Mathematics and Statistics

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On The Convexity of A Generalized q-Numerical Range’’
(2005-01) Musundi, Sammy Wabomba
For a given $q in kom$ with $|q| le 1$, we study the $C$-numerical range of a Hilbert space operator where $C$ is an operator of the form [ left( begin{array}{ccc} qI_n & sqrt{1-|q|^2}I_n \ 0_n & 0_n end{array} right) oplus 0. ] Some known results on the $q$-numerical range are extended to this set.
Unconditional Banach Space Ideal Property’’
(Scientific Advances Publishers, 2012) Musundi, Sammy 1; Shem, Aywa 2; Fourie, Jan 3; Matuya, John Wanyonyi 4
Abstract Let Lw ′ denote the assignment which associates with each pair of Banach spaces X , Y , the vector space Lw ′ ( X , Y ) and K ( X , Y ) be the space of all compact linear operators from X to Y. Let T ∈ Lw ′ ( X , Y ) and suppose (Tn ) ⊂ K ( X , Y ) converges in the dual weak operator topology (w′) of T. Denote by K u ((Tn )) the finite number given by K u ((Tn )) := sup { max { Tn , T − 2Tn }} . n∈N ′ The u-norm on Lw ( X , Y ) is then given by T u := inf { K u ((Tn )) : T = w′ − lim Tn , n Tn ∈ K ( X , Y )}. ′ It has been shown that ( Lw ( X , Y ) . u ) is a Banach operator ideal. We find ′ conditions for K ( X , Y ) to be an unconditional ideal in ( Lw ( X , Y ) . u ) .
Equivalent Banach operator ideal norms’’
(2012-01) Musundi, S. Wambomba; Shem, Aywa; Fourie, Jan
Let X, Y be Banach spaces and consider the w'-topology (the dual weak operator topology) on the space (L(X, Y) of bounded linear operators from X into X with the uniform operator norm. L w' (X, Y) is the space of all T ∈ L(X, Y) for which there exists a sequence of compact linear operators (Tn) ⊂ K(X, Y) such that T = w' - lim nT n. Two equivalent norms, on L w'(X, Y) are considered. We show that and Banach operator ideals.
On Similarity and Quasisimilarity equivalence relations
(2012-01) Sitati, Isaiah Nalianya; Musundi, Sammy Wabomba; Nzimbi, Benard Mutuku; Kirimi, Jacob
Similarity and unitary equivalence can be shown to be of equivalence relations. We discuss a result showing that two similar operators have equal spectra (i.e. point and approximate point spectrum). More so, unitary equivalence results for invariant subspaces and normal operators are proved. For similar normal operators, we state the Fuglede – Putnam –Rosenblum theorem that makes proofs for similar normal operators more simplified. It is also noted that direct sums and summands are preserved under unitary equivalence. Furthermore, we show that the natural concept of equivalence between Hilbert Space operators is unitary equivalence which is stronger than similarity. By introducing the notion of quasisimilarity of operators which is the same as similarity in finite dimensional spaces, but in infinite dimensional spaces, it is a much weaker relation, we further show that quasisimilarity is an equivalence relation. We also link invariant subspaces and hyperinvariant subspaces with quasisimilarity where it is seen that similarity preserves nontrivial invariant subspaces while quasisimilarity preserves nontrivial hyperinvariant subspaces.
Magnetic field and hall current effect on Mud free convection flow past a vertical rotating flat plate
(2012-11) Kirimi, Jacob; Musundi, Sammy Wabomba*; Kinyanjui, Anthony Muthondu; Lunani, Murwayi Alice
In this study, effects of Hall current and rotation on an electrically conducting rarefied gas are investigated . A constant , strong magnetic field is applied at an angle φ to both the electric field and the direction of flow of the fluid. The study seeks to find the effects of Hall current and angle of application of the magnetic field on velocity and temperature profiles . The flow of the gas is considered as unsteady and restricted to laminar domain. The set of equations which describe the flow are a combination of the Navier-Stokes equations of fluid dynamics, generalized Ohm’s law, Maxwell’s equations, equation of continuity and equation of energy. These equations are solved numerically using the finite difference method. Numerical results of velocity and temperature profiles are analyzed using tables and graphs for Hall current parameters m* ranging from 0.0 to 1.0 and parameter λ for angle of application of the magnetic field ranging from 0.0 to 1.0.
