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The Colorful Demise of a Sun-Like Star

Sunday, February 25, 2007

The Colorful Demise of a Sun-Like Star
The planetary nebula in this image is called NGC 2440. The white dwarf at the center of NGC 2440 is one of the hottest known, with a surface temperature of nearly 200,000 degrees Celsius (400,000 degrees Fahrenheit). The nebula's chaotic structure suggests that the star shed its mass episodically. During each outburst, the star expelled material in a different direction. This can be seen in the two bow tie-shaped lobes. The nebula also is rich in clouds of dust, some of which form long, dark streaks pointing away from the star. NGC 2440 lies about 4,000 light-years from Earth in the direction of the constellation Puppis.

NGC 2440 – click for 1241×1207 image

See also: here

Cytokine storms

Friday, February 23, 2007

Question: What do the following life-threatening medical problems have in common: avian flu, SARS, and anthrax? If you guessed "cytokine storms", you're correct. What was your clue?

In view of the medical problems in which cytokine storms have been implicated, the topic is obviously of high importance. However, because they represent a malfunction in the immune system, which is quite complex, cytokine storms are presently not at all well understood, so it's hard to make definitive statements about them. Indeed, cytokine storms are more of a symptomatic condition and could occur in varying forms, involving a number of different mechanisms. "Storm" may be an appropriate metaphor, acknowledging a variety of mechanisms in a variety of circumstances. What different examples have in common is certain components of the immune system becoming seriously out of control and causing life-threatening problems.

As background for this discussion, you might want to read (or reread) our previous article on T cells. One of the key players in this drama is the subtype of T cell known as helper T cells (also known synonymously as Th or CD4+ cells). As the name implies, Th cells assist other types of immune system cells in performing their function, so a number of other cell types may also be involved in a cytokine storm.

Immune system cells (as well as various other cell types) communicate among each other with chemical messages known as cytokines, which are proteins or peptides (small proteins). The list of known cytokines is large and continually growing. To make matters worse, the nomenclature is not well-standardized and consistent, but some examples you may have come across (if you ever read medical literature) are various kinds of interleukins, interferons, and tumor necrosis factor (TNF).

Although cytokine storms are not well understood, here's a general overview of what happens. Th cells appear to play a central role. The reason they are called "helper" cells is that they produce cytokines which in turn affect the behavior of other types of immune system cells. Which cytokines can be produced, and when, are quite variable, depending on circumstances. Likewise, which other immune system cells are affected, and in what ways, depends on the circumstances and the cytokines in the immediate vicinity. All these variables is what accounts for the complexity of the process.

One possible effect of a cytokine is to cause proliferation of a particular type of cell. Some types of cells can even generate cytokines that cause proliferation of the same type of cell. A cytokine that does this is called an autocrine. Th cells provide an important example. Immediately after a Th cell becomes activated (upon enountering an antigen), the cell secretes interleukin-2 (IL-2), which acts on the Th cell to make it (and its progeny) divide rapidly. Clearly, such a positive feedback loop has the potential to start a runaway chain reaction, so there are also mechanisms that eventually slow it down or stop it – if everything goes as it should.

Of course, Th cells also affect other cell types, such as B cells – the sort of immune system cell that produces antibodies. Cytokines from Th cells activate B cells and cause them to proliferate and begin to produce their antibodies. Other cytokines from Th cells may cause a different type of T cell – cytotoxic T cells (Tc for short) – to be activated and proliferate. At the name implies, Tc cells produce toxins that kill body cells (if they've been infected by a virus associated with an antigen that activates the Tc cell).

Other cytokines can activate additional immune system cell types, such as macrophages and neutrophils. All this activity can lead to inflammation, which may cause a variety of problems of its own.

Clearly, the immune system can do a lot of damage if it slips out of control. It's surprising problems don't crop up much more often than they actually do. In some of the research to be mentioned below, a few of the mechanisms which keep things in check will be noted. It may be possible to harness some of these as therapies to treat conditions where cytokine storms play a part – such as avian flu.

In some of the news articles to be discussed below we'll learn about a few specific examples of known cytokine storms. Here are some additional general links dealing with cytokines and cytokine storms:

Avian flu and Spanish (1918) flu

By now, is there anyone who isn't aware of avian flu, caused by the H5N1 influenza virus? You have probably also heard or read about another type of influenza virus (H1N1), which was responsible for the 1918 Spanish flu that may have killed as many as 50 million people. (An influenza virus is said to be of type HxNy depending on the variants present of two virus coat proteins, haemagglutinin (H) and neuraminidase (N).) In 2004 it was discovered that a normal flu virus modified to look like the 1918 virus could cause similar severe symptoms in mice, by inducing a cytokine storm. Background: here, here, here.

Just a year later, in 2005, the entire 1918 virus was recreated in the laboratory. It proved to be, as expected, quite lethal to mice. Background: here, here, here, here, here, here, here, here, here.

But the important question remained: Why was the virus so lethal? In September 2006 reports of the research appeared that gave a partial result:

1918 flu virus's secrets revealed
Dr John Kash, lead author of the study and assistant professor of microbiology at the University of Washington, said: "What we think is happening is that the host's inflammatory response is being highly activated by the virus, and that response is making the virus much more damaging to the host.

"The host's immune system may be overreacting and killing off too many cells, and that may be a key contributor to what makes this virus more pathogenic."

Dr Christopher Basler, a co-author from Mount Sinai School of Medicine, New York, said: "Our next step is to repeat these experiments, but deconstruct what the immune system is doing so that we can understand why it is reacting so strongly, yet failing to fight the infection."

More: here, here, here, here.

But notice that this report is still a little vague. It talks about a strong "inflammatory response" and how the immune system is "overreacting". As we know, the immune system is quite complex and has a large repertoire of responses for combating an infection. Is it possible to be more specific about what the system seems to be doing here? And does a similar effect occur not only in mice, but also in humans, or at least primates similar to humans?

Answers to these questions appeared in January of this year, with details of experiments performed on macaque monkeys.

