http://www.youtube.com/watch?v=OmSbdvzbOzY&feature=related

In November, La Recherche (also) published a special issue on the power of mathematics. While this issue contains a load of interesting papers on the various facets of current mathematics, some of which being edited reprints of earlier papers, and includes a good interview of Wendelin Werner, I find it quite significant that none of those papers ever mentions statistics! It sounds as if statistics was not part of mathematics for the editors of this issue, especially when considering the section on the applications of mathematics that includes character recognition and computer intensive methods. I understand that Gödel’s theorem and the theory of proofs may be more appealing to the layman than machine learning or bootstrap, but I still resent this exclusion from the mathematical “pantheon”! (Ironically, or not!, one of the few statistics books included in the bibliography is Py’s “Statistiques sans formules mathematiques“!)

I have previously written about the Cauchy-Schwarz Inequality.

Here is another proof of this fun inequality.  We start at a similar point to the previous proof by applying the fact that the square of any number is always non-negative.  Suppose is an arbitrary real number, then for

.

This would be true for whatever real numbers we used, but there is a reason for using these particular real numbers in this particular way. This is because we can construct a very useful sum,

,

which with some rearrangement can be seen to be a polynomial in ,

This is a quadratic equation.  For a quadratic expression,

,

the minimum value over all possible real values of occurs when

.

If we apply this information to the quadratic expression, which we have previously constructed, we get that the minimum should occur at

.

If we substitute this value into the quadratic expression, we further see that

,

and with some algebraic manipulation that,

This is the Cauchy-Schwarz Inequality.  However, it may bother some readers that it does not look exactly like the form of the Cauchy-Schwarz equality that I have given.  If so, note that if we take the square root of both sides,

But,

and

I recently read The Cauchy-Schwarz Master Class by Michael Steele which, in addition to teaching the reader about inequalities, has a lot of fun and varied things to say about the Cauchy-Schwarz inequality.

First things, first, the Cauchy-Schwarz inequality in it’s simplest form says the following:

This holds for any pair of lists of numbers, and , for .  We know that the square of any number is greater than or equal to zero. Thus, for two real numbers,

and

;

but with some algebra,

We can easily apply this knowledge to the lists and for .

Here we apply a rather entertaining trick. Suppose, we had two lists of numbers which we defined using the original lists as follows:

This would change the situation in this way,

.

So, it follows that

.

In summary,

.

This concludes the proof of the Cauchy-Schwarz inequality.

simple mathematics…

Sorprende que en el año 2009 todavía no se tuviera un modelo matemático sencillo en Ecología capaz de explicar el “misterio de los misterios” de Darwin, el origen de las especies. El sueco Pim Edelaar, miembro de la Estación Biológica de Doñana del CSIC en Sevilla, y sus colaboradores lo publican hoy en Science. Un modelo simple que explica cómo la selección natural y la selección sexual trabajan en conjunto para lograr la adaptación local y el aislamiento reproductivo que conduce a una nueva especie, incluso bajo un flujo de mutaciones genéticas importante. Las hembras prefieren los machos cuyos ornamentos sexuales mejor indican lo bien que están adaptados al medio. Un mecanismo de retroalimentación que no había sido descrito con anterioridad de forma tan sencilla y elocuente. El artículo técnico es G. Sander van Doorn, Pim Edelaar, Franz J. Weissing, “On the Origin of Species by Natural and Sexual Selection,” Science Express, Published Online November 26, 2009. El nuevo artículo es la culminación del trabajo que el primer autor, Gerrit Sander van Doorn, postdoc en el Instituto Santa Fe, Nuevo México, EE.UU., y actualmente en la Universidad de Berna, Suiza, desarrolló en su tesis doctoral en 2004, “Sexual selection and sympatric speciation,” PhD Thesis, 2004, PDF 24,30 Mb, bajo la dirección de Franz J. Wessing, y en especial del capítulo 8 de la misma. 

