Central Limit Theorem in View of Subspace Convex-Cyclic Operators

In our work we have defined an operator called subspace convex-cyclic operator. The property of this newly defined operator relates eigenvalues which have eigenvectors of modulus one with kernels of the operator. We have also illustrated the effect of the subspace convex-cyclic operator when we let it function in linear dynamics and joining it with functional analysis. The work is done on infinite dimensional spaces which may make linear operators have dense orbits. Its property of measure preserving puts together probability space with measurable dynamics and widens the subject to ergodic theory. We have also applied Birkhoff’s Ergodic Theorem to give a modified version of subspace convex-cyclic operator. To work on a separable infinite Hilbert space, it is important to have Gaussian invariant measure from which we use eigenvectors of modulus one to get what we need to have. One of the important results that we have got from this paper is the study of Central Limit Theorem. We have shown that providing Gaussian measure, Central Limit Theorem holds under the certain conditions that are given to the defined operator. In general our work is theoretically new and is combining three basic concepts dynamical system, operator theory and ergodic theory under the measure and statistics theory.

Frame-cyclic operators and their properties

Motivated by frame-vector for a unitary system, we study a class of cyclic operators on a separable Hilbert space which is called frame-cyclic operators. The orbit of such an operator on some vector, namely frame-cyclic vector, is a frame. Some properties of these operators on finite- and infinite-dimensional Hilbert spaces and their relations with cyclic and hypercyclic operators are established. A lower and upper bound for the norm of a self-adjoint frame-cyclic operator is obtained. Also, construction of the set of frame-cyclic vectors is considered. Finally, we deal with Kato’s approximation of frame-cyclic operators and discuss their frame-cyclic properties.

Automated design and parallelization of programs for heterogeneous platforms using algebra-algorithmic tools

Methods and software tools for automated design and generation of OpenCL programs based on the algebra of algorithms are proposed. OpenCL is a framework for developing parallel software that executes across heterogeneous platforms consisting of general-purpose processors and/or hardware accelerators. The proposed approach consists in using high-level algebra-algorithmic specifications of programs represented in natural linguistic form and rewriting rules. The developed software tools provide the automated design of algorithm schemes based on a superposition of Glushkov algebra constructs that are considered as reusable components. The tools automatically generate code in a target programming language on the basis of the specifications. In most computing problems, a large part of hardware resources is utilized by computations inside loops, therefore the use of automatic parallelization of cyclic operators is most efficient for them. However, the existing automatic code parallelizing tools, such as Par4All, don’t account the limited amount of accelerator’s onboard memory space while real-life problems demand huge amounts of data to be processed. Thus, there is a need for the development of a parallelization technique embracing the cases of massive computational tasks involving big data. In the paper, a method and a software tool for semi-automatic parallelization of cyclic operators based on loop tiling and data serialization are developed. The parallelization technique uses rewriting rules system to transform programs. The framework for parallelization of loops for optimization of computations using graphics processing units allows semi-automatic parallelization of sequential programs. The approach is illustrated on an example of developing a parallel OpenCL image convolution program.

Examples of cyclical operators

In this paper, we give examples of cyclic operators defined on various types of sets, in order to illustrate some results in the extremely rich literature following the seminal paper [Kirk, W. A., Srinivasan, P. S. and Veeramani, P., Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory, 4 (2003), No. 1, 79 – 89]. All examples which are presented enrich the list of cyclic operators and give a subject to future studies of this type of operators.