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17 votes
2 answers
945 views

Isoperimetric-like inequality for non-connected sets

The classical isoperimetric inequality can be stated as follows: if $A$ and $B$ are sets in the plane with the same area, and if $B$ is a disk, then the perimeter of $A$ is larger than the perimeter ...
Guillaume Aubrun's user avatar
3 votes
1 answer
478 views

Find recurrence in Pascal-like triangle of polynomials

Consider an infinite, upper-triangular Toepliz-matrix, with first row $x_1,x_2,\dots,x_n.$ Then there is a sequence of determinants obtained from the $m \times m-$ sub-matrices with upper left corner ...
Per Alexandersson's user avatar
1 vote
0 answers
1k views

Is there an upper bound to strongly connected components

Hey, given $G = (V,E)$ a directed Graph. Is there any way to calculate the maximum number of strongly connected components $k$, only based on $n$ nodes and $e$ edges?
Mika's user avatar
  • 11
5 votes
1 answer
191 views

Expressing a element of a Matrix subgroup in terms of subgroup generators

I'm no (computational) algebraist, and my searches have been pretty unyielding (probably due to the vast amounts written on the key words), but perhaps someone may know if this is possible, and if so, ...
philiph's user avatar
  • 153
7 votes
2 answers
572 views

Pairs of Permutations up to Simultaneous Conjugation

The conjugacy classes of $S_n$ are the cycle types since if $\tau = (\dots)(\dots)\dots(\dots)$, the conjugation $\tau \mapsto \sigma \tau \sigma^{-1}$ permutes the labels in the cycles of $\tau$. ...
john mangual's user avatar
  • 22.6k
0 votes
1 answer
258 views

Number of points on a complex sphere with pairwise inner product restriction

Considered the following inner products: $(1)$ $\langle x,y \rangle = \sum_{t=1}^{n}x_{t}y_{t}$ $(2)$ $\langle x,y \rangle_{c} = \sum_{t=1}^{n}x_{t}\bar{y}_{t}$ consider the following surfaces: $\...
Turbo's user avatar
  • 13.7k
4 votes
4 answers
606 views

A question about the additive group of a finitely generated integral domain

Let $R$ be an integral domain of characteristic 0 finitely generated as a ring over $\mathbb{Z}$. Can the quotient group $(R,+)/(\mathbb{Z},+)$ contain a divisible element? By a "divisible element" I ...
Sidney Raffer's user avatar
2 votes
1 answer
256 views

A Volterra-type equation

Consider the following integral equation $\phi(x) = f(x) + \frac{1}{x}\int_0^x N(x,y)\phi(y)\;dy$, where $f$ and $N$ are continuous and bounded functions. Are solutions $\phi$ of the above equation ...
Jeff's user avatar
  • 565
2 votes
1 answer
663 views

Different cuspidal automorphic representations with same representations at infinity

Let us fix a representation $\pi_\infty$ of GL(n,$\mathbb R$). Let us fix a character $\chi$ of K, where K is a compact subgroup of $GL(n,\mathbb A_{finite})$. $$K=\Pi_{v<\infty}K_v$$ $K_v$ is $...
7-adic's user avatar
  • 3,764
3 votes
2 answers
730 views

What's known about complete split primes in Q(E[p])?

Let E be an elliptic curve over $\mathbf{Q}$, and p be a prime of good reduction for E such that the Galois representation $\bar\rho_p$ of $\mathbf{Q}$ on the p-torsion of E surjects onto Aut(E[p]). ...
Tommaso Centeleghe's user avatar
12 votes
1 answer
516 views

Are there nonobvious cases where equations have finitely many algebraic integer solutions?

Let $X$ be a scheme of finite type over $\mathbb{Z}$. Let $R$ be the ring of algebraic integers. My intuition is that $X(R)$ is practically always infinite. More specifically, suppose that $X$ is ...
David E Speyer's user avatar
0 votes
2 answers
526 views

Grobner fan linked to associahedra?

