SOME NEW IDENTITIES OF ROGERS-RAMANUJAN TYPE

In this paper,we establish two transformation formulas for nonterminating basic hypergeometric series by using Carlitz's inversions formulas and Jackson s transformation formula.In terms of application,by specializing certain parameters in the two transformations,four Rogers-Ramanujan type identities associated with moduli 20 are obtained.

COMPLETE KAHLER METRICS WITH POSITIVE HOLOMORPHIC SECTIONAL CURVATURES ON CERTAIN LINE BUNDLES(RELATED TO A COHOMOGENEITY ONE POINT OF VIEW ON A YAU CONJECTURE)

In this article,we study Kahler metrics on a certain line bundle over some compact Kahler manifolds to find complete Kahler metrics with positive holomorphic sectional(or bisectional)curvatures.Thus,we apply a strategy to a famous Yau conjecture with a co-homogeneity one geometry.

GLOBAL WEAK SOLUTIONS FOR AN ATTRACTION-REPULSION CHEMOTAXIS SYSTEM WITH p-LAPLACIAN DIFFUSION AND LOGISTIC SOURCE

This paper is concerned with the following attraction-repulsion chemotaxis system with p-Laplacian diffusion and logistic source:■The system here is under a homogenous Neumann boundary condition in a bounded domainΩ ■ R^(n)(n≥2),with χ,ξ,α,β,γ,δ,k_(1),k_(2)> 0,p> 2.In addition,the function f is smooth and satisfies that f(s)≤κ-μs~l for all s≥0,with κ ∈ R,μ> 0,l> 1.It is shown that(ⅰ)if l> max{2k_(1),(2k_(1)n)/(2+n)+1/(p-1)},then system possesses a global bounded weak solution and(ⅱ)if k_(2)> max{2k_(1)-1,(2k_(1)n)/(2+n)+(2-p)/(p-1)} with l> 2,then system possesses a global bounded weak solution.

SINGULAR CONTROL OF STOCHASTIC VOLTERRA INTEGRAL EQUATIONS

This paper deals with optimal combined singular and regular controls for stochastic Volterra integral equations,where the solution X^(u,ξ)(t)=X(t)is given X(t)=φ(t)+∫_(0)^(t) b(t,s,X(s),u(s))ds+∫_(0)^(t)σ(t,s,X(s),u(s))dB(s)+∫_(0)^(t)h(t,s)dξ(s).by Here d B(s)denotes the Brownian motion It?type differential,ξdenotes the singular control(singular in time t with respect to Lebesgue measure)and u denotes the regular control(absolutely continuous with respect to Lebesgue measure).Such systems may for example be used to model harvesting of populations with memory,where X(t)represents the population density at time t,and the singular control processξrepresents the harvesting effort rate.The total income from the harvesting is represented by J(u, ξ) = E[∫_(0)^(t) f_(0)(t,X(t), u(t))dt + ∫_(0)^(t)f_(1)(t,X(t))dξ(t) + g(X(T))] for the given functions f0,f1 and g,where T>0 is a constant denoting the terminal time of the harvesting.Note that it is important to allow the controls to be singular,because in some cases the optimal controls are of this type.Using Hida-Malliavin calculus,we prove sufficient conditions and necessary conditions of optimality of controls.As a consequence,we obtain a new type of backward stochastic Volterra integral equations with singular drift.Finally,to illustrate our results,we apply them to discuss optimal harvesting problems with possibly density dependent prices.

GLOBAL STABILITY OF LARGE SOLUTIONS TO THE 3D MAGNETIC BéNARD PROBLEM

In this paper,we consider the 3D magnetic Bénard problem.More precisely,we prove that the large solutions are stable under certain conditions.And we obtain the equivalent condition with respect to this stability condition.Finally,we also establish the stability of 2 D magnetic Bénard problem under 3D perturbations.

BOUNDS FOR MULTILINEAR OPERATORS UNDER AN INTEGRAL TYPE CONDITION ON MORREY SPACES

In this paper,we study a boundedness property of the Adams type for multilinear fractional integral operators with the multilinear L^(r′,α)-Hörmander condition and their commutators with vector valued BMO functions on a Morrey space and a predual Morrey space.Moreover,we give an endpoint estimate for multilinear fractional integral operators.As an application,we obtain the boundedness of multilinear Fourier multipliers with limited Sobolev regularity on a Morrey space.

