0
votes
0answers
3 views

Differentiability of geodesics in Alexandrov subspaces of Riemannian manifolds

Let $M$ be a smooth Riemannian manifold. Let $X\subset M$ be a closed path connected subset which has curvature bounded below in the sense of Alexandrov with respect to the induced intrinsic metric. ...
0
votes
0answers
15 views

Decomposable elements in cohomology ring

I feel this must be well known, but I don't have enough knowledge on the matter. So, if there is a literature on this, then please provide a reference. Let $E$ be a ring spectrum, and $X$ a pointed ...
1
vote
0answers
28 views

How many lines of exactly n points can be placed in a discrete, square grid of size m x m?

Per the title, I'm seeking the definition of a function $f(n, m)$ which evaluates to the number of lines made from exactly $n$ points which can be placed on a two-dimensional discrete, square grid of ...
-7
votes
0answers
14 views

Student transcripts lost, you can apply for student academic qualifications

business type Main items: Diplomas, transcripts, embassy certification, certified by the Ministry of Education, student card - OFFER like. Transact real embassy notarization (ie, returned overseas ...
-4
votes
0answers
19 views

The transpose of a matrix times its transpose is the matrix's transpose times itself [on hold]

Somebody please help me to show the steps to prove this is true: $R$ is a matrix and T is transpose: $(R^TR)^T = R^TR^{TT} = R^TR$
2
votes
0answers
28 views

Ideals of $L^1(G)$

I want to study closed ideals structure of $L^1(G)$. Is there a good paper or book which it characterized closed ideals and maximal ideals of $L^1(G)$?
0
votes
0answers
38 views

An integral form of sum $\sum_{n\geq 0} \frac{f^{(n)}(0) g^{(n)}(0)}{(n!)^2}$ for two real analytic at 0 functions? Fourier|Taylor series parallels

Thinking of parallels between Fourier series and Taylor series, you might find out that the integral $\frac 1 {2 \pi}\int\limits_0^{2 \pi} f(e^{it})\,\overline{g(e^{it})} \,dt=\langle f,\, g\rangle$ ...
5
votes
0answers
105 views

Errata on Rezk's paper

I am reading this paper of Rezk's http://arxiv.org/abs/0901.3602 A cartesian presentation of weak n-categories, and as it is pointed out in the introduction, it contained a wrong statement (2.19 in ...
2
votes
0answers
72 views

étale cohomology of rings of integers of number fields and Shafarevich-Tate groups

Let $K$ be a number field, $A$ an abelian variety over $K$. Let $\mathcal{O}$ be the ring of integers of $K$, $\mathcal{A}$ the Néron model abelian scheme of $A$ over $\text{Spec}(\mathcal{O})$. For ...
-1
votes
0answers
37 views

Positivity of holomorphic bisectional curvature and existence of rational curves

Let $X$ be a compact Kähler manifold such that $K_X$ is nef. When there is no rational curve on $X$? Motivation: Siu-Yau by the positive bisectional curvature assumption, produced a rational ...
7
votes
0answers
155 views

How much mathematics has been formally verified?

That's a vague question so allow me to tighten it up a bit. I recently noticed that there is a formal machine verified proof of the Central Limit Theorem (CLT) implemented with Isabelle. This ...
5
votes
0answers
47 views

Multiplication in universal enveloping algebra in terms of PBW isomorphism

Let $\mathfrak g$ be a Lie algebra. Consider the multiplication map $m:\mathfrak g\otimes U(\mathfrak g)\to U(\mathfrak g)$ and $i:S(\mathfrak g)\to U(\mathfrak g)$ -- Poincare-Birkhoff-Witt ...
6
votes
1answer
107 views

Main term in the number of sign changes of $\psi(x) - x$

Define $N_\Delta(T)$ to be the number of sign changes of $\psi(x) - x$ in the interval $[1, T]$. Landau's Theorem says $N_\Delta(T)$ is $\Omega(\log T)$ [1]. But perhaps that estimate is too crude. ...
8
votes
0answers
78 views

Does the ring $R = \mathbb{Z}[X^{\pm1}]$ of Laurent polynomials over $\mathbb{Z}$ satisfy $SL_2(R) = E_2(R)$?

Let $R = \mathbb{Z}[X^{\pm1}]$ be the ring of Laurent polynomials on one indeterminate over $\mathbb{Z}$. Let $E_2(R)$ be the subgroup of $GL_2(R)$ generated by the matrices that differ from the ...
2
votes
0answers
55 views

Gaussian Integrals and Pseudo-Anosov Maps

The hep-th section of arXiv if often filled with beautiful semi-rigorous computations on Mathematics. However sometimes it is very difficult to understand what is being stated. Here I take from: ...
2
votes
0answers
28 views

Is the class of commutative generalized Euclidean rings stable under quotient and localization?

