In this paper, we investigate the normalized torsion units of the integral group ring of the direct product of the symmetric group $S_5$ and the cyclic group $C_3$. As a consequence, we confirm the Zassenhaus's conjecture about this group.
In the theory of formations of finite soluble groups, Bryce and Cossey proved an important theorem: A soluble local formation $\frak{F}$ is a Fitting class if and only if every value of the canonical formation function $F$ of $\frak{F}$ is a Fitting class. In this paper, basing on the theory of $\sigma$-groups, we generalized the results of Bryce and Cossey. We proved that A $\sigma$-local formation $\frak{F}$ is a Fitting class if and only if every value of the canonical $\sigma$-local definition $F$ of $\frak{F}$ is a Fitting class.
Let $\mathscr{H}$ be complex separable infinite-dimensional Hilbert spaces. Given the operators $A\in\mathscr{B}(\mathscr{H})$ and $B\in\mathscr{B}(\mathscr{H}),$ we define $M_{X}:= \begin{bmatrix}\begin{smallmatrix} A& X\\ 0& B \end{smallmatrix}\end{bmatrix}$ where $X\in \mathscr{S}(\mathscr{H})$ is a self-adjoint operator. In this paper, a necessary and sufficient condition is given for $M_{X}$ to be a left (right) Fredholm operator for some $X\in\mathscr{S}(\mathscr{H})$. Moreover, it is shown that \[\begin{array}{l} \bigcap\limits_{X\in\mathscr{S}(\mathscr{H})} \sigma_{*}(M_{X}) =\bigcap\limits_{X\in\mathscr{B}(\mathscr{H})} \sigma_{*}(M_{X})\cup\Delta, \end{array} \] where $\sigma_{*}$ is the left (right) essential spectrum. Finally, we further characterize the perturbation of the left (right) essential spectrum for Hamiltonian operators.
We consider the growth of solutions of differential-difference equation $f^{n}(z)+q(z){\rm e}^{Q(z)}f^{(k)}(z+c)=p_{1}{\rm e}^{\alpha_{1}z}+p_{2}{\rm e}^{\alpha_{2}z}$ and $f^{n}(z)+q(z){\rm e}^{Q(z)}\Delta_{c}f=p_{1}{\rm e}^{\lambda{z}}+p_{2}{\rm e}^{-\lambda{z}}$, where $n\geq{1}$ and $k\geq{1}$ are two integers, $q(z)$ is a non-zero polynomial and $Q(z)$ is a non-constant polynomial. $c,\lambda,\alpha_{1},\alpha_{2},p_{1}$ and $p_{2}$ are non-zero constants, $\alpha_{1}\neq{\alpha_{2}}$. In particular, we show that exponential polynomial solutions satisfying certain conditions must reduce to rather specific forms, which is an improvement of previous results.
We investigate and obtain a necessary and sufficient condition for three Hankel operators on the Hardy spaces to be commuting, that is, suppose $f, g$ and $u$ are nonconstant functions in $\bigcap_{q>1}H^{q}$, then $H_{\bar{f}},H_{\bar{g}}$ and $H_{\bar{u}}$ commute if and only if a nontrivial linear combination of any two functions of $f, g, u$ is constant.
For $\alpha>-1$, let $A_\alpha^2(\mathbb{B}_N)$ be the weighted Bergman space on the unit ball $\mathbb{B}_N$ in $\mathbb{C}^N$. We prove that the multiplication operator $M_{z^n}$ is quasi-similar to $\bigoplus_1^{\prod_{i=1}^N n_i}M_z$ on $A_\alpha^2(\mathbb{B}_N)$ for the multi-index $n=(n_1,n_2,\ldots,n_N)$.
In this paper, we obtain gradient estimates for positive solutions of two nonlinear parabolic equations as follows $\frac{\partial u}{\partial t}=\Delta_V u+au\log u$ and $(\Delta_V-\frac{\partial}{\partial t})u(x,t)+p(x,t)u^\beta(x,t)+q(x,t)u(x,t)=0$ where $\alpha,\, \beta\in\mathbb{R}$, $\Delta_V(\cdot):=\Delta+\langle V,\nabla(\cdot)\rangle$ on the complete Riemannian manifold with Bakry--Emery Ricci curvature bounded below. Due to the introduction of $\Delta_V$, the Laplacian comparison theorem is replaced by the $V$-Laplacian comparison theorem in the process of proving the gradient estimates. Applications of these estimates yield Harnack inequalities and Liouville type theorem.
We investigate the differentiability and compactness of the $C_0$-semigroup generated by the $k/G:N$ redundant system with finite repair time. We show that the $C_0$-semigroup is eventually differentiable and eventually compact when the repair time is finite. However, this is not true for the case when the repair time is infinite.
