All Questions

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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, ...
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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 $n$) ...
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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 $$s(f) = \prod_p \frac{1-\frac{n_f(p)}p}{1-\frac1p},$$ ...
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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 ...
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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 ...
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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?
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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 ...
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 $t \gg 1$: $$\int_M e^{n \mathbf{e}} ... 0answers 82 views System of congruences I have a system of n congruences. the generic m_th congruence of the system (m = 1,\dots,n) is in the form: (p_n\#)^2(\displaystyle\sum_{\substack{i=1\\i\neq ... 0answers 39 views Differentiability of the function \frac{x}{1+\|x\|} [on hold] Is the function f:\mathbb{R}^n\to\mathbb{R} given by$$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 ... 0answers 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 ... 0answers 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 ... 1answer 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 ... 0answers 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 ... 1answer 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 ... 0answers 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. 1answer 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? 0answers 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 ... 0answers 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 ... 0answers 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 ... 0answers 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}$$... 0answers 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 ... 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 ... 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}) ... 1answer 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 ... 5answers 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 ... 2answers 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 ... 0answers 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 ... 0answers 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 ... 0answers 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 ... 0answers 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 ... 0answers 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 ... 0answers 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 ... 0answers 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 ... 1answer 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 ... 0answers 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 0answers 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 ... 0answers 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 ... 0answers 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) ... 1answer 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 ... 0answers 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 ... 0answers 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 ... 2answers 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 ... 0answers 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 ... 0answers 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}, $$in which ... 1answer 74 views Orthogonal polynomials with respect to the lognormal distribution I am currently doing some inspection on the orthogonal polynomials with respect to the lognormal distribution. Does anyone already work on that or know some cool references? All the best, Pierre-O. 0answers 127 views “Graph Individualization”[ reference request] [on hold] Context: I am studying Weisfeiler Lehman method(WL method) and have clear idea about 1 and 2 dimensional WL method. I was wondering about the individualization process described below- ... 1answer 189 views The “\infty”-column in the periodic table of n-categories A monoid is the same as a category with a single object. A monoidal category is the same as a bi-category with a single object. A commutative monoid is the same as a bi-category with a single object ... 0answers 75 views Recursions which define polynomials? Let k be a positive integer and let$$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 ...
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 $P$?