1918 Killer Flu Tested on Monkeys
The macaque experiment was supposed to last 21 days, but after eight days the monkeys were so sick – feverish, in pain, and struggling to breathe – that ethical guidelines forced the researchers to euthanize them.

"There was some surprise that it was that nasty," University of Washington virologist and study co-author Michael Katze said. "It was the robustness of the immune system that helped victimize them."

The virus is very good at replicating itself, said Peter Palese, chairman of the microbiology department at Mount Sinai School of Medicine in New York. Its effect on the immune system "triggers what one refers to as a cytokine storm," he said. Cytokines transmit messages among cells in the immune system. Palese wasn't part of the study but has worked on the resurrected virus before.

No other flu virus is deadly to monkeys, and the speed in its spread and the overwhelming immune system response is similar to those in the H5N1 bird flu, Kawaoka said.

So in fact, what this 1918 H1N1 virus appears to be doing is raising a cytokine storm that winds up destroying a lot of the victim's lungs. And that seems to be the same thing that happens with humans who have contracted the H5N1 avian flu virus. Nasty stuff.

More: here, here, here, here, here.

Obviously, there is an urgency to discovering countermeasures to lethal influenza viruses like H1N1 and H5N1. One biotech company has already announced testing of a drug, in rodents, which may be able to control excessive immune system reactions caused by flu virus infection. The account mentions one particular cytokine, IL-6 (interleukin-6), which has been associated with other immune system disorders, and which the experimental drug appears able to control.

ImmuneRegen's Viprovex Demonstrates Immune Response Potential In Treatment Of Avian Influenza And Spanish Flu
Cytokine storm occurs when an infected individual's immune system remains activated against the virus beyond the point of being helpful to where the immune response turns deadly. Persistent, highly elevated levels of pro- and anti-inflammatory cytokines induce a complex, dysregulated condition resulting in massive pulmonary inflammation and fluid accumulation, vascular dysfunction and eventually shock and death. Thus, in cytokine storm, the body's immune system fights to rid itself of the virus, but somehow escapes from the normal controls that prevent an overzealous immune system from killing its owner.

As noted in the Nature publication, there are other disease conditions in which a hyperactive immune system is involved, and other drugs under development for treating those conditions might be beneficial in treating a pandemic influenza infection that could trigger cytokine storm. Specifically mentioned as central to regulation of the immune system, inflammation and hematopoiesis is the cytokine interleukin-6 (IL-6).

Normal production and release of IL-6 is integral to functioning immune and hematopoietic systems, activating lymphocytes and increasing B cell antibody production, but its generation has also been implicated in a number of other diseases, such as rheumatoid arthritis, multiple sclerosis, Alzheimer's Disease and AIDS dementia.

That's encouraging. But it's necessary to remember that studies of the efficacy and safety of new drugs in humans normally take the better part of a decade to perform. Even if this particular drug, or others like it, works in humans, we're hardly out of the woods yet as far as avian flu is concerned. In particular, the human immune system isn't necessarily all that similar to another mammal's, as the next section will demonstrate.

Additional references:

TGN1412 drug trial

Undoubtedly you recall the story about a rather disasterous initial human trial of a new drug that, ironically, was intended to to treat rheumatoid arthritis and other autoimmune disorders. The trial took place in March 2006, and a report on what seems to have happened came out in August:

Mystery over drug trial debacle deepens
Doctors who saved the lives of six men who nearly died in a UK drug safety trial in March have revealed full clinical details of what happened during the first 30 days.

However, far from explaining how the drug caused multiple organ failure in all six men, the results have simply added to confusion over how the drug affected their bodies.

The healthy volunteers were given an experimental antibody drug called TGN1412 in its first human trial to test for safety on 13 March 2006 in London. Within the hour the six men injected with the drug were reportedly writhing in pain. Two others who were given a placebo were unaffected.

The biggest mystery is why the men’s white blood cells – called lymphocytes and monocytes – vanished completely just hours after the drug was injected. This is the opposite of the effect observed in animal trials.

Far from having a calming effect on the immune system, as intended and as was observed in animal tests, TGN1412 seems to have provoked a cytokine storm:
All [experimental subjects] initially suffered from a so-called “cytokine storm” – a flood of inflammation-triggering chemicals pumped into the blood by activated white blood cells. This storm is what eventually led to multiple organ failure, the report says.

The experimental subjects received a dose 500 times smaller than used in animals. But that wasn't enough of a safety margin.
The injected antibody was unusual, because it was capable on its own of provoking lymphocytes, called T-cells, into becoming as active as they would be if they had to fight an infection. It normally takes two signals, not just the one provided by the antibody, to awaken T-cells. The antibody – known as a superagonist – was designed to be able to activate any type of T-cell without requiring the usual secondary signal. It works by binding to a receptor called CD28 on the T-cell surface.

In earlier trials on animals, the antibody initially triggered multiplication of T-cells, but a specific subset, called regulatory T-cells, ended up multiplying fastest and taking control. The regulatory T-cells calmed the immune system. TeGenero hoped that this immune “calming” process offered potential therapeutic benefit, perhaps easing symptoms of diseases like rheumatoid arthritis, where normal T-cells attack the body’s own tissues.

Unfortunately, the reverse happened. Around 60 to 90 minutes after the men received their injections, their bodies were flooded by a surge of inflammatory chemicals called cytokines, which combat severe infections like those seen in patients with blood poisoning. The cytokines caused severe inflammation.

Note, in particular, that the antibody was expected to activate a special type of T-cells, regulatory T cells (Treg). Until just a few years ago, immunologists weren't even sure Treg cells existed. Fortunately, they do, because their purpose in life is to suppress immune system activation, to prevent activity from getting out of control. What is "supposed" to happen is that Treg cells provide a needed negative feedback to the system, to counteract the positive feedback loop driven by other types of T cells. Treg cells are being actively studied at present, and we'll write more about them in another article. Unfortunately, in the TGN1412 trial, things didn't go as expected.

