Chen Jingrun was born on 22 May,1933 in Fu Zhou in Fu Jian province of China.Chen's father Chen Yuanjun was a clerk in postoffice and his mother was passed away in 1947.Chen's family was comparatively poor since the income of his father was lower and the population of his family was comparatively large.

We study the Hindmarsh-Rose burster which can be described by the differential system
ẋ=y-x³ + bx² + I-z, ẏ=1-5x²-y, ż=μ(s(x-x0)-z),
where b, I, μ, s, x₀ are parameters. We characterize all its invariant algebraic surfaces and all its exponential factors for all values of the parameters. We also characterize its Darboux integrability in function of the parameters. These characterizations allow to study the global dynamics of the system when such invariant algebraic surfaces exist.

In this paper, some topological concepts and definitions are generalized to cone metricspaces. It is proved that every cone metric space is first countable topological space and that sequentiallycompact subsets are compact. Also, we define diametrically contractive mappings and asymptoticallydiametrically contractive mappings on cone metric spaces to obtain some fixed point theorems byassuming that our cone is strongly minihedral.

In this paper,the authors establish the regularity in generalized Morrey spaces of solutions to parabolic equations with VMO coeffcients by means of the theory of singular integrals and linear commutators.

In this paper,some feasibly suffcient conditions are obtained for the global asymptotic stability of a positive steady state of a predator-prey system with stage structure for the predator by using the theory of competitive systems,compound matrices and stability of periodic orbits,and then the work of Wang [4] is improved.

We define the notion of evolutes of curves in a hyperbolic plane and establish the relationships between singularities of these subjects and geometric invariants of curves under the action of the Lorentz group. We also describe how we can draw the picture of an evolute of a hyperbolic plane curve in the Poincaré disk.

By making full use of the estimates of solutions to nonstationary Stokes equations and the method discussing global stability, we establish the global existence theorem of strong solutions for Navier-Stokes equatios in arbitrary three dimensional domain with uniformly C³ boundary, under the assumption that‖a‖_L ²(Θ)+‖f‖_L ¹(0,∞;L²(Θ)) or‖▽a‖_L ²(Θ)+‖f|_L ²(0,∞;L²(Θ)) small or viscosityv large. Here a is a given initial velocity and f is the external force. This improves on the previous results. Moreover, the solvability of the case with nonhomogeneous boundary conditions is also discussed.

Recall that the semigroups S and R are said to be strongly Morita equivalent if there exists a unitary Morita context (S, R.,_S P_R,R Q_S ,〈〉, ) with 〈〉 and surjective. For a factorisable semigroup S, we denote ζ_S = {(s₁, s₂) ∈S×S|ss₁ = ss₂, ∀s∈S}, S' = S/ζ_S and US-FAct = {_S M∈S-Act |SM = M and SHom_S (S, M) ≌M}. We show that, for factorisable semigroups S and M, the categories US-FAct and UR-FAct are equivalent if and only if the semigroups S' and R' are strongly Morita equivalent. Some conditions for a factorisable semigroups to be strongly Morita equivalent to a sandwich semigroup, local units semigroup, monoid and group separately are also given. Moreover, we show that a semigroup S is completely simple if and only if S is strongly Morita equivalent to a group and for any index set I, S⊗SHom_S (S, Ц_i∈I S) →Ц_i∈I S, s⊗t·f →(st)f is an S-isomorphism.

The present paper investigates the fractional derivatives of Weierstrass function, proves that there exists some linear connection between the order of the fractional derivatives and the dimension of the graphs of Weierstrass function.

