This document summarizes research on the uniform convergence of the Chord method for solving generalized equations. The Chord method is an iterative method for finding solutions to equations of the form y ∈ f(x) + F(x), where f is a function and F is a set-valued mapping. The authors prove that under certain conditions, including F being pseudo-Lipschitz and the derivative of f being continuous, the Chord method converges uniformly for small variations in the parameter y. They obtain this result in two different ways. The document also provides relevant definitions and preliminaries on generalized equations, set-valued mappings, and convergence properties.
Helmholtz equation (Motivations and Solutions)Hassaan Saleem
Solutions of Helmholtz equation in cartesian, cylindrical and spherical coordinates are discussed and the applications to the problem of a quantum mechanical particle in a cubical box is discussed.
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.
Matrix Transformations on Paranormed Sequence Spaces Related To De La Vallée-...inventionjournals
In this paper, we determine the necessary and sufficient conditions to characterize the matrices which transform paranormed sequence spaces into the spaces 푉휎 (휆) and 푉휎 ∞(휆) , where 푉휎 (휆) denotes the space of all (휎, 휆)-convergent sequences and 푉휎 ∞(휆) denotes the space of all (휎, 휆)-bounded sequences defined using the concept of de la Vallée-Pousin mean.
A Non Local Boundary Value Problem with Integral Boundary ConditionIJMERJOURNAL
ABSTRACT: In this article a three point boundary value problem associated with a second order differential equation with integral type boundary conditions is proposed. Then its solution is developed with the help of the Green’s function associated with the homogeneous equation. Using this idea and Iteration method is proposed to solve the corresponding linear problem.
Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch)
TITLE: Path integral action of a particle in the noncommutative plane and the Aharonov-Bohm effect
Helmholtz equation (Motivations and Solutions)Hassaan Saleem
Solutions of Helmholtz equation in cartesian, cylindrical and spherical coordinates are discussed and the applications to the problem of a quantum mechanical particle in a cubical box is discussed.
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.
Matrix Transformations on Paranormed Sequence Spaces Related To De La Vallée-...inventionjournals
In this paper, we determine the necessary and sufficient conditions to characterize the matrices which transform paranormed sequence spaces into the spaces 푉휎 (휆) and 푉휎 ∞(휆) , where 푉휎 (휆) denotes the space of all (휎, 휆)-convergent sequences and 푉휎 ∞(휆) denotes the space of all (휎, 휆)-bounded sequences defined using the concept of de la Vallée-Pousin mean.
A Non Local Boundary Value Problem with Integral Boundary ConditionIJMERJOURNAL
ABSTRACT: In this article a three point boundary value problem associated with a second order differential equation with integral type boundary conditions is proposed. Then its solution is developed with the help of the Green’s function associated with the homogeneous equation. Using this idea and Iteration method is proposed to solve the corresponding linear problem.
Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch)
TITLE: Path integral action of a particle in the noncommutative plane and the Aharonov-Bohm effect
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...Rene Kotze
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russia)
TITLE: Dynamical Groups, Coherent States and Some of their Applications in Quantum Optics and Molecular Spectroscopy
Some new exact Solutions for the nonlinear schrödinger equationinventy
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.
NITheP WITS node Seminar by Dr Dr. Roland Cristopher F. Caballar (NITheP/UKZN)
TITLE: "One-Dimensional Homogeneous Open Quantum Walks"
ABSTRACT: In this talk, we consider a system undergoing an open quantum walk on a one-dimensional lattice. Each jump of the system between adjacent lattice points in a given direction corresponds to a jump operator, with these jump operators either commuting or not commuting. We examine the dynamics of the system undergoing this open quantum walk, in particular deriving analytically the probability distribution of the system, as well as examining numerically the behavior of the probability distribution over long time steps. The resulting distribution is shown to have multiple components, which fall under two general categories, namely normal and solitonic components. The analytic computation of the probability distribution for the system undergoing this open quantum walk allows us to determine at any instant of time the dynamical properties of the system.
The finite difference method can be considered as a direct discretization of differential equations but in finite element methods, we generate difference equations by using approximate methods with piecewise polynomial solution. In this paper, we use the Galerkin method to obtain the approximate solution of a boundary value problem. The convergence analysis of these solution are also considered.
Stochastic Gravity in Conformally-flat SpacetimesRene Kotze
The National Institute for Theoretical Physics, and the Mandelstam Institute for Theoretical Physics, School of Physics, would like to invite to its coming talk in the theoretical physics seminar series, entitled:
"Stochastic Gravity in Conformally-flat Spacetimes"
to be presented by Prof. Hing-Tong Cho (Tamkang University, Taiwan)
Abstract: The theory of stochastic gravity takes into account the effects of quantum field fluctuations onto the classical spacetime. The essential physics can be understood from the analogous Brownian motion model. We shall next concentrate on the case with conformally-flat spacetimes. Our main concern is to derive the so-called noise kernels. We shall also describe our on-going program to investigate the Einstein-Langevin equation in these spacetimes.
Dates: Tuesday, 17th February 2015
Venue: The Frank Nabarro lecture theatre, P216
Time: 13.20 - 14.10 - TODAY
In this paper we consider the initial-boundary value problem for a nonlinear equation induced with respect to the mathematical models in mass production process with the one sided spring boundary condition by boundary feedback control. We establish the asymptotic behavior of solutions to this problem in time, and give an example and simulation to illustrate our results. Results of this paper are able to apply industrial parts such as a typical model widely used to represent threads, wires, magnetic tapes, belts, band saws, and so on.
A generalized bernoulli sub-ODE Method and Its applications for nonlinear evo...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.
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.
