Soap Film and Minimal Surface

soap_filmThe geometry of soap films and soap bubbles

If we dip two wire rings into a solution of soapy water, then what is the surface formed by the soap film ? This question is known as the minimal surface problem, or the Plateau problem. This post proposes to discuss a simple solution based on variational calculus and elementary computations.

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Arithmetic Gas


Prime numbers can be considered as the building blocks of natural numbers. For example, 12 can be factored in a unique way as 2^{2} \cdot 3 where 2 and 3 are primes. Thus, prime numbers can be compared to elementary particles, which are the building blocks of matter in physics. The concept of arithmetic gas provides an interesting connection between statistical physics and number theory, for which the Riemann zeta function plays the role of a partition function.

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Connes’ Distance Function

In classical geometry, the distance between two points p and q is given by the length of any shortest path from p to q. However, this definition is not valid in quantum mechanics, where the concept of path between two points is not well defined.  The idea introduced by Alain Connes in noncommutative geometry consists of defining a spectral distance d ( p , q ) from values taken by operator observables rather than from classical coordinates. In this way, the concept of geometrical point is not used, which allows the spectral distance to be applied to both classical and quantum spaces. Continue reading

Volterra Series

Vito Volterra (1860-1940) was one of the founding fathers of functional analysis. At the turn of the twentieth century, he introduced the notion of functions of lines, which are defined over a functional space. He studied their derivative and obtained results on integral equations. Practically, a Volterra series is a polynomial functional expansion similar to a Taylor series that provides an approximation of weakly nonlinear systems. One of the first application to nonlinear system analysis is due to Wiener in the 1940s, who developed a method for determining the nonlinear response to a white noise input. Nowadays, this approach is widely used for system identification in many domains such as electrical engineering or biological sciences.

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Brownian Motion

Brownian motion is a phenomenon discovered by the botanist Robert Brown in 1827. He observed that small pollen grains suspended in water describe very irregular movements. The motion is due to the impacts of incessantly moving molecules of water on the pollen grains. The process was explained by Einstein in 1905 as a consequence of thermal energy, then after by Langevin in 1908 through the concept of stochastic differential equation. Continue reading

Heisenberg inequality on the real line

A quantum mechanical principle discovered by Werner Heisenberg states that it is not possible to simultaneously determine the position and momentum of a particle. In fact, this principle is deeply mathematical and independent from the experimental precision of the instruments. We illustrate how it can be expressed for functions on the real line.

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Nonlinear Pendulum


A simple pendulum consists of a single point of mass m attached to a rod of length l and of negligible weight. We denote by \theta the angle measured between the rod and the vertical axis. By applying the Newton’s law of dynamics we obtain the equation of motion

\displaystyle m l^2 \ddot{\theta} + m g l \sin \theta = 0

It can be simplified by putting \omega = \sqrt{g / l}

\displaystyle \ddot{\theta} + \omega^2 \sin \theta = 0

We simplify the equations by normalizing \omega = 1 without loss of generality.

\displaystyle \ddot{\theta} + \sin \theta = 0

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Derivation in Algebra


It is well known that the derivative is an operation on functions which is linear

\displaystyle \begin{array}{c} \left(f+g\right)'=f'+g'\\ \left(\lambda f\right)'=\lambda f'\end{array}

and satisfies the Leibniz rule

\displaystyle \left(fg\right)'=f'g+fg'

In fact, these two properties are purely algebraic and can be generalized with no reference to infinitesimal calculus. This approach has become common in modern mathematics since it allows profitable interactions between multiples domains such as algebra, analysis and geometry.

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