Using the character of generalized Kato decomposition, this paper discusses the sufficient and necessary conditions for which Browder's theorem and Weyl's theorem hold from the angle of generalized Kato spectrum for a bounded linear operator.

For a Finsler manifold with the weighted Ricci curvature bounded from below, we give Cheng type and Mckean type comparison theorems for the first eigenvalue of Finsler Laplacian. When the weighted Ricci curvature is nonnegative, we also obtain Calabi-Yau type volume growth theorem. These generalize and improve some recent literatures. Especially, by using the relationship of the counterparts between a Finsler metric and its reverse metric, we remove some restrictions on the reversibility.

This paper is concerned with the necessary and sufficient conditions that a class of bounded 2×2 block operator matrices are Fredholm operators or Weyl operators. Some necessary and sufficient conditions are given under which the essential spectrum and the Weyl spectrum of the block operator matrix coincide with the essential spectrum and the Weyl spectrum of its entries.

In this paper, we study the phase separation phenomena of the limit profile as the coupling constant tending to minus infinity for some Bose-Einstein condensation system with subcritical exponent in a general smooth bounded domain via variational methods and elliptic equations theories.

In this paper, the binary Bell polynomials to construct bilinear forma, bilinear Bäcklund transformation, Lax pair of the KdV-shallow water waves equation. Through bilinear Bäcklund transformation, some soliton solutions are presented. Moreover, the infinite conservation laws are also derived by Bell polynomials, all conserved densities and fluxes are given with explicit recursion formulas.

By the generalized Hasimoto transformation, we deduce an equivalent system of the Landau-Lifshitz-Gilbert equation on hyperbolic space $\mathbb{H}$². Based on this equivalent model, we prove the global existence of the Landau-Lifshitz-Gilbert equation with the small initial condition. Until now, we have not seen a paper discussing the explicit dynamic solution of the complete equation with a damping term on this target. We construct an explicit small data global solution by the equivalent system obtained in this paper. An self-similar finite blowup solution is also presented for the equivalent system. In the previous paper[25], we constructed a finite time blowup solution without Gilbert damping on $\mathbb{H}$². The question of whether a solution of the complete equation with a Gilbert term can develop a finite time blowup from $\mathbb{H}$² and smooth initial data is not clear. The self-similar finite time blowup solution we presented here is a finite energy solution on the entire spacial domain. Our result gives a positive answer to this question.

In this paper, we study the existence of the ground state solutions for the following Schrödinger-Maxwell equations

$\left\{\begin{array}{ll} -\triangle u+V(x)u-(K(x)+\alpha)\phi u=\beta|u|^{4}u+b(x)|u|^{p-1}u, &(x,u)\in(\mathbb{R}^{3},\mathbb{R}),\\ \triangle\phi=(K(x)+\alpha)u^{2},& (x,u)\in(\mathbb{R}^{3},\mathbb{R}) \end{array} \right.$

where β is a positive constant. Under some assumptions on V, K and b(x), by using the variational method and critical point theorem, we prove that such a class of equations has at least a ground state solution for α < 0 and p ∈ (3, 4).

In this paper, we prove the existence of periodic solutions with high frequencies of some semi-linear Klein-Gordon equations. We only assume the nonlinearities are C^k regular and without smallness. Using Nash-Moser iteration, we obtained some periodic solutions in Sobolev space.

Dynamical analysis and explicit solutions for generalized (3+1)-dimension Kadomtsev-Petviashvili equation have been carried out. The singular solution is obtained by the ansatz method, the bifurcation phase portraits and corresponding explicit solution are also constructed by the approach of dynamical analysis.

In this manuscript, we study the blow-up phenomena of solutions for a class of viscous Cahn-Hilliard equation with gradient dependent potentials and source. We establish a criterion for blow-up and determine the upper bound for blow-up time by the energy method, the differential inequality and the derivative formula of the product; We determine the lower bounds for blow-up time by the differential inequality and the derivative formula of the product.

