Frequent Questions
17,981 questions
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votes
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Probability that SDE visits any point
This is a reference request question.
Statement: I am interested in an SDE of the form
\begin{equation}\fbox{1}~~~
{\rm d}X_t = f(X_t)\,{\rm d} t + g(t) \, {\rm d} B_t
\end{equation}
Where we ...
- 101
0
votes
1
answer
189
views
Terminology "upper" Ahlfors regular measure
Let $(X,d)$ be a metric space and $m$ be a Borel measure on $(X,d)$. The measure $m$ is called Ahlors regular if $m(B(x,r))\asymp r^q$ for some $q>0$ and each $x\in X$. Is there a name for ...
- 5,405
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votes
0
answers
195
views
Dirac's theorem and the 1-factorization conjecture
Let $G=(V,E)$ be a simple, undirected graph. A matching is a subset $M\subseteq E$ such that all members of $M$ are pairwise disjoint; moreover we call $M$ perfect if $\bigcup M = V$.
The 1-...
- 48.9k
0
votes
1
answer
365
views
Integral in a σ−convex set.
Having had no (proper) answer to this question, I formulate the remaining case as a new question as follows. With $I=[0,1]$, let $E$ be a separable (real) Banach space, and let $\gamma:I\to E$ be ...
- 3,584
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votes
0
answers
118
views
Maximal elements for ideals and subrings ordered by inclusion with fixed number of minimal generating polynomials
Let $R=\mathbb{R}[X_1,\dots,X_n]$, and
$$\mathfrak{I}_d=\{ \text{ideals for which there is minimal generating system with $d$ elements} \}\setminus \{\text{ ideals generated by $d$ monomials}\}$$
...
- 1,256
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votes
2
answers
2k
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non discrete valuation ring [closed]
Hi,
I am looking for examples of non-discrete valuation rings. Could you help me?
Thanks
- 141
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votes
0
answers
119
views
"Infima" and "suprema" in the homomorphism preorder on hypergraphs on $\omega$
$\newcommand{Po}{{\cal P}(\omega)}$
$\newcommand{lh}{\leq_{\text{hom}}}$
If $H_i = (V_i, E_i)$ are hypergraphs for $i = 1,2$, then a map $f:V_1 \to V_2$ is said to be a (hypergraph) homomorphism if $f(...
- 48.9k
0
votes
0
answers
235
views
Lebesgue measure of a neighbourhood of a curve
Let $\Omega\subseteq\mathbb{R}^N$ be an open, bounded and with smooth boundary (e.g. Lipschitz boundary or more if necessary).
For any function $\phi:\Omega\to\mathbb{R},\ \phi\in C^1(\overline{\Omega}...
- 1,759
0
votes
1
answer
252
views
A non integrable distribution arising from a Lie algebra of vector fields
Is there an example of a $n$ dimensional manifold $M$ and a natural number $k<n$ with a Lie subalgebra $L$ of $\chi^{\infty}(M)$ with the following property:
For every $x\in M$ the space $\{V_x ...
- 366
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votes
0
answers
174
views
3D generalization of Gaussian q-binomial coefficient
It is known that the coefficient of $q^t$ in Gaussian binomial coefficient $\binom{m+n}m_q$ equals the number of permutations of the multiset $\{0^m, 1^n\}$ with $t$ inversions.
Is there a closed ...
- 34.3k
0
votes
0
answers
643
views
A new generalization of the dimension?
During my research, I came a cross on these notions :
Definition 1: A structure $S$, is a pair $(X, \mathcal T)$ with $X$ a set and $\mathcal T$ a set of subsets of $X$, stable by arbitrary ...
- 4,074
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votes
0
answers
83
views
Deriving "quasi-theta" functions from theta functions' zeros
I've been trying to sift the zeros of \begin{align}\vartheta(z)&=e^{-\pi z^2}\prod_{k\ge1}(1+e^{-(2k-1)\pi+2\pi z})(1+e^{-(2k-1)\pi-2\pi z})\\
&=\prod_{k\ge1}(1+e^{-(2k-1)\pi+2i\pi z})(1+e^{-(...