We study the quasi-minimizers to nonhomogeneous energy functional on metric measure spaces. Assuming that the metric spaces satisfy doubling condition and Poincar\'{e} inequality, local boundedness for quasi-minimizers is obtained by establishing Caccioppoli inequality and De Giorgi iteration.
We first obtain the existence and uniqueness of the both weak and strong solutions by using the Faedo--Galerkin method, as well as the continuous dependence on initial values. Then the existence of a bounded absorbing set is used to characterize the dissipativity of the dynamical system $(X_{0},S(t))$ associated with the problem. Next the asymptotic smoothness of the dynamical system $(X_{0},S(t))$ is demonstrated when $p>0$ in the nonlinear damping term $|u_{t}|^{p}u_{t}$; and while $p=0$ we get the quasi-stability of $(X_{0},S(t))$. Finally, based on the above conclusions, we achieve the existence of the finite dimensional global attractor and the generalized exponential attractor for Kirchhoff type suspension bridge equations with polynomial damping and polynomial nonlinearity. We extend and partially improve the results of published theses before in this paper.
We establish some generalized Orlicz isoperimetric inequalities of convex bodies by using the classical Popoviciu's inequality and Orlicz--Minkowski inequality. The new Orlicz isoperimetric inequality in special case yields the classical isoperimetric inequality, $L_{p}$-isoperimetric inequality and Orlicz isoperimetric inequality, respectively.
The relation between the octonionic Heisenberg group and the octonionic Siegel half space is investigated; the Hardy spaces over the octonionic Siegel half spaces are studied, of which the characterization through boundary limits is given.
This paper considers spacelike complete stationary genus $0$ surfaces with regular flat embedded ends in $\mathbb{R}^4_1 $. Let $k$ be the number of regular flat embedded ends. We prove that when $k=1$, the surface is the plane; we also show that there exist no spacelike stationary genus $0$ surfaces with $k$ regular flat embedded ends when $k=2,3$. We give $2$-family of examples of spacelike complete stationary genus $0$ surfaces with $k$ regular flat embedded ends in $\mathbb{R}^4_1 $, when $k \ne 1,2,3,5,7$.
This paper studies the optimal investment time of a company, the selection of wage strategy of the company and the agent's optimal effort degree under information asymmetry and principal agent conflicts. A company has an option to invest in a project. Because of technical limitations, the shareholders entrust an agent to manage the project investment problem. The project generates two parts of value. One part can be informed by shareholders, and the other part is only observed by the agent and the distribution of which is related to the degree of efforts of the agent. The shareholders should choose the best time of investment and the salary level to maximize the expected value of the company under the condition that the agent does not lie about the income of the project. The agent should choose the best degree of effort to maximize his net discounted salary (discounted wage$-$discounted effort). Because one party's strategy will affect the choice of the other's strategy, it is actually a game with restricted conditions. In this paper, we will get the optimal strategies and value functions of both sides for different distributions of project value.
In this paper, a single mode method for calculating the second-order partial derivatives of a simple eigen-triplet of quadratic eigenvalue problems is presented. This method only needs the information of the eigen-triplet whose second-order partial derivatives are to be obtained; and to calculate the second-order partial derivative of the eigenvector, it only needs to solve a system of linear equations of order $n-1 $ ($n$ is the size of quadratic eigenvalue problem). Most of all, the condition number of the coefficient matrix is exactly the ratio of its maximum to its minimum non-zero singular value. Three numerical examples are given and computed for the single mode method. Numerical results show that, compared with the Nelson method, the accuracy of the second derivative of eigenvalue calculated by the single mode method is almost the same, while the accuracy of second derivatives of eigenvector is higher; and the CPU time of single mode method is slightly less than that of the Nelson method, the latter is about $1.2$ times as much as the former.
In this paper, we first introduce the concepts of weak $c$-ideals, weak $c$-simple Lie superalgebras and weak $c$-ideals complementability of Lie superalgebras, and investigate some structural properties related to weak $c$-ideals over the fields with characteristics other than $2$, $3$; Secondly, the necessary and sufficient conditions for a Lie superalgebras to be a weak $c$-simple Lie superalgebras are given. By using $\rm{Frattini}$ ideals, the sufficient and necessary conditions for a weak $c$-ideal of a Lie superalgebra to be its ideal are given, and the necessary and sufficient conditions for its quotient subalgebra to have subideal complement are given; Finally, by using some weak $c$-ideals of a class of Lie superalgebras over fields with characteristic zero, the sufficient conditions for the class of Lie superalgebras to be solvable are given.