Separation of Variables in the Kovalevskaya–Goryachev–Chaplygin Gyrostat

We construct separation variables for the Kovalevskaya–Goryachev–Chaplygin gyrostat for arbitrary values of the parameters. We show that different separation variables can be constructed for the same integrable system if different integrals of motion are chosen.     

A geometric approach to the separability of the Neumann-Rosochatius system

We study the separability of the Neumann-Rosochatius system on the n-dimensional sphere using the geometry of bi-Hamiltonian manifolds. Its well-known separation variables are recovered by means of a separability condition relating the Hamiltonian with a suitable (1,1) tensor field on the sphere. This also allows us to iteratively construct the integrals of motion of the system.     

Toda Chains in the Jacobi Method

We use the Jacobi method to construct various integrable systems, such as the Stäckel systems and Toda chains, related to various root systems. We find canonical transformations that relate integrals of motion for the generalized open Toda chains of types B _n, C _n, and D _n.     

Existence and construction of shifted lattice rules with an arbitrary number of points and bounded weighted star discrepancy for general decreasing weights

We study the problem of constructing shifted rank-1 lattice rules for the approximation of high-dimensional integrals with a low weighted star discrepancy, for classes of functions having bounded weighted variation, where the weighted variation is defined as the weighted sum of Hardy–Krause variations over all lower-dimensional projections of the integrand. Under general conditions on the weights, we prove the existence of rank-1 lattice rules such that for any δ>0, the general weighted star discrepancy is O(n^−1+δ) for any number of points n>1 (not necessarily prime), any shift of the lattice, general (decreasing) weights, and uniformly in the dimension. We also show that these rules can be constructed by a component-by-component strategy. This implies in particular that a single infinite-dimensional generating vector can be used for integrals in any number of dimensions, and even for infinite-dimensional integrands when they have bounded weighted variation. These same lattices are also good with respect to the worst-case error in weighted Korobov spaces with the same types of general weights. Similar results were already available for various special cases, such as general weights and prime n, or arbitrary n and product weights, but not for the most general combination of n composite, general weights, arbitrary shift, and star discrepancy, considered here. Our results imply tractability or strong tractability of integration for classes of integrands with finite weighted variation when the weights satisfy the conditions we give. These classes are a strict superset of those covered by earlier sufficient tractability conditions.     

On summability of eigenfunction expansions of piecewise smooth functions

We consider two forms of eigenfunction expansions associated with an arbitrary elliptic differential operator with constant coefficients and order m, that is the multiple Fourier series and integrals. For the multiple Fourier integrals, we prove the convergence of the Riesz means of order s?>?(N???3)/2 of piecewise smooth functions of N?≥?2 variables. The same result is proved in the case of the N?≥?3 dimensional multiple Fourier series.     

A special set of eigenvectors for the hyperbolic Sutherland systems

We construct the integrals of motion for Sutherland hyperbolic quantum systems of particles with internal degrees of freedom (su(n) spins) interacting with an external field of the Morse potential of an arbitrary strength τ ² . These systems are confined if certain constraints are imposed on τ, the pair coupling constant λ, and the number of particles. The ground state is described by a wave function of the Jastrow form.     

From Jacobi problem of separation of variables to theory of quasipotential Newton equations

Our solution to the Jacobi problem of finding separation variables for natural Hamiltonian systems H = ½p ² + V(q) is explained in the first part of this review. It has a form of an effective criterion that for any given potential V(q) tells whether there exist suitable separation coordinates x(q) and how to find these coordinates, so that the Hamilton-Jacobi equation of the transformed Hamiltonian is separable. The main reason for existence of such criterion is the fact that for separable potentials V(q) all integrals of motion depend quadratically on momenta and that all orthogonal separation coordinates stem from the generalized elliptic coordinates. This criterion is directly applicable to the problem of separating multidimensional stationary Schrödinger equation of quantum mechanics. Second part of this work provides a summary of theory of quasipotential, cofactor pair Newton equations $ \ddot q $ = M(q) admitting n quadratic integrals of motion. This theory is a natural generalization of theory of separable potential systems $ \ddot q $ = ??(q). The cofactor pair Newton equations admit a Hamilton-Poisson structure in an extended 2n + 1 dimensional phase space and are integrable by embedding into a Liouville integrable system. Two characterizations of these systems are given: one through a Poisson pencil and another one through a set of Fundamental Equations. For a generic cofactor pair system separation variables have been found and such system have been shown to be equivalent to a Stäckel separable Hamiltonian system. The theory is illustrated by examples of driven and triangular Newton equations.     

Bi-Hamiltonian Aspects of the Separability of the Neumann System

The Neumann system on the two-dimensional sphere is used as a tool to convey some ideas on the bi-Hamiltonian standpoint on separation of variables. We show that from this standpoint, its separation coordinates and its integrals of motion can be found systematically.     

Vassiliev invariants and finite-dimensional approximations of the euler equation in magnetohydrodynamics

We consider Hamiltonian systems that correspond to Vassiliev invariants defined by Chen’s iterated integrals of logarithmic differential forms. We show that Hamiltonian systems generated by first-order Vassiliev invariants are related to the classical problem of motion of vortices on the plane. Using second-order Vassiliev invariants, we construct perturbations of Hamiltonian systems for the classical problem of n vortices on the plane. We study some dynamical properties of these systems.     

