GABOR ANALYSIS OF THE SPACES M（p,q,w）（R~d） AND S（p,q,r,w,ω）（R~d）

Let g be a non-zero rapidly decreasing function and w be a weight function. In this article in analog to modulation space, we define the space M(p, q, w)(Rd) to be the subspace of tempered distributions f ∈ S′(Rd) such that the Gabor transform Vg(f) of f is in the weighted Lorentz space L(p, q, wdμ) (R2d). We endow this space with a suitable norm and show that it becomes a Banach space and invariant under time frequence shifts for 1 ≤ p, q ≤∞. We also investigate the embeddings between these spaces and the dual space of M(p, q, w)(Rd). Later we define the space S(p, q, r, w, ω)(Rd) for 1 < p < ∞, 1 ≤ q ≤∞. We endow it with a sum norm and show that it becomes a Banach convolution algebra. We also discuss some properties of S(p, q, r, w, ω)(Rd). At the end of this article, we characterize the multipliers of the spaces M(p, q, w)(Rd) and S(p, q, r, w, ω)(Rd).     

POLLUTION, GOVERNMENT EXPENDITURE, TAXES AND STOCHASTIC GROWTH

This paper studies a stochastic endogenous growth model with pollution. It introduces government expenditure and exogenous pollution abatement technology to eliminate pollution and proves that under appropriate equilibrium conditions the main economic indexes （including economic growth rate, the optimal government expenditure rate） in the centrally planned economy and decentralized economy can be expressed by the parameters of the model uniquely. The optimal tax policy is analyzed ,and the optimal pollution is derived.     

(0,1, …, m - 2, m) INTERPOLATION FOR THE LAGUERREABSCISSAS

A necessary and sufficient condition of regularity of (0,1,…, m - 2, m) interpo-lation on the zeros of the Laguerre polynomials Ln(α) (x) (α≥ -1) in a manageable form is established. Meanwhile, the explicit representation of the fundamental polynomials, when they exist, is given. Moreover, it is shown that, if the problem of (0,1,…, m - 2, TO) interpolation has an infinity of solutions, then the general form of the solutions is f0(x) Cf1(x) with an arbitrary constant C.     

ON A CLASS OF INHOMOGENEOUS,ENERGY-CRITICAL,FOCUSING, NONLINEAR SCHRōDINGER EQUATIONS

In this paper, we study the Cauchy problem of the inhomogeneous energy-critical Schrōdinger equation： iаtu=-△u-k（x）｜u｜4/N-2u,N≥3. Using the potential well method, we establish some new sharp criteria for blow-up of solutions in tile nonradial case. In particular, our conclusion in some sense improves on the results in [Kenig and Merle, invent. Math. 166, 645-675 （2006）], where only the radial case is considered in dimensions 3. 4. 5.     

GABOR ANALYSIS OF THE SPACES M(p,q,w)(R~d) AND S(p,q,r,w,ω)(R~d)

Let g be a non-zero rapidly decreasing function and w be a weight function. In this article in analog to modulation space, we define the space M(p, q, w)(Rd) to be the subspace of tempered distributions f ∈ S′(Rd) such that the Gabor transform Vg(f) of f is in the weighted Lorentz space L(p, q, wdμ) (R2d). We endow this space with a suitable norm and show that it becomes a Banach space and invariant under time frequence shifts for 1 ≤ p, q ≤ ∞. We also investigate the embeddings between these spaces and the...     

The Spectrum of （gv,g, 3, λ）-dDF in Zgv

In this paper, several recursive constructions for directed difference family and perfect directed difference family are presented by means of difference matrix and incomplete difference matrix. Finally the necessary and sufficient conditions for the existence of a （gv, g, 3, λ）-directed difference family in Zgv are established. As a consequence, the necessary and sufficient conditions for the existence of a cyclic directed group divisible design with block size three and type gv are obtained.     

