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Carnival of Mathematics will be here on April 6

Wednesday, March 28, 2007

Yes, right here on Friday, April 6. This is your personal invitation to suggest blog posts on mathematics at any level – your own or someone else's you especially liked. Posts on theoretical computer science or mathematics in other sciences are also welcome.

For more information, see here.

To contribute, just send me a note here: carnival AT scienceandreason.net. Please put "Carnival of Mathematics" in the Subject line, and respond by 6PM PDT on April 5. You can also use this form.

Even if you have nothing to submit, check back here April 6 to see what bloggy mathematicians are up to.

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P53 protein and tanning

Sunday, March 25, 2007

At first this may seem an odd coincidence, but maybe it isn't. P53 is the protein which plays a critical role in preventing runaway division in cells with damaged DNA, and hence inhibiting cancer. Unless p53 itself becomes faulty – which happens in the majority of cancerous cells. It does its job by stopping the cell division cycle if damaged DNA is detected.

But apparently p53 is also implicated in tanning of human skin by the sun.

'Guardian Of The Genome' Protein Found To Underlie Skin Tanning
A protein known as the "master watchman of the genome" for its ability to guard against cancer-causing DNA damage has been found to provide an entirely different level of cancer protection: By prompting the skin to tan in response to ultraviolet light from the sun, it deters the development of melanoma skin cancer, the fastest-increasing form of cancer in the world.

In a study in the March 9 issue of the journal Cell, researchers at Dana-Farber Cancer Institute report that the protein, p53, is not only linked to skin tanning, but also may play a role in people's seemingly universal desire to be in the sun -- an activity that, by promoting tanning, can reduce one's risk of melanoma.

"The number one risk factor for melanoma is an inability to tan; people who tan easily or have dark pigmentation are far less likely to develop the disease," says the study's senior author, David E. Fisher, MD, PhD, director of the Melanoma Program at Dana-Farber and a professor in pediatrics at Children's Hospital Boston. "This study suggests that p53, one of the best-known tumor-suppressor proteins in our body, has a powerful role in protecting us against sun damage in the skin."

Of course, people who tan easily or have dark pigmentation may also be less inclined to spend time in the sun for the purpose of acquiring a tan, so any other factors in an individual that might be responsible for tannning or dark pigmentation would also indirectly reduce the statistical liklihood of melanoma.

However, the research shows that p53 does influence tanning directly.

Other reports:

Update 8/3/08: There is related news about this here.

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E8

Friday, March 23, 2007

Hi there, mathematicians. I assume you've heard the big news about E8 by now:

248-dimension maths puzzle solved
An international team of mathematicians has detailed a vast complex numerical "structure" which was invented more than a century ago.

Mapping the 248-dimensional structure, called E8, took four years of work and produced more data than the Human Genome Project, researchers said.

Lie groups aren't a specialty of mine, so I'll hold off writing about this (for now), but you might be interested to read what a few other are saying.


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News dump: inflammation

I've been collecting news stories on a variety of interesting topics, but don't quite have the time to discuss all of them right away. So what I'll do is put up the list with short summaries. And hope to come back with more details later. It's sort of like spring cleaning time.

First up: inflammation. It's one of the still not well understood "features" of our complex immune system.

Inflammation and cancer


February 20, 2007 – Antibody signal may redirect inflammation to fuel cancer
The body's normally protective inflammation response can drive some precancerous tissues to become fully malignant. The inflammation apparently stimulates B cells to send signals that trigger progams for stimulating cancerous cell growth and increased blood supply to tumors.

January 25, 2007 – Molecular Link Between Inflammation And Cancer Discovered
There is much evidence that chronic inflammation can promote cancer, but the cause of this relationship is poorly understood. New research shows a linkage, involving the protein called p100, between cellular pathways for response to infection and cellular growth.

December 21, 2006 – What Cures Your Aches Might Prevent Cancer: Seeking To Prevent Cancer Using Anti-inflammatory Medication
Some of the same biological processes that cause inflammation may also be involved in developing cancer. This suggests investigating whether drugs that prevent inflammation also serve to lessen the risk of cancer. Attention is focused on nonsteroidal anti-inflammatory drugs like aspirin.

April 4, 2006 – Inflammation and Drugs to Control this Activity Studied in a Variety of Tumor Sites
Several studies presented at the 97th Annual Meeting of the American Association for Cancer Research show a relationship between chronic inflammation and cancer. The COX-2 enzyme, produced as part of the inflammatory response, seems to be associated with development of some types of cancer. Drugs that inhibit COX-2 (nonsteroidal anti-inflammatory drugs) may also inhibit cancer.

