ON AN INFINITE FAMILY IN π_S*

In this paper, a new family of homotopy elements in the stable homotopy groups of spheres represented by h1 hn hm γs in the Adams spectral sequence is detected, where n- 2 ≥ m ≥ 5 and 3 ≤ s p.     

TANGENT UNIT-VECTOR FIELDS:NONABELIAN HOMOTOPY INVARIANTS AND THE DIRICHLET ENERGY

Let O be a closed geodesic polygon in S~2 . Maps from O into S~2 are said to satisfy tangent boundary conditions if the edges of O are mapped into the geodesics which contain them. Taking O to be an octant of S 2,we compute the infimum Dirichlet energy E(H) for continuous maps satisfying tangent boundary conditions of arbitrary homotopy type H . The expression for E (H ) involves a topological invariant - the spelling length - associated with the (non-abelian) fundamental group of the n-times punctured two-sphere, π 1 (S 2 ? {s 1 , ··· , s n }, ?). The lower bound for E(H) is obtained from combinatorial group theory arguments, while the upper bound is obtained by constructing explicit representa- tives which, on all but an arbitrarily small subset of O, are alternatively locally conformal or anticonformal. For conformal and anticonformal classes (classes containing wholly con- formal and anticonformal representatives respectively), the expression for E (H ) reduces to a previous result involving the degrees of a set of regular values s 1 , ··· , s n in the target S 2 space. These degrees may be viewed as invariants associated with the abelianization of π 1 (S 2 ? {s 1 , ··· , s n }, *). For nonconformal classes, however, E(H) may be strictly greater than the abelian bound. This stems from the fact that, for nonconformal maps, the number of preimages of certain regular values may necessarily be strictly greater than the absolute value of their degrees.This work is motivated by the theoretical modelling of nematic liquid crystals in confined polyhedral geometries. The results imply new lower and upper bounds for the Dirichlet energy (one-constant Oseen-Frank energy) of reflection-symmetric tangent unit- vector fields in a rectangular prism.     

EXISTENCE OF ENTROPY SOLUTIONS TO TWO-DIMENSIONAL STEADY EXOTHERMICALLY REACTING EULER EQUATIONS

We are concerned with the global existence of entropy solutions of the twodimensional steady Euler equations for an ideal gas, which undergoes a one-step exothermic chemical reaction under the Arrhenius-type kinetics. The reaction rate function φ(T) is assumed to have a positive lower bound. We first consider the Cauchy problem(the initial value problem), that is, seek a supersonic downstream reacting flow when the incoming flow is supersonic, and establish the global existence of entropy solutions when the total variation of the initial data is suffciently small. Then we analyze the problem of steady supersonic, exothermically reacting Euler flow past a Lipschitz wedge, generating an additional detonation wave attached to the wedge vertex, which can be then formulated as an initial-boundary value problem. We establish the global existence of entropy solutions containing the additional detonation wave(weak or strong, determined by the wedge angle at the wedge vertex) when the total variation of both the slope of the wedge boundary and the incoming flow is suitably small. The downstream asymptotic behavior of the global solutions is also obtained.     

UNIQUENESS, ERGODICITY AND UNIDIMENSIONALITY OF INVARIANT MEASURES UNDER A MARKOV OPERATOR

Let X be a compact metric space and C（X） be the space of all continuous functions on X. In this article, the authors consider the Markov operator T ： C（X）N C（X）N defined by
for any f = （f1,f2,… ,fN）, where （pij） is a N x N transition probability matrix and {wij } is an family of continuous transformations on X. The authors study the uniqueness, ergodicity and unidimensionality of T＊-invariant measures where T＊ is the adjoint operator of T.     

WELL-POSEDNESS OF A COMPRESSIBLE GAS-LIQUID MODEL FOR DEEPWATER OIL WELL OPERATIONS

The main purpose of this paper is two-fold:(i) to generalize an existence result for a compressible gas-liquid model with a friction term recently published by Friis and Evje [SIAM J. Appl. Math., 71(2011), pp. 2014–2047];(ii) to derive a uniqueness result for the same model. A main ingredient in the existence part is the observation that we can consider weaker assumptions on the initial liquid and gas mass, and still obtain an existence result. Compared to the above mentioned work, we rely on a more refined application of the estimates provided by the basic energy estimate. Concerning the uniqueness result, we borrow ideas from Fang and Zhang [Nonlinear Anal. TMA, 58(2004), pp. 719–731] and derive a stability result under appropriate constraints on parameters that determine rate of decay toward zero at the boundary for gas and liquid masses, and growth rate of masses associated with the friction term and viscous coefficient.     