Numerical integrations over an arbitrary quadrilateral region

In this paper, double integrals over an arbitrary quadrilateral are evaluated exploiting finite element method. The physical region is transformed into a standard quadrilateral finite element using the basis functions in local space. Then the standard quadrilateral is subdivided into two triangles, and each triangle is further discretized into 4 × n² right isosceles triangles, with area , and thus composite numerical integration is employed. In addition, the affine transformation over each discretized triangle and the use of linearity property of integrals are applied. Finally, each isosceles triangle is transformed into a 2-square finite element to compute new n² extended symmetric Gauss points and corresponding weight coefficients, where n is the lower order conventional Gauss Legendre quadratures. These new Gauss points and weights are used to compute the double integral. Examples are considered over an arbitrary domain, and rational and irrational integrals which can not be evaluated analytically.     

Random Flights in Higher Spaces

We consider in this paper random flights in ℝ ^d performed by a particle changing direction of motion at Poisson times. Directions are uniformly distributed on hyperspheres S ₁ ^d . We obtain the conditional characteristic function of the position of the particle after n changes of direction. From this characteristic function we extract the conditional distributions in terms of (n+1)−fold integrals of products of Bessel functions. These integrals can be worked out in simple terms for spaces of dimension d=2 and d=4. In these two cases also the unconditional distribution is determined in explicit form. Some distributions connected with random flights in ℝ³ are discussed and in some special cases are analyzed in full detail. We point out that a strict connection between these types of motions with infinite directions and the equation of damped waves holds only for d=2. Related motions with random velocity in spaces of lower dimension are analyzed and their distributions derived.     

On the power of standard information for <Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript> approximation in the randomized setting

We study approximation of multivariate functions from a general separable reproducing kernel Hilbert space in the randomized setting with the error measured in the L _∞ norm. We consider algorithms that use standard information consisting of function values or general linear information consisting of arbitrary linear functionals. The power of standard or linear information is defined as, roughly speaking, the optimal rate of convergence of algorithms using n function values or linear functionals. We prove under certain assumptions that the power of standard information in the randomized setting is at least equal to the power of linear information in the worst case setting, and that the powers of linear and standard information in the randomized setting differ at most by 1/2. These assumptions are satisfied for spaces with weighted Korobov and Wiener reproducing kernels. For the Wiener case, the parameters in these assumptions are prohibitively large, and therefore we also present less restrictive assumptions and obtain other bounds on the power of standard information. Finally, we study tractability, which means that we want to guarantee that the errors depend at most polynomially on the number of variables and tend to zero polynomially in n ⁻¹ when n function values are used.     

Integrable systems on the sphere associated with genus three algebraic curves

New variables of separation for few integrable systems on the two-dimensional sphere with higher order integrals of motion are considered in detail. We explicitly describe canonical transformations of initial physical variables to the variables of separation and vice versa, calculate the corresponding quadratures and discuss some possible integrable deformations of initial systems.     

Separation of Variables in the Anisotropic Shottky–Frahm Model

We construct separated coordinates for the completely anisotropic Shottky–Frahm model on an arbitrary coadjoint orbit of SO(4). We find explicit reconstruction formulas expressing dynamical variables in terms of the separation coordinates and write the equations of motion in the Abel-type form.     

Movable Singularities of Solutions of Difference Equations in Relation to Solvability and a Study of a Superstable Fixed Point

We review applications of exponential asymptotics and analyzable function theory to difference equations in defining an analogue of the Painlevé property for them, and we sketch the conclusions about the solvability properties of first-order autonomous difference equations. If the Painlevé property is present, the equations are explicitly solvable; otherwise, under additional assumptions, the integrals of motion develop singularity barriers. We apply the method to the logistic map x _n+1=ax _n (1–x _n ), where we find that the only cases with the Painlevé property are a=–2,0,2, and 4, for which explicit solutions indeed exist; otherwise, an associated conjugation map develops singularity barriers.     

Linear information for approximation of the Itô integrals

We study optimal approximation of stochastic integrals in the Itô sense when linear information, consisting of certain integrals of trajectories of Brownian motion, is available. Upper bounds on the nth minimal error, where n is the fixed cardinality of information, are obtained by the Wagner–Platen algorithm and are O(n ^???3/2) or O(n ^???2), depending on considered class of integrands. We also show that Ω(n ^???2) is a lower bound which holds even for very smooth integrands.     

Resultants and contour integrals

Resultants are important special functions used to describe nonlinear phenomena. The resultant R_{r1 ?rn} R_{r_1 \ldots r_n } determines a consistency condition for a system of n homogeneous polynomials of degrees r ₁, ..., r _n in n variables in precisely the same way as the determinant does for a system of linear equations. Unfortunately, there is a lack of convenient formulas for resultants in the case of a large number of variables. In this paper we use Cauchy contour integrals to obtain a polynomial formula for resultants, which is expected to be useful in applications.     

Invariant tori for commuting Hamiltonian PDEs

We generalize to some PDEs a theorem by Eliasson and Nekhoroshev on the persistence of invariant tori in Hamiltonian systems with r integrals of motion and n degrees of freedom, r?n. The result we get ensures the persistence of an r-parameter family of r-dimensional invariant tori. The parameters belong to a Cantor-like set. The proof is based on the Lyapunov-Schmidt decomposition and on the standard implicit function theorem. Some of the persistent tori are resonant. We also give an application to the nonlinear wave equation with periodic boundary conditions on a segment and to a system of coupled beam equations. In the first case we construct 2-dimensional tori, while in the second case we construct 3-dimensional tori.