All Questions
153,420
questions
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3
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Simultaneous Concentration of $\sum_{i = 1}^{n} X_i^2$ and $\sum_{i = 1}^{n} X_i$ with $X_i$ iid. Poisson
Consider $n$ independent Poisson(1)-distributed random variables $(X_i)_{1 \leq i \leq n}$.
This is a (hopefully more interesting) follow-up question to Super-exponential concentration for $\frac{\...
- 99
0
votes
0
answers
6
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Reference request: amplification argument for hyperlinear groups
Let us define a group $G$ to be a hyperlinear group if it satisfies the conclusion of Theorem 3.6. in these notes by Vladimir Pestov. It is well-known that one can use the so-called amplification ...
- 6,084
-2
votes
0
answers
11
views
How do you find the volume of a curvilinear body bounded by equations through an integral? [closed]
guys. Say me pls, I have two equations that constrain a figure on the (x , y) axis.
y = 1 - x^2
y = 0
graph
Here is my assignment: Find the volume of the body formed by rotating a figure around the x-...
- 1
1
vote
0
answers
31
views
Does the existence of “strong Lebesgue averages” imply absolute continuity?
Let $f: \mathbb R \to \mathbb R$ be a measurable function. Suppose there exist some function $g \in L^1 (\mathbb R)$, and a measurable set $E \subset \mathbb R$ of full measure such that
$$\lim_{r \to ...
- 4,842
0
votes
0
answers
36
views
A possible variant of Zagier's one-sentence proof for Fermat's sum of two squares theorem?
Is it possible to modify Zagier's one-sentence proof of Fermat's sum of two squares theorem (see here) to prove certain non-trivial cases of Jacobi's four-square theorem (see here)?
Let $p$ be a prime ...
- 21
0
votes
0
answers
19
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Does the complex interpolation space $(L^1(\mathbb{R}),W^{2,1}(\mathbb{R}))_{\frac{1}{2}}$ continuously embed into $L^\infty(\mathbb{R})$?
The complex interpolation space between $(L^p(\mathbb{R}),W^{2,p}(\mathbb{R}))_\theta$ with interpolation parameter $\theta=\frac{1}{2}$ is known to be $W^{1,p}(\mathbb{R})$ for $1<p<\infty$. As ...
- 919
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votes
0
answers
21
views
Sequence that sums up to A224071
Let $a(n)$ be A224071 (i.e., number of Schroeder paths of semilength $n$ in which there are no $(2,0)$-steps at level $1$). Here
$$
a(n) = \frac{1}{2(n+1)}\sum\limits_{k=0}^{n}(k+1)((-1)^{\left\...
- 3,464
6
votes
3
answers
436
views
Real-world examples of unweighted directed graphs
Social Networks, Internet Traffic, Citation Networks, Transportation Networks, and Biological Networks exemplify real-world instances of unweighted directed graphs. Within these domains, the Graph ...
- 149
0
votes
0
answers
5
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Bounded density for determinant of GOE
Let $M$ a random GOE matrix, i.e. $M=(M_{i,j})$ is a symmetric matrix and the $M_{i,j},i\leq j$ are independent centred Gaussien entries with variance 1, except on the diagonal where the variance is $...
- 1,268
23
votes
5
answers
4k
views
Situation with Artemov's paper?
Artemov's paper on Goedel's theorem has been on the arxiv since 2019. There was a (less than fully friendly) discussion of this on FoM. At stackexchange, I found only a brief mention at this MSE ...
- 15.4k
4
votes
1
answer
130
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Group Completion of a monoid (Braid groups)
Let $B_n$ be the braid group on $n$ strands, $B_{\infty}$ the direct limit of braid groups.
For a discrete group $G$, we let $BG$ to be the classifying space of $G$.
After reading this question, I was ...
- 140
0
votes
1
answer
14
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Is that possible to upper bound the absolute value of the difference of two Kullback-Leibler divergences given the constrains below?
Given that
$D[p_1||p_3]\leq \epsilon$, $D[p_3||p_1]\leq \epsilon$, $D[p_2||p_4]\leq \epsilon$ and $D[p_4||p_2]\leq \epsilon$,
is that possible to place an upper bound on the quantity $\left \vert D[...
0
votes
1
answer
25
views
Orthogonality in Hilbert algebras and congruence
Consider a finite-dimensional Hilbert space $V$ (say, over $\mathbb{C}$) and a finite-dimensional Hilbert algebra $A$ (i.e., Hilbert space with a compatible associative unitary algebra structure). ...
