Mathematics has a new friend: biology

In 2004 Joel E. Cohen wrote an article in PLoS Biology entitled "Mathematics Is Biology's Next Microscope, Only Better; Biology Is Mathematics' Next Physics, Only Better." This short article, which has been on my "To Read" list for a long time, is a brief history and assessment of the contributions that each field has and continues to make in the other.

As the title suggests, half of Cohen's claim is that the problems in modern biology will fuel the development of new mathematics. These mathematics should address the fundamental questions posed by biologists, such as "How does it work?" and "What is it for?" There are six such questions, and they are further divided according to the many, many orders of magnitude of space and time that are spanned by biological processes: from molecular biology to global ecosystems and from ultrafast photosynthesis to evolution. Scale-dependent systems and emergent phenomena are the primary themes in modern biological problems.

To illustrate his idea of mathematics, Cohen paints a picture of a tetrahedron with the topics of data structures, theories and models, algorithms, and computers and software located at the four vertexes. Each topic affects the others and has certain strengths for addressing the problems of biology. No weaknesses in the current state of mathematics are mentioned per-se, but any real weakness is likely the notion that, for some problems, the appropriate mathematics simply don't yet exist.

For Cohen, mathematics is decidedly applied mathematics. I doubt he has much to say about topics with no direct relevance for biological applications.

The article is divided into past, present, and future. Cohen first goes into a brief review of the historical interplay between math and biology, starting with what I think is an excellent example: William Harvey's discovery of the circulation of the blood. Just enough background is given to appreciate how novel and unexpected this discovery was. Notably, empiricism, aided by calculations, was in its infancy during Harvey's time. Cohen then pays homage to several other co-developments of math and biology, some of which are nicely summarized in the article's Table 1.

For present matters, Cohen notes that issues of emergence and complexity should lead to great discoveries in mathematics. What is notable here is that emergence in biological systems at one particular level of organization is driven by events at both lower and higher levels. For example, both the genes of an organism and evolution determine many aspects of species. Cohen also provides an example of recent research that marries ideas from statistics, hierarchical clustering, and cancer cell biology. This example is a bit difficult to follow, but I think it is a good analogy of the interplay he is discussing. (To be fair, I was reading the article in an airplane flying through some turbulence, so it was difficult to give this section my full attention.)

The article finishes with a future outlook for his thesis and very briefly presents some ethical problems and opportunities for the continued correspondence between the two fields. I didn't find this section terribly insightful.

This article and those similar to it can't help but make me feel like the Age of Physics is near an end. The problems that occupy most practical people's minds today seem to be concerned with complexity. Physics, which is concerned with constructing models based on the simplest possible assumptions, is by its very nature a difficult tool for understanding phenomena that emerge from the entangled interactions of many heterogeneous parts. Biology just happens to be one field that can push forward our understanding of complex systems. Computation, information science, and neuroscience are other fields that will help further mathematics.

Physics will always be important, but the domain of natural phenomena in which it finds itself useful is lessening as the Information Age comes into full swing.

A primary school method for multiplication of numbers

I found this link that was posted by a friend on Facebook this morning. It's a pictorial description of an algorithm for computing the multiplication of two-digit numbers. I haven't had time yet to think about it, but so far as I can see now it's useful only for computing the product of numbers with one or two digits. Still, it's pretty cool.

In 2011 I had a calendar in my office on mental math algorithms called Lightning Calculation, which I believe I also mentioned in an earlier post. Check this out if you wish to train your brain in the powers of mental computation!

What is the origin of 1/f noise?

Though I've come across it numerous times, I only today questioned the physical origin of the seemingly ubiquitous 1/f noise. I was both surprised and excited to learn that there is, as of now, no general theory of 1/f noise; it has been applied to signals of various types but eludes an interpretation outside of the signal's source.

In contrast, white noise and Brownian noise have well understood theories and are related through an integration/differentiation.

There are some references to 1/f noise and its relation to self-organized criticality, but I'm going to refrain on making a comment about this until I read more.

Amplitude and phase: what are they really?

For sinusoidal signals, the idea of amplitude and phase are pretty clear. The amplitude is one half of the peak to valley distance of the signal and phase coincides with the position of the signal at a point in time or space. By signal, I mean any physical quantity that is undergoing simple harmonic motion, e.g. a mass on a spring. In optics, the amplitude and phase of plane waves are well-defined as well: the amplitude is related to the irradiance and the phase typically refers to the position of the wavefronts with respect to some reference.

