mm-1064 === Subject: are C_r fields C_r? any progress? Some time around Ô52, Serge Lang generalized Artins notion of quasi-algebraically closed fields (C_1) and defined C_r fields: A field k is said to satisfy property (C_r) when any homogeneous polynomial of degree d with coefficients in k in n+1 variables satisfying n >= d^r has a non-trivial zero. (Or, in arithmetic geometry terminology, any hypersurface in P^n over k with degree d has a k-point if n >= d^r.) (Note that a C_0 field is merely an algebraically closed field. Finite fields are C_1 by a theorem of Chevalley, the field of rational functions, or Laurent series, in r variables over an algebraically closed fields, are C_r by results of Tsen and Lang. The field Q_p of p-adic numbers is not C_2, contrary to what was believed for some time: Terjanian gave some counterexamples to that effect.) Under a certain technical assumption (viz., the existence of normic forms of order r and of all degrees), Lang proved that the existence of a non-trivial zero generalizes to families of polynomials, provided the inequality is satisfied for the sum of the degrees-to-the-r. Precisely, let us define: A field k is said to satisfy property (C_r) when any s homogeneous polynomials of degrees d_1,...,d_s with coefficients in k in n+1 variables satisfying n >= d_1^r + ... + d_s^r have a common non-trivial zero. (Or, in arithmetic geometry terminology, any intersection of s hypersurfaces in P^n over k with degrees d_1,...,d_s has a k-point if n >= d_1^r + ... + d_s^r.) (Im not sure that my notation is perfectly standard. Perhaps what other people call C_r is not exactly what is written above. But let us continue with this notation.) So Lang proves that, under a certain technical assumption which I wont recall, a C_r field is actually C_r. Later (around Ô57), Nagata showed that the technical assumption is not necessary provided all the d_j are equal. However, the question of whether the technical assumption is necessary in full generality remained open (as far as I know). My question is: has any progress been made on this question since then? Is there now a known example of a C_r field which is not C_r, or a proof that all C_r fields are C_r? Perhaps if we restrict to the (most interesting) r=1 case? At any rate, what would the educated guess be? Does it help in any way if we assume the (hypersurfaces defined by the) f_j to be in complete intersection (I cant see a way to reduce the general problem to that case, but it seems like a reasonable assumption to make)? (Basically, Id like to formulate the conjecture that, over a C_1 field k, any smooth projective (geometrically) separably rationally connected variety has a k-point: this is known when k is finite or when it is the field of rational functions over an algebraically closed field, by results of Esnault on the one hand, and Graber, Harris, de Jong and Starr on the other. At the very least, it is necessary, for the conjecture to be sensible, for the field to be C_1, so it would be embarrassing if there were already a known example of a C_1 field that is not C_1.) of Maths) and Nagata (in Kyoto University something-or-other) if necessary. -- David A. Madore (david.madore@ens.fr, http://www.dma.ens.fr/~madore/ ) === Subject: Paper published by Geometry and Topology Received-SPF: Received-SPF: pass (mailbox2.ucsd.edu: domain of gt@maths.warwick.ac.uk designates 137.205.233.100 as permitted sender) receiver=mailbox2.ucsd.edu; client_ip=137.205.233.100; envelope-from=gt@maths.warwick.ac.uk; Originator: bergv@math.uiuc.edu (Maarten Bergvelt) The following paper has been published: URL: http://www.maths.warwick.ac.uk/gt/GTVol8/paper37.abs.html Title: Commensurations of the Johnson kernel Author(s): Tara E Brendle, Dan Margalit Abstract: Let K be the subgroup of the extended mapping class group, Mod(S), generated by Dehn twists about separating curves. Assuming that S is a closed, orientable surface of genus at least 4, we confirm a conjecture of Farb that Comm(K), Aut(K) and Mod(S) are all isomorphic. More generally, we show that any injection of a finite index subgroup of K into the Torelli group I of S is induced by a homeomorphism. In particular, this proves that K is co-Hopfian and is characteristic in I. Further, we recover the result of Farb and Ivanov that any injection of a finite index subgroup of I into I is induced by a homeomorphism. Our method is to reformulate these group theoretic statements in terms of maps of curve complexes. Secondary: 20F38, 20F36 Keywords: Torelli group, mapping class group, Dehn twist Proposed: Walter Neumann Seconded: Shigeyuki Morita, Joan Birman Author(s) address(es): Department of Mathematics, Cornell University 310 Malott Hall, Ithaca, NY 14853, USA and Department of Mathematics, University of Utah 155 S 1440 East, Salt Lake City, UT 84112, USA Email: brendle@math.cornell.edu, margalit@math.utah.edu === Subject: Teiji Takagi and principalization Originator: bergv@math.uiuc.edu (Maarten Bergvelt) I couldnt get an answer on sci.math, so I hope this is suitable for here: Some sources say that Takagi proved not only that the maximal unramified extension of a L/K number field K has a Galois group corresponding to the class group, but that principalization occurs in this field--all the ideals of K extend to principal ideals of L. Others say it had to wait for Artin and Furtwangler to get the proof of this. Does anyone have the straight dope? === Subject: Re: Teiji Takagi and principalization Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > I couldnt get an answer on sci.math, so I hope this is suitable for here: > Some sources say that Takagi proved not only that the maximal > unramified extension of a L/K number field K has a Galois group > corresponding to the class group, but that principalization occurs in > this field--all the ideals of K extend to principal ideals of L. > Others say it had to wait for Artin and Furtwangler to get the proof > of this. Does anyone have the straight dope? in Cassells and Froelich, _Algebraic Number Theory_ (Thompson, 1967). He says (p. 273): With the help of this [general reciprocity] law, Artin could also reduce the principal divisor theorem, enunciated by Hilbert and not yet proved by Takagi, to a pure group-theoretical proposition, which was then proved by Furtwaengler. Ive looked at Artins paper, Idealklassen in Oberkoerpern und allgemeines Reziprozitaetsgesetz (Collected Papers 159-164), and its clear that he did not know of any earlier proof. William C. Waterhouse Penn State === Subject: Normal subgroups of surface groups? Epigone-thread: thulgonstrix Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Let G be a surface group, and H be a non-trivial normal subgroup of G. How can I prove that H is of finite index in G ? (This is a conjecturally true for any one-relator group) === Subject: Re: Normal subgroups of surface groups? Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Let G be a surface group, and H be a non-trivial normal subgroup of G. > How can I prove that H is of finite index in G ? > (This is a conjecturally true for any one-relator group) You cant, because this is not true. For instance, any surface group (other than the 2-sphere or projective plane) has a surjection to the integers, whose kernel is of infinite index. The simplest of course would be the torus, where the obvious Z subgroup is normal, and of infinite index. DR === Subject: Re: Normal subgroups of surface groups? Epigone-thread: thulgonstrix Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Youre right. Here is the restatement of the question: Show that every finitely generated normal subgroup of a non-abelian surface group (with or without boundary) is of finite index. JJ >> Let G be a surface group, and H be a non-trivial normal subgroup of G. >> How can I prove that H is of finite index in G ? >> (This is a conjecturally true for any one-relator group) >You cant, because this is not true. For instance, any surface group >(other than the 2-sphere or projective plane) has a surjection to the >integers, whose kernel is of infinite index. The simplest of course >would be the torus, where the obvious Z subgroup is normal, and of >infinite index. === Subject: rapidly converging rational sqrt Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Below is a description of an algorithm which, with each iteration, will double the number of significant digits in the computation of a rational square root approximation. I do not know if this algorithm is new, but I found it interesting nonetheless. Lisp code for implementing this algorithm can be found at: http://thegreves.com/david/sqrt/sqrt.html If by convention we say that: s = isqrt(C) d = C - s^2 Then the square root of C can be expressed as as the infinite continued fraction: d s + ----------------- d 2s + ------------ d 2s + ------- 2s + .. We designate the tail of this continued fraction using e(n) = nth error term and we say that the nth error term of the continued fraction representation has the form: d e(n) = ---------- 2s + e(n+1) Although we sometimes drop the subscript on the error term for notational convenience. A pretty good first order approximation for a square root can be computed as follows (even allowing e(1) to be zero): d sqrt(C) ~= s + -------- 2s + e(1) Without proof, we claim that a generalized expression for a partial evaluation of our continued fraction can be represented as: A + Be sqrt(C) = s + ------- C + De It is easy to see that the first order approximation given above is an instance of this expression when A = d, B = 0, C = 2s, D = 1. The generalized representation is useful for representing the result of evaluating some number of sucessive terms in the continued fraction representation. Assuming that the above representation is the result of evaluating n terms of the continued fraction for sqrt(C), then the n+1 term would be computed by substituting the next error term into the error expression in the representation. A + B(d/(2s + e)) s + --------------- C + D(d/(2s + e)) A(2s + e) + Bd s + -------------- C(2s + e) + Dd (A2s + Bd) + Ae s + ----------------- (C2s + Dd) + Ce Which, we observe, is once again in the general representational form. If the evaluation of the first n terms produced A + Be ------ C + De and the evaluation of the next m terms were to produce W + Xe ------ Y + Ze Then the evaluation of the first n+m terms would be: W + Xe A + B(------) Y + Ze A(Y + Ze) + B(W + Xe) ------------ = --------------------- W + Xe C(Y + Ze) + D(W + Xe) C + D(------) Y + Ze (AY + BW) + (AZ + BX)e ---------------------- (CY + DW) + (CZ + DX)e Because the continued fraction representation of the square root is uniform, the evaluation of any n sucessive terms will always produce the same result. We can take advantage of this fact to refine our first order approximation by substituting A,B,C, and D in for W,X,Y and Z in the above expression. The will double the number of terms we have evaluated. Of course, this procedure can be repeated again and again, with each iteration of the algorithm doubling the number of significant digits in our representation. Here is an example run computing the sqrt of 1973 for 1 to 10 iterations of the algorithm. Note that after iteration 6 we have more than 100 significant digits. 1 : 44.41845521141241485670222336460609176198432078139056676519727 541447114766739 49363834982650044981364863 2 : 44.41846462881467219505665982727838995343272680250852496023827 788068578570854 49370627285274658896791048 3 : 44.41846462902561876427524312325572040240985914842970903422305 486056596616470 16608600579504981095676558 4 : 44.41846462902561876438107965740906053956833272941354961112468 508573375443791 08155113103260155775597834 5 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 26805283703947239542091268 6 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 7 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 8 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 9 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 10 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 Dave === Subject: Re: rapidly converging rational sqrt Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Below is a description of an algorithm which, with > each iteration, will double the number of significant > digits in the computation of a rational square root > approximation. > Here is an example run computing the sqrt of 1973 for 1 to 10 > iterations of the algorithm. Note that after iteration 6 we have > more than 100 significant digits. > 1 : 44.4184 | 55211412414856702223364606091761984320781390566765197275414471 147667394936383 4982650044981364863 > 2 : 44.41846462 | 88146721950566598272783899534327268025085249602382778806857857 085449370627285 274658896791048 > 3 : 44.418464629025618764 | 27524312325572040240985914842970903422305486056596616470166086 005795049810956 76558 > 4 : 44.4184646290256187643810796574090605395 | 68332729413549611124685085733754437910815511310326015577559783 4 > 5 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 | 26805283703947239542091268 > 6 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 > 7 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 > 8 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 > 9 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 > 10 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 43686917147360058911830087 I dont wish to denigrate your algorithm unduly, but square root algorithms with this rate of convergence have been known for millennia. The iteration x_{n+1} = 1/2 (x_n + C/x_n), which is believed to have been known to the ancient Babylonians, yields the following output if we take C = 1973 and start it at 44 (since your algorithm presupposes we know isqrt(C) this seems a fair comparison.) I have added vertical bars to indicate the correct portion of each decimal expansion; I have done the same for the quoted output from your algorithm above. 1 : 44.4 | 20454545454545454545454545454545454545454545454545454545454545 454545454545454 54545454545454545455 2 : 44.4184646 | 73597060396753412870066745738272983092630061164213121235377566 920160933975208 72578432056559 3 : 44.418464629025618786 | 74355237890368384233529296754366053971048032852669727540929455 186628564646861 810 4 : 44.4184646290256187643810796574090605 | 45224167043182294394699285292428539812171115734428794096905254 72 5 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066 | 216106505407447827420345844 6 : 44.41846462902561876438107965740906053959497442704659903610246 205761940066180 436869171473600589118301... So like your algorithm this also produces 100 digits in 6 iterations, and it is easy to prove (since this is a special case of Newtons method for a general function) that convergence is quadratic in general. In short, your algorithm is interesting but it doesnt outperform the standard algorithms for square roots. Yours, David Loefßer (student, Trinity College, University of Cambridge, UK) === Subject: Laplaces method and a Double Integral Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Hello all, I wonder if anyone can show me how to evaluate the asymptotics (as N) gets large of this integral: int_0^R int_{-u}^0 r^3 [1-c_1 (c_2 n - c_3 r^2)^2]^N dn dr Ive looked in several books on Laplaces method (Erdelyi, Wong, Bleistein). If we define f(n,r) = -log[1-c_1 (c_2 n - c_3 r^2)^2] Then the integral is of the form int int r^3 exp(-N f(n,r)) dn dr where f takes on its mininum value of 0 at the pt (0,0). Unfortunately the Hessian at that pt is zero....because f_{rr}(0,0) = 0. All the explicit expansions I have seen require that the Hessian be non-zero. Does anyone know of another reference to try that might have this worked out? Jim PS: Im not a mathematician....just a humble plodding engineer! === Subject: Re: Laplaces method and a Double Integral Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >Hello all, >I wonder if anyone can show me how to evaluate the asymptotics (as N) >gets large of this integral: >int_0^R int_{-u}^0 r^3 [1-c_1 (c_2 n - c_3 r^2)^2]^N dn dr >Ive looked in several books on Laplaces method (Erdelyi, Wong, Bleistein). >If we define >f(n,r) = -log[1-c_1 (c_2 n - c_3 r^2)^2] >Then the integral is of the form int int r^3 exp(-N f(n,r)) dn dr >where f takes on its mininum value of 0 at the pt (0,0). Unfortunately >the Hessian at that pt is zero....because f_{rr}(0,0) = 0. All the >explicit expansions I have seen require that the Hessian be non-zero. Do the substitutions t=r^2, s=-n help? The integral becomes (1/2)int_0^{R^2} int_0^u t [1 - c_1(c_2 s + c_3 t)^2]^N ds dt For the corresponding f(s,t), a quick calculation gave me f_{tt}(0,0) not= 0, but you should check that yourself. Dan -- Dan Luecking Department of Mathematical Sciences University of Arkansas Fayetteville, Arkansas 72701 To reply by email, change Look-In-Sig to luecking === Subject: Re: Laplaces method and a Double Integral Originator: israel@math.ubc.ca (Robert Israel) Dans substitutions are OK, a nice trick, but I wonder if the origin (0,0) is the main contributing point. I expect also contributions from the boundaries, but this depends on the values of c_1, c_2, c_3, u and R. Nico M. Temme, http://homepages.cwi.nl/~nicot/ C W I: Centrum voor Wiskunde en Informatica Kruislaan 413, NL-1098 SJ Amsterdam Tel +31 20 592 4240 P.O. Box 94079, NL-1090 GB Amsterdam Fax +31 20 592 4199 === > Subject: Re: Laplaces method and a Double Integral >Hello all, >I wonder if anyone can show me how to evaluate the asymptotics (as N) >gets large of this integral: >int_0^R int_{-u}^0 r^3 [1-c_1 (c_2 n - c_3 r^2)^2]^N dn dr >Ive looked in several books on Laplaces method (Erdelyi, Wong, Bleistein). >If we define >f(n,r) = -log[1-c_1 (c_2 n - c_3 r^2)^2] >Then the integral is of the form int int r^3 exp(-N f(n,r)) dn dr >where f takes on its mininum value of 0 at the pt (0,0). Unfortunately >the Hessian at that pt is zero....because f_{rr}(0,0) = 0. All the >explicit expansions I have seen require that the Hessian be non-zero. > Do the substitutions t=r^2, s=-n help? The integral becomes > (1/2)int_0^{R^2} int_0^u t [1 - c_1(c_2 s + c_3 t)^2]^N ds dt > For the corresponding f(s,t), a quick calculation gave me > f_{tt}(0,0) not= 0, but you should check that yourself. > Dan > -- > Dan Luecking Department of Mathematical Sciences > University of Arkansas Fayetteville, Arkansas 72701 > To reply by email, change Look-In-Sig to luecking === Subject: Re: Laplaces method and a Double Integral Originator: israel@math.ubc.ca (Robert Israel) Re: Laplaces method and a Double Integral Since Nico thinks that the asymptotic form of the integral has also contributions not only from the neighborhood of (0,0), I derived an asymptotic approximation given in the following. It shows that only the neighborhood of (0,0) is important. Ciao Karl Breitung Schellingstr. 21 D-80799 Munich, Germany AN ASYMPTOTIC APPROXIMATION FOR THE TWO-DIMENSIONAL INTEGRAL: ,- R ,- 0 3 [ 2 2 ]N I(N)= | | r [ 1-a(bn-cr ) ] dn dr, N --> oo - 0 - -u Here we write instead of the original form: a=c , b=c , c=c . 1 2 3 1/2 2 1/2 Making the substitutions r --> v=a cr and n --> z=-a bn transforms this into: 2 ,- cR ,- bu / v 3/2 [ 2 ]N d r d n I(N)=- | | | ----- | [ 1-(z+v) ] --- ---- dz dv - 0 - 0 | 1/2 | d v d z a c / 2 ,- cR ,- bu / v 3/2 [ 2 ]N 1 1/2 -1/2 1 = | | | ----- | [ 1-(z+v) ] -(a cv) ----- dz dv= - 0 - 0 | 1/2 | 2 1/2 a c / a b 1/2 2 1/2 ,- a cR ,- a bu [ 2 ]N K | | v [ 1-(z+v) ] dz dv - 0 - 0 3/2 2 -1 with K=(2a bc ) . Now we will consider only the integral over a triangle 1/2 2 1/2 (0,0), (d,0) and (0,d) with 00 and K >0 are constants. Then: 1 2 ,- d ,- d-v [ 2 ]N I(N) sim K | | v [ 1-(z+v) ] dz dv - 0 - 0 In this triangle, we make the variable transformation (v,z) --> (w,y) with w=z+v, y=z-v. Then: ,- d [ ,- w w-y [ 2 ]N ] I(N) sim K | | | --- [ 1-w ] |det (J(w,y))| dy | dw= - 0 [ - -w 2 ] ,- d [ ,- w w-y [ 2 ]N ] K | | | --- [ 1-w ] dy | dw (*) - 0 [ - -w 2 ] J(w,y) is the Jacobian of the inverse transformation with its determinant equal to 1/2. The integral in the brackets is: ,- w w-y [ 2 ]N [ 2 ]N ,- w / w y [ 2 ]N ,- w w | --- [ 1-w ] dy=[ 1-w ] | | - - - | dy=[ 1-w ] | - dy= - -w 2 - -w 2 2 / - -w 2 [ 2 ]N 2 [ 1-w ] w 2 Inserting this into equ. (*) and writing f(w)=log (1-w ) gives: ,- d 2 I(N) sim K | w exp (Nf(w)) dw - 0 This we can evaluate using the generalized Laplace method (derived in [2], p. 37, see also [1], p. 48). Here we derive the result directly by approximating 2 2 f(w) by its second order Taylor expansion at zero, i.e. f(w)=-2w +o(w ) and 2 then making the substitution w --> x=2N w . This gives: ,- dN x -x dw ,- dN x -x 1 -1/2 I(N)sim K | -- e --dx= K | -- e ------ x dx = - 0 2N dx - 0 2N +--+ 2|2N -3/2 K ,- dN 1/2 -x N ----- | x e dx +-+ - 0 4|2 For this we get the asymptotic form replacing dN by oo: -3/2 K ,- oo 1/2 -x -3/2 K I(N)sim N ----- | x e dx sim N ----- Gamma(3/2)= +-+ - 0 +-+ 4|2 4|2 +---+ +---+ -3/2 K |pi -3/2 |pi N --- ----- = N ---------------- , N --> oo +-+ 2 +-+ 3/2 2 4|2 16|2 a bc If necessary this approximation can be refined by deriving a second term in the asymptotic expansion of I(N). Bibliography: [1] K. Breitung. Asymptotic Approximations for Probability Integrals. Springer, Berlin, 1994. Lecture Notes in Mathematics, Nr.1592. [2] A. Erdelyi. Asymptotic Expansions. Dover, New York, 1956. === Subject: groups Epigone-thread: geikreizhan Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Let $G$ be a finitely generated subgroup of Homeo(E), where $E$ is an arbitrary metric space, such that all orbits of $G$ are finite. We suppose that there exists $pin N$ such that for all $gin G$ we have $g^p = 1$. The group $G$ is it finite? === Subject: Re: groups Epigone-thread: geikreizhan Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >Let $G$ be a finitely generated subgroup of Homeo(E), where $E$ is an >arbitrary metric space, such that all orbits of $G$ are finite. >We suppose that there exists $pin N$ such that for all $gin G$ we >have $g^p = 1$. The group $G$ is it finite? The answer is no. This was essentially known as the Burnside problem, and solved in the negative by Novikov and Adjan in 1968: there exist infinite two-generator groups identically satisfying x^n=1 with n any sufficiently large odd integer. (The additional condition stipulating that G should be a subgroup of Homeo(E) adds nothing to the problem, for every group can be so represented: choose E = G, with the discrete metric where distinct points always have distance 1, and let G operate on itself by left translations; these, of course, are homeomorphisms in the given metric.) === Subject: Re: groups Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >>Let $G$ be a finitely generated subgroup of Homeo(E), where $E$ is >>an arbitrary metric space, such that all orbits of $G$ are finite. >>We suppose that there exists $pin N$ such that for all $gin G$ we >>have $g^p = 1$. The group $G$ is it finite? >The answer is no. This was essentially known as the Burnside >problem, and solved in the negative by Novikov and Adjan in 1968: >there exist infinite two-generator groups identically satisfying >x^n=1 with n any sufficiently large odd integer. (The additional >condition stipulating that G should be a subgroup of Homeo(E) adds >nothing to the problem, for every group can be so represented: choose >E = G, with the discrete metric where distinct points always have >distance 1, and let G operate on itself by left translations; these, >of course, are homeomorphisms in the given metric.) But among the hypotheses you have that the orbits of G are finite. This implies that G is residually finite and thus, by Zelmanovs positive solution to the restricted Burnside problem, G is indeed finite. Andreas === Subject: Re: groups Epigone-thread: geikreizhan Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Sorry for my oversight! Of course, Andreas is right; I overlooked the requirement that the orbits should be finite. Regretfully, Peter === Subject: Re: groups Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >>Let $G$ be a finitely generated subgroup of Homeo(E), where $E$ is an >>arbitrary metric space, such that all orbits of $G$ are finite. >>We suppose that there exists $pin N$ such that for all $gin G$ we >>have $g^p = 1$. The group $G$ is it finite? >The answer is no. This was essentially known as the Burnside >problem, and solved in the negative by Novikov and Adjan in 1968: >there exist infinite two-generator groups identically satisfying x^n=1 >with n any sufficiently large odd integer. (The additional condition >stipulating that G should be a subgroup of Homeo(E) adds nothing to >the problem, for every group can be so represented: choose E = G, with >the discrete metric where distinct points always have distance 1, and >let G operate on itself by left translations; these, of course, are >homeomorphisms in the given metric.) But the orbits of G will not be finite in this example. G having a faithful representation as mappings of a set with all orbits finite means that G has a collection of normal subgroups of finite index whose intersection is the identity. I dont know if this is true for any of the known examples in the Burnside problem. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Fall Pacific NW Geometry Seminar at U of Oregon Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Final announcement: PACIFIC NORTHWEST GEOMETRY SEMINAR University of Oregon Eugene, OR SCHEDULE Saturday, November 6 10:30 - 11:00 Morning Reception 11:00 - 12:15 Charles Doran (University of Washington) Mirror Symmetry, K-Theory, and Toric Geometry 12:15 - 2:00 Lunch 2:00 - 3:15 Mutao Wang (Columbia) Mean curvature ßows of Lagrangian submanifolds 3:15 - 4:00 Break 4:00 - 5:15 Lei Ni (UCSD) Ancient Solutions of the K.8ahler-Ricci Flow 7:00 Party at Botvinniks Sunday, November 7 8:30 - 9:00 Morning Reception 9:00 - 10:15 John Lott (University of Michigan) Ricci curvature for metric-measure spaces 10:15 - 10:45 Break 10:45 - 12:00 David Auckly (Kansas State University) The structure of maps into homogenous spaces and the Faddeev and Skyrme models Note: Each speakers time allotment includes 15 minutes for a discussion of Open Problems related to his topic. The talks will be in 110 Fenton Hall (D-7 on the campus map). The receptions and breaks will be right outside 110 Fenton. ----------------------------------------------------------- For general information about the PNGS, visit the PNGS web site: http://www.math.washington.edu/~lee/PNGS It contains up-to-date information about this meeting, travel and lodging information, general information about the PNGS, and a historical record of all PNGS meetings and speakers. ----------------------------------------------------------- For more information about this meeting, contact the organizers: Boris Botvinnik (botvinn@math.uoregon.edu) Jim Isenberg (jim@newton.uoregon.edu) === Subject: Non linear hyperbolic PDEs : ill posedness ? Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Hi everybody I am studying the well-posedness of the Cauchy problem for systems of PDE of the form : d U/dt = A d U/dx, with unknown vector U(x,t) In the linear case (A is a matrix function of (x,t)), it is well known that the problem is well-posed (existence of a unique solution depending continuously on the initial data) iff A is diagonalisable with real eigenvalues for all x and t. Here is my question : In the non linear case (A is a function of U, x and t) does one know such a system where the matrix A is not always diagonalisable but wich is still well-posed ? Michael === Subject: Partitioning 4 space with ultraskew lines, and the three body problem. Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Suppose we are in 4 space. Two lines are skew if they do not lie in the same plane(2 space). Skewness is a binary relation on lines. Now two skew lines will lie in the same 3 space. But it is possible for three lines to not lie in the same 3 space. This is a ternary relation on lines. I dont know if theres a name for this relation, so lets call it ultraskewness until we find out. 1) Can one foliate, or at least partition, 4 space with lines that are tripletwise ultraskew? I mean EACH 3 line subset of the partition must not lie in the same 3 space. 2) In the three-body problem, one could approximate the trajectory of each of the 3 masses as a straight lines until they became close enough to each other for their gravitation to have an appreciable effect. I think much of the work on this problem assumes all three trajectories lie in the same plane.(The restricted 3-body problem.) But some work has been where the trajectories are not coplanar. Wouldnt it be fun to explore the three-body problem when the trajectories are not cospatial? Richard Peterson, CSU Sacramento === Subject: Re: Partitioning 4 space with ultraskew lines, and the three body problem. ath: nntpswitch.com Originator: israel@math.ubc.ca (Robert Israel) > Suppose we are in 4 space. Two lines are skew if they do not lie in > the same plane(2 space). Skewness is a binary relation on lines. Now > two skew lines will lie in the same 3 space. But it is possible for > three lines to not lie in the same 3 space. This is a ternary relation > on lines. I dont know if theres a name for this relation, so lets > call it ultraskewness until we find out. > 1) Can one foliate, or at least partition, 4 space with lines that > are tripletwise ultraskew? I mean EACH 3 line subset of the partition > must not lie in the same 3 space. > 2) In the three-body problem, one could approximate the trajectory > of each of the 3 masses as a straight lines until they became close > enough to each other for their gravitation to have an appreciable > effect. I think much of the work on this problem assumes all three > trajectories lie in the same plane.(The restricted 3-body problem.) > But some work has been where the trajectories are not coplanar. > Wouldnt it be fun to explore the three-body problem when the > trajectories are not cospatial? > Richard Peterson, CSU Sacramento I dont know about the three-body problem, or about foliations, but I think one can partition 4-space into ultraskew lines without much trouble. The proof is via a transfinite induction of c (the cardinality of the continuum) many steps; at step k one considers the k-th point p in a fixed enumeration of 4-space, and one has already constructed a collection L of |k|-many (fewer than c) lines. If p is in the union of L there is nothing to do. Otherwise one need only find a unit tangent vector u at p so that the line through p in direction u is (a) disjoint from each line in L and (b) ultraskew to every pair of lines in L. Since p lies on no line in L (a) is satisfied so long as u does not lie in any plane containing both p and a line in L. Also, (b) is satisfied so long as u does not lie in any translate containing p of an (affine) 3-space generated by a pair of lines in L. So we need a point on the unit 3-sphere in R^4 not lying in any of a collection of fewer than c many subspaces of R^4 of dimension at most 3. Without loss of generality we may assume all the subspaces have dimension 3, so each has a perp that meets the 3-sphere in at most 2 points. As we have fewer than c subspaces, there is a point w on the 3-sphere not in the perp of any of them. Then the perp P of the line spanned by w is a 3-dimensional space meeting each of the spaces we want to avoid in a subspace of dimension at most 2. Thus it suffices to find a point on the intersection of the unit 3-sphere with P (i.e., a unit 2-sphere) not in any of a collection of fewer than c subspaces of P of dimension at most 2. By a similar argument we can drop the dimension once again, and then we need to find a point on the unit circle avoiding fewer than c lines through the origin. As the circle has c points, and each line meets it in two point, that is easy. Well, its late and Im hurrying, but I think this argument holds water. Bob Beaudoin === Subject: Integral recurrence relation Originator: bergv@math.uiuc.edu (Maarten Bergvelt) I have encountered the following integral in some research in the physical sciences int |u-A|^(2a) |u-B|^(2b) Exp[-|u|^2] du where u, A and B are cartesian vectors in 3 dimensions and the integral is to performed over all space. This seems like quite a straightforward integral but the best I can do is to write it as a triple infinite series in A^2, B^2 and |A-B|^2 (which quickly truncates, depending on the values of a and b). I was wondering if anyone has any suggestions as how I might produce a more useful formulation. Even more useful would be a suggestion as to how I might derive a recurrence relation to generate integrals of higher values of a and b or if it is possible to prove or disprove the existence of such a relation. Darragh === Subject: Re: Integral recurrence relation Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >I have encountered the following integral in some research in the >physical sciences > int |u-A|^(2a) |u-B|^(2b) Exp[-|u|^2] du >where u, A and B are cartesian vectors in 3 dimensions and the >integral is to performed over all space. This seems like quite a >straightforward integral but the best I can do is to write it as a >triple infinite series in A^2, B^2 and |A-B|^2 (which quickly >truncates, depending on the values of a and b). I was wondering if >anyone has any suggestions as how I might produce a more useful >formulation. Let your integral be F(a,b) (for nonnegative integers a,b). Consider the exponential generating function f(s,t) = sum_{a=0}^infinity sum_{b=0}^infinity F(a,b) s^a t^b/(a! b!) = int_{R^3} exp(s |u-A|^2) exp(t |u-B|^2) exp(-|u|^2) du = int_{R^3} exp(-(1-s-t) |u|^2 - 2 u.(sA+tB) + s|A|^2 + t|B|^2) du = exp(s|A|^2 + t|B|^2 + |sA+tB|^2/(1-s-t)) int_{R^3} exp(-(1-s-t) |u-(sA+tB)/sqrt(1-s-t)|^2) du = exp(s|A|^2 + t|B|^2 + |sA+tB|^2/(1-s-t)) (pi/(1-s-t))^(3/2) for |s|+|t| < 1. Then F(a,b) can be obtained from the coefficients of the bivariate Taylor series for f(s,t) around (0,0). Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Paper published by Geometry and Topology Originator: bergv@math.uiuc.edu (Maarten Bergvelt) The following paper has been published: URL: http://www.maths.warwick.ac.uk/gt/GTVol8/paper38.abs.html Title: Noncommutative localisation in algebraic K-theory I Author(s): Amnon Neeman, Andrew Ranicki Abstract: associative rings, a long exact sequence in algebraic K-theory. The main result goes as follows. Let A be an associative ring and let A-->B be the localisation with respect to a set sigma of maps between finitely generated projective A-modules. Suppose that Tor_n^A(B,B) vanishes for all n>0. View each map in sigma as a complex (of length 1, meaning one non-zero map between two non-zero objects) in the category of perfect complexes D^perf(A). Denote by the thick subcategory generated by these complexes. Then the canonical functor D^perf(A)-->D^perf(B) induces (up to direct factors) an equivalence D^perf(A)/--> D^perf(B). As a consequence, one obtains a homotopy fibre sequence K(A,sigma)-->K(A)-->K(B) (up to surjectivity of K_0(A)-->K_0(B)) of Waldhausen K-theory spectra. consequences of the main theorem in a form more suitable for the applications to surgery. For example if, in addition to the vanishing of Tor_n^A(B,B), we also assume that every map in sigma is a monomorphism, then there is a description of the homotopy fiber of the map K(A)-->K(B) as the Quillen K-theory of a suitable exact category of torsion modules. Secondary: 19D10, 55P60 Keywords: Noncommutative localisation, $K$--theory, triangulated category Proposed: Bill Dwyer Seconded: Thomas Goodwillie, Gunnar Carlsson Author(s) address(es): Centre for Mathematics and its Applications The Australian National University Canberra, ACT 0200, Australia and School of Mathematics, University of Edinburgh Edinburgh EH9 3JZ, Scotland, UK Email: Amnon.Neeman@anu.edu.au, a.ranicki@ed.ac.uk === Subject: monoidal enriched natural transformations Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Hi. I have a problem which, it seems to me, requires the notion of enriched natural transformation between enriched monoidal functors, but I havent been able to find a good reference for it. Ive taken a look at the books of Borceux (Handbook of categorical algebra), Kelly (Basic notions of enriched category) and the appendix in Levines Mixed motives, but they all end just where I need them, or even long before that point. More precisely, the situation Ive encountered seems to be the following. I have: - Two monoidal (symmetric) categories: V and W . - A couple of monoidal (symmetric) functors between them: S, T : V ---> W - A monoidal natural transformation between S and T : And here is where my problems begin. There is a well-known notion of what is a V-functor between V-categories and what a V-natural transformation is. My first need is to understand what an S-functor between a V-category C and a W-category D should be: F : C ---> D I didnt find this thing in the literature, but I expect it ought to be something like a V-functor, but with a family of morphisms in W lambda_{XY} : S[X,Y] ---> [FX,FY] for every pair of objects X, Y in C . (Here the square brackets [,] stand for the objects in V and W of morphisms of C and D , and Im leaving aside units and commutative isomorphisms for the moment.) This seems reasonable to me, since (a) is the situation I have in the real world and (b) if I put S = id_V , I find the definition of a V-functor. Next, I would need the notion of an omega - natural transformation and I think this should be something like a V-natural transformation between an S-functor F : C ---> D and a T-functor G: C ---> D, but placing at the beginning of the commutative diagram which defines a V-natural transformation an arrow like omega_{[X,Y]} : S[X,Y] ---> T[X,Y] . Assuming that this is ok, I should also need to understand what might be the definition of a monoidal omega - natural transformation. That is to say, C is a monoidal V-category, D is a monoidal W-category, F is a monoidal S-functor and G a monoidal T-functor: what is a monoidal natural transformation between F and G , over Ive drawn a couple of commutative diagrams that should appear in the definition of such a construct, but I feel I could be forgetting a dozen more. Any references for it? Unfortunately for me, Kellys book ends before this point: it explicitely says: is our decision not to discuss the Ôchange of base-category given by a symmetric monoidal functor V ---> W. Has someone else done the job after Kellys book? Agust.92 Roig === Subject: Re: monoidal enriched natural transformations Epigone-thread: tixslongcax Content-Length: 6836 Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >Hi. >I have a problem which, it seems to me, requires the notion of >enriched natural transformation between enriched monoidal functors, >but I havent been able to find a good reference for it. >Ive taken a look at the books of Borceux (Handbook of categorical >algebra), Kelly (Basic notions of enriched category) and the appendix >in Levines Mixed motives, but they all end just where I need them, >or even long before that point. >More precisely, the situation Ive encountered seems to be the >following. I have: >- Two monoidal (symmetric) categories: > V and W . >- A couple of monoidal (symmetric) functors between them: > S, T : V ---> W >- A monoidal natural transformation between S and T : >And here is where my problems begin. >There is a well-known notion of what is a V-functor between >V-categories and what a V-natural transformation is. >My first need is to understand what an S-functor between a >V-category C and a W-category D should be: > F : C ---> D >I didnt find this thing in the literature, but I expect it ought to >be something like a V-functor, but with a family of morphisms in W > lambda_{XY} : S[X,Y] ---> [FX,FY] >for every pair of objects X, Y in C . (Here the square brackets >[,] stand for the objects in V and W of morphisms of C and D >, and Im leaving aside units and commutative isomorphisms for the >moment.) >This seems reasonable to me, since (a) is the situation I have in the >real world and (b) if I put S = id_V , I find the definition of a >V-functor. >Next, I would need the notion of an omega - natural transformation >and I think this should be something like a V-natural transformation >between an S-functor F : C ---> D and a T-functor G: C ---> D, >but placing at the beginning of the commutative diagram which defines >a V-natural transformation an arrow like > omega_{[X,Y]} : S[X,Y] ---> T[X,Y] . >Assuming that this is ok, I should also need to understand what might >be the definition of a monoidal omega - natural transformation. >That is to say, C is a monoidal V-category, D is a monoidal >W-category, F is a monoidal S-functor and G a monoidal T-functor: >what is a monoidal natural transformation between F and G , over >Ive drawn a couple of commutative diagrams that should appear in the >definition of such a construct, but I feel I could be forgetting a >dozen more. Any references for it? >Unfortunately for me, Kellys book ends before this point: it >explicitely says: is our decision not to discuss the Ôchange of >base-category given by a symmetric monoidal functor V ---> W. Has >someone else done the job after Kellys book? If you havent done so already, I recommend that you get on the categories mailing list: categories@mta.ca where you would surely get a reply and advice about the literature. Alternatively, you might write Max Kelly (at the University of Sydney) or Ross Street (Macquarie University) directly. I wish I had suitable references at hand, but here are some remarks on your query. First, a monoidal functor S: V --> W induces a 2-functor S_{*}: V-Cat --> W-Cat, making straightforward use of the monoidal structure on S. If C is a V-category and D is a W-category, then what you call an S-functor is undoubtably the same as a W-functor of the form F: S_{*}C --> D. Next, a monoidal natural transformation omega: S --> T induces a 2-natural transformation between 2-functors omega_{*}: S_{*} --> T_{*} and in particular provides, for each V-category C, a W-functor of the form omega_{*}(C): S_{*}C --> T_{*}C This too is straightforward, using just the data and equations for an m.n.t. Then what you call an omega-natural transformation from F to G is undoubtably the same as a W-transformation of the form F --> G(omega_{*}(C)) where F: S_{*}C --> D and G: T_{*}C --> D are W-functors, and the target on the right is a composite of W-functors. Now to define a monoidal omega-natural transformation, you want to do a jazzed-up version of the above definitions. Heres what youll need (minimally): -- V, W braided monoidal categories -- S, T braided monoidal functors of the form V --> W -- omega a m.n.t. of the form S --> T (You can of course replace braided by symmetric, but you lose some generality in doing so.) Since V is braided monoidal, V-Cat is a monoidal 2-category, and monoidal V-categories are the same as (pseudo-)monoids in V-Cat as a monoidal 2-category. Indeed, the 2-category Mon(V-Cat) whose objects are monoidal V-categories, whose objects are monoidal V-functors, and whose 2-cells are monoidal V-transformations, is definable purely in terms of the monoidal 2-category structure on V-Cat, and therefore, the desired change of base induced by S: V --> W, Mon(V-Cat) --> Mon(W-Cat), requires only a monoidal 2-functor S_{*}: V-Cat --> W-Cat to get off the ground. The point of demanding that S be braided monoidal is so that the 2-functor S_{*} is in fact monoidal. So: under these hypotheses, a monoidal S-functor (to use your terminology) should be the same as a monoidal W-functor of the form F: S_{*}C --> D where C is a monoidal V-category and D is a monoidal W-category. Finally, if omega: S --> T is an m.n.t., there is a monoidal W-functor omega_{*}(C): S_{*}C --> T_{*}D and what you call a monoidal omega-natural transformation should just be a monoidal W-transformation of the form F --> G(omega_{*}(C)). Notice that it is unnecessary to write down a whole bunch of commutative diagrams to define these constructs: the data and axioms inherent in ordinary enriched notions and in braided monoidal notions do the work for you. However, this approach does involve some machinery of higher-dimensional categories (monoidal 2-categories, monoidal 2-functors), which is probably why Kelly didnt touch this in his book -- the relevant notions hadnt yet been formulated properly. If you want to follow up on this machinery, you might want to look at Coherence for Tricategories by Gordon, Power & Street. The full-ßedged definition of monoidal 2-category can be found there, and is shown to be equivalent in an appropriate sense to so-called Gray-monoids, which are much simpler (indeed, for V braided monoidal, V-Cat *is* a Gray-monoid). Also look at the havent already done so. Todd Trimble === Subject: Re: monoidal enriched natural transformations Originator: bergv@math.uiuc.edu (Maarten Bergvelt) vas dir: >>Assuming that this is ok, I should also need to understand what might >>be the definition of a monoidal omega - natural transformation. [...] >>Unfortunately for me, Kellys book ends before this point: it >>explicitely says: is our decision not to discuss the Ôchange of >>base-category given by a symmetric monoidal functor V ---> W. Has >>someone else done the job after Kellys book? >If you havent done so already, I recommend that you get on the >categories mailing list: > categories@mta.ca >where you would surely get a reply and advice about the literature. >Alternatively, you might write Max Kelly (at the University of >Sydney) or Ross Street (Macquarie University) directly. >I wish I had suitable references at hand, but here are some remarks >on your query. [...] >Todd Trimble Agust.92 Roig === Subject: weighted tree generation Originator: bergv@math.uiuc.edu (Maarten Bergvelt) For a given undirected graph G(V,E) I would like to generate all possible rooted trees. How many will there be? If there are a lot, at least I would like to generate a set of different ones or make a single change to a given tree. Secondly, is there any standard method to assign weights to an undirected graph such that for a given root, the all shortest paths algorithm (e.g. dijkstra) will yield a given tree? (I guess that you could assign low values to the links that are elements of the tree and high values for all others, but can it be proven that it will always yield the given tree?) Diego diego at aulignac dot com www.aulignac.com === Subject: fundamental bounds on the elements of covariance matrices Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Let x and y be n-tuplets of normally distributed random variables, and let C = cov(x,y) be the nxn covariance matrix of x and y. Let z be some fixed n-tuplet whose elements are all strictly positive. 