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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Jacobson-style Galois theory on perfect closure

Promoted from stack.exchange since I received no response: Jacobson developed a 'Galois' correspondence for purely inseperable extensions of exponent 1 (only consisting of pth roots) $K/k$, where he ...
Oddly Asymmetric's user avatar
0 votes
1 answer
360 views

First and last order statistics and their ratio for $\chi^2_{m}$ random samples

Let $X_1, \dots, X_n \sim_{iid} \chi^2_{m}$ be a random sample from a chi-squared distribution with $m$ degrees of freedom (d.f.). I was wondering if there's any known result for the order statistics $...
Learning math's user avatar
1 vote
1 answer
313 views

$L^p$ compactness for a sequence of functions from compactness of product with 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(...
Zac's user avatar
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1 answer
113 views

Restricted partition problem into parts with a given set of prime factors

I need a reference for the following question: Let $\mathcal{P}$ be a finite set of $k$ primes and let $f(n)$ be the number of partitions of $n$ into parts whose prime factors are restricted to the ...
Vlad Matei's user avatar
7 votes
0 answers
431 views

Reference for the multiprojective Nullstellensatz?

Didn't get a single comment in over a day at math.SE, so maybe the question is more appropriate here. I'm looking for a reference to a generalization of Hilbert's Nullstellensatz to the ...
Igor Makhlin's user avatar
  • 3,493
8 votes
0 answers
380 views

Reference for sets of locally finite perimeter on Riemannian manifolds

I am looking for a reasonably complete reference for Ennio De Giorgi's theory of sets of locally finite perimeter (also christened by him as Caccioppoli sets, after Renato Caccioppoli's pioneering ...
Pedro Lauridsen Ribeiro's user avatar
5 votes
1 answer
178 views

Hadwiger number of a graph: Question about the original article from 1943

I am analyzing Hadwiger's original article (Hadwiger, Hugo (1943), "Uber eine Klassifikation der Streckenkomplexe", Vierteljschr. Naturforsch. Ges. Zurich, 88: 133–143) for my work related ...
LawrenceMatthewS.'s user avatar
1 vote
1 answer
96 views

Reference to log-transition-density of a diffusion process

Consider the scalar diffusion $X=(X_t)_{t\geq 0}$ given by $$\mathrm{d}X_t \ = \ b(X_t)\,\mathrm{d}t \ + \ \sigma(X_t)\,\mathrm{d}B_t, \qquad X_0=\xi,$$ with $b$, $\sigma$ smooth, $\xi$ absolutely ...
cts12's user avatar
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1 vote
0 answers
25 views

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 ...
dog_mu's user avatar
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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 ...
David Galvin's user avatar
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-3 votes
1 answer
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Basis for space of continuous, surjective monotone functions on $\mathbb{R}$ [closed]

$\DeclareMathOperator\CM{CM}$ I recently came across Okhezin - Study of families of monotone continuous functions on Tychonoff spaces describing monotone functions on general topological spaces and I ...
ABIM's user avatar
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4 votes
1 answer
2k 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 ...
Paichu's user avatar
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0 answers
87 views

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 ...
Calvin McPhail-Snyder's user avatar
1 vote
0 answers
126 views

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

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 ...
sdey's user avatar
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12 votes
2 answers
608 views

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 ...
6 votes
1 answer
1k 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\...
lisyarus's user avatar
  • 165
5 votes
0 answers
163 views

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 $G(\mathbb{Q}_p)$-...
Pol van Hoften's user avatar
9 votes
0 answers
250 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 ...
aduh's user avatar
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3 votes
2 answers
336 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 ...
Nick Addington's user avatar
6 votes
1 answer
192 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 ...
Mare's user avatar
  • 25.8k
4 votes
0 answers
174 views

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 $\mathcal{C}$ ...
Ehud Meir's user avatar
  • 4,969
1 vote
1 answer
409 views

$L^p$ 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(...
Zac's user avatar
  • 161
6 votes
1 answer
490 views

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.
Kamel's user avatar
  • 73
9 votes
1 answer
200 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 $e^{-tA},e^{-t(A+B)}$...
folouer of kaklas's user avatar
6 votes
2 answers
383 views

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 ...
jg1896's user avatar
  • 2,683
9 votes
2 answers
236 views

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 ...
soerenssen's user avatar
15 votes
0 answers
415 views

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 ...
0 votes
0 answers
173 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 + ...
twofiveone's user avatar
9 votes
0 answers
352 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 $X_\xi$ ...
Jeff Strom's user avatar
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1 vote
1 answer
4k 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 $M$ ...
GA316's user avatar
  • 1,219
4 votes
1 answer
238 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 ...
Meisam Soleimani Malekan's user avatar
2 votes
0 answers
53 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 ...
Hilario Fernandes's user avatar
7 votes
2 answers
602 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 ...
Q_p's user avatar
  • 824
1 vote
0 answers
125 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....
user2002's user avatar
  • 181
2 votes
0 answers
139 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 - ...
John Rognes's user avatar
  • 8,692
4 votes
1 answer
191 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 ...
Mare's user avatar
  • 25.8k
9 votes
1 answer
655 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 ...
Paul Cusson's user avatar
  • 1,735
9 votes
1 answer
487 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....
Arbutus's user avatar
  • 335
5 votes
2 answers
370 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....
Ruth-NO's user avatar
  • 125
5 votes
2 answers
218 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....
user avatar
2 votes
0 answers
203 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?
peyoterain's user avatar
2 votes
0 answers
209 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 ...
user2002's user avatar
  • 181
2 votes
0 answers
68 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 ...
modnar's user avatar
  • 501
9 votes
2 answers
286 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^{\...
Ehud Meir's user avatar
  • 4,969
13 votes
1 answer
686 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 ...
Dor's user avatar
  • 723
2 votes
3 answers
247 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 ...
G G's user avatar
  • 41
3 votes
1 answer
149 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. ...
DJA's user avatar
  • 425
1 vote
0 answers
99 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 ...
spaceman's user avatar
  • 575
4 votes
1 answer
346 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 $n\geq3$? ...
Eric Yuan's user avatar
3 votes
1 answer
294 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 ...
Daron's user avatar
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