In this paper, we establish a new isoperimetric inequality for the mixture of convex and star bodies. Our result in special case yields the classical isoperimetric inequality, and which is an improvement and modification of a previous result.

This paper is to investigate the connectedness properties of quasi-metric space, and show that connectedness properties of quasi-metric space are preserved under quasimöbius maps.

In this paper, we will study four-dimensional shrinking gradient Ricci solitons with half positive isotropy curvature (half-PIC). We will show that, the bound of the traceless Ricci curvature $\overset{\circ }{\mathop{Ric}}\, $ will control the bound of the self-dual part of the Weyl tensor W₊ or the antiself-dual part W_-. In particular, we will give a new and simpler proof of the following theorem:Any oriented four-dimensional Einstein manifold with half-PIC must be half conformally flat, and therefore isometric to S⁴ or CP² with standard metric. A more general result on shrinking gradient Ricci solitons was gave.

Let B(H) be the set of all bounded linear operators on a complex Hilbert space H. Using the block operator technique, some necessary and suffcient conditions for the existence of nonnegative {1, 3}-, {1, 4}-, {1, 3, 4}-inverses for an operator A∈ B(H) with closed range is given in this paper, and these sets are completely descibed. Moreover, it is showed that the existence of nonnegative {1, 3}-, {1, 4}-inverse of an operator A is equivalent to existence of its nonnegative {1, 2, 3}-, {1, 2, 4}-inverse, respectively.

In this paper, we obtain an upper bound of the number of zeros of Abelian integral for a kind of Hamiltonian systems. The Abelian integral has k + 2 generators which satisfy two different Picard-Fuchs equations. Finally, we present two examples to illustrate an application of the theoretical result.

In this paper, we consider the blow-up phenomenon to a type of Newtonian filtration equation with variable source subject to homogeneous Neumann boundary condition. We give two methods to determine the lower bound for blow-up time of solution in $\Omega \in {{\mathbb{R}}^{3}}$ if the solutions blow up by energy estimation method and diferential inequality technique. Moreover, the effectiveness of these methods are also discussed.

We study the oscillatory behavior of a class of second-order generalized EmdenFowler-type nonlinear delay functional differential equations in this paper. By using a couple generalized Riccati transformation and some necessary analytic techniques, we establish some new oscillation criteria for the equations under both the cases canonical form and noncanonical form, which deal with some cases not covered by existing results in the literature. Three examples are given to illustrate the main results of this article.

In this paper, we discuss the existence of solutions for nonlinear boundary problem of first-order impulsive differential inclusions. In the presence of a lower solution α and an upper solution β in the reverse order β ≤ α, we establish the existence results by using Martelli's fixed point theorem with upper and lower solutions method. We find that if we give different definitions of lower and upper solutions in the reverse order, we can also get the existence results.

In this paper, we investigate the generalized Kadomtsev-Petviashvili equation for the evolution of nonlinear, long waves of small amplitude with slow dependence on the transverse coordinate. By virtue of the Hirota's bilinear form and the extended homoclinic test approach, new exact periodic solitary wave solutions for the (3+1)-dimensional generalized KadomtsevPetviashvili equation are obtained, which is different from those in previous literatures. With the aid of symbolic computation, the properties and characteristics for these new exact periodic wave solutions are presented with some figures.

In this paper, we study the weak solutions for the systems of multifluid flows, which includes the system of isentropic gas dynamics in Eulerian coordinates and a system arising from river flows. There are more linearly degenerate fields compared with single-component system, and singularities in these linearly degenerate fields emerge when considering the corresponding vanishing viscosity system. we obtain the existence of global solutions for the system of multifluid flows by analyzing the uniform BV estimates in linearly degenerate fields, coupled with the compensated compactness method and the vanishing viscosity method.

In this paper, the existence of ground state, oscillation solution and soliton solution of a class of nonlinear Schrödinger equations in plasma are considered.

