Integral Polynomials Research Articles

Algebraic Birkhoff conjecture for billiards on Sphere and Hyperbolic plane

We consider a convex curve γ lying on the Sphere or Hyperbolic plane. We study the problem of existence of polynomial in velocities integrals for Birkhoff billiard inside the domain bounded by γ . We extend the result by S. Bolotin (1992) and get new obstructions on polynomial integrability in terms of the dual curve Γ . We follow a method which was introduced by S. Tabachnikov for Outer billiards in the plane and was applied later on in our recent paper to Birkhoff billiards with the help of a new the so called Angular billiard.

Integral polynomials with small discriminants and resultants

Let n∈N be fixed, Q>1 be a real parameter and Pn(Q) denote the set of polynomials over Z of degree n and height at most Q. In this paper we investigate the following counting problems regarding polynomials with small discriminant D(P) and pairs of polynomials with small resultant R(P1,P2):(i)given0≤v≤n−1and a sufficiently large Q, estimate the number of polynomialsP∈Pn(Q)such that0<|D(P)|≤Q2n−2−2v;(ii)given0≤w≤nand a sufficiently large Q, estimate the number of pairs of polynomialsP1,P2∈Pn(Q)such that0<|R(P1,P2)|≤Q2n−2w. Our main results provide lower bounds within the context of the above problems. We believe that these bounds are best possible as they correspond to the solutions of naturally arising linear optimisation problems. Using a counting result for the number of rational points near planar curves due to R. C. Vaughan and S. Velani we also obtain the complementary optimal upper bound regarding the discriminants of quadratic polynomials.

The geodesic flow of a generic metric does not admit nontrivial integrals polynomial in momenta

Any smooth geodesic flow is locally Liouville integrable with smooth integrals. We show that generically this fails if we require, in addition, that the integrals are polynomial (or, more generally, analytic) in momenta. Consequently we obtain that a generic real-analytic metric does not admit, even locally, a real-analytic integral.

Polynomial conservation laws for the Lorentz gas and the Boltzmann–Gibbs gas

The problem of conditions ensuring the existence of first integrals that are polynomials in the momenta (velocities) is considered for certain multidimensional billiard systems which play an important role in non-equilibrium statistical mechanics. These are the Lorentz gas, a particle in a Euclidean space with (not necessarily convex) scattering domains, and the Boltzmann–Gibbs gas, a system of small identical balls in a rectangular box which collide elastically with one another and the walls of the box. The ergodic properties of such systems are only partially understood: some problems are still waiting for solution, and in certain cases (for instance, when the scatterers are non-convex) the system is known not to be ergodic. An approach to showing the absence of a non-trivial polynomial first integral with continuously differentiable coefficients is developed. The known first integrals for integrable problems in dynamics are mostly polynomials in the momenta (or functions of polynomials). The investigation of multidimensional billiards with non-compact configuration space, when there is no hope for ergodic behaviour, is of particular interest. Applications of the general results on the absence of non-trivial polynomial integrals to problems in statistical mechanics are discussed. Bibliography: 62 titles.

On local description of two-dimensional geodesic flows with a polynomial first integral**dedicated to the 55th birthday of our friend E V Ferapontov.

In this paper we present a construction of multiparametric families of two-dimensional metrics with a polynomial first integral of arbitrary degree in momenta. Such integrable geodesic flows are described by solutions of some semi-Hamiltonian hydrodynamic-type system. We give a constructive algorithm for the solution of the derived hydrodynamic-type system, i.e. we found infinitely many conservation laws and commuting flows. Thus we were able to find infinitely many particular solutions of this hydrodynamic-type system by the generalized hodograph method. Therefore infinitely many particular two-dimensional metrics equipped with first integrals polynomial in momenta were constructed.

An improved error bound on Gauss quadrature

In this paper, the refined estimates on aliasing errors on integration of Chebyshev polynomials by Gauss quadrature are present, which, together with the asymptotic formulae on the coefficients of Chebyshev expansions, deduces an improved convergence rate for Gauss quadrature. The convergence orders are attainable for some functions of finite regularities.

A problem of Zagier on quadratic polynomials and continued fractions

For non-square [Formula: see text] (mod 4), Don Zagier defined a function [Formula: see text] by summing over certain integral quadratic polynomials. He proved that [Formula: see text] is a constant function depending on [Formula: see text]. For rational [Formula: see text], it turns out that this sum has finitely many terms. Here we address the infinitude of the number of quadratic polynomials for non-rational [Formula: see text], and more importantly address some problems posed by Zagier related to characterizing the polynomials which arise in terms of the continued fraction expansion of [Formula: see text]. In addition, we study the indivisibility of the constant functions [Formula: see text] as [Formula: see text] varies.

Skorohod Integral at Vacuum State on Guichardet-Fock Spaces

In this paper, we define expectation of f∈F, i.e. E(f)=f(?), according to Wiener-Ito-Segal isomorphic relation between Guichardet-Fock space F and Wienerspace W. Meanwhile, we derive a formula for the expectation of random Hermite polynomial in Skorohod integral on Guichardet- Fock spaces. In particular, we prove that the anticipative Girsanov identities under the condition E(Hx(δ(x),‖x‖2)),n≥1 on Guichardet-Fock spaces.

Inequalities involving the integrals of polynomials and their polar derivatives

Inequalities involving the integrals of polynomials and their polar derivatives

Théorème de Chebotarev et complexité de Littlewood

Resume. The effective version of Chebotarev’s density theorem under the Generalized Riemann Hypothesis and the Artin conjecture (cf. Iwaniec and Kowalski’s Analytic Number Theory, §5.13) involves a numerical invariant of a subset D of a finite group G that we call the Littlewood Complexity of D. We study this invariant in detail. Using this study, and an application of the large sieve, we give improved versions of two standard problems related to Chebotarev : the bound on the first prime in a Frobenian set, and the asymptotics of the set of primes with given Frobenius in an infinite family of Galois extensions. We then give concrete applications to the problem of the factorization of an integral polynomial modulo primes, to the Lang-Trotter conjecture for abelian surfaces, and to the conjecture of Koblitz, with in all three cases better bounds that previously known.

Integrable quintic polynomial potential and its generalizations

We make a detailed analysis for an integrable quintic polynomial potential. By Benenti’s approach the second integral is derived, which ensures integrability. The type of separable coordinate and coordinate transformation are deduced according to a recently established systematic procedure. Bi-Hamiltonian and Lax representations are also obtained. At last we realize that it is the quintic analogue of classical Hénon–Heiles cubic system (case 1:3) and a quartic system (case 1:6:1). Another generalization is also performed.

Asymptotics behavior of integrals of the legendre polynomials and their sums

Asymptotic representations of the integrals Pnk(cosθ) of the Legendre polynomials and their sums with respect to the first subscript from n + 1 to infinity are investigated. Here, $${P_{n0}}\left( x \right) = {P_n}\left( x \right) = \int\limits_{ - 1}^x {{P_{n,k - 1}}\left( y \right)dy.}$$ It is shown that the asymptotic behavior of Pnk as n → ∞ is similar to that of Pn. However, the summand of order n–k–m–1/2 can be represented as a linear combination of m, rather than one, cosines of the form $$\cos \left[ {\left( {n + {s_1} + \frac{1}{2}} \right)\theta + \left( {{s_2} + \frac{1}{2}} \right)\frac{\pi }{2}} \right],$$ where s1 and s2 are integers depending only on k and m. For the sum Pnk with respect to the first subscript from 0 to ∞, closed expressions are obtained. The asymptotic behavior of sums from n + 1 to ∞ is described. Its form differs from that of Pnk only by the additional multiplier cotθ/2. As a byproduct, a Mehler–Dirichlet-type integral for Pnk(cosθ) is obtained.

An integrable Hénon–Heiles system on the sphere and the hyperbolic plane

We construct a constant curvature analogue on the two-dimensional sphere and the hyperbolic space of the integrable Hénon–Heiles Hamiltonian given byH=12( p12+p22)+Ω(q12+4q22)+α(q12q2+2q23), ?>where and α are real constants. The curved integrable Hamiltonian so obtained depends on a parameter which is just the curvature of the underlying space, and is such that the Euclidean Hénon–Heiles system is smoothly obtained in the zero-curvature limit . On the other hand, the Hamiltonian that we propose can be regarded as an integrable perturbation of a known curved integrable anisotropic oscillator. We stress that in order to obtain the curved Hénon–Heiles Hamiltonian , the preservation of the full integrability structure of the flat Hamiltonian under the deformation generated by the curvature will be imposed. In particular, the existence of a curved analogue of the full Ramani–Dorizzi–Grammaticos (RDG) series of integrable polynomial potentials, in which the flat Hénon–Heiles potential can be embedded, will be essential in our construction. Such an infinite family of curved RDG potentials on and will also be explicitly presented.