- 149
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votes
1
answer
199
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Prove that $(v^Tx)^2-(u^Tx)^2 < 1-(u^Tv)^2$ for any unit vectors $u$, $v$, $x$
Let $u,v,x \in \mathbb R^d$ be three unit vectors. I found a very complicated proof that $(v^Tx)^2-(u^Tx)^2 \leq 1-(u^Tv)^2$.
That is $\lVert uu^T-vv^T\rVert^2_2 = 1-(u^Tv)^2$, or that $f(v,x)\leq f(v,...
- 143
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votes
0
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111
views
Game on a square grid (part II)
Related to this question, where there the solution was unexpected for us.
Let $n,m$ be positive integers, $n \le m \le n^2/2$.
The board is $n \times n$ square grid.
Phase 1:
Two players, $A,B$ make $...
- 25.4k
0
votes
2
answers
328
views
Closed form for a binomial product sum
Is there any closed formula for the binomial product sum
\begin{align*}
\sum\limits_{\substack{i_1> i_2> \cdots > i_k\\i_1, i_2, \cdots, i_k \in \{n-j+1, n-j+2, \cdots, n-1\}}}\binom{n}{i_1}\...
0
votes
0
answers
162
views
One solution is known, how to find another one
This question is driven by pure curiosity as for all practical purposes I can generate numerical or series solutions of the given ODE
\begin{align}
p(a,u) [a'(u)]^2+q(a,u)a'(u)+a(u) &= 0,\\
a(0) &...
- 492
0
votes
1
answer
357
views
Metric space that is not a subspace of $\mathbb{R}^n$? [closed]
What are some simple examples of metric spaces that cannot be subspaces of $\mathbb{R}^n$? I've heard there is an example with $4$ points, where two points lie between the other two, but I cannot ...
- 1,465
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votes
0
answers
156
views
A stalk criterion for unit map to be an isomorphism on étale site
Let $f: X \to Y$ be a morphism of schemes and $\mathcal{F}$ sheaf of sets/Abelian groups on the small étale site $Y_{ét}$. Assume we manage somehow to show thatat every geometric point $\overline{y} \...
- 6,038
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votes
1
answer
227
views
Laplace transform injectivity for different values of $p$
Let $y\in L^{2}(0,1)$ and let $\widetilde{y}$ be its extension on $(0,\infty
).$ Assume that there exist $p_{0},p_{1}\in
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
,$ $p_{0}\neq ...
- 617
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votes
1
answer
281
views
journal to submit mathematic books' review
it has been asked to me to write a review on a book about the history of mathematics in Italy between the two world wars.
The book is a non-technical one. I would like to know which journal accepts ...
- 101
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votes
0
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114
views
Recontruction of the weak topolgy from the scalar product on a subset of a Hilbert Space
Let $M$ be a set a let $K:M\times M\to\mathbb{C}$ be a positive definite kernel. By a version of Moore-Aronszajn Theorem, there is a unique (up to the unitary euivalence) Hilbert Space $X$, and a map $...
- 5,529
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0
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362
views
can we generalize the cauchy principal value and hadamard's finite part for multiple integrals $ n>1 $
I have seen in wikipedia, that how the Cauchy's pricnipal value and Hadamard's finite part work in dimension one
My question is when can we (or if negative answer why can not ) generalize the ...
- 113
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votes
0
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128
views
Approximation Property: Characterization
Problem
Given a Banach space $E$. Denote compact sets by $\mathcal{C}$, compact operators by $\mathcal{C}(X,Y)$, and finite rank operators by $\mathcal{F}(X,Y)$.
Suppose it has the approximation ...
- 598
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votes
1
answer
403
views
Are there overwhelmingly more finite posets than finite groups? [closed]
A function $f:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ overwhelms $g:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ if for any $k\in \mathbb{Z}_{\geq 1}$ the inequality $f(n)\leq g(n+k)$ holds only for ...