On Characterization of u-ideals determined by sequences’’
(2013) Matuya, wanyonyi john; Makila, Patrick; Simiyu, Achiles; Shem, Aywa; Musundi, Sammy
The area of ideals is important in the study of Analysis, algebra, Geometry and Computer science. The various types of ideals have been studied, for example m ideals and h ideals. The m ideals defined on real Banach spaces are referred to as u - ideals. The natural examples of u - ideals with respect to their biduals, are order continuous Banach lattices. Using the approximation property, we shall study properties of u - ideals and their characterization. We define the set of compact operators K (X) on X to be u - ideals given that X is a separable reflexive Banach space with approximation property if and only if there is a sequence (Tn ) of finite rank of operators with lim 2 1 n n →∞ I T − = and lim n n →∞Tx x = . We shall show that u -ideals containing no copies of sequences 1  are strict u - ideals
On Almost Similarity Operator Equivalence Relation
(2013-06) Musundi, S. Wabomba 1*; Sitati, N. Isaiah 2; Nzimbi, B. Mutuku 3; Murwayi, A. Wunani 4
We consider the almost similarity property which is a new class in operator theory and was first introduced by A. A. S. Jibril. We establish that almost similarity is an equivalence relation. Some results on almost similarity and isometries, compact operators, hermitian, normal and projection operator are also shown. By characterization of unitary equivalence operators in terms of almost similarity we prove that operators that are similar are almost similar. We also claim that quasi-similarity implies almost similarity under certain conditions (i.e. if the quasiaffinities are assumed to be unitary operators). Furthermore, a condition under which almost similarity of operators implies similarity is investigated. Lastly, we show that two bounded linear operators of a Banach algebra on a Hilbert space are both completely nonunitary if they are contractions which are almost similar to each other.
Rapid, Economical and selective Compexometric Determination of Iron (III) in its synthetic alloys using 3 hydroxy-3- phynyl-1-(2,4,6-tribromophynyl)
(2013-09) Ochieng, Ombaka; Musundi, Sammy Wabomba; Gitonga, L. K .; Kibara, D.
The present study describes a simple, selective, rapid and economical method for the determination of iron (III) in its synthetic alloys using 3-hydroxy -3- phenyl-1- (2,4,6- tribro mophenyl) triazene as metallochromic indicator in the 𝑃𝐻 and temperature range of 2.5-3.0 and 20 −600𝑐 respectively. The colour and shape of the synthesized indicator was light yellow shinning needles having melting point of 590𝑐. It was crystallized from ethanol. The results of elemental study showed that, the values of C, H, N obtained experimentally agrees very well with those obtained theoretically. The colour at the end point changes from violet to light yellow using EDTA as a titrant. There is no interference in either determination from common metal and anion ions other than Pb(II), Cr(II), Mo(VI), Mn(II), U(vI), Cu(II), Cd(II), 𝐹−, 𝑃𝑂43−, 𝐶2𝑂42−, 𝐻𝑃𝑂42−. Reproducible and accurate results are obtained for 5.59 - 1.12mg of Iron with relative error less than ±1.79% and standard deviation not more than 0.10%. The results of the test method and reference method (Atomic absorption spectrophotometric) showed that, there is no statistical difference in the results by the two methods.
``Transitivity Action of An(n=5,6,7,8) On Unordered and Ordered Pairs’’
(2013-09) Sitati, N. I; Kinyanjui, J.N; Musundi, S.W; Rimberia, J.; Makila, P.
In this paper, we study some transitivity action properties of the alternating group 𝐴𝑛(𝑛=5,6,7 ,8) acting on unordered and ordered pairs from the set 𝑋𝑋 = {1,2,…,𝑛𝑛} through determination of the number of disjoint equivalence classes called orbits. When 𝑛𝑛≤ 8, the alternating group acts transitively on both X (2) and X[2]. Mathematics Subject Classification: 20BO5, 06A75, 06F15.
Suborbital Graphs and their properties for unordered pairs in An(n=5,6,7,8) through rank and subdegrees determination’’
(2013-12-18) Kinyanjui, J. N. 1; Musundi, S. W. 2*; Rimberia, J. 3; Mwenda, E. 4; Ruto, P. K. 5
In this paper, through computation of the rank and subdegrees of alternating group ( =5,6,7) on unordered triples we construct the suborbital graphs corresponding to the suborbits of these triples. When ( ≥ 5) acts on unordered pairs the suborbital graphs corresponding to the non-trivial suborbits are found to be connected, regular and have undirected edge except when n=6. Further, we investigate properties of the suborbital graphs constructed.
