In this paper, parabolic Marcinkiewicz integral operators along surfaces on the product domain Rⁿ × R^m (n,m ≥ 2) are introduced. L^p bounds of such operators are obtained under weak conditions on the kernels.  

We give a characterization of structurally stable diffeomorphisms by making use of the notion of L^p-shadowing property. More precisely, we prove that the set of structurally stable diffeomorphisms coincides with the C¹-interior of the set of diffeomorphisms having L^p-shadowing property.    

The main aim of this paper is to study whether the Gromov hyperbolicity is preserved under some transformations on Riemann surfaces (with their Poincaré metrics). We prove that quasiconformal maps between Riemann surfaces preserve hyperbolicity; however, we also show that arbitrary twists along simple closed geodesics do not preserve it, in general.  

We construct a class of new Lie algebras by generalizing the one-variable Lie algebras generated by the quadratic conformal algebras (or corresponding Hamiltonian operators) associated with Poisson algebras and a quasi-derivation found by Xu. These algebras can be viewed as certain twists of Xu’s generalized Hamiltonian Lie algebras. The simplicity of these algebras is completely determined. Moreover, we construct a family of multiplicity-free representations of these Lie algebras and prove their irreducibility.  

A defensive k-alliance in a graph is a set S of vertices with the property that every vertex in S has at least k more neighbors in S than it has outside of S. A defensive k-alliance S is called global if it forms a dominating set. In this paper we study the problem of partitioning the vertex set of a graph into (global) defensive k-alliances. The (global) defensive k-alliance partition number of a graph Γ = (V,E), (ψ_k^gd (Γ)) ψ_k^d(Γ), is defined to be the maximum number of sets in a partition of V such that each set is a (global) defensive k-alliance. We obtain tight bounds on ψ_k^d(Γ) and ψ_k^gd(Γ) in terms of several parameters of the graph including the order, size, maximum and minimum degree, the algebraic connectivity and the isoperimetric number. Moreover, we study the close relationships that exist among partitions of Γ₁ × Γ₂ into (global) defensive (k₁ + k₂)-alliances and partitions of Γ_{em into (global) defensive kem}-alliances, i ∈ {1, 2}.  

In this article, we study certain oscillating multipliers related to Cauchy problem for the wave equations on the Euclidean space and on the torus. We obtain that, at the end point, these operators are bounded from the L^p spaces to certain block spaces.  

In this paper, we consider the relations among L-fuzzy sets, rough sets and n-ary polygroup theory. Some properties of (normal) TL-fuzzy n-ary subpolygroups of an n-ary polygroup are first obtained. Using the concept of L-fuzzy sets, the notion of υ-lower and T-upper L-fuzzy rough approximation operators with respect to an L-fuzzy set is introduced and some related properties are presented. Then a new algebraic structure called (normal) TL-fuzzy rough n-ary polygroup is defined and investigated. Also, the (strong) homomorphism of υ-lower and T-upper L-fuzzy rough approximation operators is studied.  

Let (Ω, Σ) be a measurable space and m0 : Σ → X₀ and m1 : Σ → X₁ be positive vector measures with values in the Banach Köthe function spaces X₀ and X₁. If 0 < α < 1, we define a new vector measure [m₀,m₁]α with values in the Calderón lattice interpolation space X₀^1-α X₁^α and we analyze the space of integrable functions with respect to measure [m₀,m₁]α in order to prove suitable extensions of the classical Stein-Weiss formulas that hold for the complex interpolation of L^p-spaces. Since each p-convex order continuous Köthe function space with weak order unit can be represented as a space of p-integrable functions with respect to a vector measure, we provide in this way a technique to obtain representations of the corresponding complex interpolation spaces. As applications, we provide a Riesz-Thorin theorem for spaces of p-integrable functions with respect to vector measures and a formula for representing the interpolation of the injective tensor product of such spaces.  

