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

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3
votes
1answer
101 views

On the spectrum of stationary Gaussian process

What is the condition for ergodicity, weakly mixing, and strongly mixing properties of Gaussian process in terms of its spectrum? In a similar way let us consider a stationary vector valued Gaussian ...
5
votes
0answers
160 views

Intuition: Smooth functions on Banach Spaces

On finite-dimensional vector spaces, we all have a reasonable idea of which functions are likely to be $C^1$ or smooth. When it comes to differentiation on Banach spaces, I find that my `intuition' ...
0
votes
0answers
87 views

forgetful on ext functor : reference request

learning about ext functors I encountered the following statement (source : https://en.m.wikipedia.org/wiki/Ext_functor) : "For $\mathbb{F}_p$ the finite field on $p$ elements, we also have ...
4
votes
0answers
208 views

Verifying a source that lacks a citation

In this German Mathematics Wikibook page, formula $0.5$ lists the following equation $$\int_0^1 \sin(\pi x) x^x (1-x)^{1-x} \ \mathrm{d}x = \frac{\pi e}{24}$$ as supposedly attributed to Ramanujan (...
1
vote
2answers
59 views

Reference for the monotonicity in $\alpha$ of the Rényi entropy

I'd like to have a reference for the property $0 \leq \alpha < \alpha' \leq \infty \implies R_\alpha(\mu) > R_{\alpha'}(\mu)$, where $R_\alpha(\mu)$ is the Rényi entropy of order $\alpha$ of a ...
1
vote
1answer
89 views

Curvature of plane curves on a surface

Let $S$ be a surface and $\gamma$ a curve on $S\subseteq \mathbb{R}^3$ obtained cutting $S$ with a plane. I wuold an upper bound for the curvature of $\gamma$. Are there papers for this topic?
4
votes
0answers
98 views

Restricted addition analogue of Freiman's $(3n-4)$-theorem

There is a well-known theorem of Freiman saying that if $A$ is a finite set of integers with $|2A| \le 3|A|-4$, then $A$ is contained in an arithmetic progression with at most $|2A|-|A|+1$ terms. Is ...
1
vote
1answer
121 views

Effects of shortening and puncturing on codes

Given a binary block code $C=[n,k]$ of codeword length $n$, and dimension $k$. Suppose I've determined these properties for it : $d_{min}$ (minimum distance), $N_{dmin}$ (number of codewords at $d_{...
1
vote
2answers
141 views

A multidimensional version of Hensel's lemma? (for more than one polynomial)

The classical Hensel's lemma is stated as follows: Let $f(x) \in \mathbb{Z}_p[x]$ and $a \in \mathbb{Z}_p$ satisfy $$ |f(a)|_p < | f'(a) |_p^2. $$ Then there is a unique $\alpha \in \mathbb{Z}_p$...
0
votes
0answers
47 views

Intersection Of Valentine Convex Sets

A set X is said to be m-convex , m integer >=2, if for each set of m points at least one of the associated line segments lies in X. A 3-convex set is sometimes also known as Valentine convex after ...
3
votes
1answer
193 views

Local Langlands Conjecture for p-adic SO(4), reference request

In section 10 of Gan-Gross-Prasad's paper "Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups" http://arxiv.org/pdf/...
1
vote
2answers
120 views

When does a stationary point process on group $G$ have $0$ or $\infty$ many points a.s.?

For $G=\mathbb{R}^d$ I know that a stationary point process $X$ either has 0 or infinitely many points, a.s. Daley and Vere-Jones refer to this as the 0-Infinity dichotomy. They hint that this fact is ...
5
votes
3answers
283 views

Nonlinear ODE: $y'=(1+axy)/(1+bxy)$

Consider the first order nonlinear ODE problem: $$ y'(x)=\frac{1+ay(x)x}{1+by(x)x}, \quad x>0 $$ where $a, b>0$ are some constants. I would like to know if these kind of equations were ...
3
votes
2answers
359 views

An integral identity evaluating the gamma function

While reading a number theory paper I encountered the identity $$ \int_{- \infty}^{\infty} (1 + x^2)^{ - \frac{z}{2} - 1} dx = \sqrt{\pi} \frac{ \Gamma(\frac{z + 1}{2}) }{\Gamma(\frac{z}{2} + 1)},$$ ...
1
vote
0answers
106 views

Is there a reference for boundedness of smooth canonically polarized varieties over Z (No…)

In Kollár's paper Quotient spaces modulo algebraic groups, Kollár mentions right above Theorem 1.8 that the stack $\mathcal M_P$ of smooth canonically polarized varieties over Spec $\mathbb Z$ with ...
5
votes
1answer
170 views

Subgroup ranks of the symmetric group

It's well known that every subgroup $G$ of $S_n$ has a generating set of size at most $n-1$ and that this generating set can be found algorithmically (by Jerrum's filter) I have heard many times a ...
5
votes
0answers
156 views

