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Submitted

MUTUKU, DRNZIMBIBERNARD, P PROFPOKHARIYALGANESH, M PROFKHALAGAIJAIRUS.  Submitted.  B.M. Nzimbi, G.P. Pokhariyal and J.M. Khalagai, Linear operators for which T* and T^2 commute, Global Journal of Pure and Applied Mathematics(GJPAM), 2012, to appear. Global Journal of Pure and Applied Mathematics(GJPAM),2012, to appear. : Global Journal of Pure and Applied Mathematics(GJPAM), 2012, to appear Abstract
Recent publications have demonstrated that the protease caspase-1 is responsible for the processing of pro-interleukin 18 (IL-18) into the active form. Studies on cell lines and murine macrophages have shown that the bacterial invasion factor SipB activates caspase-1, triggering cell death. Thus, we investigated the role of SipB in the activation and release of IL-18 in human alveolar macrophages (AM), which are the first line of defense against inhaled pathogens. Under steady-state conditions, AM are a more important source of IL-18 than are dendritic cells (DC) and monocytes. Cytokine production by AM and DC was compared after both types of cells had been infected with a virulent strain of Salmonella enterica serovar Typhimurium and an isogenic sipB mutant, which were used as an infection model. Infection with virulent Salmonella led to marked cell death with features of apoptosis while both intracellular activation and release of IL-18 were demonstrated. In contrast, the sipB mutant did not induce such cell death or the release of active IL-18. The specific caspase-1 inhibitor Ac-YVAD-CMK blocked the early IL-18 release in AM infected with the virulent strain. However, the type of Salmonella infection did not differentially regulate IL-18 gene expression. We concluded that the bacterial virulence factor SipB plays an essential posttranslational role in the intracellular activation of IL-18 and the release of the cytokine in human AM.

2020

Muhati, LN, Khalagai JM.  2020.  On Unitary Invariance of Some Classes of Operators in Hilbert Spaces. Pure Mathematical Sciences. 9(1):45-52. AbstractWebsite

Itis a known fact in operator theory that two similar operators have equal spectra but they do not necessarily have to belong to the same class of operators. However,under the stronger relation of unitary equivalence it can be shown that two unitarily equivalent operators may belong to the same class of operators. In this paper we endeavor to exhibit some results on some pairs of operators which may belong to the same class under not only unitary equivalence but also isometric and co-isometric equivalence

Luketero, SW, Khalagai JM.  2020.  On unitary equivalence of some classes of operators in Hilbert spaces. International Journal of Statistics and Applied Mathematics. 5(2):35-37. AbstractWebsite

It is a well-known result in operator theory that whenever two operators are similar then they have equal spectra even though they do not have to belong to the same class of operators. However under a stronger relation of unitary equivalence it can be shown that two unitarily equivalent operators may belong to the same class of operators. In this paper we endeavor to exhibit results on such classes of operators which belong to same class under unitary equivalence.

2018

MUTURI, NE, Khalagai JM, Pokhariyal GP.  2018.  Separation axioms on function spaces defined on bitopological spaces. Journal of Advanced Studies in Topology. 9(2):113-118. AbstractWebsite

In this paper, we introduce separation axioms on the function space p− Cω(Y, Z) and study how they relateto separation axioms defined on the spaces (Z, δi) for i = 1, 2, (Z, δ1, δ2), 1 − Cς(Y, Z) and 2 − Cζ(Y, Z). Itis shown that the space p − Cω(Y, Z) is pT◦, pT1, pT2 and pregular, if the spaces (Z, δ1) and (Z, δ2) are bothT0, T1, T2 and regular respectively. The space p − Cω(Y, Z) is also shown to be pT0, pT1, pT2 and pregular,if the space (Z, δ1, δ2) is p − T0, p − T1, p − T2 and p-regular respectively. Finally, the space p − Cω(Y, Z) isshown to be pT0, pT1, pT2 and pregular, if and only if the spaces 1 − Cς(Y, Z) and 2 − Cζ(Y, Z) are both T0,T1, T2, and only if the spaces 1 − Cς(Y, Z) and 2 − Cζ(Y, Z) are both regular respectively.

EN Muturi, Khalagai JM, Pokhariyal GP.  2018.  Splitting and Admissible Topologies Defined on the Set of Continuous Functions Between Bitopological Spaces. International Journal of Mathematical Archive (IJMA). 9(1) AbstractWebsite

In this paper, p-splitting, p-admissible, s-splitting and s-admissible topologies on the sets p−C(Y, Z) and s−C(Y, Z) are defined and their properties explored. exponential functions are introduced in function spaces and s-splitting and s-admissible topologies defined on s-C(Y, Z) compared using these mappings.