In "Computing Grobner Fans" by Fukuda/Jensen/Thomas on page 2210 in Table 1 are the numbers (1,20,120,300,330,132) for some statistics on Grobner fans for Grass(2,5). This is a vector found in A126216,...
Tom Copeland's user avatar
  • 9,937
1 vote
0 answers
162 views

CH for tilings of the plane

Given any set of tiles (jordan curves) that can tile the plane, how to prove that the number of possible tilings using tiles from this set is either in bijection with the real numbers or a (possibly ...
Hermite's user avatar
  • 77
71 votes
3 answers
9k views

What exactly is the relation between string theory and conformal field theory?

Maybe it would be helpful for me to summarize the little bit I think know. A 2D CFT assigns a Hilbert space ${\cal H}$ to a circle and an operator $$A(X): {\cal H}^{\otimes n}\rightarrow {\cal H}^{\...
Minhyong Kim's user avatar
  • 13.5k
11 votes
1 answer
459 views

Do indiscernibility embeddings exist for an initial segment of an inner model of many measurable cardinals?

Background I am interested in elementary embeddings from a model of set theory into itself. One way of producing such elementary embeddings is when the model is generated by indiscernibles; this idea ...
Norman Lewis Perlmutter's user avatar
7 votes
3 answers
954 views

Simple basis for Barnes-Wall lattices in dimension `$2^n$`

I'm searching for a "simple" description of the basis of the Barnes-Wall lattices in (real) dimension $2^n$, if possible in a basis of minimal vectors, so that I can do some calculations. Can ...
2 votes
2 answers
912 views

Question about Godel's 2nd Theorem

Let Con(PA) be the sentence of arithmetic which translates as "Peano Arithmetic is consistent." Then according to Godel's 2nd incompleteness theorem, assuming PA is consistent then PA can neither ...
Keshav Srinivasan's user avatar
1 vote
1 answer
353 views

multigrid boundary conditions and variable operator

Hi, I have two question about solving an elliptic equation using multigrids. First, in case of Neumann boundary conditions : Does the error equation, which is solved on the coarse grid, have the ...
Nico's user avatar
  • 41
2 votes
0 answers
396 views

Do inverse images respect flabby sheaves?

Let $i:Y\to X$ be a closed embedding of varieties, and let $S$ be a flabby \'etale (or Nisnevich) sheaf of abelian groups on $X$. Is $i^*S$ flabby also? I am mostly interested in the case when $S=i_{x*...
Mikhail Bondarko's user avatar
5 votes
0 answers
500 views

Categorifying idempotent relations

Generalizing partial orders: A relation $R$ is transitive if $R \circ R \subseteq R$ and interpolative if $R \subseteq R \circ R$. It is idempotent if $R \circ R = R$. Interpolativeness means that ...
0 votes
0 answers
263 views

Is the absolute value of the j-invariant bounded from below on an annulus

Let $j:\mathbf{H}\to \mathbf{C}$ be the $j$-invariant. It's a modular function for $\Gamma(1) = \textrm{PSL}_2(\mathbf{Z})$. For $\epsilon>0$ small, let $B(\epsilon)$ be the image of the strip $$\{...
Taicho's user avatar
  • 225
9 votes
1 answer
843 views

$\Pi_0^1$-weakly indescribable cardinals are exactly the regulars

Hi, I'm not sure if I should ask here or over at math.stackexchange.com, but I think here it's a bit more fitting. This question stems from a homework problem: Definition: Given some class of formulas ...
Apostolos's user avatar
  • 341
20 votes
2 answers
3k views

D-modules and Algebraic Solutions of PDEs

I am not certain if this is a complete question and I fear it might be shot down. Anyway, I try to pose it. My question is in connection to using D-modules to study PDEs (and systems of PDEs). When ...
Ongaro Nyang''s user avatar
14 votes
0 answers
463 views

Cohomological interpretations of quadratic form invariants over rings?

