# All Questions

114,972
questions

**0**

votes

**1**answer

9 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?
...

**-5**

votes

**0**answers

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)
...

**4**

votes

**0**answers

73 views

### On the density map of the abundancy index

Let $σ$ be the sum-of-divisors function. Let $σ(n)/n$ be the abundancy index of $n$. Consider the density map $$f(x) = \lim_{N \to \infty} f_N(x) \ \ \text{ with } \ \ f_N(x) = \frac{1}{N} \#\{ 1 \...

**0**

votes

**0**answers

8 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

**1**answer

99 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, ...

**3**

votes

**0**answers

11 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

**0**answers

60 views

### General class of functions satisfying growth condition on a given functional

The question is inspired by Abel Plana summation formula :
Is there a general class of functions $f$ that are positive valued on the positive real axis, and which satisfy the following
$$\int_0^\infty ...

**50**

votes

**5**answers

4k views

### Category theory and set theory: just a different language, or different foundation of mathematics?

This is a question to research mathematicians, as well as to those concerned with the history and philosophy of mathematics.
I am asking for a reference. In order to make the reference request as ...

**-1**

votes

**0**answers

44 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 }...

**0**

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 ...

**-3**

votes

**0**answers

31 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 ...

**1**

vote

**0**answers

16 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 ...

**3**

votes

**0**answers

27 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 $...

**5**

votes

**1**answer

117 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 ...

**0**

votes

**0**answers

32 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

**0**answers

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-...

**2**

votes

**0**answers

16 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

**1**answer

263 views

### What is the symmetry group of this configuration?

This configuration appear as problem 3845 in Crux Mathematicorum. I see it is very beautiful. This configuration are generalization of Pascal theorem and Brianchon theorem:
Consider six points $A_1$, $...

**2**

votes

**0**answers

66 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 ...

**1**

vote

**0**answers

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 \...

**10**

votes

**0**answers

88 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 ...

**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 ...

**12**

votes

**6**answers

4k views

### What is the best way to study Rational Homotopy Theory

I studied basic algebraic topology elements:
fundamental group, higher homotopy groups, fibre bundles, homology groups, cohomology groups, obstruction theory, etc.
I want to study Rational Homotopy ...

**6**

votes

**1**answer

275 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 $...

**0**

votes

**0**answers

72 views

+50

### A set of questions on continuous Gaussian Free Fields (GFF)

As I said in my previous posts, I'm trying to teach myself some rigorous statistical mechanics/statistical field theory and I'm primarily interested in $\varphi^{4}$, but I know that the absense of ...

**3**

votes

**1**answer

154 views

### Stokes's Theorem with singularities on projective line

Let $X$ be a complex manifold and $\omega\in \Omega(X\times \mathbb{P}^1)$ a form. I met the following identity:
$$\int_{\mathbb{P}^1}(\partial_z\bar{\partial}_z\omega) \log|z|^2=\int_{\mathbb{P}^1}\...

**0**

votes

**0**answers

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 $...

**0**

votes

**0**answers

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 $...

**1**

vote

**1**answer

213 views

### Integration by parts on manifold with corners

Suppose that $M$ is a compact manifold with corners, where each boundary hypersurface is an embedded submanifold. Then, do we have an integration by parts identity? i.e.
\begin{align*}
\int_M g(\nabla ...

**5**

votes

**0**answers

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 ...

**-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

75 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 ...

**1**

vote

**0**answers

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_\...

**0**

votes

**0**answers

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 $x_{i,j} \in \{ 0,1\}^{1 \times 10}$, $i\in \{1,2,3,4,5\}$, $j \in \{1,2,3\}$ (...

**57**

votes

**11**answers

9k views

### Each mathematician has only a few tricks

The question "Every mathematician has only a few tricks" originally had approximately the title of my question here, but originally admitted an interpretation asking for a small collection ...

**9**

votes

**1**answer

214 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 ...

**3**

votes

**3**answers

98 views

### Convergence of conditional measures for a convergent sequence of probabilities whose projection is constant

Setting
Suppose $\mu_n$ is a sequence of probability measures on $[0,1]\times [0,1]$ converging to a limit probability $\mu$ meaning that
$$ \lim_{n\to+\infty}\int f(x,y)d\mu_n(x,y) = \int f(x,y)d\mu(...

**-2**

votes

**1**answer

130 views

### Polynomials of minimum degree that interpolate primes in intervals

Given an interval $[a,b]$ what is the minimum degree of univariate polynomials in $\mathbb Q[x]$ that passes through all primes between $a$ and $b$ (denoted by $\mathbb P[a,b]$ with total number of ...

**1**

vote

**2**answers

177 views

### Under what conditions are two orientation-reversing involutions of a compact surface equivalent?

Let $M$ be a compact, connected, orientable surface and $\varphi_1,\varphi_2$ be two orientation-reversing involutions (i.e., diffeomorphisms for which $\varphi^2=Id$) such that the fixed-point set ...

**2**

votes

**1**answer

80 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 ...

**1**

vote

**1**answer

154 views

### Trotter-Kato approximation theorem for uniformly continuous approximants

Let
$E$ be a $\mathbb R$-Banach space
$(T_n(t))_{t\ge0}$ and $(T(t))_{t\ge0}$ be strongly continuous contraction semigroups on $E$ with generators $(\mathcal D(A_n),A_n)$ and $(\mathcal D(A),A)$, ...

**18**

votes

**3**answers

974 views

### Surprising properties of closed planar curves

In https://arxiv.org/abs/2002.05422 I proved with elementary topological methods that a smooth planar curve with total turning number a non-zero integer multiple of $2\pi$ (the tangent fully turns a ...

**9**

votes

**1**answer

593 views

### Nodal curve in a smooth variety with injective map on fundamental groups

Let $C$ be the nodal curve obtained by gluing together the points $0$ and $1$ of $\mathbb{A}^1_{\mathbb{C}}$. The topological fundamental group of $C$ is isomorphic to $\mathbb{Z}$.
One can find an ...

**12**

votes

**5**answers

6k views

### A G-delta-sigma that is not F-sigma?

A subset of $\mathbb{R}^n$ is
$G_\delta$ if it is the intersection
of countably many open sets
$F_\sigma$ if it is the union of countably many closed sets
$G_{\delta\sigma}$ if it is the union
of ...

**1**

vote

**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 ...

**0**

votes

**0**answers

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 ...

**11**

votes

**0**answers

181 views

+100

### Identity involving zonal polynomials and $\operatorname O(N)$ irrep dimensions

$\DeclareMathOperator\U{U}\DeclareMathOperator\O{O}$Schur functions $s_\lambda(x)$ with $\lambda\vdash n$ are simultaneously the irreducible characters of the unitary group $\U(N)$ and proportional to ...

**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 ...

**4**

votes

**0**answers

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-...