The document discusses determining matrices for single-delay autonomous linear neutral control systems. It derives expressions for the determining matrices through a series of lemmas, theorems, and corollaries. The determining matrices constitute an optimal way to determine controllability and compactness of control systems. The paper establishes valid expressions for determining matrices of these types of systems and explores their relationships to partial derivatives of the systems.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document discusses the duality between initial and terminal function problems for single-delay autonomous linear control systems. It presents theorems from previous work by Ukwu and Garba that provide solution matrices and indices of control systems. The paper establishes that the initial and terminal function problems are topological duals of each other by outlining transformations that can be used to obtain the solution matrix for one problem from the solution matrix of the other problem. This is achieved through key operations including changing variables, integrals, and exponential functions in the solution matrix formulas. The transformations effectively show that results from one problem can be realized from the other, demonstrating the duality between the initial and terminal function problems.
A new six point finite difference scheme for nonlinear waves interaction modelAlexander Decker
1) The document presents a new six point finite difference scheme for modeling nonlinear wave interaction using the coupled 1D Klein-Gordon-Zakharov system of equations.
2) The scheme is derived by discretizing the system equations with finite differences. It is shown to be equivalent to a multi-symplectic integrator.
3) Numerical simulations are presented applying the new six point scheme to solve the Klein-Gordon-Zakharov equations modeling the interaction of Langmuir waves and ion sound waves in plasma.
Catalan Tau Collocation for Numerical Solution of 2-Dimentional Nonlinear Par...IJERA Editor
Tau method which is an economized polynomial technique for solving ordinary and partial differential
equations with smooth solutions is modified in this paper for easy computation, accuracy and speed. The
modification is based on the systematic use of „Catalan polynomial‟ in collocation tau method and the
linearizing the nonlinear part by the use of Adomian‟s polynomial to approximate the solution of 2-dimentional
Nonlinear Partial differential equation. The method involves the direct use of Catalan Polynomial in the solution
of linearizedPartial differential Equation without first rewriting them in terms of other known functions as
commonly practiced. The linearization process was done through adopting the Adomian Polynomial technique.
The results obtained are quite comparable with the standard collocation tau methods for nonlinear partial
differential equations.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document discusses the duality between initial and terminal function problems for single-delay autonomous linear control systems. It presents theorems from previous work by Ukwu and Garba that provide solution matrices and indices of control systems. The paper establishes that the initial and terminal function problems are topological duals of each other by outlining transformations that can be used to obtain the solution matrix for one problem from the solution matrix of the other problem. This is achieved through key operations including changing variables, integrals, and exponential functions in the solution matrix formulas. The transformations effectively show that results from one problem can be realized from the other, demonstrating the duality between the initial and terminal function problems.
A new six point finite difference scheme for nonlinear waves interaction modelAlexander Decker
1) The document presents a new six point finite difference scheme for modeling nonlinear wave interaction using the coupled 1D Klein-Gordon-Zakharov system of equations.
2) The scheme is derived by discretizing the system equations with finite differences. It is shown to be equivalent to a multi-symplectic integrator.
3) Numerical simulations are presented applying the new six point scheme to solve the Klein-Gordon-Zakharov equations modeling the interaction of Langmuir waves and ion sound waves in plasma.
Catalan Tau Collocation for Numerical Solution of 2-Dimentional Nonlinear Par...IJERA Editor
Tau method which is an economized polynomial technique for solving ordinary and partial differential
equations with smooth solutions is modified in this paper for easy computation, accuracy and speed. The
modification is based on the systematic use of „Catalan polynomial‟ in collocation tau method and the
linearizing the nonlinear part by the use of Adomian‟s polynomial to approximate the solution of 2-dimentional
Nonlinear Partial differential equation. The method involves the direct use of Catalan Polynomial in the solution
of linearizedPartial differential Equation without first rewriting them in terms of other known functions as
commonly practiced. The linearization process was done through adopting the Adomian Polynomial technique.
The results obtained are quite comparable with the standard collocation tau methods for nonlinear partial
differential equations.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
On Convergence of Jungck Type Iteration for Certain Contractive Conditionsresearchinventy
In this article we prove the strong convergence result for a pair of nonself mappings using Jungck S- iterative scheme in Convex metric spaces satisfying certain contractive condition. The results are the generalization of some existing results in the literature
STUDIES ON INTUTIONISTIC FUZZY INFORMATION MEASURESurender Singh
This document discusses studies on measures of intuitionistic fuzzy information. It begins with introductions and definitions related to fuzzy sets, intuitionistic fuzzy sets, and measures of fuzzy entropy. It then discusses special t-norm operators and proposes a measure of intuitionistic fuzzy entropy based on these t-norms. The measure is defined using a function of the membership, non-membership, and hesitancy degrees of an intuitionistic fuzzy set. Several desirable properties of such a measure are outlined, including sharpness, maximality, resolution, symmetry, and valuation. The document provides mathematical foundations and definitions to propose and analyze a measure of intuitionistic fuzzy entropy.
Stability of Iteration for Some General Operators in b-MetricKomal Goyal
This document discusses the stability of the Jungck-Mann iteration scheme for approximating fixed points of operators in b-metric spaces. It defines the Jungck-Mann iterative procedure and establishes conditions for its stability. Specifically, it proves that if two operators S and T satisfy a contractive condition in a b-metric space, and Sx converges to a coincidence point p of S and T using the Jungck-Mann scheme, then any perturbed sequence Sy will also converge to p as the perturbations go to zero. The stability result relies on deriving inequalities relating the distances between iterates.
Computational Motor Control: Optimal Control for Deterministic Systems (JAIST...hirokazutanaka
This is lecure 2 note for JAIST summer school on computational motor control (Hirokazu Tanaka & Hiroyuki Kambara). Lecture video: https://www.youtube.com/watch?v=lNH1q4y1m-U
An order seven implicit symmetric sheme applied to second order initial value...Alexander Decker
This document presents a new implicit five-step numerical method of order seven for solving second-order initial value problems of ordinary differential equations directly, without reducing them to systems of first-order equations. The method is derived by imposing conditions on the local truncation error to achieve seventh-order accuracy. Sample linear and nonlinear test problems are solved to demonstrate the applicability and accuracy of the new method compared to existing schemes. The coefficients of the method are determined to be α0 = -1, α1 = -5, α2 = 10, α3 = -10, α4 = 5, β0 = -1/12, β1 = -7/12, β2 = 26/12, β3 =
1. The document discusses stochastic approximations for mortality projection models like the Lee-Carter (LC) and Cairns-Blake-Dowd (CBD) models.
2. It presents comonotonic approximations that provide conservative estimates of survival probabilities and numbers of survivors in a portfolio, avoiding complex simulations.
3. Numerical results show the conservative comonotonic approximations are close to the true values, providing useful tools for practitioners.
Stochastics Calculus: Malliavin Calculus in a simplest wayIOSR Journals
We present the theory of Malliavin Calculus by tracing the origin of this calculus as well as giving a
simple introduction to the classical variational problem. In the work, we apply the method of integration-byparts
technique which lies at the core of the theory of stochastic calculus of variation as provided in Malliavin
Calculus. We consider the application of this calculus to the computation of Greeks, as well as discussing the
calculation of Greeks (price sensitivities) by considering a one dimensional Black-Scholes Model. The result
shows that Malliavin Calculus is an important tool which provides a simple way of calculating sensitivities of
financial derivatives to change in its underlying parameters such as Delta, Vega, Gamma, Rho and Theta
It is a new theory based on an algorithmic approach. Its only element
is called nokton. These rules are precise. The innities are completely
absent whatever the system studied. It is a theory with discrete space
and time. The theory is only at these beginnings.
Stability analysis for nonlinear impulsive optimal control problemsAI Publications
We consider the generic stability of optimal control problems governed by nonlinear impulsive evolution equations. Under perturbations of the right-hand side functions of the controlled system, the results of stability for the impulsive optimal control problems are proved given set-valued theory.
The document classifies all possible Uq(sl2)-module algebra structures on the quantum plane. It produces a complete list of these structures and describes the isomorphism classes. The composition series of the representations under these structures are computed to classify the structures.
This dissertation defense presentation summarizes Sawinder Pal Kaur's PhD dissertation on the existence and properties of positive solutions to certain nonlinear eigenvalue problems involving multi-dimensional arrays. The presentation outlines Kaur's work on proving the existence of a unique positive solution for nonlinear transformations in two and three variables, including the discretization of the Gross-Pitaevskii equation. It also summarizes the properties of these solutions, such as continuity and asymptotic behavior. The proofs utilize tools like Kronecker products, Perron-Frobenius theory, and the Kantorovich fixed point theorem.
This document contains lecture notes on exponential growth and decay from a Calculus I class at New York University. It begins with announcements about an upcoming review session, office hours, and midterm exam. It then outlines the topics to be covered, including the differential equation y=ky, modeling population growth, radioactive decay including carbon-14 dating, Newton's law of cooling, and continuously compounded interest. Examples are provided of solving various differential equations representing exponential growth or decay. The document explains that many real-world situations exhibit exponential behavior due to proportional growth rates.
A current perspectives of corrected operator splitting (os) for systemsAlexander Decker
This document discusses operator splitting methods for solving systems of convection-diffusion equations. It begins by introducing operator splitting, where the time evolution is split into separate steps for convection and diffusion. While efficient, operator splitting can produce significant errors near shocks.
The document then examines the nonlinear error mechanism that causes issues for operator splitting near shocks. When a shock develops in the convection step, it introduces a local linearization that neglects self-sharpening effects. This leads to splitting errors.
To address this, the document discusses corrected operator splitting, which uses the wave structure from the convection step to identify where nonlinear splitting errors occur. Terms are added to the diffusion step to compensate for
The document summarizes different types of derivatives:
Simple derivatives involve a single input and output. Implicit derivatives are taken for equations with two or more variables, treating one as the independent variable. An example finds derivatives of u with respect to v and v with respect to u for the equation 2u^2 - v^3 = 2 - uv. The derivatives are related by the reciprocal relationship in differential notation.
The document discusses neural-oscillator models of quantum-decision making. It begins with outlining stochastic resonance (SR) theory and its structure involving random variables. It then introduces the oscillator model for representing neural activity, with neurons modeled as coupled oscillators. Their phases and interactions encode stimulus and response information. The document explores how SR theory can be combined with the oscillator model to represent response selection and conditioning. Finally, it discusses how the models may exhibit quantum-like behaviors such as nondeterminism, contextuality, and possibly nonlocality, despite being classical systems.
Computational Motor Control: State Space Models for Motor Adaptation (JAIST s...hirokazutanaka
This is lecure 3 note for JAIST summer school on computational motor control (Hirokazu Tanaka & Hiroyuki Kambara). Lecture video: https://www.youtube.com/watch?v=dtpgJLRt90M
Existence Theory for Second Order Nonlinear Functional Random Differential Eq...IOSR Journals
This document presents an existence theory for solutions to second order nonlinear functional random differential equations in Banach algebras. It begins by introducing the type of random differential equation being studied and defining relevant function spaces. It then states several theorems and lemmas from previous works that will be used to prove the main results. The paper goes on to prove that under certain Lipschitz conditions and boundedness assumptions on the operators defining the equation, the random differential equation has at least one random solution in the given function space. It also shows that the set of such random solutions is compact. The results generalize previous existence theorems to the random case.