ON THE ASYMPTOTIC BEHAVIOR OF HOLOMORPHIC ISOMETRIES OF THE POINCARE DISK INTO BOUNDED SYMMETRIC DOMAINS

In this article we bounded symmetric domains study holomorphic isometries of the Poincare disk into Earlier we solved the problem of analytic continuation of germs of holomorphic maps between bounded domains which are isometrics up to normalizing constants with respect to the Bergman metric, showing in particular that the graph 170 of any germ of holomorphic isometry of the Poincar6 disk A into an irreducible bounded symmetric domain Ω belong to C^N in its Harish-Chandra realization must extend to an affinealgebraic subvariety V belong to C × C^N = C^N＋1, and that the irreducible component of V ∩ （△ × Ω） containing V0 is the graph of a proper holomorphic isometric embedding F ： A→ Ω. In this article we study holomorphie isometric embeddings which are asymptotically geodesic at a general boundary point b ∈ δ△. Starting with the structural equation for holomorphic isometrics arising from the Gauss equation, we obtain by covariant differentiation an identity relating certain holomorphic bisectional curvatures to the boundary behavior of the second fundamental form σ of the holomorphie isometric embedding. Using the nonpositivity of holomorphic bisectional curvatures on a bounded symmetric domain, we prove that ‖σ‖ must vanish at a general boundary point either to the order 1 or to the order 1/2, called a holomorphie isometry of the first resp. second kind. We deal with special cases of non-standard holomorphic isometric embeddings of such maps, showing that they must be asymptotically totally geodesic at a general boundary point and in fact of the first kind whenever the target domain is a Cartesian product of complex unit balls. We also study the boundary behavior of an example of holomorphic isometric embedding from the Poincare disk into a Siegel upper half-plane by an explicit determination of the boundary behavior of holomorphic sectional curvatures in the directions tangent to the embedded Poincare disk, showing that the map is indeed asymptotically totally geodesic at a general boundary point and of the first kind. For the metric computation we make use of formulas for symplectic geometry on Siegel upper half-planes.     

THE NAVIER-STOKES EQUATIONS WITH THE KINEMATIC AND VORTICITY BOUNDARY CONDITIONS ON NON-FLAT BOUNDARIES

We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in R^n with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure p can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolute boundary conditions. Based on this, we first establish the well-posedness for an appropriate local linearized problem with the absolute boundary conditions and the initial condition （without the incompressibility condition）, which establishes a velocity mapping. Then we develop apriori estimates for the velocity mapping, especially involving the Sobolev norm for the time-derivative of the mapping to deal with the complicated boundary conditions, which leads to the existence of the fixed point of the mapping and the existence of solutions to our initial-boundary value problem. Finally, we establish that, when the viscosity coefficient tends zero, the strong solutions of the initial-boundary value problem in R^n（n ≥ 3） with nonhomogeneous vorticity boundary condition converge in L^2 to the corresponding Euler equations satisfying the kinematic condition.     

MULTIPLICATION OPERATORS ON INVARIANT SUBSPACES OF FUNCTION SPACES

Let M? be the operator of multiplication by?on a Hilbert space of functions analytic on the open unit disk. For an invariant subspace F for the multiplication operator Mz, we derive some spectral prope...     

ON THE BASIC REPRODUCTION NUMBER OF GENERAL BRANCHING PROCESSES

Under a very general condition （TNC condition） we show that the spectral radius of the kernel of a general branching process is a threshold parameter and hence plays a role as the basic reproduction number in usual CMJ processes. We discuss also some properties of the extinction probability and the generating operator of general branching processes. As an application in epidemics, in the final section we suggest a generalization of SIR model which can describe infectious diseases transmission in an inhomogeneous population.     

A CLASS OF MULTIWAVELETS AND PROJECTED FRAMES FROM TWO-DIRECTION WAVELETS

This article aims at studying two-direction refinable functions and two-direction wavelets in the setting Rs, s 1. We give a sufficient condition for a two-direction refinable function belonging to L2(Rs). Then, two theorems are given for constructing biorthogonal(orthogonal) two-direction refinable functions in L2(Rs) and their biorthogonal(orthogonal) two-direction wavelets, respectively. From the constructed biorthogonal(orthogonal)two-direction wavelets, symmetric biorthogonal(orthogonal) multiwaveles in L2(Rs) can be obtained easily. Applying the projection method to biorthogonal(orthogonal) two-direction wavelets in L2(Rs), we can get dual(tight) two-direction wavelet frames in L2(Rm), where m ≤ s. From the projected dual(tight) two-direction wavelet frames in L2(Rm), symmetric dual(tight) frames in L2(Rm) can be obtained easily. In the end, an example is given to illustrate theoretical results.     

NORMALLY DISTRIBUTED PROBABILITY MEASURE ON THE METRIC SPACE OF NORMS

In this paper we propose a method to construct probability measures on the space of convex bodies. For this purpose, first, we introduce the notion of thinness of a body. Then we show the existence of ...     