THE FIELD ALGEBRA IN HOPF SPIN MODELS DETERMINED BY A HOPF*-SUBALGEBRA AND ITS SYMMETRIC STRUCTURE

Denote a finite dimensional Hopf C*-algebra by H, and a Hopf *-subalgebra of H by H1. In this paper, we study the construction of the field algebra in Hopf spin models determined by H1 together with its symmetry. More precisely, we consider the quantum double D(H, H_(1)) as the bicrossed product of the opposite dual Hopˆ of H and H1 with respect to the coadjoint representation, the latter acting on the former and vice versa, and under the non-trivial commutation relations between H1 and Ĥ we define the observable algebra AH1. Then using a comodule action of D(H, H1) on AH1, we obtain the field algebra FH1, which is the crossed product AH1⋊D(H,H_(1)), and show that the observable algebra AH1 is exactly a D(H, H1)-invariant subalgebra of FH1. Furthermore, we prove that there exists a duality between D(H, H1) and AH1, implemented by a*-homomorphism of D(H, H_(1)).

THE LEAST SQUARES ESTIMATOR FOR AN ORNSTEIN-UHLENBECK PROCESS DRIVEN BY A HERMITE PROCESS WITH A PERIODIC MEAN

We consider the least square estimator for the parameters of Ornstein-Uhlenbeck processes dY_(s)=(∑_(j=1)^(k)μ_(j)φ_(j)(s)-βY_(s))ds+dZ_(s)^(q,H),driven by the Hermite process Z_(s)^(q,H)with order q≥1 and a Hurst index H∈(1/2,1),where the periodic functionsφ_(j)(s),,j=1,...,κare bounded,and the real numbersμ_(j),,j=1,...,κtogether withβ>0 are unknown parameters.We establish the consistency of a least squares estimation and obtain the asymptotic behavior for the estimator.We also introduce alternative estimators,which can be looked upon as an application of the least squares estimator.In terms of the fractional Ornstein-Uhlenbeck processes with periodic mean,our work can be regarded as its non-Gaussian extension.

THE ENDPOINT ESTIMATE FOR FOURIER INTEGRAL OPERATORS

Let T_(a,φ)be a Fourier integral operator defined by the oscillatory integral T_(a,φ)u(x)=1/(2π)^(n)∫_(R^(n))^e^(iφ(x,ξ))a(x,ξ)(u)(ξ)dξ,where a∈S_(e,δ)^(m)andφ∈Φ^(2),satisfying the strong non-degenerate condition.It is shown that if0<(e)≤1,0≤δ<1 and m≤e^(2)-n/2,thenT_(α,φ)is a bounded operator from L^(∞()R^(n))to BMO(R^(n)).

ON F(p,q,s)SPACES

The family of spaces F(p,q,s)was introduced by the author in 1996.Since then,there has been great development in the theory of these spaces,due to the fact that these spaces include many classical function spaces,and have connections with many other areas of mathematics.In this survey we present some basic properties and recent results on F(p,q,s)spaces.

APPLICATION OF MULTI-DIMENSIONAL OF CONFORMABLE SUMUDU DECOMPOSITION METHOD FOR SOLVING CONFORMABLE SINGULAR FRACTIONAL COUPLED BURGER'S EQUATION

In this article,several theorems of fractional conformable derivatives and triple Sumudu transform are given and proved.Based on these theorems,a new conformable triple Sumudu decomposition method(CTSDM)is intrduced for the solution of singular two-dimensional conformable functional Burger's equation.This method is a combination of the decomposition method(DM)and Conformable triple Sumudu transform.The exact and approximation solutions obtained by using the suggested method in the sense of conformable.Particular examples are given to clarify the possible application of the achieved results and the exact and approximate solution are sketched by using Matlab software.

POSITIVE MAPS CONSTRUCTED FROM PERMUTATION PAIRS

A property(C) for permutation pairs is introduced. It is shown that if a pair{π_1, π_2} of permutations of(1,2,…,n) has property(C),then the D-type map Φ_(π_1,π_2) on n× n complex matrices constructed from {π_1,π_2} is positive. A necessary and sufficient condition is obtained for a pair {π_1,π_2} to have property(C),and an easily checked necessary and sufficient condition for the pairs of the form {π~p,π~q} to have property(C) is given, whereπ is the permutation defined by π(i) = i + 1 mod n and 1≤ p < q≤ n.

