Charles Wahogo | Karatina Univerisity (original) (raw)
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Papers by Charles Wahogo
Mathematical theory and modeling, 2014
Normal Form is a theory that applies in the neighbourhood of an orbit of a vector field map. The ... more Normal Form is a theory that applies in the neighbourhood of an orbit of a vector field map. The theory provides an algorithmic way to generate a sequence of non-linear coordinate changes that eliminate as much non-linearity as possible at each order (where order refers to terms in Taylors series about an orbit).The normal form is intended to be the simplest form into which any system of the intended type can be transformed by changing the coordinates in a prescribed manner. Interestingly the form of non-linear that cannot be eliminated by such coordinate changes is determined by the structure of the linear part of the vector field map. This section consists of some background knowledge, theorems and definitions necessary for understanding the concept of normal form for local dynamical systems. We briefly discuss the concept of ring of invariants and module of equivariants, and use the Groebner basis methods to compute a Groebner basis for the ideal of relations among the basic inva...
Mathematical theory and modeling, 2014
The subject of dynamical systems is concerned with: i. Given an matrix describe the beh... more The subject of dynamical systems is concerned with: i. Given an matrix describe the behavior , in a neighborhood of the origin the solutions of all systems of differential equations having a rest point at the origin with the linear part Ax ii. Describe the behavior (near origin) of all systems close to a system of the type described in i above. The normal form is intended to be the ‘’simplest ‘’form in which any system of intended type can be transformed by changing the coordinates in a prescribed manner. But if a normal form is thought as the ‘’simplest’’ form into which a system can be placed there might be disagreement as to what is considered simplest. A systematic policy for deciding what accounts for simplest is called normal form style. The important normal form styles are Semisimple, Innerproduct and SL(2) Or Triad styles. The unfolding of the normal form is intended to be the simplest form in which all systems close to the original system can be transfo...
Aspects of an organism’s defense to infections are the main problems of practical immunology. Und... more Aspects of an organism’s defense to infections are the main problems of practical immunology. Understanding the regularities in immune response provide the researchers and clinicians new powerful tools for the simulation of immune system in order to increase its efficiency in the struggle against antigen invasion. Such general regularities are revealed, as a rule, on the basis of analysis of the main components of an organism’s vital activities along with the system of immune defense. In this connection the construction of models of immune response to an antigen irritant seems to be a right tactic in the cognition of above regularities, that is why this monograph is dedicated to the analysis of the facts accumulated in immunology as a united system on the basis of logical concepts and mathematical models. Immune response to an infection by a pathogen is supported by different populations of cells (macrophages, B cells, CD4 T cells, CD8 T cells. . .). Among the many pathogen type, we...
Abstract: This section presents a mathematical simulation that includes two different effectorres... more Abstract: This section presents a mathematical simulation that includes two different effectorresponses that fight a viral infection independently: CTL (responsible for cell mediated immune responses) and antibodies (responsible for humoral immune responses).Since it is assumed that both responses rely on antigenic stimulation, the simulationscapture the competition dynamics.This is because the virus population is a resource that both CTL and antibodies require for survival. Competition can result either in the exclusion of one branch of the immune system, or both branches may coexist. We have examined and simulated five different immune dynamics of immune system responses; pathogen free, immunity free, CTL dominant, Antibody dominant and relative equal CTL and Antibody responses. Analytically we have established the minimum conditions required for either of the two branches of immune system to dominate or to coexist equally. We have explored how these competition dynamics can influ...
Mathematical Theory and Modeling, 2014
Mathematical Theory and Modeling, 2014
Mathematical theory and modeling, 2014
Normal Form is a theory that applies in the neighbourhood of an orbit of a vector field map. The ... more Normal Form is a theory that applies in the neighbourhood of an orbit of a vector field map. The theory provides an algorithmic way to generate a sequence of non-linear coordinate changes that eliminate as much non-linearity as possible at each order (where order refers to terms in Taylors series about an orbit).The normal form is intended to be the simplest form into which any system of the intended type can be transformed by changing the coordinates in a prescribed manner. Interestingly the form of non-linear that cannot be eliminated by such coordinate changes is determined by the structure of the linear part of the vector field map. This section consists of some background knowledge, theorems and definitions necessary for understanding the concept of normal form for local dynamical systems. We briefly discuss the concept of ring of invariants and module of equivariants, and use the Groebner basis methods to compute a Groebner basis for the ideal of relations among the basic inva...
Mathematical theory and modeling, 2014
The subject of dynamical systems is concerned with: i. Given an matrix describe the beh... more The subject of dynamical systems is concerned with: i. Given an matrix describe the behavior , in a neighborhood of the origin the solutions of all systems of differential equations having a rest point at the origin with the linear part Ax ii. Describe the behavior (near origin) of all systems close to a system of the type described in i above. The normal form is intended to be the ‘’simplest ‘’form in which any system of intended type can be transformed by changing the coordinates in a prescribed manner. But if a normal form is thought as the ‘’simplest’’ form into which a system can be placed there might be disagreement as to what is considered simplest. A systematic policy for deciding what accounts for simplest is called normal form style. The important normal form styles are Semisimple, Innerproduct and SL(2) Or Triad styles. The unfolding of the normal form is intended to be the simplest form in which all systems close to the original system can be transfo...
Aspects of an organism’s defense to infections are the main problems of practical immunology. Und... more Aspects of an organism’s defense to infections are the main problems of practical immunology. Understanding the regularities in immune response provide the researchers and clinicians new powerful tools for the simulation of immune system in order to increase its efficiency in the struggle against antigen invasion. Such general regularities are revealed, as a rule, on the basis of analysis of the main components of an organism’s vital activities along with the system of immune defense. In this connection the construction of models of immune response to an antigen irritant seems to be a right tactic in the cognition of above regularities, that is why this monograph is dedicated to the analysis of the facts accumulated in immunology as a united system on the basis of logical concepts and mathematical models. Immune response to an infection by a pathogen is supported by different populations of cells (macrophages, B cells, CD4 T cells, CD8 T cells. . .). Among the many pathogen type, we...
Abstract: This section presents a mathematical simulation that includes two different effectorres... more Abstract: This section presents a mathematical simulation that includes two different effectorresponses that fight a viral infection independently: CTL (responsible for cell mediated immune responses) and antibodies (responsible for humoral immune responses).Since it is assumed that both responses rely on antigenic stimulation, the simulationscapture the competition dynamics.This is because the virus population is a resource that both CTL and antibodies require for survival. Competition can result either in the exclusion of one branch of the immune system, or both branches may coexist. We have examined and simulated five different immune dynamics of immune system responses; pathogen free, immunity free, CTL dominant, Antibody dominant and relative equal CTL and Antibody responses. Analytically we have established the minimum conditions required for either of the two branches of immune system to dominate or to coexist equally. We have explored how these competition dynamics can influ...
Mathematical Theory and Modeling, 2014
Mathematical Theory and Modeling, 2014