The ultimate bound, positively invariant set and globally exponentially attractive set of a disk dynamo system are investigated via constructing a Lyapunov function. Firstly, we derive a four-dimensional ellipsoidal bound for this system. Secondly, the boundedness of the system is applied to the complete chaos synchronization. Finally, the corresponding numerical simulations are performed.

Let G=(V,E) be a connected graph. The direct product (also named Kronecker product, tensor product and cross product) G₁×G₂ has vertex set V(G₁×G₂) = V(G₁)×V(G₂) and edge set E(G₁×G₂) ={(u₁, v₁)(u₂, v₂): u₁u₂ ∈ E (G₁), v₁v₂ ∈ E(G₂)}. We define the n-double of a simple graph G as the graph D_n[G] = G×T_n. The total graph T_n on n vertices is the graph associated to the total relation (where every vertex is adjacent to every vertex). It can be obtained from the complete graph K_n by adding a loop to every vertex. In this paper, we study the (edge)connectivity, super (edge)connectivity of D_n[G].

An edge cut S of a connected graph G is called a restricted edge cut if G-S contains no isolated vertices. The minimum cardinality of all restricted edge cuts is called the restricted edge connectivity λ'(G) of G. A graph G is optimally restricted edge-connected, for short λ'-optimal, if λ'(G)=ξ(G), where ξ(G) is the minimum edge degree of G. A graph is said to be super restricted edge-connected, for short super -λ', if every minimum restricted edge cut isolates an edge. Let G be a connected bipartite graph of order n, minimum degree δ(G)≥ 2 and minimum edge degree ξ(G). In this paper, we prove that: (a) If ξ(G)≥ ((n/2)-2)(1+1/(δ(G)-1)), then G is λ'-optimal; (b) If ξ(G)> ((n/2)-2)(1+1/(δ(G)-1)), then G is super -λ' with the exception of the complete bipartite graph K_2,n-2,n≥ 6, and the Cartesian product graph K_n/4,n/4×K₂ with n≥ 8 and n is divisible by 4. Moreover, we demonstrate that these results are best possible.

Removing the assumption that measurement error variance or reliability ratio is known, we propose the adjust weighted LS estimators (AWLSE) for the estimated parameters of varying-coefficients EV model with replicated observations as well as the variance of model error. Under some mild conditions, we conclude that all of these estimators are strongly consistence and coefficient function estimators are asymptotically normal.

This paper will discuss a Holling-Tanner predator-prey ecological model with diffusion and cross-diffusion. By means of maximum principle and Harnack inequality, a priori estimates are first established. Furthermore, the degree theory is utilized to obtain the existence and non-existence of non-constant positive solutions. The results indicate that non-constant positive solutions are created under some conditions.

Conformal mapping solution of anisotropic semi-infinite crack problem is investigated in this paper. Firstly, the principal theory of anisotropic plate plane problem is introduced simply. Subsequently, elastic problem about semi-infinite cracks problem using the technique of conformal mapping with complex variable function method. The analytic solutions of the SIFs at the crack tip are obtained for anisotropic semi-infinite cracks with intensive loading at arbitrary point of the crack plane. Finally, a special case in point, The analytic solutions of the SIFs at the crack tip are obtained for anisotropic semi-infinite cracks with intensive loading at the crack surface. Thereby, the validity of results is verified. The results show that: the method is simple and practical.

Based on the fixed point theorem of condensing mapping, the impulsive functions not using compacting and extra conditions, the author obtains the existence results of initial value problems for impulsive equations by the method of extending interval on infinite interval and essentially improves the results.

In this paper, we study the credibility estimator of risk premium under the new-type generalized weighted premium principle. Use of a loss function method, the new generalized weighted premium principle is defined as the optimal estimate of the risk under the new-type generalized weighted loss function. Under this type of loss function, we constrain the estimator of risk premium to linear combination of empirical estimate and derive the credibility estimator of risk premium by minimizing the mean square error. We also prove the consistency of the credibility estimator. Finally, the numerical example is given under Esscher principle to verify the results of the paper. In addition, we also compare the credibility estimator under exponential principle with previous research. The results show that the previous estimator is the special case of this paper.

For Dirichlet class D_p in the unit ball B_n of Cⁿ, we get the inclusion relation between Dirichlet class and Hardy space, and then obtain the exact orders of best polynomial approximations and of upper bounds for deviations of Fejér means in the metric of Hardy space H^p.