Application of Banach space ideal properties in image transmission over wireless network
(2014-04) Musundi, Sammy Wabomba; Ochieng, Ombaka; Njogu, Muriuki; Kinyua, Charles
The Banach space operator ideals and nuclear maps have a large class of morphisms which behave as if they were part of a compact closed category, that is, they allow one to transfer variables between the domain and the codomain. We use the concept of nuclearity in functional analysis to establish application aspect of Banach space ideal properties in the transmission of image over wireless network based on the embedded system.
Distribution of Spectrum in a Direct Sum Decomposition of Operators into Normal and Completely Non normal parts;
(Modern Scientific Press Company, Florida, USA, 2014-09-05) Mwenda, E.1; Musundi, S. W.2*; Nzimbi, B. M. 1; Marani, V. N. 3; Loyford, N. 4
We discuss the distribution of spectra of a direct sum decomposition of an arbitrary operator into normal and completely non normal parts. We utilize the fact that any given operator 𝑇∈𝐵(𝐻) can be decomposed into a direct summand 𝑇=𝑇1⊕𝑇2 with 𝑇1 and 𝑇2 are the normal and completely non normal parts respectively. This canonical decomposition is preferred to other forms of decomposition such as Polar and Cartesian decompositions because these two do not transfer certain properties (for instance the spectra, numerical range, and numerical radius) from the original /decomposed operator to the constituent parts. This is presumably done since these parts are simpler to deal with.
A note on quasi-similarity of operators in Hilbert spaces
(2015-01) Isaiah, N. Sitati 1; Musundi, Sammy W. 2*; Nzimbi, Bernard M. 3; Kikete, W. Dennis 3
In this paper we introduce the notion of Quasi-similarity of bounded linear operators in Hilbert Spaces. We do so by defining a quasi-affinity from one Hilbert Space H to K. Some results on quasi-affinities are also discussed. It has already been shown that on a finite dimensional Hilbert Space, quasi similarity is an equivalence relation that is; it is reflexive, symmetric and also transitive. Using the definition of commutants of two operators, we give an alternative result to show that quasi similarity is an equivalence relation on an infinite dimensional Hilbert Space. Finally, we establish the relationship between quasi similarity and almost similarity equivalence relations in Hilbert Spaces
Analysis of Volatility of Real Exchange Rate and Exports in Kenya using the Garch Model: 2005:2012
(2015-07) Mustapha, Wasseja; Musundi, Sammy Wabomba; Njoroge, Elizabeth; Ngugi, Mwenda
The real exchange rate has proven to be an important factor in international trade because it is expected that exports respond to real exchange rate movements with respect to the characteristics of the importing and exporting countries. Exchange rate volatility increases uncertainty of profits on contracts denominated in foreign currency and subsequently dampens trade and economic growth. This study investigated how real exchange rate volatility affected exports of key Kenyan commodities to the European Union and United Kingdom, namely; tea, coffee and horticulture to the European Union. The presence of exchange rate volatility was determined using the GARCH model. A Bounds testing and Autoregressive Distributed Lag model was used to establish the presence of a long run relationship between exchange rate volatility and commodity exports. Findings revealed that exchange rate volatility affected tea exports to the UK and horticulture exports to the European Union. Foreign income played an important role in explaining tea and coffee exports to the UK and EU respectively.
A NOTE ON QUASI-SIMILARITY OF OPERATORS IN HILBERT SPACES
(International Journal of Mathematical Archive, 2015-07-23) SAMMY W. MUSUNDI; ISAIAH N.SITATI; BERNARD M. NZIMBI; KIKETE W. DENNIS
In this paper we introduce the notion of Quasi-similarity of bounded linear operators in Hilbert Spaces. We do so by defining a quasi- affinity from one Hilbert Space H to K. Some results on quasi- affinities are also discussed. It has already been shown that on a finite dimensional Hilbert Space, quasi similarity is an equivalence relation that is; it is reflexive, symmetric and also transitive. Using the definition of commutants of two operators, we give an alternative result to show that quasi similarity is an equivalence relation on an infinite dimensional Hilbert Space. Finally, we establish the relationship between quasi similarity and almost similarity equivalence relations in Hilbert Spaces using hermitian and normal operators.
Modeling and Forecasting Kenyan GDP Using Autoregressive Integrated Moving Average (ARIMA) Models
(Science Publishing Group, 2016-04-13) Musundi, Sammy Wabomba; M’mukiira, Peter Mutwiri; Mungai, Fredrick
The Gross Domestic Product (GDP) is the market value of all goods and services produced within the borders of a nation in a year. In this paper, Kenya’s annual GDP data obtained from the Kenya National Bureau of statistics for the years 1960 to 2012 was studied. Gretl and SPSS 21 statistical softwares were used to build a class of ARIMA (autoregressive integrated moving average) models following the Box-Jenkins method to model the GDP. ARIMA (2, 2, 2) time series model was established as the best for modeling the Kenyan GDP according to the recognition rules and stationary test of time series under the AIC criterion. The results of an in-sample forecast showed that the relative and predicted values were within the range of 5%, and the forecasting effect of this model was relatively adequate and efficient in modeling the annual returns of the Kenyan GDP. Finally, we used the fitted ARIMA model to forecast the GDP of Kenya for the next five years.
