Browsing by Author "Musundi, Sammy Wabomba"

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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.
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.
The Banach Numerical Range for Finite Linear Operators
(Modern Scientific Press, Florida, USA, 2019-01) Ohuru, Priscah M.; Musundi, Sammy Wabomba; Ombaka, Ochieng
The numerical range has been studied extensively in Hilbert spaces. Properties of the numerical range such as non-emptiness, containment of the spectrum and in particular, convexity have been proved and results have been given in these spaces. Furthermore, comparison of the numerical ranges with the spectra have been established. In this study, we consider the Banach space numerical range for a linear operator based on the definition by Lumer (1961) and establish its properties in relation to the above stated. Properties of the corresponding Banach numerical radius and spectrum are also discussed.
Correlation Between Electromagnetic Wave Equation and Einstein Theory of Relativity in Derivation of Schrödinger Equation and Hilbert Space Operators "
(2020-03) Mbatha, M. Elizabeth; Musundi, Sammy Wabomba; Kamweru, Paul
Operators in Hilbert space have properties which are useful in the study of mathematical abstract areas such as approximation theory, Banach Fixed point theory, the spectral theory as well as Quantum Mechanics. Schrödinger equation is a fundamental entity with many applications in Quantum Mechanics. This equation was initially derived by applying the knowledge of electromagnetic wave function and Einstein theory of relativity. Later, it was derived by applying the knowledge of Newtonian mechanics. It was also derived by extending the wave equation for classical fields to photons and simplified using approximations consistent with generalized non-zero rest mass. However, from the existing literature no study has been done on deriving Schrödinger equation using properties of Hilbert space operators. In this study, Hilbert space operators that include unitary operators, self adjoint operators and compact operators, norms of linear operators, Hilbert Schmidt operator, normal operators together with Lebesque Integral, Neumann Integral and spectrum are used in place of the existing concepts of electromagnetic wave function, Einstein theory of relativity and approximation consistent with generalized non zero mass to derive the Schrödinger equation. The derivation of Schrödinger equation and its application using Hilbert space operators enhances a better understanding of the concept of Schrödinger equation. The results of this work can further find use in quantum mechanics as well as in mathematical operator theory.
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
(Science Publishing Group, 2016-01) Musundi, Sammy Wabomba
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. (PDF) Modeling and Forecasting Kenyan GDP Using Autoregressive Integrated Moving Average (ARIMA) Models. Available from: https://www.researchgate.net/publication/304340496_Modeling_and_Forecasting_Kenyan_GDP_Using_Autoregressive_Integrated_Moving_Average_ARIMA_Models [accessed Dec 04 2019].
Modeling the Impact of Soil Porosity on Nitrate Leaching to Groundwater Using the Advection Dispersion Equation
(2019-04) Jimrise, Ochwach; Okong’o, Mark; Musundi, Sammy Wabomba
Nitrogen is a vital nutrient that enhances plant growth which has motivated the intensive use of nitrogen-based fertilizers to boost crop productivity. However, Pollution by nitrate is a globally growing problem due to the population growth, increase in the demand for food and inappropriate Nitrogen application. The c o m p l e x i t i e s and challenges in quantifying nitrate leaching h a v e led to development of a range of measurement and modeling techniques. However, most of them are not widely applied due to their inaccuracy. This calls for new approaches in which nitrate leaching can be analysed in order to give better understanding of nitrate fate and transport process for proper management of groundwater. This s t u d y has d e v e l o p e d a mathematical model to a n a l y s e n i t r a t e leaching into groundwater from the advection-dispersion equation. The advection-dispersion equation is modified by incorporating soil porosity and tran sfo rmed to a second order o rd ina ry differential equation by Laplace and solved. Simulations showing the variation of soil porosity is presented using the MATLAB software. The study has shown that nitrate leaching to groundwater is directly proportional to soil porosity such that more porous soil will allow more nitrate to reach to the groundwater within a short time leading to faster contamination of groundwater. The results is useful to farmers, policy makers, researchers and the general public for the purpose of understanding movement of nitrates through the soil and also provide science-based input into best alternative mathematical model which can be used to analyse leaching of nitrate into groundwater.
On Analytical Approach to Semi-Open/Semi-Closed Sets
(International Journal of Discrete Mathematics., 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.
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.
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.
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.
On The Norm of an Elementary Operator of Finite Length in A C*Algebra”
(2018-08) Kawira, Esther; Kingangi, Denis; Musundi, Sammy Wabomba
Properties of elementary operators have been studied over the past years especially the norm aspect. Various results have been obtained on elementary operators of different lengths using different approaches. In this paper, we determine the norm of an elementary operator of length n in a C*algebra using finite rank operators.We will review known results on Jordan and general elementary operators which are useful in getting our result. Maths classification: 47C15, 47A30
On the Norm of Basic Elementary Operator in a Tensor Product
(2018-01) Muiruri, Peter G; Kingangi, Denis; Musundi, Sammy Wabomba
In this paper, we determine the norm of an elementary operator in a tensor product. More precisely, we investigate the bounds of the norm of a basic elementary operator in a tensor product. We employ the techniques of tensor products and finite rank operators to express the norm of an elementary operator in terms of its coefficient operators. We also show that the norm of a basic elementary operator on is expressible in terms of the norms of basic elementary operators on and B(HOK) is expressible in terms of the norms of basic elementary operators on B(H) and B(K) .
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.