In December, another report came out that offered some hypotheses about why things went awry:

Horror clinical trial in test tube recreation
The new results presented today suggest that to send immune cells berserk, the antibody has to be tethered to a “surface” in the body rather than be free-floating. The team was only able to replicate the excessive cytokine response in the lab that the patients had experienced by effectively sticking the antibodies to a surface.

But that doesn't seem to be the final word on the subject. About a month ago there was a further report:

Researchers propose reason for severe side-effects of Northwick Park clinical trial
The research shows that stimulating the molecule CD28 on cells that mediate the immune response, known as T cells, can have an adverse effect if these immune cells have been activated and altered by infection or illness in the past.

The scientists found that when they artificially stimulated CD28 on these previously activated 'memory' T cells, this caused the cells to migrate from the blood stream into organs where there was no infection, causing significant tissue damage. CD28 is an important molecule for activating T cell responses and the TGN1412 drug tested on the human volunteers strongly activates CD28.

Around 50% of adult human T cells are memory cells, having been activated by infections and illnesses during the course of a person's life. However, animal models, such as those used to test TGN1412 before tests were carried out on humans, do not have many memory T cells because they are deliberately kept in a sterile environment where they are shielded from infections.

Here we see that yet another kind of T cells has been implicated – memory T cells. Undoubtedly there is still more of this story to be discovered. The immune system is a maze of twisty little passages...

Septic shock

One thing that is clear enough is that there must be negative feedback loops in the immune system, as well as positive ones. The latter enable the system to react quickly to serious infections. The former are needed to keep the system itself from spiraling out of control.

The immune system disorder known as septic shock is another example of the system gone berserk. As mentioned in the Wikipedia article, our old friend, the cytokine IL-6, is among those implicated in this condition. Recent research (last November) has identified a specific gene that seems tasked to protect against septic shock:

Gene Tied to Out-of-Control Immune Response
A gene called auf1 seems to protect against septic shock in mice, a new study finds. Animals lacking the gene were more likely to undergo shock, suggesting that the gene helps keep the immune system's response to infections in check. Researchers hope to discover whether different forms of auf1 and related genes make people more likely to suffer autoimmune disease or life-threatening reactions to infections such as anthrax or flu.

Infectious organisms trip specialized immune cells in the body and cause them to pump out proteins called cytokines, which produce inflammation and other hallmarks of infection, such as chills and fever. The body must carefully regulate its cytokine response, however, because "if it isn't turned off it can lead to septic shock and rapid death," says microbiologist Robert Schneider of New York University. Septic shock, which causes 9 percent of deaths in the U.S. each year, occurs when the immune reaction to a bacterial infection grows out of control, shutting down organs and sending blood pressure plummeting. Researchers think similar effects contribute to death from anthrax and pandemic flu.

It appears that the gene codes for a protein which may interfere with messenger RNA that leads to production of cytokines IL-1β and TNFα – thereby putting a brake on production of those cytokines if they are about to run amok.

Undoubtedly, there are a number of chapters yet to be written in the story of how cytokines (and there are many more than have been mentioned here) are regulated. Stay tuned.

More: New study finds on/off switch for septic shock


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Philosophia Naturalis #7 is coming up

Wednesday, February 21, 2007

It will be up Thursday, March 1, to be more precise, at Geek Counterpoint. But you don't have a lot of time to get your suggestions in – they need to be in to the editor by Monday, February 26. You can send them to the editor as described here, or email them to carnival AT scienceandreason.net.

Do it now. You'll be glad you did.

Alzheimer's disease becomes clearer

One of the interesting things about Alzheimer's disease is that brains of disease victims show, at autopsy, two distinct abnormalities in many of their neurons. But a good explanation of the relationship between these abnormalities has been lacking. Perhaps an explanation is now at hand.

'Missing Link' In Process Leading To Alzheimer's Disease Identified
In Alzheimer’s disease, two kinds of abnormal structures accumulate in the brain: amyloid plaques and neurofibrillary tangles. The plaques contain fibrils that are made from protein fragments called “beta-amyloid peptides.” The tangles also are fibrous, but they are made from a different substance, a protein called “tau.” In the new U.Va. study, the researchers found a deadly connection between beta-amyloid and tau, one that occurs before they form plaques and tangles, respectively.

According to George Bloom, the senior author of the study and a professor of biology and cell biology at U.Va., this connection causes the swiftest, most sensitive and most dramatic toxic effect of beta-amyloid found so far. What makes it most remarkable, though, is that it requires a form of amyloid that represents the building blocks of plaques, so called “pre-fibrillar beta-amyloid,” and it only happens in cells that contain tau. Even though they account for just ~10 percent of the cells in the brain, nerve cells are the major source of tau, which likely explains why they are specifically attacked in Alzheimer’s disease.

Here's the spoiler for the plot of this movie: Tau and pre-fibrillar beta-amyloid together disrupt the network of microtubules within a brain cell. The microtubules make up the transportation network within the cell. Without this network, neurons cannot maintain their synapses with other neurons. Without synapses, the memories that are stored in the connections are lost. And the cell itself is likely to die too.
Synapses are connections between nerve cells, and in the brain they are the structural basis of memory and cognition. When nerve cells in the brain lose their microtubules they also lose the ability to replace worn out synapse parts, and synapses therefore disappear. The loss of synapses, and consequent loss of memories and cognitive skills, cannot be reversed, and can lead directly to nerve cell death.


Numbers - rational and irrational, real and imaginary

Wednesday, February 14, 2007

Algebraic number theory, which is the subject we are laying the groundwork for writing about, is the theory of numbers that are solutions of certain types of polynomial equations. So we need to have a little chat about different types of numbers we may encounter. Much of this will be familar to people who have paid attention in high school and college math classes. But even for these folks, the review may help refresh some memories.

The most intuitively "natural" sort of numbers are the "counting numbers": 1, 2, 3, etc. The precise mathematical term for them, in fact, is natural numbers. 0 is also considered a natural number, though the concept was invented in India and seems to have been unknown to the ancient Greeks.

