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8 views

Can a hypergeometric function have two differently sized matrices as input?

I'm currently trying to understand how to calculate the densities of certain Gaussian random matrix ensembles and I'm having trouble with Theorem 7.2.1 from [1]. As far as I can tell, the ...
  • 439
0 votes
0 answers
8 views

Mean Value Theorem for Dirichlet series of prime support?

Let $\{a_n\}_{1\leq n\geq N}$, $a_n\in \mathbb{C}$. Let $F(s) = \sum_{n=1}^N a_n n^{-s}$. By a mean-value theorem (Montgomery-Vaughan, 1973), $$\int_0^T |F(i t)|^2 = \sum_{n=1}^N |a_n|^2 (T + O(n)).$$ ...
  • 17.1k
1 vote
0 answers
12 views

Proof of the projection formula (for cohomology of $\mathbf{P}V$)

Let $V\to X$ be a vector bundle (over say a scheme). Then the cohomology of its projectivisation is $$\text{H}^*(\mathbf{P}V)\ =\ \text{H}^*(X)[t]/(t^{n+1}+c_1(V)t^n+\cdots+c_n(V))$$ as an algebra, ...
  • 4,224
2 votes
0 answers
11 views

Finite domination and Poincaré duality spaces

Here are some definitions: A space is homotopy finite if it is homotopy equivalent to a finite CW complex. A space finitely dominated if it is a retract of a homotopy finite space. A space $X$ is a ...
  • 17.7k
1 vote
0 answers
13 views

Fourier-Mukai kernels for Fano threefolds

Let $Y_1$ and $Y_1'$ be index two degree one Fano threefolds. Suppose we have a Fourier-Mukai equivalence $\Phi_P : \mathrm{D}^b(Y_1) \to \mathrm{D}^b(Y_1')$. Can anything be said about the kernel $P$,...
  • 233
-1 votes
0 answers
3 views

Choice of approximate posterior in variational inference with positive support

I have a simple probabilistic graphical model: $z \longrightarrow x$ where $z_i \sim Exp\left(\lambda_i\right)$ where subscript $i$ denotes the $i$th dimension and $x|z \sim \mathcal{N}\left(f\left(z\...
1 vote
1 answer
23 views

Before focal point, the locally distance function is smooth

I'm reading Eschenburg's paper Local convexity and nonnegative curvature — Gromov's proof of the sphere theorem recently. And I meet a little question: In the proof of Lemma 7.4, He let $d$ be the ...
1 vote
0 answers
17 views

Under what conditions a continues linear map maps a closed subspace to a closed subspace

Are there natural conditions that ensure that a continuous linear map $\phi:V\to W$ between TVS maps any closed subspace $L\subset V$ to a closed subspace in $W$. It is obviously satisfied if $W$ is ...
  • 2,287
1 vote
0 answers
20 views

Presentation complex and arbitrary $2$-dimensional CW-complex with same fundamental group

Given a finite group $G$, consider a presentation $P$ of $G$ and consider $X_P$, the presentation complex. Now let $Y$ be any $2$-dimensional CW-complex with $\pi_1(Y)=G$. Is there any relation ...
0 votes
1 answer
31 views

Approximation from above of subharmonic functions

Let $u$ be a subharmonic function on $\mathbb{C}$. It is known that the convolution with standard mollifier gives a sequence $u_{\epsilon}$of subharmonic functions with the property: $u_{\epsilon}\in ...
  • 1
-5 votes
0 answers
25 views

How would I be able to work this question out? [closed]

A stuffed toy in the shape of an ice cream cone consists of a hemisphere attached to the base if t=a cone in such that both the hemisphere and the cone have the same radius. the ratio of the volume of ...
  • 1
0 votes
0 answers
54 views

Integral over $\Bbb C$ [closed]

Is this correct? Let $a\in\Bbb C^*$ and $f\in L^1(\Bbb C^2)$ $$\int_{\Bbb C^2}|f(a(z,w))|dz dw={1\over |a|^2}\int_{\Bbb C^2}|f((z',w'))|dz' dw'$$ where $dz$ is the usual measure Lebesgue on $\Bbb C$ ...
-4 votes
0 answers
72 views

