mm- Subject: Re: Math > Yes, most numbers look like male genitals, penises or balls. > Look at this 00 > 1 Pablo, I must correct you. You mean numerals, not numbers. Furthermore, your Þrst example would look (just very slightly) more convincing if viewed using a font with proportional spacing. Now: since the kind of thing to which you allude is in the eye of the beholder, and therefore subject neither to proof nor disproof, I suggest the following exercise: look at the Arabic form of numerals (0x0660 to 0x0669 in UniCode). You will Þnd at least one such numeral which does not have the property which you claim. (Alternatively, if you think that they all do too, when was the last time you had an eye test?) Lastly, please note that the correct plural of penis is penes - from the Latin, obviously. (And, once again, to avoid accusations of sexism, note that, switching to Greek, the correct plural of clitoris is clitorides ...) Best wishes, John johnDOTmorrisonATtescoDOTnet -- I note from what you say that you are clearly a master of the ancient and honourable Japanese art of bullshido. - Fred W. Enga PS Just why _do_ people post such things in mathematical newsgroups? --- Outgoing mail is certiÞed Virus Free. Checked by AVG anti-virus system (http://www.grisoft.com). === Subject: recurrsion limit, maple, MATLAB, I get the error on the MATLAB command window: ??? Error using ==> sym/maple Error, (in collect/series) too many levels of recursion I assume the error is in maple and I was wondering if there is any way to increase the recursion limit in maple. I think to increase it in MATLAB you change the property RecursionLimit of the root (function handle zero) === Subject: Re: recurrsion limit, maple, MATLAB, > I get the error on the MATLAB command window: > ??? Error using ==> sym/maple > Error, (in collect/series) too many levels of recursion > I assume the error is in maple and I was wondering if there is any way > to increase the recursion limit in maple. I think to increase it in > MATLAB you change the property RecursionLimit of the root (function > handle zero) There have been suggestions that I was getting the recursion limit error because of user error. I don¹t think this is the case because I am not using recursion. I once Þxed the error by increasing the recursion limit on MATLAB but then I got it later again. So maybe that was just a coincidence and I did something else without realizing it. But I do notice that if I Þrst simplify a certain part of the expression before I do the sum I don¹t get the recursion limit error. But I do recall reading something in the maple help about a parameter that effects how hard maple tries to do things. Maybe the maple function collect is one of the functions effected by this parameter. Moreover maybe the maple function collects uses recursion and the size of the stack needed grows exponentially with the length of the expression. I could see this being the case since collect is the reverse of expand and this could mean trying to factor an expression. Maybe if the resources grow exponentially with the length of the expression there is not much maple can do. But it would be nice to be able to set the recursion limit, just incase the computer has enough virtual memory to tackle the problem. P.S. If people are interested I will try to Þnd a simpliÞed way to recreate the error. Also note that maybe the error wouldn¹t occur in newer versions of maple or MATLAB. I am accessing the maple Kernal by using the symbolic package of MATLAB 6.1.0.450 (R12.1) === Subject: Re: recurrsion limit, maple, MATLAB, *snip* > P.S. If people are interested I will try to Þnd a simpliÞed way to > recreate the error. Also note that maybe the error wouldn¹t occur in > newer versions of maple or MATLAB. I am accessing the maple Kernal by > using the symbolic package of MATLAB 6.1.0.450 (R12.1) If you can Þnd some relatively compact reproduction steps for this problem, please send them to support@mathworks.com. If it is a bug and we have a test case with which we can reproduce it, we can try to Þgure what¹s going on and, if it is a problem in the Toolbox code, how to Þx it. -- Steve Lord slord@mathworks.com === Subject: Re: recurrsion limit, maple, MATLAB, > I get the error on the MATLAB command window: > ??? Error using ==> sym/maple > Error, (in collect/series) too many levels of recursion > I assume the error is in maple and I was wondering if there is any way > to increase the recursion limit in maple. I think to increase it in > MATLAB you change the property RecursionLimit of the root (function > handle zero) It probably means a non-ending recursion. Say you deÞne f(n) = 2*f(n-1) but give no starting value, then computing f(3) will give you that error: too many levels of recursion. