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most recent 30 from 2018-10-17T02:59:20Z

Ellipsoid minimizing Banach-Mazur distance to convex body

Tue, 07/17/2018 - 05:50

Given a (symmetric) convex body $K \subset \mathbb{R}^n$ (equivalently, given a norm on $\mathbb{R}^n$), there is a unique ellipsoid of maximal volume in $K$, called the John ellipsoid. The John ellipsoid can be described as a ``canonical ellipsoid'' associated to a convex body, and reading around there seem to be a few other notions of canonical ellipsoid.

On a recent research project of mine, it turned out to be important to associate a different type of ellipsoid to $K$, namely the ellipsoid which minimizes the Banach-Mazur distance to $K$, suitably normalized. More directly, let $E$ be the ellipsoid contained in $K$ which minimizes the value $\lambda \geq 1$ for which $K \subset \lambda E$. In my paper I proved a volume ratio inequality for this ellipsoid $E$ for dimension two and applied it to a problem on quasiconformal mappings (the preprint is at ).

At the time, I asked a few convex geometry people what was known about this ellipsoid, or if it had been studied before, as the John ellipsoid has been. They didn't really have anything to say on the matter, nor did I find anything in standard references, so I didn't dwell on it. Now, however, there are some junior mathematicians working on thesis projects, etc. who have been talking to me and want to use my work. So this is making me want to revisit the question of attribution. I'm curious about the following:

  1. Has the ellipsoid minimizing Banach-Mazur distance to a convex body in $\mathbb{R}^n$ ever appeared or been studied in an important/useful/systematic way?

  2. Is there a best name to give to the "ellipsoid which minimizes Banach-Mazur distance to $K$" without having to say this every time? One candidate could be simply ``Banach-Mazur ellipsoid'', but the answer to 1. might suggest a different name.

Paul Mahlo's Original Large Cardinal Paper

Tue, 07/17/2018 - 05:34

It is well known that Paul Mahlo (1883-1971) developed a systematic hierarchy of inaccessible cardinals of the type $\pi_{a,b}$ where $\pi_{1,b}$ enumerates the strongly innacessible cardinals, $\pi_{2,b}$ enumerate the fixed points of $\pi_{1,b}$ and so on. My question is, where can I find the original paper of Mahlo in English? If this doesn't exist, are there any good expositional articles on this accessible online?

Proof of an inequality $s_m(n) \le f_m(n)$

Tue, 07/17/2018 - 01:26

For fixed $m = 0, 1, 2, ...$ $$f_m(k) = \prod_{j=1}^{m}(k+j).$$ Some examples of $f_m(k)$ are as following: $$f_0(k) = 1, \quad f_1(k) = (k+1), \quad f_2(k) = (k+1)(k+2).$$

The $s_m(n)$ is defined as following: $$s_m(n) = \sin\left(\frac{t}{2}\right)\sum_{k=0}^nf_m(k)\sin(k+0.5)t,\qquad t\in[0,\pi].$$

The $s_m(n)$ can also be defined as following: $$s_m(n) = \sum_{j=0}^n\frac{(-4)^j}{(2j+1)!}\left(\sum_{k=j}^n\frac{f_m(k)(2k+1)(k+j)!}{(k-j)!}\right)x^{2j+2},\qquad x\in[0,1].$$

I want to prove $$|s_m(n)| \le f_m(n), \forall x ~or ~t$$

I am sure the inequality holds but I am unable to prove it. I used MATLAB and verified the inequality for some values of $m$ and $n$ as presented below:

\begin{array}{ccccccccc} n & \max(s_0(n)) & f_0(n) & \max(s_1(n)) & f_1(n) & \max(s_2(n))& f_2(n) & \max(s_3(n)) & f_3(n)\\ 0 & 1.00 & 1 & 1.00 & 1 & 2.00 & 2 & 6.00 & 6 \\ 1 & 1.00 & 1 & 1.53 & 2 & 4.17 & 6 & 18.00 & 24 \\ 2 & 1.00 & 1 & 2.07 & 3 & 8.00 & 12 & 42.00 & 60 \\ 3 & 1.00 & 1 & 2.60 & 4 & 12.46 & 20 & 78.30 & 120 \\ 4 & 1.00 & 1 & 3.13 & 5 & 18.03 & 30 & 132.00 & 210 \end{array}

Any help will be greatly appreciated.

