This paper investigates a stochastic phage-bacteria model in which the death rate of phage is influenced by noise. The stochastic asymptotical stability of the boundary equilibrium is studied. It shows that the solution of the stochastic model spirals around the positive equilibrium of the corresponding deterministic model. Finally, numerical simulations are presented to illustrated the theoretical results.

By using the classical Schaefer's fixed point theorem, a class of nonlocal boundary value problems for nonlinear fractional differential equations with $p$-Laplacian operator are studied. Under some suitable assumptions, some new results on the existence of solutions are obtained. An example is given to show the applicability of the results.

By the invariant subspace method, the Euler equations for Chaplygin gas are reduced to finite-dimensional dynamical systems of first-order ordinary differential equations. Many interesting exact solutions, including finite-time blowup solution, global solution and time-periodic solution, are obtained. The unusual blowup interfaces of some solutions are described.

In this paper, a general feedforward neural network is constructed, in which the activation function is not assumed to be odd, and the threshold values and direction weight values are different from the known choices. In addition, lower bounds of approximation errors of the proposed neural networks are discussed. Some examples of numerical experiments are listed to demonstrate the theoretical results.

A new method to construct two types of $\alpha$-curves based on C-B${\rm\acute{e}}$zier curve is given. The sufficient conditions for $\alpha$-curves of curvature monotony are deduced. The result shows that Quasi-Cubic B${\rm\acute{e}}$zier curve and quadratic B${\rm\acute{e}}$zier curve are special cases of $\alpha$-curves. The advantage of $\alpha$-curve is that it only has three control points and the shape of $\alpha$-curve can be adjusted by a shape parameter, therefore $\alpha$-curve is simpler and more powerful than C-B${\rm\acute{e}}$zier curve. The first type $\alpha$-curve has zero curvature at start point, and a pair of these $\alpha$-curves can be used in constructing S-shaped or C-shaped $G^{2}$ transition curves for separated circles, the ratio of two radii has no restriction. The second type $\alpha$-curve can be used in constructing a single transition curve with no curvature extreme for separated circles, and the endpoint curvature of this $\alpha$-curve degenerate to zero when specific shape parameter is selected. Test examples are given to show the effectiveness of these two types of $\alpha$-curves.

LI-ideals is an important tool for studying the structure characteristics of lattice implication algebras. In this paper, the theory of LI-ideals in lattice implication algebras was further studied by using the methods and principles of algebra and logic. Firstly, the notions of extended LI-ideals and stable LI-ideals of a LI-ideal $A$ associated to a subset $M$ of lattice implication algebra $L$ are introduced and some of their basic properties are investigated. Secondly, some lattice theory characteristics about some types sets of extended LI-ideals in a lattice implication algebra $L$ are discussed. It's proved that the set of all stable LI-ideals associated to a given subset $M\subseteq{L}$ and the set of all extended LI-ideals of LI-ideals $A$ associated to any subset of $L$ both form complete Heyting algebras. Thirdly, some properties of extended LI-ideals in quotient and product lattice implication algebras are given. Finally, some equivalent characterizations of ILI-ideals are obtained by means of extended LI-ideals in lattice implication algebras.

This paper studies the boundedness of multilinear Hardy-Littlewood maximal operator, multilinear fractional maximal operator, multilinear singular and multilinear fractional integral operators on $B_\sigma$-Morrey spaces.

Let $G$ be a simple graph, and $s\leqslant3$ be an integer. In this paper, if $G$ is a connected graph with order $n$ and $K_{1,s}$-matching number $m(G)$, such that $n>(s+1)m(G)$, then the $m(G)$-th largest Laplacian eigenvalue $\mu_{m(G)}>s+1$. And this result also holds for signless spectrum. As an application, some $Q$-eigenvalue conditions which can determine the Hamiltonicity of a graph are listed.

The credibility premiums based on the quantile and trimming under the exponential premium principle are discussed, respectively. Thus the credibility estimates of the future claim under the single contract and multi contracts are obtained, and the corresponding credibility premiums are derived, which generalizes the classical credibility theory.

A new model has been developed for pricing the VIX option under a continuous Markov-modulated version of the stochastic volatility model. This paper supposes that the model parameters depend on the states of a continuous time observable Markov chain process, which is the state of an observable macroeconomics factor. The VIX call option pricing formula has also been derived in this paper. Compared with the conventional stochastic volatility model, the pricing formula derived in this paper concludes the regime switching risk premium. The last part is the Monte Carlo simulation and some explanations for the numerical results.

In this paper, with aid of large deviation formulas established in strong topology generated by Holder norm for $k$-dimensional Brownian motion, the paper obtained the functional sample path properties for subsequence's $C$-$R$} increments of $k$-dimensional Brownian motion in Holder norm, by which one can get the functional laws of iterated logarithm for $k$-dimensional Brownian motion.

A test and estimation of change point in Logistic regression model is discussed based on empirical likelihood method. The log-likelihood ratio test statistic is conducted by establishing a change-point model. In large-sample case, the empirical log-likelihood ratio test statistic is proved to have the same extreme value distribution as that with classical parametric log-likelihood. Under some mild conditions, the maximum empirical likelihood estimation of change point is also shown to be consistent. The simulation results and a real example demonstrate the feasibility of the proposed approach.