Contact manifolds and pseudodifferential operators

By way of background, I am currently trying to understand the theory of pseudodifferential operators in the context of contact geometry. I have some knowledge of pseudodifferential operators on ...
4
votes
0answers
135 views

Surjectivity of some evaluation map on global sections of a positive vector bundle

Let $X$ be a smooth complex projective manifold, let $E \rightarrow X$ be a Hermitian vector bundle and let $L \rightarrow X$ be a positive Hermitian line bundle. Let $H^0(X,E \otimes L^d)$ denote the ...
5
votes
0answers
179 views

Topology on the space of Borel measures

Let $ B $ be the set of all measures $ \phi $ of $ \mathbf{R}^{n} $ such that every open set is $ \phi $-measurable (sometimes these measures are called Borel measures). Note the measures in $ B $ are ...
0
votes
0answers
55 views

Does anyone have a reference for a proof of expansion for this construction?

http://people.seas.harvard.edu/~salil/pseudorandomness/expanders.pdf "Construction 4.26: p-cycles with inverse chords.... The proof of expansion relies on the “Selberg 3/16 Theorem” from number ...
0
votes
1answer
89 views

Some questions related to the unitary operators

A unitary operator is a surjective linear operator between complex inner product spaces, which preserves the inner product. What is the name of the analogue for the real case? Orthogonal operator ...
8
votes
1answer
272 views

When is bar-cobar duality an equivalence?

Let $A$ be an augmented differential graded algebra over a field $k$. I will write $BA$ for its bar construction (whose homology is $Tor^A(k, k)$). This is a co-augmented differential graded ...
14
votes
4answers
698 views

Groups of matrices in which all elements have all eigenvalues equal in modulus

I am writing a research article in which I need to use the following fact: if $G$ is a subgroup of $GL_3(\mathbb{R})$ which is irreducible in the sense that no proper nontrivial subspace of $\mathbb{R}...
4
votes
1answer
1k views

Blow-up in family

Let $\pi \colon X \to T$ be a flat projective morphism, and let $Y$ be a closed sub-scheme of $X$ which is flat over $T$. We can assume that everything is defined over the complex numbers, and $T$ is ...
0
votes
0answers
140 views

Graph Coloring: Two adjacent vertices share same color

Consider, subgraphs $G_1, G_2,...... G_x$ of graph $G$. Each subgraph has $k$ vertices. Now, Fix subgraph $G_1$ and consider another subgraph $G_k$ where $1 <k \le x$. The edge set ...
5
votes
0answers
173 views

Extension operators for topological vector space-valued smooth functions on closed sets

There are many known results about extension theorems for real-valued functions on closed sets, with varying levels of differentiability and so on, all very roughly following the Whitney approach. For ...
5
votes
3answers
525 views

References - Voevodsky motives are the derived category of Nori motives?

First I would like to know if this has been worked out, and if the answer is affirmative I would like to know some references.
4
votes
1answer
121 views

What is the stationary distribution for the contact process on the half line?

The contact process is a well-studied Markov process. I'm just concerned with the one-dimensional nearest-neighbor version here. The state space is $\eta\in\{0,1\}^\mathbb Z$, and for state $\eta$ at ...
1
vote
0answers
66 views

Reference for a simple fact in measure theory (semi-algebras)

What is the textbook where the following simple fact can be found: A measure defined on a semi-algebra S can be extended to a sigma-algebra generated by S. In the texbooks that I have looked into ...
7
votes
0answers
193 views

Jets of sections of vector bundles expressed by symmetrized iterated covariant derivatives - who did it first?

The (non-unique) bundle isomorphism between the bundle $J^r E$ of $r$-th order jets of sections of a vector bundle $\pi:E\rightarrow M$ and the direct sum $$\bigoplus^r_{k=0}\vee^kT^*M\otimes E\...
4
votes
0answers
113 views

Convex polytopes as “products” of lower dimensional polytopes of the same family

This MO answer on enumerative geometry details the sense in which an associahedron is a product of lower dimensional associahedra, and the comments in this MSE-Q indicate the same is true for ...
4
votes
2answers
229 views

Historical reference request on Nilpotent groups

From Wikipedia: "Abelian groups were named after Norwegian mathematician Niels Henrik Abel by Camille Jordan because Abel found that the commutativity of the group of a polynomial implies that the ...
11
votes
3answers
2k views

Has Dedekind's proof of existence of infinite sets been analyzed by historians?

