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1answer
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
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) ...
4
votes
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
5answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
6answers
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
1answer
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
0answers
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
1answer
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
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 $...
0
votes
0answers
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
1answer
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
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 ...
-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
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
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_\...
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 $x_{i,j} \in \{ 0,1\}^{1 \times 10}$, $i\in \{1,2,3,4,5\}$, $j \in \{1,2,3\}$ (...
57
votes
11answers
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
1answer
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
3answers
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
1answer
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
2answers
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
1answer
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
1answer
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
3answers
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
1answer
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
5answers
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
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 ...
0
votes
0answers
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
0answers
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
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 ...
4
votes
0answers
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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