Dual Spaces of Generalized Cesaro Sequence Space and Related Matrix Mappinginventionjournals
In this paper we define the generalized Cesaro sequence spaces 푐푒푠(푝, 푞, 푠). We prove the space 푐푒푠(푝, 푞, 푠) is a complete paranorm space. In section-2 we determine its Kothe-Toeplitz dual. In section-3 we establish necessary and sufficient conditions for a matrix A to map 푐푒푠 푝, 푞, 푠 to 푙∞ and 푐푒푠(푝, 푞, 푠) to c, where 푙∞ is the space of all bounded sequences and c is the space of all convergent sequences. We also get some known and unknown results as remarks.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
M. Visinescu: Hidden Symmetries of the Five-dimensional Sasaki - Einstein SpacesSEENET-MTP
Balkan Workshop BW2013
Beyond the Standard Models
25 – 29 April, 2013, Vrnjačka Banja, Serbia
M. Visinescu: Hidden Symmetries of the Five-dimensional Sasaki - Einstein Spaces
Crack problems concerning boundaries of convex lens like formsijtsrd
The singular stress problem of aperipheral edge crack around a cavity of spherical portion in an infinite elastic medium whenthe crack is subjected to a known pressure is investigated. The problem is solved byusing integral transforms and is reduced to the solution of a singularintegral equation of the first kind. The solution of this equation is obtainednumerically by the method due to Erdogan, Gupta , and Cook, and thestress intensity factors are displayed graphically.Also investigated in this paper is the penny-shaped crack situated symmetrically on the central plane of a convex lens shaped elastic material. Doo-Sung Lee"Crack problems concerning boundaries of convex lens like forms" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-3 , April 2018, URL: http://www.ijtsrd.com/papers/ijtsrd11106.pdf http://www.ijtsrd.com/mathemetics/applied-mathamatics/11106/crack-problems-concerning-boundaries-of-convex-lens-like-forms/doo-sung-lee
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...Rene Kotze
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russia)
TITLE: Dynamical Groups, Coherent States and Some of their Applications in Quantum Optics and Molecular Spectroscopy
Some new exact Solutions for the nonlinear schrödinger equationinventy
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.
NITheP WITS node Seminar by Dr Dr. Roland Cristopher F. Caballar (NITheP/UKZN)
TITLE: "One-Dimensional Homogeneous Open Quantum Walks"
ABSTRACT: In this talk, we consider a system undergoing an open quantum walk on a one-dimensional lattice. Each jump of the system between adjacent lattice points in a given direction corresponds to a jump operator, with these jump operators either commuting or not commuting. We examine the dynamics of the system undergoing this open quantum walk, in particular deriving analytically the probability distribution of the system, as well as examining numerically the behavior of the probability distribution over long time steps. The resulting distribution is shown to have multiple components, which fall under two general categories, namely normal and solitonic components. The analytic computation of the probability distribution for the system undergoing this open quantum walk allows us to determine at any instant of time the dynamical properties of the system.
The finite difference method can be considered as a direct discretization of differential equations but in finite element methods, we generate difference equations by using approximate methods with piecewise polynomial solution. In this paper, we use the Galerkin method to obtain the approximate solution of a boundary value problem. The convergence analysis of these solution are also considered.
Stochastic Gravity in Conformally-flat SpacetimesRene Kotze
The National Institute for Theoretical Physics, and the Mandelstam Institute for Theoretical Physics, School of Physics, would like to invite to its coming talk in the theoretical physics seminar series, entitled:
"Stochastic Gravity in Conformally-flat Spacetimes"
to be presented by Prof. Hing-Tong Cho (Tamkang University, Taiwan)
Abstract: The theory of stochastic gravity takes into account the effects of quantum field fluctuations onto the classical spacetime. The essential physics can be understood from the analogous Brownian motion model. We shall next concentrate on the case with conformally-flat spacetimes. Our main concern is to derive the so-called noise kernels. We shall also describe our on-going program to investigate the Einstein-Langevin equation in these spacetimes.
Dates: Tuesday, 17th February 2015
Venue: The Frank Nabarro lecture theatre, P216
Time: 13.20 - 14.10 - TODAY
In this paper we consider the initial-boundary value problem for a nonlinear equation induced with respect to the mathematical models in mass production process with the one sided spring boundary condition by boundary feedback control. We establish the asymptotic behavior of solutions to this problem in time, and give an example and simulation to illustrate our results. Results of this paper are able to apply industrial parts such as a typical model widely used to represent threads, wires, magnetic tapes, belts, band saws, and so on.
A generalized bernoulli sub-ODE Method and Its applications for nonlinear evo...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.
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.
Dual Spaces of Generalized Cesaro Sequence Space and Related Matrix Mappinginventionjournals
In this paper we define the generalized Cesaro sequence spaces 푐푒푠(푝, 푞, 푠). We prove the space 푐푒푠(푝, 푞, 푠) is a complete paranorm space. In section-2 we determine its Kothe-Toeplitz dual. In section-3 we establish necessary and sufficient conditions for a matrix A to map 푐푒푠 푝, 푞, 푠 to 푙∞ and 푐푒푠(푝, 푞, 푠) to c, where 푙∞ is the space of all bounded sequences and c is the space of all convergent sequences. We also get some known and unknown results as remarks.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
M. Visinescu: Hidden Symmetries of the Five-dimensional Sasaki - Einstein SpacesSEENET-MTP
Balkan Workshop BW2013
Beyond the Standard Models
25 – 29 April, 2013, Vrnjačka Banja, Serbia
M. Visinescu: Hidden Symmetries of the Five-dimensional Sasaki - Einstein Spaces
Crack problems concerning boundaries of convex lens like formsijtsrd
The singular stress problem of aperipheral edge crack around a cavity of spherical portion in an infinite elastic medium whenthe crack is subjected to a known pressure is investigated. The problem is solved byusing integral transforms and is reduced to the solution of a singularintegral equation of the first kind. The solution of this equation is obtainednumerically by the method due to Erdogan, Gupta , and Cook, and thestress intensity factors are displayed graphically.Also investigated in this paper is the penny-shaped crack situated symmetrically on the central plane of a convex lens shaped elastic material. Doo-Sung Lee"Crack problems concerning boundaries of convex lens like forms" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-3 , April 2018, URL: http://www.ijtsrd.com/papers/ijtsrd11106.pdf http://www.ijtsrd.com/mathemetics/applied-mathamatics/11106/crack-problems-concerning-boundaries-of-convex-lens-like-forms/doo-sung-lee
A Programmable Logic Controller (PLC) or Programmable Controller is an electronic device used for Automation of industrial processes, such as control of machinery on factory assembly lines.
A Probabilistic Algorithm for Computation of Polynomial Greatest Common with ...mathsjournal
In the earlier work, Knuth present an algorithm to decrease the coefficient growth in the Euclidean algorithm of polynomials called subresultant algorithm. However, the output polynomials may have a small factor which can be removed. Then later, Brown of Bell Telephone Laboratories showed the subresultant in another way by adding a variant called 𝜏 and gave a way to compute the variant. Nevertheless, the way failed to determine every 𝜏 correctly.
