October 2020 - mathfun@mailman.xmission.com

Re: [math-fun] average angle between 2 4D vectors?
by Henry Baker 27 Oct '20

27 Oct '20

Perhaps I should have used 'typical'? Your argument works in any dimension, but the % of small angles falls off pretty rapidly. Let me define 'equator' as those points on the hypersurface with angle pi/2. Consider the 'band' within +- 5 degrees of the equator. What we want is the ratio of the *hypersurface area* of this band to the hypersurface area of the entire hypersphere. BTW, does Archimedes' 'hat box' theorem work in dimensions other than 3 ? At 09:15 AM 10/27/2020, Tom Duff wrote: >Picking random vectors is problematic -- what's the distribution? >If we settle on random unit vectors, wlog we can make one of them (0,0,0,1). >Then the cosine of the angle between them is the w coordinate of the other. >Positive and negative w's are equally likely, and >(arccos(w)+arccos(-w))/2 = pi/2 for all -1<=w<=1, >so the average is pi/2. > >On Tue, Oct 27, 2020 at 12:02 PM Henry Baker <hbaker1(a)pipeline.com> wrote: >> >> Pick 2 random vectors in 4D. >> >> What is the angle between them, in maximum likelihood ?

1 0

Re: [math-fun] average angle between 2 4D vectors?
by Dan Asimov 27 Oct '20

27 Oct '20

PS I absent-mindedly wrote the 3-volume instead of the area of the set A(t). Of course the correct formula is vol_2(A(t)) = 4π sin(t)^2. The maximum is still when sin(t) = 1, i.e., when t = π/2. Ah, the joys of aging! —Dan

1 0

Re: [math-fun] average angle between 2 4D vectors?
by Dan Asimov 27 Oct '20

27 Oct '20

Henry Baker wrote: ----- Pick 2 random vectors in 4D. What is the angle between them, in maximum likelihood ? ----- The usual definition of "maximum likelihood" is the parameter with the maximum probability. But for any given angle, its probability is 0. Also, since angle between two vectors depends only on their directions, let's assume we're considering only unit vectors. Then all such vectors form a unit 3-sphere S^3. We may as well fix one of the vectors, say as u = (1,0,0,0) in S^3. The vectors v forming an angle t with u form the set A(t) = {v ∊ S^3 | <u,v> = cos(t)} where <u,v> is dot product. It's easy to see that A(t) is a 2-dimensional sphere of radius = sin(t), and so its 2-volume is vol_2(A(t)) = 4 π sin(t)^3/3. It makes sense to interpret the "maximum likelihood" angle as that angle maximizing this volume. Clearly that occurs when sin(t) = 1 and t = π/2. —Dan

1 0

[math-fun] average angle between 2 4D vectors?
by Henry Baker 27 Oct '20

27 Oct '20

Pick 2 random vectors in 4D. What is the angle between them, in maximum likelihood ?

2 1

[math-fun] Some questions about lattices
by Dan Asimov 27 Oct '20

27 Oct '20

A lattice L of dimension n is a subgroup of R^n of rank n, i.e., it has a set of generators that form a basis for R^n as a vector space. The quotient R^n / L has finite volume (and will always be an n-dimensional flat torus, T^n). Examples include the lattice Z^n in all dimensions, the triangular lattice Z[w] in R^2 = C (where w = exp(2πi/3)), and the bcc and fcc lattices in R^3. Also in all dimensions there are the A_n lattice, defined as A_n = {(x_0,...,x_n) ∊ Z^(n+1) | ∑ x_j = 0} and D_n = {(x_1,...,x_n) ∊ Z^n | ∑ x_j ≡ 0 (mod 2)}. I have some questions about lattices that I don't seem to be able to find answers to in the literature: Question 1: ----------- Given a lattice L, what are the sublattices K ⊂ L that are *similar* to L: rotated and uniformly scaled copies of L ? Question 1a: ------------ And what are their *indices* in L, i.e., the sizes |L/K| of the quotient groups L/K ? (In Z^2 = Z[i], these lattices are q Z[i] for any nonzero Gaussian integer q ∊ Z[i], and their indices will be any integer of the form a^2 + b^2. In Z[w] these lattices are q Z[w] for any Eisenstein integer q and their indices will be any integer of the form a^2 + ab + b^2.) * * * For any lattice L ⊂ R^n, the *Voronoi cell* V(p) of p ∊ L of L is the set of points of R^n closer to p than to any other point (or tied): V(p) = {x ∊ R^n | ‖x - p‖ ≤ ‖x - q‖ for all q ∊ L} V(p) is a closed and bounded n-dimensional polytope, and all V(p) are congruent to each other by the symmetry of L. Question 2: ----------- For which lattices do the various Voronoi cells V(p) have the "nice intersection property" (NIP) that for all p, q ∊ L, p ≠ q, we have: V(p) ∩ V(q) is either a common (n-1)-dimensional face or else it is empty? For instance the triangular lattice Z[w] has the NIP (the V(p) are regular hexagons). But Z^2 does not, since V(p) ∩ V(q) may be a common vertex. * * * A lattice L ⊂ R^n is carried to itself by some distance-preserving mappings R^n —> R^n that take the origin to itself; these are called its "symmetries" and they form a group, denoted by Sym(L). Thus Sym(L) is a subgroup of the orthogonal group O(n). (For instance, every lattice has the symmetry x |—> -x.) If every lattice L' with symmetries containing Sym(L) in fact has the same symmetries: Sym(L') ⊃ Sym(L) ⟹ Sym(L') = Sym(L), then L is called "maximally symmetric". In 2D both Z^2 and Z[w] are maximally symmetric; in 3D the maximally symmetric lattices are Z^3, fcc, and bcc. Question 3: ----------- What is known about which lattices are maximally symmetric in each dimension? Presumably these also include the D_4 lattice in 4D, the E_8 lattice in 8D (defined as E_8 = {(x_1,...,x_8) ∊ Z^8 ∪ (Z^8 + (1/2,...,1/2))) | 𝝨x_j ≡ 0 (mod 2)}, and the Leech lattice in 24D. —Dan

