John S. Lew

Nonnegative solutions of a nonlinear recurrence

Abstract Orthonormal polynomials with weight ¦τ¦ ϱ exp (−τ 4 ) have leading coefficients with recurrence properties which motivate the more general equations ξ m ( ξ m − 1 + ξ m + ξ m + 1 ) = γ m 2 , m = 1, 2,…, where ξ o is a fixed nonnegative value and γ 1 , γ 2 ,… are positive constants. For this broader problem, the existence of a nonnegative solution is proved and criteria are found for its uniqueness. Then, for the motivating problem, an asymptotic expansion of its unique nonnegative solution is obtained and a fast computational algorithm, with error estimates, is given.

Asymptotic expansion of a class of integral transforms with algebraically dominated kernels

Abstract A technique is developed here which yields the asymptotic expansion, in the two limits λ → 0+ and λ → +∞, for a class of functions defined by integrals I(λ) = ∝0∞ h(λt)f(t) dt where h(t) and f(t) are algebraically dominated near both 0+ and +∞. The integral I(λ) is first expressed as a contour integral of Barnes type through the Parseval theorem for Mellin transforms. The integrand is found to involve M[h; z] and M[f, 1 − z], the Mellin transforms of h and f evaluated at z and 1 − z respectively. Under rather general conditions, these Mellin transforms can be analytically continued to meromorphic functions in a suitable half plane, with poles that can be located and classified. The desired asymptotic expansions of I(λ) are then obtained by systematically moving the contour of its Barnes integral representation to the right for λ → +∞ and to the left for λ → 0+. Each term in either of these expansions is just the residue corresponding to a pole of M[h; z]M[f; 1 − z]. Under somewhat stronger conditions, the expansions obtained originally for real λ are extended to complex λ and shown valid in appropriate sectors. Several examples are considered to illustrate the range of these results.

An improved regional correlation algorithm for signature verification which permits small speed changes between handwriting segments

If two nearly coincident accelerometers on a pen axis measure orthogonal acceleration components perpendicular to this axis, then the regional correlation algorithm for signature verification divides these data into plausible segments, it compares each segment with a corresponding reference, and it combines the results into a global similarity index. The presently used intersegment distance permits certain natural data transformations: (1) translating a segment by small integer multiples of the sampling interval: (2) moving the pen with uniformly larger amplitude throughout a segment; and (3) rotating the pen about its axis between any two segments. We propose a new intersegment distance which permits further natural transformations: (4) translating a segment by a fraction of the sampling interval: and (5) writing at slightly different uniform speed within each segment. The new distance, like the old one, is the minimum of a certain function. We describe an algorithm which computes this minimum.

Polynomial Indexing of integer Lattice-Points I. General Concepts and Quadratic Polynomials*

Abstract Denoting the nonnegative integers by N and the signed integers by Z, we let S be a subset of Zm for m = 1, 2,… and f be a mapping from S into N. We call f a storing function on S if it is injective into N, and a packing function on S if it is bijective onto N. Motivation for these concepts includes extendible storage schemes for multidimensional arrays, pairing functions from recursive function theory, and, historically earliest, diagonal enumeration of Cartesian products. Indeed, Cantors 1878 denumerability proof for the product N2 exhibits the equivalent packing functions f Cantor (x, y) = { either x or y} + (x + y)(x + y + 1) 2 on the domain N2, and a 1923 Fueter-Polya result, in our terminology, shows fCantor the only quadratic packing function on N2. This paper extends the preceding result. For any real-valued function f on S we define a density S ÷ f = lim n→∞ ( 1 n )#{S ⋔ f −1 ([−n, +n])} , and for any packing function f on S we observe the fact S ÷ f = 1. Using properties of this density, and invoking Davenports lemma from geometric number theory, we find all polynomial storing functions with unit density on N, and exclude any polynomials with these properties on Z, then find all quadratic storing functions with unit density on N2, and exclude any quadratics with these properties on Z × N, Z2. The admissible quadratics on N2 are all nonnegative translates of fCantor. An immediate sequel to this paper excludes some higher-degree polynomials on subsets of Z2.

Asymptotic Expansion of Laplace Convolutions for Large Argument and Tail Densities for Certain Sums of Random Variables

An asymptotic series

Some counterexamples in multidimensional scaling

\sum _{m,n = 0}^\infty a_{mn} t^{a(m)} (\log t)^n

Polynomial enumeration of multidimensional lattices

with

Kinematic theory of signature verification measurements

\operatorname{Re} [a(m)]

An enlarged family of packing polynomials on multidimensional lattices

either increasing or decreasing is called a Mellin series respectively near either

Diagonal polynomials for small dimensions

0 +