Product Description
This book contains discussions of hundreds of open questions in number theory, organized into 185 different topics. They represent numerous aspects of number theory and are organized into six categories: prime numbers, divisibility, additive number theory, Diophantine equations, sequences of integers, and miscellaneous. To prevent repetition of earlier efforts or duplication of previously known results, an extensive and up-to-date collection of references follows each problem. In the second edition, extensive new material has been added, and corrections have been included throughout the book. This volume is an invaluable supplement to any course in number theory. ... Read more

Customer Reviews (2)

An excursion into the labyrinths carved by numbers
Number theory is the most enigmatic of disciplines, in that the problems are so easy to state and understand and yet so hard to resolve. Furthermore, when solved, the proofs are sometimes surprisingly easy. Inthis collection, Guy has put together a truly fascinating survey of what iscurrently (un)known about numbers.
Each page is an excursion into theextensive labyrinths carved out by numbers. Approximately once a month, Iscan it looking for new avenues to explore. Invariably, I see something,sketch out some possible proof routes and then end in frustration. Atypical result of working in number theory.
Whether you are anamateur or professional, if you have an interest in number theory, you willlike this book. Perhaps you will be able to make some progress towardsresolving some of these problems. It is certainly possible, as no field hashad more positive contributions from amateurs than number theory. EvenFermat fit the definition of an amateur.

Outstanding source of intersting problems in number theory
Another of my 10 favorite books.A constant source of inspiration ... Read more

Product Description
Victor Klee and Stan Wagon discuss some of the unsolved problems in number theory and geometry, many of which can be understood by readers with a very modest mathematical background. The presentation is organized around 24 central problems, many of which are accompanied by other, related problems. The authors place each problem in its historical and mathematical context, and the discussion is at the level of undergraduate mathematics. Each problem section is presented in two parts. The first gives an elementary overview discussing the history and both the solved and unsolved variants of the problem. The second part contains more details, including a few proofs of related results, a wider and deeper survey of what is known about the problem and its relatives, and a large collection of references. Both parts contain exercises, with solutions. The book is aimed at both teachers and students of mathematics who want to know more about famous unsolved problems. ... Read more

Customer Reviews (3)

good, but out of date
This is a very cool book. It's designed to be used by people with a range of mathematical knowledge and talent. There are easier problems given along with the very hard unsolved ones, and every topic is discussed in two separate places, once at an elementary level and once at a deeper level for people with more training.

My big misgiving about this book is that it's way out of date. They produced a new edition in 1996, but it hasn't been updated since then. For example, problem 1.2 (a conjecture about illuminable polygons) was solved in 1995, but that fact didn't make it into the 1996 edition. Fermat's last theorem, problem 13 in the book, was also been solved in 1995, but that was also too late to make it into the book. Normality of pi, problem 21, has not been cracked, but significant progress has been made in the last decade. Other problems, like the various conjectures surrounding different versions of Conway's angel problem, would presumably have been good ones to include in the book if it had been updated since 1996.

So if you're going to use this book, you'd better realize that you're missing quite a lot of information, and you'll need to supplement it with more information gleaned from the internet, e.g., from Wikipedia's article on "Unsolved problems in mathematics."

Were to Find a Math Problem
I admit I didn't read the entire book. But don't think that discredits my review. You see this is a special math book... One that you don't read cover to cover. You simply skim through the book and pick a unsolved problem that interest you. So you see it isn't read cover to cover like a textbook. And that is what makes this such an excellent math book. The book isn't about remembering rules it is about problem solving. And the organization of the book helps in gathering facts and understanding how others have approached the problem.

Unsolved problems is part of what mathematics are based on. Most of the content is easy to understand at undergraduate level. For fun I recommend only reading the problem's description and do your own research and later compare it to the second section of the book. I have worked on Prime numbers and have made some progress. (Just check my profile.) Math work does not get done without math problems. So if you are looking for a learning experience this is an excellent place to start.

Essential background information on old, unsolved problems
If progress towards the solution of the unsolved problems is to be continuous, then those in the field must be reminded on a regular basis. Furthermore, it is pretty clear that many of these problems will not be solved using "traditional" approaches, but by some sort of back or side door method. Therefore, those in other areas of mathematics need to see the unsolved problems presented. One never knows when someone working in an area that appears to be unrelated will read a problem and realize that what they are doing can be used to obtain a solution.
This work not only presents the problems, but also gives enough background to allow the non-specialist to understand them. Several leading theorems and their proofs are given, all in a clear, concise style. Many exercises are given at the end of sections, with hints toward solution at the back of the book.
The methods used would also allow this book to be used as a text in an advanced undergraduate colloquium or beginning graduate seminar in the golden oldies of mathematics. The competent undergraduate will have no difficulty comprehending the majority of the material.
An excellent addition to any collection, this book is a key. For it is clear that many of the unsolved problems will be defeated by one who never started with that intent, but saw the problem and had the correct leap of insight.

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

Product Description
In 1859, Bernhard Riemann, a little-known thirty-two year old mathematician, made a hypothesis while presenting a paper to the Berlin Academy titled"On the Number of Prime Numbers Less Than a Given Quantity."Today, after 150 years of careful research and exhaustive study, the Riemann Hyphothesis remains unsolved, with a one-million-dollar prize earmarked for the first person to conquer it.

Alternating passages of extraordinarily lucid mathematical expositionwith chapters of elegantly composed biography and history, PrimeObsession is a fascinating and fluent account of an epicmathematical mystery that continues to challenge and excite the world.Amazon.com Review
Bernhard Riemann was an underdog of sorts, a malnourished son of a parson who grew up to be the author of one of mathematics' greatest problems. In Prime Obsession, John Derbyshire deals brilliantly with both Riemann's life and that problem:proof of the conjecture, "All non-trivial zeros of the zeta function have real part one-half." Though the statement itself passes as nonsense to anyone but a mathematician, Derbyshire walks readers through the decades of reasoning that led to the Riemann Hypothesis in such a way as to clear it up perfectly. Riemann himself never proved the statement, and it remains unsolved to this day. Prime Obsession offers alternating chapters of step-by-step math and a history of 19th-century European intellectual life, letting readers take a breather between chunks of well-written information. Derbyshire's style is accessible but not dumbed-down, thorough but not heavy-handed. This is among the best popular treatments of an obscure mathematical idea, inviting readers to explore the theory without insisting on page after page of formulae.

