# All Questions

114,972
questions

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votes

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4 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 ...

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votes

**1**answer

15 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?
...

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13 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 ...

**3**

votes

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13 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

**1**answer

20 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 ...

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34 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 ...

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vote

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34 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-...

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17 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 ...

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vote

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12 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 \...

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votes

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94 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 ...

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votes

**1**answer

31 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 ...

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28 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 $...

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26 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 $...

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45 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 }...

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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 ...

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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

**0**answers

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

**1**answer

77 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 ...

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32 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

**1**answer

75 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 ...

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67 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 ...

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votes

**1**answer

81 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 ...

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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

**1**answer

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

**2**answers

101 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, ...

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35 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-...

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45 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 ...

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votes

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215 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 ...

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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

**1**answer

279 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 $...

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votes

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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^{...

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votes

**1**answer

118 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

**1**answer

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 ...

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votes

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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....

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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 ...

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votes

**1**answer

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 ...

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vote

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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 ...

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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 $...

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**1**answer

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 ...

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votes

**1**answer

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 ...

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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:
$$dx=x\left[\mu\left(1-\frac{x}{\tilde{x}}\right)dt+\sigma dW\right]$$...

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68 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 ...

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66 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 ...

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71 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 ...

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**1**answer

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.

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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

**1**answer

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 ...

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70 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.

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votes

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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^{...

**2**

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76 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 ...