1) Are all the elements of the n-vector, Cz, non-negative? 2) Is zCz > 0? Where can I read more about bounds on the elements of covariance matrices? === Subject: Re: fundamental bounds on the elements of covariance matrices Originator: bergv@math.uiuc.edu (Maarten Bergvelt) See Morris L. Eatons book with multivariate in the title. I was about to answer that any matrix whatsoever could be in the role of C, but I find Im a bit rusty. -- Mike Hardy > Let x and y be n-tuplets of normally distributed random variables, and let > C = cov(x,y) be the nxn covariance matrix of x and y. > Let z be some fixed n-tuplet whose elements are all strictly positive. > 1) Are all the elements of the n-vector, Cz, non-negative? > 2) Is zCz > 0? > Where can I read more about bounds on the elements of covariance matrices? === Subject: Re: binary vector packings Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Could someone please help me identify the following problem? > Consider binary arrays {u_i} of length n with k 1s (and n-k 0s). How > many can we choose such that all pairwise inner products > sum_i u_i v_i < t ? > Equivalently, what is the maximum number of k-subsets of the n-set with > pairwise intersections less than t elements? > Does this problem, or some equivalent, have a name? Any references? Looks like your problem is more or less equivalent to finding the maximum size of a certain constant weight code (aka fixed weight code). Searching with those buzzwords should lead you to sources of known results, tables of upper bounds etc. Jyrki Lahtonen, Turku, Finland === Subject: two fibrations Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Hallo, consider an isometric action of a compact Lie group G on a connected Riemannian manifold M with a fixed point p. I am interested in the topology (especially the homology) of the space of paths in M, starting at p and ending in some fixed orbit N=Gq, denoted by P(M,ptimes N). Even for arbitrary p and submanifolds N, the end point map P(M,ptimes N)to N; c mapsto c(1) is a fibration with fibre P(M,ptimes q), the space of paths from p to q (which ist homotopy equivalent to the space of loops on M). So I can deduce some information on the homology from this fibration, e.g. by using the Leray-Serre spectral sequence. Furthermore, now restricting to the case of N being some orbit, we have P(M,ptimes Gq)=P(M,ptimes q)times_{G_q} G (twisted product - G_q is the isotropy group at q), which can be easily seen by regarding the mapping P(M,ptimes q)times Gto P(M,ptimes Gq); (c,g)mapsto gc (well-defined since p is fixed). Summarizing, we have two fibrations: i) P(M,ptimes q)to P(M,ptimes Gq)to Gq and ii) G_qto P(M,ptimes q)times G to P(M,ptimes Gq). Now my question: Does the existence of such two fibrations give any new relation between the topology of these spaces? Is there some method of extracting information from such two similar-looking fibrations? Oliver Goertsches === Subject: RA Positions at UNR Originator: bergv@math.uiuc.edu (Maarten Bergvelt) The Computer Vision Laboratory (CVL) at the University of Nevada, Reno (UNR) invites applications for research assistant positions starting in Spring2005/Fall 2005. Preference will be given to students who want to pursue a PhD degree in Computer Vision. Active research areas within CVL include object recognition, visual motion analysis, face detection and recognition, biometrics, tracking and pose estimation of human body/head/hand/eye-gaze, surveillance and activity recognition. CVL is currently funded by NSF, NASA, ONR, and Ford Motor Company. We are also collaborating with several government and industry laboratories. For more information, please visit http://www.cs.unr.edu/CVL Requirements: You must have a first degree in either an Engineering subject, in Mathematics, in Physics, or in Computer Science. Good Mathematical background, programming skills in C or C++, and familiarity with Unix/Linux/Windows are necessary. Prior familiarity with Image Processing, Computer Vision, Pattern Recognition, and Machine Learning is desirable. Good communication and writing skills in English are essential. Interested students should send their CV by regular mail, e-mail, or fax to Dr. George Bebis (bebis@cs.unr.edu) or Dr. Mircea Nicolescu (mircea@cs.unr.edu) Dr. George Bebis Department of Computer Science & Engineering University of Nevada Reno, NV 89557, USA phone: (775) 784-6463 email: bebis@cs.unr.edu http://www.cs.unr.edu/~bebis Dr. Mircea Nicolescu Department of Computer Science & Engineering University of Nevada Reno, NV 89557, USA phone: (775) 784-4356 email: mircea@cs.unr .edu http://www.cs.unr.edu/~mircea === Subject: A type of regular graph Originator: bergv@math.uiuc.edu (Maarten Bergvelt) For a graph with verticies the integers mod n, we may draw an edge between a and b iff b=a+i for some i in a set of residues mod n. Clearly such a graph is a regular graph, is it possible to characterize it further with known graph-theoretic properties? === Subject: Re: A type of regular graph 3QLpj-NoP*NzsIC,boYU]bQ]Hy<#4ga3$21: Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > For a graph with verticies the integers mod n, we may draw an edge > between a and b iff b=a+i for some i in a set of residues mod n. > Clearly such a graph is a regular graph, is it possible to > characterize it further with known graph-theoretic properties? These graphs are known as circulants, e.g. see . That doesnt answer your question, but it should at least help in searching for an answer. -- David Eppstein Computer Science Dept., Univ. of California, Irvine http://www.ics.uci.edu/~eppstein/ === Subject: building 3d shapes Originator: bergv@math.uiuc.edu (Maarten Bergvelt) hi world, im looking for toolkits of segments / magnets / plastic caps to build 3d shapes like the 1- or 2-skeleta of tetrahedra, octahedra, etc. i want to play around with the cell decomposition of the lattice a_3, so ill need quite a bunch of these. i quickly searched on the net for some way of buying these online, with no success. does anybody know where i can find them? tia, laurent -- Laurent Bartholdi laurent.bartholdiepßch EPFL, IGAT, B.89timent BCH T.8el.8ephone: +41 21-6930380 CH-1015 Lausanne, Switzerland Fax: +41 21-6930385 === Subject: Re: building 3d shapes Originator: israel@math.ubc.ca (Robert Israel) > hi world, > im looking for toolkits of segments / magnets / plastic caps to build 3d > shapes like the 1- or 2-skeleta of tetrahedra, octahedra, etc. > i want to play around with the cell decomposition of the lattice a_3, so > ill need quite a bunch of these. > i quickly searched on the net for some way of buying these online, with no > success. does anybody know where i can find them? One possibility: http://www.zometool.com/ -- http://hertzlinger.blogspot.com === Subject: Re: building 3d shapes Originator: israel@math.ubc.ca (Robert Israel) > hi world, > im looking for toolkits of segments / magnets / plastic caps to > build 3d shapes like the 1- or 2-skeleta of tetrahedra, octahedra, > etc. > i want to play around with the cell decomposition of the lattice a_3, > so ill need quite a bunch of these. > i quickly searched on the net for some way of buying these online, > with no success. does anybody know where i can find them? > tia, laurent Or try Geomag and Supermag (dont know if these are the same), e.g. at http://www.toymagnets.com/geomag/index.cfm or http://www.geomags.com/. Ive seen them sold in Swiss toyshops as well. Christian Graf === Subject: Re: building 3d shapes Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > hi world, > im looking for toolkits of segments / magnets / plastic caps to build 3d > shapes like the 1- or 2-skeleta of tetrahedra, octahedra, etc. > i want to play around with the cell decomposition of the lattice a_3, so > ill need quite a bunch of these. > i quickly searched on the net for some way of buying these online, with no > success. does anybody know where i can find them? > tia, laurent > -- > Laurent Bartholdi laurent.bartholdiepßch > EPFL, IGAT, B.89timent BCH T.8el.8ephone: +41 21-6930380 > CH-1015 Lausanne, Switzerland Fax: +41 21-6930385 For magnets look at http://www.supermagnete.ch/magnets.php?at=Z Hugo Pfoertner === Subject: Re: building 3d shapes Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > .... > im looking for toolkits of segments / magnets / plastic caps to build 3d > shapes like the 1- or 2-skeleta of tetrahedra, octahedra, etc. > i want to play around with the cell decomposition of the lattice a_3, so > ill need quite a bunch of these. > i quickly searched on the net for some way of buying these online, with no > success. does anybody know where i can find them? .... Is http://www.korthalsaltes.com/ any use to you? Ken Pledger. === Subject: natural numbers as coequalizer (Re: Those Naughty Category Theorists) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) begin{quote} Incidentally for those who think that by defining numbers as lengths of strings Im making some sort of obscure automata theoretic point with my computer science hat on, let me just point out that not only can this representation be made entirely respectable mathematically, but via a slicker mathematical name than any other Ive seen proposed in this thread so far: the free monoid (N,+,0) on one generator. Just as punchy, and as clear to a category theorist as free monoid is to an algebraist: the coequalizer of the elements of (the ordinal) *2. The set-theoretic explanation of this is a bit clumsy, but here goes. The scene is Cat, the category of all small categories. The diagram to be coequalized is the left half of 0 ----> F *1 *2 ----> Coeq(0,1) ----> 1 with the coequalizer Coeq(0,1) and its coequalizing arrow F shown on the right. The ordinal *2 is the one-nonidentity-arrow category {0->1}, the ordinal *1 is the evident {0}, the elements 0,1 form the set Hom(*1,*2). The coequalizer of 0 and 1 creates Coeq(0,1), a copy of *2 which identifies 0 and 1. This has the side effect of looping the nonidentity arrow back on itself. Since we are in Cat, we now have to specify a composition law for this arrow with itself in the least constraining way, i.e. Coeq(0,1) has to be universal. Clearly we need all composites f, ff, fff, etc. Identifying any two of these is an unwanted constraint, so we leave them all unidentified. We now have a monoid whose arrows 1,f,ff,fff,... represent the natural numbers 0,1,2,3,..., composition represents addition, and the identity arrow represents 0. (Represent is meaningful only in this set-theoretic view.) The functor F takes both objects of *2 to the object of Coeq(0,1), and takes the nonidentity arrow of *2 to f or 1, the generator of (N,+,0). This construction may seem a bit contrived until you look at how category theoretic foundations are typically organized. (Good reading: McLarty, Axiomatizing a Category of Categories, J.Symbolic Logic, 56:4(Dec91).) Ordinal constructions involving the four ordinals up to *3, along with the product *2 x *2, are at the heart of this organization, and the above construction of the natural numbers as a monoid in Cat is not only slick but very natural and in that setting. end{quote} Can we construct in a similar manner integers, rational, reals and complex numbers? David, === Subject: Re: Schlomilchs series Epigone-thread: yendwholgrimp Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Hi--Im a newbie to special functions, I hope somebody can help me out. Im doing a problem in scattering, and it would be really helpful if I could sum the following series: sum_{n=1}^inf J_0(n x + b) where b is a real number. If b=0 then this is a bread & butter Schlomilch series (see e.g. Gradshteyn and Ryzhik 5th ed. 8.521). I tried expanding J_0(n x + b) via the addition theorem. But I didnt get too far. I was wondering if this was a known series...hoping the experts have some help! :) === Subject: Re: Schlomilchs series Epigone-thread: yendwholgrimp Content-Length: 782 Originator: rusin@vesuvius There is a sort of obvious way to do it if b is an integer...take your original sum, replace x by 2x and then subtract the two infinite sums, etc. For arbitrary b Im really not sure. >Hi--Im a newbie to special functions, I hope somebody can help me >out. Im doing a problem in scattering, and it would be really >helpful if I could sum the following series: > sum_{n=1}^inf J_0(n x + b) >where b is a real number. >If b=0 then this is a bread & butter Schlomilch series (see e.g. >Gradshteyn and Ryzhik 5th ed. 8.521). >I tried expanding J_0(n x + b) via the addition theorem. But I didnt >get too far. >I was wondering if this was a known series...hoping the experts have >some help! :) === Subject: Re: Schlomilchs series Content-Length: 1302 Originator: rusin@vesuvius Please dont top-post. Im putting the original question here where it belongs: >>Hi--Im a newbie to special functions, I hope somebody can help me >>out. Im doing a problem in scattering, and it would be really >>helpful if I could sum the following series: >> sum_{n=1}^inf J_0(n x + b) >There is a sort of obvious way to do it if b is an integer...take >your original sum, replace x by 2x and then subtract the two infinite >sums, etc. Sorry, I dont understand this. You seem to be saying take the original sum, J_0(x+b) + J_0(2x+b) + J_0(3x+b) + ... and subtract J_0(2x+b) + J_0(4x+b) + J_0(6x+b) + .... But that will just give you J_0(x+b) + J_0(3x+b) + J_0(5x+b) + ... and I dont see how that helps, or what difference b being an integer makes. On the other hand, if b is an integer multiple of x, you can say something: if F(x,b) is the sum, F(x,b+x) = F(x,b) - J_0(x+b) so F(x, kx) = F(x,0) - sum_{j=1}^k J_0(jx) for positive integers k Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: On a property of Bernoulli numbers Epigone-thread: clephoystald Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >Let A(n,k)=(n^(2k)-1)*B(2k) where B(2k) denotes the 2k-th Bernoulli >number. Then I suspect the existence of a minimal positive rational >value, depending on n, r(n)=P(n)/Q(n) with (P(n),Q(n))=1 and such >that : >for any k>0 r(n)*A(n,k) is an integer value. >r(2)=2, r(3)=3/4 .... >P(n) appears to be the largest square-free divisor of n but I didnt >observation when n is a power of 2 : >for p prime, if 2^p-1 and (2^p+1)/3 are both primes then >Q(2^p)=(4^p-1)/3 (converse doesnt hold). >Can anyone confirm theorically the existence of r(n) and the formula >for P(n)? If so, what is the formula for Q(n)? Update : I found that r(n)=rad(n^3-n)/(n^2-1) where rad(n) is the square-free kernel of n, the largest square-free divisor of n. This explains why P(n)=rad(n) and we have Q(n)=(n^2-1)/rad(n^2-1). I cant say if this property of Bernoullis numbers is known. Studying r(n) I came across something looking as an integer formulation of Agohs conjecture : p is prime iff p divides (p^p-p)*B(p-1)-1 and I unearthed this amusing connection with 3-smooth numbers (numbers of form 2^i*3^j i,j>=0) : fractional part of ((n^(2k)-1)*B(2k)) is constant for any k>0 iff n is a 3-smooth number. Benoit Cloitre === Subject: Paper published by Algebraic and Geometric Topology Originator: bergv@math.uiuc.edu (Maarten Bergvelt) The following paper has been published: Algebraic and Geometric Topology URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-42.abs.html Title: A class of tight contact structures on Sigma_2 x I Author(s): Tanya Cofer Abstract: We employ cut and paste contact topological techniques to classify some tight contact structures on the closed, oriented genus-2 surface times the interval. A boundary condition is specified so that the Euler class of the of the contact structure vanishes when evaluated on each boundary component. We prove that there exists a unique, non-product tight contact structure in this case. Secondary: 53C15 Keywords: Tight, contact structure, genus-2 surface Author(s) address(es): Department of Mathematics, Northeastern Illinois University 5500 North St Louis Avenue, Chicago, IL 60625-4699, USA Email: T-Cofer@neiu.edu URL: http://www.neiu.edu/~tcofer/ === Subject: This week in the mathematics arXiv (18 Oct - 22 Oct) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Here are this weeks titles in the mathematics arXiv, available at: http://front.math.ucdavis.edu/ http://front.math.ucdavis.edu/submissions This week in the mathematics arXiv may be freely redistributed with attribution and without modification. Titles in the mathematics arXiv (18 Oct - 22 Oct) ------------------------------------------------- AC: Commutative Algebra ----------------------- math.AC/0410375 Kiran S. Kedlaya: Finite automata and algebraic extensions of function fields math.AC/0410340 Claudia Polini, Bernd Ulrich: A formula for the core of an ideal AG: Algebraic Geometry ---------------------- math.AG/0410469 Frederic Campana: Fibres multiples des surfaces math.AG/0410458 Samuel Boissiere: Chern classes of the tangent bundle on the Hilbert scheme of points on the affine plane math.AG/0410444 S. Kaplan, E. Liberman, M. Teicher: Braid Monodromy Computation of Real Singular Curves math.AG/0410442 Nickolas Michelacakis, Apostolos Thoma: On the geometry of complete intersection toric varieties math.AG/0410432 Qi Zhang: On projective varieties with nef anticanonical divisors hep-th/0410055 Volker Braun, Burt A. Ovrut, Tony Pantev, Rene Reinbacher: Elliptic Calabi-Yau Threefolds with Z_3 x Z_3 Wilson Lines math.AG/0410408 Ivan Cheltsov: Double cubics and double quartics hep-th/0410018 A. Klemm, M. Kreuzer, E. Riegler, E. Scheidegger: Topological String Amplitudes, Complete Intersection Calabi-Yau Spaces and Threshold Corrections math.AG/0410401 Gabor Szekelyhidi: Extremal metrics and K-stability math.AG/0410394 Ana Cristina Lopez: Relative Jacobians of elliptic fibrations with reducible fibers math.AG/0410393 Ana Cristina Lopez: Simpson Jacobians of reducible curves math.AG/0410392 variety math.AG/0410388 M. E. Kazaryan, S. K. Lando: Towards the Intersection Theory on Hurwitz Spaces math.AG/0410383 Philibert Nang, Kiyoshi Takeuchi: Addendum to the paper Characteristic Cycles of Perverse Sheaves and Milnor Fibers math.AG/0410379 Seongchun Kwon: Transversality properties on the moduli space of genus 0 stable maps to a smooth rational projective surface and their real enumerative implications math.AG/0410378 Silvano Baggio: Equivariant K-Theory of Smooth Toric Varieties math.AG/0410360 Tyler J. Jarvis, William E. Lang, Nansen Petrosyan, Gretchen Rimmasch, Julie Rogers, Erin D. Summers: Classification of Singular Fibres on Rational Elliptic Surfaces in Characteristic Three hep-th/0410170 Bjorn Andreas, Daniel Hernandez Ruiperez: U(n) Vector Bundles on Calabi-Yau Threefolds for String Theory Compactifications math.AG/0410349 Igor Burban, Bernd Kreussler: On a relative Fourier-Mukai transform on genus one fibrations math.AG/0410346 Oliver Lorscheid: Completeness and compactness for varieties over local fields AP: Analysis of PDEs -------------------- math.AP/0410475 Zhongwei Shen: Bounds of Riesz Transforms on $L^p$ Spaces for Second Order Elliptic Operators math.AP/0410462 Fernando Cardoso, Georgi Vodev: Weighted L^p decay estimates of solutions to the wave equation with a potential math.AP/0410452 A.G.Ramm: Existence of a solution to a nonlinear equation math.AP/0410451 A.G.Ramm: A singular perturbation problem math.AP/0410443 Arnaud Debussche, Cyril Odasso: Ergodicity for the weakly damped stochastic non-linear Schrodinger equations math.AP/0410441 Giuseppe Da Prato, Arnaud Debussche, Luciano Tubaro: Coupling for some partial differential equations driven by white noise math.AP/0410431 Burak Erdogan, Wilhelm Schlag: Dispersive estimates for Schr{o}dinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: I math.AP/0410416 Dian Palagachev, Lubomira Softova: Fine regularity for elliptic systems with discontinuous ingredients math.AP/0410415 Dian K. Palagachev, Lubomira G. Softova: Apriori estimates and precise regularity for parabolic systems with discontinuous data math.AP/0410380 Fabian Waleffe: On some dyadic models of the Euler equations math.AP/0410344 Isabelle Gallagher, Thierry Gallay, Pierre-Louis Lions: On the uniqueness of the solution of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity AT: Algebraic Topology ---------------------- math.AT/0410405 Scott O. Wilson: Partial Algebras Over Operads of Complexes and Applications math.AT/0410398 R. Brown, H.K. Kamps, T. Porter: A homotopy double groupoid of a Hausdorff space II: a van Kampen theorem math.AT/0410374 Norio Iwase, Donald Stanley, Jeffrey Strom: Implications of the Ganea Condition math.AT/0410367 Z. Fiedorowicz, R. M. Vogt: Topological Hochschild Homology of $E_n$-Ring Spectra math.AT/0410363 A.D.R. Choudary, A. Dimca, S. Papadima: Some Analogs of Zariski Theorem on Nodal Line Arrangements math.AT/0410342 Nicholas J. Kuhn: Goodwillie towers and chromatic homotopy: an overview CA: Classical Analysis and ODEs ------------------------------- math.CA/0410439 Jose L. Lopez, Nico M. Temme: Convergent Asymptotic Expansions of Charlier, Laguerre and Jacobi Polynomials math.CA/0410436 Jose L. Lopez, Nico M. Temme: Multi-point Taylor Expansions of Analytic Functions math.CA/0410395 Sever Silvestru Dragomir: Some Inequalities for Functions of Bounded Variation with Applications to Landau Type Results CO: Combinatorics ----------------- math.CO/0410471 Michiel Hazewinkel: Word Hopf algebras math.CO/0410466 Charles F. Dunkl: Hook-lengths and Pairs of Compositions math.CO/0410455 David E Speyer: Tropical Linear Spaces math.CO/0410429 Jens Christian Claussen: Time-evolution of the Rule 150 cellular automaton activity from a Fibonacci iteration math.CO/0410425 Joseph E. Bonin, Omer Gimenez: Multi-Path Matroids math.CO/0410410 Annalies Vuong, M. Ian Wyckoff: Conditions for Weighted Cover Pebbling of Graphs math.CO/0410404 F. Bonetto, H. Matzinger: Fluctuations of the Longest Common Subsequence in the Asymmetric Case of 2- and 3-Letter Alphabets math.CO/0410382 B.M. Kim, Y. Rho: Van der Waerdens Theorem on Homothetic copies of {1,1+s, 1+s+t} math.CO/0410373 Ira M. Gessel, Louis H. Kalikow: Hypergraphs and a functional equation of Bouwkamp and de Bruijn math.CO/0410366 Michiel Hazewinkel: Explicit polynomial generators for the ring of quasi-symmetric functions over the integers math.CO/0410361 Howard Kleiman: The Floyd-WarshallAlgorithm and the Asymmetric TSP nlin.AO/0407024 Fatihcan M. Atay, Tuerker Biyikoglu, Juergen Jost: On the synchronization of networks with prescribed degree distributions math.CO/0410347 Svante Linusson, Johan Waestlund: Completing a k-1 assignment math.CO/0410345 Svante Linusson, John Shareshian, Volkmar Welker: Complexes of graphs with bounded matching size CT: Category Theory ------------------- math.CT/0410412 Dominic Verity: Complicial Sets CV: Complex Variables --------------------- math.CV/0410445 P. Ebenfelt, L. P. Rothschild: Transversality of CR mappings math.CV/0410420 Rostyslav O. Hryniv, Yaroslav V. Mykytyuk: Asymptotics of zeros for some entire functions math.CV/0410399 Vladimir V. Kisil, Debapriya Biswas: Elliptic, Parabolic and Hyperbolic Analytic Function Theory--0: Geometry of Domains math.CV/0410390 Franc Forstneric, Joerg Winkelmann: Holomorphic discs with dense images math.CV/0410386 Franc Forstneric, Christine Laurent-Thiebaut: Stein compacts in Levi-ßat hypersurfaces math.CV/0410376 Bertrand Deroin: Laminations dans les esapces projectifs complexes math.CV/0410362 Young-Heon Kim: Holomorphic extensions of determinants of Laplacians math.CV/0410353 J. J. Kohn: Superlogarithmic estimates on pseudoconvex domains and CR manifolds math.CV/0410343 Mikhail Sodin: Zeroes of Gaussian analytic functions math.CV/0410341 Fedor Nazarov, Mikhail Sodin: Coarse equidistribution of the argument of entire functions of finite order DG: Differential Geometry ------------------------- math.DG/0410461 Josef Janyv{s}ka: Natural connections given by general linear and classical connections math.DG/0410460 Tom Mestdag, Bavo Langerock: A Lie algebroid framework for non-holonomic systems math.DG/0410456 Mikhail G. Katz, Yuli B. Rudyak: Lusternik-Schnirelmann category and systolic category of low dimensional manifolds nlin.SI/0407057 Paolo Lorenzoni, Marco Pedroni: On the bi-Hamiltonian structures of the Camassa-Holm and Harry Dym equations math.DG/0410435 Isabel Fernandez, Francisco J. Lopez: Relative parabolicity of zero mean curvature surfaces in $R^3$ and $R_1^3$ math.DG/0410434 Michael Schulze: On the resolvent of the Laplacian on functions for degenerating surfaces of finite geometry math.DG/0410418 Jian Song, Ben Weinkove: On the convergence and singularities of the J-ßow with applications to the Mabuchi energy math.DG/0410413 2-surfaces of prescribed mean curvature DS: Dynamical Systems --------------------- math.DS/0410464 I.Dynnikov, S.Novikov: Topology of quasiperiodic functions on the plane physics/0410160 R. Ball: The case of the trapped singularities math.DS/0410417 Charles Favre, Mattias Jonsson: Eigenvaluations math.DS/0410384 N. Haydn, Y. Lacroix & S. Vaienti: Hitting and return times in ergodic dynamical systems math.DS/0410355 Marco Lenci: Typicality of recurrence for Lorentz gases nlin.CD/0410019 Sylvie Oliffson Kamphorst, Sonia Pinto de Carvalho: The First Birkhoff Coefficient and the Stability of 2-Periodic Orbits on Billiards FA: Functional Analysis ----------------------- math.FA/0410427 W. B. Johnson, N. L. Randrianarivony: $ell_p$ (p>2) does not coarsely embed into a Hilbert space math.FA/0410422 Assaf Naor, Yuval Peres, Oded Schramm, Scott Sheffield: Markov chains in smooth Banach spaces and Gromov hyperbolic metric spaces math.FA/0410403 Stefan Bildea, Dorin Ervin Dutkay, Gabriel Picioroaga: MRA Super-wavelets math.FA/0410391 Roberto Giambo, Fabio Giannoni, Paolo Piccione: Orthogonal Geodesic Chords, Brake Orbits and Homoclinic Orbits in Riemannian Manifolds math.FA/0410351 Frederic Bayart, Catherine Finet, Daniel Li, Herve Queffelec: Composition operators on the Wiener-Dirichlet algebra math.FA/0410348 Wojciech Czaja: Remarks on Naimarks duality GM: General Mathematics ----------------------- math.GM/0410377 Jacky Cresson: Non-differentiable variational principles GT: Geometric Topology ---------------------- math.GT/0410476 J.-F. Lafont: Strong Jordan separation and applications to rigidity math.GT/0410474 Brent Everitt, John Ratcliffe, Steven Tschantz: The smallest hyperbolic 6-manifolds math.GT/0410433 Carlo Petronio: Complexity of 3-orbifolds math.GT/0410381 Suhyoung Choi: Drilling cores of hyperbolic 3-manifolds to prove tameness math.GT/0410370 Jerzy Dydak, Michael Levin: Extension of maps to the projective plane math.GT/0410369 Michael Levin: Rational acyclic resolutions math.GT/0410368 Michael Levin: Universal acyclic resolutions for arbitrary coefficient groups math.GT/0410358 Robion Kirby, Paul Melvin: Local surgery formulas for quantum invariants and the Arf invariant math.GT/0410356 Eaman Eftekhary: Filtration of Heegaard Floer homology and gluing formulas HO: History and Overview ------------------------ math.HO/0410411 Tommaso Toffoli: Maxwells daemon, the Turing machine, and Jaynes robot math.HO/0410397 V.G.Gurzadyan: Kolmogorov and Aleksandrov in Sevan Monastery, Armenia, 1929 MG: Metric Geometry ------------------- math.MG/0410440 Andreas Balser, Alexander Lytchak: Centers of convex subsets of buildings math.MG/0410437 Andreas Balser, Alexander Lytchak: Building-like spaces math.MG/0410421 Alexander Lytschak, Viktor Schroeder: Affine functions on CAT(kappa) spaces MP: Mathematical Physics ------------------------ quant-ph/0410131 Xiong-Jun Liu, Hui Jing, Xin Liu, Mo-Lin Ge: Dynamical Symmetry and Its Applications In Electromagnetically Induced Transparency math-ph/0410046 Michiel Hazewinkel, Hugo H Torriani: Coherence and uniqueness theorems for averaging processes in statistical mechanics hep-th/0410199 A.P. Balachandran, A. Pinzul: On Time-Space Noncommutativity for Transition Processes and Noncommutative Symmetries hep-th/0008117 M. Hssaini, M. Kessabi, B. Maroufi, M.B.Sedra: Central extended D=2 N=4 SU(2) Liouville self interacting model and explicit hyperkahler metric math-ph/0410045 Alexei F. Cheviakov: Plasma equilibrium equations in coordinates connected with magnetic surfaces. Exact equilibrium solutions gr-qc/0410069 Antonio Lopez-Pinto: Nonstandard spin 2 field theory physics/0410127 A. Figotin, J. H. Schenker: Hamiltonian treatment of time dispersive and dissipative media within the linear response theory math-ph/0410044 Daniel Peralta-Salas: A geometric approach to the equilibrium shapes of self-gravitating ßuids hep-th/0410013 Patrick Dorey, Adam Millican-Slater, Roberto Tateo: Beyond the WKB approximation in PT-symmetric quantum mechanics cond-mat/0410435 F. Guerra: Mathematical aspects of mean field spin glass theory math-ph/0410043 Volodymyr Sushch: On some discrete model of the magnetic Laplacian math-ph/0410042 Jochen Bruening, Vladimir Geyler, Konstantin Pankrashkin: Continuity of integral kernels related to Schrodinger operators on manifolds math-ph/0410041 O.M. Kiselev, S.G. Glebov, V.A. Lazarev: Resonant pumping in nonlinear Klein-Gordon equation and solitary packets of waves math-ph/0410040 G.Giachetta, L.Mangiarotti, G.Sardanashvily: Geometric and Algebraic Topological Methods in Quantum Mechanics hep-th/0408241 S. Meljanac, A. Samsarov: Matrix oscillator and Calogero-type models math-ph/0410039 Nasser Saad, Richard L. Hall, Qutaibeh D. Katatbeh: Study of anharmonic singular potentials NT: Number Theory ----------------- math.NT/0410428 L.A.Gutnik: On the difference equation of the Poincare type math.NT/0410409 A. Agboola: Galois modules and p-adic representations math.NT/0410387 C. S. Rajan: Recovering modular forms and representations from tensor and symmetric powers math.NT/0410372 Mark van Hoeij: Solving conics over Q(t1,..,tk) OA: Operator Algebras --------------------- math.OA/0410449 Kenneth Davidson, Elias Katsoulis: Nest representations of directed graph algebras math.OA/0410426 Benjam{i}n Itza-Ortiz: Eigenvalues, K-theory and Minimal Flows math.OA/0410400 Marius Dadarlat: On the topology of the Kasparov groups and its applications OC: Optimization and Control ---------------------------- math.OC/0410467 Antonios Armaou, Ioannis G. Kevrekidis: Equation-free optimal switching policies for bistable reacting systems using coarse time-steppers PR: Probability --------------- math.PR/0410465 Federico Camia: Scaling Limit and Critical Exponents for Two-Dimensional Bootstrap Percolation math.PR/0410459 Florent Benaych-Georges: Taylor expansions of R-transforms, application to supports and moments math.PR/0410457 Catherine Donati-Martin: Large deviations for Wishart processes math.PR/0410453 Patrick Cheridito, Freddy Delbaen, Michael Kupper: Dynamic monetary risk measures for bounded discrete-time processes math.PR/0410447 Michail Loulakis: On the Symmetry of the Diffusion Coefficient in Asymmetric Simple Exclusion math.PR/0410430 Yuval Peres, David Revelle: Scaling limits of the uniform spanning tree and loop-erased random walk on finite graphs math.PR/0410414 Robert C. Dalang, Carl Mueller, Lorenzo Zambotti: Hitting properties of s.p.d.e.s with reßection math.PR/0410402 David J. Aldous, Lea Popovic: A critical branching process model for biodiversity math.PR/0410371 Harry Kesten, Vladas Sidoravicius: A phase transition in a model for the spread of an infection math.PR/0410359 Bela Bollobas, Oliver Riordan: A short proof of the Harris-Kesten Theorem cond-mat/0410309 V. Sood, S. Redner, D. ben-Avraham: First Passage Properties of the Erdos-Renyi Random Graph QA: Quantum Algebra ------------------- math.QA/0410470 Michiel Hazewinkel: Symmetric functions, noncommutative symmetric functions, and quasisymmetric functions II math.QA/0410468 Michiel Hazewinkel: Symmetric functions, noncommutative symmetric functions, and quasisymmetric functions math.QA/0410463 S. Sinelshchikov, A. Stolin, L. Vaksman: A Quantum Analogue of the Bernstein Functor math.QA/0410450 J.E. McClure: On the chain-level intersection pairing for PL manifolds math.QA/0410448 Christian Blohmann: Reconstruction of universal Drinfeld twists from representations math.QA/0410446 Haisheng Li, Gaywalee Yamskulna: On certain vertex algebras and their modules associated with vertex algebroids math.QA/0410407 K. Szlachanyi: Monoidal Morita equivalence math.QA/0410396 S. Sinelshchikov, L. Vaksman: Quantum groups and non-commutative complex analysis math.QA/0410389 Harald Grosse, Stefan Schraml: The Eigenfunctions of the q-Harmonic Oscillator on the Quantum Line math.QA/0410365 Michiel Hazewinkel: The primitives of the Hopf algebra of noncommutative symmetric functions math.QA/0410364 Michiel Hazewinkel: Hopf algebras of endomorphisms of Hopf algebras math.QA/0410350 Henrique Bursztyn, Stefan Waldmann: Hermitian star products are completely positive deformations RA: Rings and Algebras ---------------------- math.RA/0410473 Gizem Karaali: A New Lie Bialgebra Structure on sl(2,1) math.RA/0410406 Michael Pinsker: The number of unary clones containing the permutations on an infinite set RT: Representation Theory ------------------------- math.RT/0410472 Paolo Bravi, Guido Pezzini: Wonderful varieties of type D math.RT/0410454 Francois Digne, Jean Michel, Raphael Rouquier: Cohomologie des varietes de Deligne-Lusztig math.RT/0410423 Calin Chindris: Quivers, long exact sequences and Horn type inequalities math.RT/0410357 Helmer Aslaksen, Mong Lung Lang: Extending $pi$-systems to bases of root systems SG: Symplectic Geometry ----------------------- math.SG/0410352 Eaman Eftekhary: Embedded curves and Gromov-Witten invariants of three-folds SP: Spectral Theory ------------------- math.SP/0410438 M.A. Kaashoek, A.L. Sakhnovich: Discrete skew selfadjoint canonical systems and the isotropic Heisenberg magnet model ST: Statistics -------------- math.ST/0410424 George Kahrimanis, Daniel Berleant: Direct pivotal predictive inference math.ST/0410419 Grace Wahba: An introduction to (smoothing spline) ANOVA models in RKHS with examples in geographical data, medicine, atmospheric science and machine learning math.ST/0410385 Mario Ruetti, Matthias Troyer, Wesley P. Petersen: A Generic Random Number Generator Test Suite math.ST/0410354 Stephane Gaiffas: Rates of convergence for pointwise curve estimation with a degenerate design -- / Greg Kuperberg (UC Davis) / Home page: http://www.math.ucdavis.edu/~greg/ / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/ / * All the math thats fit to e-print * === Subject: Question about Gomory-Cut Originator: israel@math.ubc.ca (Robert Israel) for an application I have to find an integer solution. I use the simplex-algorithm and the Gomory-cut. It works fine, but it is to slowly. There are multiply opportunities to make a Gomory-cut. Which Gomory-cuts should I take, to get a solution in shortest time ? Where can I get information about that (in the internet) ? Ulrich === Subject: Combinatorial inequality Epigone-thread: wunloifo Originator: israel@math.ubc.ca (Robert Israel) Can anyone help me in deriving this inequality which appears to be true? {C(n,k)/C(a,k)}*(m/n)^k <={((n-m)/(n-a))^(n-a)}*{(n/a)^a} where k=(a-m)/(1-m/n); and 0Can anyone help me in deriving this inequality which appears to be >true? > {C(n,k)/C(a,k)}*(m/n)^k <={((n-m)/(n-a))^(n-a)}*{(n/a)^a} > where k=(a-m)/(1-m/n); and 1 Here C(n,k) means the binomial coefficient and ^ means >exponentiation,* denotes ordinary multiplication. === Subject: Conformal Mapping Question Originator: israel@math.ubc.ca (Robert Israel) Id like to find the explicit formula of a bijective conformal mapping from the following region, say K, K={ x+iy in C mid x>0, y > arccos(e^{-x}) } to the interior of the unit disk (K is just the subregion in the first quadrant of the complex plane bounded below by the graph of e^{x}cos(y)=1). Ive tried basic ones and looked into Dictionary of Conformal Representations by H. Kober, but so far nothing works for me. Any