March 29, 2006 – Studies link cancer, inflammatory disease
The tumor necrosis factor (TNF) protein, which is involved in an inflammatory response, normally promotes death of damaged or infected cells, but overproduction can lead to automimmune diseases like rheumatoid arthritis. TNF may also stimulate production of the epidermal growth factor (EGF) protein, which may lead to tumor growth. So drugs that inhibit excessive TNF production may help inhibit cancer.


Inflammation and cardiovascular disease


March 4, 2007 – Treatment For Gum Disease Could Also Help The Heart
Periodontitis is a common inflammatory disease of the gums, caused by bacterial infection of gum tissue. This clinical trial is the first to demonstrate that relief of inflammation in the mouth, through intensive treatment of periodontitis, results in improved function of the arteries. The mechanism by which periodontitis affects endothelial function in the body is still uncertain. The periodontitis might trigger a low grade inflammatory response throughout the body that has a detrimental effect on the vascular wall.

February 22, 2007 – C-Reactive Protein Liver Protein Induces Hypertension, Researchers Find
Research with genetically engineered mice has shown that artificially elevated levels of C-reactive protein (CRP), normally a marker of inflammation, lead directly to higher blood pressures. It was found that the initiating mechanism is a lack of the nitric oxide in the artery wall. The lack of nitric oxide affected proteins responsible for activity of angiotensin II, which regulates blood pressure via arterial constriction.

December 30, 2006 – Inflammatory Genes Linked To Salt-sensitive Hypertension
Inflammation, a part of the immune response implicated in diseases such as cancer, Alzheimer’s and diabetes, may also help translate stress into high blood pressure. When stress activates the sympathetic nervous system, the body increases production of interleukin 6, a pro-inflammatory factor, which ultimately leads to production of other inflammatory factors such as C reactive protein. There is evidence suggesting that in salt-sensitive hypertension there are increased levels of inflammation factors such as interleukin 6 and C-reactive protein.

May 5, 2006 – Periodontitis May Increase C-reactive Protein Levels In Pregnancy
Pregnant women with periodontitis had 65 percent higher C-reactive protein (CRP) levels compared to periodontally healthy women. CRP levels are a marker of systemic inflammation and are associated with periodontal disease. CRP could amplify the inflammatory response. Alternatively, periodontal disease and CRP may share a common risk factor for predisposing individuals to a hyperinflammatory response.


Inflammation and obesity


February 19, 2007 – Obesity Finding: Chemical Pathway Causes Mice To Overeat And Gain Weight
"Knockout" mice bred to lack what is known as an E3 receptor cannot process prostaglandin E2, which is normally produced in the context of inflammation. Such mice, apparently as a consequence, do not develop a fever response. They also tend to overeat and accumulate body fat. It may be that an inability to respond to inflammation inhibits mechanisms which would otherwise control overeating.

April 10, 2006 – Research Provides Clues To Obesity's Cause And Hints Of New Approach For Curbing Appetite
New research suggests obesity is due at least in part to an attraction between leptin, the hormone that signals the brain when to stop eating, and C-reactive protein, which has been associated with heart disease. CRP not only binds to leptin but it impairs leptin's role in controlling appetite. Since fat cells produce CRP, these results may help explain why obese people have so much trouble controlling appetite and losing weight.

March 10, 2006 – Research Shows Fat Fuels Inflammation Killer
The protein sE-selectin is produced in response to inflammation in the walls of arteries. It is measured in greater amounts in people who have higher levels of body fat, indicating a higher level of inflammation in their arteries. It is known that such inflammation can directly trigger thrombosis, heart disease, strokes and diabetes.

September 16, 2005 – First Link Found Between Obesity, Inflammation And Vascular Disease
This research shows that human fat cells produce C-reactive protein (CRP), which is linked to both inflammation and an increased risk of heart disease and stroke. It would explain why higher levels of CRP are measured in overweight individuals, and why such individuals have a higher risk of cardiovascular problems. This study is the first to show how body fat participates in the inflammatory process that leads to cardiovascular disease.


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Groups and rings

Monday, March 19, 2007

In our previous installment of the series on algebraic number theory, we took a little detour into Diophantine equations in order to provide some motivation for the theory itself. Prior to that, we had looked at different types of numbers, to give a perspective on the sorts of objects the theory deals with.

In those discussions, we touched (explicitly or implicitly) on abstract algebraic structures called groups, rings, and fields. All three of these concepts are incredibly important in the theory of algebraic numbers, and to a very large extent in the rest of modern mathematics as well. Today we'll deal with groups and rings. In the next installment we'll take up fields and Galois theory. This won't be in any great detail – just the basic concepts. Further depth will be introduced later, as it becomes necessary.

