Relative equilibrium and collapse configurations of four point vortices

Relative equilibrium configurations of point vortices in the plane can be related to a system of polynomial equations in the vortex positions and circulations. For systems of four vortices the solution set to this system is proved to be finite, so long as a number of polynomial expressions in the vortex circulations are nonzero, and the number of relative equilibrium configurations is thereby shown to have an upper bound of 56. A sharper upper bound is found for the special case of vanishing total circulation. The polynomial system is simple enough to allow the complete set of relative equilibrium configurations to be found numerically when the circulations are chosen appropriately. Collapse configurations of four vortices are also considered; while finiteness is not proved, the approach provides an effective computational method that yields all configurations with a given ratio of velocity to position.      

La Methode d1Horace pour l'Interpolation à Plusieurs Variables

At how many points is it possible to prescribe values to a polynomial of given degree, together with derivatives up to order, say, t-1? We solve this problem in case of polynomials in two variables for t=2 and t=3 and in case of polynomials in three variables for t=2. Proofs develop in the frame of modern projective geometry.     

Point vortices and classical orthogonal polynomials

Stationary equilibria of point vortices in the plane and on the cylinder in the presence of a background flow are studied. Vortex systems with an arbitrary choice of circulations are considered. Differential equations satisfied by generating polynomials of vortex configurations are derived. It is shown that these equations can be reduced to a single one. It is found that polynomials that are Wronskians of classical orthogonal polynomials solve the latter equation. As a consequence vortex equilibria at a certain choice of background flows can be described with the help of Wronskians of classical orthogonal polynomials.     

THE STATIONARY DIFFUSION PROCESSES GENERATED BY SECOND ORDER DIFFERENTIAL OPERATIONS AND THEIR PROBABILITY CURRENTS

In this paper,we are concerned with the stationary Markov processes generated by second order differential operators under the local boundary conditions, It is proved that all those processes have constnt probability currents, known as circulations of the processes, and hence the processes are called single circulation processes. The invariant measures and the circulation values of those processes are calculated in all cases of boundary classification. It is shown that thr circulation value is an elementary characteristic of irreversible stationary Markov processes and that all the reversible Markov processes in the same problem are just the special ones of the single circulation processes whose circulation values are equal to zero and whose ergodic limits in the sense of weak convergence are not trivial.     

Point vortices and polynomials of the Sawada-Kotera and Kaup-Kupershmidt equations

Rational solutions and special polynomials associated with the generalized K ₂ hierarchy are studied. This hierarchy is related to the Sawada-Kotera and Kaup-Kupershmidt equations and some other integrable partial differential equations including the Fordy-Gibbons equation. Differential-difference relations and differential equations satisfied by the polynomials are derived. The relationship between these special polynomials and stationary configurations of point vortices with circulations Γ and −2Γ is established. Properties of the polynomials are studied. Differential-difference relations enabling one to construct these polynomials explicitly are derived. Algebraic relations satisfied by the roots of the polynomials are found.     

Relations Satisfied by Point Vortex Equilibria with Strength Ratio −2

Relations satisfied by the roots of the Loutsenko sequence of polynomials are derived. These roots are known to correspond to families of stationary and uniformly translating point vortices with two vortex strengths in ratio ?2. The relations are analogous to those satisfied by the roots of the Adler–Moser polynomials, corresponding to equilibria with ratio ?1. The proof uses an analysis of the differential equation that these polynomial pairs satisfy.     

Stationary Configurations of Point Vortices on a Cylinder

In this paper we study the problem of constructing and classifying stationary equilibria of point vortices on a cylindrical surface. Introducing polynomials with roots at vortex positions, we derive an ordinary differential equation satisfied by the polynomials. We prove that this equation can be used to find any stationary configuration. The multivortex systems containing point vortices with circulation Γ₁ and Γ₂ (Γ₂ = ?μΓ₁) are considered in detail. All stationary configurations with the number of point vortices less than five are constructed. Several theorems on existence of polynomial solutions of the ordinary differential equation under consideration are proved. The values of the parameters of the mathematical model for which there exists an infinite number of nonequivalent vortex configurations on a cylindrical surface are found. New point vortex configurations are obtained.     

