Related papers: Convexity and Aigner's Conjectures

Markov chains are used to give a purely probabilistic way of understanding the conjugacy classes of the finite symplectic and orthogonal groups in odd characteristic. As a corollary of these methods one obtains a probabilistic proof of…

The notion of a successful coupling of Markov processes, based on the idea that both components of the coupled system ``intersect'' in finite time with probability one, is extended to cover situations when the coupling is unnecessarily…

The paper considers a stochastic differential equation of Duffing type with Markov coefficients. The existence of unpredictable solutions is considered. The unpredictability is a property of bounded functions characterized by unbounded…

We expose in full detail a constructive procedure to invert the so--called "finite Markov moment problem". The proofs rely on the general theory of Toeplitz matrices together with the classical Newton's relations.

The theory of ``Markov-up'' processes is being developed. This is a new class of stochastic processes with ``partial'' markovian features; it could also be called ``one-sided Markov''. Such a behavior may be found in the real world and in…

In this paper, we use a variety of classical and new research methods for ternary exponential Diophantine equations and extensive use of computer calculations to study the conjecture of R. Scott and R. Styer which asserts that for any fixed…

Following McShane, we employ the stable norm on the homology of the modular torus to investigate the Markov ordering on the set of relatively prime integer pairs $(q,p)$ with $q\ge p\ge0$. Our main theorem is a characterization of slopes…

Let $\mathcal{T}_3$ be the three-rowed strip. Recently Regev conjectured that the number of standard Young tableaux with $n-3$ entries in the "skew three-rowed strip" $\mathcal{T}_3 / (2,1,0)$ is $m_{n-1}-m_{n-3}$, a difference of two…

The sign coherence phenomenon is an important feature of c-vectors in cluster algebras with principal coefficients. In this note, we consider a more general version of c-vectors defined for arbitrary cluster algebras of geometric type and…

Gabrielov numbers describe certain Coxeter-Dynkin diagrams of the 14 exceptional unimodal singularities and play a role in Arnold's strange duality. In a previous paper, the authors defined Gabrielov numbers of a cusp singularity with an…

This paper contributes to the conjecture of R. Scott and R. Styer which asserts that for any fixed relatively prime positive integers $a,b$ and $c$ all greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ in positive…

We search for triangular numbers that are multiples of other triangular numbers. It is found that for any positive non-square integer multiplier, there is an infinity of multiples of triangular numbers that are triangular numbers and…

Given any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction…

We prove that the three-state toric homogenous Markov chain model has Markov degree two. In algebraic terminology this means, that a certain class of toric ideals are generated by quadratic binomials. This was conjectured by Haws, Martin…

The Markov commutator associated to a finite Markov kernel P is the convex semigroup consisting of all Markov kernels commuting with P. Its interest comes from its relation with the hypergroup property and with the notion of Markovian…

It is conjectured that for any fixed relatively prime positive integers $a,b$ and $c$ all greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ in positive integers $x,y$ and $z$, except for specific cases. We develop…

We give a conjectural formula for the characteristic number of rational cuspidal curves in the projective plane by extending the idea of Kontsevich's recursion formula (namely, pulling back the equality of two divisors in the four pointed…

In this paper we classify all Markov traces on Iwahori-Hecke algebras associated with the finite Coxeter groups of type B. We then use these traces for constructing Jones-type invariants for oriented knots inside a solid torus, and finally…

The famous linear diophantine problem of Frobenius is the problem to determine the largest integer (Frobenius number) whose number of representations in terms of $a_1,\dots,a_k$ is at most zero, that is not representable. In other words,…

An important unsolved problem in Diophantine number theory is to establish a general method to effectively find all solutions to any given $S$-unit equation with at least four terms. Although there are many works contributing to this…