Let $R$ be an associative ring with indentity and let $E_n(R)$ be the subgroup of $GL_n(R)$ generated by matrices obtained from the identity matrix by replacing an off-diagonal entry by some $r∈R$. ...
2
votes
0answers
72 views

An elementary question about metrics on the real plane [on hold]

Given the metric $d_p$ on the real plane, i.e. $$ d_p(x,y) = d_p((x_1, y_1), (x_2, y_2)) = [|x_1 - x_2|^p+ |y_1 - y_2|^p]^{1/p} $$ for which values of $p$ ($\geq 1$) is it true that the following ...
1
vote
0answers
47 views

Irreducible binary quartic form with prescribed $I$ and $J$-invariants

Let $F(x,y) = ax^4 + bx^3y + cx^2y^2 + dxy^3 + ey^4$ be a binary quartic form with integer coefficients. It is well-known that $F$ has two algebraically independent invariants under the action of ...
0
votes
0answers
139 views

Learning roadmap for algebraic number theory

I have read some elementary number theory from David Burton's text and I know groups and rings from Herstein's book Topics in Algebra and some field theory from different sources online. I am ...
5
votes
0answers
42 views

What is the precise relationship between primitive Hida families and the connected components of the ordinary locus of the eigencurve?

In the references I've found discussing this question, I have not found any statements that I can understand and that are as precise as I would like. I'm more familiar with Hida families than with the ...
1
vote
0answers
44 views

A non-surjective coboundary map induced by a central extension

Let $k$ be a number field and $$ 1\to A \to B \to C \to 1$$ be a central extension of finite groups over $\mathcal{O}_k$ (the ring of integers of $k$), with $B$ non-commutative. Consider the induced ...
3
votes
0answers
39 views

$p$-forms acting on spinors

I'm trying to understand the paper by Atiyah, Hitchin and Singer called: ''Self-duality in four dimensional Riemannian geometry", available here. I'm stuck at the point where it explains how the ...
3
votes
1answer
128 views

Breaking the RSA encryption based on a $(e,N)$ given an integer $w \neq 0$ such that $e^w = 1 \mod(N)$?

In his book 'Forcing with Random Variables and Proof Complexity' Jan Krajíček claims (p.154) that it is possible to break the RSA encryption with public key $(e,N)$ if one has has an integer $w \neq ...
0
votes
0answers
40 views

Addition Operations in Complete Lattices

Given a complete bounded lattice $L$, what do we know about the possibility of defining an addition operation $+$ that, broadly speaking, behaves like arithmetical addition? By this, I mean that as ...
0
votes
0answers
27 views

Expression for Joint-PDF of Langevin equation?

How to derive exact or approximate analytical expression for time-dependent joint-PDF (velocity-coordinate PDF) for Langevin equations of Brownian motion? Langevin equations is: $\dot{x}=v$ ...
-6
votes
0answers
24 views

Center of Gravity Question [on hold]

What's the value of x? G is the center of the gravity of this triangle.
3
votes
0answers
64 views

Partial formality of A-infinity structure implies formality

Let $A$ be a (finite dimensional, unital, associative) $k$-algebra, where $k$ is a (algebraically closed) field. Let $M$ be a (finite dimensional) $A$-module. Then, it is known that ...
-4
votes
0answers
36 views

Prove that OB^2 + PC^2 = OP^2 [on hold]

Suppose you have A,B,C,D four points in harmonic range and O, P are the midpoints of AB and CD respectively. Prove that Prove that $OP^2=PC^2+OB^2.$ I would guess that this is a sort of corollary to ...
-2
votes
0answers
39 views

Ideal of Operators [on hold]

If H is infinite dimensional Hilbert space, $A,B\in B(H)$ and $A+B=1$, then either rang $A$ or rang $B$ contains a subspace with the same dimension as $H$ ? (this is the Lemma 17.3 of A Course in ...
0
votes
0answers
42 views

GI-hard problems that would benefit from efficient algorithm for GI

The exponential time hypothesis (ETH) states that 3SAT can not be solved in subexponential time. As far as I know, it is not known whether an efficient algorithm for graph isomorphism problem (GI) ...
5
votes
0answers
107 views

horocycle flow and the prime number theorem

Looking at Zagier's Eisenstein Series and the Riemann Zeta Function, we get a proof of the prime number theorem using horocycles. I would really love it if there were a geometric proof like this. ...
-2
votes
0answers
66 views

Turning $\{0<\dots<n\}$ into a measure space [on hold]

I'm looking for references who define interesting (i.e. nontrivial, i.e. not the Borel subsets of the order topology on $\{0<\dots < n\}$, which is discrete) $\sigma$-algebras on finite ...
4
votes
0answers
86 views

How can a sequence of functions become a basis of L^2

Let $\{\varphi_n(x)\}_{n=1}^\infty$ be an orthonormal basis of $L^2(\Omega)$, where $\Omega$ is a bounded domain in $\mathbb R^d$ with sufficiently smooth boundary. I want to find the assumption on a ...
4
votes
0answers
114 views

A metric on $Homeo([0,1])$

One can define a metric on the set $Homeo([0,1])$ by setting $dist(f,g) =$ measure of support of $f^{-1}g$, that is the measure of the set of points $x$ where $f(x)\ne g(x)$. Was this metric studied ...
-1
votes
1answer
159 views

