Doc Madhattan: August 2014

Aidan Dwyer and a new fotovoltaic design

Aidan Dwyer, was one of twelve students to receive the 2011 Young Naturalist Award from the American Museum of Natural History in New York for creating an innovative approach to collecting sunlight in photovoltaic arrays. Dwyer’s investigation into the mathematical relationship of the arrangement of branches and leaves in deciduous trees led to his discovery that these species utilized the Fibonacci Sequence in their branch and leaf design. Dwyer transformed this organic concept into a photovoltaic array based upon the Fibonacci pattern of an oak tree and conducted experiments comparing his design to conventional solar panel arrays. After computer analysis, Dwyer discovered that his Fibonacci tree design surpassed the performance of conventional methods in sunlight collection and utilized the greatest quantity of PV panels within the least amount of physical space, making it a versatile and aesthetically pleasing solution for confined and obstructed urban areas.

The discover of Morniel Mathaway

Following Deutsch and Lockwood⁽¹⁾, there are two types of time paradoxes: inconsistency paradox and knowledge paradox.
An example of the first type is the grandfather paradox, introduced by the french writer René Barjavel in Le voyageur imprudent (1943 - Future Times Three).
An example of the second type is The discover of Morniel Mathaway, a radio science fiction drama by William Tenn. It was originally transmitted by the show X Minus One by NBC:

A professor of art history from the future travels by time machine some centuries into the past in search of an artist whose works are celebrated in the professor's time. On meeting the artist in the flesh, the professor is surprised to find the artist’s current paintings talentlessly amateurish. The professor happens to have brought with him from the future a catalogue containing reproductions of the paintings later attributed to the artist, which the professor has come to see are far too accomplished to be the artist's work. When he shows this to the artist, the latter quickly grasps the situation, and, by means of a ruse, succeeds in using the time machine to travel into the future (taking the catalogue with him), where he realizes he will be welcomed as a celebrity, so stranding the professor in the "present". To avoid entanglements with authority the critic assumes the artist's identity and later achieves fame for producing what he believes are just copies of the paintings he recalls from the catalogue. This means that he, and not the artist, created the paintings in the catalogue. But he could not have done so without having seen the catalogue in the first place, and so we are faced with a causal loop.

The solar efficiency of Superman

In the last saga of the JLA by Grant Morrison, World War III, Superman, leaping against the bomb inside Mageddon says:

The way in which Superman gets the powers, or the way in which them is explained, however, is changed over time. Following Action Comics #1, the debut of the character, Jerry Siegel, combining genetics and evolution, says that on his planet of origin

the physical structure of the inhabitants had advanced millions of years compared to ours. Reaching maturity, people of that race earned a titanic force!

In Superman #1, however, Siegel focuses attention on the different gravity between Earth and Krypton, with the latter with a greater radius than aour planet and therefore with a greater severity. Such a claim is also in Ports of Call by Jack Vance. In order to verify it, we must start from the definition of the density: \[\rho = \frac{M}{V}\] where $M$ is the mass, $V$ the volume of the object, or, in our case, of the planet.

