Questions tagged [reference-request]

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

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Properties of differentiable functions on non-locally-bounded fields

I was reading some results on the structure of non-locally-bounded topological fields, and I was wondering what is known about differentiable functions on them. In particular, on the complex numbers ...
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Commuting algebra of a Zariski closure

Suppose that $\rho: G \to \text{GL}(V)$ is a rational representation, where $V$ is a finite dimensional complex vector space and $G$ is an algebraic group. Suppose that $H$ is a subgroup of $G$ (not ...
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Asymptotics of the number of minimal strongly connected digraphs

Is anything known about the number of minimal strongly connected digraphs on $n$ labeled nodes? (``Minimal’’ meaning that on the deletion of any arc, strong connectivity is lost.) Some values are ...
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Basis for space of continuous, surjective monotone functions on $\mathbb{R}$

$\DeclareMathOperator\CM{CM}$ I recently came across Okhezin - Study of families of monotone continuous functions on Tychonoff spaces describing monotone functions on general topological spaces and I ...
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Positivity of mixed derivatives of log-density of a diffusion

Consider the scalar diffusion $X=(X_t)_{t\geq 0}$ defined by the time-homogeneous SDE $$\mathrm{d}X_t \ = \ b(X_t,t)\mathrm{d}t \ + \ \sigma(X_t,t)\mathrm{d}B_t$$ with $b$, $\sigma$ smooth and $B=(B_t)...
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1answer
96 views

How to mathematically characterize a feedback loop in odes?

I have a biological system that exhibits a feedback type of behavior. The diagram is a schematic of the system of odes. In this system, the total amount of $x_1, x_2, x_3$ is conserved; however, there ...
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What is the explicit relationship between the shape parameters and the holonomy of a hyperbolic ideal triangulation?

Let $K$ be a hyperbolic knot in $S^3$. One way to describe the hyperbolic structure is to give a discrete, faithful representation $\pi_1(S^3 \setminus K) \to \operatorname{PSL}_2(\mathbb C)$, the ...
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56 views

On the dual version of an isomorphism of Spectral sequence term (from Cartan and Eilenberg)

So I'm trying to take spectral sequences as a black box for application in Commutative Algebra and I admit that I haven't really gone through (or understand) all the proofs of all the isomorphisms ...
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Suggestion for framing a course in Representation theory + Spectral graph theory

I am going to give a course in spectral graph theory to graduate students. I want to learn and teach the connection between the spectral graph theory and the representation theory of finite groups. I ...
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1answer
112 views

Explicit computation of spinor norm

I've asked this on math.stackexchange, unsuccessfully. I hope this question is appropriate for mathoverflow. Let $V$ be a finite-dimensional vector space over a field $K$ with $\operatorname{char}K\...
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When is $G(\mathbb{Z}_p)$ topologically generated by maximal tori?

Let $G/\mathbb{Z}_p$ be a connected reductive group, and let $T_1, \cdots, T_n \subset G_{\mathbb{Q}_p}$ be maximal tori. Suppose that we have precisely one maximal torus in each $G(\mathbb{Q}_p)$-...
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124 views

Does a generalization of Tietze's extension theorem hold for set-valued functions?

Let $X$ be a normal topological space. Tietze's extension theorem says that if $A \subset X$ is closed, then a continuous function $f: A \to \mathbb R^n$ can be extended to a continuous function whose ...
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2answers
110 views

Reference request: excess normal bundle and derived pullback

Consider a fiber square $\require{AMScd}$ \begin{CD} X' @>i'>> Y'\\ @V g V V @VV f V\\ X @>>i> Y, \end{CD} where $i$ and $i'$ are regular immersions, and consider the ...
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1answer
94 views

Quiver and relations of Schur algebras

Assume that the Schur algebra $S(n,r)$ with $n \geq r$ is not representation-finite. Question: For which $n$, $r$ is the quiver and relations of the blocks of $S(n, r)$ explicitly known? I just ...
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additivity of trace with respect to short exact sequences

Let $\mathcal{C}$ be an abelian rigid symmetric monoidal category over a field $K$. Assume that the endomorphism ring of the tensor unit in $\mathcal{C}$ is $K$. If $X$ is an object in $\mathcal{C}$ ...
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$L^p$ compactness for a sequence of functions from compactness of cut-off

Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
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Mori's cone theorem

I need the proof (reference) of Mori’s theorem about this implication : Let $X$ be a projective complex manifold. If $X$ contains no rational curves, then $K_K$ is nef.
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1answer
127 views

Literature request: Schatten class difference of semigroups

Let $\mathcal{H}$ be a Hilbert space and $A,B$ two operators on it (not necessarily self-adjoint) such that $A, A+B$ are generators of strongly continuous one parameter semigroups $e^{-tA},e^{-t(A+B)}$...
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1answer
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Survey of recent developments of the Gelfand-Kirillov dimension

