The discovery of infinite products byWallis and infinite series by Newton marked the
beginning of the modern mathematical era. The use of series allowed Newton to find
the area under a curve defined by any algebraic equation, an achievement completely
beyond the earlier methods ofTorricelli, Fermat, and Pascal. The work of Newton and
his contemporaries, including Leibniz and the Bernoullis, was concentrated in mathematical
analysis and physics.
Abstract In 1963 Atiyah and Singer proved the famous Atiyah-Singer Index Theorem, which states, among other things, that the space of elliptic pseudodiﬀerential operators is such that the collection of operators with any given index forms a connected subset. Contained in this statement is the somewhat more specialized claim that the index of an elliptic operator must be invariant under suﬃciently small perturbations.
The discipline known as Mathematical Logic will not speciﬁcally be deﬁned within this text. Instead,
you will study some of the concepts in this signiﬁcant discipline by actually doing mathematical logic. Thus,
you will be able to surmise for yourself what the mathematical logician is attempting to accomplish.
Consider the following three arguments taken from the disciplines of military science, biology, and
set-theory, where the symbols (a), (b), (c), (d), (e) are used only to locate speciﬁc sentences....
The usual index theorems for holomorphic self-maps, like for instance the classical holomorphic Lefschetz theorem (see, e.g., [GH]), assume that the ﬁxed-points set contains only isolated points. The aim of this paper, on the contrary, is to prove index theorems for holomorphic self-maps having a positive dimensional ﬁxed-points set. The origin of our interest in this problem lies in holomorphic dynamics.
That is all: just a computer procedure to approximate a real root. From
the narrow perspective of treating mathematics as a tool to solve real life
problems, this is of course suﬃcient. However, from the point of view of
mathematics, shouldn’t a student be interested in roots of polynomials in
general? Fourth degree? Odd degree? Other roots, once one is found?
Rational roots? Total number of roots?
Not every detail need be explained, but even the average student will
have his life improved by the mere knowledge that there are such questions,
often with answers, e.g.
A classic in its field, Professor Milne-Thomson's university text and reference book has long been one of the basic works. This is the complete reprinting of the revised (1966) edition which brings the subject up to date, including a complete and probably unique chapter on conical flow around sweptback wings. A wealth of problems, illustrations and cross-references add to the book's value as a text and a reference.
This paper is the fourth in a series where we describe the space of all embedded minimal surfaces of ﬁxed genus in a ﬁxed (but arbitrary) closed 3manifold. The key is to understand the structure of an embedded minimal disk in a ball in R3 . This was undertaken in [CM3], [CM4] and the global version of it will be completed here; see the discussion around Figure 12 for the local case and [CM15] for some more details. Our main results are Theorem 0.1 (the lamination theorem) and Theorem 0.2 (the one-sided curvature estimate). ...
Chapter 8 provides knowledge of sampling methods and central limit theorem. When you have completed this chapter, you will be able to: Explain under what conditions sampling is the proper way to learn something about a population, describe methods for selecting a sample, define and construct a sampling distribution of the sample mean,...
Chapter 8 - Sampling methods and the central limit theorem. When you have completed this chapter, you will be able to: Explain why a sample is often the only feasible way to learn something about a population, describe methods to select a sample, define sampling error, describe the sampling distribution of the sample mean,...
Abstract. This book has no equal. The priceless treasures of elementary geometry are
nowhere else exposed in so complete and at the same time transparent form. The short
solutions take barely 1.5 − 2 times more space than the formulations, while still remaining
complete, with no gaps whatsoever, although many of the problems are quite difficult. Only
this enabled the author to squeeze about 2000 problems on plane geometry in the book of
volume of ca 600 pages thus embracing practically all the known problems and theorems of
This book contains the basics of linear algebra with an emphasis on nonstandard and neat proofs of known theorems. Many of the theorems of linear algebra obtained mainly during the past 30 years are usually ignored in text-books but are quite accessible for students majoring or minoring in mathematics. These theorems are given with complete proofs. There are about 230 problems with solutions.
This book reports initial efforts in providing some useful extensions in financial
modeling; further work is necessary to complete the research agenda.
The demonstrated extensions in this book in the computation and modeling
of optimal control in finance have shown the need and potential for further
areas of study in financial modeling. Potentials are in both the mathematical
structure and computational aspects of dynamic optimization. There are needs
for more organized and coordinated computational approaches.
When the market is not complete, there is a need to create new securities in order
to complete the market. One approach is to create derivative securities on the existing
securities such as European-type options.
A European call option written on a security gives its holder the right( not obligation)
to buy the underlying security at a prespecied price on a prespecied date; whilst a
European put option written on a security gives its holder the right( not obligation) to
sell the underlying security at a prespecied price on a prespecied date.
This work is intended to survey the basic theory that underlies the multitude of
parameter-rich models that dominate the hydrological literature today. It is concerned
with the application of the equation of continuity (which is the fundamental theorem of
hydrology) in its complete form combined with a simplified representation of the
principle of conservation of momentum. Since the equation of continuity can be
expressed in linear form by a suitable choice of state variables and is also parameterfree,
it can be readily formulated at all scales of interest.
A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle of length at least ﬁve or the complement of one. The “strong perfect graph conjecture” (Berge, 1961) asserts that a graph is perfect if and only if it is Berge. A stronger conjecture was made recently by Conforti, Cornu´jols and Vuˇkovi´ — that every Berge graph either falls into e s c one of a few basic classes, or...
We prove the topological (or combinatorial) rigidity property for real polynomials with all critical points real and nondegenerate, which completes the last step in solving the density of Axiom A conjecture in real one-dimensional dynamics. Contents 1. Introduction 1.1. Statement of results 1.2. Organization of this work 1.3. General terminologies and notation 2. Density of Axiom A follows from the Rigidity Theorem 3. Derivation of the Rigidity Theorem from the Reduced Rigidity Theorem
This article concludes the comprehensive study started in [Sz5], where the ﬁrst nontrivial isospectral pairs of metrics are constructed on balls and spheres. These investigations incorporate four diﬀerent cases since these balls and spheres are considered both on 2-step nilpotent Lie groups and on their solvable extensions. In [Sz5] the considerations are completely concluded in the ball-case and in the nilpotent-case. The other cases were mostly outlined. In this paper the isospectrality theorems are completely established on spheres. ...
Note immediately one diﬀerence between linear equations and polynomial equations:
theorems for linear equations don’t depend on which ﬁeld k you are working over,
but those for polynomial equations depend on whether or not k is algebraically closed
and (to a lesser extent) whether k has characteristic zero. Since I intend to emphasize
the geometry in this course, we will work over algebraically closed ﬁelds for the major
part of the course.
This book is based on lectures delivered over the years by the author at the
Universit´e Pierre et Marie Curie, Paris, at the University of Stuttgart, and at
City University of Hong Kong. Its two-fold aim is to give thorough introductions
to the basic theorems of differential geometry and to elasticity theory in
The treatment is essentially self-contained and proofs are complete.
This book proves a number of important theorems that are commonly
given in advanced books on Commutative Algebra without proof, owing
to the difficulty of the existing proofs. In short, we give homological
proofs of these results, but instead of the original ones involving simplicial
methods, we modify these to use only lower dimensional homology
modules, that we can introduce in an ad hoc way, thus avoiding simplicial
theory. This allows us to give complete and comparatively short
proofs of the important results we state below.