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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132 views

Log-concavity of sequence related to overpartitions

The number $p_1(n)$ of overpartitions of $n$ is generated by $$\sum_{n\geq0}p_1(n)\,q^n=\prod_{k=1}^{\infty}\frac{1+q^k}{1-q^k}.$$ Let $t\in\mathbb{N}$. Now, extend this to construct a family of ...
4
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1answer
250 views

Perverse sheaves on the complex affine line

Show that a perverse sheaf on $\mathbb{A}^1(\mathbb{C})$ (the complex plane with the analytic topology) is a bounded complex $A$ of sheaves of $\mathbb{Q}$-vector spaces with constructible cohomology ...
4
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2answers
214 views

Relation of the first Hochschild cohomology and the outer automorphism group

Let $R$ be a ring. Qeustion: Is it true that the first Hochschild cohomology of $R$ is zero if and only if the outer automorphism group of $R$ is finite? (It is not true, by the two answers. Is it ...
3
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0answers
91 views

p-adic density of the image of a polynomial

Let $P(x)$ be a polynomial with integer coefficients, and let $p$ be a prime number. Recently, a user of MO proved that the limit $$\delta_p(P) := \lim_{n \to \infty} \frac{|\{P(x) \bmod p^n : x = 1,\...
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0answers
90 views

Rigidity of the TMF-valued equivariant elliptic genus

Let me preface this question by saying that I wrote it at least in part to understand its statement. As such, I hope that the reader will excuse any mistakes. $\DeclareMathOperator{\ind}{ind}\...
3
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1answer
451 views

Where can I find a rigorous proof of this statement in the literature? : $\sum_{n=1}^\infty \frac{\mu(n)}{n} = 0$

In Wolfram MathWorld site at Moebius Function topic there is identity number 10, which states that $\sum_{n=1}^\infty \frac{\mu(n)}{n} = 0$. Could you help me find a rigorous proof of this statement ...
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0answers
66 views

Invariance of categories of sheaves (on simplicial presheaves) under (local) weak equivalence

Let $\mathcal{C}$ be a Grothendieck site (secretly in my head I am thinking of Hausdorff topological spaces with open covers; if I am daring I might be thinking of the big etale site on complex ...
5
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1answer
227 views

Copy of paper by M. Naimi on squarefree friable integers

I am trying to get a hold on a copy of a 1988 paper by M. Naimi. It appears in MR as MR0950949 and in zbMATH as Zbl 0669.10066. Its title is "Les entiers sans facteurs carré $\le x$ dont leurs ...
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0answers
67 views

Two-variable generating functions over coprime pairs

I am studying a sequence $(\alpha_{p,q})$ indexed by a pair of coprime integers; this sequence arises naturally in the study of a particular set of spaces in geometric topology, but unfortunately the ...
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0answers
42 views

Statistical characteristics of low complexity subshifts

I am looking for calculations of statistical characteristics (variance, entropy, etc.) of the $n$-dimensional distributions of the invariant measures of low complexity subshifts (e.g., the Sturmian or ...
2
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1answer
118 views

Is a line associated with antipodal points (the fact, it is the generalization of Simson line) known?

First time, I found a line associated with antipodal points, detail: Let $ABC$ be a triangle, $(C)$ is circumconic of $ABC$. $P$ and $P'$ are two antipodal points. Construct three lines through $P'$ ...
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0answers
81 views

Infinite dimensional topological quantum field theories?

A topological quantum field theory is a functor $Z:Cob(n) \to Hilb$ mapping from the $n$-dimensional cobordisms to $Hilb$, the category of Hilbert spaces with morphisms being bounded linear operators. ...
1
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2answers
135 views

What to call a continuous function with preimage preserving nowhere-density?

Currently I am reading some basic literature on descriptive set theory and boolean algebras. And this comes out a lot, for example in results like: Let $X$ and $Y$ be topological spaces, and $f:X \to ...
3
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1answer
121 views

Pairs of elements in $F_n$ with distinct translation lengths

Let $F_n$ be a free group of rank n and consider all possible non-degenerate length functions on $F_n$. Could I be directed to a reference that give two non-trivial non-power-conjugate elements $g,h \...
17
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4answers
748 views

Combinatorial description of the Mandelbrot set

I have a very naïve question: can one find anywhere a combinatorial description of the Mandelbrot set? Let me try to be a bit more precise: is it possible to encode each of its "bulbs" by ...
2
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0answers
90 views

understanding the definition of subgroup of the Grothendieck-Teichmuller group

Definition. Let $\widehat{G T}^{1}$ be the set of elements $f$ in the derived subgroup $\hat{F}_{2}^{\prime}$ of $\hat{F}_{2}$ such that $x \mapsto x$ and $y \mapsto f^{-1} y f$ extends to an ...
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60 views

Generalized polynomials and a $4$-point cross-ratio function on the non-degenerate complex $4$-quadric

The Grassmannian $Gr_1(\mathbb{C}^2)$ is another name for $\mathbb{P}^1$. If one endows $\mathbb{C}^2$ with a complex symplectic form, or if one prefers (since this will allow us to generalize in ...
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4answers
505 views