Modeling and Forecasting Kenyan GDP Using Autoregressive Integrated Moving Average (ARIMA) Models
(2016-04-13) Musundi, Sammy Wabomba *; M’mukiira, Peter Mutwiri; Mungai, Fredrick
The Gross Domestic Product (GDP) is the market value of all goods and services produced within the borders of a nation in a year. In this paper, Kenya's annual GDP data obtained from the Kenya National Bureau of statistics for the years 1960 to 2012 was studied. Gretl and SPSS 21 statistical softwares were used to build a class of ARIMA (autoregressive integrated moving average) models following the Box-Jenkins method to model the GDP. ARIMA (2, 2, 2) time series model was established as the best for modeling the Kenyan GDP according to the recognition rules and stationary test of time series under the AIC criterion. The results of an in-sample forecast showed that the relative and predicted values were within the range of 5%, and the forecasting effect of this model was relatively adequate and efficient in modeling the annual returns of the Kenyan GDP. Finally, we used the fitted ARIMA model to forecast the GDP of Kenya for the next five years.
On Almost Similarity and Metric Equivalence of operators. Accepted for publication
(2016-06) Nzimbi, B. M. 1; Luketero, S. W.1; Sitati, I. N. 2; Musundi, S.W. 3; Mwenda, E 3
In this paper, we investigate almost-similarity relation, metric equivalence and other relations of operators on Hilbert spaces. We characterize almost-similar, metrically equivalent operators. Roughly speaking, we consider two operators to be “equivalent” if they are “close” to each other in some sense. We establish some spectral invariants with respect to each of these equivalence relations.
Assessment of Water Quality in Boreholes and Wells in Waa Location, Kwale County –Kenya”.
(CODEN (USA), 2017) Kilwake, J. Wanjala1; Mwakio, Tole1; Musundi, Sammy 2*
Water from boreholes and dug wells is extensively used in Kwale County, especially by rural communities living away from established market centers, where piped water is commonly available. The study aimed to assess the quality of water in boreholes and dug wells found in Waa location of Kwale County – Kenya. Selection of the boreholes and dug wells was carried out using purposive sampling and simple random sampling. All the seventy one boreholes and wells in Waa location were visited and inspected to determine their sanitary condition and functionality. Twenty eight samples of water that were collected in duplicate from 14 boreholes and dug wells (30% of total number) were analyzed for faecal coliform (Escherichia. coli), total coliform count, pH, total dissolved solids, turbidity, colour, total hardness, salinity, chloride content, electrical conductivity, total alkalinity, Ca2+ and Mg2+ using 3M Petrifilm™ method, pH meter, HACH digital titrator, Total dissolved solids/Conductivity meter, and DR 2000 (HACH) spectrophotometer at KIMAWASCO laboratory. The study revealed that 32% of the boreholes and dug wells have either permanently or temporarily failed to discharge good quality drinking water to the local community reliably. This state has been attributed to negligence from the relevant authorities and agencies in terms of water quality monitoring and low level of community involvement in the development of these water projects. The County government of Kwale and water resource providers should build the capacity of the community in water resource management, introduce desalination and water treatment plants to provide safe drinking piped water.
On Analytical Approach to Semi-open/Semi-closed sets in Metric Spaces”
(Science Publishing Group, 2017-03-03) Musundi, Sammy Wabomba; Kinyili, Musyoka; Ohuru, Priscah Moraa
The concept of open and closed sets has been extensively discussed on both metric and topological spaces. Various properties of these sets have been proved under the underlying spaces. However, scanty literature is available about semi-open /semi-closed sets on these spaces. For instance, little effort has been made in introducing these sets as clopen sets in topological spaces but no literature exists of the same under metric spaces. In this paper, with reference to the already existing definitions and properties of open and closed sets in metric spaces as well as in topological spaces we shall present definitions of semi-open/ semiclosed sets and furthermore prove basic properties of these sets on metrics spaces. The results of the study will provide a deeper understanding as well as extension knowledge for the concept of open and closed sets to their somewhat counter-intuitive terms of semi- open /semi-closed.

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