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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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13 views

Distributivity of direct sum over maximal tensor product

Let $A,B$ and $C$ be $C^{*}$-algebras.Does the following identity always holds: $(A \oplus B) \otimes^{max} C \cong (A \otimes^{max}C) \oplus (B \otimes^{max} C)$ My intuition is that this should ...
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41 views

Approximation of classifying space BG of compact Lie group G by finite CW complexes

The classifying space of a topological group $G$ is usually constructed as follows: one constructs a sequence of spaces $E_1G$, $E_2G$, $E_3G$, … with a free $G$-action such that the (homotopy) ...
3
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2answers
110 views

Example of a smooth projective family of varieties in characteristic $p$ where the Hodge numbers jump

Let $\mathbb F$ be an algebraic closure of the field of order $p$. Let $S=\textrm{Spec}(\mathbb F[[z]])$ with special point $s$ and generic point $\eta$. I'm looking for an example of a smooth ...
2
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0answers
54 views

A symmetric function that appears in the coefficients of a power expansion

Let's say we have the expression $$\sum_{k_1=1}^\infty\sum_{k_2=1}^\infty\sum_{k_3=1}^\infty\sum_{k_4=1}^\infty\sum_{k_5=1}^\infty x^{k_1+k_2+k_3+k_4+k_5} f(k_1,k_2+k_3,k_4+k_5)$$ where $f(a,b,c)$ is ...
3
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2answers
61 views

Why we use Caputo fractional derivative in application?

I'm working on some papers which use Caputo fractional evolution equation as application for thier main result: For example: $$\left\{\begin{matrix} ^CD^{\sigma}_tx(t)+Ax(t)=&f(t,x(t),\int_{...
13
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1answer
293 views

Smooth proper variety over $\mathbb Q$ with everywhere bad reduction

$\newcommand{\Spec}{\operatorname{Spec}}$ Cross-post from Math.SE, hopefully people more knowledgeable in the field will see the question here on MO. It is a well-known fact that a smooth projective ...
2
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0answers
28 views

Enrichment of lax monoidal functors between closed monoidal categories

Let $\mathscr C,\mathscr D$ be (right) closed monoidal categories. Then both of them can be considered as enriched over themselves via their internal homs, which I will denote by $\textbf{Maps}$. Now ...
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79 views

Is there an affine embedding X for every normal singularity, so that Pic(X)=0?

More formal: Let a normal algebraic singularity be given by a local ring $R$ of finite type. Is there always an affine variety $X$ with a point $x$, so that $\widehat{\mathcal{O}}_{X,x} \cong \...
2
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2answers
199 views

Applications of flat submanifolds to other fields of mathematics

Developable surfaces in $\mathbb{R}^{3}$ have lots of applications outside geometry (e.g., cartography, architecture, manufacturing). I am a curious about potential or actual applications to other ...
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53 views

Sobolev embedding in complete manifold

Let $(M,g)$ be a complete Riemannian $m$-manifold, with bounded geometry and $m\geq2$. Suppose that $(M,g)$ admits a bounded geometry. Q Can we show that for $k-\frac{m}{p}\geq l-\frac{m}{q}$, we ...
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1answer
128 views

Reference request: Dynamical systems

I’m currently reading Brin and Stuck’s Introduction to Dynamical Systems, and I think I like the field a lot so far. I haven’t finished it quite yet, but what are some other good textbooks I can read ...
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83 views

Characterisation of special Cohen-Macaulay modules

Let $A$ be an $R$-order of Krull dimension $d$ that is an isolated singularity, see for example section 3 of https://arxiv.org/pdf/0803.2841.pdf for the relevant definitions. With $D_d(-):=Hom_R(-,R)$...
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0answers
76 views

Geometric meaning of residue constraints

$\DeclareMathOperator\Res{Res}$I have been reading Kontsevich and Soibelman's "Airy structures and symplectic geometry of topological recursion" (https://arxiv.org/abs/1701.09137) and am having ...
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3answers
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Does anyone recognize this inequality?

In some paper the authors make use of the following inequality without further explanation: Let $x\in\mathbb{R}^n$ with $x_1\le\cdots\le x_n$ and $\alpha\in[0,1]^n$ with $\sum_{i=1}^n \alpha_i=N\in\{1,...
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Why is the set of lifts of a p-divisible group canonically the same as the set of lines that span $M(G)/FM(G)$?

Let $M$ be the Dieudonne module of a p-divisible group $G_0$ over $k$, and let a lift of $G_0$ to $A$ be a p-divisible group $G$ over $A$ such that $G \otimes_A k \simeq G_0$. Let $\omega_G$ be the ...
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2answers
836 views

William Thurston's quote?

Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding. Is this from Thurston? If yes, where and when it has been said. I've checked "ON PROOF AND ...
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0answers
50 views

Regularity minimizer

I am not an expert in calculus of variations and I am getting pretty lost in the vast literature. I've been studying the following functional $$ \int_\Omega (|\nabla f|+|\nabla g|)^2 dxdy $$ where $\...
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0answers
34 views

Ellipticity-type condition

An elliptic operator $L=\mathrm{div}(A(x)\nabla u)$, is called uniformly elliptic if $$C^{-1}\mathrm{Id} \le A(x) \le C \mathrm{Id}$$ If $A$ depends also on $u$, what is the condition $$C^{-1} + C^...
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0answers
69 views

Has this result about the number of permutations of a given cycle type (or centralizers) been proved?

I was playing with the cardinality of conjugacy classes of the symmetric groups, which we know is the number of permutations of a given cycle type and there is a natural one-to-one correspondence ...
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2answers
222 views

Can we do better than Azuma-Hoeffding when the variance is small?

The Azuma-Hoeffding Inequality says that if $X_1,X_2, \ldots$ is a martingale and the differences are bounded by constants, $\|X_i - X_{i-1}\| \le 1$ say, then we should not expect the difference $\|...
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2answers
230 views

Number of integer partitions modulo 3

Motivated by Parity of number of partitions of $n!/6$ and $n!/2$, I asked my computer (and my FriCAS package for guessing) for an algebraic differential equation for the number of integer partitions ...
5
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0answers
128 views

Can one use forcing as a step to prove the Keisler-Shelah isomorphism theorem?

Can one use forcing (perhaps at the expense of using larger ultrafilters) to prove the Keisler-Shelah isomorphism theorem? My idea is to use an ultrafilter $U$ on a set $I$ such that if $M=V^{I}/U$, ...
1
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1answer
122 views

Does $\delta$-invariant give a sufficient condition for flatness for plane curve singularities?

Let $\pi:\mathcal{C}\to S$ be a morphism of schemes such that $\mathcal{C} \subset \mathbb{C}^2 \times S$ with the inclusion map commuting with the natural projection to $S$ and for all $s \in S$, $\...
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0answers
89 views

Doubts related Shifting from Pure to Applied math [closed]

I am a second year (Pure) Math and (Theoretical) Physics undergraduate in India. I want to do a masters in Applied/Computational Science or Math, for which I have apply after next 7 months. I have ...
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0answers
102 views

Definition of the surface measure in some books

I am studying PDEs and in some books (Folland, Introduction to Partial Differential Equations and Evans, Partial Differential Equations), I found an integral integrated by the surface measure on a $C^...
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0answers
38 views

Name for a pair of lattices one of which having theta series with coefficients a subsequence of another lattice's theta series coefficients

Is there a name for a pair of lattices which have the property given in the title? The following example of a pair captures the property mentioned above: $$(i)\ 1 + 80q^3 + 270q^4 + 432q^5 + 960q^6 + ...
17
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2answers
602 views

Motivation behind Analytic Number Theory

I am an undergraduate student of mathematics and recently took an introductory course in analytic number theory, where the instructor roughly followed Apostol's first text on the subject. I have now ...
2
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1answer
47 views

Intensity and compensator for a jump process

Set-up and assumptions. Let $(\mathscr{F}_t, t \geq 0)$ be a right-continuous complete filtration. Let $(X_t, t\geq 0 )$ be a pure jump $\mathbb{R}$-valued process with unit jumps, that is, $$ X_t = \...
4
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1answer
84 views

“Twisted” direct sums of Banach spaces

It is hard to provide motivation, so I just want to state this definition "as is". Suppose I have a Banach space $E$ and two commuting, injective operators $R_0, R_1$ on $E$ which satisfy the ...
16
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1answer
308 views

Where to find some subset of Khovanskii's 15 proofs of the BKK theorem?

(I asked this question on MSE, but someone suggested it would be better asked here.) I'm a fan of the Bernstein-Khovanskii-Kushnirenko theorem (that the number of solutions in $(\mathbb{C}^*)^n$ to a ...
2
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0answers
137 views

Translation of Soergel's 1990 paper on category O

Is there any English translation for the folowing paper of Soergel? Kategorie $\mathcal{O}$, perverse Garben, und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), 421-445,...
5
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1answer
127 views

Moore Graphs and Finite Projective Geometry

In a comment on a blog post from 2009 about the hypothetical Moore graph(s) of degree 57 and girth 5, Gordon Royle offered the following observation (reproduced here in full for the sake of ...
3
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1answer
141 views

Cofibrations of functors

Let $\cal M$ and $\cal N$ be model categories, $S,T:\cal M\to N$ functors, and $\alpha:S\to T$ a natural transformation. Say that $\alpha$ is a <blank> cofibration if for any cofibration $i:A\to B$...
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0answers
48 views

anomalous primes and CM elliptic curves

Let $E$ be an elliptic curve defined over a number field $F$ and suppose $E$ has CM by $\mathcal{O}_K$ where $K$ is an imaginary quadratic field. What can we say about the non-anomalous primes of such ...
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318 views
+50

Are nearby points in an algebraic curve necessarily connected?

