This document provides an abstract for a thesis on using concepts from nonstandard analysis to study Taylor series approximations. Specifically, it aims to:
1) Define an approximation factor relating the Taylor series remainder to the next nonzero term.
2) Prove properties of the approximation factor for different index ranges: standard n, unlimited indices inside/outside the convergence disk.
3) Consider the analyticity of the approximation factor in special and general cases.
The thesis contains background on nonstandard analysis, theorems on approximating series, and studies the approximation factor for Taylor series in terms of standard and nonstandard analysis.
An Analysis and Study of Iteration Proceduresijtsrd
In computational mathematics, an iterative method is a scientific technique that utilizes an underlying speculation to produce a grouping of improving rough answers for a class of issues, where the n th estimate is gotten from the past ones. A particular execution of an iterative method, including the end criteria, is a calculation of the iterative method. An iterative method is called joined if the relating grouping meets for given starting approximations. A scientifically thorough combination investigation of an iterative method is typically performed notwithstanding, heuristic based iterative methods are additionally normal. This Research provides a survey of iteration procedures that have been used to obtain fixed points for maps satisfying a variety of contractive conditions. Dr. R. B. Singh | Shivani Tomar ""An Analysis and Study of Iteration Procedures"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-4 , June 2019, URL: https://www.ijtsrd.com/papers/ijtsrd23715.pdf
Paper URL: https://www.ijtsrd.com/mathemetics/computational-science/23715/an-analysis-and-study-of-iteration-procedures/dr-r-b-singh
The International Journal of Engineering and Science (The IJES)theijes
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.
An Analysis and Study of Iteration Proceduresijtsrd
In computational mathematics, an iterative method is a scientific technique that utilizes an underlying speculation to produce a grouping of improving rough answers for a class of issues, where the n th estimate is gotten from the past ones. A particular execution of an iterative method, including the end criteria, is a calculation of the iterative method. An iterative method is called joined if the relating grouping meets for given starting approximations. A scientifically thorough combination investigation of an iterative method is typically performed notwithstanding, heuristic based iterative methods are additionally normal. This Research provides a survey of iteration procedures that have been used to obtain fixed points for maps satisfying a variety of contractive conditions. Dr. R. B. Singh | Shivani Tomar ""An Analysis and Study of Iteration Procedures"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-4 , June 2019, URL: https://www.ijtsrd.com/papers/ijtsrd23715.pdf
Paper URL: https://www.ijtsrd.com/mathemetics/computational-science/23715/an-analysis-and-study-of-iteration-procedures/dr-r-b-singh
The International Journal of Engineering and Science (The IJES)theijes
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.
On algorithmic problems concerning graphs of higher degree of symmetrygraphhoc
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry. The complexity of computing the adjacency matrices of a graph Gr on the vertices X such that
Aut GR = G depends very much on the description of the geometry with which one starts. For example, we
can represent the geometry as the totality of 1 cosets of parabolic subgroups 2 chains of embedded
subspaces (case of linear groups), or totally isotropic subspaces (case of the remaining classical groups), 3
special subspaces of minimal module for G which are defined in terms of a G invariant multilinear form.
The aim of this research is to develop an effective method for generation of graphs connected with classical
geometry and evaluation of its spectra, which is the set of eigenvalues of adjacency matrix of a graph. The
main approach is to avoid manual drawing and to calculate graph layout automatically according to its
formal structure. This is a simple task in a case of a tree like graph with a strict hierarchy of entities but it
becomes more complicated for graphs of geometrical nature. There are two main reasons for the
investigations of spectra: (1) very often spectra carry much more useful information about the graph than a
corresponding list of entities and relationships (2) graphs with special spectra, satisfying so called
Ramanujan property or simply Ramanujan graphs (by name of Indian genius mathematician) are important
for real life applications (see [13]). There is a motivated suspicion that among geometrical graphs one
could find some new Ramanujan graphs.
An approach to Fuzzy clustering of the iris petals by using Ac-meansijsc
This paper proposes a definition of a fuzzy partition element based on the homomorphism between type-1 fuzzy sets and the three-valued Kleene algebra. A new clustering method
based on the C-means algorithm, using the defined partition, is presented in this paper, which will
be validated with the traditional iris clustering problem by measuring its petals.
ON ALGORITHMIC PROBLEMS CONCERNING GRAPHS OF HIGHER DEGREE OF SYMMETRYFransiskeran
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry.
