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8 views

A normal proper model of an abelian variety with geometrically integral special fiber smooth at the reduction of the origin

Let $A$ be an abelian variety over $\mathbb{Q}_p$. Does there exist a proper flat morphism $X\to \mathrm{Spec}\:\mathbb{Z}_p$ such that the special fiber is geometrically integral and smooth at the ...
0
votes
0answers
9 views

Self avoiding walks and context free languages

Let $G$ be an infinite, locally finite, connected graph whose arcs (oriented edges) are labelled by letters in a finite alphabet $\Sigma$ such that arcs starting in the same vertex are labelled by ...
0
votes
1answer
21 views

“Arithmetically diverse” infinite binary string

For $a,b \in \omega$ with $a > 0$, let $f_{a,b}: \omega\to\omega$ be defined by $n \mapsto an+b$. What is an example of an infinite binary string $b:\omega\to\{0,1\}$ with the following property? ...
1
vote
0answers
14 views

Mean square estimate for the Kloosterman sums

For $m,n\in \mathbb{N}$, denote the Kloosterman sum $$S(m,n;c)=\sum_{a\bmod c}e\left( \frac{ma+n \overline{a}}{c}\right),$$where $\overline{a}$ denotes the multiplicative inverse of $a\bmod c$. Does ...
5
votes
0answers
20 views

Is $\beta^{*}(w_{2k-2}) = 0$ for an open orientable $2k$-manifold?

This question is motivated by the vector field question I asked recently. Panagiotis Konstantis answered this question for odd manifolds and I am trying to figure out the even case. Let $M$ be a ...
1
vote
1answer
25 views

What is the appropriate notion of Weakly Equivalent or Morita Equivalent categories internal to a category of generalized Smooth Spaces?

Let $G$ and $H$ be Lie Groupoids. We know that there are two notions of equivalences of Lie Groupoids: Strongly Equivalent Lie Groupoids: (My terminology) A homomorphism $\phi:G \rightarrow H$ of ...
0
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0answers
39 views

Does category of finitely generated torsion $G$-modules has enough injectives?

Let $G$ be a profinite group. Then the category of discrete $G$-modules have enough injectives. Now I have a category of finitely generated and torsion $G$-modules with continuous $G$-action. Does ...
1
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0answers
36 views

Fourier transform on finite groups in characteristic $p>0$

Is there a Fourier theory for finite groups in characteristic $p>0$? Assume that $p$ divides the order $|G|$ of finite groups (or just work with $p$-groups), i.e., in a modular representation-...
2
votes
1answer
22 views

Isometric embedding of the modular surface

Is there an isometric embedding of the modular surface $X(1)=PSL(2,\mathbb{Z})\backslash \,\mathbb{H}$ into the Euclidean 3-space? For all I know this may be an open problem but I am also curious if ...
1
vote
0answers
14 views

Uniqueness for a transport-diffusion equation with low integrable drift

Consider the equation $$ \frac{\partial f}{\partial t} + u \cdot \nabla f - \Delta f = 0 $$ in $(0,T) \times \mathbb R^N$, with initial condition $$ f \vert_{t=0} = f_0 $$ for some given $f_0 \colon \...
10
votes
0answers
102 views

Consequences of Gromov's Conjecture

In Peter Petersen words, Gromov Betti number estimate is considered one of the deepest and most beautiful results in Riemannian geometry; which asserts that Theorem (Gromov 1981). There is a constant ...
1
vote
1answer
33 views

ratio between a polygon bounded in another polygon

Let A be a convex polygon with area SA. Construct a new polygon B by orderly connecting the midpoints of the segments of A. Denote the area of B by SB. Claim : the ratio SB/SA is constant for all ...
3
votes
0answers
30 views

Depth of modules and regular sequences of endomorphisms

Let $(R, \mathfrak{m})$ be a Noetherian local ring and $M$ is a finitely generated $R$-module of depth $t$. It is well-known that every maximal regular sequence of $M$ has length $t$. Recalling that $...
0
votes
0answers
29 views

Integer partitions into restricted parts

Given a linear diophantine equation $$x_1+\dots+x_n=m\leq nn'$$ how many solutions does it have with each $x_i\in[0,n']\cap\mathbb Z$? Looking for asymptotics that parametrizes well with both $n$ and $...
-1
votes
0answers
49 views

Find the integral of $\int\limits_0^{2\pi } {Q\left( {f(\theta )} \right)Q\left( {g(\theta )} \right)d\theta } $?