El origen de una especie (especiación) require una interacción entre procesos genéticos (diversificación genética) y procesos ecológicos (aislamiento reproductivo). El nuevo modelo matemático consiste en un sistema de dos ecuaciones diferenciales acopladas, que omitiremos, que describen cómo la selección sexual, las preferencias de las hembras por ciertos caracteres ornamentales de los machos, se acopla con la selección natural, la presencia de genes beneficiosos para la adaptación de la especie al medio, permitiendo resolver satisfactoriamente el problema de la divergencia de las especies. El modelo teórico es lo sencillo y permite un análisis dinámico (cualitativo y cuantitativo) detallado utilizando la técnica del plano de fases. El modelo muestra que las hembras prefieren a los machos cuyos ornamentos sexuales son los que mejor indican lo bien que están adaptados al medio. Esta preferencia sexual refuerza la selección natural por un mecanismo similar a un sistema de control retroalimentado. Sin este mecanismo, modelos anteriores son incapaces de explicar de forma sencilla la divergencia entre especies.

PS: Noticia en Europa Press y comentarios en Menéame.

Saloni..

13 Years. Kid. Mischievous. Loved ice-creams. Loved Cadbury. Loved Rains. Loved to Play. A typical child. And like every other child, she hated Mathematics. Her mother, Neha, was a typical mother. She wanted her child to do all things which she herself was unable to do. Neha lost her husband a year ago. In 10 years of their married life, she had seen so much happiness that the loss of her husband came to her as a disaster. Since then, she took up the charge to bring up Saloni and wanted to fulfill her every dream. Saloni was her only focus and both mother and daughter had a great time. Neha managed to get few girls to stay as Paying Guests (PG, as they say) who shared their rooms and paid something. More than the rent, Neha and Saloni actually wanted company.

But also, living in Mumbai is tough if you have got adjusted to a lifestyle and then you suddenly don’t have access to it. Saloni was a kid, but strangely, she understood. She never cried for Cadbury in the shop, never cried for Masala Corn in the mall, neither she was fond of the junk food. They were lucky that they had an apartment in Mumbai and they managed somehow. I knew Neha was searching for a job. I worked before my marriage and I am trying now. But since I had a break of nearly 10 years in my career, I am not able to get into one now. I am a Cost Accountant… Nick, Do you have any openings in your company? Neha would continue..

She wanted Saloni to become and engineer or a doctor and would keep asking me how I studied and always asked me to help Saloni in her subjects. I would politely say yes, but knew that it is not possible. Which kid would be awake at 11:30 PM at night to study Mathematics. And which Program Manager would have the luxury to return home before 11:30. Program Manager who is forcefully single.

Sunday Morning. No office. One day when I live for myself and my family. Pains of yesterday were gone. I was feeling great. I wanted to thank Neha. And I wanted to cook something for my neighbours, my only extended family in Mumbai. Saloni loved sweets. The easiest thing to cook was Custard Kheer. I was no cooking expert. But with Google around and with Mom on the phone, no one needs to be. In minutes, I was looking up the internet for recipes. Its good that the STD call tariffs have come down drastically and Internet speeds on a Wireless Datacard have shot up in India.

3 table spoons of Custard powder, ½ cup condensed milk cream, 2 table spoons of sugar…Mix up the contents…Boil Milk and pour… And lo..The custard is ready. I put in the refrigerator and am done. In 30 minutes, amazing custard is ready!

As I rang the bell and waited for Saloni to open the door, I was trying to think if I put sugar in the custard. Saloni saw me and was very happy seeing me. Bhaiyaa, Aap mere liye kuch leke aaye hain? And when she saw the custard, she was overjoyed! I loved her smile.  This is the family which saved my life yesterday and gives me a feel of home away from home. Neha came out of her room, greeted me and asked me if I have sometime today to teach Saloni. I had all the time in the day. Lol. Salonis smile faded. She brought her Maths text book and said – Uncle!! Aaj thoda sa hi padhna hai.. Aaj na meri tabiyat na .. itna jyada thik nahin hai! I was smiling. Mathematics can make me Uncle and Chocolates/Custard can make me Bhaiyaa.

