**10**

votes

**1**answer

169 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

**1**answer

167 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

**2**answers

293 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

**1**answer

93 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

**0**answers

37 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

**0**answers

18 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?

**11**

votes

**2**answers

263 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

**1**answer

282 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

**0**answers

82 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

**0**answers

39 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

**0**answers

91 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

**0**answers

93 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 ...

**9**

votes

**1**answer

281 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

**0**answers

114 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 ...

**-1**

votes

**1**answer

133 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

**0**answers

46 views

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

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.

**1**

vote

**1**answer

46 views

### Does order-preserving equal continuous? [on hold]

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?

**3**

votes

**0**answers

30 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

**0**answers

38 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

**0**answers

104 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

**0**answers

60 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

**0**answers

93 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

**0**answers

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

**1**answer

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 $(i_{1},..,i_{k})$ ...

**3**

votes

**1**answer

109 views

### Kernel of the natural map between group $C^*$-algebras

Let $\Gamma$ be a discrete group. We can form two $C^*$-algebras: the universal (or full) and reduced, to be denoted by $C^*_u(\Gamma)$ and $C^*_r(\Gamma)$ (respectively). Both of them are completions ...

**25**

votes

**5**answers

1k views

### Why should we care about “higher infinities” outside of set theory?

Let's say you are a prospective mathematician with some addled ideas about cardinality.
If you assumed that the natural numbers were finite, you'd quickly vanish in a puff of logic. :)
If you ...

**6**

votes

**2**answers

359 views

### Geodesics on SO(3)

I have two 3D rotations about the origin, represented as
$3 \times 3$ orthogonal matrices $M_1$ and $M_2$
(specified by numerical entries),
and I would like to interpolate (and compute)
a continuous ...

**1**

vote

**0**answers

33 views

### C^1 Sard related question

Let $X$ be a $k+1$ rectifiable set with finite $k+1$ Hausdorff measure in $\mathbb{R}^{n+1}$ and set $Z=\{x\in X;\;\ \textrm{s.t} \;\; e_{n+1}\perp T_xX \}$, where $T_xX$ is the approximate tangent ...

**2**

votes

**0**answers

40 views

### Coupling Marginals of Distributions on the Sphere

Given a distribution $P_X$ on $\mathbb{R}$, when does there exist a coupling (i.e. joint distribution) $P_{X^n}$ of $X_1,...,X_n$, each distributed according to $P_X$, such that $\sum X_i^2 = n$ ...

**2**

votes

**0**answers

59 views

### Listing all Lattice Points in a Box

Let $B := [-1,1]^n$ be an $n$-dimensional box. Moreover, let $v_1,\ldots,v_n \in \mathbb{R}^n$ form a basis of $\mathbb{R}^n$, where the entries of the $v_i$ are explicitly irrational. We can assume ...

**1**

vote

**0**answers

29 views

### About the partial expectation polynomials in the Interlacing-I paper and perfect matchings

I am thinking of the polynomials $f_{s_1,s_2,..,s_k}$ as in the definition 4.3 in this paper http://annals.math.princeton.edu/wp-content/uploads/annals-v182-n1-p07-p.pdf
In the use of these ...

**-4**

votes

**0**answers

57 views

### How to rewrite square root without exponents? [on hold]

I'm trying to understand what taking the square root really is, intuitively. A number to the power of two is simply multiplying the number with itself, and I know the square root is the reversal ...

**5**

votes

**0**answers

179 views

### Why do Kashiwara and Schapira require a base ring of finite global dimension?

In the book "Sheaves on Manifolds" by Kashiwara and Schapira, they work always with sheaves of $R$-modules, where $R$ is a ring of finite global dimension.
Why do they do this, what care ...

**2**

votes

**0**answers

105 views

### $\frac{1}{2}<\sigma<1$, is $f(n) = \Bigl| \,1+ \frac{1}{2^{\sigma + i n}} + \cdots + \frac{1}{n^{\sigma + i n}} \Bigr|$ from $O(\log n)$?

We have $\frac{1}{2} < \sigma < 1$ and
$$
f(n) = \Bigl|\, 1+ \frac{1}{2^{\sigma + i n}} + \cdots + \frac{1}{n^{\sigma + i n}} \Bigr|
$$
. My goal is proving this statement that $|f(n)|$ is ...

**3**

votes

**1**answer

104 views

### Classifying space for homology endomorphisms supported on a graph?

Let $X$ be a reasonable topological space (say one that has the homotopy type of a finite CW complex) and consider a subset $\Gamma$ of $X \times X$ so that the projection $p:\Gamma \to X$ onto the ...

