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A tribute to Paul Erdos, the wandering mathematician once described as the prince of problem solvers & the absolute monarch of problem posers, examines within the context of his personality & lifestyle the legacy of open problemshe left to the world of mathematics after his death in 1996.DLC: Graph theory. ... Read more

Customer Reviews (2)

Immediate throw away for non-professional Mathematicians
I got a book by Ron Graham and Fan Chung.
It is amazingly bad.
It is a book on graphs without one graph diagram
or graph matrix.
I buy these books for learning.
I'm left with not being able to give an honest review
because it would be politically suicide in Mathematics to
cross Ron Graham who has been president of both Mathematics societies.
The book is a very badly written one, too,
with a few mostly ancient( from as far back as the 1930's )
problems in graph theory.
I have to go to MathWorld to get an idea of what the graphs might look like!

Since it appears that the graphs are the dog that wags the matrices,
I thought I might get further with some better understanding.
It appears this book is the wrong place to get it.

One reason as I see it that Fan Chung and Ron Graham
don't answer emails much is that they really aren't all that nice, good or smart:
their "meal ticket" was Erdos and he is dead now.
That's not a "pretty" conclusion or one I wanted, you know?
Tattle tale stories about the great man aren't going to save this one.

Even for me this may take years to get my money's worth out of.
I looked at the review by Charles Ashbacher and I had to set the record straight.
He is a friend of a friend who is also into integer sequences.
Somebody has to be honest here.

Only one of many such books that could be written
What is incredible about this book is not that there are over one hundred pages of unsolved problems posed all or in part by Paul Erdos. The amazing thing is that the word graphs could be replaced by several other mathematical words or phrases and a similar book could be written. Erdos was an expert in so many areas of mathematics and perhaps his greatest ability was in putting forward just the right questions to just the right people. There is very little explanation of the problem proposals, the authors rely on a great deal of the listing of references to fill in the details. Therefore, anyone interested in exploring the problems in greater detail should be prepared to spend some time and effort in tracking down the relevant articles. Fortunately, the authors themselves did a great deal of that, as there are complete references for every problem that appears.
The range of problems is a demonstration of the depth of his understanding of graph theory, and also a demonstration of how little is still unresolved. I put forward no pretense to understanding any more than a few of the problems in this book. However, that did not alter my interest in the problems, as I was able to understand the fundamentals of almost all of them. Reading this book is one of the most educational experiences that I have had in the past year and I encourage all mathematicians at the level of slightest interest in graph theory and above to read it.

Published in Journal of Recreational Mathematics, reprinted with permission. ... Read more

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The paperback of the critically-acclaimed popular science book by a writer who is fast becoming a celebrity mathematician.Prime numbers are the very atoms of arithmetic. They also embody one of the most tantalising enigmas in the pursuit of human knowledge. How can one predict when the next prime number will occur? Is there a formula which could generate primes? These apparently simple questions have confounded mathematicians ever since the Ancient Greeks.In 1859, the brilliant German mathematician Bernard Riemann put forward an idea which finally seemed to reveal a magical harmony at work in the numerical landscape. The promise that these eternal, unchanging numbers would finally reveal their secret thrilled mathematicians around the world. Yet Riemann, a hypochondriac and a troubled perfectionist, never publicly provided a proof for his hypothesis and his housekeeper burnt all his personal papers on his death.Whoever cracks Riemann's hypothesis will go down in history, for it has implications far beyond mathematics. In business, it is the lynchpin for security and e-commerce.In science, it has critical ramifications in Quantum Mechanics, Chaos Theory, and the future of computing. Pioneers in each of these fields are racing to crack the code and a prize of $1 million has been offered to the winner. As yet, it remains unsolved.In this breathtaking book, mathematician Marcus du Sautoy tells the story of the eccentric and brilliant men who have struggled to solve one of the biggest mysteries in science. It is a story of strange journeys, last-minute escapes from death and the unquenchable thirst for knowledge. Above all, it is a moving and awe-inspiring evocation of the mathematician's world and the beauties and mysteries it contains. ... Read more

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A page-tunring who's not dunnit
The Music Of The Primes by Marcus du Sautoy is not a book for the faint-hearted. The author may be a populariser of mathematics, but certainly in this book there is plenty of substance that would maintain the interest of the specialist and also enough technicality to cause the general reader to pause. The book is a brilliant piece of work, however, so all must resist any temptation to skip. The Music Of The Primes is a glittering account, superbly paced, of an unfinished story. From the very first page it demands to be read, so much so that like me you will probably finish it in two sittings at most.

