Newest Questions
159,098 questions
-4
votes
1
answer
108
views
An integral similar to the Delta function [closed]
I have an integral on the form
$\int_{-\infty}^{\infty} e^{-k \omega' |\tau|} e^{i \tau(\omega'-\omega)} d\tau$
that I would like to simplify (or basically solve). This indeed comes from a problem ...
- 3
15
votes
1
answer
1k
views
(Very) Large numbers, Chaitin's incompletness theorem and a specific upper bound
Chaitin's incompleteness theorem roughly saying states that for any theory $S$ there exists universal constant $L$ that for any string $\sigma$ one cannot prove (within this theory) that $K(\sigma)>...
- 9,340
41
votes
2
answers
3k
views
Is number of different sums monotone?
Suppose you have a set $S$ consisting of $n$ different integers.
Let $$W_k = \#\biggl\{x\in\Bbb Z\colon \text{there exists } T \subseteq S,\, \#T=k,\, \sum_{a \in T} a = x\biggr\}.$$
My question is: ...
- 673
3
votes
1
answer
199
views
Subgroups of top cohomological dimension
Let $G$ be a geometrically finite group, i.e. there exists a finite CW complex of type $K(G,1)$.
By Serre's Theorem, every finite-index subgroup $H$ of $G$ satisfies $cd(H)=cd(G)$, but what about the ...
- 843
2
votes
0
answers
89
views
Is a finite flat groupoid of affine schemes equivalent to an action groupoid of a finite flat group scheme?
Let $\mathscr X$ be an algebraic stack with a finite locally free presentation $\pi:X\to\mathscr X$ where $X$ is an affine scheme. Is it possible to find a presentation $\mathscr X=[Y/H]$ where $Y$ is ...
- 21
2
votes
1
answer
175
views
Optimization over permutation
The Problem
This is the problem I am working on: Given a set $X = \{x_1, x_2, \cdots , x_n\}$ in a metric space, find an optimal ordering $\pi : X \rightarrow X$ that maximizes the following objective ...
- 43
3
votes
0
answers
222
views
Number of partitions of set restricted by sum of square of part size
Let $p_1^{a_1}p_2^{a_2}\cdots$ denotes the integer partition of $n$, i.e. $a_1p_1+a_2p_2+\cdots=n$. Or equivalently $m_1+m_2+\cdots=n$. It is known that the number of partitions of set $\{x_1,x_2,\...
- 405
9
votes
2
answers
484
views
Connected geometric thickness two
A graph $G = (V,E)$ has geometric thickness two if there exists an embedding $\varphi: V \rightarrow \mathbb{R}^2$ and a decomposition $E = E_1\cup E_2$ such that $G_1 = (V,E_1)$ and $G_2 = (V,E_2)$ ...
- 479
5
votes
0
answers
185
views
Regularity of convergent flow of parabolic PDE (Fokker-Planck equation)
Consider the divergence-type 2nd order linear PDE on $\mathbb{R}^d$
$$\partial_t u_t = Lu_t := \nabla\cdot(u_t\,\nabla V)+\Delta u_t,$$
representing the Fokker-Planck evolution equation for the ...
- 91
3
votes
1
answer
200
views
Finite subgroup of $\operatorname{Sp}(2n,K)$
Let $G$ be the algebraic group $\operatorname{Sp}(2n, K)$ where $K$ is an algebraically closed field of characteristic not $2$. There is a quaternion subgroup $Q$ such that $Q/Z(G)$ is elementary ...
- 501
2
votes
0
answers
166
views
What is the topology on the space of differential forms $\Omega^2(M)$?
I have posted this question on MSE one year ago, but till now I did not received an answer. Therefore I have decided to post it here.
I have difficulty in understanding the meaning of "A ...
- 191
2
votes
1
answer
254
views
A question about equivalence of weighted Sobolev space norm in S. Benzoni-Gavage and D. Serre's book
This question may not be at the research level, but it has really bothered me for a long time. The following space is used for handling initial boundary value problem for first order hyperbolic ...
