# All Questions

104,371
questions

**2**

votes

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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**

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**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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**

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**0**answers

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

**1**answer

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

**1**answer

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**

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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

**0**answers

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 ...

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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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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

**2**answers

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

**1**answer

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**

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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**

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**0**answers

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**

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**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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**

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140 views

### Basic set theory question [closed]

Let $\lambda$ be some cardinal, is it true that $\lambda^{\lambda} \leq 2^{2^{\lambda}}$?

**0**

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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**

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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

**1**answer

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**

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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

**2**answers

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

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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**

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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**

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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**

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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

**1**answer

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

**4**answers

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

**2**answers

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

**0**answers

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**

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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

**2**answers

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**

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**0**answers

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

**1**answer

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**

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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

**2**answers

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

**0**answers

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**

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**0**answers

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**

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**0**answers

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

**1**answer

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**

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**0**answers

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 ...