Customer Reviews (90)

entertaining and informative
This is a very well-written account of the most famous unsolved problem in mathematics, the Riemann Hypothesis. The author has organized it into odd-numbered chapters which have most of the math (targeted at the reader with only a high-school level background in math and no calculus background), and even-numbered chapters which have most of the history. The history is interesting enough that the math chapters could be skipped if the reader is so inclined. The math chapters are as gentle as possible given the formidable nature of the topic so many readers will find them readable. Somewhat more ambitious readers might find this book a good stepping-off point for further investigations, but be forewarned that truly comprehending this material will require a solid grounding in calculus and complex analysis, so if your background is only high-school math you probably have a minimum of two years equivalent of college math to get through first.

The Riemann Hypothesis relates to number theory, specifically the distribution of prime numbers, which is fairly understandable at the elementary level to the non-mathematically trained, so even without understanding the math behind the Hypothesis itself, most readers can still come away with an understanding of what the Hypothesis is trying to say and why it's so important.

What I found most interesting is the way in which seemingly completely unrelated areas of math and science have been found to have connections with the Hypothesis. This is true of a lot of mathematical propositions, and is perhaps the most mysterious aspect of math in general. As Shakespeare once wrote, "there's more in heaven and earth than is dreamt of in your philosophy, Horatio!" Lots more!

History of a mystery
Prime numbers hold an unmistakable appeal. The basic concept seems so simple - once you have the idea of multiplying some numbers together to make others, the question seems natural: which numbers can not be formed by multiplying others? Those are the primes.

So, starting from such a simple statement, it's quite amazing that so much subtlety inheres in the primes, and that they occur with such a maddening mix of predictability and randomness, and that they materialize in places so seemingly unrelated to their definition. This book traces one thread of that inquiry into the primes, the Riemann Hypothesis. With a little mathematical magic, the primes transform into function with wild behavior - but with a startling regularity of its own. At least, it seems regular, but "seeming" isn't good enough for mathematicians. It must be proven. Despite a hundred and fifty years of attack, the Hypothesis remains firmly defended against any effort at explanation. Derbyshire traces the history of that attack, and gives us glimpses into the baffling personalities of those attackers.

Although I enjoyed this book, it might not hold great appeal unless the math itself holds your interest. There's a balance to maintain when tracing the history of a phenomenon like this, between the personalities of those seeking out this mathematical beast and the beast itself. Derbyshire includes plenty of math, despite his efforts to keep this popular. I could have wished for a bit more of the human side of the quest, and I'm sure other readers will wish the same. Even so, it makes a fascinating story and a clear statement of the problem that has become such a consuming passion for so many of mathematics' finest minds.

- wiredweird

Math's Greatest Mystery for Everyone
I just started reading this book and I literally can't put it down. The book presents in a relatively simple language the hypothesis that has plagued mathematicians for such a long time. The author explains crucial math concepts like infinity, limit and function for people whose math background is not strong. The alternation of chapters is very appropriate -- you read the math first, and then the historical context of that math. For example, I never really thought of the fact that Gauss counted primes using... a paper and a pencil (well, there were no computers in his time); this made me appreciate the math even more.

Personally, I would recommend this book to anyone who is interested in math, no matter how strong his/her background is.

Becoming obsessed
I am becoming obsessed with the Riemann Hypothesis. Better, I'm obsessed to, at least, trying to understand what it is about. After peeking up some books, some more mathematically inclined then others I can surely say that this was, by far, the one that gave me a better insight. It is very delightful to read a book such as this one.

Kindled an Obsession in Me
I browsed a bookstore in 2007 and bought it after a skimming. Since then, I have read and re-read it many times. I vowed to some day understand everything in it. I bought a copy of Maple and recently of Mathematica, so I could reproduce every mathematical expression in the book and then some. My first victory was when I reproduced "The Derbyshire Spiral", as I began to call the logo on the cover. I have now reproduced Derbyshire's elucidation of the random Hermitian matrix and its strange behavior, as well as most of the other math in the book.

I could go on describing other adventures in math that were inspired by this book. Having flunked as many math courses as I ever passed,long ago,the clear presentations by Derbyshire were a pleasant surprise! The historical context makes for highly interesting reading as well whenever the math gets to be too much.

If the idea of prime numbers has any attraction to a casual reader at all, this book is a must buy.

Mauri Pelto ... Read more

Product Description
This is the only scientific review of ball lightning in printat the present time with an up-to-date coverage of theory. This bookprovides a critical review of eyewitness observations of balllightning and of evidence of its interaction with the environment.Ball lightning is probably not hazardous, but may be a precursor ofordinary lightning. The conclusion is that ball lightning has lowerenergy than generally assumed and that some theories are thusredundant. After critically reviewing the current theory,recommendations are provided for future research. This specialist bookdraws new conclusions about the characteristics of ball lightning andrelates these to current theories. ... Read more

Customer Reviews (2)

excellent
This book is by far presents the best un-bias scientific analysis of ball lightning since the 1900's. The book is worth the price. I was hesitant at first to make the big purchase, but I don't regret it one bit any more. Must buy if you want scientific analysis of ball lightning without all the crap from other sources.