Suppose A is a unital C^*-algebra and r >1. In this paper, we define a unital C^*-algebra C_cb^*(A, r) and a completely bounded unital homomorphism α_r: A → C_cb^* (A, r) with the property that C_cb^*(A, r) = C^*(α_r(A)) and, for every unital C^*-algebra B and every unital completely bounded homomorphism φ: A → B, there is a (unique) unital *-homomorphism π: C_cb^* (A, r) → B such that φ = π ○ α_r. We prove that, if A is generated by a normal set {t_λ: λ ∈ Λ}, then C_cb^* (A, r) is generated by the set {α_r(t_λ): λ ∈ Λ}. By proving an equation of the norms of elements in a dense subset of C_cb^*(A, r) we obtain that, if B is a unital C^*-algebra that can be embedded into A, then C_cb^*(B, r) can be naturally embedded into C_cb^*(A, r). We give characterizations of C_cb^*(A, r) for some special situations and we conclude that C_cb^*(A, r) will be "nice" when dim(A) ≤ 2 and "quite complicated" when dim(A) ≥ 3. We give a characterization of the relation between K-groups of A and K-groups of C_cb^*(A, r). We also define and study some analogous of C_cb^* (A, r).

A brief summary of the development on Kadison's famous problems (1967) is given. A new set of problems in von Neumann algebras is listed.

Toroidal Leibniz algebras are the universal central extensions of the iterated loop algebras ġ⊕C[t₁^±1 ,...,t_v^±1 ] in the category of Leibniz algebras. In this paper, some properties and representations of toroidal Leibniz algebras are studied. Some general theories of central extensions of Leibniz algebras are also obtained.

A graph has the unique path property UPP_n if there is a unique path of length n between any ordered pair of nodes. This paper reiterates Royle and MacKay's technique for constructing orderly algorithms. We wish to use this technique to enumerate all UPP₂ graphs of small orders 3² and 4². We attempt to use the direct graph formalism and find that the algorithm is inefficient. We introduce a generalised problem and derive algebraic and combinatoric structures with appropriate structure. Then we are able to design an orderly algorithm to determine all UPP₂ graphs of order 3², which runs fast enough. We hope to be able to determine the UPP₂ graphs of order 4² in the near future.

In this paper, general modular theorems are obtained for meromorphic functions and their derivatives. The related criteria for normality of families of meromorphic functions are proved.

The set of probability functions is a convex subset of L¹ and it does not have a linear space structure when using ordinary sum and multiplication by real constants.Moreover,difficulties arise when dealing with distances between densities.The crucial point is that usual distances are not invariant under relevant transformations of densities.To overcome these limitations,Aitchison’s ideas on compositional data analysis are used,generalizing perturbation and power transformation,as well as the Aitchison inner product,to operations on probability density functions with support on a finite interval.With these operations at hand,it is shown that the set of bounded probability density functions on finite intervals is a pre-Hilbert space.A Hilbert space of densities,whose logarithm is square-integrable,is obtained as the natural completion of the pre-Hilbert space.

We prove some fixed point theorems in partially ordered sets,providing an extension of the Banach contractive mapping theorem.Having studied previously the nondecreasing case,we consider in this paper nonincreasing mappings as well as non monotone mappings.We also present some applications to first-order ordinary differential equations with periodic boundary conditions,proving the existence of a unique solution admitting the existence of a lower solution.

The study of the convergent rate (spectral gap) in the L²-sense is motivated from several different fields: probability, statistics, mathematical physics, computer science and so on and it is now an active research topic. Based on a new approach (the coupling technique) introduced in[7] for the estimate of the convergent rate and as a continuation of[4],[5],[7-9],[23] and[24], this paper studies the estimate of the rate for time-continuous Markov chains. Two variational formulas for the rate are presented here for the first time for birth-death processes. For diffusions, similar results are presented in an accompany paper[10]. The new formulas enable us to recover or improve the main known results. The connection between the sharp estimate and the corresponding eigenfunction is explored and illustrated by various examples. A previous result on optimal Markovian couplings[4] is also extended in the paper.

We first apply non-negative matrix theory to the matrix K=D+A,where D and A are the degree-diagonal and adjacency matrices of a graph G,respectively,to establish a relation on the largest Laplacian eigenvalue λ₁(G) of G and the spectral radius ρ(K) of K.And then by using this relation we present two upper bounds for λ₁(G) and determine the extrernal graphs which achieve the upper bounds.