Does the permanent equals the determinant over $\mathbb{F}_{2^n}$? [closed]

It is known that the permanent equals the determinant modulo $2$. We got numerical evidence that this generalizes to $\mathbb{F}_{2^n}$. Is it true that for all square matrices with entries from $\...
  • 23.6k
0 votes
1 answer
35 views

Analytic formula for minimizer of $f(x) := \sqrt{(x-a)^\top S(x-a)}+ r \|x\|_2$

Let $S$ be a positive-definite $n \times n$ matrix and define $\|z\|_S := \sqrt{x^\top S x}$ for any $x \in \mathbb R^n$. Let $a$ be a fixed vector in $\mathbb R^n$ and $r \ge 0$, and consider the ...
  • 5,870
0 votes
0 answers
31 views

Can we have such an infinite descending sequence of functions with prior ones inside their successors?

Let $M$ be some non-well-founded model of $\sf ZF$, can we have a sequence $(S_n)_{n \in \mathbb N}$ of nonempty sets in $M$, where each $S_n \subset \mathcal P(S_{n+1})$; and such that there exists ...
0 votes
0 answers
18 views

Subordinated non-deterministic Gaussian process is non-deterministic

Let $X = \{ X(k), k \in \mathbb{Z} \}$ be a strictly stationary, Gaussian time series whose spectral density $f_X$ exists. Furthermore, let $X$ be non-deterministic, i.e. $$ \mathbb{E}\big[ \vert X(n +...
2 votes
0 answers
18 views

Topology of the Malcev-Neumann group ring

For a ring $R$ and a group $G$ the group ring $R[G]$ consist of maps from $G$ to $R$ with finite support. It was shown that if the group is fully ordered them this ring can be embedded in a division ...
2 votes
0 answers
31 views

Can a punctured ball $(B\setminus\{0\})\subset\mathbb{C}^n$ be foliated by complete leaves?

Recently Antonio Alarcón proved that in the case of the unit ball $B\subset\mathbb{C}^n$ for $n\geq 2$ every smooth closed complex submanifold of dimension $q\leq n$, $V\subset\mathbb{C}^n$ defines a ...
2 votes
0 answers
26 views

How to get Bakry Emery Criterion $ \Phi'(t)=\frac{d}{dt}\int \Gamma_1(P_t f)d\pi=-\int\Gamma_2(P_tf)d\pi? $

I am reading Bakry Emery Criterion https://terrytao.wordpress.com/2013/02/05/some-notes-on-bakry-emery-theory/. Let a function $H\in C^2(R)$ and define the infinitesimal generator: $$ Lf=\Delta f-\...
  • 194
4 votes
0 answers
37 views

Univalence for weakly Tarski universes

In Martin-Löf type theory, a weakly Tarski universe is a type $\mathcal{U}$ with a type family $\mathcal{T}(A)$ indexed by terms $A:\mathcal{U}$, which is closed under identity types, dependent ...
0 votes
0 answers
14 views

Sum over lattice points in homogeneously expanding domains

In his book Algebraic Number Theory (2nd ed., Thm 2 in p.128), Lang proves the following (well-known) auxiliary result. Let $D\subset\mathbb{R}^N$ with $(N-1)$-Lipschitz parametrizable boundary. Let $...
  • 2,959
12 votes
0 answers
155 views

A new (?) way of composing monads

By composition of monads, I mean given two monads $S$ and $T$, making their composite $S T$ into a monad. Or more generally, given two monoid $X$ and $Y$ in a non-symetric monoidal category, making $X ...
  • 33.7k
0 votes
0 answers
24 views

Matrix optimization to find ideal embedding

Basically, I am trying to find the embeddings so I can approximate $K \approx M(\vec{\phi})$. The embeddings are for each one of my samples $\vec{\phi}(x_i) \in \mathbb{R}^D$ so I thought it should ...
0 votes
0 answers
17 views

Netting edges of a graph

Consider a weighted directed graph. Is there any known algorithm to compute the associated graph with "netted edges"? By edge netting I mean a function such that the (sub-) branch ...
  • 101
3 votes
0 answers
54 views

Definition of Radon measure on Takesaki's first volume

Consider the following theorem from Takesaki's first volume "Theory of operator algebras": In $(i)$, it is claimed that $L^\infty(\Gamma,\mu)$ is an abelian von Neumann algebra. How does ...
  • 435
1 vote
1 answer
54 views

Existence of a family of sets with some properties

Is it possible to find an example of a family $\mathcal{F}$ of $n$ finite distinct non-empty sets, a universe of maximum size $n/4$, with at least $\lfloor \frac{2}{3}{n \choose 2} \rfloor$ unordered ...
  • 398
5 votes
0 answers
76 views

What is the Balmer spectrum of the p-complete stable homotopy category?