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: recurrsion limit, maple, MATLAB, > I get the error on the MATLAB command window: > ??? Error using ==> sym/maple > Error, (in collect/series) too many levels of recursion > > I assume the error is in maple and I was wondering if there is any way > to increase the recursion limit in maple. I think to increase it in > MATLAB you change the property RecursionLimit of the root (function > handle zero) > It probably means a non-ending recursion. Say you deÞne > f(n) = 2*f(n-1) but give no starting value, then > computing f(3) will give you that error: too many levels > of recursion. I think it might also have something to do with how MATLAB or maple manages memory. Because I can rut it one time and it works run it again and it doesn¹t work, run it again and it works again. Go Þgure. === Subject: int(exp(x)*erf(x),x = a .. b) using MATLAB 6.1.0.450 (R12.1) to access the maple kernel I don¹t get a solution. for: int(exp(x)*erf(x),x = a .. b) Does anyone know if this is solvable? It looks like you could do parts a few times, and then equate terms. Since: >> int(sym(Œerf(x)¹)) ans = x*erf(x)+1/pi^(1/2)*exp(-x^2) What about the integral: int(exp(x^2)*erf(x),x = a .. b) For this I could let u=erf(x) and do substitution. Right? Moreover, If I Þgure out how to do the integral myself, how do I teach maple to do it? === Subject: Re: int(exp(x)*erf(x),x = a .. b) > using MATLAB 6.1.0.450 (R12.1) to access the maple > kernel I don¹t get a solution. for: > int(exp(x)*erf(x),x = a .. b) > Does anyone know if this is solvable? It looks like you could do parts > a few times, and then equate terms. Since: >nt(sym(Œerf(x)¹)) > ans = > x*erf(x)+1/pi^(1/2)*exp(-x^2) > What about the integral: > int(exp(x^2)*erf(x),x = a .. b) > For this I could let u=erf(x) > and do substitution. Right? > Moreover, If I Þgure out how to do the integral myself, how do I > teach maple to do it? If this is any help, on Mathematica 5.01 I get In[1]:= Integrate[Erf[x]*E^x, {x, a, b}] Out[1]= (-E^(1/4))*Erf[1/2 - a] - E^a*Erf[a] + E^(1/4)*Erf[1/2 - b] + E^b*Erf[b] and In[2]:= Integrate[Erf[x]*E^x^2, {x, a, b}] Out[2]= (1/Sqrt[Pi])*((-a^2)*HypergeometricPFQ[{1, 1}, {3/2, 2}, a^2] + b^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2]) where HypergeometricPFQ is deÞned here --Urijah Kaplan === Subject: Re: int(exp(x)*erf(x),x = a .. b) Macsyma gives the indeÞnite integral as %E^X*ERF(X)-%E^(1/4)*ERF(X-1/2) but refuses to do the exp(x^2). My Þrst assumption is that the stack overþow on your previous message is user error. Unless you supply a way to recreate it, you can¹t expect anyone to help. I have reduced the cc¹s to the most relevant newsgroups. > using MATLAB 6.1.0.450 (R12.1) to access the maple > kernel I don¹t get a solution. for: > int(exp(x)*erf(x),x = a .. b) > Does anyone know if this is solvable? It looks like you could do parts > a few times, and then equate terms. Since: >nt(sym(Œerf(x)¹)) > ans = > x*erf(x)+1/pi^(1/2)*exp(-x^2) > What about the integral: > int(exp(x^2)*erf(x),x = a .. b) > For this I could let u=erf(x) > and do substitution. Right? > Moreover, If I Þgure out how to do the integral myself, how do I > teach maple to do it? === Subject: Re: int(exp(x)*erf(x),x = a .. b) If you meant exp(-x^2)*erf(x), that integrates to sqrt(%pi)*erf(x)^2/4. > Macsyma gives the indeÞnite integral as > %E^X*ERF(X)-%E^(1/4)*ERF(X-1/2) > but refuses to do the exp(x^2). > My Þrst assumption is that the stack overþow on your previous > message is user error. Unless you supply a way to recreate > it, you can¹t expect anyone to help. > I have reduced the cc¹s to the most relevant newsgroups. > using MATLAB 6.1.0.450 (R12.1) to access the maple > kernel I don¹t get a solution. for: > int(exp(x)*erf(x),x = a .. b) > Does anyone know if this is solvable? It looks like you could do parts > a few times, and then equate terms. Since: >nt(sym(Œerf(x)¹)) > ans = > x*erf(x)+1/pi^(1/2)*exp(-x^2) > What about the integral: > int(exp(x^2)*erf(x),x = a .. b) > For this I could let u=erf(x) > and do substitution. Right? > Moreover, If I Þgure out how to do the integral myself, how do I > teach maple to do it? === Subject: Solving transcedental equations? Currently I am working on reÞning a program in maple as part of my research. We are trying to streamline our eccentricity calculation, seeing as it takes many iterations to complete. We can reduce the lhs of our equation to just a number, but the rhs is of the form: f:=int(cos(E)*cos(E-e_new*sin(E))*(1-e_new*cos(E)),E=0..2*Pi) /Pi/(-3/2*e_new ) We are trying to solve for little e_new, and integrating over big E. Is there a quick and easy way to do this in maple? Any suggestions would be greatly appreciated -Peter === Subject: Re: Solving transcedental equations? > Currently I am working on reÞning a program in maple as part of my > research. We are trying to streamline our eccentricity calculation, > seeing as it takes many iterations to complete. We can reduce the lhs of > our equation to just a number, but the rhs is of the form: > f:=int(cos(E)*cos(E-e_new*sin(E))*(1-e_new*cos(E)),E=0..2*Pi) /Pi/(-3/2*e_new) > We are trying to solve for little e_new, and integrating over big E. Is > there a quick and easy way to do this in maple? > Any suggestions would be greatly appreciated > -Peter I¹m not sure I understand what it is you ultimately want to do, but maybe it will help to have a closed for for the integral expression above. In[1]:= InputForm[ff = Integrate[Cos[ee]*Cos[eenew*Sin[ee]]*(1-eenew*Cos[ee]), {ee,0,2*Pi}, Assumptions->Element[eenew,Reals]] / (Pi*(-3/2*eenew))] Out[1]//InputForm= (4*BesselJ[1, Abs[eenew]])/(3*Abs[eenew]) This