In this paper, we will give a probabilistic algorithm to determine the variant 𝜏 correctly in most cases by adding a few steps instead of computing 𝑡(𝑥) when given 𝑓(𝑥) and𝑔(𝑥) ∈ ℤ[𝑥], where 𝑡(𝑥) satisfies that 𝑠(𝑥)𝑓(𝑥) + 𝑡(𝑥)𝑔(𝑥) = 𝑟(𝑥), here 𝑡(𝑥), 𝑠(𝑥) ∈ ℤ[𝑥]
Fixed Point Results for Weakly Compatible Mappings in Convex G-Metric Spaceinventionjournals
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.
Matrix Transformations on Some Difference Sequence SpacesIOSR Journals
The sequence spaces 𝑙∞(𝑢,𝑣,Δ), 𝑐0(𝑢,𝑣,Δ) and 𝑐(𝑢,𝑣,Δ) were recently introduced. The matrix classes (𝑐 𝑢,𝑣,Δ :𝑐) and (𝑐 𝑢,𝑣,Δ :𝑙∞) were characterized. The object of this paper is to further determine the necessary and sufficient conditions on an infinite matrix to characterize the matrix classes (𝑐 𝑢,𝑣,Δ ∶𝑏𝑠) and (𝑐 𝑢,𝑣,Δ ∶ 𝑙𝑝). It is observed that the later characterizations are additions to the existing ones
A Mathematical Model for the Hormonal Responses During Neurally Mediated Sync...IJRES Journal
The purpose of this study is to find a Mathematical model for the participation of central serotonergic activity in neurocardiogenic syncope by comparing cortisol and prolactin plasma levels in patients with positive and negative tilt test by using Multivariate Normal Distribution.
A Mathematical Model for the Hormonal Responses During Neurally Mediated Sync...irjes
The purpose of this study is to find a Mathematical model for the participation of central serotonergic activity in neurocardiogenic syncope by comparing cortisol and prolactin plasma levels in patients with positive and negative tilt test by using Multivariate Normal Distribution.
Richard's entangled aventures in wonderlandRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
Cancer cell metabolism: special Reference to Lactate PathwayAADYARAJPANDEY1
Normal Cell Metabolism:
Cellular respiration describes the series of steps that cells use to break down sugar and other chemicals to get the energy we need to function.
Energy is stored in the bonds of glucose and when glucose is broken down, much of that energy is released.
Cell utilize energy in the form of ATP.
The first step of respiration is called glycolysis. In a series of steps, glycolysis breaks glucose into two smaller molecules - a chemical called pyruvate. A small amount of ATP is formed during this process.
Most healthy cells continue the breakdown in a second process, called the Kreb's cycle. The Kreb's cycle allows cells to “burn” the pyruvates made in glycolysis to get more ATP.
The last step in the breakdown of glucose is called oxidative phosphorylation (Ox-Phos).
It takes place in specialized cell structures called mitochondria. This process produces a large amount of ATP. Importantly, cells need oxygen to complete oxidative phosphorylation.
If a cell completes only glycolysis, only 2 molecules of ATP are made per glucose. However, if the cell completes the entire respiration process (glycolysis - Kreb's - oxidative phosphorylation), about 36 molecules of ATP are created, giving it much more energy to use.
IN CANCER CELL:
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
introduction to WARBERG PHENOMENA:
WARBURG EFFECT Usually, cancer cells are highly glycolytic (glucose addiction) and take up more glucose than do normal cells from outside.
Otto Heinrich Warburg (; 8 October 1883 – 1 August 1970) In 1931 was awarded the Nobel Prize in Physiology for his "discovery of the nature and mode of action of the respiratory enzyme.
WARNBURG EFFECT : cancer cells under aerobic (well-oxygenated) conditions to metabolize glucose to lactate (aerobic glycolysis) is known as the Warburg effect. Warburg made the observation that tumor slices consume glucose and secrete lactate at a higher rate than normal tissues.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
What is greenhouse gasses and how many gasses are there to affect the Earth.moosaasad1975
What are greenhouse gasses how they affect the earth and its environment what is the future of the environment and earth how the weather and the climate effects.
This presentation explores a brief idea about the structural and functional attributes of nucleotides, the structure and function of genetic materials along with the impact of UV rays and pH upon them.
(May 29th, 2024) Advancements in Intravital Microscopy- Insights for Preclini...Scintica Instrumentation
Intravital microscopy (IVM) is a powerful tool utilized to study cellular behavior over time and space in vivo. Much of our understanding of cell biology has been accomplished using various in vitro and ex vivo methods; however, these studies do not necessarily reflect the natural dynamics of biological processes. Unlike traditional cell culture or fixed tissue imaging, IVM allows for the ultra-fast high-resolution imaging of cellular processes over time and space and were studied in its natural environment. Real-time visualization of biological processes in the context of an intact organism helps maintain physiological relevance and provide insights into the progression of disease, response to treatments or developmental processes.
In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...University of Maribor
Slides from:
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...
Uniformity of the Local Convergence of Chord Method for Generalized Equations
1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. II (Jan - Feb. 2015), PP 16-23
www.iosrjournals.org
DOI: 10.9790/5728-11121623 www.iosrjournals.org 16 | Page
Uniformity of the Local Convergence of Chord Method for
Generalized Equations
M. H. Rashid1
, A. Basak2
and M. Z. Khaton3
1,2
Department of Mathematics, Faculty of Science, University of Rajshahi, Rajshahi-6205, Bangladesh
3
Department of Mathematics, Sapahar Government Degree College, Naogaon-6560, Bangladesh
Abstract: Let 𝑋 be a real or complex Banach space and 𝑌 be a normed linear space. Suppose that 𝑓: 𝑋 → 𝑌 is
a Frechet differentiable function and 𝐹: 𝑋 ⇉ 2 𝑌
is a set-valued mapping with closed graph. Uniform
convergence of Chord method for solving generalized equation 𝑦 ∈ 𝑓 𝑥 + 𝐹 𝑥 … … . (∗), where 𝑦 ∈ 𝑌 a
parameter, is studied in the present paper. More clearly, we obtain the uniform convergence of the sequence
generated by Chord method in the sense that it is stable under small variation of perturbation parameter 𝑦
provided that the set-valued mapping 𝐹 is pseudo-Lipschitz at a given point (possibly at a given solution).
Keywords: Chord method, Generalized equation, Local convergence, pseudo-Lipschitz mapping, Set-valued
mapping.
AMS (MOS) Subject Classifications: 49J53; 47H04; 65K10.