The Erdös-Sós conjecture says that a graph G on n vertices and number of edges e(G) > n(k - 1)/2 contains all trees of size k. In this paper we prove a sufficient condition for a graph to contain every tree of size k formulated in terms of the minimum edge degree ξ(G) of a graph G defined as ξ(G) = min{d(u) + d(v) - 2 : uv ∈ E(G)}. More precisely, we show that a connected graph G with maximum degree Δ(G) ≥ k and minimum edge degree ξ(G) ≥ 2k - 4 contains every tree of k edges if d_G(x) + d_G(y) ≥ 2k - 4 for all pairs x, y of nonadjacent neighbors of a vertex u of d_G(u) ≥ k.  

Let F be a family of functions holomorphic on a domain D ⊂ C. Let k ≥ 2 be an integer and let h be a holomorphic function on D, all of whose zeros have multiplicity at most k -1, such that h(z) has no common zeros with any f ∈ F. Assume also that the following two conditions hold for every f ∈ F :  Then F is normal on D.  

In the present paper we propose a new hierarchical management scheme within suitable extensions of 2 -adic numbers, in which the subfields are treated as classes, and the communications are realized by computation on elliptic curves. The security of our system is much stronger than that of the traditional public key cryptographic systems.  

In this paper, the structure of the critical group of the graph K_m× C_n is determined,where m, n ≥ 3.  

Let G = Spec A be an affine K-group scheme and à = {w ∈ A*: dimK A*·w < ∞, dimK w· A* < ∞}. Let <-, -> : A* × Ã → K, <w,w> := tr(ww), be the trace form. We prove that G is linearly reductive if and only if the trace form is non-degenerate on A*.  

Let H be a complex Hilbert space of dimension greater than 2, and B(H) denote the Banach algebra of all bounded linear operators on H. For A, B ∈ B(H), define the binary relation A ≤_* B by A*A = A*B and AA* = AB*. Then (B(H), “≤_*”) is a partially ordered set and the relation “≤_*” is called the star order on B(H). Denote by B_s(H) the set of all self-adjoint operators in B(H). In this paper, we first characterize nonlinear continuous bijective maps on B_s(H) which preserve the star order in both directions. We characterize also additive maps (or linear maps) on B(H) (or nest algebras) which are multiplicative at some invertible operator.  

The notion of quasi-biorthogonal frame wavelets is a generalization of the notion of orthogonal wavelets. A quasi-biorthogonal frame wavelet with the cardinality r consists of r pairs of functions. In this paper we first analyze the local property of the quasi-biorthogonal frame wavelet and show that its each pair of functions generates reconstruction formulas of the corresponding subspaces. Next we show that the lower bound of its cardinalities depends on a pair of dual frame multiresolution analyses deriving it. Finally, we present a split trick and show that any quasi-biorthogonal frame wavelet can be split into a new quasi-biorthogonal frame wavelet with an arbitrarily large cardinality. For generality, we work in the setting of matrix dilations.  

We obtain appropriate sharp bounds on Triebel-Lizorkin spaces for rough oscillatory integrals with polynomial phase. By using these bounds and using an extrapolation argument we obtain some new and previously known results for oscillatory integrals under very weak size conditions on the kernel functions.  

Let k be a positive integer. A Roman k-dominating function on a graph G is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has at least k neighbors with label 2. A set {f₁, f₂, . . . , f_d} of distinct Roman k-dominating functions on G with the property that ∑_i=1^d f_i(v) ≤ 2 for each v ∈ V (G), is called a Roman k-dominating family (of functions) on G. The maximum number of functions in a Roman k-dominating family on G is the Roman k-domatic number of G, denoted by d_kR(G). Note that the Roman 1-domatic number d_1R(G) is the usual Roman domatic number d_R(G). In this paper we initiate the study of the Roman k-domatic number in graphs and we present sharp bounds for d_kR(G). In addition, we determine the Roman k-domatic number of some graphs. Some of our results extend those given by Sheikholeslami and Volkmann in 2010 for the Roman domatic number.  

