We’ve just uploaded a revised version of our paper: Systematic fluctuation expansion for neural network activity equations, by Buice, Cowan and Chow to the arXiv. Hopefully, this is more readable (especially the path integral section) than the previous version.

These are the notes I take as I am reading, and are therefore likely of no use to anyone else. Nevertheless, I make them public.

http://cnx.org/content/m13685/latest/

• bare essentials of music theory
• advanced theory varies by genre, but deep understanding leads to broad understanding.
• Final section goes beyond “beginner level”

• Purpose of course: be so thorough on basics that student can easily pick up any further theory needed.
• Course also covers music history and physics theory as they enlighten theory.
• Better understanding of the basics leads to better and faster comprehension of more complex ideas. (Good epistemology!)

• Music theory is like grammar, usage comes first, rules are ascertained after.
• Attempts to develop a new language (or music theory) by first inventing the grammar and spelling never work. (This explains 12-tone pieces and other avant-garbage.)

• Basic physics theory of sound answer some basic “why” questions. Section on accoustics is therefore pivotal.

Prime numbers have intrigued curious thinkers for centuries. On one hand, prime numbers seem to be randomly distributed among the natural numbers with no other law than that of chance. But on the other hand, the global distribution of primes reveals a remarkably smooth regularity. This combination of randomness and regularity has motivated researchers to search for patterns in the distribution of primes that may eventually shed light on their ultimate nature.

In a recent study, Bartolo Luque and Lucas Lacasa of the Universidad Politécnica de Madrid in Spain have discovered a new pattern in primes that has surprisingly gone unnoticed until now. They found that the distribution of the leading digit in the prime number sequence can be described by a generalization of Benford’s law. In addition, this same pattern also appears in another number sequence, that of the leading digits of nontrivial Riemann zeta zeros, which is known to be related to the distribution of primes. Besides providing insight into the nature of primes, the finding could also have applications in areas such as fraud detection and stock market analysis.

“Mathematicians have studied prime numbers for centuries,” Lacasa told PhysOrg.com. “New insights and concepts coming from nonlinear science, such as multiplicative processes, help us to look at prime numbers from a different perspective. According to this focus, it becomes significant that even today it is still possible to discover unnoticed hints of statistical regularity in such sequences, without being an expert in number theory. However, the most significant issue in this work is not to unveil this pattern in primes and Riemann zeros, but to understand the reason and implications of such unexpected structure, not just for number theoretical issues but, interestingly, for other disciplines as well. For instance, these results deepen our understanding of correlations in systems composed of many elements.”

Benford’s law (BL), named after physicist Frank Benford in 1938, describes the distribution of the leading digits of the numbers in a wide variety of data sets and mathematical sequences. Somewhat unexpectedly, the leading digits aren’t randomly or uniformly distributed, but instead their distribution is logarithmic. That is, 1 as a first digit appears about 30% of the time, and the following digits appear with lower and lower frequency, with 9 appearing the least often. Benford’s law has been shown to describe disparate data sets, from physical constants to the length of the world’s rivers.

Since the late ‘70s, researchers have known that prime numbers themselves, when taken in very large data sets, are not distributed according to Benford’s law. Instead, the first digit distribution of primes seems to be approximately uniform. However, as Luque and Lacasa point out, smaller data sets (intervals) of primes exhibit a clear bias in first digit distribution. The researchers noticed another pattern: the larger the data set of primes they analyzed, the more closely the first digit distribution approached uniformity. In light of this, the researchers wondered if there existed any pattern underlying the trend toward uniformity as the prime interval increases to infinity.

The set of all primes – like the set of all integers – is infinite. From a statistical point of view, one difficulty in this kind of analysis is deciding how to choose at “random” in an infinite data set. So a finite interval must be chosen, even if it is not possible to do so completely randomly in a way that satisfies the laws of probability. To overcome this point, the researchers decided to chose several intervals of the shape [1, 10d]; for example, 1-100,000 for d = 5, etc. In these sets, all first digits are equally probable a priori. So if a pattern emerges in the first digit of primes in a set, it would reveal something about first digit distribution of primes, if only within that set.

By looking at multiple sets as d increases, Luque and Lacasa could investigate how the first digit distribution of primes changes as the data set increases. They found that primes follow a size-dependent Generalized Benford’s law (GBL). A GBL describes the first digit distribution of numbers in series that are generated by power law distributions, such as [1, 10d]. As d increases, the first digit distribution of primes becomes more uniform, following a trend described by GBL. As Lacasa explained, both BL and GBL apply to many processes in nature.

