Reducibility of hyperplane arrangements

Certain problems on reducibility of central hyperplane arrangements are settled. Firstly, a necessary and sufficient condition on reducibility is obtained. More precisely, it is proved that the number of irreducible components of a central hyperplane arrangement equals the dimension of the space consisting of the logarithmic derivations of the arrangement with degree zero or one. Secondly, it is proved that the decomposition of an arrangement into a direct sum of its irreducible components is unique up to an isomorphism of the ambient space. Thirdly, an effective algorithm for determining the number of irreducible components and decomposing an arrangement into a direct sum of its irreducible components is offered. This algorithm can decide whether an arrangement is reducible, and if it is the case, what the defining equations of irreducible components are.     

Bernstein–Sato polynomials of hyperplane arrangements

We show that the Bernstein–Sato polynomial (that is, the b-function) of a hyperplane arrangement with a reduced equation is calculable by combining a generalization of Malgrange’s formula with the theory of Aomoto complexes due to Esnault, Schechtman, Terao, Varchenko, and Viehweg in certain cases. We prove in general that the roots are greater than \(-2\) and the multiplicity of the root \(-1\) is equal to the (effective) dimension of the ambient space. We also give an estimate of the multiplicities of the roots in terms of the multiplicities of the arrangement at the dense edges, and provide a method to calculate the Bernstein–Sato polynomial at least in the case of 3 variables with degree at most 7 and generic multiplicities at most 3. Using our argument, we can terminate the proof of a conjecture of Denef and Loeser on the relation between the topological zeta function and the Bernstein–Sato polynomial of a reduced hyperplane arrangement in the 3 variable case.     

The diffeomorphic types of the complements of arrangements in CP~3Ⅱ

For any arrangement of hyperplanes in CP~3,we introduce the soul of this arrangement. The soul,which is a pseudo-complex,is determined by the combinatorics of the arrangement of hyper- planes.In this paper,we give a sufficient combinatoric condition for two arrangements of hyperplanes to be diffeomorphic to each other.In particular we have found sufficient conditions on combinatorics for the arrangement of hyperplanes whose moduli space is connected.This generalizes our previous result on hyperplane point arrangements in CP~3.     

On the freeness of Böröczky line arrangements

Periodica Mathematica Hungarica - In the present note, we focus on the freeness and some combinatorial properties of line arrangements in the projective plane having only double and triple points....     

On the Sylvester–Gallai and the orchard problem for pseudoline arrangements

We study a non-trivial extreme case of the orchard problem for 12 pseudolines and we provide a complete classification of pseudoline arrangements having 19 triple points and 9 double points. We have also classified those that can be realized with straight lines. They include new examples different from the known example of Böröczky. Since Melchior’s inequality also holds for arrangements of pseudolines, we are able to deduce that some combinatorial point-line configurations cannot be realized using pseudolines. In particular, this gives a negative answer to one of Grünbaum’s problems. We formulate some open problems which involve our new examples of line arrangements.     

An inductive approach to Coxeter arrangements and Solomon’s descent algebra

In our recent paper (Douglass et al. (2011)), we claimed that both the group algebra of a finite Coxeter group W as well as the Orlik–Solomon algebra of W can be decomposed into a sum of induced one-dimensional representations of centralizers, one for each conjugacy class of elements of W, and gave a uniform proof of this claim for symmetric groups. In this note, we outline an inductive approach to our conjecture. As an application of this method, we prove the inductive version of the conjecture for finite Coxeter groups of rank up to 2.     

Sums of Squares of Integers

We find a formula for the sum of the first n squares:Sn = 12 22 32 42 … n2.We haveS1 = 1, S2 = 12 22 = 5, S3 = l2 22 32 = 14,S4 = 30, S5 = 55, S6 = 91.and so on. To make things a little simpler, we will also use the sum of the squares of the first zero terms , S0 = 02 = 0. Arranging these values in a table,we find thatSince the third differenes △3Sn are constant,these data values must follow a cubic pattern;that is, the formula for the sum of the squares of the first n integers is a cubic function,     

On the Growth of Components of Meromorphic Solutions of Systems of Complex Differential Equations

This paper investigates the problem of the growth of the components of meromorphic solutions of a class of a system of complex algebraic differential equations, and generalized some of N. Toda＇s results concerning the growth of differential equations to the case of systems of differential equations. The paper considers the existence of admissible solutions of the system of differential equations.     