The next most "natural" sort of number includes the negatives of all natural numbers. Collectively all natural numbers and their negatives are known as integers. Mathematicians use the symbol Z to denote the set of all integers. The idea of negative numbers seems to have existed in China before 400 CE. The Chinese had specific tools for reckoning with negative quantities (e. g. debts), but they had even less algebra than the Greeks. For their part, the Greeks seem to have had no concept of negative quantities as such. Negative numbers may have made their first appearance in the written record in the work of the Indian mathematician Brahmagupta early in the 7th century CE. He seems to have been the first to use both 0 and negative numbers systematically, and even recognized that a negative number could be the root of a quadratic equation. (For instance, both +2 and -2 are solutions of x2-4=0.) But since it was not easy to see a negative number of tangible things or count with negative numbers on one's fingers, suspicion of them as mere fictions was widespread for centuries in the West. (Just as many today still regard "imaginary" numbers with deep suspicion.)

If the concept of symbolic equations involving unknown quantities had been more well understood, negative numbers would have been accepted much more readily. They provide a means, after all, of solving even the simplest equations, such as x+1=0, a first degree equation in which all the coefficients are natural numbers.

The operation of division is the inverse of multiplication, and so the reciprocal of a nonzero number n is 1/n -- 1 divided by n. Negative numbers are merely formed using subtraction (the inverse of addition), since -n = 0-n. So it's curious that the Greeks didn't think of negative numbers, though they, and other ancient people, did embrace fractions readily. One assumes this is because fractions arise naturally in geometry, measurement, commerce, and so forth. Fractions are just ratios of two natural numbers, in the form a/b for positive integers a and b, and so they are called rational numbers. If the numbers involved were allowed to be negative as well, the rational numbers too could be negative. Mathematicians use Q for the set of all such rational numbers. If the Greeks had been more capable of thinking abstractly in terms of solutions to equations, it would have been easy to define rational numbers as possible solutions to any linear equation of the form ax+b=0, where a (≠ 0) and b are integers.

Geometry was the most developed form of mathematics in ancient Greece, so it was natural to think of numbers (apart from simple counting) as the lengths of lines, areas of circles, volumes of solids, etc. In other words, it was easy to perceive that arithmetic rules of working with counting numbers behaved in the same way as rules for adding and subtracting the lengths of lines, or computing areas and volumes by multiplication and division. It looked as though, perhaps, all numbers of any consequence should be rational. It thus came as a shocking revelation to the classical Greeks that there were "numbers" that could occur as lenghts of lines in a geometric figure which could not be rational numbers. A proof of this was discovered by followers of Pythagoras, specifically that the length of the diagonal of a square whose sides had 1 unit of length could not be a rational number. In modern notation this length is simply √2.

The proof that √2 is not rational is simple. Suppose it were rational. Then √2 = a/b for natural numbers a and b. Hence a2 = 2b2. We may suppose that the fraction is in lowest terms, so that a and b have no whole number factors in common. (Otherwise, just divide those out.) a2 is clearly an even whole number, so a must be even also. (If 2 divides a2, it has to divide a as well, by the rule of unique factorization into prime numbers, also known as the fundamental theorem of arithmentic. As we shall see, this can be proven fairly easily.) So a is divisible by 2; say a = 2A. Then 4A2 = 2b2, hence 2A2 = b2. It follows that b2 is even, so b must be also. But that is a contradiction, since we could assume a and b had no factor in common. This contradiction means that the original assumption that √2 was rational must be false. QED.

This was so shocking to its discoverers that everyone who learned of it was pledged to secrecy. After all, "not rational", or "irrational" meant to the Greeks (just as in English) "unreasonable". This linguistic fluke suggested that the whole field of endeavor of Greek mathematicians was deeply flawed, so it would be devastating to their prestige if this notion became widely known.

In truth, there is nothing inherently contradictory or unreasonable about "irrational" numbers. They simply are not ratios of integers, but they can occur as solutions of polynomial equations with rational coefficients: for example, ±√2, which are solutions of x2-2 = 0. Numbers of this sort are called algebraic numbers, for obvious reasons. This class of algebraic numbers is the principal subject dealt with in algebraic number theory.

Algebraic numbers clearly exist, since the length of the diagonal of a unit square is certainly a meaningful concept. We've just seen the proof that some algebraic rational numbers are not rational. What are they then? In some sense, answering this question is what the subject of algebraic number theory is largely about. The theory attempts to say what they are in terms of mathematical properties they have. We will be spending most of our time on this issue.

Before we dive into that, let's look at the broader context. Recall the result Gauss proved in his thesis, the fundamental theorem of algebra. This theorem is about the roots of a polynomial equation of the form
anxn + an-1xn-1 + ... + a1x + a0 = 0
where n is a positive integer, x is an "unknown", an ≠ 0, and for 0≤j≤n all aj are rational (symbolically, ajQ). Such an equation, as we noted, is said to be of degree n. This can be simplified a little, because if an ≠ 0, then we can divide both sides of the equation by it, without affecting any of the solutions of the equation (known as roots), and therefore assume that the coefficient of xn is 1. The polynomial in such an equation is said to be monic. For simplicity, we often write the polynomial part of an equation as f(x) so that the equation is f(x) = 0.

What Gauss actually proved is that the polynomial in this equation factors completely into polynomials which have degree at most 2 -- even if the coefficients are any real numbers, not necessarily rationals. Intuitively, a real number is one that can be represented in decimal form as a whole number plus a fractional part that is an infinite series of decimal digits. The decimal digits in the fractional part may or may not repeat in groups, from a certain point on. (For instance, .000123123123... is a repeating decimal, while the fractional parts of the numbers √2 and π never repeat.) It is not hard to show that a number is rational if, and only if, its fractional part is a repeating decimal. Thus the rational numbers form a subset of all real numbers. The set of all real numbers is denoted by R.

