In this paper, utilizing the method of operator theory, some properties and perturbation of g-franes in Hilbert C^*-modules are discussed. some characterizations of sums of g-frames in Hilbert C^*-modules are obtained. Moreover, it is shown that these results extend and improve the existing results.

Through the establishment of a new space, we prove an embedding theorem. Furthermore, using that embedding theorem and mountain pass theorem, we obtain the existence of nontrivial solutions of a biharmonic problem in the new space.

Based on a 2 × 2 spectral problem, the corresponding hierarchy of evolution equations is derived. According to the property of its 2 × 2 Lenard pair of operators, it can be checked that the hierarchy is a generalized Hamilton system and possesses Bi-Hamilton structure and Multi-Hamilton structure. Furthermore, its Liouville integrability is also evidenced. What's more, this hierarchy, in special cases, can reduce to the general TD hierarchy, TD hierarchy, general C-KdV hierarchy, C-KdV hierarchy, etc. In the end, the one to one relation between the Hamilton functionals and the conservation densities are provided too.

A vector-valued Ekeland's variational principle is introduced, where the objective function is from a complete metric space into a cone-ordered topological linear space. This result is a generalization of the Ekeland's variational principle obtained by Zhong Chengkui.

In this paper, the existence, uniqueness and decay properties for the strong solutions of the MHD equations in spacePLⁿ∩PL^p, (1<p<n,n≥2) are studied by applying the semi-group theory and the method used by Kato^[1].

Let X and Y be Banach spaces, A∈B(X),B∈B(Y), C∈B(Y,X), M_C=(^A₀ ^C _B) be the operator matrix acting on the Banach space X⊕Y. In this paper, we give out 20 kind spectra structure of M_C, decide 18 kind spectra filling-in-hole properties of M_C and present some interesting examples of these problems.

The model of age-independent branching processes with static immigration in random environments is introduced. Giving the renewal equation of conditional generation function and considering the Kolmogorov equation for the special case. Moreover, we obtain the moments through studying the renewal equation and consider the extinction probability simply. Finally, we give an open problem.

Let f(z)=h(z)+g(z) be a harmonic mapping of the unit disk U. In this paper, the sharp coefficient estimates for bounded planar harmonic mappings are established, the sharp coefficient estimates for normalized planar harmonic mappings with |h(z)|+|g(z)|≤M are also provided. As their applications, Landau's theorems for certain biharmonic mappings are provided, which improve and refine the related results of earlier authors.

The authors prove that the commutators of Multipliers not only has the (M_p^q(Rⁿ), Lip_(β-(n/q))(Rⁿ)) boundedness but also has the (M_p^q(Rⁿ), BMO(Rⁿ)) boundedness. Furthermore, the boundedness of Multipliers and commutators on the generalized Morrey space L^p,ρ(Rⁿ) is established.

First of all, the necessary and sufficient conditions concerning the exponential stability of degenerate C₀-semigroup are obtained via functional analysis and operator theory in Hilbert space. Then, the exponential stability for a family of degenerate C₀-semigroup is discussed, the necessary and sufficient conditions are given in the light of degenerate C₀-semigroup.

Let Ω be a function on the unit sphere and b a radial function, and ρ be a complex parameter with Re (ρ)>0. Let Ψ be in C²([0,∞)), convex, and increasing function with Ψ(0) = 0. The parametric Marcinkiewicz operator μ^ρ_{Ω, b} with nonhomogenous rough kernel and the rough Marcinkiewicz operator μ^ρ_{Ω, Ψ, b} related to a surface of revolutions are considered in this paper, we prove the boundedness of these Marcinkiewicz operators on Hardy spaces and weak Hardy spaces under the minimum smooth conditions for the rough kernels Ω and b. The results in this paper extend as well as improve previously known results.

Let a be a positive integer with a>1, f(a) be a polynomial of a with nonnegative integer coefficients, and f(1)=2^rp+4, where r is a positive integer with r>1, p=2^l-1 is a Mersenne prime. In this paper, the finiteness of positive integer solution (x,n) of the equation

(a-1)x²+f(a)=4aⁿ

is discussed, and prove that if f(a)=91a+9, then the equation has only the positive integer solutions (x,n)=(3,3), (11,3) and (3,4) for a=5,7 and 25 respectively.