COMPOSITION TYPE OPERATORS FROM HARDY SPACES TO μ-BLOCH SPACES ON THE UNIT BALL

Let φ be a holomorphic self-map of Bn and ψ ∈ H（Hn）. A composition type operator is defined by Tψ,φ（f） = ψf o φ for f ∈ H（Bn）, which is a generalization of the multiplication operator and the composition operator. In this article, the necessary and sufficient conditions are given for the composition type operator Tψ,φ to be bounded or compact from Hardy space HP（Bn） to μ-Bloch space Bμ（Bn）. The conditions are some supremums concerned with ψ,φ, their derivatives and Bergman metric of Bn. At the same time, two corollaries are obtained.     

STABILIZATION OF MARKOVIAN JUMP LINEAR SYSTEMS VIA RANDOM COMMUNICATION TIME DELAYED AND IMPULSIVE CONTROLLERS

This paper deals with the stabilization problems for a networked control system via time-delayed and impulsive controllers. The random communication delays in the model signal are modeled as a Markov chain. First, we introduce a hybrid controller with delay and impulses for the networked control systems. Then, some sufficient conditions are proposed for the design of a hybrid controller such that the closed-loop system is stochastically stable.     

ON THE EXISTENCE OF SINGULAR DIRECTIONS OF HOLOMORPHIC MAPS FROM THE UNIT DISK INTO Pn（C）

This article proves the existence of Julia directions of value distribution of holomorphic mapping f from the unit disk into the n-dimensional complex projective space P n(C) under the assumption lim sup r→1-T(r,f)/log 1/1-r=+∞ for hypersurfaces in general position.A heuristic principle concerning the existence of Julia directions of holomorphic mappings from the unit disk into P n(C) is given also.     

SUBORDINATION PROPERTIES OF MULTIVALENT FUNCTIONS INVOLVING AN EXTENDED FRACTIONAL DIFFERINTEGRAL OPERATOR

The object of this article is to investigate inclusion, radius, and other various properties of subclasses of multivalent analytic functions, which are defined by using an extended version of the Owa-Srivastava fractional differintegral operator Ω(λ,p).     

ON PROPERTIES OF MEROMORPHIC SOLUTIONS FOR COMPLEX DIFFERENCE EQUATION OF MALMQUIST TYPE

In this paper, we present the properties on zeros, fixed points, poles, Borel exceptional value of finite order transcendental meromorphic solutions of complex difference equation of Malmquist typewhere n（∈ N） 〉 2, P（f（z）） and Q（f（z）） are relatively prime polynomials in f（z） with rational coefficients a8 （s = 0, 1,…,p） and bt （t = 0, 1,… ,q） such that aoapbq 7≠ O, and also consider the existence and the forms on rational solutions of this type of difference equations. Some examples are also listed to show that the assumptions of theorems, in certain senses, are the best possible.     

STRONG COMPACTNESS OF APPROXIMATE SOLUTIONS TO DEGENERATE ELLIPTIC-HYPERBOLIC EQUATIONS WITH DISCONTINUOUS FLUX FUNCTION

Under a non-degeneracy condition on the nonlinearities we show that sequences of approximate entropy solutions of mixed elliptic-hyperbolic equations are strongly precompact in the general case of a Caratheodory flux vector. The proofs are based on deriving localization principles for H-measures associated to sequences of measurevalued functions. This main result implies existence of solutions to degenerate parabolic convection-diffusion equations with discontinuous flux. Moreover, it provides a framework in which one can prove convergence of various types of approximate solutions, such as those generated by the vanishing viscosity method and numerical schemes.     

CONSTRUCTION OF OPTIMAL BLOCKING SCHEMES FOR ROBUST PARAMETER DESIGNS

Robust parameter design (RPD) is an important issue in experimental designs. If all experimental runs cannot be performed under homogeneous conditions, blocking the units is effective. In this paper, w...     

GENERALIZED DERIVATIONS ON PARABOLIC SUBALGEBRAS OF GENERAL LINEAR LIE ALGEBRAS

Let P be a parabolic subalgebra of a general linear Lie algebra gl(n, F) over a field F, where n ≥ 3, F contains at least n different elements, and char(F) = 2. In this article, we prove that generalized derivations, quasiderivations, and product zero derivations of P coincide, and any generalized derivation of P is a sum of an inner derivation, a central quasiderivation, and a scalar multiplication map of P. We also show that any commuting automorphism of P is a central automorphism, and any commuting derivation of P is a central derivation.     

REGULARITY OF THE SOLUTIONS FOR NONLINEAR BIHARMONIC EQUATIONS IN RN

The purpose of this article is to establish the regularity of the weak solutions for the nonlinear biharmonic equation
{△^2u ＋ a（x）u = g（x, u）
u∈ H^2（R^N）,
where the condition u∈ H^2（R^N） plays the role of a boundary value condition, and as well expresses explicitly that the differential equation is to be satisfied in the weak sense.