Irrational numbers such as √2 are examples of real numbers which are not rational. However, √2 is an algebraic number because it is a root of x2-2=0. Yet not all real numbers are algebraic. In fact, there are vastly more real numbers than there are algebraic numbers. In some sense, there are just as many rational numbers (or even integers) as there are algebraic numbers, because the set of all algebraic numbers can be put into a 1-to-1 correspondence with either Z or Q. (Algebraic numbers may actually be "complex", as will be discussed shortly, but for now just think about real algebraic numbers.) All of these sets are subsets of R which are strictly smaller than R, because they cannot be put into 1-to-1 correspondence with R. (The argument is simple. If A is the set of all (real) algebraic numbers, it has a 1:1 correspondence with positive integers, which means one can, in principle write down all members of A in some order. Now we can define a new number r as the number whose nth decimal digit is one more than the corresponding digit of the nth member of A in the list (or 0 if that digit is 9). r is therefore a real number which cannot appear anywhere in the list, since it differs from every one of them in at least one place. So the supposed list of all real algebraic numbers can't exist.)

A real number which is not algebraic is said to be transcendental. Curiously, even though there is a vast quantity of transcendental numbers, it is quite difficult to prove that specific numbers, such as π, are transcendental. In fact, it was not until 1873 that a "familiar" number (e, the base of the natural logarithms in this case) was shown to be transcendental, by Charles Hermite. π was shown to be transcendental in 1882, by C. F. Lindemann.

Let's return to Gauss' fundamental theorem of algebra. It is now possible to prove something more general than what Gauss showed. Namely, we can work in any algebraic structure called a field. We'll explain more carefully later what that structure is, but for now just accept that it is any system of objects (like numbers) that allow the arithmetic operations of addition, subtraction, multiplication, and division (except division by 0) following rules just like those of rational or real numbers. In this case, let F be any field, and f(x) be a monic polynomial with coefficients in F. Then it is possible to construct a slightly larger field E that contains F be adding certain new elements which are defined by simple polynomial relations. For instance, if F=Q, we can add or adjoin another element which we will denote by α and which has the property that α2=2. This new field, which we denote by F(α), consists of all possible sums and differences of α with elements of F, as well as all products and quotients of such expressions. There are standard ways to do this construction rigorously and to prove that the result E=F(α) is a field, called an extension field. F is said to be a subfield of E. (Note that as sets, F⊆E.) You may think of the extension E, if you wish, as a collection of formal expressions of sums, differences, products, and quotients involving α and elements of F, always simplified by using the relationship α2=2 whenever possible. That is, always replace α2 by 2 whenever it occurs.

Given all this, it can be shown that there is one root of f(x)=0 in some extension field of the field F that contains the coeffiecients of f(x). Call this root α, so that f(α)=0. With polynomials, there is a process very much like long division of integers which allows one to compute the quotient of f(x) divided by x-α, yielding another polynomial g(x) = f(x)/(x-α). This algorithm guarantees the coefficients of g(x) are in E=F(α) if the coefficients of f(x) are. (In particular, if the coefficients are actually in F.) Consequently, f(x) = (x-α)g(x), where g(x) is monic and has degree exactly one less than that of f(x). We can repeat this process with g(x), and so after exactly n steps, we will arrive at a complete factorization of f(x) into linear factors with coefficients (the constant terms) that are in some extension field of F. We might have to adjoin n different symbols (the roots of f(x)), but at least it can be done. (In fact, it can be shown there is a single additional element θ, called a primitive element, or a generator of the field, which is the only element that needs to be adjoined to F to produce an extension field E=F(θ) in which f(x) splits into linear factors. In other words, this field E contains all the roots of f(x)=0.)

Note that unlike other sorts of numbers we considered before, the "numbers" in an extension field of Q may be somewhat abstract objects, such as formal expressions. They certainly can't be just expressions involving radicals, if the degree of the lowest degree polynomial they satisfy is 5 or more (as Abel and Galois proved). Nevertheless, as long as they are elements of R, i. e. real algebraic numbers, they still "make sense", say, as the length of a geometric object.

The real numbers themselves are rather abstract objects when one tries to construct them rigorously. There are ways to do this other than using decimal expansions. One such method, involving set theory, is called the method of Dedekind cuts, after its inventor Richard Dedekind (1831-1916). (Dedekind's name will come up again, because he was one of the primary contributors to algebraic number theory.) More generally, we can adjoin to Q all possible limits of sequences {an} of rational numbers to form the completion of Q considered as a metric space. We won't attempt to describe these abstract constructions further. The point is that once one goes beyond the field Q of rationals, larger fields consist of objects which are somewhat more abstract -- and to an extent arbitrary, subject only to the rules which define a field.

A perfect example of this is the field of complex numbers, which is obtained by adjoining the element i to the field R of real numbers, subject only to the relation i2=-1. So we can say that i=√-1. What is i "actually"? It doesn't matter. The only thing one needs to know is i2=-1. This should not be cause for suspiciousness or skepticism about such imaginary numbers. Their existence is just as secure as any other abstract object of modern mathematics. If we adjoin i to R the field C = R(i) of complex numbers is what we get.

Another way to describe C is as the set of all "numbers" of the form a+bi with a,b∈R, i. e. C = {a+bi | a,b∈R}. Addition and multiplication are defined on this set by the rules (a+bi)+(c+di) = (a+c)+(b+d)i, and (a+bi)×(c+di) = (ac-bd)+(bc+ad)i. This is very much as if i were an "unknown" symbol like x, except that we always simplify expressions by using the relation i2=-1.

There are other ways to think of this field. For instance, we can take it to be the set of all ordered pairs {(a,b) | a,b∈R} where addition and multiplication are given by (a,b)+(c,d) = (a+c,b+d) and (a,b)×(c,d) = (ac-bd,bc+ad), as suggested by the preceding paragraph. In this notation, it is apparent that C is "nothing but" the Cartesian plane R×R with a peculiar sort of multiplication. (Indeed, topologically, C and R×R are the same.)