- 23
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votes
1
answer
86
views
Regular maps on hyperbolic plane for large number of vertices
I want to generate large regular maps of a tiling on hyperbolic space. How I can start doing that?
0
votes
2
answers
261
views
Representations of modular lattices, extension to cellular sheaves
There are various "representation theorems" for lattices such as Birkhoff's Representation Theorem that states that every finite distributive lattice is isomorphic to a quasi-sublattice of the lattice ...
- 147
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votes
1
answer
203
views
LDP for Marchenko Pastur with k/n tending to 0
I am interested in the determinant of $W = X * X'$, where $X \in \mathbb{R}^{k \times n}$ is a matrix with each row drawn IID from some sub-Gaussian distribution on $\mathbb{R}^{n}$. (I am aware of ...
- 435
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votes
0
answers
161
views
Superharmonic extension
We know the following classic result. If $K\subset\mathbb{R}^m$ ($m>1$) is a compact set and $u$ is superharmonic on a neighborhood of $K$, then we can extend $u$ to a superharmonic function $\...
- 411
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votes
2
answers
525
views
Is acyclic ZF consistent?
I need to know if the following system is consistent, because I want to use it in presenting automorphisms over stratified versions of it.
The system I'd label as "Acyclic ZF", which is $\...
- 11.3k
0
votes
2
answers
508
views
K3 surface with a non-symplectic involution: a basic question
Let $X$ be a K3 surface (algebraic, complex). An involution $\sigma:X\rightarrow X$ is called non-symplectic if it acts trivially on $H^{2,0}(X)=\Bbb{C}\omega_X$ $\ $ (where $\omega_X$ is any nowhere ...
- 761
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votes
1
answer
412
views
Given a generating function with "zeros", can one derive the function for ONLY the "zeros"?
If I have an generating function (GF) --- ordinary or exponential --- defining a series with at least one coefficient equal to zero, is there a general method to find the "inverse GF", i.e., the GF ...
- 1,055
0
votes
1
answer
2k
views
Are Chow groups a birational invariant?
Let us work in the category of smooth, projective varieties (say, over an algebraically closed field $k$). If $X$ and $X'$ are birational, then do they have the same Chow groups? Is there at least a ...
- 179
0
votes
1
answer
477
views
Is my application of Faà di Bruno's formula correct?
Suppose I have a function $f$ from $\mathbb R^d$ to $\mathbb R$, and denote $g = \exp \circ f$.
I want to express the derivatives of the function $g$ in term of the derivatives of $f$ and vice versa, ...
- 686
0
votes
1
answer
233
views
Is the random point $(C,S)$ the same as $(1,0)+(\cos U,\sin U)=(1 + \cos U,\sin U)$, with $U$ a uniform r.v.?
Suppose that $\theta_1$ and $\theta_2$ are independent and identically distributed (i.i.d.) random variables and that $\theta_j$ has probability density function (PDF) $f_j = \frac{1}{2\pi}$ ($i.e.$, ...
0
votes
0
answers
759
views
On sets of coprime integers in intervals
Briefly,
Question: Is it "good enough" to use least prime factor in choosing a maximal set of coprime integers in an interval?
The post title comes from a 1993 paper of Erdos and Sarkozy. They ...
- 13k
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votes
0
answers
125
views
Is Burgess' ST theory an algebraic set theory? If it is, what kind of algebras are described by this theory?
I found the mention of Burgess' ST set theory in the article https://en.wikipedia.org/wiki/General_set_theory about George Boolos' generalized set theory (GST). It sounds like this: "ST is GST ...
- 889
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votes
1
answer
249
views
About uniform continuity
Is there a definition Df(g) of uniform continuity of g, without using the notion of metric?
Let $(E,d_E)$ and $(F, d_F)$ metrics spaces, $f$ continuous fonction of $E$ to $F$
We must have :
Df$(f)$ ...
