# Questions tagged [reference-request]

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

9,291 questions

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### 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) ...

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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 ...

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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 ...

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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_{...

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**1**answer

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 ...

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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 \...

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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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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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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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2k views

### 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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67 views

### 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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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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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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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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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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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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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 ...

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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 $M=V^{I}/U$, ...

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**1**answer

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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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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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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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 + ...

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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 ...

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**1**answer

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 = \...

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**1**answer

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 ...

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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 ...

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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,...

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**1**answer

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 ...

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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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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

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### 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 ...

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**1**answer

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 ...

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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) ...

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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 ...

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**1**answer

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 ...

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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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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=...

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**1**answer

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 ...

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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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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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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 $U$, ...

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**1**answer

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 ...

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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 ...

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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. $...

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**1**answer

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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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 ...