In this paper a
kind of integral operator $I^{\beta}_{\alpha}f(z)$ related to the
parameters $\alpha, \beta$ in an open unit disk is investigated. Firstly,
the bi-univalent function classes $\mathfrak{H}_{\Sigma_{q}}^{\alpha,\beta}(\lambda;\phi)$ and $\mathfrak{L}_{\Sigma_{q}}^{\alpha,\beta}(\mu,\lambda;\phi)$ involving this integral operator and the $q$-derivative operator are defined by applying the subordination principle of analytic functions. Then, the upper bounds of the first two coefficients $a_{2}$ and $ a_{3}$ of the two classes of bi-univalent analytic functions are estimated, and the corresponding Fekete-Szegö inequalites for these classes are obtained.
This paper presents some new classes of $n$-cycle permutations over finite fields. At first, we present a concise criterion for Dickson polynomials over finite fields being $n$-cycle permutations.
Then, we give a necessary and sufficient condition for linearized polynomials over finite fields being involutions.
Finally, some interesting new classes of $n$-cycle permutations are demonstrated by considering polynomials of different forms.
The generalized inverse and partial order of tensor are important components of tensor theory.
In this paper, we introduce the T-CS inverse of third-order tensor,
obtain some characterizations and properties of it under the T-product,
and apply it to
introduce a new binary relation:
the ${\textcircled{S}}$ order, which is equivalent to the T-star order under the set of i-EP tensors.
Based on the ${\textcircled{S}}$ order, we further introduce the T-CS partial order and give some characterizations of it.
In this paper, we study the existence and multiplicity of positive solutions for the following double singular problem with $p(x)$-Laplace operator
\begin{equation*}
\left\{
\begin{array}{ll}
-\Delta _{p(x)}u+V(x)|u|^{p(x)-2}u=\mu\frac{|u|^{s(x)-2}u}{|x|^{s(x)}}+\lambda h(x)u^{-\gamma(x)}&\quad \text{in}\quad \Omega,\\
u=0&\quad \text{on}\quad \partial\Omega.
\end{array}%
\right.
\end{equation*}
Due to the presence of singular term $u^{-\gamma(x)}$ and singular potential $|x|^{-s(x)}$ in the equation, it is more difficult to deal with the existence of positive solutions. By using the decomposition of Nehari manifold and some refined estimates, we show that there admits at least two positive solutions for the double singular problem.
This paper studies the inverse eigenvalue problem for a class of quaternion conjugate symplectic tensors under the Einstein product. Firstly, the properties and characteristic structures of conjugate symplectic tensors are obtained by using the transformation operator of quaternion tensors. Secondly, for the given ${I_1}{I_2} \cdots {I_N}$ characteristic pairs of quaternion tensors, a quaternion self-conjugated symplectic tensor $\mathcal{S}$ is found to include all the given characteristic pairs. As an application,
we give a necessary and sufficient condition for the existence of conjugate symplectic tensor solutions and the expression of solutions to the quaternion tensor equation $\mathcal{B}\ast_{N}\mathcal{S} = \mathcal{D}$. The feasibility of the proposed method is showed with numerical examples.
In this paper we consider the gradient estimates on the positive solutions to the elliptic equation $\Delta v+v^r-v^s= 0,$ defined on a complete Riemannian manifold $(M,\,g)$,
where $r$ and $s$ are two real constants.
When$(M,\,g)$ satisfies $Ric \geq -(n-1)\kappa$ (where $n\geq2$ is the dimension of $M$ and $\kappa$ is a nonnegative constant), we employ the Nash-Moser iteration technique to derive a Cheng-Yau type gradient estimate for the positive solutions to the above equation under some suitable geometric and analysis conditions.
Moreover, it is shown that when the Ricci curvature of $M$ is nonnegative, this elliptic equation does not admit any positive solutions except for $v\equiv 1$ if\ $r<s$ and $1<r<\frac{n+3}{n-1}~\mbox{or}~1<s<\frac{n+3}{n-1}.$
In this paper we investigate the necessary and sufficient conditions such that both a-Weyl's theorem and the
property $(WE)$ hold for a bounded linear operator, and study the stability of a-Weyl's theorem and
the property $(WE)$ under quasi-nilpotent
or compact perturbations. As an application, the relative stability of special operators is studied.
In order to improve the continuous support capability of the resilient system under complex cyber attacks, this paper first
builds a class of cyber attack model based on the principle of virus epidemical dynamics to simulate the complex cyber threat environment in the resilient system,
and deeply studies the dynamic characteristics of the model. Then, a variant target model of the resilient system under cyber attacks is constructed, and the local stability of the resilient system is analyzed by
using the center manifold theorem and the bifurcation theory.
Finally, with the Matlab software, the dynamic epidemical process of cyber attacks and the dynamic variant process of the target of the resilient system are displayed
in some numerical portraits.
This paper constructs a linear, second-order, unconditionally energy stable, semi-discrete time stepping scheme for the modified phase field crystal equation with periodic boundary conditions.
The unique solvability, unconditionally energy stability and unconditionally temporal convergence of order 2 of the numerical scheme are showed by introducing a Lagrange multiplier to deal with the nonlinear terms and adopting the second-order
Crank-Nicolson method to discrete time. Numerical experiments are given in the last section to validate the efficiency of the proposed scheme.
In this paper, we study the method for solving the special weighted linear complementarity problem. Based on a weighted smoothing function, we reformulate the problem as a system of smooth nonlinear equations and then propose a predictor-corrector smoothing Newton method to solve it. Under some suitable conditions, we show that the algorithm has the global and local quadratic convergence properties. In particular, when the solution set is nonempty we show that the merit function sequence converges to zero. Numerical experiments demonstrate that our algorithm is effective.