1. Introduction
Let 𝑋 be a real or complex Banach space and 𝑌 be a normed linear space. We are concerned with the problem of
finding the solution 𝑥∗
of parameterized perturbed generalized equation of the form
𝑦 ∈ 𝑓 𝑥 + 𝐹 𝑥 , (1.1)
where 𝑦 ∈ 𝑌 is a parameter, 𝑓: 𝑋 → 𝑌 is a single-valued function and 𝐹: 𝑋 ⇉ 2 𝑌
is set-valued mapping with
closed graph.
The Generalized equation problems were introduced by S.M. Robinson [1, 2] as a general tool for
describing, analyzing, and solving different problem in a unified manner. Typical examples are systems of
inequalities, variational inequalities, linear and non-linear complementary problems, systems of nonlinear
equations, equilibrium problems, etc.; see for example [1--3].
It is remark that when 𝐹 = 0 , (1.1) is an equation. When F is positive orthand in 𝑅 𝑛
, (1.1) is a system
of inequalities. When 𝐹 is the normal cone to a convex and closed set in 𝑋, (1.1) represents variational
inequalities.
Newton-type method can be considered to solve this generalized equation when the single-valued
function involved in (1.1) is differentiable. Such an approach has been used in many contributions to this
subject; see for example [4--6].
To solve (1.1), Dontchev [4] introduced a Newton type method of the form
𝑦 ∈ 𝑓 𝑥 𝑘 + 𝛻𝑓 𝑥 𝑘 𝑥 𝑘+1 − 𝑥 𝑘 + 𝐹 𝑥 𝑘+1 , 𝑘 = 0,1, . . ., (1.2)
where 𝛻𝑓 𝑥 𝑘 is the Frechet derivative of 𝑓 at the point 𝑥 𝑘 and obtained the stability of the method (1.2).
A large number of iterative methods have been presented for solving (1.1). Pietrus [5] showed the stability of
this method under mild conditions. Other achievements on this topic can be found in [6, 7].
Marinov [8] associated the following method name as “Chord method” for solving the generalized equation
(1.1):
𝑦 ∈ 𝑓 𝑥 𝑘 + 𝐴 𝑥 𝑘+1 − 𝑥 𝑘 + 𝐹 𝑥 𝑘+1 , (1.3)
where 𝐴 ∈ 𝐿 𝑋, 𝑌 . It should be noted that when 𝐴 = 𝛻𝑓 𝑥 𝑘 , the method (1.3) reduces to the well-known
Newton-type method (1.2).
In the present paper, we are intended to present a kind of convergence of the sequence generated by the
Chord method defined by (1.3) which is uniform in the sense that the attraction region does not depend on small
variations of the value of the parameter 𝑦 near 𝑦∗
and for such values of 𝑦 the method finds a solution 𝑥 of
(1.1) whenever 𝐹 is pseudo-Lipschitz at (𝑥∗
, 𝑦∗
) and the Frechet derivative of 𝑓 is continuous. We will prove
the uniformity of the local convergence of the Chord method defined by (1.3) in two different ways.
This work is organized as follows: In Section 2, we recall few preliminary results that will be used in
the subsequent sections. In Section 3, we prove the existence and uniform convergence of the sequence
generated by the Chord method defined by (1.3) for solving generalized equation (1.1). Finally in Section 4, we
will give conclusion of the major results obtained in this study.
2. Uniformity of the Local Convergence of Chord Method for Generalized Equations
DOI: 10.9790/5728-11121623 www.iosrjournals.org 17 | Page
II. Notations And Preliminary Results
Throughout this work we suppose that 𝑋 is a real or complex Banach space and 𝑌 is a normed linear space. All
the norms are denoted by ∙ . The closed ball centered at x with radius 𝑟 > 0 denoted by 𝐵𝑟(𝑥) and the set of
linear operators denoted by 𝐿(𝑋, 𝑌).
The following definition is taken from [7, 9].
Definition 2.1: Let 𝐹: 𝑋 ⇉ 2 𝑌
be a set valued mapping. Then the Domain of 𝐹 is denoted by dom 𝐹, and is
define by
dom 𝐹 = 𝑥 ∈ 𝑋 ∶ 𝐹 𝑥 ≠ ∅ .
The inverse of 𝐹 is denoted by 𝐹−1
, and is defined by
𝐹−1
𝑦 = 𝑥 ∈ 𝑋: 𝑦 ∈ 𝑓 𝑥 ,
and
The Graph of 𝐹 is denoted by gph 𝐹, and is define by
gph 𝐹 = 𝑥, 𝑦 ∈ 𝑋 × 𝑌 ∶ 𝑦 ∈ 𝐹 𝑥 .
The following definition of distance from a point to a set and excess are taken from [7, 9].
Definition 2.2: Let 𝑋 be Banach space 𝐴 be subset of 𝑋. Then the distance from a point 𝑥 to a set 𝐴 is denoted
by dist 𝑥, 𝐴 , and is define by
dist (𝑥, 𝐴) = inf 𝑥 − 𝑎 ∶ 𝑎 ∈ 𝐴 .
Definition 2.3: Let 𝑋 be Banach space and 𝐴, 𝐵 ⊆ 𝑋. The excess 𝑒 from the set 𝐴 to a set 𝐵 is given by
𝑒 𝐵, 𝐴 = sup 𝑥 − 𝐴 ∶ 𝑥 ∈ 𝐵 .
The following definition is taken from [9].
Definition 2.4: Let 𝐹: 𝑋 ⇉ 2 𝑌
be a set-valued mapping. Then 𝐹 is said to be pseudo Lipschitz-around 𝑥0, 𝑦0 ∈
gph 𝐹 with constant 𝑀 if there exist positive constants 𝛼, 𝛽 > 0 such that
𝑒 𝐹 𝑥1 ∩ 𝐵𝛽 𝑦0 , 𝐹 𝑥2 ≤ 𝑀 𝑥1 − 𝑥2 ,
for every 𝑥1, 𝑥2 ∈ 𝐵𝛼 𝑥0 . When 𝐹 is single-valued, this corresponds to the usual concept of Lipschitz
continuity.
The definition of Lipschitz continuity is equivalent to the definition of Aubin continuity, which is given below:
A set-valued map 𝛤: 𝑌 ⇉ 2 𝑋
is Aubin continuous at 𝑦0, 𝑥0 ∈ gph 𝛤 with positive constants 𝛼, 𝛽 and 𝑀 if for
every 𝑦1, 𝑦2 ∈ 𝐵𝛽 𝑦0 and for every 𝑥1 ∈ 𝛤 𝑦1 ∩ 𝐵𝛼 𝑥0 , there exists an 𝑥2 ∈ 𝛤 𝑦2 such that
𝑥1 − 𝑥2 ≤ 𝑀 𝑦1 − 𝑦2 .
