This tag is used if a reference is needed in a paper or textbook on a specific result.

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**1**answer

29 views

### Homotopy type of some lattices with top and bottom removed

The only reaction to this question on math.SE was 21 views in 4 days, so I decided to repost it here. I am not changing anything.
There was an interesting question on MO which OP removed by some ...

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votes

**0**answers

11 views

### $H$ self-adjoint with mass gap, $P≥0,Ω∈D(P),H+λP$ self-adjoint $⟹$ for $λ$ small, $H+λP$ has gap?

Suppose $H$ is a self-adjoint operator on a Hilbert space having a simple isolated least eigenvalue $0$ with gap $1$ ( $H\Omega = 0$, $\Vert \Omega\Vert = 1$ ), $P$ is a non-negative symmetric ...

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**0**answers

38 views

### Closure property of completely monotone functions [on hold]

A $C^{\infty}$ function $f(x_1, \dots, x_n)$ defined on $(0,\infty)^n$ is said to be completely monotone if
$$
(-1)^{k}\frac{\partial^k f}{\partial x_{i_1} \cdots \partial x_{i_k}} \geq 0
$$
...

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votes

**0**answers

110 views

### Where can I find basic “computations” of equivariant stable homotopy groups?

I am new to this subject; so please correct me if I will say something wrong or if you don't like my notation. In particular, I don't know whether it is reasonable to consider an infinite group $G$ (...

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**0**answers

53 views

### Reference request for a well-known lemma in Parabolic Vector Bundle

In the paper- "Moduli Space of parabolic vector bundles on a curve" - Usha N Bhosle, Indranil Biswas-Beitr Algebra Geom (2012), 53:437-449, DOI: 10.1007/s13366-011-0053-7, Lemma $2.1$ is being ...

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votes

**1**answer

127 views

### Poincaré–Bendixson theorem on the torus

I was reading the paper A Generalization of a Poincaré-Bendixson Theorem to Closed Two-Dimensional Manifolds by Arthur J. Schwartz which proves the following theorem:
THEOREM. Let $M$ be a ...

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votes

**0**answers

20 views

### Algorithm for Longest Common Subtour

For a new kind of heuristic for the TSP I need to calculate the longest subtour, that is common to a set $T_1,\ ...,\ T_m$ of tours, that are "good" approximations of the optimal tour $T_{opt}$.
By a ...

**6**

votes

**1**answer

240 views

### The current status of the conjecture on algebraic matroids

Can anyone point out some articles for the conjecture: the dual of an algebraic matroid is algebraic?
Thank you!

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votes

**0**answers

91 views

### Selmer and free rank of Elliptic Curves

If I am not mistaken, the equality of the $p$-Selmer rank and the free rank of an elliptic curve are conjectured to be equal.
This is one of the many equivalent formulations of the Birch and ...

**0**

votes

**1**answer

67 views

### Heat kernel upper bounds on a complete Riemannian manifold

Let $M$ be a complete Riemannian manifold, and $p(t, x, y)$ denotes its heat kernel. I am trying to find sufficient conditions for when the following holds:
$$ p(t, x, y) \leq Ct^{-n/2}, \forall x, y, ...

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votes

**1**answer

105 views

### Chevalley restriction theorem for non-split Cartan

Let $G$ be a reductive group over a field $k$ with maximal torus $H$. Let $\mathfrak{g}$ and $\mathfrak{h}$ denote the corresponding Lie algebra. If $k$ is algebraically closed, we have a theorem of ...

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votes

**1**answer

151 views

### What kind of set theory is obtained from the canonical models of K?

Consider the minimal normal modal logic $K$ (axioms = classical propositional logic + $(\Box(p\land q)\leftrightarrow\Box p\land\Box q)$ + $(\Box\top)$, nothing else).
Its canonical model with no ...

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votes

**1**answer

198 views

### Reference for push-pull formula in cohomology

I would like a precise reference for the following fact.
Assume that
$$
\begin{array}{ccc}
M\times_B N & \stackrel{f'}{\to} & N \newline
\quad\downarrow g' & & \quad\downarrow g \...

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votes

**2**answers

612 views

### Algebraic Geometry for Topologists

As someone who is
familiar with algebraic topology, say, at the level of Hatcher's book, and
familiar with homological algebra and categories and applications in topology
but has no idea what a ...

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votes

**2**answers

181 views

### Examples of Sets with Positive Upper Density

While reading the statement of Roth's theorem I started asking myself what are examples of sets of positive upper density? It's not hard to come up with a few:
Flip a coin with probability $\mathbb{...

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votes

**0**answers

19 views

### Criteria for existence of stable principal submatrices of a stable matrix?

Let $A$ be an $n\times n$ real matrix. Suppose $A$ is stable, that is, all the eigenvalues of $A$ have strictly negative real part.
Question: What are some results about existence of stable ...

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vote

**1**answer

63 views

### Every $W^{1,p}$ has a representative in ACL

Let $\Omega:=(0,1)^n$ and define $ACL_i(\Omega)$ as the set of all Borel functions $u:\Omega\to\mathbb{R}$ such that
$$ t\mapsto u(x_1,\dots,x_{i-1},t,x_{i+1},\dots,x_n) $$
is $AC$ for a.e. $(x_1,\...