Stability of Solution of a Class of Quasilinear Elliptic Equations

In this paper, using capacity theory and extension theorem of Lipschitz functions we first discuss the uniqueness of weak solution of nonhomogeneous quasilinear elliptic equationsin space W(θ,p)(Ω), which is bigger than W1,p(Ω). Next, using revise reverse Holder inequality we prove that if ωc is uniformly p-think, then there exists a neighborhood U of p, such that for all t ∈U, the weak solutions of equation corresponding t are bounded uniformly. Finally, we get the stability of weak solutions on exponent p.     

Vertices of π-Irreducible Characters of Groups of Odd Order

If G is a group of odd order and χ ∈ Irr(G) lifts an irreducible Brauer character ?, then we associate to χ a canonical pair (Q, δ) up to G-conjugacy, where Q is a vertex of ? and δ ∈ Irr(Q) is a linear character of Q. We show that (Q, δ) is a Navarro vertex for χ. We also discuss examples.     

Estimates of groups of deviations of Faber sums and strong summability of faber series on classes of ψ-integrals

We establish upper bounds for a group of *-deviations of Faber sums on the classes of -integrals in a complex plane introduced by Stepanets.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 4, pp. 451–461, April, 2004.     

Dynamics of Polynomial Automorphisms of C~N

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The Degree of Unsolvability of Models

The purpose of this paper is to describe the complexity of models by theirdegrees of unsolvability,J.Richter defined the degree of a structure to be deg ()=sup{deg(),deg(R_i),i=1,…,n},Where is a model for afinite language L={R_i,i=1,…,n}and the universe of is a subset of ω. Shepointed out that, according to her definition, there can be models which areisomorphic but their degrees are different.Also,her discussions are restricted tofinite languages and models whose universes are subsets of ω.     

Correlation of zeros of automorphic L-functions

We compute the n-level correlation of normalized nontrivial zeros of a product of L-functions:L(s,π1)···L(s,πk), where πj, j=1,...,k, are automorphic cuspidal representations of GLmj(QA). Here the sizes of the groups GLmj(QA) are not necessarily the same. When these L(s,πj) are distinct, we prove that their nontrivial zeros are uncorrelated, as predicted by random matrix theory and verified numerically. When L(s,πj) are not necessarily distinct, our results will lead to a proof that the n-level correlation of normalized nontrivial zeros of the product L-function follows the superposition of Gaussian Unitary Ensemble (GUE) models of individual L-functions and products of lower rank GUEs. The results are unconditional when m1,...,mk 4,but are under Hypothesis H in other cases.     

Extension of a Theorem of Carleson-Duren

1. Introduction A theorem of Carleson,as generalized by Duren characterizes those positive measure μ on the unit disc U={z∈C:|z|<1} for which the H~p norm domiates the L~q(μ) norm of elements of H~p. Later on, Hasting proved an analogous results with H~p replaced by A~p, the Bergman space of fuctions f     

Generllzation of Direct Product and Generation of Sets of Orthogonal Functions

Two new kinds of direct product of matrices are defined. Their properties are investigated. Direct products of matrix and set of continuous functions are also defined. Many complete sets of orthogonal functions, such as those sets given by Walsh [2], Paley [3], Chrestenson[4], and Watari [5], may be generated by these newkinds of direct product. Direct products are also applicable to the generation of sets of piecewise orthogonal functions.     

Characterizations of connected orthogonality graphs of projections of Rickart ∗-rings

In this paper, we study the orthogonality graphs (see Definition 1.2) of ortholattices. We provide a graph theoretic condition for an ortholattice to be orthomodular. We prove that, the orthogonality graphs of two orthomodular lattices are isomorphic if and only if the lattices are isomorphic. As an application, it is proved that the zero-divisor graph of a Rickart ?-ring is obtained by successively duplicating the vertices of the orthogonality graph of the lattice of projections in the ring. We characterize the finite Rickart ?-rings for which the orthogonality graph of projections is connected.