When doing computations with spectra, we first reduce to working at a prime p by using the arithmetic fracture theorem: (the homotopy groups of) a spectrum of finite type can be recovered from its ...
0 votes
1 answer
27 views

Method for (binary) optimization under constraints

I would like to know if there is a method to solve the Problem. Problem: Maximize the following function: $$f(p_{1,i},p_{2,i},\dotsc,p_{m,i})=\sum_{i=1}^{n}\begin{bmatrix}p_{1,i} & p_{2,i} & \...
  • 1
0 votes
0 answers
19 views

Turing reaction diffusion equations and neural networks

Suppose you have a certain Turing-type reaction-diffusion equations that describe the formation of a some 2D pattern. I wonder if there are published works that trace some sort of strong connection ...
0 votes
0 answers
81 views

Can we have a proper class of infinitely descending infinite ordinals?

Working in $\sf ZF-Reg.$, can we have a transitive model $M$ of $\sf ZF$ such that there exists a proper class (i.e. a subset of $M$ that is not an element of $M$) of infinitely descending infinite ...
0 votes
0 answers
29 views

Term for degrees realizing least possible first n jumps

Is there a term for (Turing) degrees which realize the least possible jump (in the following sense) for the first n jumps. That is degrees which satisfy for all $0 < m \leq n$: $$X^m \equiv_T X \...
  • 1,635
0 votes
1 answer
117 views

Understanding the definition of left homotopy as given in Quillen’s Homotopical algebra book

Given two topological spaces $X,Y$, and two maps $f,g:X\rightarrow Y$, there is a notion of homotopy between $f$ and $g$. It is given by a continuous map $H:X\times I\rightarrow Y$ such that the ...
4 votes
2 answers
387 views

Do non-zero derivatives imply tangent lines (and vice versa)?

Let $\gamma : \mathbb{R} \rightarrow \mathbb{R}^2$ be any continuous function, with image given by $C_\gamma$. We can say that $\gamma$ has an image tangent at $t \in \mathbb{R}$ if there exists $\...
6 votes
0 answers
78 views

Automorphisms of Lubotzky–Phillips–Sarnak graphs

For the Lubotzky–Phillips–Sarnak (LPS) graph $X^{p,q}$, what is its automorphism group? These graphs are not just ($p+1$)-regular but are Cayley graphs for $G=\mathrm{PSL}_2(\mathbb{F}_q)$, so clearly ...
  • 415
2 votes
0 answers
41 views

Derivative of anti-self-dual forms on Kähler space

I am puzzled if we can establish differential relations about anti-self-dual 2-forms on the Kähler space similar to ones for self-dual forms? Let $(\mathcal{M},g,J,\omega = J^{(1)})$ be a Kähler space....
2 votes
0 answers
36 views

Cross-ratio for projective lines over division rings

If one considers a projective line over a field $k$, then the cross-ratio $(w,x;y,z)$ is a well-known geometric tool. But what if $k$ is not commutative, that is, if $k$ is a division ring ? Is there ...
  • 3,467
0 votes
1 answer
31 views

When a polygonal line become a loop in hyperbolic plane?

Suppose we have a 5 tuple of positive real numbers $(l_1,l_2,m_1,m_2,m_3)$, with $m_i \in (0,\pi)$ for all $i$. Now fix a point $v_1$ in the hyperbolic plane. Then consider a geodesic of length $l_1$ ...
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0 votes
0 answers
8 views

Dual equivalence of minimum feedback-vertex sets and cycle packing

it is known that the duals of feedback-set problems are set-packing problems; in the context of digraphs the feedback set are either a minimal set of vertices or edges that hit every oriented cycle; ...
  • 11.3k
0 votes
1 answer
27 views

Skip-free random walks: recurrence and transience

Let us define a one dimensional random walk: for all $n\in\mathbb{N}$ $$ X_n:=\sum_{i=1}^nZ_i $$ with $Z_i$ i.i.d. random variables taking values in $\{-1,0,1,2,\dots\}$. This process is sometimes ...
  • 1
4 votes
0 answers
75 views

Is $C^r(M)$ non-isomorphic to $C^s(N)$ for $r\neq s$ and nontrivial manifolds $M,N$?

