We generalize the inverse inequality of Marcinkiewicz function in Stein [On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz, Trans. Amer. Math. Soc., 1958, 88: 430-466] to general cases. The main result of this paper is that, for any basis of spherical harmonic functions of order k in n-dimensional Euclidean space, we obtain the general inverse inequalities for Marcinkiewicz function, i.e. there exist a constant C_p does not depend on f such that ||f||_p ≤ C_p∑_j^N=1 ||μj(f)||_p, where {μj(f)}_j^N =1 are the Marcinkiewicz functions of f generated by these spherical harmonic functions. Moreover, for any n-variable homogeneous harmonic polynomial Q(x) of order k, and the Poisson kernel P_t(x), we have Q(D)P_t(x) =C_n,k((tQ(x))/((|x|²+t²)/(n+2k+1/2)).

We consider linear product operator DC_φ acting on the Zygmund space Z = {f ∈H(D) : sup_z∈D(1-|z|²)|f" (z)|< ∞}, where C_φ is the composition operator and D is the differentiation operator. The boundedness and compactness of the operator DC_φ on the Zygmund space Z and the little Zygmund space Z₀ are established in terms of the function theorectic property of the symbol φ.

Let a ≥ 2 be a positive integer. In this paper, we will prove that if a = 2, then the equation X²-(a²+1)Y⁴ = 3-4a has only one positive integer solution (X, Y) = (20, 3); if a = 3, then the equation has only two coprime positive integer solutions (X, Y) = (1, 1), (79, 5); if a ≥ 4 and 4a + 1 is a nonsquare positive integer, then the equation has at most four coprime positive integer solutions (X, Y); if a ≥ 4 and 4a + 1 is a square, then the equation has at most five coprime positive integer solutions (X, Y).

Given a positive function F on H_n⁺ which satisfies a convexity condition, we introduce an extrinsic curvature Θ_F for spacelike hypersurfaces with regular Weingarten operator in Lⁿ⁺¹. For certain F, we show that the only spacelike hypersurfaces having constant Θ_F and F-support function bounded between two negatives are the spacelike Wulff shapes. In case F= 1, we also get the same uniqueness result for the spacelike hypersurfaces with Hⁱ/Hⁿ =const.

We give a new definition of symplectic matrix, and discuss its properties. Using methods of symplectic matrix for fourth order self-adjoint boundary conditions, we obtain the fundamental forms of self-adjoint boundary conditions, such that all other forms can be generated by the symplectic transformations of the fundamental form.

We prove the generalized Dunkl-Williams inequality and its inverse in the frame-work of pre-Hilbert C^*-modules in a new approach and characterize the equality case.

We study the oscillatory hyper-Hilbert transform H_n,α,βf(x) = ƒ₀¹f(x - Γ(t))e^it-βt^-1-αdt, along the curve Γ(t) = (t^p₁, t^p₂, …, t^p_n), where 0 < p₁ < p₂ <…< p_n and α > β > 0. For 0 ≤ γ < (nα)/((n+1)(p₁+α)), we prove that H_n,α,β is bounded from L_γ^p (Rⁿ) to L^p(Rⁿ) whenever |1/p-1/2|< (β-(n+1)[α-(β+p₁)γ])/(2β), and that H_n,α,β is bounded from L^γ2 (Rⁿ) to L²(Rⁿ) whenever β ≥ (α-γp₁)/(γ+ (1/n+1)).

Firstly, the concept of up-covering relationship is introduced in join-semilattice, and based on up-covering relationship the concepts of enriched join-semi-lattice, up-coverage and C-filter of enriched join-semi-lattice are proposed, it is proved that the family C Fil(S) consisting of C-filters of enriched join-semi-lattice S is a co-frame, the properties of simple-up-set-valued mapping u : S → C Fil(S) are investigated. Secondly, it is proved that the subset A of direct product П_λ∈Γ A_λ of family {A_λ |λ ∈ Γ} of co-frames whose elements have finite non-zero coordinates is an enriched joinsemi- lattice, and further it is proved that A is the co-product object of family {A_λ|λ ∈ Γ} of co-frames in the join-semi-lattice category. Thirdly, an up-coverage C^* of A is defined by up-covering relationship of A'_λs. Moreover, by the simple-up-set-valued mapping u : A → C^* Fil(A) and the standard embedding mapping q_λ : A_λ → П_λ∈Γ A_λ it is proved that the co-frame C^* Fil(A) derived by up-coverage C^* of the enriched joint-semi-lattice A is the co-product object of family {A_λ|λ∈ Γ} of co-frames in co-frame category.