It is almost two decades since the now classical books by McConnell and Robinson's [ Noncommutative Noetherian rings. With the cooperation of L. W. Small. Revised edition. Graduate Studies in ...
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Conjecture of van der Holst and Pendavingh related to bound for Colin de Verdière invariant

In their 2009 paper (“On a graph property generalizing planarity and flatness”. In: Combinatorica 29.3 (May 2009), pp. 337–361. issn: 1439-6912. doi: 10.1007/s00493-009-2219-6.), van der Holst and ...
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Grothendieck dessins d'enfants - current surveys or text you can recommend?

I was recommended this forum to be the leading site for algebraic geometry, so I would like to ask you a question about Grothendieck dessins d´enfants. My background is in maps on surfaces (graph ...
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73 views

Generalization of elementary symmetric polynomials

The elementary symmetric polynomials (ESPs) are defined as - \begin{align*} E_{1}^{1} &= X_1, \\ E_{1}^{2} &= X_1 + X_2, \\ E_{2}^{2} &= X_1 X_2, \\ E_{2}^{3} &= X_1 X_2 + X_1 X_3 + ...
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155 views

When is an increasing union a colimit?

Let's consider a diagram $\Phi: \lambda \to \mathcal{T}_*$ $$ X_0 \to X_1 \to \cdots \to X_\xi \to X_{\xi+1} \to \cdots $$ of pointed spaces, indexed by some ordinal $\lambda$, in which each $X_\xi$ ...
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1answer
67 views

Relation between the eigenvalues of a block matrix and the eigenvalues of its diagonal blocks

Consider the $(m+n) \times (m+n)$ block matrix $$M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$$ I need references where they are talking about the relation between the eigenvalues of $M$ ...
4
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1answer
195 views

Latest progress on Tarski numbers

Two questions, the first: What is the smallest non negative integer that we do not know yet is the Tarski number of a group? The second question is the same as in the title: What is the latest ...
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34 views

Free resolutions of universal enveloping algebras for simple, finite dimensional Lie algebras

I'm currently studying Anick's resolution on the context of universal enveloping algebras for certain Lie algebras, namely some of the smallest cases: $A_1,A_2,A_3,B_2,G_2$, and so on. What are some ...
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108 views

Reference request for the explicit formula for $\sum_{n\leq x} \Lambda(n)n^{-s}$

Denote by $\Lambda(n)$ th e von Mangoldt function, which is equal to $\log p$ if $p\geq 2$ is a prime, and $0$ otherwise. Let $\rho$ denote a complex zero of the Riemann $\zeta$-function. If i recall ...
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47 views

Angular excitations and Schrodinger operators with radial potential in N-dimensions

Can someone please explain the following in mathematical language? "First of all, angular excitations only push the energy up, never down, so it is enough to analyze spherically symmetric s-waves....
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56 views

Reference for the $3$-series of an elliptic formal group law

The $3$-series of the formal group law of the Weierstrass curve $y^2 = x^3 + a_2 x^2 + a_4 x$ begins $$ [3](z) = 3 z - 8 a_2 z^3 + (24 a_2^2 - 96 a_4) z^5 - (72 a_2^3 - 288 a_2 a_4) z^7 + (216 a_2^4 - ...
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1answer
88 views

Computation of the Lusztig a-function

See for example https://www.sciencedirect.com/science/article/pii/0021869387901542 for the definition of the Lusztig a-function. Question 1: Is there a table for the values of Lusztig's a-function ...
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1answer
491 views

Learning from unsuccessful attempts at the Poincaré conjecture

This is reposted from MSE, but perhaps it is more appropriate to post it here. Let me know if I'm wrong. Among the many unsuccessful attempts at solving the Poincaré conjecture, I'm wondering if there ...
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106 views

Proof of a 'well-known' result of Shimura on periods of modular forms

It is often noted in the literature that there are certain complex periods that allow one to normalize the modular symbol associated to a modular form in such a way that its coefficients are algebraic....
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2answers
232 views

Conjecture about minimal number of edge crossings in complete bipartite graphs

I am interested in the status of the conjecture about the minimum number of edge crossings $cr(K_{m,n})$ in a drawing of the complete bipartite graph $K_{m,n}$. The Wikipedia article https://en....
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2answers
163 views

Reference for Cochran-Orr-Teichner's filtrations on knot concordance

I would be interested in recommendations for easily accessible/somehow simply readable texts to Cochran-Orr-Teichner's filtrations on knot concordance: Tim D. Cochran, Kent E. Orr, and Peter Teichner....
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12 views

