FIXED/PREASSIGNED-TIME SYNCHRONIZATION OF QUATERNION-VALUED NEURAL NETWORKS INVOLVING DELAYS AND DISCONTINUOUS ACTIVATIONS: A DIRECT APPROACH

The fixed-time synchronization and preassigned-time synchronization are investigated for a class of quaternion-valued neural networks with time-varying delays and discontinuous activation functions. Unlike previous efforts that employed separation analysis and the real-valued control design, based on the quaternion-valued signum function and several related properties, a direct analytical method is proposed here and the quaternion-valued controllers are designed in order to discuss the fixed-time synchronization for the relevant quaternion-valued neural networks. In addition, the preassigned-time synchronization is investigated based on a quaternion-valued control design, where the synchronization time is preassigned and the control gains are finite. Compared with existing results, the direct method without separation developed in this article is beneficial in terms of simplifying theoretical analysis, and the proposed quaternion-valued control schemes are simpler and more effective than the traditional design, which adds four real-valued controllers. Finally, two numerical examples are given in order to support the theoretical results.

GRADIENT ESTIMATES AND ENTROPY FORMULAE FOR WEIGHTED p-HEAT EQUATIONS ON SMOOTH METRIC MEASURE SPACES

Let （M,g, e^-fdv） be a smooth metric measure space. In this paper, we con- sider two nonlinear weighted p-heat equations. Firstly, we derive a Li-Yau type gradient estimates for the positive solutions to the following nonlinear weighted p-heat equationand f is a smooth function on M under the assumptionthat the m-dimensional nonnegative Bakry-Emery Ricci curvature. Secondly, we show an entropy monotonicity formula with nonnegative m-dimensional Bakry-Emery Ricci curva- ture which is a generalization to the results of Kotschwar and Ni [9], Li [7].

Boundedness of Linear Commutators on Herz-Morrey Spaces on Spaces of Homogeneous Type

Under certain weak local size conditions, the boundedness of linear commutators on Herz-Morrey spaces on spaces of homogeneous type are studied. In addition, the boundedness of Hardy-Littlewood maximum commutators on Herz-Morrey spaces on spaces of homogeneous type are obtained.

THE BOUNDEDNESS FOR COMMUTATORS OF ANISOTROPIC CALDERóN-ZYGMUND OPERATORS

Let T be an anisotropic Calderón-Zygmund operator andφ:R^n×[0,∞)→[0,∞)be an anisotropic Musielak-Orlicz function withφ(x,·)being an Orlicz function andφ(·,t)being a Muckenhoupt A∞(A)weight.In this paper,our goal is to study two boundedness theorems for commutators of anisotropic Calderon-Zygmund operators.Precisely,when b∈BMOw(R^n,A)(a proper subspace of anisotropic bounded mean oscillation space BMO(R^n,A)),the commutator[b,T]is bounded from anisotropic weighted Hardy space H^1ω(R^n,A)to weighted Lebesgue space L^1ω(R^n)and when b∈BMO(R^n)(bounded mean oscillation space),the commutator[b,T]is bounded on Musielak-Orlicz space L^φ(R^n),which are extensions of the isotropic setting.

VAR AND CTE BASED OPTIMAL REINSURANCE FROM A REINSURER'S PERSPECTIVE

In this article,we study optimal reinsurance design.By employing the increasing convex functions as the admissible ceded loss functions and the distortion premium principle,we study and obtain the optimal reinsurance treaty by minimizing the VaR(value at risk)of the reinsurer's total risk exposure.When the distortion premium principle is specified to be the expectation premium principle,we also obtain the optimal reinsurance treaty by minimizing the CTE(conditional tail expectation)of the reinsurer's total risk exposure.The present study can be considered as a complement of that of Cai et al.[5].

A Game Theoretic Approach for a Minimal Secure Dominating Set

The secure dominating set(SDS),a variant of the dominating set,is an important combinatorial structure used in wireless networks.In this paper,we apply algorithmic game theory to study the minimum secure dominating set(Min SDS) problem in a multi-agent system.We design a game framework for SDS and show that every Nash equilibrium(NE) is a minimal SDS,which is also a Pareto-optimal solution.We prove that the proposed game is an exact potential game,and thus NE exists,and design a polynomial-time distributed local algorithm which converges to an NE in O(n) rounds of interactions.Extensive experiments are done to test the performance of our algorithm,and some interesting phenomena are witnessed.

Restricted Arc-Connectivity of Harary Digraphs

The h-restricted arc-connectivity of a digraph is an important parameter to measure fault-tolerance of interconnection networks.This paper determines that the h-restricted arc-connectivity of the Harary digraph D=G(n;1,2,.:,k)is equal to n/2 for 2≤h≤n/2,k=2 and n iseven,andλ_(h)(D)=g(k-1)for 2<h≤g and 3≤k≤n/2,where g is the girth of D.As consequences,the super restricted arc-connectedness of Harary digraph D is obtained immediately.In particular,for k=2 and n is even or 3≤k<n/2and n can be divided by k,it can be determined that distinct positive(respectively,negative)Ah-superatoms of D are vertex disjoint for 2≤h≤g.