Total-coloring of graph G is to color the vertices and the edges of the graph properly. To this end, we must use at least Δ+1 colors to color the graph properly. In this paper, we use discharging method to verify that every simple planar graph with maximum degree 8 and without specified subgraph is 9-totally colorable.

In this paper, we investigate the traveling wave for an axonal transport arising from neuroscience and try to understand mathematically how the neural signal transfers through the traveling wave.

In this paper, we considered the existence of solutions to the first boundary value of the nonlinear Sturm-Liouville equation:
where p(x) is a segmentation constant in interval [0,1].

A second-order semi-linear equation's singularly perturbed boundary value problem with nonlinear and infinite boundary value conditions is studied in this paper. By making use of the method of boundary layer functions, the correction functions of the left and right boundary layers, including the exponential and algebra-tic types, are constructed. Thus, the uniformly valid asymptotic solution is derived. Then, based on the theory of the differential inequality, existence of solutions of the boundary value problem is proved and the error estimation between the reduced and the exact solutions is given. By comparing with the numerical integration solution, a typical example is deduced for verifying the correctness of the theoretical result.

A class of singular perturbation of three-pointed value problem of high-order nonlinear differential equation is examined in this paper.Relative boundary conditions are supplemented according to the different order power of small parameter. Using the method of boundary functions, the asymptotic solution of this problem is shown and proved to be uniformly effective.The existence and uniqueness of the solution for the system has also been derived.Numerical result is presented, which supports the theoretical result.

In this paper, we introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of a system of generalized variational inequalities and the set of common fixed points of a finite family of strictly pseudo-contractive mappings which solves some variational inequality in a real Banach space. Our results improve and extend the corresponding results announced by many others.

A radial basis function collocation method combined with difference is developed, which based on the collocation method and radial basis function interpolation for Two-dimensional water movement equation in unsaturated soil. This method first uses difference method to deal with the nonlinear term, and then solves the equation by using implicit scheme of the equation. Moreover, the existence and uniqueness of the solution to the method is established. According to the numerical computing example results, it indicates that the new radial basis function has the best simulated precision than MQ function and Gauss function. Furthermore, as compared with other methods, this algorithm is practical and effective.

C(m,2)is a circulant graph obtained from C_m (v₁v₂…v_mv₁) by adding edges v_iv_i+2 (i=1,…,m,i+2 (mod m)). A single(double) suspension of C(m,2) is the graph which obtained from C(m,2) by adding one vertex x (two vertices x and y) and the edges xv (the edges xv, yv) and each v ∈ V(G). In this paper, we have proved that the crossing number of one and two suspensions of C(2m-1,2) are m,2m,respectively. And we extend the earlier results to the Cartesian products of C(m,2)×P_n, showing that the crossing number of cartesian product of P_n with circulant graph C(9,2) is 10n.

In this paper, we are investigated the existence and successive iteration of positive solutions for impulsive boundary value problems with p-Laplacian on time scales. On the same time, we give the example to show the main result.

This paper considers a special Max-Min problem that is the distributionally robust optimization problem which is different from stochastic programming and robust optimization. In this kind of problem, the uncertain variable's probability distribution often can't be accurately acquired but with some known information about its probability, such as the first-order, second-order and support set information. Then the distributionally robust optimization problem is to find the worst-case solution under all possible distributions. In general to find the solution for this problem is NP hard. In this paper, we suppose a special case where the decision-maker gets across some parts of information about the uncertain distributions, for example the first-order, support set and affine first-order information. By applying the duality of the semi-infinite programming, the distributionally robust optimization problem can be equivalently reformulated as a linear optimization problem, and then it can be solved by some well-established linear programming approach. To verify the effectiveness of the method, we discuss an applications to portfolio management problem with transaction costs.

The exact marginal posterior distributions of the parameters and the predictive distribution of the future responses in the multivariate linear regression model with singular matrix variate elliptically contoured distributed errors under noninformative prior are derived, with respect to the Hausdorff measure. Similar results are obtained under singular matrix variate normal distribution and normal-inverse Wishart conjugate prior. The marginal posterior distribution of the parameter matrix of regression coefficients and the predictive distribution of the future responses are double singular matricvariate t distributions, both in the noninformative and conjugate prior cases. The results give inference robustness with respect to departures from the underlying distribution assumption in the direction of elliptically contoured distributions under noninformative prior.

