0
votes
0answers
5 views

Learning the exponents in a sum of two modular roots of unity

$\newcommand{\Z}{\mathbb{Z}}$ Suppose that $n$ is a large and known integer (say, with 100 digits) and that you are given access to a function $$f(x) = x^a + x^b$$ with unknown exponents $a,b \in ...
0
votes
0answers
6 views

Rank of the Jacobian of a family of hyperelliptic curves of genus 2

Assume tha $C$ be the hyperelliptic curve $y^2 = (x-a_1)\cdots (x-a_5)$ of genus $g=2$ and $a_i \in \mathbb{Z}$ and we know that the integers $a_i$ has the form $a_i= d_1^2 - d_i^2$ for some positive ...
0
votes
0answers
25 views

Does $\mathbb Z \times \mathbb Z$ mod the obvious $\mathbb Z$ action have more structure than just a set?

$\mathbb Z$ acts on the lattice $\mathbb Z \times \mathbb Z$ by adding an element to itself n times. I am studying some function arising from symplectic geometry which happens in my case to be ...
0
votes
0answers
6 views

Basic question about discrete minimal surfaces

Let $P$ be a convex polygon with $n > 3$ vertices $v_1, \ldots, v_n \in \mathbb{R}^2$, let $x$ be a point in the interior of $P$, and let $u$ be a function with prescribed values at the vertices of ...
0
votes
0answers
23 views

Tangent space of Hilbert scheme

We have the following theorem: Let $X$ be a projective scheme over an algebraically closed field $k$, and $Y \subset X$ a closed subscheme with Hilbert polynomial $P$. Then$$T_{[Y]}\text{Hilb}_P (X) ...
0
votes
0answers
8 views

Avoiding the range of a bivariate integer function or Diophantine function

I have a bivariate integer function $f(x,y)=5+23x+7y+30xy$ where $x,y \geq 0$ and are integers. The lattice points of this function, or its range contain a large number of values. I'm trying to see if ...
1
vote
1answer
20 views

Automorphism group of the affine groups AGL(n,q), ASL(n,q)

I have a question. The automorphism group of the linear groups $GL(n,q)$, the group of linear transformations of $V = \mathbb{F}_q^n$, and $SL(n,q)$, the subgroup of $GL(n,q)$ consisting of elements ...
0
votes
0answers
60 views

How do mathematicians find the underlying idea? [on hold]

While reading through the lecture notes here (http://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/week2.pdf , page 22, last paragraph), I came across the following " Thus there must be some ...
2
votes
1answer
45 views

Trivial zeroes of the Riemann Zeta function are simple

The trivial zeroes of the Riemann Zeta function are located on $-2\mathbb N^*$ and they are simple. It is not difficult to see that, but the proof I have in mind is using the fact that ...
1
vote
0answers
22 views

Number of linearly bisected subsets in finite vector space $F_2^n$

We consider the $n$-dimenstional finite vector space $\mathbb{F}_2^{n}$ over the finite field of two elements. For a subset $A\subseteq \mathbb{F}_2^{n}$ of even size $|A|=2m$ and a linear form ...
0
votes
0answers
19 views

Bounding Expected Value of a piecewise function [on hold]

Let X and Y to be two independent random variables with known pdfs. Get a bound for the expected value of the following expression in terms of $E[X]$, $E[Y]$, VAR[X] and VAR[Y]: \begin{equation} ...
2
votes
0answers
14 views

Changing the sign in the definition of the cocommutator of a coboundary Lie bialgebra

A Lie bialgebra is a Lie algebra additionally equipped with a 1-cocycle $\delta: {\mathfrak g}\to \Lambda^2 {\mathfrak g}$ that satisfies the co-Jacobi identity. Non-trivial Lie bialgebras can be ...
2
votes
0answers
52 views

What are the restrictions in the ramification behavior of a Galois extension of number fields imposed by the Galois group of the extension?