Gradient-descent like scheme for composition of functions with separate objectives

Let $\{x_n\}_{n=1}^N$ be points in $\mathbb{R}^d$, $\{y_n\}_{n=1}^N,\{z_n\}_{n=1}^N$ be points in $\mathbb{R}^D$ and let $f:\mathbb{R}^d\times \mathbb{R}^k\rightarrow \mathbb{R}^d$, $g:\mathbb{R}^d\...
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67 views

Poincaré Recurrence Theorem for flows

Someone knows a book that have the proof of the Poincaré Recurrence Theorem for flows when the set have finite volume and when don't?
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82 views

Fourier mode decomposition and eigenvalues of Schroedinger operators with radial potential in N-dimensions

In the study the stability of minimal hipersurfaces $\Sigma \subset \mathbb{R}^{N+1}$ one is lead to study the Morse index of a Schroedinger operator $J := \Delta_g + |A|^2$ (usually called Jacobi ...
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40 views

Linearly dependent points and the uniform position theorem

One proof of the uniform position theorem (as stated in p. 109 or p. 113 in Section III.1 of "Geometry of Algebraic Curves") uses a monodromy argument. While this gives us something even ...
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2answers
139 views

Schur Weyl duality for the supergroup $\text{GL}(m|n)$

Let $G$ be the supergroup $\text{GL}(m|n)$. It has a tautological representation $V= \mathbb{C}^{m|n}$. For every natural number $d$ we have a natural map $$\Phi_d:\mathbb{C} S_d\to \text{End}_G(V^{\...
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32 views

Prove compactness of approximate (finite differences) solutions of heat equation

Let $\{u^h\}$ be the family of solutions of the discretized heat equations on the interval $[0,1]$ (uniform grid of size $h>0$). $$\begin{cases} u^h_t - \Delta_hu^h = 0\\ u^h(t,0) = u^h(t,1) = 0 \\...
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65 views

Reference request: Cohen-Macaulay representations

Question: Are there good references of Cohen-Macaulay (CM) representation theory with concrete calculations of examples. In particular, I look for ones which contatin the classification of CM modules ...
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1answer
430 views

Identity involving the probability that a random walk stays below a curve

I'm looking for a direct proof of the following identity: Let $W_n$ be a simple random walk with $W_0=0$. For all $x>0$ we have $$ \lim _{N\to \infty} \sqrt{N} \cdot \mathbb P \Big( \forall n \le ...
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3answers
174 views

Diophantine equation of a factorial type

I'm interested in nontrivial solutions of Diophantine equations of the type $$a^2b^3 = \frac{c!}{(c-k)!} $$ For various values of k fixed, and of course $a,b,c \in \mathbb{Z^+}$ Does anyone have any ...
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0answers
47 views

Reference request - random regular graphs vs random graphs w/ degree sequence

There are some properties that are easily studied for random d-regular graphs, but that are very hard to extend to random graphs with a given degree sequence (e.g. whether a graph is w.h.p. ...
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0answers
58 views

Dislocations and Random Matrix Theory

Does anyone have a good reference book that works as a good starting point for an analyst to learn about the connection between Random Matrix Theory and Dislocations? Thank you for your help. By ...
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0answers
24 views

Hitting Time-Analogue for Chaotic Systems

Let a topologically mixing dynamical map $f$ on $\mathbb{R}^n$, and define the dynamical system with initial value $x \in \mathbb{R}^n$ by $$ x_{t+1}^x = f(x_t^x),\, x_0^x=x . $$ Fix $y\in\mathbb{R}^n$...
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1answer
173 views

Blow up the diagonal of a symmetric product space

Consider the complex blow-up of the diagonal $\triangle\subset Sym^2(\mathbb{C}^n)$. It is known that the blown-up space is smooth when $n=1,2$. I was wondering if that is still smooth for $n\geq3$? ...
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1answer
55 views

Reference for Function-Valued Random Variables?

Question: Are there any good references for facts about function-valued random variables? In particular for facts like the following: Let $X$ be a topological space, $Y$ be a random variable with ...
5
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1answer
180 views

A graph similar to the Bruhat graph, what is it called?

The weak Bruhat graph (or 1-skeleton of the permutohedron) $B_n$ can be constructed as follows: the vertices of $B_n$ are the permutations of the tuple $(1,...,n)$, two are joined by an edge, if they ...
3
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1answer
260 views

Reference to a Classical Regularity Theorem

(Edited) I need a reference to the following result: If $u \in H^2(B_1^+) \cap {\rm Lip}(B_1^+)$ satisfies \begin{cases} {\rm div}(F(x,u,\nabla u)) = F_0(x,u,\nabla u) \quad & {\rm in} \ B_1^+ ...

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