I am trying to find the integral of the following function: $\int\limits_0^{2\pi } {Q\left( { - (e\cos (a + \theta )\sqrt{x} + g} \right))Q\left( { - (f\sin (a + \theta )\sqrt{x} + h} \right))d\theta }...
5
votes
0answers
22 views

A question regarding an analog of Young symmetrizer: the product row and column preserving subgroups without sign representation

Consider a rectangular Young diagram $\lambda$ with $n = pq$ boxes, with $p$ rows and $q$ columns. If $C$ is the column preserving subgroup of $\lambda$ and $R$ is the row preserving subgroup, then we ...
0
votes
0answers
18 views

Reconstruction of a binary vector from any two rows of a binary matrix

Let $ a = (a_1,a_2, \ldots,a_{10})\in \{ 0,1\}^{10 \times 1}$ be a binary column vector of length $10$. How to find $x_{i,j} \in \{ 0,1\}^{1 \times 10}$, $i\in \{1,2,3,4,5\}$, $j \in \{1,2,3\}$ (...
-1
votes
0answers
24 views

Removing training examples from SVM [migrated]

If we have a SVM that already classifies a training set. Is it possible to remove examples from the training set and still produce the same SVM?
3
votes
1answer
86 views

Computation on homotopy colimit cocomplete triangulated categories

I have a couple of questions about dealing with homotopy (co)limits cocomplete triangulated categories. Question I:The first one concerns a comment by Peter Arndt in this discussion about derived ...
-3
votes
0answers
34 views

Calculating a fuzzy crisp value from a linguistic fuzzy weight [closed]

I am struggling to find a clear source of information on-line that will help me understand how to convert a fuzzy weight for a linguistic preference to a crisp value. For instance, below we have a ...
2
votes
1answer
79 views

Behavior of invariants under reduction mod p

Let $R$ be a finitely generated $\mathbb{Z}$-algebra with an [edit: linear algebraic] action of $G(\mathbb{Z})$ where $G$ is a split simply-connected semisimple group. Then for any prime $p$ we have a ...
2
votes
0answers
69 views

Full measure properties for Zariski open subsets in $p$-adic situation

Let $F$ be a $p$-adic field and let $X$ be a smooth integral variety over $F$ (I am chiefly interested in the case when $X$ is a connected reductive group over $F$). Let $U$ be a non-empty open subset ...
2
votes
1answer
82 views

Measure theory on abstract Boolean ring

Since a σ-algebra in measure theory is indeed an algebra over $\mathbb{Z}_2$ with addition given by symmetric difference and multiplication given by intersection, does it mean we can put measure on ...
-5
votes
0answers
143 views

A reform of logic to secure naive set theory? [closed]

Set-theoretic paradoxes noted by Russell and others led to attempts to produce a consistent set theory as a foundation for mathematics. (Stanford Encyclopedia of Philosophy. Inconsistent Mathematics) ...
2
votes
1answer
77 views

Chromatic number of square of a tree

What is an upper bound on the chromatic number of the square of a tree on $n$ vertices? Note that the power of the graph is considered in this sense. If the tree were a path, then it is easy to see ...
4
votes
2answers
107 views

A generalization of strong primes

In this post we denote the sequence of prime numbers as $p_k$ for integers $k\geq 1$. I don't know if the following definition is in the literature. Definition. We define the $\theta$-strong primes, ...
4
votes
0answers
36 views

Frechet-Urysohn quotient of second countable locally compact Hausdorff space

In this paper from 2010 https://cmuc.karlin.mff.cuni.cz/pdf/cmuc1001/arhangav.pdf Arhangelskii asks if there is a quotient of a second countable locally compact Hausdorff space which is Frechet-...
0
votes
0answers
46 views

Relation between the $L^2$ norm of the Poisson bracket of $f$ and $g$ and their $H^1$ norms

Let $f,g\in H^1(\Omega)$ where $\Omega$ is a sufficiently nice bounded domain in $\mathbb{R}^2$. If $\{\cdot,\cdot\}:H^1(\Omega)\times H^1(\Omega)\to L^2(\Omega)$ is the Poisson bracket, is there some ...
9
votes
1answer
217 views

Is there an orientable prime manifold covered by a non-prime manifold?

A manifold is called prime if whenever it is homeomorphic to a connected sum, one of the two factors is homeomorphic to a sphere. Is there an example of a finite covering $\pi : N \to M$ of closed ...
0
votes
0answers
32 views

On isoclinism classes of finite p-groups

With reference to James, Rodney, The groups of order (p^6) ((p) an odd prime)., Math. Comput. 34, 613-637 (1980). ZBL0428.20013., My question is can we get isoclinism class $\phi_2$ for a finite p-...
6
votes
1answer
287 views

The largest group acting on a non-orientable surface of genus 5

Let $N_5$ denote the non-orientable surface of genus 5. In Conder's database https://www.math.auckland.ac.nz/~conder/BigSurfaceActions-Genus2to101-ByGenus.txt we can see that the biggest finite group $...
-2
votes
0answers
23 views

Characteristic function of a “incomplete” binomial random variable

For some fixed integer $r > 0$ consider a "incomplete" binomial random variable: $$P(X=k)=\cases{{(2r-1)-k\choose{r-1}} p^r(1-p)^{r-k}, & for $k=r,\ldots,1$\\ {(2r-1)-k\choose{r-1}}p^{...
5
votes
1answer
122 views