Marcus du Sautoy regularly refers to prime numbers as the atoms of our number system. I have some reservations with this metaphor, but I was willing to live with it. For the uninitiated, a prime number has just two factors, one and itself. It cannot be exactly divided by anything else. The list begins 2, 3, 5, 7, 11, 13 and continues ad infinitum. The sieve of Eratosthenes established that fact a couple of thousand years ago. But, despite many lifetimes of trying, we have never successfully been able to predict whether a particular number would be prime, or conversely, exactly where the next prime number might be. They seem to be distributed randomly throughout our number system, all odd except for that initial 2, the odd-one-out pair that spoils it for every other even.

The Music Of The Primes relates how mathematicians have closed in on the mystery of how these numbers occur without, as yet, managing to crack the complete code.Marcus du Sautoy describes some of the great contributions to the understanding of prime numbers. The names Fermat, Gauss and Euler figure regularly. But it is the great name of Riemann that emerges as the lynchpin of this story, his Conjecture being the unsolved problem that currently occupies many a brain, the one million dollars in prize money offered for its solution oiling the machinations.

Riemann turned the search for prime numbers on its head when he used complex numbers to reposition the problem. Complex numbers, by the way, are at least in part imaginary and, though they don't exist, no self-supporting bridge would stand up without them. His now famous Conjecture was that evidence of the existence of prime numbers would line up in a predictable way in a four-dimensional space created when one two-dimensional complex number was plotted against another, the latter being the solution to a particular power series called a zeta function equated to zero. His problem was that he couldn't prove that things lined up in precisely the way he predicted. He had strong hunches that he was right, but, lacking proof, a conjecture is what it remained. And people have been trying to prove it for a century and a half.

Prime numbers are now big business, of course. Public-private key encryption now oils the wheels of internet commerce and the security it offers is based on the possession of quite huge, quite astronomically large prime numbers. Find a few new ones and you could make a very good living. If you want to taste the complexity of the task, then spend no more than five minutes finding the two tree-digit factors of 8051. Imagine then the work involved in identifying two 200 digit prime numbers that combine to a Rivest, Shamir and Adelman security key. Reading this superb book will provide further insight.

It will also illustrate very well the value of pure research conducted by specialist academics. When the accountants complain that programmes have no apparent immediate application, it is worth remembering how advances in human knowledge made over two hundred years ago are only just finding wide application in fields completely unenvisaged by their inventors. Without the knowledge they developed in their apparent vacuum, of course, the modern-day application may never have been conceived. Just imagine where the human race might be two centuries from now if Kurt Gödel's ideas have become the basis for all mathematics. Read this book and then imagine.
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This volume is produced from digital images created through the University of Michigan University Library's large-scale digitization efforts. The Library seeks to preserve the intellectual content of items in a manner that facilitates and promotes a variety of uses. The digital reformatting process results in an electronic version of the original text that can be both accessed online and used to create new print copies. The Library also understands and values the usefulness of print and makes reprints available to the public whenever possible. This book and hundreds of thousands of others can be found in the HathiTrust, an archive of the digitized collections of many great research libraries. For access to the University of Michigan Library's digital collections, please see http://www.lib.umich.edu and for information about the HathiTrust, please visit http://www.hathitrust.org ... Read more

Customer Reviews (1)

A collection of difficult problems that will test you
This book was one of the first of the collections of problems by Florentin Smarandache that I read. As the title implies, it contains only a listing of problems, with solutions the province of the reader. I spent a great deal of time working with these problems and managed to make progress on several of them. Like all good mathematics problems, complete solutions are elusive, difficult and fun to pursue. I enjoyed them so much that it was part of the motivation for me to write three books on the problems posed by Smarandache, which are known as Smarandache notions.
The problems are primarily in the area of number theory, which as the luminary Paul Erdos often said, makes them easy to state and understand, but hard to prove. If you are interested in problems that will frustrate and fascinate you, check out the ones in this book. ... Read more

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Second edition sold 2241 copies in N.A. and 1600 ROW.

New edition contains 50 percent new material.