Groups



We begin with groups. As with most other sorts of algebraic systems, groups are defined abstractly in terms of sets of elements satisfying certain axioms. The axioms for a group are not the simplest that an interesting mathematical system can have -- monoids and semigroups have somewhat weaker axioms. But groups are just about the simplest objects that occur commonly in algebraic number theory.

A group G is a mathematical system consisting of a set of elements and one operation between any two elements of the set. If "∘" denotes the operation, in a group there are three requirements:

  1. the operation should be associative: x∘(y∘z) = (x∘y)∘z for all x, y, z in G;
  2. there should be an identity element "e": e∘x = x∘e = x for all x in G;
  3. every element of G should have an inverse: x∘x-1 = x-1∘x = e.


These axioms can be stated in slightly different ways, but we don't need to get into that.

Note one respect in which these group rules are different from the usual rules of either addition or multiplication in arithmetic: the commutative property x∘y = y∘x is not required for elements of a group, though it might hold for some or even all elements. If it does hold for all group elements, the group is said to be commutative or abelian (after Niels Abel). In the theory of algebraic numbers, whenever groups consist of actual algebraic numbers they will necessarily be commutative, since the rules of arithmentic (both addition and multiplication) still hold. But we will encounter groups that are defined in different ways that definitely won't be commutative. Some of the hardest problems of the theory, in fact, occur in the non-commutative cases.

For a nontrivial example of a commutative group that's important in algebraic number theory, just look at the set of all units, as we discussed in reference to Pell's equation. As you recall, we denoted by Z[√n] the set of numbers of the form a+b√n, where a and b are integers, and n is a positive integer that's not a perfect square. (Z[√n] is in fact a ring, as we'll define the term in a moment.)

Within that set, consider the subset of numbers such that the equation a2 - nb2 = ±1 holds. In other words, the "norm" of a+b√n, N(a+b√n), as defined by the left hand side of the equation, has the value ±1. We noted that this condition is necessary and sufficient for a number in the subset to have a multiplicative inverse. We called such numbers units of the ring Z[√n]. Note that 0 is not a unit, but 1 is, and that the existence of a multiplicative inverse of any unit makes the set of units into a commutative group under multiplication (with ordinary addition being irrelevant in this group – indeed, the sums and differences of units are not units).

Another thing to note is that the requirement for an identity element is a requirement for a solution to a certain simple equation, and we have seen this in action several times. For instance, the natural numbers N (nonzero integers) do not form a group under addition, because there is in general no solution to an equation of the form x+a = 0 with arbitrary a∈N. But if we extend N to the integers Z by "adjoining" all negative numbers, we have in effect simply included all formal solutions of equations x+a=0 for each a∈N and gotten lucky in that the enlarged domain satisfies the group axioms without difficulty.

Very much the same thing happened with respect to the operation of multiplication when we passed from the integers Z to the rationals Q. Again, with respect to multiplication, Z satisfies the group axioms except for the existence of inverses. That is, we are not able to solve the equation xa = 1 for arbitrary a∈Z. In fact, a solution exists only for a = ± 1. But in defining the rational numbers Q in effect we just formally adjoined the inverses (reciprocals) 1/a for each a∈Z (except a=0). The resulting group with the operation of multiplication consists of all nonzero elements of Q. This group is sometimes denoted by Q×.

If we wanted to preserve the additive structure of Z at the same time as providing multiplicative inverses, in order to construct the ring Q, we would need to have been a little subtler. This process is a standard one. It is called constructing a field of fractions, and we will come back to it.

Rings



Let's look at rings next, since they are the next major level up in axiom complexity. A ring is a mathematical system which has two distinct binary operations: "+" and "×", which are intended to be rather like the addition and multiplication of ordinary arithmetic. If R is a ring, then it satisfies the axioms for a commutative group with respect to addition. With respect to multiplication, R must satisfy the associative axiom. Sometimes rings are not required to have a multiplicative identity element, but most in fact do. Inverse elements, however, do not typically exist, even if there is a multiplicative identity element. Addition in a ring is always commutative, but multiplication need not be. If the multiplication is commutitive, the ring is a commutative ring. The rings that occur in algebraic number theory are commutative rings if they consist of ordinary algebraic numbers, but a few important cases of rings (matrix rings for example) aren't commutative.

In addition to the requirements on the operations of addition and multiplication seprately, they must satisfy a compatibility condition, known as the distributive law of multiplication with respect to addition:

  • for all a, b, c in R, a(b + c) = ab + ac

If multiplication in R isn't commutative, the same thing must hold for multiplication on the right as well. One consequence of this axiom is that if 0 is the additive identity element, a0=0a=0 for all a∈R.

