in this paper, we introduce the concepts of $2$-isometry, collinearity, $2$%-lipschitz mapping in $2$-fuzzy $2$-normed linear spaces. also, we give anew generalization of the mazur-ulam theorem when $x$ is a $2$-fuzzy $2$%-normed linear space or $im (x)$ is a fuzzy $2$-normed linear space, thatis, the mazur-ulam theorem holds, when the $2$-isometry mapped to a $2$%-fuzzy $2$-normed linear space...

We consider models of random Schrr odinger operators which exist thanks to a cohomological theorem in ergodic theory. Examples are ergodic Schrr odinger operators with random magnetic uxes on discrete two-dimensional lattices or non-periodic situations like Penrose lattices.

Using the ratio ergodic theorem for a measure preserving transformation in a σ-finite measure space we give a straightforward proof of Derriennic’s reverse maximal inequality for the supremum of ergodic ratios.

In this paper, we give a new proof of a result of R. Jones showing almost everywhere convergence of spherical means of actions of R on L(X)-spaces are convergent for d ≥ 3 and p > d d− 1 . This is done by adapting the proof of the spherical maximal theorem by Rubio de Francia so as to obtain directly the ergodic theorem.

In a recent work [3], the authors established new results about general linear Mahler systems in several variables from perspective of transcendental number theory, such as multivariate extension Nishioka’s theorem. Working with functions and different transformations leads to complications, including need prove vanishing theorem use tools ergodic Ramsey theory Diophantine approximation (e.g., ...

The aim of this thesis is to formulate and prove quantum extensions of the famous Shannon-McMillan theorem and its stronger version due to Breiman. In ergodic theory the Shannon-McMillan-Breiman theorem is one of the fundamental limit theorems for classical discrete dynamical systems. It can be interpreted as a special case of the individual ergodic theorem. In this work, we consider spin latti...

As a cornerstone of functional analysis, Hahn–Banach theorem constitutes an indispensable tool of modern analysis where its impact extends beyond the frontiers of linear functional analysis into several other domains of mathematics, including complex analysis, partial differential equations and ergodic theory besides many more. The paper is an attempt to draw attention to certain applications o...

In this note we intend to present basic ergodic theory. We begin with the notion of a measure preserving transformation. We then define ergodicity and provide examples. Finally we sketch Birkhoff’s Ergodic Theorem and elucidate it with some examples.

Ergodic theory studies the long-term averaging properties of measurepreserving dynamical systems. In this paper, we state and present a proof of the ergodic theorem due to George Birkhoff, who observed the asymptotic equivalence of the time-average and space-average of a point x in a finite measure space. Then, we examine a number of applications of this theorem in number-theoretic problems, in...