Applied Sciences

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CERTAIN INEQUALITIES FOR (a,B)- NORMAL TRANSLOID OPERATORS ON HILBERT SPACES
(International Journal of Academic Studies, 2015-11-29) Nyaluke Kiprono Wesley” , N. B. Okelo , Omolo Ongati-
Let H be an infinite dimensional complex Hilbert space and N.(H) denote the class of (a, 2)-normal transaloid operators. In this paper, we characterize (a, B)- normal transaloid operators and later determine norm and numerical radii inequalities for (a, 2)-normal transaloid operators.
CHANNEL STRUCTURED METAMATERIALS FOR SUPER RESOLUTION IMAGING
(ASME, 2019-07-14) ohn Birir1, Michael J. Gatari and Prabhu Rajagopal
ABSTRACT When inspection is required to be conducted from a remote distance, due for example to safety concerns, guided ultrasonic wave testing technique is the preferred choice. Guided waves are usually low frequency hence are not subjected to much attenuation relative to the bulk waves. As a result, guided waves can travel a longer distance. One of the limitations of guided waves however, is that the low frequency (longer wavelength) used lead to lower resolution capabilities due to the inherent diffraction limits of λ/2 (where λ is wavelength). Guided waves are therefore generally used as screening tools to locate areas of interest. A higher resolution technique is then employed to further investigate and characterize the features of defects in the identified areas. To overcome this challenge of resolution, a technique is proposed that increases the resolution capability of guided waves beyond the diffraction limit. Simulation using commercial finite element software is used to optimize variables involved in the proposed method. The simulation is then validated with experiments. In the present work a resolution of λ/72 is demonstrated experimentally.
Pressure Distribution of a Horizontal Well in an Oil Reservoir Subject to Simultaneous Single Edged and Bottom Water Drive Mechanisms
(IOSR Journal of Mathematics (IOSR-JM), 2023-08-10) Mutili Peter Mutisya, Awuor Kennedy Otieno, Oyoo Daniel Okang’a and Adewole Stephen Ezizanami
In this study the pressure distribution in an oil reservoir with a horizontal well is investigated. A horizontal well with single-edged and constant bottom pressure is outlined. A reservoir bounded with two constant pressure boundaries, like an edge and bottom water, requires that the production engineer should adhere diligently to a production schedule, developed by a reservoir engineer, for clean oil production to be possible. This means that arbitrary production practices through selection of production rates could lead to production of these external fluids. This can mar the economics of the project. Production schedules or plans show acceptable rates, well design and production time that can guarantee only clean oil production. In this study, pressure behaviour of a horizontal well drilled and completed in a reservoir subject to with simultaneous single-edged and bottom water drives is investigated in detail. All possible flow periods or patterns that can be exhibited by the well are determined. Fluid flow in oil reservoirs in real time is governed by a heterogenous diffusivity equation, which describes reservoir pressure as a function of reservoir, fluid and wellbore properties. To solve this unsteady state problem, Green’s functions were deployed to represent the boundaries of the reservoir selected for study. The Green’s functions selected are for flow from start of transient to late time, when all the external boundaries are felt. Newman product rule was used to derive a dimensionless pressure expression for the reservoir system oil flow. The source of pressure transient was production throughout. All the resulting integrals were performed numerically. MATLAB programming was used to plot the curves by applying spline functions interpolation. Influence of reservoir, fluid and wellbore properties on reservoir pressure was investigated in real time. To assist interpretation, dimensionless pressure derivatives were also computed. Near wellbore problems, like skin and wellbore storage, which affect well performance only at very early time, were not considered in the study. From the results, 𝑃𝐷 and 𝑃𝐷 ′ vary directly with ℎ𝐷 and inversely as 𝐿𝐷. The 𝑃𝐷 ′ gradually reduces to zero when 𝑃𝐷 begins to exhibit a constant trend. 𝑃𝐷 ′ vary inversely with ℎ𝐷 and 𝑦𝑒𝐷 at all flow times. The number of flow periods varies with reservoir size, well length and production time. The time at which the 𝑃𝐷 ′ starts to exhibit a downward trend is the external fluid breakthrough time. The breakthrough time is affected by well design. Longer wells exhibit delayed breakthrough time because of lower pressure drawdown associated with increased well length. If production rate is sustained for any particular well design, the well will completely water-out. Finally, infinite conductivity 𝑥𝐷 = 0.732 and uniform flux condition do not really affect 𝑃𝐷 and 𝑃𝐷 ′ at early time.