should translate to Maple in a straightforward manner. I¹m guessing you later want to solve expressions of the form ff = number for eenew. This could be done with a numerical univariate root Þnder. The one caution is that you have a decaying oscillating function so for smallish values there might be multiple inverses and you would need to decide which is of interest. Another means of attack that might be suitable for approximation purposes would be to work with a series expansion of the integrand, integrating termwise to approximate the value of your lhs. But offhand I see no reason for the closed form involving BesselJ to cause any trouble. Daniel Lichtblau Wolfram Research === Subject: Re: Solving transcedental equations? How nice of you of answer a question about Maple. Too bad the original poster couldn¹t ask in the Mathematica forum where all such questions are censored! Caesar Garcia > > Currently I am working on reÞning a program in maple as part of my > research. We are trying to streamline our eccentricity calculation, > seeing as it takes many iterations to complete. We can reduce the lhs of > our equation to just a number, but the rhs is of the form: > > f:=int(cos(E)*cos(E-e_new*sin(E))*(1-e_new*cos(E)),E=0..2*Pi) /Pi/(-3/2*e_new) > > We are trying to solve for little e_new, and integrating over big E. Is > there a quick and easy way to do this in maple? > > Any suggestions would be greatly appreciated > -Peter > I¹m not sure I understand what it is you ultimately want to do, but > maybe it will help to have a closed for for the integral expression > above. > In[1]:= InputForm[ff = > Integrate[Cos[ee]*Cos[eenew*Sin[ee]]*(1-eenew*Cos[ee]), {ee,0,2*Pi}, > Assumptions->Element[eenew,Reals]] / (Pi*(-3/2*eenew))] > > Out[1]//InputForm= (4*BesselJ[1, Abs[eenew]])/(3*Abs[eenew]) > This should translate to Maple in a straightforward manner. > I¹m guessing you later want to solve expressions of the form ff = > number for eenew. This could be done with a numerical univariate root > Þnder. The one caution is that you have a decaying oscillating > function so for smallish values there might be multiple inverses and > you would need to decide which is of interest. > Another means of attack that might be suitable for approximation > purposes would be to work with a series expansion of the integrand, > integrating termwise to approximate the value of your lhs. But offhand > I see no reason for the closed form involving BesselJ to cause any > trouble. > Daniel Lichtblau > Wolfram Research === Subject: Re: Solving transcedental equations? > Currently I am working on reÞning a program in maple as part of my > research. We are trying to streamline our eccentricity calculation, > seeing as it takes many iterations to complete. We can reduce the lhs of > our equation to just a number, but the rhs is of the form: f:=int(cos(E)*cos(E-e_new*sin(E))*(1-e_new*cos(E)),E=0..2*Pi) /Pi/(-3/2*e_new ) > We are trying to solve for little e_new, and integrating over big E. Is > there a quick and easy way to do this in maple? > Any suggestions would be greatly appreciated > -Peter You can plot the right side as a function of e_new by using: >plot(evalf(Int(cos(E)*cos(E-e_new*sin(E))*(1-e_new*cos(E)),E =0..2*Pi))/Pi/( -3/2*e_new), e_new=-1..1, t=-10..10); This shows a graphs something like -1/e_new. Then for example if f is negative you can Þnd a solution for e_new in the interval 0 to 1 or (0 to something else if the absolute value of f is very small) using the following: >for f from 1 to 10 do > -f,fsolve(-f=evalf(Int(cos(E)*cos(E-e_new*sin(E))*(1-e_new* cos(E)),E=0..2* Pi))/Pi/(-3/2*e_new),e_new=0..1); > od; -1, 0.5835322740 -2, 0.3205784253 -3, 0.2182653575 -4, 0.1649689565 -5, 0.1324571567 -6, 0.1106018453 -7, 0.09491654116 -8, 0.08311754546 -9, 0.07392233968 -10, 0.06655595836 === Subject: Re: orders of permutations > Well, I am a student in a computer math class. My teacher totally > fell > behind on schedule and ended up assigning us a programming assignment > during Þnals week. Therefore, I cannot consult him for questions. > Well, > There are two parts to the problem. I have completed part 1, which is > creating a program that lists all permutations of 1,2,..,n in such a > way > that consecutive permutations differ by a single transposition using > the > Johsnon-Trotter algorithm. However, there is a second part to the > question > which I think he meant to make trivial but I am not too familiar with > the > terms and logic behind his question so I am confused... It states: > permutation > of 1,2,3,...,n can have. Run it for as large a value of n as you can > and > tabulate the largest possible orders as a function of n. > Any and all feedback/insight/help would be greatly appreciated! > - John Landau¹s function: http://www.research.att.com/projects/OEIS?Anum=A000793 Rob Pratt === Subject: Re: orders of permutations > The order of a permutation is the minimum number of times you can > apply the permutation before the permutation is an identity. I believe > in symmetry group four (all permutations of (1,2,3,4) ) the largest > order is three. But the identity has order 1, so your count is off by 1. There is always a permutaton of