By combining the variational iteration method and homotopy perturbation method, a new method named variational homotopy perturbation iteration method is proposed for solving an inverse problem of determining an unknown parameter in a linear parabolic equation.Using this method, a rapid convergent sequence tending to the exact solution of the inverse problem can be obtained. To show the effciency and the reliability of the proposed method, some interesting examples are presented.

A graph is 1-embedded on a surface if it can be drawn on the surface so that each edge is crossed by at most one other edge. χ'(G) and Δ(G) denote the chromatic index and the maximum degree of G, respectively. Let G be 1-embedded on a surface of Euler characteristic χ(Σ)≥0. The paper shows that Δ(G)=χ'(G) if Δ(G)≥8 and G contains no 4-cycles or Δ(G)≥7 and g(G)≥4, where, g(G) denotes the length of the shortest cycle in G.

The (strong) oscillation problems for a class of quasilinear parabolic systems with impulse perturbation and delay effect are investigated. By using a new technique of treating quasilinear diffusion term and impulsive delay differential inequalities, some new sufficient conditions are established for the (strong) oscillation of all solutions of such systems under Neumann boundary value condition. The obtained results fully indicate that the oscillations of the systems are caused by impulse perturbation and delay effect.

The problem with which we shall be concerned relates to the following situation: Two agents jointly process a set of jobs received from customers. Each agent has a single machine and each job will be processed by one of the two machines without interruption. Both agents will get their profits and pay processing costs after finishing the jobs assigned to them. We will find an optimal bipartition of all jobs such that the multiplication of the two agents' net profits is maximized. We discuss three models with different classical scheduling objectives as the processing costs. For each model, we show the complexity, analyze the optimality and develop dynamic programming algorithm.

In this paper, we consider a class of p-Laplace equation with singular potential and its perturbation in
-Δ_pu-(μ|u|^p-2u/|x|^p)=λ((u^p*(t)-2)/(|x|^t))u+βf(x,u), x∈R^N, u∈D₀¹, p(R^N),
where N ≥ 3, D₀^1,p(R^N) is C₀^∞(R^N) closure, Δ_pu=-div(|∇u|^p-2∇u), 2 < p < N, 0 ≤ μ < μ=(N-p)^p/p^p, λ > 0, 0 ≤ t < p, p^*(t)=p(N-t)/(N-p) is called Hardy-Sobolev critical parameter. By using concentration-compactness principle and minimax procedure,infinitely many solutions are obtain.

D₃□P_n is the cartesian product of a dipole graph D₃ and a path P_n. We derive a recursion for the genus distribution of the graph D₃□P_n, with the aid of skillfully introducing a new kind of edges-adding operations and combining the partial genus distribution of the graph.

Based on an adaptive conjugacy condition and CG_DESCENT conjugate gradient method, a modified THREECG conjugate gradient method is proposed for solving unconstrained problems, in which the sufficient descent condition is satisfied independent of the search condition used and the convexity of the objective functions at each iteration. Under mild condition, we show the proposed method converges globally. Numerical results illustrate that our method can efficiently solve the test problems and therefore is promising.

In this paper, we focus on a class of regular higher-order differential operators with finite transmission conditions and coupled boundary conditions. The eigenvalue problems and the completeness of eigenfunctions are investigated. First, follwing a combination of a new inner product relating the transmission conditions, we transform the eigenvalue problems of the original operator into those of the symmetric differential operator in the new Hilbert space. Then, applying piece-wise fundamental solutions of the differential equation, we clarify that eigenvalues satisfying the eigenequation coincide with the zeros of the entire function. Moreover, on this problem, the existence of countably many eigenvalues is proved as well as the necessary and sufficient conditions of eigenvalues are established. Furthmore, by the spectral theorem of compact operator and properties of inverse operator, Green's function is constructed and the completeness of eigenfunctions is demostrated.