The integers Z are the most obvious example of a ring. For any field F containing Q, there is also the concept of a ring of integers of the field F. This ring is a direct generalization of Z, and it is one of the central objects of study. One wants to know as much as possible about the structure of such rings, because this knowledge has extensive practical application to the study of Diophantine equations, as we shall see. Elements of a such a ring of integers are called, simply, algebraic integers.

In many respects, of groups, rings, and fields, it is rings which are most interesting. They have the complexity due to possessing two operations, but the freedom of a less restrictive set of axioms than fields. This results in many more special situations, though not all of the strong theorems about fields (such as Galois theory) apply to rings.

We'll have a lot more to say about abstract ring theory, but in the next installment the theory of fields and Galois theory will be reviewed.

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MicroRNA

Wednesday, March 14, 2007

MicroRNA (miRNA) is a short (about 21 to 23 nucleotides) single-stranded RNA molecule that is now recognized as playing an important role in gene regulation – even though the term has been in use only since 2001. It is similar to, but distinct from, another type of short RNA, known as small interfering RNA (siRNA).

Although miRNA and siRNA both have gene regulation functions, there are subtle differences. MiRNA may be slightly shorter than siRNA (which has 20 to 25 nucleotides). MiRNA is single-stranded, while siRNA is formed from two complementary strands. The two kinds of RNA are encoded slightly differently in the genome. And the mechanism by which they regulate genes is slightly different.

MiRNA attaches to a piece of messenger RNA (mRNA) – which is the master template for building a protein – in a non-coding part at one end of the molecule. This acts as a signal to prevent translation of the mRNA into a protein. SiRNA, on the other hand, attaches to a coding region of mRNA, and so it physically blocks translation.

In addition to the Wikipedia articles, here's another handy source of information on miRNA.

There have been several research results reported recently that illustrate some of the important functional roles of miRNA.

MicroRNA and cancer



The importance of gene regulation by miRNA is not trivial. As the following article notes, "microRNAs found in mammals regulate over a third of the human genome, as shown in a 2005 study by the lab of Whitehead Member and Howard Hughes Medical Institute Investigator David Bartel and colleagues." (Reference: MicroRNAs Have Shaped The Evolution Of The Majority Of Mammalian Genes)

Since either overexpression or underexpression of certain genes can cause cancer, it's not surprising that miRNA should have significant cancer-related effects.

MicroRNA helps prevent tumors
Looking to find a promising target for an individual microRNA, Christine Mayr, a postdoctoral researcher in the Bartel lab, picked Hmga2, a gene that is defective in a wide range of tumors.

In these tumors, the protein-producing part of the Hmga2 gene is cut short and replaced with DNA from another chromosome. Biologists have mostly focused on the shortened protein as the possible reason that the cells with this DNA swap became tumors. But this DNA swap removes not only the gene's protein-producing regions but also those areas that don't code for protein. And these non-protein-producing regions contain the elements that microRNAs recognize.

It turns out that in the non-protein-producing region, Hmga2 has seven sites that are complementary to the let-7 microRNA, a microRNA expressed in the later stages of animal development. Mayr wondered whether loss of these let-7 binding sites, and therefore loss of regulation by let-7 of Hmga2, might cause over-expression of Hmga2 that in turn would result in tumor formation.

This turned out to be a very good guess:
Overall, the results highlight a new mechanism for cancer formation. Hmga2, and perhaps certain other genes that are normally regulated by microRNAs, can help give rise to tumors if a mutation in the gene disrupts the microRNA's ability to regulate it.


MicroRNA and stem cells



It's not news to anyone that research into stem cells is a very active area these days. It turns out that miRNA may play a key role in keeping stem cells from differentiating prematurely into normal body cells.

Master Switches Found For Adult Blood Stem Cells
Johns Hopkins Kimmel Cancer Center scientists have found a set of "master switches" that keep adult blood-forming stem cells in their primitive state. Unlocking the switches' code may one day enable scientists to grow new blood cells for transplant into patients with cancer and other bone marrow disorders.

The scientists located the control switches not at the gene level, but farther down the protein production line in more recently discovered forms of ribonucleic acid, or RNA. MicroRNA molecules, once thought to be cellular junk, are now known to switch off activity of the larger RNA strands which allow assembly of the proteins that let cells grow and function.