The following nonhomogeneous Kirchhoff type equations with critical exponent

$\begin{eqnarray*}\left\{\begin{array}{ll}-(a+b\int_{\Omega}|\nabla u|^2{\rm d}x)\Delta u=|u|^{4}u+\lambda f(x), & \ \ x \in \Omega, \\u=0, & \ \ x\in\partial\Omega, \end{array}\right. \end{eqnarray*} $

where $\Omega$ is a smooth bounded domain in ${\Bbb R} ^{3}, $ $a, b, \lambda>0$ are parameters and $f\in L^{\frac{6}{5}}(\Omega)$ is nonzero and nonnegative, are considered. By the variational method, the second positive solution is obtained which completes and improves the resulsts of [3] and [5].

This paper concerns the parabolic-parabolic Keller-Segel type model. By making use of the Neumann heat semigroup, the asymptotic inequality on the gradient of $\rho$ depending on the chemotaxis signal $\chi$ is derived. Meanwhile, the convergence results on $\|\rho(\cdot,t)\|_{L^1(\Omega)}, \|n(\cdot,t)\|_{C^{\theta }(\bar{\Omega})}$ and $\|c(\cdot,t)\|_{L^\infty(\Omega)}$ have also been obtained. It reveals that mass of sperm will tend to the initial difference between sperm and eggs mass in the process of evolution, eggs are all fertilized and the concentration of the chemical substance will be also exhausted eventually. At the same time, the depletion of chemical substance is accompanied by complete fertilization of eggs. It illustrates that concentration of the chemical substance plays a relevant role in the fertilization process of corals.

In this paper, we shall strengthen our results on the W^{1, p} convergence rates for homogenization problems for solutions of partial differential equations with rapidly oscillating Neumann boundary data. Such a problem raised due to its importance for higher order approximation in homogenization theory, which gives rise to the so-called boundary layer phenomenon. Our techniques are based on integral representation of the solutions as well as analysis of oscillatory integrals, in conjunction with Fourier expansion of the oscillating periodic function.

In this paper, we consider the 3D generalized Oldroyd-B type models with fractional Laplacian dissipation (-△)^η₁u and (-△)^η₂τ in the corotational case. By using energy method, for η₁ ≥ 5/4 and η₂ ≥ 5/4, we obtain the global regularity of classical solutions when the initial data (u₀, τ₀) are sufficiently smooth.

In this paper, we consider the regularity of weak solutions to the incompressible NS equations and MHD equations in the Triebel-Lizorkin space and multiplier space respectively. By using Littlewood-Paley decomposition and energy estimate methods, we proved that if horizontal velocity ũ=(u₁, u₂, 0) satisfies

$ \nabla_{h}\tilde{u}\in L^{p}(0,T; \dot{F}^{0}_{q,\frac{2q}{3}}(\mathbb{R} ^{3})), ~~~~~\frac{2}{p}+\frac{3}{q}=2,~~~~\frac{3}{2}<q\leq\infty, $

then the weak solution is actually the unique strong solution on[0, T). For MHD equations, we prove that if horizontal velocity and magnetic field satisfies

$ (\tilde{u},\tilde{b})\in L^{\frac{2}{1-r}}(0, T;\dot{X}_{r}(\mathbb{R} ^{3})),\hspace{0.2cm}r\in[0, 1), $

or horizontal gradient satisfies

$(\nabla_{h}\tilde{u},\nabla_{h}\tilde{b})\in L^{\frac{2}{2-r}}(0, T;\dot{X}_{r}(\mathbb{R} ^{3})),\hspace{0.2cm}r\in[0, 1], $

then the weak solution is actually unique strong solution on[0, T).

In this paper, the lifespan estimate to the Cauchy problem of the semi-linear wave equation utt u_tt-△u=(1+|x|²)^α|u|^p in ${\mathbb{R}}$ⁿ is studied. The upper bound of lifespan is improved for the cases n=2, 1 < p ≤ 2 and n=1, p > 1, by using the improved Kato's type lemma.

In this paper, with the help of the wilson element, new semi-discrete and fullydiscrete schemes are proposed for parabolic integro-differential equation. Based on the properties of the element, through defining a new bilinear form, without using the technique of extrapolation and interpolated postprocessing, in the norm which is stronger than the usual H¹-norm, the convergence results with order O(h²)/O(h²+τ) for the primitive solution are obtained for the corresponding schemes, respectively. The above results are just one order higher than the usual error estimates for the wilson element. Here, h and τ are parameters of the subdivision in space and time step, respectively. Finally, numerical results are provided to confirm the theoretical analysis.