**-5**

votes

**0**answers

31 views

### How to convert a row of matrix as ordered pairs in matlab? [on hold]

I have an n*3 matrix.Now i have to plot each row as a point with the elements in first, second,third columns as x,y,z coordinates in 3-d space.can anyone help me do this in matlab?
Thanks in advance

**1**

vote

**0**answers

74 views

### Profinite groups, directed sets and $H^1$

Usually whenever one reads the definition of profinite group, one starts with an ordered set $I$ which is directed, meaning that for every $i,j\in I$ there is some $k\in I$ such that $i\leq k$ and ...

**5**

votes

**0**answers

120 views

### Tensorization of Orlicz norm?

Associated with a convex function $\phi:[0,\infty)\mapsto[0,\infty)$ satisfying $\lim_{x\to 0} \frac{\phi(x)}{x} = 0, \lim_{x\to\infty}\frac{\phi(x)}{x} = \infty,$ the Orlicz norm of a random variable ...

**5**

votes

**0**answers

146 views

### Relationship between the syntomic cohomology of Kato and of Fontaine-Messing

Fix a prime $p$ and let $X$ be a $\mathbb{Z}_{p}$-scheme. Write $X_{n}:=X\otimes\mathbb{Z}/p^{n}$ and $\phi:X_{1}\rightarrow X_{1}$ for the absolute Frobenius. Let $X\hookrightarrow Z$ be a (suitable) ...

**4**

votes

**1**answer

119 views

### An extension of $K$-theory to topological $^*$-algebras

What I have in mind is the following: a (sequence of) functor(s) $K_\bullet$ on the category of topological $^*$-algebras (with values in the category of commutative groups) that satisfies (among ...

**-3**

votes

**0**answers

24 views

### Permutations of given length [migrated]

Given the counts of each of three letters a, b, and c. I want to find all permutations of a given length where each letter can occur at most times as the given count.
I am only interested in the ...

**2**

votes

**0**answers

77 views

### Convergence rate of Pearson correlation matrix

I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let ...

**4**

votes

**2**answers

133 views

### Natural examples of $\bf\Sigma^0_3$ equivalence relations

I have been reading about Borel equivalence relations and I have noticed that while $\bf\Sigma^0_3$ equivalence relations are mentioned, there is a conspicuous absence of natural examples (other than ...

**1**

vote

**0**answers

58 views

### Comparing Dirichlet energy and area of a Surface-immersion

Let $(F,g)$ be a closed Surface, $(M,h)$ a Riemannian 3-Manifold and $f: F \to M$ a smooth immersion. Denote by $f^*(h)$ the pullback metric on $TF$ induced by $f$ and let $dV_g$ and $dV_{f^*(h)}$ be ...

**0**

votes

**0**answers

56 views

### Singular integral equation

Investigating a control problem for heat equation I stacked on solution of this integral equation which seems to be singular:
$$
\int_0^1\mathcal{K}(\tau)u(\tau)d\tau=\mathcal{M},
$$
in which ...

**2**

votes

**1**answer

74 views

### Orthogonal polynomials with respect to the lognormal distribution

I am currently doing some inspection on the orthogonal polynomials with respect to the lognormal distribution. Does anyone already work on that or know some cool references?
All the best,
Pierre-O.

**1**

vote

**0**answers

127 views

### “Graph Individualization”[ reference request] [on hold]

Context: I am studying Weisfeiler Lehman method(WL method) and have clear idea about 1 and 2 dimensional WL method. I was wondering about the individualization process described below-
...

**4**

votes

**1**answer

189 views

### The “$\infty$”-column in the periodic table of n-categories

A monoid is the same as a category with a single object.
A monoidal category is the same as a bi-category with a single object.
A commutative monoid is the same as a bi-category with a single object ...

**6**

votes

**0**answers

75 views

### Recursions which define polynomials?

Let $k$ be a positive integer and let
$$h(n,k,q)=\frac{1-(1+q^{k})q^{2k(n-1)+1}+q^{2}}{1-q^{2n-1}}h(n-1,k,q)-\frac{(1-q^{k(2n-3)})(1-q^{2k(n-1)})q^2}{(1-q^{2n-1})(1-q^{2n-3})}h(n-2,k,q)$$
with ...

**5**

votes

**1**answer

71 views

### Path-connected interval topologies

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