Research Inventy : International Journal of Engineering and Scienceresearchinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
This document provides tips for having a fun and safe trip. It recommends planning ahead by booking travel early, bringing essential contact information, and having photos of children in case they get lost. Safety tips include being wary of strangers claiming to be officials, never getting in unknown locals' cars, and staying calm when problems arise. Financial tips include choosing alternate destinations to save money, using hotel ice for filtered water, and asking about local rates.
USING GOOGLE’S KEYWORD RELATION IN MULTIDOMAIN DOCUMENT CLASSIFICATIONIJDKP
The document describes a new method for multi-domain document classification using keyword sequences extracted from documents. It introduces the Word AdHoc Network (WANET) system which uses Google's Keyword Relation and a new similarity measurement called Google Purity to classify documents into domains based on their extracted 4-word keyword sequences, without requiring pre-established keyword repositories. Experimental results showed the classification was accurate and efficient, allowing cross-domain classification and management of knowledge from different sources.
On Convergence of Jungck Type Iteration for Certain Contractive Conditionsresearchinventy
In this article we prove the strong convergence result for a pair of nonself mappings using Jungck S- iterative scheme in Convex metric spaces satisfying certain contractive condition. The results are the generalization of some existing results in the literature
STUDIES ON INTUTIONISTIC FUZZY INFORMATION MEASURESurender Singh
This document discusses studies on measures of intuitionistic fuzzy information. It begins with introductions and definitions related to fuzzy sets, intuitionistic fuzzy sets, and measures of fuzzy entropy. It then discusses special t-norm operators and proposes a measure of intuitionistic fuzzy entropy based on these t-norms. The measure is defined using a function of the membership, non-membership, and hesitancy degrees of an intuitionistic fuzzy set. Several desirable properties of such a measure are outlined, including sharpness, maximality, resolution, symmetry, and valuation. The document provides mathematical foundations and definitions to propose and analyze a measure of intuitionistic fuzzy entropy.
Stability of Iteration for Some General Operators in b-MetricKomal Goyal
This document discusses the stability of the Jungck-Mann iteration scheme for approximating fixed points of operators in b-metric spaces. It defines the Jungck-Mann iterative procedure and establishes conditions for its stability. Specifically, it proves that if two operators S and T satisfy a contractive condition in a b-metric space, and Sx converges to a coincidence point p of S and T using the Jungck-Mann scheme, then any perturbed sequence Sy will also converge to p as the perturbations go to zero. The stability result relies on deriving inequalities relating the distances between iterates.
Computational Motor Control: Optimal Control for Deterministic Systems (JAIST...hirokazutanaka
This is lecure 2 note for JAIST summer school on computational motor control (Hirokazu Tanaka & Hiroyuki Kambara). Lecture video: https://www.youtube.com/watch?v=lNH1q4y1m-U
An order seven implicit symmetric sheme applied to second order initial value...Alexander Decker
This document presents a new implicit five-step numerical method of order seven for solving second-order initial value problems of ordinary differential equations directly, without reducing them to systems of first-order equations. The method is derived by imposing conditions on the local truncation error to achieve seventh-order accuracy. Sample linear and nonlinear test problems are solved to demonstrate the applicability and accuracy of the new method compared to existing schemes. The coefficients of the method are determined to be α0 = -1, α1 = -5, α2 = 10, α3 = -10, α4 = 5, β0 = -1/12, β1 = -7/12, β2 = 26/12, β3 =
1. The document discusses stochastic approximations for mortality projection models like the Lee-Carter (LC) and Cairns-Blake-Dowd (CBD) models.
2. It presents comonotonic approximations that provide conservative estimates of survival probabilities and numbers of survivors in a portfolio, avoiding complex simulations.
3. Numerical results show the conservative comonotonic approximations are close to the true values, providing useful tools for practitioners.
Stochastics Calculus: Malliavin Calculus in a simplest wayIOSR Journals
We present the theory of Malliavin Calculus by tracing the origin of this calculus as well as giving a
simple introduction to the classical variational problem. In the work, we apply the method of integration-byparts
technique which lies at the core of the theory of stochastic calculus of variation as provided in Malliavin
Calculus. We consider the application of this calculus to the computation of Greeks, as well as discussing the
calculation of Greeks (price sensitivities) by considering a one dimensional Black-Scholes Model. The result
shows that Malliavin Calculus is an important tool which provides a simple way of calculating sensitivities of
financial derivatives to change in its underlying parameters such as Delta, Vega, Gamma, Rho and Theta
It is a new theory based on an algorithmic approach. Its only element
is called nokton. These rules are precise. The innities are completely
absent whatever the system studied. It is a theory with discrete space
and time. The theory is only at these beginnings.
Stability analysis for nonlinear impulsive optimal control problemsAI Publications
We consider the generic stability of optimal control problems governed by nonlinear impulsive evolution equations. Under perturbations of the right-hand side functions of the controlled system, the results of stability for the impulsive optimal control problems are proved given set-valued theory.
The document classifies all possible Uq(sl2)-module algebra structures on the quantum plane. It produces a complete list of these structures and describes the isomorphism classes. The composition series of the representations under these structures are computed to classify the structures.
This dissertation defense presentation summarizes Sawinder Pal Kaur's PhD dissertation on the existence and properties of positive solutions to certain nonlinear eigenvalue problems involving multi-dimensional arrays. The presentation outlines Kaur's work on proving the existence of a unique positive solution for nonlinear transformations in two and three variables, including the discretization of the Gross-Pitaevskii equation. It also summarizes the properties of these solutions, such as continuity and asymptotic behavior. The proofs utilize tools like Kronecker products, Perron-Frobenius theory, and the Kantorovich fixed point theorem.
This document contains lecture notes on exponential growth and decay from a Calculus I class at New York University. It begins with announcements about an upcoming review session, office hours, and midterm exam. It then outlines the topics to be covered, including the differential equation y=ky, modeling population growth, radioactive decay including carbon-14 dating, Newton's law of cooling, and continuously compounded interest. Examples are provided of solving various differential equations representing exponential growth or decay. The document explains that many real-world situations exhibit exponential behavior due to proportional growth rates.
A current perspectives of corrected operator splitting (os) for systemsAlexander Decker
This document discusses operator splitting methods for solving systems of convection-diffusion equations. It begins by introducing operator splitting, where the time evolution is split into separate steps for convection and diffusion. While efficient, operator splitting can produce significant errors near shocks.
The document then examines the nonlinear error mechanism that causes issues for operator splitting near shocks. When a shock develops in the convection step, it introduces a local linearization that neglects self-sharpening effects. This leads to splitting errors.
To address this, the document discusses corrected operator splitting, which uses the wave structure from the convection step to identify where nonlinear splitting errors occur. Terms are added to the diffusion step to compensate for
The document summarizes different types of derivatives:
Simple derivatives involve a single input and output. Implicit derivatives are taken for equations with two or more variables, treating one as the independent variable. An example finds derivatives of u with respect to v and v with respect to u for the equation 2u^2 - v^3 = 2 - uv. The derivatives are related by the reciprocal relationship in differential notation.
The document discusses neural-oscillator models of quantum-decision making. It begins with outlining stochastic resonance (SR) theory and its structure involving random variables. It then introduces the oscillator model for representing neural activity, with neurons modeled as coupled oscillators. Their phases and interactions encode stimulus and response information. The document explores how SR theory can be combined with the oscillator model to represent response selection and conditioning. Finally, it discusses how the models may exhibit quantum-like behaviors such as nondeterminism, contextuality, and possibly nonlocality, despite being classical systems.
Computational Motor Control: State Space Models for Motor Adaptation (JAIST s...hirokazutanaka
This is lecure 3 note for JAIST summer school on computational motor control (Hirokazu Tanaka & Hiroyuki Kambara). Lecture video: https://www.youtube.com/watch?v=dtpgJLRt90M
Existence Theory for Second Order Nonlinear Functional Random Differential Eq...IOSR Journals
This document presents an existence theory for solutions to second order nonlinear functional random differential equations in Banach algebras. It begins by introducing the type of random differential equation being studied and defining relevant function spaces. It then states several theorems and lemmas from previous works that will be used to prove the main results. The paper goes on to prove that under certain Lipschitz conditions and boundedness assumptions on the operators defining the equation, the random differential equation has at least one random solution in the given function space. It also shows that the set of such random solutions is compact. The results generalize previous existence theorems to the random case.
Research Inventy : International Journal of Engineering and Scienceresearchinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
This document provides tips for having a fun and safe trip. It recommends planning ahead by booking travel early, bringing essential contact information, and having photos of children in case they get lost. Safety tips include being wary of strangers claiming to be officials, never getting in unknown locals' cars, and staying calm when problems arise. Financial tips include choosing alternate destinations to save money, using hotel ice for filtered water, and asking about local rates.
USING GOOGLE’S KEYWORD RELATION IN MULTIDOMAIN DOCUMENT CLASSIFICATIONIJDKP
The document describes a new method for multi-domain document classification using keyword sequences extracted from documents. It introduces the Word AdHoc Network (WANET) system which uses Google's Keyword Relation and a new similarity measurement called Google Purity to classify documents into domains based on their extracted 4-word keyword sequences, without requiring pre-established keyword repositories. Experimental results showed the classification was accurate and efficient, allowing cross-domain classification and management of knowledge from different sources.
Buy Hawaii Diet Cookbook 2013 (updated2) by Dr.Terry Shintani (Paperback) online at Lulu. Visit the Lulu Marketplace for product details, ratings, and reviews.
This document provides a body maintenance manual to help men ages 18-45 maintain the health of their "body machine." The manual includes sections to record medical history and health goals, establish a preventative maintenance schedule for checkups and vaccines, get tips on maintaining sexual health and weight, and includes important contact information. The overall goal is to help men travel down the road to a healthier lifestyle.
The document discusses the principles of the HawaiiDiet program. It promotes an approach focused on eating satisfying whole foods without calorie counting. The diet emphasizes foods from local supermarkets that can be enjoyed for life. It also highlights the program's emphasis on health for all people regardless of attributes. The HawaiiDiet led to significant weight loss and health improvements for a group of participants in 3 weeks based on following the program's pyramid and SMI guidelines for whole food selections.
Este documento resume los diferentes tipos de virus informáticos como virus troyanos, gusanos, de sobreescritura, de enlace, múltiples, residentes, mutantes y de macros. También describe los principales antivirus como AVG, Norton, Microsoft Security, Avira, Kaspersky, Panda y Avast, y ofrece consejos sobre precauciones contra virus como mantener el antivirus activado y no abrir archivos adjuntos de remitentes desconocidos.