- 187
2
votes
2
answers
167
views
Example of random walk in a random environment (RWRE) saying things on the environment
I was wondering if anyone is aware of works/articles/examples where random walks in a random environment (RWRE) are actually used for obtaining information on the random environment.
To clarify a bit, ...
- 59
5
votes
1
answer
294
views
Words which are not inverted by any endomorphism
Let $w$ be a word in a free group $F_2$ of two generators $x_1, x_2$ such that there does not exist any endomorphism of free group which takes $w$ to $w^{-1}$. Let $w_1, w_2$ be two words in the same ...
- 355
3
votes
1
answer
298
views
Pointwise convergence and disjoint sequences in $C(K)$
Let $K$ be a Hausdorff compact space and let $C(K)$ be the space of continuous real-valued functions on $K$. A sequence $(h_n)$ in $C(K)$ is called almost disjoint if there is a sequence $(g_n)$ with ...
- 5,529
2
votes
0
answers
72
views
Doubt on regularity at "Minimal solutions of a semilinear elliptic equations with a dynamical boundary condition"
In the paper Minimal solutions of a semilinear elliptic equations with a dynamical boundary condition by Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami, in Chapter 2 there is a construction of a ...
4
votes
0
answers
191
views
K-theory of toric varieties
Let $X$ be a smooth projective toric variety over $\mathbb{C}$. Is there a good presentation for the K-theory ring $K_0(X)$ in terms of the corresponding fan, analogous to the presentation of the Chow ...
- 3,446
3
votes
1
answer
185
views
$l$-adic cohomology of hyperplane arrangements
Consider an arrangement of hyperplanes given by the faces of a simplex. Let's consider it as a scheme (a non-regular scheme) and let's also work over a finite field. Has the rational $l$-adic ...
- 5,901
5
votes
2
answers
623
views
Embedding large countable ordinals into the complex plane
Consider large countable ordinals (e.g. $\epsilon_0$ which is not "large", but still interesting).
These are countable sets, so they inject into the complex plane ( or even the real line).
...
- 593
6
votes
1
answer
354
views
Signed measures and poset inequalities
Consider a triangulated ball $D$, and assume that $\omega$ is an assignment of real weights
to the simplices of $D$, including the empty one, such that for every maximal simplex $F$ and every simplex $...
- 752
2
votes
1
answer
335
views
Combinatorial meaning of a binomial expansion
Let $F$ be a generating function $F(x) = \sum_{i=0}^\infty f_i x^i$, and
suppose that we can do operations formally without worrying about
convergence issues.
Define the coefficients
\begin{gather*}
...
- 5,230
5
votes
2
answers
885
views
Why isn't $S^1$ contractible in homotopy type theory?
$\newcommand\base{\mathit{base}}\newcommand\unique{\mathit{unique}}\DeclareMathOperator\transport{transport}\newcommand\loop{\mathit{loop}}\DeclareMathOperator\refl{refl}$In the context of homotopy ...
- 119
1
vote
0
answers
66
views
How can I calculate the derivative of an integral with respect to a parameter if Leibniz's formula gives a divergent integral?
We are working on the problem related to a magnetic field in an axially symmetric magnetic plasma trap. Let's express the vector potential through the magnetic flux function
\begin{gather}
\label{1:01}...
- 241
0
votes
0
answers
429
views
What is the exact asymptotic bound on the following sum of polynomials?
I am trying to describe the asymptotic growth of the function $$f(n) = \sum_{k = 1}^{n-1} \frac{k^{2n - 4k - 3}(n^2 -2nk + 2k^2)}{(n-k)^{2n-4k-1}}$$ as $n \rightarrow \infty$. Plotting $f(n)$ for the ...
11
votes
0
answers
717
views
John-type theorems: trading structure for accuracy?
Given two symmetric convex bodies $B, B'$ in ${\bf R}^d$, define the Banach-Mazur distance $d(B,B')$ between them to be the least constant $\tau \geq 1$ such that
$$ B \subset TB' \subset \tau B$$
for ...