Let M be the multilinear maximal function and = (b₁, . . . , b_m) be a collection of locally integrable functions. Denote by M and [,M] the maximal commutator and the commutator of M with , respectively. In this paper, the multiple weighted strong and weak type estimates for operators M and [,M] are studied. Some characterizations of the class of functions b_j are given, for which these operators satisfy some strong or weak type estimates.

In this paper we provide a complete description of linear biseparating maps betweenspaces lip0(X^α,E)of Banach-valued little Lipschitz functions vanishing at infinity on locally com-pact H¨older metric spaces X^α=(X,d^α_X)with 0<α<1.Namely,it is proved that any linear bijection T:lip0(X^α,E)→lip0(Y^α,F)satisfying that ‖Tf()‖_F‖Tg(y)‖_F=0 for all y∈Y if and only if ‖f(x)‖_E‖g(x)‖_E=0 for all x ∈X,is a weighted composition operator of the form Tf(y)=h(y)(f(φ(y))),whereφis a homeomorphism from Y onto X and h is a map from Y into the set of all linear bijections from E onto F.Moreover,T is continuous if and only if h(y)is continuous for all y ∈Y.In this case,φbecomes a locally Lipschitz homeomorphism and h a locally Lipschitz map from Y^α into the space of all continuous linear bijections from E onto F with the metric induced by the operator canonical norm.This enables us to study the automatic continuity of T and the existence of discontinuous linear biseparating maps.

We first determine in this paper the structure of the generalized Fitting subgroup F^*(G) of the finite groups G all of whose defect groups (of blocks) are conjugate under the automorphism group Aut(G) to either a Sylow p-subgroup or a fixed p-subgroup of G. Then we prove that if a finite group L acts transitively on the set of its proper Sylow p-intersections, then either L/O_p(L) has a T.I. Sylow p-subgroup or p = 2 and the normal closure of a Sylow 2-subgroup of L/O₂(L) is 2-nilpotent with completely descripted structure. This solves a long-open problem. We also obtain some generalizations of the classic results by Isaacs and Passman on half-transitivity.

We show that for a class of second order divergence form elliptic equations on an infinite strip with the Dirichlet boundary condition, the space of fixed order exponential growth solutions is of finite dimension. An optimal estimation of the dimension is given. Examples also show that the finiteness property may not be true if one drops some of the conditions we make in our result.

It has been conjectured that there is a hamiltonian cycle in every finite connected Cayley graph. In spite of the difficulty in proving this conjecture, we show that almost all Cayley graphs are hamiltonian. That is, as the order n of a group G approaches infinity, the ratio of the number of hamiltonian Cayley graphs of G to the total number of Cayley graphs of G approaches1.

The uniqueness of the ADS spacetime among all static vacuum spacetimes with the same conformal infinity is proved for dimension n ≤ 7. For dimension n > 7, the same result is established under the spin assumption.

This paper studies the optimal controls of stochastic systems of functional type with end constraints. The systems considered may be degenerate and the control region may be nonconvex. A stochastic maximum principle is derived. The method is based on the idea that stochastic systems are essentially infinite dimensional systems.

In the present paper we establish a sharp asymptotic formula for

with an error term (σ is a suitable positive constant) uniformly inq and T,which improves the best result hitherto given.

A proper [h]-total coloring c of a graph G is a proper total coloring c of G using colors of the set [h] = {1, 2, . . . ,h}. Let w(u) denote the sum of the color on a vertex u and colors on all the edges incident to u. For each edge uv ∈ E(G), if w(u) = w(v), then we say the coloring c distinguishes adjacent vertices by sum and call it a neighbor sum distinguishing [h]-total coloring of G. By tndi (G), we denote the smallest value h in such a coloring of G. In this paper, we obtain that G is a graph with at least two vertices, if mad(G) < 3, then tndi (G) ≤ k + 2 where k = max{Δ(G), 5}. It partially confirms the conjecture proposed by Pilśniak and Woźniak.

