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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Explicit formula for Fibonacci numbers; compositions of $n$

A Fibonacci-type sequence is a sequence with two seed-values, $F_1$ and $F_2$, and which, for all $n>2$, abides by the recurrence relation $F_n = F_{n-1} + F_{n-2}$. If $F_1 = F_2 = s$, then the $n$...
2 votes
1 answer
59 views

Differentiability of the fixed points of a family of contraction maps

Given a general Banach space $B$ and a one-parameter family of contractions $F_t:B\to B$ which is defined for all $t \in (a,b)$. $F_t$ depends continuously on $t$ (in the sense $\lim_{\varepsilon\to 0}...
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5 votes
1 answer
139 views

Question about and good reference for Kahn and Markovic result

As far as I understand, the celebrated result of Kahn and Markovic about quasi-Fuchsian immersions of surfaces in hyperbolic 3-manifolds has the following corollary: Let $M$ be a compact hyperbolic $3$...
13 votes
1 answer
385 views

Applications of equivariant homotopy theory to representation theory

Equivariant homotopy theory focuses on spaces together with some group action on them. Jeroen van der Meer and Richard Wong have a paper where they use equivariant methods to compute the Picard group ...
5 votes
1 answer
222 views

Sum of three squares as class numbers and Waldspurger's formula

It is known that the number of ways to express $n \in \mathbb{Z}_{\geq 0}$ as a sum of three squares (let's denote it as $r_3(n)$) can be expressed as Hurwitz-Kronecker class number (certain weighted ...
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5 votes
1 answer
264 views

Making a submanifold transverse to a vector field by an isotopy

Let $M$ be a smooth manifold, $N\subset M$ be a smooth closed hypersurface not bounding a compact submanifold, and $X$ be a smooth nowhere-zero vector field on $M$. I would like to learn what is known ...
0 votes
0 answers
22 views

Size of the second largest strongly connected component in random directed graphs with fixed in- and out-degree sequences

It has been known for long (Molloy and Reed 1995) that in a supercritical undirected configuration model, that is when $E[D(D-2)]>0$, $D$ degree of a uniform vertex, the size of the second largest ...
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4 votes
1 answer
193 views

When are filtered colimits of (trivial) cofibrations still (trivial) cofibrations?

Let $\mathcal{M}$ be a locally finitely presentable model category, cofibrantly generated by two sets $\mathcal{I}$ and $\mathcal{J}$ of cofibrations and trivial cofibrations with presentable domain ...
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3 votes
1 answer
111 views

General form of bounded linear functionals on Banach spaces

It is a famous result due to Riesz that every bounded linear functional $f$ on a Hilbert space $\mathcal{H}$ is of the form $f(x)=\langle x,z \rangle$ for a unique $z\in H$. On p.188 of Introductory ...
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2 votes
1 answer
130 views

Could certain closed covering determine a coherent sheaf?

We know that a coherent sheaf on a scheme is determined by its restriction on certain open coverings (satisfying compatibility condition). Now I wonder how about a closed covering. To do so I started ...
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1 vote
0 answers
70 views

Keller's cubing conjecture but with arbitrary cubes of side $1$

These days I have been reading about Keller's cube tyling conjecture, which asks if in any covering of $\mathbb{R}^n$ by translates of $[0,1]^n$ with disjoint interiors there are two cubes sharing one ...
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3 votes
1 answer
93 views

Coordinate principal bundle over a curve

I am trying to understand more about geometric interpretation of vertex algebras following "Vertex Algebras and Algebraic Curves" by Ben-Zvi and Frenkel, but I am in trouble with the ...
3 votes
0 answers
53 views

General references on dynamics of continuous, piecewise linear interval maps

I want to find a general reference on topological and measurable dynamics of continuous piecewise linear interval maps. I am particularly interested in cases with only three pieces. I know there are ...
5 votes
1 answer
192 views

Which finite projective planes can have a symmetric incidence matrix?

As the title says. Which finite projective planes admit a symmetric incidence matrix? I am not an expert in the field at all, but I consulted with one. He claimed that $PG(2, \mathbb F_q)$ can always ...
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4 votes
1 answer
165 views

Shafarevich's conjecture on Galois groups over fields ramified at finitely many places

Let $K$ be a number field i.e. a finite extension of $\mathbb{Q}$, $\overline{K}$ a fixed separable closure of $K$, and $G_K:=\mathrm{Gal}(\overline{K}/K)$ the absolute Galois group of $K$. Let $S$ be ...
9 votes
1 answer
405 views

Are any embeddings $[0,1]\to\mathbb{R}^3$ topologically equivalent?

Suppose we are given embeddings $f_1,f_2:[0,1]\to\mathbb R^3$. Does there exist a homeomorphism $g:\mathbb R^3\to\mathbb R^3$ such that $g\circ f_1=f_2$? This question seems to be classical eighty ...
1 vote
1 answer
185 views

Complex Borel measures: does $\mu_n \to \mu$ weakly imply $|\mu|(\Theta) \le \liminf_n |\mu_n|(\Theta)$ for every open subset $\Theta$?

