New results on the conjecture of Rhodes and on the topological conjecture

The Conjecture of Rhodes, originally called the “type II conjecture” by Rhodes, gives an algorithm to compute the kernel of a finite semigroup. This conjecture has numerous important consequences and is one of the most attractive problems on finite semigroups. It was known that the conjecture of Rhodes is a consequence of another conjecture on the finite group topology for the free monoid. In this paper, we show that the topological conjecture and the conjecture of Rhodes are both equivalent to a third conjecture and we prove this third conjecture in a number of significant particular cases.     

Divisibility properties of values of partial zeta functions at non-positive integers

We conjecture a new bound on the exact denominators of the values at non-positive integers of imprimitive partial zeta functions associated with an Abelian extension of number fields. At s?=?0, this conjecture is closely connected to a conjecture of David Hayes. We prove the new conjecture assuming that the Coates–Sinnott conjecture holds for the extension.     

Cn中有界域的Bergman核函数的零点问题

多复变数空间Cn中有界域的Bergman核函数的零点问题集中表现为陆启铿猜想.陆启铿猜想是波兰数学家M.Skwarczynski对陆启铿1966年的一篇文章中关于Bergman核函数的零点问题而命名的,至今已经40年了.该猜想已写入了多复变函数论的多本专著,引起很多数学家的兴趣而研究之,已经成为多复变函数论中的一个活跃的研究方向.本文简述了陆启铿猜想的最初含意,综述了迄今为止关于有界域的Bergman核函数有无零点的各种研究成果以及所用的思想和方法.特别对近来出现的陆启铿猜想的新研究领域进行了较详细的阐述并在最后提出了关于陆启铿猜想的6个Open Problems,希望国内的年轻数学家对陆启铿猜想感到兴趣而研究之.     

Some phases of oscillatory integrals

The first example of a phase is presented for which Arhold’s conjecture on the validity of uniform estimates for oscillatory integrals with maximal singularity index is true, while his conjecture on the semicontinuity of the singularity index is false. A rough upper bound for the Milnor number such that the latter conjecture fails is obtained. The corresponding counterexample is simpler than Varchenko’s well-known counterexample to Arnold’s conjecture on the semicontinuity of the singularity index. This gives hope to decrease codimension and the Milnor number for which the conjecture on the semicontinuity of the singularity index fails.     

Tate''s Conjecture, Algebraic Cycles and Rational K-Theory in Characteristic p

The purpose of this article is to discuss conjectures on motives, algebraic cycles and K-theory of smooth projective varieties over finite fields. We give a characterization of Tate's conjecture in terms of motives and their Frobenius endomorphism. This is used to prove that if Tate's conjecture holds and rational and numerical equivalence over finite fields agree, then higher rational K-groups of smooth projective varieties over finite fields vanish (Parshin's conjecture). Parshin's conjecture in turn implies a conjecture of Beilinson and Kahn giving bounds on rational K-groups of fields in finite characteristic. We derive further consequences from this result.     

Fibonacci-like behavior of the number of numerical semigroups of a given genus

We conjecture a Fibonacci-like property on the number of numerical semigroups of a given genus. Moreover we conjecture that the associated quotient sequence approaches the golden ratio. The conjecture is motivated by the results on the number of semigroups of genus at most 50. The Wilf conjecture has also been checked for all numerical semigroups with genus in the same range.     

Proof of Ding’s conjecture on maximal stable sets and maximal cliques in planar graphs

X. Deng et al. proved Chvātal＇s conjecture on maximal stable sets and maximal cliques in graphs. G. Ding made a conjecture to generalize Chvátal＇s conjecture. The purpose of this paper is to prove this conjecture in planar graphs and the complement of planar graphs.     

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.     

Asymptotic behaviour and bifurcation in competitive Lotka-Volterra Systems

A conjecture about global attraction in autonomous competitive Lotka-Volterra systems is clarified by investigating a special system with a circular matrix. Under suitable assumptions, this system meets the condition of the conjecture but Hopf bifurcation occurs in a particular instance. This shows that the conjecture is not true in general and the condition of the conjecture is too weak to guarantee global attraction of an equilibrium. Sufficient conditions for global attraction are also obtained for this system.     

Equivariant Yamabe problem and Hebey-Vaugon conjecture

In their study of the Yamabe problem in the presence of isometry groups, E. Hebey and M. Vaugon announced a conjecture. This conjecture generalizes T. Aubin's conjecture, which has already been proven and is sufficient to solve the Yamabe problem. In this paper, we generalize Aubin's theorem and we prove the Hebey-Vaugon conjecture in dimensions less or equal to 37.     