Heat flow in anharmonic crystals with internal and external stochastic baths: a convergent polymer expansion for a model with discrete time and long range interparticle interaction

We investigate a chain of oscillators with anharmonic on-site potentials, with long range interparticle interactions, and coupled both to external and internal stochastic thermal reservoirs of Ornstein–Uhlenbeck type. We develop an integral representation, a` la Feynman–Kac, for the correlations and the heat current. We assume the approximation of discrete times in the integral formalism (together with a simplification in a subdominant part of the harmonic interaction) and develop a suitable polymer expansion for the model. In the regime of strong anharmonicity, strong harmonic pinning, and for the interparticle interaction with integrable polynomial decay, we prove the convergence of the polymer expansion uniformly in volume (number of sites and time). We also show that the two-point correlation decays in space such as the interparticle interaction. The existence of a convergent polymer expansion is of practical interest: it establishes a rigorous support for a perturbative analysis of the heat flow problem and for the computation of the thermal conductivity in related anharmonic crystals, including those with inhomogeneous potentials and long range interparticle interactions. To show the usefulness and trustworthiness of our approach, we compute the thermal conductivity of a specific anharmonic chain, and make a comparison with related numerical results presented in the literature.

On some time marching schemes for the stabilized finite element approximation of the mixed wave equation

Abstract In this paper we analyze time marching schemes for the wave equation in mixed form. The problem is discretized in space using stabilized finite elements. On the one hand, stability and convergence analyses of the fully discrete numerical schemes are presented using different time integration schemes and appropriate functional settings. On the other hand, we use Fourier techniques (also known as von Neumann analysis) in order to analyze stability, dispersion and dissipation. Numerical convergence tests are presented for various time integration schemes, polynomial interpolations (for the spatial discretization), stabilization methods, and variational forms. To analyze the behavior of the different schemes considered, a 1D wave propagation problem is solved.

A characterization of almost universal ternary inhomogeneous quadratic polynomials with conductor 2

An integral quadratic polynomial (with positive definite quadratic part) is called almost universal if it represents all but finitely many positive integers. In this paper, we provide a characterization of almost universal ternary quadratic polynomials with conductor 2.

On polynomial integrability of the Euler equations on so(4)

In this paper we prove that the Euler equations on the Lie algebra s o ( 4 ) with a diagonal quadratic Hamiltonian either satisfy the Manakov condition, or have at most four functionally independent polynomial first integrals.

Schur Coefficients of the Integral Form Macdonald Polynomials

In this paper, we consider the combinatorial formula for the Schur coefficients of the integral form of the Macdonald polynomials. As an attempt to prove Haglund's conjecture that $\Biggl<\frac{J_{\mu}(X;q,q^k)}{(1-q)^{|\mu|}},s_{\lambda}(X)\Biggr> \in \mathbb{N}[q]$, we have found explicit combinatorial formulas for the Schur coefficients in one row case, two column case and certain hook shape cases [Yoo12]. A result of Egge-Loehr-Warrington [ELW] gives a combinatorial way of getting Schur expansion of symmetric functions when the expansion of the function in terms of Gessel's fundamental quasi symmetric functions is known. We apply this result to the combinatorial formula for the integral form Macdonald polynomials of Haglund [Hag] in quasi symmetric functions to prove the Haglund's conjecture in more general cases.

The representation of integers by positive ternary quadratic polynomials

An integral quadratic polynomial is called regular if it represents every integer that is represented by the polynomial itself over the reals and over the p-adic integers for every prime p. It is called complete if it is of the form Q(x+v), where Q is an integral quadratic form in the variables x=(x1,…,xn) and v is a vector in Qn. Its conductor is defined to be the smallest positive integer c such that cv∈Zn. We prove that for a fixed positive integer c, there are only finitely many equivalence classes of positive primitive ternary regular complete quadratic polynomials with conductor c. This generalizes the analogous finiteness results for positive definite regular ternary quadratic forms by Watson [18,19] and for ternary triangular forms by Chan and Oh [8].

Duality in spaces of polynomials of degree at most n

We study spaces of continuous polynomials of degree at most n between Banach spaces. Using symmetric tensor products we show that any polynomial of degree at most n has a natural linearisation and that the space of all scalar-valued polynomials of degree at most n has an isometric predual. We introduce the spaces of integral and nuclear polynomials of degree at most n and endow them with norms that allows us to develop a duality theory for spaces of polynomials of degree at most n.