I would like a result of the following form: For every algebraic curve $C$ in $\mathbb{R}\mathbf{P}^{n-1}$, there exists an explicit and easy-to-compute $\epsilon=\epsilon(C)>0$ such that ...
21
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1answer
510 views

The Euler-Mascheroni constant and entropy

I would like to know if I have discovered or merely rediscovered the following pretty fact. A partition of $[0,1]$ into intervals of lengths $p_{i, i=1\ldots n}$ induces a probability distribution ...
3
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0answers
54 views

Laplacian variational problem with asymptotically quadratic term

Consider the functional $$J= \int_\Omega |\nabla u|^2 - \int_\Omega F(u),$$ where $\Omega$ is a bounded smooth domain. The problem has been solved for example if $F$ is (1) subquadratic, or (2) ...
2
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0answers
120 views

Where can I find a copy of this paper of Chowla and Vijayaraghavan?

Does anyone know where I can find a copy of Chowla and Vijayaraghavan's paper, ''On the largest prime divisors of numbers''? The relevant literature say it was published in the Journal of the Indian ...
5
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1answer
92 views

Hessian formula for the sub manifold distance

I am working on some problems involving foliations and group actions and would be very nice to consider the second derivatives for the distance function of an orbit or a leaf. So my question is: does ...
2
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0answers
81 views

Reference Request: Local Central Limit Type Theorem (CLT) for the Cycle

I am looking for a reference for a local CLT for the usual SRW on the cycle -- in continuous-time, ideally. I know the statement for a SRW (and a reference, say Lawler and Limic; Random Walk: A Modern ...
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0answers
54 views

Reference for convergence to a Poisson Point Process

Edited after comment by Ofer Zeitouni I have a sequence of discrete time stochastic processes $\big((S_n(i))_{i \geq 1}\big)_{n\geq 1}$ such that for every $n$, $i$, \begin{equation}S_n(i)=\sum_{j=...
2
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1answer
196 views

Does the Hopf construction work for $S^0$?

Given a $0$-connected $H$-space $X$, we can prove that left and right multiplication are (weak) homotopy equivalences. This allows us to construct the Hopf fibrations for $S^1$, $S^3$ and $S^7$. I ...
3
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0answers
76 views

Degeneration of cycle class map

Let $f:\mathcal{X} \to \Delta$ be a flat family of projective varieties, smooth over the punctured disc $\Delta^*$ and the central fiber is a simple normal crossings divisor. Let $\mathcal{Z} \subset \...
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0answers
17 views

Injectivity of covariance operator and full support of Gaussian measure

Let $\gamma$ be a Gaussian measure on a separable Banach space $\mathcal X$ with the covariance operator $C\colon \mathcal X^* \to \mathcal X$ defined dy $$ \langle Cf, g\rangle = \int_{\mathcal X} \...
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3answers
268 views

Reference Request: Structure constants for G2

Let $G$ be a split semisimple real Lie group in characteristic zero, and let $B=TU$ be a Borel subgroup with unipotent radical $U$ and Levi $T$. Fix an ordering on the roots $\Phi^+$ of $T$ in $U$, ...
3
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1answer
179 views

On the convergence of $\sum_{n=1}^{\infty} \frac{\lambda(n)}{n}$ and the Prime Number Theorem

Let $\lambda$ be the Lioville function of number theory. I've heard several times that if $L=\sum_{n=1}^{\infty} \frac{\lambda(n)}{n} =O(1)$ then $L=0$ (the Prime Number Theorem). How can this be ...
3
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0answers
48 views

system of Euler like ode's

I am interested in solving some linear elliptic system like $$ -\Delta \phi(x) + \frac{C_1 \psi(x)}{|x|^\beta} =f(x)$$ $$ -\Delta \psi(x) + \frac{C_2 \phi(x)}{|x|^\alpha} =g(x)$$ in $B_1$ (the ...
4
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0answers
113 views
+50

mixing time of Young tower

Given Young tower $(\Delta, m, F)$ with base $\Delta_0$, return map $R$, distortion of $F^R$, g.c.d($R$)$=1$. This system is exact, so mixing, and implies: given fixed $\epsilon$, exists $N$, s.t. $...
5
votes
1answer
120 views

Where can I find this result of Ingham?

Sometime ago, I read somewhere (should be in Titschmarsh) that, if $N(\sigma, T)$ denotes the number of zeros of the Riemann zeta function $\zeta(s)$ with $\Re(s)\geq \sigma>1/2$ up to height $T>...
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0answers
58 views

Increase in rank of elliptic curves

I expect answers to these questions are known, or at least partial answers are known: Let $E$ be a rank 0 elliptic curve defined over $\mathbb{Q}$ and let $p$ be an odd prime. Is it possible that the ...