The Bateman–Horn conjecture: Heuristic,history, and applications

The Bateman–Horn conjecture is a far-reaching statement about the distribution of the prime numbers. It implies many known results, such as the prime number theorem and the Green–Tao theorem, along with many famous conjectures, such the twin prime conjecture and Landau’s conjecture. We discuss the Bateman–Horn conjecture, its applications, and its origins.     

Class numbers of cyclic 2-extensions and Gross conjecture over ℚ

The Gross conjecture over ? was first claimed by Aoki in 1991. However, the original proof contains too many mistakes and false claims to be considered as a serious proof. This paper is an attempt to find a sound proof of the Gross conjecture under the outline of Aoki. We reduce the conjecture to an elementary conjecture concerning the class numbers of cyclic 2-extensions of ?.     

Sphere packings, I

We describe a program to prove the Kepler conjecture on sphere packings. We then carry out the first step of this program. Each packing determines a decomposition of space into Delaunay simplices, which are grouped together into finite configurations called Delaunay stars. A score, which is related to the density of packings, is assigned to each Delaunay star. We conjecture that the score of every Delaunay star is at most the score of the stars in the face-centered cubic and hexagonal close packings. This conjecture implies the Kepler conjecture. To complete the first step of the program, we show that every Delaunay star that satisfies a certain regularity condition satisfies the conjecture.     

Counterexamples to the Connectivity Conjecture of the Mixed Cells

In [4] a conjecture concerning the connectivity of mixed cells of subdivisions for Minkowski sums of polytopes was formulated. This conjecture was, in fact, originally proposed by Pedersen [3]. It turns out that a positive confirmation of this conjecture can substantially speed up the algorithm for the ``dynamical lifting' developed in [4]. In the mean time, when the polyhedral method is used for solving polynomial systems by homotopy continuation methods [2], the positiveness of this conjecture also plays an important role in the efficiency of the algorithm. Very unfortunately, we found that this conjecture is inaccurate in general. In Section 1 a counterexample is presented for a general subdivision. In Section 2 another counterexample shows that even restricted to ``regular' subdivisions induced by liftings, this conjecture still fails to be true. Received September 18, 1996, and in revised form February 17, 1997.     

Class numbers of cyclic 2-extensions and Gross conjecture over Q

The Gross conjecture over Q was first claimed by Aoki in 1991.However,the original proof contains too many mistakes and false claims to be considered as a serious proof.This paper is an attempt to find a sound proof of the Gross conjecture under the outline of Aoki.We reduce the conjecture to an elementary conjecture concerning the class numbers of cyclic 2-extensions of Q.     

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.     

Modular Representation Theory of Blocks with Trivial Intersection Defect Groups

We show that Uno's refinement of the projective conjecture of Dade holds for every block whose defect groups intersect trivially modulo the maximal normal p-subgroup. This corresponds to the block having p-local rank one as defined by Jianbei An and Eaton. An immediate consequence is that Dade's projective conjecture, Robinson's conjecture, Alperin's weight conjecture, the Isaacs–Navarro conjecture, the Alperin–McKay conjecture and Puig's nilpotent block conjecture hold for all trivial intersection blocks. Presented by A. Verschoren Mathematics Subject Classification (2000) Primary 20C20. Charles W. Eaton: Current address: School of Mathematics, University of Manchester, Sackville Street, PO Box 88, Manchester M60 1QC, U.K. e-mail: charles.eaton@manchester.ac.uk This research was supported in part by the Marsden Fund of New Zealand via grant #9144/3368248.     

Primes between consecutive squares

A well known conjecture about the distribution of primes asserts that between two consecutive squares there is always at least one prime number. The proof of this conjecture is quite out of reach at present, even under the assumption of the Riemann Hypothesis. The aim of this paper is to provide a bound for the exceptional set for this conjecture, unconditionally and under the assumption of some classical hypothesis. We also provide a conditional proof of the conjecture assuming an hypothesis about the behavior of Selberg's integral in short intervals.     

The Alperin and dade conjectures for the simple Conway’s third group

This paper is part of a program to study Alperin’s weight conjecture and Dade’s conjecture on counting ordinary characters in blocks for several finite groups. The classifications of radical subgroups and certain radical chains and their local structures of the simple Conway’s third group have been obtained by using the computer algebra system CAYLEY. The Alperin weight conjecture and the Dade final conjecture have been confirmed for the group.     

HECKE OPERATOR AND PELLIAN EQUATION CONJECTURE(Ⅱ)

It is proved that the famous Ankeny-Artin-Chowla conjecture on Pellian equationis equivalent to a conjecture on Hecke operator.