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Questions tagged [reference-request]

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

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Integration theory for functions and values with values in topological rings

I am curious whether somebody ever tried to generalize the classical theory of Lebesgue integral to functions and measures with values in Hausdorff topological rings. The generalization of a measure ...
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33 views

When is information maximized? [closed]

Consider a system $S$ with rule set $R$. Under what $R$ is information maximized? It's a simple question, theres nothing to clarify.
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Formula expressing symmetric polynomials of eigenvalues as sum of determinants

The trace of a matrix is the sum of the eigenvalues and the determinant is the product of the eigenvalues. The fundamental theorem of symmetric polynomials says that we can write any symmetric ...
3
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50 views

On Ext-duals of injective modules for commutative rings

Let $R$ be a commutative noetherian ring and $I=E(R/p)$ the injective hull of the module $R/p$ for a prime ideal $p$. Question: Is there a (more) explicit description of the $R$-modules $Ext_R^i(I,R)$...
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2answers
709 views

Reference request: the theory of currents

I am a graduate student and want to study the theory of currents. What is a good reference for a beginner? I should be familiar with the theory of distributions or generalized functions on $\mathbb R^...
4
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1answer
131 views

A rather non-$F_\sigma$ Borel set

I asked this question at MSE a week ago, but received no answer, so I cross-post it here. I obtained a negative answer to this MSE question provided each metric space $X$ such that $|X|=\frak c$ and ...
5
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146 views

Reference for Grothendieck's theorem on representation of unramified functors

In the Exposé 294 of the Bourbaki Seminar of the year 1964-1965, Murre gives an outline of proof of a theorem of Grothendieck giving necessary and sufficient conditions of representability by a scheme ...
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72 views

A question related to the union-closed sets conjecture

Let $f(n)$ denote the maximum possible cardinality of a collection $\mathcal F$ of nonempty sets which is closed under unions ($X,Y\in\mathcal F\implies X\cup Y\in\mathcal F$) and is such that no ...
7
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229 views

Do generalizations of the identity $\sum_{n=k+2}^{\infty} \binom{n-1}{k} (\zeta(n) -1) = 1 $ exist?

On p. 263 of Borwein's paper entitled “Computational Strategies for the Riemann zeta function”, the following identity is stated: $$\sum_{n=k+2}^{\infty} \binom{n-1}{k} (\zeta(n) -1) =1 . \qquad \...
8
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1answer
249 views

Generalising the Union-closed sets conjecture from lattice to a larger class of posets

(edit: I decided to simplify the question and only pose it for bounded posets first) The Union-closed sets conjecture is equivalent for lattices P to: There exists a join-irreducible element $a$ with ...
2
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1answer
109 views

Do identities exist for the binomial series $\sum_{k=m+1}^{n+1} \binom{k}{m} \binom{n+1}{k-1} $?

While examining the product of two upper triangular matrices, I've found that the $(m,n)$'th entry of the resulting matrix amounts to: $$\sum_{k=m+1}^{n+1} \binom{k}{m} \binom{n+1}{k-1} $$ when $n \...
5
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1answer
97 views

Reference on internal categories and externalization

I'm looking for a reference on internal categories and externalization of internally defined notions. The nlab has a stub on externalization (more details are available under small fibration) and the ...
2
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1answer
129 views

Prove / disprove: If $1 \le n < N$ and $A$ is an $N \times n$ matrix with iid from $\mathcal N(0,1)$, then $s_\min(A) \ge c\sqrt{N}$ w.p $1-2e^{-N}$

Let $1 \le n < N$ be integers and $A$ be a random $N\times n$ matrix with iid entries from $\mathcal N(0,1)$. This paper (Rudelson and Vershynin) claims in the paragraph just before formula (3.4) ...
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Lax CD(K, $\infty)$ space in the sense of Sturm

In K.T. Sturm's "On the geometry of metric measure spaces. I", Definition 4.5, he introduces a "lax" version of the usual CD(K,$\infty$) lower bound. Namely, one allows for $\...
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1answer
53 views

Diagonal terms in the Kochen Stone inequality

In a paper in Lecture Notes in Mathematics vol. 1874, Yan states the Kochen-Stone theorem in the following form, where $A_n$ is a sequence of events such that $\sum_{n=1}^\infty P(A_n) = \infty$: $$ ...
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optimal abscissae and weights for Legendre expansion inversion

I have asked this question a while ago on MSE, without any response. I thought that perhaps someone here has a reference. I have the Legendre expansion of a suitable function $$f(x) = \sum_{n=0}^{\...
7
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4answers
197 views