Resumen de la Decision de la Sala Constitucional del 07/10/2014 que declaró la Inconstitucionalidad (parcial) del artículo 46 del Codigo Organico Tributario 2001
Este documento resume las características principales de Facebook e Instagram. Explica que Facebook es un sitio de redes sociales creado por Mark Zuckerberg para estudiantes de Harvard que luego se abrió a más usuarios, mientras que Instagram es una aplicación para compartir fotos con filtros y efectos. También enumera algunas ventajas como comunicarse con amigos, conocer gente nueva y entretenimiento; y desventajas como posibles problemas de privacidad, adicción y uso excesivo del tiempo en estas plataformas.
This document contains Spanish proverbs and their English translations. It offers advice around making the most of opportunities, valuing friendship, diligence, perseverance, finding solutions, avoiding hastiness and superficial judgments. The proverbs counsel that a gift should be appreciated for what it is, helping friends in difficult times is important, it is best to complete tasks without delay, early risers are rewarded, where there is determination solutions can be found, patience leads to rewards, challenges contain benefits, lost love is preferable to none, setbacks contain blessings, something in hand is better than possibilities elsewhere, rushing leads to mistakes, and one should consider the full nature rather than surface impressions alone.
The document is a storyboard for a Barry M nail varnish advertisement. It follows four female characters: Izzy, Pia, Natalie, and Ellie. The storyboard shows the characters playing chess at a cafe, running into town smiling, waiting at a crosswalk while admiring their nail colors, relaxing on a bridge where they spot Ellie biking, and sneaking into a private regatta event for a picnic where they apply makeup and paint their nails. The final shot is an extreme close-up of the Barry M logo.
El documento discute varios riesgos que enfrentan los niños y adolescentes al usar Internet, incluyendo la exposición a material pornográfico y de contenido inapropiado, el acoso cibernético, los intentos de seducción sexual por parte de adultos, y la violencia y adicción a los juegos de azar en línea. También ofrece consejos sobre cómo navegar de manera más segura en Internet.
The document provides an analysis of three short films:
1. "Young Offender" by Isabel Anderton represents the ignorance of a racist male teenager in a young offenders institute and uses lighting and sound to portray his isolation and loneliness.
2. "Young Offender" by Justin Edgar represents the problems with gang culture in Britain and realistically portrays the representation of power and misguidance through the characters and their actions.
3. "Light 'Em Up" by Phil Stoole and Damien Wasylkiw represents the contrasts of lifestyle in Britain by showing two teenagers relaxing as the city of London experiences an apocalypse in the background.
Invited plenary talk given by Prof. Farid El-Daoushy at Inter. Radiation Protection Congress, Afrirpa-2010, Sept. 2010, Nairobi, Kenya.
This talk summarizes the sources, processes and fate of environmental radioactivity (natural and artificial) at the earth\'s surface. It links environmental radioactivity to radiation and environmental protection issues in Africa. The role of global and regional processes on the transport and accumulation of anthropogenic waste in complex ecosystems, e.g. the Nile Basin, are described. These issues help fulfilling the African mission and vision for achieving their golals: ONE PEOPLE, ONE GOAL, ONE FAITH.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This paper provides alternative optimal expressions for determining matrices of a class of double-delay autonomous linear differential systems. The paper extends previous work by resolving ambiguities in permutations and analyzing computational complexity and cardinalities of determining matrices. Key results include theorems and corollaries that provide expressions for determining matrices ( )kQ jh in terms of sums of permutation products involving system matrices A, A1, A2. The expressions are proven to be consistent with previous work and fill gaps in the prior literature.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Necessary and Sufficient Conditions for Oscillations of Neutral Delay Differe...inventionjournals
In this paper, we discuss the oscillatory behavior of all solutions of the first order neutral delay difference equations with several positive and negative coefficients ( ) ( ) ( ) ( ) 0 , i i j j k k I J K x n p x n r x n q x n , o n n (*) where I, J and K are initial segments of natural numbers, pi , rj , qk are positive numbers, i , j are positive integers and k is a nonnegative integer for iI, jJ and kK. We establish a necessary and sufficient conditions for the oscillation of all solutions of (*) is that its characteristic equation ( 1) 1 0 j i k i j k I J K p r q has no positive roots . AMS Subject Classifications : 39A10, 39A12.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Simple Exponential Observer Design for the Generalized Liu Chaotic Systemijtsrd
In this paper, the generalized Liu chaotic system is firstly introduced and the state observation problem of such a system is investigated. Based on the time-domain approach with differential and integral equalities, a novel state observer for the generalized Liu chaotic system is constructed to ensure the global exponential stability of the resulting error system. Besides, the guaranteed exponential convergence rate can be precisely calculated. Finally, numerical simulations are presented to exhibit the effectiveness and feasibility of the obtained results. Yeong-Jeu Sun"Simple Exponential Observer Design for the Generalized Liu Chaotic System" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-1 , December 2017, URL: http://www.ijtsrd.com/papers/ijtsrd7126.pdf http://www.ijtsrd.com/engineering/engineering-maths/7126/simple-exponential-observer-design-for-the-generalized-liu-chaotic-system/yeong-jeu-sun
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Convergence Theorems for Implicit Iteration Scheme With Errors For A Finite F...inventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Common Fixed Theorems Using Random Implicit Iterative Schemesinventy
This document summarizes research on common fixed point theorems using random implicit iterative schemes. It defines random Mann, Ishikawa, and SP iterative schemes. It also defines modified implicit random iterative schemes associated with families of random asymptotically nonexpansive operators. The paper proves the convergence of two random implicit iterative schemes to a random common fixed point. This generalizes previous results and provides new convergence theorems for random operators in Banach spaces.
This document summarizes research on generalized Cantor sets and functions where the standard construction is modified. It introduces Cantor sets defined by an arbitrary base where the intervals removed at each stage are not all the same length. It also defines irregular or transcendental Cantor sets generated by transcendental numbers like e. The key findings are:
1) There exists a unique probability measure for generalized Cantor sets that generates the cumulative distribution function.
2) The Holder exponent of generalized Cantor sets is shown to be logn/s where n is the base and s is the number of subintervals.
3) Lower and upper densities are defined for the measure on generalized Cantor functions and their properties are
Exponential State Observer Design for a Class of Uncertain Chaotic and Non-Ch...ijtsrd
The document presents an exponential state observer design for uncertain chaotic and non-chaotic systems. It introduces a class of uncertain nonlinear systems and explores the state observation problem for such systems. Using an approach with integral and differential equalities, an exponential state observer is established that guarantees global exponential stability of the error system. Numerical simulations exhibit the feasibility and effectiveness of the observer design.
The document provides definitions and formulas for inverse trigonometric functions. It begins with basic introductions and definitions. It then lists key formulas for inverse trigonometric functions such as sin-1, cos-1, tan-1, etc. along with proofs of each formula. The document also discusses inverse trigonometric identities and relations involving addition or subtraction of inverse trigonometric functions.
This document presents an analysis of the exponential distribution under an adaptive type-I progressive hybrid censoring scheme for competing risks data. Maximum likelihood and Bayesian estimation methods are used to estimate the distribution parameter. Specifically, maximum likelihood estimators are derived for the exponential distribution parameter. Bayesian estimators are also obtained for the parameter based on squared error and LINEX loss functions using gamma priors. Asymptotic confidence intervals and Bayesian credible intervals are proposed. A simulation study is conducted to evaluate the performance of the estimators.
The document is a thesis on fuzzy implications submitted for a Master of Science in Applied Mathematics. It discusses fuzzy sets and fuzzy logic as extensions of classical set theory and logic to account for imprecise and uncertain information. It defines fuzzy implications and various basic fuzzy implications functions. It also discusses properties of fuzzy implication lattices, including meet, join and other properties like commutativity, associativity and absorption. An example is provided to demonstrate these properties for specific fuzzy implication functions.
Existence of Solutions of Fractional Neutral Integrodifferential Equations wi...inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Some Continued Mock Theta Functions from Ramanujan’s Lost Notebook (IV)paperpublications3
Abstract: Ramanujan’s lost notebook contains many results on mock theta functions. In particular, the lost notebook contains eight identities for tenth order mock theta functions. Previously many authors proved the first six of Ramanujan’s tenth order mock theta function identities. It is the purpose of this paper to prove the seventh and eighth identities of Ramanujan’s tenth order mock theta function identities which are expressed by mock theta functions and also a definite integral. The properties of modular forms are used for the proofs of theta function identities and L. J. Mordell’s transformation formula for the definite integral.Keywords: Mock Theta Functions from Ramanujan’s Lost Notebook.
Title: Some Continued Mock Theta Functions from Ramanujan’s Lost Notebook (IV)
Author: MOHAMMADI BEGUM JEELANI SHAIKH
ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Paper Publications
Discover the latest insights on Data Driven Maintenance with our comprehensive webinar presentation. Learn about traditional maintenance challenges, the right approach to utilizing data, and the benefits of adopting a Data Driven Maintenance strategy. Explore real-world examples, industry best practices, and innovative solutions like FMECA and the D3M model. This presentation, led by expert Jules Oudmans, is essential for asset owners looking to optimize their maintenance processes and leverage digital technologies for improved efficiency and performance. Download now to stay ahead in the evolving maintenance landscape.
Null Bangalore | Pentesters Approach to AWS IAMDivyanshu
#Abstract:
- Learn more about the real-world methods for auditing AWS IAM (Identity and Access Management) as a pentester. So let us proceed with a brief discussion of IAM as well as some typical misconfigurations and their potential exploits in order to reinforce the understanding of IAM security best practices.
- Gain actionable insights into AWS IAM policies and roles, using hands on approach.
#Prerequisites:
- Basic understanding of AWS services and architecture
- Familiarity with cloud security concepts
- Experience using the AWS Management Console or AWS CLI.
- For hands on lab create account on [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
# Scenario Covered:
- Basics of IAM in AWS
- Implementing IAM Policies with Least Privilege to Manage S3 Bucket
- Objective: Create an S3 bucket with least privilege IAM policy and validate access.
- Steps:
- Create S3 bucket.
- Attach least privilege policy to IAM user.
- Validate access.
- Exploiting IAM PassRole Misconfiguration
-Allows a user to pass a specific IAM role to an AWS service (ec2), typically used for service access delegation. Then exploit PassRole Misconfiguration granting unauthorized access to sensitive resources.
- Objective: Demonstrate how a PassRole misconfiguration can grant unauthorized access.
- Steps:
- Allow user to pass IAM role to EC2.
- Exploit misconfiguration for unauthorized access.
- Access sensitive resources.
- Exploiting IAM AssumeRole Misconfiguration with Overly Permissive Role
- An overly permissive IAM role configuration can lead to privilege escalation by creating a role with administrative privileges and allow a user to assume this role.
- Objective: Show how overly permissive IAM roles can lead to privilege escalation.
- Steps:
- Create role with administrative privileges.
- Allow user to assume the role.
- Perform administrative actions.
