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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8
votes
1answer
362 views

$2$-adic valuations: a tale of two $q$-series

Let $\nu_p(n)$ denote the $p$-adic valuation of $n$, i.e. the highest power of $p$ dividing $n$. Consider the following two $q$-series formed by infinite products $$\prod_{n\geq1}\left(\frac{1+q^n}{1-...
1
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0answers
85 views

Only Zariski-closed subsets of compact Lie groups with nonempty interior have nonzero measure

In this question, the following fact was used by the respondent A Zariski-closed subset of a compact Lie group $G$ with nonzero Haar measure contains a coset of $G^0$, the connected component of $G$ ...
0
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0answers
160 views

Total sum of characters over partitions with distinct parts

In my earlier quest, we looked at $\chi_{\mu}^{\lambda}=$value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu$ and $\lambda$ are (unrestricted) partitions of $n$. Then, ...
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0answers
33 views

Precise definition of locally closed complex curve

In Stein Manifold and Holomorphic Mappings, by Forstnerič, I refer to Definition 8.9.9: An exposed point is a point belonging to a certain subset $\Sigma$ of $\Bbb C^2$, enjoying certain properties. ...
6
votes
2answers
302 views

Simplicial set construction of the classifying space

What would be the best book, article, or otherwise to reference for the specific construction of the classifying space for a discrete group $G$ which goes as follows?: Regard $G$ as a category with ...
0
votes
1answer
82 views

Sampling uniformly in a ball of radius $\epsilon$ in the space of dicrete r.v. of m modalities for the total variation metric

I am looking for some reference or an algorithm that allows to sample uniformly in the ball centered at a discrete random variable of n modalities in the TV distance. For the record for 2 discrete ...
4
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0answers
130 views

Strictification for closed monoidal categories

The strictification theorem for monoidal categories states that every monoidal categorically is monoidally equivalent to a strict monoidal category. Is there a strictification theorem for closed ...
1
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0answers
37 views

Identities on the Whittaker function $W_{-\kappa,\mu}(z)$?

As in (for example) [Magnus, W., Oberhettinger, F., Soni, R.: Formulas and Theorems for the Special Functions of Mathematical Physics, Grundlehren der mathematischen Wissenschaften 52, Springer, ...
2
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0answers
52 views

Where can I find a proof of the main properties of Weyl Curvature for semi-Riemannian manifolds?

Most of the references I've seen deal with Riemannian geometry, rather than semi-Riemannian geometry. Chens monograph, Pseudo-Riemannian Geometry, $\Delta$-Invariants and Applications is one of the ...
1
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0answers
120 views

$L_p$ estimate in mixed boundary problem for elliptic equation

Let $Q$ be convex polygon, $\Gamma$ be a portion of boundary $\partial Q$ and $H^1_\Gamma(Q)=\lbrace u\in H^1(Q): u|_\Gamma=0\rbrace$. For $f\in (L_2(Q))^2$ consider the problem $$ \int_Q A(x)\nabla u ...
1
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0answers
128 views

Alternative Mersenne numbers

Let $\ b\in\mathbb Z,\ $ and $\ |b|>1.\ $ Call $$ M_b(n)\ :=\ \frac{b^n-1}{b-1} $$ to be $n$-th Mersenne number mod $b$. The necessary condition for $\ M_b(n)\ $ to be a prime is that $\ n\ $ is a ...
8
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0answers
221 views

Does Borel fixed-point theorem hold for Deligne-Mumford stacks?

Let $X$ be a proper Deligne-Mumford stack over $\mathbb{C}$ with an action by a complex torus $T$. Let $X^T$ denote the fixed locus. Question: Is the following statement true? ...
4
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0answers
189 views

What to do with antique/older mathematics books? Throw away or something else? [duplicate]

My father, who held 4 post graduate degrees and was a lifetime student, passed away recently. He has an entire bookcase full of older mathematics books, including some on related topics such as ...
1
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0answers
51 views

Two types of the Germain prime siblings

Let $\ p\ $ and $\ q:=2\cdot p+1\ $ be primes — they are called Germain prime siblings. Such a pair belongs to the first type $\ \Leftarrow:\Rightarrow\ \frac{q^2-1}8\equiv\pm1\mod8,\ $ and to the ...
2
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0answers
115 views

Product and coproduct in derived category

I'm sure this is either a standard result or false, but I don't have enough experience with the derived category to decide either way. I have tried looking in Kashiwara-Schapira's Sheaves on Manifolds ...
2
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0answers
44 views

Relations between LR coefficients and cores and quotients of partitions

I have a formula for certain coefficients in terms of Littlewood-Richardson coefficients and $p$-cores and $p$-quotients of partitions ($p$ is a prime). I would like to obtain some positivity ...
0
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0answers
63 views

Expected diameter of a random point set

General problem: For a point set $S\subset X$ in a metric space $(X,d)$, let $\text{diam}(S)=\max_{x,y\in S}d(x,y)$. Given a distribution $P$ on $X$ and $m$ i.i.d. points $x_1,\ldots,x_m\sim P$, what ...
3
votes
1answer
292 views