Studying class field theory, I have come across the following Proposition: Proposition. Let $K/E$ be an extension of number fields so that there is no nontrivial unramified subextension $F/E$ with ...
2
votes
0answers
44 views

homology coalgebra of iterated loop space of spheres

In the paper On the homology of configuration spaces, Section 4.1, a basis for the graded vector space $H_*(\Omega^m S^m;\mathbb{Z}_2)$ is given as $$ u_0, u_1, Q_I u_0, Q_I u_1 $$ where $0,1$ denote ...
1
vote
0answers
38 views

A compact Alexandrov space with curvature bounded below has curvature bouneded above?

For a compact Riemannian manifold, Since the curvature tensor is continuous, we know that the sectional curvature is bounded, i.e. bounded above and below. Now let $M$ be a compact Alexandrov space ...
3
votes
0answers
55 views

Understanding Segal's definition of conformal field theory

I have a fundamental problem in understanding Segal's definition of a conformal field theory: On the one hand his monoidal CFT-functor is a formalization of the fact that, physically, the integrand ...
1
vote
0answers
61 views

$Ext$-algebra generated by $Hom$ and $Ext^1$ as $A_\infty$-algebra?

In [Keller: A-infinity algebras in representation theory, Proposition 1(b)], Keller states that for an associative algebra the $Ext$-algebra of the simples is generated by $Ext^1(S,S)$ as an ...
0
votes
0answers
11 views

MLE of Gamma when only given observations [on hold]

i've been given 10 observations of X, where X is a random variable. the observations are 141 16 46 40 351 259 317 1511 107 567 and now given they are gamma distributed, how do you find the MLE using ...
-2
votes
0answers
90 views

Elementary question of Group cohomology [on hold]

Let $G$ be a finite group. Assume $G$ acts on finite abelian module $M$ such that $(|G|,|M|)=1$. Question: Why $H^i(G,M) = 0$ for $i > 0$? Pierre MATSUMI
2
votes
0answers
36 views

Relationship of ${\cal P}(\omega)/fin$ and ${\cal L}$

Define ${\cal L}$ as in this question: the set of functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $f(n)\leq f(n+1)\leq f(n)+1$, where two functions are considered equal if they differ at at ...
-3
votes
0answers
90 views

Flower Arrangements [on hold]

We have a $n\times m$ grid with $k$ flowers (not necessarily distinct). The grid is assumed to have horizontal and vertical symmetry. What is the value of $A(n,m,k)$, where $A(n,m,k)$ is the number of ...
11
votes
2answers
177 views

Brouwer's theorem for the Cauchy reals

Brouwer famously proved, using principles motivated by intuitionistic choice sequences, that every function $\mathbb{R}\to \mathbb{R}$ is continuous. In Sheaves in geometry and logic (section VI.9), ...
2
votes
1answer
93 views

On the search for an explicit form of a particular integral

Let $f$ be integrable over the interval $(0, 1)$, and $$I_n = \int_0^{1} x^n f(x) \, \mathrm{d}x.$$ Suppose $f(x) = f(1-x)$; we can then show that $$I_n = \sum_{k=0}^{n} \binom{n}{k} (-1)^k \, ...
-2
votes
0answers
38 views

are signs of coefficients arbitrary in 01-integer programming? [on hold]

looking to understand more about full coverage integer programming i wondered if my research should include forms of the problem with the possibility for negative coefficients i went looking for an ...
4
votes
0answers
72 views

An inequality from the “Interlacing-1” paper

This question is in reference to this paper, http://annals.math.princeton.edu/wp-content/uploads/annals-v182-n1-p07-p.pdf (or its arxiv version, http://arxiv.org/abs/1304.4132) For the argument to ...
3
votes
0answers
57 views

Generalized identities of (soluble) groups

Let $G$ be a group. Let us say that $G$ satisfies a generalized identity of degree $n$ if there exist $a_1,a_2,\dots a_n \in G$ such that $$x^{a_1}x^{a_2}\dots x^{a_n}=1,$$ for all $x\in G$. ...
-1
votes
0answers
26 views

