Aug 2024
Editors:
Claudio Bonanno, Alfonso Sorrentino, Corinna Ulcigrai
Provides an overview of some modern aspects of the theory of dynamical systems
Written by world's leading experts in the field
Gives special emphasis to the connections with other areas of mathematical research
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2347)
Part of the book sub series: C.I.M.E. Foundation Subseries (LNMCIME)
This book provides an overview of recent advances in the theory of dynamical systems, with a particular emphasis
on their connections to other areas of mathematical research, including number theory, geometry, mathematical
physics, complex analysis, and celestial mechanics. Compiling the lecture notes from some of the contributions
presented at the C.I.M.E. school "Modern Aspects of Dynamical Systems" held in Cetraro in August 2021,
the contributions are the following: omogeneous dynamics and Diophantine problems by Manfred Einsiedler, ffective
ergodic theory for translation flowby Giovanni Forni, ntegrability and rigidity for convex billiards by Vadim Kaloshin,
olomorphic dynamics?Eby Jasmin Raissy and xponentially small phenomena and its role in the dynamics?Eby
Tere Martinez-Seara.
These notes are suitable for graduate students and young researchers interested in an introduction to some of
the modern research areas within the field of dynamical systems.
Textbook
Aug 2024
Studies Diophantine equations systematically, in order
Hundreds of equations are left as exercises, solutions provided
Contains a large collection of open problems of all levels of difficulty
This book proposes a novel approach to the study of Diophantine equations: define an appropriate version of
the equation size, order all polynomial Diophantine equations by size, and then solve the equations in order.
Natural questions about the solution set of Diophantine equations are studied in this book using this approach.
Is the set empty? Is it finite or infinite? Can all integer solutions be parametrized? By ordering equations by size,
the book attempts to answer these questions in a systematic manner. When the size grows, the difficulty
of finding solutions increases and the methods required to determine solutions become more advanced.
Along the way, the reader will learn dozens of methods for solving Diophantine equations, each of which is
illustrated by worked examples and exercises. The book ends with solutions to exercises and a large collection
of open problems, often simple to write down yet still unsolved.
The original approach pursued in this book makes it widely accessible. Many equations require only high school
mathematics and creativity to be solved, so a large part of the book is accessible to high school students,
especially those interested in mathematical competitions such as olympiads. The main intended audience is
undergraduate students, for whom the book will serve as an unusually rich introduction to the topic of
Diophantine equations. Many methods from the book will be useful for graduate students, while Ph.D. students
and researchers may use it as a source of fascinating open questions of varying levels of difficulty.
Textbook
Aug 2024
Written clearly to make complex mathematics accessible
It provides many examples and solved exercises
A rigorous, complete and self-contained presentation
Part of the book series: UNITEXT (UNITEXT, volume 165)
Part of the book sub series: La Matematica per il 3+2 (UNITEXTMAT)
This book provides a concise yet rigorous introduction to probability theory. Among the possible approaches
to the subject, the most modern approach based on measure theory has been chosen: although it requires
a higher degree of mathematical abstraction and sophistication, it is essential to provide the foundations
for the study of more advanced topics such as stochastic processes, stochastic differential calculus
and statistical inference. The text originated from the teaching experience in probability and applied mathematics
courses within the mathematics degree program at the University of Bologna; it is suitable for second- or
third-year students in mathematics, physics, or other natural sciences, assuming multidimensional differential
and integral calculus as a prerequisite. The four chapters cover the following topics: measures and probability spaces;
random variables; sequences of random variables and limit theorems; and expectation and conditional distribution.
The text includes a collection of solved exercises.
Textbook
Aug 2024
Rigorous, comprehensive, and self-contained presentation
Written clearly to make complex mathematics accessible
Comprehensive overview of stochastic process theory with brief mentions of its most significant applications
Part of the book series: UNITEXT (UNITEXT, volume 166)
Part of the book sub series: La Matematica per il 3+2 (UNITEXTMAT)
This book offers a modern approach to the theory of continuous-time stochastic processes and stochastic calculus.
The content is treated rigorously, comprehensively, and independently. In the first part, the theory of Markov processes
and martingales is introduced, with a focus on Brownian motion and the Poisson process. Subsequently, the theory of
stochastic integration for continuous semimartingales was developed. A substantial portion is dedicated to stochastic
differential equations, the main results of solvability and uniqueness in weak and strong sense, linear stochastic equations,
and their relation to deterministic partial differential equations. Each chapter is accompanied by numerous examples.
This text stems from over twenty years of teaching experience in stochastic processes and calculus within master's
degrees in mathematics, quantitative finance, and postgraduate courses in mathematics for applications and mathematical
finance at the University of Bologna. The book provides material for at least two semester-long courses in scientific
studies (Mathematics, Physics, Engineering, Statistics, Economics, etc.) and aims to provide a solid background for
those interested in the development of stochastic calculus theory and its applications. This text completes the journey
started with the first volume of Probability Theory I - Random Variables and Distributions, through a selection of
advanced classic topics in stochastic analysis.
Textbook
Oct 2024
Provides a unified introduction to elementary and analytic number theory
Includes algorithms in Python for a computational approach to number theory
Contains over 200 exercises with hints
Part of the book series: Springer Undergraduate Mathematics Series (SUMS)
This introductory text is designed for undergraduate courses in number theory, covering both elementary number
theory and analytic number theory. The book emphasises computational aspects, including algorithms and their
implementation in Python.