The constant 𝑀 is called the modulus of Aubin continuity.
The following definition of continuity is taken from the book [10].
Definition 2.5: A map 𝑓: 𝛺 ⊆ 𝑋 → 𝑌 is said to be continuous at 𝑥 ∈ 𝛺 if for every 𝜀 > 0, there exist a 𝛿 > 0
such that
𝑓 𝑥 − 𝑓 𝑥 < 𝜀, for all 𝑥 ∈ 𝛺 for which 𝑥 − 𝑥 < 𝛿.
Definition 2.6: A map 𝑓: 𝛺 ⊆ 𝑋 → 𝑌 is said to be Lipschitz continuous if there exists constant 𝑐 such that
𝑓 𝑥 − 𝑓 𝑦 ≤ 𝑐 𝑥 − 𝑦 , for all 𝑥 and 𝑦 in the domain of 𝑓.
The following definition of linear convergence is taken from the monogram [11].
Definition 2.7: Let {𝑥 𝑛 } be a sequence which is converges to the number 𝑥. Then the sequence {𝑥 𝑛 } is said to
be converges linearly to 𝑥, if there exists a number 0 < 𝑐 < 1 such that
𝑥 𝑛+1 − 𝑥 ≤ 𝑐 𝑥 𝑛 − 𝑥 .
The following Lemma is a version of fixed point theorem, which is taken from [7].
Lemma 2.1: Let 𝛷: 𝑋 ⇉ 2 𝑋
be a set valued mapping and let 𝜂0 ∈ 𝑋, 𝑟 > 0 and 0 < 𝜆 < 1
be such that
(a) dist 𝜂0, 𝛷 𝜂0 < 𝑟 1 − 𝜆 and
(b) 𝑒 𝛷 𝑥1 ∩ 𝐵𝑟 𝜂0 , 𝛷 𝑥2 ≤ 𝜆 𝑥1 − 𝑥2 , for all 𝑥1, 𝑥1 ∈ 𝐵𝑟 𝜂0 .
Then 𝛷 has a fixed point in 𝐵𝑟 𝜂0 , that is, there exists 𝑥 ∈ 𝐵𝑟 𝜂0 such that 𝑥 ∈ 𝛷 𝑥 . If 𝛷 is single-valued,
then 𝑥 is the unique fixed point of 𝛷 in 𝐵𝑟 𝜂0 .
III. Uniform Convergence Of Chord Method
This section is devoted to study the stability of the Chord method for solving generalized equations
(1.1) involving set-valued mapping and parameters. Let 𝑓: 𝑋 → 𝑌 be a single-valued function which is Frechet
differentiable on an open set in 𝑋 and 𝐹: 𝑋 ⇉ 2 𝑌
be a set-valued mapping with closed graph. Let 𝑦 ∈ 𝐹 𝑥 and
𝐹−1
be pseudo-Lipschitz around 𝑦 , 𝑥 . Then through Theorem 2.1 in [12], we have that 𝑓 + 𝐹 −1
is pseudo-
Lipschitz around 𝑦 + 𝑓(𝑥 , 𝑥 ). Let 𝑥 ∈ 𝑋 and define the mapping 𝑃𝑥 : 𝑋 ⇉ 2 𝑌
by
𝑃𝑥 ∙ = 𝑓 𝑥 + 𝐴 ∙ −𝑥 + 𝐹 ∙ .
Moreover, the following equivalence is obvious, for any 𝑧 ∈ 𝑋 and 𝑦 ∈ 𝑌,
𝑧 ∈ 𝑃𝑥
−1
𝑦 ⇔ 𝑦 ∈ 𝑓 𝑥 + 𝐴 𝑧 − 𝑥 + 𝐹(𝑧). (3.1)
3. Uniformity of the Local Convergence of Chord Method for Generalized Equations
DOI: 10.9790/5728-11121623 www.iosrjournals.org 18 | Page
In particular, 𝑥∗
∈ 𝑃 𝑥∗
−1
𝑦∗
for each 𝑥∗
, 𝑦∗
∈ gph 𝑓 + 𝐹 . Let 𝑎 > 0, 𝑏 > 0. Throughout the whole paper,
we suppose that 𝐵𝑎 𝑥∗
⊆ 𝛺 ∩ dom 𝐹 and that 𝑥∗
, 𝑦∗
∈ gph 𝑃 𝑥∗ , for every 𝑥∗
, 𝑦∗
∈ gph 𝑓 + 𝐹 . Then by
Lemma 3.1 in [12], we have that the mapping 𝑃𝑥∗
−1
∙ is pseudo-Lipschitz around 𝑦∗
, 𝑥∗
with constant 𝑀, that
is,
𝑒 𝑃𝑥∗
−1
𝑦1 ∩ 𝐵𝑎 𝑥∗
, 𝑃𝑥∗
−1
𝑦2 ≤ 𝑀 𝑦1 − 𝑦2 , for any 𝑦1, 𝑦2 ∈ 𝐵𝑏 𝑦∗
. (3.2)
Choose 𝐴 − 𝛻𝑓 𝑥∗
> 0 and set
𝑟 = min 𝑏 − 2𝑎 𝐴 − 𝛻𝑓 𝑥∗
,
𝑎 1−𝑀 𝐴−𝛻𝑓 𝑥∗
4𝑀
. (3.3)
Then 𝑟 > 0 ⟺ 𝐴 − 𝛻𝑓 𝑥∗
< min
𝑏
2𝑎
,
1
𝑀
. (3.4)
The following Lemma is useful for proving the existence of a sequence generated by the method (1.3). The
proof is analogous to the proof of Rashid et al. [7].
Lemma 3.1 Suppose that 𝑃𝑥∗
−1
∙ is pseudo-Lipschitz around 𝑦∗
, 𝑥∗
with constant 𝑀. Let 𝑥 ∈ 𝐵 𝑎
2
𝑥∗
and let
𝐴 ∈ 𝐿 𝑋, 𝑌 be such that 𝑀 𝛻𝑓 𝑥∗
− 𝐴 < 1. Suppose that
Sup
𝑥∈𝐵 𝑎
2
𝑥∗
𝛻𝑓 𝑥 − 𝛻𝑓 𝑥∗
≤ 𝐴 − 𝛻𝑓 𝑥∗
≤ min
𝑏
2𝑎
,
1
𝑀
.