The linear complexity and the k-error linear complexity of a sequence have been used as important measures of keystream sequence strength. By studying linear complexity of binary sequences with period 2ⁿ, it is proposed that the computing of k-error linear complexity should be converted to finding error sequences with minimal Hamming weight. Based on Games-Chan algorithm, 3-error linear complexity distribution of 2ⁿ-periodic binary sequences with linear complexity 2ⁿ is discussed. For k=3,4, the complete counting functions on the k-error linear complexity of 2ⁿ-periodic binary sequences with linear complexity 2ⁿ are derived. Based on those results, the counting functions for the number of all 2ⁿ-periodic binary sequences with given 3-error linear complexity can be obtained. Generally, the complete counting functions on the k-error linear complexity of 2ⁿ-periodic binary sequences with linear complexity 2ⁿ-m can be obtained using a similar approach.

In this paper we present some new existence results for singular boundary value problems for the impulsive one-dimensional p-Laplacian. Our nonlinearity may be singular in its dependent variable.

In this paper, the relationship between the number of square-integrable solutions of the linear Hamiltonian systems with real-values spectral parameter λ and point spectrum of linear Hamiltonian operators with arbitrary deficiency index d is discussed. If for all λ in an open interval I, there always exist d linearly independent square-integrable solutions, then the point spectrum of each self-adjoint extension of the minimal operator is nowhere dense in I.

This paper deals with the linear and nonliner scalarization of generalized cone-convex maps under the variable domination structures. Firstly, in the variable orderings topology vector spaces, it is shown that the nearly-cone-convexity of the vector-valued maps can be characterized by means of the extreme directions of the positive polar cone. Secondly, a nonlinear scalariztion funciton is introduced for a variable domination structure. This nonlinear function is then applied to characterize the nearly-cone-convex vector-valued mappings.

In this paper ,based on WZ theory, a new proof for a truncating binomial series (the identity for a_m,n^(K) (x), where n ≥ 2, 0≤ x ≤ 1) in one paper of Peter Paule and Carsten Schneider was given, at the same time, it was pointed out that another truncating binomial series(the identity for a_m,1^(K) (x), where n = 1, 0 ≤ x < 1) in the paper mentioned above is wrong, and a corresponding correct formula was given, where, for K, m, n ∈ N, K ≥ m + 1 ≥ n, a_m,n^(K) (x) =.

A family of non-self maps satisfying a Lipschitz type condition with variable coefficients in a complete metrically convex space was considered and a convergent sequence {x_n} was constructed by using the given boundary condition and the given mappings, and then the fact that the unique limit of {x_n} is the unique common fixed point of the given mappings was proved. Finally, more general results were given. Our main theorems generalize and improve many common fixed point theorems of contractive type mappings.

Let (G,u,v) be a double-rooted connected graph with roots u and v, and let G+e be the graph obtained by joining the roots u and v of (G,u,v) with an edge e. For the case that u and v are both 2-valent, Gross showed a relationship between the genus distribution of G+e and the partitioned genus distribution of (G,u,v). In this paper, we generalize one of the two roots to arbitrarily high valence, and derive the genus distribution of G+e from the partitioned genus distributions of (G,u,v).

In general, linear network code is based on the finite fields. The aim of this paper is to extend the linear network code based on finite fields to that based on general R-modules in theory, where R is a given ring. A class of important R-modules is Z-modules. The simplest example is the integer ring Z. Another large class of important R-modules is general number fields or algebraic number fields which contain a given subring R. For example, integer ring Z rational number field Q real number field R complex number field C. After generalization of linear network codes, the network can transmit message (signal) which takes value in real or complex number fields, hence the applied field is wider.

By means of the theory of time scales, Riccati transformation technique, the averaging functions technique and inequalities, the oscillation behavior for the second order half-linear neutral delay dynamic equations with damping on time scales is studied. Some sufficient conditions are obtained for oscillation of this equation. The results in this paper extend and improve some known results.

In this paper, we propose statistical analysis of Weibull distribution with Type-I and Type-Ⅱ censored samples of the step-stress accelerated life testing under CE model. The likelihood function of the scale parameter with normal stress level is obtained using accelerated coefficient and acceleration equations to convert scale parameters under various accelerated stress levels into scale parameters under normal stress level. Scale parameters and accelerated coefficient are chosen as the conjugate prior and noninformative prior when choosing priors of parameters. Beta distribution and Gamma distribution are chosen as the priority when shape parameter m<1 and m>1, respectively. Bayesian estimation of this model under square loss is proposed using Gibbs sampling and Slice sampling. Finally, statistical results show that the Bayesian estimation is efficient through Monte Carlo simulation.