Symbolic Computation via Gröbner BasisIJERA Editor
The purpose of this paper is to find the orthogonal projection of a rational parametric curve onto a rational parametric surface in 3-space. We show that the orthogonal projection problem can be reduced to the problem of finding elimination ideals via Gröbnerbasis. We provide a computational algorithm to find the orthogonal projection, and include a few illustrative examples. The presented method is effective and potentially useful for many applications related to the design of surfaces and other industrial and research fields.
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.
EMU M.Sc. Thesis Presentation
Thesis Title: "Dark Matter; Modification of f(R) or WIMPS Miracle"
Student: Ali Övgün
Supervisor: Prof. Dr. Mustafa Halilsoy
On algorithmic problems concerning graphs of higher degree of symmetrygraphhoc
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry. The complexity of computing the adjacency matrices of a graph Gr on the vertices X such that
Aut GR = G depends very much on the description of the geometry with which one starts. For example, we
can represent the geometry as the totality of 1 cosets of parabolic subgroups 2 chains of embedded
subspaces (case of linear groups), or totally isotropic subspaces (case of the remaining classical groups), 3
special subspaces of minimal module for G which are defined in terms of a G invariant multilinear form.
The aim of this research is to develop an effective method for generation of graphs connected with classical
geometry and evaluation of its spectra, which is the set of eigenvalues of adjacency matrix of a graph. The
main approach is to avoid manual drawing and to calculate graph layout automatically according to its
formal structure. This is a simple task in a case of a tree like graph with a strict hierarchy of entities but it
becomes more complicated for graphs of geometrical nature. There are two main reasons for the
investigations of spectra: (1) very often spectra carry much more useful information about the graph than a
corresponding list of entities and relationships (2) graphs with special spectra, satisfying so called
Ramanujan property or simply Ramanujan graphs (by name of Indian genius mathematician) are important
for real life applications (see [13]). There is a motivated suspicion that among geometrical graphs one
could find some new Ramanujan graphs.
An approach to Fuzzy clustering of the iris petals by using Ac-meansijsc
This paper proposes a definition of a fuzzy partition element based on the homomorphism between type-1 fuzzy sets and the three-valued Kleene algebra. A new clustering method
based on the C-means algorithm, using the defined partition, is presented in this paper, which will
be validated with the traditional iris clustering problem by measuring its petals.
ON ALGORITHMIC PROBLEMS CONCERNING GRAPHS OF HIGHER DEGREE OF SYMMETRYFransiskeran
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry.
Symbolic Computation via Gröbner BasisIJERA Editor
The purpose of this paper is to find the orthogonal projection of a rational parametric curve onto a rational parametric surface in 3-space. We show that the orthogonal projection problem can be reduced to the problem of finding elimination ideals via Gröbnerbasis. We provide a computational algorithm to find the orthogonal projection, and include a few illustrative examples. The presented method is effective and potentially useful for many applications related to the design of surfaces and other industrial and research fields.
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.
EMU M.Sc. Thesis Presentation
Thesis Title: "Dark Matter; Modification of f(R) or WIMPS Miracle"
Student: Ali Övgün
Supervisor: Prof. Dr. Mustafa Halilsoy
Master Thesis on the Mathematial Analysis of Neural NetworksAlina Leidinger
Master Thesis submitted on June 15, 2019 at TUM's chair of Applied Numerical Analysis (M15) at the Mathematics Department.The project was supervised by Prof. Dr. Massimo Fornasier. The thesis took a detailed look at the existing mathematical analysis of neural networks focusing on 3 key aspects: Modern and classical results in approximation theory, robustness and Scattering Networks introduced by Mallat, as well as unique identification of neural network weights. See also the one page summary available on Slideshare.
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.
