Pages: 548
ISBN: 978-981-98-0726-0 (hardcover)
Metrical Fixed Point Theory, originating from the 1922 Banach Fixed Point Theorem, is one of the most dynamic areas within Operator Equations Theory. This book aims to discuss the foundational aspects of this theory, focusing on questions of existence, uniqueness, and approximation in operator equations ? whether explicit or implicit, anticipative or non-anticipative ? across standard, ordered, and relational metric spaces. Key themes include implicit methods for analyzing metrical contractions, factorial techniques for reducing coincidence point problems to standard fixed point ones, homotopical fixed point results in gauge spaces with ordered metric space parameters, and constant class reduction of PPF-dependent fixed point results.
The book is structured into four chapters. Chapter 1 provides an overview of essential preliminary concepts. Chapter 2 delves into various contraction classes within bi-relational, local Branciari, and ordered metric spaces. Chapter 3 applies maximal techniques to address the discussed questions, and Chapter 4 explores additional topics, including contractive-type conditions derived from self and non-self maps. Through this structure, the book offers a comprehensive view of the core aspects and applications of Metrical Fixed Point Theory.
Preliminaries:
Dependent Choice Principles
Conv-Cauchy Structures
Admissible Functions
Classes of Natural Progressions
Geometric-Asymptotic Relations
Topological Structures
Banach Contractions and Relatives
Classes of Contractive Maps:
Anticipative Rhoades Contractions in bi-Relational Metric Spaces
Functional Contractions in Local Branciari Ordered Metric Spaces
Matkowski Regular Maps in Ordered Metric Spaces
Fixed Points via Multistep Type Iterations
Maximal Techniques:
Caristi-Kirk Results in Banach Lattices
Tarski Fixed Point Results over Separable Structures
Homotopic Metric Interval Maps in Gauge Spaces Varia:
Geometric Functional Contractions in Ordered Quasi-Metric Spaces
Factorial Jungck Contractions in Ordered Metric Spaces
Kannan Spectral Contractive Maps in Relational Metric Spaces
PPF Dependent Fixed Points and the Unexpected Way to Nirvana
Meir-Keeler Maps in Order Pseudometric Spaces
Graduates and researchers in metrical fixed point theory and operator equations.
Pages: 200
ISBN: 978-981-98-0848-9 (hardcover)
ISBN: 978-981-98-0937-0 (softcover)
In China, many excellent mathematics students take an active part in various mathematical contests, and each year, the best six senior high school students are selected to form the IMO National Team to compete in the International Mathematical Olympiad. In the past ten years, China's IMO Team has achieved outstanding results ? they won the first place almost every year.
The authors of this book are esteemed coaches of the China national team, including Xiao Liang, Qu Zhenhua, Xiong Bin, Yu Hongbing, Yao Yijun, Wang Xinmao, Li Ting, Ai Yinghua, Fu Yunhao, Wang Bin, He Yijie, and Zhang Sihui. Contributing to the translation efforts were Chen Haoran and Zhao Wei.
The materials of this book come from a book (in Chinese) on Forward to IMO: A Collection of Mathematical Olympiad Problems (2023). It is a collection of problems and solutions of the major mathematical competitions in China. It provides a glimpse of how the China national team is selected and formed.
China Mathematical Competition (First Round)
China Mathematical Competition (Second Round)
China Mathematical Olympiad
The 21st Chinese Girls' Mathematical Olympiad
The 19th China Southeastern Mathematical Olympiad
China Mathematical Olympiad Hope Alliance Summer Camp
China National Team Selection Test
The 14th Romanian Master of Mathematics Competition
The 12th European Girls' Mathematical Olympiad
International Mathematical Olympiad
Senior high school student and maths teacher; undergraduate of department of mathematics; amateurs interested in mathematics.
Pages: 212
ISBN: 978-981-98-0742-0 (hardcover)
ISBN: 978-981-98-0816-8 (softcover)
Drawing from their collective experience as math enthusiasts, the authors, who are co-founders of Mathematical Circus, have compiled a collection of mathematical activities centered around a standard deck of cards. This book presents a range of self-working card tricks, each rooted in mathematical principles, explained in a clear and straightforward manner. Designed to be both educational and entertaining, the book makes these mathematical concepts accessible to readers of all backgrounds.
Nice and Easy
Counting the Cards
Shuffle and Deal
Secret Codes
Gilbreath Galore
Order in the Ranks!
Short Biographies
High school and university students, math educators at all levels, educational staff at science centers and museums, and magic practitioners and enthusiasts.