The standard approach to classifying of quadratic forms over $\Bbb Q$ is to use the Hasse (local-global) principle together with a system of standard invariants of quadratic forms over the local ...
Jonathan Hanke's user avatar
2 votes
0 answers
168 views

Bundles of (co)chain complexes

A stupid question: does anybody know a good book or something in which invariants of a vector bundle of finite-dimensional acyclic (co)chain complexes (which, I believe is equivalent to a acyclic (co)...
gshar's user avatar
  • 291
8 votes
1 answer
1k views

Serre's Analogue of the Weil Conjectures for Non-Compact Kahler Manifolds

The classical Riemann Hypothesis concerns the locations of zeroes of the Riemann zeta-function, or more generally the Dedekind zeta-functions of number fields. Its analogue for varieties defined over ...
Abtan Massini's user avatar
5 votes
5 answers
826 views

For any $n$, does there exist a number field with at least $n$ solutions to the unit equation

Let $n$ be a positive integer. Does there exist a number field $K$ such that the number of solutions of the unit equation $$a+b =1, \quad a,b\in O_{K}^\ast$$ is at least $n$? Can we write down such a ...
Taicho's user avatar
  • 225
3 votes
1 answer
490 views

Are Weierstrass points algebraic

Let $X$ be a compact connected Riemann surface of genus $g>0$. Suppose that $X$ can be defined over a number field (as an algebraic curve). Then, is it clear that each Weierstrass point of $X$ is ...
Taicho's user avatar
  • 225
4 votes
1 answer
424 views

Inverse of Kleisli star, or "extension operator"

While thinking about monads in the theory of denotational semantics, I have made an observation about the Kleisli category that I would like to check Suppose $F : \mathcal D \to \mathcal C$, $G : \...
Tom Ellis's user avatar
  • 2,775
12 votes
1 answer
3k views

Is the degree of a finite morphism stable by base change

Let $f:X\longrightarrow Y$ be a finite morphism of schemes of degree $n$. Let $S\to Y$ be a morphism of schemes. Is the degree of the finite morphism $X\times_Y S \longrightarrow S$ equal to $n$? If ...
Taicho's user avatar
  • 225
1 vote
0 answers
230 views

Lower bound for intersection number

The base scheme is an algebraically closed field. Let $X\to \mathbf{P}^1$ be an arithmetic surface over $\mathbf{P}^1$ and let $P$ be a section of $X\to \mathbf{P}^1$. Let $D$ be an effective (edited)...
Taicho's user avatar
  • 225
0 votes
2 answers
807 views

Structural definition of "product" in set theory

At first sight there is no abstract (= structural) definition of "product" in set theory. E.g. the Cartesian product of sets $A$ and $B$ is defined as the set of all ordered pairs $(x,y)$, $x \in A$, $...
Hans-Peter Stricker's user avatar
10 votes
1 answer
1k views

Use of games to approximate solutions to Partial Differential Equations

Hi there, Hopefully the mathematics community can help me out this one, I'm currently studying my senior capstone at my college, and decided to do some research on a chapter in Stanley Farlow's book "...
DaveNine's user avatar
  • 203
8 votes
2 answers
1k views

Is the free product of arbitrarily many copies of `${\mathbb{Z}}$` and `${\mathbb{Z}}/2$` residually nilpotent?

A group is residually nilpotent if the intersection of the terms in its lower central series is the trivial group. Is the free product of arbitarily (possibly infinite) many abelian groups residually ...
user avatar
7 votes
1 answer
800 views

sheaf valued functors: how much can you prove about them just using category theory

This question is related to my earlier question. I have learnt a lot since I asked that question and I think I can phrase my thoughts more clearly now. To someone who is comfortable with category ...
Daniel Barter's user avatar
2 votes
1 answer
571 views

Moschovakis Coding Lemma

I trying to study the Coding Lemma (in descriptive Set Theory) and there is a small point in the proof that I don't understand. Let me first recall the version I'm studying ( there are different ...
Rachid Atmai's user avatar
  • 3,756
13 votes
3 answers
1k views

Random Reidemeister moves to unknot

Suppose one has a link diagram of the unknot, and applies random Reidemeister moves until the unknot is reached. Surely it requires an exponential number of moves, exponential in, say, the crossing ...
Joseph O'Rourke's user avatar
2 votes
1 answer
277 views

Tree graph restructuring.

