Newest Questions
159,084 questions
2
votes
1
answer
370
views
Tests for determining membership of exponential family
Provided an arbitrary random variable $X$ with probability density/mass function $f$, are there any tests to determine if $f$ forms an exponential family?
Certainly, if $f$ can be written in the form
$...
1
vote
0
answers
84
views
Number of polyhedral covers of a triangulation of $S^2$
For a given triangulation (combinatorial Type I. or Type II.) of a $2$-sphere, what is the number of unique polygonal covers with $n$ polygons where ($n$ goes from $2$ to $N$)?
Under polygonal cover, ...
- 183
14
votes
0
answers
241
views
Unitary group of a von Neumann algebra: is it a retract of $U(H)$?
Let $M\subset B(H)$ be a properly infinite von Neumann algebra (the case I care about is $M=$ hyperfinite $\mathrm{III}_1$).
Consider the unitary groups $U(M)$ and $U(H)$ in their strong operator ...
- 43.2k
1
vote
0
answers
120
views
Large Tate-Shafarevich group of an elliptic curve with the form $E_{p,n}:y^2=x^3+p^nx$
Let $p$ be a prime number and $n$ be positive integer.
Let $E_{p,n}:y^2=x^3+p^nx$ be an elliptic curve.
LMFDB reads in the case $(p,n)=(73,3)$ , $\#Sha(E_{p,n})=64$.
This is the biggest size of $Sha(...
- 1,541
3
votes
0
answers
70
views
Is every finite spectrum $X$ $K(h)$-locally equivalent to a finite spectrum $Y$ with $\dim (K(h)_\ast Y) = \dim ((H\mathbb F_p)_\ast Y)$?
Let $X$ be a finite spectrum and $K = K(h)$ be the $h$th Morava $K$-theory at the prime $p$. Then $\dim_{K_\ast} K_\ast X$ is increasing in $h$, and eventually constant at $\dim H_\ast(X,\mathbb F_p)$....
- 64k
7
votes
1
answer
178
views
Homogeneous metric connections on 3-dimensional Lie groups
Let $G$ be a 3-dimensional unimodular Lie group equipped with a left-invariant metric $q$. Call $P_{SO}$ its oriented orthonormal frame bundle.
Considering the moduli space of connections $\mathscr{B}$...
- 73
6
votes
1
answer
213
views
Is there a subgroup of a non-abelian $p$-group $G$ with a large nilpotency class?
Let $G$ be a non-abelian $p$-group ($p\ne2$). Does there exist a group $H\subset G$ such that both 1, 2 are satisfied?
$|H| = |G|/p$.
$c(H)\geq c(G) - 1$.
- 291
7
votes
1
answer
275
views
Does the inner automorphism group of the fundamental group of a closed aspherical manifold always have an element of infinite order?
Let $\pi_1$ be the fundamental group of a closed aspherical manifold of dimension $n$. In particular, $\pi_1$ is finitely presented, torsion-free and its cohomology is finitely generated and satisfies ...
- 73
4
votes
0
answers
246
views
Dynamical obstruction for a vector field to have a Harmonic divergence
Let $(M,g)$ be an analytic Riemannian manifold and $X$ be an analytic vector field on $M$. Can we always have a volume form $\Omega$ such that $\operatorname{Div}_{\Omega} X$ is a harmonic ...
- 366
12
votes
0
answers
168
views
Can the optimal packing density in $\mathbb{Z}^d$ be irrational?
For a finite $S \subset \mathbb{Z}^d$, let $d_p(S)$ be its optimal packing density. That is, the maximal lower asymptotic density of $A+S$, where $A \subset \mathbb{Z}^d$ is such that $(a_1+S)\cap (...
- 1,209
1
vote
0
answers
154
views
The space of ergodic elements of a topological or Lie group
Let $G$ be a compact topological group with normalized Haar measure $\mu$. An element $g\in G$ is an ergodic element if the mapping $L_g:G \to G $ with $x\mapsto gx$ is an ergodic map. The ...
- 366
0
votes
1
answer
87
views
Is the $2$-point function translation invariant for general Gaussian meaures?
Let us consider the real Hilbert space $H:=L^2\bigl(\mathbb{R}^n, \mathbb{R}^n\bigr)$ and "any" centered Gaussian measure $d\mu$ on it.