PS: Please refer to this question. I asked for inductive proof so that I could use induction steps in the above inequality. But I did not get one.

Homotopy theories of operads

Mon, 07/16/2018 - 22:45

I know of three homotopy theories of colored operads.

  • The (derived) localization category of Berger-Moerdijk's model structure on the category of strict simplicial (or topological) operads, with weak equivalences given by strict maps $O\to O'$ which induce weak equivalences on spaces of operations.
  • The "dendroidal" notion of infinity-operad, introduced by Moerdijk and Weiss and further studied in this paper by Cisinski and Moerdijk.
  • Lurie's infinity-operads, which are infinity categories fibered over the nerve of the category $\mathrm{Set}^*$ of pointed sets, satisfying certain Segal-style properties.

If all is well in the world, these three homotopy theories (viewed as e.g. $\infty$-categories) should be equivalent (or there should be a good reason for them not to be). But I can't find references for any equivalences between them. This paper seems to compare (1) and (2), via a Quillent adjunction, which it does not show is an equivalence. The obvious functor (1) $\implies$ (3) is written down in Lurie's Higher Algebra, but it is not (as far as I can tell) shown to be an equivalence.

Is more known about comparisons between these homotopy theories? I'm specifically interested in the comparison between (1) and (3). Are the corresponding $\infty$-categories equivalent, perhaps under additional restrictions? Is the functor (1) $\implies$ (3) fully faithful? Are there known examples where this functor is not an equivalence?

Integral representations of finite groups and lattice point geometry

Mon, 07/16/2018 - 13:39

This contains both a reference request, and a specific problem. Let $K$ be a finite group, and let $\theta: K \to {\rm GL}(d,{\bf Z})$ be a (faithful) group representation over the integers. Consider the following property of $\theta(K)$ (not merely of $\theta$): there exists a finite subset $S$ of $ H = {\bf Z}^d$ such that

(0) $0 \in S$;

(i) $\cup_{n \geq 1} nS = H$ [here $nS$ means the set of sums of $n$ elements of $S$];

(ii) $S$ is $\theta$-invariant;

(iii) if we set $C$ to be the convex hull (in Euclidean space, ${\bf R}^d$), of $S$, then $0$ belongs to the interior of $C$ and the natural action of $\theta$ on $C$ is transitive on the set of facets of $C$ [a facet is a face of codimension one; the transitivity condition is that every facet can be moved to all other facets by elements of $\theta(K)$];

(iv) for each facet $F$ of $C$, let $S_F =S \cap {\rm cvx } \{F,0\}$; we require that $S_F$ generate $ H$ as a group, and $C_F:=\cup_{n\geq 1} nS_F$ is a simplicial cone.

Despite this ridiculously complicated set of conditions, there are plenty of natural examples.

Example 1 Take $K ={\bf Z}_2^d$ acting as the group of diagonal $\pm1$ matrices on ${\bf Z}^d$, and set $S = \{0; \pm e_i \}$ where the $e_i$ vary over the standard basis. Then $C$ is just a $d$-dimensional version of the octohedron, and the facets correspond to the intersection with the orthants, and it is clear that $\theta$ acts transitively on them. The simplicial condition is obvious. [Here, $K$ acts far from transitively on the extreme points of $C$, but this does not matter.]

Example 2 Let $K = \cal S_{d+1}$, the full permutation group on $d+1$ symbols, or any subgroup which acts transitively on the symbols. Then $K$ acts on ${\bf Z}^{d+1}$ by permuting the standard basis elements, and leaves $v= (1,1,\dots, 1)$ invariant. So we obtain the quotient action of $K$ on ${\bf Z}^{d+1}/v{\bf Z} $ identified with ${\bf Z}^{d}$. With $S$ being the orbit of $e_1$ in ${\bf Z}^{d}$ together with $0$, that is, $S = \{0, e_1, e_2,\dots, e_d; -\sum e_i \}$, then it is easy to see that $C$ is a simplex, and the simplicial condition holds, and transitivity is obvious.

