Related papers: On circular distributions and a conjecture of Cole…
About the relation between a counterexample in the theory of numbers constructible by ruler and compass, due to Ian Stewart, and Alhazen's problem concerning the circular mirror.
This is a survey paper on Geometric Manin's conjecture which was proposed by Brian Lehmann and the author. We introduce Geometric Manin's conjecture (GMC) and review some recent progress on this conjecture.
In this work we generalize the concept of injective module and develop a theory of divisibility for modules over a general ring, which provides a general and unified framework to study Kummer-like field extensions arising from commutative…
The union-closed sets conjecture, attributed to P\'eter Frankl from 1979, states that for any non-empty finite union-closed family of finite sets not consisting of only the empty set, there is an element that is in at least half of the sets…
Let $p$ be an odd prime. In the paper we collect the author's various conjectures on congruences modulo $p$ or $p^2$, which are concerned with sums of binomial coefficients, Lucas sequences, power residues and special binary quadratic…
I review the classical theory of likelihood based inference and consider how it is being extended and developed for use in complex models and sampling schemes.
Combinatorics is a powerful tool for dealing with relations among objectives mushroomed in the past century. However, an more important work for mathematician is to apply combinatorics to other mathematics and other sciences not merely to…
This paper examines the classical matching distribution arising in the "problem of coincidences". We generalise the classical matching distribution with a preliminary round of allocation where items are correctly matched with some fixed…
We prove a general result on irregularities of distribution for Borel sets intersected with bounded measurable sets or affine half-spaces.
One way of defining probability distributions for circular variables (directions in two dimensions) is to radially project probability distributions, originally defined on $\mathbb{R}^2$, to the unit circle. Projected distributions have…
We formulate conjectures regarding percolation on planar triangulations suggested by assuming (quasi) invariance under coarse conformal uniformization.
On the basis of perturbed Kolmogorov backward equations and path integral representation, we unify the derivations of the linear response theory and transient fluctuation theorems for continuous diffusion processes from a backward point of…
It has been conjectured that the sequence $(3/2)^n$ modulo $1$ is uniformly distributed. The distribution of this sequence is signifcant in relation to unsolved problems in number theory including the Collatz conjecture. In this paper, we…
We are interested in comparing probability distributions defined on Riemannian manifold. The traditional approach to study a distribution relies on locating its mean point and finding the dispersion about that point. On a general manifold…
The "Modularity Conjecture" is the assertion that the join of two nonmodular varieties is nonmodular. We establish the veracity of this conjecture for the case of linear idempotent varieties. We also establish analogous results concerning…
This paper considers random (non-Hermitian) circulant matrices, and proves several results analogous to recent theorems on non-Hermitian random matrices with independent entries. In particular, the limiting spectral distribution of a random…
Consider the empirical spectral distribution of complex random $n\times n$ matrix whose entries are independent and identically distributed random variables with mean zero and variance $1/n$. In this paper, via applying potential theory in…
In the present paper we prove the following conjecture in Kingman, J.F.C., Random walks with spherical symmetry, Acta Math.,109, (1963), 11-53. concerning a famous Raikov's theorem of decomposition of Poisson random variables: "If a radial…
We study the convergence of distributions on finite paths of weighted digraphs, namely the family of Boltzmann distributions and the sequence of uniform distributions. Targeting applications to the convergence of distributions on paths, we…
We study certain polyadicly continuous sequences from point of view the probability theory.