Concerning the stability problem of functional equations, we introduce a general (m, n)- Cauchy-Jensen functional equation and establish new theorems about the Hyers-Ulam stability of general (m, n)-Cauchy-Jensen additive mappings in C^*-algebras, which generalize the results obtained for Cauchy-Jensen type additive mappings.  

For a double array of blockwise M-dependent random variables {X_mn,m ≥ 1, n ≥ 1}, strong laws of large numbers are established for double sums ∑_i=1^m∑_j=1ⁿ X_ij , m ≥ 1, n ≥ 1. The main results are obtained for (i) random variables {X_mn,m ≥ 1, n ≥ 1} being non-identically distributed but satisfy a condition on the summability condition for the moments and (ii) random variables {X_mn,m ≥ 1, n ≥ 1} being stochastically dominated. The result in Case (i) generalizes the main result of Móricz et al. [J. Theoret. Probab., 21, 660-671 (2008)] from dyadic to arbitrary blocks, whereas the result in Case (ii) extends a result of Gut [Ann. Probab., 6, 469-482 (1978)] to the bockwise M-dependent setting. The sharpness of the results is illustrated by some examples.  

Let (M,g(t)), 0 ≤ t ≤ T, be an n-dimensional closed manifold with nonnegative Ricci curvature, |Rc| ≤ (C/t) for some constant C > 0 and g(t) evolving by the Ricci flow  In this paper, we derive a differential Harnack estimate for positive solutions to parabolic equations of the type u_t = Δu - aulogu - bu on M × (0, T], where a > 0 and b ∈ R.  

Let {X_n; n ≥ 1} be a sequence of independent and identically distributed U[0,1]-distributed random variables. Define the uniform empirical process F_n(t) = n^-12∑_i=1ⁿ(I_{{Xi≤t} - t)}, 0 ≤ t ≤ 1, ‖F_n‖ = sup_0≤t≤1 |F_n(t)|. In this paper, the exact convergence rates of a general law of weighted infinite series of E{‖F_n‖ - εg^s(n)}+ are obtained.  

The main aim of this paper is to prove that for any 0 < p ≤ 2/3 there exists a martingale f ∈ H_p such that Marcinkiewicz-Fejér means of the two-dimensional conjugate Walsh-Fourier series of the martingale f is not uniformly bounded in the space L_p.  

In two real Banach spaces, we shall present two conditions, under one of which each nonexpansive mapping must be an isometry.  

We apply discrete Littlewood-Paley-Stein theory, developed by Han and Lu, to establish Calderón-Zygmund decompositions and interpolation theorems on weighted Hardy spaces H_w^p for w ∈ A_∞ in both the one-parameter and two-parameter cases.  

In this paper we propose a class of new large-update primal-dual interior-point algorithms for P_*(κ) nonlinear complementarity problem (NCP), which are based on a class of kernel functions investigated by Bai et al. in their recent work for linear optimization (LO). The arguments for the algorithms are followed as Peng et al.'s for P_*(κ) complementarity problem based on the self-regular functions [Peng, J., Roos, C., Terlaky, T.: Self-Regularity: A New Paradigm for Primal-Dual Interior- Point Algorithms, Princeton University Press, Princeton, 2002]. It is worth mentioning that since this class of kernel functions includes a class of non-self-regular functions as special case, so our algorithms are different from Peng et al.'s and the corresponding analysis is simpler than theirs. The ultimate goal of the paper is to show that the algorithms based on these functions have favorable polynomial complexity.  

A Fejér-Riesz inequality for the weighted Besov spaces B_p,q is given. Some characterizations of functions in B_p,q in terms of their Taylor coefficients are obtained.  

In this paper, we establish an estimate for the solutions of small-divisor equation of higher order with large variable coefficient. Then by formulating an infinite-dimensional KAM theorem which allows for multiple normal frequencies and unbounded perturbations, we prove that there are many periodic solutions for the coupled KdV equation subject to small Hamiltonian perturbations.  

Let y = y(x) be a function defined by a continued fraction. A lower bound for |Λ| = |β₁y₁ + β₂y₂ + α| is given, where y₁ = y(x₁), y₂ = y(x₂), x₁ and x₂ are positive integers, α, β₁ and β₂ are algebraic irrational numbers.  