Discretizing a line segment with pixels which satisfies the Pythagorean theorem

There are plenty of line drawing algorithms to discretize line segments using pixels. The Bresenham's algorithm gives a line where the number of pixels in the segment is the same as its width (in x-...
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1answer
66 views

Lower-bound on smallest singular-value of rectangular random matrix

Let $X$ be a random $N \times n$ matrix with iid entries from $\mathcal N(0, 1)$ and with $n/N =: \lambda(N,n) \le \lambda_0$, for some $\lambda_0 \in (0, 1)$. That is, $X$ is genuinely rectangular (...
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43 views

Approximating step functions using step functions

Let $f$ be any BV function over $\mathbb{T}$. Let the Fourier series partial sum be $S_n$ which is constructed using the first $n$ Fourier series coefficients. We know that $s_n \to f$ pointwise at ...
5
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1answer
145 views

Schwänzl and Vogt, Cofibration and fibration structures in enriched categories

In [Schwänzl and Vogt, Strong cofibrations and fibrations in enriched categories], the authors refer to an earlier preprint, [Schwänzl and Vogt, Cofibration and fibration structures in enriched ...
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45 views

Is Fourier dimension finitely stable?

Let $A,B$ be compact subsets of $\mathbb R$. Let $a=\mathrm{dim}_F(A)$, $b=\mathrm{dim}_F(B)$ be their Fourier dimensions, respectively. My questions are as follows: Is it true that $\mathrm{dim}_F(A\...
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Graph theory: Closed neighourhoods and generalized clustering coefficients

The neighbourhood of node $v$ in graph $G$ is the subgraph of $G$ induced by all vertices adjacent to $v$. The number of edges between neighbours divided by the number of pairs of neighbours is ...
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31 views

Sections of (affine) toric varieties by linear subspaces

I am looking for a reference that expands on the following statement: "Sections of toric varieties by linear subspaces defined by coordinates or differences of coordinates are binomial schemes.&...
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3answers
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Cardioid-looking curve, does it have a name?

The curve, given in polar coordinates as $r(\theta)=\sin(\theta)/\theta$ is plotted below. This is similar to the classical cardioid, but it is not the same curve (the curve above is not even ...
3
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1answer
64 views

“Classical” proof that maximal minors form a Grobner basis under diagonal term order

Let $R= k[x_{11} , x_{12} \dotsc , x_{nm}]$ denote the coordinate ring of a generic $n \times m$ matrix, $M$. It is well known that under the standard diagonal term order, the ideal of maximal minors ...
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1answer
115 views

Reference book for Galois extension [closed]

I need a reference for field extension and Galois extension (like an introduction) please. Thank you.
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116 views

Counting special paths on a certain rectangle integer grid (binary matrix)

Crossposting from MSE after getting no answers. The bounty on the MSE question is still open, but not for long. Be advised that the comments of the MSE question regard an obsolete version, and that ...
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0answers
82 views

invariant theory for non-polynomial functions (eg Hilbert spaces)

I am looking for references regarding the study of group invariant functions that are not polynomials. In particular, I have a (nice) group $G$ and I am interested in what can be said about the $G$-...
5
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1answer
175 views

Finite order automorpisms of affine Kac-Moody Lie algebras

It is known that for a finite order automorphism $\phi$ of a complex semisimple Lie algebra $L$, the fixed point subalgebra $L^{\phi}$ is a reductive Lie algebra and the centralizer of a Cartan ...
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4answers
514 views

Self-contained formalization of random variables?

I have not been able to find any formalization of random variables that supports construction of new random variables dependent on previously constructed ones. In what I have found, a random variable $...
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2answers
921 views

Roadmap to learning the classification of finite simple groups

I want to learn the classification of finite simple groups. But it is often commented that it is a theorem spanning tens of thousands of pages of research papers. So it is quite intimidating to an ...
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0answers
43 views

Reference request for invariance principles

In various places, an example being https://projecteuclid.org/download/pdf_1/euclid.aoap/1034625254, the authors consider a discrete-time process (real-valued, say) $(X_n)_{n \in \mathbb{N}}$, define ...
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47 views

Reference request: convergence of cadlag stochastic processes at $t=\infty$

Let $D\equiv D([0,\infty))$ be the space of cadlag functions (right continuous with left limits) on $[0,\infty)$. Consider a sequence of stochastic processes $\big(X^n\equiv (X^n(t))_{t\ge 0}\big)_{n\...
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2answers
670 views

Vector-Valued Stone-Weierstrass Theorem?