Differential equations for the elementary 3-symmetric Chebyshev polynomials

We continue the study of a “compound model of a generalized oscillator” and related elementary 3-symmetric Chebyshev polynomials. For these polynomials, we obtain second-order differential equations which are of Fuchs type and have 13 singular points. In the considered simplest case, the obtained results give us an answer to a more general question: What changes in the differential equations for polynomials of the Askey–Wilson scheme when the Jacobi matrix related to these polynomials is perturbed by a diagonal matrix with a complex diagonal? Bibliography: 8 titles.     

Interpolation polyn?miale sur un ordre d’un corps de nombres

Résumé  Dans ce texte, nous introduisons des polyn?mes d’interpolation sur un ordre d’un corps de nombres qui généralisent les polyn?mes bin?miaux usuels. Comme application, on montre, en utilisant ces polyn?mes, qu’une fonction entière envoyant un ordre d’un corps de nombres dans l’anneau des entiers de ce corps et dont les croissances analytique et arithmétique sont faibles est un polyn?me, étendant ainsi au cas non discret les résultats antérieurs dans ce domaine.

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In this paper, we introduce interpolation polynomials on an order of a number field which generalize the usual binomial polynomials. As an application of these polynomials we prove that any entire function on an order of a number field with integer values in this field and which has slow analytic and arithmetic growth is a polynomial. This results extends earlier results in this area to the non discrete case.

     

Real polynomial iterative roots in the case of nonmonotonicity height ≥ 2

It is known that a strictly piecewise monotone function with nonmonotonicity height ≥ 2 on a compact interval has no iterative roots of order greater than the number of forts. An open question is: Does it have iterative roots of order less than or equal to the number of forts? An answer was given recently in the case of "equal to". Since many theories of resultant and algebraic varieties can be applied to computation of polynomials, a special class of strictly piecewise monotone functions, in this paper we investigate the question in the case of "less than" for polynomials. For this purpose we extend the question from a compact interval to the whole real line and give a procedure of computation for real polynomial iterative roots. Applying the procedure together with the theory of discriminants, we find all real quartic polynomials of non-monotonicity height 2 which have quadratic polynomial iterative roots of order 2 and answer the question.     

Invariance of Milnor numbers and topology of complex polynomials

We give a global version of Lê-Ramanujam μ-constant theorem for polynomials. Let , , be a family of polynomials of n complex variables with isolated singularities, whose coefficients are polynomials in t. We consider the case where some numerical invariants are constant (the affine Milnor number μ(t), the Milnor number at infinity λ(t), the number of critical values, the number of affine critical values, the number of critical values at infinity). Let n=2, we also suppose the degree of the is a constant, then the polynomials and are topologically equivalent. For we suppose that critical values at infinity depend continuously on t, then we prove that the geometric monodromy representations of the are all equivalent. Received: January 14, 2002     

Uniform approximation on [?1, 1] via discrete de la Vallée Poussin means

Starting from the function values on the roots of Jacobi polynomials, we construct a class of discrete de la Vallée Poussin means, by approximating the Fourier coefficients with a Gauss?CJacobi quadrature rule. Unlike the Lagrange interpolation polynomials, the resulting algebraic polynomials are uniformly convergent in suitable spaces of continuous functions, the order of convergence being comparable with the best polynomial approximation. Moreover, in the four Chebyshev cases the discrete de la Vallée Poussin means share the Lagrange interpolation property, which allows us to reduce the computational cost.     

Finite dimensional characteristic functions of Brownian rough path

The Brownian rough path is the canonical lifting of Brownian motion to the free nilpotent Lie group of order 2: Equivalently, it is a process taking values in the algebra of Lie polynomials of degree 2, which is described explicitly by the Brownian motion coupled with its area process. The aim of this article is to compute the finite dimensional characteristic functions of the Brownian rough path in ?^d and obtain an explicit formula for the case when d = 2     

On the roots of the orthogonal polynomials and residual polynomials associated with a conjugate gradient method

In this paper we explore two sets of polynomials, the orthogonal polynomials and the residual polynomials, associated with a preconditioned conjugate gradient iteration for the solution of the linear system Ax = b. In the context of preconditioning by the matrix C, we show that the roots of the orthogonal polynomials, also known as generalized Ritz values, are the eigenvalues of an orthogonal section of the matrix C A while the roots of the residual polynomials, also known as pseudo-Ritz values (or roots of kernel polynomials), are the reciprocals of the eigenvalues of an orthogonal section of the matrix (C A)^?1. When C A is selfadjoint positive definite, this distinction is minimal, but for the indefinite or nonselfadjoint case this distinction becomes important. We use these two sets of roots to form possibly nonconvex regions in the complex plane that describe the spectrum of C A.     