Molecular Characterization of Anisotropic Musielak–Orlicz Hardy Spaces and Their Applications

Let A be an expansive dilation on Rn and φ ： Hn×[0, ∞）→[0, ∞） an anisotropic Musielak-Orlicz function. Let HAφ（R^n） be the anisotropic Hardy space of Musielak-Orlicz type defined via the grand maximal function. In this article, the authors establish its molecular characterization via the atomic characterization of HAφ（R^n）. The molecules introduced in this article have the vanishing moments up to order s and the range of s in the isotropic case （namely, A ：= 2In×n） coincides with the range of well-known classical molecules and, moreover, even for the isotropic Hardy space HP（R^n） with p∈[（0, 1] （in this case, A ：= 2In×n,φ（x, t） ：= t^p for all x ∈ R^n and t∈[0,∞））, this molecular characterization is also new. As an application, the authors obtain the boundedness of anisotropic Caldeon-Zygmund operators from HA^φ（Hn） to L^φ（R^n） or from HA^φ（Hn） to itself.

THE ENDPOINT ESTIMATE FOR FOURIER INTEGRAL OPERATORS

Let T_(a,φ)be a Fourier integral operator defined by the oscillatory integral T_(a,φ)u(x)=1/(2π)^(n)∫_(R^(n))^e^(iφ(x,ξ))a(x,ξ)(u)(ξ)dξ,where a∈S_(e,δ)^(m)andφ∈Φ^(2),satisfying the strong non-degenerate condition.It is shown that if0<(e)≤1,0≤δ<1 and m≤e^(2)-n/2,thenT_(α,φ)is a bounded operator from L^(∞()R^(n))to BMO(R^(n)).

Necessary and Sufficient Condition for Adjoint Uniqueness of the Graph (∪_(i∈A)P_i)∪(∪_(j∈B)U_j)

For a graph G, let h(G;x) = h(G) and [G]h denote the adjoint polynomial and the adjoint equivalence class of G, respectively. In this paper, a new application of [G]h is given. Making use of [G]h, we give a necessary and sufficient condition for adjoint uniqueness of the graph H such that H = G, where H = ( i∈A Pi) ( j∈B Uj), A ■ A = {1,2,3,5} {2n|n ∈ N,n ≥ 3}, B ■ B = {7,2n|n ∈ N,n ≥ 5} and G = aP1 a0P2 a1P3 a2P5 ( in=3aiP2i).

Anisotropic weak Hardy spaces of Musielak- Orlicz type and their applications

Anisotropy is a common attribute of the nature, which shows different characterizations in different directions of all or part of the physical or chemical properties of an object. The anisotropic property, in mathematics,can be expressed by a fairly general discrete group of dilations where A is a real matrix with all its elgenvalues A satisfy . The aim of this article is to study a general class of anisotropic function spaces, some properties and applications of these spaces. Let be an anisotropic p-growth function with . The purpose of this article is to find an appropriate general space which includes weak Hardy space of Fefferman and Soria, weighted weak Hardy space of Quek and Yang, and anisotropic weak Hardy space of Ding and Lan. For this reason, we introduce the anisotropic weak Hardy space of Musielak-Orlicz type and obtain its atomic characterization. As applications, we further obtain an interpolation theorem adapted to and the boundedness of the anisotropic Calder6n-Zygmund operator from. It is worth mentioning that the superposition principle adapted to the weak Musielak-Orlicz function space, which is an extension of a result of E. M. Stein, M. Taibleson and G. Weiss, plays an important role in the proofs of the atomic decomposition of and the interpolation theorem.

The Skew-commutator Relations and Grobner-Shirshov Bases of Quantum Group of Type c3

In this paper,we give a Grobner-Shirshov basis of quantum group of type C3 by using the Ringel-Hall algebra approach.For this,first we compute all skew-commutator relations between the isoclasses of indecomposable reprersentations of Ringel-Hall algebras of type C3 by using an“inductive”method.Precisely,we do not use the traditional way of computing the skew-commutative relations,that is first compute all Hall polynomials then compute the corresponding skew-commutator relations;contrarily,we compute the“easier”skew-commutator relations which corresponding to those exact sequences with middile term indecomposable or the split exact sequences first,then“inductive”others from these“easier”ones and this in turn gives Hall polynomials as a byproduct.Then we prove that the set of these relations is closed under composition.So they constitutes a minimal Grobner-Shirshov basis of the positive part of quantum group of type C3.Dually,we get a Grobner-Shirshov basis of the negative part of quantum group of type C3.And finally we give a Grobner-Shirshov basis for the whole quantum group of type C3.