**1**

vote

**0**answers

75 views

### Induced structure of topological group [closed]

If we consider a closed Jordan curve $\mathcal{C}$, I know that it's homeomorphic to the circle $S^1$. Now I take an homeomorphism $\phi:S^1\longrightarrow\mathcal{C}$ and this homeomorphism induces a ...

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votes

**0**answers

88 views

### A color interpolation lemma

I need the following "color interpolation lemma". Actually I know a way to prove it, but I'm not very satisfied with that proof.
Lemma. Let $G=(V,E)$ be a (properly) colored graph with colors $1, \...

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votes

**1**answer

143 views

### Asymptotics on the number of ways to pair off $\{1, 2, \dots, 2n\}$ into primes

Given $S = \{1, 2, \dots, 2n\}$, we can always pair off elements into $n$ pairs such that each sum to a prime. The proof of this fact is easy and follows from Bertrand's postulate.
Now, let $\gamma(n)...

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vote

**0**answers

53 views

### Is the category of prederivators cartesian closed?

The question is in the title. ${\bf PDer} = Fun({\bf Cat}^\text{op}, {\bf CAT})$ is obviously cartesian since $\bf CAT$ is. The usual argument for presheaf categories does not apply directly since 1-...

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**0**answers

288 views

### Follow-up to Steinberg's problem (12) in his 1966 ICM talk?

Steinberg's lecture at the 1966 ICM in Moscow here surveyed his work on regular elements of semisimple algebraic groups, while also formulating a number of then-open questions as "problems" (with ...

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votes

**0**answers

88 views

### The number of monotone Boolean functions

In the paper "The number of monotone Boolean functions" A. D. Korshunov calculates an asymptotic number of the number of monotone Boolean functions
(see https://en.wikipedia.org/wiki/Dedekind_number)...

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votes

**0**answers

56 views

### An idea of two sided invertible matrix [closed]

Can you suggest me a reading material that expand on the idea of two matrices that are inverse to each other but which are not square matrices?
Here's what I think of, take $A$ a matrix of order $n\...

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votes

**0**answers

21 views

### Existence of solution to first order pde [closed]

Let $U : = \big(-\frac 12, \frac 12 \big)^2 \setminus B_R(0)$ for some sufficiently small $R > 0$.
I would like to prove the existence of a solution $\rho = \rho (x_1, x_2)\in C^1(U)$ to the ...

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votes

**3**answers

361 views

### linear independence of $\sin(k \pi / m)$

I have tried searching the literature for a result like the following, but have not found anything.
For a positive integer $m$, is it known that
$$\{ \sin (k \pi / m): 1 \leq k \leq m/2, (k,m)=1 \}$$
...

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votes

**0**answers

82 views

### Constructible sheaves on general stratified spaces

I am not an expert in the field, so my question might be rather standard. Let $X$ be a compact metric space. Assume that $X=\cup_{i=1}^NS_i$ is a finite disjoint union of locally closed topological ...

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votes

**0**answers

24 views

### Interpolation inequality for fractional Sobolev spaces

In Theorem 5.2 of the book
Robert A. Adams and John J. F. Fournier, MR 2424078 Sobolev spaces, ISBN: 0-12-044143-8.
is stated that, for any integers $0 \leq j \leq m$ and $u \in W^{m,p}(\Omega)$ (...

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votes

**0**answers

36 views

### Software for matching theorems to inputted conditions/hypotheses

Many times I find myself going through analysis books, wikipedia and papers, looking for what is known for my functions/objects at hand.
So is there any software that at least tries to move in that ...

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vote

**1**answer

72 views

### Reference on Probability theory on functional spaces (in special Hilbert spaces)

Currently, I am working on some sort of stochastic optimization problems defined over function spaces.
I am familiar with standard probability theory (R. Durrett, ''Probability: Theory and Examples")...

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vote

**0**answers

109 views

### Vector bundles and equivariant vector spaces

It seems commonly accepted that most of the results of equivariant geometry for vector spaces yield analog result for vector bundles.
In so far as I understand it, the reason for that is the ...

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votes

**1**answer

157 views

### When the Ratio of Two Factors is a power of $2$ (i.e. $\lfloor{\frac{a}{b}}\rfloor = 2^{s-2}$)

We can write, $n!= 2^s \times a \times b$ where $gcd(a,b)=1$ and $2^{s+1} \nmid n!$.
Problem: Is there infinite number of $n$ when $\lfloor{\frac{a}{b}}\rfloor = 2^{s-2}$?
Question:
1.How ...

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votes

**0**answers

192 views

### Tree property using side conditions

The following problems were asked during the high and low forcing workshop:
Question 1. Can one force tree property at $\kappa^{++}$ for $\kappa$ singular using side conditions?
Question 2. ...