Next, denote a generic element of $H$ by the column ...
- 3,487
2
votes
0
answers
70
views
Zero sets of integral power series that converge on disks
Fix a radius $r \leq 1$. I'm interested in any necessary conditions, or any sufficient conditions, for a subset $S$ of $B(0,r)$, the origin-centered open disk of radius $r$, for $S$ to be the set of ...
- 17.6k
5
votes
1
answer
159
views
Do these $p$-groups have the same nilpotency class?
Let $G$ be a $p$-group, $\{e\}\not= H\subseteq G$ be a subgroup of $G$ such that $G' = H'$. Is it true that $c(G) = c(H)$, where $c(\cdot)$ denotes the nilpotency class of a group?
- 291
1
vote
0
answers
41
views
The boundedness of dynamical systems discretized from Hamiltonian systems
Let $H(p,q) = T(q) + U(p)$ be a Hamiltonian function that defines a Hamiltonian system, i.e.,
\begin{align}
&\frac{dp}{dt} = \frac{\partial H}{\partial q}(p,q) = \frac{dT}{dq},\\
&\frac{dq}{dt}...
- 47
2
votes
1
answer
111
views
Are there interesting examples of unitary fusion categories where a tensor product of two simple objects is simple?
Let $\mathcal{C}$ be a unitary fusion category. Is it true that the tensor product of any two simple objects is simple ? If not, are there interesting (nontrivial) examples of such a $\mathcal{C}$ ?
- 21
2
votes
0
answers
72
views
Fourier Transform Of Fractional Laplacian [closed]
Fourier Transform Of Fractional Laplacian
I dont know why we have the last 2 inequality and why it occurs the characteristics ball B1
1
vote
0
answers
54
views
Minimal F-semi-norms
There are conflicting terminologies in the literature on this subject, so let me define an F-semi-norms on a real vector space $E$ to be a subadditive function $\rho:E\to[0,+\infty)$ such that $\rho\...
- 5,529
1
vote
0
answers
67
views
Are the lower elementary functions closed under limited recursion?
The lower elementary functions (also called Skolem elementary functions) are functions generated from the successor, modified subtraction, projection functions by the operations of composition and ...
- 1,874
13
votes
1
answer
1k
views
Apéry's constant $\zeta(3)$ fastest convergent series
UPDATE Feb.02.2024
The series below, Eq.(3) for computing and Eq.(2) for verifying, were applied by Andrew Sun on Dec.22.2023 to get over $2\cdot10^{12}$ decimal digits and break the number of ...
- 2,836
1
vote
0
answers
74
views
Automorphic images of cones in free group
Let $F_2$ be the free group with basis $\{a,b\}$, with corresponding word metric $d$. For $x\in F_2$, the cone $C(x)$ is $C(x):=\{y\in F_2\mid d(1,y)=d(1,x)+d(x,y)\}$, that is, the set of elements ...
- 3,740
2
votes
0
answers
114
views
Differential graded modules and the Serre-Swan theorem
I am thinking about how connections combine with a modification of the Serre-Swan theorem, which relates vector bundles to projective modules.
If $E \rightarrow B$ is a vector bundle, or even just any ...
user30211
1
vote
2
answers
181
views
Solution of $\Delta f -\frac{1}{2}hf = 0$ behaves asymptotically as $f(x) = 1 - C/|x|$
Let $f: \mathbb{R}^{3} \to \mathbb{R}$ be the solution of the following PDE:
$$\Delta f -\frac{1}{2}h f = 0$$
where $h \in C_{c}^{\infty}(\mathbb{R}^{3})$ (compactly supported an smooth) and $f$ ...
- 3,535
8
votes
1
answer
724
views
How "correct" is Knuth's fast addition $(a,b) \mapsto (a \oplus b) \oplus ((a\land b) \ll 1)$?
Donald Knuth suggested a bitwise approximation for addition on the non-negative integers that is very fast on common processors:
$(a,b)\mapsto (a\oplus b) \oplus ((a\land b) \ll 1)$,
where $a,b$ are ...
- 48.9k
12
votes
1
answer
471
views
Is the Grothendieck construction a homotopy pullback?
