An investigation is made of the continuity, the compactness and the spectrum of the Cesàro operator $\mathsf{C}$ when acting on the weighted Banach sequence spaces $\ell _{p}(w)$, $1<p<\infty$, for a positive decreasing weight $w$, thereby extending known results for $\mathsf{C}$ when acting on the classical spaces $\ell _{p}$. New features arise in the weighted setting (for example, existence of eigenvalues, compactness) which are not present in $\ell _{p}$.

For a normalized analytic function $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$, the estimate of the integral means

$$\begin{eqnarray}L_{1}(r,f):=\frac{r^{2}}{2{\it\pi}}\int _{-{\it\pi}}^{{\it\pi}}\frac{d{\it\theta}}{|f(re^{i{\it\theta}})|^{2}}\end{eqnarray}$$

$f(z)$

$\mathbb{D}\setminus \{0\}$

${\rm\Delta}(r,f)$

$\mathbb{D}_{r}:=\{z\in \mathbb{C}:|z|<r\}$

$f$

$0<r\leq 1$

$L_{1}(r,f)$

${\rm\Delta}(r,z/f)$

$r$

$f$

$m$

In a recent paper, Soundararajan has proved the quantum unique ergodicity conjecture by getting a suitable estimate for the second order moment of the so-called ‘Hecke multiplicative’ functions. In the process of proving this he has developed many beautiful ideas. In this paper we generalize his arguments to a general $k$th power and provide an analogous estimate for the $k$th power moment of the Hecke multiplicative functions. This may be of general interest.

We study the isomorphism problem of vertex-transitive cubic graphs which have a transitive simple group of automorphisms.

The Bishop property (♗), introduced recently by K. P. Hart, T. Kochanek and the first-named author, was motivated by Pełczyński’s classical work on weakly compact operators on $C(K)$-spaces. This property asserts that certain chains of functions in said spaces, with respect to a particular partial ordering, must be countable. There are two versions of (♗): one applies to linear operators on $C(K)$-spaces and the other to the compact Hausdorff spaces themselves. We answer two questions that arose after (♗) was first introduced. We show that if $\mathscr{D}$ is a class of compact spaces that is preserved when taking closed subspaces and Hausdorff quotients, and which contains no nonmetrizable linearly ordered space, then every member of $\mathscr{D}$ has (♗). Examples of such classes include all $K$ for which $C(K)$ is Lindelöf in the topology of pointwise convergence (for instance, all Corson compact spaces) and the class of Gruenhage compact spaces. We also show that the set of operators on a $C(K)$-space satisfying (♗) does not form a right ideal in $\mathscr{B}(C(K))$. Some results regarding local connectedness are also presented.

In this paper, we consider a neural network model for solving the generalized nonlinear complementarity problem (denoted by GNCP) over a polyhedral cone. The neural network is derived from an equivalent unconstrained minimization reformulation of the GNCP, which is based on the penalized Fischer–Burmeister function ${\it\phi}_{{\it\mu}}(a,b)={\it\mu}{\it\phi}_{\mathit{FB}}(a,b)+(1-{\it\mu})a_{+}b_{+}$. We establish the existence and the convergence of the trajectory of the neural network, and study its Lyapunov stability, asymptotic stability and exponential stability. It is found that a larger ${\it\mu}$ leads to a better convergence rate of the trajectory. Simulation results are also reported.

In an extrapolation argument, we prove certain $L^{p}\,(1<p<\infty )$ estimates for nonisotropic Marcinkiewicz operators associated to surfaces under the integral kernels given by the elliptic sphere functions ${\rm\Omega}\in L(\log ^{+}L)^{{\it\alpha}}({\rm\Sigma})$ and the radial function $h\in {\mathcal{N}}_{{\it\beta}}(\mathbb{R}^{+})$. As applications, the corresponding results for parametric Marcinkiewicz integral operators related to area integrals and Littlewood–Paley $g_{{\it\lambda}}^{\ast }$-functions are given.

For a locally compact group $G$ with left Haar measure and a Young function ${\rm\Phi}$, we define and study the weighted Orlicz algebra $L_{w}^{{\rm\Phi}}(G)$ with respect to convolution. We show that $L_{w}^{{\rm\Phi}}(G)$ admits no bounded approximate identity under certain conditions. We prove that a closed linear subspace $I$ of the algebra $L_{w}^{{\rm\Phi}}(G)$ is an ideal in $L_{w}^{{\rm\Phi}}(G)$ if and only if $I$ is left translation invariant. For an abelian $G$, we describe the spectrum (maximal ideal space) of the weighted Orlicz algebra and show that weighted Orlicz algebras are semisimple.

In this paper we study the Lorentzian surfaces with finite type Gauss map in the four-dimensional Minkowski space. First, we obtain the complete classification of minimal surfaces with pointwise 1-type Gauss map. Then, we get a classification of Lorentzian surfaces with nonzero constant mean curvature and of finite type Gauss map. We also give some explicit examples.