One of the requirements of a field is that division by any element of the field except 0 is always possible -- that is, all nonzero elements have a multiplicative inverse. What is 1/(a+bi), the inverse of a+bi? First, we use the notation (a+bi)* = a-bi for the operation of complex conjugation. a-bi is said to be the complex conjugate of a+bi. This is used quite frequently. Next we note that (a+bi)×(a+bi)* = a2 + b2, a non-negative real number that is 0 if and only if a=b=0. So the square root of this is a real number, and we use the notation |a+bi| = √((a+bi)×(a+bi)*). This is called the norm of the complex number a+bi. It follows that if a+bi≠0, then its inverse is given by 1/(a+bi) = (a+bi)*/|a+bi|2.

Just a little more teminology and we can move on. The set of all polynomials in one variable that have coefficients in a field F is denoted by F[x]. A polynomial f(x)∈F[x] is said to be irreducible over F if it has no factors other than 1 and itself belonging to F[x]. An irreducible polynomial is completely analogous to a prime number in the integers. Suppose an element α is a member of some extension E of F. f(x)∈F[x] is said to be a minimal polynomial for α if f(α) = 0 and this is true of no polynomial in F[x] that has degree less than that of f(x). It's easy to show that when f(α)=0, f(x) is a minimal polynomial for α if and only if f(x) is irreducible over F. α is said to have degree n over F if n is the degree of its minimal polynomial. The degree of an extension E⊇F, denoted by [E:F], can be defined in various ways, but what it boils down to is that [E:F] is the degree of a primitive element θ such that E=F(θ). In some ways, a better definition of the degree [E:F] comes about when we regard E as a vector space over F. This is a concept from linear algebra. In these terms, [E:F] is the dimension of E as a vector space over F.

Given all that, we note that [C:R]=2 and that i is a primitive element for the extension CR. C has the fairly special property of being algebraically closed. This means that any polynomial in C[x] factors completely into linear factors in C[x]. In other words, there are no irreducible polynomials in C[x] having degree more than 1, and all roots of any f(x)∈C[x] actually lie in C itself. These facts follow from Gauss' fundamental theorem of algebra. (C does have extensions of infinite degree, such as the field of rational functions of one variable, but we won't go into that now.)

In order to avoid ambiguity, whenever discussing extension fields of some field F, we always assume the extensions are subfields of some fixed algebraically closed field that contains F. A smallest such field is known as an algebraic closure of F. For instance, C is an algebraic closure of R. Q has an algebraic closure (the field of all algebraic numbers) contained in C that is of infinite degree over Q, but much smaller than C itself. (For instance, the algebraic closure of Q contains no transcendental numbers.)

We've now given an overview, in fairly concrete terms, of the kind of numbers that occur in algebraic number theory. The next installment will be a discussion of Diophantine equations. These can be understood in very elementary terms, but actually solving them in many cases requires algebraic number theory, and is one of the principal motivations of the theory.


Why opposites attract romantically

Tuesday, February 13, 2007

Just in time for Valentine's day:

New Study Is First To Link Romantic Relationships To Genes
New research suggests that choosing a mate may be partially determined by your genes. A study published in Psychological Science has found a link between a set of genes involved with immune function and partner selection in humans.

Vertebrate species and humans are inclined to prefer mates who have dissimilar MHC (major histocompatibility complex) genotypes, rather than similar ones. This preference may help avoid inbreeding between partners, as well as strengthen the immune systems of their offspring through exposure to a wider variety of pathogens.

The study investigated whether MHC similarity among romantically involved couples predicted aspects of their sexual relationship. “As the proportion of the couple’s shared genotypes increased, womens' sexual responsivity to their partners decreased, their number of extra-pair sexual partners increased and their attraction to men other than their primary partners increased, particularly during the fertile phase of their cycles,” says Christine Garver-Apgar, author of the study.

On the other hand... While this may apply to non-human mammals like mice and chimps, there could be trouble in our species with parters who are too opposite. If she prefers classical music, luxury cruising, a neat and tidy house, and eventually wants a large family, while he prefers hip hop, mountain climbing, a relaxed attitude towards domestic order, and doesn't care that much for kids, there could be trouble brewing.

Even if they have great sex together.

Carnival of Mathematics

Monday, February 12, 2007

The first edition of the brand new Carnival of Mathematics was posted on Friday (OK, so I'm a little late) at Abstract Nonsense. It's good.

A brief history of algebra

Thursday, February 8, 2007

I'm planning to write a series of posts about one of my favorite topics in mathematics: algebraic number theory. We'll start out very gently, with material anyone who's studied high school algebra can easily appreciate. Before long, the climb will become somewhat steeper, but I hope it can be followed by anyone who has read popularizations of mathematical subjects, such as recent books on the Riemann Hypothesis. (Which is a subject that connects up, eventually, with algebraic number theory.) If there are any terms or names mentioned here for which you'd like more detail, I suggest looking in Wikipedia, although I won't litter the narrative with explicit links.

The place to begin, naturally, is with a brief history of algebra itself.

The word "algebra" comes from Arabic: al-jebr because the subject was studied and written about in something like the modern sense, by scholars who spoke Arabic in what is now the Middle East, in the 9th century CE. Although classical Greeks and various of their predecessors and contemporaries had investigated problems we now call "algebraic", these investigations became known to speakers of European languages not from the classical sources but from Arabic writers. So that is why we use a term derived from Arabic.

Muhammad ben Musa al-Khwarizmi seems to have been the first person whose writing uses the term al-jebr. As he used it, the term referred to a technique for solving equations by performing operations such as addition or multiplication to both sides of the equation – just as is taught in first-year high school algebra. al-Khwarizmi, of course, didn't use our modern notation with Roman letters for unknowns and symbols like "+", "×", and "=". Instead, he expressed everything in ordinary words, but in a way equivalent to our modern symbolism.

The word al-jebr itself is a metaphor, as the usual meaning of the word referred to the setting or straightening out of broken bones. The same metaphor exists in Latin and related languages, as in the English words "reduce" and "reduction". Although they now usually refer to making something smaller, the older meaning refers to making somethng simpler or straighter. The Latin root is the verb ducere, to lead – hence to re-duce is to lead something back to a simpler from a more convoluted state. In elementary algebra still one talks of "reducing" fractions to lowest terms and simplifying equations.