Is there a proof (maybe formulated by Feferman) which says that a proof about the (in)consistency of ZFC is unachievable? [on hold]

Is there a proof (maybe formulated by Feferman) which says that a proof about the (in)consistency of ZFC is unachievable? A professor said it to me a long time ago, but I don't have any references. ...
0
votes
0answers
56 views

$\mathbb{Z}/4\mathbb{Z}$ is indecomposable in the homotopy category of chain complexes of abelian groups [on hold]

I want to understand the accepted answer to this question. The answer is supposed to work for the homotopy category of chain complexes of abelian groups too. (i.e. it shows that that category is not ...
1
vote
0answers
25 views

A question about Fourier transform of function of the type $(1+P(x))^{z}$

Let $$f= (1+P(x))^{z},$$ where $P(x)\ge 0$ is a real polynomial in $\mathbb{R}^n$ of degree $2m>0$, and $z=a+ib$ with $a<0$. I want to study the behavior near the origin of the Fourier ...
0
votes
0answers
34 views

Dual space of vector fields with null divergence

Let us consider $\mathscr D(\mathbb R^3;\mathbb R^3)$ the space of smooth compactly supported vector fields in $\mathbb R^3$ and let us define $\widetilde{\mathscr D}=\mathscr D(\mathbb R^3;\mathbb ...
5
votes
0answers
122 views

Is there an explicit construction of Tate-Beilinson residue?

Background: Tate's construction of abstract residues is generalized in Beilinson's constructon as follows: Let $E$ be a unital, cubically decomposed $k$-algebra, i.e., (a) there are two sided ideals ...
0
votes
0answers
35 views

Explicit solution of recursive function for $F(n,a)=F(n-1,a)+F(n-(1+a),a), F(b,a)=1, b={1,2,3…,a+1}$ [on hold]

For the most common cases, $a=0$ and $a=1$, the explicit solutions are generally known as: $$F(n,0)=2^{n-1}$$ $$F(n,1)=\dfrac{\phi^n-(-\phi)^{-n}}{\sqrt{5}}$$ Is it possible to derive similar ...
3
votes
1answer
199 views

Irreducible polynomial $p_{n}(x)=\sum_{k=0}^{n}\dfrac{x^k}{k!}$ for all positive integers $n$

Let $n$ be a positive integer greater than $1$, and define the polynomial $$p_{n}(x)=\sum_{k=0}^{n}\dfrac{x^k}{k!}$$ Is $p_{n}(x)$ irreducible in $\mathbf{Q}[x]$? I can show it when $n$ is a ...
5
votes
1answer
126 views

Regular minimal model of $X_0(p^2)$

Consider the compactified modular curve $X_0(p^2)$ and the corresponding algebraic curve over $\mathbb{Q}$. My questions are the following: Where do the cusps of $X_0(p^2)_{\mathbb{Q}}$ live? That ...
1
vote
1answer
85 views

Basis for the Orlicz Space

Does the Orlicz space (https://en.wikipedia.org/wiki/Birnbaum-Orlicz_space) has unconditional Schauder basis? Can we find such a orthonormal basis like the Hermitian polynomials in $L^2$?
3
votes
3answers
133 views

A question about intuition of fluid limit in queuing system

This is a question about intuition in understanding the fluid limit queuing system. Assume we have a sequence of queuing systems $\{S^N\}_{N=1}^{\infty}$ with N servers and each server has unit ...
12
votes
2answers
390 views

1-st cohomology of multiplicative group in a vector space

Let $\mathbb k$ be a field of characteristic $p$ and let $\mathbb k_n$ be a 1-dimensional representation of $\mathbb k^\times$, where the action is given by $t\circ v= t^n v$. Is it known what are the ...
0
votes
0answers
139 views

exactness of induction functor [on hold]

I am studying induction functor of algebraic groups. In particular, let $G$ be a reductive group scheme defined over an algebraically closed field $k$ of prime characteristic and $T$ a maximal torus ...
3
votes
0answers
168 views

the origin of the differential of Spectral Sequence? [on hold]

I'm wondering why the differential in homological Spectral Sequences $(E^*_{p,q},d^r)$ is defined as; $d^r_{p,q}: E^{r}_{p,q} \rightarrow E^{r}_{p-r,q+r-1}$ . From Weibel's homological algebra(p122), ...
-3
votes
0answers
77 views

Finding Bijection between Permutation Set [on hold]

$\beta$ is a subset of symmetric group $S_n$ which acts on $n$ elements of set $X$. Permutations of $\beta$ acts on $k $ elements of $X$ only. Set $L$ is a set of $n$ labels which labels ...
3
votes
0answers
49 views

Lattice points in regular simplex

Suppose we are given a regular (closed) simplex $S$ in a vector space $V$ of dimension $n$, whose vertices have integer values. Then for a lattice $L$, is there a sufficient criterion, for $S$ to ...
10
votes
2answers
657 views

Witness to a failure of Fubini/Tonelli

Is it provable in ZFC that there is a subset of the plane all of whose vertical cross sections have Lebesgue measure zero and all of whose horizontal cross sections are complements of sets of Lebesgue ...

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