Note on Halphen’s system

The Ramanujan Journal - In this paper, we construct extensions of the differential field obtained from Halphen’s system of ordinary differential equations (ODEs) using quasi-modular forms of...     

Stein’s method on Wiener chaos

We combine Malliavin calculus with Stein’s method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process. Our approach generalizes, refines and unifies the central and non-central limit theorems for multiple Wiener–Itô integrals recently proved (in several papers, from 2005 to 2007) by Nourdin, Nualart, Ortiz-Latorre, Peccati and Tudor. We apply our techniques to prove Berry–Esséen bounds in the Breuer–Major CLT for subordinated functionals of fractional Brownian motion. By using the well-known Mehler’s formula for Ornstein–Uhlenbeck semigroups, we also recover a technical result recently proved by Chatterjee, concerning the Gaussian approximation of functionals of finite-dimensional Gaussian vectors.     

A Note on Lewin’s Problem

The parameter l(G) for a primitive digraph G introduced by Lewin is the minimum positive integer k for which there are walks of both lengths k and k + 1 from some vertex u to some vertex v. We obtain upper bounds on l(G) if G is primitive ministrong, or G is just primitive and not necessarily ministrong, or G is primitive symmetric. We also discuss the numbers attainable as l(G).AMS Subject Classification (2000): 05C20, 15A48Partially supported by the National Natural Science Foundation of China (19771040) and the Guangdong Provincial Natural Science Foundation of China (990447).     

A Note on Schnyder’s Theorem

We give an alternate proof of Schnyder’s Theorem, that the incidence poset of a graph G has dimension at most three if and only if G is planar.     

Note on Sus̆kevic̆’s problem on zero divisors

We solve an old Sus?kevic?’s problem on right zero divisors in the ring T(∞,K) of infinite ?×? upper triangular matrices over a field K. In the solution we use a notion of strong linear independence.     

A Note on a Generalisation of Ore’s Condition

The purpose of this note is to study the problem of cyclability under the condition called regional Ores condition. As a consequence, we get the hamiltonicity of a graph G for which Ores condition holds in each of k vertex subsets partitioning V(G) separately (regionally), provided that the graph is k connected.The work was done while two last authors were visiting L R I. This stay was partially supported by french-polish programme POLONIUM     

A Note on Shen’s Conjecture on Groups with Given Same-Order Type

Mathematical Notes - Let $$G$$ be a group. Define an equivalence relation $$sim$$ on $$G$$ as follows: for $$x,y in G$$ , $$x sim y$$ if $$x$$ and $$y$$ have same order. The set of sizes of...     

Eddington’s statistical theory

The density-particle correlation, described in a previous paper of the series, is used to define the «uncertainty of the physical origin» considered by Eddington, and to relate it to the Hubble constant.     

Herman’s theory revisited

We prove that a C ^2+α -smooth orientation-preserving circle diffeomorphism with rotation number in Diophantine class D _δ , 0≤δ<α≤1, α−δ≠1, is C ^1+α−δ -smoothly conjugate to a rigid rotation. This is the first sharp result on the smoothness of the conjugacy. We also derive the most precise version of Denjoy’s inequality for such diffeomorphisms.     

Note on the Permutations which Preserve Buck’s Measure Density

In the first part some conditions under which a permutation preserves Buck’s measure density, are derived. In the second part is constructed a countable system of permutations which has the ergodic property on the system of Buck’s measurable sets. Received: March 27, 2007., Accepted: August 25, 2008.     

Note on Vector Translation with Respect to Thompson’s Metric

It is known that vector translations are contractive with respect to Thompson’s part metric. Here, we give a simple proof, based on a representation of Thompson’s metric through positive functionals. Moreover, we use contractivity of translations to prove a fixed point result for mappings that are Lipschitz continuous with respect to Thompson’s metric with Lipschitz constant r>1. The case r = 1 for order preserving or order reversing mappings has been recently studied by Lawson and Lim. We apply our result to a nonlinear boundary value problem.     

A Note on Toeplitz’ Conjecture

In 1911, Toeplitz made a conjecture asserting that every Jordan curve in $\mathbb{R}^{2}$ contains four points forming the corners of a square. Here Conjecture C is presented, which states that the side length of the largest square on a closed curve that consists of edges of an n×n grid is at least $1/\sqrt{2}$ times the side length of the largest axis-aligned square contained inside the curve. Conjecture C implies Toeplitz’ conjecture and is verified computationally for n≤13.