- 114k
3
votes
1
answer
222
views
asymptotics of numbers represented by certain indefinite binary quadratic forms
A postdoc wrote to me, asking about about the asymptotic number of elements represented integrally by, say, $x^2 - n^3 y^2$ in the set of numbers $-N$ to $N.$ Actually, he included $x^2 - n^5 ...
- 25.7k
1
vote
0
answers
181
views
Non-trivial homotopy, but vanishing homology
I wonder if there are examples of 5-dimensional manifolds with vanishing integral second homology group, but non-vanishing second homotopy group? Or is it impossible by some Hurewicz theorem type of ...
- 129
3
votes
1
answer
390
views
When does a fibre bundle induce a long exact sequence in homotopy groups of mapping spaces?
Given a smooth fibre bundle of (compact, say) manifolds $F \to E \to M$, with $\pi : E \to M$ the projection, and $i_p : F \to E$ the inclusion of $F$ into any fibre $\pi^{-1}(p)$, we get, for any ...
- 1,763
4
votes
2
answers
378
views
A possible measure-theoretic pathology
Let $S$ be a nonempty closed subset of the open unit square $(0,1)^2 = X \times Y$
that has the following "shadow property":
For any aligned open square $C = A \times B$ that intersects $S$, ...
- 41
1
vote
0
answers
54
views
Expected value for minimum denominator of arbitrarily chosed rational out of a ball of fixed radius to complex plane
So I have a research problem which states that we compute the probability mass function of the random variable which returns the smallest denominator of a reduced fraction in a randomly chosen real ...
- 111
5
votes
2
answers
625
views
Reconstruction of second-order elliptic operator from spectrum
Let $M$ be a compact smooth manifold, $(\lambda_n)_{n=1}^{\infty}$ be a square-summable monotonically increasing sequence of non-negative numbers, and let $(f_k)_{k=1}^{\infty}$ be continuous ...
- 364
1
vote
0
answers
49
views
Find an order-embedding of $S_3\times{\bf2}\times{\bf k}$ into a product of $3$ chains, one of size at most $k$
A function $f:P\to Q$ from a poset $(P,\le_P)$ to a poset $(Q,\le_Q)$ is an order-embedding if, for all $p,p'\in P$, $p\le_P p'$ if and only if $f(p)\le_Q f(p')$.
We partially order the Cartesian ...
- 1,644
1
vote
1
answer
188
views
Practical calculation of Canterbury approximants
I'm looking for references on how to compute Canterbury approximants numerically from a practical point of view. The references on Canterbury approximants that I am aware of all appear rather abstract ...
- 3,065
-3
votes
1
answer
81
views
What is the asymptotic behavior of the Levy distribution $P (x)$ when the independent variable $x$ approaches $0$ [closed]
What is the asymptotic behavior of the Levy distribution
$$P(x)=\frac{1}{\pi}\int_{0}^\infty \exp(-\gamma q^\alpha)\cos qx\,dq$$
when the independent variable $x$ approaches $0$?
- 7
1
vote
1
answer
208
views
Weak-star convergence implies trace-norm convergence
By definition, if bounded operators $a_i$ converge to $0$ in the weak*-star topology, then $\operatorname{tr} a_it \to 0$ for any trace-class $t$.
Does this also hold for the trace-norm instead of the ...
- 896
0
votes
1
answer
109
views
Expanding on a step in the calculation of ζ(f,χ,s) = Λ(s)
I'm trying to understand the derivation here:
Above, f is defined as a product of Schwartz-Bruhat functions which are their own Fourier transform.
I was hoping someone could spell out the second ...
user30211
1
vote
1
answer
131
views
Is the projective dimension of finite torsion-free modules over regular ring of dimension $n$ smaller that $n$?
Let $R$ be a Noetherian regular integral domain of Krull dimension $n$. Let $M$ be a finite torsion-free $R$-module. Is this true that $M$ has projective dimension $<n$ ?
This would be a ...
- 1,479
6
votes
3
answers
385
views
Poisson and homotopy Poisson operads
$\newcommand{\Pois}{\mathit{Pois}}\newcommand{\Comm}{\mathit{Comm}}\newcommand{\Lie}{\mathit{Lie}}$For my thesis, I'm trying to understand the Poisson operad (I'll call it $\Pois$) and its homotopy ...
