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.

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).

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.

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.

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 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.

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.

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.

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.

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.

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.

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.

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.

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 .

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.

The asymptotic expansions of the trace of the heat kernel for small positive t, where {λ_v} are the eigenvalues of the negative Laplacian in Rⁿ (n=2 or 3), are studied for a general annular bounded domain Ω with a smooth inner boundary and a smooth outer boundary , where a finite number of piecewise smooth Robin boundary conditions on the components Γ_j (j=1,..., k) of and on the components Γ_j(j=k+1,...,m) of are considered such that and and where the coeffcients γ_j(j=1,...,m) are piecewise smooth positive functions. Some applications of Θ(t) for an ideal gas enclosed in the general annular bounded domain Ω are given. Further results are also obtained.

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) ≡ +∞.

We know that an operator T acting on a Banach space satisfying generalized Weyl's theorem also satisfies Weyl's theorem. Conversely we show that if all isolated eigenvalues of T are poles of its resolvent and if T satisfies Weyl's theorem, then it also satisfies generalized Weyl's theorem. We give also a similar result for the equivalence of a-Weyl's theorem and generalized a-Weyl's theorem. Using these results, we study the case of polaroid operators, and in particular paranormal operators.

This article is concerned with a class of hyperbolic inverse source problem with memory term and nonlinear boundary damping.Under appropriate assumptions on the initial data and parameters in the equation,we establish two results on behavior of solutions.At first we proved stability of solutions when the integral overdetermination tends to zero as time goes to infinity and finally a blow-up result is established for certain solution with positive initial energy.

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.

Let X={X(t) ∈ R^d, t ∈ R^N} be a centered space-anisotropic Gaussian random field whose components satisfy some mild conditions. By introducing a new anisotropic metric in R^d, we obtain the Hausdorff and packing dimension in the new metric for the image of X. Moreover, the Hausdorff dimension in the new metric for the image of X has a uniform version.

Let D and D′ be domains in real Banach spaces of dimension at least 2. The main aim of this paper is to study certain arc distortion properties in the quasihyperbolic metric defined in real Banach spaces. In particular, when D′ is a QH inner ψ-uniform domain with ψ being a slow (or a convex domain), we investigate the following: For positive constants c, h,C,M, suppose a homeomorphism f : D → D′ takes each of the 10-neargeodesics in D to (c, h)-solid in D′. Then f is C-coarsely M-Lipschitz in the quasihyperbolic metric. These are generalizations of the corresponding result obtained recently by Väisälä.  

A signed graph is a graph with a sign attached to each edge. This paper extends some fundamental concepts of the Laplacian matrices from graphs to signed graphs. In particular, the relationships between the least Laplacian eigenvalue and the unbalancedness of a signed graph are investigated.

In this paper we obtain some new identities containing Fibonacci and Lucas numbers. These identities allow us to give some congruences concerning Fibonacci and Lucas numbers such as L_2mn+k ≡ (-1)^(m+1)n L_k (mod L_m), F_2mn+k ≡ (-1)^(m+1)n F_k (mod L_m), L_2mn+k ≡ (-1)^mn L_2mn+k(mod F_m) and F_2mn+k ≡ (-1)^mn F_k (mod F_m). By the achieved identities, divisibility properties of Fibonacci and Lucas numbers are given. Then it is proved that there is no Lucas number L_n such that L_n = L₂k_tL_mx² for m > 1 and k ≥ 1. Moreover it is proved that L_n = L_mL_r is impossible if m and r are positive integers greater than 1. Also, a conjecture concerning with the subject is given.  

In this paper, we consider the Neumann problem for parabolic Hessian quotient equations. We show that the k-admissible solution of the parabolic Hessian quotient equation exists for all time and converges to the smooth solution of elliptic Hessian quotient equations. Also solutions of the classical Neumann problem converge to a translating solution.