Mathematics is a unique aspect of human thought

Mathematics is a unique aspect of human thought, and its history differs in essence from all other histories.
As time goes on, nearly every field of human endeavor is marked by changes which can be considered as correction and/or extension. Thus, the changes in the evolving history of political and military events are always chaotic; there is no way to predict the rise of a Genghis Khan, for example, or the consequences of the short-lived Mongol Empire. Other changes are a matter of fashion and subjective opinion. The cave-paintings of 25,000 years ago are generally considered great art, and while art has continuously-even chaotically-changed in the subsequent millennia, there are elements of greatness in all the fashions. Similarly, each society considers its own ways natural and rational, and finds the ways of other societies to be odd, laughable, or repulsive.
But only among the sciences is there true progress; only there is the record one of continuous advance toward ever greater heights.
And yet, among most branches of science, the process of progress is one of both correction and extension. Aristotle, one of the greatest minds ever to contemplate physical laws, was quite wrong in his views on falling bodies and had to be corrected by Galileo in the 1590s. Galen, the greatest of ancient physicians, was not allowed to study human cadavers and was quite wrong in his anatomical and physiological conclusions. He had to be corrected by Vesalius in 1543 and Harvey in 1628. Even Newton, the greatest of all scientists, was wrong in his view of the nature of light, of the achromaticity of lenses, and missed the existence of spectral lines. His masterpiece, the laws of motion and the theory of universal gravitation, had to be modified by Einstein in 1916.
Now we can see what makes mathematics unique. Only in mathematics is there no significant correction-only extension. Once the Greeks had developed the deductive method, they were correct in what they did, correct for all time. Euclid was incomplete and his work has been extended enormously, but it has not had to be corrected. His theorems are, every one of them, valid to this day.
Ptolemy may have developed an erroneous picture of the planetary system, but the system of trigonometry he worked out to help him with his calculations remains correct forever.
Each great mathematician adds to what came previously, but nothing needs to be uprooted. Consequently, when we read a book like A History Of Mathematics, we get the picture of a mounting structure, ever taller and broader and more beautiful and magnificent and with a foundation, moreover, that is as untainted and as functional now as it was when Thales worked out the first geometrical theorems nearly 26 centuries ago.
Nothing pertaining to humanity becomes us so well as mathematics. There, and only there, do we touch the human mind at its peak.

Isaac Asimov from the foreword to the second edition of A History of Mathematics by Carl C. Boyer and Uta C. Merzbach

Maryam Mirzakhani and Riemann surfaces

http://t.co/ZAdRPeiy8b Maryam Mirzakhani wins #FieldsMedal with Riemann surfaces

Maryam Mirzakhani has made several contributions to the theory of moduli spaces of Riemann surfaces. In her early work, Maryam Mirzakhani discovered a formula expressing the volume of a moduli space with a given genus as a polynomial in the number of boundary components. This led her to obtain a new proof for the conjecture of Edward Witten on the intersection numbers of tautology classes on moduli space as well as an asymptotic formula for the length of simple closed geodesics on a compact hyperbolic surface. Her subsequent work has focused on Teichmüller dynamics of moduli space. In particular, she was able to prove the long-standing conjecture that William Thurston's earthquake flow on Teichmüller space is ergodic.
Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces".

Riemann surfaces are one dimensional complex manifolds introduced by Riemann: in some sense, his approach is a cut-and-paste procedure.

He imagined taking as many copies of the open set as there are branches of the function and joining them together along the branch cuts. To understand how this works, imagine cutting out sheets along the branch curves and stacking them on top of the complex plane. On each sheet, we define one branch of the function. We glue the different sheets to each other in such a way that the branch of the function on one sheet joins continuously at the seam with the branch defined on the other sheet. For instance, in the case of the square root, we join each end of the sheet corresponding to the positive branch with the opposite end of the sheet corresponding to the negative branch. In the case of the logarithm, we join one end of the sheet corresponding to the $2 \pi n$ branch with an end of the $(2n+1) \pi n$ sheet to obtain a spiral structure which looks like a parking garage.

A more formal approach to the construction of Riemann surfaces is performed by Hermann Weyl, and the work by Maryam Mirzakhani puts in this line of research.

Some papers:
Mirzakhani M. (2007). Weil-Petersson volumes and intersection theory on the moduli space of curves, Journal of the American Mathematical Society, 20 (01) 1-24. DOI: http://dx.doi.org/10.1090/s0894-0347-06-00526-1
Mirzakhani M. (2006). Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Inventiones mathematicae, 167 (1) 179-222. DOI: http://dx.doi.org/10.1007/s00222-006-0013-2 (pdf)
Mirzakhani M. (2008). Growth of the number of simple closed geodesics on hyperbolic surfaces, Annals of Mathematics, 168 (1) 97-125. DOI: http://dx.doi.org/10.4007/annals.2008.168.97 (pdf)

The equation of happiness

\[H(t) = w_0 + w_1 \sum_{j=1}^t \gamma^{t-j} CR_j + w_2 \sum_{j=1}^t \gamma^{t-j} EV_j + w_3 \sum_{j=1}^t \gamma^{t-j} RPE_j\] I don't know if my intuition is correct, but the equation from Rutledge et al. reminds me of a neural network, or more correctly a sum of three different neural networks. In every case, this could became an important step in order to mathematically describe our brain.

