Q-operator and factorised separation chain for Jack polynomials

Applying Baxter's method of the Q-operator to the set of Sekiguchi's commuting partial differential operators we show that Jack polynomials P_λ^(1/g) (χ₁, …, χ_n) …, χ_n) are eigenfunctions of a one-parameter family of integral operators Q_z. The operators Q_z are expressed in terms of the Dirichlet-Liouville n-dimensional beta integral. From a composition of n operators Q_zk we construct an integral operator S_n factorising Jack polynomials into products of hypergeometric polynomials of one variable. The operator S_n admits a factorisation described in terms of restricted Jack polynomials P_λ^(1/g) (x₁, …, x_k, 1, … 1). Using the operator Q_z for z = 0 we give a simple derivation of a previously known integral representation for Jack polynomials.     

Finite free resolutions of length two and koszul generalized complex

Abstract

In 1956, Ehrenfeucht proved that a polynomial f ₁(x ₁) + · + f _n (x _n ) with complex coefficients in the variables x ₁, …, x _n is irreducible over the field of complex numbers provided the degrees of the polynomials f ₁(x ₁), …, f _n (x _n ) have greatest common divisor one. In 1964, Tverberg extended this result by showing that when n ≥ 3, then f ₁(x ₁) + · + f _n (x _n ) belonging to K[x ₁, …, x _n ] is irreducible over any field K of characteristic zero provided the degree of each f _i is positive. Clearly a polynomial F = f ₁(x ₁) + · + f _n (x _n ) is reducible over a field K of characteristic p ≠ 0 if F can be written as F = (g ₁(x ₁)) ^p  + (g ₂(x ₂)) ^p  + · + (g _n (x _n )) ^p  + c[g ₁(x ₁) + g ₂(x ₂) + · + g _n (x _n )] where c is in K and each g _i (x _i ) is in K[x _i ]. In 1966, Tverberg proved that the converse of the above simple fact holds in the particular case when n = 3 and K is an algebraically closed field of characteristic p > 0. In this article, we prove an extension of Tverberg's result by showing that this converse holds for any n ≥ 3.     

Estimates for the first and second derivatives of the Stieltjes polynomials

Let w_λ(x)(1−x²)^λ−1/2 and P_n^(λ) be the ultraspherical polynomials with respect to w_λ(x). Then we denote E_n+1^(λ) the Stieltjes polynomials with respect to w_λ(x) satisfyingIn this paper, we give estimates for the first and second derivatives of the Stieltjes polynomials E_n+1^(λ) and the product E_n+1^(λ)P_n^(λ) by obtaining the asymptotic differential relations. Moreover, using these differential relations we estimate the second derivatives of E_n+1^(λ)(x) and E_n+1^(λ)(x)P_n^(λ)(x) at the zeros of E_n+1^(λ)(x) and the product E_n+1^(λ)(x)P_n^(λ)(x), respectively.     

Divergence of Infinite-Variance Nonradial Solutions to the 3D NLS Equation

We consider solutions u(t) to the 3d NLS equation i? _t u + Δu + |u|² u = 0 such that ‖xu(t)‖ _{L ²}  = ∞ and u(t) is nonradial. Denoting by M[u] and E[u], the mass and energy, respectively, of a solution u, and by Q(x) the ground state solution to ?Q + ΔQ + |Q|² Q = 0, we prove the following: if M[u]E[u] < M[Q]E[Q] and ‖u ₀‖ _{L ²} ‖?u ₀‖ _{L ²}  > ‖Q‖ _{L ²} ‖?Q‖ _{L ²} , then either u(t) blows-up in finite positive time or u(t) exists globally for all positive time and there exists a sequence of times t _n  → + ∞ such that ‖?u(t _n )‖ _{L ²}  → ∞. Similar statements hold for negative time.     

Decompositions of Quotient Rings and m-Power Commuting Maps

Let R be a semiprime ring with symmetric Martindale quotient ring Q, n ≥ 2 and let f(X) = X ⁿ h(X), where h(X) is a polynomial over the ring of integers with h(0) = ±1. Then there is a ring decomposition Q = Q ₁ ⊕ Q ₂ ⊕ Q ₃ such that Q ₁ is a ring satisfying S _2n?2, the standard identity of degree 2n ? 2, Q ₂ ? M _n (E) for some commutative regular self-injective ring E such that, for some fixed q > 1, x ^q  = x for all x ∈ E, and Q ₃ is a both faithful S _2n?2-free and faithful f-free ring. Applying the theorem, we characterize m-power commuting maps, which are defined by linear generalized differential polynomials, on a semiprime ring.     