This paper discusses the (T,r,Q) inventory replenishment strategy for the perishable items with fixed-lifetime during an infinite period. On the condition that a random demand as well as stock out are permitted, after setting a maximum shortages, we formulate a model to determine the optimal replenishment policy for the perishable items. Basing on the idea of multivariate extremes and implicit function theorem, the model is processed with the shortage being backlogge, then both the optimal batch quantities and order point are obtained. In a numerical study, the optimal batch and the optimal stockout quantities are obtained by Matlab software, besides, sensitivity analysis for the parameters are also given in the example.

It is pointed out that the main results in 《The Optimization Condition of Henig Proper Efficient Solution for Set-valued Optimization Problem》 are special cases in 《On Strict Efficiency in Set-valued Optimization with Nearly Cone-subconvexlikeness》.

A multi-group model of a vector-borne disease with discrete delay in the vector and the host is investigated. It is shown that if the reproduction number of the model R₀ < 1, the unique disease-free equilibrium is shown to be globally asymptotically stable. It is proved that the strain i with the largest reproduction number is locally asymptotical stabile. Under certain conditions, it is shown In this case, the strain i dominance equilibrium is globally asymptotically stable. In this case,the competitive exclusion principle is valid.

This paper establishes the oscillation criteria for second-order dynamic equations with p-Laplacian, damping and oscillatory coefficients on time scales of the form
(r(t)φα(x^Δ(t)))^Δ+p(t)φα(x^Δ(t))+q(t)f(x^σ(t))=0.
Two examples are included to show the significance of the results.

Guo (Discrete Appl. Math. 95 (1999) 273-277) proposed the concept of outpath. An outpath of a vertex x (an arc xy, respectively) in a digraph is a path starting at x (an arc xy, respectively) such that x dominates the endvertex of the path only if the endvertex also dominates x. A k-outpath is an outpath of length k. In this article, the following results are proved: Let D be an almost regular c-partite tournament. If each partite set contains at least two vertices, then every arc of D has a (k-1)- or k-outpath for each k∈{3,4,...,|V(D)|-1}. Furthermore, if D is an almost regular c-partite (c≥8) tournament with the partite sets V₁,V₂,...,V_c such that |V₁|=|V₂|=...=|V_c|, then every arc of D is contained in a k- or (k+1)-cycle for each k∈{3,4,..., |V(D)|-1}.

In this paper, a state-dependent impulsive model is proposed for the algae control, which involves state-dependent impulse and Holling type I functional response function. We investigate the existence and uniqueness of order 1 periodic solution of the system by using the qualitative theory of ordinary differential equation and the properties of the Lambert W function and Poincare map. Finally, numerical simulations are carried out to illustrate the theoretical results.

In this paper, using the control technology of the fractional function, we obtain boundedness of a sort of multilinear operators generalized by Bochner-Riesz on the Morrey Spaces; also we establish it's continous injections from Morrey spaces to Lipschtz and from Morrey spaces to BMO spaces.

In this paper, we generalized the alternating Ramanujan's circular summation formula and give some applications. We also generalized the results of Boon et al and obtain some new identities of theta functions.

By connecting each vertex of a graph G to each vertex of a graph H, join graph, denoted by G ∨ H, is obtained. In this paper, we have proved that the crossing number of S₅ ∨ C_n is Z(6,n)+4?n/2?+3, where Z(m,n)=?m/2??m-1/2??n/2??n-1/2? and both m and n are nonnegative integers.

Some characterizations on ε-strictly efficient elements for binary set-valued functions with saddle points are considered in Hausdorff locally convex linear topological spaces. Under the hypothesis of near cone-subconvexlikeness (near cone-subconcavelikeness), by applying separation theorem for convex sets, necessary optimality conditions of loose saddle points on ε-strictly efficient elements for binary set-valued functions are established. Sufficient optimality conditions are also obtained with help of scalarization theorem. In particular, for ε=0 the sufficient and necessary optimality conditions of loose saddle points on strictly efficient elements for binary set-valued functions are gained.