A common question in the social science of well-being asks, "How happy do you feel on a scale of 0 to 10?" Responses are often related to life circumstances, including wealth. By asking people about their feelings as they go about their lives, ongoing happiness and life events have been linked, but the neural mechanisms underlying this relationship are unknown. To investigate it, we presented subjects with a decision-making task involving monetary gains and losses and repeatedly asked them to report their momentary happiness. We built a computational model in which happiness reports were construed as an emotional reactivity to recent rewards and expectations. Using functional MRI, we demonstrated that neural signals during task events account for changes in happiness.

Rutledge R.B., Skandali N., Dayan P. & Dolan R.J. (2014). A computational and neural model of momentary subjective well-being., Proceedings of the National Academy of Sciences of the United States of America, PMID: http://www.ncbi.nlm.nih.gov/pubmed/25092308

via design & trends

Generalized Venn diagram for genetics

A generalized Venn diagram with three sets $A$, $B$ and $C$ and their intersections. From this representation, the different set sizes are easily observed. Furthermore, if individual elements (genes) are contained in more than one set (functional category), the intersection sizes give a direct view on how many genes are involved in possibly related functions. During optimization, the localization of the circles is altered to satisfy the possibly contradictory constraints of circle size and intersection size.

For the purpouse of the paper, the researchers used polygons instead of circles. In order to compute the polygons' area, they used the simple formula: \[A = \sum_{k=1}^L x_k (y_{k+1} - y_k)\] where $L$ is the number of the edges of the polygon, and $y_{L+1} := y_1$.

Kestler, H., Muller, A., Gress, T., & Buchholz, M. (2004). Generalized Venn diagrams: a new method of visualizing complex genetic set relations Bioinformatics, 21 (8), 1592-1595 DOI: 10.1093/bioinformatics/bti169

Turing's morphogenesis and the fingers' formation

On today Science's issue it is published a paper about the application of Turing's morphogenesis to the formation of fingers. In this period I'm not able to download the papers, so I simple publish the editor's summaries. First of all I present you the incipit of the paper by Aimée Zuniga, Rolf Zeller⁽²⁾

Alan Turing is best known as the father of theoretical computer sciences and for his role in cracking the Enigma encryption codes during World War II. He was also interested in mathematical biology and published a theoretical rationale for the self-regulation and patterning of tissues in embryos. The so-called reaction-diffusion model allows mathematical simulation of diverse types of embryonic patterns with astonishing accuracy. During the past two decades, the existence of Turing-type mechanisms has been experimentally explored and is now well established in developmental systems such as skin pigmentation patterning in fishes, and hair and feather follicle patterning in mouse and chicken embryos. However, the extent to which Turing-type mechanisms control patterning of vertebrate organs is less clear. Often, the relevant signaling interactions are not fully understood and/or Turing-like features have not been thoroughly verified by experimentation and/or genetic analysis. Raspopovic et al.⁽¹⁾ now make a good case for Turing-like features in the periodic pattern of digits by identifying the molecular architecture of what appears to be a Turing network functioning in positioning the digit primordia within mouse limb buds.

And now the summary of the results:

Most researchers today believe that each finger forms because of its unique position within the early limb bud. However, 30 years ago, developmental biologists proposed that the arrangement of fingers followed the Turing pattern, a self-organizing process during early embryo development. Raspopovic et al.⁽¹⁾ provide evidence to support a Turing mechanism (see the Perspective by Zuniga and Zeller). They reveal that Bmp and Wnt signaling pathways, together with the gene Sox9, form a Turing network. The authors used this network to generate a computer model capable of accurately reproducing the patterns that cells follow as the embryo grows fingers.

(1) Raspopovic, J., Marcon, L., Russo, L., & Sharpe, J. (2014). Digit patterning is controlled by a Bmp-Sox9-Wnt Turing network modulated by morphogen gradients Science, 345 (6196), 566-570 DOI: 10.1126/science.1252960
(2) Zuniga, A., & Zeller, R. (2014). In Turing's hands--the making of digits Science, 345 (6196), 516-517 DOI: 10.1126/science.1257501

via phys.org

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