: Re: Mathematical consistency, courage> In his post Decker claimed to mirror my argument using a quadratic> instead of a cubic, where he has> (5a_1(x) + 7)(5a_2(x) + 7) = 7(25x^2 + 30x + 2) > where his a's are roots of > a^2 - (x - 1)a + 7(x^2 + x).Noting that at x=0, a=0 or a=-1, you can usea_2(x) = b_2(x) - 1 to get(5a_1(x) + 7)(5b_2(x) + 2) = 7(25x^2 + 30x + 2)and multiplying out gives25 a_1(x) b_2(x) + 10 a_1(x) + 35 b_2(x) + 14 = 7(25x^2 + 30x) + 14so you can subtract 14 from both sides to get25 a_1(x) b_2(x) + 10 a_1(x) + 35 b_2(x) = 7(25x^2 + 30x)and the constant terms are gone.Now my position has been for a while now that since 7 and 2 on theleft are factors of 2(7) on the right, it stands to reason that they*remain* factors of it without regard to the value of x, but peoplelike Rick Decker disagree because then it follows that you can divideboth sides by 7 first to have(5a_1(x)/7 + 1)(5b_2(x) + 2) = 25x^2 + 30x + 2and multiply out to get25 a_1(x) b_2(x)/7 + 10 a_1(x)/7 + 5 b_2(x) + 2 = 25x^2 + 30x + 2and once again subtract both sides to get25 a_1(x) b_2(x)/7 + 10 a_1(x)/7 + 5 b_2(x) = 25x^2 + 30xand again the constant terms are gone.Now, look back at what came before as I had25 a_1(x) b_2(x) + 10 a_1(x) + 35 b_2(x) = 7(25x^2 + 30x)and notice what happens now when I divide both sides by 7, as then Ihave25 a_1(x) b_2(x)/7 + 10 a_1(x)/7 + 5 b_2(x) = 25x^2 + 30x.Is that a parlor trick to all of you?I've wondered and wondered how simple algebra could be such adifficult thing for so many of you, but I'm at a === posts revived our faded hopes that you were going tobask in the glory of your recent achievements and work behind the scenes.But, like that damned blow-up clown with sand in his feet, here you are again!--There are two things you must never attempt to prove: the unprovable -- and the === calculating problems.BBC BASIC64.SeriCalc4S.pgms1..donpgms.txt .Pgms.PGMS1.sysacapps05.06.00 12:0322.01.04 23:12give me maths calculating problems.BBC BASIC64.SeriCalc4S.sci.math,nz.general,comp.sys.acorn.apps, alt.math.recreati onalon-line encyclopedia of integer sequences.If you have a problem concerning research/maths calculating e.g.I may be able to recommend one of the (following) programs.email me immediately or post. (100 LINES plaintext please.)I may be able to reply in 3 days. At least let you knowif I have tackled that sort of calc.I will snail mail a 3.5 in floppy diskettein your desired format, Acorn 1.6MB etc.email: don.lotto@paradise.net.nzenquiries regarding / help with / copies of- any of these files or BBC BASIC64 programsplease ring ring +64 (4 ) 389 6820. call minder.I desire to release my programs freely public domain.previous upload programsPD57xprogrms.LotoDaBase.PackdDir, etc.27xPD-pgmsSiev89/hint.tFileName is often a t.ext version of prog.most files are Acorn / txt / Qbasic or / BBC. BASIC V-64.filetypes.toshiba T-2100 laptop 486 Win95,Acorn A4000/A5000, Risc OS 3.11 ( 1993.)cheersREM > DonMcD.Pgms.ProgramsPD.Descript1, n.b. pt1.###23/5/1995. 27.07.96, 03.11.96many programs have undergone continuous upgradePrograms package, by Don McDonald. ProgramsPD.Upload to 'The Bridge' BBS, Wellington N.Z.Unpack with PackDir, available from BBS. (If compressed set type of archive to PackDDir. Drag to PackDir on icon bar?)(63/*3 Hut chi son Rd, Wellington2, New Zealand. PH: 64 +4 + 389 6820.)*.CSD .pgms.programspd20/80 (Basic)Calculate odds of lower tier lottery prizes,Lotto 6/40, Keno 20/80, etc.a2p-0 (Basic)Factorise Mersenne (Prime?) numbers.a2p999 (Basic) Factorise Generalised Mersenne nos., a^p+-b^n, base 1000.Abundaint (BASIC). Artificial intelligence A.I. numbers program. You can ask for Primes, Perfects, Odds, Squares, (or SquareFrees), Abundants, Multiperfects, (Booleans), and starting anywhere.ANALOGX (Basic) Analogue and digital clock. Sweep Second hand. Time adjust.BirthRanLi. BirthWhite Birthtext (Basic.) Personalised Birthday Cake/Card. Teletext colour graphics. BBC Basic 2. !Host 65 or Bas V. Enter first name, age, day.month. Maximum 30 items of data. B'day standard. Randomises & repeats the order of lines of data.BodyMsIndx (BASIC). Your Healthy weight. 