Still a mysterious phenomenon
Ball lightning obviously is an extremely rare event. As in the case of other rare phenomena, the investigator is dependent on reports of casual witnesses, because systematic observing programs are not possible. Thereports collected over many years include reliable as well as doubtfulinformation. Stenhoff gives a good sample of reports, photos anddescriptions. These clearly show the range of observed phenomena and thequestions which remain open. The summary of sightings and the review oftheories to explain the variety of observations makes the book veryvaluable, particularly as there is no comparable recent publication on thistopic. I also appreciate the list of references (up to 1998 publications)which allows to look for details regarding reports as well as theoreticalpapers. All together: highly recommended. ... Read more

Product Description
Mathematicians and non-mathematicians alike have long been fascinated by geometrical problems, particularly those that are intuitive in the sense of being easy to state, perhaps with the aid of a simple diagram. Each section in the book describes a problem or a group of related problems. Usually the problems are capable of generalization of variation in many directions. The book can be appreciated at many levels and is intended for everyone from amateurs to research mathematicians. ... Read more

Customer Reviews (1)

A wide expanse of problems in geometry
After the counting numbers, geometry is the oldest branch of mathematics and no doubt the first one that required abstract thinking. Even so, thereis always a certain "concreteness" about it in the sense that diagrams canalmost always be constructed. The range of problems that fall under thegeometric umbrella is extremely wide and some even have practical uses.
This book is a testament to the wide range of problems that are geometricin nature. One of my favorites is known as the "worm problem." To be moreprecise, the question is, "find the convex set of least area where anycontinuous curve of length one can be placed in it." This type of problemhas ramifications in optimal packings, where a single type of containerneeds to be constructed for all possible ways an object can fold. Otherproblems such as tiling and dissection; packing and covering andcombinatorial geometry are also covered.
However, the best part ofthe book may be the extensive references. Every problem is followed by alist of references, so if you wish to take a crack at it, you will havelittle difficulty in locating the work done to the date of publication.
This is one of those books that always seems to beckon me when it lieson my bookshelf. Every once in awhile I pull it off and browse through it,admiring the skill and breadth of mathematicians in their pursuit of truth.It should be in every academic library.

Published in Journal ofRecreational Mathematics, reprinted with permission ... Read more

Product Description
This book provides clear presentations of more than sixty important unsolved problems in mathematical systems and control theory. Each of the problems included here is proposed by a leading expert and set forth in an accessible manner. Covering a wide range of areas, the book will be an ideal reference for anyone interested in the latest developments in the field, including specialists in applied mathematics, engineering, and computer science.

The book consists of ten parts representing various problem areas, and each chapter sets forth a different problem presented by a researcher in the particular area and in the same way: description of the problem, motivation and history, available results, and bibliography. It aims not only to encourage work on the included problems but also to suggest new ones and generate fresh research. The reader will be able to submit solutions for possible inclusion on an online version of the book to be updated quarterly on the Princeton University Press website, and thus also be able to access solutions, updated information, and partial solutions as they are developed. ... Read more

Product Description
Featuring Original Contributions from Dr. Stephen Hawking

Unfold the mysteries that vex the greatest minds in science

Gain extensive knowledge of the most challenging scientific problems and learn from more than 60 of the world’s foremost scientists—among them, 40 Nobel laureates! Expand your horizons with a wide range of advanced scientific theories and techniques on problems concerning:

Permanently storing nuclear waste or eliminating it altogether
Harvesting energy from a reaction similar to that of the sun
Earthquake prediction
The creation of the universe
Comprehension of free will
The mystery of dark matter
The cosmological constant problem
The construction of a consistent quantum theory of gravity
And much more

Science has reached dazzling heights of discovery, transforming civilization in the process. And yet, some of the most fundamental questions remain unsolved! In The World’s 20 Greatest Unsolved Problems, John Vacca—together with more than 60 of the world’s most highly respected scientists—explains these problems in detail and describes the intellectual and technological hurdles to be overcome in order to solve them.

This book is indispensable for science buffs, teachers, students, and scientists who want to keep pace with the latest developments. The World’s 20 Greatest Unsolved Problems delves deep into mysteries such as the creation of the universe, dark matter, the quantum theory of gravity, protein folding, free will, consciousness, earthquake prediction, Fullerenes, the quantum mechanical vacuum, storing or eliminating nuclear waste, and more. No other resource explains science’s most compelling dilemmas with such clarity and authority, and nowhere else can you share the expertise of so many brilliant minds! You’ll find

Complex topics made intelligible, as only experts in their fields can
Coverage of the key problems expected to dominate the next 40 years of scientific research
The World’s 20 Greatest Unsolved Problems is must reading for anyone teaching science or performing scientific research. It also will fascinate the moderately technical reader or scientific novice. ... Read more

Customer Reviews (31)

An embarrassment to NASA
But hey, at least he didn't drive cross country wearing diapers to abduct his ex-lover's current companion.

Jes, John Vacca apparently was an actual NASA employee before his "early retirement" (clue #1 something may be amiss...why would you put "early" in front of "retirement" on your own bio?).Yes, there are "contributions from Stephen Hawking".Unfortunately, the vast majority of the "contributions" to the book came from a crackpot pseudo-science website called "Jupiter Scientific" ("Official site of the book "The Bible According to Einstein"....woah...and wasn't Einstein Jewish anyway? ha just kidding).In an internet quack science version of "Bum Fight", Jupiter Scientific apparently figured this out itself and went on the offensive.You can see its press release on the internet.Of course, I don't know who to believe, but after reading the section in the book regarding "free energy", Vacca's credibility is about zero.Indeed, the first sentence of this section is "you either believe in it or you don't."Say what?I thought this was supposed to be a book about science.

This book is great if you're a social anthropologist looking for a current reflection of the general public's complete misunderstanding of general scientific principles at the beginning of the 21st century, kind of like Galan's anatomy is good for an understanding of how much the Romans knew about the human body (but not so good if you actually want to do successful heart surgery).It is also an embarassment to our public school system that even students on the bottom half of the bell curve can't figure out within about 5 minutes of scanning it that it's complete poppycock.

The World's 20 Greatest Unsolved Problems
20 Greatest Unsolved Problems covers the fields of Astronomy and Cosmology, Physics and Astrophics, Biology and Paleontology, Neuroscience, Geology, Chemestry, and Energy.
It should be required reading for anyone interested in acquiring a better understanding of the current ideas of science as a whole.
The price was right as shipping was far more than used market price and in almost pristine condition.

Wide Range of Topics/Wide range of explanations
I really enjoyed reading this book.I am not very proficient in some of the areas covered.I thought the book did a great job of explaining the topics and then giving me some of the explanations that are possible and then finally coming to a conclusion.The book will definitely make you think and understand how little human beings really know about the universe they reside in.I feel after reading this book I know what questions should be asked and I now I have a better grasp of some of the world's greatest mysteries.

