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- ItemOpen AccessDisappearance of Criticality and Ignition Times of an Exothermic Chemical Reaction in a Reactor(2015-08-21) Adeleye, Olabode EmmanuelThis study determined the criticality, disappearance of criticality and ignition times in some limiting cases for an unsteady state energy balance equation. It also obtained the critical and transitional values of modified Semenov's number Ф, reduced temperature excess and activation energy parameter for three cases: (I) convective only, (II) radiative only and (III) both convective and radiative heat losses considering Arrhenius, bi-molecular and sensitized kinetics cases. This was with view of incorporating both convective and radiative heat losses in an exothermic chemical reaction with pre-exponential factor. The transient variation of temperature equation was solved under physically reasonable assumptions, employing; Semenov's sufficient and necessary conditions for thermal explosion for case I and the modified El-Sayed definitions were used for cases II and III. Moreover, a parameter a being the ratio of the convective heat transfer coefficient (I), to the radiative hear transfer coefficient Ф2, was Introduced into case III to allow for extension of the study to accommodate both convective and radiative heat losses. Subsequently, the ignition times were obtained using a three-term regular perturbation, standard techniques and a numerical procedure. The critical and transitional values of the parameters θ, β and Ф for Arrhenius, bimolecular and sensitized kinetics in case I were investigated. The results showed that in the special case II, the modified Semenov's number at critical point was obtained, while the dimensionless critical temperature obtained was a polynomial of degree 5. Moreover, a transcendental equation involving; the transitional values of θ, β and the reaction order n, was obtained in the form of a polynomial of degree 4. In the event of case III, the modified Semenov's number at critical point was obtained in terms of the other parameters, while the dimensionless critical temperature was obtained as a transcendental equation relating α, βc and n in a polynomial of degree 5. A transcendental equation incorporating; a, θtr, βtr and n in a polynomial of degree 4 was obtained as the point at which transition takes place. In addition, the ignition time obtained numerically for n = 0, β = 0.012, Ф1 =55 and Ф2= 70, was 1.35 X 10-13 seconds. This showed that the reaction was very fast. For the first case, (a) Фtr decreased steadily as n increased provided n ≤ 1, (b) θtr decreased as n increased if n ≤ 0 and increased sharply for 0
- ItemOpen AccessResults of Fixed Points for Some Contractive Maps in Banach Spaces(2015-08-06) Alfred, Olufemi BosedeThis study investigated the approximation of fixed points in relation to the notion of approximate operator and their various contractive maps in Banach spaces on Picard, Mann and Ishikawa iterations with the aim of establishing new results, extending and generalizing some of the existing results in literature. This study also investigated some new convergence conditions with the aim of getting some entirely new convergence results on Picard, Mann and Isikawa iterations in Banach spaces. The notions of approximate operator, continuity and monotonicity of functions, concepts of α-contractions, comparison functions, Zamfirescu mappings, asymptotically non-expansive mappings, asymptotically demicontractive mappings and multi-valued weakly Picard mappings were all employed. Some properties of metric spaces, normed Iinear spaces and Banach spaces were also used. In some of the methods of this study, 'T' was assumed to be a selfmap of a Banach space E while U was assumed to be an approximate operator of T. The estimate ||p — q|| was computed, where p is a fixed point of T and q is an assumed fixed point of U. The following contractive definition ||T x — Ty || ≤ 2δ || x — T x || + δ ||x — y|| , ∀x , y ∈ Ε (i) was also used, where Ε is a normed linear space and T is a selfmap of E satisfying the conditions of a Zamfirescu mapping and 0 < δ < 1. Some new parameters were introduced and the following contractive definitions d(Tx,Ty) ≤ δ [d(x ,Tx ) + d(x,Ty) + d(x , y)], ∀x , y ∈ Κ (ii) and d(Tx,Ty) + d( x , T x ) + d(y,Ty)< 2δd(x,'Ty) +θ d( x ,y) , ∀x , y ∈ Κ (iii) were used, where K is a closed and convex subset of a complete metric space E and T is a selfmap of K with 2 < θ < 5 and for some suitable δ ∈ [0, 1]. The following ϕ−contractive-type definition ||T x — Ty || ≤ ϕ(||x — y|| )+L||x — Tx||, ∀x , y ∈ Ε (iv) was also used, where L > 0 is a constant,ϕ :ℜ+→ ℜ+ is a comparison function, E