In this paper, the Cauchy problem of the MHD equations with damping is studied. When $\beta \ge 1$ and initial data satisfy ${u_0}$, ${b_0} \in {L^2}({{\mathbb{R} ^3}})$, the Galerkin method is used to prove the global weak solution of the equations. When the initial data satisfy ${u_0} \in H_0^1 \cap {L^{\beta + 1}}({{\mathbb{R} ^3}})$, ${b_0} \in H_0^1({{\mathbb{R} ^3}})$, it is possible to obtain a unique local strong solution for the equation group.

A boundary value problem to a class of elliptic equations with lower order terms and degenerate coercivity is studied in this paper. With help of De Giorgi iterative technique and Boccardo-Brezis's test function, the L^∞ estimate to weak solutions of the problem is obtained. Based upon the uniform L^∞ bound, the existence of bounded solution is proved.

The aim of this paper is to study the nonlinear biharmonic equations of the following form $\triangle^2u=f(|x|, u, |\nabla u|)(x\in \mathbb{R} ^N, N>2)$. The Sufficient and necessary condition for the existence of positive entire solutions is proved, and some properties of the solutions are obtained.

A predator-prey model with cross diffusion under homogeneous Dirichlet boundary conditions is investigated. Firstly, the existence of positive solutions can be established by the Leray-Schauder degree theory. Secondly, we consider that the existence of positive solutions of the regular perturbation system and the singular perturbation system when m=β is sufficiently large, respectively, and moreover, we show that the positive solutions of the singular perturbation system will blow up along the continuum at a^* by the bifurcation theory. Finally, the multiplicity results of positive solutions of system is also considered.

In this paper, we investigate the stability of the trapezoidal method for a class of neutral differential equation with variable delay and obtain the asymptotic estimation of numerical solution with the aid of a functional inequality. The asymptotic estimation is more accurate than asymptotic stability in describing the behaviours of the numerical solution, and gives the upper bound estimates of the numerical solution for the nonstable case.

In this paper, we consider the asymptotic stability in the p-th moment of mild solutions of impulsive neutral stochastic functional differential equations driven by fractional Brownian motion in a real separable Hilbert space. A fixed point approach is used to achieve the required result. A practical example is provided to illustrate the viability of the abstract result of this work.

In this paper, we study the collision and intersection local times of the solution to stochastic heat equation with additive fractional noise. We mainly prove its existence and smoothness properties through local nondeterminism and chaos expansion.

In this paper, the exponential tracking control for a star-shaped network of Eulerbernoulli beams with unknown internal disturbance is studied. The problem is transformed into the stabilization of the error system between the objective network and the active network. The idea of sliding-mode control is used to design a nonlinear feedback control law. The solvability of the error system is obtained via monotone operator theory under an appropriately chosen space norm. The error system is proved to be exponentially stabilized at any decay rate by a suitable Lyapunov functional. So the objective network can track the active network exponentially at any designated rate.

In this paper, we consider the numerical solution of boundary integral equations (BIE) arise in the study of the 2D scattering of a time-harmonic acoustic incident plane wave. Fast multipole method (FMM) is a very efficient and popular algorithm for the rapid solution of boundary value problems. However, when the FMM method is used for high frequency acoustic wave problems, it will give rise to the computation of oscillatory integrals. The standard quadrature methods are exceedingly difficult to calculate these oscillatory integrals and the computation cost steeply increases with the frequency. We apply the boundary element method (BEM) to discretize the BIE and use the FMM to accelerate the solutions of BEM. Oscillatory integrals are calculated by using efficient Clenshaw-Curtis Filon (CCF) methods. The effectiveness and accuracy of the proposed method are tested by numerical examples.

This paper is concerned with a linear quadratic optimal control problem for meanfield backward stochastic differential equations driven by a Poisson random martingale measure and a Brownian motion. Firstly, by the classic convex variation principle, the existence and uniqueness of the optimal control is obtained. Secondly, the optimal control is characterized by the stochastic Hamilton system which turns out to be a linear fully coupled mean-field forward-backward stochastic differential equation with jumps by the duality method. Thirdly, in terms of a decoupling technique, the stochastic Hamilton system is decoupled by introducing two Riccati equations and a MF-BSDE with jumps. Then an explicit representation for the optimal control is obtained.