Error prone, biased and uninteresting
This book was a very disappointing gift.

I found Vacca's explanations of science's current puzzles to be tedious and frustrating.Most of the topics are explained in tedious detail, yet not enough detail to permit real understanding.As an example, it is perhaps impossible to explain the first few milliseconds after the Big Bang without heavy use of mathematics - yet Vacca tries, using extensive, vague and (to me) uninteresting prose.He repeats this performance, attempting to deal with issues ranging from quantum gravity to protein folding to free will.

The frustration is made worse by the inclusion of many grammatical and technical errors.There were more of these errors than I've seen in any comparable volume. He also makes excessive use of boldly-highlighted mid-page "Notes" boxes - which really should be relegated to footnotes.

Finally, Vacca's self-conferred mantle of scientific objectivity is destroyed in the chapter on "Free Energy."He seems to believe it's possible to suck all the energy the earth needs from the cosmic ether.So much for the laws of thermodynamics!His "Notes" in this chapter consist of jabs at the "naysayers and skeptics" with little clue that these "naysayers" comprise over 99% of the world's scientists.

If a layman wants a technical, yet non-mathematical explanation of cosmology and quantum physics, Hawking is much, much better.If a layman wants a well written and wide-ranging overview of science, Bill Bryson's _A Short History of Nearly Everything_ is much more engaging.

Fascinating despite some repetition and some fuzziness
The problems range from dark matter and dark energy through attempts to reconcile gravity and quantum mechanics to problems associated with DNA and proteins, to neuroscientific concerns about free will and consciousness to what to do about nuclear fusion and its waste.

There are other books on cutting edge problems in science that I have read, e.g., John Malone's Unsolved Mysteries of Science: A Mind-Expanding Journey through a Universe of Big Bangs, Particle Waves, and Other Perplexing Concepts (2001) or The Next Fifty Years: Science in the First Half of the Twenty-First Century (2002) edited by John Brockman; but there is only one other that is anywhere near as ambitious as this work.That book would beMagic Universe: The Oxford Guide to Modern Science (2003) by Nigel Calder.

To compare these two books for the reader I would say that Calder's book is not only longer but covers more ground, is better edited and relies on a greater range of scientific authority.But Vacca's book has the virtue of narrowing in on just where the scientific action is while he does a good job of presenting the various opinions.That is, insofar as I, personally, can tell.To be honest, much of the material in all these books is above my level of expertise.Consequently I take most of what I read at face value.Clearly I cannot choose between cosmological models of inflation and quintessence.Nor do I have any firsthand experience with the complications of protein folding, etc.But neither will most readers.However we needn't be critical readers.It is enough to read appreciably about the wonders of science and how such wonders inform our beliefs and enrich our lives.

As for the repetition in the book and the typos and the other errors pointed out by other readers, it is good to understand that Vacca wrote this book by himself (although he interviewed and relied on the work of many scientists) and probably did so in a first draft/correct it mode (judging again from the repetition and some of the unpolished prose).Let's face it, life is short and a book like this needs to be written fast or it will become outdated before it hits the book stores.Furthermore, although he had editors to check for technical errors, editors to check his spelling and such, and had the benefit of the professionals at Prentice Hall, it is in the nature of a book like this that no single person with the exception of the author can really be close enough to the content to adequately edit it.

Now I want to look at a couple of the problems that Vacca discusses.

He talks about traveling back into the past and asserts that the usual paradoxes relating to killing your grandfather before your parents were born, etc. can be overcome by having you go back to a past in a parallel universe.Relying on the many worlds interpretation of quantum mechanics (which he seems to favor), Vacca finds this reasonable.The problem, however, is that Vacca has already in a previous chapter made it clear that there is no interaction between postulated parallel universes, so he ends up justifying travel to the past by making it doubly difficult: not only do you have to violate causality but you have to go to a parallel universe to do so!I imagine he would say in response that by going to another universe you actually avoid the violation of causality since you do not in any way affect the universe you are in.However to go back in time in the other universe you have to be in that universe.

Vacca suggests that the dark matter that cosmologists are now utterly convinced exists because of its gravitation presence is perhaps an example of a parallel universe.Actually he takes the opposite perspective and asks if a parallel universe exists (parallel to the dark matter) and answers that it does.It is us. (p.115)

Since gravity that makes us aware of the existence of that dark universe (and remember there is no evidence of any information about the dark matter via the electromagnetic force or the weak or strong nuclear forces) could it not make them aware of us?(Assuming there is somebody there to be aware.)Perhaps some day we will communicate with other universes through some type of gravitational mechanism.(Huh?--Well, maybe.)

On free will Vacca uses basically three authorities, Timothy O'Connor, Miroslav Backonja, and Paul J. Bertics, and from them constructs what he sees as the current understanding by neuroscience.I wasn't even aware that neuroscientists had a position on free will.I thought it was a purely philosophic or religious question.The opposing camps of naturalism (no such thing as free will) and libertarians (humans have free will) are reconciled in the neuroscientific community through the idea of "compatibilism," a word I encountered here for the first time.What it means is that the lack of free will (which most neuroscientists, Buddhists and myself, among others see as obvious) is made compatible with the societal and human psychological need to believe in free will (for punishment and criminal deterrence) by realizing that in an Orwellian way we can say that free will does not exist, but in order for society to run smoothly we must pretend that it does.Vacca discusses the ramifications from this doublethink and concludes that whether free will is an illusion or not depends on your point of view.Your free will is obvious, but that of others has to be taken on their say so.

Here's an example of Vacca's sometimes strikingly expressive prose: "As much as free will exposes humans to the threat of unlimited retaliation for wrong-doing, it nevertheless compensates them by making them the lords of their little domains, the micro-gods of their minds." (p. 394) ... Read more

Product Description
The definitive account of the Everests of mathematics--the seven unsolved problems that define the state of the art in contemporary math.