n elements that has order n: just rotate the elements. In S4: 1 2 3 4 2 3 4 1 > I wonder if n-1 works. Or maybe the answer is in terms of a factorial. The order of a permutation is the least common multiple of the cycle lengths. So we can have a permutation of 5 elements that has order 6: 1 2 3 4 5 2 1 4 5 3 In disjoint cycle notation that is (1 2) (3 4 5) I don¹t know a formula for the highest order that can be obtained with n elements. === Subject: Re: orders of permutations > I don¹t know a formula for the highest order that can be obtained with n > elements. To state it as simple sentence, it is the largest divisor of n! that can be factored in such a way that the sum of factors in less than or equal to n. That paper for which Edwin gave the abstract should be highly accessible. === Subject: Re: orders of permutations > I don¹t know a formula for the highest order that can be obtained with n > elements. The paper abstracted below gives an approximation. Apparently no one has yet found an exactly formula. --Edwin Miller, William The maximum order of an element of a Þnite symmetric group. Amer. Math. Monthly 94 (1987), no. 6, 497--506. elements by $G(n)$. The goal of this paper is to summarize what is known about $G(n)$ and then present a proof of one of the premier results, namely, that (1) $log G(n)simsqrt{nlog n}$. The proof of (1) has an aesthetic quality that greatly enhances its appeal: It applies a deep, number-theoretic result to a question from group theory, yet in such a way that a general reader can appreciate the details. The deep result from number theory is the prime number theorem. The remainder of the paper is organized as follows. In Section 2, we review what is known about $G(n)$. Next (Section 3), we discuss some basic notions concerning $G(n)$. Beginning in Section 4, we turn to proving (1). The Þrst step is to make a connection between $G(n)$ and the prime numbers. This connection leads us, in a natural way, to consider a function $F(n)$ that approximates $G(n)$ but is simpler to handle. We then show, in Sections 5 and 6, respectively, that $log G(n)simlog F(n)$ and that $log F(n)simsqrt{nlog n}$, from which (1) is evident. Finally, we offer some concluding remarks in Section 7. The arguments of Sections 4 and 5 employ a variety of simple ideas. It is not until Section 6 that we must invoke the prime number theorem. === Subject: Re: Maple question |>Does anyonw knows how to Þnd solution of a set of multi-variable |>polynomials using Maple? Is it ok using Groebner package? How to use |>it? It might work, or might not. The more variables and the higher the degrees, the less likely it is to work (even if it does work, you might get a really horrible-looking solution). Here¹s an example using the Groebner package. Suppose you want to solve the system of equations x^2 + y*z - 3 = 0 x*y*z - x + 4 = 0 x + y - 1 = 0 > with(Groebner): gsolve([x^2 + y*z - 3, x*y*z - x + 4, x + y - 1],[x,y,z]); /[[ 2 ] ] { [[5 z - 8 z + 13, 7 y + 5 z - 18, 7 x - 5 z + 11], plex(x, y, z), {z - 1}], [[z - 1, y + 1, x - 2], plex(x, y, z), {}] } / So there are two cases: one where 5*z^2-8*z+13=0, and then y and x are obtained by solving the second and third equations of the Þrst list, and one where z=1,y=-1,z=2. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: goldbech conjecture proof by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i5GIAYj14302; >Use of non-ASCII characters in a newsgroup posting make this unreadable >to me, and perhaps to others. Masoud Sheykhi >Sarcheshmeh Copper complex,Kerman , Iran >Technical_inspection@nicico.com >Abstract >Goldbach¹s conjecture says that; >Every even number greater than 2 is the sum of two primes. >In this paper I introduce the conclusion of my research during several >years on goldbach¹s conjecture . I believe that I have solved and I >have turned it to a theorem as follows. >Proof >First we know that ; 4 = 2+2 . Suppose that the conjecture is true for >all even integers n = 2k , 2 of natural numbers ; > N = {1,2,3,...}.Now , we set ; k = L+1 . when 2L-1 is a prime number >then ; 2(L+1) = (2L-1) +3 . Otherwise we consider all representations >of each n = 2q , 2 primes that obtaining from the representations of all even numbers n >= 2q , 2 pj,s let Ij = <2(L+1) - pj > be the principal ideal generated by the >integer 2(L+1) - pj . Let Q be the intersection of all above Ij,s . >Since Var(Q) = {P in Spec(Z); Q is a subset of P} is a non-empty set >,where Spec(Z) is the set of all prime ideals of the commutative ring >of integer ; Z , there exist some prime ideals say ;P1 = ,P2 = > ,..., Pn = where n in N and p1(k),p2(k),...,pn(k) are prime numbers and such that Var(Q) = >{P1 ,P2,...,Pn}, n in N . But by commutative algebra , for each Pi ,i >= 1,2,...,n there exist some natural coefÞcient say ; c1(i) ,c2(i) >,...,cm(i) such that < c1(i)*pi(k)> ,< c2(i) * pi(k)> ,..., < cm(i)* >pi(k)> are the subsets of Pi . Let A = {cj(i)^r * pi(k)^s ;r = 0, 1, >2, 3,...; s = 0, 1 , 2 ,3,...; i = 1, 2 , ...,n , n in N ; j = 1 , 2 >, ...,m(i) , m(i)in N} be ordered by the relation R where m R n if >and only if there exists a non-negative integer; k such that n = km . >therefore we have partially ordered the