- 4,074
0
votes
1
answer
292
views
Volume of randomly changing sphere follows beta distribution
We are given $X,X_1,\ldots,X_N$ independent and identically distributed $k$-dimensional vectors. For a given query point $X_q\in\mathbb{R}^k$ assume without loss of generality that $X_1,\ldots,X_m$ ...
- 329
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votes
0
answers
113
views
How much a probability distribution is non-uniform in a convex subspace of $\mathbb{R}^d$?
I know a number of (standard and well known) ways to measure the distance between two probability distributions and, more in general, to quantify how much one is far from another.
Could you please ...
- 1,677
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votes
0
answers
144
views
Number of open tours by a biased rook on a specific $f(n)\times 1$ board which end on a $k$-th cell from the right
We have a simple structure - biased rook of the two types.
Biased rook of the first kind which make open tours on a specific $f(n)\times 1$ board where $f(n) = \left\lfloor\log_2{2n}\right\rfloor + 1$ ...
- 4,938
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votes
1
answer
475
views
uniqueness for Poisson equation in R^d with mildly regular data
I'm interested in Poisson's equation $-\Delta u=f$ set in the whole space $R^d$ (let's say $d\geq 3$ for simplicity) when $f$ has very little integrability, specifically $f\in L^{1+\varepsilon}$ for ...
- 5,380
0
votes
0
answers
140
views
Polynomial generated with primitive element modulo p
This question is equivalent to the question "Normal basis in cyclotomic number fields" that I asked recently. I am posing this question because maybe in this format somebody can have an answer:
Let $...
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0
answers
97
views
Uniqueness of the solution to some SDE of state-dependent coefficient
This is a continuation of my question posted in Uniqueness of the solution to some SDE
Consider
$$X_t=X_0 + t + \int_0^t \frac{\sigma(s,X_s)}{1+m(s)}dW_s,\quad \forall t\ge 0,\quad\quad\quad (\ast)$$
...
- 1,334
0
votes
2
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291
views
A question about the inverse theorem function in $\mathbb{R}^n$
Let $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a continuously differentiable mapping. We assume that the set $$\{x\in\mathbb{R}^{n};j(f)(x)=0\}$$ is a hypersurface of $\mathbb{R}^{n}$, where $j(f)(x)$ ...
0
votes
4
answers
457
views
Confining a polytope to one side of an affine hyperplane
Judging whether one convex polytope is inside of another when both are expressed as a system of linear inequalities seems not to be an easy problem.
This answer on math.stackexchange.com claims the ...
- 2,251
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votes
0
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123
views
A quantity associated with an algebraic variete
Let $P:\mathbb{C}^{n}\to \mathbb{C}$ be an irreducible homogenous polynomial.
Is there a geometric or algebra geometric interpretation for the following quantity:
The maximum number $k$ such that ...
- 366
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votes
1
answer
1k
views
Are absolute Galois groups condensed?
Let $k^{s}$ be a separable closure of a field $k$. Is $Gal(k^s/k)$ a condensed group in the sense of condensed mathematics? If condensed, is it always solid?
- 27
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votes
3
answers
3k
views
When is the union of embedded smooth manifolds a smooth manifold?
Suppose we have k embeddings of one single smooth manifold into one other, such that the intersections are manifolds,too. What are sufficient conditions, such that the union of those embeddings is a ...
- 137
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votes
1
answer
330
views
Sum of the first m terms of the expansion $(x+y)^n$
Let $S(x, y, m, n) = \sum\limits_{i=0}^m \binom{n}{i}x^i y^{n-i}$, where $0 < m < n$. I want to derive the relation between $S(x, y, m, n)$ and $S(x, y, m, n-1)$.
Is there any formulas I can use?...
- 113
-1
votes
1
answer
218
views
Function on quadratic numbers
Let $\mathbb{N}$ denote the set of the positive integers. We consider the following function $f:\mathbb{N}\times \mathbb{N}\to \mathbb{Q}$: $$f(a,b)=\frac{a^2+b^2}{1+ab} \text{ for all } a,b\in\mathbb{...
- 48.9k