We define the generalized orders and generalized types of Laplace-Stieltjes transforms which convergence in the right half-plane. Some interesting relationships on the maximum modulus M_u(σ, F) = sup{|ƒ₀^x e^-(σ+it)ydν(y)|: x ∈ (0,+∞), t ∈ R}, the maximum term μ(σ, F) = max_n∈N{A_n^*e^-λnσ}, the index of maximum term υ(σ, F)= max_k{λ_k|μ(σ, F) = A_n^*e^-λnσ} and the coefficients of such transforms are obtained, which briefly extend some results of Dirichlet series.

We consider the Hyers-Ulam stability of sesquilinear forms on unit linked left l-modules over C^*-algebras and C^*-algebras with real rank zero respectively.

We present the definition of generalized e-ω-concave operator, under the assumption that A is not a cone mapping, we obtain the existence and uniqueness of the fixed point, and the monotone iterative sequence of the fixed point for generalized e-ω-concave operator. Finally, we apply the main results to a class of Hammerstein integral equations.

Let M be a semifinite von Neumann algebra. For 0 < p < ∞, 0 < q ≤ ∞, we define the noncommutative weighted Lorentz spaces Λ_ω^p,q (M) and its associate spaces Λ_ω^p,q (M)'. Subsequently, we give some properties of the spaces Λ_ω^p,q(M) and Λ_ω^p,q(M)'. As an application, the dual spaces of Λ_ω^p,q(M) is presented for 0 < p < ∞.

The genus distribution of a double-rooted graph whose one root has arbitrary degree after self-pasting at root vertices have been derived, by applying vertexdeleting, edge-addition theorem, multiple production rules and self-amalgamation the-orem. And analogous results of “two roots are two-degree” in literature [Genus distribution of graph amalgamations: self-pasting at root-vertices, Aust. J. Comb., 2011, 49: 19-38] had been provided by Gross have been generalized.

We establish two inequalities concerning the weakly convergent sequence WCS(X) and Jordan-von Neumann type constants C'_t(X), C__∞(X), which enable us to obtain some sufficient conditions for normal structure. The results obtained in this paper represent an extension as well as refinement of previous known results. Meanwhile, the Jordan-von Neumann type constants C'_t(X), C__∞(X) for the Bynum space l_2,∞ are computed, and are used to show that our results are sharp.

In this paper, by constructing a partial sum of ζ(k) on the Riemann Zeta function, using the ζ_n(k) power series function, Binomial expression and Cauchy product formula, some important sums are calculated. By extending the power series function from one variable to multivariate, some relationships of high order sums on Riemann Zeta function are established. By using the relationship between the logarithmic function and Stirling numbers of the first kind, and some other related equations, 18 seven-order sums of Riemann Zeta function and some other high order sums are obtained.

The purpose of this paper is to find out the deep and rich structure of the set of Parseval frame wavelets with special dilation matrices (M-PFW). We study the all kinds of properties of M-PFW, semi-orthogonal M-PFW and MRA M-PFW based on shift-invariant space and the dimension function. We investigate various subclass of the set of M-PFWs. Several constructive examples illustrating the various possibilities are given.

In this paper, an extragradient-type method proposed by Wang, Xiu and Zhang [Modified extragradient-type method for variational inequalities and verification of the existence of solutions, J. Optim. Theory Appl., 2003, 119: 167-183] for solving variational inequalities in Euclidean spaces is extended to Banach spaces. The operator involved in the variational inequality is not necessarily uniformly continuous. The main advantage lies in that the proposed algorithm is well defined no matter whether the variational inequality problem has a solution or not. Furthermore, the existence of the solution to the variational inequality problem can be verified through the behavior of the generated sequence. Under some suitable assumptions, the sequence generated by the proposed method is strongly convergent to the solution of the variational inequality, which is closest to the initial iterate in the sense of Bregman distance. The main results presented in this paper generalize and improve the recent ones in the literature.

We study the multiplicity of positive solutions to a p-Kirchhoff equation with sign-changing weight functions. These positive solutions are obtained by the variational methods and the decomposition of the Nehari manifold.

Based on anisotropic Sobolev space and compensated compactness methods, we obtain the homogenization of nonlinear elliptic equations with general anisotropic diffusivity.

Different type of continuous functions of unbounded variation have been discussed. According to the local structure, all continuous functions have been classified. Meanwhile, certain examples of different kinds of fractal functions have been given. Properties of those fractal functions have also been discussed. Definition of singular continuous functions has been given and an equivalent theorem which can be used to describe whether a function is a singular continuous function or not has been set up. Based on definition of singular continuous functions, we make research on definition of local fractal functions and fractal functions. Fractal functions are composed of singular fractal functions, irregular fractal functions and regular fractal functions. Finally, definition of fractal functions corresponding to discontinuous functions has been explored.