Since a miRNA molecule can attach itself to a mRNA molecule if there is a match in only about seven consecutive nucleotides, it wouldn't be surprising if one miRNA could regulate the translation of many different proteins. And indeed, this is one of the findings of the research:
To identify the key microRNAs, Georgantas sifted through thousands of RNA pieces with a custom-built, computer software program. Its algorithms let the software, fed data from samples of blood and bone marrow from healthy donors, match RNA pairs. The outcome was a core set of 33 microRNAs that match with more than 1,200 of the larger variety RNA already known to be important for stem-cell maturation.

Just as important for the persistence of a species are stem cells for germline cells – eggs and sperm.

MicroRNA Pathway Essential For Controlling Self-renewal Of Stem Cells
"The findings were interesting to us because they demonstrated that the microRNA pathway is essential for controlling self-renewal or maintenance of two types of stem cells – germline stem cells and somatic stem cells," said Dr. Jin. "In the future, the small RNAs responsible for stem cell regulation could potentially be used to control stem cell functions in vivo and stem cell expansion in vitro."


Editing of microRNA



We have noted that a single miRNA can affect the expression of a large set of genes. It turns out that relatively minor editing of the miRNA after initial transcription can cause it to affect a completely different set of genes:

Killing the messenger RNA — But which one?
Now, a new study led by researchers at The Wistar Institute shows that these microRNAs can undergo a kind of molecular editing with significant physiological consequences. A single substitution in their sequence can redirect these microRNAs to target and silence entirely different sets of genes from their unedited counterparts. Further, errors in the editing can lead to serious health problems.

"What we found was that, in certain cases, edited versions of these microRNAs are being produced that differ from the unedited versions by only a single nucleotide change," says Kazuko Nishikura, Ph.D., a professor in the Gene Expression and Regulation Program at Wistar and senior author on the study. "These edited microRNAs are not encoded in the DNA, which means that at least two versions can being produced by one gene.

If there's one conclusion to be derived from all this recent research, it is that a relative handful of miRNA species – several hundred have been identified so far, compared with 25,000 or so protein-coding genes in humans – are capable of drastically influencing all kinds of cellular processes. That's a lot of "leverage". Applied at the right times and places, miRNA could provide therapies for a large number of diseases. But of course, the ability of a single miRNA to affect so many different genes means that they have to be targeted very, very carefully. It will be interesting to see how this plays out.

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Adolescent sex

Handheld Computer Study Captures Emotional State Of Sexually Active Adolescents
Through a unique and innovative data-gathering method, researchers at Children's Hospital Boston have gained new insight into adolescents' sexual behavior and how sex affects their moods. Their findings appear on-line in the Journal of Adolescent Health.

Using handheld personal digital assistants (PDAs), the researchers closely tracked 67 sexually active youth, aged 15 to 21, from an adolescent medicine clinic at an urban children's hospital. Each participant was given a PDA that beeped them at random, four to six times a day, asking them a series of questions about any recent sexual activity and their emotional state and feelings.

Kind of surprising that kids these days are so open about their sex lives. That's a good sign.
The 67 adolescents reported on a total of 266 unique sexual intercourse reports, 94 percent of which were with a main partner and only 49 percent involved the use of a condom.

Don't tell this to the Pope. He might croak. Then again, that wouldn't be such a bad idea... However, he might just be ecstatic that fewer than half the reports involved a condom...
The findings suggest adolescents of both sexes tended to feel more positive and less negative after engaging in sex than at times after they had not.

What's amazing is how much trouble social scientists will go to in order to document the obvious... But I suppose this finding will disappoint those who think sex with a "main partner" should be an occasion for anything but positive feelings.

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Philosophia Naturalis #8 call for submissions

Tuesday, March 13, 2007

Philosophia Naturalis #8 will be hosted at Sujit Datta's blog metadatta on Thursday, March 29.

If you've recently written or read a noteworthy blog article on physics, math, astrophysics, chemistry, Earth science, advanced technology, or any other physical science topic, read the call for submissions for details of how to submit it for inclusion in the carnival.

Sujit needs the submissions a little early – by March 23 – so don't delay. Send something in now.

Posterity will thank you.

No Single Gene For Eye Color, Researchers Prove

Monday, March 12, 2007

Since I wrote a little about eye color before, I think I ought to clarify some things. The earlier article was really about how recessive and dominant genes can have an indirect effect on social behavior via evolution. But the truth about the genetics of eye color is actually somewhat more complicated:

No Single Gene For Eye Color, Researchers Prove
A study by researchers from The University of Queensland's Institute for Molecular Bioscience (IMB) and the Queensland Institute of Medical Research is the first to prove conclusively that there is no single gene for eye colour.

Instead, it found that several genes determine the colour of an individual's eyes, although some have more influence than others.


Update 8/3/08: There is more recent news on this subject here.

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Feeling hot tonight?