We prove a generalized Ul'yanov type inequality for the classical moduli of smoothness on the unit sphere, which has important applications in imbedding theory, spherical polynomial approximation and the theory of interpolation in function spaces on the sphere. Our proof is based on several new estimates on spherical harmonic expansions, which seem to be of independent interest.

We have classified completely the space-like hypersurfaces with parallel conformal second fundamental forms in the conformal space. In this work, we study the time-like case and classify the Type I time-like hypersurfaces with parallel conformal second fundamental forms.

In this paper we introduce the concept and operations of the function system matrix, contruct the complete family of fractals by the function system matrix, and we determine the Hausdorff dimension and the Box-dimension of the complete family of similar fractals under the open set conditions.

The analytical characteristics of a class of fractal interpolation functions (FIFs) with function vertical scaling factors, including smoothness, stability and sensitivity, are studied in this work. The results on smoothness of such FIFs are presented, and their stability is proved. In addition, the sensitivity and moments for the class of FIFs are discussed. It is shown that if a small perturbation occurs in the iterated function systems (IFSs) generating the class of FIFs, then a small perturbation is also produced in the corresponding FIFs and their moments. The upper estimates of the perturbation errors are given.

In this paper, firstly we give a condition for two semi-frame sequences such that both of them be frames for Hilbert space H, which is related to the perturbation theory of frames. Then we give some conditions on dilation and translation parameters and the generating functions such that wavelet frames for L²(R) are generated. These results generalize the classical similar results in literatures. We also discuss the Bessel sequence and add some other sufficient conditions.

The unitary equivalence of multiplication operator M_φ on the Dirichlet space D is studied, where φ is a Blaschke product of order two. The main result shows that for a multiplier ψ of D, the multiplication operator M_ψ is unitarily equivalent to M_φ if and only if ψ(z)=φ(θz) for some constant θ with |θ|=1. This is very different from the corresponding result in the Hardy space and in the Bergman space.

The holomorphic functions of several complex variables are closely related to the so-called isotonic Dirac system in which different Dirac operators in the half dimension act from the left and from the right on the functions considered. In this paper we mainly study the boundary properties of the isotonic Cauchy type integral operator over the smooth surface in Euclidean space of even dimension with values in a complex Clifford algebra. We obtain Privalov theorem inducing Sokhotskii-Plemelj formula as the special case for the isotonic Cauchy type integral operator with Hölder density functions taking values in a complex Clifford algebra, and show that Privalov theorem of the classical Bochner-Martinelli type integral and the classical Sokhotskii- Plemelj formula over the smooth surface of Euclidean space for holomorphic functions of several complex variables may be derived from it.

This paper establishes an existence and uniqueness result for the solution to a one-dimensional backward stochastic differential equation (BSDE for short) whose generator satisfies Constantin’s condition in y and is uniformly continuous in z, which generalizes some known results.

We investigate the regular cryptogroup construction of the Rees matrix semigroup over a Clifford semigroup with identity; secondly prove that the two conditions are equivalent on a Rees matrix semigroup S over a Clifford semigroup with identity: (1) the congruence ρ of S is a completely simple semigroup congruence; (2) there is an order-preserving bijection from the congruences of S to the admissible triples of S; finally prove that the lattice of completely simple congruences on a Rees semigroup over a Clifford semigroup with identity is semimodular.

In this paper, the principle of finite kernel is used to complete the successive difference substitution. Then a complete algorithm for deciding positive semi-definite polynomial is presented. This algorithm can be applied further to compute the global optimization of rational function. Being different from any other common methods of numerical optimization, the method in this paper gets accurate symbolic solution.

An analytic expression of von Koch curve has been given. Based on this complex-valued function, we give estimation of fractal dimension of its fractional calculus. Graphs of Weyl-Marchaud fractional derivative of this function have been given. Such function can also be transferred into certain self-affine fractal function. Finally, we set up the linear connection between fractal dimension of this function and order of fractional calculus. Graphs and numerical results of certain examples have been shown.