We mainly discuss the average sampling and reconstruction of signals in weighted reproducing kernel subspaces of L_ν^p . First, the sampling stability for two kinds of average sampling functionals are established. Then, we give a general iterative algorithm based on probability measure, which provides a unified treatment for iterative approximation projection algorithm and iterative frame algorithm. Finally, the asymptotic pointwise error estimate is presented for reconstructing a signal from its average samples corrupted by white noise.

Suppose that k ≥ 2 is an integer. In this note, we prove that every graph with m edges admits a partition V₁, V₂, . . ., V_k such that e(V₁, V₂, . . ., V_k) ≥ (k-1)/k·m + (k-1)/(2k)·(√2m + 1/4-1/2)-((k-2)²)/(8k), and max{e(V_i) : 1 ≤ i ≤ k} ≤ m/(k²) + (k-1)/(2k²) (√2m + 1/4-1/2)+ 3/8-(7k-4)/(8k²). This improves the main result of [Fan G., Hou J., Zeng Q., A bound for judicious k-partitions of graphs, Discrete Appl. Math., 2014, 179: 86-99].

We study the generalized Roper-Suffridge operator (F(z₁)+f'(z₁)Σ_{k =2}ⁿakz_k^pk, f' (z₁)^1/(p2)z₂, . . ., f' (z₁)^1/(pn)z_n)', and prove that it can preserve the properties of almost spirallikeness of type β and order α, spirallikeness of type β and order α, and strongly spirallikeness of type β on the following Reinhardt domain in complex Euclidean space Ω_n,p2,...,pn = {z = (z₁, . . ., z_n) ∈ Cⁿ : |z₁|² +Σ_k=2ⁿ |z_k|^pk < 1, pk ∈ N⁺, k = 2, . . ., n} under some special conditions, respectively.

Batch codes were first proposed by Ishai et al., an (n,N, k,m)-combinatorial batch code was defined by Peterson et al. as a purely combinatorial version of batch codes. It is a system consisting of m subsets of an n-element set such that any k elements can be retrieved by reading at most one (or in general, t) elements from each subset and the number of total elements in m subsets is N. For given parameters n, k,m, the minimum N, denoted by N(n, k,m), is of particular interest, which has both theoretical interests and strong practical motivation. So far, for k ≥ 5, m+3 ≤ n <, precise values of N(n, k,m) have not been established except for some special settings of the parameters. In this article, we obtain N(m+3, 5,m) = m+11 (m ≥ 7), N(9, 5, 6) = 18, N(m+3, 6,m) = m+13 (m ≥ 8), N(10, 6, 7) = 21. Our results partly answer an open problem of Paterson et al.

We investigate boundedness and compactness of the generalized composition operators between weighted Bloch-type spaces. Some sufficient and necessary conditions for the boundedness and compactness of these operators are obtained.

We study a class of not-finitely graded Lie algebras related to generalized Heisenberg-Virasoro algebras HV,and we determine the concrete structures of the derivations of HV.To compute the derivations Der HV of HV,we can decompose them into a sum of the inner derivations ad HV and the derivations (Der HV)₀ whose degree is zero.

We construct indecomposable quadratic lattices of rank 3,4,5 and 6 over a suitable Hasse domain of the rational function field F₁₉(x),which,in particular,solves the problem proposed by Gerstein about the existence of indecomposable quadratic lattices of rank 5 over global function fields.

Let F be the familiar class of normalized univalent functions in the unit disk.Fekete and Szegö proved the well-known result max_f∈F|a₃-λa₂²|=1+2e^-2λ/1-λ for λ∈[0,1].We investigate the corresponding problem for the class of starlike mappings defined on the bounded starlike circular domain in Cⁿ.The proofs of these results use some restrictive assumptions,which in the case of one complex variable are automatically satisfied.

The main purpose of this paper is using the analytic methods and the properties of Gauss sums to study the computational problem of one kind hybrid power mean of two-term exponential sums and two-term character sums,and give an exact expression for it.As its applications,we obtained an asymptotic formula for the hybrid power mean and a sharp asymptotic formula for the mean value of Dirichlet L-functions weighted by the two-term exponential sums and two-term character sums.

