On a criterion for matchability in hypergraphs

A strengthened version of a previous conjecture of the author is considered. The former version of the conjecture was that if every subset of a given setA of vertices in a hypergraph is connected to a set of edges with ‘large’ matching number, thenA is matchable. Here we suggest that it is possible to assume only large fractional matching numbers. We prove the conjecture in the case ∣A∣ = 2, and also a fractional version of the conjecture.     

Maximum graphs with a unique minimum dominating set

We present a conjecture on the maximum number of edges of a graph that has a unique minimum dominating set. We verify our conjecture for some special cases and prove a weakened version of this conjecture in general.     

On a Conjecture for Higher-Order Szegő Theorems

We disprove a conjecture of Simon for higher-order Szeg? theorems for orthogonal polynomials on the unit circle and propose a modified version of the conjecture.     

Probabilistic analysis of a parallel algorithm for finding maximal independent sets

We consider a natural parallel version of the classical greedy algorithm for finding a maximal independent set in a graph. This version was studied in Coppersmith, Raghavan, and Tompa and they conjecture there that its expected running time on random graphs of arbitrary edge density of O (log n). We prove that conjecture.     

A version of the volume conjecture

We propose a version of the volume conjecture that would relate a certain limit of the colored Jones polynomials of a knot to the volume function defined by a representation of the fundamental group of the knot complement to the special linear group of degree two over complex numbers. We also confirm the conjecture for the figure-eight knot and torus knots. This version is different from S. Gukov's because of a choice of polarization.     

A counterexample to the bipartizing matching conjecture

The bipartizing matching conjecture (BMC) is a rather new approach to the nowhere zero 5-flow conjecture (NZ5FC) and the cycle double cover conjecture (CDCC). We show that the BMC is wrong in its actual version by constructing a counterexample. The construction arises from the investigation of the problem to cover the vertices of a graph by two induced Eulerian subgraphs. Finally, we state a modified version of the BMC which has the same impact on the NZ5FC and CDCC.     

Multiple recurrence of Markov shifts and other infinite measure preserving transformations

We discuss the concept of multiple recurrence, considering an ergodic version of a conjecture of Erdős. This conjecture applies to infinite measure preserving transformations. We prove a result stronger than the ergodic conjecture for the class of Markov shifts and show by example that our stronger result is not true for all measure preserving transformations.     

On a question of Miles Reid

A conjecture of Miles Reid states that the relative canonical algebra for a pencil of curves of genus greater than one is always generated in degrees at most three (1-2-3 conjecture). We give some explicit counter-examples to the conjecture. Received: 10 December 1998/ Revised version: 15 April 1999     

La conjecture de Manin g��om��trique pour une famille de quadriques intrins��ques

We prove a version of Manin??s conjecture for a certain family of intrinsic quadrics, the base field being a global field of positive characteristic. We also explain how a very slight variation of the method we use allows to establish the conjecture for a certain generalized del Pezzo surface.     

Some notes on distribution of Hecke eigenvalues for Maass cusp forms

In this paper, we get some results on the distribution of Hecke eigenvalues for Maass cusp forms. We consider the diagonal version of Sato–Tate conjecture, a central limit theorem, and a quantitative result on the Ramanujan conjecture.     

The composition conjecture for Abel equation

The composition conjecture for the Abel differential equation states that if all solutions in a neighborhood of the origin are periodic then the indefinite integrals of its coefficients are compositions of a periodic function. Several research articles were published in the last 20 years to prove the conjecture or a weaker version of it. The problem is related to the classical center problem of polynomial two-dimensional systems. The conjecture opens important relations with classical analysis and algebra. We give a widely accessible exposition of this conjecture and verify the conjecture for certain classes of coefficients.     

On a conjecture of Martin on the parameters of completely regular codes and the classification of the completely regular codes in the biggs-smith graph

In this paper we study a conjecture of Martin [7] on the paramaters of completely regular codes in distance-regular graphs. We show that this conjecture is not true in general, but for most classical graphs it is true. we also show that there is a counterexample in the Biggs-Smith graph for a weakened version of Martin's conjecture. Furthermore we classify the completely regular codes in the Biggs-Smith graph.     

Solvable Three-Body Problem and Painlevé Conjectures

For a special choice of the three interparticle coupling constants in the three-body version of a many-body problem in the plane that was recently investigated, the general solution of the equations of motion can be written in closed form (and is remarkably simple). We also discuss another analogous three-body problem and obtain two third-order highly nonlinear autonomous ODEs whose general solutions, we conjecture, are entire. In other words, we conjecture that these ODEs feature (a strong version of) the Painlevé property.     

Tamagawa numbers of diagonal cubic surfaces, numerical evidence

A refined version of Manin's conjecture about the asymptotics of points of bounded height on Fano varieties has been developed by Batyrev and the authors. We test numerically this refined conjecture for some diagonal cubic surfaces.

     

Labelled cospan categories and properads

We prove Steinebrunner's conjecture on the biequivalence between (coloured) properads and labelled cospan categories. The main part of the work is to establish a 1-categorical, strict version of the conjecture, showing that the category of properads is equivalent to a category of strict labelled cospan categories via the symmetric monoidal envelope functor.     

Petits points et conjecture de Bogomolov

This paper is devoted to the statement known as the Bogomolov conjecture on small points. We present the outline of Zhang's proof of the generalized version of the conjecture. An explicit bound for the height of a non-torsion variety of an abelian variety is obtained in the frame of Arakelov theory. Some further developments are mentioned.     

An algorithm for testing Leopoldt's conjecture

Based on an elementary version of Leopoldt's conjecture due to Iwasawa and Sands we develop an algorithm for testing this conjecture in an arbitrary algebraic number field for any prime p. Using this algorithm we are able to prove Leopoldt's conjecture for several pure fields of degree 5 and 7. We also discuss relations with class numbers.     

Homological symbols and the Quillen conjecture

We formulate a “correct” version of the Quillen conjecture on linear group homology for certain arithmetic rings and provide evidence for the new conjecture. In this way we predict that the linear group homology has a direct summand looking like an unstable form of Milnor K-theory and we call this new theory “homological symbols algebra”. As a byproduct, we prove the Quillen conjecture in homological degree two for the rank two and the prime 5.     

Proof of semialgebraic covering mapping cylinder conjecture with semialgebraic covering homotopy theorem

In this paper we prove the semialgebraic version of Palais' covering homotopy theorem, and use this to prove Bredon's covering mapping cylinder conjecture positively in the semialgebraic category. Bredon's conjecture was originally stated in the topological category, and a topological version of our semialgebraic proof of the conjecture answers the original topological conjecture for topological G-spaces over “simplicial” mapping cylinders.     

On deletions of largest bonds in graphs

A well-known conjecture of Scott Smith is that any two distinct longest cycles of a k-connected graph must meet in at least k vertices when k≥2. We provide a dual version of this conjecture for two distinct largest bonds in a graph. This dual conjecture is established for k?6.