A certain type of combinatorial identity, involving de Montmort numbers

It would lead me too far to explain how I stumbled upon the somewhat obscure identities $$\sum_{m=0}^n \binom{n}{m} (1-m)^m m^{n-m}=(-1)^n d_n, \quad \sum_{m=0}^n \binom{n}{m} (1-m)^{m-1} m^{n-m}=1,$$...
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0answers
145 views

Homotopy fixed points of involutive automorphisms of discrete groups

$\DeclareMathOperator\Map{Map}$Precis: I am looking for a reference/explanation of a (general) computation of homotopy fixed points $(BG)^{h\Gamma}$ of a finite group $\Gamma$ acting on a discrete (...
3
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2answers
174 views

Bounded density for diffusions with diffusion coefficients bounded away from $0$

Consider a diffusion given by $$X_t=\int_0^t a(s,X_s)\,dW_s$$ for $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and $a$ is a smooth enough positive function bounded away from $...
20
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0answers
781 views

A mysterious paper of Stallings that was supposed to appear in the Annals

In Stallings's paper Stallings, John, Groups with infinite products, Bull. Amer. Math. Soc. 68 (1962), 388–389. he briefly discusses how to prove "several generalizations" of Brown's ...
3
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1answer
125 views

Reference request: Stallings-Epstein-Waldhausen construction

I am looking for a reference for the Stallings-Epstein-Waldhausen construction (constructing an incompressible surface in a 3-manifold from a nontrivial splitting of the fundamental group). I know of ...
17
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0answers
594 views

Reference request for a proof of the two-square Theorem

One can show (see below for a sketch of a proof) that every odd prime number $p$ can be written in exactly $(p+1)/2$ different ways as $$p=a\cdot b+c\cdot d$$ with $a,b,c,d\in\mathbb N$ satisfying $\...
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0answers
78 views

Furtwängler's family of irreducible polynomials

In the question Examples of nice families of irreducible polynomials over Z, user trew mentions a family of irreducible polynomials over the integers of the following form: $$ p(x) = x^4 \prod_{i=1}^{...
14
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3answers
512 views

Height functions on $\mathcal{M}_g(\overline{\Bbb{Q}})$ defined via dessins d'enfants?

Belyi's theorem establishes a correspondence between smooth projective curves defined over number fields and the so called dessins d'enfants which are bipartite graphs embedded on an oriented surface ...
5
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2answers
214 views

Which result guarantees convergence of solution of an ODE to a set of non-compact, non-isolated equilibrium?

Consider a continuous ODE, $$\dot x = f(x), f \in C^1$$ $\dot x = 0$ for all $x \in K \subset \mathbb{R}^n$, where we assume that $K$ is a closed but unbounded set of non-isolated equilibrium. For ...
4
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1answer
95 views

Gromov hyperbolicity for (non-geodesic) metrics on the upper-half plane invariant with respect to SL(2, R) action

$\DeclareMathOperator\SL{SL}$Let $d$ be a metric on the upper-half plane $\mathbb H = \{(x,y) : y > 0\}$ which is invariant with respect to the action of $\SL(2, \mathbb R)$ to $\mathbb H$ which is ...
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0answers
89 views

Books on limiting properties of matrices with growing size

This question has been posted on Math-Se previously. I am studying asymptotic properties of the Projection Matrix $$ H_n=X'(X'X)^{-1}X $$ By the Gerschgorin disc theorem, the bounds on the ...
15
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3answers
1k views

Theoretical results on neural networks

With this question I'd like to have a recollection of theoretical rigorous results on neural networks. I'd like to have results that have been settled, as opposed to hypothesis. As an example, this ...
2
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0answers
51 views

Characterization of inverse limits of finite-dimensional convex cones

Consider a countable inverse system $C_1\substack{f_1 \\ \leftarrow} C_2 \substack{f_2 \\ \leftarrow} C_3 \substack{f_3 \\ \leftarrow} \ldots$ where the $C_i$ are finite-dimensional convex cones of ...
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0answers
152 views

Monotonicity of the cycle index polynomial under restriction

The cycle index (polynomial) of the symmetric group $\mathfrak{S}_n$ is given by the formula: $$Z(\mathfrak{S}_n)(x_1,\dots,x_n)=\sum_{1j_1+2j_2+\cdots+nj_n=n}\prod_{k=1}^n\frac{x_k^{j_k}}{k^{j_k}j_k!}...
1
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1answer
108 views

Proving an estimate for the Neumann problem on $\mathbb{R}^3 \setminus B_1$ in Weighted Sobolev spaces

Let $M := \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball. It is known that for every $g \in H^{\frac{1}{2}} (\partial M)$, and for an appropriate $\delta$, there exists a unique solution $u$ ...
4
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0answers
336 views

Deligne's letter to Soulé from 1985

There is a famous letter of Deligne to C. Soulé in which, apparently, Deligne first formulated the conjecture on the existence of an abelian category of mixed motives, extending Grothendieck's pure ...
1
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0answers
69 views