- 220
4
votes
1
answer
208
views
Representation of a number as a product of $\sqrt{n^2 + 1} + n$
Question. Do there exist two multisets $A, B$ consisting of positive integer numbers such that $|A|$ and $|B|$ have different parity and
$$
\prod_{n\in A}(n + \sqrt{n^2 + 1}) = \prod_{m\in B}(m + \...
- 577
6
votes
0
answers
182
views
Conditions for metrisability
If a normal, first countable space is the union of countably many open metrisable subspaces, must that space be metrisable?
Partial answers, which I proved in the 1980's, include:
(0) The answer is ...
- 69
1
vote
1
answer
117
views
When is the Tor-dimension of $R/(r)$ strictly smaller than that of $R$?
Let $R$ be a ring (commutative with unit) which I assume Noetherian and regular. In particular, the homological dimension of $R$ is the same as its Krull dimension.
I am looking for results in ...
- 1,479
1
vote
0
answers
47
views
Homology groups of moduli of parabolic bundles with fixed determinant
I am looking for the Homology groups of the moduli space of stable parabolic bundles over a smooth projective curve with fixed determinant.
In particular, what is the second homology group of such ...
- 195
1
vote
0
answers
42
views
Higher topos theory lemma 5.2.3.1 and Joints of marked simplicial sets
So here is the proof of the 5.2.3.1:
I have two question about this proof.
A.in the last paragraph it says:
It suffices to prove that there exists an extension of $\bar{k_0}$ to a map $\bar{k}:X\...
- 429
7
votes
0
answers
102
views
A group, all of whose non-trivial mapping tori are finitely presentable?
By a mapping tori of $G$, I mean a semidirect product $G\rtimes\mathbb{Z}$, and by a trivial mapping tori I mean one isomorphic to $G\times\mathbb{Z}$.
If $G$ is finitely generated but not finitely ...
- 2,821
8
votes
1
answer
326
views
The Parity Principle and $\mathbf{C}_2$ (choice for $2$-sets)
The Parity Principle states that
if $X\neq \emptyset$ is a set, then there is $\mathcal B\subseteq \mathcal P(X)$ such that whenever $a,b\in \mathcal P(X)$ with $a\mathbin\Delta b = \{x\}$ for some $...
- 48.9k
0
votes
1
answer
111
views
Finding first and second moments of (extended) projected normal distribution on a 3D unit sphere
I am interested in understanding the statistics of a specific distribution defined on the 3D unit sphere $\mathbb{S}^2$. The distribution in question arises from taking a 3D vector sampled from a ...
- 1
6
votes
1
answer
249
views
Existence of adjoint operators on manifolds
Let $(M,g)$ be an oriented Riemannian manifold and $V$ a finite-rank vector bundle equipped with a non-degenerate bundle metric $\langle\cdot,\cdot\rangle_{V}$. This bundle metric, in turn, gives rise ...
- 1,429
2
votes
0
answers
118
views
polynomials with no repeated factors
Assume that $F(x_1,\ldots, x_n)$ is a polynomial with integer coefficients that is "square-free" over $\mathbb Q$, i.e. it does not have repeated polynomial factors whose coefficients are in ...
- 3,062
1
vote
0
answers
37
views
Both-way flows in a directed graph
Let $G$ be a finite directed graph, and let $s,t$ be two distinct vertices.
Problem $1(s,t)$. Find the maximum number of mutually edge-disjoint directed paths from $s$ to $t$. OK, I didn't think of ...
- 37.7k
3
votes
1
answer
1k
views
Sequence of $L^2$ functions converging to zero weakly s.t. $|f_n|^2$ converges to 1 weak-star?
I am trying to construct a sequence $\{f_n\} \in L^2([0,1])$ with $f_n \geq 0$ a.e. such that $f_n \to 0$ weakly in $L^2$ (meaning $\int f_n dx \to 0$ for all $f \in L^2([0,1]))$ and such that for all ...
- 31