Example 3 Let $d = 2$, and let $\theta : K \to {\rm GL}(2,{\bf Z})$ be one to one (faithful). Provided $K$ has more than two elements, then an $S$ satisfying (0--iv) exists.

For the last, the only group which requires more than a few sentences is $K = C_2 \times C_2$, for which there exist two outer conjugacy classes of faithful representions on ${\bf Z}^2$.

The existence of a set $S$ with the properties (0--iv) is obviously an invariant of the integral representation $\theta$ of $K$, in fact an outer invariant: it is an invariant of the image of $K$, $\theta(K) \subset {\rm GL}(d,{\bf Z}^{d})$. This type of thing (using integral geometry to obtain invariants for integral representations) must have been done before.

First question [finally] A reference request for analyzing integral representations of finite groups by lattice point geometry.

Second question Are there more examples than the ones I described, or more generally, can useful necessary/sufficient conditions be derived for the existence of such a set $S$?

Apparently, it is necessary that the trivial representation not be a subrepresentation of $\theta$, and multiplicities should be avoided.

The motivation comes from studying random walks and suitable weight functions on the semidirect product groups ${\bf Z}^{d}\times_{\theta} K$; when such an $S$ exists, the random walks (weight functions) have nicer properties than when no such $S$ exists.

Random 3-manifolds in $R^4$

Mon, 07/16/2018 - 10:50

Consider following program:

  1. Generate random 3-manifold embedded in $R^4$.
  2. Perform its triangulation.
  3. Put it to Regina and calculate what manifold it is.

Assuming that we have good algorithm for random submanifolds in point 1. then we can conclude which 3-manifolds of of complexity 5,6,7,8 etc are embeddable in $R^4$. For example from this paper I can see that there are 175 3-manifolds of complexity 7. Those which were not obtained in this process we can assume are not embeddable in $R^4$ with some probability.

Possible choices for algorithm in point 1 are:

a) zero of four variables polynomial;

b) random embedded 1-surgery;

c) gluing cubes;

d) drilling small hole cubes in big cube;

e) boundary of regular neighborhood of 2-complex in $R^4$ (added 2018-08-23)

The questions are:

A. What are achievements in finding good polynomial of four variables hoping to obtain interesting 3-manifold as its zero ?

B. What could be the algorithm for finding random loop in $M$ embedded in $R^4$ to perform embedded surgery ?

C. Is Regina accepting command line execution with some input in TXT file containing triangulation and producing result (or LOG) in other TXT file ?

Related questions are:

D. What could be other ideas for producing random 3-submanifolds of $R^4$ ?

E. How could we generate random slice knots and what manifolds we obtain by repeating 1-surgery on slice knots ?

F. Is it known which 2-dimensional CW-complexes are embeddable in $R^4$ ? Such CW-complex can be seen as few words in set of generators which are forming bouquet of circles. I am hoping all 3-manifolds embeddable in 4-space are boundaries of regular neighborhoods of some 2-complex.

EDIT 2018-07-30

Regarding last question. I have been able to find embedding of 2-complex with one word in 4-space. So I thought to use this as starting point. Assuming that this 2-complex is defined as 2-skeleton in $\mathbb R^4$. Related question is

F2. Is it known algorithm for finding regular neighborhood of 2-skeleton in $R^4$ ? If I have it then I find its triangulated boundary as 3-manifold I want. The 3-simplex belongs to boundary when it belongs to only one 4-simplex.

EDIT 2018-08-23

In this question I found reference to the paper:

Dranišnikov, A. N.; Repovš, Dušan, Embedding up to homotopy type in Euclidean space, Bull. Aust. Math. Soc. 47, No. 1, 145-148 (1993). ZBL0796.57011.