Homotopy group action and equivariant cohomology theories

Many of the introductory notes on generalized equivariant cohomology theories assume that one is working over the category of $G$-spaces or $G$-spectra. However, one thing that concerns me is that the ...
3
votes
1answer
178 views

Recommendations for mathematical essayists

I was wondering if people had recommendations for mathematical essays (by this I mean essays on a mathematical topic, not necessarily essays written by mathematicians). A person who I used to find ...
-2
votes
0answers
34 views

Find Angle in Triangle [closed]

In triangle ABC, angle a = 56 degrees and angle B = 50 degrees. The altitude from B to AC is extended until it intersects the line through A that is parallel to BC; that intersection is called point K....
0
votes
1answer
119 views

Injectivity of analytic functions

Suppose $f : \mathbb{R} \rightarrow \mathbb{R}^n$ is a real analytic function on $(a, \infty)$. I have two questions: Suppose $||f(x)|| \rightarrow \infty$ as $x \rightarrow \infty$. I know without ...
0
votes
1answer
61 views

how to prove the binomial equation below [closed]

I tried to open up all binomial expressions but things got more complicated. I could not find an appropriate solution.I'm just stuck and trying to find a solution for like 2 hours.I would be very ...
1
vote
0answers
45 views

Determining the irreducible invariant subspaces of a permutation action by computing eigenspaces of a matrix

Let $\Sigma\subseteq\mathrm{Sym}(n)$ be a permutation group on $N:=\{1,...,n\}$. My goal is to determine the irreducible invariant subspaces of the permutation action of $\Sigma$ on $\Bbb R^n$, and I ...
0
votes
0answers
35 views

Flat function with a spectral gap

I am looking for a sequence of functions $f_n,n\geq 1$ in $L^2(\mathbb R)$ such that $f_n$ is equal to $1$ on $[-n,n]$ and $\hat{f_n}$ vanishes on $[-1,1]$. Actually, I would also like $f_n$ to be $...
3
votes
1answer
45 views

Permanent of a Kronecker product of matrices

It is well known that $\det(A \otimes B) = \det(A)^m \det(B)^n$ when $A$ and $B$ are square matrices of size $n$ and $m$ where $\otimes$ denotes the Kronecker product. Question: Is there a similar ...
8
votes
1answer
102 views

Continuous version of the fundamental theorem of invariant theory for the orthogonal group

A standard result in the invariant theory of the orthogonal group states the following. Theorem Let $(E, \langle .,. \rangle)$ be an n-dimensional euclidean vector space, let $f : E^m \rightarrow {\bf ...
0
votes
0answers
14 views

Moments of Logistic SDE's solution

On this article starting from equation $(30)$ it's presented a derivation of the first moment for the solution the logistic SDE: $$dx=x\left[\mu\left(1-\frac{x}{\tilde{x}}\right)dt+\sigma dW\right]$$...
0
votes
0answers
69 views

Is this model of converting integers to Gray code correct?

The model shown in the figure converts all numbers that have k digits in the binary system to Gray code without any calculation, but I have no proof that guarantees this claim. Here is some ...
5
votes
0answers
67 views

Are groups with the Haagerup property hyperlinear?

In his 2008 paper Hyperlinear and Sofic Groups: A Brief Guide, Pestov asked (Open Question 9.5) whether every group with the Haagerup property is hyperlinear (or sofic). Has this question been ...
5
votes
0answers
72 views

Left Kan extensions of “strong” monoidal functors

Consider the 2-category $\mathsf{MonCat}$ where objects are monoidal categories, 1-cells are strong monoidal functors, and 2-cells are monoidal natural transformations. Given arrows $f: \mathsf{C} \to ...
6
votes
1answer
139 views

Example of homeomorphism of $3$-manifolds

How can we see that the following $3$-manifolds are homeomorphic? I couldn't use the moves of Kirby calculus.
1
vote
0answers
56 views

A Fredholm equation with non-separable kernel

I'm trying to solve this form of Fredholm equation: $$ g(v)=f_1(v)+\int\limits_{0}^{v_\mathrm{th}} g(v_s)\frac{e^{-\tfrac{\big[(v-v_\mathrm{init})-(1-v_\mathrm{leak})(v_s-v_\mathrm{init})\big]^2}{2v_\...
8
votes
1answer
346 views

Off-diagonalize a matrix

Consider a self-adjoint matrix $M$ that has block form $$M = \begin{pmatrix} M_{11} & M_{12} \\ M_{12}^* & M_{11} \end{pmatrix}.$$ I am wondering if there exists any criterion to decide if ...
-6
votes
0answers
71 views

A math quiz that i never understood [closed]

If ((x^x)^x)=2 then what is x^2 equal to. Never managed to solve it after hours of trying, when my friend explained it to me it didn't feel right.
0
votes
0answers
66 views

General asymptotic result in additive combinatorics (sums of sets)

Let $S_1,\cdots,S_k$ be $k$ infinite sets of positive integers. Let $N_i(z)$ be the numbers of elements in $S_i$ that are less or equal to $z$. Let us further assume that $$N_i(S) \sim \frac{a_i z^{...

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