In this paper, we study the existence of the steady state solutions for a classical Schrödinger equation in nonlinear optical lattices by means of gradient flow method. We first establish the existence of a global solution of the governing parabolic equation. Then we prove the convergence of the global solution to an equilibrium (i.e., a steady state solution in optical lattices model) as time goes to infinity. Furthermore, we provide an estimate on the convergence rate by using the Łojasiewicz-Simon inequality.

In this paper, we use a new method to improve the corresponding result of Chen and Sung (2014)^[5] and Qiu and Chen (2014)^[6] by truncating the WOD random variables into five parts.

In this paper, we investigate a type of complex-valued neural networks with discontinuous activation functions. By using Filippov differential inclusion theory, Leray-Schauder alternative theorem and Lyapunov function, we obtain the sufficiet conditions for the global exponential periodicity of the neural network. The simulation shows the effectiveness of the results.

The proper orthogonal decomposition (POD) has been known as an efficient reduction technique in simulating physical process which governed by partial differential equations. In this paper, the basic properties of time coefficients in POD method are studied. Based on the theories and matrix analysis, by introducing the cross-correlation matrix (CROM) of the whole signals and the covariance matrix of the fluctuating parts of signals (in which the mean values have been cancelled, COVM), we derive the influence mechanism of mean values on the POD method in theory. There are fewer relevant studies in previous works. Finally, numerical examples are provided for verification and demonstration the influence mechanism in POD process.

Deliberation is an effective method to solve complex problems. In the process of deliberation, real-time analysis and evaluation of group consensus will help to activate group thinking and promote the convergence of group thinking. In this paper, a deliberation dialogue frameworks(DDF) based on IBIS is proposed, in which argumentation and uncertainty reasoning are introduced to calculate the certainty-factors of propositions. Firstly, the basic IBIS model is extended to multi-level argumentation structure with propositions about issues not only be put forward but also be argued. The process of argumentation about proposition is described as dialogue tree, and the all the deliberation is described as dialogue forest. Then, the uncertain of argument's premise and the argument's strength were quantified with certainty-factor, and the dialogue forest is mapped into fuzzy petri net, called FPND. Using the parallel computing ability of petri nets, the final certainty-factors of proposition are calculated by matrix iterative operation. Finally, an example is used to verify the validity and rationality of the proposed method.

This paper considers a M/G/1 queueing system with p-entering discipline and Min(N, D, V)-policy, in which the customers who arrive during multiple vacations enter the system with probability p(0 < p ≤ 1). By using the total probability decomposition technique and the Laplace transform, we discuss the transient distribution of queue length at any time t which started from an arbitrary initial state, and obtain the expressions of the Laplace transform of transient queue-length distribution. Moreover, we obtain the recursion expressions of the steady-state queue length distribution. Meanwhile, we discuss the optimal capacity design by combining the steady-state queue length distribution and numerical example. Finally, the explicit expression of the long-run expected cost rate is derived under a given cost structure. And by through numerical calculation, we determine the optimal control policy (N^*, D^*) for minimizing the long-run expected cost per unit time.

In this paper, we propose and study a delayed SVEIR epidemic model with vaccination and saturation incidence. The existence and local stability of equilibria are addressed. By using Lyapunov functionals and Lyapunov-LaSalle invariance principle, it shows that if the basic reproduction number is less than or equal to one, the disease-free equilibrium is globally asymptotically stable and the disease will disappear; and if the basic reproduction number is greater than one, the endemic equilibrium is globally asymptotically stable and the disease will persist. Some numerical simulations are performed to illustrate our analytic results.

Due to the influence of age or physical fitness, resulting in a number of susceptible to infection into the latent, some directly into patients, study of epidemic model so it is necessary to enter a latent or infected population proportion, but this kind of model is rarely taken into account the influence of age, study on the age structure of the infectious disease model of this change on the proportion of the total population in infected groups or latent, the expressions of the threshold parameters related to the population growth index are obtained, the existence and the local asymptotic stability conditions of the disease-free equilibrium and endemic equilibrium are discussed, then, the use of these conditions to control the spread of disease has important theoretical and practical significance.

In this paper, we study the dividend problem with Parisian delay for the classical risk model with debit interest. By cutting the excursion, we get the expression for the expected discounted dividend payments.