Classical non-steady boundary layer equations are fundamental nonlinear partial differential equations in the boundary layer theory of fluid dynamics. In this paper, we introduce various schemes with multiple parameter functions to solve these equations and obtain many families of new explicit exact solutions with multiple parameter functions. Moreover, symmetry transformations are used to simplify our arguments. The technique of moving frame is applied in the three-dimensional case in order to capture the rotational properties of the fluid. In particular, we obtain a family of solutions singular on any moving surface, which may be used to study turbulence. Many other solutions are analytic related to trigonometric and hyperbolic functions, which reflect various wave characteristics of the fluid. Our solutions may also help engineers to develop more effective algorithms to find physical numeric solutions to practical models.  

During the last ten to fifteen years, a lot of progress has been achieved in the study of complex operator spaces. In this paper, we show that a corresponding theory can be developed for real operator spaces. With some appropriate modifications, many complex results still hold for real operator spaces.

The purpose of this paper is to study the canonical foliations of an almost cosymplectic or almost Kenmotsu manifold M in a unified way. We prove that the canonical foliation F defined by the contact distribution is Riemannian and tangentially almost Köhler of codimension 1 and that F is tangentially K?hler if the manifold M is normal. Furthermore, we show that a semi-invariant submanifold N of such a manifold M admits a canonical foliation F_N which is defined by the antiinvariant distribution and a canonical cohomology class c(N) generated by a transversal volume form for F_N . In addition, we investigate the conditions when the even-dimensional cohomology classes of N are non-trivial. Finally, we compute the Godbillon-Vey class for F_N .

Zhao and Ho asked in a recent paper that for each T₀ space X, whether KB(X) (the set of all irreducible closed sets of X whose suprema exist) is the canonical k-bounded sobrification of X in the sense of Keimel and Lawson. In this paper, we construct a counterexample to give a negative answer. We also consider the subcategory Top_κ of the category Top₀ of T₀ spaces, and prove that the category KBSob of k-bounded sober spaces is a full reflective subcategory of the category Top_κ.

We generalize the Chinese Remainder Theorem, use it to study number theory models, compare and analyse several number theory theorems in non-standard number theory models.

Let {X,X_n; n ≥ 0} be a sequence of independent and identically distributed random variables with EX = 0, and assume that EX²I(|X| ≤ x) is slowly varying as x→∞, i.e., X is in the domain of attraction of the normal law. In this paper, a self-normalized law of the iterated logarithm for the geometrically weighted random series ∑_n=0^∞βⁿX_n (0 < β < 1) is obtained, under some minimal conditions.

In order to cope with some difficulties due to the fact that the derivative of a constant is not zero with the commonly accepted Riemann-Liouville definition of fractional derivative, one (Jumarie) has proposed recently an alternative referred to as (local) modified Riemann-Liouville definition, which directly, provides a Taylor’s series of fractional order for non differentiable functions. We examine here in which way this calculus can be used as a framework for a differential geometry of fractional order. One will examine successively implicit function, manifold, length of curves, radius of curvature, Christoffel coefficients, velocity, acceleration. One outlines the application of this framework to Lagrange optimization in mechanics, and one concludes with some considerations on a possible fractional extension of the pseudo-geodesic of thespecial relativity and of the Lorentz transformation.

We describe the relationship between the fuzzy sets and the algebraic hyperstructures. In fact, this paper is a continuation of the ideas presented by Davvaz in (Fuzzy Sets Syst., 117: 477- 484, 2001) and Bhakat and Das in (Fuzzy Sets Syst., 80: 359-368, 1996). The concept of the quasicoincidence of a fuzzy interval value with an interval-valued fuzzy set is introduced and this is a natural generalization of the quasi-coincidence of a fuzzy point in fuzzy sets. By using this new idea, the concept of interval-valued (α, β)-fuzzy sub-hypermodules of a hypermodule is defined. This newly defined interval-valued (α, β)-fuzzy sub-hypermodule is a generalization of the usual fuzzy sub-hypermodule. We shall study such fuzzy sub-hypermodules and consider the implication-based interval-valued fuzzy sub-hypermodules of a hypermodule.

We are concerned with the following Dirichlet problem:
(P)
where f(x, t) ∈C (Ω×R), f(x, t)/t is nondecreasing in t∈R and tends to an L^∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0,
(AR)
is no longer true, where As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞.