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votes
0answers
46 views

Isoartinian and isosimple modules

I'm reading this article by A. Facchini and Z. Nazemian, in wich they discuss modules with chain conditions up to isomorphism. A couple of the main concepts are the following: Definition We say that ...
4
votes
0answers
81 views

The space of $k$ differential forms as a Fréchet space

Given a smooth manifold $M$, can define define seminorms on $\Gamma(U,\bigwedge^kT^{\ast}M)$ for $U$ a coordinate open set by the following: $p^{s}_L(u = \sum_{I}u_I dx_I) = \sup_{x \in M}\max_{|I|=p, ...
6
votes
1answer
178 views

$(-2)$-curves in complex $3$-folds

Let $X$ be a smooth complex $3$-fold, and let $C \subset X$ be an embedded smooth rational curve whose normal bundle $N_{C/X}$ is isomorphic to $\mathscr{O}(-1) \oplus \mathscr{O}(-1)$. Is it true ...
0
votes
1answer
91 views

Linear equation with summation

I've got that kind of linear equation which I can't solve. I tried everything. $\delta_{n0} + A_n = D_{0n} - \sum_{m = -\infty}^{\infty}{(A_mD_{mn} + B_mE_{mn})}$ $B_n = E_{0n} - \sum_{m = -\infty}^{...
1
vote
1answer
115 views

Lipschitz constant of exponential map

I asked before this question on MSE but I was not able to work out the details on my own. Suppose $M$ is a smooth compact Riemannian manifold, take $p \in M$ and consider the map $$ T_pM \ni v \...
4
votes
0answers
138 views

Fibered surfaces degenerating to Frobenius

Let $R$ be a DVR with algebraically closed, positive characteristic residue field $k$. Let $X\rightarrow Spec(R)$ and $C\rightarrow Spec(R)$ be smooth projective morphisms of relative dimension 2 and ...
2
votes
1answer
59 views

Fermat stationary point theorem - a generalization exists?

Let $f:E\to\mathbb{R}$ a functional (here $E$ is a normed vector space). Is it true that if $x_0\in E$ is a local minimum for $f$, then all the directional derivatives are 0? We have the derivative ...
2
votes
1answer
123 views

Characterization of nilpotent adjacency matrices [on hold]

Let $\theta$ be the adjacency matrix of a simple graph (symmetric and zeros on the diagonal). What is the characterization of those $\theta$ which satisfy $$\theta^2 \equiv 0 \pmod{2}$$ i.e. which $\...
4
votes
0answers
132 views

Change of variables for $p$-adic integral

Say $p$ is an odd prime. Suppose I have a measure $\mu$ on $\mathbf{Z}_p$. As in II.4.3 in Colmez - Fonctions d'une variable $p$-adique, I can restrict $\mu$ to $1+p\mathbf Z_p$, and there is a ...
0
votes
0answers
34 views

Bilinear Strichartz estimates for the Schrodinger equation

Let $d\geq 2$, $I\subset\mathbb{R}$ a time interval and $t_0\in I$. Let $u,v$ be solutions to the free Schrodinger equation on $\mathbb{R}^d$, i.e. $(i\partial_t+\Delta)u=(i\partial_t+\Delta)v=0.$ Let ...
1
vote
0answers
72 views

Gradient of squared riemannian distance on complete manifold

Let $\theta: M \times M \to \mathbb{R}$ the squared distance function $\theta(x,y)=d(x,y)^{2}$ on complete Riemannian manifold $M$. I would like to calcule the gradient of $d^{2}$, where $d^{2}_{y}(x)=...
5
votes
1answer
86 views

Representation-finite quivers over dual numbers

Given a Dynkin quiver $Q$ and a field $K$. Question 1: For which such $Q$ are there only finitely many indecomposable representations over the dual numbers $K[x]/(x^2)$? Note that those ...
14
votes
1answer
446 views

Simplicial set of permutations

Let $S_k$ be the set of all permutations of $k+1$ elements $0,1,...,k$. introduce boundary maps $d_i : S_k \rightarrow S_{k-1}$ by deleting from permutation $\eta$ element $\eta(i)$ and monotone ...
7
votes
1answer
147 views

Integration with values in a topological vector space

Is there a general theory of integration of functions with values in a topological vector space (not necessarily locally convex)? Browsing through mathoverflow posts, I came across a discussion ...
4
votes
0answers
63 views

Reflexive object and infinite products

The category CPO of cpos and continuous functions has a reflexive object, i.e. an object $A$ such that $A\times A\simeq A$ and $A\simeq A^A$. Since CPO has countable products, my question is whether ...
5
votes
1answer
142 views

Is there a name for a “stable” physical measure?

Let $M$ be a Riemannian manifold with volume measure $\lambda$, let $f \colon M \to M$ be a continuous map, and let $\mu$ be a probability measure on $M$ with compact support. Definition. The ...
3
votes
2answers
172 views

Tiling of genus 2 surface by 8 pentagons

In theses these notes, Example 5.6, it is said that there is a "symmetric tiling of a genus 2 surface by 8 right-angled hyperbolic pentagons". Question 1: What does this tiling look like? Question 2:...
16
votes
2answers
503 views

Constructive proof of existence of free algebras for infinitary equational theories

Is it constructively true that all (not necessarily finitary) equational theories $T = (\Sigma, E)$ have an initial model? The usual proof for finitary equational theories I know constructs first ...
0
votes
1answer
85 views

Optimal Sobolev regularity for $(-\Delta)^{-s}$ on domains

In the paper Chen, Huyuan; Véron, Laurent, Semilinear fractional elliptic equations involving measures, (J. Differ. Equations 257, No. 5, 1457-1486 (2014). ZBL1290.35305) in the proof of Proposition ...
3
votes
0answers
44 views

Set-theoretic solutions of YBE for $n=3$

Is there a list of all set theoretic solutions $S:X \times X \to X \times X$ of the YBE for $X=\{1,2,3\}$? Or is it known how many solutions there are? I mean, $S_9$ is big but maybe not too big to ...
1
vote
0answers
101 views

Is it possible to get an interesting statement about even perfect numbers from the equation $1/\operatorname{rad}(n)=1/2-2\varphi(n)/\sigma(n)$?

It is well known that the problem concerning even perfect numbers is to prove or refute if there are infinitely many of them. Few weeks ago I wrote the following conjecture, where $\varphi(n)$ denotes ...
3
votes
0answers
155 views

Diffeomorphism classification of Grassmannian manifolds

Is anything known about the diffeomorphism classification of Grassmannian manifolds? I know that there are some results on projective spaces (for example in Lopez de Medrano's "Involutions on ...
-1
votes
1answer
123 views

Does possible to get an interesting statement about odd perfect numbers studying these equations?

I would like to ask a question about identities that involve odd perfect numbers, since I am curious to know if you can deduce a statement with a good mathematical content through these. The ...
2
votes
1answer
183 views

Purity and skyscraper sheaves

In "The Geometry of moduli spaces of sheaves" a coherent sheaf $\mathcal{F}$ is defined to be pure of dimension $d$ if dim$(\mathcal{E})=d$ for all non-trivial proper subsheaves $\mathcal{E} \subset \...
5
votes
1answer
173 views

Tannaka duality for closed monoidal categories

I asked this some time ago at mathstackexchange, and people there explained to me the mathematical part of what I was asking, but the question about references remains open. In my impression, people ...
-5
votes
0answers
140 views

Basic set theory question [closed]

Let $\lambda$ be some cardinal, is it true that $\lambda^{\lambda} \leq 2^{2^{\lambda}}$?
0
votes
0answers
17 views

Existence of closed-loop stochastic control?

In a controlled system, for example,$dX_t=b(t,X_t,u_t)dt+\sigma(t,X_t,u_t)dW_t$, where $W_t$ is standard Brownian motion, $u_t$ is the controls strategies. If I want to find a kind of feed back ...
2
votes
0answers
139 views

Kuga-Satake in characteristic $p$ [on hold]

Have Kuga-Satake correspondences been investigated in characteristic $p$? (I'm being intentionally vague about what this would mean.)
3
votes
1answer
137 views

Origin of the theorem related to the integral transform pair

The development of Fast Fourier transform is attributed to Cooley & Tukey, both have written a lot about it is historical development. Both Cooley and Tukey call it a re-discovery rather. However,...
2
votes
0answers
81 views

Automorphisms of a neighborhood of a negative curve

Let $X$ e a smooth complex surface and let $C\subset X$ be a smooth rational curve with negative self intersection. Is there any known description of the automorphisms of a infinitesimal ...
2
votes
2answers
138 views

Is there an abstract proof of Kleene's Recursion Theorem in a typed lambda-calculus?

I have written out a proof using lambda-functions that formalizes the exposition of Kleene's Recursion Theorem statement and proof in Michael Sipser's book "Introduction to the Theory of Computation." ...
-2
votes
0answers
32 views

Literature on generalized higher moment probability distributions

Given any probability distribution $P=\{p_1,p_2,\dots,p_n\}$, one can consider a related normalized probability distribution $P_k=\{\frac{p_1^k}{Z_k},\frac{p_2^k}{Z_k},\dots,\frac{p_n^k}{Z_k}\}$ where ...
6
votes
0answers
166 views

A cohomology associated to a (not necessarily integrable) distribution (Hilbert 16th problem and dynamical Lefschetz trace formula 2)

Let $(M,D)$ be a pair consisting of a manifold $M$ and a distribution $D$ on $M$. The de Rham complex $\Omega^*(M)$ has the following subcomplex $$\Omega^*(M,D)=\{ \alpha \in \Omega^*(M)\mid \alpha_{|...
6
votes
0answers
72 views

Finding the maximal component of a vector in sublinear time

Given a vector $u \in \Bbb R^n$, finding the value of the largest component of $u$ needs linear time in $n$. However, what if we additionally know that $u$ lies in some linear subspace $U \subset \Bbb ...
5
votes
0answers
90 views

Open subfunctor of Quot Functor induced by open immersion

Let $f: X \rightarrow S$ be a morphism of noetherian schemes and $\mathfrak{Q}uot_{\mathcal{E}/X/S}$ be the functor parametrizing families of quotients of $\mathcal{E}$ in the category of locally ...
3
votes
1answer
218 views

Prove an existing formula for a limit of a specific sum

Prove that$$\lim_{n\to\infty}\frac1n\sum_{i_1,i_2,...i_k=1}^n\lambda_1^{|i_1-i_2-s_1|}\lambda_2^{|i_2-i_3-s_2|}...\lambda_k^{|i_k-i_1-s_k|}$$is equal to$$\sum_{j=1}^k\lambda_j^{S+k-1}\prod_{l=1,l\ne j}...
8
votes
4answers
627 views

The tensor product of two monoidal categories

Given two monoidal categories $\mathcal{M}$ and $\mathcal{N}$, can one form their tensor product in a canonical way? The motivation I am thinking of is two categories that are representation ...
1
vote
2answers
202 views

Equivalence between categories of affine schemes over $X$ and representable functors $\operatorname{Points}(x) \to \operatorname{Sets}$

I am reading Strickland - Formal schemes and formal groups. For a functor $X\colon \operatorname{Rings}\to \operatorname{Sets}$, he defines (2.14) the category of $\operatorname{Points}(X)$ in the ...
4
votes
0answers
94 views

Continuity of the Green function with respect to the measure

Let $G$ be a finitely generated group and let $\mu$ be a finite measure on $G$. Define the Green function as $$G(\mu)=\sum_{n\geq 0}\mu^{*n}(e),$$ where $\mu^{*n}$ is the $n$th convolution power of $\...
0
votes
0answers
23 views

Coexistence of different solutions in a nonlinear matrix equation

I've faced a system of first-order nonlinear matrix equation, and I have tried to use perturbation method to approach the solutions. The equation has the form: \begin{align} \mathbf{F}(\mathbf{x},\...
6
votes
2answers
139 views

Classification of minimal sets of properties proving a group is Abelian

Let $S$ be a non-empty, possibly infinite, set of integers, all of which are greater than $1$. For a given group $G$, let $S[G]$ denote the collection of statements $$ \forall (n \in S, a \in G, b\in ...
1
vote
0answers
105 views

Topological invariants of a certain “stratified” manifold, with pieces of different “dimensions”

Disclaimer: I don't fully understand what I'm talking about in the question below. I'm still trying to figure out the right question to ask. Quotations and question marks in brackets mean that I'm not ...
2
votes
1answer
103 views

In which space are we solving the Kähler Ricci flow?

The Kähler Ricci flow on a compact Kähler manifold are formulated as $\frac{\partial}{\partial t}w(t) = -Ric(w)$, $w(0) = w_0$, where $w(t)$ is a family of Kähler metrics and $w_0$ is the initial ...
5
votes
0answers
213 views
+100

Obstructions to locally trivial deformations

Let $X$ be a complex projective variety. If $X$ is smooth, then the first-order infinitesimal deformations are given by $H^1(X, T_X)$ and the obstructions are in $H^2(X, T_X)$. Now assume that $X$ is ...
2
votes
2answers
118 views

Rate of convergence of mollifiers // Sobolev norms

Following up to the question raised here, I am searching for a reference (or a simple argument) to establish (in the whole space) the following (suggested) equivalence : Given $N_1$ and $N_2$ two (...
1
vote
0answers
111 views

Is every $(n-1)$-connected $n$-manifold embeddable in $\mathbb{R}^{n+1}$ homeomorphic to $\mathbb{S}^{n}$? [migrated]

Let $M^n$ be a compact, topological $n$-manifold which is a subspace of $\mathbb{R}^{n+1}$. If $M^n$ is $(n-1)$-connected (i.e. $\pi_i$ vanishes for $i<n$), does it have to be homeomorphic to the $...
4
votes
0answers
76 views

Left passage probability of $SLE_8$?

Schramm's formula on left passage probabilities of $SLE_k$ is stated for $k \in [0,8)$ in theorem 2 here. However, after the statement he remarks that the formula simplifies to $1/2$ for $k = 8$. It ...
4
votes
0answers
36 views

When can two elementary end extensions of models of PA be uniquely amalgamated?

$\DeclareMathOperator{Cod}{Cod}$ $\DeclareMathOperator{Scl}{Scl}$ $\DeclareMathOperator{Def}{Def}$ $\DeclareMathOperator{Lt}{Lt}$ Background: All of the background to this question can be found in ...
5
votes
1answer
283 views

Identification problem: Does this group have a name?

I've encounter a group with properties that are very familiar, but I cannot say what group is it. Consider the variables $(t,x,y,z)$, an affine transformation $M \in A(3)$ on the last three variables ...
2
votes
0answers
96 views

Is this a typo in Ihara's “On discrete subgroups of the two by two projective linear group over p-adic fields”?

In Eq. (9'') on p. 227 of Ihara's paper "On discrete subgroups of the two by two projective linear group over p-adic fields" (link), where the second line says $$"\log Z_{\Gamma}(0,\chi)=1",$$ is this ...

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