This paper considers when one can invert general recursive operators which map a class of functions F to F . In this regard, we study four different notions of inversion. We additionally consider enumeration of operators which cover all general recursive operators which map F to F in the sense that for every general recursive operator Ψ mapping F to F , there is a general recursive operator in ...

We will construct a set C containing binary lsm functions and closed under a minimalist set of operations related to T2-constructibility [3]. Later, in section 3, we will relate this to ppsω-definability. For concision we will use vector notation such as x = (x1, · · · , xn), (x,y) = (x1, · · · , xn, y1, · · · , ym), 1 = (1, · · · , 1), (i, j,x) = (i, j, x1, · · · , xn), and x · y = x1y1 + · · ...

We prove the existence of a weak solution to a two-dimensional resonant 3 × 3 system of conservation laws with BV initial data. Due to possible resonance (coinciding eigenvalues), spatial BV estimates are in general not available. Instead, we use an entropy dissipation bound combined with the time translation invariance property of the system to prove existence based on a two-dimensional compen...

The main purpose of this paper is to obtain an explicit expression of a family of matrix valued orthogonal polynomials {Pn}n, with respect to a weight W , that are eigenfunctions of a second order differential operator D. The weight W and the differential operator D were found in [12], using some aspects of the theory of the spherical functions associated to the complex projective spaces. We al...

Commutative semifields in odd characteristic can be equivalently described by planar functions (also known as PN functions). We describe a method to construct a semifield which is canonically associated to a planar function and use it to derive information on the nuclei directly from the planar function. This is used to determine the nuclei of families of new commutative semifields of dimension...

We study special functions on euclidean spaces from the viewpoint of riemannian symmetric spaces. Here the euclidean space En = G/K where G is the semidirect product Rn · K of the translation group with a closed subgroup K of the orthogonal group O(n). We give exact parameterizations of the space of (G,K)–spherical functions by a certain affine algebraic variety, and of the positive definite on...

In this paper, we obtain asymptotic formulas for an infinite class of rank generating functions. As an application, we solve a conjecture of Andrews and Lewis on inequalities between certain ranks.

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We study configurations of immersed curves in surfaces and surfaces in 3-manifolds. Among other results, we show that primitive curves have only finitely many configurations which minimize the number of double points. We give examples of minimal configurations not realized by geodesics in any hyperbolic metric.