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Many of the Smarandache Notions translated into Chinese by a group of university students. ... Read more

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Chapters: P Versus Np Problem, Aanderaa-karp-rosenberg Conjecture, Computational Complexity of Mathematical Operations, One-Way Function, Unique Games Conjecture, Computational Complexity of Matrix Multiplication, P = Bpp Problem, P = Pspace Problem, Barendregt-Geuvers-Klop Conjecture, Nc = P Problem, Np = Co-Np Problem. Source: Wikipedia. Pages: 80. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: The relationship between the complexity classes P and NP is an unsolved problem in theoretical computer science, and is considered by many theoretical computer scientists to be the most important problem in the field. The Clay Mathematics Institute, which is dedicated to increasing and disseminating mathematical knowledge, has included it in its list of Millennium Prize Problems; anyone who provides a satisfactory solution to the problem may be entitled to a million dollar prize. In essence, the question P = NP? asks: if 'yes'-answers to a 'yes'-or-'no'-question can be verified "quickly" can the answers themselves also be computed "quickly"? The theoretical notion of "quick" used here is that of an algorithm that runs in polynomial time, which usually but not always corresponds to an algorithm that is fast in practice. Consider the subset sum problem, an example of a problem which is easy to verify but whose answer is suspected to be theoretically difficult to compute. Given a set of integers, does some nonempty subset of them sum to 0? For instance, does a subset of the set add up to 0? The answer "yes, because add up to zero", can be quickly verified with three additions. However, finding such a subset in the first place could take more time. The information needed to verify a positive answer is also called a certificate. Given the right certificates, "yes" answers to our problem can be verified ...More: http://booksllc.net/?id=6115 ... Read more

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Purchase includes free access to book updates online and a free trial membership in the publisher's book club where you can select from more than a million books without charge. Chapters: Accretion Disc, Ball Lightning, Magnetic Monopole, List of Unsolved Problems in Physics, Pioneer Anomaly, Cp Violation, Flyby Anomaly, Unified Field Theory, Introduction to M-Theory, Vacuum Catastrophe, Fredkin Finite Nature Hypothesis, Hessdalen Light. Excerpt:Artist's conception of a binary star system with one black hole and one main sequence star Accretion disc jets: Why do the discs surrounding certain objects, such as the nuclei of active galaxies , emit radiation jets along their polar axes? These jets are invoked by astronomers to do everything from getting rid of angular momentum in a forming star to reionizing the universe (in AGNs ), but their origin is still not understood. : An accretion disc is a structure (often a circumstellar disk ) formed by diffuse material in orbital motion around a central body. The central body is typically a young star, a protostar , a white dwarf , a neutron star , or a black hole . Gravity causes material in the disc to spiral inward towards the central body. Gravitational forces compress the material causing the emission of electromagnetic radiation. The frequency range of that radiation depends on the central object. Accretion discs of young stars and protostars radiate in the infrared; those around neutron stars and black holes in the x-ray part of the spectrum. Accretion disc physics In the 1940s, models were first derived from basic physical principles. In order to agree with observations, those models had to invoke a yet unknown mechanism for angular momentum redistribution. If matter is to fall inwards it must lose not only gravitational energy but also lose angular momentum . Since the total angular momentum of the disc is conserved, the angular momentum loss of the mass falling into the center has to be compensated by an angul... ... Read more

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This digital document is an article from Science and Its Times, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 4046 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.The histories of science, technology, and mathematics merge with the study of humanities and social science in this interdisciplinary reference work. Essays on people, theories, discoveries, and concepts are combined with overviews, bibliographies of primary documents, and chronological elements to offer students a fascinating way to understand the impact of science on the course of human history and how science affects everyday life. Entries represent people and developments throughout the world, from about 2000 B.C. through the end of the twentieth century. ... Read more

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New improved results of the research in Chinese language on Smarandache s codification used in computer programming, smarandacheials, totient and congruence functions, sequences, irrational constants in number theory, but also multi-spaces and geometries. ... Read more

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This scarce antiquarian book is a selection from Kessinger Publishing's Legacy Reprint Series. Due to its age, it may contain imperfections such as marks, notations, marginalia and flawed pages. Because we believe this work is culturally important, we have made it available as part of our commitment to protecting, preserving, and promoting the world's literature. Kessinger Publishing is the place to find hundreds of thousands of rare and hard-to-find books with something of interest for everyone! ... Read more