Optimization problem with an integral in the objective function [on hold]

I would like to find $x$ that solves the following optimization problem: $min \;L(x) = \int\limits_{-24}^{48} c(t)\cdot f(x,t) dt$ sa $ 0 \leq x \leq 24$ with $c(t) = 5 + cos(\frac{\pi\cdot ...
1
vote
0answers
72 views

Distribution of a transformed Gaussian random vector

Let $B$ be a random vector with $b_{i} \backsim N(m_i,\sigma_i) $ and $Y$ another random vector with $y_i \backsim N(r_i,\psi_i)$. Let $A$ be a symmetric and non-singular square matrix. What is ...
-1
votes
0answers
79 views

Curvature in geometry-interpretation

Previously this question was asked on stack exchange: the answer contained only reference to the wikipedia page which I already read (as mentioned in my post). So here is the question: The are ...
3
votes
0answers
65 views

Vanishing natural transformation and strong generator

Let $X$ be a smooth projective variety (over the field of complex numbers). Let $T$ be strong generator of $D^b(X)$ : this means that every object in $D^b(X)$ can be obtained in a finite number of ...
6
votes
1answer
136 views

For a 3-manifold $Y$, when does $Y\times S^{1}$ admits a Riemannian metric with positive scalar curvature?

Let $Y$ be an orientable, smooth 3-manifold and let $X=Y\times S^{1}$. My question is that: when does $X$ admits a Riemannian metric with positive scalar curvature? An obvious case is when $Y$ ...
1
vote
0answers
59 views

Properties of Coefficients of Order Polynomials

I am working on a problem involving determining the order polynomial $\Omega_P(k)$ of a partial order $P$, which counts the number of order-preserving transformations/maps from $P$ to the $k$-chain ...
15
votes
0answers
403 views

A Linear Order from AP Calculus

In teaching my calculus students about limits and function domination, we ran into the class of functions $$\Theta=\{x^\alpha (\ln{x})^\beta\}_{(\alpha,\beta)\in\mathbb{R}^2}$$ Suppose we say that ...
3
votes
0answers
53 views

Compensated compactness for system of conservation laws?

As far as I knew, the method of compensated compactness can be used only for one-dimensional scalar and $2\times 2$ systems of conservation laws, i.e. $u_t+f(u)_x=0$. But if I understood correctly ...
2
votes
0answers
31 views

When do positively invariant subset contain a given set?

Non-triviality of the Conley index for an isolating neighborhood $N$ and a flow $\varphi$ can be used to prove non-emptyness of the related isolated invariant set. In particular, if $N$ doesn't ...
-3
votes
0answers
44 views

How to find $B$ by solving the following linear system: $s_k$ $B$ ${s_k}^T$ $=1,$ [on hold]

How to find $B$ by solving the following linear system: $s_k$ $B$ ${s_k}^T$ $=1,$ $\qquad$ for $k=1 ... ,p$. Where $s_k$ is a $1\times3$ row_vector from the matrix $S= [s_1 ... ...
7
votes
0answers
91 views

Finding combinatorial models / statistics

In many cases in combinatorics and especially algebraic combinatorics with some representation theory, the main problem is about finding the correct statistic on a mathematical object. For example, ...
4
votes
1answer
143 views

Construction of coherent sheaf such that $\text{Proj}\,\text{Sym}\,(\mathcal{F}) = \text{Sym}^n X$

Let $X$ be a smooth projective curve. How do I construct a coherent sheaf $\mathcal{F}$ on $\text{Pic}^n X$ (i.e., the component of the Picard scheme of $X$ parametrizing line bundles of degree $n$) ...
4
votes
1answer
226 views

Bateman-Horn, continued even further

As before, consider the "singular series", which shows up in the Bateman-Horn conjecture: for an irreducible polynomial $f,$ this is equal to $$ s(f) = \prod_p \frac{1-\frac{n_f(p)}p}{1-\frac1p}, $$ ...
4
votes
1answer
71 views

symmetric measurable 2-cocycles on compact abelian groups vanish?