set A by the divisibility >relation . we deÞne a map f: A->A such that for each a = cj(i)^r * >pi(k)^s in A ; f(cj(i)^r * pi(k)^s) = cj(i)^r-1 * pi(k)^s , if r = 1, >2 ,3,...,and s = 0, 1 , 2, 3,..., f(cj(i)^r * pi(k)^s) = 1 ,if r = 0 >and s= 0,1,2,3,... Then ; f(a)R a . Now ,Since (A,R) be a nonempty >partially ordered set such that every totally ordered subset of A has >a greatest lower bound and f(a)Ra , for every a in A ; there exist h in A >such that f(h) = h , hence by deÞnition of f , for some j ,i ,r ,s we >have ; h = cj(i)^r * pi(k)^s such that ; cj(i)^r * pi(k)^s = f(h) = >cj(i)^r-1 * pi(k)^s . Thus ; for some j ,i ; cj(i) =1 and such that; >f(1) = 1= f(cj(i)) = cj(i) . Therefore some Ij is a prime ideal or >2(L+1)- pj is a prime number for some j . Thus ; n = 2(L+1) = [2(L+1) - pj ] + pj . But by the induction hypothesis ; pj is a prime number >, hence inductively goldbach¹s conjecture is proven. === Subject: Re: goldbech conjecture proof by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i5GCV0612987; >Maoud Sheykhi >Sarcheshmeh Copper complex,Kerman , Iran >Technical_inspection@nicico.com >Abstract >Goldbach¹s conjecture says that; >Every even number greater than 2 is the sum of two primes. >In this paper I introduce the conclusion of my research during several >years on goldbach¹s conjecture . I believe that I have solved and I >have turned it to a theorem as follows. >Proof >First we know that ; 4 = 2+2 . Suppose that the conjecture is true for >all even integers n = 2k , 2 .89ÅÛk.89ÅÛL where k and L belong to the set >of natural numbers ; > N = {1,2,3,.89¥Ï}.Now , we set ; k = L+1 . when 2L-1 is a prime number >then ; 2(L+1) = (2L-1) +3 . Otherwise we consider all representations >of each n = 2q , 2 .89ÅÛq.89ÅÛL as the sum of two primes .Let pj,s be all >primes that obtaining from the representations of all even numbers n >= 2q , 2 .89ÅÛq.89ÅÛL by the induction hypothesis .For each pj of above >pj,s let Ij = <2(L+1) - pj > be the principal ideal generated by the >integer 2(L+1) .89¥ã pj . Let Q be the intersection of all above Ij,s . >Since Var(Q) = {Pë[Micro] Spec(Z); Q is a subset of P} is a non-empty set >,where Spec(Z) is the set of all prime ideals of the commutative ring >of integer ; Z , there exist some prime ideals say ;P1 = ,P2 = > ,.89¥Ï, Pn = are prime numbers and such that Var(Q) = >{P1 ,P2,.89¥Ï,Pn}, në[Micro] N . But by commutative algebra , for each Pi ,i >= 1,2,.89¥Ï,n there exist some natural coefÞcient say ; c1(i) ,c2(i) >,.89¥Ï,cm(i) such that < c1(i)*pi(k)> ,< c2(i) * pi(k)> ,.89¥Ï, < cm(i)* >pi(k)> are the subsets of Pi . Let A = {cj(i)^r * pi(k)^s.89ä[Hyphen]r = 0, 1, >2, 3,.89¥Ï ; s = 0, 1 , 2 ,3,.89¥Ï ; i = 1, 2 , .89¥Ï ,n , në[Micro] N ; j = 1 , 2 >, .89¥Ï ,m(i) , m(i)ë[Micro] N} be ordered by the relation R where m R n if >and only if there exists a non-negative integer; k such that n = km . >therefore we have partially ordered the set A by the divisibility >relation . we deÞne a map f: A.89ºâA such that for each a = cj(i)^r * >pi(k)^s ë[Micro] A ; f(cj(i)^r * pi(k)^s) = cj(i)^r-1 * pi(k)^s , if r = 1, >2 ,3,.89¥Ï and s = 0, 1 , 2, 3,.89¥Ï. f(cj(i)^r * pi(k)^s) = 1 ,if r = 0 >and s= 0,1,2,3,.89¥Ï . Then ; f(a)R a . Now ,Since (A,R) be a nonempty >partially ordered set such that every totally ordered subset of A has >a greatest lower bound and f(a)Ra , for every aë[Micro]A ; there exist hë[Micro]A >such that f(h) = h , hence by deÞnition of f , for some j ,i ,r ,s we >have ; h = cj(i)^r * pi(k)^s such that ; cj(i)^r * pi(k)^s = f(h) = >cj(i)^r-1 * pi(k)^s . Thus ; for some j ,i ; cj(i) =1 and such that; >f(1) = 1= f(cj(i)) = cj(i) . Therefore some Ij is a prime ideal or >2(L+1)- pj is a prime number for some j . Thus ; n = 2(L+1) = [2(l+1) >.89¥ã pj] + pj . But by the induction hypothesis ; pj is a prime number >, hence inductively goldbach¹s conjecture is proven..89öÊ My comments to goldbech conjecture proof Would please add the deleted statement where në[Micro] N and p1(k), p2(k),.89¥Ï pn(k)in the proof after the statement there exist some prime ideals say ;P1 = ,P2 = ,.89¥Ï, Pn = and before the statementare prime numbers and such that Var(Q) = >{P1 ,P2,.89¥Ï,Pn}, në[Micro] N . Also, consider the name Masoud Sheykhi instead of Maoud Sheykhi Masoud Sheykhi === Subject: Re: goldbech conjecture proof by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i5GCV0l12968; >Masoud Sheykhi >Sarcheshmeh Copper complex,Kerman , Iran >Technical_inspection@nicico.com >Abstract >Goldbach¹s conjecture says that; >Every even number greater than 2 is the sum of two primes. >In this paper I introduce the conclusion of my research during several >years on goldbach¹s conjecture . I believe that I have solved and I >have turned it to a theorem as follows. >Proof >First we know that ; 4 = 2+2 . Suppose that the conjecture is true for >all even integers n = 2k , 2 .89ÅÛk.89ÅÛL where k and L belong to the set >of natural numbers ; > N = {1,2,3,.89¥Ï}.Now , we set ; k = L+1 . when 2L-1 is a prime number >then ; 2(L+1) = (2L-1) +3 . Otherwise we consider all representations >of each n = 2q , 2 .89ÅÛq.89ÅÛL as the sum of two primes .Let pj,s be all >primes that obtaining from the representations of all even numbers