Then 𝑃𝑥
−1
∙ is pseudo-Lipschitz around 𝑦∗
, 𝑥∗
with constant
𝑀
1−𝑀 𝐴−𝛻𝑓 𝑥∗ ,
that is, 𝑒 𝑃𝑥
−1
𝑦1 ∩ 𝐵 𝑎
2
𝑥∗
, 𝑃𝑥
−1
𝑦2 ≤
𝑀
1−𝑀 𝐴−𝛻𝑓 𝑥∗ 𝑦1 − 𝑦2 for any 𝑦1, 𝑦2 ∈ 𝐵𝑟 𝑦∗
.
Proof: Let 𝑦1, 𝑦2 ∈ 𝐵𝑟 𝑦∗
and 𝑥′
∈ 𝑃𝑥
−1
𝑦1 ∩ 𝐵 𝑎
2
𝑥∗
. (3.5)
We need to prove that there exists 𝑥′′
∈ 𝑃𝑥
−1
𝑦2 such that
𝑥′
− 𝑥′′
≤
𝑀
1 − 𝑀 𝐴 − 𝛻𝑓 𝑥∗
𝑦1 − 𝑦2 .
To finish this, we will verify that there exists a sequence 𝑥 𝑘 ⊂ 𝐵𝑎 𝑥∗
such that
𝑦2 ∈ 𝑓 𝑥 + 𝐴 𝑥 𝑘−1 − 𝑥 + 𝛻𝑓 𝑥∗
𝑥 𝑘 − 𝑥 𝑘−1 + 𝐹 𝑥 𝑘 (3.6)
and 𝑥 𝑘 − 𝑥 𝑘−1 ≤ 𝑀 𝑦1 − 𝑦2 𝑀 𝐴 − 𝛻𝑓 𝑥∗ 𝑘−2
for each 𝑘 = 2,3,4,… (3.7)
We prove by induction on 𝑘. Denote
𝑧𝑖 = 𝑦𝑖 − 𝑓 𝑥 − 𝐴 𝑥′
− 𝑥 + 𝑓 𝑥∗
+ 𝛻𝑓 𝑥∗
𝑥′
− 𝑥∗
for each 𝑖 = 1,2. (3.8)
Note by (3.5) that 𝑥 − 𝑥′
≤ 𝑥 − 𝑥∗
+ 𝑥∗
− 𝑥′
≤ 𝑎. Following from (3.5) and using (3.3) with the
relation 𝑟 ≤ 𝑏 − 2𝑎 𝐴 − 𝛻𝑓 𝑥∗
, we have that
𝑧𝑖 − 𝑦∗
≤ 𝑦𝑖 − 𝑦∗
+ 𝑓 𝑥 − 𝑓 𝑥∗
− 𝛻𝑓 𝑥∗
𝑥 − 𝑥∗
+ 𝐴 − 𝛻𝑓 𝑥∗
𝑥 − 𝑥′
≤ 𝑟 + 𝛻𝑓 𝑥∗
+ 𝑡 𝑥 − 𝑥∗
− 𝛻𝑓 𝑥∗
𝑥 − 𝑥∗
𝑑𝑡
1
0
+ 𝐴 − 𝛻𝑓 𝑥∗
𝑥 − 𝑥′
≤ 𝑟 + 𝐴 − 𝛻𝑓 𝑥∗
𝑥 − 𝑥∗
+ 𝐴 − 𝛻𝑓 𝑥∗
𝑥 − 𝑥′
= 𝑟 + 𝐴 − 𝛻𝑓 𝑥∗
𝑥 − 𝑥∗
+ 𝑥 − 𝑥′
≤ 𝑟 + 𝐴 − 𝛻𝑓 𝑥∗
𝑎
2
+ 𝑎 ≤ 𝑏.
So we get 𝑧𝑖 ∈ 𝐵𝑏 𝑦∗
for each 𝑖 = 1,2. Setting 𝑥1 = 𝑥′
. Then 𝑥1 ∈ 𝑃𝑥
−1
𝑦1 by (3.5), and it comes from (3.1)
that 𝑦1 ∈ 𝑓 𝑥 + 𝐴 𝑥1 − 𝑥 + 𝐹 𝑥1 , which can be written as
𝑦1 + 𝑓 𝑥∗
+ 𝛻𝑓 𝑥∗
𝑥1 − 𝑥∗
∈ 𝑓 𝑥 + 𝐴 𝑥1 − 𝑥 + 𝐹 𝑥1 + 𝑓 𝑥∗
+ 𝛻𝑓 𝑥∗
𝑥1 − 𝑥∗
.
By the definition of 𝑧1 in (3.8), we obtain that
𝑧1 ∈ 𝑓 𝑥∗
+ 𝛻𝑓 𝑥∗
𝑥1 − 𝑥 + 𝐹 𝑥1 .
Hence 𝑥1 ∈ 𝑃𝑥∗
−1
𝑧1 by (3.1), and then 𝑥1 ∈ 𝑃𝑥∗
−1
𝑧1 ∩ 𝐵𝑟 𝑥∗ 𝑥∗
by (3.5). Noting that 𝑧1, 𝑧2 ∈ 𝐵𝑏 𝑦∗
, and
from (3.2) we claim that their exists 𝑥2 ∈ 𝑃𝑥∗
−1
𝑧2 such that
𝑥2 − 𝑥1 ≤ 𝑀 𝑧1 − 𝑧2 = 𝑀 𝑦1 − 𝑦2 . (3.9)
Setting 𝑥1 = 𝑥′
. By the definition of 𝑧2 in (3.8), we have that
𝑥2 ∈ 𝑃𝑥∗
−1
𝑧2 = 𝑃𝑥∗
−1
𝑦2 − 𝑓 𝑥 − 𝐴 𝑥1 − 𝑥 + 𝑓 𝑥∗
+ 𝛻𝑓 𝑥∗
𝑥1 − 𝑥∗
, which, together with (3.1), implies
that
𝑦2 ∈ 𝑓 𝑥 + 𝐴 𝑥1 − 𝑥 + 𝛻𝑓 𝑥∗
𝑥2 − 𝑥1 + 𝐹 𝑥2 . (3.10)
Therefore (3.10) and (3.9) shows respectively that (3.6) and (3.7) are true for the constructed points 𝑥1, 𝑥2. We
assume that 𝑥1, 𝑥2, 𝑥3, … . , 𝑥 𝑛 are constructed such that (3.6) and (3.7) are true for 𝑘 = 2,3, … . , 𝑛. We need to
construct 𝑥 𝑛+1 such that (3.6) and (3.7) are also true for 𝑘 = 𝑛 + 1. For this purpose, we can write
𝑧𝑖
𝑛
= 𝑦2 − 𝑓 𝑥 − 𝐴 𝑥 𝑛+𝑖−1 − 𝑥 + 𝑓 𝑥∗
+ 𝛻𝑓 𝑥∗
𝑥 𝑛+𝑖−1 − 𝑥∗
, for each 𝑖 = 0,1.