Pages: 250
ISBN: 978-981-98-0182-4 (hardcover)
ISBN: 978-981-98-0301-9 (softcover)
Why are there more poor people with small bank accounts than rich people with big bank accounts? Why are there more small planets and stars than big ones in the cosmos? And why are there more small rivers than big rivers, and more harmless tremors than devastating earthquakes? Empirical examinations of real-life data overwhelmingly confirm the existence of such uneven size proportions in favor of the small, leaving us only the question: Why?
This law of nature carries through a vast list of topics and disciplines, confirming it to be nearly universal. In the biological world there are only approximately two million whales and over three billion birds; in number theory there are more small prime numbers than there are big ones; in census data there are more villages than towns, more towns than cities, and more cities than metropolises; in history there have been more minor conflicts than great wars.
In a landmark, first-ever study on the distribution of relative sizes, this book discusses several real-life case studies such as those above in extensive detail, and presents three distinct explanations for the phenomenon. Readers of all disciplines and levels of expertise will find an easily accessible yet original exploration that concludes with a numerical quantification demonstrating precisely by how much the relatively small is more numerous than the relatively big.
Empirical Evidence of the Small is Beautiful Phenomenon:
The Small is Beautiful Phenomenon
Small Life Forms Decisively Outnumber Big Creatures
Small Molecules Outnumber Big Molecules in the Chemical World
Small Atoms Outnumber Big Atoms in the Composition of Chocolate
Slow Spinning Pulsars Outnumber Fast Ones Throughout the Universe
Small Planets Outnumber Big Planets in the Milky Way Galaxy
Small Rivers Outnumber Big Rivers Worldwide
Small Stars Outnumber Big Stars in the Universe
Small Cities and Towns Outnumber Big Metropolises
Small Wars Outnumber Big Wars in History
The Poor Vastly Outnumbers the Rich in Global Wealth Distribution
Small Companies Outnumber Big Corporations
Small Bills Outnumber Big Bills in Revenue and Expense Data
Cheap Items Outnumber Expensive Items in Catalogs and Price Lists
Small House Numbers Outnumber Big House Numbers in Address Data
Global Carbon Dioxide Pollution Data in Favor of the Small
Small Families Outnumber Big Families Regarding Number of Children
Small Meteorites Outnumber Big Meteorites
Small Volcano Eruptions Outnumber Big Volcano Eruptions
Small Countries Outnumber Big Countries
Small Prime Numbers Slightly Outnumber Big Prime Numbers
Four-Dice Multiplication Game at the Casino in Favor of the Small
Three-Dice Selection Game at the Casino in Favor of the Small
Proper Criteria for Sizes Balancing the Chances of Big and Small
Exceptions and Counter Examples to the Phenomenon
Seven-Dice Addition Game at the Casino in Favor of the Medium
Beauty is in the Eye of the Beholder
Causes and Explanations of the Phenomenon:
Partitioning as a Cause of the Small is Beautiful Phenomenon
Random Dependent Partition is Always in Favor of the Small
Random Real Partition is Always in Favor of the Small
Numerous & Distinct Parts in Partitions are Necessary Conditions
Conclusion: Random Partitions and the Small is Beautiful Phenomenon
Multiplication Processes Lead to the Small is Beautiful Phenomenon
Exponential Growth Series and the Small is Beautiful Phenomenon
Explanation of the Small is Beautiful Phenomenon in Growth Series
The Fibonacci Series and the Small is Beautiful Phenomenon
Growth Model for Planets, Stars, Cities, and Bank Accounts
Data Aggregation Leads to the Small is Beautiful Phenomenon
Three-Dice Selection Game as Data Aggregation or Probability Scheme
Thirty-Dice Selection Game is Consistently in Favor of the Small
Chains of Statistical Distributions & the Small is Beautiful Phenomenon
The Prevalence of the Lognormal Distribution Favoring the Small
The Prevalence of the Exponential Distribution Favoring the Small
Discovery of Size Pattern: The General Law of Relative Quantities:
The Quest for Numerical Consistency in All Data Sets
Division of Data along Small, Medium, and Big Sizes
Constructing a Set of Infinitely Expanding Histograms
Numerical Consistency in Bin Schemes for 15 Real-Life Data Sets
The Quest for the General Mathematical Expression for All D&F Cases
The Postulate on Relative Quantities
Application of the Postulate Via Generic Bin Schemes on K/X
The General Law of Relative Quantities (GLORQ)
Saturation in GLORQ-Inducing Processes Precludes Extreme Skewness
GLORQ is Number System Invariant
Benford's Law as Direct Consequence of the General Law:
The Physical Clues Leading to the Discovery of Benford's Law
The First Digit on the Left Side of Numbers
Benford's Law and the Predominance of Low Digits
Empirical Tests and Confirmation of Compliance with Benford's Law
The Theoretical Clues Leading from Benford's Law to GLORQ
The Base Invariance Principle in Benford's Law
The Scale Invariance Principle in Benford's Law
Integral Powers of Ten (IPOT)
Benford's Law as a Special Case and Direct Consequence of GLORQ
Benford & GLORQ as a Subset of the Small is Beautiful Phenomenon
Physical Order of Magnitude of Data (POM)
A Robust Measure of Physical Order of Magnitude (CPOM)
Two Essential Requirements for Benford and GLORQ Behavior
Sum of Squares Deviation Measure (SSD)
A Critique on the Mixture of Distribution Model in Benford's Law
The Random and the Deterministic Flavors in Benford's Law
Tugs of War between Addition and Multiplication of Random Variables
Quantitative Partition Models and Benford's Law
Chains of Statistical Distributions are Nearly Always Benford
Development Pattern within the Expanding Histograms of Benford
Appendices:
Conceptual Explanation of Histograms
Conceptual Explanation of Powers and Logarithms
Glossary of Frequently Used Abbreviations
Bibliography
The book is suitable for professionals and experts in the fields of mathematics, statistics, and the sciences, as well as university students of these disciplines. It is also suitable for the interested layman, non-expert, and educated general public.