Parameter resolution is a criterion for measuring whether two adjacent signals can be distinguished under the given noise conditions, it provides an evaluation of the "ruler" for the measurement of sensitive parameters, effective precision and accuracy. This paper proposes a definition of the parameter resolution of EM algorithm, which is based on the EM algorithm, and the idea of Fisher linear discriminant criterion, two-component Gaussian mixed model is taken as an example to verify it. Experiments show that when two normal distributions with a variance of 0.1 have a mean distance greater than 0.206, the EM algorithm can tell the differences between the two distributions under a confidence of 90%, by constructing the connection between experimental results and theoretical derivation, the scale factor graphs with different confidence levels are obtained. The proposed resolution of the parameters provides a quantitative indicator for the accuracy measurement and also provides a new solution for the differentiation of similar signals.

In this paper, the pricing method and parameter calibration of jump-diffusion model are investigated. First, the risk-neutral characteristic function of jump-diffusion model is derived under the mean correctiong equivalent martingale measure. The option under jump-diffusion model is priced by using the COS pricing method. Then, the pricing error of the COS algorithm is analyzed and the effectiveness of the COS pricing method is verified through numerical experiment. Subsequently, the parameters of the jump-diffusion model are calibrated by the relative entropy regularization method. Numerical experiments demonstrate the accuracy and reliability of the proposed method. Finally, the calibration method is tested by analyzing the S&P500 market data. The results show that the values of calibrated parameter are qualitatively for each maturity. Moreover, the results indicate a better fitting to the market data for the Merton jump-diffusion model in comparison to the Black-Scholes model.

In order to measure the error of E-Bayesian estimation, this paper the definition of E-MSE(expected mean square error) is introduced based on the definition of E-Bayesian estimation. For parameter of Poisson distribution, under different loss functions (including:squared error loss, K-loss and weighted squared error loss), the formulas of E-Bayesian estimation and formulas of E-MSE are given respectively. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and a real data set have been analysed for illustrative purposes, results are compared on the basis of E-MSE, the results show that the proposed method is feasible and convenient for application.

In this paper, we show a stochastic predator-prey model with pulse input in a polluted environment, the existence and uniqueness of the positive global solution and the boundedness of expectation of the system are all proved, the sufficient conditions for the existence and boundedness of periodical solution are obtained, and it is globally attractive with probability 1, and the threshold of population extinction and persistence in the mean are obtained too. Finally, some numerical simulations are carried out to illustrate the main results.

In view of the side effects, the technique relying on diseased pest releases as a valuable non-chemical tool is getting much more essentiality in pest management. Inspired by Xiang (2009) and Bhattacharyya et al (2006), the present thesis firstly focuses on a susceptible and infected pest model for pest management, which possesses multiple dynamic behaviors but does not eradicate susceptible individuals. For eliminating the pests, human impulsive interventions are embroiled in this model. Then the sufficient conditions for the global asymptotic stability of the susceptible pest-eradication periodic solution are established by unlimited pulse interventions. However, the strategy driving susceptible pests to extinction is unadvisable from ecological and economical aspects since the appropriate amount of pests in the field is beneficial for conservation of natural enemies and maintaining the crop overcompensation after pest injury. Hence, three different optimal problems involving different pest control tactics are deliberated in order to diminish the susceptible population at the terminal time and keep this in balance with the cost of the intervention (control). Subsequently, by time scaling and translation transformation techniques, the gradients for the cost functional on durations, fractions of susceptible pests killed due to chemical sprays as well as the number of infected pest released at each impulsive intervention moment are computed, which are vital to capture the optimal control strategy for pest regulation. Finally, on the basis of simulations, the strategy of alternative integrated control at unfixed time is validated to be the most effective compared with the other two policies. In addition, by comparing our optimal strategy with pest-extinction one, it is revealed that our strategy is more desirable.