Let $\Omega$ be a metric space, $C_b(\Omega)$ the space of all real-valued bounded continuous functions on $\Omega$, and $\mathcal{M}(\Omega)$ the space of all finite signed Borel measures on $\Omega$...
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4 votes
2 answers
169 views

Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation

I'm reading a proof of below theorem from this paper. Theorem A.3. Let $\Omega$ be a locally compact normal Hausdorff space. Let $\left\{\mu_n\right\} \cup\{\mu\} \subset \mathcal{M}(\Omega)$ and ...
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2 votes
1 answer
56 views

Results in Computational Geometry Utilizing Doubling Dimension of a Metric Space

According to Wikipedia, However, many results from classical harmonic analysis and computational geometry extend to the setting of metric spaces with doubling measures. My question is: what are some ...
6 votes
3 answers
832 views

Discontinuous functions without removable discontinuities

A function $f:\mathbb{R}\rightarrow \mathbb{R}$ has a removable discontinuity at a given real $x$ in case the left and right limits are equal but not to the function value, i.e. $f(x+)=f(x-)$ but $f(x)...
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25 votes
1 answer
1k views

A simple proof of the fundamental theorem of Galois theory

Update. It's now on the arXiv. Some time ago I found my "own" proof of the fundamental theorem of Galois theory. You can find a pdf with the proof (link removed, see arXiv). It is quite ...
7 votes
0 answers
102 views

Hölder continuity of spectrum of matrices

Endow $\mathbb{C}^{d \times d}$ with the norm induced by the Euclidean norm on $\mathbb{C}^d$. It is well-known (to those who know it well, I guess) that the spectrum $\sigma(A)$ of a matrix $A \in \...
1 vote
1 answer
228 views

Good references to study Baker theory

I am studying diophantine equations and I need the theory of Bakers, Can you advise me about good books, or lectures on Baker theory?
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2 votes
1 answer
120 views

A stronger version of paracompactness

Given a topological space $(X,\tau)$, recall that a cover $\mathcal{U}$ of $X$ is locally finite if for every point $x\in \mathcal{U}$ has a neighborhood $U$ that intersects finitely many elements of $...
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2 votes
0 answers
81 views

Question concerning relationships between different $p$-modular systems and Brauer character values

Let $(K,\mathcal{O},k)$ be a large enough $p$-modular system, where $\mathcal{O}$ is a complete discrete valuation ring of characteristic zero with unique maximal ideal $J(\mathcal{O})$, algebraically ...
8 votes
1 answer
109 views

Free idempotent monad associated to a monad

Let $C$ be a category. There is a full subcategory $\text{IdemMnd}(C) \hookrightarrow \text{Mnd}(C)$ of the category of monads on $C$ spanned by the idempotent monads. Given a monad $T$ on $C$, ...
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7 votes
2 answers
444 views

An elementary inequality of operators

Suppose $a,b$ are two positive-definite linear operators on (say) $\mathbb R^n$. For $p\in(0,1)$, do we then have $(a+b)^p\leq a^p+b^p$ (with respect to the Loewner order)?
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0 votes
0 answers
42 views

Reference request: truncated SVD and congruence transformations

I am interested in how the singular value decomposition relates to congruence transformations. Suppose $X$ is positive definite of full rank $n$, and $X_k$ is the best rank $k$ approximation of $X$ in ...
3 votes
1 answer
138 views

A linearly distributed version of the balls into bins problem

Some years ago, I found a paper with all the formulas for the balls into bins problem when the "areas" (i.e., probabilities to capture a ball) of the bins are all different. However, the ...
0 votes
0 answers
77 views

Newton and affine curvature

This is a reference request for the following modern formulation of one of the central results of mathematical physics—Newton’s deduction of the inverse square law from Kepler’s description of the ...
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12 votes
1 answer
344 views

Is there a more modern account of the main results of "Adding Dependent Choice" by D. Pincus?

I would like to read Pincus' article Adding dependent choice, where he proves, among other things, the consistency of the theory $\mathsf{ZF+DC+O+\neg AC}$, where $\mathsf{DC}$ stands for Dependent ...
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2 votes
1 answer
137 views

Proof of generalized Siegel's mean value formula in geometry of numbers

Let $\mu$ be the Haar measure defined on the space of unimodular lattices, identified with $\text{SL}(d,\mathbb R)/\text{SL}(d,\mathbb Z)$. The classical Siegel's formula in geometry of numbers states ...
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0 votes
1 answer
68 views

Ito-Levy decomposition for $\alpha$-stable processes?