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.  

The present paper deals with oblique derivative problems for second order nonlinear equations of mixed type with degenerate hyperbolic curve, which include the Tricomi problem as a special case. Firstly the formulation of the problems for the equations is given, next the representation and estimates of solutions for the above problems are obtained, finally the existence of solutions for the problems is proved by the successive iteration of solutions of the equations and the fixed-point principle. In this paper, we use the complex analytic method, namely the new partial derivative notations, elliptic complex functions in the elliptic domain and hyperbolic complex functions in the hyperbolic domain are introduced, such that the second order equations of mixed type with degenerate curve are reduced to the first order mixed complex equations with singular coefficients, and then the advantage of complex analytic method can be applied.  

In this paper we characterize semirings all of whose p-injective semimodules are injective. We also classify monoids all of whose p-injective acts are injective.  

By using the upper and lower solution method and fixed point theory, we investigate some nonlinear singular second-order differential equations with linear functional boundary conditions. The nonlinear term f(t, u) is nonincreasing with respect to u, and only possesses some integrability. We obtain the existence and uniqueness of the C[0,1] positive solutions as well as the C¹[0, 1] positive solutions.  

For a bipartite graph G on m and n vertices, respectively, in its vertices classes, and for integers s and t such that 2 ≤ s ≤ t, 0 ≤ m - s ≤ n - t, and m + n ≤ 2s + t - 1, we prove that if G has at least mn - (2(m - s) + n - t) edges then it contains a subdivision of the complete bipartite K(_s,t) with s vertices in the m-class and t vertices in the n-class. Furthermore, we characterize the corresponding extremal bipartite graphs with mn - (2(m - s) + n - t + 1) edges for this topological Turan type problem.  

In this paper, some properties of order topology and bi-Scott topology on a poset are obtained. Order-convergence in posets is further studied. Especially, a sufficient and necessary condition for order-convergence to be topological is given for some kind of posets.  

It is shown that the curvature function satisfies a nonlinear evolution equation under the general curve shortening flow and a detailed asymptotic behavior of the closed curves is presented when they contract to a point in finite time.  

In this paper we prove that the boundedness and compactness of Toeplitz operator with a BMO_α¹ symbol on the weighted Bergman space A_α²(B_n) of the unit ball is completely determined by the behavior of its Berezin transform, where α > -1 and n ≥ 1.  

The current paper is devoted to the study of stochastic stability of FitzHugh-Nagumo systems in infinite lattice perturbed by Gaussian white noise. We first study the dynamics of stochastic FitzHugh-Nagumo systems, then prove the existence and uniqueness of their equilibriums, which mix exponentially. Finally, we investigate asymptotic behavior of equilibriums when the size of noise gets to zero.  

Without the linear growth condition, by the use of Lyapunov function, this paper establishes the existence-and-uniqueness theorem of global solutions to a class of neutral stochastic differential equations with unbounded delay, and examines the pathwise stability of this solution with general decay rate. As an application of our results, this paper also considers in detail a two-dimensional unbounded delay neutral stochastic differential equation with polynomial coefficients.  

The first goal of this paper is to establish some properties of the ridge function representation for multivariate polynomials, and the second one is to apply these results to the problem of approximation by neural networks. We find that for continuous functions, the rate of approximation obtained by a neural network with one hidden layer is no slower than that of an algebraic polynomial.  

We study profiles of positive solutions for quasilinear elliptic boundary blow-up problems and Dirichlet problems with the same equation:
-εΔ_pu = f(x, u) in Ω,
where 1 < p < ∞, ε > 0 is a small parameter,
 
where ω > 0, a(x) is a continuous function satisfying 0 < a(x) < 1 for x ∈ Ω, Ω is a bounded smooth domain in R^N. We will see that the profile of a minimal positive boundary blow-up solution of the equation shares some similarities to the profile of a positive minimizer solution of the equation with homogeneous Dirichlet boundary condition.