suggestions are greatly appreciated. -- So Okada Ph.D. Student of Math at UMass Amherst okada@math.umass.edu === Subject: This week in the mathematics arXiv (25 Oct - 29 Oct) Originator: israel@math.ubc.ca (Robert Israel) Here are this weeks titles in the mathematics arXiv, available at: http://front.math.ucdavis.edu/ http://front.math.ucdavis.edu/submissions This week in the mathematics arXiv may be freely redistributed with attribution and without modification. Titles in the mathematics arXiv (25 Oct - 29 Oct) ------------------------------------------------- AC: Commutative Algebra ----------------------- math.AC/0410598 Kamran Divaani-Aazar, Amir Mafi: Associated primes of local cohomology module math.AC/0410585 Nicholas Baeth: A Krull-Schmidt Theorem for One-dimensional Rings of Finite Cohen-Macaulay Type math.AC/0410535 Anurag K. Singh, Uli Walther: On the arithmetic rank of certain Segre products math.AC/0410497 Juan C. Migliore, Uwe Nagel, Tim Romer: The Multiplicity Conjecture in low codimensions math.AC/0410478 Carlos DAndrea, Laurent Buse: Properness and inversion problems by means of matrices AG: Algebraic Geometry ---------------------- math.AG/0410604 Elizabeth S. Allman, John A. Rhodes: Phylogenetic ideals and varieties for the general Markov model math.AG/0410602 E. Carlini: Codimension one decompositions and Chow varieties math.AG/0410600 J. C. Sierra, L. Ugaglia: On double Veronese embeddings in the Grassmannian G(1,N) math.AG/0410584 Carolina Araujo: Rational curves of minimal degree and characterizations of ${mathbb P}^n$ math.AG/0410572 Israel Moreno Mej{i}a: The trace of an automorphism on H^0(J,O(nTheta)) math.AG/0410558 Ivan Cheltsov: Birationally superrigid cyclic triple spaces math.AG/0410554 Meirav Amram, David Goldberg: Higher degree Galois covers of CP^1 x T math.AG/0410547 D. A. Stepanov: Non-rational divisors over non-Gorenstein terminal singularities math.AG/0410540 Pan Peng: A simple proof of Gopakumar-Vafa conjecture for local toric Calabi-Yau manifolds math.AG/0410537 Eduardo Esteves, Steven Kleiman: The compactified Picard scheme of the compactified Jacobian math.AG/0410527 Cristiano Bocci: Special effect varieties and (-1)-curves math.AG/0410526 Shihoko Ishii: Arcs, valuations and the Nash map math.AG/0410524 Boris E. Kunyavskii, Louis H. Rowen, Sergey V. Tikhonov, Vyacheslav I. Yanchevskii: Division algebras that ramify only on a plane quartic curve math.AG/0410520 Laurent Manivel, Emilia Mezzetti: On linear spaces of skew-symmetric matrices of constant rank math.AG/0410518 Elena Drozd: Curves on a nonsingular Del Pezzo Surface in $P^4_k$ math.AG/0410513 Kalle Karu: The cd-index of fans and lattices AP: Analysis of PDEs -------------------- math.AP/0410581 G. Olafsson, A. Pasquale: Support properties and Holmgrens uniqueness theorem for differential operators with hyperplane singularities math.AP/0410564 James Nolen, Jack Xin: A Variational Principle Based Study of KPP Minimal Front Speeds in Random Shears math.AP/0410546 Plamen Stefanov, Gunther Uhlmann: Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map math.AP/0410538 J. Colliander, W. Staubach: $L^2$ blowup solutions of cubic NLS on $R^2$ concentrate a fixed amount of mass math.AP/0410525 Gianni Dal Maso, Rodica Toader: On a notion of unilateral slope for the Mumford-Shah functional math.AP/0410499 Hans Lindblad, Jacob Sterbenz: Global Stability for Charge Scalar Fields on Minkowski Space AT: Algebraic Topology ---------------------- math.AT/0410589 Nora Ganter: Smash products of E(1)-local spectra at an odd prime math.AT/0410552 Javier Turiel: Polynomials Maps and Even Dimensional Spheres math.AT/0410503 Kathryn Hess, Ran Levi: An algebraic model for the loop homology of a homotopy fiber CA: Classical Analysis and ODEs ------------------------------- math.CA/0410548 Marc Artzrouni: A new family of periodic functions as explicit roots of a class of polynomial equations math.CA/0410542 Projections And Universal Encoding Strategies math.CA/0410508 Stephen Semmes: Potpourri, 8 math.CA/0410490 Stephen Semmes: Potpourri, 7 math.CA/0410489 Stephen Semmes: Potpourri, 6 math.CA/0410483 A. A. Bolibruch, S. Malek, C. Mitschi: On the generalized Riemann-Hilbert problem with irregular singularities CO: Combinatorics ----------------- math.CO/0410592 S. Ole Warnaar: Hall--Littlewood functions and the A_2 Rogers--Ramanujan identities quant-ph/0410226 P. Blasiak, A. Horzela, K. A. Penson, A. I. Solomon: Deformed Bosons: Combinatorics of Normal Ordering math.CO/0410550 Ewa Krot: Further develpoements in finite fibonomial calculus math.CO/0410529 math.CO/0410482 Michael Anshelevich: Orthogonal polynomials with a resolvent-type generating function CT: Category Theory ------------------- math.CT/0410555 Alan Robinson: Partition complexes, duality and integral tree representations gr-qc/0410104 J. Daniel Christensen, Louis Crane: Causal sites as quantum geometry CV: Complex Variables --------------------- math.CV/0410599 Laurent Gendre: Inegalites de Markov tangentielles locales sur les courbes algebriques singulieres de R^n math.CV/0410578 Dmitri Prokhorov, Alexander Vasilev: Optimal control in Bombieris and Tammis conjectures math.CV/0410509 David E. Barrett: A ßoating body approach to Feffermans hypersurface measure DG: Differential Geometry ------------------------- math.DG/0410610 Francisco Martin Cabrera: SU(3)-structures on hypersurfaces of manifolds with $G_2$-structure hep-th/0410183 Anton Alekseev, Thomas Strobl: Current Algebras and Differential Geometry math.DG/0410579 Jorge Lauret: A canonical compatible metric for geometric structures on nilmanifolds math.DG/0410575 Joseph H.G. Fu: Structure of the unitary valuation algebra math.DG/0410561 math.DG/0410559 Tomasz S. Mrowka, Yann Rollin: Legendrian knots and monopoles math.DG/0410557 A. V. Kiselev, G. Manno: On the symmetry structure of the minimal surface equation math.DG/0410553 Anton Deitmar: A prime geodesic theorem for higher rank II: singular geodesics math.DG/0410551 Eduardo Martinez: Classical field theory on Lie algebroids: Variational aspects math.DG/0410512 Maks A. Akivis, Vladislav V. Goldberg, Arto V. Chakmazyan: Induced connections on submanifolds in spaces with fundamental groups math.DG/0410511 Maks A. Akivis, Vladislav V. Goldberg: Dually degenerate varieties and the generalization of a theorem of Griffiths--Harris math.DG/0410498 Boris S. Kruglikov, Vladimir S. Matveev: Strictly non-proportional geodesically equivalent metrics have $h_text{top}(g)=0$ math.DG/0410494 George Papadopoulos: Spin Cohomology math.DG/0410493 Abdenago Barros G. Pacelli Bessa: Estimates of the first eigenvalue of minimal hypersurfaces of $mathbb{S}^{n+1} math.DG/0410487 Hiroshi Iritani: Quantum D-modules and equivariant Floer theory for free loop spaces math.DG/0410484 Aleksis Raza: An application of Guillemin-Abreu theory to a non-abelian group action DS: Dynamical Systems --------------------- math.DS/0410580 I. Binder, M. Braverman, M. Yampolsky: Filled Julia sets with empty interior are computable math.DS/0410517 Le Van Hien: Stability of Solutions of Fuzzy Differential Equations math.DS/0410507 Topologies on the group of homeomorphisms of a Cantor set math.DS/0410506 Topologies on the group of Borel automorphisms of a standard Borel space math.DS/0410505 Sergey Bezuglyi, Anthoni H. Dooley, Konstantin Medynets: The Rokhlin lemma for homeomorphisms of a Cantor set math.DS/0410504 Sergey Bezuglyi, Konstantin Medynets: Smooth automorphisms and path-connectedness in Borel dynamics math.DS/0410500 Ara Basmajian, Mahmoud Zeinalian: Maximal Convergence Groups and Rank One Symmetric Spaces math.DS/0410481 Pavlos B. Konstadinidis: The Real 3x+1 Problem FA: Functional Analysis ----------------------- math.FA/0410596 Ralf Meyer: Embeddings of derived categories of bornological modules math.FA/0410573 Jorge Antezana, Gustavo Corach, Demetrio Stojanoff: Spectral shorted operators math.FA/0410571 Massimo Fornasier, Holger Rauhut: Continuous Frames, Function Spaces, and the Discretization Problem math.FA/0410567 A. Brudnyi: Contractibility of Maximal Ideal Spaces of Certain Algebras of Almost Periodic Functions math.FA/0410549 Massimo Fornasier: Banach frames for alpha-modulation spaces math.FA/0410501 V.Yaskin: The Busemann-Petty problem in hyperbolic and spherical spaces math.FA/0410496 A.Koldobsky, V.Yaskin, M.Yaskina: Modified Busemann-Petty problem on sections of convex bodies math.FA/0410491 T. Banks, T. Constantinescu, Nermine El-Sissi: Tensor algebras and displacement structure. IV. Invariant kernels math.FA/0410479 A.G.Ramm: Dynamical systems method (DSM) for nonlinear equations in Banach spaces GM: General Mathematics ----------------------- math.GM/0410556 Joao R. Cardoso: An Explicit Formula for the Matrix Logarithm GR: Group Theory ---------------- math.GR/0410593 Henrik Baarnhielm: The Schreier-Sims algorithm for matrix groups math.GR/0410590 Edith Adan-Bante: Products of characters with few irreducible constituents math.GR/0410583 Edith Adan-Bante: Products of characters and derived length II math.GR/0410582 Edith Adan-Bante: Squares of characters and groups of odd order math.GR/0410539 Daniel Farley, Lucas Sabalka: Discrete Morse theory and graph braid groups math.GR/0410533 Stephen DeBacker: Parametrizing nilpotent orbits via Bruhat-Tits theory math.GR/0410516 J.Mostovoy, J.M. Perez-Izquierdo: Dimension filtration on loops math.GR/0410515 Jacob Mostovoy: On the notion of lower central series for loops GT: Geometric Topology ---------------------- math.GT/0410606 Greg Friedman: Knot spinning math.GT/0410603 R. C. Penner, Dennis Sullivan: The Structure and Singularities of Arc Complexes math.GT/0410595 Pascal Hubert, Samuel Lelievre: Noncongruence subgroups in H(2) math.GT/0410570 Andras Nemethi: On the Heegaard Floer homology of S^3_{-p/q}(K) math.GT/0410565 Brooke Brennan, Thomas W. Mattman, Roberto Raya, Dan Tating: Ribbonlength of torus knots math.GT/0410541 Ensil Kang, J. Hyam Rubinstein: Ideal triangulations of 3--manifolds I: spun normal surface theory math.GT/0410495 Dror Bar-Natan: Khovanovs Homology for Tangles and Cobordisms KT: K-Theory and Homology ------------------------- math.KT/0410597 Ralf Meyer: Combable groups have group cohomology of polynomial growth LO: Logic --------- math.LO/0410523 Fredrik Engstrom: Expansions, omitting types, and standard systems MG: Metric Geometry ------------------- math.MG/0410566 Piotr W. Nowak: On coarse embeddability into $ell_p$-spaces and a conjecture of Dranishnikov MP: Mathematical Physics ------------------------ quant-ph/0410201 Kazuyuki Fujii: Jaynes-Cummings Model and a Non-Commutative Geometry : A Few Problems Noted math-ph/0410062 David Damanik, Daniel Lenz, Gunter Stolz: Lower Transport Bounds for One-Dimensional Continuum Schrodinger Operators math-ph/0410061 P. Di Francesco, P. Zinn-Justin: Razumov-Stroganov sum rule: a proof based on multi-parameter generalizations math-ph/0410060 A.W.Beckwith: How false vacuum synthesis of a universe sets initial conditions which permit the onset of variations of a nucleation rate per Hubble volume per Hubble time math-ph/0410059 Manfred Requardt: Supersymmetry on Graphs and Networks math-ph/0410058 Vitaly V. Bulatov, Yuriy V. Vladimirov, Vasily A. Vakorin: Weak Singularity for Two-Dimensional Nonlinear Equations of Hydrodynamics and Propagation of Shock Waves math-ph/0410057 Joseph V. Pule, Andre F. Verbeure, Valentin A. Zagrebnov: Models with Recoil for Bose-Einstein Condensation and Superradiance math-ph/0410056 Petko Nikolov, Tihomir Valchev: Description of all conformally invariant differential operators, acting on scalar functions astro-ph/0404408 Ing-Guey Jiang, Li-Chin Yeh: On the Chaotic Orbits of Disc-Star-Planet Systems math-ph/0410055 A. A. Hujeirat: Problem-orientable numerical algorithm for modelling multi-dimensional radiative MHD ßows in astrophysics -- the hierarchical solution scenario math-ph/0410054 A.A. Hujeirat: A method for enhancing the stability and robustness of explicit schemes in astrophysical ßuid dynamics hep-th/0410172 Beatriz Gato-Rivera: The Adapted Ordering Method in Representation Theory math-ph/0410053 Alexander Rybko, Senya Shlosman: Poisson Hypothesis for Information Networks II. Cases of Violations and Phase Transitions math-ph/0410052 A.C.D.van Enter, E.A.Verbitskiy: On the Variational Principle for Generalized Gibbs Measures math-ph/0410051 Xavier Gracia, Ruben Martin: Time-dependent singular differential equations math-ph/0410050 Nicolae Cotfas: Systems of orthogonal polynomials defined by hypergeometric type equations math-ph/0410049 Joachim Kupsch, Subhashish Banerjee: Ultracoherence and Canonical Transformations math-ph/0410048 Sebastian Bauer: Post-Newtonian approximation of the Vlasov-Nordstrom system hep-th/0410212 C. Chryssomalakos, E. Okon: Generalized Quantum Relativistic Kinematics: a Stability Point of View cond-mat/0410424 Sandeep Tyagi: New series representation for Madelung constant quant-ph/0410151 S. Twareque Ali, F. Bagarello: Some Physical Appearances of Vector Coherent States and CS Related to Degenerate Hamiltonians math-ph/0410047 Volodymyr Sushch: Discrete model of Yang-Mills equations in Minkowski space hep-th/0410109 S. Odake, R. Sasaki: Equilibrium Positions, Shape Invariance and Askey-Wilson Polynomials hep-th/0410102 S. Odake, R. Sasaki: Shape Invariant Potentials in Discrete Quantum Mechanics NA: Numerical Analysis ---------------------- math.NA/0410488 Paul Sablonniere: Recent Results on Near-Best Spline Quasi-Interpolants NT: Number Theory ----------------- math.NT/0410563 Dragos Ghioca: The Mordell-Lang Theorem for Drinfeld modules math.NT/0410536 construction of some Galois modules math.NT/0410531 Takashi Taniguchi: A mean value theorem for the square of class numbers of quadratic fields math.NT/0410522 A. Ivic, E. Kratzel, M. Kuhleitner, W.G. Nowak: Lattice points in large regions and related arithmetic functions: Recent developments in a very classic topic math.NT/0410519 Howard Kleiman: Bounds for the Solutions of Cubic Diophantine Equations math.NT/0410502 Avner Ash, David Pollack, Warren Sinnott: A_6-extensions of Q and the mod p cohomology of GL(3,Z) math.NT/0410477 Bernd C. Kellner: Some remarks on Kurepas left factorial OA: Operator Algebras --------------------- math.OA/0410607 P. S. Muhly, M. Skeide, B. Solel: Representations of B^a(E) math.OA/0410601 Marek Bozejko, Wlodzimierz Bryc: On a class of free Levy laws related to a regression problem math.OA/0410594 Kenley Jung: Some free entropy dimension inequalities for subfactors math.OA/0410587 Wei Wu: Non-commutative metric topology on matrix state space math.OA/0410534 Todd Kemp: Strong hypercontractivity in non-commutative holomorphic spaces math.OA/0410492 Gelu Popescu: Unitary invariants in multivariable operator theory math.OA/0410480 Marius Ionescu, Yasuo Watatani: $C^{ast}$-Algebras associated with Mauldin-Williams Graphs PR: Probability --------------- math.PR/0410569 Aaron Abrams, Henry Landau, Zeph Landau, James Pommersheim, Eric Zaslow: Random Multiplication Approaches Uniform Measure in Finite Groups math.PR/0410560 Elchanan Mossel, Ryan ODonnell, Oded Regev, Jeffrey Steif, Benjamin Sudakov: Non-interactive correlation distillation, inhomogeneous Markov chains, and the reverse Bonami-Beckner inequality math.PR/0410545 Ravi Montenegro: Vertex and edge expansion properties for rapid mixing math.PR/0410544 Frederik Herzberg: The fairest price of an asset in an environment of temporary arbitrage math.PR/0410543 Frederik Herzberg: On measures of unfairness and an optimal currency transaction tax math.PR/0410532 Anna Rudas: Random tree growth with general weight function math.PR/0410514 Jianjun Tian, Xiao-Song Lin: Colored Genealogical Trees and Coalescent Theory math.PR/0410510 Anna Karczewska: Properties of convolutions arising in stochastic Volterra equations math.PR/0410485 Jacques Franchi, Yves Le Jan: Relativistic Diffusions and Schwarzschild Space cond-mat/0410543 Federico Camia: A Note on Edwards Hypothesis for Zero-Temperature Ising Dynamics QA: Quantum Algebra ------------------- math.QA/0410605 D. Shklyarov, S. Sinelshchikov, L. Vaksman: Fock Representations and Quantum Matrices math.QA/0410562 Vasiliy Dolgushev, Pavel Etingof: Hochschild cohomology of quantized symplectic orbifolds and the Chen-Ruan cohomology math.QA/0410530 S. Sinelshchikov, L. Vaksman: Quantum Groups and Bounded Symmetric Domains math.QA/0410528 Michel Van den Bergh: Double Poisson algebras math.QA/0410486 Vladimir D. Lyakhovsky: On a class of skew classical r-matrices with large carrier RA: Rings and Algebras ---------------------- math.RA/0410591 Aaron Lauve: NSym into Q_{infty} is not a Hopf Map math.RA/0410576 Friedrich Wehrung, Jiri Tuma: Congruence lifting of diagrams of finite Boolean semilattices requires large congruence varieties math.RA/0410521 Jerzy Matczuk: Ore Extensions over Duo Rings RT: Representation Theory ------------------------- math.RT/0410588 Konstantin Styrkas: Regular representation on the big cell and big projective modules in the category O SG: Symplectic Geometry ----------------------- math.SG/0410609 Joa Weber: Noncontractible periodic orbits in cotangent bundles and Floer homology math.SG/0410608 Weimin Chen: Pseudoholomorphic curves in four-orbifolds and some applications math.SG/0410568 Eugene Lerman: Gradient ßow of the norm squared of a moment map SP: Spectral Theory ------------------- math.SP/0410577 Sergio Albeverio, Alexander K. Motovilov: Operator integrals with respect to a spectral measure and solutions to some operator equations ST: Statistics -------------- math.ST/0410586 Rasa Karapandza, Milos Bozovic: You Can Fool Some People Sometimes math.ST/0410574 Igor Podlubny: A note on comparison of scientific impact expressed by number of citations in different fields of science -- / Greg Kuperberg (UC Davis) / Home page: http://www.math.ucdavis.edu/~greg/ / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/ / * All the math thats fit to e-print * === Subject: 3rd cohomology of loop group Originator: israel@math.ubc.ca (Robert Israel) Id like to know the 3rd cohomology group of the loop group OmegaE_8 over E_8, H^3(OmegaE_8,H) , with H some abelian group. Does anyone know where I could find respective information? === Subject: Re: Open questions related to periodic continued fractions Originator: israel@math.ubc.ca (Robert Israel) Diana Mecum asked: > I am starting research for a thesis on continued fractions, and want > to look at open questions related to periodic continued fractions. Is > anyone aware of current open questions of interest? If n is not a perfect square, let p(n) be the length of the period of the s.c.f. of sqrt(n). In terms of n, how large can p(n) be? Its been shown that p(n) = O(sqrt(n) log(n)). It follows from some generalization of the Riemann hypothesis that p(n) = O(sqrt(n) log(log(n))). It seems likely that p(n)/sqrt(n) is unbounded, but I dont think its even been shown that it doesnt tend to 0. The first 23 record-setting values of p(n)/sqrt(n) are shown below: n p(n) p(n)/sqrt(n) 2 1 0.70711 3 2 1.15470 7 4 1.51186 43 10 1.52499 46 12 1.76930 211 26 1.78991 331 34 1.86881 631 48 1.91085 919 60 1.97922 1726 88 2.11818 4846 152 2.18349 7606 194 2.22445 10399 228 2.23583 10651 234 2.26736 10774 238 2.29292 18379 322 2.37517 19231 332 2.39407 32971 438 2.41217 48799 544 2.46260 61051 614 2.48497 78439 696 2.48510 82471 716 2.49323 111094 834 2.50219 See http://www.research.att.com/projects/OEIS?Anum=A003285 for some references. Dean Hickerson dean@math.ucdavis.edu === Subject: Re: Open questions related to periodic continued fractions > I am starting research for a thesis on continued fractions, and want > to look at open questions related to periodic continued fractions. Is > anyone aware of current open questions of interest? As you may know, the outstanding problem in this area is to improve on known conditions for the length of the period of the continued fraction expansion of sqrt{D} to be odd (D > 0, D not a square, equivalent to x^2 - Dy^2 = -1 having solutions). The following might be a current summary of known conditions: B. D. Beach and H. C. Williams, A Numerical Investigation of the Diophantine Equation x^2 - dy^2 = -1, Congressus Numerantium VI, Proceedings of the Third Southeastern Conference on Combinatorics, Graph Theory and Computing, Utilitas Mathematica Publishing Inc., Winnipeg, Canada, 1972, pages 37 to 52. A less well known problem is as follows. As far as I know, this is an open problem, but I cannot be sure. For D > 0 not a square, D congruent to 1 modulo 4, let L1(D) and L4(D) denote the lengths of the periods of the continued fraction expansions of sqrt{D} and (1+sqrt{D})/2 respectively. The question is whether it is possible to have L1(D) = L4(D) + 4 when L4(D) == 3 (mod 6). As far as I can tell, this is not prohibited by results in the literature. I have empirical evidence that it is not possible based on testing those D up to 30 billion that have L4(D) <= 255. It is not hard to show this for one particular case, namely if L4(D) = 3 then L1(D) cannot be 7. See discussion under the heading ``Periods of Continued Fractions in April and May of 2000 in the archives of the Number Theory Listserver at http://listserv.nodak.edu/archives/nmbrthry.html for related comments and some references that might be of interest. John Robertson === Subject: Re: Open questions related to periodic continued fractions Received-SPF: Received-SPF: none (mailbox4.ucsd.edu: domain of news@newsread1.news.pas.earthlink.net does not designate permitted sender hosts) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > I am starting research for a thesis on continued fractions, and want > to look at open questions related to periodic continued fractions. Is > anyone aware of current open questions of interest? > As you may know, the outstanding problem in this area is to improve on known > conditions for the length of the period of the continued fraction expansion of > sqrt{D} to be odd (D > 0, D not a square, equivalent to x^2 - Dy^2 = -1 having > solutions). The following might be a current summary of known conditions: > B. D. Beach and H. C. Williams, A Numerical Investigation of the > Diophantine Equation x^2 - dy^2 = -1, Congressus Numerantium VI, > Proceedings of the Third Southeastern Conference on Combinatorics, > Graph Theory and Computing, Utilitas Mathematica Publishing Inc., > Winnipeg, Canada, 1972, pages 37 to 52. > A less well known problem is as follows. As far as I know, this is an open > problem, but I cannot be sure. For D > 0 not a square, D congruent to 1 modulo > 4, let L1(D) and L4(D) denote the lengths of the periods of the continued > fraction expansions of sqrt{D} and (1+sqrt{D})/2 respectively. The question > is whether it is possible to have L1(D) = L4(D) + 4 when L4(D) == 3 (mod 6). > As far as I can tell, this is not prohibited by results in the literature. I > have empirical evidence that it is not possible based on testing those D up to > 30 billion that have L4(D) <= 255. It is not hard to show this for one > particular case, namely if L4(D) = 3 then L1(D) cannot be 7. > See discussion under the heading ``Periods of Continued Fractions in April > and May of 2000 in the archives of the Number Theory Listserver at > http://listserv.nodak.edu/archives/nmbrthry.html > for related comments and some references that might be of interest. > John Robertson === Subject: Re: Open questions related to periodic continued fractions Epigone-thread: drermestrin Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Regarding the maximal element in the CF of sqrt(n) (say M(n)) . Recently I noticed M(n)-2*sqrtint(n) = 0 or 1 and I suspect this behaviour : sum_{i non square <= n} 2*sqrt(i)-M(i) = c*n+o(n) where c is >0 and = 0.8... I dont know references on this subject. Should exist some. B. Cloitre === Subject: Re: Open questions related to periodic continued fractions Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Regarding the maximal element in the CF of sqrt(n) (say M(n)) . > Recently I noticed M(n)-2*sqrtint(n) = 0 or 1 and I suspect this > behaviour : > sum_{i non square <= n} 2*sqrt(i)-M(i) = c*n+o(n) where c is >0 > and = 0.8... > I dont know references on this subject. Should exist some. Im not sure what ``sqrtint(n) is. The maximum partial quotient in the continued fraction expansion of sqrt(n) (for n not a square) is 2 times the integer part of the square root of n. This is proved in most references that consider the continued fraction expansion of sqrt(n), e.g., Mollins Fundamental Number Theory with Applications, or Niven, Zuckerman, and Montgomery. Also, (1/n) Sum_{i=1}^{n} [sqrt(i) - int(sqrt(i))] tends to 1/2. John Robertson === Subject: Re: Open questions related to periodic continued fractions Epigone-thread: drermestrin Originator: bergv@math.uiuc.edu (Maarten Bergvelt) >Regarding the maximal element in the CF of sqrt(n) (say M(n)) . >Recently I noticed M(n)-2*sqrtint(n) = 0 or 1 and I suspect this >behaviour : >sum_{i non square <= n} 2*sqrt(i)-M(i) = c*n+o(n) where c is >0 and = >0.8... >I dont know references on this subject. Should exist some. >B. Cloitre May be the following remarks are all well known: observe for instance sqrt(41) giving sequence (6;1/2,1/2,1/12,...) the period (1/2,1/2,1/12 ..)is linked to function (1/2,1/2,1/12,x) or (5x+62)/(2x+25) with two fixed points ,the positive -6+sqrt(41) is related to our continuous fraction. f(x)=(5x+62)/(2x+25) is easily iterated (sci.math 20 oct), Friendly yours,Alain. === Subject: Re: Open questions related to periodic continued fractions Epigone-thread: drermestrin Content-Length: 2687 Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Diana, in case it can help. Is anything known about the continued fraction of log_2(3)? (that is, of the logarithm of 3, base 2) Is it bounded,...!?? Maybe there are some results on the continued fractions of logarithms.. I conjecture, with the standard notations, that a_{n+1}/q_n <= 1/4, for that number. (One more thing, is log_2(3) algebraic or trascendent?..!) Jose >> I am starting research for a thesis on continued fractions, and want >> to look at open questions related to periodic continued fractions. Is >> anyone aware of current open questions of interest? >> As you may know, the outstanding problem in this area is to improve on >known >> conditions for the length of the period of the continued fraction >expansion of >> sqrt{D} to be odd (D > 0, D not a square, equivalent to x^2 - Dy^2 = -1 >having >> solutions). The following might be a current summary of known conditions: >> B. D. Beach and H. C. Williams, A Numerical Investigation of the >> Diophantine Equation x^2 - dy^2 = -1, Congressus Numerantium VI, >> Proceedings of the Third Southeastern Conference on Combinatorics, >> Graph Theory and Computing, Utilitas Mathematica Publishing Inc., >> Winnipeg, Canada, 1972, pages 37 to 52. >> A less well known problem is as follows. As far as I know, this is an >open >> problem, but I cannot be sure. For D > 0 not a square, D congruent to 1 >modulo >> 4, let L1(D) and L4(D) denote the lengths of the periods of the continued >> fraction expansions of sqrt{D} and (1+sqrt{D})/2 respectively. The >question >> is whether it is possible to have L1(D) = L4(D) + 4 when L4(D) == 3 (mod >6). >> As far as I can tell, this is not prohibited by results in the literature. >> have empirical evidence that it is not possible based on testing those D >up to >> 30 billion that have L4(D) <= 255. It is not hard to show this for one >> particular case, namely if L4(D) = 3 then L1(D) cannot be 7. >> See discussion under the heading ``Periods of Continued Fractions in >April >> and May of 2000 in the archives of the Number Theory Listserver at >> http:// listserv.no dak.edu/archives/nmbrthry.html for related comments and some references that might be of interest. >> John Robertson === Subject: Re: Open questions related to periodic continued fractions Received-SPF: Received-SPF: none (mailbox3.ucsd.edu: domain of news@nntp.itservices.ubc.ca does not designate permitted sender hosts) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Diana, > in case it can help. > Is anything known about the continued fraction of log_2(3)? (that >is, of the logarithm of 3, base 2) > Is it bounded,...!?? > Maybe there are some results on the continued fractions of >logarithms.. I doubt that this will help. Very little is known about the continued fractions of closed-form numbers apart from rationals and quadratic irrationals, and AFAIK there is no prospect, with currently available mathematical techniques, of being able to prove whether the continued fraction of a number such as this has bounded elements. IMHO this is not a problem to give to a student starting her thesis. > I conjecture, with the standard notations, that a_{n+1}/q_n <= 1/4, >for that number. > (One more thing, is log_2(3) algebraic or trascendent?..!) Transcendental, by the Gelfond-Schneider Theorem. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Open questions related to periodic continued fractions Originator: bergv@math.uiuc.edu (Maarten Bergvelt) > Diana, > in case it can help. > Is anything known about the continued fraction of log_2(3)? (that > is, of the logarithm of 3, base 2) > Is it bounded,...!?? No, its unbounded. But, so far, nobody knows how to prove that. > Maybe there are some results on the continued fractions of > logarithms.. > I conjecture, with the standard notations, that a_{n+1}/q_n <= 1/4, > for that number. How does that compare to almost all continued fractions? > (One more thing, is log_2(3) algebraic or trascendent?..!) It is transcendental. -- G. A. Edgar edgar at math.ohio-state.edu === Subject: Re: Galois group of a given quartic equation Epigone-thread: twomprendlex Originator: israel@math.ubc.ca (Robert Israel) The pending issue is whether the quartic Q(x) = 0 may have the Galois group isomorphic to Z4 for a and b different. No progress has been posted. One way to rule out D4 in the Z4/D4 case is given below. (If Q(x) is irreducible over Z, the discriminant is not a square and the cubic resolvent R(t) = 0 of the quartic has one and only one integral root t0, then the Galois group G is either Z4 or D4.) Let R(t) = (t - t0) r(t) where r(t) is a monic irreducible quadratic with integral coefficients. Further, let the discriminant of r(t) be D. Let d be the squarefree part of D and E the splitting field of R(t) = 0. Then E = Q[Sqrt[d]] and therefore easy to determine. Then put E to use as follows: If Q(x) is reducible over E then G is Z4; otherwise G is D4. May be this can be used to settle the case whether G may be Z4 or not. Kent Holing === Subject: matrix derivative Epigone-thread: zoypryrwhou Originator: israel@math.ubc.ca (Robert Israel) I am currently having a matrix derivative problem What is derivative of trace{(A+F*B*F)^(-1)} with respect to matrix F where () is the transpose of a matrix. A and B is diagonal matrices. I searched online and was only able to find the derivative of d trace{(F*B*F)^(-1)}/d F =-2*B*F*(F*B*F)^(-2) without knowing how they get it. Moreover, all these matrix derivative problems seem to be difficult forme. Could anyone be kind enough to give me some good references on this topic. === Subject: Re: matrix derivative Originator: israel@math.ubc.ca (Robert Israel) > I am currently having a matrix derivative problem > What is derivative of > trace{(A+F*B*F)^(-1)} > with respect to matrix F > where () is the transpose of a matrix. > A and B is diagonal matrices. > I searched online and was only able to find the derivative of > d trace{(F*B*F)^(-1)}/d F > =-2*B*F*(F*B*F)^(-2) > without knowing how they get it. > Moreover, all these matrix derivative problems seem to be > difficult forme. Could anyone be kind enough to give me > some good references on this topic. d A denotes the matrix differential for the matrix A. x = trace{ (A+F*B*F)^(-1) } d x = trace{ d ( (A+F*B*F)^(-1) ) } = trace{ - (A+F*B*F)^(-1) ( d (A+F*B*F) ) (A+F*B*F)^(-1) } = trace{ - (A+F*B*F)^(-2) ( d (F*B*F) ) } = trace{ - (A+F*B*F)^(-2) ( d F *B*F + F*B* d F ) } = - ( trace{ (A+F*B*F)^(-2) d F *B*F } + trace{ (A+F*B*F)^(-2) F*B* d F } ) = - trace{ B * F * (A+F*B*F)^(-2) d F } - trace{ (A+F*B*F)^(-2) F*B* d F } = - trace{ (A+F*B*F)^(-2) * F * B * d F} - trace{ (A+F*B*F)^(-2) F*B* d F } (Note that B is symmetric, and B=B; also, (A+F*B*F) is also symmetric) = -2 trace{ (A+F*B*F)^(-2) * F * B * d F} Therefore, by the first identification theorem, the differential is as given above. Recall that trace( (d x / d F) * d F ) = d x I find working with matrix differential much easier than other alternatives. The book recommended by Peter has a great account on how to manipulate matrix differential. It also provides the theorectical justification. Hope that helps. === Subject: Re: matrix derivative Originator: israel@math.ubc.ca (Robert Israel) >I am currently having a matrix derivative problem >What is derivative of >trace{(A+F*B*F)^(-1)} >with respect to matrix F >where () is the transpose of a matrix. >A and B is diagonal matrices. >I searched online and was only able to find the derivative of >d trace{(F*B*F)^(-1)}/d F >=-2*B*F*(F*B*F)^(-2) >without knowing how they get it. >Moreover, all these matrix derivative problems seem to be >difficult forme. Could anyone be kind enough to give me >some good references on this topic. Derivatives are not appropriate for functions of more than one variable, and you are asking for derivatives with respect to a matrix. Differentials satisfy the usual properties, but the failure of commutativity is quite important. So if q is a differential function of an argument in a locally ßat space, the Frechet derivative is dq(x, m) = lim ((q(x+em) - q(x))/e) as e -> 0. Using this, and the result that d(X^(-1)) = - X^(-1) dX X^(-1), we get that the differential you seek is trace (-2*(A+F*B*F)^(-1)*F*B*dF*(A+F[CapitalOTilde ]*B*F)^(-1)); this uses the results about trace being invariant under permutation and transposition. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: matrix derivative Originator: israel@math.ubc.ca (Robert Israel) >I am currently having a matrix derivative problem >What is derivative of >trace{(A+F*B*F)^(-1)} >with respect to matrix F >where () is the transpose of a matrix. >A and B is diagonal matrices. >I searched online and was only able to find the derivative of >d trace{(F*B*F)^(-1)}/d F >=-2*B*F*(F*B*F)^(-2) >without knowing how they get it. >Moreover, all these matrix derivative problems seem to be >difficult forme. Could anyone be kind enough to give me >some good references on this topic. some reference which might help: Magnus, J.R., Neudecker, H. Matrix Differential Calculus with Applications in Statistics and Econometrics. paul fackler has some notes at http://www4.ncsu.edu/~pfackler/MATCALC.ps hth peter === Subject: standard probability spaces Originator: israel@math.ubc.ca (Robert Israel) let $(Omega,F)$ be a standard probability space and $X:[0,t]times Omegato E$ a stochastic process with values in a Polish space $E$ and RCLL trajectories. Is it true that $(Omega,F^X)$ is standard where $F^X$ denotes the $sigma$-algebra generated by $X$? J. p.s.: Please reply to email as well. Thx. === Subject: Average number of vectors on a plane Originator: israel@math.ubc.ca (Robert Israel) Hi there, I try to solve the following problem: Given N points random (lets say with mean m) distributed on a plane(2D). Each of these points can be the beginning or the end of a vector of length less than R. Under the condition that the beginning of each vector must have distance greater than R from the end of all the other vectors, what is the average number of vectors I can have on the given plane? (e.g. if (a_i) is the beginning point and (b_i) is the end point of the i-th vector, what is the mean{i} under the constrains ||a_i-b_i||<=R and ||a_i-b_j||>R [for i != j] , where ||x-y|| denotes the Euclidean diatance) Any ideas on which direction I have to look or about any related work it would be very grateful. Thanos === Subject: Two papers published by AGT Originator: israel@math.ubc.ca (Robert Israel) The following two papers have been published: (1) The conjugacy problem for relatively hyperbolic groups by Inna Bumagin URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-43.abs.html (2) Mp-small summands increase knot width by Jacob Hendricks URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-44.abs.html Full details follow: (1) Algebraic and Geometric Topology URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-43.abs.html Title: The conjugacy problem for relatively hyperbolic groups Author(s): Inna Bumagin Abstract: Solvability of the conjugacy problem for relatively hyperbolic groups was announced by Gromov [Hyperbolic groups, MSRI publications 8 (1987)]. Using the definition of Farb of a relatively hyperbolic group in the strong sense [B Farb, Relatively hyperbolic groups, Geom. Func. Anal. 8 (1998) 810-840], we prove this assertion. We conclude that the conjugacy problem is solvable for fundamental groups of complete, finite-volume, negatively curved manifolds, and for finitely generated fully residually free groups. Secondary: 20F10 Keywords: Negatively curved groups, algorithmic problems Received: 5 May 2002 Author(s) address(es): Department of Mathematics and Statistics, Carleton University 1125 Colonel By Drive, Herzberg Building Ottawa, Ontario, Canada K1S 5B6 Email: bumagin@math.carleton.ca (2) Algebraic and Geometric Topology URL: http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-44.abs.html Title: Mp-small summands increase knot width Author(s): Jacob Hendricks Abstract: Scharlemann and Schultens have shown that for any pair of knots K_1 and K_2, w(K_1 # K_2) >= max{w(K_1),w(K_2)}. Scharlemann and Thompson have given a scheme for possible examples where equality holds. Using results of Scharlemann-Schultens, Rieck-Sedgwick and Thompson, it is shown that for K= #_{i=1}^n K_i a connected sum of mp-small knots and K any non-trivial knot, w(K # K)>w(K). Secondary: 57M27 Keywords: Thin position, knot width Author(s) address(es): Department of Mathematics, University of Arkansas, Fayetteville, AR 72701, USA Email: jghendr@uark.edu === Subject: An Algebraic Structure Originator: israel@math.ubc.ca (Robert Israel) I have found a concrete and non trivial example of the following algebraic structure: Let V an non empty set where are defined two binary operations +,*: V->VxV and there exists a non empty subset D such that: (1) +,* are associative and commutative. (2) * is distributive with respect to +. (3) * has a neutral element: 1. (4) each element in (V-D) has an multiplicative inverse in (V-D) . (5) forall v in (V-D) exists w in (V-D) such that v+w is in D ( existence of additive pseudo-inverse). (6) forall v,v in V : v*v in D->v is in D or v is in D. It is something which look like as a field and it is a field when D contains only one element. If this stucture doesnt have a name I would call it pseudo-field (anyway do not stick on this detail). Can please anybody provide me reference sources for this algebraic structure, if any? Roberto Volpe PS I hope now is more readable of the previous version made with cut and paste by my LaTeX note. === Subject: Re: An Algebraic Structure Originator: israel@math.ubc.ca (Robert Israel) Roberto Volpe in litteris scripsit: > Let V an non empty set where are defined two binary operations +,*: V->VxV > and there > exists a non empty subset D such that: > (1) +,* are associative and commutative. > (2) * is distributive with respect to +. > (3) * has a neutral element: 1. > (4) each element in (V-D) has an multiplicative inverse in (V-D) . > (5) forall v in (V-D) exists w in (V-D) such that v+w is in D ( existence > of additive pseudo-inverse). > (6) forall v,v in V : v*v in D->v is in D or v is in D. Any (commutative) local ring satisfies these axioms and more (essentially, the converse of (6) and the existence of additive inverses not just pseudo-inverses wrt D) if D is the maximal ideal. I believe your axioms are, as they stand, too weak to make much of (for example, V = the set of natural numbers, and D = all of V, seems to work; and you can always replace a pair (V,D) which works by taking D to be all of V). -- David A. Madore (david.madore@ens.fr, http://www.dma.ens.fr/~madore/ ) === Subject: Re: An Algebraic Structure Originator: israel@math.ubc.ca (Robert Israel) David Madore ha scritto nel messaggio > I believe your axioms are, as they stand, too weak to make much of > (for example, V = the set of natural numbers, and D = all of V, seems > to work; and you can always replace a pair (V,D) which works by taking > D to be all of V). You are right I was vague when I mentioned in my original post that I found a non trivial example. For me a non trivial example means that D is a proper subset of V. Most likely I have found a local ring . Roberto Volpe === Subject: Re: Please help me solve this riddle by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iA76F7x15014; >what runs but doesnt walk, >what has a month but doesnt talk, >what has a bed but doesnt sleep, >what has a head but doesnt weep. A RIVER === Subject: Re: Precalculus Help! by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iA76F8J15033; help! solve for x: 0.5(a^x-a^-x)3, a>0 solve for x: log(base 5)(x+1)<2 solve for x:(base b)=(b) log(b)(3x+2)+log(b)64+log(b)2-log(b)(3x-2) what