Computer Science
Active and Programmable Networks
Active safety systems
Ad Hoc & Sensor Network
Ad hoc networks for pervasive communications
Adaptive, autonomic and context-aware computing
Advance Computing technology and their application
Advanced Computing Architectures and New Programming Models
Advanced control and measurement
Aeronautical Engineering,
Agent-based middleware
Alert applications
Automotive, marine and aero-space control and all other control applications
Autonomic and self-managing middleware
Autonomous vehicle
Biochemistry
Bioinformatics
BioTechnology(Chemistry, Mathematics, Statistics, Geology)
Broadband and intelligent networks
Broadband wireless technologies
CAD/CAM/CAT/CIM
Call admission and flow/congestion control
Capacity planning and dimensioning
Changing Access to Patient Information
Channel capacity modelling and analysis
Civil Engineering,
Cloud Computing and Applications
Collaborative applications
Communication application
Communication architectures for pervasive computing
Communication systems
Computational intelligence
Computer and microprocessor-based control
Computer Architecture and Embedded Systems
Computer Business
Computer Sciences and Applications
Computer Vision
Computer-based information systems in health care
Computing Ethics
Computing Practices & Applications
Congestion and/or Flow Control
Content Distribution
Context-awareness and middleware
Creativity in Internet management and retailing
Cross-layer design and Physical layer based issue
Cryptography
Data Base Management
Data fusion
Data Mining
Data retrieval
Data Storage Management
Decision analysis methods
Decision making
Digital Economy and Digital Divide
Digital signal processing theory
Distributed Sensor Networks
Drives automation
Drug Design,
Drug Development
DSP implementation
E-Business
E-Commerce
E-Government
Electronic transceiver device for Retail Marketing Industries
Electronics Engineering,
Embeded Computer System
Emerging advances in business and its applications
Emerging signal processing areas
Enabling technologies for pervasive systems
Energy-efficient and green pervasive computing
Environmental Engineering,
Estimation and identification techniques
Evaluation techniques for middleware solutions
Event-based, publish/subscribe, and message-oriented middleware
Evolutionary computing and intelligent systems
Expert approaches
Facilities planning and management
Flexible manufacturing systems
Formal methods and tools for designing
Fuzzy algorithms
Fuzzy logics
GPS and location-based app
On New Root Finding Algorithms for Solving Nonlinear Transcendental EquationsAI Publications
In this paper, we present new iterative algorithms to find a root of the given nonlinear transcendental equations. In the proposed algorithms, we use nonlinear Taylor’s polynomial interpolation and a modified error correction term with a fixed-point concept. We also investigated for possible extension of the higher order iterative algorithms in single variable to higher dimension. Several numerical examples are presented to illustrate the proposed algorithms.
A PROBABILISTIC ALGORITHM OF COMPUTING THE POLYNOMIAL GREATEST COMMON DIVISOR...ijscmcj
In the earlier work, subresultant algorithm was proposed to decrease the coefficient growth in the Euclidean algorithm of polynomials. However, the output polynomial remainders may have a small factor which can be removed to satisfy our needs. Then later, an improved subresultant algorithm was given by representing the subresultant algorithm in another way, where we add a variant called 𝜏 to express the small factor. There was a way to compute the variant proposed by Brown, who worked at IBM. Nevertheless, the way failed to determine each𝜏 correctly.
APPROXIMATIONS; LINEAR PROGRAMMING;NON- LINEAR FUNCTIONS; PROJECT MANAGEMENT WITH PERT/CPM; DECISION THEORY; THEORY OF GAMES; INVENTORY MODELLING; QUEUING THEORY
My invited talk at the 2018 Annual Meeting of SIAM (Society of Industrial and...Anirbit Mukherjee
This is a slightly expanded version of the talk I gave at the 2018 ISMP (International Symposium on Mathematical Programming). This SIAM talk has some more introductory material than the ISMP talk.
In this paper, the underlying principles about the theory of relativity are briefly introduced and reviewed. The mathematical prerequisite needed for the understanding of general relativity and of Einstein field equations are discussed. Concepts such as the principle of least action will be included and its explanation using the Lagrange equations will be given. Where possible, the mathematical details and rigorous analysis of the subject has been given in order to ensure a more precise and thorough understanding of the theory of relativity. A brief mathematical analysis of how to derive the Einstein’s field’s equations from the Einstein-Hilbert action and the Schwarzschild solution was also given.
A Nonstandard Study of Taylor Ser.Dev.-Abstract+ Intro. M.Sc. Thesis
1. -1-
A NONSTANDARD STUDY
ON THE
TAYLOR SERIES DEVELOPMENT
A THESIS
SUBMITTED TO THE COLLEGE OF SCIENCE
UNIVERSITY OF SALAHADDIN-ERBIL
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE
IN MATHEMATICS
BY
IBRAHIM OTHMAN HAMAD
B.Sc. (MATHEMATICS)-1992
JUNE JOZARDAN
2000 2700
2. -3-
Abstract
The main aim of the present work is to use some concepts of
nonstandard analysis given by Robinson, A. [26] and axiomatized by
Nelson, E.[23] to associate to the classical formula of the remainder
majoration, that plays an incontestable role, with an approximation of this
remainder. More precisely, we try to find a connection between the
remainder and the development of the next nonzero term by defining
)(
)(
)( 1
xT
xR
x
n
n
n
−
=Φ as an approximation factor of )(xTn w.r.t )(xf for a
Taylor series ∑
∞
=0
)(
k
n xT . Moreover under certain conditions we shall prove
the following:
i) The approximation factor 1)( ≅Φ xn for limited n.