- Differentiation between PassRole vs AssumeRole
Try at [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
Generative AI Use cases applications solutions and implementation.pdfmahaffeycheryld
Generative AI solutions encompass a range of capabilities from content creation to complex problem-solving across industries. Implementing generative AI involves identifying specific business needs, developing tailored AI models using techniques like GANs and VAEs, and integrating these models into existing workflows. Data quality and continuous model refinement are crucial for effective implementation. Businesses must also consider ethical implications and ensure transparency in AI decision-making. Generative AI's implementation aims to enhance efficiency, creativity, and innovation by leveraging autonomous generation and sophisticated learning algorithms to meet diverse business challenges.
https://www.leewayhertz.com/generative-ai-use-cases-and-applications/
Software Engineering and Project Management - Introduction, Modeling Concepts...Prakhyath Rai
Introduction, Modeling Concepts and Class Modeling: What is Object orientation? What is OO development? OO Themes; Evidence for usefulness of OO development; OO modeling history. Modeling
as Design technique: Modeling, abstraction, The Three models. Class Modeling: Object and Class Concept, Link and associations concepts, Generalization and Inheritance, A sample class model, Navigation of class models, and UML diagrams
Building the Analysis Models: Requirement Analysis, Analysis Model Approaches, Data modeling Concepts, Object Oriented Analysis, Scenario-Based Modeling, Flow-Oriented Modeling, class Based Modeling, Creating a Behavioral Model.
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODELijaia
As digital technology becomes more deeply embedded in power systems, protecting the communication
networks of Smart Grids (SG) has emerged as a critical concern. Distributed Network Protocol 3 (DNP3)
represents a multi-tiered application layer protocol extensively utilized in Supervisory Control and Data
Acquisition (SCADA)-based smart grids to facilitate real-time data gathering and control functionalities.
Robust Intrusion Detection Systems (IDS) are necessary for early threat detection and mitigation because
of the interconnection of these networks, which makes them vulnerable to a variety of cyberattacks. To
solve this issue, this paper develops a hybrid Deep Learning (DL) model specifically designed for intrusion
detection in smart grids. The proposed approach is a combination of the Convolutional Neural Network
(CNN) and the Long-Short-Term Memory algorithms (LSTM). We employed a recent intrusion detection
dataset (DNP3), which focuses on unauthorized commands and Denial of Service (DoS) cyberattacks, to
train and test our model. The results of our experiments show that our CNN-LSTM method is much better
at finding smart grid intrusions than other deep learning algorithms used for classification. In addition,
our proposed approach improves accuracy, precision, recall, and F1 score, achieving a high detection
accuracy rate of 99.50%.
Gas agency management system project report.pdfKamal Acharya
The project entitled "Gas Agency" is done to make the manual process easier by making it a computerized system for billing and maintaining stock. The Gas Agencies get the order request through phone calls or by personal from their customers and deliver the gas cylinders to their address based on their demand and previous delivery date. This process is made computerized and the customer's name, address and stock details are stored in a database. Based on this the billing for a customer is made simple and easier, since a customer order for gas can be accepted only after completing a certain period from the previous delivery. This can be calculated and billed easily through this. There are two types of delivery like domestic purpose use delivery and commercial purpose use delivery. The bill rate and capacity differs for both. This can be easily maintained and charged accordingly.
Redefining brain tumor segmentation: a cutting-edge convolutional neural netw...IJECEIAES
Medical image analysis has witnessed significant advancements with deep learning techniques. In the domain of brain tumor segmentation, the ability to
precisely delineate tumor boundaries from magnetic resonance imaging (MRI)
scans holds profound implications for diagnosis. This study presents an ensemble convolutional neural network (CNN) with transfer learning, integrating
the state-of-the-art Deeplabv3+ architecture with the ResNet18 backbone. The
model is rigorously trained and evaluated, exhibiting remarkable performance
metrics, including an impressive global accuracy of 99.286%, a high-class accuracy of 82.191%, a mean intersection over union (IoU) of 79.900%, a weighted
IoU of 98.620%, and a Boundary F1 (BF) score of 83.303%. Notably, a detailed comparative analysis with existing methods showcases the superiority of
our proposed model. These findings underscore the model’s competence in precise brain tumor localization, underscoring its potential to revolutionize medical
image analysis and enhance healthcare outcomes. This research paves the way
for future exploration and optimization of advanced CNN models in medical
imaging, emphasizing addressing false positives and resource efficiency.
Design and optimization of ion propulsion dronebjmsejournal
Electric propulsion technology is widely used in many kinds of vehicles in recent years, and aircrafts are no exception. Technically, UAVs are electrically propelled but tend to produce a significant amount of noise and vibrations. Ion propulsion technology for drones is a potential solution to this problem. Ion propulsion technology is proven to be feasible in the earth’s atmosphere. The study presented in this article shows the design of EHD thrusters and power supply for ion propulsion drones along with performance optimization of high-voltage power supply for endurance in earth’s atmosphere.
Rainfall intensity duration frequency curve statistical analysis and modeling...bijceesjournal
Using data from 41 years in Patna’ India’ the study’s goal is to analyze the trends of how often it rains on a weekly, seasonal, and annual basis (1981−2020). First, utilizing the intensity-duration-frequency (IDF) curve and the relationship by statistically analyzing rainfall’ the historical rainfall data set for Patna’ India’ during a 41 year period (1981−2020), was evaluated for its quality. Changes in the hydrologic cycle as a result of increased greenhouse gas emissions are expected to induce variations in the intensity, length, and frequency of precipitation events. One strategy to lessen vulnerability is to quantify probable changes and adapt to them. Techniques such as log-normal, normal, and Gumbel are used (EV-I). Distributions were created with durations of 1, 2, 3, 6, and 24 h and return times of 2, 5, 10, 25, and 100 years. There were also mathematical correlations discovered between rainfall and recurrence interval.
Findings: Based on findings, the Gumbel approach produced the highest intensity values, whereas the other approaches produced values that were close to each other. The data indicates that 461.9 mm of rain fell during the monsoon season’s 301st week. However, it was found that the 29th week had the greatest average rainfall, 92.6 mm. With 952.6 mm on average, the monsoon season saw the highest rainfall. Calculations revealed that the yearly rainfall averaged 1171.1 mm. Using Weibull’s method, the study was subsequently expanded to examine rainfall distribution at different recurrence intervals of 2, 5, 10, and 25 years. Rainfall and recurrence interval mathematical correlations were also developed. Further regression analysis revealed that short wave irrigation, wind direction, wind speed, pressure, relative humidity, and temperature all had a substantial influence on rainfall.
Originality and value: The results of the rainfall IDF curves can provide useful information to policymakers in making appropriate decisions in managing and minimizing floods in the study area.
Rainfall intensity duration frequency curve statistical analysis and modeling...
D023031047
1. International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org ǁ Volume 2 ǁ Issue 3 ǁ March 2014 ǁ PP-31-47
www.ijmsi.org 31 | P a g e
The structure of determining matrices for single-delay
autonomous linear neutral control systems
Ukwu Chukwunenye
Department of Mathematics,University of Jos, P.M.B 2084 Jos, Plateau State, Nigeria.
ABSTRACT: This paper derived and established the structure of determining matrices for single – delay
autonomous linear neutral differential systems through a sequence of lemmas, theorems and corollaries and the
exploitation of key facts about permutations. The proofs were achieved using ingenious combinations of
summation notations, the multinomial distribution,change of variables technique and compositions of signum
and max functions. The paper also pointed out some induction pitfalls in the derivation of the main results.
The paper has extended the results on single–delay models, with more complexity in the structure of the
determining matrices.
KEYWORDS: Delay, Determining, Neutral, Structure, Systems.
I. INTRODUCTION
The importance of determining matrices stems from the fact that they constitute the optimal
instrumentality for the determination of Euclidean controllability and compactness of cores of Euclidean targets.
See Gabasov and Kirillova (1976) and Ukwu (1992, 1996 and 2013a). In sharp contrast to determining matrices,
the use of indices of control systems on the one hand and the application of controllability Grammians on the
other, for the investigation of the Euclidean controllability of systems can at the very best be quite
computationally challenging and at the worst, mathematically intractable. Thus, determining matrices are
beautiful brides for the interrogation of the controllability disposition of single-delay neutral control systems.
See (Ukwu 2013a).
However up-to-date review of literature on this subject reveals that there is currently no correct result on
the structure of determining matrices for single-delay neutral systems. This could be attributed to the severe
difficulty in identifying recognizable mathematical patterns needed for inductive proof of any claimed result
and induction pitfalls in the derivation of determining matrices. This paper establishes valid expressions for
the determining matrices in this area of acute research need.
II. ON DETERMINING MATRICES AND CONTROLLABILITY OF SINGLE-DELAY
AUTONOMOUS NEUTRAL CONTROL SYSTEMS
We consider the class of neutral systems:
1 0 1( ) ( ) ( ) , 0 (1)
d
x t A x t h A x t A x t h Bu t t
dt
where
A A A1 0 1, , are n n constant matrices with real entries and B is an n m constant matrix with the real
entries. The initial function is in , 0 , n
C h R equipped with sup norm. The control u is in
10, , n
L t R . Such controls will be called admissible controls. 1
for, 0,n
x t x t h t t R . If
1
, , n
x C h t R , then for t t 0 1, we define , 0 , n
t C hx R by
x s x t s s ht , , 0 .
2.1 Existence, Uniqueness and Representation of Solutions
If A 1 0 and is continuously differentiable on , 0h , then there exists a unique function : ,x h
which coincides with on , 0h , is continuously differentiable and satisfies (1) except possibly at the points
2. The structure of determining matrices for single-delay…
www.ijmsi.org 32 | P a g e
; 0,1, 2,...jh j . This solution x can have no more derivatives than and continuously differentiable if and
only if the relation:
1 0 1
0 (0) 0 (2)A h A A h Bu
is satisfied. See Bellman and Cooke (1963) and theorem 7.1 in Dauer and Gahl (1977) for complete discussion
on existence, uniqueness and representations of solutions of (1).
2.2 Preliminaries on the Partial Derivatives
( , )
, 0,1,
k
k
X t
k
Let t t, , 0 1 for fixed t, let t, be the unique function satisfying the matrix differential
equation:
1 0 1
, , , , (3)
subject to:
; (4)
, =
0,
0 , ; 0,1,...,
n
t t h A t A t h A
I t
t
t
t t jh j
By Tadmore (1984 , p. 80), ,t is analytic on 1 , , 0,1,....t j h t jh j and hence
, ist C
on these intervals.
The left and right-hand limits of ,t exist at t jh , so that ,t is of bounded
variation on each compact interval. Cf. Banks and Jacobs (1973), Banks and Kent (1972) and Hale (1977). We
maintain the notation
k k
t t, , , as in the double-delay systems.
2.3 Investigation of the Expressions and Structure of Determining Matrices for System (1)
In this section we establish the expressions and the structure of the determining matrices for system (1), as well
as their relationships with
( )
( , )k
X t through a sequence of lemmas, theorems and corollaries and the
exploitation of key facts about permutations.