On convergence of entire functions

Suppose we have a sequence of entire functions $f_n$ such that $$\text{$f_n(z)\to0$ for each natural $z$}\tag{1}$$ (as $n\to\infty$). Is it possible to give general additional conditions on the ...
7
votes
2answers
438 views

Total sum of squares of characters of the symmetric group $\mathfrak{S}_n$

In my earlier MO post, I proposed the double sum $\sum_{\mu\vdash n}\sum_{\lambda\vdash n}\chi_{\mu}^{\lambda}$ regarding characters of the symmetric group $\mathfrak{S}_n$. Soon after, I started ...
2
votes
2answers
150 views

Name of an inductively defined sequence of graphs

Let $G_k$ be the graph obtained by applying the following procedure k-times: Start with a graph with single vertex $v$ (Call this graph $H$) Add a vertex $u$ such that $u$ is not adjacent to any ...
2
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0answers
58 views

Does the following corollary of Mackey's tensor product theorem hold for smooth representations?

Let $G$ be a locally profinite group, and let $H$ be a closed subgroup of $G$. Let $\sigma$ be a smooth representation of $G$, and let $\tau$ be a smooth representation of $H$ (henceforth, every ...
4
votes
1answer
367 views

Total sum of characters of the symmetric group $\frak{S}_n$

Let $\chi_{\mu}^{\lambda}$ denote a value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu, \lambda\vdash n$. When $\mu=(n)$, then it's known that $$\sum_{\lambda\vdash n}\...
2
votes
1answer
127 views

Distance formula for continued fractions

In the book Neverending fractions from Borwein, van der Poorten, Shallit and Zudilin, there is the so called distance formula (Theorem 2.45, p. 43) stated: $$\alpha_1\alpha_2\cdot...\cdot\alpha_n=\...
0
votes
0answers
37 views

Reference request: Pascal type octagon theorem

I am looking for a reference to a generalisation of the celebrated hexagon theorem of Pascal which states that if $A$, $B$, $C$, $A_1$, $B_1$ and $C_1$ are the (distinct) vertices of a hexagon in ...
15
votes
1answer
429 views

Reference request: Moore graphs

It is clear that the term Moore graph was coined by Hoffman and Singleton in their paper On Moore graphs with diameters $2$ and $3$, where they write E. F. Moore has posed the problem of describing ...
2
votes
0answers
26 views

Reduction of the general Lauricella hypergeometric function $F_B$ for identical parameters and variables

The Lauricella function $F_B^{n}$ of $n$ variables is defined as $$F_B^{(n)}(a_1, \ldots, a_n, b_1, \ldots, b_n, c; x_1, \ldots x_n) = \sum_{k_1, \ldots, k_n = 0}^\infty \frac{1}{(c)_{k_1 + \ldots + ...
3
votes
0answers
91 views

Quasi-crystaline generalization of elliptic functions

I came across some meromorphic function, call it $f(z)$, which is "quasicrystalline" in the following sense: one can write $f$ as: $$ f(z)=\frac{\sum_i a_i e^{i(q_{i,x}x+q_{i,y}y)}}{\sum_i ...
1
vote
0answers
82 views

A random process with conserved momentum: 'particle decay'?

Consider a particle $p_1$ moving at unit speed along a straight line in $\mathbf{R}^2$, directed by some vector $v_1 \in \mathbf{S}^1$. Equid this particle with a Poisson clock $\tau_1$, with ...
9
votes
3answers
1k views

Should every modern day mathematician care about category theory? [closed]

As far as I know, category theory is used mainly in topology. I have a dislike towards category theory, similar to my dislike of Bourbakism, and want to avoid it as much as I can. However, the head of ...
6
votes
1answer
214 views

A strong Borel selection theorem for equivalence relations

In Kechris' book "Classical Descriptive Set Theory" there is the following theorem (12.16): Let $X$ be a Polish space and $E$ an equivalence relation such that every equivalence class is ...
6
votes
2answers
469 views

Counting $\pm 1$ and $0$'s in the character tables of $\frak{S}_n$

Let $\chi_{\mu}^{\lambda}$ denote a value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu, \lambda\vdash n$. When $\mu=(n)$, then it's known that $$\sum_{\lambda\vdash n}\...
2
votes
0answers
134 views

The variety of $\mathbb{C}[t]_{< d}$-points on a variety

(This was posted to https://math.stackexchange.com/q/4244260/799193 where it did not receive an answer.) Let $X \subseteq \mathbb{C}^n$ be an affine variety defined by $f_i(x_1, \ldots, x_n)=0, 1 \le ...
2
votes
0answers
94 views

"higher" micro-support

Recall that for a sheaf $F$ on an analytic manifold $X$ the micro-support consists of those $\omega\in T^*X$ for which there exists a $C^1$ function $f$ defined around $\pi(\omega)$ with $f(\pi(\omega)...
4
votes
0answers
75 views

Associative rings with "big" commutative subrings

Let $A$ be an associative ring and $R\subset A$ be a commutative subring. Suppose that every element of $A$ has the form $urv$ where $r\in R$ and $u, v\in A^*$ are invertible. A basic example is $A=...
15
votes
4answers
2k views

When did Grothendieck join Bourbaki? [closed]

Bourbaki listed Grothendieck as a third-generation member. Nevertheless, it does not provide details on when he joined and when he left. Concerning his departure, there is a Letter from October 9, ...
2
votes
1answer
165 views

Is $g(v)=\mathbb{E}[f(v+W)]$ a differentiable function of $v$ when $f$ is continuous and $W$ is multivariate normal?

