0
votes
0answers
156 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
2answers
261 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 ...
3
votes
1answer
101 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
29 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
29 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
85 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 ...
1
vote
0answers
53 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 ...
4
votes
0answers
93 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 ...
4
votes
1answer
207 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
13 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
106 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
4
votes
0answers
67 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 ...
-2
votes
0answers
114 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 ...
14
votes
3answers
336 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), ...
3
votes
1answer
147 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
44 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
86 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 ...
4
votes
0answers
83 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
90 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 ...
4
votes
2answers
174 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 given finite number ...
7
votes
2answers
261 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
72 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 ...
16
votes
0answers
470 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
74 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
39 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
45 views

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

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 ... ...
10
votes
1answer
210 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, ...
5
votes
1answer
177 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
3answers
358 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
95 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
40 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
19 views

what is formula of cumulative distribution function of hypergeometric distribution? [closed]

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?
11
votes
2answers
271 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
295 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
87 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
39 views

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

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
103 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 ...
6
votes
0answers
97 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 ...
10
votes
1answer
413 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
117 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 ...
0
votes
1answer
207 views

Non-standard numbers and exponential form of Zeta function

Basic idea For a long time I was looking for a numerical system that would allow to compare infinite sets. In contrast to Cantor's approach that empathizes the possibility of on-to-one correspondence ...
-4
votes
0answers
46 views

Are all derivatives of sinc function bounded on real axis? [closed]

It seems that all derivatives of sinc function (sinc(x)=sin(x)/x) are bounded on real axis. Is it true or no? Thanks in advance.
2
votes
1answer
52 views

Does order-preserving equal continuous? [closed]

Let $P,Q$ be posets and endow them with the interval topology $\tau_i(P)$ and $\tau_i(Q)$ respectively. Is it true that if $f: P\to Q$ is order-preserving, then it is continuous, and vice versa?
4
votes
1answer
46 views

Path-connected Hausdorff interval topologies

Let $(P,\leq)$ be a poset with more than $1$ point such that the interval topology $\tau_i(P)$ is path-connected and $T_2$. Does this imply that $[0,1]$ order-embeds into $P$? (This is a follow-up ...
1
vote
0answers
39 views

Minkowski spacetime in Newman Penrose formalism

I have a rather basic question for which (surprisingly!) I cannot find a short and clear answer anywhere: I'm currently looking at the Newman Penrose (NP) formalism (I use primarily Chandrasekhar's ...
5
votes
0answers
109 views

Indecomposable representations of a wreath product

If $G$ is a finite group, we know the irreducible representations of $G ≀ S_n$ (over $\mathbb Q$) are classified by partitions of $n$ 'decorated' by an irrep of $G$. I'm wondering to what extent the ...
4
votes
0answers
68 views

“Edge Density” of Infinite Planar Graphs

Given an infinite planar graph $G$, let's denote by $\{H_1,H_2,\dots,H_m\}$ all the labeled graphs on $n$ vertices that appear as subgraphs of $G$. Also let $$d_n=\frac{\sum_{i=1}^m \#E(H_i)}{nm}$$ ...
3
votes
0answers
98 views

Elementary examples on sheaf extension

Let $V\subset\mathbb{P}^n$ be a projective variety and $C_V$ its conormal subvariety in $T^\ast\mathbb{P}^n$. Denote by $\mathscr{O}_{C_V}$ its structure sheaf, then when will the condition ...
3
votes
0answers
71 views

Representing rational homotopy class by geometric objects

Given a smooth manifold $M$, we can study its rational homotopy type by looking at differential forms. I am wondering if there is a way to represent each $rational$ homotopy class by geometric ...
3
votes
1answer
56 views

convex hull of the set of permutations with one cycle

is there a way to describe the convex hull of the set of permutation matrices with exactly one non-trivial cycle? or maybe I should ask for the convex hull of cycle matrices : let $(i_{1},..,i_{k})$ ...

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