characterizes exponential functions? === Subject: Re: Precalculus Help! >help! >solve for x: >0.5(a^x-a^-x)3, a>0 [snip more problems] >what characterizes exponential functions? Were not averse to helping with homework, but it is not help simply to give you answers that you could get by reading your textbook. Please show us what youve done to solve these problems, and then we can give you specific, focused help where you got stuck. -- Stan Brown, Oak Road Systems, Tompkins County, New York, USA http://OakRoadSystems.com/ And if youre afraid of butter, which many people are nowa- days, (long pause) you just put in cream. --Julia Child === Subject: Re: Precalculus Help! memogoez@yahoo.com wants help: >help! >solve for x: >0.5(a^x-a^-x)3, a>0 That is an expression. What is the relation? What is the expression supposed to equal? G C === Subject: Prob1. For x as belonging to the set [-10, 12], f (x) = [x^4][(x-4)^7]. On which two intervals is the function increasing? Find the region in which the function is positive. Where does the function achieve abs. minimum? For the first portion, I find f, which is [4(x^3) (x-4)^7] + [(7(x^4)(x-4)^6]. I set this to zero. X equals 0 and 4, I think. (I couldnt actually get any neat quadratic out of this or anything, I simply assumed that so long as all the multiplicative coefficients had an X, a zero would set it all to zero, and since they all had a multiplicative factor of x-4, an x of 4 would set them all to zero. I test the various parts of f. I find that -Inf to -10 increases, -10 to 0 increases, 0 to 4 decreases, and 4 to 12 increases. I put in the answer -10 to 0, and 4 to 12, into webworks. I then set the function itself to 0, to find where it crosses the x-axis, in search of where the function is positive. I test the various parts of the function. I get -10 to 0 neg, 0 to 4 pos, 4 to 12 pos, 12 to inf. pos. Testing the various values, abs. min is at -10. So, the answers I put in: Two increasing intervals: -10 to 0, 4 to 12. Positive region of function: 0 to 12. Absolute min. locale: -10. Webworks says: -10 to 0 is correct. Of 4 to 12, the 12 is correct, the 4 is not. Of the 0 to 12, the 12 is correct, the 0 is not. Abs. min of -10 is correct. Whered I go wrong? === Subject: Re: Prob1. alt.math.undergrad: >For x as belonging to the set [-10, 12], f (x) = [x^4][(x-4)^7]. >On which two intervals is the function increasing? Find the region in which >the function is positive. Where does the function achieve abs. minimum? >For the first portion, I find f, which is [4(x^3) (x-4)^7] + >[(7(x^4)(x-4)^6]. THANK YOU for showing your work! It makes it so much easier to help you. I agree with your derivative, and also with your solution for critical points as far as it goes: >I set this to zero. X equals 0 and 4, I think. (I couldnt actually get any >neat quadratic out of this or anything, I simply assumed that so long as all >the multiplicative coefficients had an X, a zero would set it all to zero, >and since they all had a multiplicative factor of x-4, an x of 4 would set >them all to zero. Thats true, but you can actually factor this equation -- and you really need to factor it, to remove any doubt that youre missing any additional roots. f(x) = 4x^3 (x-4)^7 + 7x^4 (x-4)^6 You see common factors of x^3 and (x-4)^6, so you rewrite f(x) = x^3 (x-4)^6 [ 4(x-4) + 7x ] = x^3 (x-4)^6 (11x-16) There are THREE zeroes of f: 0, 4, and 16/11. Since all three are in the stated domain [-10, 12], all must be considered. >I test the various parts of f. I find that -Inf to -10 increases, Actually theres no need to test that interval since its not in the stated domain. >-10 to 0 increases, Correct. On that interval x^3 is negative but so is (11x-16); therefore f(x) is positive on that interval and f(x) is increasing. (Theres no need to consider the (x-4)^6 factor of f(x) since its never negative.) > 0 to 4 decreases, Here we disagree. x=16/11 is a critical point. positive and therefore f(x) increases. >and 4 to 12 increases. Yup. Summary: f(x) increases on (-10, 0) decreases on (0, 16/11) increases on (16/11, 12) >I put in the answer -10 to 0, and 4 to 12, into webworks. I dont know what that is. The function achieves absolute minimum either at x=-10 or x=12 (the endpoints of the closed interval) or at x=16/11 (where the function stops decreasing and starts increasing). Substitution shows that f(- 10) is far smaller than f(16/11) or f(12). >Testing the various values, abs. min is at -10. Correct. You lucked out because though you missed f(16/11) = 0 where f(16/11) = about -3100, the function value at the left end of the interval is on the order of -10^12. >I then set the function itself to 0, to find where it crosses the x-axis, in >search of where the function is positive. I test the various parts of the >function. I get -10 to 0 neg, 0 to 4 pos, 4 to 12 pos, 12 to inf. pos. Look again at the function: x^4 (x-4)^7. x^4 is never negative, so the function is positive or negative where (x-4)^7 is positive or negative. But (x-4^ is negative for x<4 and positive for x>4. Therefore I conclude f(x) is negative on [-10,0) f(0) = 0 f(x) is negative on (0,4) f(4) = 0 f(x) is positive on (4,12) -- Stan Brown, Oak Road Systems, Tompkins County, New York, USA http://OakRoadSystems.com/ And if youre afraid of butter, which many people are nowa- days, (long pause) you just put in cream. --Julia Child === Subject: Re: Prob1. days. My association with the Department is that of an alumnus. >For x as belonging to the set [-10, 12], f (x) = [x^4][(x-4)^7]. >On which two intervals is the function increasing? Find the region in which >the function is positive. Where does the function achieve abs. minimum? >For the first portion, I find f, which is [4(x^3) (x-4)^7] + >[(7(x^4)(x-4)^6]. >I set this to zero. X equals 0 and 4, I think. It is not that hard: you have f(x) = 4x^3(x-4)^7 + 7x^4(x-4)^6. Factor out x^3(x-4)^6 to get f(x) = x^3(x-4)^6*[ 4(x-4) + 7x] = x^4(x-4)^6*[4x-16+7x] = x^3(x-4)^6*[11x - 16] which is zero at x=0, x=4, and x = 16/11; all three lie in your interval. >(I couldnt actually get any >neat quadratic out of this or anything, I simply assumed that so long as all >the multiplicative coefficients had an X, a zero would set it all to zero, >and since they all had a multiplicative factor of x-4, an x of 4 would set >them all to zero. I have no idea what you are saying here. But I dindt try too hard. >I test the various parts of f. I find that -Inf to -10 increases, -10 to 0 >increases, 0 to 4 decreases, and 4 to 12 increases. f(x) = x^3(x-4)^6[11x-16]. Note that (x-4)^6 is always greater than or equal to 0. 11x-16 is positive on (16/11, infinity) and negative on (-infinity,16/11). x^3 is negative on (-infinity,0) and positive on (0,infinity). You are only interested in [-10,12]. The derivative is then positive on [-10,0) (two negative factors, one positive); it is negative on (0,16/11) (one negative, two positives), positive on (16/11,4) and on (4,12]. Thus the function f(x) is increasing on [-10,0] and on [16/11,12]. It is decreasing on [0,16/11] The critical points are x=0, x=4, and x=16/11. To find the absolute extremes, we plug in -10, 0, 4, 16/11, and 12 into f, and compare the values. Largest value is the absolute maximu, smallest value the absolute minimum. [.snip.] >Whered I go wrong? You didnt try hard enough to figure out where f(x) was zero. -- Its not denial. Im just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Prob1. >>For the first portion, I find f, which is [4(x^3) (x-4)^7] + >>[(7(x^4)(x-4)^6]. >>(I couldnt actually get any >>neat quadratic out of this or anything, I simply assumed that so long as >>all >>the multiplicative coefficients had an X, a zero would set it all to zero, >>and since they all had a multiplicative factor of x-4, an x of 4 would set >>them all to zero. > I have no idea what you are saying here. But I dindt try too hard. Yes, I was unclear, and used poor terminology. Well, the two portions of the polynomial have the terms of 7x^4 and 4x^3 outside the parenthesis, so any x of 0 would set everything to 0 (since these terms would become 0, and multiply everything in the parenthesis out to 0.) Both terms *inside* the parenthesis were x-4, so any x of 4 would become 4-4=0, which would make everything *outside* the parenthesis multiply to zero. So, any X of 0 or 4 would make everything 0. Also, thank you for your help. === Subject: Re: Prob1. days. My association with the Department is that of an alumnus. >For the first portion, I find f, which is [4(x^3) (x-4)^7] + >[(7(x^4)(x-4)^6]. >(I couldnt actually get any >neat quadratic out of this or anything, I simply assumed that so long as >all >the multiplicative coefficients had an X, a zero would set it all to zero, >and since they all had a multiplicative factor of x-4, an x of 4 would set >them all to zero. >> I have no idea what you are saying here. But I dindt try too hard. >Yes, I was unclear, and used poor terminology. Well, the two portions of the >polynomial have the terms of 7x^4 and 4x^3 outside the parenthesis, so any x >of 0 would set everything to 0 (since these terms would become 0, and >multiply everything in the parenthesis out to 0.) >Both terms *inside* the parenthesis were x-4, so any x of 4 would become >4-4=0, which would make everything *outside* the parenthesis multiply to >zero. >So, any X of 0 or 4 would make everything 0. In other words: you have two expressions adding; you figured out how to make BOTH expressions equal to zero, and decided that this was the only way in which the sum could be equal to zero. Not just wrong, but damn wrong. It only gives you SOME zeros. -- Its not denial. Im just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Prob1. alt.math.undergrad: >For the first portion, I find f, which is [4(x^3) (x-4)^7] + >[(7(x^4)(x-4)^6]. >Both terms *inside* the parenthesis were x-4, so any x of 4 would become >4-4=0, which would make everything *outside* the parenthesis multiply to >zero. >So, any X of 0 or 4 would make everything 0. True. But just because you find _some_ zeroes does not prove you have found _all_. The only choice is to factor, or use another method such as numeric solution (which has its own perils). -- Stan Brown, Oak Road Systems, Tompkins County, New York, USA http://OakRoadSystems.com/ And if youre afraid of butter, which many people are nowa- days, (long pause) you just put in cream. --Julia Child === Subject: Re: Please help me solve this riddle by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iA81mrk17074; me? well, the riddle is: soft as petal that falls from the tree as i dry and the wetter Ill be. well, anything pop up in your mind, then === Subject: Re: Please help me solve this riddle > me? well, the riddle is: soft as petal that falls from the tree as i > dry and the wetter Ill be. well, anything pop up in your mind, then Please post riddles in the rec.puzzles news group rather than a mathematical group. Ken Pledger. === Subject: Re: Equation Analysis by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iA82dHR21246; i need you help for more equations!! asap === Subject: Tangent line by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iA8Gfad30232; How do I go about trying to solve this problem? The tangent like to the curve y=x^3-6x^2-34x-9 has slope 2 at two points on the curve. Find the two points. === Subject: Re: Tangent line by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAF2RA714568; >How do I go about trying to solve this problem? >The tangent like to the curve y=x^3-6x^2-34x-9 has slope 2 at two >points on the curve. Find the two points. this is a good question! first note that a tangent line on a cubic graph is going to touch the graph twice, no more. we should start by setting up equations 2x+b=y x^3-6x^2-34x-9=y so then=> x^3-6x^2-34x-9=2x+b x^3-6x^2-36x-(9+b)=0; where b makes the graph only have two solutions that are distinct. -3(x^3-6x^2-(6^2)x)=-3(9+b) x^3+3(6x^2)+3(6^2x)+6^3=-3(9+b)+6^3+4x^3 (x+6)^3=4x^3+(6^3-27-3b) denote c as -(6^3-27-3b)/4 {(x+6)^3}/4=x^3-c it is necessary at this moment to consider the fact that when x^3-c is going to be factored, an (x+6) must be divisible by it (because we want to bring it down to a quadratic in order to have two solutions) so 4(x+6)(x^2-6x+36)= (x+6)^3 or 4(x^2-6x+36)= x^2+12x+36 3x^2-36x+108=0 3(x-6)(x-6)=0 so x=6 -matt overduin === Subject: Re: Tangent line >How do I go about trying to solve this problem? >The tangent like to the curve y=x^3-6x^2-34x-9 has slope 2 at two >points on the curve. Find the two points. What is x when dy/dx = 2? === Subject: Re: Tangent line > How do I go about trying to solve this problem? > The tangent like to the curve y=x^3-6x^2-34x-9 has slope 2 at two > points on the curve. Find the two points. The slope at x is dy/dx = 3x^2 - 12x - 34. The slope is 2 when 3x^2 - 12x - 34 = 2. I.e. 3x^2 - 12x -36 = 0 So 3(x + 2)(x - 6) = 0 and x = -2 or x = 6. === Subject: Re: Tangent line by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iA8Hol804598; >How do I go about trying to solve this problem? >The tangent like to the curve y=x^3-6x^2-34x-9 has slope 2 at two >points on the curve. Find the two points. The derivative at x is the slope of the tangent at x. y = 3x^2 - 12x - 34 = 2 3x^2 - 12x - 36 = 0 x^2 - 4x - 12 = 0 x = [4 +- sqrt(4^2 + 4*12)]/2 = [4 +- sqrt(36)]/2 = (4 +- 6)/2 = = 5 or -1 === Subject: Re: Tangent line >How do I go about trying to solve this problem? >The tangent like to the curve y=x^3-6x^2-34x-9 has slope 2 at two >points on the curve. Find the two points. Do you know what the slope of a curve is, and how to compute it? Look in your book, find how to compute the slope of a curve, and set it equal to 2. Continue from there. --Lynn === Subject: Limit Definition by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iA8Gfai30228; please help me solve this problem, i have no idea what to do... Using the limit definition of derivative, show that the derivative of 1/(3x^2) is -2/(3x^3) === Subject: Re: Limit Definition > please help me solve this problem, i have no idea what to do... > Using the limit definition of derivative, show that the derivative of > 1/(3x^2) is -2/(3x^3) f(x) = 1/(3*x^2) f(x) = lim_{h->0} (f(x+h)-f(x))/h === Subject: Re: Limit Definition >please help me solve this problem, i have no idea what to do... >Using the limit definition of derivative, show that the derivative of >1/(3x^2) is -2/(3x^3) Please review the limit definition of derivative in your textbook. Pay particular attention to the example you find there; it gives you a model to follow in solving this problem. General rule: When you have no idea what to do it almost surely means that you need to study your textbook before you do anything else. Somewhere between the time I went to college and now, students were coned into believing that the lecture is enough for them to start right in doing problems. Thats very far from true. You might like to have a look at my How to Succeed in Math at . -- Stan Brown, Oak Road Systems, Tompkins County, New York, USA http://OakRoadSystems.com/ And if youre afraid of butter, which many people are nowa- days, (long pause) you just put in cream. --Julia Child === Subject: Re: Limit Definition by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iA8HThD02341; >please help me solve this problem, i have no idea what to do... >Using the limit definition of derivative, show that the derivative of >1/(3x^2) is -2/(3x^3) f(x) = lim [f(x + h) - f(x)]/h = lim [1/(3(x + h)^2) - 1/(3x^2)]/h h->0 h->0 = lim [x^2 - (x + h)^2]/[h*3x^2*(x + h)^2] = h->0 = lim [x^2 - x^2 - 2hx - h^2]/[h*3x^2*(x + h)^2] = h->0 = lim [-h*(2x + h)]/[h*3x^2*(x + h)^2] = h->0 = -lim (2x + h)/[3x^2*(x + h)^2] = -2x/[3x^2*x^2] = -2/(3x^3) h->0 === Subject: Re: Limit Definition >please help me solve this problem, i have no idea what to do... >Using the limit definition of derivative, show that the derivative of >1/(3x^2) is -2/(3x^3) Start with f(x) = 1/(3x^2). Compute (f(x + h) - f(x) )/ h. Simplify it. Let h -> 0. If you have sufficiently simplified your expression, you can likely just put h = 0 in the last step. --Lynn === Subject: Question about even and odd numbers by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iA8IfL109058; hi I know how to solve this,but question is indirectly related to this and similar problems Three consecutive even integers are such that the square of the first plus the square of the third is 136. find the three integers. It can be solved in two ways: 1-With use of 2n notation for even number (2n-2)(2n-2)+(2n+2)^2=136 n=4 or 2-that we dont use 2n notation for even numbers 136 = (x-2)(x-2)+(x+2)(x+2) x = 4 QUESTIONS: -When is the notation 2x for even or 2x+1 for odd numbers absolutely necessary? -if I instead of (x-2) and (x+2) use x and (x+4) equation cant be solved.Is there something as a rule that says that both variables have to be expressed by third variable or something similar? thank you === Subject: Re: Question about even and odd numbers > hi > I know how to solve this,but question is indirectly related to this > and similar problems > Three consecutive even integers are such that the square of the first > plus the square of the third is 136. find the three integers. > It can be solved in two ways: > 1-With use of 2n notation for even number > (2n-2)(2n-2)+(2n+2)^2=136 > n=4 > or > 2-that we dont use 2n notation for even numbers > 136 = (x-2)(x-2)+(x+2)(x+2) > x = 4 > QUESTIONS: > -When is the notation 2x for even or 2x+1 for odd numbers absolutely > necessary? It is helpful for some problems. Is it ever *absolutely* necessary? > -if I instead of (x-2) and (x+2) use x and (x+4) equation cant be > solved. Lets try: The numbers are x, x+2, x+4, so 136 = x^2 + (x + 4)^2 = 2x^2 + 8x + 16 which gives x = -10 and the numbers are -10, -8, -6 or x = 6 and the numbers are 6, 8, 10. Check: in the first case (-10)^2 + (-6)^2 = 136 in the second case 6^2 + 10^2 = 136. I think youve made a mistake. === Subject: Re: Question about even and odd numbers >I know how to solve this,but question is indirectly related to this >and similar problems >Three consecutive even integers are such that the square of the first >plus the square of the third is 136. find the three integers. >-When is the notation 2x for even or 2x+1 for odd numbers absolutely >necessary? Never, as far as I know. As a general rule, most word problems are easiest to solve when the you assign a variable as the quantity to be found, or one of the quantities to be found. So ordinarily you would make consecutive even _or_ odd numbers x, x+2, x+4 or x-2, x, x+2, or similar. As I say, there may be an exception where this makes things harder, but I cant think of one off hand. >-if I instead of (x-2) and (x+2) use x and (x+4) equation cant be >solved. Sure it can! x^2 + (x+4)^2 = 136 x^2 + x^2 + 8x + 16 = 136 2x^2 + 8x - 120 = 0 x^2 + 4x - 60 = 0 (x+10)(x-6) = 0 x = 6 or -10 The three numbers are either 6, 8, 10 or -10, -8, -6. Check: 6^2 + 10^2 = 136 or (-10)^2 + (-6)^2 = 136 If you dont consider that negative numbers can be even or odd, you would exclude the second solution. -- Stan Brown, Oak Road Systems, Tompkins County, New York, USA http://OakRoadSystems.com/ And if youre afraid of butter, which many people are nowa- days, (long pause) you just put in cream. --Julia Child === Subject: Re: Question about even and odd numbers Its quite amazing. After a two hour gap, two replies come simultaneously. Just like buses. === Subject: Re: Question about even and odd numbers > hi > I know how to solve this,but question is indirectly related to this > and similar problems > Three consecutive even integers are such that the square of the first > plus the square of the third is 136. find the three integers. > It can be solved in two ways: > 1-With use of 2n notation for even number > (2n-2)(2n-2)+(2n+2)^2=136 > n=4 > or > 2-that we dont use 2n notation for even numbers > 136 = (x-2)(x-2)+(x+2)(x+2) > x = 4 > QUESTIONS: > -When is the notation 2x for even or 2x+1 for odd numbers absolutely > necessary? > -if I instead of (x-2) and (x+2) use x and (x+4) equation cant be > solved.Is there something as a rule that says that both variables have > to be expressed by third variable or something similar? > thank you First of all, you second method results in the solution x = 8, not x = 4. Therefore, the two numbers whos squared add up to 136 are 6 and 10. Now to answer your second question: You are wrong in saying the equation cannot be solved. You get x^2 + (x + 4)^2 = 168 Try this again, and youll find that it does have a solution. In fact it has two solutions x = 6 and x = -10. Both of these are valid, because: using x = 6, we have 6^2 + 10^2 = 136 using x = -10, we have (-10)^2 + (-6)^2 = 136 Ill leave your first question for the time being, and will come back in a day or so if nobody else has answered it. === Subject: Re: Question about even and odd numbers > I know how to solve this,but question is indirectly related to this > and similar problems > Three consecutive even integers are such that the square of the first > plus the square of the third is 136. find the three integers. > It can be solved in two ways: > 1-With use of 2n notation for even number > (2n-2)(2n-2)+(2n+2)^2=136 > n=4 > or > 2-that we dont use 2n notation for even numbers > 136 = (x-2)(x-2)+(x+2)(x+2) > x = 4 No, x = 8. > QUESTIONS: > -When is the notation 2x for even or 2x+1 for odd numbers absolutely > necessary? Never: you can always add evenness or oddness as an extra requirement. But explicitly writing an even number as even can save you work later on if, for instance, you have to divide the number by 2. > -if I instead of (x-2) and (x+2) use x and (x+4) equation cant be > solved. [...] Sure it can: its just a quadratic. x^2 + (x + 4)^2 = 136 2x^2 + 8x + 16 = 136 x^2 + 4x + 8 = 68 x^2 + 4x - 60 = 0 x = [-4 +/- sqrt(16 + 4*60)]/2 = -2 +/- 8 x = 6 or x = -10 Brian === Subject: Proving that two topologies are equivalent Question about proving the equivalence of the following two topologies defined on a vector space X. The square topology defined by metric: d((x1,x2,.,xn),(y1,y2,.,yn))=max{|x1-y1|, .,|xn-yn|}. The usual topology - the minimal topology such that every linear functional on X is continuous. I want to know if my sketch of proof makes sense: Start by proving that every linear functional on X is continuous with respect to the square topology. Actually, I think you just need to show that the (usual) basis elements (bi) of the dual space X* are continuous. This should be fairly trivial since: bi((x1,.,xn))=xi bi<-(U)=(X,X,.,U,.,X) (inverse image of an open set U is open) So, this would show that the usual topology is contained in the square topology. To prove the converse, is it enough to show that every open ball in the square topology is contained in the usual topology? I am thinking that every open set in the square topology can be considered to be a possibly infinite union of such open balls, and so by the definition of a topology, this set must also belong to the usual topology. I.e. the square topology is contained in the usual topology. It looks like this equivalence proof will work for any metric topology (not just the square one) as long as the set (X,X,...,U,....,X) is open for some open U in R. Is that correct? === Subject: Re: Proving that two topologies are equivalent >Question about proving the equivalence of the following two topologies >defined on a vector space X. Evidently X is a finite-dimensional real vector space... >The square topology defined by metric: > d((x1,x2,.,xn),(y1,y2,.,yn))=max{|x1-y1|, .,|xn-yn|}. >The usual topology - the minimal topology such that every linear functional >on X is continuous. >I want to know if my sketch of proof makes sense: >Start by proving that every linear functional on X is continuous with >respect to the square topology. Actually, I think you just need to show that >the (usual) basis elements (bi) of the dual space X* are continuous. Why is that? > This >should be fairly trivial since: >bi((x1,.,xn))=xi >bi<-(U)=(X,X,.,U,.,X) (inverse image of an open set U is open) bi<-(U) means the inverse image of U under bi? Ok... but (X,X,.,U,.,X) is not the right notation for what you have in mind - you meant the cartesian product X x X x X x ... x U x ... x X. >So, this would show that the usual topology is contained in the square >topology. Yes. >To prove the converse, is it enough to show that every open ball in the >square topology is contained in the usual topology? Yes. Now how do you _do_ that? >I am thinking that every >open set in the square topology can be considered to be a possibly infinite >union of such open balls, and so by the definition of a topology, this set >must also belong to the usual topology. >I.e. the square topology is contained in the usual topology. >It looks like this equivalence proof will work for any metric topology (not >just the square one) as long as the set (X,X,...,U,....,X) is open for some >open U in R. Is that correct? ************************ David C. Ullrich === Subject: Re: Proving that two topologies are equivalent >Question about proving the equivalence of the following two topologies >defined on a vector space X. > Evidently X is a finite-dimensional real vector space... Yes, that is correct (I should have been more specific) >The square topology defined by metric: > d((x1,x2,.,xn),(y1,y2,.,yn))=max{|x1-y1|, .,|xn-yn|}. >The usual topology - the minimal topology such that every linear functional >on X is continuous. >I want to know if my sketch of proof makes sense: >Start by proving that every linear functional on X is continuous with >respect to the square topology. Actually, I think you just need to show that >the (usual) basis elements (bi) of the dual space X* are continuous. > Why is that? The book Im reading (which talks abut finite dimensional spaces only) explains that: 1. For any topological space X, the sum(fi) of a finite set of continuous functions f1,...fn:X->R is again continuous. 2. For any basis of a finite dimensional vector space V with some topology T, we have that f.91V* continuous if and only if the vectors of the dual basis are continuous. > This >should be fairly trivial since: >bi((x1,.,xn))=xi >bi<-(U)=(X,X,.,U,.,X) (inverse image of an open set U is open) > bi<-(U) means the inverse image of U under bi? Ok... > but (X,X,.,U,.,X) is not the right notation for what you > have in mind - you meant the cartesian product > X x X x X x ... x U x ... x X. >So, this would show that the usual topology is contained in the square >topology. > Yes. >To prove the converse, is it enough to show that every open ball in the >square topology is contained in the usual topology? > Yes. Now how do you _do_ that? Well for the square metric, an open ball can be considered as a finite intersection of the n sets of form: X x X x X x ... x B x ... x X. Which we have already shown to belong to the usual topology. This is easy for the square metric. Other metrics such as the Euclidean metric I might have a harder time with? Is there another method? >I am thinking that every >open set in the square topology can be considered to be a possibly infinite >union of such open balls, and so by the definition of a topology, this set >must also belong to the usual topology. >I.e. the square topology is contained in the usual topology. >It looks like this equivalence proof will work for any metric topology (not >just the square one) as long as the set (X,X,...,U,....,X) is open for some >open U in R. Is that correct? > ************************ > David C. Ullrich === Subject: Re: Proving that two topologies are equivalent >>Question about proving the equivalence of the following two topologies >>defined on a vector space X. >> Evidently X is a finite-dimensional real vector space... >Yes, that is correct (I should have been more specific) >>The square topology defined by metric: >> d((x1,x2,.,xn),(y1,y2,.,yn))=max{|x1-y1|, .,|xn-yn|}. >>The usual topology - the minimal topology such that every linear >functional >>on X is continuous. >>I want to know if my sketch of proof makes sense: >>Start by proving that every linear functional on X is continuous with >>respect to the square topology. Actually, I think you just need to show >that >>the (usual) basis elements (bi) of the dual space X* are continuous. >> Why is that? >The book Im reading (which talks abut finite dimensional spaces only) >explains that: >1. For any topological space X, the sum(fi) of a finite set of continuous >functions f1,...fn:X->R is again continuous. >2. For any basis of a finite dimensional vector space V with some topology >T, we have that f.91V* continuous if and only if the vectors of the dual basis >are continuous. >> This >>should be fairly trivial since: >>bi((x1,.,xn))=xi >>bi<-(U)=(X,X,.,U,.,X) (inverse image of an open set U is open) >> bi<-(U) means the inverse image of U under bi? Ok... >> but (X,X,.,U,.,X) is not the right notation for what you >> have in mind - you meant the cartesian product >> X x X x X x ... x U x ... x X. >>So, this would show that the usual topology is contained in the square >>topology. >> Yes. >>To prove the converse, is it enough to show that every open ball in the >>square topology is contained in the usual topology? >> Yes. Now how do you _do_ that? >Well for the square metric, an open ball can be considered as a finite >intersection of the n sets of form: >X x X x X x ... x B x ... x X. >Which we have already shown to belong to the usual topology. Seems fine. >This is easy for the square metric. Other metrics such as the Euclidean >metric I might have a harder time with? >Is there another method? >>I am thinking that every >>open set in the square topology can be considered to be a possibly >infinite >>union of such open balls, and so by the definition of a topology, this >set >>must also belong to the usual topology. >>I.e. the square topology is contained in the usual topology. >>It looks like this equivalence proof will work for any metric topology >(not >>just the square one) as long as the set (X,X,...,U,....,X) is open for >some >>open U in R. Is that correct? >> ************************ >> David C. Ullrich ************************ David C. Ullrich === Subject: Re: Proving that two topologies are equivalent > Question about proving the equivalence of the following two topologies > defined on a vector space X. > The square topology defined by metric: > d((x1,x2,.,xn),(y1,y2,.,yn))=max{|x1-y1|, .,|xn-yn|}. > The usual topology - the minimal topology such that every linear functional > on X is continuous. > I want to know if my sketch of proof makes sense: > Start by proving that every linear functional on X is continuous with > respect to the square topology. Actually, I think you just need to show that > the (usual) basis elements (bi) of the dual space X* are continuous. This > should be fairly trivial since: > bi((x1,.,xn))=xi > bi<-(U)=(X,X,.,U,.,X) (inverse image of an open set U is open) > So, this would show that the usual topology is contained in the square > topology. > To prove the converse, is it enough to show that every open ball in the > square topology is contained in the usual topology? I am thinking that every > open set in the square topology can be considered to be a possibly infinite > union of such open balls, and so by the definition of a topology, this set > must also belong to the usual topology. > I.e. the square topology is contained in the usual topology. > It looks like this equivalence proof will work for any metric topology (not > just the square one) as long as the set (X,X,...,U,....,X) is open for some > open U in R. Is that correct? Actually, isnt it enough to show that each open set in each topology contains some open set in the other topology? === Subject: Re: Proving that two topologies are equivalent >> Question about proving the equivalence of the following two topologies >> defined on a vector space X. >> The square topology defined by metric: >> d((x1,x2,.,xn),(y1,y2,.,yn))=max{|x1-y1|, .,|xn-yn|}. >> The usual topology - the minimal topology such that every linear functional >> on X is continuous. >> I want to know if my sketch of proof makes sense: >> Start by proving that every linear functional on X is continuous with >> respect to the square topology. Actually, I think you just need to show that >> the (usual) basis elements (bi) of the dual space X* are continuous. This >> should be fairly trivial since: >> bi((x1,.,xn))=xi >> bi<-(U)=(X,X,.,U,.,X) (inverse image of an open set U is open) >> So, this would show that the usual topology is contained in the square >> topology. >> To prove the converse, is it enough to show that every open ball in the >> square topology is contained in the usual topology? I am thinking that every >> open set in the square topology can be considered to be a possibly infinite >> union of such open balls, and so by the definition of a topology, this set >> must also belong to the usual topology. >> I.e. the square topology is contained in the usual topology. >> It looks like this equivalence proof will work for any metric topology (not >> just the square one) as long as the set (X,X,...,U,....,X) is open for some >> open U in R. Is that correct? >Actually, isnt it enough to show that each open set in each topology >contains some open set in the other topology? No, in fact that condition holds for _any_ two topologies on a set (because the empty set is open). Whats enough is to show that for every open O in the first topology and every x in O there exists V open in the second topology with x in V and V contained in O. (Which of course is the same as showing that O is open in the second topology). ************************ David C. Ullrich === Subject: Re: Proving that two topologies are equivalent >> Question about proving the equivalence of the following two topologies >> defined on a vector space X. >> >> >> >> The square topology defined by metric: >> >> d((x1,x2,.,xn),(y1,y2,.,yn))=max{|x1-y1|, .,|xn-yn|}. >> >> >> >> The usual topology - the minimal topology such that every linear >> functional >> on X is continuous. >> >> >> >> I want to know if my sketch of proof makes sense: >> >> >> >> Start by proving that every linear functional on X is continuous with >> respect to the square topology. Actually, I think you just need to show >> that >> the (usual) basis elements (bi) of the dual space X* are continuous. This >> should be fairly trivial since: >> >> >> >> bi((x1,.,xn))=xi >> >> >> >> bi<-(U)=(X,X,.,U,.,X) (inverse image of an open set U is open) >> >> >> >> So, this would show that the usual topology is contained in the square >> topology. >> >> >> >> To prove the converse, is it enough to show that every open ball in the >> square topology is contained in the usual topology? I am thinking that >> every >> open set in the square topology can be considered to be a possibly >> infinite >> union of such open balls, and so by the definition of a topology, this set >> must also belong to the usual topology. >> >> I.e. the square topology is contained in the usual topology. >> >> >> >> It looks like this equivalence proof will work for any metric topology >> (not >> just the square one) as long as the set (X,X,...,U,....,X) is open for >> some >> open U in R. Is that correct? >Actually, isnt it enough to show that each open set in each topology >contains some open set in the other topology? > No, in fact that condition holds for _any_ two topologies on a set > (because the empty set is open). Whats enough is to show that for > every open O in the first topology and every x in O there exists > V open in the second topology with x in V and V contained in O. > (Which of course is the same as showing that O is open in the > second topology). > ************************ > David C. Ullrich Been a long time. Replace open sets with neighborhoods. === Subject: Sets, functions, and ordered pairs help by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iA9CYIf03222; I dont know what this question means. Please explain to me what a set, ordered pair, function, elements and range mean. Those words are in this questions context by the way. THank you very much. THE QueSTION This set of ordered pairs defines a function: {(1,2), (2,6), (3,2)}. How many different elements are in the range of this function? === Subject: Re: Sets, functions, and ordered pairs help > I dont know what this question means. Please explain to me what a set, > ordered pair, function, elements and range mean. Those words are in > this questions context by the way. THank you very much. > THE QueSTION > This set of ordered pairs defines a function: {(1,2), (2,6), (3,2)}. > How many different elements are in the range of this function? You could have a look on the mathworld website, which is a pretty good maths encyclopedia: http://mathworld.wolfram.com/search/ just search for set, element, etc.. Then if you have specific points you dont understand you could ask here again. Mike. === Subject: Re: Sets, functions, and ordered pairs help alt.math.undergrad: >Please explain to me what a set, >ordered pair, function, elements and range mean. What does your textbook say? Which parts of your textbooks definitions do you need help with? No, Im not just being obstructive. You will find if you develop the habit of actually using your textbook that you will do MUCH better in the course. -- Stan Brown, Oak Road Systems, Tompkins County, New York, USA http://OakRoadSystems.com/ And if youre afraid of butter, which many people are nowa- days, (long pause) you just put in cream. --Julia Child === Subject: Re: Sets, functions, and ordered pairs help > I dont know what this question means. Please explain to me what a set, > ordered pair, function, elements and range mean. Those words are in > this questions context by the way. THank you very much. > THE QueSTION > This set of ordered pairs defines a function: {(1,2), (2,6), (3,2)}. > How many different elements are in the range of this function? In, perhaps, a better-known notation, call the function f: f(1) = 2, f(2) = 6, f(3) = 2 so the two different elements in the range of f are 2 and 6. {X, Y, Z} is an example of a set, it is collection containing X, Y and Z. X, Y and Z are the elements of the set. In your question the elements are the ordered pairs (1, 2), (2, 6) and (3, 2). A function is a set of ordered pairs (subject to some conditions that the set in the question does indeed satisfy). The range of a function is the set of its values, the values are the second elements of the ordered pairs; in the question they are 2, 6 and 2. === Subject: Re: Sets, functions, and ordered pairs help > I dont know what this question means. Please explain to me what a set, > ordered pair, function, elements and range mean. Those words are in > this questions context by the way. THank you very much. > THE QueSTION > This set of ordered pairs defines a function: {(1,2), (2,6), (3,2)}. > How many different elements are in the range of this function? Ok, lets start with something more basic: What is a function? What is the range? What is the domain? You need to know what a term means before you can answer a question. Do you? -- Will Twentyman email: wtwentyman at copper dot net === Subject: Ways to compute percentages by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iA9EDnf12627; hi Im just learning percentages and would like to know how many ways there are to compute how much is for example 60% out of 150.Im not looking for any extra fancy methods using rocket science math since I wouldnt understand it anyway I can think of 3 ways - (150/150) * (60/100)=150/150 * (3/5)=450/750=90/150 - 150 * 60%=150 * 0.6=90 - x/150=60/100 x=(150*60)/100 thank you === Subject: Re: Ways to compute percentages > hi > Im just learning percentages and would like to know how many ways > there are to compute how much is for example 60% out of 150.Im not > looking for any extra fancy methods using rocket science math since I > wouldnt understand it anyway > I can think of 3 ways > - (150/150) * (60/100)=150/150 * (3/5)=450/750=90/150 > - 150 * 60%=150 * 0.6=90 > - x/150=60/100 > x=(150*60)/100 > thank you a percentage is essentially a fractional quantity which has 100 as the denominator. so that 50% is really 50/100, which if you top and bottom gives 1/2. when you see something like 20% of 50 or whatever, then read the Ôof as meaning multiply, so that 20% of 50 really means 20/100 x 50, which is 1/5 x 50 which equals 10 === Subject: Re: Ways to compute percentages by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAB0VlB10158; >a percentage is essentially a fractional quantity which has 100 as the >denominator. so that 50% is really 50/100, which if you hcfs >top and bottom gives 1/2. >when you see something like 20% of 50 or whatever, then read the Ôof as >meaning multiply, so that 20% of 50 really means 20/100 x 50, which is 1/5 x >50 which equals 10 But I dont see a difference.We basically divided 150 in 100 equal parts,and 20% are 20 of those equal parts === Subject: Re: Ways to compute percentages >>a percentage is essentially a fractional quantity which has 100 as > the >>denominator. so that 50% is really 50/100, which if you > hcfs >>top and bottom gives 1/2. >>when you see something like 20% of 50 or whatever, then read the Ôof > as >>meaning multiply, so that 20% of 50 really means 20/100 x 50, which > is 1/5 x >>50 which equals 10 > But I dont see a difference.We basically divided 150 in 100 equal > parts,and 20% are 20 of those equal parts im not sure what you mean, but if you simply imagine a percentage as being a fraction then you cant go wrong. if you know how to handle fractions, then percentage calculations are a sinch. as i said , all a percentage is is a fraction with 100 as its denominator, so that 75% is 75/100 ( or 3/4) , 50% is 50/100 (or 1/2) , 25% is 1/4 etc. you can even have Ôtop heavy percentages so something like 150% is 150/100 or 3/2. I just think you are unnecessarily confusing yourself === Subject: Re: Ways to compute percentages > hi > Im just learning percentages and would like to know how many ways > there are to compute how much is for example 60% out of 150.Im not > looking for any extra fancy methods using rocket science math since I > wouldnt understand it anyway > I can think of 3 ways > - (150/150) * (60/100)=150/150 * (3/5)=450/750=90/150 > - 150 * 60%=150 * 0.6=90 > - x/150=60/100 > x=(150*60)/100 > thank you Your second and third method produce 90 which is correct for 60% of 150, but your first method produces 0.6. If I were you I would avoid the first method. === Subject: Re: Ways to compute percentages Bugger these computers. Did I just accidentally send my half-finished reply? If so, ignore it. Ho-hum, lets try again anyway... > hi > Im just learning percentages and would like to know how many ways > there are to compute how much is for example 60% out of 150. To be correct, one would say 60% of 150 not 60% out of 150. > Im not > looking for any extra fancy methods using rocket science math since I > wouldnt understand it anyway Dont worry, I dont think there IS any rocket science math to do with percentages. Hey, even *I* understand them. (I think!) > I can think of 3 ways > - (150/150) * (60/100)=150/150 * (3/5)=450/750=90/150 Im assuming that the minus signs at the beginning of the lines are some sort of bullet point? (Very bad choice!). Although the above is arithmetically correct I really cant see what youre trying to achieve here. > - 150 * 60%=150 * 0.6=90 There is essentially only ONE way to calculate 60% of 150, and this is it. The answer is 150 * 60/100, or 150 * 0.6, which is 90, exactly as you say. > - x/150=60/100 > x=(150*60)/100 This is a correct but (to my mind) unnecessarily roundabout way of calculating the same result. Its just a rearrangement of your last method. For, if x is the answer to the question what is 60% of 150 then, as we know, x satisfies the equation x = 150 * 60/100. Rearranging this gives x/150 = 60/100. True, but as far as calculating percentages is concerned, unnecessary. (Although, as an afterthought, I suppose the equation x/150 = 60/100 has some merit in that it aids understanding of what percentages mean. If x is 60% of 150, then the ratio of x to 150 is the same as the ratio of 60 to 100.) > thank you Youre welcome, it helps with my rehab. === Subject: Re: Ways to compute percentages You can start by understanding certain few things: % means of 100 equal parts; of means multiply by, but not for the phrase in of 100 equal parts. 