Results in Mathematics最新文献

This is an exposition on supercyclicity and weak supercyclicity, especially designed to advance further developments in weakly supercyclicity, which is a recent research field showing significant momentum during the past two decades. For operators on a normed space, the present paper explores the relationship among supercyclicity and three versions of weak supercyclicity, namely, weak l-sequential, weak sequential, and weak supercyclicities, and their connection with strong stability, weak stability, and weak quasistability. A survey of the literature on weak supercyclicity is followed by an analysis of the interplay between supercyclicity and strong stability, as well as between weak supercyclicity and weak stability. A description of the spectrum of weakly l-sequentially supercyclic operators is also given.

An operator (Tin {mathcal {L(H)}}) is said to be C-normal if there exists a conjugation C on ({{mathcal {H}}}) such that the commutator ([(CT)^{#}, CT]) equals zero, where ([R,S]:=RS-SR) and (R^{#}) is a Hermitian adjont operator of R as in (1). If there exists a conjugation C with respect to which (Tin mathcal {L(H)}) is C-normal, then T is called a conjugation-normal operator. In this paper, we study properties of conjugation-normal operator matrices. In particular, we focus on the conjugation-normality of the component operators of operator matrices which are conjugation-normal.

In the present paper we considered the problems of studying the best approximation order and inverse approximation theorems for families of neural network (NN) operators. Both the cases of classical and Kantorovich type NN operators have been considered. As a remarkable achievement, we provide a characterization of the well-known Lipschitz classes in terms of the order of approximation of the considered NN operators. The latter result has inspired a conjecture concerning the saturation order of the considered families of approximation operators. Finally, several noteworthy examples have been discussed in detail

In this paper, we characterize the extendibility of the normal curvature at singularities of frontals and give a representation formula for the class of frontals with this property. We introduce the relative normal curvature, which allows us to study the classical normal curvature, the asymptotic curves and the lines of curvature through singularities. Also, we provide representation formulas for wavefronts near all types of singularities and subclasses, such as wavefronts with extendable Gaussian curvature, bounded Gaussian curvature, and extendable principal curvature, among others.

We give new representations of various power sums, including the signed/alternating sum of powers of consecutive integers as well as odd numbers. The generalized Eulerian power sums (E_{n,m}(z):=sum _{k=1}^n k^mz^k) are also considered.

The goal of this paper is to develop a dilation theory on the regular ({{mathcal {U}}})-twisted polyball (textbf{B}_{{mathcal {U}}}({{mathcal {H}}})), which is the set of all k-tuples (T:=(T_1,ldots , T_k)) of row contractions (T_i:=[T_{i,1}cdots T_{i,n_i}]) on a Hilbert space ({{mathcal {H}}}) satisfying certain positivity condition on the defect operator (Delta _T(I)) and ({{mathcal {U}}})-commutation relations, where ({{mathcal {U}}}subset B({{mathcal {H}}})) is an appropriate set of unitary operators. The role of operator model is played by a standard multi-shift (textbf{S}:=(textbf{S}_1,ldots , textbf{S}_k)) with (textbf{S}_i:=[textbf{S}_{i,1}cdots textbf{S}_{i,n_i}]), which is a k-tuple of doubly (Iotimes {{mathcal {U}}})-commuting pure row isometries on the Hilbert space (ell ^2({{mathbb {F}}}_{n_1}^+times cdots times {{mathbb {F}}}_{n_k}^+)otimes {{mathcal {H}}}), where ({{mathbb {F}}}_{n_i}^+) is the unital free semigroup with (n_i) generators and each (textbf{S}_{i,j}) is a weighted shift with operator-valued weights. It is shown that many of the classical results concerning the dilation theory of contractions on Hilbert spaces have analogues for ({{mathcal {U}}})-twisted polyballs. This includes: Sz.-Nagy dilation theorem, von Neumann inequality, Itô and Brehmer dilations for commuting isometries and contractions, respectively, and Beurling characterization of the invariant subspaces for the unilateral shift on the Hardy space (H^2).

We present a characterization for the rotational soliton for the curve shortening flow (CSF) on the revolution surfaces of (mathbb {R}^3). Furthermore, we describe the behavior of such curves by showing that the two ends of each open curve are asymptotic to a parallel geodesic.

Motivated by recent investigations of Sophie Grivaux and Étienne Matheron on the existence of invariant measures in Linear Dynamics, we introduce the concept of locally bounded orbit for a continuous linear operator (T:Xlongrightarrow X) acting on a Fréchet space X, and we use this new notion to construct (non-trivial) T-invariant probability Borel measures on ((X,mathscr {B}(X))).