In this paper there is construction of embedding of any 2-complex in $R^4$ up to homotopy type. It is described as simpler proof of Stallings theorem from 1965. Therefore I am planning to convert that construction to simplicial complex in $R^4$. Next construct its regular neighborhood, its boundary will be 3-manifold which I would like to recognize using Regina or other software.

What is not clear for me is why embedding of 2-complex in $R^4$ listed as open issue number 5.3 on Kirby's open problem list.


A translation between formal and rigid geometry

Mon, 07/16/2018 - 09:31

The following lemma is in Bosch's book "Lectures on Formal and rigid geometry" p198.

Lemma Let $K$ be a non-archimedean field and $R$ its valuation ring. Let $X= \mathrm{Spf}A$ be an affine admissible formal $R$ scheme. Then there are canonical bijections between (1) the set of non-open prime ideals $\mathfrak{p}\subset A$ with $\dim A/\mathfrak p =1$ and (2) the set of maximal ideals in $A\otimes_RK$.

Now let's look at a easy case: $A=R\langle \zeta_1,\dots,\zeta_n\rangle$. Then $A\otimes_RK$ is simply the Tate algebra $T_n:=T_n(K)$. Then we know the set of maximal ideals in $T_n$ can be described by the points of the unit ball $\mathbb B^n(K)$(assume $K$ is algebraically closed): $\mathfrak{m}_x=\{f\in T_n\mid f(x)=0\}$ The question is that what the prime ideal does $\mathfrak m_x$ correspond by the above lemma?

Quadratic integer program over a sphere

Mon, 07/16/2018 - 03:26

Given vector $a \in \mathbb{R}^K$ and symmetric and positive definite matrix $M \in \mathbb{R}^{K \times K}$,

\begin{align} \underset{\vec{x}}{\text{minimize}}\quad &(\vec{x}-a)^\dagger M(\vec{x}-a)\\ \text{subject to}\quad & \| \vec{x} \|^2 = n \\ & \vec{x} \in \mathbb{Z}^K \end{align}

Has any work been done on this? My thinking is that the solution can somehow be approximated with the LLL algorithm.

An explicit correspondence for reductions of modular curves $Y(N)$

Mon, 07/16/2018 - 01:19

Let $Y(N)$ be the modular curve associated with the principal congruence subgroup $\Gamma(N) \subset \mathrm{SL}(2, \mathbb{Z})$ of level $N \in \mathbb{N}$. It is well known that this curve has a plane affine model over a quadratic extension of $\mathbb{Q}$. Consider its reduction $Y_p(N) \subset \mathbb{A}^{\!2}_{x,\,y}$ over $\mathbb{F}_{p^2}$, where $p$ is a prime, $p \nmid N$. We know that points of $Y_p(N)(\overline{\mathbb{F}_p})$ correspond (up to an isomorphism) to triples $(E, P, Q)$, where $E$ is an elliptic curve over $\overline{\mathbb{F}_p}$ and $P$, $Q$ form a basis of the $N$-torsion subgroup $E[N]$. Finally, under this correspondence all triples with supersingular elliptic curves belong $Y_p(N)(\mathbb{F}_{p^2})$.

Is there a way to explicitly compute a bijective map between points of $Y_p(N)$ (pairs (x, y) satisfying some modular equation) and triples $(E, P, Q)$? May be, is there a birational map between $Y_p(N)$ and the curve $$V = \{ (j, \pi(P, Q)) \mid K_{j}(\pi_j(P, Q)) = 0; P,Q \rm{ \ form \ a \ basis \ of \ } E[N] \}?$$ Here $K_{j}$ is the Kummer surface for the direct square $E_{j}^2$ of the elliptic curve $E_j$ with $\mathrm{j}$-invariant $j$ and $\pi_j\!: E_{j}^2 \to K_j$ is the natural $2$-sheeted covering associated with the involution $[-1]$ on $E_j^2$.