The essence of the study of algebra, then, is solving or "reducing" equations to the simplest possible form. The emphasis is on finding and describing explicit methods for performing this simplification. Such methods are known as algorithms – in honor of al-Khwarizmi. Different types of methods can be used. Guessing at solutions, for instance, is a method. One can often, by trying long enough, guess the exact solution of a simple equation. And if one has a guess that is close but not exact, by changing this guess a little one can get a better solution by an iterative process of successive approximation. This is a perfectly acceptable method of "solving" equations for many practical purposes – so much so that it is the method generally used by computers (where irrational numbers can be specificed only approximately anyhow). Some approximation methods are fairly sophisticated, such as "Newton's method" for finding the roots of polynomial equations – but they're still based essentially on guessing an initial rough answer.

Another method for finding solutions of equations is by means of geometric construction. One can construct geometric figures in which the length of a certain line segment is a solution to some given equation. This works well, for example, when square roots are needed, since the hypotenuse of a right triangle has a length which is the square root of the sum of squares of the other two sides of the triangle. That is, if the lengths of the sides are a, b, and c, then a2 + b2 = c2 and hence c = √(a2 + b2). If a and b are whole numbers, so is the sum of their squares. Algorithms for finding the approximate square root of a whole number were known, so c could be computed approximately. However, with a geometric construction, c could be found simply be measuring the length of the right line segment. For future reference, note that an interesting problem is finding two numbers a and b such that for some given number d, d = a2 + b2. This is because if d is given, finding a and b enables one to find the square root of d by a geometric construction.

In addition to such approximation and geometric methods, al-Khwarizmi was interested in methods for finding exact solutions by a sequence of steps that could be described in cookbook style – algorithms. This can be done completely successfully for linear equations of the form ax + b = c (in modern notation) using just the arithmetic operations of addition, subtraction, multiplication, and division. Just as important, the operations can be performed symbolically – not just on particular numbers, but on symbols that stand for "any number". And so, one seeks to express the solution of a given equation in a symbolic form.

An interesting question is that of when the idea of representing equations in symbolic form arose. It's not easy to answer such a question, in part because symbolic representations were used before their full and considerable utility was recognized. For instance, Greek geometers labeled the lines in their figures with single letters, so it was natural to write what we now recognize as the Pythagorean theorem in the form a2 + b2 = c2. But the importance of this representation was somewhat blurred, since the distinction between a line and the length of a line was not fully appreciated. In fact, although Greeks and other early mathematicians (e. g. in India) used symbolic equations, al-Khwarizmi did not. (Hence it is likely he didn't know of Greek mathematics and much of his work was original, if not always as advanced as that of the Greeks.)

In modern notation, polynomial equations can be classified in terms of the highest power of any variable which occurs in them. We call an equation linear if the highest power is one, because its graph is a straight line. If there is just one variable, such an equation has the most general form ax + b = 0. If the highest power is two, the equation is called "quadratic", and has the form ax2 + bx + c = 0. (Why does the Latin prefix quad, usually associated with the number 4, occur here? Simply because the word for "square" is quadra in Latin.) In spite of lacking a symbolic representation of equations, al-Khwarizmi effectively did know the quadratic forumula which says that there are two solutions of the last equation, that can be written as x = {-b±√(b2-4ac)}/2a. He also realized that the equation has solutions at all in terms of "real" numbers only if the quantity we now call the discriminant, b2-4ac, is not negative.

The highest power of an unknown which occurs in a given polynomial equation is known as the degree of the equation. Although al-Khwarizmi doesn't seem to have studied equations of degree 3, called cubic equations, a more famous successor, who lived about 250 years later, did: Omar Khayyam (ca. 1050-1123), a Persian. Like his predecessor, Khayyam did not work with symbolic expressions for the equations. But he was able to produce solutions using geometric constructions (involving conic sections), provided a positive solution exists. He also thought, mistakenly, that such solutions couldn't be found by algebraic (algorithmic) methods of the sort al-Khwarizmi used.

The next substantial advance in solving equations began when scholars in Western Europe began to study and appreciate the work of people like al-Khwarizmi and Khayyam. Most notable among these scholars was Leonardo of Pisa, more commonly known as Fibonacci (ca. 1180-1250). He showed that algorithmic (as opposed to geometric) methods could be used to find solutions of some cubic equations. Fibonnacci had a much more obscure contemporary, Jordanus Nemorarius, of whom little is known apart from several books attributed to him on arithmetic, mechanics, geometry, and astronomy. He made a more systematic use of letters to stand for "variable" (not necessarily unknown) quantities in equations, but the importance of this technique was still not widely appreciated

With the advent of the Renaissance, progress in mathematics began to speed up. One of the first notable names was a German, Regiomontanus (1436-76). Though he produced less original work than others, he was widely read in the classic works of both the Greeks and the Muslim world. In particular, he had studied the Arithmetic of Diophantus of Alexandria (who was active around 250 CE) in the original Greek, and even considered publishing a Latin translation, though he never got around to it. Diophantus was in some respects more advanced than any other mathematician before the Renaissance, and among the problems he considered were what are now called Diophantine equations. The relevance of such problems will be explained in due time.

Somewhat more original than Regiomontanus was a Frenchman, Nicolas Chuquet, who died around 1500. He used expressions involving nested radicals farily close to the modern style, such as √(14-√180)), to represent solutions of 4th degree equations.

The real breakthrough came in the work of several Italians in the 16th century. In 1545 Girolamo Cardano (1501-76) published explicit algebraic solutions (that is, using arithmetic operations plus extraction of roots) of both cubic and quartic (4th degree) equations. Cardano, however, did not discover the solutions himself. The result for cubics was known before 1541 by Niccolo Tartaglia (ca. 1500-57), though apparently discovered even earlier by Scipione del Ferro (ca. 1465-1526). Cardano admitted he had not discovered the solution, but apparently he did break a promise to Tartaglia to keep the results a secret. (Just as now, precedence in publishing new scientific results was a matter of great prestige.) As for the quartic, Cardano states that the solution was discovered by Ludovico Ferrari (1522-65), though at his [Cardano's] request.