JACQUES-LOUIS LIONS,1928-2001

Jacques-Louis Lions was born in the heart of Provence in the charming city of Grasse,much renowned for its perfume industry and its historical center.In spite of his young age, he had the courage and determination to join the FrenchResistance at the end of 1943, as a soldier in the FFI (French Forces of the Interior). Therehe met Andree, his wife and life-long companion.     

INSTANTANEOUS SHRINKING AND LOCALIZATION OF FUNCTIONS IN Yλ （m,p,q,N） AND THEIR APPLICATIONS

61. IntroductionAs well knoWn, the solutionS of semilinear Parabolic equations at = An -- ac, 0 < q < 1have the ISS (the thetantaneous shrinking of the support) ~erty (see 11--5]).Such PrOPerty is d~d as follows.Defiallion 1.1. A hoction u E La(Q) is sold to haw the ISS p~rty, ijjor augr > 0, there hots a POsiboe n~ber R = R(T) such thatu(z, t) = 0for a. e. (z,t) e (R" \ Ba) x (T, co), tDhere AN is the N-dimensional Euclidean space,and Q = AN X (0, co).UP to the preseDt, there…     

REGULARITY OF (0,1,…,r-2,r)INTERPOLATION ON THE UNIT CIRCLE

The object of this paper is to show regularity of(0,1,...,r-2,r) interpolation on the set obtained by projecting vertically the zeros of (1-x2)pn(x)(λ≥1/2)onto the unit circle,where Pn(x)stands for the nth ultraspherical polynomial.     

ON THE MODIFIED NONLINEAR SCHRDINGER EQUATION IN THE SEMICLASSICAL LIMIT:SUPERSONIC,SUBSONIC,AND TRANSSONIC BEHAVIOR

The purpose of this paper is to present a comparison between the modified nonlinear Schro¨dinger (MNLS) equation and the focusing and defocusing variants of the (unmodified) nonlinear Schr¨odinger (NLS) equation in the semiclassical limit. We describe aspects of the limiting dynamics and discuss how the nature of the dynamics is evident theoretically through inverse-scattering and noncommutative steepest descent methods. The main message is that, depending on initial data, the MNLS equation can behave either like the defocusing NLS equation, like the focusing NLS equation (in both cases the analogy is asymptotically accurate in the semiclassical limit when the NLS equation is posed with appropriately modified initial data), or like an interesting mixture of the two. In the latter case, we identify a feature of the dynamics analogous to a sonic line in gas dynamics, a free boundary separating subsonic flow from supersonic flow.     

Mild Solution of Stochastic Equations with Levy Jumps： Existence, Uniqueness, Regularity and Stability

The existence and uniqueness of mild solution to stochastic equations with jumps are established, a stochastic Fubini theorem and a type of Burkholder-Davis- Gundy inequality are proved, and the two formulas are used to study the regularity property of the mild solution of a general stochastic evolution equation perturbed by Levy process. Then the authors prove the moment exponential stability, almost sure exponential stability and comparison principles of the mild solution. As applications, the stability and comparison principles of stochastic heat equation with Levy jump are given.     

An exact, closed-form solution for equilibrium of trveling, sagged, elastic cables under uniformly distributed loading

IntroctionThquilibrium of sagged, elastic cable s under its own weight loading is needed in tiledesign of cable structures. In 1851 RO'htsLll first investigated a uniform, inextensible suspendedchain hanging freely under its own weight. Ro.th[Zj considered a model of a heterogeneouschain hanging in the form of a cycloid, and application of this model to the uniform chainreduced to the Rohrs model when the sag ratio is small. However, the chain was modeled asinextensible. Pugsley['] develope…     

MULTI-DIMENSIONAL GEOMETRIC BROWNIAN MOTIONS, ONSAGER-MACHLUP FUNCTIONS, AND APPLICATIONS TO MATHEMATICAL FINANCE

1 IntroductionThe solution of the following stochastic differential equationis called the geometric Brownian motion, where a(t), b(t) are deterministic functions of ti ac isa Brownian motion, and ddt is the It6 integral. This equation has been successfully applied tothe financial problems such as modeling the prices of stocks, sc.e for example, [81, [7], [14], [13],[19], [21]. When the initial condition is given, i.e. xo = x, the solution isIt is known that in the financial market, it is als…     