The process of obtaining necessary and sufficient conditions for the Euclidean controllability of (1) in
chapter four will be initiated in the rest of chapter three as follows:
i. Obtaining a workable expression for the determining matrices of
(1): 1
for : 0, 1, 2,.. (5)k
Q jh j t jh k
ii.Showing that:
1 1
, 1 (6)
kk
k
t jh t Q jh
for j t jh k: , , , ,....,1 0 0 1 2
iii.Showing that Q t 1 is a linear combination of:
0 1 1
, ,..., , 0, ,...., 1 (7)n
Q s Q s Q s s h n h
We now define the determining equation of the n n matrices, Q sk .
For every integer k and real number s, define Q sk by:
1 0 1 1 1
(8)k k k k
Q s A Q s h A Q s A Q s h
for k s h h 0 1 0 2, ,...; , , ,..... subject to Q In0 0 , the n n identity matrix and
0 for 0 or 0k
Q s k s .
2.4 Preliminary Lemma on Determining matrices ( ),kQ s sR
In (1), assume that A 1 0, then for j k, 1,
i. Q Ak
k
0 0
3. The structure of determining matrices for single-delay…
www.ijmsi.org 33 | P a g e
ii. Q jh A
j
0 1
iii.
Q h A A A A A A Ak
k
r
k
k r r
0 1
0
1
0
1
1 1 0 0
iv.
Q jh A A A A A A Aj
r
j
r j r
1 1 0
0
1
1 0 1 1 1
1
v.
1 1 0, 1
kk k
t t A
vi.
1 1, 0
k
t t
vii. 1 1 1, j
t jh t A
Remarks: Notice that Ar
0 can switch places with
Ak r
0
1
in (iii); similarly for Ar
1 and
A j r
1
1
in
(iv), since r and 1k r have the same range for fixed k; similarly in (iv), r and 1j r have the same
range for fixed j.
Proof
i.From (8) we have
Q A Q h A Q A Q h
A Q
k k k k
k
0 0
0 0 0
1 0 1 1 1
0 1
Now, Q A Q A I An1 0 0 0 00 0 . Assume that Q Ak
k
0 0 for some k 1. Then,
1 1 1 0 1 1
0 0
1
0
by the induction hypotheses
0 0 0 0 0k k k k k
k
k
Q A Q h A Q A Q h A Q
A A
A
Hence, Q Ak
k
0 0 for all k 12, ,.....; this proves (i).
0 1 0 0 1 1 1
1 0
ii 1 1
1 0 0
Q jh A Q j h A Q jh A Q j h
A Q j h
Hence, Q h A Q A0 1 0 10
The rest of the proof is by the principle of mathematical induction: Assume that 0 1
for some 1
j
Q jh A j
.
Then
Q j h A Q jh A Q j h A Q j h
A Q jh A A Aj j
0 1 0 0 1 1 1
1 0 1 1 1
1
1 1 1
0 0
,
by induction hypothesis. So 0 1
for 1, 2,......
j
Q jh A j
iii From (8),
1 1 1 0 0 1 0
1 0 0 1 1
( ) 0 0
.
Q h A Q A Q h A Q
A A A A A
.
Plugging in the right-hand side of iii yields
1 1
1 1
1 0 1 0 1 1 0 0 0 1 1 1 0
0
( ) r r
r
Q h A A A A A A A A A A A A
.
Therefore the formula (iii) is valid for k 1.
Assume the validity of (iii) for some k 1. Now
4. The structure of determining matrices for single-delay…
www.ijmsi.org 34 | P a g e
Q h A Q A Q k A Q
A A A A A A A A A A A A A
A A A A A A A A A A A A
A A A A A
k k k k
k k
r
k
k r r k
k
r
k
k r r k
k
r
k
k r
1 1 1 0 1
1 0
1
0 0 1 0
0
1
0
1
1 1 0 0 1 0
0
1
1 0
0
1
0
1
1 1 0 0 1 1 0 0
0
1
1
0
1
0 1
0 0
1 0 0 0 1 1 0 0
0
1
1
0
0 1 1 0 0 1
A A A A A A A
A A A A A A A Q h
r k k k
k
r
k
k r r
k
.
Therefore (iii) is true for k 1 and so true for k 1 2, ,.....
iv Q h A A A A A1 0 1 1 0 1 .
Plug in the right-hand side of (iv) to obtain
A A A A A A A A A A A A 1 0 1
0
0 1 1 1
0
1 0 0 1 1
It follows that (iv) is true for 1.j Assume that (iv) is valid for some j 1. Now
1 1 1 0 0 1 0
1
1
1 1 0 1 0 1 1 1
0
1
0 1 1 1
1
11 1
1 0 1 0 1 1 1 0 1 1 1
0
1
1 0 1 0 1 1 1
1
by the induction hypothesis
1 1
j
j rj r
r
j j
j
j rj r j
r
j
j r j
r
Q j h A Q jh A Q j h A Q jh
A A A A A A A A
A A A A
A A A A A A A A A A A
A A A A A A A
1 0 1 1 1
0
1
1 0 1 0 1 1 1 1
0
1
r r j r
r
j
j r j r
r
A A A A A
A A A A A A A Q j h
So, the formula (iv) is valid for j 1. This completes the proof that (iv) holds for all j 1 2, ,...
v. For k 1,
1
1
1
1
1 1 1 1 1
1 1 0 1 1 1 1 1 0 1 0
,
, 0. 0
, ,
, ,
k
t
t
t
t t t t t
t t A t h t A h t A A A A
by (3), noting that for sufficiently close to t h t1 1, ; so t h t1 1
.
Assume that
1 1 0, 1
kk k
t t A
for some k 1. Then
1
1 1 1
1
1 1 0 1 1 1 1 1 1
1 1
0 0 0
, ,
( , ) , ,
1 0 0 1
k k
t
k k k
k kk k
X t t X t
X t t A X t h t A X t h t A
A A A
Therefore,
1 1 0, for all 1, 2,...
kk
X t t A k
.
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1
1 1
1 1 1
lim
vi. , , 0
k k
t
t t h
X t t X t
, since t1
vii Let j be a nonnegative integer such that t jh1 0 . We integrate (3) and apply (4). Thus
1 1
1 1 1 1 0 1 10 0
, , , ,
t jh t jh
X t X h t A d X t A X h t A d
Hence,
1
1 1 1 1 1 1 1 1
1 0 1 10
, 1 , 0, ,
, ,
t jh
t jh t t j h t A t h t A
t A h t A d
Similarly
1
1 1 1 1 1 1 1 1
( )
1 0 1 10
1, 1 0, ,
, ,
, ,
t jh
X t jh t X t t j h t X t h t A
t A h t A d
A
Therefore
1
1
1 1 1 1 1 1 1 1 1 1
1 0 1 1
1 1
, , 0,
, , , ,
t jh
t jh
X t jh t X t j h t A X t jh t X t j h t A
X t A X h t A d
since lim
0 0
a
a
f t dt
, for any bounded integrable function f, and 1 0 1 1, ,t A h t A
is bounded and integrable. Therefore we have deduced that
1 1 1 1 1 1 1 1 1
1 1, , , ,X t jh t X t jh t X t j h t X t j h t A
for any 1j , j integer.
For j 1, we have
1 1 1 1 1 1 1 1 1 1 1, , , , nX t h t X t h t X t t X t t A I A A
by (vi).
The rest of the proof is by induction on j.
Assume the validity of (vii) for j 1,..., p for some p 1.
Then
1 1 1 1 1 1
1 1 1 1 1 1 1
1 , 1 , 1 ,
, , p
X t p h t X t p h t X t p h t
X t ph t X t ph t A A A
by the induction hypothesis. So 1
1 1 11 , p
t p h t A
and hence (vii) is valid.
2.5 Lemma on 1
( ) and ( )k
Q h Q jh as sums of products of permutations
For any nonnegative integers and ,j k
(i)
1 11
1 11 1(1),0( ) 0( 1),1(1)( , , ) ( , , )
( ) k k
k k k k
k v v v v
v v P v v P
Q h A A A A
(ii)
1 1 1
1 1 11( ),0(1) 1( 1), 1(1)
1
( , , ) ( , , )
( ) jj
jj j j
v v v v
v v P v v P
Q jh A A A A
Proof of (i)
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By lemma 2.4,
1
1
0 1 0 1 0 1 1 0 0
0
and
k
k rj k r
k
r
Q jh A Q h A A A A A A A
1
1 2 1
1 1(1),0(0)
1 2 11(1),0(1) 0(1 1),1(1)
0 1 1
1 0 1 1 0 1 0 1 1 0 1
( , )
0, 1, ; 0
1 ;
v
v v v
v P
v v P v P
k j Q h A rhs A A
k Q h A A A A A rhs A A A A A A A A
So, the lemma is valid for {0,1}.k
Assume the validity of the lemma for 2 ,k p for some integer p . Then,
1
1
0 1 0 1 1 0 0
0
p
p rp r
p
r
Q h A A A A A A A
1 1 1
1 1 11(1),0( ) 0( 1),1(1)( , , ) ( , , )
p
pp p p
pv v v
v v P v v P
vA A A A
1 1 0
1 1 0 0
0
1
1 1
1 11
0 1 0 1 1 0 0
0
1
0 1 0 1 1 0 0
0
1
1
0 1 0 0 1 1 0 0
0
1
1 (1 )
0 1 0 0 1 1 0 0
0
1
(1
0 1 0
0
Now p
p
p rp r
r
p
p p r r
r
p
p p p r r
r
p
p p p r r
r
p
p p
r
A A A
A A A A
A
Q h A A A A A A A
A A A A A A A
A A A A A A A A
A A A A A A A A
A A A
1 1 0
)
1 1 0 0 0
r r p
A A AA A A A A
0 1 1
0 1 0 1
1 1 1
1 1 11(1),0( ) 0( 1),1(1)
1 1 1
1 1 11(1),0( ) 0( 1),1(1)
1
0 0
( , , ) ( , , )
1
0 0
( , , ) ( , , )
pp
pp p p
pp
pp p p
p p
v v v v
v v P v v P
p p
v v v v
v v P v v P
A A A A A A A
A A A A A A A A
A A
A A
0 11 2
1 2 1(1),0( 1)
1 2
1 2 1(1),0( 1) ( , , )( , , )
(with a leading ) (with a leading )
p
p p
p
p p
v v
v v P
v v
v v P
A AA AA A
0 11 1 1 1
1 1 1 10( 1 1),1(1) 0( 1 1),1(1)( , , ) ( , , )
(with a leading ) (with a leading )p p
p pp p
v v v v
v v P v v P
A A A A A A
1 2 1 1
1 2 1 11(1),0( 1) 0( ),1(1)( , , ) ( , , )
p p
p pp p
v v v v
v v P v v P
A A A A
Therefore, part (i) of the lemma is valid for 1k p , and hence valid for every nonnegative integer k .
Proof of (ii)
Using similar reasoning (ii) can be established by induction; but let us do it differently by using the expression
for 1
( )Q jh from lemma 2.4.
