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Vol 54, No 1 (2019)
- Year: 2019
- Articles: 6
- URL: https://journals.rcsi.science/1068-3623/issue/view/14096
Differential Equations
A Moment Condition and Non-synthetic Diagonalizable Operators on the Space of Functions Analytic on the Unit Disk
Abstract
Examples are given of (continuous, linear) operators on the space of functions analytic on the open unit disk in the complex plane having the monomials as eigenvectors, but which fail spectral synthesis (that is, which have closed invariant subspaces which are not the closed linear span of any collection of eigenvectors).
1-7
Real and Complex Analysis
SWW Sequences and the Infinite Ergodic Random Walk
Abstract
This article is concerned with demonstrating the power and simplicity of sww (special weakly wandering) sequences. We calculate an sww growth sequence for the infinite measure preserving random walk transformation. From this we obtain the first explicit eww (exhaustive weakly wandering) sequence for the transformation. The exhaustive property of the eww sequence is a “gift” from the sww sequence and requires no additional work. Indeed we know of no other method for finding explicit eww sequences for the random walk map or any other infinite ergodic transformation. The result follows from a detailed analysis of the proof of Theorem 3.3.12 in the book S.Eigen, A.Hajian, Y.Ito, V.Prasad, Weakly Wandering Sequences in Ergodic Theory (Springer, Tokyo, 2014) as applied to the random walk transformation from which an sww growth sequence is obtained.We explain the significance of sww sequences in the construction of eww sequences.
20-27
On Continuous Selections of Set-valued Mappings with Almost Convex Values
Abstract
In this paper, it is proved that through each point of the graph of a continuous setvalued mapping with almost convex and star-like values can be passed a continuous selection of that mapping.
28-37
Weighted Norm Inequalities for Area Functions Related to Schrödinger Operators
Abstract
Let L = −Δ + V be a Schrödinger operator, where Δ is the Laplacian operator on ℝn, and V is a nonnegative potential belonging to certain reverse Hölder class. In this paper, we establish some weighted norm inequalities for area functions related to Schrödinger operators and their commutators.
38-50
Some Formulas for the Generalized Analytic Feynman Integrals on the Weiner Space
Abstract
In this paper, we define a new concept of analytic Feynman integral on theWiener space, which is called the generalized analytic Feynman integral, to explain various physical circumstances. Furthermore, we evaluate the generalized analytic Feynman integrals for several important classes of functionals.We also establish various properties of these generalized analytic Feynman integrals. We conclude the paper by giving several applications involving the Cameron-Storvick theorem and quantum mechanics.
8-19
Probability Theory and Mathematical Statistics
Joseph Mecke’s Last Fragmentary Manuscripts - a Compilation
Abstract
Summarizing results from Joseph Mecke’s last fragmentary manuscripts, the generating function and the Laplace transform for nonnegative random variables are considered. The concept of thickening of a random variable, as an inverse operation to thinning (which is usually applied to point processes) is introduced, based on generating functions, and a characterization of thickable random variables is given. Further, some new relations between exponential distributions and their interpretation in terms of Poisson point processes are derived with the help of the Laplace transform.
51-60