COSET DIAGRAMS FOR A HOMOMORPHIC IMAGE OF △（3, 3, k）

Let q be a prime power. By PL（Fq） the authors mean a projective line over the finite field Fq with the additional point ∞. In this article, the authors parametrize the conjugacy classes of nondegenerate homomorphisms which represent actions of △（3, 3, k） = （u, v： u^3 = v^3 = （uv）^k = 1〉on PL（Fq）, where q ≡ ±1（modk）. Also, for various values of k, they find the conditions for the existence of coset diagrams depicting the permutation actions of △（3, 3, k） on PL（Fq）. The conditions are polynomials with integer coefficients and the diagrams are such that every vertex in them is fixed by （u^-v^-）^k. In this way, they get △（3, 3, k） as permutation groups on PL（Fq）.     

Herz-type Triebel-Lizorkin Spaces, Ⅰ

Let s ∈R,0〈β≤∞, 0〈 q, p〈 ∞ and-n/q〈α. In this paper the authors introduce the Herz-type Triebel-Lizorkin spaces,Kq^α,pFβ^s（R^n）andKq^α,pFβ^s（R^n）which are the generalizations of the well-known Herz-type spaces and the inhomogeneous Triebel-Lizorkin spaces, Some properties on these Herz-type Triebel Lizorkin spaces are also given.     

EXISTENCE AND INVARIANCE FOR CONVECTION, DISPERSION, REACTION SYSTEMS WITH PRESSURE

In this paper, we consider the initial boundary value problem of Convection-Dispersion-Reaotion system which arise in solute transport or thermal transport in groundwater. Also, we prove the existence of the classical solutions of these systems.     

C^s＋αp,q-forms Solution of Э^-b-equation and Its L^sp,q Estimates

In this paper, by using holomorphic support function of strictly pseudoconvex domain on Stein manifolds and the kernel defined by DEMAILY J P and Laurent Thiebaut, we construct two integral operators Tq and Sq which are both belong to C^s αp,q-1(ЭD) and obtain integral representation of the solu-tion of ( p, q )-form Э^-b-equation on the boundary of pseudoconvex domain in Stein manifolds and the L^sp,q extimates for the solution.     

The spectrum of (gv, g, 3, λ)-DF in Z_(gv)

In this paper, the necessary and sufficient conditions for the existence of a (gv, g, 3, λ)- difference family in Zgv are established. As a consequence, the existence spectrum of a cyclic (3, λ)-GDD of type gv without short orbits is determined.     

Subcritical, Critical and Supercritical Size Distributions in Random Coagulation-Fragmentation Processes

We consider the asymptotic probability distribution coagula-tion-fragmentation process in the thermodynamic limit of the size of a reversible random We prove that the distributions of small, medium and the largest clusters converge to Gaussian, Poisson and 0-1 distributions in the supercritical stage （post-gelation）, respectively. We show also that the mutually dependent distributions of clusters will become independent after the occurrence of a gelation transition. Furthermore, it is proved that all the number distributions of clusters are mutually independent at the critical stage （gelation）, but the distributions of medium and the largest clusters are mutually dependent with positive correlation coefficient in the supercritical stage. When the fragmentation strength goes to zero, there will exist only two types of clusters in the process, one type consists of the smallest clusters, the other is the largest one which has a size nearly equal to the volume （total number of units）.     

DECOMPOSITION, REPRESENTATION AND EXTENSION OF THE HARMONIC FUNCTIONS

In this paper, by using the probabilistic method, we obtained a decomposition theorem for the harmonic functions, generalized and strengthened some theorems: Doob's theorem on the limit behaviors of the superharmonic functions, Bocher's theorem on the representation of the harmonic functions and the extension theorem of the harmonic functions etc.