STABILITY OF ε-ISOMETRIES ON L_∞-SPACES

In this article,we discuss the stability ofε-isometries for L∞,λ-spaces.Indeed,we first study the relationship among separably injectivity,injectivity,cardinality injectivity and universally left stability of L∞,λ-spaces as well as we show that the second duals of universally left-stable spaces are injective,which gives a partial answer to a question of Bao-Cheng-Cheng-Dai,and then we prove a sharpen quantitative and generalized Sobczyk theorem which gives examples of nonseparable L∞-spaces X(but not injective)such that the couple(X,Y)is stable for every separable space Y.This gives a new positive answer to Qian's problem.

OPTIMAL ERROR ESTIMATES OF A DECOUPLED SCHEME BASED ON TWO-GRID FINITE ELEMENT FOR MIXED NAVIER-STOKES/DARCY MODEL

Although the two-grid finite element decoupled scheme for mixed Navier-Stokes/ Darcy model in literatures has given the numerical results of optimal convergence order, the theoretical analysis only obtain the optimal error order for the porous media flow and a non-optimal error order for the fluid flow. In this article, we give a more rigorous of the error analysis for the fluid flow and obtain the optimal error estimates of the velocity and the pressure.

THE POINTWISE ESTIMATES OF SOLUTIONS FOR A NONLINEAR CONVECTION DIFFUSION REACTION EQUATION

This paper studies the time asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in one dimension.First,the pointwise estimates of solutions are obtained,furthermore,we obtain the optimal Lp,1≤ p ≤ ＋∞,convergence rate of solutions for small initial data.Then we establish the local existence of solutions,the blow up criterion and the sufficient condition to ensure the nonnegativity of solutions for large initial data.Our approach is based on the detailed analysis of the Green function of the linearized equation and some energy estimates.

A NECESSARY AND A SUFFICIENT CONDITION FOR THE EXISTENCE OF THE POSITIVE RADIAL SOLUTIONS TO HESSIAN EQUATIONS AND SYSTEMS WITH WEIGHTS

In this article, we consider the existence of positive radial solutions for Hessian equations and systems with weights and we give a necessary condition as well as a sufficient condition for a positive radial solution to be large. The method of proving theorems is essentially based on a successive approximation technique. Our results complete and improve a work published recently by Zhang and Zhou（existence of entire positive k-convex radial solutions to Hessian equations and systems with weights. Applied Mathematics Letters,Volume 50, December 2015, Pages 48–55）.

OPTIMAL CONDITIONS OF GLOBAL EXISTENCE AND BLOW-UP FOR A NONLINEAR PARABOLIC EQUATION

According to the variational analysis and the potential well argument, we get the optimal conditions of global existence and blow-up for a type of nonlinear parabolic equations. Furthermore, we give its application in the instability of the steady states.

ON POINTS CONTAIN ARITHMETIC PROGRESSIONS IN THEIR LROTH EXPANSION

For any x ∈ （0, 1] （except at most countably many points）, there exists a unique sequence {dn（x）}n≥1 of integers, called the digit sequence of x, such that x =∞ ∑j=1 1/d1（x）（d1（x）-1）……dj-1（x）（dj-1（x）-1）dj（x）. The dexter infinite series expansion is called the Liiroth expansion of x. This paper is con- cerned with the size of the set of points x whose digit sequence in its Liiroth expansion is strictly increasing and contains arbitrarily long arithmetic progressions with arbitrary com- mon difference. More precisely, we determine the Hausdorff dimension of the above set.

EKELAND'S PRINCIPLE FOR SET-VALUED VECTOR EQUILIBRIUM PROBLEMS

In this paper, we introduce a concept of quasi C-lower semicontinuity for setvalued mapping and provide a vector version of Ekeland＇s theorem related to set-valued vector equilibrium problems. As applications, we derive an existence theorem of weakly efficient solution for set-valued vector equilibrium problems without the assumption of convexity of the constraint set and the assumptions of convexity and monotonicity of the set-valued mapping. We also obtain an existence theorem of ε-approximate solution for set-valued vector equilibrium problems without the assumptions of compactness and convexity of the constraint set.

IDENTIFYING AN UNKNOWN SOURCE IN SPACE-FRACTIONAL DIFFUSION EQUATION

In this paper, we identify a space-dependent source for a fractional diffusion equation. This problem is ill-posed, i.e., the solution （if it exists） does not depend continuously on the data. The generalized Tikhonov regularization method is proposed to solve this problem. An a priori error estimate between the exact solution and its regularized approximation is obtained. Moreover, an a posteriori parameter choice rule is proposed and a stable error estimate is also obtained, Numerical examples are presented to illustrate the validity and effectiveness of this method.