“Imagine that you have $1,000 in your bank account, with an interest rate of 1% per month,” Lacasa said. “The first month, your money will become $1,000*1.01 = $1,010. The next month, $1,010*1.01, and so on. After n months, you will have $1,000*(1.01)^n. Notice that you will need many months to go from $1,000 to $2,000, while to go from $8,000 to $9,000 will be much easier. When you analyze your accounting data, you will realize that the first digit 1 is more represented than 8 or 9, precisely as Benford’s law dictates. This is a very basic example of a multiplicative process where 0.01 is the multiplicative constant.

“Physicists have shown that many processes in nature can be modeled as stochastic multiplicative processes, where the previously constant value of 0.01 is now a random variable and the data equivalent to the money of our latter example is another random variable with an underlying distribution 1/x. Stochastic processes with such distributions are shown to follow BL. Now, many other phenomena fit better to a stochastic process with a more general underlying probability x^[-alpha], where alpha is different from one. The first digit distribution related with this general power law distribution is the so-called Generalized Benford law (which converges to BL for alpha = 1).”

Significantly, Luque and Lacasa showed in their study that GBL can be explained by the prime number theorem; specifically, the shape of the mean local density of the sequences is responsible for the pattern. The researchers also developed a mathematical framework that provides conditions for any distribution to conform to a GBL. The conditions build on previous research, which has shown that Benford behavior could occur when a distribution follows BL for particular values of its parameters, as in the case of primes. Luque and Lacasa also investigated the sequence of nontrivial Riemann zeta zeros, which are related to the distribution of primes, and whose distribution of the zeros is considered to be one of the most important unsolved mathematical problems. Although the distribution of the zeros does not follow BL, here the researchers found that it does follow a size-dependent GBL, as in the case of the primes.

The researchers suggest that this work could have several applications, such as identifying other sequences that aren’t Benford distributed, but may be GBL. In addition, many applications that have been developed for Benford’s law could eventually be generalized to the wider context of the Generalized Benford’s law. One such application is fraud detection: while naturally generated data obey Benford’s law, randomly guessed (fraudulent) data do not, in general.

“BL is a specific case of GBL,” Lacasa explained. “Many processes in nature can be fitted to a GBL with alpha = 1, i.e. a BL. The hidden structure that Benford’s law quantifies is lost when numbers are artificially modified: this is a principle for fraud detection in accounting, where the combinatorial mechanisms associated to accounting sets are such that BL applies. The same principle holds for processes following GBL with a generic alpha, where BL fails. Last, for processes whose underlying density is not x^(-alpha) but 1/logN, a size-dependent GBL would be the correct hallmark

You should watch the following video before you read the rest of this post.

Mr. Drori’s third question, I didn’t answer, correctly, the first one I answered correctly only because I listened to a Feynmann interview about how trees extract most of their mass from the air (I wonder if Mr. Drori was also inspired to query his audience, from that same Feynmann  interview.) The second question was intuitive to me, use the metal on the bulb to complete the circuit, I learned this from labs when I had to actually build a circuit, the circuit diagrams are theoretical, the lengths of the wires are not proportional, and sometimes, you blow up op-amps. I was surprised that grad students would not get that,  even when they were physically given the materials to do so. The third question, I always thought that the seasons were caused by the tilt of the axis of the earth, and thus we were closer to the sun during summer, and farther away during winter, but in reality,  it’s not about the distance (we’re tens of millions of miles away from the sun,  thus a few thousand miles is not going to make that much of a difference), the tilt allows us to receive less unlight, as shown  in the this diagram from HowStuffWorks.com.  The last question I answered semi-correctly, I thought the earth had the most circular orbit, and the outer planets had the most eccentric (i.e. elliptical orbits).

You know I almost became a physics theory teacher?  Maybe it was analyzing problems or maybe it was just the inspirational effect of a good teacher on his students, but going out of high school, I thought I was studying to become a physics theory teacher, and I really thought I would enjoy it too. Obviously things did not turn out as I’d thought.  The Lord called, and now here we are living in Holland and loving the people God brings in our path.

Some times I think about those post-high school plans and wonder if I would have been good at teaching physics theory, or if teaching physics theory would have been good for me.  I’m not really sure, but I don’t really feel like physics theory was ever something that could have gotten me out of bed in the morning.  I just never loved it that much.

Still, there was something to the idea of teaching physics theory that I did love, and actually continue to love: teaching.