I really can't understand why this wasn't terribly obvious. Here's an update of a story I wrote about before.

New Measure Of Sexual Arousal Found For Both Men And Women
According to a new study published in the latest issue of The Journal of Sexual Medicine and conducted in the Department of Psychology of McGill University, thermography shows great promise as a diagnostic method of measuring sexual arousal. It is less intrusive than currently utilized methods, and is the only available test that requires no physical contact with participants.


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Oddest science story of the week

Thursday, March 8, 2007

Public 'lack knowledge about sex'
The public have a worrying lack of knowledge and understanding about sex, a survey suggests.

A poll of 495 people by the Family Planning Association found some thought exercise or urinating after intercourse could prevent pregnancy.

And, unfortunately, these are the ones most likely to have babies...

Diophantine equations

Sunday, March 4, 2007

Remember, our eventual goal is to talk more about algebraic number theory. But first let's take a short detour to say a little about plain old-fashioned number theory, which has been a subject of investigation for over 1800 years. Primarily, what this has been about is the study of what are called Diophantine equations. In them we'll find some motivation for later topics.

Diophantine equations get their name from Diophantos of Alexandria. Diophantos flourished in the 3rd century CE and wrote a highly regarded treatise he called the Arithmetica. Much of this dealt with solving algebraic equations in several unknowns, with the added restriction that solutions had to be rational or integral. This was especially important to the Greeks because of the discovery by Pythagoreans centuries earlier that many solutions of algebraic equations, even something as simple as the square root of 2, were not rational numbers. Then, as now, the term "irrational" had very negative connotations (though no longer in modern times as regards numbers).

In any case, early mathematicians were very interested in finding solutions of simple algebraic equations that are rational or integral numbers. This is often a very natural condition to impose on a particular problem. The equation, for example, may apply to things which ordinarily occur in whole number amounts, such as living animals.

A Diophantine equation, then, is an algebraic equation for which rational or integral solutions are sought. (Systems of such equations are also considered.) An algebraic equation is one that involves only polynomial expressions in one or more variables. What makes the equation "Diophantine" is that the coefficients of the polynomials should be rational numbers (or often, integers), and further that only rational (or integer) solutions are sought. Since these are the only conditions, not much in the way of a general theory has developed, though a lot is known about many more specialized cases.

The closest thing to a general theory is to be found in algebraic geometry, which deals with the geometric properties of solution sets of algebraic equations, usually over an "algebraically closed" field of numbers, such as the complex numbers. Algebraic geometry, indeed, turns out to be very helpful in solving problems with Diophantine equations, and a great deal of very deep and beautiful theory has been developed. However, a general theory that can deal with finding integral or rational solutions of arbitrary algebraic equations, or even with determining whether such solutions exist, is still a long way off.

In 1970 Yuri Matiyasevich gave a negative solution of the so-called 10th Hilbert problem by showing that there is no general decision procedure to determine whether solutions of an arbitrary Diophantine equation exist. Subsequently Matiyasevich showed that equations with as few as 9 variables lack a decision procedure. It is possible there may be equations with even fewer variables that lack such a procedure. However, the methods used in Wiles' proof of Fermat's Last Theorem indicate that all equations in 3 variables should be decidable. It is at present an open question what the least number of variables of an undecidable equation might be.

Nevertheless, we do have some very illuminating theory. For instance, there is a very celebrated conjecture stated by Louis Mordell in 1922 which says, roughly, that if P(x,y) is a polynomial with rational coefficients, then the equation P(x,y)=0 has only finitely many rational solutions (provided P(x,y) also has a "genus" greater than 1). This was an open problem until a proof was finally given by Gerd Faltings in 1983. The result can be interpreted geometrically as saying that a surface which contains all complex number solutions of the equation has only finitely many points on it with rational coordinates.

The most famous Diophantine equation of all, of course, is Fermat's equation: xn + yn = zn. Andrew Wiles finally proved in 1995 that, if n is 3 or more, this equation has no nonzero integral solutions -- even though Fermat had conjectured as much more than 350 years earlier. The final solution turned out to require extremely sophisticated tools from algebraic geometry (such as the theory of elliptic curves), and much more besides from a great deal of the remainder of modern mathematics.

Note that Fermat's problem is really about solutions of an infinite set of equations (one for each value of n). Technically speaking, if n were regarded as one of the variables in the equation, it would no longer be algebraic. Many other famous Diophantine problems have this feature, where one of the "variables" occurs as an exponent.

