Related papers: Pendulum: separatrix splitting
Fragmentation processes are part of a broad class of models describing the evolution of a system of particles which split apart at random. These models are widely used in biology, materials science and nuclear physics, and their asymptotic…
It is shown that the asymptotic spectra of finite-time Lyapunov exponents of a variety of fully chaotic dynamical systems can be understood in terms of a statistical analysis. Using random matrix theory we derive numerical and in particular…
We give an elementary introduction, through illustrative examples but without proofs, to one of the basic consequences of the Langlands programme, namely the law governing the primes modulo which a given irreducible integral polynomial…
In this short note, we prove positivity of the Lyapunov exponent for 1D continuum Anderson models by leveraging some classical tools from inverse spectral theory. The argument is much simpler than the existing proof due to…
We propose a new analytical method to solve for nonexactly soluble Schrodinger equation via expansions through some existing quantum numbers. Successfully, it is applied to the rational non-polynomial oscillator potential. Moreover, a…
Recent results for rotations expressed as polynomials of spin matrices are derived here by elementary differential equation methods. Structural features of the results are then examined in the framework of biorthogonal systems, to obtain an…
We exhibit explicit expressions, in terms of components, of discriminants, determinants, characteristic polynomials and polynomial identities for matrices of higher rank. We define permutation tensors and in term of them we construct…
In an earlier paper, we discussed the probability that the determinant of a matrix undergoes the least change upon perturbation of one of its elements, provided that most or all of the elements of the matrix are chosen at random and that…
We develop a Melnikov framework for the Kuramoto Sivashinsky (KS) equation under weak deterministic and stochastic forcing. By treating KS as an infinite dimensional dynamical system, we derive a Melnikov functional that measures splitting…
The hyperdeterminant of a polynomial (interpreted as a symmetric tensor) factors into several irreducible factors with multiplicities. Using geometric techniques these factors are identified along with their degrees and their…
We study a generalization of the discriminant of a polynomial, which we call the tolerant. The tolerant differs by multiplication by a square from the duplicant, which was discovered in recent work on $\mathbb{P}^1$-loop spaces in motivic…
We prove that for any non-trivial perturbation depending on any two independent harmonics of a pendulum and a rotor there is global instability. The proof is based on the geometrical method and relies on the concrete computation of several…
We consider the symmetric group $S_n$-module of the polynomial ring with $m$ sets of $n$ commuting variables and $m'$ sets of $n$ anti-commuting variables and show that the multiplicity of an irreducible indexed by the partition $\lambda$…
We have obtained the exact asymptotics of the determinant $\det_{1\leq r,s\leq L}[\binom{r+s-2}{r-1}+\exp(i\theta)\delta_{r,s}]$. Inverse symbolic computing methods were used to obtain exact analytical expressions for all terms up to…
We exploit the analogy between dynamics of inertial particle pair separation in a random-in-time flow and the Anderson model of a quantum particle on the line in a spatially random real-valued potential. Thereby we get an exact formula for…
We study modal separability for fixpoint formulae: given two mutually exclusive fixpoint formulae $\varphi,\varphi'$, decide whether there is a modal formula $\psi$ that separates them, that is, that satisfies…
This paper studies the dynamics and integrability of a variable-length coupled pendulum system. The complexity of the model is presented by joining various numerical methods, such as the Poincar\'e cross-sections, phase-parametric diagrams,…
We give abstract versions of the large deviation theorem for the distribution of zeros of polynomials and apply them to the characteristic polynomials of Hermitian random matrices. We obtain new estimates related to the local semi-circular…
We present a complete framework for determining the asymptotic (or logarithmic) efficiency of estimators of large deviation probabilities and rate functions based on importance sampling. The framework relies on the idea that importance…
This contribution is dedicated to the exploration of exponential operator splitting methods for the time integration of evolution equations. It entails the review of previous achievements as well as the depiction of novel results. The…