Then, by the inductive assumption, we have that
4. Uniformity of the Local Convergence of Chord Method for Generalized Equations
DOI: 10.9790/5728-11121623 www.iosrjournals.org 19 | Page
𝑧0
𝑛
− 𝑧1
𝑛
= 𝐴 − 𝛻𝑓 𝑥∗
𝑥 𝑛 − 𝑥 𝑛−1
≤ 𝐴 − 𝛻𝑓 𝑥∗
𝑥 𝑛 − 𝑥 𝑛−1
≤ 𝑦1 − 𝑦2 𝑀 𝐴 − 𝛻𝑓 𝑥∗ 𝑛−1
. (3.11)
Since 𝑥1 − 𝑥∗
≤
𝑎
2
and 𝑦1 − 𝑦2 ≤ 2𝑟 by (3.5), it follows from (3.7) that
𝑥 𝑛 − 𝑥∗
≤ 𝑥 𝑘 − 𝑥 𝑘−1 + 𝑥1 − 𝑥∗
𝑛
𝑘=2
≤ 2𝑀𝑟 𝑀 𝐴 − 𝛻𝑓 𝑥∗ 𝑘−2
+
𝑎
2
𝑛
𝑘=2
≤
2𝑀𝑟
1 − 𝑀 𝐴 − 𝛻𝑓 𝑥∗
+
𝑎
2
.
From (3.3), we see 𝑟 ≤
𝑎 1−𝑀 𝐴−𝛻𝑓 𝑥∗
4𝑀
, and so 𝑥 𝑛 − 𝑥∗
≤ 𝑎. (3.12)
Accordingly, 𝑥 𝑛 − 𝑥 ≤ 𝑥 𝑛 − 𝑥∗
+ 𝑥∗
− 𝑥 ≤
3𝑎
2
. (3.13)
Furthermore, using (3.5) and (3.13), one has, for each 𝑖 = 0,1, that
𝑧𝑖
𝑛
− 𝑦∗
≤ 𝑦2 − 𝑦∗
+ 𝑓 𝑥 − 𝑓 𝑥∗
− 𝛻𝑓 𝑥∗
𝑥 − 𝑥∗
+ 𝐴 − 𝛻𝑓 𝑥∗
𝑥 − 𝑥 𝑛+𝑖−1
≤ 𝑟 + 𝐴 − 𝛻𝑓 𝑥∗
𝑥 − 𝑥∗
+ 𝐴 − 𝛻𝑓 𝑥∗
𝑥 − 𝑥 𝑛+𝑖−1
= 𝑟 + 𝐴 − 𝛻𝑓 𝑥∗
𝑥 − 𝑥∗
+ 𝑥 − 𝑥 𝑛+𝑖−1
≤ 𝑟 + 𝐴 − 𝛻𝑓 𝑥∗ 𝑎
2
+
3𝑎
2
= 𝑟 + 2 𝐴 − 𝛻𝑓 𝑥∗
𝑎.
By the assumption of 𝑟 in (3.3), we have that 𝑧𝑖
𝑛
∈ 𝐵𝑏 𝑦∗
for each 𝑖 = 0,1. Since assumption (3.6) holds for
𝑘 = 𝑛 we have
𝑦2 ∈ 𝑓 𝑥 + 𝐴 𝑥 𝑛−1 − 𝑥 + 𝛻𝑓 𝑥∗
𝑥 𝑛 − 𝑥 𝑛−1 + 𝐹 𝑥 𝑛 , which can be written as
𝑦2 + 𝑓 𝑥∗
+ 𝛻𝑓 𝑥∗
𝑥 𝑛−1 − 𝑥∗
∈ 𝑓 𝑥 + 𝐴 𝑥 𝑛−1 − 𝑥 + 𝛻𝑓 𝑥∗
𝑥 𝑛 − 𝑥 𝑛−1
+𝐹 𝑥 𝑛 + 𝑓 𝑥∗
+ 𝛻𝑓 𝑥∗
𝑥 𝑛−1 − 𝑥∗
,
that is, 𝑧0
𝑛
∈ 𝑓 𝑥∗
+ 𝛻𝑓 𝑥∗
𝑥 𝑛 − 𝑥∗
+ 𝐹 𝑥 𝑛 by the assumption of 𝑧0
𝑛
. This, jointly with (3.1) and (3.12),
gives 𝑥 𝑛 ∈ 𝑃𝑥∗
−1
𝑧0
𝑛
∩ 𝐵𝑎 𝑥∗
. Utilize (3.2) again, their exists an element 𝑥 𝑛+1 ∈ 𝑃𝑥∗
−1
𝑧1
𝑛
such that
𝑥 𝑛+1 − 𝑥 𝑛 ≤ 𝑀 𝑧0
𝑛
− 𝑧1
𝑛
≤ 𝑀 𝑦1 − 𝑦2 𝑀 𝐴 − 𝛻𝑓 𝑥∗ 𝑛−1
, (3.14)
where the last inequality holds by (3.11). By the assumption of 𝑧1
𝑛
, we get
𝑥 𝑛+1 ∈ 𝑃𝑥∗
−1
𝑧1
𝑛
= 𝑃𝑥∗
−1
𝑦2 − 𝑓 𝑥 − 𝐴 𝑥 𝑛 − 𝑥 + 𝑓 𝑥∗
+ 𝛻𝑓 𝑥∗
𝑥 𝑛 − 𝑥∗
, which, together with (3.1),
implies that
𝑦2 ∈ 𝑓 𝑥 + 𝐴 𝑥 𝑛 − 𝑥 + 𝛻𝑓 𝑥∗
𝑥 𝑛+1 − 𝑥 𝑛 + 𝐹 𝑥 𝑛+1 .