Asymptotic analysis of the Hermite polynomials from their differential–difference equation

We analyze the Hermite polynomials H _n (x) and their zeros asymptotically, as n → ∞ We obtain asymptotic approximations from the differential–difference equation which they satisfy, using the ray method. We give numerical examples showing the accuracy of our formulas.     

On Kernel polynomials and self-perturbation of orthogonal polynomials

Given an orthogonal polynomial system {Q _n (x)} _n=0 ^∞, define another polynomial system by where α _n are complex numbers and t is a positive integer. We find conditions for {P _n (x)} _n=0 ^∞ to be an orthogonal polynomial system. When t=1 and α₁≠0, it turns out that {Q _n (x)} _n=0 ^∞ must be kernel polynomials for {P _n (x)} _n=0 ^∞ for which we study, in detail, the location of zeros and semi-classical character. Received: November 25, 1999; in final form: April 6, 2000?Published online: June 22, 2001     

A characterization of some q-orthogonal polynomials

We show that the only orthogonal polynomials satisfying a q-difference equation of the form π(x)D _q P _n (x) = (α _n x + β _n )P _n (x) + γ _n P _n−1(x) where π(x) is a polynomial of degree 2, are the Al-Salam Carlitz 1, little and big q-Laguerre, the little and big q-Jacobi, and the q-Bessel polynomials. This is a q-analog of the work carried out in [1]. 2000 Mathematics Subject Classification Primary—33C45, 33D45     

Intersective polynomials and the primes

Intersective polynomials are polynomials in Z[x] having roots every modulus. For example, P₁(n)=n² and P₂(n)=n²−1 are intersective polynomials, but P₃(n)=n²+1 is not. The purpose of this note is to deduce, using results of Green and Tao (2006) [8] and Lucier (2006) [16], that for any intersective polynomial h, inside any subset of positive relative density of the primes, we can find distinct primes p₁,p₂ such that p₁−p₂=h(n) for some integer n. Such a conclusion also holds in the Chen primes (where by a Chen prime we mean a prime number p such that p+2 is the product of at most 2 primes).     

Continuous symmetrized Sobolev inner products of order N (I)

Given a symmetrized Sobolev inner product of order N, the corresponding sequence of monic orthogonal polynomials {Qn} satisfies that Q2n(x)=Pn(x²), Q2n+1(x)=xRn(x²) for certain sequences of monic polynomials {Pn} and {Rn}. In this paper, we deduce the integral representation of the inner products such that {Pn} and {Rn} are the corresponding sequences of orthogonal polynomials. Moreover, we state a relation between both inner products which extends the classical result for symmetric linear functionals.     

Inverse Problems for Time-Dependent Singular Heat Conductivities: Multi-Dimensional Case

We consider an inverse boundary value problem for the heat equation ? _t u = div (γ? _x u) in (0, T) × Ω, u = f on (0, T) × ?Ω, u| _t=0 = u ₀, in a bounded domain Ω ? ? ⁿ , n ≥ 2, where the heat conductivity γ(t, x) is piecewise constant and the surface of discontinuity depends on time: γ(t, x) = k ² (x ∈ D(t)), γ(t, x) = 1 (x ∈ Ω?D(t)). Fix a direction e* ∈ 𝕊 ^n?1 arbitrarily. Assuming that ?D(t) is strictly convex for 0 ≤ t ≤ T, we show that k and sup {e*·x; x ∈ D(t)} (0 ≤ t ≤ T), in particular D(t) itself, are determined from the Dirichlet-to-Neumann map : f → ?_ν u(t, x)|_{(0, T)×?Ω}. The knowledge of the initial data u ₀ is not used in the proof. If we know min_0≤t≤T (sup _x∈D(t) x·e*), we have the same conclusion from the local Dirichlet-to-Neumann map. Numerical examples of stationary and moving circles inside the unit disk are shown. The results have applications to nondestructive testing. Consider a physical body consisting of homogeneous material with constant heat conductivity except for a moving inclusion with different conductivity. Then the location and shape of the inclusion can be monitored from temperature and heat flux measurements performed at the boundary of the body. Such a situation appears for example in blast furnaces used in ironmaking.     

