Browsing by Author "Oduor, Owino Maurice."

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On generation of measurable covers for measurable sets using multiple integral of functions
(200-09-20) Olwamba, Otanga Levi.; Aywa, Shem.; Oduor, Owino Maurice.
A fundamental notion in many areas of mathematics, including optimization, probability, variational problems, functional analysis and operator theory, is that of an integral functional. By this is meant an expression of the form If(x) = ~ f(s,x(s))p(ds), x E X, where X is a linear space of measurable functions defined on a measure space (S,A,~) and having values in a linear space E. The function f: S x E § R is the associated Inte~rand. Classically, only finite integrands on S x R n were studied, usually under the assumption that f(s,x) was continuous in x and measurable in s (the Carathgodory condition). However, from the modern point of view it is essential to admit possibly infinite values for f and If, since it is in this way that important kinds of constraints can most efficiently be represented. Such integrands require a distinctly new theoretical approach, where questions of measurability and the existence of measurable selections are prominent and are reflected in a concept of "normality". The purpose of these notes is to provide a relatively thorough treatment of the most common case in applications, that where E = R n. While many of the results have extensions in one way or another beyond this case, as indicated to some extent in the text, these are often more complicated technically and may require further restrictions. For example, it is only for R n that one presently knows how to develop a complete theory without assuming that the measurable space is complete, an assumption which appears to be awkward in some situations. In treating Inflnite-dimenslonal spaces E, there are the usual problems of the multiplicity of topologies and dualities which must be ironed out. It is desirable, therefore, to have available a full and consistent exposition of the details in the basic case of E = R n, freeing one from the need to search for auxilliary results through sequences of papers wlth varying frameworks.
Perfect repetition codes of ideals of the polynomial ring 2 [ ] ( 1) n n F x mod x − for error control in computer applicatons
(2016) Fanuel, Olege.; Colleta, Okaka Akinyi.; Oduor, Owino Maurice.; Aywa, Shem.
The study of perfect codes has attracted a lot of interest among researchers in coding theory in view of the fact that many authors have indicated that this type of codes is rare. These codes are considered the best for theoretical and practical reasons. In this paper we demonstrate the determination of perfect codes from ideals of polynomial rings and characterize them for error control in computer applications. GAP software has been used to generate these codes and to confirm that they are indeed perfect. The Mathematical Structure of the generating polynomial ring has been fully discussed and the corresponding perfect codes have been characterized. Keywords: Polynomial Ring, Ideals, Perfect codes, Error Control.