ii) The approximation factor ∑
∞
=
≅Φ
0
)(
n
n
nn zcx for unlimited index inside the
convergence disc where )(
ω
+ω
=
a
a
c n
n
o
.
iii) The approximation factor ∑
∞
=
−≅Φ
1
)(
n
n
n
n
z
d
x for unlimited index outside
the convergence disc, where )(
ω
−ω
=
a
a
d n
n
o
.
We also consider the analyticity of the approximation factor. It is
proved that the approximation factor ∑=Ψ n
n
n
u
u
c
u)(o
where )(
ω
+ω
=
a
a
c n
n
o
,
and
)0(
)(
)( )(
)(
ω
ω
ω
ω=Ψ
f
u
f
u .
3. -4-
Contents
Page No.
List of Symbols 10
Introduction 11
Chapter One Basic Concepts
1-1 Backgrounds and Definitions 16
1-2 An Asymptotic Approximation of Series 23
Chapter Two A Nonstandard Approximation Using
Taylor Series
2-1 Introduction 29
2-2 A Shadow Determination of an Approximation Factor
for Standard n and 0≅ε . 33
2-3 A Shadow Determination of an Approximation Factor
for Unlimited Index Inside the Convergence Disc. 34
2-4 A Shadow Determination of an Approximation Factor
for Unlimited Index Outside the Convergence Disc. 40
2-5 Motivation of a General Study of an Approximation
Factor for Unlimited Index. 44
Chapter Three Analyticity of the Approximation Factor
3-1 Analyticity of the Approximation Factor (Special Case) 47
3-2 Analyticity of the Approximation Factor (General Case) 51
References 57
4. -5-
Table of Symbols
Symbols Description
Infinitely Close
)(≅< Less than but not Infinitely Close
st
∀ For All Standard
fin
∀ For All Finite
finst
∀ For All Standard Finite
)(xnΦ Approximation Factor of Order n of x
)(xTn Term of Degree n as a function of x
xo
Shadow of x
)(xm Monad of x
)(xgal Galaxy of x
N Set of Standard Natural Numbers
N Set of unlimited Natural Numbers
εγβα ,,, Infinitesimal Numbers
ω Infinitely Large Number
∞
C Space of Every Where Differentiable Functions
5. -6-
Introduction
The primary idea about nonstandard analysis goes back to the problem of
determining the slope of a tangent of a curve, limits and derivatives in which
Newton, I(1642-1727) and Leibniz(1646-1716) worked out to approximate the
tangent of a curve by a line which intersects the curve at two points such that the
distance between them is infinitely small. Such distance, was named by Newton
small quantity (or infinitesimal)[4] [9] [29].
Many attempts had been done to establish the foundations of infinitesimals.
Leibniz and his followers were never been able to state with sufficient precision
just what rules were supposed to govern their new system including infinitely
small as well as infinitely large quantities with no contradiction in these rules, until
Abraham Robinson in (1961), presented a complete and satisfactory solution of
Leibniz’s problem by formulating the ideal quantities (i.e., infinitely small and
infinitely large) in precise mathematical structures under the name (Nonstandard
Analysis). More precisely, Robinson showed that there exists a proper extension,
say *R, of the field of real numbers R which in a certain sense have the same
formal properties as R and is non-Archimedean [4] [26] [29].
Later on some mathematicians tried to reconstruct nonstandard analysis
models using the set theory [13] [23] [29], such as Luxembourg (1962), and
Nelson, E. (1977) who presented a great and illustrative construction using the
axiomatic set theory of ZFC (Zermelo-Fraenckel Set Theory with Axiom of
Choice)[31] . Today, the nonstandard analysis regarded as a technique rather
than a subject. There are problems, the mathematicians were unable to prove
them using conventional methods, while they can be proved using nonstandard
methods, such as Bernstein Robinson theorem [4].
