Related papers: On Stanley's theorem and its Generalizations
This article provides a proof of a generalization of Schur's theorem on the partition regularity of the equation x+y=z, which involves a divisibility condition. This generalization will be utilized to prove the existence of 'small'…
Refined versions, analytic and combinatorial, are given for classical integer partition theorems. The examples include the Rogers-Ramanujan identities, the Gollnitz-Gordon identities, Euler's odd=distinct theorem, and the Andrews-Gordon…
Steinitz's theorem states that if a point $a \in \mathrm{int\,conv\,} X$ for a set $X \subset \mathbb{R}^d$, then $X$ contains a subset $Y$ of size at most $2d$ such that $a \in \mathrm{int\,conv\,}Y$. The bound $2d$ is best possible here.…
We continue the study on sheaves of rings on finite posets. We present examples where the ring of global sections coincide with toric faces rings, quotients of a polynomial ring by a monomial ideal and algebras with straightening laws. We…
We give a simple proof of Strassen's theorem on stochastic dominance using linear programming duality, without requiring measure-theoretic arguments. The result extends to generalized inequalities using conic optimization duality and…
A new general all terminal network reliability factorization theorem is stated. We relegate the proof to a forthcoming second part paper.
This note contains a Stone-style representation theorem for compact Hausdorff spaces.
In this work, we state and prove versions of the linear and bilinear $T(b)$ theorems involving quantitative estimates, analogous to the quantitative linear $T(1)$ theorem due to Stein.
We present a simple extension of Lindeberg's argument for the Central Limit Theorem to get a general invariance result. We apply the technique to prove results from random matrix theory, spin glasses, and maxima of random fields.
The generalized quantum Stein's lemma provides an explicit expression for the optimal error exponent when distinguishing many independent and identically distributed (iid) copies of a given bipartite state from the set of separable…
We discuss various known generalizations of the classical Hartogs' extension theorem on Stein spaces with arbitrary singularities and present an analytic proof based on d-bar methods.
Schur's partition theorem states that the number of partitions of n into distinct parts congruent 1, 2 (mod 3) equals the number of partitions of n into parts which differ by >= 3, where the inequality is strict if a part is a multiple of…
A well-known generalisation of positional numeration systems is the case where the base is the residue class of $x$ modulo a given polynomial $f(x)$ with coefficients in (for example) the integers, and where we try to construct finite…
In this paper, we prove two identities on the partition of a totally positive algebraic integer over a totally real number field which are the generalization of the Sylvester Theorem and that of the Rogers-Ramanujan Identities.…
We extend the celebrated Stone's theorem to the framework of distributional regression. More precisely, we prove that weighted empirical distribution with local probability weights satisfying the conditions of Stone's theorem provide…
By a re-examination of MacMahon's original proof of his celebrated theorem on the distribution of the major indices over permutations, we give a reformulation of his argument in terms of the structure of labeled partitions. In this…
We prove a general divisibility theorem that implies, e.g., that, in any group, the number of generating pairs (as well as triples, etc.) is a multiple of the order of the commutator subgroup. Another corollary says that, in any associative…
We find a canonical form for pure states of a general multipartite system, in which the constraints on the coordinates (with respect to a factorisable orthonormal basis) are simply that certain ones vanish and certain others are real. For…
We generalise Euler's partition theorem involving odd parts and different parts for all moduli and provide new companions to Rogers-Ramanujan- Andrews-Gordon identities related to this theorem.
We prove the conjectured limiting normality for the number of crossings of a uniformly chosen set partition of [n] = {1,2,...,n}. The arguments use a novel stochastic representation and are also used to prove central limit theorems for the…