60 % of 150 means (60/100) multiplied by 150. G C >Im just learning percentages and would like to know how many ways >there are to compute how much is for example 60% out of 150.Im not >looking for any extra fancy methods using rocket science math since I >wouldnt understand it anyway >I can think of 3 ways >- (150/150) * (60/100)=150/150 * (3/5)=450/750=90/150 >- 150 * 60%=150 * 0.6=90 >- x/150=60/100 > x=(150*60)/100 >thank you === Subject: Does anyone have any tips for writing mathematical symbols in plain text? E.g. set symbols, integrals, formulas, etc. Im having a hard time writing these. === Subject: Re: Does anyone have any tips for writing mathematical symbols in plain text? >E.g. set symbols, integrals, formulas, etc. >Im having a hard time writing these. INTEGRAL(x=a to x = b)(f(x)dx) SUM(n=0 to +oo)[S(n)] formulas ---just write them ; / for fractions and division; ^ for exponents; * for multiplication === Subject: Re: Does anyone have any tips for writing mathematical symbols in plain text? > E.g. set symbols, integrals, formulas, etc. > Im having a hard time writing these. If you want to write them in news groups, then no method is perfect, but you may like to consider the alt-algebra.help FAQ http://aah.ryan-usa.com/node15.html or (perhaps better, but in French) the fr.sci.maths FAQ http://giromini.org/maths/formules-maths.html Ken Pledger. === Subject: Re: Does anyone have any tips for writing mathematical symbols in plain text? days. My association with the Department is that of an alumnus. >E.g. set symbols, integrals, formulas, etc. >Im having a hard time writing these. The standard (and remember, one of the great things about standards is that there are so many to choose from) is either pseudo-TeX notation, or else in words. For example, in TeX, int is the control sequence to produce the integral symbol; _ is a control sequence indicating that the next term will be a subscript, and ^ for a superscript. So int_0^3 would be interpreted as the integral, from 0 to 3. -- Its not denial. Im just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: Re: Does anyone have any tips for writing mathematical symbols in plain text? >>E.g. set symbols, integrals, formulas, etc. >>Im having a hard time writing these. > The standard (and remember, one of the great things about standards > is that there are so many to choose from) is either pseudo-TeX > notation, or else in words. > For example, in TeX, int is the control sequence to produce the > integral symbol; _ is a control sequence indicating that the next > term will be a subscript, and ^ for a superscript. So > int_0^3 > would be interpreted as the integral, from 0 to 3. Or if you are really anal, you could write $$int_{0}^{3}f(x),dx$$ ;=) DM === Subject: Re: Academy123.com by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iA9N2jJ01512; >Found this on the chatboard at Teachers.net >Math teachers beware! >There is no substitute for a good teacher who can guide her >students toward understanding. >Academy123 asks us to believe students can learn math by >watching a video. Score Kaplan thinks computers are a >suitable replacement for teachers. >Whats next? >Academy123 is solely about making money. >The six members that make up its management team and advisory panel >have no educational background whatsoever. >They hire young teachers for a cheap rate to do all the work for them. >If they can get 1000 students from around the world to buy a >subscription, the company earns $40,000 per month. Thats what many >teachers make in a whole year. That is such crap!! Its a business, duh. And if they have a six-man management team and they get 1000 people to sign up and pay per year each member of the six man management team is making 80k per year. Ill let you know a little secret: most management gets paid much much more. And considering most management people actually have people to manage, they probably arent bringing in too much cash. Sounds like they are trying to basically make up their costs and make a business out of this and I dont see anything wrong with that. I went on their website and they never professed to be anything BUT a business. Christ on a cracker people need to chill. === Subject: Re: Precalculus Help! by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iA9Nnl305006; The volume of a can (V = pie*r(squared *h) is 50 cubic inches. Express the area of thr surface of the can as a function of radius r. Notice that the surface area consists of two circular regions(top and bottom) each with area pie*r(squared) and a curved side with area 2(pie)r*h. Help...anyone === Subject: Re: Precalculus Help! > The volume of a can (V = pie*r(squared *h) is 50 cubic inches. > Express the area of thr surface of the can as a function of radius r. > Notice that the surface area consists of two circular regions(top and > bottom) each with area pie*r(squared) and a curved side with area > 2(pie)r*h. Help...anyone This should be a new posting? As you say, the volume of a cylinder is given by V = pi*r^2*h Notational points: It is pi, not pie. r(squared *h) means nothing. To make any sense it would have to be r(squared)*h. r squared would normally be written as r followed by a superscript 2, but the 1970s-style newsgroup technology doesnt support superscripts so you write r^2 instead. (The symbol ^ means raise to the power of.) In this problem you know that V = 50, so 50 = pi*r^2*h (Eqn 1) You also know that the surface area, by convention denoted S, is given by S = 2*pi*r^2 + 2*pi*r*h (Eqn 2) Equation 2 gives S in terms of r and h, but what you want is an expression for S in terms of r only (you want to get rid of the h). To achieve this you must use the information in equation 1. If you are still having problems then post back explaining what you have tried and where you are stuck. === Subject: Re: Precalculus Help! >The volume of a can (V = pie*r(squared *h) is 50 cubic inches. >Express the area of thr surface of the can as a function of radius r. >Notice that the surface area consists of two circular regions(top and >bottom) each with area pie*r(squared) and a curved side with area >2(pie)r*h. Help...anyone Why dont you show us how far you got with this problem? It may or may not be homework, but it looks like homework. If we just hand you a solution, youll read it but you probably wont be any better equipped to solve the next problem that comes along. -- Stan Brown, Oak Road Systems, Tompkins County, New York, USA http://OakRoadSystems.com/ And if youre afraid of butter, which many people are nowa- days, (long pause) you just put in cream. --Julia Child === Subject: Simple derivative question. How does sin4x/x become 4cos4x? In general, how do handle derivatives for CosNx or SinNx? === Subject: Re: Simple derivative question. alt.math.undergrad: >How does sin4x/x become 4cos4x? Sigh. Is it _really_ that hard to use parentheses to make your meaning clear? >In general, how do handle derivatives for CosNx or SinNx? Chain Rule. -- Stan Brown, Oak Road Systems, Tompkins County, New York, USA http://OakRoadSystems.com/ And if youre afraid of butter, which many people are nowa- days, (long pause) you just put in cream. --Julia Child === Subject: Re: Simple derivative question. > How does sin4x/x become 4cos4x? I think you mean d(sin4x)/dx. > In general, how do handle derivatives for CosNx or SinNx? They are of the form f(g(x)) where f is sin or cos (dont use capital letters) and g is multiplying by N. d(f(g(x)))/dx = f(x)g(x) This is called the chain rule. === Subject: Re: Simple derivative question. > How does sin4x/x become 4cos4x? > I think you mean d(sin4x)/dx. > In general, how do handle derivatives for CosNx or SinNx? > They are of the form f(g(x)) where f is sin or cos (dont use > capital letters) and g is multiplying by N. > d(f(g(x)))/dx = f(x)g(x) you mean d(f(g(x)))/dx = f(g(x))g(x) > This is called the chain rule. === Subject: Re: Simple derivative question. How does sin4x/x become 4cos4x? > I think you mean d(sin4x)/dx. In general, how do handle derivatives for CosNx or SinNx? > They are of the form f(g(x)) where f is sin or cos (dont use > capital letters) and g is multiplying by N. > d(f(g(x)))/dx = f(x)g(x) > you mean d(f(g(x)))/dx = f(g(x))g(x) === Subject: Re: Simple derivative question. >How does sin4x/x become 4cos4x? Its doesnt algebraically but by differentiating both numerator and denominator. . Maybe youre talking about limits of quotients and lHospitals rule? If so, read the theorem called lHospitals rule. >In general, how do handle derivatives for CosNx or SinNx? === Subject: Re: Simple derivative question. >>How does sin4x/x become 4cos4x? > Its doesnt algebraically but by differentiating both numerator > and denominator. . Maybe youre talking about limits of quotients > and lHospitals rule? If so, read the theorem called lHospitals > rule. >>In general, how do handle derivatives for CosNx or SinNx? The sarcasm is not neccasary. There is nothing inappropriate about trying to The email address is miedvied at si dot rr dot com. === Subject: Re: Simple derivative question. > How does sin4x/x become 4cos4x? > In general, how do handle derivatives for CosNx or SinNx? Sin(Nx) is a function of a function - to evaluate it given x, you would first multiply x by N (first function), then take the sin of the result (second function). This may also be referred to as the composite of two functions in your text book. The derivative of the composite of two functions turns out to be basically the product of the derivatives of the original functions. There are different ways this can be expressed, depending on your preferred notation... E.g. if h(x) = f(g(x)) is the composite function, then h(x) = f(g(x))g(x) is the derivative. [#1] See, this is just f multiplied by g, but being careful to evaluate f at the appropriate point g(x), not at x. Applying this to the case h(x) = sin(Nx) we have g: x--->Nx so g: x--->N is the derivative of g (i.e. the derivative is a constant) f: y--->Sin(y) so f: y--->cos(y) is the derivative of f and so h(x) = f(g(x))g(x) = cos(g(x))g(x) = cos(Nx)*N Maybe easier to follow is to introduce a substitution for the result of applying the first function to x, i.e. let u = Nx so if y = sin(Nx), then y = sin(u) and the rule for the derivative of a function of a function says dy/dx = (dy/du)(du/dx) [#2] = cos(u) * N = cos(Nx) * N (The rules [#1] and [#2] are saying exactly the same thing, but expressed in different notations.) Other examples of applying the above rule (so you can try out the above to check whether you understand properly): if f(x) = sin(x^3), then f(x) = cos(x^3) * (3x^2) if f(x) = sin(sin(x)), then f(x) = cos(sin(x)) * cos(x) if f(x) = cos(-x), then f(x) = -sin(-x) * (-1) = sin(-x) = -sin(x) etc. hope this helps Mike. === Subject: Re: Simple derivative question. > How does sin4x/x become 4cos4x? It doesnt. d(sin(4x)/x) = (1/x^2)[4x cos(4x) - sin(4x)] > In general, how do handle derivatives for CosNx or SinNx? Chain rule again. cos(nx) = cos(u), u=nx d(cos(nx))/dx = d(cos(u))/du * du/dx = -sin(u) * d(nx)/dx = -sin(u) * n = -n sin(u) = -n sin(nx) Following a similar approach youll get d(sin(nx))/dx = n cos(nx) Or, if youve gotten to complex numbers, you could use: e^(iu) = cos(u) + isin(u) So cos(nx) + isin(nx) = e^(inx) Now take the derivative of both sides: [cos(nx) + isin(nx)] = [e^(inx)] [cos(nx)] + i[sin(nx)] = in e^(inx) [cos(nx)] + i[sin(nx)] = in[cos(nx) + isin(nx)] [cos(nx)] + i[sin(nx)] = i[n*cos(nx) + i*n*sin(nx)] [cos(nx)] + i[sin(nx)] = in cos(nx) - n sin(nx) and since for two complex numbers to be equal their real and imaginary parts must separately be equal, we have [cos(nx)] = -n sin(nx) [sin(nx)] = n cos(nx) -- Rich Carreiro rlcarr@animato.arlington.ma.us === Subject: Re: Simple derivative question. >> How does sin4x/x become 4cos4x? > It doesnt. > d(sin(4x)/x) = (1/x^2)[4x cos(4x) - sin(4x)] So, in this instance, wed be doing: [ x(sin4x) - x(sin4x) ] / x^2 -- Presuming quotient rule Finding the derivative of sin4x using the chain rule would be: u = 4x Derivative becoming cos u * du, or cos4x * 4, 4cos(4x). So, we have: [ x (4cos(4x)) - (1)(sin4x) ] / x^2 [ 4xcos(4x) - sin(4x) ] / x^2 Is that how you got that answer? === Subject: Re: Simple derivative question. > So, in this instance, wed be doing: > [ x(sin4x) - x(sin4x) ] / x^2 -- Presuming quotient rule > Finding the derivative of sin4x using the chain rule would be: > u = 4x > Derivative becoming cos u * du, or cos4x * 4, 4cos(4x). > So, we have: > [ x (4cos(4x)) - (1)(sin4x) ] / x^2 > [ 4xcos(4x) - sin(4x) ] / x^2 > Is that how you got that answer? Yes. -- Rich Carreiro rlcarr@animato.arlington.ma.us === Subject: Re: Simple derivative question. >> How does sin4x/x become 4cos4x? > It doesnt. > d(sin(4x)/x) = (1/x^2)[4x cos(4x) - sin(4x)] My textbook, apparently, says that it does. Thus my ... profound confusion. (Id been using the chain rule to try and follow their steps, but obviously, Ive been unable to reproduce the result. They dont go into great detail, because its just one step in a larger problem.) The problem is this, as an example of IHopitals rule. lim as x approaches 0: ((1-cosx)sin4x) / x^3 (cosx.) = [1-cosx / x^2] [sin4x/x] [1/cosx] (all lim. as x-->0) = [sinx/2x] [4cos4x/1] [1/cosx] (all lim. as x-->0) = (1/2)(4)(1) = 2 === Subject: Re: Simple derivative question. >> How does sin4x/x become 4cos4x? > It doesnt. > d(sin(4x)/x) = (1/x^2)[4x cos(4x) - sin(4x)] > My textbook, apparently, says that it does. Thus my ... profound confusion. > (Id been using the chain rule to try and follow their steps, but obviously, > Ive been unable to reproduce the result. They dont go into great detail, > because its just one step in a larger problem.) > The problem is this, as an example of IHopitals rule. > lim as x approaches 0: ((1-cosx)sin4x) / x^3 (cosx.) > = [1-cosx / x^2] [sin4x/x] [1/cosx] (all lim. as x-->0) > = [sinx/2x] [4cos4x/1] [1/cosx] (all lim. as x-->0) > = (1/2)(4)(1) = 2 The relevant application of LHopitals rule here is in finding lim (x -> 0) [ sin4x / x ] In using LHopitals rule, where we have, as here, a quotient approaching a limit where both numerator and denominator are zero, we differentiate the numerator and denominator - *separately* - and evaluate the derivatives at the limit, if possible. d/dx (sin4x) = 4 cos 4x d/dx (x) = 1 So lim (x -> 0) [ sin4x / x ] = lim (x -> 0) (4 cos 4x) / 1 which we can evaluate at x=0 straight away, giving us 4. At no point do we differentiate (sin 4x / x). The problem also uses LHopital to obtain lim (x -> 0) [ (1 - cos x) / x^2 ], the first term of the overall product. d/dx (1 - cos x) = sinx d/dx (x^2) = 2x So lim (x -> 0) [ (1 - cos x) / x^2 ] = lim (x -> 0) sin x / 2x = (1/2) lim (x -> 0) sin x / x which latter limit I am assuming your text has already demonstrated to be 1, since the quoted expression continues directly to give (1/2). -- Larry Lard Replies to group please === Subject: Re: Simple derivative question. > In using LHopitals rule, where we have, as here, a quotient > approaching a limit where both numerator and denominator are zero No, you have a quotient (which may or may not have a limit) in which both numerator and denominator *approach* zero. > we > differentiate the numerator and denominator - *separately* - and > evaluate the derivatives at the limit, if possible. I do not know what evaluate the derivatives at the limit is supposed to mean. We are in a situation where x -> 0. You seem to be confusing 0 with the limit. Its quite simple: If the quotient of derivatives has a limit as x -> 0, then the original quotient has the same limit. === Subject: Re: Simple derivative question. >> How does sin4x/x become 4cos4x? > It doesnt. > d(sin(4x)/x) = (1/x^2)[4x cos(4x) - sin(4x)] > My textbook, apparently, says that it does. No it doesnt. > Thus my ... profound confusion. You are certainly confused, and thats because you havent been keeping up, reading your notes or following the text. Youre ßailing away. > (Id been using the chain rule to try and follow their steps, but obviously, > Ive been unable to reproduce the result. They dont go into great detail, > because its just one step in a larger problem.) > The problem is this, as an example of IHopitals rule. > lim as x approaches 0: ((1-cosx)sin4x) / x^3 (cosx.) > = [1-cosx / x^2] [sin4x/x] [1/cosx] (all lim. as x-->0) Youve been on the group long enough to know how to write this stuff correctly. Put in some parentheses. The last expression should be something like [(1-cos(x))/x^2] [sin(4x)/x] [1/cos(x)] (1). > = [sinx/2x] [4cos4x/1] [1/cosx] (all lim. as x-->0) > = (1/2)(4)(1) = 2 You have the product of 3 expressions in (1). Theres no trouble with the last one. The first 2 are of the form 0/0 as x -> 0. Now go read LHospitals Rule carefully to see how to proceed. === Subject: Re: Simple derivative question. > How does sin4x/x become 4cos4x? >> It doesnt. >> d(sin(4x)/x) = (1/x^2)[4x cos(4x) - sin(4x)] >> My textbook, apparently, says that it does. > No it doesnt. >> Thus my ... profound confusion. > You are certainly confused, and thats because you havent been keeping > up, > reading your notes or following the text. Youre ßailing away. >> (Id been using the chain rule to try and follow their steps, but >> obviously, >> Ive been unable to reproduce the result. They dont go into great >> detail, >> because its just one step in a larger problem.) >> The problem is this, as an example of IHopitals rule. >> lim as x approaches 0: ((1-cosx)sin4x) / x^3 (cosx.) >> = [1-cosx / x^2] [sin4x/x] [1/cosx] (all lim. as x-->0) > Youve been on the group long enough to know how to write this stuff > correctly. Put in some parentheses. The last expression should be > something > like > [(1-cos(x))/x^2] [sin(4x)/x] [1/cos(x)] (1). >> = [sinx/2x] [4cos4x/1] [1/cosx] (all lim. as x-->0) >> = (1/2)(4)(1) = 2 > You have the product of 3 expressions in (1). Theres no trouble with the > last one. The first 2 are of the form 0/0 as x -> 0. Now go read > LHospitals Rule carefully to see how to proceed. LHopitals rule is putting the derivative over the derivative. However, as the derivative of Ôx is Ô1, its pretty much the derivative of sin4x. Which they still end up turning into 4cos4x. === Subject: Re: Simple derivative question. > How does sin4x/x become 4cos4x? It doesnt. > d(sin(4x)/x) = (1/x^2)[4x cos(4x) - sin(4x)] My textbook, apparently, says that it does. >> No it doesnt. > Thus my ... profound confusion. >> You are certainly confused, and thats because you havent been keeping >> up, >> reading your notes or following the text. Youre ßailing away. > (Id been using the chain rule to try and follow their steps, but > obviously, > Ive been unable to reproduce the result. They dont go into great > detail, > because its just one step in a larger problem.) > The problem is this, as an example of IHopitals rule. > lim as x approaches 0: ((1-cosx)sin4x) / x^3 (cosx.) > = [1-cosx / x^2] [sin4x/x] [1/cosx] (all lim. as x-->0) >> Youve been on the group long enough to know how to write this stuff >> correctly. Put in some parentheses. The last expression should be >> something >> like >> [(1-cos(x))/x^2] [sin(4x)/x] [1/cos(x)] (1). > = [sinx/2x] [4cos4x/1] [1/cosx] (all lim. as x-->0) > = (1/2)(4)(1) = 2 >> You have the product of 3 expressions in (1). Theres no trouble with the >> last one. The first 2 are of the form 0/0 as x -> 0. Now go read >> LHospitals Rule carefully to see how to proceed. > LHopitals rule is putting the derivative over the derivative. However, > as the derivative of Ôx is Ô1, Er, my bad. x ^ -1 would become -x ^ -2. === Subject: Need formula for radius NNTP-Proxy-Relay: library1-aux.airnews.net I havent touched a trig book in 50 years, so Im struggling. This is a practical problem. I have a cord distance of 23.06 millimeters; The distance from the center of the cord to the circumference is .227 mm. What is the formula for the radius? Tim (remove .gov for direct reply) === Subject: Re: Need formula for radius > .... > I have a cord distance of 23.06 millimeters; > The distance from the center of the cord to the circumference is .227 mm. > What is the formula for the radius? > .... First, youve posted this to at least two news groups, once here and twice in the other. If you do that, its helpful to cross-post (e.g. to alt.math.undergrad, alt.math.recreational) so that people dont waste time duplicating one anothers efforts. Also, your measurements vary slightly in two of the versions: 23.06 and .227 mm, or 23 and .277 mm. Anyway, lets say the chord has length 2t and the sagitta (arrow) has length s. No doubt someone will suggest using the so-called Theorem of Pythagoras, but its quicker to use the intersecting chords theorem Euclid III.35. You have a chord and diameter crossing each other at a point which breaks the chord into lengths t and t, and the diameter into lengths s and (2r - s). Therefore s(2r - s) = t^2, so r = (s + (t^2)/s)/2. Your measurements above give r = (0.227 + ((11.53)^2)/0.227)/2 = 293 mm. Your alternative measurements give r = (0.277 + ((11.5)^2)/0.277)/2 = 239 mm. Ken Pledger. === Subject: Re: Need formula for radius NNTP-Proxy-Relay: library1-aux.airnews.net I didnt know about the cross-posting. My computer said the other posting was refused because of an error, So, I tried this group. Tim > .... > I have a cord distance of 23.06 millimeters; > The distance from the center of the cord to the circumference is .227 mm. > What is the formula for the radius? > .... > First, youve posted this to at least two news groups, once here > and twice in the other. If you do that, its helpful to cross-post > (e.g. to alt.math.undergrad, alt.math.recreational) so that people dont > waste time duplicating one anothers efforts. Also, your measurements > vary slightly in two of the versions: 23.06 and .227 mm, or 23 and .277 > mm. > Anyway, lets say the chord has length 2t and the sagitta (arrow) > has length s. No doubt someone will suggest using the so-called Theorem > of Pythagoras, but its quicker to use the intersecting chords theorem > Euclid III.35. You have a chord and diameter crossing each other at a > point which breaks the chord into lengths t and t, and the diameter into > lengths s and (2r - s). > Therefore s(2r - s) = t^2, > so r = (s + (t^2)/s)/2. > Your measurements above give > r = (0.227 + ((11.53)^2)/0.227)/2 = 293 mm. > Your alternative measurements give > r = (0.277 + ((11.5)^2)/0.277)/2 = 239 mm. > Ken Pledger. === Subject: Re: Need formula for radius >This is a practical problem. >I have a cord distance of 23.06 millimeters; >The distance from the center of the cord to the circumference is .227 mm >What is the formula for the radius? Check in a geometry or trig book. What is your distance of 23.03 millimeters? You asked for the radius. The formula C=2pr where p means pi is not even necessary because you already know the radius. You said from the center to the circumference. If this is not the radius then you need to restate your problem. Algebryonic === Subject: Optimising numerical integration Hi I have an n-fold multiple integral S ... S f(x_1, x_2 ... x_n) dx_1 dx_2 ... dx_n n (the number of variables involved) depends on the scenario but would typically be, say, 10. All the integrals are definite integrals and the bounds are known. Some of the inner bounds depend on some of the outer variables. For example, the bounds on x_1 depend on x_n. The function f is horrendous and there is no chance of doing the integration analytically. Therefore I need to fall back on numerical integration. I can write a program to do this, no problem. What I end up with is n nested levels of loops. Lets say I need 10,000 iterations per loop to get decent accuracy. The total number of iterations is then 10,000^n. Say n = 10, and 10,000 iterations take one second to compute. The total time would then be about 10^28 years, by which time the universe will have ended and there will be no-one around to appreciate my ground-breaking result. Does anyone know of any general techniques to speed up this computation enough to make it feasible? Alternatively, does anyone know for sure that its impossible? (btw, f cant be expressed as a product of functions of subsets of the xs). matt === Subject: Re: Optimising numerical integration > Hi > I have an n-fold multiple integral > S ... S f(x_1, x_2 ... x_n) dx_1 dx_2 ... dx_n > n (the number of variables involved) depends on the scenario but would > typically be, say, 10. > All the integrals are definite integrals and the bounds are known. > Some of the inner bounds depend on some of the outer variables. For > example, the bounds on x_1 depend on x_n. > The function f is horrendous and there is no chance of doing the > integration analytically. Therefore I need to fall back on numerical > integration. > I can write a program to do this, no problem. What I end up with is n > nested levels of loops. > Lets say I need 10,000 iterations per loop to get decent accuracy. > The total number of iterations is then 10,000^n. Say n = 10, and > 10,000 iterations take one second to compute. The total time would > then be about 10^28 years, by which time the universe will have ended > and there will be no-one around to appreciate my ground-breaking > result. > Does anyone know of any general techniques to speed up this > computation enough to make it feasible? Alternatively, does anyone > know for sure that its impossible? Consider Monte-Carlo Integration, Google should turn up all the references you need. RonL -- Ignorance is the most delightful of the sciences .. . === Subject: Re: Optimising numerical integration > Hi > I have an n-fold multiple integral > S ... S f(x_1, x_2 ... x_n) dx_1 dx_2 ... dx_n > n (the number of variables involved) depends on the scenario but would > typically be, say, 10. > All the integrals are definite integrals and the bounds are known. > Some of the inner bounds depend on some of the outer variables. For > example, the bounds on x_1 depend on x_n. > The function f is horrendous and there is no chance of doing the > integration analytically. Therefore I need to fall back on numerical > integration. > I can write a program to do this, no problem. What I end up with is n > nested levels of loops. > Lets say I need 10,000 iterations per loop to get decent accuracy. > The total number of iterations is then 10,000^n. Say n = 10, and > 10,000 iterations take one second to compute. The total time would > then be about 10^28 years, by which time the universe will have ended > and there will be no-one around to appreciate my ground-breaking > result. > Does anyone know of any general techniques to speed up this > computation enough to make it feasible? Alternatively, does anyone > know for sure that its impossible? > Consider Monte-Carlo Integration, Google should turn up > all the references you need. > RonL technique and tried it on a special test case of the 10-fold integration where I know that the answer is exactly one. Unfortunately, even with 100,000,000 trials I get answers around +/-0.03 of the true value. Although this is actually better than I expected, it still isnt accurate enough for my purposes. I really need at least six-figure accuracy, quickly, reliably and repeatably (the calculation must be able to be performed on an ad hoc basis as the various parameters are changed by the user). I think the problem is that the integration space is just so vast that to get this accuracy I need an infeasible number of calculations whichever way I do it. Im not sure theres any way round it. But many problems ... === Subject: Re: Optimising numerical integration > Hi I have an n-fold multiple integral S ... S f(x_1, x_2 ... x_n) dx_1 dx_2 ... dx_n n (the number of variables involved) depends on the scenario but would > typically be, say, 10. All the integrals are definite integrals and the bounds are known. > Some of the inner bounds depend on some of the outer variables. For > example, the bounds on x_1 depend on x_n. The function f is horrendous and there is no chance of doing the > integration analytically. Therefore I need to fall back on numerical > integration. I can write a program to do this, no problem. What I end up with is n > nested levels of loops. Lets say I need 10,000 iterations per loop to get decent accuracy. > The total number of iterations is then 10,000^n. Say n = 10, and > 10,000 iterations take one second to compute. The total time would > then be about 10^28 years, by which time the universe will have ended > and there will be no-one around to appreciate my ground-breaking > result. Does anyone know of any general techniques to speed up this > computation enough to make it feasible? Alternatively, does anyone > know for sure that its impossible? > Consider Monte-Carlo Integration, Google should turn up > all the references you need. > RonL > technique and tried it on a special test case of the 10-fold > integration where I know that the answer is exactly one. > Unfortunately, even with 100,000,000 trials I get answers around > +/-0.03 of the true value. Although this is actually better than I > expected, it still isnt accurate enough for my purposes. I really > need at least six-figure accuracy, quickly, reliably and repeatably > (the calculation must be able to be performed on an ad hoc basis as > the various parameters are changed by the user). > I think the problem is that the integration space is just so vast that > to get this accuracy I need an infeasible number of calculations > whichever way I do it. Im not sure theres any way round it. But many > problems ... In the case of Monte-Carlo Integration the problem would be that the variation in the integrand is large rather than the integration space is vast. The error of a Monte-Carlo estimate depends on the sample size and the variation in the integrand rather than the size of the integration space (which is why it can be of help for high dimensional integrals). The accuracy of Monte-Carlo integration can be improved using variance reduction techniques (I seem to remember sitting in a seminar of Martin Beals where he presented a method which reduced the standard error in the integral in the toy problem he was using as an example to 0). However I would suspect that an error of less than 1 part in 10^6 might well be beyond all practical Monte-Carlo techniques unless the problems have a structure that allows the removal of almost all of the variation in the integrand. RonL -- Ignorance is the most delightful of the sciences .. === Subject: Ôinverse iteration for complex eigenvalues hello, i have been using the Ôinverse algorithm to find eigenvalues of a real matrix, but it seems to only converge satisfactorily for real eigenvalues. for complex eigenvalues it seems to give spurious and widely converging results. the algorithm can be found here http://www.cs.utk.edu/~dongarra/etemplates/node96.html im sure im using the algorithm correctly for the complex case, and correctly carrying out the complex number arithmetic, but i just cannot get it to converge to the correct complex eigenvalue. is it the case that this is only suited for real eigenvalues? === Subject: Sufficient to show bases generate the same topology? Hello all, In my topology class we have been assigned to write, as papers, certain problems assigned as homework and then trade for editing before turning them in. The current assignment was to show that the dictionary order topology on R x R is the same as the product topology on R_d x R where R_d is the real numbers with the discrete topology. Now, in reading a classmates paper, I have begun to question my own. I showed that any open set in R x R contains a basis element of R_d x R. Is this sufficient to show they generate the same topology? By a lemma from our text, this shows that the basis of R_d x R is a basis for the dictionary order topology on R x R. Do I need to Ôprove what the basis elements for the dictionary order topo I thought my argument had sufficiently answered the question, but my classmate starts by proving the intervals ( a x b, a x d ) where a x b denotes a point in R x R, form a basis for the dictionary order topology on R x R and then goes on to show that the basis elements are actually equivalent to those of R_d x R. But is this necessary? The fact that my argument is shorter leads me to question its validity. Should I? bbscott === Subject: Re: Sufficient to show bases generate the same topology? alt.math.undergrad: > Hello all, > In my topology class we have been assigned to write, as > papers, certain problems assigned as homework and then > trade for editing before turning them in. The current > assignment was to show that the dictionary order > topology on R x R is the same as the product topology on > R_d x R where R_d is the real numbers with the discrete > topology. > Now, in reading a classmates paper, I have begun to > question my own. I showed that any open set in R x R > contains a basis element of R_d x R. Is this sufficient > to show they generate the same topology? Not quite. You need to show that if V is open in R x R with the lexicographic order topology, and (x, y) is any point of V, then there is a basis element of R_d x R containing (x, y) and contained in V. In other words, you have to show that V is actually a union of basis elements of R_d x R. When youve done this, youll have shown that every set open in the lexicographic order topology on R x R is also open in R_d x R. This still doesnt show that the topologies are the same, however, since its still possible that the topology on R_d x R could contain sets that arent open in the lex. ord. top. on R x R. Thus, you also have to show that if V is any open set in R_d x R, and (x, y) is any point of V, then there is an open set in the lex. ord. top. on R x R containing (x, y) and contained in V. > By a lemma > from our text, this shows that the basis of R_d x R is a basis for the > dictionary order topology on R x R. No, it doesnt; youve misunderstood the lemma. If you copy it out in full, we can probably figure out what youre missing. [...] > I thought my argument had sufficiently answered the question, but my > classmate starts by proving the intervals > ( a x b, a x d ) > where a x b denotes a point in R x R, form a basis for the dictionary > order topology on R x R and then goes on to show that the basis elements > are actually equivalent to those of R_d x R. But is this necessary? Its about the most straightforward argument possible, and any correct argument is probably going to do essentially the same work, though the details might be a little different. [...] Brian === Subject: Re: Sufficient to show bases generate the same topology? >>Now, in reading a classmates paper, I have begun to >>question my own. I showed that any open set in R x R >>contains a basis element of R_d x R. Is this sufficient >>to show they generate the same topology? > Not quite. You need to show that if V is open in R x R with > the lexicographic order topology, and (x, y) is any point of > V, then there is a basis element of R_d x R containing > (x, y) and contained in V. In other words, you have to show > that V is actually a union of basis elements of R_d x R. > When youve done this, youll have shown that every set open > in the lexicographic order topology on R x R is also open in > R_d x R. This still doesnt show that the topologies are > the same, however, since its still possible that the > topology on R_d x R could contain sets that arent open in > the lex. ord. top. on R x R. Thus, you also have to show > that if V is any open set in R_d x R, and (x, y) is any > point of V, then there is an open set in the lex. ord. top. > on R x R containing (x, y) and contained in V. My apologies for the lack of clarity, the lemma is as follows: Let X be a topological space. Suppose C is a collection of open sets of X such that for each open set U of X and each x in U, there is an element C of C such that x in C and C is a subset of U. Then C is basis for the topology of X. Of course, I should have realized I had to show the converse too... I showed it for open sets in R x R containing basis elements of R_d x R but not the other way around. Thatll learn me for oversimplifying! >>By a lemma >>from our text, this shows that the basis of R_d x R is a basis for the >>dictionary order topology on R x R. > No, it doesnt; youve misunderstood the lemma. If you copy > it out in full, we can probably figure out what youre > missing. The lemmas above. I missed the crux of it; I used it to prove the first half of your argument (that each point of an open set in R x R is in a basis element of R_d x R which is contained in that open set) but neglected to realize the fact that two bases generate the same topology does not mean they are the same (?). >>I thought my argument had sufficiently answered the question, but my >>classmate starts by proving the intervals >>( a x b, a x d ) >>where a x b denotes a point in R x R, form a basis for the dictionary >>order topology on R x R and then goes on to show that the basis elements >>are actually equivalent to those of R_d x R. But is this necessary? > Its about the most straightforward argument possible, and > any correct argument is probably going to do essentially the > same work, though the details might be a little different. I see. I will rethink my argument. > Brian === Subject: Re: Sufficient to show bases generate the same topology? > >>Now, in reading a classmates paper, I have begun to >question my own. I showed that any open set in R x R >contains a basis element of R_d x R. Is this sufficient >to show they generate the same topology? >> Not quite. You need to show that if V is open in R x R with >> the lexicographic order topology, and (x, y) is any point of >> V, then there is a basis element of R_d x R containing >> (x, y) and contained in V. [...] > My apologies for the lack of clarity, the lemma is as follows: > Let X be a topological space. Suppose C is a collection of open sets of > X such that for each open set U of X and each x in U, there is an > element C of C such that x in C and C is a subset of U. Then C is > basis for the topology of X. [...] >By a lemma >from our text, this shows that the basis of R_d x R is a basis for the >dictionary order topology on R x R. >> No, it doesnt; youve misunderstood the lemma. If you copy >> it out in full, we can probably figure out what youre >> missing. Note that in the lemma C is a collection of *open* sets in X. If you show that each basis element of R_d x R is open in R x R with the lex. ord. top., and then show (as you did) > that each point of an open set in R x R is in a > basis element of R_d x R which is contained in that open set then you can appeal to the lemma to say that the basis for R_d x R is also a basis for the lex. ord. top. on R x R. [...] Brian === Subject: Any good texts on Philosphy of Mathematics Any good texts? === Subject: Re: Any good texts on Philosphy of Mathematics > Any good texts? You might want to look at _Thinking About Mathematics_, by Shapiro. ISBN 0192893068 -- Replace Roman numerals with digits to reply by email === Subject: Re: Any good texts on Philosphy of Mathematics > Any good texts? For example: Philosophy of Mathematics, an Anthology edited by Dale Jacquette, published by Blackwell. Philosophy of Mathematics: Selected Readings edited by Paul Benacerraf and Hilary Putnam, published by Cambridge University Press. Frege: Philosophy of Mathematics Michael Dummett, published by Duckworth. The Philosophy of Mathematics Stephan Korner, Introduction to Mathematical Philosophy Bertrand Russell, published by Allen & Unwin. === Subject: Proving that two topologies are equivalent (Revisited) I am reading a book, and it contains the following exercise which I would be glad to have some help on: Show that the diamond metric: dd((x1,...,xn),(y1,...,yn)) = |x1-y1|+...