In 2000, the Clay Foundation announced a historic competition: whoever could solve any of seven extraordinarily difficult mathematical problems, and have the solution acknowledged as correct by the experts, would receive $1 million in prize money. There was some precedent for doing this: In 1900 the mathematician David Hilbert proposed twenty-three problems that set much of the agenda for mathematics in the twentieth century. The Millennium Problems--chosen by a committee of the leading mathematicians in the world--are likely to acquire similar stature, and their solution (or lack of it) is likely to play a strong role in determining the course of mathematics in the twenty-first century. Keith Devlin, renowned expositor of mathematics and one of the authors of the Clay Institute's official description of the problems, here provides the definitive account for the mathematically interested reader. ... Read more

Customer Reviews (26)

Great
Exactly as described, smooth/easy transaction, exactly as described, fast shipping, would do business with again!

As others have noted...
...the task before KD -- to explain the Millennium Problems to a lay readership, in so few pages no less -- was just shy of impossible.That KD managed to convey the essence of 5 of 7 of the MPs is much to his credit.I have read longer treatments of a few of these problems that did not do so able a job of explaining them.By the time he turned to the problems in the final two chapters in MP, KD all but threw up his hands, acknowledging that they defy simple (intuitive) explanation.These last two chapters may have something of interest for the specialist reader (who would probably turn elsewhere for information); for the non-specialist reader, they are pretty much a total wash.KD, himself, suggests that readers skip them.For my part, I skimmed them in the hope that I might glean some small insight into the problems under discussion.Alas, no go.Whatever, the first 5 chapters in KD are gems of popular mathematics exposition.For non-mathematicians interested in cutting-edge mathematics, MP is well worth the read.

Good introduction to the current frontier given inevitable constraints
As time passes math tends to get further and further out of the reach of most people (in fact as one gathers from the book that mathematicians themselves lose track of their colleagues techniques and progress to a non-trivial degree).This book is an attempt to shrink some of that distance and is an entertaining summary of the spirit of the current (though one problem has been solved) most noted problems in math- to a large degree, the modern day equivalent of Hilbert's list from a century ago.

It is always tough for authors to strike a balance between describing the problem and its history versus going describing the various paths that a solution might be found on.This book minimizes the solution potentials and their descriptions, for good reason, as the problems are modern day math and would require specific graduate training to even attempt parts of the solution (exept maybe the P NP problem) and instead focuses on more of the historical side of things and how the questions were motivated and became prominent.

To me its always nice for a working mathematician to spend the time to try to describe the spirit of a problem.This book does a great job where it can, as the problems are spelled out very nicely- towards the end of the book, the goal of heuristically describing the problem becomes close to impossible (one learns the author needed weeks to understand the problem himself).I recommend this to all who want to try to an elevated view of current problems and what mathematicians are working on (and solving).

God is clever, so we threw some money at these problems...
This book is absolutely fascinating.Monsieur Devlin gives a wonderful presentation of these interesting problems, and although it is a little cumbersome (because of the math), he delves well into each topic and manages to reveal the juicy red meat of each, which I lavishly devoured piece by ... okay, so it's a cool book.Seriously, it's ten bucks.Buy it.

An excellent though difficult-to-digest read.
Keith Devlin has done a good job of explaining these seven extremely difficult problems.Granted, I am a mathematics professor, but I enjoyed the book and will rely on it when I teach my History of Mathematics course.Devlin is not able to make the Hodge Conjecture understandable in the last chapter, but that's a tall order; he frankly admits that the problem is difficult for even a professional mathematician to understand, and I doubt that many expositors could have done better with the Hodge Conjecture. ... Read more

Product Description
An in-depth look at the most intriguing puzzles in science today

In this illuminating book, professors Arthur Wiggins and Charles Wynn explore what they believe are the five biggest science problems:

• Physics: Why do some particles have mass, while others have none?
• Chemistry: By what series of chemical reactions did atoms form the first living things?
• Biology: What is the complete structure and function of the proteome?
• Geology: Is accurate, long-range weather forecasting possible?
• Astronomy: Why is the universe expanding faster and faster?

Wiggins and Wynn carefully explain each of these problems, then discuss the theories that address them. Some of the many topics covered include string theory, the human genome, chaos theory, and protein folding.

Featuring humorous illustrations from renowned science cartoonist Sidney Harris, this book invites you to explore the events that led to these problems and the cutting-edge efforts being made to solve them.The authors also provide Idea Folders, which contain additional details about the unsolved problems, and Resources for Digging Deeper, such as books, periodicals, and Web sites. ... Read more

Customer Reviews (4)

5 Unsolved Problems - Or is it 100's

This book is a GREAT introduction to anyone interested in science, wishing to become better informed about scientific issues, or wanting to know what many of our best scientists are working on right now.It is not particularly deep and is therefore a good primer for the non-scientifically oriented reader.

The authors open with a careful discussion of the scientific process.Most of us make our daily life decisions on the basis of tradition, word of mouth, and limited anecdotal experience - and refer more important decisions to a professional.How enlightening it is to be reminded of the rigorous testing, the meticulous standards and procedures, the relentless re-examination in the face of new data, and the necessary grilling by peers that is part of the day to day activity of scientists.Science uses a different language from that used in everyday speech - that language being mathematics."Because mathematical skills require a great deal of effort to acquire, explaining scientific hypotheses to people not trained in mathematics requires translation of mathematical concepts into conversational language.Unfortunately, the meaning of the hypothesis may suffer in the process."

The five unsolved problems are then discussed, one each from the fields of physics, chemistry, biology, geology, and astronomy.Then 27 more are asked in an "appendix" type chapter and given only half-page answers.I like these sorts of books because they help me to maintain general scientific literacy.

For every bit of knowledge we gain, new questions invariably arise - that never-ending quest being the nature of science.I personally think we'll still be looking for the TOE (Theory of Everything) in 100 years, but reading about unsolved problems will notify me if string theory is abandoned, if parallel universes are found, or if the complete structure and function of the proteome is ascertained (fat chance).I might add that the running cartoon commentary by Sidney Harris is superb!

Fine introduction for the general reader
This is written by two guys who teach physics (Wiggins) and chemistry (Wynn) at the college level in a reader-friendly manner in which each of the five unsolved problems are presented, explained, and critiqued.Each section is then concluded with indications of how these problems might be solved.There are some nice cartoons by Sidney Harris to augment the text.

The five "biggest" problems are (from different disciplines and not without controversy):

1. The nature and origin of mass.(Why do some particles have mass while others do not?)

It is obvious that we really do not understand the nature of mass from our inability to form a unified theory involving gravity, a theory that would unify quantum mechanics and relativity.Indeed I think physicists are just whistling in the dark when they talk about particles and fields.It's clear to even this casual observer that the real nature of particles/waves, particles/fields is not really understood, and perhaps cannot be understood in anything other than a once or twice removed mathematical sense.We can write equations that describe what we observe, but the intrinsic nature of all phenomena remains veiled.We avoid infinities in the mathematics of physics as a long-observed and much beloved rule (something like Occam's Razor) with the result that we (necessarily) "construct" limits on the physical world like those named after Max Planck.Beyond (or "below," or "under" or "smaller than," etc.) those limits is potentially a whole universe of physics much like what might be beyond the Big Bang in cosmology.

Authors Wiggins and Wynn acknowledge that the Standard Model of physics has a "dark side" (p. 30) and that the long-sought Higgs field particle may be a "mathematic convenience." (p. 31)From my point of view everything in physics (and this includes all of string theory) that has not met with experimental proof is possibly a "mathematical convenience."This is not to denigrate physics or physicists.On the contrary.What physicists have accomplished toward an understanding of the world in which we live stands as one of humankind's most glorious achievements.The problem is that (as quoted from J.B.S. Haldane on page 159) "the universe is not only queerer than we suppose, but queerer than we can suppose."In other words don't hold your breath for the dreamed-of "Theory of Everything."And if it arrives, don't imagine that "everything" really is "everything."

2.How did lifeless chemical reactions become life?

The authors present some of the history (Stanley Miller's primordial soup experiment; panspermia, etc.), outline the problems, tell us a little about DNA and RNA, and finish with how the puzzle might be solved and by whom.I would observe that imbedded within this question is a theoretical bugaboo that first needs to be resolved.We have to agreeably define what "life" is before we can hope to make a distinction between very complex but "lifeless" self-replicating molecules and molecules forming living organisms.As such, the problem is one of definition as much as anything else.Clearly if we left out our notion of things living as opposed to things not living, we might discover a step-by-step continuum without a clear demarcation point.

3. How do proteins control cells and tissues?("What is the complete structure and function of the proteome?")

The authors note that since the genome has been mapped and sequenced, "the unsolved problem" in biology "has shifted" to "How do protein molecules built from directions provided by [the]...genomes contribute to the structure and function of organisms?" (p. 71)A very complex problem indeed, but at least it is a practical problem and not a theoretical one, and as such (unlike some others in this book) is one that conceivably can be solved through a whole lot of hard work.

This is about cells and how they function.The authors reprise the genome mapping and sequencing story, and then point to "Protemics: The Next Frontier."

4. Can we predict the weather?(Or, how accurate can our weather predictions be?)

This of course is about complexity theory and why that famous butterfly in the Sahara continues to influence the formation of hurricanes in the Carribean.

Quick answer: accuracy will continue to fall off as the square of the distance in time.Just joking, but clearly the more lengthy the forecast, the more uncertain it will continue to be.

5.Why is the universe expanding faster and faster?

Ah, yes.What IS the nature of Einstein's fudge factor that has recently returned?I love this one.The real question is what IS all that dark energy and dark matter out there?As the authors point out only 4%(!) (see page 129) of all the matter in the universe is accounted for in terms of things seen.Seventy-three percent is in the form of dark energy and another 23% in the form of dark matter.It is amazing to realize that over 90% of what exists is stuff we know next to nothing about!

A nice part of the book are the "folders" at the back in which many other interesting issues are briefly presented.The problems in the "Problem Folder" are organized according to disciplines, "Physics Problems," Chemistry Problems," etc.There are sixteen ideas in the "Ideas Folder," including such things as anti-matter, protein folding, chaos theory, global warming, and so on.

Bottom line: Wiggings and Wynn do a good job of introducing the general reader to what scientists at the horizon are working on.It's really amazing to realize how far we've come as knowledge-seeking creatures, and then to get a glimpse of how incredibly much there is we don't know.

A Fun Way to Discover the Intriguing
We teach subjects not related to the science Wiggins and Wynn address - except in the ways that really matter. For example, reviewer Winston teaches American Government; he finds this book profoundly "usable" in that its point includes the notion(disused in some high political places) that problems can be treated by thinking, and shouldn't be expected to be solved by wish fulfillment and received authority.In this sense the book is a continuation of the earlier "Quantum Leaps in The Wrong Direction," only this time the scientific problems are "harder," as they are particular (albeit big!) quests on which scientists have been working. The pleasure scientists take in the pursuit is communicated in a way that leaves no doubt by the authors, and it is marvelous, breath-taking and invigorating. This is just one of the delights of the book.Another is that there is an unsolved mystery for everyone! The chapter sub-headings are a great device and most welcome. The pictures are fascinating and the cartoons are right-on and funny. Reviewer Friedler, who teaches creative and expository writing, finds the book accessible and readable in its well-timed understatement and clever analogy, and in its style and conceptual clarity. It flows and the reader understands.This is not an easy thing, given the mighty topics. We highly recommend this delightful book for its captivating material and entertaining reading.

A survey course in Science
This is like a condensed survey course of various facets of Science. The authors cut to the chase by focusing on one major unsolved problem in each discipline. Their breadth of knowledge is impressive and their presentations are comprehensible for laymen like me. I was particularly attracted to the question "How was the first living thing formed?" and also "Why is the universe expanding faster and faster?". The accompanying cartoons are well done. The ideas folder at the end touches on many current news topics like greenhouse gases and genetic engineering.An interesting read for those who wonder "Why?". ... Read more

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The field of astrophysics is in the midst of a technologically driven renaissance, as fundamental discoveries are being made with astonishing frequency. In the last decade, new detectors in space, on earth, and deep underground have, when coupled with the computational power of modern computers, revolutionized our knowledge and understanding of the astronomical world. This is a great time for a student of any age to become acquainted with the remarkable universe in which we live. This volume is a collection of essays, originally presented orally to a diverse group of students and professionals, which reveal the most fertile areas for future study of astronomy and astrophysics. The emphasis of this work is on the clear description of the current state of our knowledge as a preparation for the future unraveling of the mysteries of the universe that appear today as most fundamental and most amenable to solution.

