Propagation of light in(2+1)-dimensional nonlinear optical media with spatially inhomogeneous nonlinearities

The(2 1)-dimensional nonlinear Schr(o)dinger(NLS)equation with spatially inhomogeneous nonlinearities is investigated,which describes propagation of light in(2 1)-dimensional nonlinear optical media with inhomogeneous nonlinearities.New types of optical modes and nonlinear effects in optical media are presented numerically.The results reveal that the regular split of beam can be obtained in (2 1)-dimensional nonlinear optical media with inhomogeneous nonlinearities,by adjusting the guiding parameter.Furthermore,the stability of beam regular split is discussed numerically,and the results reveal that the beam regular split is stable to the finite initial perturbations.     

Kazhdan Constants of Hyperbolic Groups

Let H be an infinite hyperbolic group with Kazhdan property (T) and let (H,X) denote the Kazhdan constant of H with respect to a generating set X. We prove that inf_X(H,X)=0, where the infimum is taken over all finite generating sets of H. In particular, this gives an answer to a Lubotzky question.     

Volume of Hyperbolic 3-Manifolds with Iterated Pseudo-Anosov Amalgamations (Dedicated to Professor Mitsuyoshi Kato on his sixtieth birthday)

Let : be a pseudo-Anosov homeomorphism. We will study an asymptotic behaviour of the volume of closed hyperbolic 3-manifolds N _n obtained from certain 3-manifolds M, M by attaching their boundaries by the n-th iteration ⁿ of .     

The Rate of Convergence for Subexponential Distributions and Densities

A distribution function F on the nonnegative real line is called subexponential if lim_x(1-F ^*n (x)/(1 - F(x)) = n for all n 2, where F ^*n denotes the nfold Stieltjes convolution of F with itself. In this paper, we consider the rate of convergence in the above definition and in its density analogue. Among others we discuss the asymptotic behavior of the remainder term R _n (x) defined by R _n (x) = 1 - F*n(x) - n(1 - F(x)) and of its density analogue r_n (x) = -(R_n (x))'. Our results complement and complete those obtained by several authors. In an earlier paper, we obtained results of the form _n(x) = O(1)f(x)R(x), where f is the density of F and R(x) = ₀ ^x (1-F(y))dy. In this paper, among others we obtain asymptotic expressions of the form R _n(x)= ₂ ⁿ R₂(x) + O(1)(-f'(x))R²(x) where f' is the derivative of f.     

The asymptotic distributions of sums of record values for distributions with lognormal-type tails

Let X1,X2,... be a sequence of i.i.d. random variables and let X(1),X(2),... be the associatedrecord value sequence. We focus on the asymptotic distributions of sums of records, Tn=∑nk=1X(k), forX1 ∈ LN(γ). In this case, we find that 2 is a strange point for parameter γ. When γ＞ 2, Tn is asymptoticallynormal, while for 2 ＞γ＞ 1, we prove that Tn cannot converge in distribution to any non-degenerate lawthrough common centralizing and normalizing and log Tn is asymptotically normal.     

Unmixed local rings with minimal Hilbert-Kunz multiplicity are regular

We give a new and simple proof that unmixed local rings having Hilbert-Kunz multiplicity equal to must be regular.

     

Algebraic obstructions and a complete solution of a rational retraction problem

For each compact smooth manifold containing at least two points we prove the existence of a compact nonsingular algebraic set and a smooth map such that, for every rational diffeomorphism and for every diffeomorphism where and are compact nonsingular algebraic sets, we may fix a neighborhood of in which does not contain any regular rational map. Furthermore is not homotopic to any regular rational map. Bearing in mind the case in which is a compact nonsingular algebraic set with totally algebraic homology, the previous result establishes a clear distinction between the property of a smooth map to represent an algebraic unoriented bordism class and the property of to be homotopic to a regular rational map. Furthermore we have: every compact Nash submanifold of containing at least two points has not any tubular neighborhood with rational retraction.