This paper deals with a class of Non-Newton fluids

with the initial and boundary condition
.
By the iterative method, the blow-up criterion for the local existence result on strong solutions for this model is discussed. and it is proved that if T_* is the maximal existence time of the strong solution (ρ, u) and T_*<T₁, then .

Let G be a finite graph and H be an subgraph of G. If V(H)=V(G) then the subgraph H is called a spanning subgraph of G. A λ-fold F-factor of G, denoted by S_λ(F,G), is a spanning subgraph of G, which can be partitioned into copies of F (called F-blocks), such that each vertex of V(G) appears exactly in λ F-blocks. A large set of λ-fold F-factors of G, denoted by LS_λ(F,G), is a partition {B_i}_i of all subgraphs of G isomorphic to F, such that each B_i forms a S_λ(F,G). For λ=1, the index 1 is often omitted. In this paper, it has been proved that there exists an LS(K_1,3,K_v,v,v) for v≡4 mod 24.

In this paper, we study the almost periodic solutions and the global exponential stability of the almost periodic solutions for a class of cellular neural networks(CNN_S) with S-type Distributed Delays. Some sufficient conditions for the almost periodic solutions and the global exponential stability of the almost periodic solutions are established by employing exponential dichotomy and Schauder fixed point theorem and building Lyapunov functions. Moreover, an example is given to illustrate the effectiveness of our results.

As one kind of complex variable function method to solve the plane problem in fracture mechanics, conformal transformation method is very practical and effective, which transforms the area in physical plane to the unit circle inside (outside) or the upper half-plane (lower half-plane) in mathematical plane by a conformal transformation. In this article, by using the complex variable function method and proposing a new conformal mapping, the fracture problem of a finite Griffith crack in a point group 6 of one-dimensional hexagonal quasicrystals strip is studied under anti-plane shear stress load in the crack surface. The analytic solution of the stress intensity factors at the crack tip is obtained.When the height of the strip tends to infinite, the present results can be degenerated to the solutions of a Griffith crack in infinite point group 6 of one-dimensional hexagonal quasicrystals.

In this paper, we discuss the existence of extreme solutions for the impulsive integro-differential equation of mixed type with impulsive integral conditions. The main tool is the method of upper and lower solutions coupled with the monotone iterative technique.

For self-similar set E_λ=E_λ/5∪(E_λ/5+λ/5)∪(E_λ/5+4/5), this paper discusses the dimension, the strong separation condition and Lipschitz equivalence of E_λ as λ tends to 3. The paper also studies the geometric structure of the parameter set {λ:E_λ doesn't satisfy the strong separation condition}.

In this paper, we concerned with the existence of positive solutions for nonlinear boundary value problem.The main results in this paper proof the existence of double positive solutions for the boundary value problem.The proofs are using a three functionals fixed point theorem in a cone. For illustrating the theoretical analysis, we also give an examples.

The incompressible viscous flow problem between two concentric rotating spheres is a simple model for the geophysical flow around the earth. Firstly, we obtain rational expresstion of eigenvalue and eigenfunction of Stokes problem by using rational polynomial form of spherical bessel function in this case. Secondly, we obtain spectral approximate solutions of Navier-Stokes equation, and if we take three model as bases functions in spectral approximate, a quasi-lorenz system is obtained. In this paper, the bifurcation phenomena and attractors are discussed.

In this paper, we emphasize on the nonconvex relaxation of the semidefinite matrix rank minimization problem and the involved properties. Firstly, we apply a nonconvex relaxation, the Schatten p-norm (0<p<1) semidefinite program, for solving the desired rank minimization problem. Next, by defining the semidefinite restricted isometry/orthogonal constant, we propose a sufficient condition for the uniqueness of the solution to (P). Finally, with the semidefinite restricted isometry property (semi-RIP), we give a sufficient condition under which the Schatten p-norm relaxation and the underlying matrix rank minimization share the unique common solution. In particular, for any 0<p<1, we obtain a uniform exact recovery condition.

In this paper, using the control technology of the fractional maximal function, we obtain boundedness of the maximal multilinear Bochner-Riesz operators on the Morrey Spaces. Also we establish it's boundedness from M^{p₁,1+p₁(λ-β/n)} spaces to BMO^λ,p₂ spaces, where, -1/p₂≤λ<1/n, 1/p₁-β/n=1/p₂.