The category of elements of a functor $F:\mathcal C\to\mathsf{Set}$ can be obtained as the strict pullback in with the forgetful functor of pointed sets $\mathsf{Set_*}\to\mathsf{Set}$:
$$
\begin{...
- 527
1
vote
1
answer
311
views
Weak convergence in $H^{1}$ implies different convergence in $L^{p}$?
Suppose I have a sequence $\{f_{n}\}_{n\in \mathbb{N}} \subset H^{1}(\mathbb{R}^{d})$ which converges weakly to $f$ in $H^{1}(\mathbb{R}^{d})$, in the sense that $\langle f_{n},\varphi \rangle_{L^{2}}+...
1
vote
1
answer
84
views
optimization over moving domains
Let $A, B$ be Banach spaces, and for any $a\in A$, $B_a\in B$ is a measurable subset. Consider the following optimization problem:
$$L(a)=\inf_{b\in B_a}\ell(b),$$
where $\ell(b)$ is a infinite-times ...
- 87
2
votes
1
answer
160
views
Measurability of two hitting times at the stopped $\sigma$-algebra
Let $\mathcal{F}=(\mathcal{F}_t)_{t\ge 0}$ be the complete filtration generated by the Brownian motion $B $, and let $a<0<b$. Define the stopping times
$\tau_a=\inf\{t\ge 0\mid B_t=a\}$ and $\...
- 503
5
votes
1
answer
212
views
Are $\infty$-categories functorially colimits of their simplices?
Let $\mathcal C$ be an $\infty$-category. If $C$ is a quasicategory modeling $\mathcal C$, then we have a coend decomposition
$$\mathcal C = \int^{[n] \in \Delta} \Delta[n] \times C_n.$$
This allows ...
- 64k
2
votes
1
answer
194
views
Minimal degree of a polynomial such that $|p(z_1)| > |p(z_2)|, |p(z_3)|, ..., |p(z_n)|$
I was investigating the behavior of $p(x)^n \mod {q(x)}$, for some polynomials $p, q \in \mathbb{C}[x]$. We'll assume $q$ is squarefree. If $q(x) = (x - z_1) (x - z_2) (x - z_3) ... (x - z_n)$ for ...
- 3,319
0
votes
1
answer
152
views
Almost Pell type equation
Consider the following Diophantine equation
$$
2x^2-Ny^2 = -1.
$$
where $N$ is an integer. Is there any result expressing the values of $N$ for which the above equation admits an integral solution?
- 8,998
1
vote
0
answers
117
views
Reduction mod 2 for orthogonal groups
Setting Let $k$ be a real quadratic field, $\mathbb Z_k$ its ring of integers. Let $n$ be an even integer $A$ a symmetric $n$-by-$n$ matrix with coefficients in $\mathbb Z_k$. Let $L$ be the lattice $\...
- 3,399
1
vote
0
answers
93
views
Basis of subgroup of free group
Let $F_2$ be a free group on $2$ generators $a, b$. We know $b$ and a conjugate of $b$, which is different from $b$, generate rank 2 free subgroup of $F_2$ and they are free generating set of the ...
- 11
0
votes
1
answer
115
views
Lifting an automorphism of a curve to an automorphism of its Jacobian
Let $C: Y^3Z = f(X,Z)$, with $f(X,Z)\in K[X,Z]$, a degree 4 homogeneous polynomial, and $K$ a field.
The curve $C$ has an order $3$ automorphism, given by sending $(x,y,z)$ to $(x,\omega y, z)$, where ...
- 2,907
1
vote
0
answers
145
views
Complexity of calculating the expectation of $\operatorname{Tr} h(A)$, $A$ is a random matrix
$A$ is a $d_1\times d_1$ random matrix. Given $\{g_i\}~(1\leq i\leq n)$ iid Gaussian variables, $f_{ij}(g_1,g_2,...,g_n)~(1\leq i,j\leq d_1)$ are degree-$d_2$ polynomials. And $f_{ij}\equiv f_{ji}~(\...
- 91
3
votes
1
answer
245
views
Integration against Eisenstein series can be regarded as a cup product
This summer, I was very fortunate and honored to attend the conference "Iwasawa 2023" at the University of Cambridge as a young Ph.D. student on Iwasawa theory. There, one of the speakers, ...