What's often taken for granted by new physics students (and even experienced ones) is that plane waves and sinusoidal signals are not real. Any real signal must be of finite duration; a sinusoid extends to infinity both forwards and backwards in time. As a result, the amplitude and phase of real signals become somewhat ambiguous.

The two concepts are retained in practice, however, because they simplify the interpretation of many physical situations, such as the effects of a filter on a signal or the output of an optical interferometer or two-slit experiment. A real signal s(t) has an analytic representation z(t) that consists of s(t) plus the signal's Hilbert transform times the imaginary number, j (I was trained as an engineer, but consider it the same as i used by physicists and mathematicians). Two things should be noted here: 1) the analytic signal is unique, and 2) the analytic signal is complex; it has a real and imaginary part. Like all complex numbers, it can also be represented by a magnitude and phase angle. The magnitude is often denoted the amplitude and the phase angle is thought of as the phase of the signal.

Unfortunately, the amplitude and phase of the analytic signal can not always be directly correlated to physically meaningful quantities. For frequency modulated signals, the time-varying amplitude and phase are independent only so long as their individual spectra are non-overlapping. Furthermore, even if the time-varying amplitude and phase are spectrally separate, it is very counter-intuitive (at least to me) to be able to represent one function of time (the real signal) with two functions of time. This last point I believe is related to the Kramers-Kronig relationship between the real and imaginary parts of a signal. So, while a time-varying amplitude and phase can represent a real modulated signal and simplify interpretation, the only thing about the signal with any physical significance is what can be measured, such as current or voltage.

Most of this post comes from thoughts I've had recently concerning how casually we in optical sciences use the terms amplitude and phase. Once an optical signal becomes incoherent, either in space or time, its amplitude and phase are said to vary randomly. This statement only really makes since once one considers that amplitude and phase are purely theoretical constructs for describing real data and that one must not place too much physical significance in their meaning.

I also learned a lot and reproduced some information from this very good article by Boashash in the Proceedings of the IEEE: http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=135376&tag=1.

More on mental calculations

Just a quick post today. I found this article in Nature Neuroscience from Wikipedia's page on mental calculation.  The abstract follows:

Calculating prodigies are individuals who are exceptional at quickly and accurately solving complex mental calculations. With positron emission tomography (PET), we investigated the neural bases of the cognitive abilities of an expert calculator and a group of non-experts, contrasting complex mental calculation to memory retrieval of arithmetic facts. We demonstrated that calculation expertise was not due to increased activity of processes that exist in non-experts; rather, the expert and the non-experts used different brain areas for calculation. We found that the expert could switch between short-term effort-requiring storage strategies and highly efficient episodic memory encoding and retrieval, a process that was sustained by right prefrontal and medial temporal areas.

It seems that individuals who are naturally gifted at mental calculation possess a sort of encoding mechanism for rapid recall of previously acquired mental facts.

Fortunately, I believe our brains are not hard-wired but able to adapt to imposed stresses and routines so that even non-prodigies can become proficient at performing tasks such as these.

http://en.wikipedia.org/wiki/Mental_calculation
http://www.nature.com/neuro/journal/v4/n1/full/nn0101_103.html

Mentat training

For Christmas I received a "Lightning Calculation" calendar. It's a wall calendar that explains techniques for performing mental calculations, such as finding what day of the week a date fell on and performing fast multiplications. In addition, there are historical bits on famous individuals who have been able to perform incredibly complicated mental calculations. I've spent at least half an hour in my office every morning reading through it and trying the practice problems (of which there are hundreds, if not over a thousand).

Something I've learned is the anchor method for multiplication of two, two-digit numbers. It may seem complicated at first, but it's actually incredibly powerful. First consider the product to be found as a product of two sums, each one "anchored" to a nearby round number. This product can be written as

(a + c)(a + d) = a^2 + ac + ad + cd
(a + c)(a + d) = a(a + c + d) + cd

where a is the anchor. From the right-hand-side of the last line above, the anchor is multiplied by itself with the sum or difference terms c and d. Then the correction term cd is added or subtracted, depending on whether the signs of c and d were the same or not, respectively.

For example, consider the product 17 * 18. This can be written as (20 - 3) * (20 - 2) = 20(20 - 3 - 2) + (2 * 3) = 306. The algorithm is easy to carry out because the first product is 20 * 15 = 300. Cool stuff.

The information for the calendar can be found here: http://myreckonings.com/wordpress/2010/11/22/a-2011-%E2%80%9Clightning-calculation%E2%80%9D-calendar/. It's very nicely done and I highly recommend it for all those mentats in training =)