17 more days in Mumbai. 17 days remaining to accept Kotler as God.

Gravedad cero. Imagina, como Newton, que la Tierra fuera hueca y te encontraras en su interior. Estarías flotando, completamente ingrávido, como los astronautas en el espacio, pero por una razón diferente. En el interior hueco de una distribución esférica de masa el campo gravitatorio es nulo. Newton lo demostró geométricamente como muestra este extracto de los Principia. Considera un punto P en el interior y dos conos con el mismo ángulo que atraviesan el cascarón. Como la ley de la gravead decae con la inversa de la distancia al cuadrado y la cantidad de masa en el cascarón contenida en cada cono depende de la distancia al cuadrado, la fuerza ejercida en P por ambos cascarones es idéntica pero de sentido opuesto. Sea cual sea P, la fuerza gravitatoria en P debida al cascarón es exactamente cero. Obviamente cualquier objeto exterior al cascarón que rompa la simetría esférica, como la Luna o el Sol en nuestro ejemplo, introducirá una fuerza gravitatoria muy débil pero matemáticamente no nula.

La demostración de Newton es geométrica e intuitiva. La clave es que la fuerza de la gravedad se proporcional al inverso del cuadrado de la distancia. La masa en el punto P puede ser cualquiera, siempre que sea puntual (su volumen es muy pequeño comparado con el de la esfera hueca). En los primeros cursos de física es habitual presentar una demostración más técnica de este teorema de Newton basada en el teorema de la divergencia de Gauss. Por ende, aplicable a la fuerza de Coulomb dentro de una distribución esférica de carga eléctrica.

En la teoría de la gravedad de Einstein, la relatividad general, el teorema de Newton o el teorema de Gauss también son aplicables aunque con una ligera salvedad. En el punto P la masa ha de ser nula, ya que por muy pequeña que sea deforma el espaciotiempo a su alrededor y la distribución esférica de masa deja de serlo, la simetría esférica se rompe (salvo que P se encuentre justo en el centro). Este resultado de la gravitación de Einstein se llama teorema de Birkhoff y es aplicable incluso al universo entero en su conjunto. Sus aplicaciones son múltiples. Por ejemplo, permite demostrar que la gravedad de la materia puede frenar la expansión del espaciotiempo debida a la Gran Explosión.

El teorema de Newton-Gauss-Birkhoff no se cumple en todas las variantes de la gravedad que han sido propuestas en las últimas décadas. Una de las más famosas es la teoría MOND, una modificación empírica de la gravedad newtoniana propuesta en origen para explicar la curvas de rotación de las galaxias sin necesidad de recurrir a la materia oscura. Para campos gravitatorios muy débiles, la teoría MOND corrige la ley inversa del cuadrado de Newton con un pequeño término de aceleración. La teoría MOND no cumple el teorema de Newton-Gauss-Birkhoff. Todo punto P dentro de una distribución esférica de masa hueca sufre una pequeñísima fuerza en dirección hacia el centro de la distribución de masas. La gravedad cero deja de serlo si la teoría MOND es correcta. Los interesados en los detalles matemáticos de la demostración pueden recurrir a Reijiro Matsuo, su PPT “Does Birkhoff’s law hold in MOND?,” 2008, o su artículo técnico De-Chang Dai, Reijiro Matsuo, Glenn Starkman, “Birkhoff’s theorem fails to save MOND from non-local physics,” ArXiv, 10 Nov 2008, last revised 16 Jun 2009.