2017

Sitati, IN, Nzimbi BM, Luketero SW, Khalagai JM.  2017.  Remarks on A-skew-adjoint, A-almost similarity equivalence and other operators in Hilbert space. Pure and Applied Mathematics Journal. 6(3):101-107. AbstractWebsite

Abstract
In this paper, notions of A-almost similarity and the Lie algebra of A-skew-adjoint operators in Hilbert space are introduced. In this context, A is a self-adjoint and an invertible operator. It is shown that A-almost similarity is an equivalence relation. Conditions under which A-almost similarity implies similarity are outlined and in which case their spectra is located. Conditions under which an A-skew adjoint operator reduces to a skew adjoint operator are also given. By relaxing some conditions on normal and unitary operators, new results on A -normal, binormal and A-binormal operators are proved. Finally A-skew adjoint operators are characterized and the relationship between A-self- adjoint and A-skew adjoint operators is given.

2016

Sitati, IN, Nzimbi BM, Luketero SW, Khalagai JM.  2016.  On A-Self-Adjoint, A-Unitary Operators and Quasiaffinities. SciencePG journals. Vol. 1(3 ):56-60. Abstract

In this paper, we investigate properties of A-self-adjoint operators and other relations on Hilbert spaces. In this
context, A is a self-adjoint and an invertible operator. More results on operator equivalences including similarity, unitary and metric equivalences are discussed. We also investigate conditions under which these classes of operators are self- adjoint and unitary. We finally locate their spectra.

Sitati, IN, Nzimbi BM, Luketero SW, Khalagai JM.  2016.  On A-Self-Adjoint, A-Unitary Operators and Quasiaffinities. SciencePG journals. Vol. 1(3 ):56-60. Abstract

In this paper, we investigate properties of A-self-adjoint operators and other relations on Hilbert spaces. In this
context, A is a self-adjoint and an invertible operator. More results on operator equivalences including similarity, unitary and metric equivalences are discussed. We also investigate conditions under which these classes of operators are self- adjoint and unitary. We finally locate their spectra.

2013

2012

Kavila, M, Khalagai JM.  2012.  On A-Commuting Operators. AbstractOn A-Commuting Operators

Two bounded linear operators A and B on a complex Hilbert space are said to ,1.- commute for ,lEe provided that: AB = ABA. In this paper we look for some properties satisfied by the operators A and B so that ,1.= 1. It is shown among other results that if one of the operators raised to some power is normal and 0 does not belong to the interior of the numerical range of the other operator then: A = 1 AMS 200 Mathematics Subject Classification 47B47 47 A30, 47B20

2010

  2010.  Basic Mathematics. AbstractWebsite

This module consists of three units which are as follows: Unit 1: (i) Sets and Functions (ii) Composite Functions This unit starts with the concept of a set. It then intoroduces logic which gives the learner techniques for distinguishing between correct and incorrect arguments using propositions and their connectives. A grasp of sets of real numbers on which we define elementary functions is essential. The need to have pictorial representations of a function necessitates the study of its graph. Note that the concept of a function can also be viewed as an instruction to be carried out on a set of objects. This necessitates the study of arrangements of objects in a certain order, called permutations and combinations. Unit 2: Binary Operations In this unit we look at the concept of binary operations. This leads to the study of elementary properties of integers such as congruence. The introduction to algebraic structures is simply what we require to pave the way for unit 3. Unit 3: Groups, Subgroups and Homomorphism This unit is devoted to the study of groups and rings. These are essentially sets of numbers or objects which satisfy some given axioms. The concepts of subgroup and subring are also important to study here. For the sake of looking at cases of fewer axiomatic demands we will also study the concepts of homomorphisms and isomorphisms. Here we will be reflecting on the concept of a mapping or a function from either one group to the other or from one ring to the other in order to find out what properties such a function has.

2008

MUTUKU, DRNZIMBIBERNARD, P PROFPOKHARIYALGANESH, M PROFKHALAGAIJAIRUS.  2008.  B.M. Nzimbi, G.P. Pokhariyal and J.M. Khalagai, A note on Similarity, Almost-Similarity and Equivalence of Operators, Far East Mathematics Journal, Vol 28, Issue 2(February 2008), 305-317.. Far East Mathematics Journal, Vol 28, Issue 2(February 2008), 305-317.. : Global Journal of Pure and Applied Mathematics(GJPAM), 2012, to appear Abstract
The almost-similar and similar relations between operators on finite-dimensional Hilbert spaces are investigated. It is shown that almost-similar operators share some properties with some other classes of operators. Various results on almost-similarity and similarity are proved. An attempt is made to classify those operators where almost-similarity implies similarity. We investigate some properties of corresponding parts of operators which enjoy these equivalence relations.

2007

Khalagai, JM, Pokhariyal GP, Nzimbi.  2007.  A note on similarity, almost -similarity and equivalence of operators. Abstract

The almost- similar and similar relations between operators on finitedimensional Hilbert spaces are investigated. It is shown that almost similar operators share some properties with some other classes of operators. Various results on almost-similarity and similarity are proved. An attempt is made to classify those operators where almost similarity implies similarity. We investigate some properties of corresponding parts of operators which enjoy these equivalence relations.

