Let $K$ be a complete infinite rank valued field and $E$ a $K$-Banach space with a countable orthogonal base. In [9] and [10] we have studied bounded (called Lipschitz) operators on $E$ and introduced the notion of a strictly Lipschitz operator. Here we characterize them, as well as compact and nuclear operators, in terms of their (infinite) matrices. This results provide new insights and also useful criteria for constructing operators with given properties.

In [17] the author considered a compound Markov renewal process $(\widetilde{S}_{N_t})$ where $((J_n,S_n))$ and $((\widetilde{J}_n,\widetilde{S}_n))$ are suitable independent Markov additive processes such that $(S_n-S_{n-1})$ are positive random variables, and $N_t=\sum _{n\geq 1}1_{S_n\leq t}$. In this paper we present the analogous results for a more general situation where we consider a unique Markov additive process $((J_n,Z_n))$ in place of $((J_n,S_n))$ and $((\widetilde{J}_n,\widetilde{S}_n))$, and $Z_n=(\widetilde{S}_n,S_n)$. Some further results are also presented; in particular we relate in terms of large deviations the sequence $((\widetilde{S} _n,S_n))$ and the process $((\widetilde{S}_{N_t},N_t))$.

Suppose that consumers have to classify $n$ items or baskets of goods according to their individual preferences or utility and such that ties are allowed. In this paper we study the number of possible classifications or outcomes $f(n)$. We obtain different representations for $f(n)$ and use singularity analysis to determine the asymptotic behaviour of $f(n)$. We also give a probabilistic interpretation of $f(n)$ and use a renewal argument to study $f(n)$ as $n\rightarrow \infty $. Assuming that each of the $f(n)$ outcomes has equal probability to occur, we study the random variable $N_{n}$ where $N_{n}$ equals the number of most preferred items, i.e. the number of items on the top of the list.

Let $A$ be a bounded linear operator acting on a Hilbert space $H$. In [32], A. Uchiyama proved that Weyl's theorem holds for class A operators with the additional condition that $\ker A|_{[TH]}=0$ and he showed that every class A operator whose Weyl spectrum equals to zero is compact and normal. In this paper we show that Weyl's theorem holds for algebraically class $A$ operator without the additional condition $\ker A|_{[TH]}=0$. This leads as to show that a class $A$ operator whose Weyl spectrum equals to zero is always compact and normal.

A number of ways of relaxing the stratification constraint for the axioms of Quine's NF are reviewed. It is shown how most of them result in inconsistency.

A new third order finite difference scheme for the solution of initial value problems for hyperbolic conservation laws is presented. The advantages of the scheme are its simplicity, third order accuracy and that it can be used for large time steps which saves more time. The scheme is proved stable for initial and initial boundary value problems for linear case. The technique of making the third order scheme oscillations free (TVD) is carried out. In this paper we extend TVD scheme to two dimension problems. The extension of the TVD scheme to nonlinear system of equations is illustrated by solving shallow water equations. Numerical results are presented and compared with exact solutions and other methods.

Curvature properties of some generalizations of contact metric manifolds are studied, with special attention to $\left(\kappa,\mu\right)$-nullity conditions in the framework of $\cal S$-manifolds.

In this paper, a Fourier spectral method for an initial boundary value problem for a class of systems of generalized Zakharov equations is proposed. Semi-discrete and fully discrete Fourier spectral schemes are given. In fully discrete case we have established a two level scheme which is convenient and saves time in real computation. An energy estimation method is used to obtain error estimates for approximate solutions.

Let $\Bbb D$ be the open unit disk in the complex plane. For $\varepsilon \gt 0$ we consider the sector $\Sigma_{\varepsilon} = \{z\; : \; |\arg z | \lt \varepsilon \}$. We will prove that for certain classes of functions $f$ in the Besov's space $B_p\left(\Bbb D\right)$ such that $f(0)=0$, the $B_p$ norm is obtained by integration over $f^{-1}(\Sigma_{\varepsilon})$.

In [7] Oprea gave an improved version of Chen's inequality for Lagrangian submanifolds of $\mathbb CP^n(4)$. For minimal submanifolds this inequality coincides with a previous version proved in [5]. We consider here those non-minimal $3$-dimensional Lagrangian submanifolds in $\mathbb CP^3 (4)$ attaining at all points equality in the improved Chen inequality. We show how all such submanifolds may be obtained starting from a minimal Lagrangian surface in $\mathbb CP^2(4)$.

In this note we provide a new insight into trans-separable spaces, i.e. those which are separable by seminorms. This approach enables getting an easy proof of the fact that in a wide class of uniform spaces, containing $(DF)$ and $(LM)$-spaces, precompact subsets are metrizable.

We give a classification of parallel surfaces in the groups of rigid motions of Minkowski plane and Euclidean plane, equipped with a general left-invariant metric. Our result completes the classification of parallel surfaces in the eight three-dimensional model geometries of Thurston and in three-dimensional unimodular Lie groups with maximal isometry group.

Let $\Omega $ and $\Pi $ be two simply connected domains in the complex plane $ \mathds{C}$ which are not equal to the whole plane $\mathds{C}$. We are concerned with the set $A(\Omega,\Pi)$ of functions $f: \Omega\to\Pi$ holomorphic on $\Omega$ and we prove estimates for $|f^{(n)}(z)|, f\in A(\Omega,\Pi), z \in \Omega$, of the following type. Let $\lambda_{\Omega}(z)$ and $\lambda_{\Pi}(w)$ denote the density of the Poincar\'{e} metric of $\Omega$ at $z$ and of $\Pi$ at $w$, respectively. Then for any pair $(\Omega,\Pi)$ where $\Omega$ is convex, $f\in A(\Omega,\Pi), z \in \Omega$, and $n\geq 2$ the inequality \[ \frac{|f^{(n)}(z)|}{n!}\leq (n+1) 2^{n-2}\frac{(\lambda_{\Omega}(z))^n}{\lambda_{\Pi}(f(z))} \] is valid.

For functions $f\in A(\Omega,\Pi)$, which are injective on $\Omega$, the validity of above inequality was conjectured by Chua in 1996.

We consider locally symmetric almost Kenmotsu manifolds showing that such a manifold is a Kenmotsu manifold if and only if the Lie derivative of the structure, with respect to the Reeb vector field $\xi$, vanishes. Furthermore, assuming that for a $(2n+1)$-dimensional locally symmetric almost Kenmotsu manifold such Lie derivative does not vanish and the curvature satisfies $R_{XY}\xi =0$ for any $X, Y$ orthogonal to $\xi$, we prove that the manifold is locally isometric to the Riemannian product of an $(n+1)$-dimensional manifold of constant curvature $-4$ and a flat $n$-dimensional manifold. We give an example of such a manifold.

The aim of this paper is to give necessary and sufficient conditions for uniform exponential instability of linear skew-product semiflows in terms of functionals on normed function spaces. We obtain the versions of some results due to Datko, Pazy, Neerven and Rolewicz for the case of instability of linear skew-product semiflows.

We study the geodesic flow on the global holomorphic sections of the bundle $\pi:{\mbox{TS}}^2\rightarrow \mbox{S}^2$ induced by the neutral Kähler metric on the space of oriented lines of ${\Bbb{R}}^3$, which we identify with ${\mbox{TS}}^2$. This flow is shown to be completely integrable when the sections are symplectic, and the behaviour of the geodesics is described.