A stellar group of astronomers and astrophysicists describes the directions and styles of work that they think are most likely to lead to progress. Bibliographical notes at the end of each presentation provide guidance for the reader who wishes to go more deeply into a given subject. Unsolved Problems in Astrophysics is a uniquely stimulating introduction to some of the most important topics in modern astrophysics. ... Read more

Customer Reviews (2)

Good book on a wealth of topics
This book provides a great amount of information on a wide variety of hot topics in contemporary astronomy, such as cosmology, AGNs, GRBs, and dark matter.My only complaint is that the material does tend to slant towardsextragalactic astronomy.However, otherwise the material is extremelyuseful for someone looking for fresh material to work on.The referencesalso provided within are great.The reading level is also easy, and isprobably similar to Scientific American or Sky & Telescope.

the information contained therein is accurate and readable!
As far as I have been able to see with cross-referencing and my own checks and double-checks, the information in this publication is accurate.The layman ought be able to understand it with minimal effort.Very readable. ... Read more

Product Description
Since 1859, when the shy German mathematician Bernhard Riemann wrote an eight-page article giving a possible answer to a problem that had tormented mathematical minds for centuries, the world's greatest mathematicians have been fascinated, infuriated, and obsessed with proving the Riemann hypothesis. They speak of it in awed terms and consider it to be an even more difficult problem than Fermat's last theorem, which was finally proven by Andrew Wiles in 1995.

In The Riemann Hypothesis, acclaimed author Karl Sabbagh interviews some of the world's finest mathematicians who have spent their lives working on the problem--and whose approaches to meeting the challenges thrown up by the hypothesis are as diverse as their personalities.

Wryly humorous, lively, accessible and comprehensive, The Riemann Hypothesis is a compelling exploration of the people who do math and the ideas that motivate them to the brink of obsession--and a profound meditation on the ultimate meaning of mathematics.
... Read more

Customer Reviews (20)

Others deserve learning about complexity too
Karl Sabbagh has done his job well. He never pretended to be a master at this topic. He pretended to give the layman as myself access to the intricate and complex mind of a genius. This is what divulgation is about.

I do not understand those hard ratings or much less comments like "which I could probably do here ultra-briefly if I had a charge number for a couple of hours". It is as if I buy a children's book on physics and complain about the depth of its concepts. Some wise pedagogue would say I was completely confused on what divulgation is about.

If you think you can enlighten our small minds on this matter dedicating a couple hours, I suggest you send a manuscript to the press. I shall be happy to buy it and rate it afterwards.

In scientific literature there ought to be room for books that can be understood by more than one hundred people. Sometimes complexity is an added feature that does not add value...

The fact that I am not an expert at math does not make me a conceptual handicapped. I think this book is a good entrance to the topic. In my case it has triggered the purchase of other deeper approaches.

Well done Karl!!!

Eh
There are a few mistakes:
1) p. 133 In the smallest n-digit primes sequence, there are incorrect 8th and 11th members.
2) p. 190 The values of the Mobius function are listed from one to ten, the 7th is incorrect (should be -1)
3) p. 206 The inline formula for the gamma function is incorrect. It reads "Gamma(n) = n * Gamma(n-1)" but should read "Gamma(n) = (n-1) * Gamma(n-1)".

There is also a fuzzy explanation which contradicts another explanation of T. H. White's "everything not forbidden is compulsory" (p. 185). Sabbagh claims that Gell-Mann says this is how the laws of physics work--that if something is predicted to exist by a theory, then it exists somewhere. Kip Thorne clearly explains in 'Black Holes & Time Warps' that this is not how the laws of physics work. From the book, "Many of the things permitted by the laws of physics are so highly improbable that in practice they will never happen." (p 137) It may be that I misunderstood Sabbagh's writing or that Sabbagh misunderstood Gell-Mann. Nevertheless there is something going on here that doesn't match up.

Other than that, the book was pretty good. It didn't quite give me the dose of mathematics I was looking for, but then again I always ask for more than I can handle. It was very readable.

Where's the Beef?
Nice to have a book to read on the mathematical history and personalities hooked on the search.Way too soft on what the search is for and actual progress in any direction.Has an appendix with some current effort at a proof but without any surrounding discussion it's hardly worth the effort.

The Riemann Hypothesis for non-mathematicians
The book is really about the search for solutions to the Riemann Hypothesis, and contains lots of stories about mathematicians, and mathematical topics. It also includes a wide variety of tangential topics, such as the resolution of previously unsolved problems: the Bierbach Conjecture and Fremat's Last Theorem.

While intended as an accessible account of the Riemann Zeta Function and the Riemann Hypothesis for the non-mathematician, it uses overly trite and simplistic descriptions of basic mathematical concepts such as prime numbers, complex numbers, and zeros of functions, while glossing over more advanced concepts, such as L-functions, Hilbert Spaces, and Random Matrices. Clearly, this was written by a mathematical novice.