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1
1 1 0 1 0 1 1 1
0
1
1 0 1 1 1 1
0 0
1
1
j
j r
r
j j
r r
r r
j r
j r j r
Q jh A A A A A A A
A A A A A A
1 1 1
1 1 11( ),0(1) 1( 1),1(1)( , , ) ( , , )
,jj
jj j j
v v v v
v v P v v P
A A A A
noting that in the first summation over r 0A occupies positions 1,2, , 1j for 0,1, ,r j
respectively. 0A leads only once and trails only once.; in the second summation 1A occupies positions
1,2, , j for 0,1, , 1r j respectively. 1A leads only once and trails only once.
Remarks: Observe that 1
( )Q jh may be deduced from ( )k
Q h by (a) replacing k by j in the expression for
( )k
Q h (b) interchanging -1 and 0 in the expression for ( )k
Q h (c) replacing 0 by -1 in the summations or
permutations involving only the indices 0 and 1.
2.6 Lemma on 1( , )X t
of uncontrolled part of (1)
1
(1)
1 1 1 1 1
0 0
1
(1)
1 1 1 1 1
0 0
1
(1)
1 1 1 1 1
0 0
(i) (( ) , ) (( ( ( )) ) , )
(ii) (( ) , ) (( ( ( )) ) , )
(iii) ( , ) ( ( ( )) ), )
j
r
i
r i
j
r
i
r i
j
r
i
r i
X t jh t X t j r i h t A A
X t jh t X t j r i h t A A
X t jh t X t j r i h t A A
Proof of (i)
(1) (1)
1 1 1 1 0 1 1 1 1 1 1
(1)
1 1 1 1 1 0 1 1 1 1
(1) 2
1 1 1
( ,( ) ) (( ) , ) (( ( 1) ) , ) (( ( 1) ) , )
(( ( 1) ) , ) [ (( ( 1) ) , ) (( ( 2) ) , ) ]
(( ( 2) ) , )
X t t jh X t jh t A X t j h t A X t j h t A
X t j h t A X t j h t A X t j h t A A
X t j h t A
(1) (1)
1 1 0 1
1 0 1 0 1
(By(v)of lemma 2.4)
0 lhs ( , ) rhs; 1 lhs (( ) , )
(( ) , )
j X t t A j X t h t
X t h t A A A A
1 1
1 1 1 1 1 0
0 0
1 1 1 1 0 1
Let us examine yield
yield
Sum the results for these contingencies to get
yield yield
(( (1 ( )) ) , ) ; 0, 0 ( ( ,( ) )
0, 1 ( ( , ) ; 1, 0 ; 1, 1 0.
r
i
r i
X t r i h t A A r i X t t h A
r i X t t A A r i A A r i
1 1
1 1 1 1 1 0 1 0 1
0 0
(( (1 ( )) ) , ) ( (( ) , )r
i
r i
X t r i h t A A X t h t A A A A
So, the lemma is valid for 1j . Let us explore 2j .
(1) (1)
1 1 0 1 1 1 1
2
1 0 1 1 1 0 1 1 1 0 1
2 lhs (( 2 ) , ) (( 2 ) , ) (( ) , ) (( ) , )
(( 2 ) , ) (( ) , ) (( ) , )
j X t h t X t h t A X t h t A X t h t A
X t h t A X t h t A X t h t A A A A A A
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2 1
1 1 1
0 0
1 1 0
1 1 1 1 1 0 1
1 1
2
0 1
Let us examine
yield
yield yield
yield
yield y
(( (2 ( )) ) , ) ;
0, 0 ( (( 2 ) , )
0, 1 ( (( ) , ) ; 1, 0 ( (( ) , ) ;
1, 1 ;
2, 0 ; 2, 0
r
i
r i
X t r i h t A A
r i X t h t A
r i X t h t A r i X t h t A A
r i A A
r i A A r i
2 1
1 1 1
0 0
2
1 1 0 1 1 1 1 1 0 1 1 1 0 1
ield
Sum the results for these contingencies to get
0.
(( (2 ( )) ) , )
( (( 2 ) , ) ( (( ) , ) ( (( ) , ) .
r
i
r i
X t r i h t A A
X t h t A X t h t A X t h t A A A A A A
So, (i) of the lemma is valid for {0,1,2}.j Assume that (i) of the lemma is valid for
3 , for some integer .j n n Then
1
(1)
1 1 1 1 1
0 0
by the induction hypothesis.(( ) , ) (( ( ( )) ) , ) ,
n
r
i
r i
X t nh t X t n r i h t A A
(1)
1 1 1 1 0 1 1 1
(1)
1 1 1
1
(1)
1 1 1 1 1
0
1
1
1 1 1 1
0 0
Now, (( ( 1) ) , ) (( ( 1) ) , ) (( ) , )
(( ) , )
(( ) , ) (( ( 1 ) ) , )
(( ( ( )) ) , ) (( ( 1
i
i
n
r
i
r i
X t n h t X t n h t A X t nh t A
X t nh t A
X t nh t A X t n i h t A
X t n r i h t A A X t n
1
1
0
) ) , ) i
i
i h t A
1 1
1 1 1
0 0
as desired. So (i) is true for ,and hence valid
for all .
(( ( 1 ( )) ) , ) ,
n
r
i
r i
j n
j
X t n r i h t A A
Proof of (ii)
(1) (1)
1 1 1 1 0 1 1 1 1 1 1
(1)
1 1 1 1 1 0 1 1 1 1
(1) 2
1 1 1
(( ) , ) (( ) , ) (( ( 1) ) , ) (( ( 1) ) , )
(( ( 1) ) , ) [ (( ( 1) ) , ) (( ( 2) ) , ) ]
(( ( 2) ) , )
X t jh t X t jh t A X t j h t A X t j h t A
X t j h t A X t j h t A X t j h t A A
X t j h t A
(1)
1 1
(1) (1)
1 1 1 1 0 1 1 1 1 1 1
1 1 0
0 lhs ( , ) 0 rhs;
1 lhs (( ) , ) (( ) , ) ( , ) ( , )
(( ) , ) 0 0
j X t t
j X t h t X t h t A X t t A X t t A
X t h t A
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1 1
1 1 1 1 1 1 0
0 0
1 1 1 1 1 0 1
Let us examine yield
yield yield yield
Sum the results for these contingenc
(( (1 ( )) ) , ) ; 0, 0 ( ( ,( ) , )
0, 1 ( ( , ) 0; 1, 0 ( , ) 0; 1, 1 0.
r
i
r i
X t r i h t A A r i X t t h t A
r i X t t A r i X t t A A r i
1 1
1 1 1 1 1 0
0 0
ies to get
(( (1 ( )) ) , ) ( (( ) , )r
i
r i
X t r i h t A A X t h t A
So, (ii) of the lemma is valid for 1j . Let us explore 2j .
(1) (1)
1 1 0 1 1 1 1
1 0 1 1 1 0 1 1 1 1 1
(1) 2
1 1
1 0 1 1
2 lhs (( 2 ) , ) (( 2 ) , ) (( ) , ) (( ) , )
(( 2 ) , ) (( ) , ) (( ) , ) ( , )
( , )
(( 2 ) , ) (( ) , ) (( ) ,
j X t h t X t h t A X t h t A X t h t A
X t h t A X t h t A X t h t A A X t t A A
X t t A
X t h t A X t h t A X t h t
1 0 1)A A
2 1
1 1 1
0 0
1 1 0 1 1 1
1 1 0 1
Let us examine
yield yield
yield yield yield
yield
Sum
(( (2 ( )) ) , ) ;
0, 0 ( (( 2 ) , ) ; 0, 1 ( (( ) , ) ;
1, 0 ( (( ) , ) ; 1, 1 0; 2, 0 0;
2, 1 0.
r
i
r i
X t r i h t A A
r i X t h t A r i X t h t A
r i X t h t A A r i r i
r i
2 1
1 1 1
0 0
1 1 0 1 1 1 1 1 0 1
the results for these contingencies to get
(( (2 ( )) ) , )
( (( 2 ) , ) ( (( ) , ) ( (( ) , ) .
r
i
r i
X t r i h t A A
X t h t A X t h t A X t h t A A
So, (ii) of the lemma is valid for {0,1,2}.j Assume that (ii) of the lemma is valid for
3 , for some integer .j n n Then
1
(1)
1 1 1 1 1
0 0
by the induction hypothesis.(( ) , ) (( ( ( )) ) , ) ,
n
r
i
r i
X t nh t X t n r i h t A A
(1)
1 1 1 1 0 1 1 1
(1)
1 1 1
1
(1)
1 1 1 1 1
0
1
1
1 1 1 1
0 0
Now, (( ( 1) ) , ) (( ( 1) ) , ) (( ) , )
(( ) , )
(( ) , ) (( ( 1 ) ) , )
(( ( ( )) ) , ) (( ( 1
i
i
n
r
i
r i
X t n h t X t n h t A X t nh t A
X t nh t A
X t nh t A X t n i h t A
X t n r i h t A A X t n
1
1
0
) ) , ) i
i
i h t A
1 1
1 1 1
0 0
as desired. So (ii) is true for 1,and
hence true for all
(( ( 1 ( )) ) , ) ,
.
n
r
i
r i
j nX t n r i h t A A
j
Proof of (iii)
The proof follows by noting that
(1) (1) (1)
1 1 1 1 1 1( , ) (( ) , ) (( ) , )X t jh t X t jh t X t jh t
and then
using (i) and (ii) to deduce that
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1
(1)
1 1 1 1 1 1 1
0 0
1 1
( )
1 1 1 1 1
0 0 0 0
( ,( )) (( ( ( )) ) , ) (( ( ( )) ) , )
(( ( ( )) ), )
j
r
i
r i
j j
r j r i r
i i
r i r i
X t t jh X t j r i h t X t j r i h t A A
X t j r i h t A A A A A
2.7 Lemma on
( )
1 1( ,(( ))k
X t t jh recursions
1
( ) ( 1)
1 1 1 1 1
0 0
( , ) (( ( ( )) ), )
j
k k r
i
r i
X t jh t X t j r i h t A A
Proof
1
( ) ( 1)
1 1 0 1 1
0
( 1) ( 1) 1 1
1 1 0 1 1 1 0 0 0 (by lemma 2.4)
If 0,then 0; lhs ( , ) ( 1) ,rhs ( , )
( , ) ( , ) ( 1)( 1) 0 ( 1)
k k k k
i
i
k k k k k k
j r X t t A X t ih t A
X t t A X t h t A A A A
( ) ( 1) ( 1)
1 1 1 1 1 0 1 1 1
( ) ( 1) 1 1
1 1 1 1 1 0 0 1
( 1) 1 1
0 1 1 1 0 0
If 1, then using lemma 2.4,
lhs ( (1 1) , ) ( , ) ( , )
( (1 1) , ) ( , ) ( 1)
( 1) [ ( , ) ( 1)
k k k
k k k k
k k k k k
j
X t h t A X t h t A X t t A
X t h t A X t h t A A A
A A X t h t A A
1 (by 2.4).A
1
( 1)
1 1 1
0 0
( 1) 0 ( 1) ( 1) 1 ( 1)
1 1 0 1 1 1 1 1 1 0 1 1 1 1 1
( 1) 1 1 1
1 1 0 0 1 0
Rhs (( ( ( )) ), )
(( ), ) ( , ) ( , ) ( , )
(( ), ) ( 1) ( 1)
j
k r
i
r i
k k k k
k k k k k
X t j r i h t A A
X t h t A A X t t A X t t A A X t h t A A
X t h t A A A A
1 1
0 1
( 1) 1 1
0 1 1 1 0 0 1
0
( 1) (( ), ) ( 1)k k k k k
A A
A A X t h t A A A
So, lhs = rhs. Therefore the lemma is valid for j = 1. Assume the validity of the lemma for 2 j m for
some integer m .
