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

**3**

votes

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**3**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**4**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**3**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**3**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**3**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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-...

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votes

**0**answers

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 ...

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votes

**3**answers

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

**0**answers

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

**1**answer

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

**0**answers

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 $\...