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Triangular arrays are a unifying thread throughout various areas of
discrete mathematics such as number theory and combinatorics. They
can be used to sharpen a variety of mathematical skills and tools,
such as pattern recognition, conjecturing, proof-techniques, and
problem-solving techniques.
While a good deal of research exists concerning triangular arrays
and their applications, the information is scattered in various
journals and is inaccessible to many mathematicians. This is the
first text that will collect and organize the information and
present it in a clear and comprehensive introduction to the topic.
An invaluable resource book, it gives a historical introduction to
Pascal's triangle and covers application topics such as binomial
coefficients, figurate numbers, Fibonacci and Lucas numbers, Pell
and Pell-Lucas numbers, graph theory, Fibonomial and tribinomial
coefficients and Fibonacci and Lucas polynomials, amongst others.
The book also features the historical development of triangular
arrays, including short biographies of prominent mathematicians,
along with the name and affiliation of every discoverer and year of
discovery. The book is intended for mathematicians as well as
computer scientists, math and science teachers, advanced high
school students, and those with mathematical curiosity and
maturity.
Like the intriguing Fibonacci and Lucas numbers, Catalan numbers
are also ubiquitous. "They have the same delightful propensity for
popping up unexpectedly, particularly in combinatorial problems,"
Martin Gardner wrote in Scientific American. "Indeed, the Catalan
sequence is probably the most frequently encountered sequence that
is still obscure enough to cause mathematicians lacking access to
Sloane's Handbook of Integer Sequences to expend inordinate amounts
of energy re-discovering formulas that were worked out long ago,"
he continued. As Gardner noted, many mathematicians may know the
abc's of Catalan sequence, but not many are familiar with the
myriad of their unexpected occurrences, applications, and
properties; they crop up in chess boards, computer programming, and
even train tracks. This book presents a clear and comprehensive
introduction to one of the truly fascinating topics in mathematics.
Catalan numbers are named after the Belgian mathematician Eugene
Charles Catalan (1814-1894), who "discovered" them in 1838, though
he was not the first person to discover them. The great Swiss
mathematician Leonhard Euler (1707-1763) "discovered" them around
1756, but even before then and though his work was not known to the
outside world, Chinese mathematician Antu Ming (1692?-1763) first
discovered Catalan numbers about 1730. A great source of fun for
both amateurs and mathematicians, they can be used by teachers and
professors to generate excitement among students for exploration
and intellectual curiosity and to sharpen a variety of mathematical
skills and tools, such as pattern recognition, conjecturing,
proof-techniques, and problem-solving techniques. This book is not
intended for mathematicians only but for a much larger audience,
including high school students, math and science teachers, computer
scientists, and those amateurs with a modicum of mathematical
curiosity. An invaluable resource book, it contains an intriguing
array of applications to computer science, abstract algebra,
combinatorics, geometry, graph theory, chess, and world series.
Praise for the First Edition beautiful and well worth the reading
with many exercises and a good bibliography, this book will
fascinate both students and teachers. Mathematics Teacher Fibonacci
and Lucas Numbers with Applications, Volume I, Second Edition
provides a user-friendly and historical approach to the many
fascinating properties of Fibonacci and Lucas numbers, which have
intrigued amateurs and professionals for centuries. Offering an
in-depth study of the topic, this book includes exciting
applications that provide many opportunities to explore and
experiment. In addition, the book includes a historical survey of
the development of Fibonacci and Lucas numbers, with biographical
sketches of important figures in the field. Each chapter features a
wealth of examples, as well as numeric and theoretical exercises
that avoid using extensive and time-consuming proofs of theorems.