Elementary examples



Diophantine equations need not be limited to equations in only one variable (such as x2-2 = 0). It's frequently more interesting to consider equations with several variables, such as the Fermat equation. Just about the simplest equation with two variables and degree higher than 1 is x2-y2 = 0. (The degree of a single term is the sum of exponents of all variables occuring in the term. The degree of the equation is the largest of the degrees of its terms.) But this equation isn't very interesting, because it can be written as a product of polynomials of first degree: (x+y)(x-y) = 0. When the polynomial in such an equation is a product of factors of degree one having suitable coefficients, the solutions consist of the solutions of any of the factors, and those are trivial.

The next simplest second degree equation in two variables is x2+y2 = 1. (The equation x2+y2 = 0 has only the trivial solution x=y=0 when excluding complex numbers.) You probably recognize that as the equation of a circle of radius 1. Any point on the circle has coordinates satisfying the equation, and vice versa. The polynomial x2+y2-1 does not factor into first degree polynomials with real (i. e., not imaginary) coefficients. Nevertheless, it has solutions for x and y that are rational numbers, and even integers, such as (x,y) = (±1,0) or (0,±1). Clearly these are the only integer solutions, because we must have -1≤x≤1, so x has to be -1, 0, or 1, and so does y.

It was Diophantos himself who discovered all the rational solutions of this equation. He found what is called a parametric solution, obtained via elementary algebra. Namely, if t is any rational number (or even complex number, for that matter), then
x=(1-t2)/(1+t2), y=2t/(1+t2)

gives a point on the circle, and the coordinates are obviously rational if t is. Conversely, suppose (X,Y) is any point on the circle. It also lies on a straight line passing through the point (-1,0), having some slope t. The equation of this line is Y=t(X+1). The line intersects the circle at a unique point besides (-1,0), and if you do the algebra, you see that point must be given by X=(1-t2) and Y=2t/(1+t2). Since t = Y/(X+1), t must be rational if X and Y are. This shows that every rational point (except (-1, 0)) on the circle has the parametric form given, for some rational t. I. e., there's a very tidy 1:1 correspondence between rational numbers and points on the circle with rational coordinates.

This is actually just a special case of a second degree equation in three variables, which was extremely important to Greek mathematicians, namely x2+y2 = z2. Geometrically, this is the equation of a circle of radius z, centered on the origin. But more importantly to the Greeks, if x and y are the lengths of the sides of a right angled triangle, the length of the hypotenuse is z. This equation was discovered by Pythagoras (or someone in his school) some time in the 6th century BCE, and (of course) it was known as the Pythagorean theorem. If the lengths of the sides of the triangle are whole numbers, it is not necessarily true that the so is the length of the hypotenuse, since all we know is z=√(x2+y2). Indeed, the length isn't necessarily even rational – a fact which was considered quite scandalous at the time.

But if the length of the hypotenuse is a whole number, then (x,y,z) is called a Pythagorean triple. Such a triple was held in mystical regard by the Pythagoreans, so the search for such triples assumed a high importance. This search amounted to finding integer solutions of what we now call an instance of a Diophantine equation in 3 variables. Note that any solution in rational numbers also yields an integral solution. This is because if (x,y,z) is a rational triple, then multiplying through by the least common multiple of the denominators gives a triple of integers. (This works because the equation is homogeneous, with all terms having the same degree.)

No lesser a man than Euclid discovered how to describe all solutions in (positive) integers of this equation. Again, this is given in parametric form:
x=(u2-w2)w, y=2uvw, z=(u2+w2)w
It's trivial to check that you get a soution of the equation for any integer u, v, w (including 0 and negative values). The proof that this gives all solutions is harder and we leave that for you to think about.

Pell's equation



With the next step up in complexity, we reach some interesting territory that's directly connected with algebraic number theory. This involves an equation of the form x2 - ny2 = 1 for some positive integer n. n is fixed and not an additional variable. Solutions of this equation, however, depend very much on the arithmetic properties of n, and in some sense help us understand these properties.

The name "Pell's equation" was conferred by Leonhard Euler, who mistakenly gave credit for it to the otherwise obscure mathematician John Pell. However, the history of the equation goes all the way back to the Greeks, who were especially concerned with the case n=2, because it shines some light on the number √2. As you recall, √2 was of great interest (or concern) to them, ever since they realized √2 is not a rational number.

If y=0, then x=±1, so let's exclude that trivial solution. Then we can rewrite the equation as (x/y)2 = n+1/y2. Suppose there exist infinitely many solutions of the equation (as is in fact the case). Taking one with y large enough, 1/y2 can be arbitrarily small. And therefore, a solution (x,y) gives us a rational number x/y as close as we like to √n. This fact was already known implicitly to the Greeks, who also understood limit arguments. So they were pleased to know that even though √2 isn't rational, it can be approximated arbitrarily well by rational numbers. And the same is true of √n if n isn't a perfect square, so that √n is not rational.

