Euler characteristic of the Milnor fibre of plane singularities

We give a formula for the Euler characteristic of the Milnor fibre of any analytic function of two variables. This formula depends on the intersection multiplicities, the Milnor numbers and the powers of the branches of the germ of the curve defined by The goal of the formula is that it use neither the resolution nor the deformations of Moreover, it can be use for giving an algorithm to compute it.

     

Morse theory, Milnor fibers and minimality of hyperplane arrangements

Through the study of Morse theory on the associated Milnor fiber, we show that complex hyperplane arrangement complements are minimal. That is, the complement of any complex hyperplane arrangement has the homotopy type of a CW-complex in which the number of -cells equals the -th betti number. Combining this result with recent work of Papadima and Suciu, one obtains a characterization of when arrangement complements are Eilenberg-MacLane spaces.

     

Extended Linial Hyperplane Arrangements for Root Systems and a Conjecture of Postnikov and Stanley

A hyperplane arrangement is said to satisfy the Riemann hypothesis if all roots of its characteristic polynomial have the same real part. This property was conjectured by Postnikov and Stanley for certain families of arrangements which are defined for any irreducible root system and was proved for the root system A _{n – 1}. The proof is based on an explicit formula [1, 2, 11] for the characteristic polynomial, which is of independent combinatorial significance. Here our previous derivation of this formula is simplified and extended to similar formulae for all but the exceptional root systems. The conjecture follows in these cases.     

On a Family of Hyperplane Arrangements Related to the Affine Weyl Groups

Let be an irreducible crystallographic rootsystem in a Euclidean space V, with ⁺ theset of positive roots. For , , let be the hyperplane . We define a set of hyperplanes . This hyperplane arrangement is significant inthe study of the affine Weyl groups. In this paper it is shown that thePoincaré polynomial of is , where n is the rank of and h is the Coxeter number of the finiteCoxeter group corresponding to .     

Complex monodromy and the topology of real algebraic sets

A relation between the Euler characteristics of the Milnorfibres of a real analytic function is derived from a simple identity involvingcomplex monodromy and complex conjugation. A corollary is the result of Costeand Kurdyka that the Euler characteristic of the local link of an irreduciblealgebraic subset of a real algebraic set is generically constant modulo 4. Asimilar relation for iterated Milnor fibres of ordered sets of functions isused to define topological invariants of ordered collections of algebraicsubsets.     

On the bilinear and cubic forms of some symplectic connected sums

Let ( M1 , M2 , N ) be three symplectic manifolds and suppose that we can do the symplectic connected sum of M1 and M2 along their submanifold N to obtain M1#NM2 . In this paper, we consider the bilinear and cubic forms of H* (M1#NM2 , Z) when dim M1#NM2 = 4, 6. Under some conditions, we get some relations of the bilinear and the cubic forms between M1#NM2 and M1■M2 .     

Faces of a Hyperplane Arrangement Enumerated by Ideal Dimension, with Application to Plane, Plaids, and Shi

The ideal dimension of a real affine set is the dimension of the intersection of its projective topological closure with the infinite hyperplane. We obtain a formula for the number of faces of a real hyperplane arrangement having given dimension and ideal dimension. We apply the formula to the plane, to plaids, which are arrangements of parallel families in general position, and to affinographic arrangements. We compare two definitions of ideal dimension and ask about a complex analog of the enumeration.     

Determinants of the Hypergeometric Period Matrices of an Arrangement and Its Dual

We fix three natural numbers k, n, N, such that n+k+1 = N, and introduce the notion of two dual arrangements of hyperplanes. One of the arrangements is an arrangement of N hyperplanes in a k-dimensional affine space, the other is an arrangement of N hyperplanes in an n-dimensional affine space. We assign weights α ₁, . . . , α _N to the hyperplanes of the arrangements and for each of the arrangements consider the associated period matrices. The first is a matrix of k-dimensional hypergeometric integrals and the second is a matrix of n-dimensional hypergeometric integrals. The size of each matrix is equal to the number of bounded domains of the corresponding arrangement. We show that the dual arrangements have the same number of bounded domains and the product of the determinants of the period matrices is equal to an alternating product of certain values of Euler’s gamma function multiplied by a product of exponentials of the weights. Supported in part by NSF grant DMS-0244579.     

关于代数微分方程的Gackstatter和Laine的猜想

在本文中，我们讨论了一类复代数微分方程其在允许解时的形状，得到了一个主要结果，它是文[3-5]的推广。     

Modified Bers operators and the duality of multiplicative holomorphic automorphic forms

The integral Bers operator, related to a reflection in some quasicircle, plays an important role in the theory of single-valued automorphic forms [1, 2]. Study was started in [3] of the normed spaces of measurable and multiplicative holomorphic automorphic forms for a Fuchsian group. In the present article we introduce some basic multiplicative modifications of the Bers operator and the corresponding bilinear pairing in connection with duality in the spaces of multiplicative holomorphic automorphic forms. Under study we obtain a universal norm estimate and establish selfadjointness for all operators.     

Lumps and their interaction solutions of a (2+1)-dimensional generalized potential Kadomtsev-Petviashvili equation

A (2+1)-dimensional generalized potential Kadomtsev-Petviashvili (gpKP) equation which possesses a Hirota bilinear form is constructed. The lump waves are derived by using a positive quadratic function solution. By combining an exponential function with a quadratic function, an interaction solution between a lump and a one-kink soliton is obtained. Furthermore, an interaction solution between a lump and a two-kink soliton is presented by mixing two exponential functions with a quadratic function. This type of lump wave just appears to a line $k_2x+k_3y+k_4t+k_5 \sim 0$. We call this kind of lump wave is a special rogue wave. Some visual figures are depicted to explain the propagation phenomena of these interaction solutions.     

基于Q-过程和随机模拟的病床安排方案

本文设计的一个新的住院病床安排方案,可用来取代传统的FCFS(先到先得)规则.患者在等候队列中的优先级将基于病情和手术的紧急程度.本文把患者的入住情况转化为经典的M/M/N排队论问题,利用可逆Q-过程的性质以及经典排队论的结果,得到各类疾病患者的平均逗留时间以及所需床位数的先验估计.根据独立Poisson过程的性质确定排队过程的Q-矩阵,从而作出相应的先验估计和具体表达式.在若干约束条件的基础上,利用C++编程求得最优病床分配方案.本文建立的评价指标体系,通过计算机随机模拟,证明了在多种情形下,该模型较FCFS(先到先得)规则更具优势.     

On the Norm of a Self-Adjoint Operator and a New Bilinear Integral Inequality

In this paper, the expression of the norm of a self-adjoint integral operator T ： L^2（0, ∞） → L^2 （0, ∞） is obtained. As applications, a new bilinear integral inequality with a best constant factor is established and some particular cases are considered.     

Critical points of the area functional of a complex closed curve on the manifold of Kähler metrics

We consider a compact complex manifold of dimension that admits Kähler metrics and we assume that is a closed complex curve. We denote by the space of classes of Kähler forms that define Kähler metrics of volume 1 on and define by . We show how the Riemann-Hodge bilinear relations imply that any critical point of is the strict global minimum and we give conditions under which there is such a critical point : A positive multiple of is the Poincaré dual of the homology class of . Applying this to the Abel-Jacobi map of a curve into its Jacobian, , we obtain that the Theta metric minimizes the area of within all Kähler metrics of volume 1 on .