Assessing the impact of optimized prevention strategies for mother-to-child HIV transmission dynamics in Kenya: a mathematical modeling study
(medRxiv, 2025-04-19) Robert Mureithi Maina, Samuel Musili Mwalili, Duncan Kioi Gathungu
HIV can be transmitted from a HIV infected mother to her child during pregnancy, delivery, or breastfeeding. According to NSDCC 2023, Kenya has estimated PMTCT coverage of 89.56% and PMTCT transmission rate of 8.6%. Even though there has been strides to address PMTCT, there is need to gear up approaches in addressing MTCT in order to significantly advance elimination. This research formulates a mathematical model to represent the dynamics of MTCT. Equilibrium points of the model are computed and the stability of HIV-free point is investigated. The numerical results show that a 50% decrease in maternal HIV transmission lowers infant infection rates by about 17.7%, whereas the same reduction in infant transmission decreases infections by nearly 39%, highlighting the greater sensitivity of infant transmission rates to direct interventions. While combination of strategies achieves the highest HIV minimization rates of up to 99.89% on infants, ART adherence alone significantly reduces transmission, particularly on infants (91.42%) while use of post-exposure prophylaxis (PEP) shows limited effectiveness when used alone(39.65%), suggesting that it should be complemented with other strategies for optimal impact. These findings emphasize the critical need for integrated interventions, where combining multiple prevention methods yields the best outcomes in reducing HIV infections on infants and moving closer to the elimination of pediatric HIV. These findings align with global recommendations from World Health Organization (WHO). This research can be used by the ministry of health to inform policy as well as recreated for other maternal infections.
Metamaterial enhanced subwavelength imaging of inaccessible defects in guided ultrasonic wave inspection
(ScienceDirect, 2024-04) John Kiprotich Birir, Michael James Gatari , Mohamed Subair Syed Akbar Ali and Prabhu Rajagopal
Detection of defects located close to design features such as welds and supports remains a challenge in guided ultrasonic wave inspection, primarily due to the diffraction limit. Although metamaterial based approaches hold promise, the best previous work in this regard required placing a sensor right above the defect location to achieve resolution. Here, a novel angled channel metamaterial concept is proposed to overcome this limitation, thus permitting placing of a sensor at an offset from the defect location. The concept is demonstrated and discussed using simulations validated by experiments. It is shown that sub-wavelength resolution of crack-like defects is possible using the angled channel metamaterial offset by a distance of up to half the wavelength. The operating physics of this problem is further discussed using simulations and analysis, bringing out the strengths and limitations of the proposed technique, highlighting the benefits for guided wave screening of hidden regions.
Structured channel metamaterials for deep sub wavelength resolution in guided ultrasonics
(AIP Publishing, 2020-06-19) John K. Birir, Michael J. Gatari and Prabhu Rajagopa
Experimental results on deep subwavelength resolution of defects are presented for the first time in the context of guided ultrasonic wave inspection of defects, using novel “structured channel” metamaterials. An Aluminum bar with side-drilled holes is used as a test sam ple, interrogated by the fundamental bar-guided symmetric mode. Simulations were conducted to optimize dimensional parameters of the metamaterial structure. Experiments using metamaterials fabricated accordingly demonstrate a resolution down to 1/72 of the operating wavelength, potentially bringing the resolution of guided wave inspection to the same range as that of bulk ultrasonics. This work has much promise for remote inspection in industry and biomedicine.
Effects of Fluid and Reservoir Characteristics on Dimensionless Pressure and Derivative of a Horizontal Well in a Bounded Oil Reservoir with Simultaneous Single Edge and Bottom Water Drive
(American Journal of Engineering Research (AJER), 2021) Mutili Peter Mutisya, Adewole Stephen Ezizanami, Awuor Kennedy Otieno and Oyoo Daniel Okang’a
This study investigates the effects of fluid, wellbore and reservoir characteristics on dimensionless pressure and dimensionless pressure derivatives at late time flow of a horizontal well in a bounded oil reservoir subjected to a single edge and bottom water drive mechanisms. The properties considered in this paper include the dimensionless well length, dimensionless reservoir width and dimensionless pay thickness. The main objective is achieved by using the source and Green’s functions together with Newman product method. Spline functions for interpolation in curve fitting was used to plot the graphs aided by MATLAB program. Results show that the dimensionless pressure increases with decrease in dimensionless reservoir width and pay thickness. The dimensionless pressure derivative potentially collapses to zero when the dimensionless pressure becomes constant. Higher oil production is indicated by larger magnitudes of dimensionless pressure derivatives. Information in this study will assist in designing and completion of horizontal wells in a bounded reservoir for prolonged enhanced oil production.