Seguramente pensarás que los efectos del incumplimiento del teorema de Birkhoff por parte de la teoría MOND son despreciables a escala galáctica y a escalas mayores, pero no es así, como nos han contado recientemente Reijiro Matsuo, Glenn Starkman, “Screening and Antiscreening of the MOND field in Perturbed Spherical Systems,” ArXiv, 18 Nov 2009. Las dificultades de la teoría MOND a la hora de poder describir el comportamiento de los cúmulos de galaxias y de los supercúmulos de galaxias (donde se requiere la presencia de materia oscura) están relacionados con este problema técnico, como nos cuentan Pedro G. Ferreira, Glenn Starkmann, “Einstein’s Theory of Gravity and the Problem of Missing Mass,” ArXiv, 6 Nov 2009.

Resulta curioso que el problema de una nueva propuesta como teoría de la gravedad sea la Gravedad Cero.

Esta la contribución de la Mula Francis a “El Carnaval de la Física en Gravedad Cero. Hoy 30 de noviembre con motivo de la primera observación por parte de Galileo de un objeto celeste con su telescopio.” He de confesar que me enteré de esta iniciativa gracias a MiGUi, que a su vez se enteró en un tweet de Ciencia Kanija. Menéame y otros foros se han hecho eco de la misma. Enhorabuena, Carlo (Ferri) y Roi (Oliva).

I'm thankful for seeing truth presented with beauty.

This is a picture to help understand an Anosov flow obtained from the cat map. It’s part of research on a technique we’ve used to analyze network traffic.

Steven Pigeon recently introduced to us the Frivolous Theorem of Arithmetic in his Harder, Better, Faster blog. As a complement, I shall present a rigorous, rather useless but quite educative proof of this theorem which will serve as a base for more inquiries in the internal structure of natural numbers.

The theorem reads as follows:

Almost all natural numbers are very, very, very large.

First and foremost, this yields the question of what is considered very, very, very large. In fact, the notion of a number being large or very large is inexistant in mathematics and irelevant to the internal structure of numbers. Hence, for the purpose of this proof, we shall choose a überlargeness threshold that we shall call M, a natural number. In fact, you can be an eccentric fellow and say 1 is already large enough for you and so be it. On the other end, maybe the excruciatingly, humongously large number:

is a number you envison as almost big (the Google calculator accepts to calculate 10! but dutifuly refuses to calculate 10!!; how unfortunate !). Regardless, the proof still holds. Let A and B be two sets such that:

Where A is the set of natural numbers smaller than or equal to the threshold and B the set of numbers we consider to be very, very, very large. To prove the theorem, we have to compare the cardinalities (the number of elements) of both sets. First, we note that the cardinality of of A is:

And the cardinality of the set of natural numbers is:

Which reads aleph-null, the first transfinite cardinal number who denotes the cardinality of countably infinite sets, i.e. sets with elements which can be counted individually, even if there is an infinite number of them. Since there exists a one-on-one relation (bijectivity) between N and B, which is of the form:

And according to the properties of cardinalities:

Meaning there is an infinity of numbers larger than M, a number chosen to be überlarge and we can thus conclude that almost all natural numbers are very, very, very large. This proof led us to an important insight on the internal structure of natural numbers: since each natural number has a succesor which also has a succesor (1 has succesor 2, which has succesor 3, ad nauseam),  no natural number can be used to express the cardinality of this set. We can’t simply say there is an “infinite” number of natural numbers, as this is too imprecise. In fact, there is an injective relation between the reals and the naturals but there can be no bijective relation between the two sets, meaning that the cardinality of the naturals is “smaller” than the cardinality of the reals. Intuitively, the cardinality of reals is also infinite, meaning we have two or more types of “infinity”. This will be the subject of a follow-up post.

If anyone has any interest in geometry or mathematics in general, please give me any feedback necessary on the proof I posted about 1/4 down the page.

We hate Journal für Kunst, Sex und Mathematik and we hate the post November 24th, 2009.

We know the arrows mean something. They just have to!

1. WE HATE that 99% of the nauseating confusion we feel upon viewing your blog has nothing to do with the fact that we don’t know German.

Ceci n'est pas une flashlight lamp.