+|xn-yn| Generates the same topology as the square metric: ds((x1,...,xn),(y1,...,yn)) = max{|x1-y1|,...,|xn-yn|} by using the lemma that the composition of continuous maps between topological spaces is again continuous. Quoting from the book Notice that this mutual inclusion argument is much less work than expressing open sets in one directly as unions of explicitly defined open sets in the other OK. So I think that an open ball in the one topological space can be mapped to an open ball in the other by a suitable bijective map. For 2-D this would be a rotation of PI/4 in the plane. I think for 3-D this is again a rotation (rotation of a cube?). If such a mapping exists, then it must be continuous in either direction since open sets map to open sets. This would be one of the continuous maps mentioned in the lemma. Lets call this map A: A:(X,dd)->(X,ds) Then we consider the inverse rotation, but this time we keep within the same topological space: B:(X,ds)->(X,ds) Assume (for the moment) that this map is continuous, we have: BA:(X,dd)->(X,ds):x->x Which from the lemma, must be continuous. So if a set is open in one of the topological spaces, it must be open in the other. So in order for this proof to work, I would need to show that a mapping always exists that converts open balls in the one topology to open balls in the other. And I would need to show that its inverse is continuous when considered as a mapping within a topological space. Im not sure if my approach is what the author had in mind, but it is the only way I can see that you would incorporate the lemma? It doesnt seem that much easier to me, since you still have to deal with the nitty details of how the metrics are defined? Am I missing something? === Subject: Re: Proving that two topologies are equivalent (Revisited) > I am reading a book, and it contains the following exercise which I would be > glad to have some help on: > Show that the diamond metric: > dd((x1,...,xn),(y1,...,yn)) = |x1-y1|+...+|xn-yn| > Generates the same topology as the square metric: > ds((x1,...,xn),(y1,...,yn)) = max{|x1-y1|,...,|xn-yn|} > by using the lemma that the composition of continuous maps between > topological spaces is again continuous. > Quoting from the book Notice that this mutual inclusion argument is much > less work than expressing open sets in one directly as unions of explicitly > defined open sets in the other [...] Im not sure what the author has in mind but I doubt it is rotations in n-space. Consider the identity map from (X,ds) to (X,dd). To show it is continuous, let e > 0, choose d = e/n then ds(x, y) < d => dd(x, y) < e. Similarly - and even easier - the identity map from (X,dd) to (X,ds) is continuous. Those two show the identity is a homeomorphism and consequently the topologies are the same. Whether that is the _desired_ solution, I dont know. -- Paul Sperry Columbia, SC (USA) === Subject: Re: Proving that two topologies are equivalent (Revisited) alt.math.undergrad: >> I am reading a book, and it contains the following exercise which I would be >> glad to have some help on: >> Show that the diamond metric: >> dd((x1,...,xn),(y1,...,yn)) = |x1-y1|+...+|xn-yn| >> Generates the same topology as the square metric: >> ds((x1,...,xn),(y1,...,yn)) = max{|x1-y1|,...,|xn-yn|} >> by using the lemma that the composition of continuous maps between >> topological spaces is again continuous. >> Quoting from the book Notice that this mutual inclusion argument is much >> less work than expressing open sets in one directly as unions of explicitly >> defined open sets in the other > [...] > Im not sure what the author has in mind but I doubt it is rotations in > n-space. Likewise. > Consider the identity map from (X,ds) to (X,dd). To show it is > continuous, let e > 0, choose d = e/n then > ds(x, y) < d => dd(x, y) < e. > Similarly - and even easier - the identity map from (X,dd) to (X,ds) is > continuous. > Those two show the identity is a homeomorphism and consequently the > topologies are the same. > Whether that is the _desired_ solution, I dont know. It fits the bit about Ôthis mutual inclusion argument, but I dont see that its using preservation of continuity under composition. Then again, I dont see how that lemma would come into play in any natural approach to the problem. Brian === Subject: Newtons Law of Cooling by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iACMhBX19918; How do you sovle the equation T(t)=s+(To-s)e^-kt for the value of k? I need the value for k. I need the equation to resemble k=.... PLEASE HELP! THANK YOU! === Subject: Re: Newtons Law of Cooling >How do you sovle the equation T(t)=s+(To-s)e^-kt for the value of k? >I need the value for k. I need the equation to resemble k=.... >PLEASE HELP! THANK YOU! First use algebra to get as much stuff that isnt k on the other side of the equal sign as possible. (T(t)-s)/(To-s)=e^-kt Now do the same thing to both sides, take the log of each side log((T(t)-s)/(To-s))=log(e^-kt) Then use rules about log(e^a)=... Ill leave the rest of this to you === Subject: Generative Grammar but not context-sensitive is the Grammar G(N, T, n P) N = {a, b, c, s} n = s T = {-, *} P = {abc -> cba, ba -> *-, cb -> -, s -> abc} not a context-sensitive grammar? If you could kindly answer this simple question, i can find out whether i got === Subject: Re: Generative Grammar but not context-sensitive alt.math.undergrad: > is the Grammar G(N, T, n P) > N = {a, b, c, s} > n = s > T = {-, *} > P = {abc -> cba, ba -> *-, cb -> -, s -> abc} > not a context-sensitive grammar? It is context-sensitive. However, the only possible derivations are s --> abc --> cba --> -a and s --> abc --> cba --> c*-, so its impossible to derive a terminal string, and the grammar generates the empty language. Thus, its a context-sensitive grammar for a context-free language. [...] Brian === Subject: Re: Generative Grammar but not context-sensitive In a book they define a context sensitive grammar as follows: P is of the form: XyZ -> XYZ where X, Y, Z element A* and y element N where A* is the set of all words over the Union of T and N and N is the set of non-terminating symbols and T is the set of terminating symbols So I wonder how abc -> cba can be of the from XyZ? Marcel >> N = {a, b, c, s} >> n = s >> T = {-, *} >> P = {abc -> cba, ba -> *-, cb -> -, s -> abc} >> not a context-sensitive grammar? > It is context-sensitive. However, the only possible > derivations are > s --> abc --> cba --> -a > > and > s --> abc --> cba --> c*-, > > so its impossible to derive a terminal string, and the > grammar generates the empty language. Thus, its a > context-sensitive grammar for a context-free language. > [...] > Brian === Subject: Re: Generative Grammar but not context-sensitive alt.math.undergrad: > N = {a, b, c, s} > n = s > T = {-, *} > P = {abc -> cba, ba -> *-, cb -> -, s -> abc} > not a context-sensitive grammar? >> It is context-sensitive. [...] > In a book they define a context sensitive grammar as follows: > P is of the form: > XyZ -> XYZ > where X, Y, Z element A* > and y element N > where A* is the set of all words over the Union of T and N > and N is the set of non-terminating symbols > and T is the set of terminating symbols > So I wonder how abc -> cba can be of the from XyZ? Sorry, youre right: its Type 0. I got to thinking about the language that it produces and lost my train of thought. Brian === Subject: help me by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAE1I8X20288; what walks on one leg in the morning? two legs in the afternoon and one leg at night? === Subject: Re: help me >what walks on one leg in the morning? >two legs in the afternoon and >one leg at night? This is NOT a riddle group! Itsa math group! === Subject: Re: help me >what walks on one leg in the morning? >two legs in the afternoon and >one leg at night? > This is NOT a riddle group! Itsa math group! To be a little more positive, I suggest you send riddles to the rec.puzzles news group. Ken Pledger. === Subject: Re: help me > what walks on one leg in the morning? > two legs in the afternoon and > one leg at night? The letter n === Subject: Pt. of Inßection / Crit Number Problem Consider a fuction with domain (-10, 10). f Ô (x) = [(x-5)^3][(x+3)^8], domain -10Consider a fuction with domain (-10, 10). >f(x) = [(x-5)^3][(x+3)^8], domain -10f(x) = [(x-5)^2][(x+3)^7](11x-31), domain -10Solving for crits, I find -3, and 5. Okay, I didnt really solve - >I just dont see the opportunity to calculate any other numbers. >Checking for positive/negative along the number line, I find that >its decreasing from -10 to 5, and increasing from 5 to 10. It depends on the meaning of it. f(x) <= 0 for -10 <= x <= 5, f(x) >= 0 for 5 <= x <= 10, f(x) does not change sign at x = -3. Therefore, f(x) (the it) is decreasing from -10 to 5 and increasing from 5 to 10. >Solving for inßection points, I find -3, 5, and 11/31. Checking on >a number line, we find that the second derivative is increasing >from -10 to -3, decreasing from -3 to 5, increasing from 5 to 10, >with no change of signs before and after 11/31 (~2.8). -3, 5, and 31/11 (not 11/31) are just possible inßection points. It is also necessary for f(x) to change signs. f(x) >= 0 for -10 <= x <= -3, f(x) <= 0 for -3 <= x <= 31/11, f(x) >= 0 for 31/11 <= x <= 10, f(x) does not change sign at x = 5. Therefore, the inßection points are -3 and 31/11. f(x) is concave down when f(x) <= 0, i.e., from -3 to 31/11. >The webworks problem asks the following: >Critical numbers from left to right (two of them.) >Function increasing on interval: >Function decreasing on interval: >X-Co-Ords of inßection points, from left to right (two of them): >And interval where function is concave down: >I answered -3, 5 for crits. 5, 10 for f increasing, -10 to 5 for f >decreasing, -3 and 5 for inßection points, and function is concave >down on -3, 5. critical points: -3, 5 f(x) increasing from 5 to 10 f(x) derceasing from -10 to 5 inßection points -3 and 31/11 f(x) is concave down from -3 to 31/11 >Webworks says that I am 90% correct, but doesnt specify where I >went wrong. Ive gone over my math a few times, but I dont see what >Im missing. I feel like I left something out by not having 11/31 >appear anywhere in the answer, but the only place I imagined it >would fit is as the second inßection point, but substituting it in >for 5 both as the second inßectoin point, and as the end-point of >the concave down interval (either one at a time, or both), still >doesnt net me the right answer. That is because you answered 11/31 instead of 31/11. === Subject: Re: Pt. of Inßection / Crit Number Problem > Consider a fuction with domain (-10, 10). > f Ô (x) = [(x-5)^3][(x+3)^8], domain -10 f Ô (x) = [(x-5)^2][(x+3)^7](11x-31), domain -10 Solving for crits, I find -3, and 5. Okay, I didnt really solve - I just > dont see the opportunity to calculate any other numbers. Checking for > positive/negative along the number line, I find that its decreasing > from -10 to 5, and increasing from 5 to 10. > Solving for inßection points, I find -3, 5, and 11/31. 31/11 [...] -- Paul Sperry Columbia, SC (USA) === Subject: Re: Pt. of Inßection / Crit Number Problem >> Consider a fuction with domain (-10, 10). >> f Ô (x) = [(x-5)^3][(x+3)^8], domain -10> f Ô (x) = [(x-5)^2][(x+3)^7](11x-31), domain -10> Solving for crits, I find -3, and 5. Okay, I didnt really solve - I just >> dont see the opportunity to calculate any other numbers. Checking for >> positive/negative along the number line, I find that its decreasing >> from -10 to 5, and increasing from 5 to 10. >> Solving for inßection points, I find -3, 5, and 11/31. > 31/11 I didnt actually make that mistake. It was a typo as I misremembered when not looking at the paper I did my work on. === Subject: Re: Pt. of Inßection / Crit Number Problem Got it. I ... still dont know what exactly I was doing wrong, but Ive got it all right now. Damn, I wish I knew where exactly I went wrong in my math. I think I just fudged my results where I checked various points before and after the inßection points. === Subject: Horiz. Asymptote In the function f(x) = (5) / [(e^-2x) - 6] Left-hand horizontal asymptote: Right-hand horizontal asymptote: Vertical asymptote: I solved for the horizontal asymptote by turning the equation into 5 / x (Using the rule of A / x^r where x approaches inf. equals 0), and got 0 (which webworks says is correct on the left-hand, but incorrect on the right.) I solved for the vertical, finding that it is, correctly, -.8958. My question is this, as I cant seem to find a strategy for it in my textbook, nor a sample problem that attacks it: a) How do I find horizontal asymptotes from the left and right hand specifically, as opposed to just generally? and b) How would I go about doing it in this problem specifically? ( A strategy would be appreciated; an answer is, for the time being at least, unneccasary. Id rather try my own hand at it; I just dont know where to begin. ) === Subject: Re: Horiz. Asymptote alt.math.undergrad: >In the function f(x) = (5) / [(e^-2x) - 6] >Left-hand horizontal asymptote: >Right-hand horizontal asymptote: >Vertical asymptote: >I solved for the horizontal asymptote by turning the equation into 5 / x I do not understand that. Whats the limit of e^(-2x as x->infinity? Its zero, right? Therefore the limit of f(x) as x->infinity is -5/6, and you have a horizontal asymptote at y = -5/6. You can use similar reasoning to find the left horizontal asymptote, if there is one. >My question is this, as I cant seem to find a strategy for it in my >textbook, nor a sample problem that attacks it: >a) How do I find horizontal asymptotes from the left and right hand >specifically, as opposed to just generally? As above: find the limits of the function as x->inf and -inf. If those limits exist, then y = those limits is/are the equation(s) of the horizontal asymptotes. >and b) How would I go about doing it in this problem specifically? As above. I find it hard to believe that your textbook has no useful examples. What are the author and title of it -- or do you just find it easier to post to Usenet than to read your textbook? -- Stan Brown, Oak Road Systems, Tompkins County, New York, USA http://OakRoadSystems.com/ And if youre afraid of butter, which many people are nowa- days, (long pause) you just put in cream. --Julia Child === Subject: Re: Horiz. Asymptote Stan Brown bitches yet again in message <... What are the author and title of it -- or do you just find it easier > to post to Usenet than to read your textbook? Since he wants to actually _communicate_ with someone about this, quite obviously it is indeed easier accomplished posting to a math NG rather than reading a textbook. What a smart ass! Apparently he has not considered that most people do not hear voices from their textbooks that they can converse with. Dont worry about him...hes on the rag again. Several recent posts of his carry the same theme about reading the book. Got a private email from one of his students once about his rants, even. It is possible he may state a thing or two of value interspersed with his ranting and raving, if you choose to muddle through the ranting and raving, that is. Youve received some good answers on horz. asymptotes of f(x). If the limit(s) exist, they are they line(s): y= lim x-->+/-oo f(x) True, thats a Ôgeneral response, but to make it specific to your case, just plug in your function: y= lim x-->+/-oo 5/((e^-2x)-6) I didnt see anyone mention vertical asymptotes, other than your stating you solved it correctly. Generally, they occur where the denominator is zero and the numerator is nonzero (be sure to reduce first, if possible.) In this case, 5 is never zero and: e^-2x - 6 = 0 e^-2x = 6 ln(e^-2x) = ln 6 -2x = ln 6 x = (-ln6) /2 note this is not an approximation Oblique (slant) asymptotes are another type. IOW, asymptotes that are neither vertical nor horizontal. Headstart for oblique asymptotes of rational functions (please ignore if appropriate): Reduce first, if possible. If the numerator is then exactly one degree higher than the denominator, an oblique asymptote exists. To find its equation, perform long division to get f(x)=quotient+remainder. It will have equation y=quotient (ignore any remainder). -- Darrell === Subject: Re: Horiz. Asymptote > alt.math.undergrad: >>In the function f(x) = (5) / [(e^-2x) - 6] >>Left-hand horizontal asymptote: >>Right-hand horizontal asymptote: >>Vertical asymptote: >>I solved for the horizontal asymptote by turning the equation into 5 / x > I do not understand that. > Whats the limit of e^(-2x as x->infinity? Its zero, right? > Therefore the limit of f(x) as x->infinity is -5/6, and you have a > horizontal asymptote at y = -5/6. > You can use similar reasoning to find the left horizontal asymptote, > if there is one. >>My question is this, as I cant seem to find a strategy for it in my >>textbook, nor a sample problem that attacks it: >>a) How do I find horizontal asymptotes from the left and right hand >>specifically, as opposed to just generally? > As above: find the limits of the function as x->inf and -inf. If > those limits exist, then y = those limits is/are the equation(s) of > the horizontal asymptotes. >>and b) How would I go about doing it in this problem specifically? > As above. > I find it hard to believe that your textbook has no useful examples. > What are the author and title of it -- or do you just find it easier > to post to Usenet than to read your textbook? Calculus And Its Applications, Strauss, Bradley, and Smith. And, by all means, when Im done with this course, youre welcome to buy it off of me used - complete with every chapters text and important points highlighted, and extensive ßashcard-format notes on everything. Sure, I dont read a thing. === Subject: Re: Horiz. Asymptote >> alt.math.undergrad: >In the function f(x) = (5) / [(e^-2x) - 6] >Left-hand horizontal asymptote: >Right-hand horizontal asymptote: >Vertical asymptote: >I solved for the horizontal asymptote by turning the equation into 5 / x >> I do not understand that. >> Whats the limit of e^(-2x as x->infinity? Its zero, right? >> Therefore the limit of f(x) as x->infinity is -5/6, and you have a >> horizontal asymptote at y = -5/6. >> You can use similar reasoning to find the left horizontal asymptote, >> if there is one. >My question is this, as I cant seem to find a strategy for it in my >textbook, nor a sample problem that attacks it: >a) How do I find horizontal asymptotes from the left and right hand >specifically, as opposed to just generally? >> As above: find the limits of the function as x->inf and -inf. If >> those limits exist, then y = those limits is/are the equation(s) of >> the horizontal asymptotes. >and b) How would I go about doing it in this problem specifically? >> As above. >> I find it hard to believe that your textbook has no useful examples. >> What are the author and title of it -- or do you just find it easier >> to post to Usenet than to read your textbook? > Calculus And Its Applications, Strauss, Bradley, and Smith. And, by all > means, when Im done with this course, youre welcome to buy it off of me > used - complete with every chapters text and important points > highlighted, and extensive ßashcard-format notes on everything. > Sure, I dont read a thing. And, I /do/ like reading it on Usenet. I find the alternative approaches explained here make my understanding of the topics much more comprehensive than they would be otherwise. === Subject: Re: Horiz. Asymptote > In the function f(x) = (5) / [(e^-2x) - 6] > Left-hand horizontal asymptote: > Right-hand horizontal asymptote: > Vertical asymptote: > I solved for the horizontal asymptote by turning the equation into 5 / x > (Using the rule of A / x^r where x approaches inf. equals 0), and got 0 > (which webworks says is correct on the left-hand, but incorrect on the > right.) > I solved for the vertical, finding that it is, correctly, -.8958. > My question is this, as I cant seem to find a strategy for it in my > textbook, nor a sample problem that attacks it: > a) How do I find horizontal asymptotes from the left and right hand > specifically, as opposed to just generally? Just consider the limits of the function as x approaches -infinity and as x approaches +infinity. If the former is a real constant, then y = that constant gives a left horizontal asymptote; if the latter is a real constant, then y = that constant gives a right horizontal asymptote. > and b) How would I go about doing it in this problem specifically? Perhaps begin by asking yourself what happens to e^(-2x) as x approaches either -oo or +oo. Then... David === Subject: Squeeze Rule This question isnt so much about my not understanding the squeeze rule, as it is, the question being one step too complicated for me to know how to apply it. Limit of a function as x approaches infinity, where the function is: [-2cos(x) +3sin(x) ]/ x Now, the way I approached it was: -1/x < our function < 1/x And sought the limit of 1/x. The thing is, I dont know what to do about the infinity. I decided that since we were using 1/x, and x was approaching infinity, what I was looking for was a horizontal asymptote (which we know is 0, because, hey, its 1/x). I suspect that 0 is the correct answer, but I was hoping this could be clarified somewhat, because Im feeling pretty shaky on it, and doubt Id want to walk into an exam like this. === Subject: Re: Squeeze Rule > This question isnt so much about my not understanding the squeeze rule, as > it is, the question being one step too complicated for me to know how to > apply it. > Limit of a function as x approaches infinity, where the function is: > [-2cos(x) +3sin(x) ]/ x > Now, the way I approached it was: > -1/x < our function < 1/x Not so, at x = pi, your function f(x) = 2/pi -5/x <= f(x) <= 5/x > And sought the limit of 1/x. The thing is, I dont know what to do about the > infinity. > I decided that since we were using 1/x, and x was approaching infinity, what > I was looking for was a horizontal asymptote (which we know is 0, because, > hey, its 1/x). Thats right, lim(x->oo) a/x = 0 > I suspect that 0 is the correct answer, but I was hoping this could be > clarified somewhat, because Im feeling pretty shaky on it, and doubt Id > want to walk into an exam like this. Review the definition for lim(x->oo) g(x) and show lim(x->oo) 1/x = 0, the old fashion way grinding it out with epsilon and N. === Subject: Now math fraud, case against algebraic integers I think most of you probably fail to see how basic and simple the mathematical argument I have is, so I thought Id show you, and remind me of your future, or lack of it, if you think this information can just be ignored. Basically I had an idea a while back to factor polynomials into non-polynomial factors by factoring with respect to something other than the polynomial variable. For instance, consider P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078 which can be factored into non-polynomial factors as P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 which looks complicated but if you multiply it all out and simplify, you get the first, and I just have an analysis method for breaking up a polynomial in a certain way to factor it--into non-polynomial factors--and here the factorization is P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) where the as are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). And its easy to get that cubic by solving for one of the as using (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) = 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3. Now then, the constant term of P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078 is 1078, which equals 7(7)(22), and it turns out all of the coefficients of P(x) have 49 as a factor, so you have P(x) = 49(300125 x^3 - 18375 x^2 - 360 x + 22) and its fairly easy to show that two of the as have 7 as a factor, algebraically, when its also easy to show that in the ring of algebraic integers, if a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) is irreducible over Q, then in the ring of algebraic integers, not one of them has 7 as a factor. Showing that algebraically two of the as have 7 as a factor is as easy as dividing that 49 from both sides and paying close attention: P(x)/49 = 300125 x^3 - 18375 x^2 - 360 x + 22 and P(x)/49 = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7)/49 where the question is, how does 49 divide through? Well, for the sake of argument, imagine you have w_1(x) w_2(x) w_3(x) = 49, and b_1(x), b_2(x), b_3(x), factors of a_1(x), a_2(x), and a_3(x), respectively, that result from dividing 49 from both sides: P(x)/49 = (5 b_1(x) + w_1(x))(5 b_2(x) + w_2(x))(5 b_3(x) + w_3(x)) BUT P(x)/49 = 300125 x^3 - 18375 x^2 - 360 x + 22 so any terms contributing to the constant term, which is 22, must be coprime to 7. Now maybe you wish to imagine that NEITHER w_1(x), w_2(x), nor w_3(x) is in any way a factor of the constant term--22--but then youre not a mathematician. Necessarily, any of them that are factors of 22 are coprime to 7, unless theyre units, but they cant all be units as they multiply to give 49. So what gives? Why do people argue with me--successfully if you believe them--when ultimately their position requires that you believe that 7 and 22 share non-unit factors? Well, you see, they DO in the ring of algebraic integers! Provably, in the ring of algebraic integers, each of the as must share non-unit factors with 7, which forces that 22 share non-unit factors with 7 as well--in the ring of algebraic integers. Its a basic contradiction--in the ring of algebraic integers. Confused? Lost? Think Im just wrong, even if you cant figure out why? Well, its algebra. Remember, algebra is a rather important part of mathematics. If you cant grasp algebra at this level, what makes you think you can be a mathematician? Give up on understanding here, and you need to give up on your math degree. Resolving the issue--as mathematics IS consistent--requires figuring out what is going wrong, and Ive explained that elsewhere so I wont here. What I will point out to you is the basic known fact that human beings resist change mightily. Worse, the result above is important enough to shoot down the way that Galois Theory is currently taught, and big enough to blow apart various arguments thought to be proofs over a hundred years plus. Your math professors dont want to deal with such a problem. Why would they? Theyre old. They have nowhere to go. What? You figure they want to go back to school and re-learn algebraic number theory? As far as theyre concerned, who cares if the math theyve taught is wrong, who cares because at the end of the day, they have bills, children to take care of, and a comfortable life that they dont want upset. But you are students. You are learning. You have your future ahead of you. The old men who teach you, will sacrifice you for their own comforts and betray the spirit of mathematics because human nature is that weak. Its up to you. Dig deeper. Find out whats true, and then decide if you wish to fake being a mathematician, or be one, a real one, like me. James Harris http://mathforprofit.blogspot.com/ === Subject: Re: Now math fraud, case against algebraic integers > Resolving the issue--as mathematics IS consistent ... How do you know mathematics is consistent? Do you have a proof of that? === Subject: Re: Now math fraud, case against algebraic integers days. My association with the Department is that of an alumnus. >How do you know mathematics is consistent? Do you have a proof of that? Theres a saying in Mexico: le estas pidiendo peras al olmo. Roughly: you are asking the elm to give you pears. -- Its not denial. Im just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) Arturo Magidin magidin@math.berkeley.edu === Subject: JSH: Minor correction Oh well, already have a minor correction to what I posted as it should be w_1(x) w_2(x) w_3(x) = 7 and if you have some math skills you should see why it should be that instead of what I had in my post! Make no mistake though, if you follow the algebra the contradiction in the ring of algebraic integers is real, where you can prove that 22 and 7 share non-unit factors in that ring. Part of the resolution is that a number can be a non-unit in the ring of algebraic integers when it properly is a unit. I say properly because the ring of algebraic integers for historical reasons is thought to be ok, when it is quirky, so that 22 and 7 can share non-unit factors in the ring. Its weird. Its quirky. Its like one of the hardest intellectual puzzles youll ever come across to figure it out and understand it all. James Harris http://mathforprofit.blogspot.com/ === Subject: Re: JSH: Minor correction >Oh well, already have a minor correction to what I posted Of course you do. The rest of us have major corrections, btw. >[...] >Its weird. Yes, the fact that you continue to post this crap is weird. Youve been finding a lot more support in alt.math.undergrad though, right? Giggle. ************************ David C. Ullrich === Subject: Re: JSH: Minor correction > Part of the resolution is that a number can be a non-unit in the ring > of algebraic integers when it properly is a unit. I say properly But you dont say what it means. What *does* it mean for a number to Ôproperly be a unit ? -- Larry Lard Replies to group please === Subject: JSH: Contradiction shown Now it turns out that having thought it all through I finally realized that I could show a direct contradiction using the objections raised against my work. Notice here Ill actually use constants where before I talked about variables being held constant. I start with P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078 which can be factored into non-polynomial factors by using P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 which looks complicated but if you multiply it all out and simplify, you get the first, and I just have an analysis method for breaking up a polynomial in a certain way to factor it--into non-polynomial factors--and here the factorization is P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) where the as are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). And its easy to get that cubic by solving for one of the as using (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) = 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3. Now Ive pointed out that setting x=0, will show that two of the as equal 0 at that point, while one equals 3, which others have argued is a special case. The cubic with x=0 is a^3 -3a^2 = 0 so two of the as equal 0, while one equals 3. Usually I arbitrarily select a_1, and a_2 to equal 0, at x=0, which is what Ill do now. Then, with P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) at x=0, you have two of the as equal 0, while one equals 3, to give P(0) = 7(7)(22) = 1078 which fits with P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078. Then for the first two factors (5a_1(x) + 7) and (5a_2(x) + 7) the 7s visible in the expressions are factors of the constant term of P(x). But P(x) is a multiple of 49 as *each* coefficient has 49 as a factor, so I divide out the 49 to get P(x)/49 = 300125 x^3 - 18375 x^2 - 360 x + 22. Now assume each of the as has some non-unit factor in common with 7, for a given x, which in fact they do in the ring of algebraic integers whenever all the as are irrational, which is a point that has been brought up repeatedly by people arguing with me. For a while I resisted that fact, but now as Ive said before I concede that they are in fact correct--each of the as, in the ring of algebraic integers does in fact share a non-unit factor with 7 when all of the as are irrational. So let w_1(x) w_2(x) w_3(x) = 49, where the ws are those factors, so a_1(x) = w_1(x) b_1(x), a_2(x) = w_2(x) b_2(x), and a_3(x) = w_3(x) b_3(x) and where w_1(x) v_1(x) = 7, w_2(x) v_2(x) = 7, and w_3(x) v_3(x) = 7, and divide through by 49 to get P(x)/49 = (5 b_1(x) + v_1(x))(5 b_2(x) + v_2(x))(5 b_3(x) + v_3(x)) and if you allow that the factors are each factors of the constant term as before, then you have v_1(0) v_2(0)(15 + v_3(0)) = 22 at x=0, and when x does not equal 0, you have v_1(x) v_2(x)(5u_3(x) + v_3(x)) = 22 where I introduce u_3(x) to handle any further weirdness with how a_3(x) behaves, where u_3(0) = 3, to agree with previous results, and remember P(x)/49 = 300125 x^3 - 18375 x^2 - 360 x + 22 and the constant term doesnt change, as, well, its constant, as its 22. So, notice, I now have that v_1(x) and v_2(x) are factors of 22. Provably, they cannot be units in the ring of algebraic integers when a_1(x), a_2(x), and a_3(x) are all irrational as thats what people argued with me about for so long. The full result was just so weird that it escaped even me for a while, as following their arguments to their logical conclusion 22 and 7 must share non-unit factors in the ring of algebraic integers. But you can also appear to prove that 22 and 7 are coprime in the ring of algebraic integers using various accepted definitions of coprimeness, like you can find algebraic integers x, and y such that 22x + 7y = 1 and claim to have proven that they are coprime. So, in the ring of algebraic integers, using whats commonly accepted you can prove that 22 and 7 are both coprime and that they share non-unit algebraic integer factors. Thats the full result. Now, of course, posters arguing with me on Usenet wouldnt follow through to the full result, probably because they werent smart enough to see it, as make no mistake, thats the kind of result thats quite big. Thats a career maker, or could have been one. For me, its just one of my results. Nice, sure, but still just one, with techniques I introduced in a couple of lines in another paper. The ring of algebraic integers is quirky. That arbitrary selection of roots of monic polynomials with integer coefficients allows you to do all kinds of wacky things, including believing that youre doing all kinds of great math, when you have nothing at all. Make no mistake, the big names in math today have just about zero reason to accept these results. Would you if you were them? But youre not them. You are students. You have the future to make your names, with real math that is actually correct. Im offering you your future. Follow the math, and then if you have what it takes, then accept what is true without holding your breath waiting for old men who are trying to stave off ruin to do the right thing. They are cowards. They will admit the truth when they no longer believe they can get away with lying. Come on, do you really think results like these could have been missed by the top mathematicians of the world? Well, maybe, maybe they dont know. But I think they do. After all, Ive contacted people all over the math world, as Ive been quite bold in pushing forward this issue, in warning about the error. I think they do know and they are waiting to see what you do. I think they are waiting to see if you will let them get away with the lies. Maybe you will...or at least, maybe youll try. LOL. Some of you probably will try because youre young, and foolish, and so eager to please them. James Harris http://mathforprofit.blogspot.com/ === Subject: Re: JSH: Contradiction shown > Now it turns out that having thought it all through I finally realized > that I could show a direct contradiction using the objections raised > against my work. > Notice here Ill actually use constants where before I talked about > variables being held constant. > I start with > P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078 > which can be factored into non-polynomial factors by using > P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 > which looks complicated but if you multiply it all out and simplify, > you get the first, and I just have an analysis method for breaking up > a polynomial in a certain way to factor it--into non-polynomial > factors--and here the factorization is > P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) > where the as are roots of > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). > And its easy to get that cubic by solving for one of the as using > (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) = > 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3. > Now Ive pointed out that setting x=0, will show that two of the as > equal 0 at that point, while one equals 3, which others have argued is > a special case. Could you explain what is meant by it being a special case? It looks no more special than setting x=1, 2, 3, 100, etc. > The cubic with x=0 is > a^3 -3a^2 = 0 > so two of the as equal 0, while one equals 3. So, you would write that as a_1(0)=0, a_2(0)=0, a_3(0)=3, without loss of generality. > Usually I arbitrarily select a_1, and a_2 to equal 0, at x=0, which is > what Ill do now. Then, with > P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) > at x=0, you have two of the as equal 0, while one equals 3, to give > P(0) = 7(7)(22) = 1078 > which fits with > P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078. > Then for the first two factors (5a_1(x) + 7) and (5a_2(x) + 7) the 7s > visible in the expressions are factors of the constant term of P(x). 