is a normed linear space and T is a selfmap of E. Modified iteration schemes with uniformly continuous asymptotically nonexpansive and uniformly continuous asymptotically demi-contractive mappings were also employed. New fixed points and convergence results were established for Picard, Mann and Ishikawa iterations in Banach spaces in the sense that for a suitable η > 0 small enough, we have ||p — q|| < 2δ ||xn — p ||+ (2δ + 1)η , 0 <δ < 1 , n = 0 , 1 , 2 , . . . . . ( v ) (1—δ ) (1—δ ) where p is a filed point of T and q is a fixed point of U. Some new convergence results were also obtained on the modified schemes in Hilbert spaces. The equivalence between the convergences of lshikawa, Mann and Picard iterations in Banach spaces was also established. The notion of weak contraction from the case of single-valued mappings to the case of multi-valued weak ϕ—contractions was also extended. In conclusion, this study generalized and extended some existing fixed points and convergence results in literature as well as the equivalence between the convergences of Ishakwa, Mann and Picard iterations in Banach spaces. This study also extended the notion of weak contraction from the case of single-valued mappings to the case of multi-valued weak ϕ— contractions which is an improvement over the existing results in literature.
- ItemOpen AccessCharacterization of Bol Loops of Small Orders.(Obafemi Awolowo University, 1986) Solarin, A. R. T.; Sharma, B. L.Finite Bol loops of small orders are characterised in this study. There exist up to isomorphism 6 non-associative Bol loops of order 8, 2 non-associative Bol loops of order 4p and every Bol loop of order 2p or p2 is a group (where p is an odd prime). Some properties of loops satisfying identities of Bol-Moufang type are discussed. The common properties between these loops and loops satisfying Bol identity are investigated. A general construction Theorem which yields Bol loops of order 2p2 is given. It is also proved that there exist only two non-isomorphic Bol loops of order 2p2. This settles some of the open problems stated by Niederreiter, H. and Robinson, K.H. concerning existence of Bol loops of orders 18, 50 and 98. It is also proved that there are 6(p + 7) and -(p + 5), non-isomorphic Bol loops of order 3p, when 31p-1 and 31p-1 respectively and that Bol loops of order 3p are isomorphic to their loop isotopes. These results are at variance with the claims of Niederreiter, H. and Robinson, K.H. Finally, it is proved that there exist a total of 472 non-isomorphic Bol loops of order 16. This result has been verified on the computer and the relevant programmes are listed in the Appendices.
- ItemOpen AccessA Thermal Explosion Theory for Parallel Reactions.(Obafemi Awolowo University, 1986) Okoya, Samuel Segun; Ayeniss, R. O.The equations for reacting fluids were considered with particular interest on the effect of pre-exponential factor. We assumed constant and variable pre-exponential factors and considered parallel reactions. Essentially our notion of blow up implies non-existence and so we consider in general uniqueness and existence of the resulting equations. We also examined the conditions for thermal explosion. We show that even where there is reactant consumption; there exist one, two or more solutions of the problem depending on the activation energy of the system when the vessel is spherical. Analytical solutions were obtained for the system and bounds on solutions were provided in cases of interest. We discussed the effects of Frank-Kamenetskii parameter, the so called parameter from the introduction of effective activation energy and the initial temperature. Important theorems were stated and proved with respect to these parameters. To emphasize these theorems we illustrated the conclusions graphically. We established that for finite activation energy in unsteady case there is no thermal explosion whether there is diffusion or otherwise. In the limit of large activation energy we compared the explosion time for two step reactions with that obtained when effective activation energy is employed. We discuss in details the conditions for the occurrence of steady solutions.
- ItemOpen AccessA Survey of the Structure Theory of Annihilator Complemented Algebras.(Obafemi Awolowo University, 1985) Benyah, Francis; Oshobi, E. O.In this Thesis we study the structure theory of some algebra, namely, Annihilator Algebras, Modular Annihilator Algebras, and Complemented Algebras. Of particular interest are the works of F. F. Bonsall, A.W. Goldie, Bruce Barnes and Freda Alexander. We also investigate the relationship between these algebras, as well as give simple examples to illustrate their structures.