Let T_σ be the bilinear Fourier multiplier operator associated with multiplier σ satisfying the Sobolev regularity that sup_k∈Z ||σk||W^s (R²ⁿ)<∞ for some s∈(n,2n].We give the boundedness of T_σ and the commutators T_σ,b generated by T_σ and b=(b₁,b₂)∈(BMO (Rⁿ))²,as well as the compactness of T_σ,b (if b₁,b₂∈CMO (Rⁿ),the BMO-closure of C_c^∞ (Rⁿ)) from L^p1,λ(ω₁)×L^p2,λ(ω₂) to L^p,λ(ν_w) for appropriate indices p₁,p₂,p∈(1,∞)(1/p=1/p₁+1/p₂) and multiple weights ω=(ω₁,ω₂)∈A_p/t (R²ⁿ).The main ingredient is to establish the multiple weighted estimates for the variants of certain multi (sub) linear maximal operators on the weighted Morrey spaces,and a sufficient condition for a subset in the weighted Morrey spaces to be a strongly precompact set,which are in themselves interesting.

Applying I-convergence and I^*-convergence of sequences in Banach space X,this paper first present a sufficient and neccessary condition for an ideal I has the additive property,then establish the relation between w-I-convergence,w-I^*-convergnence and uni-w-I^*-convergence,also the connection between w-I-convergence and convergence,finally we define I-A-statistical convergence which is the generalization of I-λ-statistical convergence,and by using subdifferential mapping to define a set of finite additive measures,we show the equivalent description of I-A-statistical convergence,this is an application of finite additive measures.

The definitions of partially weakly increasing property of a pair of random mappings (F,G) with respect to g and partially weakly increasing property of (F,G) are introduced and the existence of common coupled random coincidence points and common coupled random fixed points for a sequence of mappings F_k:Ω×X×X→X,k=1,2,...and g:Ω×X→X and h:Ω×X→X under various contractive conditions in complete and separable partially ordered metric spaces are studied.Many new results are obtained,which generalize some results in the corresponding literatures.

The paper studied the deficiency and value distribution of meromorphic functions concerning difference.We mainly proved:let c be a nonzero finite complex number,let f(z) be a transcendental entire function in C,and let n be a positive integer.If Δ_cⁿf(z)≠0,then either f(z) takes on every complex value infinitely often,or Δ_cⁿf(z) takes on every non-zero complex value infinitely often.

Many authors discussed the problem of existence and growth of complex difference equations,and obtained some intererting results.Using Nevanlinna theory of the value distribution of meromorphic functions,we investigate the expression of meromorphic solutions of a type of system of comples difference equations,and extend some results of solutions of complex difference equations to systems of complex difference equations.

We using the technique of space decompositions,the representation of the Drazin inverse for multiplicative combination aP+bQ+cPQ+dQP+ePQP associated with two idempotents P and Q is obtained under the condition PQP=QPQ.

We focus on symmetric partially balanced incomplete block designs with automorphism group M₁₂ acting on 396 points.To begin with,we proved that there is no non-trivial 2-(396,k,λ) symmetric design admitting M₁₂ be an automorphism group.Then up to isomorphism,we obtained three SPBIB designs with v=396 and k=80.Finally,we gave the complete classification of SPBIB designs with v=396 and automorphism group M₁₂.

We use the Schauder fixed point theorem and the upper and lower solutions method to investigate the existence and uniqueness of positive solutions for a class of fractional p-Laplacian boundary value problems,and establish the iterative sequences for the unique solution.

Let M₂ be the algebra of all 2×2 complex matrices and Φ:M₂→M₂ be a linear map.A 1-1 correspondence between the set of 3×3 complex orthogonal matrices and the set of similarity transformations of M₂ is established,which then is applied to show that Φ preserves the determinant (resp.the spectrum,the peripheral spectrum) of Lie products if and only if there exist a scalar c∈{±1,±i},matrices S,T∈M₂ with T invertible,such that Φ(A)=cTAT^-1+tr(SA) I for all A∈M₂.

The crossing numbers of K_m^-□P_n were successively determined for 4 ≤ m ≤ 6 by Kleš? et al.In this paper,the crossing numbers of K_m^--2K₂ are obtained by the construction method for 4 ≤ m ≤ 12 and m≠10,12.On the basis,the crossing numbers of K_m^-□P_n for 4 ≤ m ≤ 9 and m≠8 can be determined.The method that we use is more general relatively.

Let H be an infinite dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H.T∈B(H) satisfies Property (ω) if σ_a(T)\σ_ea(T)=π₀₀(T),where σ_a(T) and σ_ea(T) denote the approximate point spectrum and essential approximate point spectrum of T respectively,π₀₀(T)={λ∈iso σ(T):0N (T-λI)<∞}.T∈B(H) is said to have the perturbation of Property (ω) if T+K satisfies Property (ω) for all compact operator K∈B(H).We prove the equivalence of the perturbation of Property (ω) for anti-diagonal operator matrix and its square.