On the Invariant Fields and Rings of Some Groups of Polynomial Automorphisms

Abstract

We consider the group G of C-automorphisms of C(x, y) (resp. C[x, y]) generated by s, t such that t(x) = y, t(y) = x and s(x) = x, s(y) = ? y + u(x) where u ∈ C[x] is of degree k ≥ 2. Using Galois's theory, we show that the invariant field and the invariant algebra of G are equal to C.     

Orthogonal Polynomial Solutions of Spectral Type Differential Equations: Magnus' Conjecture

Let τ=σ+ν be a point mass perturbation of a classical moment functional σ by a distribution ν with finite support. We find necessary conditions for the polynomials {Q_n(x)}^∞_n=0, orthogonal relative to τ, to be a Bochner–Krall orthogonal polynomial system (BKOPS); that is, {Q_n(x)}^∞_n=0 are eigenfunctions of a finite order linear differential operator of spectral type with polynomial coefficients: L_N[y](x)=∑^N_i=1 ℓ_i(x) y⁽ⁱ⁾(x)=λ_ny(x). In particular, when ν is of order 0 as a distribution, we find necessary and sufficient conditions for {Q_n(x)}^∞_n=0 to be a BKOPS, which strongly support and clarify Magnus' conjecture which states that any BKOPS must be orthogonal relative to a classical moment functional plus one or two point masses at the end point(s) of the interval of orthogonality. This result explains not only why the Bessel-type orthogonal polynomials (found by Hendriksen) cannot be a BKOPS but also explains the phenomena for infinite-order differential equations (found by J. Koekoek and R. Koekoek), which have the generalized Jacobi polynomials and the generalized Laguerre polynomials as eigenfunctions.     

Nonlinear functional equations satisfied by orthogonal polynomials

Let c be a linear functional defined by its moments c(xⁱ)=c_i for i=0,1,…. We proved that the nonlinear functional equations P(t)=c(P(x)P(αx+t)) and P(t)=c(P(x)P(xt)) admit polynomial solutions which are the polynomials belonging to the family of formal orthogonal polynomials with respect to a linear functional related to c. This equation relates the polynomials of the family with those of the scaled and shifted family. Other types of nonlinear functional equations whose solutions are formal orthogonal polynomials are also presented. Applications to Legendre and Chebyshev polynomials are given. Then, orthogonality with respect to a definite inner product is studied. When c is an integral functional with respect to a weight function, the preceding functional equations are nonlinear integral equations, and these results lead to new characterizations of orthogonal polynomials on the real line, on the unit circle, and, more generally, on an algebraic curve.     

Some Pólya-Type Irreducibility Criteria for Multivariate Polynomials

We provide irreducibility criteria for multivariate polynomials with coefficients in an arbitrary field that extend a classical result of Pólya for polynomials with integer coefficients. In particular, we provide irreducibility conditions for polynomials of the form f(X)(Y ? f ₁(X))…(Y ? f _n (X)) + g(X), with f, f ₁, ?, f _n , g univariate polynomials over an arbitrary field.     

Polynomial interpolation and sums of powers of integers

In this note, we revisit the problem of polynomial interpolation and explicitly construct two polynomials in n of degree k + 1, P_k(n) and Q_k(n), such that P_k(n) = Q_k(n) = f_k(n) for n = 1, 2,…?, k, where f_k(1), f_k(2),…?, f_k(k) are k arbitrarily chosen (real or complex) values. Then, we focus on the case that f_k(n) is given by the sum of powers of the first n positive integers S_k(n) = 1^k + 2^k + ??? + n^k, and show that S_k(n) admits the polynomial representations S_k(n) = P_k(n) and S_k(n) = Q_k(n) for all n = 1, 2,…?, and k ≥ 1, where the first representation involves the Eulerian numbers, and the second one the Stirling numbers of the second kind. Finally, we consider yet another polynomial formula for S_k(n) alternative to the well-known formula of Bernoulli.