1078 factors as 7*7*2*11, yes. > But P(x) is a multiple of 49 as *each* coefficient has 49 as a factor, > so I divide out the 49 to get > P(x)/49 = 300125 x^3 - 18375 x^2 - 360 x + 22. Ok. > Now assume each of the as has some non-unit factor in common with 7, > for a given x, which in fact they do in the ring of algebraic integers > whenever all the as are irrational, which is a point that has been > brought up repeatedly by people arguing with me. For a while I > resisted that fact, but now as Ive said before I concede that they > are in fact correct--each of the as, in the ring of algebraic > integers does in fact share a non-unit factor with 7 when all of the > as are irrational. > So let w_1(x) w_2(x) w_3(x) = 49, where the ws are those factors, so > a_1(x) = w_1(x) b_1(x), > a_2(x) = w_2(x) b_2(x), > and > a_3(x) = w_3(x) b_3(x) > and where > w_1(x) v_1(x) = 7, w_2(x) v_2(x) = 7, and w_3(x) v_3(x) = 7, > and divide through by 49 to get > P(x)/49 = > (5 b_1(x) + v_1(x))(5 b_2(x) + v_2(x))(5 b_3(x) + v_3(x)) > and if you allow that the factors are each factors of the constant > term as before, then you have > v_1(0) v_2(0)(15 + v_3(0)) = 22 > at x=0, At this point you have made no restrictions on the behavior of the ws and vs for values other than x. For example, it would be nice if the ws and vs were functions from the algebraic integers to the algebraic integers. > and when x does not equal 0, you have > v_1(x) v_2(x)(5u_3(x) + v_3(x)) = 22 > where I introduce u_3(x) to handle any further weirdness with how > a_3(x) behaves, where u_3(0) = 3, to agree with previous results, and > remember Care to offer any properties of u_3(x), such as domain and range? > P(x)/49 = 300125 x^3 - 18375 x^2 - 360 x + 22 > and the constant term doesnt change, as, well, its constant, as its > 22. > So, notice, I now have that v_1(x) and v_2(x) are factors of 22. > Provably, they cannot be units in the ring of algebraic integers when > a_1(x), a_2(x), and a_3(x) are all irrational as thats what people > argued with me about for so long. > The full result was just so weird that it escaped even me for a while, > as following their arguments to their logical conclusion 22 and 7 must > share non-unit factors in the ring of algebraic integers. Or, more likely, you have somewhere *left* the ring of algebraic integers because you werent careful to make sure you stayed in it. > But you can also appear to prove that 22 and 7 are coprime in the ring > of algebraic integers using various accepted definitions of > coprimeness, like you can find algebraic integers x, and y such that > 22x + 7y = 1 > and claim to have proven that they are coprime. They are coprime in the ring of integers, which is a subring of the algebraic integers. Therefor, they *are* coprime in the ring of algebraic integers. Why do you use the phrase appear to prove? > So, in the ring of algebraic integers, using whats commonly accepted > you can prove that 22 and 7 are both coprime and that they share > non-unit algebraic integer factors. > Thats the full result. You havent demonstrated the non-unit algebraic integer factor(s). -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: JSH: Contradiction shown > Now it turns out that having thought it all through I finally realized > that I could show a direct contradiction using the objections raised > against my work. ... it just gets better and better ... > Notice here Ill actually use constants where before I talked about > variables being held constant. As usual, you are incredibly vague -- all of the upcoming blather takes place (presumably) in some ring. Just _which_ ring might that be ?? > I start with > P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078 > which can be factored into non-polynomial factors by using > P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 > which looks complicated but if you multiply it all out and simplify, > you get the first, and I just have an analysis method for breaking up > a polynomial in a certain way to factor it--into non-polynomial > factors--and here the factorization is > P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) > where the as are roots of > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). So, for each x in some unspecified ring, you get a factorization of the ring element P(x). It used to be that the a_i(x)s were supposed to be algebraic integers - is that still the case ? > And its easy to get that cubic by solving for one of the as using > (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) = > 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3. > Now Ive pointed out that setting x=0, will show that two of the as > equal 0 at that point, while one equals 3, which others have argued is > a special case. ... as it manifestly _is_ ... on the face of it ... > The cubic with x=0 is > a^3 -3a^2 = 0 > so two of the as equal 0, while one equals 3. > Usually I arbitrarily select a_1, and a_2 to equal 0, at x=0, which is > what Ill do now. Then, with > P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) > at x=0, you have two of the as equal 0, while one equals 3, to give > P(0) = 7(7)(22) = 1078 > which fits with > P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078. > Then for the first two factors (5a_1(x) + 7) and (5a_2(x) + 7) the 7s > visible in the expressions are factors of the constant term of P(x). > But P(x) is a multiple of 49 as *each* coefficient has 49 as a factor, > so I divide out the 49 to get > P(x)/49 = 300125 x^3 - 18375 x^2 - 360 x + 22. > Now assume each of the as has some non-unit factor in common with 7, > for a given x, which in fact they do in the ring of algebraic integers > whenever all the as are irrational, which is a point that has been > brought up repeatedly by people arguing with me. For a while I > resisted that fact, but now as Ive said before I concede that they > are in fact correct--each of the as, in the ring of algebraic > integers does in fact share a non-unit factor with 7 when all of the > as are irrational. > So let w_1(x) w_2(x) w_3(x) = 49, where the ws are those factors, so ... that is: you can find, for each x (whatever these are), elements w_1(x), w_2(x) and w_3(x) in your unspecified ring with this property. And you can find a bunch of other elements having the properties listed next. (In a later posting, you said that Ô49 above ought to be replaced by Ô7 -- I dont see why thats so - in fact, I think that you do _not_ want to do that. Do you still think the above is in error ??) > a_1(x) = w_1(x) b_1(x), > a_2(x) = w_2(x) b_2(x), > and > a_3(x) = w_3(x) b_3(x) > and where > w_1(x) v_1(x) = 7, w_2(x) v_2(x) = 7, and w_3(x) v_3(x) = 7, > and divide through by 49 to get > P(x)/49 = > (5 b_1(x) + v_1(x))(5 b_2(x) + v_2(x))(5 b_3(x) + v_3(x)) > and if you allow that the factors are each factors of the constant > term as before, then you have > v_1(0) v_2(0)(15 + v_3(0)) = 22 Actually ... v_1(0) v_2(0) (5 b_3(0) + v_3(0)) = 22 > at x=0, and when x does not equal 0, you have > v_1(x) v_2(x)(5u_3(x) + v_3(x)) = 22 Nope -- care to (attempt to) provide a *proof* of this unsupported assertion ?? After all, you are in _no_ way comparing constant terms when you take this amazing leap -- whats the constant term in 5 b_i(x) + v_i(x) ?? > where I introduce u_3(x) to handle any further weirdness with how > a_3(x) behaves, where u_3(0) = 3, to agree with previous results, and > remember > P(x)/49 = 300125 x^3 - 18375 x^2 - 360 x + 22 > and the constant term doesnt change, as, well, its constant, as its > 22. > So, notice, I now have that v_1(x) and v_2(x) are factors of 22. Not hardly ... > Provably, they cannot be units in the ring of algebraic integers when > a_1(x), a_2(x), and a_3(x) are all irrational as thats what people > argued with me about for so long. > The full result was just so weird that it escaped even me for a while, > as following their arguments to their logical conclusion 22 and 7 must > share non-unit factors in the ring of algebraic integers. > But you can also appear to prove that 22 and 7 are coprime in the ring > of algebraic integers using various accepted definitions of > coprimeness, like you can find algebraic integers x, and y such that > 22x + 7y = 1 > and claim to have proven that they are coprime. 22*1 + 7*(-3) = 1 appears to show them to be coprime in any superring of the ring of integers (or, at least, those where thats an appropriate test of coprimeness -- which definitely includes the ring of algebraic integers). > So, in the ring of algebraic integers, using whats commonly accepted > you can prove that 22 and 7 are both coprime and that they share > non-unit algebraic integer factors. No - you havent done that. So ... no need to concern yourself with the possible inconsistency of mathematics ... > Thats the full result. No - just another episode in Usenets longest-running saga of error and arrogance ... > Now, of course, posters arguing with me on Usenet wouldnt follow > through to the full result, probably because they werent smart enough > to see it, as make no mistake, thats the kind of result thats quite > big. Yeah ... pretty much _no_ one posting on (say) sci.math has the necessary intellectual equipment to follow in your footsteps ... gathering up the (few) pearls that you have deigned to cast before the swine that inhabit that region ... yeah ... thats it ... > Thats a career maker, or could have been one. For me, its just one > of my results. Nice, sure, but still just one, with techniques I > introduced in a couple of lines in another paper. > The ring of algebraic integers is quirky. That arbitrary selection of > roots of monic polynomials with integer coefficients allows you to do > all kinds of wacky things, including believing that youre doing all > kinds of great math, when you have nothing at all. This is one of your more bizarre idiosyncracies ... the collection of algebraic integers form a ring -- we can prove many theorems about the properties of that ring and of the algebraic integers. These concepts are *useful* - thats why we study them and pass the results along to the next generation. If you want to study some *other* subring of the complex numbers, whos stopping you ?? Why all of this whining about the algebraic integers -- they are what they are and most of the things you say in this mode are just plain stupid ... > Make no mistake, the big names in math today have just about zero > reason to accept these results. Would you if you were them? If you could prove any of what you say, youd get more acceptance. As it is, you cant even provide a coherent *statement* of the things that you then go on _not_ to prove. > But youre not them. > You are students. You have the future to make your names, with real > math that is actually correct. So be sure to build your mathematical career on real math that is actually correct. Theres a litmus test for that, yknow: *proof*. You need to practice proving things and understanding proofs that others have constructed. When youve reached a modest level of competence in this endeavor, youll be able to judge whether James Harris has anything on offer that you want to buy. Your decision on that question _will_ say a lot about whether youre a mathematician ... > Im offering you your future. Follow the math, and then if you have > what it takes, then accept what is true without holding your breath > waiting for old men who are trying to stave off ruin to do the right > thing. > They are cowards. They will admit the truth when they no longer > believe they can get away with lying. > Come on, do you really think results like these could have been missed > by the top mathematicians of the world? *Chortle* > Well, maybe, maybe they dont know. But I think they do. After all, > Ive contacted people all over the math world, as Ive been quite bold > in pushing forward this issue, in warning about the error. > I think they do know and they are waiting to see what you do. > I think they are waiting to see if you will let them get away with the > lies. > Maybe you will...or at least, maybe youll try. LOL. Some of you > probably will try because youre young, and foolish, and so eager to > please them. ... ludicrous ... simply ludicrous ... > James Harris > http://mathforprofit.blogspot.com/ === Subject: Re: JSH: Contradiction shown > Now it turns out that having thought it all through I finally realized > that I could show a direct contradiction using the objections raised > against my work. > ... it just gets better and better ... Meaning? Im actually curious as to what you mean by that statement as your start. > Notice here Ill actually use constants where before I talked about > variables being held constant. As usual, you are incredibly vague -- all of the upcoming blather > takes place (presumably) in some ring. Just _which_ ring might > that be ?? Im not vague as I noted that the variables are algebraic integers. Since Im pointing out a problem with the ring of algebraic integers its always kind of dicey to say everything is in that ring, when I know the ring has problems. But I can just say that the variables are all algebraic integers at the start, and that is ok, for the mathematician as the manipulations are all algebraic. > I start with P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078 which can be factored into non-polynomial factors by using P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 which looks complicated but if you multiply it all out and simplify, > you get the first, and I just have an analysis method for breaking up > a polynomial in a certain way to factor it--into non-polynomial > factors--and here the factorization is P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) where the as are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). So, for each x in some unspecified ring, you get a factorization > of the ring element P(x). It used to be that the a_i(x)s were > supposed to be algebraic integers - is that still the case ? Im almost ready to stop here, as you betray that you either have NO UNDERSTANDING of the subject at hand, or are playing some odd manipulative game, as the as are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) = 0 where all the coefficients are algebraic integers, so necessarily they MUST be algebraic integers. Its not about what I SAY or what anyone says, but about what follows mathematically. Ill let you go on for a bit more, but Im already unsettled by your post, as it bugs me that people like you keep bouncing around, screwing around with the details, making little asides like how you began your post, and basically JUST POSTING TO DISAGREE rather than caring about what is true. Its obnoxious. > And its easy to get that cubic by solving for one of the as using (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) = 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3. Now Ive pointed out that setting x=0, will show that two of the as > equal 0 at that point, while one equals 3, which others have argued is > a special case. ... as it manifestly _is_ ... on the face of it ... > The cubic with x=0 is a^3 -3a^2 = 0 so two of the as equal 0, while one equals 3. Usually I arbitrarily select a_1, and a_2 to equal 0, at x=0, which is > what Ill do now. Then, with P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) at x=0, you have two of the as equal 0, while one equals 3, to give P(0) = 7(7)(22) = 1078 which fits with P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078. Then for the first two factors (5a_1(x) + 7) and (5a_2(x) + 7) the 7s > visible in the expressions are factors of the constant term of P(x). But P(x) is a multiple of 49 as *each* coefficient has 49 as a factor, > so I divide out the 49 to get P(x)/49 = 300125 x^3 - 18375 x^2 - 360 x + 22. Now assume each of the as has some non-unit factor in common with 7, > for a given x, which in fact they do in the ring of algebraic integers > whenever all the as are irrational, which is a point that has been > brought up repeatedly by people arguing with me. For a while I > resisted that fact, but now as Ive said before I concede that they > are in fact correct--each of the as, in the ring of algebraic > integers does in fact share a non-unit factor with 7 when all of the > as are irrational. So let w_1(x) w_2(x) w_3(x) = 49, where the ws are those factors, so ... that is: you can find, for each x (whatever these are), elements > w_1(x), w_2(x) and w_3(x) in your unspecified ring with this property. > And you can find a bunch of other elements having the properties > listed next. (In a later posting, you said that Ô49 above ought to > be replaced by Ô7 -- I dont see why thats so - in fact, I think > that you do _not_ want to do that. Do you still think the above is > in error ??) I made this post to clear that area up, as there was an earlier posting where I did have a problem with the ws, but thats my fault for a messy clean-up. Theres no error there. As for the ws, how can you talk about an unspecified ring when I talk about accepting an objection having to do with the as having factors in the RING OF ALGEBRAIC INTEGERS? The whole issue is about a problem with the ring of algebraic integers!!! Either youre just completely out of it, or youre playing mindgames, trying to be annoyingly obtuse. The ws are algebraic integer functions. The assumption is that youre in the ring of algebraic integers. > a_1(x) = w_1(x) b_1(x), a_2(x) = w_2(x) b_2(x), and a_3(x) = w_3(x) b_3(x) and where w_1(x) v_1(x) = 7, w_2(x) v_2(x) = 7, and w_3(x) v_3(x) = 7, and divide through by 49 to get P(x)/49 = (5 b_1(x) + v_1(x))(5 b_2(x) + v_2(x))(5 b_3(x) + v_3(x)) and if you allow that the factors are each factors of the constant > term as before, then you have v_1(0) v_2(0)(15 + v_3(0)) = 22 > Actually ... v_1(0) v_2(0) (5 b_3(0) + v_3(0)) = 22 Oh, a nitpick as I went ahead and substutited for b_3(0), since b_3(0) = 3. The other substitutions are v_1(0) = 1, v_2(0) = 1. Agree? > at x=0, and when x does not equal 0, you have v_1(x) v_2(x)(5u_3(x) + v_3(x)) = 22 > Nope -- care to (attempt to) provide a *proof* of this > unsupported assertion ?? After all, you are in _no_ way > comparing constant terms when you take this amazing > leap -- whats the constant term in 5 b_i(x) + v_i(x) ?? > The proof is easy. Now, yes, I CAN provide a proof, at which point you should agree with me, right? Im stopping here to challenge you here as Im not interested in your running away from this point. After all, if I cant prove the individual links in my own argument, what basis for my claims would I have? The basis for the proof is the separability between factors of the constant term from other factors. Proving that separability is rather easy. What I want the undergrads to see is whether or not you play by the rules theyre learning supposedly rule math society. James Harris === Subject: Re: JSH: Contradiction shown > Now it turns out that having thought it all through I finally realized > that I could show a direct contradiction using the objections raised > against my work. > ... it just gets better and better ... > > Meaning? Its good to stumble across your threads again ?? After all, they provide a good deal of amusement to those of us who have followed your long career on sci.math ... you seem to have abandoned us, though ... Thinking youll have better luck winning converts among those who havent quite as much mathematical experience/maturity ?? > Im actually curious as to what you mean by that statement as your > start. Well, I hope the above response satisfies your curiosity ... > Notice here Ill actually use constants where before I talked about > variables being held constant. > As usual, you are incredibly vague -- all of the upcoming blather > takes place (presumably) in some ring. Just _which_ ring might > that be ?? Im not vague as I noted that the variables are algebraic integers. No ... you didnt (which prompted my comment). I snipped _none_ reading whats up above. At one point, you used to start off various versions of this screed by saying that everything took place in a commutative ring -- thats about as useless as what you do here, but at least it acknowledges that specifying some context _is_ important (and, unfortunately, something youre incapable of doing with any success). [*] That way, a copy of your post will survive, even if all of your contributions mysteriously vanish from the archives ... > Since Im pointing out a problem with the ring of algebraic integers > its always kind of dicey to say everything is in that ring, when I > know the ring has problems. > But I can just say that the variables are all algebraic integers at > the start, and that is ok, for the mathematician as the manipulations > are all algebraic. > I start with P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078 which can be factored into non-polynomial factors by using P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 which looks complicated but if you multiply it all out and simplify, > you get the first, and I just have an analysis method for breaking up > a polynomial in a certain way to factor it--into non-polynomial > factors--and here the factorization is P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) where the as are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). > So, for each x in some unspecified ring, you get a factorization > of the ring element P(x). It used to be that the a_i(x)s were > supposed to be algebraic integers - is that still the case ? Im almost ready to stop here, as you betray that you either have NO > UNDERSTANDING of the subject at hand, or are playing some odd > manipulative game, as the as are roots of > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) = 0 > where all the coefficients are algebraic integers, so necessarily they > MUST be algebraic integers. Since you never said what sorts of Ôxs youre considering, it was an honest question. > Its not about what I SAY or what anyone says, but about what follows > mathematically. You may not want to make things quite that cut and dried, since youre not real good at figuring out what follows mathematically. > Ill let you go on for a bit more, but Im already unsettled by your > post, as it bugs me that people like you keep bouncing around, > screwing around with the details, making little asides like how you > began your post, and basically JUST POSTING TO DISAGREE rather than > caring about what is true. I actually have/had some questions that I thought to raise -- posting just to disagree would be even more time-wasting than trying to get a straight answer from you ... > Its obnoxious. Good I didnt do it, then ... > And its easy to get that cubic by solving for one of the as using (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) = 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3. Now Ive pointed out that setting x=0, will show that two of the as > equal 0 at that point, while one equals 3, which others have argued is > a special case. > ... as it manifestly _is_ ... on the face of it ... The cubic with x=0 is a^3 -3a^2 = 0 so two of the as equal 0, while one equals 3. Usually I arbitrarily select a_1, and a_2 to equal 0, at x=0, which is > what Ill do now. Then, with P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) at x=0, you have two of the as equal 0, while one equals 3, to give P(0) = 7(7)(22) = 1078 which fits with P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078. Then for the first two factors (5a_1(x) + 7) and (5a_2(x) + 7) the 7s > visible in the expressions are factors of the constant term of P(x). But P(x) is a multiple of 49 as *each* coefficient has 49 as a factor, > so I divide out the 49 to get P(x)/49 = 300125 x^3 - 18375 x^2 - 360 x + 22. Now assume each of the as has some non-unit factor in common with 7, > for a given x, which in fact they do in the ring of algebraic integers > whenever all the as are irrational, which is a point that has been > brought up repeatedly by people arguing with me. For a while I > resisted that fact, but now as Ive said before I concede that they > are in fact correct--each of the as, in the ring of algebraic > integers does in fact share a non-unit factor with 7 when all of the > as are irrational. So let w_1(x) w_2(x) w_3(x) = 49, where the ws are those factors, so > ... that is: you can find, for each x (whatever these are), elements > w_1(x), w_2(x) and w_3(x) in your unspecified ring with this property. > And you can find a bunch of other elements having the properties > listed next. (In a later posting, you said that Ô49 above ought to > be replaced by Ô7 -- I dont see why thats so - in fact, I think > that you do _not_ want to do that. Do you still think the above is > in error ??) I made this post to clear that area up, as there was an earlier > posting where I did have a problem with the ws, but thats my fault > for a messy clean-up. > Theres no error there. So, then, your later post was an error ?? (It really was an honest question -- I can see where Ô49 is much more useful than Ô7 here, couldnt see why youd want to change it ... ) > As for the ws, how can you talk about an unspecified ring when I talk > about accepting an objection having to do with the as having factors > in the RING OF ALGEBRAIC INTEGERS? Hopefully, your confusion on this point has been cleared up -- you never specified a ring, so its an unspecified ring ... > The whole issue is about a problem with the ring of algebraic > integers!!! > Either youre just completely out of it, or youre playing mindgames, > trying to be annoyingly obtuse. > The ws are algebraic integer functions. The assumption is that > youre in the ring of algebraic integers. > a_1(x) = w_1(x) b_1(x), a_2(x) = w_2(x) b_2(x), and a_3(x) = w_3(x) b_3(x) and where w_1(x) v_1(x) = 7, w_2(x) v_2(x) = 7, and w_3(x) v_3(x) = 7, and divide through by 49 to get P(x)/49 = (5 b_1(x) + v_1(x))(5 b_2(x) + v_2(x))(5 b_3(x) + v_3(x)) and if you allow that the factors are each factors of the constant > term as before, then you have v_1(0) v_2(0)(15 + v_3(0)) = 22 Actually ... v_1(0) v_2(0) (5 b_3(0) + v_3(0)) = 22 > > Oh, a nitpick as I went ahead and substutited for b_3(0), since b_3(0) > = 3. Whys that ?? You know that w_3(0) b_3(0) = a_3(0) = 3 -- does that guarantee that b_3(0) = 3 ?? (I dont see how ... ) > The other substitutions are v_1(0) = 1, v_2(0) = 1. > Agree? No - actually, I dont. Can you convince me that you arent jumping to unwarranted conclusions here ?? Since weve agreed that we have a_1(0) = a_2(0) = 0, you _can_ conclude what I said up above (which is [mostly] what you said originally), but you dont know what v_1(0) or v_2(0) are ... > at x=0, and when x does not equal 0, you have v_1(x) v_2(x)(5u_3(x) + v_3(x)) = 22 > Nope -- care to (attempt to) provide a *proof* of this > unsupported assertion ?? After all, you are in _no_ way > comparing constant terms when you take this amazing > leap -- whats the constant term in 5 b_i(x) + v_i(x) ?? > > The proof is easy. Now, yes, I CAN provide a proof, at which point > you should agree with me, right? If you can provide a proof, then (sure) I will agree with you ... publicly ... right here ... in front of God and everybody ... > Im stopping here to challenge you here as Im not interested in your > running away from this point. Im not running away ... so bring it on. > After all, if I cant prove the individual links in my own argument, > what basis for my claims would I have? Funny ... you asking that question ... huh ... > The basis for the proof is the separability between factors of the > constant term from other factors. Proving that separability is rather > easy. I can hardly wait ... > What I want the undergrads to see is whether or not you play by the > rules theyre learning supposedly rule math society. > James Harris === Subject: Re: JSH: Contradiction shown > Now it turns out that having thought it all through I finally realized > that I could show a direct contradiction using the objections raised > against my work. > ... it just gets better and better ... > Meaning? > Its good to stumble across your threads again ?? After all, they > provide a good deal of amusement to those of us who have followed > your long career on sci.math ... you seem to have abandoned us, though ... > Thinking youll have better luck winning converts among those who > havent quite as much mathematical experience/maturity ?? Its quieter here, until sci.math posters like yourself come around making noise, and then acting as if youre actually in charge. You follow me poster. And besides, Im helping out the undergrads because theyre the ones potentially who have a future. Posting here is just idle amusement while I wait on publication, but at least it can be something of a public service! > Im actually curious as to what you mean by that statement as your > start. > Well, I hope the above response satisfies your curiosity ... > Notice here Ill actually use constants where before I talked about > variables being held constant. > As usual, you are incredibly vague -- all of the upcoming blather > takes place (presumably) in some ring. Just _which_ ring might > that be ?? > Im not vague as I noted that the variables are algebraic integers. No ... you didnt (which prompted my comment). I snipped _none_ > reading whats up above. At one point, you used to start off various > versions of this screed by saying that everything took place in > a commutative ring -- thats about as useless as what you do here, > but at least it acknowledges that specifying some context _is_ > important (and, unfortunately, something youre incapable of doing > with any success). Oh yeah, youre right. Ok, so I made a mistake there with that accusation. So, then, now, the variables that I start with are algebraic integers. I doubt thatll satisfy you as youre here to disagree, but at least I can try to be helpful. > [*] That way, a copy of your post will survive, even if all of > your contributions mysteriously vanish from the archives ... I remove posts that I wish to control in the future as to whether or not they can be seen easily without my wishing, and also to possibly limit the kind of copyright violations I get from true turds like Dik Winter, who just copied off of Usenet like copyright law doesnt apply to him. I doubt any of the post are actually lost just because I asked Google not to display them. But if one of you dips later try to post copyrighted material, especially from posts that I specifically requested not be seen, then I can whack you with greater legal force. > Since Im pointing out a problem with the ring of algebraic integers > its always kind of dicey to say everything is in that ring, when I > know the ring has problems. But I can just say that the variables are all algebraic integers at > the start, and that is ok, for the mathematician as the manipulations > are all algebraic. I start with P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078 which can be factored into non-polynomial factors by using P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 which looks complicated but if you multiply it all out and simplify, > you get the first, and I just have an analysis method for breaking up > a polynomial in a certain way to factor it--into non-polynomial > factors--and here the factorization is P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) where the as are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). > So, for each x in some unspecified ring, you get a factorization > of the ring element P(x). It used to be that the a_i(x)s were > supposed to be algebraic integers - is that still the case ? > Im almost ready to stop here, as you betray that you either have NO > UNDERSTANDING of the subject at hand, or are playing some odd > manipulative game, as the as are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) = 0 where all the coefficients are algebraic integers, so necessarily they > MUST be algebraic integers. Since you never said what sorts of Ôxs youre considering, > it was an honest question. Bull. Thats it, Im jumping down a bit. trying to be annoyingly obtuse. The ws are algebraic integer functions. The assumption is that > youre in the ring of algebraic integers. a_1(x) = w_1(x) b_1(x), a_2(x) = w_2(x) b_2(x), and a_3(x) = w_3(x) b_3(x) and where w_1(x) v_1(x) = 7, w_2(x) v_2(x) = 7, and w_3(x) v_3(x) = 7, and divide through by 49 to get P(x)/49 = (5 b_1(x) + v_1(x))(5 b_2(x) + v_2(x))(5 b_3(x) + v_3(x)) and if you allow that the factors are each factors of the constant > term as before, then you have v_1(0) v_2(0)(15 + v_3(0)) = 22 Actually ... v_1(0) v_2(0) (5 b_3(0) + v_3(0)) = 22 > Oh, a nitpick as I went ahead and substutited for b_3(0), since b_3(0) > = 3. Whys that ?? You know that w_3(0) b_3(0) = a_3(0) = 3 -- does > that guarantee that b_3(0) = 3 ?? (I dont see how ... ) Really? Hmmm...that explains a lot. > The other substitutions are v_1(0) = 1, v_2(0) = 1. Agree? No - actually, I dont. Can you convince me that you arent jumping > to unwarranted conclusions here ?? Since weve agreed that we have > a_1(0) = a_2(0) = 0, you _can_ conclude what I said up above (which > is [mostly] what you said originally), but you dont know what v_1(0) > or v_2(0) are ... LOL. If you cant get that then you dont have a clue whats going on. Youre not showing a basic understanding of very basic algebra. So continuing is a waste of time. Its like, if I were trying to explain a problem with a car to someone who I thought was a mechanic, and then they didnt know what a spark plug was, what would be the point of continuing? James Harris === Subject: Re: JSH: Contradiction shown > But if one of you dips later try to post copyrighted material, > especially from posts that I specifically requested not be seen, then > I can whack you with greater legal force. If someone copies your postings in order to make serious comments about them, and gives you proper attribution, you have no legal leg to stand on. In fact, the comment can be as simple as . === Subject: Re: JSH: Contradiction shown > Now it turns out that having thought it all through I finally realized > that I could show a direct contradiction using the objections raised > against my work. > ... it just gets better and better ... > Meaning? > Its good to stumble across your threads again ?? After all, they > provide a good deal of amusement to those of us who have followed > your long career on sci.math ... you seem to have abandoned us, though ... > Thinking youll have better luck winning converts among those who > havent quite as much mathematical experience/maturity ?? > Its quieter here, until sci.math posters like yourself come around > making noise, and then acting as if youre actually in charge. > You follow me poster. > And besides, Im helping out the undergrads because theyre the ones > potentially who have a future. > Posting here is just idle amusement while I wait on publication, but > at least it can be something of a public service! > > Im actually curious as to what you mean by that statement as your > start. > Well, I hope the above response satisfies your curiosity ... > > Notice here Ill actually use constants where before I talked about > variables being held constant. > As usual, you are incredibly vague -- all of the upcoming blather > takes place (presumably) in some ring. Just _which_ ring might > that be ?? > Im not vague as I noted that the variables are algebraic integers. > No ... you didnt (which prompted my comment). I snipped _none_ > reading whats up above. At one point, you used to start off various > versions of this screed by saying that everything took place in > a commutative ring -- thats about as useless as what you do here, > but at least it acknowledges that specifying some context _is_ > important (and, unfortunately, something youre incapable of doing > with any success). > Oh yeah, youre right. Ok, so I made a mistake there with that > accusation. > So, then, now, the variables that I start with are algebraic integers. > I doubt thatll satisfy you as youre here to disagree, but at least I > can try to be helpful. [*] That way, a copy of your post will survive, even if all of > your contributions mysteriously vanish from the archives ... I remove posts that I wish to control in the future as to whether or > not they can be seen easily without my wishing, and also to possibly > limit the kind of copyright violations I get from true turds like Dik > Winter, who just copied off of Usenet like copyright law doesnt apply > to him. James, why dont you try reading about copyright law before shooting your mouth off. I recommend starting at http://www.templetons.com/brad/copymyths.html In particular read section 4 which deals with fair use. In section 4, paragraph 4: Note that most inclusion of text in Usenet followups is for commentary and reply, and it doesnt damage the commercial value of the original posting (if it has any) and as such it is fair use. Basically this means that since a reply does not change the original post, it is fair use. In section 4, paragraph 6: Facts and ideas cant be copyrighted This means you can not claim any copyright status on any concepts you put in your papers. For example your ÔObject Ring mentioned in other threads can not be copyrighted. BTW, my two quotes fall under fair use since I cited the source. > I doubt any of the post are actually lost just because I asked Google > not to display them. > But if one of you dips later try to post copyrighted material, > especially from posts that I specifically requested not be seen, then > I can whack you with greater legal force. > Since Im pointing out a problem with the ring of algebraic integers > its always kind of dicey to say everything is in that ring, when I > know the ring has problems. But I can just say that the variables are all algebraic integers at > the start, and that is ok, for the mathematician as the manipulations > are all algebraic. I start with P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078 which can be factored into non-polynomial factors by using P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 which looks complicated but if you multiply it all out and simplify, > you get the first, and I just have an analysis method for breaking up > a polynomial in a certain way to factor it--into non-polynomial > factors--and here the factorization is P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) where the as are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). > So, for each x in some unspecified ring, you get a factorization > of the ring element P(x). It used to be that the a_i(x)s were > supposed to be algebraic integers - is that still the case ? > Im almost ready to stop here, as you betray that you either have NO > UNDERSTANDING of the subject at hand, or are playing some odd > manipulative game, as the as are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) = 0 where all the coefficients are algebraic integers, so necessarily they > MUST be algebraic integers. > Since you never said what sorts of Ôxs youre considering, > it was an honest question. Bull. Thats it, Im jumping down a bit. > trying to be annoyingly obtuse. The ws are algebraic integer functions. The assumption is that > youre in the ring of algebraic integers. a_1(x) = w_1(x) b_1(x), a_2(x) = w_2(x) b_2(x), and a_3(x) = w_3(x) b_3(x) and where w_1(x) v_1(x) = 7, w_2(x) v_2(x) = 7, and w_3(x) v_3(x) = 7, and divide through by 49 to get P(x)/49 = (5 b_1(x) + v_1(x))(5 b_2(x) + v_2(x))(5 b_3(x) + v_3(x)) and if you allow that the factors are each factors of the constant > term as before, then you have v_1(0) v_2(0)(15 + v_3(0)) = 22 Actually ... v_1(0) v_2(0) (5 b_3(0) + v_3(0)) = 22 > Oh, a nitpick as I went ahead and substutited for b_3(0), since b_3(0) > = 3. > Whys that ?? You know that w_3(0) b_3(0) = a_3(0) = 3 -- does > that guarantee that b_3(0) = 3 ?? (I dont see how ... ) > Really? > Hmmm...that explains a lot. > The other substitutions are v_1(0) = 1, v_2(0) = 1. Agree? > No - actually, I dont. Can you convince me that you arent jumping > to unwarranted conclusions here ?? Since weve agreed that we have > a_1(0) = a_2(0) = 0, you _can_ conclude what I said up above (which > is [mostly] what you said originally), but you dont know what v_1(0) > or v_2(0) are ... > LOL. If you cant get that then you dont have a clue whats going > on. > Youre not showing a basic understanding of very basic algebra. > So continuing is a waste of time. Its like, if I were trying to > explain a problem with a car to someone who I thought was a mechanic, > and then they didnt know what a spark plug was, what would be the > point of continuing? > James Harris === Subject: Re: JSH: Contradiction shown > Now it turns out that having thought it all through I finally realized > that I could show a direct contradiction using the objections raised > against my work. > ... it just gets better and better ... > Meaning? > Its good to stumble across your threads again ?? After all, they > provide a good deal of amusement to those of us who have followed > your long career on sci.math ... you seem to have abandoned us, though ... > Thinking youll have better luck winning converts among those who > havent quite as much mathematical experience/maturity ?? Its quieter here, until sci.math posters like yourself come around > making noise, and then acting as if youre actually in charge. You follow me poster. And besides, Im helping out the undergrads because theyre the ones > potentially who have a future. Posting here is just idle amusement while I wait on publication, but > at least it can be something of a public service! > Im actually curious as to what you mean by that statement as your > start. > Well, I hope the above response satisfies your curiosity ... > > Notice here Ill actually use constants where before I talked about > variables being held constant. > As usual, you are incredibly vague -- all of the upcoming blather > takes place (presumably) in some ring. Just _which_ ring might > that be ?? > Im not vague as I noted that the variables are algebraic integers. > No ... you didnt (which prompted my comment). I snipped _none_ > reading whats up above. At one point, you used to start off various > versions of this screed by saying that everything took place in > a commutative ring -- thats about as useless as what you do here, > but at least it acknowledges that specifying some context _is_ > important (and, unfortunately, something youre incapable of doing > with any success). Oh yeah, youre right. Ok, so I made a mistake there with that > accusation. So, then, now, the variables that I start with are algebraic integers. I doubt thatll satisfy you as youre here to disagree, but at least I > can try to be helpful. > [*] That way, a copy of your post will survive, even if all of > your contributions mysteriously vanish from the archives ... > I remove posts that I wish to control in the future as to whether or > not they can be seen easily without my wishing, and also to possibly > limit the kind of copyright violations I get from true turds like Dik > Winter, who just copied off of Usenet like copyright law doesnt apply > to him. > James, why dont you try reading about copyright law before shooting > your mouth off. I recommend starting at > http://www.templetons.com/brad/copymyths.html > In particular read section 4 which deals with fair use. > In section 4, paragraph 4: Note that most inclusion of text in Usenet > followups is for commentary and reply, and it doesnt damage the > commercial value of the original posting (if it has any) and as such > it is fair use. > Basically this means that since a reply does not change the original > post, it is fair use. > In section 4, paragraph 6: Facts and ideas cant be copyrighted > This means you can not claim any copyright status on any concepts you > put in your papers. For example your ÔObject Ring mentioned in other > threads can not be copyrighted. > BTW, my two quotes fall under fair use since I cited the source. Sigh. A sci.math poster copied an entire post of mine off of Usenet onto his