1
( ) ( 1)
1 1 1 1 1
0 0
Then (( ), ) (( ( ( )) ), )
m
k k r
i
r i
X t mh t X t m r i h t A A
( )
1 1
( 1) ( 1)
1 1 0 1 1 1
( )
1 1 1
1 1
( 1) ( 1) 1
1 1 1 1 1
0 0 0
Now (( ( 1) ), )
(( ( 1) ), ) ( , )
( , )
(( ( 1 ) ), ) (( ( ( )) , )
k
k k
k
m
k k r
i i
i r i
X t m h t
X t m h t A X t mh t A
X t mh t A
X t m i h t A X t m r i h t A A
1 1 1
( 1) ( 1) 1 1
1 1 1 1 1
0 1 0
(( ( 1 ) ), ) (( ( ( 1 )) , )
m
k k r
i i
i r i
X t m i h t A X t m r i h t A A
1 1 1
( 1) ( 1)
1 1 1 1 1
0 0 0
1 1 1
( 1) ( 1)
1 1 1 1 1
0 0 0
(( ( 1 ) ), ) (( ( 1 ( )) , )
(( ( 1 ) ), ) (( ( 1 ( )) , )
m
k k r
i i
i r i
m
k k r
i i
i r i
X t m i h t A X t m r i h t A A
X t m i h t A X t m r i h t A A
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(We used the change of variables 1r r in the r summand, 1r r in the limits).r
So the lemma is valid for 1j m . This completes the desired proof.
The following major theorem gives explicit computable expressions for ( ),k
Q jh based on painstakingly and
rigorously observed ( )k
Q jh patterns for various ,j k contingencies.
III. THEOREM ON EXPLICIT COMPUTABLE EXPRESSION FOR DETERMINING
MATRICES OF (1)
Let andj k be nonnegative integers. If 1, thenj k
1
1
1
1
1
1 11( ),0( ) 1( ),1( )
1 1( ),0( ),1( )
( , , ) ( , , )
( , , )
Q ( )
j r
k
k
v
r
j
jj k j k j k k
j r r j k r k r
v
v
j kv v
v v P v v P
v v P
v
jh
A A
A A A A
1 1
1 11( ),0( ) 0( ),1( )
1
1 1( ),0( ),1( )
( , , ) ( , , )
1
1 ( , , )
If 1, then
Q ( )k
j k k
j k j k k k j j
k r
k r r r k j j r
v v v v
v v P v v P
j
v v
r v v P
k j
jh
A A A A
A A
Proof
Case j k .
0 0 1
1
1 1 0 1 0 1 1 1
0
1
1 0 1 0 1 1
0
( 1)
If 0, then ( ) (0) , by ( 4); 0 ( ) ( ) (by (ii) of lemma 2.4).
1, 1 ( ) ( ) ( ) ,(by (iv) of lemma 2.4)
k n k
j
j r
k
r
j
j
r
r
j r
j r j r
j k Q jh Q I k Q jh Q jh A
j k Q jh Q jh A A A A A A A
A A A A A A
1
1 1
0
1 1 1
1 1( 1),1(1)1 1 1( ),0(1)
( 1)
( , , )( , , )
j
r
jj
j jj j
j r
v v v v
v v Pv v P
A A A A A A
1 3 1 2
1 2 11 3
2 1 2 0 1 1 1
1 1
0 0 1
1 3 1 21(1),0(2) 0(1),1(1)
1 3 1 2 1 21(2),0(1) 1(1),1(1) 1(1),0(
( , , ) ( , )
( , , ) ( , ) ( , )
If 2, then ( ) (2 ) ( ) (2 ) ( )k
v v v v
v v vv v
v v P v v P
v v P v v P v v P
j k Q jh Q h A Q h A Q h A Q h
A A A A A A
A A A A A A A A
2 11
11) 1(1)
v v
v P
A A A
1 4 1 3
1 4 1 2 1 2
2
1
2 1
1
1 4 1 31(2),0(2) 1(1),0(1),1(1)
1 4 1 2 1 21(2),0(2) 1(2 2),1(2) 1( ),0( ),1(2 )
( , , ) ( , , )
( , , ) ( , ) ( , , )
r
v v v v
v v
r r s r r r
v v v v
v v P v v P
v v P v v P v v P
A A A A A
A A A A A A
Therefore the theorem is valid for , {0, 1, 2}.j k In particular it is valid for 0 2.k j
1
The cases : ( ) and ( ) have been established already. We need only prove the remaining cases.k
Q jh Q h
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Case :j k Assume that the theorem is valid for all pairs of triples
, , ( ); , , ( ) for which ,for some 3, and 2.k k
j k Q jh j k Q jh j k j k j k
Then apply the induction
principle to the right-hand side of the determining equation:
0 1 1 1 1
([ 1] ) ([ 1] ) ( ) ( ),noting that 1 ,
1 1, 1.
k k k k
Q j h A Q j h A Q jh A Q jh j k j k
j k j k
,
Therefore:
1
1
1 1
1 1
1 1 11( 1),0( 1) 1( 1 ( 1)),1( 1)
1
1 1 1( 1 ( 1),0( ),1( 1 )
1
1 1(
0 0
1 1
0
1
1 1
( , , ) ( , , )
( , , )
( , , )
Q ([ 1] )
j
jj k
jj k j k j k k
j r
j r r j k r k r
j k
j
k
v
k
v
r
v v v
v v v
v v P v v P
v
v v P
v
v v P
j h
A A A A A A
A A A
A A A A A A
1
1
1 1 1( ),0( 1) ( 1)),1( 1)
1
1 11( ),0( ) 1( ),1( )
1
1
1
1
1 1
1
1 1( ( 1),0( ),1( 1 )
1
( , , )
( , , ) ( , , )
( , , )
( , , )
j r
j k j k j k k
jj k
jj k j k j k k
k r
v v
s
v
j r r j k r k r
j r
j r
v v P
v v v
v v P v v P
v
v v P
v
v v
A A A
A A A A A A
A A A
1
1 1( ),0( ),1( )
k
r r j k r k rP
1 1 11
1 1 1 11( 1),0( ) 1( 1 ),1( )
1 1
1 1 1( 2 ),0( ),1( 1 )
0
1 { 1,1}
0 01 1
1 1 1( 2 ),1( 1)
( , , ) ( , , )
( , , )( , , )
(9)
(10)
jj k
iL iL
j jj k j k k
j r r j k r k r
i i
r
j j r
j j k k
v v v v
v v P v v P
v v
v v P
v v
v v P
A A A A
A A A A A A
1
2
1
2
1 1
1
1
1
1
1 1
1 11 1( ),0( 1) 1( 1 ),0( ),1( 1 )
1
1 1( ),0( ),1( )
( , , ) ( , , )
( , , )
(11)
(12)
k
k
r
k
r
j rj k
j rj k j k r j k r k r
j r
j r r j k r k r
vv v v
v v P v v P
v v
v v P
A A A A A A
A A A
1 1 11
1 1 1 11( 1),0( ) 1( 1 ),1( )( , , ) ( , , )
The expression 9 yields:
(13)jj k
j jj k j k k
v v v v
v v P v v P
A A A A
1 1
1
1 1( 1 ),0( ),1( )
1
1 ( , , )
The expression:
is immediate from (12)
(14)j r
L
j r r j k r k r
k
r
v v
v v P
A A
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1 1 1 1
1 1 1 11( 2 ),1( 1) 1( 2 ),0( ),1( 1 )
1 1
1 1
2
0 0
1
0
( , , ) ( , , )
( , , )
We use the change of variables technique in the expression 10 to get:
(15)j j r
j j rj k k r j k r k r
j
j
k
r
v v v v
v v P v v P
v v
v v
A A A A A A
A A A
1 1 1
1 1 11( 2 ),1( 1) 1( 1 2 ),0( 1),1( 1 ( 1))
1 1 1 1 1
1 1 1 1 11( 2 ),1( 1) 1( 1 ),0( 1),1( )
1
0
2
1
0 0
2
( , , )
( , , ) ( , , )
j r
j rj k k r j k r k r
j j r
j j rj k k r j k r k r
k
r
k
r
v v
P v v P
v v v v
v v P v v P
A A A
A A A A A A
1 1 1 1 1
1 1 1 1 11( 2 ),1( 1) 1( 1 ),0( 1),1( )
1 1 1 1 1 1 1
1 1 1 1 1 1 11(1 1 ),0(1 1),1( 1)
1
0 0
1
0 0
( , , ) ( , , )
( , , ) ( , , )
j j r
j j rj k k r j k r k r
j j r
j j rj k k
k
v
r
v v v
v v P v v P
v v v v
v v P v v P
A A A A A A
A A A A A A
1( 1 ),0( 1),1( )
1 1
0
1 1 1( 1 ),0( ),1( )
1
1
1
1 ( , , )
(16)
r j k r k r
j r
L
j r r j k r k r
k
r
k
r
v v
v v P
A A
11 1
1 11 1( ),0( 1) 1( 1 ),0( ),1( 1 )
1 1
1 1 1( ),0( 1)
2
1 1
1
1 1 1
( , , ) ( , , )
( , , )
We are left with expression 11 , which yields:
(17)j rj k
j rj k j k r j k r k r
jj k
j k j k
k
v v
r
v v
v v P v v P
v v v
v v P
v
A A A A A A
A A A A A A
1 1( 1 ),0( ),1( 1 )
1 1
1 11( 1 1 ),0( 1),1( 1 ( 1)) 1( 1 ),0( ),1( 1 )
1 1
1 1 1( 1
1
1
1
1 1
1
( , , )
( , , ) ( , , )
( , , )
r
j r r j k r k r
j r j r
j r j rk j k k k k r j k r k r
j r
j r r j
k
r
k
r
v v P
v v v v
v v P v v P
v v
v v P
A A A A A A
A A
1
),0( ),1( )
1
1
(18)
L
k r k r
k
r
The resulting expressions (13), (14), (16) and (18) add up to yield:
11
1
1
1
1 11 1( 1),0( ) 1( 1 ),1( )
1
1 1 1( 1 ),0( ),1( )
( , , ) ( , , )
( , , )
Q ([ 1] )
(19)
j
k
j k v v
k
r
jj k j k j k k
j r
j r r j k r k r
v v
v
v v P v v P
v
v v P
j h
A A A A
A A
This concludes the inductive proof for the case .j k
We proceed to furnish the inductive proof for the case .k j
Theorem 3 is already proved for , {1,2,3,4.}, .j k k j
Therefore the theorem is valid for , {0, 1, 2, 3,4}.j k Assume that the theorem is valid for all pairs of
triples , , ( ); , , ( ) for which ,for some 3, and 4.k k
j k Q jh j k Q jh j k j k j k
Then apply the induction
principle to the right-hand side of the determining equation:
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1 0 1 1 1
( ) ( ) ([ 1] ) ([ 1] ),noting that k 1 ,
1 1, 1. Therefore:
k k k k
Q jh A Q jh A Q j h A Q j h j k j
k j k j
1 1
1 11( ),0( ) 0( )),1( )
1
1 1( ),0( ),1( )
1 11
1 1 11 0( ( 1)),1( 1)
1
0 0
1
0
1
1 1
( , , ) ( , , )
( , , )
( , , ) ( , , )
Q ( )
kj k
j k j k k k j j
k r
j r r r k j j r
j k k
jj k k j j
k
v v v
j
r
v
v v P v v P
v v
v v P
v v v v
v v v v P
jh
A A A A A A
A A A
A A A A A A
1( 1),0( )
1
1 1( ),0( ( 1)),1( 1 )
1 1 1
1 11( 1),0( 1) 0( 1 ( 1)),1( 1)
1 1
1 1 1( ),0(
1
1
1
1 1
1
( , , )
( , , ) ( , , )
( , , )
j k
k r
k r r r k j j r
j k k
jj k j k k j j
k r
k r r r
j r
r
P
v v
v v P
v v v v
v v P v v P
v v
v v P
A A A
A A A A A A
A A A
1 ( 1)),1( 1 )
2
1 k j j r
j
r
1 11 1
1 1 11 1( ),0( 1) 0( 1 ),1( )