In this paper, we study the traceability of locally in(out)-semicomplete digraphs and extended locally in(out)-semicomplete digraphs by multi-insertion technique. First, we show that for a connected locally in-semicomplete digraph D of order n, if for every dominated pair of non-adjacent vertices {x, y}, either d(x)≥n-1 and d(y)≥n-2, or d(x)≥n-2 and d(y)≥n-1, then D is traceable. At the same time, we show for a connected locally in-semicomplete digraph D of order n, if for every dominated pair of non-adjacent vertices {x, y}, min{d⁺(x)+d^-(y), d^-(x)+d⁺(y)} ≥n-1, then D is traceable. Second, we show that for a connected extended locally in-semicomplete digraph D of order n, if D is satisfied with the following two conditions: (1) for every dominated pair of non-adjacent vertices {u, v}, d(u)≥n-1, d(v)≥n-1; (2) for every dominated pair of non-adjacent vertices {x, y}, either d(x)≥n-1 and d(y)≥n-2 or d(x)≥n-2 and d(y)≥n-1, then D is traceable. Finally, by the properties of reversing digraphs, we generalize these three results to locally out-semicomplete digraphs and extended locally out-semicomplete digraphs.

In this paper, the authors propose and consider a reaction-diffusion model with stage structure and nonlocal spatial effect, which models the interaction between the two species, the adult members of which are in competition. In the distributed delay type model, the maturation time of each species obeys some probability distribution. Only the adult members are involved competition and the immature members are in the absence of competition. By using the method of upper-lower solutions due to Redlinger, dynamical behaviors of model are studied. In addition, sharp local stability and global stability criteria are established for the coexistence equilibrium as well as the extinction equilibrium. Effects of stage structure with distributed types of delays and nonlocal spatial regulation on behaviors of species are discussed, which indicates the reaction-diffusion model will not appear Turing instability.

In this short note, we show a uniqueness result of the energy solutions for the Cauchy problem of Schrödinger flow in the whole space R² provided there is a smooth solution in the energy class.

In this paper, we discuss the problems to form the Riesz basis in L²[0,π] from the solutions of initial value problem and boundary problem of integral-differential equation. The necessary and sufficient condition for initial solutions forming Riesz basis is obtained, and by selecting the functions from the root subsets of multiple eigen-values, the Riesz basis from the eigen- and associated functions of the boundary value problem is obtained also.

By using Mawhin's continuation theorem of coincidence degree theory and differential inequality skills, we establish the existence of eight positive periodic solutions for delay Lotka-Volterra competition patch systems with harvesting terms. An example is given to show the effectiveness of the obtained results.

The application of fractional model can be more accurately to describe the mechanical and physical behavior of a complex system. With the fractional calculus used successfully in many areas of science and engineering, the traditional theories and methods of analytical mechanics need to be extended to the systems with fractional calculus. Transformation is a vital tool in the study of analytical mechanics. This paper focuses on studying the theory of transformation for a fractional mechanical system. Based upon the definition of Caputo fractional derivative, the Lagrangian and the Hamiltonian of a mechanical system is defined, and the fractional Hamilton principle is established under the exchange relationship of Hölder, and the fractional Hamilton canonical equations are deduced by means of variational calculation from the fractional Hamilton principle. The theory of canonical transformation for the fractional systems is established, and four basic forms of fractional canonical transformation are given, and some examples are given to illustrate the application of the results and the role played by a generating function in the canonical transformation.

In this paper, we firstly construct a family of error-correcting pooling designs with the incidence matrix of two types of subspaces of singular unitary space over finite fields, and exhibit their disjunct properties. then we show that the new construction gives better ratio of efficiency than the former ones under conditions. At last, we obtain how the parameters influence the test efficiency by analyzing the relationship between the related parameters and the numbers of the new design's columns and rows. It is beneficial for us to choose an applicable pooling design along with our need.

In this paper, we introduce IF approximating spaces which is a type of IF topological spaces associate with IF relations and obtain some decision conditions that every IF topological space is an IF approximating space.