YOU can understand the Riemann Hypothesis....after this commercial break.
I'm halfway through this book and am getting very frusturated.The beginning chapters did a very good job laying out the foundations of the RH.You begin to think you may, finally, be able to understand what the RH is about.However, I am realizing that less and less of the book is actually about the RH or the RZF the farther I progress.An entire chapter will waste space describing a mathematician (physical appearance, quirky behavorial traits), his unrelated contributions to math, and either a vague description of the work he is doing to prove the RH or some opinion on whether it will be solved or not. The book has devolved into irritating tangents unrelated to my understanding of the RH/RZF...clever math poems, crap about Fermats theorem, etc.I would even be forgiving if he was describing the life and times of Riemann himself.There are plenty of cute little math stories here, but I didnt waste $14.00 on a book about the RH to NOT read about the RH.I want every single chapter to build on the previous, start slow, explain the concepts, and raise the bar.If you get stuck, too bad...research the web and pick the book up later.Thats what I wanted here, and this book is turning out to be a letdown.
There is very good information for someone completely new to the concept of the RH or RZF, but this book is 4 chapters of useful material spread out into an entire book filled with math-related lore.I appreciate what Sabbagh is trying to do here, but lets be realistic.This is a very advanced mathematical concept; you can't hand-hold someone through this from start to finish.At some point he should challenge the reader and escalate the difficulty of the book, but this does not appear to happen.Instead we get a smoke-screen...a cliffhanger.Would I recommend this book?Only to those who have absolutely no understanding of what the RH/RZF is and want a springboard to something else. ... Read more

Product Description
This timely volume reviews recent progress in our understanding of all aspects of stellar structure and evolution, with special emphasis on currently unsolved problems. It covers every stage in the life of a star, from birth to death, as well as the fundamental processes that affect stellar evolution. Each chapter is written by a leading world expert, based on presentations at an international conference held at the Space Telescope Science Institute.A complete understanding of stellar evolution is important in its own right, constituting a vital piece in the more general puzzle of understanding how galaxies form and evolve. This volume presents the most comprehensive and up-to-date survey available of this crucial topic in astrophysics. ... Read more

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Unsolved Problems in Number Theory contains discussions of hundreds of open questions, organized into 185 different topics. They represent numerous aspects of number theory and are organized into six categories: prime numbers, divisibility, additive number theory, Diophantine equations, sequences of integers, and miscellaneous. To prevent repetition of earlier efforts or duplication of previously known results, an extensive and up-to-date collection of references follows each problem. For the third edition, the author has added new problems and figures. ... Read more

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UNSOLVED PROBLEMS OF SCIENCE UNSOLVED PROBLEMS OF SCIENCE by A. W. HASLETT Sometime Foundation Scholar of Kings College, Cambridge LONDON G. BELL AND SONS, LTD 1935 CONTENTS Authors Note page ix Chapter I. THE UNENDING QUEST The ignorance of science its cost the progress of science practical achieve ments knowledge not fully applied discoveries seldom revolutionary problems can be discovered as well as facts different types of ignorance of detail of principle of the nature of things the limitations of science their relation to religion a coming difficulty page I Chapter II. THE CREATION OF THE UNIVERSE Difficulty of creation from nothingness an alternative approach astro nomers survey of space one universe among many the scattering of the universes all the universes once one the Relativity interpretation space itself expanding the Concertina Universe the Explosive Crea tion Universe Creation by Toppling connection with the birth of the stars a non-Relativity picture leads to similar conclusions an open choice page i Chapter III. ARE THERE OTHER WORLDS THAN OURS Martian tales discovery of the canals the requirements of life on the earth, water and oxygen temperature limitations possibilities of adapta tion when the suns heat fails conditions on Mars it undoubtedly has an atmosphere the canals discussed evidence of plant life living on Mars the mystery of Venus planets of other stars their low birth-rate connection with the expanding universe life a rare accident or a de liberate creation page 40 Chapter IV. THIS CHANGING WORLD Mountains point the way our not-so-solid earth seas on the mountain tops Scandinavia tilting perhaps England too coastal erosion mountain growth to-day a radium furnace within the earth earthquakes and mountain chains the forecasting problem the worlds changing geography even continents may not be fixed the Wegener theory of drift evolution and the geologist attractiveness of the Wegener picture attempted measurements of drift a new synthesis page 60 VI CONTENTS Chapter V. OUR WEATHER CAULDRON Economic importance of weather the long-range problem the scale of natures operations the ocean boiler house anti-cyclones and depressions ten million million tons of air shifted annually country omens in perspec tive the quest for cycles sunspot changes world weather relationships a gigantic survey India-South America-Siberia-Honolulu building up a forecasting formula a 4-to-i chance many records to be examined ignorance above ground level the missing control factors page 91 Chapter VI. MESSAGES FROM SPACE Messages are always being received cosmic rays meteors space not empty our own atmosphere as a border land two lines of evidence point to warmth discovery of the cosmic cloud chalk , salt and ordinary air theory that the cosmic cloud is hot meteors from beyond the sun discovery of cosmic rays welcomed in balloons pursued to the bottom of lakes more penetrating X-rays recreation in space annihilation of matter an alternative picture the earth as a magnet effect on incoming rays two international surveys both explanations may be right page 119 Chapter VII. THE ORIGIN OF MAN Evolution proved man can scarcely be an exception yet his ancestry so far irrecoverable arguments from anatomy from blood sera from embryo development search for missing links Neanderthal man Java ape-man Piltdown man in England Pekin man no single line of ascent the pro blem of Africa our ape relations difficulties of parallel evolution hopes of further finds page 147 Chapter VIII... ... Read more

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Philosophers and historians of the French Revolution have seen Rousseau's influence as the decisive link between the doctrines of the Enlightenment and the practice of its revolutionary disciples. Professor Leigh here addresses the bibliographical foundations of that question, without which all attempts to settle it in the past have lacked authority. Introducing the most advanced techniques to identify variant and pirate editions of Rousseau's writings, he establishes that there were at least 28 separate imprints and an additional 12 reprints of the Contrat Social in collective editions between 1762 and 1783. Professor Leigh shows also that Rousseau's life and thought excited a fascination and interest in the last years of the Ancien Regime which was nursed by the publishers of his Oeuvres, who sought to satisfy an apparently ceaseless demand by extending their editions, while at the same time attempting to ward off both their creditors and their imitators. ... Read more

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Although the Milky Way is the most studied and best understoodgalactic system, there are many fundamental questions about our Galaxythat remain unanswered. This book concentrates on those questionswhich have the widest applicability in all of astrophysics, and forwhich answers are most likely to be forthcoming in the next few years.Is the Milky Way a barred spiral, and if so, what are its properties?Is the disk of the Milky Way axisymmetric and what does the answertell us about its dynamical history? Is there a black hole at thecenter of the Galaxy? How far does the Galaxy extend? How much darkmatter is there in the Milky Way system? And more. ... Read more