- 42.2k
8
votes
2
answers
857
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Commutation relations between covariant and Lie derivatives
I am currently working on extrinsic riemannian geometry and I am looking for a sort of commutation relation between the covariant and Lie derivatives.
To be more precise : considering an hypersurface $...
- 1
2
votes
3
answers
205
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Fundamental group of a generalized connected sum
Let $M$ and $N$ be two $n$ dimensional connected closed manifolds with $n \ge 3$, and let $S$ be a $(n-1)$ dimensional closed submanifold common to $M$ and $N$. Consider the connected sum of $M$ and $...
- 24.2k
0
votes
0
answers
42
views
Convexifying a Non-Linear Fractional Function
I am working on a problem that involves a non-convex, non-linear fractional function:
$$
Y(X_1, X_2) = \frac{X_1 + X_2}{\alpha X_1 + \beta X_2}
$$
Where $X_{1}$, $X_{2}$, $Y$ are decision variables ...
1
vote
1
answer
323
views
A sequence and majorization
For two positive vectors $a,b$ such that $a\prec b$, we know that there is an $m$ sequence of vectors $c^{(i)}$ such that $$a\prec c^{(1)}\prec \ldots \prec c^{(m)}\prec b$$ where each vector in the ...
CommunityBot
- 1
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votes
1
answer
74
views
An asymptotic integral with complex phase
Suppose that $D\subset \mathbb R^2$ is the closed unit disk and that $f\in C^{\infty}(D)$. Assume that for all $\lambda \in (1,\infty)$ there holds
$$ \left|\int_D f(x^1,x^2)\, e^{\lambda (x^1+ix^2)}\,...
- 5,096
9
votes
0
answers
169
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Are these comparison morphisms between Čech and Grothendieck cohomology the same?
For better or for worse, there is more than one approach to comparing Čech cohomology $\smash{\check{H}}^\bullet(\mathfrak{U},X;\mathscr{F})$ of a sheaf $\mathscr{F}$ on a space $X$ w.r.t the cover $\...
- 721
1
vote
1
answer
78
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Prékopa-Leindler style inequality?
Does anyone know a simple proof of the following Prékopa-Leindler style inequality:
If we have $f_1,f_2,g_1,g_2$ strictly positive functions on $\mathbb{R}$ such that, for any $x_1,x_2 \in \mathbb{R}$,...
- 103k
0
votes
0
answers
14
views
Small total variation distance between sums of random variables in finite Abelian group implies close to uniform?
Let $\mathbb{G}$ be $\mathbb Z_p$. Let X,Y,Z be independent random variables in $\mathbb G$.
For a small $\epsilon$ we have $dist_{TV}(X+Y,Z+Y)<\epsilon$.
Assuming X and Z are disjoint, I'd like to ...
- 1
0
votes
0
answers
9
views
Strict positive definite function gradient tuple
I have a (Gaussian) random function (aka "stochastic process" or "random field") $(f(t))_{t\in \mathbb{R}^d}$. I now want to consider the vector valued random function $g=(f, \...
- 357
1
vote
0
answers
14
views
System of Local Isos gives System of Local Epis
Suppose that $W$ is a system of local isomorphisms on a presheaf topos $\mathbf{Pre}(\mathcal{C})$. We say a map in $W$ is a $W$-local isomorphism, and we say that a map of presheaves $f: X \to Y$ is ...
0
votes
0
answers
44
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Calculation of the distance of cocycles (the telescope formula for the difference of the nth iterates)
Let $T: X \to X$ be a Lipschitz continuous on a compact metric space $(X, d_1)$. Assume that $Y$ is a Banach algebra and we consider the metric $d_2$ in the space $Y$. Let $f:X \to Y$ be a uniform ...
- 1,011
4
votes
0
answers
55
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If matrices $A$ and $B$ are normal with $\sigma(A),\sigma(B)\subseteq \mathbb{R}\cup \mathbb{T}$, does $\text{rank}([A,B])=\text{rank}([A^*,B])$?
Here $\mathbb{T}=\{z\in\mathbb{C}: |z|=1\}$ denotes the unit circle in the complex numbers.
This holds, if we have $\sigma(A)\subseteq \mathbb{R}$ or $\sigma(A)\subseteq \mathbb{T}$ (independent of $B$...