Pages: 464
ISBN: 978-981-98-0810-6 (hardcover)
This volume is a reprint of the special issue of the Journal of Algebra and its Applicationstitled Recent Advances in Non-Commutative Ring Theory. It features twenty-four articles highlighting the latest developments in noncommutative rings, modules over them, and applications to algebraic coding theory. The aim of publishing this special issue in book form is to provide a comprehensive resource on the current state-of-the-art in noncommutative ring theory, offering valuable insights to emerging researchers in the field.
Part-1:
Representability of Relatively Free Affine Algebras over a Noetherian Ring (Alexei Kanel-Belov, Louis Rowen, and Uzi Vishne)
The Question of Arnold on Classification of Co-artin Subalgebras in Singularity Theory (V V Bavula)
Structural Properties of The Quantized Matrix Algebra Dq(n) established by Means of Grobner-Shirshov Basis Theory (Lina Niu and Rabigul Tuniyaz)
On Some Notions of Algebraically Closed ƒÐ-fields (Masood Aryapoor)
On Three Types of Galois Extensions which are Azumaya (Lianyong Xue)
Multiplicative Lattices, Idempotent Edomorphisms, and Left Skew Braces (Alberto Facchini and Mara Pompili)
On Some Generalizations of Abelian Rings (T Y Lam)
E-solid Rings, Strongly IC Rings and the Jacobson Radical (Xavier Mary)
Strongly ƒÎ-regular Elements and Drazin-Inverses (Dimple Rani Goyal, Anjana Khurana, and Dinesh Khurana)
Units and Linear Independence (Sergio R Lopez-Permouth, Jeremy Moore, Nick Pilewski, and Steve Szabo)
Fixed Points and Orbits in Skew Polynomial Rings (Adam Chapman and Elad Paran)
On the Structure of Ideals in a Family of Skew Polynomial Rings (Ehsan Shahoseini, Reza Dastbasteh, Hai Q Dinh, and Hamed Mousavi)
Part-2:
Small Sets without Unique Products in Torsion-free Groups (Pace P Nielsen and Lindsay Soelberg)
Generating Symmetric and Alternating Groups (Stefan Catoiu and R?zvan Diaconescu)
Abelian Groups with C2 (Stefan Catoiu and R?zvan Diaconescu)
Part-3:
A Characterization of Goldie Extending Trivial Morita Context (Sara Moradiani, Ahmad Moussavi, and Masoome Zahiri)
On Hypercyclic Rings (Christian Lomp, Mohamed Yousif, and Yiqiang Zhou)
When are Baer Modules Extending? (Fatma Azmy F Ebrahim, Syed Tariq Rizvi, and Cosmin S Roman)
Relatively Polyform Modules (Xiaoxiang Zhang, Gangyong Lee, and Nguyen Khanh Tung)
Part-4:
On Skew-Commuting Generalized Skew Derivations in Prime and Semiprime Rings (Luisa Carini and Vincenzo De Filippis)
Jordan-type Derivations on Trivial Extension Algebras (Mohammad Ashraf, Md Shamim Akhter, and Mohammad Afajal Ansari)
On ă-k-skew Lie Product involving Commuting Additive Maps on Prime *-Algebras (Abdul Nadim Khan, Shakir Ali, Mohd Arif Raza, and Husain Alhazmi)
Part-5:
Crossed Product Codes (A Duarte, A Pereira, and C Polcino Milies)
On Constacyclic Codes of length 9ps over ??pm and their Optimal Codes (Hai Q Dinh, Hieu V Ha, Nhan T V Nguyen, and Nghia T H Tran)
Researchers working in noncommutative ring theory.