This pdf by David Joyce notes that in paragraph 66 of his famous essay, Dedekind claims to prove the existence of an infinite set. The proof exploits the assumption that there exists a set $S$ of all ...
2
votes
0answers
36 views

Karhunen-Loeve expansion convergence rate for Gaussian Proccess

Consider A Gaussian Procces $X(t):\mathbb{R} \to \mathbb{R}$ with and $\mathbb{E} \left[ X \right] = 0$. Consider also its KL expansion $X(t) = \sum\limits_{k=0}^{\infty} Z_k e_k (t)$, with $Z_k$ ...
1
vote
0answers
85 views

Internal language type of power objects

It is a basic fact that in a category with finite limits the following are equivalent Each object $X$ has a (membership relation to a) power object $PX\times X\hookleftarrow\ni_X$ subject to the ...
2
votes
0answers
63 views

Characters on lattices and isogenies of Abelian varieties

Let $V:=\mathbb{C}^g$ and $\Lambda \subset V$ be a lattice, i.e. a discrete subgroup of rank $2g$. Then $A:=V/ \Lambda$ is a complex torus of dimension $g$. We moreover assume that $A$ is algebraic, ...
9
votes
1answer
364 views

Real-valued measurable cardinals

A cardinal $\kappa$ is real-valued measurable if there is a probability measure on the $\sigma$-algebra of all subsets of $\kappa$ which is zero on singletons and additive on disjoint families of ...
2
votes
0answers
64 views

Which matroids have not unique unimodular representation?

Matroid $M$ is represented by real vectors, and we know that any base of $M$ generates the same lattice (this is called unimodular representation, I guess.) If we change the sign of any vector, we ...
10
votes
1answer
388 views

The geometric median of a solid triangle

Let $\Omega\subset \mathbb R^n$ be a compact subset of dimension $n$. Define the geometric median on $\Omega$ as the point $m_{\Omega}\in \mathbb R^n$ such that the integral $\int_{\Omega}|x-m_{\Omega}...
2
votes
0answers
115 views

question about currents

I have a question in the field of currents: Let M be an n-dimensional smooth manifold, and let T be a k-current (induced by a k dimensional sub-manifold), I would like to approximate it by a series of ...
2
votes
3answers
406 views

Learning roadmap for algebraic number theory

I have read some elementary number theory from David Burton's text and I know groups and rings from Herstein's book Topics in Algebra and some field theory from different sources online. I am ...
1
vote
0answers
58 views

A non-surjective coboundary map induced by a central extension

Let $k$ be a number field and $$ 1\to A \to B \to C \to 1$$ be a central extension of finite groups over $\mathcal{O}_k$ (the ring of integers of $k$), with $B$ non-commutative. Consider the induced ...
3
votes
0answers
60 views

Lattice points in regular simplex

Suppose we are given a regular (closed) simplex $S$ in a vector space $V$ of dimension $n$, whose vertices have integer values. Then for a lattice $L$, is there a sufficient criterion, for $S$ to ...
2
votes
0answers
110 views

Semistability of a sheaf on nodal curve

Suppose $X$ is a projective, connected, nodal curve (can be reducible) over an algebraically closed field $k$ of arbitrary characteristic. Let $F$ be a pure sheaf on $X$ and denote by $\pi^{*}(F)$ its ...
0
votes
0answers
66 views

Reference request: Uniformly totally bounded classes of compact metric spaces are Gromov-Hausdorff precompact

The following Theorem can be found for instance here (Theorem 7.4.15): Theorem. (author ?) Any uniformly totally bounded class $\mathfrak X$ of compact metric spaces is pre-compact in the Gromov-...
3
votes
0answers
51 views

Real interpolation of weighted Sobolev spaces with different weights

Let $\Omega \subseteq \mathbb{R}^n$ be open and let $w_0$ and $w_1$ be measurable and almost everywhere positive and finite functions defined on $\Omega$. Let $L^2_{w_0}(\Omega)$ be the weighted ...
5
votes
3answers
229 views

Text for studying group representations in the context of (abstract) harmonic analysis

I would like to study elements of representation theory as I often encounter it when reading texts on harmonic analysis. I was therefore curious if someone could recommend a book for this. When ...
2
votes
0answers
37 views

Points of continuity of a lower semicontinuous function have non empty interior

Let $X\subset \mathbb R^d$ having non empty interior and let $f:X\to\mathbb{R}$ be lower semicontinuous. I know that the set of discontinuities of such a function is contained in a meager set, and ...
2
votes
1answer
93 views

Smooth dependence on the initial condition of the integral of an ODE

I am considering an ODE $\dot{x}=f(x)$, with $x\in\mathbb{R}^d$ and $d<\infty$. $f$ is a $C^k$ function and I denote by $\Phi_t x$ be the value of the solution at time $t$. I assume that my ODE ...
7
votes
0answers
154 views

$\Bbb A^1$-Localisation of Schemes, and $\Bbb A^1$-Rigid Schemes

Question 1: Are there some publications or preprints that provide $\Bbb A^1$-fibrant replacements of certain classes of (smooth) schemes? Of course, smooth schemes that are $\Bbb A^1$-fibrant are $\...