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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4
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0answers
414 views

Does this category has a computable algebraic K-theory

This question is inspired by this post. I decided to make some order in my very limited knowledge in algebraic K-theory. Let me start with the commutative case: $R$ is a commutative ring. $Perf(R)$ ...
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82 views

Character degrees in induced blocks

Let $G$ be a finite group and $U\leq H\leq G$ a chain of subgroups. Presume that $p$ is a prime dividing the order of $U$. Suppose that $b_1$ is a $p$-block of $U$ and $b_2$ a $p$-block which is ...
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141 views

Infinite composition of continuous functions

Let $f_n:\mathbb{R}\rightarrow \mathbb{R}$ be a sequence of functions and define $F_n:= f_n\circ \dots\circ f_1$. Then $F_n$ is continuous. However, the pointwise limit need not be (consider Mateusz'...
4
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1answer
205 views

Arthur's Simple Trace Formula

In Deligne–Kazhdan–Vigneras's "Représentations des groupes réductifs sur un corps local," they use the Simple Trace Formula to prove cases of the local Jacquet–Langlands correspondence ...
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0answers
61 views

Counting unions of unlabelled connected graphs

My question can be stated as follows: let $X$ be a hereditary family of unlabelled graphs closed under disjoint unions. Suppose we know, for each $n$, the number $c_n$ of connected graphs in X on $n$ ...
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118 views

Checking existence of proofs of fixed length

This question is a continuation of a related previous question (check here). Let $\mathcal{L}$ be a recursive first-order theory with the Hilbert-Ackerman's proof calculus, and such that the ...
2
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1answer
191 views

Are orbit polytopes of rotation subgroup of Coxeter group combinatorially equivalent?

Suppose that $G\subset O(d)$ is a finite reflection (finite Coxeter) group. For any $v\in \mathbb{R}^d$ which is not fixed by any non-trivial $g\in G$, one can consider the orbit polytope (Coxeter) ...
5
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2answers
315 views

Reference on Fourier analysis on compact groups

I am looking for a reference for Fourier analysis on compact (Lie) groups. The kind of theorems I would like the book to cover/do are the Peter-Weyl theorem, define Fourier transforms and use the ...
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0answers
83 views

bijection mapping a transversal to a transversal

The following must certainly be a standard result, so what I'm looking for is a reference, or the name of this theorem. I don't have any combinatorics books at my fingertips, but I could see this ...
3
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1answer
77 views

regularity of the solutions of Prandtl equation on the segment

Let $p(x)$ be a positive measurable function on $(-1,1)$. Consider the Prandtl equation $$ u(x)-\frac{p(x)}\pi \int_{-1}^1 \frac{u'(t)}{t-x}dt=p(x)h_0(x),\quad u(1)=u(-1)=0.\quad\quad(\star) $$ What ...
3
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1answer
124 views

On $(2,3)$-generation of finite simple classical groups

A group $G$ is called $(a,b)$-generated if $G=\langle x,y\rangle$ for some $x,y\in G$ with $|x|=a$ and $|y|=b$. I know some of the histories on this problem. For example, in this early paper in 1996 ...
13
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1answer
297 views

A royal road to Coulomb branches of 3D $\mathcal{N}=4$ gauge theories

So, I've been very interested recently with the developements of the (now mathematically precise) theory of Coulomb branches - in particular because of its recent applications on representation theory ...
3
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95 views

Trees of prescribed ordinal

My question is very imprecise, as I know very little about descriptive set theory. In a problem I am thinking about I have a family of well-founded trees (finite sequences on $\cup_n X^n$ closed under ...
3
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2answers
146 views

Searching for theorems characterizing when $O_p(G)$ is trivial / non-trivial

Let $G$ be a finite group. Let $p$ be a prime. Let $O_p(G)$ be the $p$-core of $G$. Are there any theorems known saying something like $O_p(G)$ is trivial, if and only if ... and $O_p(G)$ is non-...
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24 views

Bounding Greens function matrix elements in terms of the diagonal elements

Consider the Hilbert space $l^2( \mathbb{Z}^2)$ and suppose that I have a unitary band matrix. I.e. $ \langle e_j , U {e_k} \rangle = 0 $ for say $\vert \vert j-k \vert \vert > 2 $ (in say taxi-cap ...
3
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43 views

How are these “Voronoi-dual” configurations called?

If $\mathscr P\subset \mathbb R^d$ is a discrete point configuration, take the Voronoi diagram of $\mathscr P$ and call $\mathscr P'$ the vertices of that diagram. I would like to know if ...
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187 views

Bijective proof of a combinatorial identity: $\sum\limits_{k=0}^n\binom nk^2 \binom k{n-m}=\binom nm \binom{n+m}m$

Identity \begin{equation} \sum_{k=0}^n\binom nk^2 \binom k{n-m}=\binom nm \binom{n+m}m \tag{1} \end{equation} was used in an answer here. As shown in that answer, (1) easily reduces to \begin{...
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57 views

When is the Kochen-Stone inequality an equality?