n >= 2q , 2 .89ÅÛq.89ÅÛL by the induction hypothesis .For each pj of above >pj,s let Ij = <2(L+1) - pj > be the principal ideal generated by the >integer 2(L+1) .89¥ã pj . Let Q be the intersection of all above Ij,s . >Since Var(Q) = {Pë[Micro] Spec(Z); Q is a subset of P} is a non-empty set >,where Spec(Z) is the set of all prime ideals of the commutative ring >of integer ; Z , there exist some prime ideals say ;P1 = ,P2 = > ,.89¥Ï, Pn = where në[Micro] N and p1(k),p2(k), ... ,pn(k) are prime numbers and such that Var(Q) = >{P1 ,P2,.89¥Ï,Pn}, në[Micro] N . But by commutative algebra , for each Pi ,i >= 1,2,.89¥Ï,n there exist some natural coefÞcient say ; c1(i) ,c2(i) >,.89¥Ï,cm(i) such that < c1(i)*pi(k)> ,< c2(i) * pi(k)> ,.89¥Ï, < cm(i)* >pi(k)> are the subsets of Pi . Let A = {cj(i)^r * pi(k)^s.89ä[Hyphen]r = 0, 1, >2, 3,.89¥Ï ; s = 0, 1 , 2 ,3,.89¥Ï ; i = 1, 2 , .89¥Ï ,n , në[Micro] N ; j = 1 , 2 >, .89¥Ï ,m(i) , m(i)ë[Micro] N} be ordered by the relation R where m R n if >and only if there exists a non-negative integer; k such that n = km . >therefore we have partially ordered the set A by the divisibility >relation . we deÞne a map f: A.89ºâA such that for each a = cj(i)^r * >pi(k)^s ë[Micro] A ; f(cj(i)^r * pi(k)^s) = cj(i)^r-1 * pi(k)^s , if r = 1, >2 ,3,.89¥Ï and s = 0, 1 , 2, 3,.89¥Ï. f(cj(i)^r * pi(k)^s) = 1 ,if r = 0 >and s= 0,1,2,3,.89¥Ï . Then ; f(a)R a . Now ,Since (A,R) be a nonempty >partially ordered set such that every totally ordered subset of A has >a greatest lower bound and f(a)Ra , for every aë[Micro]A ; there exist hë[Micro]A >such that f(h) = h , hence by deÞnition of f , for some j ,i ,r ,s we >have ; h = cj(i)^r * pi(k)^s such that ; cj(i)^r * pi(k)^s = f(h) = >cj(i)^r-1 * pi(k)^s . Thus ; for some j ,i ; cj(i) =1 and such that; >f(1) = 1= f(cj(i)) = cj(i) . Therefore some Ij is a prime ideal or >2(L+1)- pj is a prime number for some j . Thus ; n = 2(L+1) = [2(l+1) >.89¥ã pj] + pj . But by the induction hypothesis ; pj is a prime number >, hence inductively goldbach¹s conjecture is proven..89öÊ --------------------------------------------------------- add in the proof after the statement P1 = ,P2 = > ,.89¥Ï, Pn = the following deleted statement: where në[Micro] N and p1(k),p2(k), ... ,pn(k) Masoud Sheykhi === Subject: Re: goldbach conjecture proof by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i5H4CeH00486; Remark: I have sent messages with subjectgoldbech conjecture proof . Now,i replace it to the subject goldbach conjecture proof i have typed wrongly the character e instead of a in typing the name ofgoldbach. sincerely Masoud Sheykhi === Subject: Re: Lambert¹s W function question xadrezus@yahoo.com says... Please Þnd below a visual script for computing LambertW_R(x) in the real domain This script can be included in Excel as a macro. Using LambertW_R(x;n) with x >-1/exp(-1) and n gives precision as 10 power(-n). Public Function LambertW_R(ByVal X As Double, ByVal Digt As Integer) Œ computation of an inital guest If X < -1 / Exp(1) Then LambertW_R = Error(1) Exit Function ElseIf X < -0.26 Then p = Sqr(2# * (Exp(1#) * X + 1)) Wx = -1# + p - p ^ 2 / 3# + 11# / 72# * p ^ 3 ElseIf X < 0.37 Then Wx = X - X ^ 2 + 3# * X ^ 3 / 2# - 8# / 3# * X ^ 4 ElseIf X < 2 Then Wx = 0.0979767105 + 0.5764369031 * X - 0.1072703232 * X ^ 2 Else sigma = 1# / Log(X) tau = Log(Log(X)) * sigma Ltau = -Log(1# - tau) Wx = (1# - tau) / sigma + Ltau - Ltau * sigma / (1# - tau + sigma) End If Œiterative solver Ewx = Exp(Wx) Residu = Wx * Ewx - X Precis = 1# / (10# ^ Digt) ŒPrecis = 0.00000001 i = 0 Do While Abs(Residu) > Precis i = i + 1 Wx = Wx - Residu / ((Wx + 1#) * Ewx - 0.5 * (Wx + 2#) * Residu / (Wx + 1#)) Ewx = Exp(Wx) Residu = Wx * Ewx - X If i > 1000000 Then LambertW_R = Error(1) Exit Function End If Loop Œ Þnal value LambertW_R = Wx End Function Now if x< -1/exp(1) you can this script Lambert_WI(x,n) Public Function LambertW_I(ByVal X As String, ByVal Digt As Integer) As String Œ identiÞng if X is numeric. If it¹s, calculating Re(X) and Im(X) LX = Len(X) If Mid(X, LX, 1) <> i Then If IsNumeric(X) Then Re = CDbl(X) Im = 0 End If ElseIf X = i Then Re = 0 Im = 1# ElseIf IsNumeric(Mid(X, 1, LX - 1)) Then AA = Mid(X, LX - 1, 1) If AA = + Or AA = - Then Re = CDbl(Mid(X, 1, LX - 2)) Im = 1# Else Re = 0 Im = CDbl(Mid(X, 1, LX - 1)) End If Else I = 2 If IsNumeric(Left(X, 1)) Then I = 1 End If J = InStr(I, X, +) If J = 0 Then J = LX K = InStr(I, X, -) If K = 0 Then K = LX If K < J Then J = K If Mid(X, J - 1, 1) = e Or Mid(X, J - 1, 1) = E Then J1 = J + 1 J = InStr(J1, X, +) If J = 0 Then J = LX K = InStr(J1, X, -) If K = 0 Then K = LX If K < J Then J = K End If Re = CDbl(Mid(X, 1, J - 1)) K = InStr(X, i) If K - J > 1 Then Im = CDbl(Mid(X, J, K - J)) Else X = Replace(X, i, 1) Im = CDbl(Mid(X, J, K + 1 - J)) End If End If Œ computing LambertW for X real and X> -1/exp(1) If Im = 0 And Re >= -1# / Exp(1#) Then LambertW_I = LambertW_R(Re, Digt) Exit Function End If Œcomputing LambertW in complex domain ABX = Sqr(Re * Re + Im * Im) ISG = 1# Pi2 = 2# * Atn(1) Pi = 2# * Pi2 If ABX < Exp(-1) Then NF = 1# Ra = 1# Ia = 0# Rb = 0# Ib = 0# IJ = -1 N1 = -1# N = 0# Precis = 1# / (10# ^ Digt) ŒPrecis = 0.00000001 Do While ABX > Precis IJ = -IJ N = N + 1# N1 = N1 + 1# NF = NF * N XX = Ra * Re - Ia * Im Ia = Ra * Im + Ia * Re Ra = XX Io = IJ * N ^ N1 / NF Rc = Io * Ra Ic = Io * Ia ABX = Sqr(Rc * Rc + Ic * Ic) Rb = Rb + Rc Ib = Ib + Ic If N > 100 Then ABX = 0 Loop Re = Rb Im = Ib Else If Im < 0 Then Im = -Im ISG = -ISG End If R = ABX If Re = 0 Then If Im = 0 Then Theta = Pi2 ElseIf Im > 0 Then Theta = Pi2 Else Theta = -Pi2 End If ElseIf Re > 0 Then If Im = 0 Then Theta = 0 Else Theta = Atn(Im / Re) End If Else If Im = 0 Then Theta = Pi Else Theta = (Pi - Atn(Abs(Im) / Abs(Re))) * Sgn(Im) End If End If Œiterative solver Xa = 0 Ya = -Theta Xb = Pi2 Yb = Xb + R - Theta If Yb < 0 Then Xb = (-1 / R / Exp(1)) If Xb > -1 Then Xb = Atn(Sqr(1# - Xb * Xb) / Xb) + Pi Yb = Xb - Tan(Xb) - Theta Else LambertW_I = error 2 to be reported Exit Function End If End If Xc = (Xa + Xb) / 2 Precis = 1# / (10# ^ Digt) CX = Cos(Xc) Yc = LambertW_R(R * CX, Digt) Residu = Xc + Yc * Tan(Xc) - Theta ŒPrecis = 0.00000001 I = 0 Do While Abs(Residu) > Precis I = I + 1 Xc = Xc - Residu * CX ^ 2 * (1 + Yc) / (CX + Yc) ^ 2 CX = Cos(Xc) Yc = LambertW_R(R * CX, Digt) Residu = Xc + Yc * Tan(Xc) - Theta If I > 1000 Then LambertW_I = unable to converge Exit Function End If Loop If Xc <> Pi2 Then R = LambertW_R(R * Cos(Xc), Digt) / Cos(Xc) End If Œ conversion to excel format Re = R * Cos(Xc) Im = ISG * R * Sin(Xc) End If If Im > 0 Then XX = Str(Re) & + & Str(Im) & i Else XX = Str(Re) & Str(Im) & i End If XX = Replace(XX, , ) Œ conversion to french decimal format Œ XX = Replace(XX, ., ,) LambertW_I = XX Œ End If End Function J. Billiotte >Hi all, > I¹ve read with interest the many applications and real-life uses of the >nearly-elementary Lambert W function (which is the functional inverse >of y = x.exp(x), as you surely know), in particular how it can be used >to give explicit closed form solution to a number of fairly common >transcendental equations in terms of it, namely: >(1) x.exp(x) = a -> x = W(a) [of course] >(2) exp(x)/x = a -> x = -W(-1/a) [if i¹m not wrong, quoting from memory] >(3) x^x = a -> x = exp(W(log(a)) [also seen as x = log(a)/W(log(a)) ] >and a number of parameterized generalizations. My question is: >can the solution to the following equation be expressed in terms >of W as well ? : >(4) cos(x)/x = a >The mere fact that it heavily resembles (2) exp(x)/x = a, coupled with >the fact that trigonometric functions sin(x) and cos(x) can be expressed >in terms of the exp(x) function applied to complex arguments makes the >possibility of solving it using W rather plausible to me (or at least not >downright preposterous), but regrettably I¹ve been unable to derive such an >expression. If it does exist then the real root of (5) cos(x) = x between >0 and 1 could then be expressed exactly in terms of W. >Could any of you shed some light on whether it¹s possible at all >or else can this be proved impossible ? -- _____________________________________________________________ Joel Billiotte joel.billiotte@ensmp.fr Centre de Geologie de l¹Ingenieur Universite de Marne-la-Vallee - Batiment IFI Cite Descartes - 5,Bd Descartes - 77454 Marne-la-Vallee cedex2 tel : +33 1 49 32 90 89 fax: +33 1 49 32 91 28 _____________________________________________________________ Ce message est emis a titre personnel et n¹engage pas l¹Ecole des Mines de Paris// This message reþects my own opinion and does not engage the Ecole des Mines de Paris. _____________________________________________________________ === Subject: A very interesting problem I have got a seemingly simple but pretty complex problem, here is it: When I am putting N identical apples randomly in M baskets, What is the expected number of apples that will fall in baskets that has more than K apples? (Assume K <= N <= M) Can anyone help me out. It is critical for one of our computer research. Sailesh === Subject: Re: A very interesting problem > I have got a seemingly simple but pretty complex problem, here is it: > When I am putting N identical apples randomly in M baskets, What is the > expected number of apples that will fall in baskets that has more than K > apples? > (Assume K <= N <= M) > Can anyone help me out. It is critical for one of our computer research. > Sailesh My guess would be M*sum(prod(k/M,k=K+1,k=j),j=K+1,j=N). Which I think could be solved using a computer. The hypothesis could also be tested with a computer. === Subject: Re: A very interesting problem Originator: bellenot@haar >> I have got a seemingly simple but pretty complex problem, here is it: >> When I am putting N identical apples randomly in M baskets, What is the >> expected number of apples that will fall in baskets that has more than K >> apples? >> (Assume K <= N <= M) >> Can anyone help me out. It is critical for one of our computer research. >> Sailesh >My guess would be M*sum(prod(k/M,k=K+1,k=j),j=K+1,j=N). >Which I think could be solved using a computer. The hypothesis could >also be tested with a computer. Let K=1, N=2, your formula gives M*sum(prod(k/M,k=2..j),j=2..2) = M*(2/M) = 2. Probability always works with distinct balls into distinct boxes, then one `forgets the distinctness¹. Otherwise one thinks oo| o|o |oo are otherwise equally likely but it fact the middle one