Suppose $f$ is a continuous function on $\mathbb{R}^n$, and $W$ has a multivariate normal distribution on $\mathbb{R}^n$. If the expectation $$g(v)=\mathbb{E}[f(v+W)]$$ is defined for all $v \in \...
4
votes
0answers
97 views

Choice of topology in the (log) crystalline site

Let $X$ be a scheme or fs log scheme over a finite field. There seem to be several slightly different definitions of the (log) crystalline site of $X/S$ available in the literature, depending on ...
7
votes
0answers
122 views

Is there a complete characterization of hyperimaginaries in $\mathsf{DLO}$?

Recall that in a first-order theory $T$, a hyperimaginary is an equivalence class of some type-definable equivalence relation $E(x,y)$ (with $x$ a possibly infinite tuple of parameters). The $E$-...
1
vote
0answers
72 views

Continuous decomposition of permutation-invariant set functions

The seminal machine learning paper Deep Sets (Zaheer et al., 2017) discusses representations of permutation-invariant functions on real tuples, or (multi)set functions. Given a countable set $X$ and a ...
3
votes
0answers
109 views

When the Leray spectral sequences for nice compactifications give the Deligne's weight ones?

Assume that $X$ is a proper smooth variety over an algebraically closed field $k$, $U=X\setminus (\cup D_i))$ where $D_i$ are closed subvarieties such that the set-theoretic intersections of all sets ...
1
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0answers
52 views

Divergence between random variables after transformation

Let $X$ and $Y$ be random variables with laws $\mu_X$, $\mu_Y$ and $d$ be some $f$-divergence (e.g. KL, total variation, Hellinger). Writing $d(X,Y)$ for the divergence between $\mu_X$ and $\mu_Y$, ...
1
vote
0answers
59 views

Solution of Riccati system of ODEs

We have following equation: $$ w(t,v) = \exp\Bigl(-\phi (t) \frac{v^2}{2}-\psi (t) v -\chi (t)\Bigr),\quad (t,v)\in [0,T]\times \mathbb{R}, $$ where $(\phi, \psi ,\chi)$ are solutions of the Riccati ...
6
votes
0answers
99 views

Explicit homotopy for Hochschild chains from natural isomorphism

Let $A,B$ be $k$-linear (possibly, dg-)categories, let $f,g:A\to B$ be two linear functors, and let $T:f\Rightarrow g$ be a natural isomorphism. If one denotes by $C_\bullet(A,A)$ the standard ...
2
votes
1answer
223 views

Central limit theorem for weak correlated random variables

I have a sequence of weak correlated continuous random variables $\{X_i\}$ with bounded variance and $\operatorname{Cov}(X_i,X_j)\rightarrow0$ for $|i-j|\rightarrow\infty$. I was able to find a ...
2
votes
2answers
259 views

Has the von Neumann entropy ever been used in classical mechanics?

After going through an application of the von Neumann entropy(from quantum information theory) to certain problems in computational neuroscience [2], it occurred to me that this entropy might have ...
7
votes
1answer
224 views

Stallings' binding tie

I came to know that the statement below could be proved using Stallings' binding tie argument, though I have no reference article proving the statement by the binding tie argument. Can anyone help me ...
0
votes
0answers
132 views

A diophantine equation involving partial sums of exponentials similar than the equation in Fermat's Last Theorem

I'm curious about the following diophantine equation from my invention: I don't know if this is in the literature, I wrote it using creativity in an attempt to write a variant of the equation in ...
2
votes
0answers
48 views

Leray-Schauder degree in Banach manifolds

The so called Leray-Schauder degree is usually defined for maps of the form $I - f$, where $f: X \to X$ is a compact map defined on a Banach space. Is there an extended definition for the setting of ...
1
vote
1answer
174 views

Stochastic Integral + conditional expectation

Let $\overline{\widehat{Z}_i} = \frac{E_i\left[ \int_{t_i}^{t_{i+1}}\widehat{Z}_sds\right] }{\Delta t_i}$ with $\widehat{Z}$ a square integrable process, $\Delta t_i := t_{i+1} - t_i$, and $E_i$ ...
2
votes
0answers
40 views

Is there a generalisation of this perturbation result about rank-one modifications of diagonal matrices?

In Theorem 1 of [1] we have the following result: Let $D$ be a real $n \times n$ diagonal matrix and consider the rank-one modification $C = D + \rho z z^T$, where $\rho > 0$ is a real scalar and $...

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