The Kochen-Stone theorem says that if $A_n$ is sequence of events with $\sum_{i=1}^{\infty} P(A_i) = \infty$, then: $$ P(A_n \mbox{ i.o.}) \ge \limsup_{n \to \infty} \frac{(\sum_{i=1}^nP(A_i))^2}{\...
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324 views

Reference for a real algebraic geometry problem [migrated]

Disclaimer: I am not a mathematician by training. I encountered the following problem in my research. Assume that I have $N$ real variables $x_1, x_2, \dots, x_N$. I am given $N$ homogeneous ...
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83 views

How can I find this unpublished manuscript?

I am looking for an unpublished manuscript that has been cited many times. Its title is "Global smooth solutions for the second-order quasilinear wave equations with the first order dissipation&...
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1answer
44 views

Relation between random graph models $G^{(B)}_{n,m}$ and $G_{n,m}$

In Frieze, Alan; Karoński, Michał, Introduction to random graphs, in Section 1.3 Pseudo-Graphs, there is a model of random multi-graphs, which is denoted as $\mathbb{G}^{(B)}_{n,m}$. Def. A random ...
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0answers
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Cheeger constant of truncated hypercube

Look at the $d$-dimensional hypercube and truncate it. This means one replaces each vertex by a cycle (of length $d$) in such a way the the new graph is 3-regular. Question 1: What is the asymptotic ...
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1answer
103 views

Question on countably homogeneous structures

A homogeneous structure is a countable first order structure $M$ over a relational language such that any isomorphism between finite substructures of $M$ can be extended to an automorphism of $M$. ...
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Algebras for products or limits of monads

If a category $C$ has limits of a certain type, then the category of monads on $C$ has the same type of limits, and these limits are computed "levelwise" (i.e. are preserved by the forgetful ...
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1answer
141 views

Convexity at a point and Jensen inequality

I am looking for a reference for the following claim: Let $\phi:\mathbb (a,b) \to \mathbb R$ be a continuous function, and let $c \in (a,b)$ be fixed. Suppose that "$\phi$ is convex at $c$". ...
6
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1answer
335 views

Who was Bickart?

The term "Bickart points" is often used for the foci of the Steiner circumellipse of a triangle. Who was Bickart, and what was the first publication to use the term?
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3answers
528 views

Is the number of solutions of $\phi(x)=n!$ bounded? If yes, what is its bound?

Pillai showed in 1929 that the function $A(n)$ giving the number solutions of the equation $\phi(x)=n$ is unbounded in (S. Pillai, On some functions connected with $\varphi(n)$, Bull. Amer. Math. Soc. ...
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1answer
111 views

Equivalence generated by Jacobian minors

Let $f,g:\mathbb{R}^m \to \mathbb{R}^n$ be two smooth functions and let $k$ be a strictly positive integer. Write $f \sim_k g$ if at each point in the domain, the determinants of all $k \times k$ ...
4
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2answers
482 views

Origin and variations of problem on $4xy-x-y$ being square

One of the forms in which the Diophantine equation in question can be found in the literature is this: Solve the equation \begin{eqnarray}z^{2} = 4xy-x-y \qquad \qquad (\ast)\end{eqnarray} in ...
6
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1answer
126 views

Does $\pi_k(M)\neq 0$ implies $\operatorname{ind}(\gamma) < k$?

Cross post from MSE. and sorry if this is an obvious question. Here is a line of proof of Theorem 1.15 from Ricci Flow and the Sphere Theorem by Simon Brendle Let us fix two points $p, q \in M$ such ...
6
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1answer
224 views

Algebras Morita equivalent with the Weyl Algebra and its smash products with a finite group

My question os motivated, naturally, by the problem of classifying symplectic reflection algebras up to Morita equivalence (a classical reference for rational Cherednik algebras is Y. Berest, P. ...
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99 views

Gauss curvature derived from unit normal vector [migrated]

I want to know more about the differential geometry of surfaces, especially Gaussian curvature. Obviously, we can get the mean curvature of a surface from the divergence of the unit normal vector of ...
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0answers
36 views

Reference request : Convergence of radial basis function interpolation or spline interpolation as points become dense, for a continuous function

Is there any proof for this. Kindly request a reference in case available or any related documents towards this. PS : I am specifically interested in the case of scattered data (irregularly placed), ...
3
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1answer
51 views

$AC^p$ curves and pointwise metric speed in abstract metric spaces?

