## Archive for the ‘**MATHEMATICS**’ Category

## A Poet of Computation Who Uncovers Distant Truths

**The theoretical computer scientist Constantinos Daskalakis has won the Rolf Nevanlinna Prize for explicating core questions in game theory and machine learning.**

Scroll down to the bottom of Constantinos Daskalakis’ web page — past links to his theoretical computer science papers and his doctoral students at the Massachusetts Institute of Technology — and you will come upon a spare, 21-line poem by Constantine Cavafy, “The Satrapy.”

Written in 1910, it addresses an unnamed individual who is “made for fine and great works” but who, having met with small-mindedness and indifference, gives up on his dreams and goes to the court of the Persian king Artaxerxes. The king lavishes satrapies (provincial governorships) upon him, but his soul, Cavafy writes, “weeps for other things … the hard-won and inestimable Well Done; the Agora, the Theater, and the Laurels” — all the things Artaxerxes cannot give him. “Where will you find these in a satrapy,” Cavafy asks, “and what life can you live without these.”

For Daskalakis, the poem serves as a sort of talisman, to guard him against base motives. “It’s a moral compass, if you want,” he said. “I want to have this constant reminder that there are some noble ideas that you’re serving, and don’t forget that when you make decisions.” …

Read more at https://www.quantamagazine.org/computer-scientist-constantinos-daskalakis-wins-nevanlinna-prize-20180801/

Read also: **The Work of Constantinos Daskalakis**

## Longest Straight Line Paths on Water or Land on the Earth

**Rohan Chabukswar, Kushal Mukherjee**

There has been some interest recently in determining the longest distance one can sail for on the earth without hitting land, as well as in the converse problem of determining the longest distance one could drive for on the earth without encountering a major body of water. In its basic form, this is an optimisation problem, rendered chaotic by the presence of islands and lakes, and indeed the fractal nature of the coasts. In this paper we present a methodology for calculating the two paths using the branch-and-bound algorithm.

Read more at https://arxiv.org/pdf/1804.07389.pdf

## Emergence of Benford’s Law in Classical Music

**Azar Khosravani, Constantin Rasinariu**

We analyzed a large selection of classical musical pieces composed by Bach, Beethoven, Mozart, Schubert and Tchaikovsky, and found a surprising connection with mathematics. For each composer, we extracted the time intervals each note was played in each piece and found that the corresponding data sets are Benford distributed. Remarkably, the logarithmic distribution is not only present for the leading digits, but for all digits.

Read more at https://arxiv.org/pdf/1805.06506.pdf

## Duality symmetries behind solutions of the classical simple pendulum

**Román Linares**

The solutions that describe the motion of the classical simple pendulum have been known for very long time and are given in terms of elliptic functions, which are doubly periodic functions in the complex plane. The independent variable of the solutions is time and it can be considered either as a real variable or as a purely imaginary one, which introduces a rich symmetry structure in the space of solutions. When solutions are written in terms of the Jacobi elliptic functions the symmetry is codified in the functional form of its modulus, and is described mathematically by the six dimensional coset group Γ/Γ(2) where Γ is the modular group and Γ(2) is its congruence subgroup of second level. In this paper we discuss the physical consequences this symmetry has on the pendulum motions and it is argued they have similar properties to the ones termed as duality symmetries in other areas of physics, such as field theory and string theory. In particular a single solution of pure imaginary time for all allowed value of the total mechanical energy is given and obtained as the S-dual of a single solution of real time, where S stands for the S generator of the modular group.

I. INTRODUCTION

The simple plane pendulum constitutes an important physical system whose analytical solutions are well known.

Historically the first systematic study of the pendulum is attributed to Galileo Galilei, around 1602. Thirty years later he discovered that the period of small oscillations is approximately independent of the amplitude of the swing, property termed as isochronism, and in 1673 Huygens published the mathematical formula for this period. However, as soon as 1636, Marin Mersenne and Rene Descartes had stablished that the period in fact does depend of the amplitude. The mathematical theory to evaluate this period took longer to be established.

The Newton second law for the pendulum leads to a nonlinear differential equation of second order whose solutions are given in terms of either Jacobi elliptic functions or Weierstrass elliptic functions …

… Read more at http://arxiv.org/pdf/1601.07891v1.pdf