- 639
11
votes
2
answers
769
views
Are topological PID's Noetherian?
Romain Giquaud has given a counterexample to the general form of the question. The bounty is for a solution for locally compact, metrizable rings. (I suspect the answer may be positive with this ...
- 42.8k
5
votes
1
answer
593
views
MREF tool and TeX formatting
For sometime I have used the very useful MathSciNet MRef tool. It allows one to input a citation (e.g. from zbMath or from MRLookup, if you don't have MathSciNet access, as I didn't for a while, and ...
- 18.7k
2
votes
1
answer
175
views
Clique number of $k$-critical graphs
A graph $G$ is called a ${\it{k}}$-${\it{critical}}$ graph if $\chi(G)=k$ and for any proper subgraph $H$ of $G$ we have $\chi(H)<k$， where $\chi(G)$ denotes the chromatic number of $G$. The ...
- 51
1
vote
0
answers
128
views
Representability of twists of projective schemes
Let $K$ be a perfect field, and let $S$ be a projective $K$-scheme. Denote by $\text{Twist}(S/K)$ the set of twists of $S$ up to $K$-isomorphism. These are (apriori) sheaves $\mathcal{F}$ on the ...
- 2,907
10
votes
0
answers
488
views
Reconstruction of commutative differential graded algebras
Let $k$ be an algebraically closed field of characteristic $0$.
Let $A,B$ be commutative differential graded algebras (cdga) over $k$ such that $H^{i}(A)=H^{i}(B) =0 \ (i>0)$.
Here, differentials ...
- 889
3
votes
1
answer
340
views
On a Poincaré inequality with weight
Let $\Omega$ be a bounded convex (non-empty) open subset of $\mathbb{R}^n$ ($\Omega$ can be as smooth as you like). Let also $p, q > 1$ be conjugate exponents.
Is it true that there exists a ...
- 1,263
0
votes
1
answer
140
views
nonlinear equation problem
Can you please help me solve the following nonlinear equation to determine the value of the vector $z$ :
$$
\boldsymbol{a}=\boldsymbol{z}^{2} \odot \boldsymbol{K}*\boldsymbol{z}^{-1}$$
Where:
$\...
- 11
1
vote
0
answers
95
views
Linear Program Optimal Value
If $f(A,b,c)$ is the optimal value of a linear program
$\min c.x$
subject to $A.x \leq b ; x \geq 0.$
Does $f(A,b,c)$ have a piecewise polynomial/rational upper bound in $(A,b,c)$ on the domain of ...
1
vote
1
answer
150
views
Relative $G$-equivariant homology groups
Let $X$ be a free $G$-CW-complex with $G$-equivariant cell filtration by
$n$-skeleta $X_0 \subset \dots \subset X_n \subset \dots \subset X$ (for
rigorous definition see
Chap. II, p. 98 in linked ...
- 5,966
7
votes
1
answer
503
views
Combinatorial consequences of de Branges's Theorem?
I'm usually not a proponent of the mentality “here is a tool, what results can we prove with it?” (as I prefer to start at the other end with a well-motivated problem), but this famously entertaining ...
- 191
3
votes
2
answers
617
views
Negative of combinatorial game
I am having problem understanding what negative of a combinatorial game $G$ exactly means in combinatorial game theory. Does it mean that if I have normal game, if I create inverse, i.e., $-G = \{-G^R ...
- 31
2
votes
0
answers
169
views
Understanding the Seiberg-Witten equations in dimension $3$
I am trying to understand the dimensional reduction of Seiberg-Witten equations from dimension $4$ to $3$, more specifically my concern is about ellipticity of the new equations in dimension $3$ under ...
- 954
2
votes
0
answers
114
views
What is the quantity $\sqrt{\frac{c^2+d^2}{a^2+b^2}}$ of a matrix with determinant one?
Suppose that $A \in \mathbb R^{2 \times 2}$ has determinant one,
$$
A = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
$$
I'm currently working on a problem where I obtained a condition on the ...
- 173
6
votes
2
answers
724
views
In the constructive theory of direct categories, is it decidable whether an arbitrary morphism is an identity or not?
I'm wondering what the legit definition of direct categories should be in constructive mathematics. I must admit I don't even know in what literature I should look for the definition. I would ...
- 366