2. WE HATE that when we click on your “sex” tag we get images like this:

We're getting all moist over here.

And this:

Yummy.

And, oh dear, this:

Oh, holy hell.

3. WE HATE your amateurishly pretentious ramblings (when they’re in English).

“D. is interested in seers. Seers see what others do not see. Swedenborg is a seer (Buddha of the North). D. questions that seers have different eyes than others. But what makes them see more, then?”

You call that pretension? Come on, you can do better.

“Realometer writes: text-as-text is a necessary fiction, an unreal principal of reality. Magnetism-as-magnetism is not possible.”

There you go.

4. WE HATE that your images are just as pretentious as your writing. What the hell is this?

No, really. We demand an explanation. The rooster demands an explanation.

Yawn.

5. WE HATE that this next image is probably going to give us weird sex dreams tonight.

Finally, a visual depiction of the beast our parents said would devour us in our sleep if we masturbated.

When we talk about markets we often use terms like equilibrium or even market force. We choose this terminology for a reason. The analogy to the well established theories of mechanics and quantum mechanics is intended and the pictures we have in mind are a pendulum or even a simple spring. Their restoring forces seem to model the market forces and therefore we frequently observe argumentations very similar to:

if prices increase, then demand decreases and vice versa finally, because of some process still to be described, the market settles down in an equilibrium (called Walrasian price equilibrium).

As a start, that sounds convincing. There just remains one big question. Is that a good picture? Or, even more to the point:

Are there any justifications for the existence of market forces?

Rather than answering this question (regular readers know my standpoint anyway) I would like to justify why this question is actually reasonable and should be asked and answered. In physics this question is answered to the positive, in economics the situation is a little blurry to say the least. I continue by comparing mechanics with economics in catchwords. Thereby pointing out similarities, but also discrepancies and, in a way, recalling ‘the story so far’.

Basic notions

Let me start with two of the fundamental notions in mechanics, namely position and momentum. In earlier posts we have identified their counterparts in economics as price and demand.

Symmetries

In mechanics the intuition is that momentum is invariant under translation of position. In economics we need demand invariance under price-scaling.

Commutation relations

These symmetries lead to commutation relations of the form in quantum mechanics and in economics (cf. here). This difference is essential and has a huge impact, albeit not immediately.

Bounded representations

Both commutation relations imply that the symmetry groups do not have representations on a finite-dimensional vector space (cf. here).

Unbounded representations

While there are no bounded representations, we get unbounded representations on the Hilbert space of square integrable functions. Momentum and demand operators are differential operators, whereas position and price are (different) multiplication operators (cf. here).

Uncertainty principle

The uncertainty principle of quantum mechanics is well-known. So far I didn’t write about that here in the blog, but in economics the commutation relations imply inequalities which can also be interpreted as some sort of uncertainty principle. I shall come back to this later.

Time evolution

As described in scientific laws to get the time evolution in quantum mechanics one chooses an action, one uses Legendre transform to obtain the energy, one derives the canonical equations and essentially plugs in the above representation to obtain Schrödingers equation governing the time evolution of a quantum system. That surely sounds more complicated than it actually is.

Why can’t we just do that for markets and obtain market equations governing their time evolution? Now, there are a couple of technical difficulties. The most prominent probably is that the Legendre transform of a market action is not invariant under time translation. Hence, in markets there is no conservation of energy. This fact alone makes the usage of a term like market force a little obscure. What is meant by force if there is no energy or at least no energy conservation?

That essentially is the programme for the rest of the year. I shall spell out the maths behind the uncertainty principle for markets and then delve into the technical details of obtaining a time evolution for markets.

Stay tuned …

Clearly, the biggest reward to passing an actuarial exam is not having to take it again.

-Krzysztof Ostaszewski, PhD, FSA, CFA, MAAA

I passed SOA Exam P. After 800+ pages of paper, 84 hours of study leave, 10 cups of latte and two review manuals, I’m done!

Many thanks to everyone who prayed and believed!