2000

M, PROFKHALAGAIJAIRUS, M PROFKHALAGAIJAIRUS.  2000.  On Quasiaffinity and Quasiaffine inverses of partial isometries. Bulletin of the Allahabad Mathematical Society Vol. 15 35-40.. : Global Journal of Pure and Applied Mathematics(GJPAM), 2012, to appear Abstract
The almost-similar and similar relations between operators on finite-dimensional Hilbert spaces are investigated. It is shown that almost-similar operators share some properties with some other classes of operators. Various results on almost-similarity and similarity are proved. An attempt is made to classify those operators where almost-similarity implies similarity. We investigate some properties of corresponding parts of operators which enjoy these equivalence relations.

1996

M, PROFKHALAGAIJAIRUS, M PROFKHALAGAIJAIRUS.  1996.  On quasiffine inverses of operators; Kenya Journal of Science. Series A 10 (2), 107 - 116.. Bulletin of the Allahabad Mathematical Society Vol. 15 35-40.. : Global Journal of Pure and Applied Mathematics(GJPAM), 2012, to appear Abstract
J. M. Khalagai,

1994

M, PROFKHALAGAIJAIRUS, M PROFKHALAGAIJAIRUS.  1994.  On intertwining operators Discovery and innovation vol. 16 no 4 p355-357.. Bulletin of the Allahabad Mathematical Society Vol. 15 35-40.. : Global Journal of Pure and Applied Mathematics(GJPAM), 2012, to appear Abstract
J. M. Khalagai,

1991

M, PROFKHALAGAIJAIRUS, M PROFKHALAGAIJAIRUS.  1991.  On Quasiaffine inverses of Operators. Proceedings of the Third Pan-African Congress of Mathematicians held in Kenya. : Global Journal of Pure and Applied Mathematics(GJPAM), 2012, to appear Abstract
J. M. Khalagai,

1988

M, PROFKHALAGAIJAIRUS, M PROFKHALAGAIJAIRUS.  1988.  Commutants and conditions on Operators implying normality. Kenya Journal of Science, Series A. vol. 9 p.43-45. : Global Journal of Pure and Applied Mathematics(GJPAM), 2012, to appear Abstract
J. M. Khalagai,

1987

M, PROFKHALAGAIJAIRUS, M PROFKHALAGAIJAIRUS.  1987.  On the Operator Equation AH = KA, Mathematics today, volume v, p. 29-36.. Kenya Journal of Science, Series A. vol. 9 p.43-45. : Global Journal of Pure and Applied Mathematics(GJPAM), 2012, to appear Abstract
J. M. Khalagai,

1985

M, PROFKHALAGAIJAIRUS.  1985.  J. M. Khalagai, "On the Operator Equation TST* = S. Unitary Solutions. Kenya J. Science Technology Series A 6(2): 157-163.. : Global Journal of Pure and Applied Mathematics(GJPAM), 2012, to appear Abstract
J. M. Khalagai,

1983

M, PROFKHALAGAIJAIRUS, M PROFKHALAGAIJAIRUS.  1983.  Further remarks on the Operator Equation AB+BA*=A*B+BA=I. Kenya J. Science Technology Series A 4(l): 31-35. : Global Journal of Pure and Applied Mathematics(GJPAM), 2012, to appear Abstract
J. M. Khalagai,

1982

M, PROFKHALAGAIJAIRUS.  1982.  .P. Duggal and J. M. Khalagai, Operator Equation AB+BA*=A*B+BA=I, India J.. Pure Applied Maths. Vol.13 (11) , 526-531.. : Global Journal of Pure and Applied Mathematics(GJPAM), 2012, to appear Abstract
J. M. Khalagai,

1981

M, PROFKHALAGAIJAIRUS.  1981.  On the operator equation AB + BA* = A*B+BA = I, Proceedings of the First African Symposium in Pure and Applied Mathematics and Mathematics Education. Nairobi, Kenya. : Global Journal of Pure and Applied Mathematics(GJPAM), 2012, to appear Abstract
J. M. Khalagai,  held in Nairobi, Kenya in
MUTUKU, DRNZIMBIBERNARD, P PROFPOKHARIYALGANESH, M PROFKHALAGAIJAIRUS.  1981.  B.M. Nzimbi, G.P. Pokhariyal and J.M. Khalagai, Characterization of C_00 Contractions and their invariant subspaces, corrected manuscript re-submitted to Opuscula Mathematica, paper under review.. Nairobi, Kenya. : Opuscula Mathematica, Abstract
J. M. Khalagai,  held in Nairobi, Kenya in
M, PROFKHALAGAIJAIRUS, M PROFKHALAGAIJAIRUS.  1981.  On the Operator Equation AB+BA*=A*B+BA=l. Maths Japon, 26 No.5 577-583. : Global Journal of Pure and Applied Mathematics(GJPAM), 2012, to appear Abstract
J. M. Khalagai,
M, PROFKHALAGAIJAIRUS.  1981.  On the operator equation AB + BA* = A*B+BA = I, Proceedings of the First African Symposium in Pure and Applied Mathematics and Mathematics Education. Nairobi, Kenya. : Global Journal of Pure and Applied Mathematics(GJPAM), 2012, to appear Abstract
J. M. Khalagai,

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