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votes

**1**answer

167 views

### Reference - Generalized Hodge conjecture for triangulated motives

GHC for triangulated motives: The Hodge conjecture holds and an object $\rm M \in Dmg$ is effective if and only if its Hodge realization is effective.
I would like to know some references on GHC ...

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votes

**1**answer

97 views

### A quotient group of a self-normalizing spherical subgroup

Let $G$ be simply connected, simple algebraic group over $\mathbb{C}$.
Let $H\subset G$ be a self-normalizing spherical subgroup of $G$,
not necessarily connected or reductive.
Here "self-normalizing" ...

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votes

**0**answers

68 views

### Realization/embedding for (weakly) finite linear categories

I am trying to determine the status of the following claim. I know how to prove this (unless I made a stupid mistake), so the question is mostly
Is it in the literature?
If not, is there something ...

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votes

**1**answer

117 views

### Density of smooth functions on Hölder spaces

The following result is often cited without reference in the context of PDEs:
Let $\varOmega \subset\mathbb R^n$ be a bounded open set with smooth boundary. If $0<\beta<\alpha<1$ then $C^\...

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votes

**0**answers

66 views

### A question on indefinite binary quadratic forms

Let $f(x,y) = ax^2 + 2bxy + cy^2$ be an irreducible, indefinite (so that $b^2 - ac > 0$) binary quadratic form. Put $d = b^2 - ac$. We say that two pairs of integers $(x_1, y_1)$ and $(x_2, y_2)$ ...

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votes

**1**answer

129 views

### Irreducibility of family of polynomials

Consider the following family of polynomials over $\mathbb{Q}$:
$$f_n = x^n - x^{n-1} - \dots - 1$$
Notice that these polynomials satisfy the recurrence
$$ f_{n+1} = x f_n - 1 $$
I would like to ...

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votes

**0**answers

103 views

### Alternative definitions of Sobolev spaces on non-compact Riemannian manifolds

SHORT VERSION: Does the Meyers-Serrin theorem hold on complete, non-compact Riemannian manifolds, i.e. $W^{k,p}(M) = H^{k,p}(M)$? My guess is that this holds for the special case $k=1$ (and all $p\geq ...

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votes

**0**answers

21 views

### Article Using Kullback Leibler Divergence to Measure Divergence of Observation from Distribution

I am currently attempting to compare an observed distribution to a theoretical distribution, and my current approach is to normalize the two and find the Kullback Leibler Divergence. I am beginning to ...

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votes

**1**answer

134 views

### A Combinatorial Identity Involving Characters of $S_n$ (Reference Request?)

It is a well-known exercise that $C_n = \chi_{(n,n)}(1)=\chi_{(n,n)}^{1^n}$ where $C_n$ is the $n$th Catalan number and $\chi_{(n,n)}^{1^n}$ is the character of the irrep $(n,n)$ on conjugacy class $1^...

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votes

**1**answer

193 views

### Numbers divisible only by primes of the form 4k+1

Let $A(N)$ denote the number of positive integers $n\le N$ composed of prime numbers $p\equiv 1\pmod 4$ only. Is there an asymptotic formula for $A(N)$ (as $N$ tends to infinity)?

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votes

**1**answer

335 views

### Higgs fields whose determinant have only simple zeros

Is the following property true for every stable holomorphic bundle of rank 2 with trivial determinant on a compact Riemann surface:
The space of trace-free Higgs fields, whose determinant have only ...

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votes

**16**answers

8k views

### Examples of algorithms requiring deep mathematics to prove correctness

I am looking for examples of algorithms for which the proof of correctness requires deep mathematics ( far beyond what is covered in a normal computer science course).
I hope this is not too broad.

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votes

**0**answers

149 views

### Regularity for a div-curl system

Let $Q = [0,1]^3$ be the unit cube in $ \mathbb{R}^3$, and let $U \subset Q$ be a simply-connected subdomain with smooth boundary. Suppose $g \: \colon Q \to \mathbb{R}^3$ is a non-negative smooth ...

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votes

**0**answers

176 views

### Equivalence of algebraic and topological monodromy representations?

Does anyone know of a reference for the following fact?
Let $M_g$ denote the moduli stack of genus g curves, let $A_g$ denote the moduli stack of abelian varieties, and let $U_g \rightarrow A_g$ ...

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votes

**0**answers

29 views

### First eigenvalue for strictly convex domains

Let $M^n$ be a compact Riemannian manifold with boundary, suppose 1). $Ric(M)\ge (n-1)$ and 2). the principle curvatures of the boundary is bounded from below by $h\ge 0$. Is there any results on the ...

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votes

**0**answers

36 views

### Fisher metric for shift-invariant probabilities

I'm just discovering what seems to be the tremendous heuristic value of the (century-old, more or less) canonical Riemannian metric (Fisher metric) on the $n$-dimensional simplex $\Sigma_n:=\{(p_i)_{i=...

**5**

votes

**1**answer

234 views

### How often does a pair of linear maps generate a Zariski-dense subgroup of $GL(d,\mathbb{R})$?

I am an analyst working on a number of problems which in some way relate to random matrix products. In this context I frequently find that the analytic properties I am interested in depend in some way ...