A generalized interior characterization of convex cone is given based on the nonemptiness of the quasi interior and relative algebraic interior for sets, consistency of the quasi interior and relative algebraic interior is proved for a convex cone, and further nonconvex separation theorems are established via quasi interior and relative algebraic interior for a convex cone. Moreover, some concrete examples are also presented to illustrate the main results.

A two species obligate Lotka-Volterra mutualism model with feedback controls is studied in this paper. By constructing suitable Lyapunov functions, sufficient condition which guarantee the global attractivity of the positive equilibrium and boundary equilibrium are obtained, respectively. Our results show that if the system without feedback controls admits a unique positive equilibrium, then restrict the feedback control variables to some range, the stability property of the positive equilibrium still retains, however, if the feedback control variables are enough large, then the boundary equilibrium become globally stable, which means the extinction of second species; If the system without feedback controls admits a unique globally stable boundary equilibrium, then feedback control variables can only influence the position of the equilibrium, and have no influence on the stability of the equilibrium.

In the B-F reserve model, the estimations of accident year mean are very critical in reserve model. However, the traditional approach assume that there are some prior estimates for accident year means which are determined by the actuary based on past experience, which has great subjectivity. If the prior estimates are choose correctly, we will get the an accurate estimate of the reserve. On the contrary, if the a prior estimate are selected incorrectly, there must be bring large errors in the reserve estimates. This paper presents an improved stochastic B-F reserve model. The ideas from credibility theory are used and the credibility estimates of accident year means are derived. In addition, the empirical Bayes approach are investigated and the estimations of structural parameters are given. Furthermore, we get empirical Bayes estimates of reserves. We use numerical simulation to verify the mean square error for Empirical Bayes estimates, and the conclusions show that this empirical Bayes estimates are valid in stochastic B-F model. Finally, practical examples of insurance company are given and the differences are compared among our empirical Bayes estimates obtained, chain ladder estimates and traditional BF estimates.

Due to the availability to describe the relationship between financial market volatility and return, GARCH-M model has been widely studied since it was proposed. The majority of traditional methods used to estimate GARCH-M type models are based on quasi maximum likelihood estimation. However, these approaches normally require strong moment conditions of the innovations, which may not be satisfied by the practical data. Hence, it makes sense to investigate how to estimate GARCH-M models under weaker moment conditions. In this article, a special parametric GARCH-M type model is considered. Different from the traditional GARCH-M model, the considered one is of conditional variance driven by past observable time series. Local estimation for model parameters is given with the basis of the quasi-maximum exponential likelihood estimation approach. Under weak moment conditions, asymptotic normality of the estimation is proved. Simulation studies demonstrate that the estimation performs well. Empirical study implies that the estimation is of certain practical value.

The oscillation for certain second-order nonlinear neutral variable delay Emden-Fowler functional dynamic equations with damping on time scales is discussed. By using the calculus theory on time scales and the generalized Riccati transformation and the inequality technique, we establish some new oscillation criteria for the equations. Our results extend and improve some known results, but also unify the oscillation of second-order Emden-Fowler damped differential equations and difference equations. Some examples are given to illustrate the main results of this article.

Let {X(t), t≥0} be a standard (zero-mean, unit-variance) stationary strongly dependent Gaussian process with correlation function r(t) and continuous sample paths. Under some conditions related to the correlation function r(t), we proved that the upcrossing point processes formed by the numbers of {X(t), t≥0} upcronssing and ε-upcronssing level u converge weakly to Cox-process, as u→∞.

In this paper, we first introduce the Levitin-Polyak well-posedness of variational inequality with variational inequality constraint. We also establish some metric characterizations of Levitin-Polyak well-posedness for variational inequality with variational inequality constraint. We give some relations between variational inequality with variational inequality constraint and mixed variational inequality. Finally, we prove that the Levitin-Polyak well-posedness of variational inequality with variational inequality constraint is equivalent to the existence and uniqueness of solutions.