I cannot contain my happiness. This is truly something I would never forget.

Gotta love those teeth. :>

Nobody could guess what I rewarded myself.

It’s finally Thanksgiving Week of 2009; a chance to take my nose out of my books. I’ve neglected this site, yet again.

If you haven’t heard from me in a while, it’s probably because of my full-time load as a MA candidate in Economics at San Francisco State. It might not be the big leagues quite yet, but it’s graduate study, and it’s been quite a ride so far. I know I was a good student in undergrad–still, the start of graduate-level economic theory and methods ain’t no joke. I never thought all my calculus, engineering, and computer science coursework from 10 years ago (that never amounted to a major; another story, another time) would come in handy in economics. It’s been a steep learning curve, but I feel I’ve disciplined myself into a studying routine that’s working this semester. Well, almost working…I’m confident about three of my classes, but feeling questionable about my fourth.

So, if you’re reading this page, and if you are considering studying Economics (or something related) at the graduate level, some advice:

• Take a FULL calculus sequence. Not the business calculus courses either. Take Calculus I–II–III with the science and engineering kids. It may feel like the stages of hell getting through it, but you’ll be set.
• Take MORE mathematics. You wince, my youngling, but it will ease the pain of further graduate study. Linear algebra, probability theory and statistics, differential equations, proof, real analysis…they’ve been recommended to me by other students.
• Use your professor’s office hours. USE THEM! Like your life depends on it. (In a sense, it does!)
• Work with your comrades-in-arms in all your classes. Don’t work alone all the time. Everyone thinks about problems differently; having multiple avenues of attack on a problem helps immensely.
• Make time for some relaxing activities, because all the Latin and Greek-letter variables will make your head spin. I actually enjoy math, and it makes MY head hurt.

References: UC Berkeley’s Economics PhD applicant page

A few math resources to help you in your journey, wherever it may lead:

• Wolfram MathWorld (math reference site)
• Wolfram Alpha (“computational knowledge engine”)
• Khan Academy (YouTube tutorial videos for free!)

Happy Thanksgiving, everyone! Back to the front line next week.

More.

Workout notes 5 miles: 1 AMT, 2 elliptical, 2 Stairmaster. On the stairmaster I am learning to do without the rails for balance.

We had a pleasant drive to Checotah, Oklahoma from Peoria. We stopped to eat at Bandana’s Barbecue just outside of St. Louis and at Maggie’s Mexican in Pryor, Oklahoma.

Posts
Football: here is a nice article about the human side of the Notre Dame football coach, Charlie Weis:

[...]
“Sunday is the most excruciating day,” Weis says, referring to the pain that he feels in both legs, “because I’ve been standing up at least four hours the day before. It’ll start feeling better by Monday night.”

The ravaged knees are the result of an accidental blindside hit Weis took during last season’s Michigan game (although he has nerve damage in his lower extremities dating back to 2002, the result of a botched gastric bypass surgery).
[...]

Weis’s catastrophically impaired limbs are just one unforeseen trauma of his encore return to South Bend. During his first go-round, as a student from 1974-78, he was anonymous and single. Now the most visible and highly compensated person on campus, he has a family: his wife, Maura, son Charlie Jr., and daughter Hannah.

“The damage to Maura and Charlie Jr. is irreparable,” says Weis, referring to the personal nature of the attacks he has been subject to for years now. “It’s watching me get hammered. I’ll never forgive the people who character-assassinated me without even knowing me. Those people did irreparable damage to my wife and son, and I’ll never forgive them.”

On Saturday, Maura Weis, for the first time since her husband was hired, opted not to attend a Notre Dame home game.

“They have the right to criticize the coach for being 6-5,” says Weis. “They have that right. It’s all the other stuff. You think I don’t know that I’m fat? Duh!”

Asked if he should be gone, where would Charlie Jr. would go to college, the coach reponded: “I know where he won’t be going to college.”