I have a graph $G(V,E)$ and a tree $T(V',E')$ where $|V|=|V'|$ and $T$ is isomorphic to a subgraph of $G$. In other words I found a spanning tree of $G$ and made one of its nodes act as the root. I ...
njvb's user avatar
  • 143
15 votes
0 answers
633 views

Are "fpqc algebraic spaces" algebraic spaces?

Suppose $F:Sch^\text{op}\to Set$ is a sheaf in the fpqc topology, has quasi-compact representable diagonal, and has an fpqc cover by a scheme. Must $F$ be an algebraic space? That is, must $F$ have an ...
Anton Geraschenko's user avatar
3 votes
1 answer
300 views

Hermit H-machines

I call an H-machine a machine that can be connected to turing machines and that takes as input a natural integer n and instantly returns the n'th digit of the mathematical constant H. Is there a ...
Hermite's user avatar
  • 77
4 votes
2 answers
2k views

Ample divisors on blown-up projective space

Let $\mathbb{P}=\mathrm{Proj}(\mathbb{C}[x_0,\ldots,x_n])$ be complex projective $n$-space. Assume I have linear subvarieties $L_1,\ldots,L_k\in\mathbb{P}$ of codimension $r_i\ge 2$, respectively. Let ...
Jesko Hüttenhain's user avatar
19 votes
2 answers
2k views

Area of distance sphere in manifold with Ricci $\ge 0$.

Let $M$ be a open complete manifold with Ricci curvature $\ge 0$. By a theorem of Calabi and Yau, the volume growth of $M$ is at least of linear. I am wondering whether the following statement is true:...
user16750's user avatar
  • 881
2 votes
0 answers
259 views

Generic Sets and Category Theory

Hi, I am interested in categorical versions of forcing techniques, and I was wondering if there is anyway that topological/set-theoretical concepts such as filters, density and generic sets can be ...
Gianluca's user avatar
1 vote
0 answers
222 views

(Non-)Surjectivity of the Maslov index

Let $V$ be a symplectic space over a field $k$ (for simplicity, the characteristic of $k$ is not $2$). The Maslov index sends a collection of $n$ lagrangian subspaces of $V$ to a quadratic space over $...
Justin Campbell's user avatar
8 votes
1 answer
841 views

Symmetrization map for universal enveloppings. What are Harish-Chandra images of symmetrization (poisson-center of S(gl_n) ) ?

For any Lie algebra symmetrization maps Poisson center of S(g) to center of U(g). Consider g=gl_n, the Poisson center of S(gl_n) is isomorphic to algebra of symmetric polynomials in eigenvalues. ...
Alexander Chervov's user avatar
5 votes
1 answer
609 views

Characterizing invertible nonnegative matrices with bounded sums

Almost a year ago, I asked in this question about obtaining a tight bound on the sum of the entries of the inverse of a strictly positive definite matrix. Denis Serre gave a nice counterexample ...
Suvrit's user avatar
  • 28.4k
0 votes
2 answers
425 views

Formula with prime-density 1 in the integers [closed]

The Hermite-grand-conjecture implies that f(k)=(2^(2^5^11^(7k+1))+1)/3 is prime for all natural numbers $k$. Is there any explicit formula that has so far been proven to produce primes for all ...
Hermite's user avatar
  • 77
13 votes
3 answers
1k views

Schemes over ℤ with a “graded existence over 𝔽₁”

Let $X$ be a (separated, and with whatever other tameness conditions are appropriate) scheme over the integers $\mathbb{Z}$. (If you don't like schemes much, imagine that $X$ is described by ...
Greg Kuperberg's user avatar
3 votes
1 answer
462 views

Base change for the Gauss-Manin sheaf

I want to see the following thing: $\ \ $If $X$ is a smooth geometrically connected scheme over a field $k$ of characteristic 0, $U\subseteq X$ is a non-empty open, $(E,\nabla)$ is an integrable ...
unknown's user avatar
  • 251
3 votes
1 answer
517 views

Study of free monoids of the recursive S. Eilenberg.

Compared to the usual treatises on recursion (eg, Rogers H. "Computability and Undecidability." McGraw-Hill, New York) the book of Samuel Eilenberg & Calvin C. Elgot "Recursiveness" treats such ...
Buschi Sergio's user avatar

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