In this paper, we use a pair of surjective multi-valued mappings satisfying quasi-contractive condition with some variable coefficients or φ-quasi-contractive condition to construct a convergent sequence, and prove that the unique limit of the sequence is the common fixed point of the two mappings and give sufficient conditions that the pair of mappings have an unique common fixed point, and then give several particular conclusions. The obtained results in this paper generalize and improve some known results.

In this paper, we study the generalized finite volume methods (GFVM), prove their structure-preserving property, establish the relationships with other structure-preserving algorithms, and give the variational property such that GFVM can be viewed as the finite element method (FEM) and mixed FEM. Finally, numerical experiment shows the GFVM is effective.

Let G:H be the two points join graph of two graphs G and H. We study the Tutte polynomial of G:H and obtain an splitting formula for T(G:H; x, y). Using the obtained result, we obtain exact analytical expressions for the number of spanning trees of two network models and the Tutte polynomial of the generalized book graph. Furthermore, we study the regular polygon chain graph and present a recursive formula for its Tutte polynomial.

Quanto options is a very important financial derivative, and its pricing model is a two-dimensional Black-Scholes equation with a mixed derivative term. The research of the equation's numerical solution is value both in theory and practice. This paper gives a new ADI difference method based on the Craig-Sneyd splitting technique (C-S ADI) for solving the quanto options pricing model. This C-S ADI method first splits the two-dimensional B-S equation into two separate one-dimensional problems and one two-dimensional problem with a mixed term. And then solve them by semi-implicit difference scheme and explicit difference scheme per time-step respectively to get the solution. The C-S ADI method has the following advantages:parallelism, unconditional stability, convergency, and the calculation of second-order in space, first-order in time.
The numerical experiments show this method is very efficient and gives better accuracy than the existent Crank-Nicolson difference scheme and the ADI method based on the Douglas-Rachford splitting technique (D-R ADI). What's more, as the natural parallel property of the C-S ADI method, it is easy to realize parallel computing, and the saving calculation time of it is nearly 1/5 of the serial Crank-Nicolson scheme's. Thus the method given by this paper can be used to solve the quanto options pricing problems effectively.

This paper presents the concept of trees including bipartite graceful tree, strongly graceful tree, strongly odd-graceful tree, edge symmetric tree and dual labelling and so on. We define the (2m+1, 1)-p-trees and prove that the tree is bipartite strongly graceful and bipartite strongly odd-graceful. Moreover, we validate the graceful labelling of (2m+1, 1)-p-tree is dual labelling. Finally, we show that the edge symmetric tree of (2m+1, 1)-p-tree is still bipartite graceful tree..

This paper focuses upon iterative algorithms for the nonzero symmetric solution of discrete-time algebraic Riccati equation arising from low gain feedback design and time delay control system. Based on the modified conjugate gradient method, the (inexact) Newton-MCG algorithm and T-MCG algorithm are proposed. The (inexact) Newton-MCG algorithm has no other limits to coefficient matrices except for the existence of nonzero symmetric solution, while T-MCG algorithm demands the existence of invertible symmetric solution. Particularly, the solution derived from T-MCG algorithm is positive definite under suitable conditions such as controllabilities of relative coefficient matrices, but the solution from (inexact) Newton-MCG algorithm is not necessarily. Numerical results illustrate the efficiency of the above algorithms.

In recent years, the cooperation system among the multi populations is widely studied. When the cooperative species have non-overlapping generations, the discrete model described by difference equation is better than the continuous model and can better reflect the relationship between biological systems. This paper has studied the existence of positive periodic solution for a class of discrete Lotka-Volterra cooperative systems with harvest terms. By means of a continuous theorem of Brouwer coincidence degree theory, through the analysis of deformation and use some estimation of inequality technique, we construct four bounded open sets. At the same time, by using the homotopy invariance, we have calculated to explain that the Brouwer degree of the four open bounded operator is not equal to zero. Last we have got the sufficient and brief conditions to the system's four positive periodic solutions.