In this paper, the dynamical behaviors of the time delayed viral infection model with latent period and CTL immune response are studied. The model describes the interaction of viral and two classes of target cells: CD4$^+$T cells and macrophages. By constructing suitable Lyapunov functionals, using the LaSalle invariance principle, it's shown that the basic repro\text{d}uctive amounts $R_{0}$ of CD4$^+$T cells and macrophages and the immune response reproductive $R_{*}$ of CD4$^+$T cells and macrophages CTL determine the global properties of the model. If $R_{0}\leq1$ , the virus is cleared. If $R_{0}>1$ , positive solutions approach to an immune-free equilibrium when $R_{*}\leq1$ , and to a positive equilibrium when $R_{*}>1$ . Thus the sufficient conditions for the global stability of the infection-free equilibrium , the immune-free equilibrium and the positive equilibrium are obtained.

A type of stochastic volatility model which includes fast-slow alternate multiple scales of high dimension Asian option pricing problem is discussed in this paper. According to Girsanov theorem and Radon-Nikodym, it realizes a transformation between expected return rate and no risk interest rate; Defining the new arithmetic average algorithm of path-dependent options and using Feynman-Kac’s formula, the Black-Scholes model is formed in which the risky assets of multiscale Asian option prices. A singular perturbation expansion is used to derive an approximation for multiscale Asian option pricing equation and the uniform valid estimation is derived.

Based on new integral estimates, the regularity of the solutions of fourth order elliptic equations in arbitrary three-dimensional domains is studied. The boundedness of the gradient of the solutions is obtained.

A forced vibration resonance phenomena with fractional derivative damping is studied in this paper.First, an asymptotic solution is constructed. Then, by the definition and properties of RiemannLiouville fractional derivative, the expression of the fractional derivative item is obtained. Then using the method of multiple scales, the frequency of each resonance is calculated. These resonances include the main resonance and the secondary resonances. For each resonance frequency, the detuning parameter is introduced. The secular terms are eliminated. After drawing the graph of the numerical solutions of amplitude and the relevant variables of initial phase when fractional derivatives are different by mathematical software, the influence of fractional derivatives on resonance is found. The first-order approximate expressions of the asymptotic solution for each resonance frequency is gotten.

The class of widely orthant-dependent (WOD, in short) random variables includes independent sequence, negatively associated sequence, negatively orthant dependent sequence and extended negatively dependent sequence as special cases. In this paper, the complete convergence for weighted sums of WOD random variables under some mild conditions is established by using the Rosenthal type inequality for WOD random variables and the truncation method. The result obtained in the paper generalizes the corresponding ones for some dependent random variables.

Basing on a law of the iterated logarithm for independent random variables sequences, an iterated logarithm theorem for WOD random variables sequences with different distributions is obtained. The proof is based on a Kolmogrov type exponential inequality.

In this paper, the asymptotic properties of $\widehat{\alpha}$ are considered, which is a least squares estimator for the parameter $\alpha$ of a weighted fractional bridge $$X_0=0,~~ \text{d}X_t=-\alpha\frac{X_t}{T-t}\text{d}t +\text{d}B^{a,b}_t, \ \ 0\leq t0, T>0$. The estimator has various convergence depending on the value of $\alpha$ as $t\rightarrow T.$ The rate of corresponding convergence is proved as well.

In this paper, the problem of ideals in negative non-involutive residuated lattices is studied by using the principle and method of universal algebras and lattice-order theory. Firstly, the notions of LI-ideals and LI-ideal generated by a non-empty subset are introduced in negative non-involutive residuated $L$, and some their properties are investigated. Secondly, lattice operations $\sqcap$ and $\sqcup$ are defined on the set $\textbf{Id}(L)$ of all LI-ideals in $L$. It is proved that $(\textbf{Id}(L), \subseteq, \sqcap, \sqcup)$ forms a distributive continuous lattice, and particularly forms a frame. Then, the notion of prime LI-ideals is introduced in $L$ and its properties are discussed. The prime LI-ideals theorem is estabLIshed in pre-LInear negative non-involutive residuated lattices. Finally, some equivalent characterizations of prime elements of LI-ideal lattice $(\textbf{Id}(L), \subseteq, \sqcap, \sqcup)$ are obtained by means of prime LI-ideals in pre-LInear negative non-involutive residuated lattice $L$.

Several important results for finite posets are proved. With these results and the specialization order of a topology, as well as relationships between topologies and orderings, on a 4-element set the total number of $T_0 $-topologies to be 219 and the total number of topologies to be 355 are obtained.

For a connected graph $G$, the least eigenvalue $q_n(G)$ of the signless Laplacian of $G$ equals zero if and only if $G$ is bipartite. $q_n(G)$ is often used to measure the non-bipartiteness of a graph $G$, and has attracted the interest of more and more researchers. This paper investigates conditions depending on $q_n(G)$ under which a graph $G$ contains a long path, and characterizes the extremal graph in which the least signless Laplacian eigenvalue attains the minimum among all the $P_t$-free non-bipartite unicyclic graphs and $P_t$-free non-bipartite connected graphs of order $n$, respectively.

Strongly symmetric self-orthogonal diagonal Sudoku square is investigated. The construction used by Lorch is extended via linear space over finite field, and a construction of such a Sudoku square is obtained. It is proved that a strongly symmetric self-orthogonal diagonal Sudoku square of order $n$ exists when $n$ is an odd prime integer.

With respect to cooperative game with fuzzy coalitions, in which the payoffs information are partially known, the definition of $E$-incompelte fuzzy games is introduced. The $w$-weighted Shapley value, which satisfies linearity and symmetry, is proposed based on the cardinality set of incomplete coalition value. By considering the marginal contribution between coalitions, the equivalent form of $w$-weighted Shapley value is provided. The study showes that the Shapley value for complete fuzzy cooperative games in accordance with the $w$-weighted Shapley value. In the frame of fuzzy coalitions, some desirable properties of $w$-weighted Shapley value, such as coalition monotonicity, zero-normalization, etc, are discussed in detail. Finally, a numerical example is illustrated to show the validity of the $w$-weighted Shapley value.

Statistical convergence is an important extension of usual convergence. In this paper, the properties of statistically sequential spaces are studied. After introducing the concepts of statistically sequentially continuous mappings, statistically sequence-covering mappings and statistically sequentially quotient mappings, the relationship among the new or old mappings and their effects in statistically sequential spaces are discussed. Also a negative answer is given to a problem about the products of statistically sequential spaces.

This paper discusses the integer invertibility for integer matrix, and provides several sufficient and necessary conditions on the existence of integer general inverse. And a new concept of prime-holding for integer matrix is firstly explored. Then it contributes two methods to construct the integer inverse.