Invariants and Chebyshev polynomials

On different compact sets from ? ⁿ , new multidimensional analogs of algebraic polynomials least deviating from zero (Chebyshev polynomials) are constructed. A brief review of the analogs constructed earlier is given. Estimates of values of the best approximation obtained by using extremal signatures, lattices, and finite groups are presented.     

Euler polynomials,Bernoulli polynomials,and Lévyʼs stochastic area formula

In 1951, P. Lévy represented the Euler and Bernoulli numbers in terms of the moments of Lévy?s stochastic area. Recently the authors extended his result to the case of Eulerian polynomials of types A and B. In this paper, we continue to apply the same method to the Euler and Bernoulli polynomials, and will express these polynomials with the use of Lévy?s stochastic area. Moreover, a natural problem, arising from such representations, to calculate the expectations of polynomials of the stochastic area and the norm of the Brownian motion will be solved.     

Twice continuously differentiable S-splines

Twice continuously differentiable S-splines consisting of fifth degree polynomials are constructed, uniqueness and existence theorems are proved, stability conditions are established for such splines. The first three coefficients of each polynomial are determined by conditions of smooth gluing, the others are determined by the least squares method. This provides the ability to smooth initial data. The peculiarity of these splines is their semilocal property, i.e., each polynomial implicitly depends on function values determining previous polynomials and does not depend on values determining subsequent polynomials. It turns out that in this case the stability conditions are fulfilled under some very strong restrictions. Under there conditions and other ones ensuring sufficient closeness of the first polynomial and its derivatives to values of the function and its derivatives it is proved that this closeness is retained on the whole given interval.     

Biorthogonal decompositions of the Radon transform by multivariate interpolation

The concept of biorthogonal and singular value decompositions is a valuable tool in the examination of ill-posed inverse problems such as the inversion of the Radon transform. By application of the theory of multivariate interpolation, e. g. the set of Lagrange polynomials with respect to the space of homogeneous spherical polynomials, we determine new biorthogonal decompositions of the Radon transform. We consider the case of functions with support in the unit ball and the case of functions with support ?^r. In both cases we assume that the functions are square integrable with respect to some weight functions. In the important special case of square integrable functions with respect to the unit ball the structure of the biorthogonal decompositions is easier in comparison with the known singular and biorthogonal decompositions. Especially the calculation of the unknown expansion coefficients can be done by using arbitrary fundamental systems (μ-resolving data set in terms of tomography with a minimum number of nodes) and simplifies essentially. The decompositions are based on a system of zonal (ridge) Gegenbauer (ultraspherical) polynomials which are used in the theory of the Radon transform and in the field of numerical algorithms for the inversion of the transform.     

Bivariate Gonarov polynomials and integer sequences

Univariate Gonarov polynomials arose from the Gonarov interpolation problem in numerical analysis.They provide a natural basis of polynomials for working with u-parking functions,which are integer sequences whose order statistics are bounded by a given sequence u.In this paper,we study multivariate Gonarov polynomials,which form a basis of solutions for multivariate Gonarov interpolation problem.We present algebraic and analytic properties of multivariate Gonarov polynomials and establish a combinatorial relation with integer sequences.Explicitly,we prove that multivariate Gonarov polynomials enumerate k-tuples of integers sequences whose order statistics are bounded by certain weights along lattice paths in Nk.It leads to a higher-dimensional generalization of parking functions,for which many enumerative results can be derived from the theory of multivariate Gonarov polynomials.     

Estimates for the orders of zeros of polynomials in some analytic functions

In the present paper, we consider estimates for the orders of zeros of polynomials in functions satisfying a system of algebraic differential equations and possessing a special D-property defined in the paper. The main result obtained in the paper consists of two theorems for the two cases in which these estimates are given. These estimates are improved versions of a similar estimate proved earlier in the case of algebraically independent functions and a single point. They are derived from a more general theorem concerning the estimates of absolute values of ideals in the ring of polynomials, and the proof of this theorem occupies the main part of the present paper. The proof is based on the theory of ideals in rings of polynomials. Such estimates may be used to prove the algebraic independence of the values of functions at algebraic points.