'Body Mass index' is the ratio kilogrammes wt/ metres tall squared. If your BMI is above 25 number, you could be overweight. Flexible input including stones, lbs, kgs, feet, inches, cm, metres, (mixed.) Numerical and graphical display.CR290DA (Data)Data file for Program SelfTst..Data concepts, Intro Business Computing course.Descript/ Descript1. (text) Descript is this file that you may be reading now. It describes the programs and text files in ProgramsPD.descript1DISPFIL (Basic) Display contents of file. IntegeReals, Strings.EgyptianF (Basic)Egyptian fractions(math representations as sums of unit fractions, 1/n.)EVENTSV .. events etc.. (basic) Brick wall diary of calendar/church events. Print address labels, etc.FACTOR6 (BASIC.) Factorise an integer or numeric expression, reducing to a positive integer (max greater than 2E9). Into prime factoincluding repeat prime factors.FACTORS (Basic)BigNum.facthcff (BASIC 64.) factorise integers bigger than 2E9. (5 byte reals, 2^52. etc.) includes sample numbers.Help1 (Text 10k). Text help for calculator program SeriCalc3./SeriCalc4 S.ILIKEPI (text.) Article on calculating pi to 15 decimal places, e.g. circle arc series. Mnics, References.Ken20dd80 (Basic)Calculate odds and returns (~60 %) of all NZ Keno 20/80 groups.KEPLER3 (Basic)Show (circular) orbits of several/selec solar system planets.LINREGTEXT (Text)Article about linear regression program PLOTPref.MNPHASWGTN (Basic)Moon Phases from Wellington, New Zealand. Illustration.MONEIL3 (Basic)R-th root scaled (non-linear) graphs of road toll, deaths, injuries(dissimilar scales, friendly numbers on axes.)PI314 (BASIC VI, etc.) Calculate pi, as done by Isaac Newton and similarly by Don McDonald. (Min. 15 decimal places.)PLOTPREF (Basic)Linear regression plot y- against- x, preferred numbers on axes.Scatter plot, fit straight line, equation, predic, residuals.POWER (Basic.)Solve Fermats Last Theorem(?). ABC Conjecture.x^a + y^b = z^c.PrimeCabi (BASIC.) Find factors up to 92,681 = SQR(2^33.) of .. Test vast numbers e.g. 2E18 and higher (Worlds largest known prime nos.) or c * a^b +i, where c,a,b, (-)i < 2E9.PRIMEPRTV1 (Basic)Number Theory program. Calculate 'primitive roots' of primes.Primes (BASIC). Primes and count primes. Starting anywhere. Input number or expression.Profile (BASIC.) Find personal properties of integeincluding square, square root, log, sum of squares, factoreciprocal, ctd fraction. Starting anywhere and continuing successively.PrtvRoots (Basic)Primitive roots, number theory.PTOCALC (BBC BASIC 2.)Printout Calculator for BBC model-B Micro.(Archimedes use SeriCalc4S, far developed.)SELFTSTbst (Basic)SelfTest (Best), learning quiz program.Books of Bible, Data concepts, type in, store.Prompt Question? Answer xxx.One mark for each correct letter (CAPS).Repeat, review, scoreboard, learn, test, speed, etc.SeriCalc3 (BASIC V, VI.) Math calculator, 40 functions. Combinations, primes, printout, divisoabundants, number-theory, DIV-MODs, base conversions, comma 000, Poisson distribution, continued fractions, poweetc.SERICalC4S (Basic)number theory calculator. Optional printout/ task edit.SHEMASDAN (BASIC). S.outhern He.misphere M.oon a.t S.unset, D.ays A.fter N.ew. Phase and position of Moon, Altitude- Azimuth. New, Crescent, Quarter, Gibbous, Full. (Mode 0 graphical disp.)SIEV7/8911 (Basic)Primes [option Factors] up to 900 million, Archimedes.Store to disk (option). Sieve. Setup. Start anywhere.See Beeblet, newsletter BBC/ Acorn Computer User Group NZ Inc.(Twin primes, prime triplets etc., the Black Key sieve.)SpCLOCKAr (Text. spooled) Article on Modular (clock/ remainder) arithmetic. Used in mathematics for proving large prime numbers. Factors.SPTEMP (Data?)For programs SelfTst, DispFil.STARCLU02 (Basic)Dynamical simulation of whirligig orbits of globular star cluster,25 x 1 solar mass stars. Colour graphic simulation.storeconst (Basic)Calculator. E.g. C.onstant(314) = PI = 3.14,159...SUNSTAR (Basic)Random sun plot. Yellow stars on blue sky. (compare dandelions)Enter birthdate, random number, name. Star pattern.TENNIS (Basic 2)Championship tennis scoreboard. Analysis of service strengths etc.Writeup in Beeblet paper magazine. (Chris Lewis).testoscli1 (Basic)Test operating system command line.Change directories, load objects.Compare !Interword, friendly loading and saving.TypEnvp9cm (Basic)Type envelopes, Typewriter.Line by line type out, indent, address envelopes, etc.WIPER1 (Basic)WIPERS (Basic)Rain on windscreen. 