Such rapid progress naturally raised the question of solutions to equations of 5th degree (quintics) and higher, either by algebraic means (using arithmetic operations and radicals) or at least by means of geometric constructions (using only straightedge and compass). Surprisingly, it was proven almost 300 years later that solutions of either sort were not possible in general, i. e. for all cases. This was done independently by two young men, Niels Henrik Abel (1802-29) in 1824 and Évariste Galois (1811-32) in 1832. Galois' result is especially important, as it is based on very novel methods of abstract algebra – the theory of groups – and in fact Galois' ideas thoroughly permeate the theory of algebraic numbers, to be discussed.

In spite of that astonishing negative result, only a few year earlier Carl Friedrich Gauss (1777-1855) had proven in his doctoral thesis of 1798 that polynomial equations of any degree n must have exactly n solutions in a certain very specific sense. This result was so important it became known as the fundamental theorem of algebra. The exact sense in which that theorem is true is the subject of the other part of this story of algebraic numbers – "numbers". That will be taken up in the next installment.

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Philosophia Naturalis #6

Thursday, February 1, 2007

What I like most about putting together an edition of a blog carnival like this is the opportunity to look carefully at a lot of good science blogging. But one thing I don't especially enjoy is trying to think up some creative "theme" around which to structure the damned thing. There have been some very creative and entertaining themes both in this carnival and in many others. But me, I'd rather just enjoy the articles themselves, and not try to dazzle everyone with some wacky theme, like maybe "Pirates of the Caribbean" or whatever. (I just picked that out of the air; apologies to anyone who's actually used that as a carnival theme.) Anyhow, let's just get down to business.

January, inevitably, is astronomy-astrophysics month, because it starts out with the big meeting of the American Astronomical Society. This year, the 209th meeting was held January 5-10 in Seattle. If (like me) you couldn't attend, there are podcasts available of some of the main plenary sessions.

If you are satisfied just to see brief reviews of some of the highlights, there are a number of those scattered around the relevant science blogs. Phil Plait at the Bad Astronomy Blog wrote up some of the better accounts. Such as this one on dark matter and large-scale structure, and this on things that go boom. See also the live blogging done at the conference by the folks from Nature Newsblog. (I'll try to highlight other meeting reports in a separate blog post.)

However, I can't leave out mention of one of the most important findings reported at the meeting – the spectacular work of the COSMOS project regarding dark matter. Two of the better reports on this come from Sean Carroll at Cosmic Variance and Rob Knop at Galactic Interactions.

At the opposite end of the spectrum, a few, um, "alternative science" proponents managed to sneak into the meeting and exhibit their theories. I'll let Rob Knop fill you in on the gory details.

Ranging a little further afield, in the celestial realms but away from the conference, we have this book review of Paul Davies' Cosmic Jackpot, offered by Alejandro Satz at Reality Conditions. The book deals with the "anthropic principle", which is somewhat controversial...

We're currently in a golden age of observational astrophysics and cosmology, thanks in large part to the many magnificient new instruments that have been deployed in recent years, both on the ground and in space. And it's only going to keep getting better, at least for awhile, due to new instuments which will come on line in the next few years. One of the most exciting of these is the James Webb Space Telescope. (Named for the NASA administrator who guided the Apollo Project, not the newly elected Senator from Virginia.) Centauri Dreams writes about the JWST here and here.

It has recently been discovered that the ancient Greeks possessed much more sophisticated astronomical instruments than had previously been supposed. Lorne Ipsum of Geek Counterpoint tells us about the Antikythera Mechanism of the ancient geeks in a podcast.

Most people, lay watchers of the night skies as well as astronomers and other scientists, find inspiration in the firmament beyond our little planet. A smaller – but very fortunate – number find inspiration in physics too. Sabine Hossenfelder of Backreaction writes eloquently about this here, guest blogging at Asymptotia. Thinking of such things while the Sun is above the horizon can lead to daydreams, and Sabine writes about some recent research in that area too, with a good explanation of fMRI, the latest toy of neuroscientists.

Meanwhile, Asymptotia's host, Clifford Johnson, explains one of the basic concepts of relativity – light cones.

In fact, it's been a big month for physicists explaining some of the basic principles of their science. Jennifer Ouellette of Cocktail Party Physics leads off with concise explanations of ten basic concepts. Jennifer is followed by Chad Orzel at Uncertain Principles, who tells us all about forces, fields, and energy.

And for more advanced physics buffs who don't quail at quantum theory, Matthew Leifer at Quantum Quandries answers the perennial question, What can decoherence do for us?

Of course, physicists these days have quite a lot of explaining to do. Like, for instance, where the heck is that Higgs boson anyhow? John Conway, writing at Cosmic Variance, tells us he's been looking for it for 20 years, and fills us in on many of the details here and here.

Physicists aren't the only scientists with a lot of explaining to do. Mathematicians, normally a reclusive lot, have quite a bit to do as well, and much of it is crucial in physics as well as science in general. Fortunately, we have Adam Gurri at Sophistpundit to tell us about Bayes' Theorem – an important tool of statistical inference in all sciences.

Next up is Arunn Narasimhan of Nonoscience, who gives us a quick introduction to Fourier series – truly a transforming experience. Finally, for the fearless mathophile, Mark Chu-Carroll at Good Math, Bad Math, tells us about fiber bundles, one of the main tools of modern mathematical physics.

Time to kick back and relax. Fractals is a mathematical topic that can be enjoyed purely on an aesthetic level. I wrote a little essay at this blog on fractal art, and then went on with some musings about the nature of art in general.

One last thing before I bid you adieu. I'll leave you with the question, posed at Memoirs of a Postgrad by Paul Baxter – What does cognitive robotics mean?

And please remember to tune in again next month, on March 1, when Lorne Ipsum at Geek Counterpoint will host Philosophia Naturalis #7. Watch his blog and the PN blog for the announcement and further details of how you can have your blog article about physical sciences and/or technology featured in the next edition of PN.

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