T he fractal natu re of quantum paths contributing to Feynman path integral formul ation of Quantum mechanics is investigated. A comp ute r simula t ion of both oneand two-dimensional qu antum harmonic oscilla.tors yields resul ts in agreement wit h rigorou s results on the Hausdorff-Besicovitch dimension for Brownian motion .

By a space, we always mean topological space. Throughout the paper, we mostly use the standard topological notions as in [1] or [3] with the exception that all spaces are assumed without any separation axioms. Especially, compactness is understood without the Hausdorff separation axiom. Some definitions (with broad references and explanations) of less standard notions (related especially to non...

Abstract. This paper presents new sequence spacesX(r, s, t, p;B) forX ∈ {l∞(p), c(p), c0(p), l(p)} defined by using generalized means and difference operator. It is shown that these spaces are complete paranormed spaces and the spaces X(r, s, t, p;B) for X ∈ {c(p), c0(p), l(p)} have Schauder basis. Furthermore, the α-, β-, γduals of these sequence spaces are computed and also obtained necessary...

Let D be a nonempty compact subset of a Banach space X and denote by S(X) the family of all nonempty bounded closed convex subsets of X. We endow S(X) with the Hausdorff metric and show that there exists a set F ⊂ S(X) such that its complement S(X) \ F is σ-porous and such that for each A ∈ F and each x̃ ∈ D, the set of solutions of the best approximation problem ‖x̃− z‖ → min, z ∈ A, is nonempty...

In this work we reproduce the characterization of G s -sets from euclidean setting [11] to more general metric spaces. These sets have Hausdorff dimension at least and are closed by countable intersections, which is particularly useful estimate so called α -approximable points.

Abstract In this paper, we study area-minimizing hypersurfaces in manifolds of Ricci curvature bounded below with Cheeger–Colding theory. Let N i {N_{i}} be a sequence smooth ≥ - n ⁢ κ 2</m:...

In this work the main objective is to extend the theory of Hausdorff measures in general metric spaces. Throughout the thesis Hausdorff measures are defined using premeasures. A condition on premeasures of ‘finite order’ is introduced which enables the use of a Vitali type covering theorem. Weighted Hausdorff measures are shown to be an important tool when working with Hausdorff measures define...

We give conditions on a general family $P_{\lambda}:\R^n\to\R^m, \lambda \in \Lambda,$ of orthogonal projections which guarantee that the Hausdorff dimension formula $\dim A\cap P_{\lambda}^{-1}\{u\}=s-m$ holds generically for measurable sets $A\subset\Rn$ with positive and finite $s$-dimensional measure, $s>m$, lower density. As an application we prove $A,B\subset\Rn$ $s$- $t$-dimensional meas...

We consider any transcendental meromorphic function f of Class S whose Julia set is a Jordan curve. We show that the Julia set of f either is an extended straight line or has Hausdorff dimension strictly greater than 1. The proof uses conformal iterated function systems and extends many earlier results of this type.