Related papers: The Euler and Springer numbers as moment sequences
Multi-time equations are evolution equations involving several time variables, one for each particle. Such equations have been considered for the purpose of making theories manifestly Lorentz invariant. We compare their status and…
Moments are expectation values of products of powers of position and momentum, taken over quantum states (or averages over a set of classical particles). For free particles, the evolution in the quantum case is closely related to that of a…
The formalism of classical and quantum mechanics on phase space leads to symplectic and Heisenberg group representations, respectively. The Wigner functions give a representation of the quantum system using classical variables. The…
We study the Euler-Frobenius numbers, a generalization of the Eulerian numbers, and the probability distribution obtained by normalizing them. This distribution can be obtained by rounding a sum of independent uniform random variables; this…
We introduce some generalizations of the Euler-Kronecker constant of a number field and study their arithmetic nature.
The classical dynamics of a particle that is driven by a rapidly oscillating potential (with frequency $\omega$) is studied. The motion is separated into a slow part and a fast part that oscillates around the slow part. The motion of the…
In this paper we generalize and improve a method for calculating the period of a classical oscillator and other integrals of physical interest, which was recently developed by some of the authors. We derive analytical expressions that prove…
The subject of our discussion is the theory of differential equations as set out in two classical Euler's textbooks "Institutiones Calculi Differentialis" and "Institutiones Calculi Integralis".
We describe both quantum particles and classical particles in terms of a classical statistical ensemble, characterized by a probability distribution in phase space. By use of a wave function in phase space both can be treated in the same…
This paper introduces a new object called the momentum tensor. Together with the velocity tensor it forms a basis for establishing the tensorial picture of classical and relativistic mechanics. Some properties of the momentum tensor are…
We survey Weber's class number problem and its variants in the spirit of arithmetic topology; we recollect some history, present a relation to certain units and generalized Pell's equation, and overview a study of the $p$-adic limits of…
At large quantum numbers, the probability densities for particle-in-a-box or simple harmonic oscillator converge to the classical result upon coarse-graining the quantum mechanical probability densities by introducing a finite resolution in…
I study the class of problems efficiently solvable by a quantum computer, given the ability to "postselect" on the outcomes of measurements. I prove that this class coincides with a classical complexity class called PP, or Probabilistic…
Anisotropic Kepler problem is investigated by perturbation method in both classical and quantum mechanics. In classical mechanics, due to the singularity of the potential, global diffusion in phase space occurs at an arbitrarily small…
By virtue of numerical arguments we study a bifurcation phenomenon occurring for a class of minimization problems associated with the quasi-linear Schrodinger equation.
For any positive integer $q$, the sequence of the Euler up/down numbers reduced modulo $q$ was proved to be ultimately periodic by Knuth and Buckholtz. Based on computer simulations, we state for each value of $q$ precise conjectures for…
The Bertrand's theorem can be formulated as the solution of an inverse problem for a classical unidimensional motion. We show that the solutions of these problems, if restricted to a given class, can be obtained by solving a numerical…
Quantum mechanics for a four-state-system is derived from classical statistics. Entanglement, interference, the difference between identical fermions or bosons and the unitary time evolution find an interpretation within a classical…
Euler's equation relates the change in angular momentum of a rigid body to the applied torque. This paper fills a gap in the literature by using Lagrangian dynamics to derive Euler's equation in terms of generalized coordinates. This is…
For the compressible Euler equations, even when the initial data are uniformly away from vacuum, solution can approach vacuum in infinite time. Achieving sharp lower bounds of density is crucial in the study of Euler equations. In this…