I am sorry for the personal troubles. And I have to admit that I was surprised that his teams didn’t play better, especially on defense. Then again, I was surprised that the previous coach (Tyrone Willingham) didn’t work out either, given his success at Stanford.

I wonder if it is Notre Dame’s special set of circumstances or something else.

Education It appears that the Obama administration is serious about mathematics and science education:

President Obama announced on Monday a campaign to enlist companies and nonprofit groups to spend money, time and volunteer effort to encourage students, especially in middle and high school, to pursue science, technology, engineering and math.

“You know the success we seek is not going to be attained by government alone,” Mr. Obama said kicking off the initiatives. “It depends on the dedication of students and parents, and the commitment of private citizens, organizations and companies. It depends on all of us.”

Mr. Obama, accompanied by students and a robot that scooped up and tossed rocks, also announced an annual science fair at the White House.

“If you win the N.C.A.A. championship, you come to the White House,” he said. “Well, if you’re a young person and you’ve produced the best experiment or design, the best hardware or software, you ought to be recognized for that achievement, too.

“Scientists and engineers ought to stand side by side with athletes and entertainers as role models, and here at the White House, we’re going to lead by example. We’re going to show young people how cool science can be.”

The campaign, called Educate to Innovate, focuses mainly on activities outside the classroom. For example, Discovery Communications has promised to use two hours of the afternoon schedule on its Science Channel cable network for commercial-free programming geared toward middle school students.

Science and engineering societies are promising to provide volunteers to work with students in the classroom, culminating in a National Lab Day in May. [...]

Speaking of mathematical literacy, my guess is that man will miss Eugene Robinson’s point here:

ntellectually, it’s simple to understand why it might make sense for women — those who have no special risk factors for breast cancer — to wait until they’re 50, rather than 40, to start getting mammograms. The analysis by the U.S. Preventive Services Task Force, which made the recommendation, looks sound. According to the panel, a whopping 10 percent of mammograms result in false-positive readings that can lead to unjustified worry and unneeded procedures, such as biopsies. In a small number of cases, women are subjected to cancer treatment or even a mastectomy they didn’t need.

This harm, the task force reasoned, outweighs the benefits of discovering relatively few cases of fast-growing, life-threatening breast cancer in women in their 40s through annual mammography. It is also true that waiting to begin regular mammograms until a woman reaches 50 — and reducing the frequency to once every two years, as the task force recommended — would save a portion of the more than $5 billion spent on mammography in the United States each year.

Most people won’t know what “10 percent false positives” means. It means that out of every 100 tests performed, 10 will show “positive” but be false! It is unlikely that there will be any “true positives” out of 100.

Or, put another way:

The probability that an abnormal mammogram is due to cancer increases with age. A large study in Northern California estimated positive predictive values for abnormal mammograms at 2 percent to 4 percent among women aged 40-49, 5 percent to 9 percent among women aged 50-59, and 7 percent to 19 percent among women aged 60 and older.3,9 Positive predictive values were also higher among women with a family history of breast cancer in two studies.3

That is, among women 40-49, a positive mammogram means that the person who had the positive reading has about 2-4 percent chance of actually having breast cancer; the similar number rises to 5-9 percent for women over 50. Keep in mind that the typical 40-49 year old woman has a 1.5 percent chance of developing breast cancer within 10 years; that is, having a positive result conveys only a tiny bit of information!

Talk: Studying and modifying the dynamics of neural networks by Larry Abbott

About the speaker: http://www.neurotheory.columbia.edu/~larry/

Comments: It’s the first time I listened to his talk, and he is really an impressive person, very smart and scientific. I haven’t started reading his book “Theoretical Neuroscience” yet, which is the text book of computational neuroscience in MIT. His experiment on neural network is quite enlightening, which hints that the chaotic world could be synchronized into an orderly output with a stimulus, no matter what inside is.  The model is quite interesting and may help to explain many mechanism of neural development or diseases.  I will read more before writing in details here.