This is an obvious continuation of an MO question. Let $r,s\in\mathbb N\cup\{\infty\}$ with $r\neq s$, and $M,N$ two connected manifolds of positive dimension (which roots out the trivial case of a ...
  • 1,001
3 votes
1 answer
116 views

Semi-orthogonal decomposition for maximally non-factorial Fano threefolds

Let $X$ be a nodal maximally non-factorial Fano threefold. If there is $1$-node and no other singularities, they by the work of Kuznetsov-Shinder https://arxiv.org/pdf/2207.06477.pdf Lemma 6.18, $D^b(...
  • 1,590
5 votes
0 answers
33 views

Generalizations of classical integrals of motion

In classical Hamiltonian mechanics, we are given the Hamiltonian $H\in C^\infty(M)$. $L\in C^\infty(M)$ is called an integral of motion (or a conserved quantity) if the Poisson bracket of $H$ and $L$ ...
0 votes
0 answers
26 views

Rates of convergence of empirical measures in Wasserstein distance

Let $X_1, X_2, \ldots$ be iid random variables in $\mathbb {R}^d$ with common distribution $\mu$, and $\mu_N = \frac 1N \sum_{k=1}^N \delta_{X_k}$, $N \geq 1$, the associated empirical measures. If $\...
5 votes
1 answer
487 views

Can deleting a random entry from an iid sequence destroy the iid property?

Let $X=(X_1,\ldots,X_n)$ be an iid sequence of random variables, and let $\nu$ be a uniformly random integer in the range $1,\ldots,n$. Then $\xi_\nu$ is a random entry of $X$. Is it always true that ...
1 vote
0 answers
26 views

Galois cohomology with coefficients in the integers of the Lubin-Tate extension

Let $K$ be a $p$-adic local field, and $L$ the Lubin-Tate extension obtained from $K$ by attaching roots of some Lubin-Tate formal $\mathcal{O}_{K}$-module with $Gal(L/K) \simeq \mathcal{O}_{K}^{\...
6 votes
1 answer
71 views

Onsager-Machlup functional when drift is time-dependent

Let $X(t)$ be a diffusion process on $\mathbb{R}^d$ generated by \begin{align} \mathcal{D} = \nabla^2 + \sum_{i=1}^d b_i(x) \frac{\partial}{\partial x_i}, \end{align} where $b_i(x) \in \mathcal{C}_b^2(...
  • 161
2 votes
0 answers
62 views

Embeddings of reductive groups over algebraically closed fields

Let $K/k$ be an extension of fields, not necessarily algebraic; let $G$ and $H$ be split, reductive groups over $K$; and let $f : H \to G$ be an embedding of groups. Do there exist split, reductive ...
  • 8,703
0 votes
0 answers
26 views

Variance of sum of combinations of a discrete set

I have a vector of weights $\vec{w}$ (size $n$) and a random boolean vector $\vec{x}$ sampled uniformly from all possible boolean vector of size $n$. Is there a shortcut to compute the variance of $\...
  • 1
4 votes
1 answer
145 views

What are the jumps in the ramification filtration of the absolute Galois group of a local field?

Let $k$ be a (complete) discretely valued field and $\ell$ a Galois extension of $k$, possibly infinite. The Galois group $\Gamma=\text{Gal}(\ell/k)$ of $\ell$ over $k$ admits a descreasing, $\mathbb ...
3 votes
1 answer
55 views

Matrix determinant lemma for non-rank-one updates

The well-known matrix determinant lemma states that for an invertible square matrix $A$ and column vectors $u,v$ one has $$ \det(A + uv^T) = \det(A)(1 + v^T A^{-1} u). $$ Is there any analogous ...

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