Is the following result true? If it is, could you plese give me a reference for it? Thanks in advance! Let $(G, \mu)$ be any compact abelian group with Haar measure $\mu$ (The case I am interested ...
0
votes
0answers
33 views

Eigenvalue bounds from eigenvalues of Schur complement

Is it possible to lower bound the minimal eigenvalue of a symmetric PSD matrix $M= \begin{pmatrix}A & C\\ C^*&B\end{pmatrix}$ from the knowledge of the eigenvalues of $M$'s Schur complement ...
0
votes
0answers
17 views

what is formula of cumulative distribution function of hypergeometric distribution? [on hold]

sorry if this question seams ridiculous but I can't understand what does F means in the CDF formula for hypergeometric distribution in this wiki page. can somebody help me?
10
votes
2answers
233 views

Quantum Hamiltonian for an Inverse Cube Force Law

If you have a nonrelativistic quantum particle in $\mathbb{R}^3$ in an attractive inverse cube force, its Hamiltonian is $$ H = -\nabla^2 - \frac{c}{r^2} $$ where I'm keeping things simple by ...
4
votes
1answer
258 views

Asymptotic $\int_M \mathrm{exp}[\mathbf{e}\left(n -\frac{t}{2\pi i}\right)] \left( 1 + \frac{5}{6} \mathbf{e}^2 \right)^{1/2} $ on quintic Calabi-Yau

Let $M = \{ G(x) = 0 \} \subseteq \mathbb{P}^4$ be a quintic Calabi-Yau and $\mathbf{e} \in H^2(M, \mathbb{Z})$ such that $\int_M \mathbf{e}^3 = 5$. Then as $t \gg 1$: $$ \int_M e^{n \mathbf{e}} ...
1
vote
0answers
70 views

System of congruences

I have a system of $n$ congruences. the generic $m$_th congruence of the system ($m = 1,\dots,n$) is in the form: $(p_n\#)^2(\displaystyle\sum_{\substack{i=1\\i\neq ...
-4
votes
0answers
38 views

Differentiability of the function $\frac{x}{1+\|x\|}$ [on hold]

Is the function $f:\mathbb{R}^n\to\mathbb{R}$ given by $$f(x)=\frac{x}{1+\|x\|},\;\;\forall x\in \mathbb{R}^n,$$ where $\|x\|=\sqrt[]{\sum_{i=1}^n{x_i}^2}$,, for all $x=(x_1, x_2,...x_n)$ in ...
6
votes
0answers
72 views

A “universally non Hypercomplete” $\infty$-topos?

My question is : Is there a classifying $\infty$-topos for $\infty$-connected objects ? Does this $\infty$-topos has a nice description (as an $\infty$-category ) ? What I mean by $\infty$-connected ...
4
votes
0answers
83 views

Rings that are $K_0$ of finite groups

Is there a simple characterisation of all rings which appear as $K_0$ of finite groups? By $K_0$ of a finite group $G$ I mean $K_0(\mathbb C[G])$ which in the same as a ring of virtual characters of ...
7
votes
0answers
160 views

Is an irreducible ideal in $R$ also irreducible in $R[x]$?

Let $R$ be a commutative Noetherian ring and $I\subset R$ an ideal that is irreducible in the sense that if $I = J_1 \cap J_2$, then $I=J_1$ or $I=J_2$. Is (the ideal generated by) $I$ irreducible in ...
3
votes
0answers
94 views

Does the Bombieri-Lang conjecture imply severe restrictions on rational points on twists of hyperelliptic curves?

According to Silverman, the Bombieri-Lang conjecture implies that the rational points of surface on general type lie on finite set of curves, except for a finite set of points. Let $f$ be univariate ...

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