Reference request: Optimal controls can be assumed to take values in a convex set

Consider the deterministic controlled system: $$\dot x(t) = Ax(t) + Bu(t), \ t \in [0, T]$$ $$x(0) = x_0$$ where $x: [0, T] \to \mathbb R^n$ is the controlled state process, $A \in \mathbb R^{n \times ...
-1
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1answer
74 views

Finding a $k$-subset which maximizes a matrix sum

Let $M\in \mathbb{R}^{N\times N}$ be a given matrix and $k\ge 2$ be a given integer. Then my question is the following optimization problem: Is there a polynomial-time solution to the following ...
3
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1answer
139 views

Well-definedness of maximum likelihood estimation

Consider a family $\{\mu_\theta:\theta\in\Theta\}$ of probability measures on a measurable space $X$. Given $x\in X$, the maximum likelihood estimate is the value of $\theta$ which maximizes the ...
1
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1answer
78 views

Square-free representation of polynomials with integer coefficients [closed]

Can every polynomial $P(x)$ with integer coefficients we represented in the form $$ P(x) = Q^2(x)R(x), $$ where $Q(x)$ and $R(x)$ are polynomials with integer coefficients such that $R(x)$ has no ...
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0answers
168 views

On connectedness of the complement

In the application of Runge type theorems on the approximation of functions with some regularity on a neighborhood of a compact, it is interesting to know whether the complement of a compact has ...
4
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0answers
217 views

A learning roadmap for topological modular forms

There are some MO posts about the uses of topological modular forms: Why should I care about topological modular forms? What can topological modular forms do for number theory? I am interested in ...
5
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0answers
77 views

D-modules as ind-coherent sheaves over positive characteristics?

There is an interpretation of D-modules over "sufficiently nice" prestacks $X$ (read: various finiteness conditions apply, perhaps even smoothness) by Gaitsgory and Rozenbylum (see chapter I....
1
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1answer
84 views

Diagonalization of octonionic Hermitian matrices of size $2\times 2$

The group $Spin(9)$ is a subgroup of $SO(16)$ and acts transitively on the unit sphere $S^{15}$. $Spin(9)$ acts naturally on the space of octonionic Hermitian $2\times 2$-matrices (I do not define ...
2
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1answer
136 views

Reference where the Siegel-Walfisz theorem for the Möbius function is proved

Let $A>0$ and $q\leq (\log N)^A$. Then there exists a constant $c$ depending on $A$ such that $\displaystyle \sum_{n\equiv a \bmod q; n\leq N}\mu(n)\ll N\exp(-c\sqrt{\log N})$. I know this result ...
2
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0answers
49 views

Geometric meaning of Catlin multi types

Can someone working in the area of several complex variables explain the geometric idea behind the Catlin multitype. I have seen the technical definition, but unable to grasp the idea behind this. ...
7
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2answers
446 views

A 2-page paper on a lower bound of Ramsey number

I'm looking for a 2-page paper on a lower bound of Ramsey number $R(a,b)$ for some constants $a$ and $b$. The paper was published in 80s or 90s. I googled it for a few days, but I cannot find the ...
3
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0answers
71 views

The behavior of an integral related to the inward normal vector near a point of the boundary of a domain

Inspired by this Q&A, I am asking for what kind of non-smooth domains $D$ the following limit $$ \lim_{r \to 0}\frac{1}{m(D \cap B(x,r))}\int\limits_{D \cap B(x,r)}\frac{z-x}{r}\,m(dz) $$ where $...
2
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1answer
170 views

asymptotic growth of a sum involving partitions

Let $\lambda\vdash n$ denote the integer partition of $n$. Define the product $\mathcal{N}(\lambda)=\lambda_1\lambda_2\cdots\lambda_r$ when $\lambda=(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_r>0)...
2
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1answer
130 views

A Riemann surface is automatically paracompact

[A question I remember from many years ago.] Definition A Riemann surface is a connected complex manifold $X$ of complex dimension one. This means that $X$ is a connected Hausdorff space that is ...
10
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0answers
195 views

What is known about differentiable and analytic structures on the long line (and half-line)?

When reading about this question which recently became active for some reason, I wanted to make a comment, as a warning regarding non-metrizable manifolds, to the effect that the every $C^\infty$ ...
2
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0answers
66 views

Regular exposable points on the boundary of compacts in Stein manifolds

Given a Stein manifold $Y$, there exists $\rho$, a $\mathscr C^2$-smooth strictly plurisubharmonic exhausting function for $Y$, such that the set of critical points $C=\{z\in Y\;:\;d\rho(z)=0\}$ is ...
1
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1answer
47 views

Lower bound on likelihood of binary outcomes

I am wondering about the following: does there exist a stochastic process $(X_n)_{n \ge 1}$ with values in $\{0,1\}$ on a probability space $(\Omega, \mathcal F, \mathbb P)$ such that for all $n \ge 1$...