On Connected Graphs with Distance Eigenvalue-1 of Multiplicity at Least n-4

Let G_(n)([-1]^(i))denote the set of all connected graphs on n vertices having distance eigenvalue-1 of multiplicity i.By using the distribution of the third largest distance eigenvalue and the second least distance eigenvalue of a connected graph,in this paper we completely characterize the graphs in G_(n)([-1]^(i)),where i=n-1,n-2,n-3 or n-4.

Trees,Unicyclic Graphs and Bicyclic Graphs with Exactly Two Q-main Eigenvalues

The signless Laplacian matrix of a graph G is defined to be the sum of its adjacency matrix and degree diagonal matrix, and its eigenvalues are called Q-eigenvalues of G. A Q-eigenvalue of a graph G is called a Q-main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this work, all trees, unicyclic graphs and bicyclic graphs with exactly two Q-main eigenvalues are determined.

Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces

Let A :=(A_1, A_2) be a pair of expansive dilations and φ : R^n×R^m×[0, ∞) → [0, ∞) an anisotropic product Musielak-Orlicz function. In this article, we introduce the anisotropic product Musielak-Orlicz Hardy space H~φ_A(R^n× R^m) via the anisotropic Lusin-area function and establish its atomic characterization, the g-function characterization, the g_λ~*-function characterization and the discrete wavelet characterization via first giving out an anisotropic product Peetre inequality of Musielak-Orlicz type. Moreover, we prove that finite atomic decomposition norm on a dense subspace of H~φ_A(R^n× R^m) is equivalent to the standard infinite atomic decomposition norm. As an application, we show that, for a given admissible triplet(φ, q, s), if T is a sublinear operator and maps all(φ, q, s)-atoms into uniformly bounded elements of some quasi-Banach spaces B, then T uniquely extends to a bounded sublinear operator from H~φ_A(R^n× R^m) to B. Another application is that we obtain the boundedness of anisotropic product singular integral operators from H~φ_A(R^n× R^m) to L~φ(R^n× R^m)and from H~φ_A(R^n×R^m) to itself, whose kernels are adapted to the action of A. The results of this article essentially extend the existing results for weighted product Hardy spaces on R^n× R^m and are new even for classical product Orlicz-Hardy spaces.

Stochastic dynamics of the transmission of Dengue fever virus bet ween mosquit oes aiul humans

Considering the impact of environmental white noise on the quantity and behavior ofvector of disease,a stochastic differential model describing the transmission of Denguefever between mosquitoes and humans,in this paper,is proposed.By using Lyapunovmethods and Ito's formula,we first prove the existence and uniqueness of a globalpositive solution for this model.Further,some sufficient conditions for the extinction andpersistence in the mean of this stochastic model are obtained by using the techniquesof a series of stochastic inequalities.In addition,we also discuss the existence of aunique stationary distribution which leads to the stochastic persistence of this disease.Finally,several numerical simulations are carried to illustrate the main results of thiscontribution.

Finite Atomic Decomposition Characterization of Variable Anisotropic Hardy Spaces

In 2011,Dekel et al.developed highly geometric Hardy spaces H^(p)(Θ),for the full range 0<p≤1,which are constructed by continuous multi-level ellipsoid coverΘof R^(n) with high anisotropy in the sense that the ellipsoids can change shape rapidly from point to point and from level to level.The authors obtain the finite atomic decomposition characterization of Hardy spaces H^(p)(Θ)and as an application,the authors prove that given an admissible triplet(p,q,l)with 1≤q≤∞,if T is a sublinear operator and uniformly bounded elements of some quasi-Banach space B for maps all(p,q,l)-atoms with q<∞(or all continuous(p,q,l)-atoms with q=∞),then T uniquely extends to a bounded sublinear operator from H^(p)(Θ)to B.These results generalize the known results on the anisotropic Hardy spaces of Bownik et al.

Solitary Wave Solutions of Delayed Coupled Higgs Field Equation

This paper is devoted to the study of the solitary wave solutions for the delayed coupled Higgs field equation{vtt-uxx-αu+βf*u|u|^(2)-2uv-τu(|u|^(2))x=0 vtt+vxx-β(|u|^(x))xx=0.We first establish the existence of solitary wave solutions for the corresponding equation without delay and perturbation by using the Hamiltonian system method.Then we consider the persistence of solitary wave solutions of the delayed coupled Higgs field equation by using the method of dynamical system,especially the geometric singular perturbation theory,invariant manifold theory and Fredholm theory.According to the relationship between solitary wave and homoclinic orbit,the coupled Higgs field equation is transformed into the ordinary differential equations with fast variables by using the variable substitution.It is proved that the equations with perturbation also possess homoclinic orbit,and thus we obtain the existence of solitary wave solutions of the delayed coupled Higgs field equation.

Commutator of Hypersingular Integral with Rough Kernels on Sobolev Spaces

In this paper, the authors give the boundedness of the commutator of hypersingular integral T γ from the homogeneous Sobolev space Lpγ （Rn） to the Lebesgue space Lp（Rn） for 1