- 151
1
vote
0
answers
36
views
Short path problem on Cayley graphs as language translation task (from "Permutlandski" to "Cayleylandski"(s) :). Reference/suggestion request
Context: Algorithms to find short paths on Cayley graphs of (finite) groups are of some interest - see below.
There can be several approaches to that task. One of ideas coming to my mind - in some ...
1
vote
1
answer
83
views
Abstract algebraic link between two problems involving polynomials and (generalized) Vandermonde matrices?
Let two integers $L\leq N$, a sequence of $L+1$ distinct real numbers $\alpha_0,\dots,\alpha_L$, and integers $k_0,\dots,k_L\in\mathbb N$ such that $N=L+\sum_{i=0}^Lk_i$.
I noticed that the two ...
6
votes
1
answer
215
views
What integers can be represented as $\textrm{lcm}(a, b) + \textrm{lcm}(b, c) + \textrm{lcm}(c, a)$?
Find all integers $n$ that can be written as $n = \textrm{lcm}(a, b) + \textrm{lcm}(b, c) + \textrm{lcm}(c, a)$ where $a$, $b$, $c$ are all integers.
- 1,321
0
votes
0
answers
20
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Extension of automorphism of shift of finite type. Is $\mathrm{Aut}(X)\subset \mathrm{Aut}(Y)$?
$\DeclareMathOperator\Aut{Aut}$Let $X$ and $Y$ be SFT and $X\subset Y$, and $\phi:X\rightarrow X$ be a homeomorphism commuting with the shift map. Is there any extension of $\phi$ on $Y$ whic is a ...
- 60.4k
1
vote
1
answer
77
views
$\operatorname{Hess}r$ is scalar matrix $\implies$ $M$ is isometric to the space form
I'm trying to prove the rigidity part of a theorem in my paper, which requires the use of the classical Hessian comparison theorem's rigidity part:
$$\DeclareMathOperator\sn{sn}\operatorname{Hess}r=\...
- 133
6
votes
0
answers
117
views
Running minimum of exponential random walks
Let $\{X_i\}$ be a collection of i.i.d. Exp$(1)$ random variables. For $k \in \mathbb{Z}_{>0}$, define
$$S_k = \sum_{i=1}^k X_i$$
and note that $\mathbb{E}[S_k] = k$.
I was wondering if there is ...
- 186
11
votes
2
answers
477
views
Cohomology for extension problems in symbolic/topological dynamics?
Context: I know essentially nothing about cohomology of any kind, but I have a problem involving classifying obstructions to extensions of certain maps or covers, and I have heard that cohomology is ...
0
votes
0
answers
22
views
Measurability of the upper and lower Bowen entropy spectra
Let $(Y,\rho)$ be a compact metric space and $T:Y\rightarrow Y$ be a continuous transformation. Let $\mathcal{M}(Y)$ be the set of all Borel probability measures on $Y.$
For any $n\in N,$ $x, y\in Y$ ...
- 39
6
votes
2
answers
166
views
Counting adjoints in the symmetric or antisymmetric square of a Lie group representation
EDIT (November 1, 2022): Over the weekend I think I found a technique to determine the exact multiplicities, according to how conjugation acts on the fundamental weights. While I haven't done the ...
- 1,364
6
votes
2
answers
173
views
Euclidean algorithm for simple closed curves
In the proof of Proposition 6.2 in Farb & Margalit, "A primer on mapping class groups", an analog of the Euclidean algorithm is used to construct a simple, closed representative (...
- 26.4k
7
votes
0
answers
143
views
+50
Word maps from $\mathrm{Cl}^{n+1}$ to $G^n$: quasi-injectivity?
Let $G = \mathrm{SL}_n$ (say). Then, for $g$ regular semisimple, the conjugacy class $\mathrm{Cl}(g)$ has dimension $= n^2-n$ as a variety, and so $\mathrm{Cl}(g)^{n+1}$ and $G^n$ have the same ...
- 19.4k
2
votes
2
answers
117
views
Compute Christoffel symbols of sphere by embedding
In his answer V. Semeria, starts by taking
$$(y_1,\dots,y_{n+1})=\left(x_1,\dots,x_n,\sum_{i=1}^{n+1}x_i^2 -R^2\right)$$
Write $(\vec{e}_1,\dots,\vec{e}_{n+1})$ the canonical basis of $\mathbb{R}^{n+1}...
- 21
2
votes
1
answer
53
views
Given a low-rank symmetric positive semidefinite matrix and a basis of its nullspace, is there a fast way to get the nonzero eigenvalues?