The synchronization of the n-dimensional second order lattices of coupled oscillators with external periodic forces under Dirichlet, Neumann, Periodic boundary conditions are studied by introducing a new norm in the phase space. For the system with Dirichlet boundary condition, if the first order partial derivatives of the nonlinear term are bounded, and the coupled coefficients are both large enough, the system will be bounded dissipative and the solutions of the system will be synchronized to each other. For the system with Neumann or Periodic boundary condition, if the variation of the out force of different subsystems and the variation of the nonlinear terms of different subsystems are both small, the system is bounded dissipative and the coupled coefficients are both large enough, then all the components of any one solution of the system will be asymptotic synchronized to each other. Moreover, for the above two cases, when the coupled coefficients c₁ → +∞, c₂ → +∞, all the components of any one solution of the systems will be synchronized to each other.

By using Ahlfors’ covering surface method, we establish the existence theory on a new singular radius of an algebroidal function in the unit disc, namely T-radius, and apply a Type-function of infinite order to extend some results of meromorphic functions.

Let R be a ring and let S be a family of left modules. A left module E is called S-injective if Ext_R¹(N,E) = 0 for any N ∈ S. In this paper it is shown that if S is a Baer family of modules, then the Baer Criterion for S-injective modules holds and that if S is a complete family of modules, then every module has an S-injective envelope. A module E is called max-injective if Ext_R¹(N,E) = 0 for any simple module N. A module E over a commutative ring R is called reg-injective if Ext_R¹(N,E) = 0 for any torsion module N. It is shown that every module has a max-injective envelope and that a commutative ring R is an SM ring if and and if the reg-injective envelope e(T/R) of T/R is Σ-reg-injective, where T = T(R) is the total quotient ring of R; if and only if every GV -torsion-free reg-injective module is Σ-reg-injective.

It is shown that every finit p-group is isomorphic to certain subgroups of U(n,Z_p) for some positive integrer n, and the maximal subgroups and sub-maximal subgroups of U(n,Z_p) are obtained.

Let k be an algebraically closed field of characteristic zero, H a semisimple Hopf algebra of dimension pq² of Frobenius type, where p, q are distinct prime numbers. This paper proves that if p > q and H^* is also of Frobenius type then H is a biproduct of a group algebra A of dimension q² and a Yetter-Drinfeld Hopf algebra R over A of dimension p. That is H ≌ R#A. As an example, this paper then proves that every semisimple Hopf algebra of dimension 63 or 68 is of Frobenius type.

A basic model of immune with Logistic growth and Holling type-II functional response has been studied. By choosing the time delay as the parameter, the stability of the positive equilibrium and the existence of the Hopf bifurcation are investigated. By using the normal form theory and the center argument, the explicit formulae which determine the stability and the direction are derived. Finally, numerical simulations supporting our theoretical results are also included.

This paper discusses the K₀-group of the operator algebra B(X) on Banach space X and gives two sufficient conditions for K₀(B(X)) = Z₂.

The present paper proved that if λ, μ are non-zero real numbers, not both negative, λ is irrational, and k is a positive integer, then there exist infinitely many primes p and pairs of primes p₁, p₂ such that [λp₁+μp₂²] = kp. In particular [λp₁+μp₂²] represents infinitely many primes.

In this paper, the estimation of the number of maximum genus nonorientable embeddings of graphs is studied, and an exponential lower bound for such number is found. Applying the theory of current graph, K_12s has at least 2^3s-2 distinct minimum genus embedding in non-orientable surfaces; K_12s+3 has at least 2^2s distinct minimum genus embedding in non-orientable surfaces; K_12s+7 has at least 2^2s+1 distinct minimum genus embedding in non-orientable surfaces.

We study Kadison-Singer lattices in the matrix algebra M_n(C), and prove that each Kadison-Singer lattice generating M₃(C) as an algebra is similar to L₀ or I-L₀, where L₀ is the KS lattice generated by a maximal nest of diagonal projections and a rank one projection matrix with nonzero entries in M₃(C), hence each Kadison-Singer algebra with trivial diagonal in M₃(C) has dimension 4. In addition, we give some examples of nonisomorphic Kadison-Singer lattices which generate M₄(C).

In the setting of homogeneous space (X, ρ, μ), it is shown that the commutators of Calderón-Zygumund operator and Potential operator with BMO function are bounded in Grand Morrey space, which contains the grand Lebesgue space and Morrey space. Moreover, all results are new even for Euclidean spaces.