2 programs. Colour, rivulets, coalesce, wind..*....ANALOGX1 WR/ Astronomy D/ CDSTAR01 LWR/ClockArith D/ HARMONY D/ HlthyWght D/ LPRINTQBAS WR/ Prime D/ ProgramsPD D/ StonWalTxt D/ testoscli1 WR/ TypeWriter D/moved pgms.mathscrea new foldersprogramspd.keno80 (odds, probability..)programspd.learngtest (Selftest.)*sp.*h. countSyntax: *Count []*count * r === is one of the form M/N ( we do not require lowest terms here ).We can form an Index Number I = ( 2^M ) * ( 3^N ).By Prime Number decomposition each I has one and onlyone M,N pair and each M,N pair generates a unique I( ie: No other M',N' can generate I ).Since the cardinality of I is the same as the Integersand is the same as the M,N pairs the rationals areonly countably infinite.tom-- We have discovered a therapy ( NOT a === What kind of math?I would like to do math problems with a prison inmate I visit. Howshould I go about determining what kind of math he will enjoy, andevaluating his level of === math problems with a prison inmate I visit. How> should I go about determining what kind of math he will enjoy, and> evaluating his level of ability?You might start with Prisoners Dilemma. Seriously, I would first ask whetherHE would enjoy the idea at all and then focus on simple real world problemsand === sum_{i=1}^{infty}frac{mu^iln{i!}}{i!}What do you want to know about it? For large positivemu, it is approximately log( === in this sum?> sum_{i=1}^{infty}frac{mu^iln{i!}}{i!}>or this one?> sum_{i=1}^{infty}frac{mu^iiln i}{i!}>i've given === sum? sum_{i=1}^{infty}frac{mu^iln{i!}}{i!}>>or this one?> sum_{i=1}^{infty}frac{mu^iiln i}{i!}>>i've given up,>steve> Nosorry, i'll rephrase that.i'm interes in this sum; would === in this sum? sum_{i=1}^{infty}frac{mu^iln{i!}}{i!}>>or this one?> sum_{i=1}^{infty}frac{mu^iiln i}{i!}>>i've given up,>steve> No> sorry, i'll rephrase that.> i'm interes in this sum; would anyone offer help?> steveNever mind Mathedman, he can be so nice sometimes...But , for the first sum, you mean: oo--- i mu ln(i!)/ ------------ ,--- i!i=1(view with fixed-width font)I cannot offer much, but...the sum equals also oo i--- i --- mu / --------- / ln(j) =--- i! ---i=1 j=1 oo oo--- --- i+j-1 mu ln(j)/ / ---------------- .--- --- (i+j-1)!i=1 j=1But this does not help you, I bet.;/Leroy === sum_{i=1}^{infty}frac{mu^iln{i!}}{i!}>>or this one?> sum_{i=1}^{infty}frac{mu^iiln i}{i!}>>i've given up,>steve> No> sorry, i'll rephrase that.> i'm interes in this sum; would anyone offer help?> steveNot that I am likely to be able to help you, but I am among those whodo not understand this particularly notation. Could you pleasetranlate it into something more intuitive (i.e ASCII in === sum_{i=1}^{infty}frac{mu^iln{i!}}{i!}>> sum_{i=1}^{infty}frac{mu^iiln i}{i!}>i'm interes in this sum; would anyone offer help?They are both entire functions of mu, but are unlikely tobe expressible in closed form. What else do you want to knowabout them?Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia === sum_{i=1}^{infty}frac{mu^iln{i!}}{i!}>>or this one?> sum_{i=1}^{infty}frac{mu^iiln i}{i!}>>i've given up,>steveThey === matrixLet B_m be elements of G (a subgroup of GL(n,C)), let B_m ---> I, let Y_m =log(B_m), which is defined for all sufficiently large m (since B_m ----> I).Now, since B_m ---> I, then Y_m ---> 0, and so ||Y_m|| ---> 0. The nextline in my text says that therefore, we can find integers k_m such thatk_m||Y_m||) ---> tfor any t in the real numbers.I guess I don't know my analysis that well, but why is this true? And whyshould it follow from the fact that ||Y_m|| ---> === of G (a subgroup of GL(n,C)), let B_m ---> I, let Y_m => log(B_m), which is defined for all sufficiently large m (since B_m ----> I).> Now, since B_m ---> I, then Y_m ---> 0, and so ||Y_m|| ---> 0. The next> line in my text says that therefore, we can find integers k_m such that> k_m||Y_m||) ---> t> for any t in the real numbers.> I guess I don't know my analysis that well, but why is this true? And why> should it follow from the fact that ||Y_m|| ---> 0?Clearly we must assume ||Y_m|| > 0 for large m. Suppose t > 0. For large m, ||Y_m|| < t, hence there will be a largest positive integer n_m such that n_m*||Y_m|| < === about Fibonacci Numbers> This might be well-known, but not to me.> Consider the sequence of ratios of neighboring Fibonacci> numbers:> 1/1, 1/2, 2/3, 3/5, 5/8, etc.> Let r_n be the nth rational on this list. What appears> to be true is this:> r_{n+2} is the rational with the smallest denominator that> is strictly between r_n and r_{n+1}> I don't immediately see how to prove this, though.You have already seen ways to do this using Pick's theorem. Here isanother approach. If you look at all the fractions in lowest termsbetween 0 and 1 whose denominator is at most n, you get somethingcalled the Farey sequence of order n. For instance the Farey sequenceof order 5 would be0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1If you look at the Farey sequence of any order it is a standard andremarkable fact that if x/y lies strictly between a/b and c/d, thenx/y = (a+c)/(b+d). There is nothing sacred in this result about usingthe numbers between 0 and 1. Your property of Fibonacci numberquotients is a simple consequence of this property of Farey fractions,using the Fibonacci recursion and the fact that consecutive Fibonaccinumbers are necessarily === Fibonacci NumbersDaniel W. Johnson says...>> Consider the sequence of ratios of neighboring Fibonacci>> numbers:>> 1/1, 1/2, 2/3, 3/5, 5/8, etc.>> Let r_n be the nth rational on this list. What appears>> to be true is this:>> r_{n+2} is the rational with the smallest denominator that>> is strictly between r_n and r_{n+1}>> I don't immediately see how to prove this, though.>I came up with a proof while arguing with a crank in another thread.>There is an old result about the area of a polygon whose vertices are>lattice points (i.e., their (x,y) coordinates are both integers): The>area is one less than the number of lattice points in the interior plus>half the number of lattice points on the boundary.>Ah, here's the name: Pick's Theorem.>http://mathworld.wolfram.com/PicksTheorem.html>http:// en.wikipedia.org/wiki/Pick%27s_theorem I had never heard of this theorem. It's a dandy littleresult.>Consider the point (x,y) where x/y is the lowest-terms representation of>the rational with the smallest denominator that is strictly between r_n>and r_{n+1}. You can show that (x,y) is an interior lattice point for>the triangle with vertices at (0,0), (3*F_n, 3*F_{n+1}), and (2*F_{n+1},>2*F_{n+2}). You can also show that the only interior lattice point for>that triangle is (F_{n+2}, F_{n+3}). The desired result follows.Did you actually compu the area of the triangle (usingHeron's === Conjecture about Fibonacci Numbers>Consider the point (x,y) where x/y is the lowest-terms representation of>the rational with the smallest denominator that is strictly between r_n>and r_{n+1}. You can show that (x,y) is an interior lattice point for>the triangle with vertices at (0,0), (3*F_n, 3*F_{n+1}), and (2*F_{n+1},>2*F_{n+2}). You can also show that the only interior lattice point for>that triangle is (F_{n+2}, F_{n+3}). The desired result follows.> Did you actually compu the area of the triangle (using> Heron's formula)?Yes, but not with Heron's formula. There is an area formula for atriangle using its coordinates in a determinant, near the bottom of thefollowing page:http://mathworld.wolfram.com/TriangleArea.htmlWith that and Pick's Theorem, you can show that the only lattice pointsinvolved with that triangle are the obvious ones.-- Daniel W. Johnsonpanoptes@iquest.nethttp://members.iquest.net/~panoptes/ === Fibonacci Numbers by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i0LKRAX16609;Suppose b/c is a rational number between 3/5 and 5/8.The difference (b/c)-(3/5) is a fraction whose denominator is 5c,namely (5b-3c)/(5c), so its value is at least 1/(5c).Similarly (5/8)-(b/c) is at least 1/(8c).Adding, (5/8)-(3/5) is at least 1/(5c)+1/(8c)=13/(40c).But (5/8)-(3/5)=1/40, so c is at least 13.You need to generalize, and prove that certain things always happen.>This might be well-known, but not to me.>Consider the sequence of ratios of neighboring Fibonacci>numbers:> 1/1, 1/2, 2/3, 3/5, 5/8, etc.>Let r_n be the nth rational on this list. What appears>to be true is this:> r_{n+2} is the rational with the smallest denominator that> is strictly between r_n and r_{n+1}>I don't immediately see how to prove this, though.>-->Daryl McCullough>Ithaca, NY> === think of M_n(C), the n x n matrices in the complex numbers asR^(2n^2), where R denotes the reals.Then, if G is a matrix lie group, g (the lie algebra of G), is a subspace ofR^(2n^2).Let D denote the orthogonal complement of g with respect to the usual innerproduct on R^(2n^2). Consider the mapf : g + D ---> GL(n,C) given byf(X,Y) = (e^X)(e^Y)(where + in the mapping denotes direct sum)Of course, we can identify g + D with R^(2n^2). Moreover, GL(n,C) is anopen subset of R^(2n^2). Thus, we can regard f as a map from R^(2n^2) toitself.Now, using the properties of the matrix exponential, we see thatd/dt (f(tX,0)) = 0 (differentia at t = 0) *d/dt (f(0,tY)) = 0 (differentia at t = 0) **Why does this show that the derivative of f at the point 0 in R^(2n^2) isthe identity? (Recall that the derivative at a point of a function fromR^(2n^2) to itself is a linear map of R^(2n^2) to itself).I am trying to use the straight up definition of differential at a point.i.e. the jacobian. But the claim says that the jacobian is the identity,right? Why is this true? How can we even calculate the Jacobian given thetwo facts * and ** above?If anyone can help I would highly appreciate it,Moshe === Let us think of M_n(C), the n x n matrices in the complex numbers as> R^(2n^2), where R denotes the reals.> Then, if G is a matrix lie group, g (the lie algebra of G), is a subspace of> R^(2n^2).> Let D denote the orthogonal complement of g with respect to the usual inner> product on R^(2n^2). Consider the map> f : g + D ---> GL(n,C) given by> f(X,Y) = (e^X)(e^Y)> (where + in the mapping denotes direct sum)> Of course, we can identify g + D with R^(2n^2). Moreover, GL(n,C) is an> open subset of R^(2n^2). Thus, we can regard f as a map from R^(2n^2) to> itself.> Now, using the properties of the matrix exponential, we see that> d/dt (f(tX,0)) = 0 (differentia at t = 0) *> d/dt (f(0,tY)) = 0 (differentia at t = 0) **> Why does this show that the derivative of f at the point 0 in R^(2n^2) is> the identity? (Recall that the derivative at a point of a function from> R^(2n^2) to itself is a linear map of R^(2n^2) to itself).> I am trying to use the straight up definition of differential at a point.> i.e. the jacobian. But the claim says that the jacobian is the identity,> right? Why is this true? How can we even calculate the Jacobian given the> two facts * and ** above?> If anyone can help I would highly appreciate it,> Moshe AdrianIf you have a differentiable function (and surely f is) then its full derivative is simply the matrix of its partial derivatives. (*) and (**) would imply that the derivative is 0, not the identity. On the other === Lie algebras, matrix exponential help!> Let us think of M_n(C), the n x n matrices in the complex numbers as> R^(2n^2), where R denotes the reals.Then, if G is a matrix lie group, g (the lie algebra of G), is asubspace of> R^(2n^2).Let D denote the orthogonal complement of g with respect to the usualinner> product on R^(2n^2). Consider the mapf : g + D ---> GL(n,C) given byf(X,Y) = (e^X)(e^Y)(where + in the mapping denotes direct sum)Of course, we can identify g + D with R^(2n^2). Moreover, GL(n,C) is an> open subset of R^(2n^2). Thus, we can regard f as a map from R^(2n^2)to> itself.Now, using the properties of the matrix exponential, we see thatd/dt (f(tX,0)) = 0 (differentia at t = 0) *> d/dt (f(0,tY)) = 0 (differentia at t = 0) **Why does this show that the derivative of f at the point 0 in R^(2n^2)is> the identity? (Recall that the derivative at a point of a function from> R^(2n^2) to itself is a linear map of R^(2n^2) to itself).I am trying to use the straight up definition of differential at apoint.> i.e. the jacobian. But the claim says that the jacobian is theidentity,> right? Why is this true? How can we even calculate the Jacobian giventhe> two facts * and ** above?If anyone can help I would highly appreciate it,Moshe Adrian>> If you have a differentiable function (and surely f is) then its fullderivative> is simply the matrix of its partial derivatives. (*) and (**) would implythat> the derivative is 0, not the identity. On the other hand, I think meant to writed/dt (f(tX,0)) = X (differentia at t = 0) *d/dt (f(0,tY)) = Y (differentia at t = 0) **I still don't understand!!! * and ** do not say anything about the partialderivatives of f (using component functions), do they?f : R^(2n^2) -----> R^(2n^2)df_0 = the matrix[ df1/dx1 .......... df2n^2/dx1 ][ df2/dx1 ......... .. ][ df3/dx1 ][ ... ][ df2n^2/dx1 ........... df2n^2/dx2n^2 ]evalua at (0,0,0,.....,0) the 2n^2-tuple.Am I missing === of a matrix)I am having some issues with the norm of a matrix. Let A be an n x nmatrix.I have seen it defined as ||A|| = sup ||Ax||/||x|| (x not equal to 0) = max||Ax|| (where ||x|| = 1)Therefore, ||I||, the norm of the identity matrix would equal 1, correct?(Using ||A|| = max ||Ax|| where ||x|| = 1)But, in my text, ||A|| is defined as||A|| =[ sum (from k,l = 1 to n) of |A_(kl)|^2 ] ^(1/2)the entries in the matrix, summing them, and taking the square root. Thenif you take the n x n identity matrix, wouldn't the norm be the square rootof === matrix)> I have seen it defined as ||A|| = sup ||Ax||/||x|| (x not equal to 0) = max> ||Ax|| (where ||x|| = 1)That's called a matrix norm induced by (associa with) a vectornorm. For each vector norm (the 1-norm, 2-norm, inf-norm, .... ) thisgives you a different matrix norm.> But, in my text, ||A|| is defined as> ||A|| => [ sum (from k,l = 1 to n) of |A_(kl)|^2 ] ^(1/2)That's the Frobenius norm, and it's also _a_ matrix norm. It's notinduced by any vector norm.V.-- email: lastname at cs === Stupid question on notation (norm of a matrix)> I am having some issues with the norm of a matrix. Let A be an n x n> matrix.> I have seen it defined as ||A|| = sup ||Ax||/||x|| (x not equal to 0) = max> ||Ax|| (where ||x|| = 1)> Therefore, ||I||, the norm of the identity matrix would equal 1, correct?> (Using ||A|| = max ||Ax|| where ||x|| = 1)That is called the operator norm> But, in my text, ||A|| is defined as> ||A|| => [ sum (from k,l = 1 to n) of |A_(kl)|^2 ] ^(1/2)That is called the Hilbert-Schmidt norm> the entries in the matrix, summing them, and taking the square root. Then> if you take the n x n identity matrix, wouldn't the norm be the square root> of n?> I hate notational issues....These are different norms. Any given space has many norms on it,perhaps === matrix) > Therefore, ||I||, the norm of the identity matrix would equal 1, correct? > (Using ||A|| = max ||Ax|| where ||x|| = 1) > That is called the operator normOr 2-norm, or Euclidean norm, or whatever. This one goes by many differentnames. To calculate it you have to calculate the singular values of thematrix. > But, in my text, ||A|| is defined as > ||A|| = > [ sum (from k,l = 1 to n) of |A_(kl)|^2 ] ^(1/2) > That is called the Hilbert-Schmidt normYou have also ||A|| = max{i}sum{j} |A_(ij)| and = max(i)sum(j) |A_(ji)|. > the entries in the matrix, summing them, and taking the square root. Then > if you take the n x n identity matrix, wouldn't the norm be the square root > of n? > I hate notational issues.... > These are different norms. Any given space has many norms on it, > perhaps even many useful norms.In numerical algebra most of the different norms are used as cheapcalculatable substitutes for the one and only true norm, the 2-norm ;-).But seriously, J.H. Wilkinson, the Algebraic Eigenvalue Problem (prettyold book) has a good discussion about vector norms, the associamatrix norms and === Theory)Recently I came across this problem in my textbook and I've been quiteinteres to know the answer to.Here it is,Chicken Nuggets used to be sold at a hamburger chain in packages of 6, 9, 20pieces. What is the largest number of pieces you could not order exactly?I'd appreciate any help on === this problem.GavinSubject: Re: Chicken Nugget Problem (Number and I've been quite> interes to know the answer to.> Here it is,> Chicken Nuggets used to be sold at a hamburger chain in packages of 6, 9, 20> pieces. What is the largest number of pieces you could not order exactly?> I'd appreciate any help on this problem.> GavinStart looking at how you can increment by 1.Example: if you have a 20 and a 6, you can replace them with 3x9 to increment 1.if you have a 20 you can replace it with 2x6 and 9 to increment 1.if you have 10x6, you can replace by 3x20 to increment by 0.Once you have several of these set up, look at what the minimum start is to be able to ALWAYS increment by 1.-- Will Twentymanemail: wtwentyman at === Theory)> Recently I came across this problem in my textbook and I've been quite> interes to know the answer to.> Here it is,> Chicken Nuggets used to be sold at a hamburger chain in packages of 6, 9, 20> pieces. What is the largest number of pieces you could not order exactly?> I'd appreciate any help on this problem.> GavinAs I recall, the answer is 43. This is a modern example of Diophantine equationsand yields to classic methods for that branch of the arts.--There are two things you must never attempt to prove: the unprovable -- and === proof...This is in relation to the chicken nuggets that I 've pos, which I cannotobtain a proof to as well. Can someone outline what they would do? a fullproof is not neccessary.1. Show that (a, (b,c)) = ((a,b), c) for any three numbers a, b, c. Definethe greatest common disvisor of a, b, c, call it (a, b, c) and show that (a,b, c) = (a, (b, c)).2. Show that (a, b, === c) = ax + by + cz for some integers x, y, z.Subject: Re: GCD 've pos, which I cannot> obtain a proof to as well. Can someone outline what they would do? a full> proof is not neccessary.> 1. Show that (a, (b,c)) = ((a,b), c) for any three numbers a, b, c. Define> the greatest common disvisor of a, b, c, call it (a, b, c) and show that (a,> b, c) = (a, (b, c)).I would attempt to do this in two stages:a) show that (a,b,c) <= (a, (b,c)) by showing that (a,b,c) is a divisor of (a,(b,c))b) show that (a,b,c) >= (a, (b,c)) by a similar argument.> 2. Show that (a, b, c) = ax + by + cz for some integers x, y, z.Extend (a,b) = ax + by for some integers x,y.-- Will === with Shell Method of Finding SolidsI have a pretty basic question concerning the shell method of findingvolumes through integration.I understand the basic concept and can figure out the simpler problemsas they are revolved around the x or y axes. However, when somethingis revolved around say x=4, or x=-1, I can never figure out what theradius of the shell should be.For example, the graph y=x^2, bounded by y=0, x=1, x=2 and revolvedaround x=4 has a radius of (4-x), but the region bordered by y=x^1/2(square root of x), x=1, x=2 revolved around the line x=-1 has aradius of (x+1).I do not understand how they came up with the radius for these twoexamples. What am I missing? It must be something simple or === Continued-Fraction-Genera Sequence Also Equals...Consider the continued fraction:[1; 1,1, 1/2,1/2, 1/3,1/3, 1/4,1/4,...,ceiling((m-2)/2)]= c(m), for m >= 3, where the total number of (rational) CF-terms is (m-1).Let c(1) = 1, and c(2) = 2, so we have:{c(j)} -> 1, 2, 2, 3/2, 7/4, 19/12, 61/36,... Now, let us take 'another' sequence {c'(j)},where:c'(1) = 1, c'(2) = 2.And c'(1+m) =(sum{j=0 to floor((m-1)/2)} c'(m- 2j)) /ceiling(m/2).In other words, c'(1+m) is the average of every-other previous c'(),the average of {c'(m),c'(m-2),c'(m-4),...,c'(1 or 2)}.So, you guessed it,each c(k) = c'(k).Also,c(1+m) = c(m)/ceiling(m/2) + c(m-1)(1 -1/ceiling(m/2)),which should (easily?) lead to a closed-form for c(m).But I have not the time now to explore further...By the way, what is the closed-form forlimit{m-> oo} === Circle...(almost)We all (should) know it is impossible to square a circle with only astraight-edge and compass and pencil.But, of course, we can approximately square a circle to any finite,but imperfect, accuracy we desire.A brute-force method might be to first list the squareroot of pi inbinary.Then keep bisecting a segment equal to the circle's radius so as toget the lengths:(radius)/2^k, for k = 0, 1, 2, ...n,where n is any arbitrarily high finite integer.Then finally construct the line-segment equal to the sum of thesegments' lengths in the previous step which correspond withthe 1's in sqrt(pi)'s binary expansion.And, finally, make a square with this last segment's side-length.So we have a square with an area (just under)(radius)^2 *pi.So, FOR FUN, what are some other more efficientapproximate-circle-squaring algorithms (using onlystraight-edge and compass) you all can come up with??(I recall reading someplace that some semi-crank from years ago cameup with a decent approximation === Circle...