The book is divided into two parts. The first part, on elementary number theory, deals with concepts such as induction,
divisibility, congruences, primitive roots, cryptography, and continued fractions. The second part is devoted to analytic
number theory and includes chapters on Dirichlet theorem on primes in arithmetic progressions, the prime number
theorem, smooth numbers, and the famous circle method of Hardy and Littlewood. The book contains many topics not
often found in introductory textbooks, such as Aubry theorem, the Tonelli hanks algorithm, factorisation methods,
continued fraction representations of e, and the irrationality of (3). Each chapter concludes with a summary and notes,
as well as numerous exercises.
Assuming only basic calculus for the first part of the book, the second part assumes some knowledge of complex analysis.
Familiarity with basic coding syntax will be helpful for the computational exercises.
Textbook
Nov 2024
Offers a thorough exploration of discrete-time martingale theory, suitable for master's students
Presents complex concepts in a clear and understandable manner with examples and exercises for active engagement
Emphasizes applications in different areas such as urn models, CLT, and branching processes
Part of the book series: Texts and Readings in Mathematics (TRIM, volume 86)
This concise textbook, fashioned along the syllabus for master and Ph.D. programmes, covers basic results on
discrete-time martingales and applications. It includes additional interesting and useful topics, providing the ability
to move beyond. Adequate details are provided with exercises within the text and at the end of chapters.
Basic results include Doob optional sampling theorem, Wald identities, Doob maximal inequality, upcrossing lemma,
time-reversed martingales, a variety of convergence results and a limited discussion of the Burkholder inequalities.
Applications include the 0-1 laws of Kolmogorov and Hewitt avage, the strong laws for U-statistics and exchangeable
sequences, De Finetti theorem for exchangeable sequences and Kakutani theorem for product martingales.
A simple central limit theorem for martingales is proven and applied to a basic urn model, the trace of a random
matrix and Markov chains. Additional topics include forward martingale representation for U-statistics, conditional Borel
antelli lemma, Azumaoeffding inequality, conditional three series theorem, strong law for martingales and the Kesten
tigum theorem for a simple branching process. The prerequisite for this course is a first course in measure theoretic
probability. The book recollects its essential concepts and results, mostly without proof, but full details have been
provided for the Radon ikodym theorem and the concept of conditional expectation.
June 2024
Pages: 400
ISBN: 978-981-12-8723-7 (hardcover)
ISBN: 978-981-12-8743-5 (softcover)
The series is edited by the head coaches of China's IMO National Team. Each volume, catering to different grades,
is contributed by the senior coaches of the IMO National Team. The Chinese edition has won the award of Top 50
Most Influential Educational Brands in China.
The series is created in line with the mathematics cognition and intellectual development levels of the students
in the corresponding grades. All hot mathematics topics of the competition are included in the volumes and are
organized into chapters where concepts and methods are gradually introduced to equip the students with necessary
knowledge until they can finally reach the competition level.
In each chapter, well-designed problems including those collected from real competitions are provided so that
the students can apply the skills and strategies they have learned to solve these problems. Detailed solutions are
provided selectively. As a feature of the series, we also include some solutions generously offered by the members
of Chinese national team and national training team.
Linear Equations with Absolute Values
Linear Inequalities with Absolute Values
Polynomial Factorization (I)
Polynomial Factorization (II)
Calculation of Rational Fractions
Partial Fractions
Polynomial Equations and Fractional Equations with Unknown Constants
Real Numbers
Quadratic Radicals
Evaluating Algebraic Expressions
Symmetric Polynomials
Proving Identities
Linear Functions
Inversely Proportional Functions
Statistics
The Sides and Angles of a Triangle
Congruent Triangles
Isosceles Triangles
Right Triangles
Parallelograms
Trapezoids
The Angles and Diagonals of a Polygon
Proportion of Segments
Similar Triangles
The Midsegment
Translation and Symmetry
The Area
Secondary school students engaged in mathematical competition, coaches
in mathematics
teaching, and teachers setting up math elective courses.
Pages: 384
ISBN: 978-981-12-8665-0 (hardcover)
ISBN: 978-981-12-8729-9 (softcover)
This book provides a comprehensive introduction to quantum mechanics from the ground up.
It is designed to be completely self-contained and assumes very little knowledge or mathematical
background on the part of students as it takes them through the major topics of quantum mechanics.
Designed to be appropriate for students across a wide range of abilities and backgrounds, this book will be
particularly helpful for students who might lack some of the mathematical background typically assumed in
an undergraduate quantum mechanics course. The book includes three "math interludes" covering such
topics as complex numbers, linear operators, vector spaces, and matrix manipulation. The book also discusses
some interesting modern applications of quantum mechanics: magnetic resonance imaging and quantum computing,
and it concludes with an introduction to relativistic quantum theory.
This second edition includes expanded and improved coverage of the Heisenberg uncertainty principle,
the use of ladder operators to solve the harmonic oscillator, as well as the treatment of the Lamb shift.
The Origins of Quantum Mechanics
Math Interlude A:
Numbers and Linear Operators
The Schrodinger Equation
Solutions of the One-Dimensional Time-Independent Schrodinger Equation
Math Interlude B: Linear Algebra
Solutions of the Three-Dimensional Time-Independent Schrodinger Equation
Math Interlude C: Matrices, Dirac Notation, and the Dirac Delta Function
Spin Angular Momentum
Time-Independent Perturbation Theory
The Variational Principle
Time-Dependent Perturbation Theory
Scattering Theory
Multiparticle Schrodinger Equation
Modern Applications of Quantum Mechanics
Relativistic Quantum Mechanics
Undergraduate students, useful for all Physics majors and some Engineering
majors; Educators of undergrad students.