This, together with (3.14), finishes the induction step, and proves the existence of sequence 𝑥 𝑘 satisfying (3.6)
and (3.7). Since 𝑀 𝐴 − 𝑓 𝑥∗
< 1, we see from (3.7) that, 𝑥 𝑘 is a Cauchy sequence, and hence it is
convergent. Let 𝑥′′
= 𝑥 𝑘𝑘→∞
𝐿𝑖𝑚
. Then, taking limit in (3.6) and noting that 𝐹 has closed graph, we get 𝑦2 ∈
𝑓 𝑥 + 𝐴 𝑥′′
− 𝑥 + 𝐹 𝑥′′
and so 𝑥′′
∈ 𝑃𝑥
−1
𝑦2 .
Furthermore, 𝑥′
− 𝑥′′
≤ lim 𝑛→∞ sup 𝑥 𝑘 − 𝑥 𝑘−1
𝑛
𝑘=2
≤ lim 𝑛→∞ 𝑀 𝐴 − 𝛻𝑓 𝑥∗ 𝑘−2
𝑀 𝑦1 − 𝑦2
𝑛
𝑘=2
=
𝑀
1−𝑀 𝐴−𝛻𝑓 𝑥∗ 𝑦1 − 𝑦2 .
This completes the proof of the lemma.
3.1 Uniformity of Linear Convergence of First Kind
This section is devoted to study the uniformity of linear convergence of the Chord method defined by (1.3).
Theorem 3.1 Let 𝑥∗
be a solution of (1.1) for 𝑦 = 0. Suppose that 𝑃𝑥∗
−1
∙ is pseudo-Lipschitz around 0, 𝑥∗
with constants 𝑎, 𝑏 and 𝑀. Let 𝑥 ∈ 𝐵 𝑎
2
𝑥∗
and 𝐴 ∈ 𝐿 𝑋, 𝑌 . Suppose that 𝛻𝑓 is continuous on 𝐵 𝑎
2
𝑥∗
with
constant 𝐴 − 𝛻𝑓 𝑥∗
such that 3𝑀′
𝐴 − 𝛻𝑓 𝑥∗
< 1 , i.e,
sup
𝑥∈𝐵 𝑎
2
𝑥∗
𝛻𝑓 𝑥 − 𝛻𝑓 𝑥∗
≤ 𝐴 − 𝛻𝑓 𝑥∗
≤ min
𝑏
2𝑎
,
1
𝑀
.
Then there exists positive constants 𝜎 and 𝑐 such that for every 𝑦 ∈ 𝐵𝑏 0 and for every 𝑥0 ∈ 𝐵𝜎 𝑥∗
there
exists a sequence 𝑥 𝑘 generated by (1.3) with initial point 𝑥0, which is convergent to a solution 𝑥 of (1.1) for 𝑦,
i.e.
𝑥 𝑘+1 − 𝑥 ≤ 𝑐 𝑥 𝑘 − 𝑥 .
8. Uniformity of the Local Convergence of Chord Method for Generalized Equations
DOI: 10.9790/5728-11121623 www.iosrjournals.org 23 | Page
+ 𝛻𝑓 𝑥∗
− 𝛻𝑓 𝑥 𝑘 + 𝛻𝑓 𝑥∗
− 𝐴 𝑥 − 𝑥 𝑘
≤ 𝑏 + 𝛻𝑓 𝑥∗
− 𝐴 𝑥 − 𝑥 𝑘 + 𝛻𝑓 𝑥∗
− 𝐴 + 𝛻𝑓 𝑥∗
− 𝐴 𝑥 − 𝑥 𝑘
= 𝑏 + 3 𝛻𝑓 𝑥∗
− 𝐴 𝑥 − 𝑥 𝑘
≤ 𝑏 + 3 𝛻𝑓 𝑥∗
− 𝐴 3𝑀′
𝛻𝑓 𝑥∗
− 𝐴 𝑥 − 𝑥 𝑘−1
≤ 𝑏 + 𝑀′
3 𝛻𝑓 𝑥∗
− 𝐴 2
𝑥 − 𝑥 𝑘−1 ≤ 𝛽.
Now, from Lemma 3.1, there exists 𝑥 𝑘+1 ∈ 𝑃𝑥 𝑘
−1
𝑦 , i.e. 𝑦 ∈ 𝑓 𝑥 𝑘 + 𝐴 𝑥 𝑘+1 − 𝑥 𝑘 + 𝐹 𝑥 𝑘+1 , such that
𝑥 𝑘+1 − 𝑥 ≤ 𝑀′ 𝑓 𝑥 − 𝑓 𝑥 𝑘 − 𝐴 𝑥 − 𝑥 𝑘
≤ 𝑀′ 𝑓 𝑥 − 𝑓 𝑥 𝑘 − 𝛻𝑓 𝑥 𝑘 𝑥 − 𝑥 𝑘
+𝑀′
𝛻𝑓 𝑥 𝑘 − 𝛻𝑓 𝑥∗
+ 𝛻𝑓 𝑥∗
− 𝐴 𝑥 − 𝑥 𝑘
≤ 𝑀′ 𝛻𝑓 𝑥∗
− 𝐴 𝑥 − 𝑥 𝑘
+𝑀′
𝛻𝑓 𝑥∗
− 𝐴 + 𝛻𝑓 𝑥∗
− 𝐴 𝑥 − 𝑥 𝑘
≤ 𝑀′ 𝛻𝑓 𝑥∗
− 𝐴 + 2 𝛻𝑓 𝑥∗
− 𝐴 𝑥 − 𝑥 𝑘
= 3𝑀′
𝛻𝑓 𝑥∗
− 𝐴 𝑥 − 𝑥 𝑘 .
Taking 𝛾 = 3𝑀′
𝛻𝑓 𝑥∗
− 𝐴 . Then from the last inequality, we have that
𝑥 𝑘+1 − 𝑥 ≤ 𝛾 𝑥 − 𝑥 𝑘 .
This completes the proof.
IV. Concluding Remarks
In this study, we have established the uniformity of the local convergence results for the Chord method
defined by (1.3) under the assumptions that 𝑃𝑥∗
−1
∙ is pseudo-Lipschitz and 𝛻𝑓 is continuous. In particular, the
uniformity of the convergence results of the Chord method defined by (1.3) are presented in two different ways
when 𝑃𝑥∗
−1
∙ is pseudo-Lipschitz and 𝛻𝑓 is continuous. These results improve and extend the corresponding
one [4].
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