For a fixed "reasonable" metric space $(X,d)$ (say complete, separable, whatever is needed...), a curve $\gamma:[0,1]\to X$ is said to be $AC^p(0,1)$ (absolutely continuous) if $$ d(\gamma(s)...
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0answers
68 views

Is there any good reference on the Bayesian view that can be helpful for reading papers on the number theory using heuristic arguments?

Nowadays there are many papers on the number theory using heuristics. I have read some of them. But I have no clear understanding of the Bayesian Probability(subjective probability). The concept of ...
7
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3answers
304 views

Homotopy group action and equivariant cohomology theories

Many of the introductory notes on generalized equivariant cohomology theories assume that one is working over the category of $G$-spaces or $G$-spectra. However, one thing that concerns me is that the ...
4
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1answer
62 views

Permanent of a Kronecker product of matrices

It is well known that $\det(A \otimes B) = \det(A)^m \det(B)^n$ when $A$ and $B$ are square matrices of size $n$ and $m$ where $\otimes$ denotes the Kronecker product. Question: Is there a similar ...
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0answers
47 views

Optimal control of nonlinear harmonic oscillator

Consider the ODE $$ \begin{cases} x''(t) + \sin (x(t)) = u(t) \\ x(0)=x_0\\ x'(0)= x_1 \end{cases} $$ and the problem of minimizing $$J(u) = \int_0^T |x(t) - \bar x|^2 dt + \int_0^T u^2(t) dt$$ for $...
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2answers
562 views

Élie Cartan's paper “Les groupes réels simples, finis et continus” of 1914

Question 1. Does Élie Cartan's paper Les groupes réels simples, finis et continus, Ann. Sci. École Norm. Sup. (3) 31 (1914), 263–355 contain a classification of $\Bbb C$-linear involutions of simple ...
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0answers
122 views

On classifying groups of order $p^5$

Can someone suggest me some source where the author has classified all non-isomorphic groups of order $p^5$ ? I need complete classification (not upto isoclinism), and also in finitely presented form ....
3
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2answers
144 views

Convergence of conditional measures for a convergent sequence of probabilities whose projection is constant

Setting Suppose $\mu_n$ is a sequence of probability measures on $[0,1]\times [0,1]$ converging to a limit probability $\mu$ meaning that $$ \lim_{n\to+\infty}\int f(x,y)d\mu_n(x,y) = \int f(x,y)d\mu(...
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0answers
136 views

Pseudoreflection groups in affine varieties

Suppose $\mathsf{k}$ is an algebraically closed field of zero characteristic. Chevalley-Shephard-Todd (C-S-T) Theorem in one of its equivalent versions is the following result: (C-S-T): Let $G$ be a ...
6
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0answers
163 views

When did the main conjecture in Vinogradov's mean value theorem first appear in literature?

Recently I was asked about the history of Vinogradov's mean value theorem that I was hoping someone here could clarify. Let me first start with some terminology. Let $J_{s, k}(X)$ be the number of $2s$...
11
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1answer
245 views

Good overviews on $\phi^{4}$-field theory?

I'm looking for nice overviews on $\phi^{4}$-field theory from the mathematical-physics point of view. To be a little more specific, here are some topics I'd like to read about: (1) What are the ...
8
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1answer
265 views

Minimum cardinality of a cofinal collection of countable subsets of a set

Setup Let $X$ be a set of cardinality $\kappa\geq \aleph_0$. Edit: Based on Todd Eisworth's suggestion: What is the minimum cardinality of a collection $\hat{X}$ of countable subsets of $X$ such that ...
3
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1answer
71 views

Directed graph minor theorems

In proving the graph minor theorem, Robertson and Seymour proved a stronger statement, namely that the directed graph minor theorem is true, using the definition A directed graph is a minor of ...
12
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1answer
237 views

Rational homotopy invariance of algebraic $K$-theory

Suppose that $R\to S$ is a 1-connected morphism of connective structured ring spectra that induces an isomorphism on rational homotopy groups. Is the induced map of (Waldhausen) K-theory spectra $$ K(...
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1answer
100 views

Faithful representation of group of order $p^4$

In the (xi) group of the classification of groups of order $p^4$ given by W.Burnside in his book, "Theory of groups of finite order". The group ($\mathbb{Z}_{p^{2}}\rtimes \mathbb{Z}_{p^{}}) ...
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2answers
90 views

Properties of the total variation norm on space of totally finite measure (from Bogachev)

Let $(X,d)$ be a metric space, $\mathcal{B}$ the Borel $\sigma$-algebra on $X$, and $\mathcal{M}(X)$ the space of totally finite measures on $\mathcal{B}$. Let $\|\mu\|_{TV}$ be the total variation ...
6
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2answers
514 views

“Well-known fact” that every irreducible 3-manifold with non-empty boundary has an incompressible surface

I have seen in several sources that this results holds, however none of them included the proof. Does anyone know where I can find one? Also, it would be great if someone could provide me with a ...