# All Questions

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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: $(R^TR)^T = R^TR^{TT} = R^TR$
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### 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 $L^1(G)$?
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### 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 $\frac 1 {2 \pi}\int\limits_0^{2 \pi} f(e^{it})\,\overline{g(e^{it})} \,dt=\langle f,\, g\rangle$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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: ...
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### 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 $r∈R$. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $p$-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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $L^2$?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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), ...
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### 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 ...
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 ...