Derivation of Black-Scholes-Merton Logistic Brownian Motion Di erential Equation with Jump Di usion Process
(JS Publication., 2019) Andanje Mulambula, D. B. Oduor and B. Kwach
Black- Scholes formed the foundation of option pricing. However, some of the assumptions like constant volatility and interest among others are practically impossible to implement hence other option pricing models have been explored to help come up with a much reliable way of predicting the price trends of options. Black-scholes assumed that the daily logarithmic returns of individual stocks are normally distributed. This is not true in practical sense especially in short term intervals because stock prices are able to reproduce the leptokurtic feature and to some extent the volatility smile . To address the above problem the Jump-Di usion Model and the Kou Double-Exponential Jump-Di usion Model were presented. But still they have not fully addressed the issue of reliable prediction because the observed implied volatility surface is skewed and tends to atten out for longer maturities; The two models abilities to produce accurate results are reduced. This paper ventures into a research that will involve Black-Scholes-Merton logistic-type option pricing with jump di usion. The knowledge of logistic Brownian motion will be used to develop a logistic Brownian motion with jump di usion model for price process. MSC: 91GXX, 91G50, 62P05, 97M30.
Volatility Estimation Using European-Logistic Brownian Motion with Jump Di usion Process
(© JSPublication., 2020) Andanje Mulambula, D. B. Oduor, and B. O. Kwach
Volatility is the measure of how we are uncertain about the future of stock or asset prices. Black-Scholes model formed the foundation of stock or asset pricing. However, some of its assumptions like constant volatility and interest among others are practically impossible to implement hence other option pricing models have been explored to help come up with a much reliable way of predicting the price trends of options. The measure of volatility and good forecasts of future volatility are crucial for implementation, evaluation of asset and derivative pricing of asset. In particular, volatility has been used in nancial markets in assessment of risk associated with short-term uctuations in nancial time-series. Constant volatility is not true in practical sense especially in short term intervals because stock prices are able to reproduce the leptokurtic feature and to some extent the volatility smile . To address the above problem the Jump-Di usion Model and the Kou Double-Exponential Jump-Di usion Model were presented. But still they have not fully addressed the issue of reliable prediction because the observed implied volatility surface is skewed and tends to atten out for longer maturities; the two models abilities to produce accurate results are reduced. This study ventures into a research that will involve volatility estimation using European logistic-type option pricing with jump di usion. The knowledge of logistic Brownian motion will be used to develop a logistic Brownian motion with jump di usion model for price process. MSC: 91GXX, 91G50, 62P05, 97M30.
A Mathematical Model for Pressure Distribution in a Bounded Oil Reservoir Subject to Single-Edged and Bottom Constant Pressure
(IOSR Journal of Mathematics (IOSR-JM), 2020-07) Mutili Peter Mutisya, Adewole Stephen Ezizanami, Awuor Kennedy Otieno and Oyoo Daniel Okang’a
Well test analysis of a horizontal well is complex and difficult to interpret. Most horizontal well mathematical models assume that horizontal wells are perfectly horizontal and are parallel to the top and bottom boundaries of the reservoir. As part of effort towards correct horizontal well test analysis, the purpose of this study is to develop a mathematical model using source and Green’s functions for a horizontal well completed in an oil reservoir at late time flow period, where the reservoir is bounded by an edge and bottom constant pressure boundaries. The purpose of the derivation is to understand the effects of well completion, well design and reservoir parameters on pressure and pressure derivative behavior of the well at late flow time, when all these external boundaries are presumed to have been felt. If the model is applied for well test analysis therefore information like reservoir natural permeability distribution, actual external boundary types and even the well completion performance will be decidable easily. Dimensionless variables were used to derive throughout the derivations. Results of the derivation show that the dimensionless pressure and dimensionless pressure derivatives increase with increase in dimensionless well length. This means that higher well productivity is achievable with extended well length when the reservoir is surrounded partially by constant pressure boundaries. Furthermore, the models show that higher directional permeabilities would also encourage higher well productivity at late flow time. The dimensionless pressure derivative will, as a result of a constant dimensionless pressure, potentially collapse gradually to zero at the moment the dimensionless pressure begins to exhibit a constant trend. Finally, the dimensionless pressure and dimensionless pressure derivatives vary inversely with the reservoir dimensionless width at late flow time.
Characterization of Inner Derivations induced by Norm-attainable Operators
(International Journal of Modern Science and Technology,, 2018) M. O. Oyake, N. B. Okelo, O. Ongati
In the present paper, results on characterization of inner derivations in Banach algebras are discussed. Some techniques are employed for derivations due to Mecheri, Hacene, Bounkhel and Anderson. Let H be an infinite dimensional complex Hilbert space and B(H) the algebra of all bounded linear operators on H. A generalized derivation δ: B(H) → B(H) is defined by δA,B(X) = AX −XB, for all X ∈ B(H) and A,B fixed in B(H). An inner derivation is defined by δA(X) = AX −XA, for all X ∈ B(H) and A fixed in B(H). Norms of inner derivations have been investigated by several mathematicians. However, it is noted that norms of inner derivations implemented by norm-attainable operators have not been considered to a great extent. In this study, we investigate properties of inner derivations which are strictly implemented by norm-attainable and we determine their norms. The derivations in this work are all implemented by norm-attainable operators. The results show that these derivations admit tensor norms of operators.

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