I have a (possibly dense) $k\times k$ real matrix $L = AA^T + B^T B$, a type of combinatorial Laplacian (self-adjoint, symmetric, positive semidefinite) of rank $(k-n)$ and possibly repeated nonzero ...
4
votes
1
answer
157
views
New vectors for representations of GSp4 with nontrivial central character
Roberts and Schmidt have developed a theory of new vectors for generic irreducible smooth representations of $\operatorname{PGSp}_4(F)$ for $F$ a nonarchimedean local field, using the "paramodular ...
- 36.2k
-1
votes
0
answers
48
views
Integrating $\int_{-\infty}^{\infty}\exp(-x^2/(1-ix))(1-ix)^{-1/2}{\rm d}x$
I want to calculate the integral
$$
I=\int_{-\infty}^{\infty}\exp\left(\frac{-x^2}{1-ix}\right)(1-ix)^{-1/2}\ {\rm d}x.
$$
I got two ways to deal this integral. The first, noting that
$$
\int_{-\infty}...
- 429
-3
votes
0
answers
37
views
find delta of each numbers in a number list such that each consecutive number differs at most by delta and first and last number also differ by delta [closed]
Given a sequences of numbers a0, a1, a2, ..., an, a0 and epsilon each number is randomly initialized (i don't know if this is relevant), find a sequences of delta d0, d1, ..., dN, such that for 0 <=...
6
votes
4
answers
611
views
What is the $\inf$ and $\sup$ of the area of quadrilateral given its sides length?
I asked this question on MSE here.
Given the length of the sides of a quadrilateral $a,b,c,d$ ( side lengths are given in order around the quadrilateral) the area of the quadrilateral is less than or ...
- 41
4
votes
2
answers
156
views
Independence number of configuration graph (consisting of all (2k-1)!! ways to partition {1,2,...,2k} into k pairs)
Let $k$ be a nonnegative integer. A configuration on $2k$ labeled points is simply a partition of the points into $k$ pairs, so for any set of $2k$ labeled points, there are $(2k-1)!!$ configurations.
...
3
votes
1
answer
283
views
Global duality theorem for 2-part
$\DeclareMathOperator\coker{coker}\DeclareMathOperator\Sha{Sha}$Let $K$ be a number field.
Let $E/K$ be an elliptic curve over $K$.
Suppose finiteness of $\Sha(E/K)$.
According to Global duality ...
CommunityBot
- 1
3
votes
0
answers
61
views
When is the intersection of cosets of a conjugacy class $0$-dimensional?
Let $G = \mathrm{SL}_n$ (say); let $K$ be a field. Let $g$ be a regular semisimple element of $G(K)$, and $\mathrm{Cl}_g$ its conjugacy class, considered as an algebraic variety. Then $\mathrm{Cl}_g$ ...
- 19.4k
7
votes
2
answers
6k
views
On a proof of the existence of tubular neighborhoods.
Studying analysis on manifolds, I have found, in the proof of the existence of tubular neighborhoods, a reference to theorem 3.1.2 in "Topologie algebrique et theorie des faisceaux" of Godement.
...
- 1,446
2
votes
0
answers
23
views
A question about the product of manifolds with nonnegative curvature operators with spheres
Let $(M,g_M)$ be a closed connected $n$-dimensional Riemannian manifold with nonnegative curvature operator. Let $S^n(R)$ be a sphere of radius $R$ with standard metric $g_0$.
My question: Is the ...
- 573
11
votes
3
answers
932
views
sums of fractional parts of linear functions of n
As $\alpha$ and $\gamma$ range uniformly over $[0,1]$, what is the typical (e.g. median or root-mean-square) order of magnitude of $C_m (\alpha,\gamma)$ := $\sum_{1 \leq k \leq m} \left( {\rm frac}(k\...
- 4,618
0
votes
1
answer
308
views
Douglas' lemma for integral operators
Douglas' lemma gives a necessary and sufficient condition for two bounded operators $A$ and $B$ on a Hilbert space $H$ to get $\operatorname{Ran} A \subset \operatorname{Ran} B$ (wikipedia article, ...
CommunityBot
- 1
3
votes
1
answer
106
views
Gaussian free field from Liouville quantum gravity?
If $\Sigma$ is a Riemann surface, there are two measures on $\text{H}^s(\Sigma)$:
the Gaussian free field $h(z)$ and
the Gaussian multiplicative chaos $\mu(z)= \lim_{\epsilon\to0} e^{\gamma h_\...
- 3,555