Analog of Favard's Theorem for Polynomials Connected with Difference Equation of 4-th Order

Orthonormal polynomials on the real line {pn (λ)} n=0 ... ∞ satisfy the recurrent relation of the form: λn−1 pn−1 (λ) + αn pn (λ) + λn pn+1 (λ) = λpn (λ), n = 0, 1, 2, . . . , where λn > 0, αn ∈ R, n = 0, 1, . . . ; λ−1 = p−1 = 0, λ ∈ C. In this paper we study systems of polynomials {pn (λ)} n=0 ... ∞ which satisfy the equation: αn−2 pn−2 (λ) + βn−1 pn−1 (λ) + γn pn (λ) + βn pn+1 (λ) + αn pn+2 (λ) = λ2 pn (λ), n = 0, 1, 2, . . . , where αn > 0, βn ∈ C, γn ∈ R, n = 0, 1, 2, . . ., α−1 = α−2 = β−1 = 0, p−1 = p−2 = 0, p0 (λ) = 1, p1 (λ) = cλ + b, c > 0, b ∈ C, λ ∈ C. It is shown that they are orthonormal on the real and the imaginary axes in the complex plane ...

Polynomials of Pellian Type and Continued Fractions

We investigate infinite families of integral quadratic polynomials {fk (X)} k∈N and show that, for a fixed k ∈ N and arbitrary X ∈ N, the period length of the simple continued fraction expansion of √fk (X) is constant. Furthermore, we show that the period lengths of √fk (X) go to infinity with k. For each member of the families involved, we show how to determine, in an easy fashion, the fundamental unit of the underlying quadratic field. We also demonstrate how the simple continued fraction ex- pansion of √fk (X) is related to that of √C, where √fk (X) = ak*X^2 +bk*X + C. This continues work in [1]–[4].

Discriminant Sets of Families of Hyperbolic Polynomials of Degree 4 and 5

∗ Research partially supported by INTAS grant 97-1644

Generalization of a Conjecture in the Geometry of Polynomials

In this paper we survey work on and around the following conjecture, which was first stated about 45 years ago: If all the zeros of an algebraic polynomial p (of degree n ≥ 2) lie in a disk with radius r, then, for each zero z1 of p, the disk with center z1 and radius r contains at least one zero of the derivative p′ . Until now, this conjecture has been proved for n ≤ 8 only. We also put the conjecture in a more general framework involving higher order derivatives and sets defined by the zeros of the polynomials.

On Representations of Algebraic Polynomials by Superpositions of Plane Waves

* The author was supported by NSF Grant No. DMS 9706883.

On a Five-Diagonal Jacobi Matrices and Orthogonal Polynomials on Rays in the Complex Plane

∗ Partially supported by Grant MM-428/94 of MESC.

MRI pressure and stress measurement in novel homogeneous soft solids

This paper presents MRI measurements of a novel semi solid MR contrast agent to pressure. The agent is comprised of potassium chloride cross linked carageenan gum at a concentration of 2% w/v, with micron size lipid coated bubbles of air at a concentration of 3% v/v. The choice for an optimum suspending medium, the methods of production and the preliminary MRI results are presented herein. The carageenan gum is shown to be ideally elastic for compressions relating to volume changes less than 15%, in contrast to the inelastic gellan gum also tested. Although slightly lower than that of gellan gum, carageenan has a water diffusion coefficient of 1.72×10-9 m2.s-1 indicating its suitability to this purpose. RARE imaging is performed whilst simultaneously compressing test and control samples and a maximum sensitivity of 1.6% MR signal change per % volume change is found which is shown to be independent of proton density variations due to the presence of microbubbles and compression. This contrast agent could prove useful for numerous applications, and particularly in chemical engineering. More generally the method allows the user to non-invasively image with MRI any process that causes, within the solid, local changes either in bubble size or bubble shape. © 2008 American Institute of Physics.

A Note on a Classical Generating Function for the Jacobi Polynomials

Mathematics Subject Classification: 33C45.

Fractional Extensions of Jacobi Polynomials and Gauss Hypergeometric Function

2000 Mathematics Subject Classification: 26A33, 33C45

Rolle's Theorem for Complex Polynomials with Real Coefficients

Let p(z) be an algebraic polynomial of degree n ¸ 2 with real coefficients and p(i) = p(¡i). According to Grace-Heawood Theorem, at least one zero of the derivative p0(z) is on the disk with center in the origin and radius cot(¼=n). In this paper is found the smallest domain containing at leas one zero of the derivative p0(z).

Hermite Polynomials and the Zero-Distribution of Riemann's Zeta Function

AMS Subject Classification 2010: 11M26, 33C45, 42A38.

On a Stratification Defined by Real Roots of Polynomials

2000 Mathematics Subject Classification: 12D10

Overdetermined Strata in General Families of Polynomials

2000 Mathematics Subject Classification: 12D10.

On the Factorization of the Poincaré Polynomial: A Survey

2000 Mathematics Subject Classification: 13P05, 14M15, 14M17, 14L30.

Dickson Polynomials that are Permutations

2000 Mathematics Subject Classification: 11T06, 13P10.