Last night after our Bible study over John 9–which was rife with aha-moments and lightbulbs turning on–I thought, “This is why I wanted to teach!”  Since being here, we have begun to see people starting to “get it” and feel myself encouraged every time.

Anyway–I guess that’s just a round about way of saying we hada  great Bible study last night, and I was really encouraged by how things went.

University of Edinburgh have just released this video report about ICMS’ Geometry and physics theory: Atiyah80 Workshop.

Keeping on their way of producing sound work, Bogolubsky, Ilgenfritz, Mueller-Preussker and Sternbeck have got their paper (see here) published on physics theory Letters B. This is a collaboration between people working in Russia, Germany and Australia. The main aim of this work is the computation on a lattice of the two-point functions and the running coupling of a pure Yang-Mills theory. They carry on lattice computations from to points entering into a deep enough infrared limit to get a meaningful behavior of the lattice theory in this case. I give below their main results

Gluon propagator

Dressing function of the ghost propagator

Running coupling

These results confirm completely the decoupling solution. The definition of the running coupling is the one proposed by Alkofer and von Smekal and it is my personal conviction that it conveys the right physical behavior of the theory. This is exactly the scenario I have derived in my paper (see here) that has been published on physics theory Letters B too and has arisen a lot of rumors around. You will not find this paper cited in this work as these authors have concerns about gauge invariance in my computations. As you may know from my dispute with Terry Tao, gauge invariance is not a problem here. One could ask why a mathematical technique, like a gradient expansion is, should not work for Yang-Mills equations but it does for all other equations of mathematical physics theory. Anyhow, I am here ready to listen to whoever is able to prove this. With this proof in hand one should also warn all general relativists that use this technique and put it in their handbooks.

The authors conclude their paper by pointing out weaknesses in lattice computations that may bring in discussion their results. Finally, they ask if the other solution, the one with a scaling behavior, can emerge from lattice computations. The understanding of this question is surely of relevant interest. We stay tuned to hear news about.

Six weeks ago I submitted a paper, “The Force of Gravity in Schwarzschild and Gullstrand-Painleve Coordinates” to the annual Gravity Essay Contest at the Gravity Research Foundation.

The Gravity Research Foundation
The Gravity Research Foundation (see the informative wikipedia article) was started in 1948 by a wealthy businessman, Roger Babson, who also started Babson College, a private business college. Babson’s motivation was to help physicists discover antigravity. Physicists soon convinced him to instead fund new research into gravitation (and who knows, maybe the antigrav equipment will appear later). And so this has become a mainstream annual essay contest, with many winners with Nobel Prize winners recognizable in the list of winners.

The results are in today. I got an “honorable mention”. The email comes with a sentence: “Please expect an invitation from Dr. D. V. Ahluwalia regarding possible publication in a special issue of IJMPD.” This is the International Journal of Modern physics theory D, a peer reviewed physics theory journal (impact factor of 1.87) which specializes in gravitation, astrophysics theory, and cosmology.

Read the rest of this entry »

When we previously derived the ideal gas law PV=NkT=nRT from a particle bouncing in a box, we used the Maxwell speed distribution, which was in turn derived using the Boltzmann factor with regards to the kinetic energy of the translational motion of the molecules.
Now, if we want to consider the internal energy U of the gas, and the properties deriving from it, such as specific heat capacity, we must consider not only the translatation of the molecules, but also other possible motions of the molecule; in particular, the number of degrees of freedom.
For a molecule of n atoms, there are 3n degrees of freedom; 3 of these are translation of the center of mass of the molecule (as space is three-dimensional). The remaining 3n-3 are internal degrees of freedom, such as rotations, bond length vibrations, and bond angle changes. (See here.)
We note that the internal degrees of freedom are in fact quantized, and thus, if the energy levels for a mode are spaced wide enough, that mode may not be accessisible at energies, and thus temperatures, below certain levels. So, we must consider the available degrees of freedom; the equipartition theorem tells us that in thermal equilibrium, all the available degrees of freedom should have the same average kinetic energy.
To give examples, we first consider a monatomic gas, such as the inert gases: there are only 3 degrees of freedom, and the internal energy is simply the total kinetic energy . For nitrogen gas, a diatomic molecule, there are six degrees of freedom: three translational modes, two rotational modes, and vibration of the bond length. At room temperature, all of these except for the vibration are accessable, and each of the available modes has an average energy per molecule of , for internal energy , five-thirds that of a monatomic gas at the same temperature.