own webpage where he inserted his own commentary including disparaging comments. I think a few others have copied posts off of Usenet as well for ßame websites. Thats not fair use. Now when some people copy your posts off of Usenet to their own websites to ßame you, against copyright law, then you might understand why I get pissed off, but probably that wont happen, right? So you mouth off at me without even understanding whats going on. James Harris === Subject: Re: JSH: Contradiction shown > ... against copyright law, ... Are your posts copyrighted? Ive not seen copyright notices on them. === Subject: Re: JSH: Contradiction shown Discussion, linux) >> ... against copyright law, ... > Are your posts copyrighted? Ive not seen copyright notices on > them. There is no need for copyright notices. By default (at least since Sonny Bono and Mickey Mouse, the copyright is good for seventy-five years after the death of the author (I think). Now, one might ask how Usenet and (especially) Googles archives are compatible with copyright law. With Usenet, it seems pretty clear that posting gives one the right to propagate it as is the convention with newsgroups. It seems considerably less obvious that Google has the right to make a public archive available, but I guess no ones challenged that yet. In any case, yes, Jamess posts are copyrighted. So are yours. -- So, at this time, Id like to assure you that I am not interested in making sure mathematicians worldwide get fired. Ive rethought my desire to go to Congress and try to get funding for mathematicians cut. -- James Harris is a reasonable man. Whew! === Subject: Re: JSH: Contradiction shown > ... against copyright law, ... > Are your posts copyrighted? Ive not seen copyright notices on them. Under US copyright law, all authors of printed (and by extension, electronically published) works have automatically copyright. No explicit notice is required. === Subject: Re: JSH: Contradiction shown > Now it turns out that having thought it all through I finally realized > that I could show a direct contradiction using the objections raised > against my work. > ... it just gets better and better ... > Meaning? > Its good to stumble across your threads again ?? After all, they > provide a good deal of amusement to those of us who have followed > your long career on sci.math ... you seem to have abandoned us, though ... > Thinking youll have better luck winning converts among those who > havent quite as much mathematical experience/maturity ?? Its quieter here, until sci.math posters like yourself come around > making noise, and then acting as if youre actually in charge. You follow me poster. And besides, Im helping out the undergrads because theyre the ones > potentially who have a future. Posting here is just idle amusement while I wait on publication, but > at least it can be something of a public service! > Im actually curious as to what you mean by that statement as your > start. > Well, I hope the above response satisfies your curiosity ... > > Notice here Ill actually use constants where before I talked about > variables being held constant. > As usual, you are incredibly vague -- all of the upcoming blather > takes place (presumably) in some ring. Just _which_ ring might > that be ?? > Im not vague as I noted that the variables are algebraic integers. > No ... you didnt (which prompted my comment). I snipped _none_ > reading whats up above. At one point, you used to start off various > versions of this screed by saying that everything took place in > a commutative ring -- thats about as useless as what you do here, > but at least it acknowledges that specifying some context _is_ > important (and, unfortunately, something youre incapable of doing > with any success). Oh yeah, youre right. Ok, so I made a mistake there with that > accusation. So, then, now, the variables that I start with are algebraic integers. I doubt thatll satisfy you as youre here to disagree, but at least I > can try to be helpful. > [*] That way, a copy of your post will survive, even if all of > your contributions mysteriously vanish from the archives ... > I remove posts that I wish to control in the future as to whether or > not they can be seen easily without my wishing, and also to possibly > limit the kind of copyright violations I get from true turds like Dik > Winter, who just copied off of Usenet like copyright law doesnt apply > to him. James, why dont you try reading about copyright law before shooting > your mouth off. I recommend starting at > http://www.templetons.com/brad/copymyths.html > In particular read section 4 which deals with fair use. In section 4, paragraph 4: Note that most inclusion of text in Usenet > followups is for commentary and reply, and it doesnt damage the > commercial value of the original posting (if it has any) and as such > it is fair use. Basically this means that since a reply does not change the original > post, it is fair use. In section 4, paragraph 6: Facts and ideas cant be copyrighted This means you can not claim any copyright status on any concepts you > put in your papers. For example your ÔObject Ring mentioned in other > threads can not be copyrighted. BTW, my two quotes fall under fair use since I cited the source. Sigh. A sci.math poster copied an entire post of mine off of Usenet > onto his own webpage where he inserted his own commentary including > disparaging comments. > I think a few others have copied posts off of Usenet as well for ßame > websites. > Thats not fair use. Section 4 paragraph 1: The Ôfair use exemption to (U.S.) copyright law was created to allow things such as commentary, parody, news reporting, research and education about copyrighted works without the permission of the author. The site in question falls under commentary and to some extent education. Section 4 paragraph 2: Fair use is usually a short excerpt and almost always attributed. The site in question is a critique of the general work, so what was copied was relevant. And what was quoted was clearly attributed to you. Section 4 paragraph 2: It should not harm the commercial value of the work -- in the sense of people no longer needing to buy it. You posted the source material freely in USENET, which I believe means that since anybody has free access (via Google) that there was no commercial value to it to begin with. If you really think that you could have made money, then you need to file a lawsuit and enforce the rights you believe you have. > Now when some people copy your posts off of Usenet to their own > websites to ßame you, against copyright law, See above. > then you might > understand why I get pissed off, but probably that wont happen, > right? > So you mouth off at me without even understanding whats going on. You dont even understand copyright law enough to know what fair use is. If you do think that the useage was was not fair use AND that your work has commercial value, you would have already contacted a lawyer or filed a lawsuit by now. If you want to seek any form of compensation, you need to do it in court, not on Usenet. And if you do decide to go to court, you will need to convince the judge that your work has commercial value. That will be especially hard since copies of your work originally posted by you are all over sci.math which anyone can access for free. > James Harris === Subject: Re: JSH: Contradiction shown > I think a few others have copied posts off of Usenet as well for ßame > websites. > Thats not fair use. What makes it not fair use? === Subject: Re: JSH: Contradiction shown > I think a few others have copied posts off of Usenet as well for ßame > websites. > Thats not fair use. > What makes it not fair use? Copyright laws recognize a legal right of writers to have some control over their work. Its necessary as otherwise, say, I could take any book published, and sell it myself, or make my own changes--to improve it--and then sell it. Part of the idea I think is to protect creative people because otherwise thered be a negative pressure against creativity. Its like if you produce something, only to have people who made no effort themselves come in and profit from your work, its not only not fair, its depressing. How many artists would never publish if they had to face copiers who could operate with impunity and even make mocking changes to a work to attack the artist? The Internet makes that possible in the new format areas, like with Usenet posts, as practically its difficult to stop someone from copying off your writing from Usenet and profiting from it on their own website. Now when I saying profiting some of you may counter that the guy probably isnt making any money off of my writing, but he must see *some* profit for himself--not necessarily monetary--in copying my work to his own website, or why would he do it? The law actually covers what hes doing, but the enforcement hasnt caught up with the Internet. Now fair use has to do with the societal need to use some bits of writing without people having to ask the authors permission, which covers a lot of areas. The usual point is to just copy what you need to make some point, which doesnt need the *entire* thing, and usually, if you can, not disparage the person youre borrowing from! Now the non-disparagement can vary, as at times, what people say is just so useful in making a point that is negative to them, but still theres an idea of civility in discussion. That is, the idea is that we are a civilized society. Copying off an entire post from Usenet and then using it to insult the writer is not only completely counter to fair use and civilized behavior, it is clearly an anti-social act. Understand fair use now? James Harris === Subject: Re: JSH: Contradiction shown > I think a few others have copied posts off of Usenet as well for ßame > websites. Thats not fair use. > What makes it not fair use? > Copyright laws recognize a legal right of writers to have some control > over their work. Its necessary as otherwise, say, I could take any > book published, and sell it myself, or make my own changes--to > improve it--and then sell it. > Part of the idea I think is to protect creative people because > otherwise thered be a negative pressure against creativity. > Its like if you produce something, only to have people who made no > effort themselves come in and profit from your work, its not only not > fair, its depressing. > How many artists would never publish if they had to face copiers who > could operate with impunity and even make mocking changes to a work to > attack the artist? > The Internet makes that possible in the new format areas, like with > Usenet posts, as practically its difficult to stop someone from > copying off your writing from Usenet and profiting from it on their > own website. > Now when I saying profiting some of you may counter that the guy > probably isnt making any money off of my writing, but he must see > *some* profit for himself--not necessarily monetary--in copying my > work to his own website, or why would he do it? > The law actually covers what hes doing, but the enforcement hasnt > caught up with the Internet. > Now fair use has to do with the societal need to use some bits of > writing without people having to ask the authors permission, which > covers a lot of areas. The usual point is to just copy what you need > to make some point, which doesnt need the *entire* thing, and > usually, if you can, not disparage the person youre borrowing from! > Now the non-disparagement can vary, as at times, what people say is > just so useful in making a point that is negative to them, but still > theres an idea of civility in discussion. That is, the idea is that > we are a civilized society. > Copying off an entire post from Usenet and then using it to insult the > writer is not only completely counter to fair use and civilized > behavior, it is clearly an anti-social act. > Understand fair use now? I know it must seem unfair to you, but the law defines fair use differently. Your understanding of copyright law is not as good as your understanding of algebra. Thats a joke, James. === Subject: Re: JSH: Contradiction shown > Sigh. A sci.math poster copied an entire post of mine off of Usenet > onto his own webpage where he inserted his own commentary including > disparaging comments. > I think a few others have copied posts off of Usenet as well for ßame > websites. > Thats not fair use. By definition, it is. === Subject: Re: JSH: Contradiction shown > Now it turns out that having thought it all through I finally realized > that I could show a direct contradiction using the objections raised > against my work. > ... it just gets better and better ... > Meaning? > Its good to stumble across your threads again ?? After all, they > provide a good deal of amusement to those of us who have followed > your long career on sci.math ... you seem to have abandoned us, though ... > Thinking youll have better luck winning converts among those who > havent quite as much mathematical experience/maturity ?? > Its quieter here, until sci.math posters like yourself come around > making noise, and then acting as if youre actually in charge. > You follow me poster. Youve got a marvellously inßated view of yourself ... I find you amusing, thats all. > And besides, Im helping out the undergrads because theyre the ones > potentially who have a future. helping them out, eh ??? > Posting here is just idle amusement while I wait on publication, but > at least it can be something of a public service! Youll let us know when that paper gets accepted, right ?? Is it the old standby (Advanced Polynomial Factorization) or have you finally admitted defeat/inability-to-repair on that one ? If it _is_ APF, did you modify it in any way before submitting it to another journal or is it still the same incoherent, error-ridden manuscript that appeared > > Im actually curious as to what you mean by that statement as your > start. > Well, I hope the above response satisfies your curiosity ... > > Notice here Ill actually use constants where before I talked about > variables being held constant. > As usual, you are incredibly vague -- all of the upcoming blather > takes place (presumably) in some ring. Just _which_ ring might > that be ?? > Im not vague as I noted that the variables are algebraic integers. > No ... you didnt (which prompted my comment). I snipped _none_ > reading whats up above. At one point, you used to start off various > versions of this screed by saying that everything took place in > a commutative ring -- thats about as useless as what you do here, > but at least it acknowledges that specifying some context _is_ > important (and, unfortunately, something youre incapable of doing > with any success). > Oh yeah, youre right. Ok, so I made a mistake there with that > accusation. > So, then, now, the variables that I start with are algebraic integers. OK, then ... so any variable-looking thing that appears below Im to assume is an algebraic integer (unless you explicitly say otherwise) ?? Just want to make sure that I understand the proof that you promised me (which, Im guessing, Ill find down below). > I doubt thatll satisfy you as youre here to disagree, but at least I > can try to be helpful. Why do you keep saying that ?? (Oh ... I understand that nonsense about your trying to be helpful and all ... I mean, why do you keep asserting that Im posting just to disagree ? I had some questions, I asked them, You promised to provide me with a proof that would clear up some of my doubts about what you claim ... ) [*] That way, a copy of your post will survive, even if all of > your contributions mysteriously vanish from the archives ... I remove posts that I wish to control in the future as to whether or > not they can be seen easily without my wishing, and also to possibly > limit the kind of copyright violations I get from true turds like Dik > Winter, who just copied off of Usenet like copyright law doesnt apply > to him. If you say so ... I suspect that you remove posts when you later think about some of the things that you say (after a heavy bout with the bottle ??) and would just as soon they werent publicly known. About the Dik Winter thing: think fair use. > I doubt any of the post are actually lost just because I asked Google > not to display them. But ... I bet you wish with all your might that those _particular_ posts _would_ disappear. Shame about there being more archives than just Google ... > But if one of you dips later try to post copyrighted material, > especially from posts that I specifically requested not be seen, then > I can whack you with greater legal force. > Since Im pointing out a problem with the ring of algebraic integers > its always kind of dicey to say everything is in that ring, when I > know the ring has problems. But I can just say that the variables are all algebraic integers at > the start, and that is ok, for the mathematician as the manipulations > are all algebraic. I start with P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078 which can be factored into non-polynomial factors by using P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 which looks complicated but if you multiply it all out and simplify, > you get the first, and I just have an analysis method for breaking up > a polynomial in a certain way to factor it--into non-polynomial > factors--and here the factorization is P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) where the as are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). > So, for each x in some unspecified ring, you get a factorization > of the ring element P(x). It used to be that the a_i(x)s were > supposed to be algebraic integers - is that still the case ? > Im almost ready to stop here, as you betray that you either have NO > UNDERSTANDING of the subject at hand, or are playing some odd > manipulative game, as the as are roots of a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) = 0 where all the coefficients are algebraic integers, so necessarily they > MUST be algebraic integers. > Since you never said what sorts of Ôxs youre considering, > it was an honest question. Bull. Thats it, Im jumping down a bit. Why the Bull. ? You never did say, in the post that I was answering. In this post, you (sort of) _do_ say -- if I made the same comment *now*, you might have a case here ... > trying to be annoyingly obtuse. The ws are algebraic integer functions. The assumption is that > youre in the ring of algebraic integers. a_1(x) = w_1(x) b_1(x), a_2(x) = w_2(x) b_2(x), and a_3(x) = w_3(x) b_3(x) and where w_1(x) v_1(x) = 7, w_2(x) v_2(x) = 7, and w_3(x) v_3(x) = 7, and divide through by 49 to get P(x)/49 = (5 b_1(x) + v_1(x))(5 b_2(x) + v_2(x))(5 b_3(x) + v_3(x)) and if you allow that the factors are each factors of the constant > term as before, then you have v_1(0) v_2(0)(15 + v_3(0)) = 22 Actually ... v_1(0) v_2(0) (5 b_3(0) + v_3(0)) = 22 > Oh, a nitpick as I went ahead and substutited for b_3(0), since b_3(0) > = 3. > Whys that ?? You know that w_3(0) b_3(0) = a_3(0) = 3 -- does > that guarantee that b_3(0) = 3 ?? (I dont see how ... ) > Really? > Hmmm...that explains a lot. Oh ... wait ... w_3(0) is simultaneously a factor of 3 and 7, so its a unit... so b_3(0) is (at least) an associate of 3. Its reasonable then to normalize things so that b_3(0) = 3. If thats it, you could just have given the reason ... to be helpful and all ... > The other substitutions are v_1(0) = 1, v_2(0) = 1. Agree? > No - actually, I dont. Can you convince me that you arent jumping > to unwarranted conclusions here ?? Since weve agreed that we have > a_1(0) = a_2(0) = 0, you _can_ conclude what I said up above (which > is [mostly] what you said originally), but you dont know what v_1(0) > or v_2(0) are ... > LOL. If you cant get that then you dont have a clue whats going > on. As noted above, you might have said something to justify your claim -- thinking about it, I see now that you _can_ (without loss of generality) assume that v_1(0) = v_2(0) = 1. > Youre not showing a basic understanding of very basic algebra. Youre demonstrating your utter inability to communicate mathematics. I grant you that the assertions above that I questioned can be justified -- note the reasonableness of _my_ stance here. Contrast that with: > So continuing is a waste of time. Its like, if I were trying to > explain a problem with a car to someone who I thought was a mechanic, > and then they didnt know what a spark plug was, what would be the > point of continuing? So ... you can _not_ provide that proof that you promised, right ?? Because, if you _could_, youd go ahead and do so ... just to rub my nose in it. Oh ... sorry ... make that: just to be helpful and provide a public service ... For the record: you still owe me that proof ... > James Harris === Subject: Re: JSH: Contradiction shown > I remove posts that I wish to control in the future as to whether or > not they can be seen easily without my wishing, and also to possibly > limit the kind of copyright violations I get from true turds like Dik > Winter, who just copied off of Usenet like copyright law doesnt apply > to him. them while you were drunk, or depressed, or delusional, or engaging in one of your periodic bouts of self-pity. I.e., you remove things because youre a coward and wont stand behind your own words. Fortunately there are other archives that preserve them, so we still can point them out to people who want to see what a truly pathetic creature you are. -- Wayne Brown (HPCC #1104) | When your tails in a crack, you improvise fwbrown@bellsouth.net | if youre good enough. Otherwise you give | your pelt to the trapper. e^(i*pi) = -1 -- Euler | -- John Myers Myers, Silverlock === Subject: Re: JSH: Contradiction shown >Notice here Ill actually use constants where before I talked about >variables being held constant. >> As usual, you are incredibly vague -- all of the upcoming blather >> takes place (presumably) in some ring. Just _which_ ring might >> that be ?? > Im not vague as I noted that the variables are algebraic integers. > Since Im pointing out a problem with the ring of algebraic integers > its always kind of dicey to say everything is in that ring, when I > know the ring has problems. > But I can just say that the variables are all algebraic integers at > the start, and that is ok, for the mathematician as the manipulations > are all algebraic. No, it isnt. Suppose I am working in the ring Z[x]. I can tell you that a and b are integers, which merely tells me they are special members of Z[x]. If you say the variables are algebraic integers you could be working in any ring that has the algebraic integers as a subring, which can have a huge impact on what words such as coprime mean. >I start with >P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078 >which can be factored into non-polynomial factors by using >P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3 >which looks complicated but if you multiply it all out and simplify, >you get the first, and I just have an analysis method for breaking up >a polynomial in a certain way to factor it--into non-polynomial >factors--and here the factorization is >P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) >where the as are roots of >a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x). >> So, for each x in some unspecified ring, you get a factorization >> of the ring element P(x). It used to be that the a_i(x)s were >> supposed to be algebraic integers - is that still the case ? > Im almost ready to stop here, as you betray that you either have NO > UNDERSTANDING of the subject at hand, or are playing some odd > manipulative game, as the as are roots of > a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x) = 0 > where all the coefficients are algebraic integers, so necessarily they > MUST be algebraic integers. Actually, as written the MUST be functions of x, which means they cannot be algebraic integers. Now they may be functions that map algebraic integers to algebraic integers, but that has not been asserted or (if necessary) proven. > Its not about what I SAY or what anyone says, but about what follows > mathematically. > Ill let you go on for a bit more, but Im already unsettled by your > post, as it bugs me that people like you keep bouncing around, > screwing around with the details, making little asides like how you > began your post, and basically JUST POSTING TO DISAGREE rather than > caring about what is true. He is pointing out ßaws in your communication. You are unclear, whether you believe it or not. >And its easy to get that cubic by solving for one of the as using >(5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) = > 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3. >Now Ive pointed out that setting x=0, will show that two of the as >equal 0 at that point, while one equals 3, which others have argued is >a special case. >> ... as it manifestly _is_ ... on the face of it ... >The cubic with x=0 is >a^3 -3a^2 = 0 >so two of the as equal 0, while one equals 3. >Usually I arbitrarily select a_1, and a_2 to equal 0, at x=0, which is >what Ill do now. Then, with >P(x) = (5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) >at x=0, you have two of the as equal 0, while one equals 3, to give >P(0) = 7(7)(22) = 1078 >which fits with >P(x) = 14706125 x^3 - 900375 x^2 - 17640 x + 1078. >Then for the first two factors (5a_1(x) + 7) and (5a_2(x) + 7) the 7s >visible in the expressions are factors of the constant term of P(x). >But P(x) is a multiple of 49 as *each* coefficient has 49 as a factor, >so I divide out the 49 to get >P(x)/49 = 300125 x^3 - 18375 x^2 - 360 x + 22. >Now assume each of the as has some non-unit factor in common with 7, >for a given x, which in fact they do in the ring of algebraic integers >whenever all the as are irrational, which is a point that has been >brought up repeatedly by people arguing with me. For a while I >resisted that fact, but now as Ive said before I concede that they >are in fact correct--each of the as, in the ring of algebraic >integers does in fact share a non-unit factor with 7 when all of the >as are irrational. >So let w_1(x) w_2(x) w_3(x) = 49, where the ws are those factors, so >> ... that is: you can find, for each x (whatever these are), elements >> w_1(x), w_2(x) and w_3(x) in your unspecified ring with this property. >> And you can find a bunch of other elements having the properties >> listed next. (In a later posting, you said that Ô49 above ought to >> be replaced by Ô7 -- I dont see why thats so - in fact, I think >> that you do _not_ want to do that. Do you still think the above is >> in error ??) > I made this post to clear that area up, as there was an earlier > posting where I did have a problem with the ws, but thats my fault > for a messy clean-up. > Theres no error there. > As for the ws, how can you talk about an unspecified ring when I talk > about accepting an objection having to do with the as having factors > in the RING OF ALGEBRAIC INTEGERS? Because you have written the as as functions of x, which means they cannot be in the ring of algebraic integers. > The whole issue is about a problem with the ring of algebraic > integers!!! The point he made is that you have a problem with notation. You cannot just put everything in the ring of algebraic integers. Functions are not there. Various other mathematical objects that you appear to be using are not there. > Either youre just completely out of it, or youre playing mindgames, > trying to be annoyingly obtuse. > The ws are algebraic integer functions. The assumption is that > youre in the ring of algebraic integers. I would guess you mean the ws are functions from the algebraic integers to the algebraic integers. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: JSH: Contradiction shown >> Now it turns out that having thought it all through I finally realized >> that I could show a direct contradiction using the objections raised >> against my work. >> ... it just gets better and better ... >> > Meaning? > Im actually curious as to what you mean by that statement as your > start. Anyone but a narcissist would recognize that immediately as ridicule. For those of us who find your ridiculous claims amusing it means, This nonsense he spouts just keeps getting funnier and funnier. Heres a Free Clue: It gets better and better is a *very* common figure of speech. Its generally used by a listener when the speaker is saying things that are more and more obviously wrong; a similar expression is, He keeps digging himself in deeper and deeper. In a more literal sense, its a sarcastic way of saying This is getting worse and worse. Im surprised such an erudite student of human nature and popular culture as yourself wouldnt know that. (In case you didnt recognize it, that last comment *also* was sarcasm and means the exact opposite of what it says.) -- Wayne Brown (HPCC #1104) | When your tails in a crack, you improvise fwbrown@bellsouth.net | if youre good enough. Otherwise you give | your pelt to the trapper. e^(i*pi) = -1 -- Euler | -- John Myers Myers, Silverlock === Subject: Re: JSH: Contradiction shown >Now it turns out that having thought it all through I finally realized >[...] You just _now_ thought it all through? Youve been explaining for years that the evil mathematicians of the world have been perpetrating a fraud - one would think that youd think it all through first, before starting on a campaign like that, lest you look silly when it turns out youre wrong. But as long as youve _now_ got it all straight... >Well, maybe, maybe they dont know. But I think they do. After all, >Ive contacted people all over the math world, as Ive been quite bold >in pushing forward this issue, in warning about the error. I got an email the other day from someone quite boldly warning me that pi = 3.125, not 3.14159 as formerly thought. You have any opinion on that one? >I think they do know and they are waiting to see what you do. >I think they are waiting to see if you will let them get away with the >lies. >Maybe you will...or at least, maybe youll try. LOL. Some of you >probably will try because youre young, and foolish, and so eager to >please them. >James Harris >http://mathforprofit.blogspot.com/ ************************ David C. Ullrich === Subject: Prove a formula with LEM by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAF2R8F14454; Hello to everybody, is my first time in the forum and I would like to ask if anybody know if I could make typical proof of this formula not-not F -| F using the derived rule law of excluded middle instead of double negation elimination rule directly. Is there any general rule that makes it possible to use LEM (derived rule)instead of double negation elimination rule. === Subject: Can somebody helo me with inequalities? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAF2R9k14556; Here is the problem that I need help with: Jim Olsen makes and sells gourmet food items. He makes 2 types of salad dressing, garlic and tofu. Each gallon of garlic dressing requires 2 qt of oil and 2 qt of vinegar. Each gallon of tofu dressing requires 3 qt of oil and 1 qt of vinegar. Jim makes a $3 profit on each gallon of garlic dressing and a $2 profit on each gallon of tofu dressing. He has 18 qt of oil and 10 qt of vinegar on hand. How many gallons of each type of dressing should he make to maximize his profits? 1) Identify the variables: Let x=the number of gallons of garlic dressing Let y=the number of gallons of tofu dressing. 2) Write a system of inequalities: === Subject: Re: Can somebody helo me with inequalities? > Here is the problem that I need help with: > Jim Olsen makes and sells gourmet food items. He makes 2 types of > salad dressing, garlic and tofu. Each gallon of garlic dressing > requires 2 qt of oil and 2 qt of vinegar. Each gallon of tofu dressing > requires 3 qt of oil and 1 qt of vinegar. Jim makes a $3 profit on > each gallon of garlic dressing and a $2 profit on each gallon of tofu > dressing. He has 18 qt of oil and 10 qt of vinegar on hand. How many > gallons of each type of dressing should he make to maximize his > profits? > 1) Identify the variables: > Let x=the number of gallons of garlic dressing > Let y=the number of gallons of tofu dressing. > 2) Write a system of inequalities: Ide change everything to gallons (or quarts) so I would not get confused. Something has to be less than or equal to 18. What? Something has to less than or equal to 10. What? Ide also write down an equation for total profit. Just to be ready. Bill === Subject: Re: order of an element in a group [re-post] by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAF2R7414342; I am confused with the following problem 1)Given a group G, h is an element of G and ord h = n.Find the order of g, such that g = h^12 and when n = 60k+1 === Subject: Re: order of an element in a group [re-post] > I am confused with the following problem > 1)Given a group G, h is an element of G and ord h = n.Find the order > of g, such that g = h^12 and when n = 60k+1 Does it help if you observe that n and 12 are relatively prime? -- Paul Sperry Columbia, SC (USA) === Subject: Help by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAF2RAS14564; General smith accused of high teason is sentenced to death by court martial. He is allowed to make a final statement, after which he will be shot it the statement it false or will be hung if the statement it true. General Smith makes his final statement and is released. The question: What could he have said? === Subject: Re: Help >General smith accused of high teason is sentenced to death by court >martial. He is allowed to make a final statement, after which he will >be shot it the statement it false or will be hung if the statement it >true. General Smith makes his final statement and is released. The >question: What could he have said? maybe what has this to do with mathematics ? === Subject: Re: HELP!!! >I just finished my first semester in precal and havent covered this >yet. How would you solve: (PS im not sure if this is calc level or >not, havent had the chance to ask my teacher yet) >[-log4 (64) + log3 (81^2) ? log3 (81) + (5^3)^2 ? 18] mod 7 Why dont you show us how far you got with this problem? It may or may not be homework, but it looks like homework. If we just hand you a solution, youll read it but you probably wont be any better equipped to solve the next problem that comes along. -- Stan Brown, Oak Road Systems, Tompkins County, New York, USA http://OakRoadSystems.com A: Maybe because some people are too annoyed by top-posting. Q: Why do I not get an answer to my question(s)? A: Because it messes up the order in which people normally read text. Q: Why is top-posting such a bad thing? === Subject: Re: HELP!!! > I just finished my first semester in precal and havent covered this > yet. How would you solve: (PS im not sure if this is calc level or > not, havent had the chance to ask my teacher yet) > [-log4 (64) + log3 (81^2) .9a log3 (81) + (5^3)^2 .9a 18] mod 7 If logN means log base N Id rather write it as log_N. So [-log4 (64) + log3 (81^2) .9a log3 (81) + (5^3)^2 .9a 18] mod 7 = [-log_4(64) + log_3(81^2) - log_3(81) + (5^3)^2 - 18] mod 7 = [-3 + log_3(81) + 5^6 - 18] mod 7 = [-3 + 4 + 15625 - 18] mod 7 = 15608 mod 7 = 5 === Subject: Re: HELP!!! > I just finished my first semester in precal and havent covered this > yet. How would you solve: (PS im not sure if this is calc level or > not, havent had the chance to ask my teacher yet) > [-log4 (64) + log3 (81^2) .9a log3 (81) + (5^3)^2 .9a 18] mod 7 > If logN means log base N Id rather write it as log_N. > So [-log4 (64) + log3 (81^2) .9a log3 (81) + (5^3)^2 .9a 18] mod 7 > = [-log_4(64) + log_3(81^2) - log_3(81) + (5^3)^2 - 18] mod 7 > = [-3 + log_3(81) + 5^6 - 18] mod 7 > = [-3 + 4 + 15625 - 18] mod 7 > = 15608 mod 7 > = 5 and 5^6 mod 7 can be evaluated in ones head 5*5 = 25 = 4 mod 7 5^3 mod 7 = 4 * 5 mod 7 = 20 mod 7 = -1 mod 7 5^3*5^3 mod 7 = (-1)*(-1) mod 7 = 1 mod 7 === Subject: Re: HELP!!! > I just finished my first semester in precal and havent covered this > yet. How would you solve: (PS im not sure if this is calc level or > not, havent had the chance to ask my teacher yet) > [-log4 (64) + log3 (81^2) [CapitalEth] log3 (81) + (5^3)^2 [CapitalEth] 18] mod 7 Since its not an equation, and has no variables, Im guessing you want to evaluate it. Start with this: what does log_4(64) mean? log_4(64)=3 because 4^3=64. What does n mod 7 mean? n mod 7 is the remainder when dividing n by 7. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Help >General smith accused of high teason is sentenced to death by court >martial. He is allowed to make a final statement, after which he will >be shot it the statement it false or will be hung if the statement it >true. General Smith makes his final statement and is released. The >question: What could he have said? > maybe what has this to do with mathematics ? Maybe he made a mathematical statement that could not be proved either true or false? === Subject: Re: Help > General smith accused of high teason is sentenced to death by court > martial. He is allowed to make a final statement, after which he will > be shot it the statement it false or will be hung if the statement it > true. General Smith makes his final statement and is released. The > question: What could he have said? This statement is false. -- Darrell === Subject: Re: Help > General smith accused of high teason is sentenced to death by court > martial. He is allowed to make a final statement, after which he will > be shot it the statement it false or will be hung if the statement it > true. General Smith makes his final statement and is released. The > question: What could he have said? > This statement is false. I wont be hung, Ill be shot. Ill be shot. === Subject: Help with inequalities! by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iAF2R9B14544; I need help with inequalities! Here is a problem that I need to solve: Jim Olsen makes and sell gourmet food items. He makes two types of salad dressing, garlic and tofu. Each gallon of garlic dressing requires 2 qt of oil and 2 qt of vinegar. Each gallon of tofu dressing requires 3 qt of oil and 1 qt of vinegar. Jim makes a $3 profit on each gallon of garlic dressing and a $2 profit on each gallon of tofu dressing. He has 18 qt of oil and 10 qt of vinegar on hand. How many gallons of each type of dressing should he make to maximize his profits? Identify the variables: Let x= the number of gallons of garlic dressing. Let y= the number of gallons of tofu dressing. Write a system of inequalities: CAN SOMEBODY PLEASE HELP ME!? === Subject: Re: Help with inequalities! x >= 0 y >= 0 2x + 3y = 18 2x + y = 10 so intersection will be bounded by: ( remember only points of integer coordinates are possible choices ) 1) x-axis, (0,0) to (5,0); 2) y-axis, (0,0) to (0,6); 3) y = -2x/3 + 6, (0,6) to (3,4); 4) y = -2x + 10, (3,4) to (5,0); profit = p(x,y) = 3x + 2y; dp/dx = 3, dp/dy = 2, so no critical points. when u look at the bounds, maximum at 1) is p(5,0) = 15; maximum at 2) is p(0,6) = 12; maximum at 3) is p(3,4) = 17; maximum at 4) is p(3,4) = 17; so maximum is (3,4) >I need help with inequalities! > Here is a problem that I need to solve: > Jim Olsen makes and sell gourmet food items. He makes two types of > salad dressing, garlic and tofu. Each gallon of garlic dressing > requires 2 qt of oil and 2 qt of vinegar. Each gallon of tofu dressing > requires 3 qt of oil and 1 qt of vinegar. Jim makes a $3 profit on > each gallon of garlic dressing and a $2 profit on each gallon of tofu > dressing. He has 18 qt of oil and 10 qt of vinegar on hand. How many > gallons of each type of dressing should he make to maximize his > profits? > Identify the variables: > Let x= the number of gallons of garlic dressing. > Let y= the number of gallons of tofu dressing. > Write a system of inequalities: > CAN SOMEBODY PLEASE HELP ME!? === Subject: Re: Help with inequalities! I did a mistake, the inequations are : x >= 0 y >= 0 2x + 3y <= 18 2x + y <= 10 >x >= 0 > y >= 0 > 2x + 3y = 18 > 2x + y = 10 > so intersection will be bounded by: ( remember only points of integer > coordinates are possible choices ) > 1) x-axis, (0,0) to (5,0); > 2) y-axis, (0,0) to (0,6); > 3) y = -2x/3 + 6, (0,6) to (3,4); > 4) y = -2x + 10, (3,4) to (5,0); > profit = p(x,y) = 3x + 2y; > dp/dx = 3, dp/dy = 2, so no critical points. > when u look at the bounds, > maximum at 1) is p(5,0) = 15; > maximum at 2) is p(0,6) = 12; > maximum at 3) is p(3,4) = 17; > maximum at 4) is p(3,4) = 17; > so maximum is (3,4) >>I need help with inequalities! >> Here is a problem that I need to solve: >> Jim Olsen makes and sell gourmet food items. He makes two types of >> salad dressing, garlic and tofu. Each gallon of garlic dressing >> requires 2 qt of oil and 2 qt of vinegar. Each gallon of tofu dressing >> requires 3 qt of oil and 1 qt of vinegar. Jim makes a $3 profit on >> each gallon of garlic dressing and a $2 profit on each gallon of tofu >> dressing. He has 18 qt of oil and 10 qt of vinegar on hand. How many >> gallons of each type of dressing should he make to maximize his >> profits? >> Identify the variables: >> Let x= the number of gallons of garlic dressing. >> Let y= the number of gallons of tofu dressing. >> Write a system of inequalities: >> CAN SOMEBODY PLEASE HELP ME!?