Reference request: explicit equivariant localization formula on toric complete intersections

Sun, 07/15/2018 - 06:51

This post is about an equivariant integration formula in a famous paper by Alexander Givental, where the author presented the formula without proof or reference. It turns equivariant localization into calculation of residues I am trying to combine it with global residue theorem to derive some vanishing result of some invariant.

$$ \int_X f(p,\lambda)=\sum_{\alpha}Res_{\alpha}\frac{f(p,\lambda)dp_1\wedge\cdots\wedge dp_k}{u_1(p,\lambda)\cdots u_n(p,\lambda)} $$ where $X$ is a toric symplectic variety, $p_k$ forms base of $H^2(X)$ (the Picard group) and $u_i(p,\lambda)=\sum_ip_im_{ij}-\lambda_j$, where $m_{ij}$ is a integer matrix describing how $p_i$ span all the invariant divisors and $\lambda_i$ is the ordinary equivariant index for torus action. The RHS sums over fixed point or pole specified by some of the $u_i=0$.

This paper is hard to read because I am not an expert on symplectic geometry but I really need to re-derive this explicit formula. Maybe some experts could kindly provide me some hints or references and any other discussion is welcomed. Thanks in advance.

Testing whether a quiver algebra is cellular with a computer

Sun, 07/15/2018 - 06:25

With a friend I made a program in the GAP-package QPA to check whether a given finite dimensional quiver algebra is quasi-hereditary. It is very slow since it has to go through all permutations of points in the quiver but in principle it works by using just linear algebra. Now a similar class as quasi-hereditary algebras are cellular algebras: A cellular algebra is quasi-hereditary iff it has finite global dimension.

Question: Is it possible to have a programm in QPA that checks whether a given finite dimensional quiver algebra is cellular (by using given commands that preferably only use linear algebra)?

I have no real experience with cellular algebras and the definitions make it look like the answer is no. But maybe there is an equivalent definition of cellular algebras that makes such a QPA-program easy?

representation of prime numbers

Sun, 07/15/2018 - 06:20

We know that primes of the form $p=4k+1$ can be written as sum of two squares, i.e., $p=x^2+y^2$ (uniquely iff $0<x<y$). However, this expression holds for composite numbers that all of the prime factors are of the form $4k+1$, or if they have prime factors of the form $4k+3$, these factors are of even power.

I am looking for some expression to represent prime numbers of the form $p=4k+1$ such that it does not hold for composite numbers? Is it possible to have such expression?

P.S.: If there are some conditions (not necessarily expressing as polynomials) that determine such prime numbers would be desirable.

Thank you.

Surjectivity of norm map on subspaces of finite fields

Sun, 07/15/2018 - 05:52

It is basic that the norm map $N:\mathbf{F}_{q^n}^* \to \mathbf{F}_q^*$ is surjective for finite fields. In fact $N(x) = x^{(q^n-1)/(q-1)}$. How well does this simple fact extend to subspaces?

A basic example is an intermediate extension $\mathbf{F}_{q^d}$. On $\mathbf{F}_{q^d}^*$ we have $$N(x) = \left(x^{(q^d-1)/(q-1)}\right)^{(q^n-1)/(q^d-1)} = \left(x^{(q^d-1)/(q-1)}\right)^{n/d}$$ since the term in the brackets is in $\mathbf{F}_q^*$ and $(q^n-1)/(q^d-1) \equiv n/d \pmod {q-1}$. So $N$ is surjective on $\mathbf{F}_{q^d}^*$ if and only if $(n/d, q-1) = 1$. In particular $N$ fails to be surjective on a subspace of dimension $n/2$ whenever $n$ is even and $(n/2, q-1) > 1$.

As a sort of converse note that if $(n,q-1)=1$ then $N$ is surjective on every one-dimensional subspace.

Is it true that if $V \leq \mathbf{F}_{q^n}$ is a $\mathbf{F}_q$-rational subspace of dimension $>n/2$ then $N$ is surjective on $V$?

