Related papers: One half log discriminant
We survey the use of dynamics of $SL(2, \R)$-actions to understand gap distributions for various sequences of subsets of $[0, 1)$, particularly those arising from special trajectories of various two-dimensional dynamical systems. We state…
We develop a new approach to the $L^p$ Dirichlet problem via $L^2$ estimates and reverse Holder inequalities. We apply this approach to second order elliptic systems and the polyharmonic equation on a bounded Lipschitz domain $\Omega$ in…
The W-polynomial is applied in two ways to questions involving the Kauffman bracket of some families of links. First we find a geometric property of a link diagram, which is less than or equal to the twist number, that bounds the Mahler…
This paper shows that various relevant dynamical systems can be described as vector fields associated to smooth functions via a bracket that defines what we call a Leibniz structure. We show that gradient flows, some dissipative systems,…
We propose a variational splitting technique for the generalized-$\alpha$ method to solve hyperbolic partial differential equations. We use tensor-product meshes to develop the splitting method, which has a computational cost that grows…
We show how the size of the Galois groups of iterates of a quadratic polynomial $f(x)$ can be parametrized by certain rational points on the curves $C_n:y^2=f^n(x)$ and their quadratic twists. To that end, we study the arithmetic of such…
Recently, a new definition for a Wigner distribution function for a one-dimensional finite quantum system, in which the position and momentum operators have a finite (multiplicity-free) spectrum, was developed. This distribution function is…
A generalisation of the classical Calogero-Moser model obtained by coupling it to the Gaudin model is considered. The recently found classical dynamical r-matrix [E. Billey, J. Avan and O. Babelon, PAR LPTHE 93-55] for the…
We establish the Lagrangian nature of the discrete isospectral and isomonodromic dynamical systems corresponding to the re-factorization transformations of the rational matrix functions on the Riemann sphere. Specifically, in the…
We introduce a simple method to estimate the system parameters in continuous dynamical systems from the time series. In this method, we construct a modified system by introducing some constants (controlling constants) into the given…
This paper reports on recent work to compute the asymptotic solution of a n-th order ordinary differential equation. Symbolic methods are used to compute the asymptotics over a large region. Application is made to the computation of the…
We construct the hyper-K\"ahler moduli space of framed monopoles over $\mathbb{R}^3$ for any connected, simply connected, compact, semisimple Lie group and arbitrary mass and charge, and hence symmetry breaking. In order to do so, we define…
We give a computationally efficient method for constructing the linear differential operator with polynomial coefficients whose space of holomorphic solutions is spanned by all the branches of a function defined by a generic algebraic…
Global pointwise estimates are obtained for quasilinear Lane-Emden-type systems involving measures in the "sublinear growth" rate. We give necessary and sufficient conditions for existence expressed in terms of Wolff's potential. Our…
The paper deals with a comprehensive theory of mappings, whose local behavior can be described by means of linear subspaces, contained in the graphs of two (primal and dual) generalized derivatives. This class of mappings includes the…
A computer-algebra aided method is carried out, for determining geometric objects associated to differential operators that satisfy the elliptic ansatz. This results in examples of Lame curves with double reduction and in the explicit…
In this article, we consider a counting problem for orbits of hyperbolic rational maps on the Riemann sphere, where constraints are placed on the multipliers of orbits. Using arguments from work of Dolgopyat, we consider varying and…
For an elliptic curve $E$ defined over a field $k\subset \mathbb C$, we study iterated path integrals of logarithmic differential forms on $E^\dagger$, the universal vectorial extension of $E$. These are generalizations of the classical…
A new method is proposed to numerically integrate a dynamical system on a manifold such that the trajectory stably remains on the manifold and preserves first integrals of the system. The idea is that given an initial point in the manifold…
We study angles of multipliers of repelling cycles for hyperbolic rational maps in $\mathbb C(z)$. For a fixed $K \gg 1$, we show that almost all intervals of length $2\pi/K$ in $(-\pi,\pi]$ contain a multiplier angle with the property that…