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5 views

A curve with geometrically integral reduction everywhere

Is there a proper flat morphism $X\to \mathrm{Spec}\:\mathbb{Z}$ such that all fibers are geometrically integral and the generic fiber is a smooth curve of positive genus?
1
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0answers
9 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
votes
0answers
26 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
vote
0answers
31 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
0answers
10 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
9 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 \...
9
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0answers
72 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 ...
0
votes
1answer
28 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
24 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
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0answers
24 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
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0answers
37 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
20 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
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0answers
15 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
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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?
2
votes
1answer
55 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
21 views

Calculating a fuzzy crisp value from a linguistic fuzzy weight

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
74 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
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0answers
65 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
79 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
137 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) ...
1
vote
1answer
74 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 ...
2
votes
0answers
87 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
32 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
43 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
211 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
30 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
253 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
21 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
112 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
177 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
117 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
44 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
33 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
44 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
99 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
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0answers
67 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
64 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
68 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
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0answers
55 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
344 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
69 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
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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^{...
6
votes
1answer
235 views

When does a (higher) Deligne-Mumford stack 'have enough geometric points'?

We take the following as our definition of spectral Deligne-Mumford stack (following Lurie): A pair $(X,O_X),$ where $X$ is an $\infty$-topos and $O_X$ is a strictly Henselian sheaf of $E_\infty$-...
2
votes
0answers
74 views

Variation of Euler characteristic when the sheaf is not flat

Let $f:X \to Y$ be a flat, projective morphism with $Y$ integral and every fiber of $f$ normal and integral. Let $F$ be a torsion-free, coherent sheaf on $X$ (not necessarily flat over $Y$). Then, is ...
2
votes
0answers
15 views

Does a compact ANR have a local equiconnecting function which connects distinct points by simple paths?

It is known that if $X$ is a (metric) ANR, then $X$ is locally equiconnected, that is, there is a neighborhood $V$ of the diagonal $\Delta X \subseteq X \times X$ and a continuous function $$f \colon ...
2
votes
1answer
144 views

Question about automorphism functor in Sernesi's “Deformations of algebraic schemes”

Let $X$ denote an algebraic scheme over $\operatorname{Spec} k$ such that its deformation functor $\operatorname{Def}_X$ has a semi-universal couple $(R,u)$, where $R$ is an Artinian $k$-algebra and $...

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