(almost) ETAsAhRbg9NqOk1rSUdrV1l6GapH11MolwIUeCXiha2roaR/ hctS7oVPqZGA4po= If you inscribe an n-sided regular polygon in the circle, andcircumscribe an n-sided regular polygon about the same circle, then youcan approximate the circumference by adding one-third the circumscribedperimeter to two-thirds the inscribed perimeter. You may use any n thatgives a constructible polygon, and the larger the n you use the moreaccurate you will be.Let's try this out with n = 4. Taking the unit length to be the circleradius, we get a circumscribed perimeter of 4 and an inscribed perimeterof 2*sqrt(2) = 2.283. Then the approxima circumference is4/3*(1+sqrt(2)) = 4/3*2.414 = 3.219. Which is sort of OK.Increase n to 6 and we get:Circumscribed perimeter = 2*sqrt(3) = 3.464Incscribed perimeter = 3Approxima circumference = 3.155 (!)Basically you are rendering x as (2 sin x + tan x)/3 for small x. Theright hand side is x + O(x^5) so you get quite accurate without x === Circle...(almost)> We all (should) know it is impossible to square a circle with only a> straight-edge and compass and pencil.I'm sorry, but you never make it clear what you mean by square acircle. Please explain.Moshe-- http://runslinux.net :: === Accessible Algorithm-MazeCome on, this is easy!...I will post the answer, if no one else gets it sooner, in the next few days.Leroy> Below is an abstract maze.> It is a bit more accessible/understandable and easier than my most > recently pos algorithm maze. > > (For example, its only math is simple addition.)> So, perhaps it would be enjoyable for some children to try to solve> the maze.> (And I originally intended to make this maze accessible to children,> but it ended up being a little harder than I had intended originally,> even being too difficult for many *adults* to understand the RULES,> let alone to get the solution.)> Follow the rules in-order so as determine which path to take,> from square (containing a symbol {ie. ascii-character})> to adjacent square to adjacent square to...,> forming a continuous path as the maze's solution.> You must move from square to adjacent square so as to> form the maze's solution-path.> By adjacent square, I mean a square immediately next> to the square you are currently on,> in the direction of above, below, left of, or right of;> but *not* diagonal to.> (So your pencil, or finger or mind, draws the path as a> rook moves in Chess.)> And, importantly, you must not leave the grid or land> on any square more than once!> (But not every square is necessarily visi by the> solution's path.)> And, if you cannot move to any adjacent square,> given the rule associa with the move,> then your path has reached a dead-end within the maze. > (View with fixed-width font.)>! ------------------------->! | # | # | P | U | O | W |>! ----+---+---+---+---+---->! | R | 3 | 7 | * | D | N |>! ----+---+---+---+---+---->! | 2 | A | 4 | 3 | % | * |>! ----+---+---+---+---+---->! | @ | 5 | 8 | + | @ | L |>! ----+---+---+---+---+---->! | 1 | R | * | T | % | E |>! ----+---+---+---+---+---->! | 3 | I | G | H | T | F |>! -------------------------> Rules (follow in-order, similar to a computer-program):> 1) Start in upper-left square.> 2) Move to adjacent square with a letter.> 3) Move to adjacent square with a number.> 4) Move down one square.> 5) Move to adjacent square with a number.> 6) Add number in square you are currenty on and the most> immediately previous number your path crossed before your> current square, and move to adjacent square with this sum.> 7) Move to an adjacent square with a *.> 8) Move to adjacent square to its adjacent square to...> so as to form a chain of squares, each containing a letter,> spelling a word.> 9) Move ____[word spelled in previous step]___ one square.> 10) Move up or down the number of squares equal to the number> of [symbols] in the entire maze, where [symbol] is the> symbol in the square currently you are at > (which you moved to in previous step).> 11) Repeat the fifth step (step-5 above) twice, then repeat> steps (6),(7),(8),(9) and (10) once each in-order.> (Skip this {11th} step the second time you encounter it.)> 12) Which square are you now on, and which symbol is in that square??> === Zeta-Function Continued FractionJust curious...Is there anything interesting that can be said aboutx = zeta(2) + 1/(zeta(3) +1/(zeta(4) + 1/(zeta(5) +....))),such as a closed-form for the limit as the number of Riemann zeta-functionsapproaches infinity?Or more generally, how about:x(y) = zeta(y) + 1/(zeta(y+1) +1/(zeta(y+2) + 1/(zeta(y+3) +....))) ??Leroy Quet