Rather than pursue the details of Pell's equation, we need to look at how it's related to algebraic number theory. Recall that the set of all rational numbers is usually denoted by Q. Mathematically, this set has the structure known as a ring, in that it has the arithmetic operations of addition, subtraction, and multiplication, which observe certain rules. In fact, Q also has the structure of a field, which also allows for division by any of its elements other than 0. We will say much more about rings and fields in the next installment, so let's dispense with the formalities for now. It's safe, for present purposes, to think of the arithmetic operations of rings and fields as the ones you are already quite familiar with.

The integers, Z, also form a ring. Let Z[√n] stand for the set of all polynomials in powers of √n with coefficients in Z, and likewise for Q[√n] (with coefficients in Q). If n is a perfect square, these sets are just Z and Q, so let's assume n isn't a prefect square. It's obvious that these are rings also. But they are not necessarily fields, since 1/√n isn't rational if √n isn't.

What may be surprising, however, is that some elements of the form a+b√n may have inverses in these rings even if b≠0 and a≠±1. For instance, let n=2, and consider the number α=3+2√2 in Z[√2]. Observe that (3+2√2)(3-2√2) = 1. Hence 1/α = 1/(3+2√2) = 3-2√2, which is also an element of Z[√2]. Thus α has an inverse in Z[√2]. Such an element is called a unit. The only units of Z itself are ±1. But it turns out that for any positive n that's not a perfect square, the ring Z[√n] has infinitely many units.

What do these units look like? Well, first we need to define one more thing. If a+b√n is in Z[√n], define the function N(a+b√n) to be the number (a+b√n)(a-b√n) = a2-nb2. (This is the same definition as the square of the norm of a complex number, where n=-1.) Clearly, N(α) is also an integer for any α∈Z[√n]. It's simple algebra to show that if α,β∈Z[√n], then N(αβ)=N(α)N(β).

Given all this, if α∈Z[√n] is a unit, with inverse 1/α, so α(1/α)=1, we must have N(α)=1/N(1/α) is an integer, so N(α) is a unit of Z, which means it must be ±1. Conversely, if N(&alpha) = N(a+b√n) = (a+b√n)(a-b√n) = ±1, then α is a unit because it has an inverse. Hence the units α of Z[√n] are precisely the α such that N(α)=±1.

This is what allows us to show that Pell's equation has an infinite number of solutions, if there are any at all besides ±1 (which we haven't actually shown here). This is because every solution of Pell's equation a2-nb2 = 1 gives us a unit α = a+b√n with N(α)=1. For any such α, N(±αk)=1 too, for any k∈Z by the above, hence all ±αk are units, and give solutions of Pell's equation. Moreover, these are all distinct, because the only rational numbers that are units are ±1 (when b=0). For any irrational unit α the absolute value |α|≠1. We may assume |α|>1 (otherwise use 1/α). Then αk are distinct numbers for all positive k∈Z because the absolute values |αk|=|α|k are all distinct.

There may be other units as well, satisfying a2-nb2 = -1. For example, if n=5, we have N(2+√5)=-1, so 2+√5 is a unit of Z[√5]. If there exists a unit β = a+b√n with N(β)=-1, then there are also an infinite number of solutions of a2-nb2 = -1, corresponding to the units βk for odd integers k. (If k is even, we get solutions of Pell's equation.)

You may think of algebraic number theory as the study of rings such as Z[√n] and certain generalizations. The serious study of Diophantine equations leads naturally to consideration of such things. Conversely, the study of particular Diophantine equations, such as Pell's, can tell us a lot about the properties of algebraic structures that come up in algebraic number theory.

Our next installment will deal with these abstract structures, such as rings and fields, in a lot more detail.

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Philosophia Naturalis #7 - tabloid journalism at its best!

Friday, March 2, 2007

The truly amazing and astounding thing about this edition of Philosophia Naturalis is that it does not contain anything about flying saucers and space aliens!

How could such a government scandal have been hidden so long? But now the truth is finally coming out: UFO science key to halting climate change: former Canadian defense minister
A former Canadian defense minister is demanding governments worldwide disclose and use secret alien technologies obtained in alleged UFO crashes to stem climate change, a local paper said Wednesday.


But wait! There's even more tabloid goodness to this story. You see, it must be Bill Clinton's fault! In fact, Limbaugh says this is all a Liberal scam! Just like global warming itself!

Whatever. After you've digested that little bit of news, head on over to Lorne's place at Geek Counterpoint to read other incredible facts!
 

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