In the present work, we use some concepts of nonstandard analysis that
are axiomatized by Nelson, E. [23]. In practice, an approximation is successful if
the error is small. For an approximation study, it is, therefore interesting to
6. -7-
arrange a mathematical theory, which permits us to express simply the notion”
be small” [15] [24].
The nonstandard analysis makes this possible, especially the axiomatic
version IST (Internal Set Theory) proposed by Nelson, E. It happens, that one
side of the extension of set theory’s language ZFC is by introducing predicate
“standard”, while the other side is by adding to ZFC some axioms concerning
the usage of the new predicate. Two of the principle consequences of these
axioms are:
1- each set defined in ZFC is standard.
2- each infinite set defined in ZFC pssesses nonstandard elements.
Any collection of real numbers with predicates (standard, infinitesimal,
limited, unlimited,…ect)is not a set in the axiomatic sense of ZFC , and it is
called an external set.
Now we point out three interferences of the external sets to our study of
approximations.
Firstly, for a given accuracy, we can form the “ external” set of numbers
for what the approximation attains that accuracy. We have used this possibility
in chapter two to describe sketches, by using a computer program written in a
Visual Basic Language Version 5, showing the approximation of a function by
it’s Taylor polynomials. We have characterized the collection of points for
which the graphs of the functions and their Taylor polynomials are conform to
the naked eye, by an external set of points such that the remainder is
infinitesimal.
Secondly, the external sets interfere to a practical level for reasoning
concerning the approximation. They are at the base of “ permanence principles”
of set statements, which enable us to understand the validity of a proposition
beyond the domain where it is proved to be effective.
Thirdly, the external sets can be used to distinguish certain speeds of the
approximation considered as the approximation of a standard real number “a ”
7. -8-
by the terms of the standard sequence Nnnu ∈}{ which converges to this number.
For example, ea = and ∑
=
=
n
k
n
k
u
0 !
1
. This can be expressed by the sentence “ for
every non limited ∈n N, nu belongs to a monad of “a ”. This sentence connects
the two external sets of unlimited positive integers and that of real numbers at an
infinitesimal distance from “a “.
In the theory of asymptotic development, we consider particularly the
functions f that are approximated by a sequence of functions NnnP ∈}{ such
that the remainder nn PfxR −=)( , satisfies the property )()( n
n xxR ο= , if
0→x , for every ∈n N. This property is in particular, satisfied in the case where
f in C
∞
, and nP its Taylor polynomial. Moreover, notice that if f is standard,
then the proposition “ monadR n
n −∈εε)( for every standard n , and 0≅ε , is
satisfied. This constitutes a connection point between the classical method and our
nonstandard method [2] [22] [25] [30].
Another correspondence is situated at the terminology level that can be
illustrated by the following example: consider a problem of the behavior of a
family of functions depending on a parameter “a ”, at infinity. It is then of a
classical use to distinguish between “numbers depending on x ” ( )(xaa = ),
susceptible to increase over every value, and also can influence the asymptotic
behavior and a “fixed number”, independent of x , and have no influence on the
asymptotic behavior. We compare this with a nonstandard distinction between
“numbers is depending on ω,ωunlimited positive” ( )(ωaa = ) and “standard
numbers”. But however, we observe an important difference. The first distinction
does not permit us to separate the two types of numbers on the real line, while the
second distinction makes it possible. For unlimited numbers )(ωa , which are
greater than the standard numbers, but a number )(xa “depends on x ” is always
going beyond the fixed numbers.
8. -9-
In this thesis, which consists of three chapters we tried to study a
nonstandard approximation, using Taylor polynomials. We have tried to make this
work a self-contained as much as possible.
Chapter one was written in order to provide the reader with general
background, notions and materials needed. It consists of two sections, the first
gives a brief description of IST and other concepts and terminologies of
nonstandard analysis needed. The second section contains some theorems and
results concerning (nonstandard approximation of series).
Chapters two and three considered to be the climax of the work of this
thesis. Chapter two consists of fife sections, in which we study the
approximation of a function using Taylor polynomials. In the first section, we
discuss the classical notion about the remainder and explaining its
disadvantages. In the second section we study the shadow of the approximation
factor for Taylor Series for standard n and 0≅ε . In the sections three and
fourth, we tried to determine the shadow of the approximation factor for Taylor
series inside and outside the disc of convergence. In the fifth section, we give a
motivation of a general study of the approximation factor. In chapter three, we
study the analyticity of the approximation factor in two cases; special and
general.