2 2

Re: [math-fun] Question on the Riemann Hypothesis
by Marek Wolf 26 Oct '20

26 Oct '20

Ken Roberts wrote: > Is there something missing in your question? > Did you mean something like this? > Consider the y values which appear as imaginary parts of zeta function zeros. > Is that set of y values, modulo 2*pi, dense in [0,2*pi] ? No, there is nothnig missing: my question conerned conjecture which means maximal violation of the Riemann Hypothesis. In connection with your remark I notice that in Journal für die reine und angewandte Mathematik 254, 100–109 (1972) (Theorem 2) P.D.T.A. Elliott assuming RH proved that the sequence \alpha\gamm_n (n = 1, 2, ...), where \gamma_n is the n-th imaginary part of nontrivial zero on the critical line, is uniformly distributed modulo 1 for every real nonzero \alpha. Marek Wolf

2 1

Re: [math-fun] Question on the Riemann Hypothesis
by Dan Asimov 26 Oct '20

26 Oct '20

[Marek, could you please format your posts so they are easier to read? Thanks.] Even taking "gamm_n" to mean "nth imaginary part of nontrivial zero on the critical line" this still makes no sense to me. BUT: Don't you mean "gamm_n" means "the (positive) imaginary part of the nth nontrivial zero on the critical line" ??? There is no good reason to use backslashes "\" before variables on math-fun; this just makes things harder to read. And there is no reason to use multiple spaces or random line breaks in the middle of a sentence. —Dan also some corrections: nothnig —> nothing conern —> concern Marek Wolf wrote: ----- Ken Roberts wrote: > Is there something missing in your question? > Did you mean something like this? > Consider the y values which appear as imaginary parts of zeta function zeros. > Is that set of y values, modulo 2*pi, dense in [0,2*pi] ? No, there is nothnig missing: my question conerned conjecture which means maximal violation of the Riemann Hypothesis. In connection with your remark I notice that in Journal für die reine und angewandte Mathematik 254, 100–109 (1972) (Theorem 2) P.D.T.A. Elliott assuming RH proved that the sequence \alpha\gamm_n (n = 1, 2, ...), where \gamma_n is the n-th imaginary part of nontrivial zero on the critical line, is uniformly distributed modulo 1 for every real nonzero \alpha. -----

1 0

[math-fun] Gaussian origami
by Allan Wechsler 25 Oct '20

25 Oct '20

Two different Math-Fun threads have cross-fertilized in my brain, leading to the following rumination. Thanks to Jim Propp, and indirectly to my wife, for the origami theme; and thanks to Tom Karzes for his Gaussian-integer puzzle (to which I think the answer is mostly "Gaussian GCD"). I propose two operations, folding and unfolding, on any set of Gaussian integers. For both operations you start by selecting a crease-line, which must be one of the reflection lines of the Gaussian integers -- that is, a horizontal or vertical line with integer or half-integer intercept, or a 45-degree diagonal with integer intercepts. To "fold" a set, you remove all the points on one side of the crease, while adding their mirror images to the other side. (The constraint on the crease line ensures that the mirror images will also be Gaussian integers). To "unfold" a set, just add all mirror images, with no erasing. The central question of the "Theory of Gaussian origami" is, what sets are achievable from what starting sets? In particular, what sets are achievable starting from a single point? I know that by repeated folding I can get a single point from any finite starting set. But, for example, starting with {0}, can I get {0, 2+i} by any number of folding and unfolding operations?

7 30

[math-fun] Question on the Riemann Hypothesis
by Marek Wolf 25 Oct '20

25 Oct '20

Does anyone know who formulated this conjecture: on the critical strip the set {x: there is a y such that zeta(x+i*y) = 0 } is dense in [0, 1] ? Thank you very much in advance, Marek Wolf

2 1

Re: [math-fun] Relative convexity
by Dan Asimov 24 Oct '20

24 Oct '20

Maybe Jim could elaborate on how the desired concept relates to convexity. —Dan ----- This is a little bit different from what I was asking about, though it's also interesting. Dan's notion is about X being convex relative to some bigger set M; I'm asking about X being convex relative to some smaller set Y. In the case where the smaller set is just a point p, the natural notion of "convex relative to {p}" might be "starlike from p". ... ... > Jim Propp wrote: > ----- > Is there a notion of "relative convexity" that would make an ordinary torus > consisting of points at small fixed distance from a large circle "convex > relative to the circle"? > > Thinking about different sorts of polyhedral tori people have come up with, > I realize that part of what I want esthetically is some kind of relative > convexity, but I don't know what it should mean! > ----- -----

4 6