The Second Edition offers new opportunities to illustrate and
expand on various problem-solving skills and techniques. In
addition, the book features: A clear, comprehensive introduction to
one of the most fascinating topics in mathematics, including links
to graph theory, matrices, geometry, the stock market, and the
Golden Ratio Abundant examples, exercises, and properties
throughout, with a wide range of difficulty and sophistication
Numeric puzzles based on Fibonacci numbers, as well as popular
geometric paradoxes, and a glossary of symbols and fundamental
properties from the theory of numbers A wide range of applications
in many disciplines, including architecture, biology, chemistry,
electrical engineering, physics, physiology, and neurophysiology
The Second Edition is appropriate for upper-undergraduate and
graduate-level courses on the history of mathematics,
combinatorics, and number theory. The book is also a valuable
resource for undergraduate research courses, independent study
projects, and senior/graduate theses, as well as a useful resource
for computer scientists, physicists, biologists, and electrical
engineers. Thomas Koshy, PhD, is Professor Emeritus of Mathematics
at Framingham State University in Massachusetts and author of
several books and numerous articles on mathematics. His work has
been recognized by the Association of American Publishers, and he
has received many awards, including the Distinguished Faculty of
the Year. Dr. Koshy received his PhD in Algebraic Coding Theory
from Boston University. Anyone who loves mathematical puzzles,
number theory, and Fibonacci numbers will treasure this book. Dr.
Koshy has compiled Fibonacci lore from diverse sources into one
understandable and intriguing volume, [interweaving] a historical
flavor into an array of applications. Marjorie Bicknell-Johnson
Pell and Pell-Lucas numbers, like the well-known Fibonacci and
Catalan numbers, continue to intrigue the mathematical world with
their beauty and applicability. They offer opportunities for
experimentation, exploration, conjecture, and problem-solving
techniques, connecting the fields of analysis, geometry,
trigonometry, and various areas of discrete mathematics, number
theory, graph theory, linear algebra, and combinatorics. Pell and
Pell-Lucas numbers belong to an extended Fibonacci family as a
powerful tool for extracting numerous interesting properties of a
vast array of number sequences. A key feature of this work is the
historical flavor that is interwoven into the extensive and
in-depth coverage of the subject. An interesting array of
applications to combinatorics, graph theory, geometry, and
intriguing mathematical puzzles is another highlight engaging the
reader. The exposition is user-friendly, yet rigorous, so that a
broad audience consisting of students, math teachers and
instructors, computer scientists and other professionals, along
with the mathematically curious will all benefit from this book.
Finally, Pell and Pell-Lucas Numbers provides enjoyment and
excitement while sharpening the reader's mathematical skills
involving pattern recognition, proof-and-problem-solving
techniques.
Volume II provides an advanced approach to the extended gibonacci
family, which includes Fibonacci, Lucas, Pell, Pell-Lucas,
Jacobsthal, Jacobsthal-Lucas, Vieta, Vieta-Lucas, and Chebyshev
polynomials of both kinds. This volume offers a uniquely unified,
extensive, and historical approach that will appeal to both
students and professional mathematicians. As in Volume I, Volume II
focuses on problem-solving techniques such as pattern recognition;
conjecturing; proof-techniques, and applications. It offers a
wealth of delightful opportunities to explore and experiment, as
well as plentiful material for group discussions, seminars,
presentations, and collaboration. In addition, the material covered
in this book promotes intellectual curiosity, creativity, and
ingenuity. Volume II features: A wealth of examples, applications,
and exercises of varying degrees of difficulty and sophistication.
Numerous combinatorial and graph-theoretic proofs and techniques. A
uniquely thorough discussion of gibonacci subfamilies, and the
fascinating relationships that link them. Examples of the beauty,
power, and ubiquity of the extended gibonacci family. An
introduction to tribonacci polynomials and numbers, and their
combinatorial and graph-theoretic models. Abbreviated solutions
provided for all odd-numbered exercises. Extensive references for
further study. This volume will be a valuable resource for
upper-level undergraduates and graduate students, as well as for
independent study projects, undergraduate and graduate theses. It
is the most comprehensive work available, a welcome addition for
gibonacci enthusiasts in computer science, electrical engineering,
and physics, as well as for creative and curious amateurs.
This is a student solutions manual for Elementary Number Theory
with Applications 1st edition by Thomas Koshy (2002). Note that the
textbook itself is not included in this purchase. From the back
cover of the textbook: Modern technology has brought a new
dimension to the power of number theory: constant practical use.
Once considered the purest of pure mathematics, number theory has
become an essential tool in the rapid development of technology in
a number of areas, including art, coding theory, cryptology, and
computer science. The range of fascinating applications confirms
the boundlessness of human ingenuity and creativity. Elementary
Number Theory captures the author's fascination for the subject:
its beauty, elegance, and historical development, and the
opportunities number theory provides for experimentation,
exploration, and, of course, its marvelous applications.
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