is twice as common as either of the others. One instead should use the sample space 11 -- 2 balls in boxes with > 1 ball 12 -- 0 21 -- 0 22 -- 2 which gives an expectation of 1 In general, there are M^N ways to assign n distinct balls to m distinct boxes. When K=N-1, only M of these `ways¹ have boxes with more than K elements so the expected value would be M*N/M^N (and checking yes we get 1 when M=N=2.) I didn¹t see a way to count the general case. -- Steven Bellenot http://www.math.fsu.edu/~bellenot Department of Mathematics real.email@line.below Florida State University bellenot at math.fsu.edu Tallahassee, FL 32306-4510 USA +1 (850) 644-7189 (FAX: 4053) === Subject: Re: A very interesting problem >>I have got a seemingly simple but pretty complex problem, here is it: >>When I am putting N identical apples randomly in M baskets, What is the >>expected number of apples that will fall in baskets that has more than K >>apples? >>(Assume K <= N <= M) >>Can anyone help me out. It is critical for one of our computer research. > My guess would be M*sum(prod(k/M,k=K+1,k=j),j=K+1,j=N). > Which I think could be solved using a computer. The hypothesis could > also be tested with a computer. The problem, as stated, is unclear. Does the Þrst apple that goes into a basket that eventually has more than (say) K=1 apple count? At the time the apples falls, the basket doesn¹t more than K apples. A clearer statement, but possibly not what you want is, What is the expected number of apples in baskets that have more than K apples? Creighton¹s guess can be shown to be incorrect. Consider the simple case of M=N=2, K=0. His formula predicts 2. The easily computed (and hopefully correct 8-) value is 3/2. 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worried ofjco > graham ofjco > javascript ofjco > bmagaj ofjco > bmagpam@yahoo.com ofjco > scolded ofjco > paulson@eyou.com ofjco > sslivens ofjco > zozzoz ofjco > skips ofjco > tempe ofjco > unlike ofjco > sigh ofjco > eadghn ofjco > ranting ofjco > agreement ofjco > thier ofjco > addressed ofjco > desch ofjco > bologna ofjco > carric ofjco > occt ofjco > ivillage ofjco > faith ofjco > afraid ofjco > beneath ofjco > revenge ofjco > laugh ofjco > hltlao@usa.net ofjco > whithout ofjco > wissen ofjco > shells ofjco > theyve ofjco > intyre ofjco > datenschutzdsb ofjco > processor ofjco > stalked ofjco > bmagnola ofjco > messageboards ofjco > vienello ofjco > lacessit ofjco > hobbies ofjco > incredibly ofjco > better ofjco > causes ofjco > operation ofjco > picw ofjco > bizarre ofjco > geraldine ofjco > warrior ofjco > newsguy ofjco > davebright ofjco > techie ofjco > more ofjco > areas ofjco > recognition ofjco > came ofjco > diet ofjco > continues ofjco > shutting ofjco > valse ofjco > godsonokoliuche ofjco > wishes ofjco > thief ofjco > grows ofjco > thoughts ofjco > update ofjco > taho@inbox.ru ofjco > nshore ofjco > jenarmstrong ofjco > nuances ofjco > terrorists ofjco > current ofjco > block ofjco > essays ofjco > qmail ofjco > irritate ofjco > newmoneya ofjco > verbs ofjco > therefore ofjco > happily ofjco > unmasked ofjco > cant ofjco > keywords ofjco > badtrans ofjco > asian ofjco > serving ofjco > surgam@hotmail.com ofjco > pelopasgr@yahoo.com ofjco > bycicle ofjco > opinions ofjco > prob ofjco > german ofjco > creditcardinformatie ofjco > admin ofjco > encounter ofjco > entry ofjco > soon ofjco > stuff ofjco > pathology ofjco > especially ofjco > unfortunately ofjco > abuse@excite.com ofjco > pity ofjco > pawns ofjco > love ofjco > messageid ofjco === Subject: Symbolic multiplication of matrices of unknown size by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i5IJiX410054; Hi all, Ok the subject sounds kinda ridiculous but more speciÞcally, this is what I want: I have two symbolic matrices F, E of size nxn each and whose entries are functions of n. I have to verify some property about the elements of P=F*E (say to check whether they are non-zero). I intended to use maple to accomplish this for ANY n, but I got stuck at the Þrst step: the statement F:=matrix(n,n,...) doesnt make any sense without giving Œn¹ an integer value. Is there any way I can go ahead deÞne such matrices F,E and perform their multiplication to obtain P (in maple or if there is any other application in which this can be done)? kranthi === Subject: Re: Symbolic multiplication of matrices of unknown size > Hi all, > Ok the subject sounds kinda ridiculous but more speciÞcally, this is > what I want: I have two symbolic matrices F, E of size nxn each and > whose entries are functions of n. I have to verify some property > about the elements of P=F*E (say to check whether they are non-zero). > I intended to use maple to accomplish this for ANY n, but I got stuck > at the Þrst step: the statement F:=matrix(n,n,...) doesnt make any > sense without giving Œn¹ an integer value. > Is there any way I can go ahead deÞne such matrices F,E and perform > their multiplication to obtain P (in maple or if there is any other > application in which this can be done)? > kranthi Maybe something like this: P := (i,j) -> (sum(F(i,k)*E(k,j),k=1..n)); and let E (and F) also be procedures where E(i,j) is the entry in row i column j of the matrix.