1
1 1( ),0( ),1( )
1 1
1 1 1( 1),0( )
0 1
1 0
1
0
1
1 1
( , , ) ( , , )
( , , )
( , , )
(20)
(21)
j k k
iL iL
jj k j k k j j
k r
k r r r k j j r
j k
j k j k
i i
j
r
v v v v
v v P v v P
v v
v v P
v v
v v P
A A A A
A A A
A A A A A
1
1 1( ),0( 1 ),1( 1 )
1 12 1
1 11 0( 2 ),0( 1) 1 1( ),0( 2 ),1( 1 )
2
1
2
1 1
1
( , , )
( , , ) ( , , )
(22)
(23)
k r
k r r r k j j r
k j k r
k k j j k r r r k j j r
j
r
j
r
v v
v v P
v v v v
v v P v v P
A
A A A A A A
1 1 11
1 1 1 11( ),0( 1) 0( 1 ),1( )( , , ) ( , , )
The expression 20 yields:
(24)jj k
j jj k k j j
v v v v
v v P v v P
A A A A
1
1
1 1
0
1 1 1( ),0( 1 ),1( )( , , )
The expression:
is immediate from (21)
(25)
j
r
k r
L
k r r r k j j r
v v
v v P
A A
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2
1 1
1
1
1 11
1 11 1( 1),0( ) 1( ),0( 1 ),1( 1 )
1 1
1 1 1( 1),0(
( , , ) ( , , )
( , , )
We use the addition and subtraction technique in (22) to get:
=
j
r
j k k r
j k j k k r r r k j j r
j k
j k j
v v v v
v v P v v P
v v
v v P
A A A A A A
A A A
1
1
1
1
1
1
1
1
1) 1( ),0( 1 ),1( 1 )
1
1 1 1( 1),0( 1 1 ),1( 1 ( 1))
1 1
1
1 1( ),0( 1 ),1( )
( , , )
( , , )
( , , )
(26)
j
r
j
r
k r
k k r r r k j j r
k j
k j j j k j j j
k r
L
k r r r k j j r
v
v v
v v P
v
v v P
v v
v v P
A A A
A A A
A A
We are left with expression (23), which, using the change of variables technique yields:
1 1
1 1 1 1( 1),0( 1
2
1 1
1
1 1
1 12 1
1 11 0( 2 ),0( 1) 1 1( ),0( 2 ),1( 1 )
2 1 1
1 1 0( 2 ),0( 1) ( , , )
( , , ) ( , , )
( , , )
=
k r r r k
j
r
k j k r
k k j j k r r r k j j r
k j k r
k k j j
v v
v v P
v v v v
v v P v v P
v v
v v P
A A A A A A
A A A A A A
2 ),1( 1 ( 1))
1 12
1 1 1 1( 1),0( 1 2 ),1( 1 ( 1))
1
1
2
1
1 1
1
1
1 1
1 1 0( 2 ),0( 1)
1 1 1
1 1 1 1 1(1 1),0(1 1 2 ),1
( , , )( , , )
( , , )
j j r
k j
k r r r k j j r
j
r
j
r
k r
k k j j
k
k k j
v v v
v v P
v
v
v v P
v
v v P
A A A A A A
A A A
1
1
( 1 (1 1))
1 1
1
1 1 1( ),0( 1 ),1( )( , , )
(27)
j
r
j
k r
L
k r r r k j j r
v v
v v P
A A
The resulting expressions (24), (25), (26) and (27) add up to yield:
1
1 1 1
1 11 1( ),0( 1) 1 0( 1 ),1( )
1 1
1 1 1( ),0( 1 ),1( )
1
1
( , , ) ( , , )
( , , )
Q ( )
(28)
k
j k k
j k j k k k j j
k r
j r r r k j j r
j
r
v v v v
v v P v v P
v v
v v P
jh
A A A A
A A
This concludes the inductive proof for the case , completing the proof of the theorem.k j
The cases andj k k j , in the preceding theorem can be unified by using a composition of the max and the
signum functions as follows:
IV. INDUCTION PITFALLS IN THE DERIVATION OF THE DETERMINING
MATRICES
In the development of ( )kQ jh for the single – delay neutral system (1) we had proposed, based on the observed
pattern for ( )k
Q jh for 0 min{ , } 2,j k that for 1,j k
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1
1
1 1
1
1 11( ),0( ) 1( ),1( )
1
1 2 1( ),0( ),1( )
( , , ) ( , , )
( , , )
Q ( ) ; 1
; 2
jk
k k r
r s
j k
jj k j k j k k
j r
r j k s r j k r s
v vv v
v v P v v P
v v
v v P
jh k
A A k
A A A A
and for 1,k j
1
1
1
1 1
1 0( ),1( )
2
1
1 1( ),0( )
1 2 1( ),0( ),1( )
( , , )( , , )
( , , )
; 1
; 2
Q ( )
j j r
v
r s
k k j j
r k j s
vk j k k
j k j k
r k j s r r k j s
v v P
v
v v v
v v P
v v P
j
A A j
jh A A A A
This formula is valid for all , :1 min{ , } 2.j k j k However it turns out to be false for all
, :min{ , } 3.j k j k This revelelation came to the fore following the discovery of some fallacy in the proof of
the expressions for ( ),kQ jh which could never be overcome; it became imperative to examine 3 (3 )Q h from
the determining equations for ( ).kQ jh The formula was found to be inconsistent with the true result from the
determining equations. Therefore the formula had to be abandoned after so much effort.
Further long and sustained research effort led to the conjecture
1
1 1
1 1( ),1( )
1
1 2 1( ),0( ),1( )
1
1 2 1( ),0( ),1( )
1 1
1 1( ),0( )
( , , )
( , , )
( , , )
( , , )
Q ( ) ; 1
; 2, ,
; 2
k
k k r
r s
j
j j k k
j r
r j k s r j k r s
j r
r j k s r j k r s
j k
j k j k
v v P
v v
v v P
v v
v v P
v v v v
v v P
jh k
A A k j k
A A j k
A A A A
1
1 max{1, 1 2 }
k k r
r s k r
as well as the analogous expression for ( )k
Q jh for 1.k j
The formula was found to be valid for , :1 min{ , } 2,j k j k as in the aborted formula. Furthermore
3
(3 )Q h obtained from the postulated formula agreed with the corresponding result (sum of 63 permutation
products) obtained directly from the determining equations. Unfortunately, the proof for arbitrary j and k could
not push through; there was no way to apply induction on the lower limit for s, not to talk of dealing with the
piece–wise nature of the third component summations.
Above pitfalls provided object lessons that integer–based mathematical claims should never be
accepted without error-free proofs by mathematical induction. There is no other way to ascertain that a pattern
will persist –no matter the number of terms for which it has remained valid.
V. CONCLUSION
The results in this article bear eloquent testimony to the fact that we have comprehensively extended
the previous single-delay result by Ukwu (1992) together with appropriate embellishments through the
unfolding of intricate inter–play multiple summations and permutation objects in the course of deriving the
expressions for the determining matrices.
By using the change of variables technique, deft application of mathematical induction principles and
careful avoidance of some induction pitfalls, we were able to obtain the structure of the determining matrices for
the single–delay neutral control model, without which the computational investigation of Euclidean
controllability would be impossible.
17. The structure of determining matrices for single-delay…
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REFERENCES
[1] Gabasov, R. and Kirillova, F. The qualitative theory of optimal processes ( Marcel Dekker Inc., New York, 1976).
[2] Ukwu, C. Euclidean controllability and cores of euclidean targets for differential difference systems Master of Science Thesis in
Applied Math. with O.R. (Unpublished), North Carolina State University, Raleigh, N. C. U.S.A., 1992.
[3] Ukwu, C. An exposition on Cores and Controllability of differential difference systems, ABACUS, Vol. 24, No. 2, 1996, pp. 276-
285.
[4] Ukwu, C. On determining matrices, controllability and cores of targets of certain classes of autonomous functional
differential systems with software development and implementation. Doctor of Philosophy Thesis, UNIJOS, 2013a (In
progress).
[5] Bellman, R. and Cooke, K. Differential-difference equations. Acad. Press, New York, (1963).
[6] Dauer, J. P. and Gahl, R. D. Controllability of nonlinear delay systems. J.O.T.A. Vol. 21, No. 1, January (1977).
[7] G. Functional differential equations of retarded and neutral types: Analytical solutions and piecewise continuous controls. J.
Differential equations, ,Vol. 51, No. 2, 1984,Pp. 151-181.
[8] Banks, H. T. and Jacobs, M. An attainable sets approach to optimal control of functional differential equations with
function space terminal conditions. Journal of differential equations, 13, 1973, 121- 149.
[9] Banks, H. T. and Kent G. A. Control of functional differential equations of retarded and neutral type to targets in function
space, SIAM J. Control, Vol. 10, November 4, 1972.
[10] Hale, J. K. Theory of functional differential equations. Applied Mathematical Science, Vol. 3, 1977, Springer-Verlag, New York.