The standard statement of the Stone-Weierstrass theorem is: Let $X$ be compact Hausdorff topological space, and $\mathcal{A}$ a subalgebra of the continuous functions from $X$ to $\mathbb{R}$ which ...
0
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0answers
73 views

Reference: Irreducible components of the Steinberg variety are conormal bundles

The (partial) Springer resolution is defined as a map $\mu: T^*\mathcal{F} \to \mathcal{N}$, where $\mathcal{F}$ is the partial flag variety consisting of $n$-step partial flags of $\mathbb{C}^d$, and ...
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75 views

Reference for an analogue of Goursat's lemma for noncommutative rings?

The following question might be a duplicate but I couldn't find anything when I searched just now. (I also would not object hugely if the question gets moved to MSE except that I do not have an ...
1
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1answer
63 views

Terminology: Co-completion of Met?

In main-stream mathematical literature, the term metric space is reserved for $(X,d)$ where $X$ is a set and $d:X\times X\rightarrow [0,\infty)$ satisfies the usual properties of a metric. However, ...
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206 views

Reference request: Goldbach's conjecture [closed]

I am looking for an article on the following aspects related to the Goldbach's conjecture: The Goldbach function $g ( 2n ) $ represents the number of different ways in which $2n$ can be expressed as ...
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152 views

References about “monoidal fibrations” in $\infty$-category theory

$\newcommand{\cat}{\mathsf} \newcommand{\fun}{\mathrm{Fun}} \newcommand{\calg}{\mathrm{CAlg}}$ Let $\cat C^\otimes,\cat D^\otimes, \cat E^\otimes$ be symmetric monoidal $\infty$-categories, and $p^\...
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1answer
374 views

Source of infection on chessboard

I am looking for the original source of the following well known problem. Seven unit cells of a 8×8-chessboard are infected. In one time unit, the cells with at least two infected neighbors (having a ...
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0answers
25 views

Moduli of continuity and Wasserstein differentiability of functions between measures

Let $X=\mathbb{R}^n$; I am also interested in the general case $X$ is a metric space but for simplicity let's focus on Euclidean space. Let $\mathcal{P}(X)$ denote the space of Borel probability ...
2
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0answers
331 views

Reference request for Miyaoka's wrong proof of FLT [duplicate]

Before Wiles and Taylor's proof of Fermat's Last Theorem, Yoichi Miyaoka famously (and wrongly) claimed to have solved the problem, at least for sufficiently large prime exponents. However, i have ...
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336 views

Philosophical underpinnings of Grothendieck's construction of the Hilbert scheme

Long ago when I was in grad school I was told that Grothendieck's construction of the Hilbert scheme is rooted in two main technical points: Castelnuovo-Mumford regularity and Mumford flattening ...
7
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1answer
187 views

Pushouts and products in categories

This has to do with the "pushout-product" construction. In a category $\mathcal{C}$, suppose we have $C\gets A\to B$ with pushout $D$ and $Y\gets W\to X$ with pushout $Z$. Then we can form ...
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0answers
43 views

What is a random eigenfunction on the hyperbolic plane?

Is there an (invariant under isometries) notion of a random eigenfunction on the hyperbolic plane, for a given eigenvalue? It is a reference request because the answer is probably positive and I even ...
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0answers
40 views

Multivarate “RKHS” Examples

I've been reading about RKHSs and Hilbert spaces of functions these days a bit these days and I haven't yet come across an example of a hilbert space $H$ whose elements are all functions $f:\mathbb{R}^...
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88 views

Homogeneous metric surfaces

I am looking for a reference for this result. Let $S$ be a metric space such that it is homeomorphic to a two-dimensional manifold, it is 2-homogeneous: given two pairs of points $(x,y)$ and $(x',y')...
3
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2answers
118 views

Request for books/articles on random polynomials

Can somebody kindly recommend me a couple of introductory books/articles on random polynomials with clear expositions of fundamental results (like the distribution of roots, expected number of real ...
0
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0answers
54 views

Log-concave sequences and triangular arrays

A non-negative sequence $\{a_i\}$ is said to be log-concave if $a_i^2 \geq a_{i+1}\,a_{i-1}$ for all $i\geq 1$. I'm interested in investigating triangular arrays $\{a(n,k)\}_{n,k\geq 0}$ such that $\{\...
2
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1answer
113 views

Cusps of hyperbolic surfaces under finite covers

The following statement seems true, but I don't know a proof or a reference for it (and I would like one). Let $\Gamma< \operatorname{PSL}(2,\mathbb R)$ be a nonuniform lattice with one cusp. We ...

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