The Ito-Levy decomposition is well-known as a characterization of Levy processes. What does it give for the specific case of $\alpha$-stable Levy processes?
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4 votes
1 answer
119 views

Source on equality-free second-order logic (nontrivially construed)

Throughout I'm only interested in the standard semantics for second-order logic, and all structures/languages are relational for simplicity. If defined naively, second-order logic without equality is ...
2 votes
1 answer
65 views

Proving that a function is a trapdoor function

I am working on a problem that involves an iterative application of a function I think might be a trapdoor function. Formally, I have a function $f:X \to X$ that can be described as $$ [x_{1,N+1}, ...,...
1 vote
0 answers
66 views

Asymptotic densities of rules of elementary cellular automata

In Tables of cellular automata, p.542, Wolfram defines the density $\delta$ of a rule to be the asymptotic density of nonzero sites when the initial configuration has density $1/2$. Wolfram quotes ...
3 votes
1 answer
72 views

Mutual information in large deviation theory

Many information theoretic quantities such as entropy and relative entropy appear in rate functions in large deviation theory (LDT). Is there any result in LDT that relates mutual information and rate ...
1 vote
1 answer
106 views

Upper bound Wasserstein distance by $\chi^2$ distance

Given two random variables $X,Y$ which are both $\mathbb{N}$-valued and have the same expected value (which is some fixed positive constant), and denote their probability mass functions by ${\bf p} = (...
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1 vote
1 answer
92 views

Strong subobject classifier for measurable spaces

Does the category of measurable spaces have a strong subobject classifier (specifically $2 = (\{0,1\}, \{\varnothing, \{0,1\}\})$? I would think the situation could be analogous to $\mathsf{Top}$, ...
2 votes
1 answer
89 views

How does the conditional Wiener measure work?

In the theorem below $P_D$ means the heat kernel in the open $D \subset \mathbb{R}^m$ and $P_m$ is the heat kernel in whole $\mathbb{R}^m.$ I know absolutely nothing about what Brownian bridges are, ...
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3 votes
1 answer
116 views

Abelianization of unit quaternions over a p-adic field

Suppose $p$ is a prime, that $F$ is a finite extension of the field $\mathbb{Q}_p$, $D$ is the division quaternion algebra over $F$ and $\mathcal{O}_D$ is the valuation ring of $D$. What is the ...
2 votes
0 answers
113 views

Do you know this formula for the scalar product in barycentric coordinates?

I've found a formula for a scalar product in barycentric coordinates which I think is pretty cool. I hope that it's new. Is it? Suppose that you have points $x_1,\dots,x_n$ sitting in general position ...
3 votes
2 answers
270 views

Binomial coefficient congruence modulo $p^n$

I am interested in the following congruence $$\binom{ap^n}{bp^n}\equiv \binom{a}{b}\pmod{p^n}$$ I am aware that by some reference in a book the above it should actually hold modulo $p^{3n}$; the ...
7 votes
1 answer
181 views

Reference request. Finiteness of the Selmer group

Let $K$ be a global field (ie either a number field or the function field of a curve over a finite field). Let $A,B$ be abelian varieties over $K$ and let $\phi:A\to B$ be an isogeny. Associated with $...
9 votes
2 answers
373 views

A category of spaces in which $\mathbb{R}_{>0}$ is not isomorphic to $\mathbb{R}$

$\def\CC{\mathbb{C}}\def\RR{\mathbb{R}}$I am writing up notes on totally positivity in flag varieties. I often have the following situation: I have a complex variety $X$ defined over $\RR$ and a real ...
2 votes
0 answers
117 views

On the generalization of a Cech-to-sheaf type spectral sequence

Let $(X,O_{X})$ be a ringed space and $\mathcal{F^{\bullet}}$ be a bounded below complex of $O_{X}$-modules on $(X,O_{X})$. On [Stacks Project 20.25.1][1] it is shown that there is a weakly convergent ...
6 votes
0 answers
52 views

Vector algebra in a Tarski space

By a Tarski space I understand a mathematical structure $(X,B,E)$ consisting of a set $X$, a ternary betweenness relation $B\subseteq X^3$ and a 4-ary equidistance relation $E\subseteq X^2\times X^2$ ...
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9 votes
0 answers
179 views

Every locally presentable $\infty$-category can be presented by a proper model category

Is there an argument in the litterature that show that every locally presentable $\infty$-category is equivalent to the localization of proper combinatorial Quillen model category ? Of course if one ...
  • 35.1k
9 votes
1 answer
139 views

Bibliography request: Entropy for continued fractions

Given a strictly positive real number $x$ we set $e(x)=\log(1+x)$ if $x$ is an integer and $$e(x)=\log(1+x)+\frac{1+\lbrace x\rbrace}{1+x}\left(e(1/\lbrace x\rbrace)-\log(1+\lbrace x\rbrace)\right)$$ ...
10 votes
3 answers
931 views

Axioms for the category of groups

Certain categories of mathematical structures have had synthetic axiom systems developed for them. One particularly well known such category is the category of sets and functions $\mathit{Set}$, which ...