Equivalently, if $\dim_{\mathbf{F}_q} V > n/2$, can we always find $x^{q-1} \in V$?

irreducible component of support of function

Sun, 07/15/2018 - 02:49

Vakil gives two equivalent definitions of associated points in his "Rising Sea":

  1. a prime ideal $p$ of a ring $A$ is called an associated prime for module $M$ if it is the annihilator of an element $ m \in M$, i.e. $p = \mathrm{Ann}(m) $
  2. a point p in $\mathrm{Spec} A$ is called an associated point for module $M$ if there is an element $ m \in M$ such that $p$ is a generic point of $\mathrm{Supp } \text{ }m$

Where $A$ is a Noetherian, $M$ is finite generated over A.

Vakil asks readers to check this equivalence, and he gives a hint:

if $p$ is an associated point, then there is an element $m$ with Support $\bar{p}$

If I prove this then I finish the exercise, but I can't.

What is the expected value of n? [on hold]

Sun, 07/15/2018 - 01:52

• A special die is designed with n faces, enumerated 1 through n with all faces being equally likely. If X is the observed number when this die is thrown, what is the expected value of X? [C]
(a) $\frac n2+1$
(b) $\frac n2$
(c) $\frac{n+1}2$
(d) $\frac{n−1}2$
Why is the answer c correct? I know that it is a discrete uniform distribution.

Reference request: Bipartite symmetric graphs are hamiltonian

Sun, 07/15/2018 - 01:23

Does anyone know whether bipartite symmetric graphs are hamiltonian? I'm not sure whether anyone have proved it before, but a nonhamiltonian symmetric bipartite graph would lead to a counterexample to the Lovasz conjecture. I would appreciate any references or ideas.

Does there exist a Lebesgue nonmeasurable set $E$ in $\mathbb{R}$ satisfies that $E\cap A$ is a Borel null set for every Borel null set $A$?

Sun, 07/15/2018 - 00:42

Let $\mathcal{B}_{\mathbb{R}}$ be the Borel $\sigma$-algebra on $\mathbb{R}$ and $\mu_L$ be the Lebesgue measure on $\mathbb{R}$.

Define a new $\sigma$-algebra $\mathcal{B}_0$ as follows: $$\mathcal{B}_0=\{A\in \mathcal{B}_{\mathbb{R}}:\mu_L(E)=0\ \text{or}\ +\infty\}.$$ I want to prove that the family of all locally measurable sets of the measure space $(\mathbb{R},\mathcal{B}_0,\mu_L|_{\mathcal{B}_0})$, that is, $$\{E\subset \mathbb{R}:E\cap A\in \mathcal{B}_0\ \text{for all $A\in \mathcal{B}_0$ such that $\mu_L(A)<\infty$}\}$$ is not the family of all subsets of $\mathbb{R}$.

So I want to ask whether there exists a Lebesgue nonmeasurable set $E$ in $\mathbb{R}$ satisfies that $E\cap A$ is a Borel null set for every Borel null set $A$.

General solution for first-order difference equation

Sat, 07/14/2018 - 22:48

I have the following first-order difference equation

$$y_{t} = \frac{x_{t}}{1-\rho L} + \epsilon_{t}$$

where $L$ denotes the backshift operator, i.e., $L(x_{t}) = x_{t-1}$. I can obtain a solution to this difference equation heuristically, but I am wondering if there is a general procedure. A solution (the solution?) is

$$y_{t} = \sum_{i = 0}^{\infty} \rho^{i}X_{t-i} + \epsilon_{t}$$

I looked in Goldberger's text and couldn't find it there. Any reference is appreciated also. Thanks.

How to visualize a Witt vector?

Sat, 07/14/2018 - 22:24

As the question title asks for, how do others "visualize" Witt vectors? I just think of them as algebraic creatures. Bonus points for pictures.

What are the internal categories in an endofunctor category

Sat, 07/14/2018 - 17:53

Take a category $C$, and take all endofunctors of $C$, so the set $E= \{ M| M: C \rightarrow C \}$. $E$ forms the objects of a category with morphisms given by all natural transformations $\mu : M \rightarrow N$ for $M,N \in E$. Let $\mathcal{C}$ be the endofunctor category as defined. What are the internal categories in $\mathcal{C}$?

Further suppose $C$ is a symmetric monoidal dagger category, in this case, what are the internal categories in $\mathcal{C}$?