On the circle polynomials of Zernike and related orthogonal sets

1954 ◽  
Vol 50 (1) ◽  
pp. 40-48 ◽  
Author(s):  
A. B. Bhatia ◽  
E. Wolf
Keyword(s):  
Unit Circle ◽  
Generating Functions ◽  
Legendre Polynomials ◽  
Phase Contrast ◽  
Diffraction Theory ◽  
Zernike Polynomials ◽  
Optical Aberrations ◽  
Orthogonal Set ◽  
Complete Orthogonal Set ◽  
Explicit Expressions

ABSTRACTThe paper is concerned with the construction of polynomials in two variables, which form a complete orthogonal set for the interior of the unit circle and which are ‘invariant in form’ with respect to rotations of axes about the origin of coordinates. It is found that though there exist an infinity of such sets there is only one set which in addition has certain simple properties strictly analogous to that of Legendre polynomials. This set is found to be identical with the set of the circle polynomials of Zernike which play an important part in the theory of phase contrast and in the Nijboer-Zernike diffraction theory of optical aberrations.The results make it possible to derive explicit expressions for the Zernike polynomials in a simple, systematic manner. The method employed may also be used to derive other orthogonal sets. One new set is investigated, and the generating functions for this set and for the Zernike polynomials are also given.

The Enumeration of Maps on the Torus and the Projective Plane

1988 ◽  
Vol 31 (3) ◽  
pp. 257-271 ◽  
Author(s):  
E. A. Bender ◽  
E. R. Canfield ◽  
R. W. Robinson
Keyword(s):  
Projective Plane ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Embedded Graphs ◽  
Asymptotic Expressions ◽  
Explicit Expressions

AbstractThe enumeration of rooted maps (embedded graphs), by number of edges, on the torus and projective plane, is studied. Explicit expressions for the generating functions are obtained. From these are derived asymptotic expressions and recurrence relations. Numerical tables for the numbers with up to 20 edges are presented.

scholarly journals Markov chains with quasitoeplitz transition matrix: first zero hitting

1989 ◽  
Vol 2 (3) ◽  
pp. 205-216
Author(s):  
Alexander M. Dukhovny
Keyword(s):  
Markov Chains ◽  
Boundary Value Problems ◽  
Unit Circle ◽  
Generating Functions ◽  
Transition Matrix ◽  
Boundary Value ◽  
Riemann Boundary ◽  
Hitting Probabilities

This paper continues the investigation of Markov Chains with a quasitoeplitz transition matrix. Generating functions of first zero hitting probabilities and mean times are found by the solution of special Riemann boundary value problems on the unit circle. Duality is discussed.

A Note on Absorption Probabilities for a Random Walk Between a Reflecting and an Absorbing Barrier

1969 ◽  
Vol 6 (01) ◽  
pp. 224-226
Author(s):  
J.C. Hardin ◽  
A.L. Sweet
Keyword(s):  
Random Walk ◽  
Generating Functions ◽  
Absorption Probabilities ◽  
Explicit Expressions

Generating functions for the absorption probabilities for a random walk on the integers {0,1, …, b}, where 0 is an absorbing barrier and b a semi-reflecting barrier have been obtained by Weesakul [1] and Neuts [2]. However, determination of explicit expressions for the absorption probabilities from the generating functions is generally quite difficult. In this note, two cases where this is possible are presented.

The random walk associated by the game of roulette

1981 ◽  
Vol 18 (04) ◽  
pp. 931-936
Author(s):  
James M. Hill ◽  
Chandra M. Gulati
Keyword(s):  
Random Walk ◽  
Power Series ◽  
Generating Functions ◽  
The Right ◽  
Lagrange's Theorem ◽  
Explicit Expressions

The random walk arising in the game of roulette involves an absorbing barrier at the origin; at each step either a unit displacement to the left or a fixed multiple displacement to the right can occur with probabilities q and p respectively. Using generating functions and Lagrange's theorem for the expansion of a function as a power series explicit expressions are deduced for the probabilities of the player's capital at the nth step.

scholarly journals Wavefront optics and retinal image quality

2013 ◽  
Vol 3 (1) ◽  
pp. 1
Author(s):  
Cristina M. Oliveira ◽  
Sandra Franco
Keyword(s):  
Vision Research ◽  
Optical Quality ◽  
Wavefront Sensing ◽  
Zernike Polynomials ◽  
Secondary Effect ◽  
General Description ◽  
Wave Optics ◽  
Optical Aberrations ◽  
Optical Components ◽  
Optical Structure

With the increasing importance of optical aberrations in both vision research and clinical practice, it becomes necessary for vision scientists to have a thorough understanding of the concepts behind wavefront optics. Therefore, in this review, we provide some basic wave optics concepts useful to understand wavefront analysis, and describe the application of Zernike polynomials in the decomposition of aberrations. A general description of the human eye optical structure is given, followed by a more detailed analysis of the optical components of the eye and optical aberrations, and their secondary effect on overall optical quality. We further provide an overview of the current corneal and ocular wavefront sensing methods.

scholarly journals On Irreducible Maps and Slices

2014 ◽  
Vol 23 (6) ◽  
pp. 914-972 ◽  
Author(s):  
J. BOUTTIER ◽  
E. GUITTER
Keyword(s):  
Generating Functions ◽  
Shortest Paths ◽  
Integrable Equations ◽  
Natural Approach ◽  
Planar Maps ◽  
Equivalent Description ◽  
Explicit Expressions

We consider the problem of enumeratingd-irreducible maps,i.e., planar maps all of whose cycles have length at leastd, and such that any cycle of lengthdis the boundary of a face of degreed. We develop two approaches in parallel: the natural approach via substitution, where these maps are obtained from general maps by a replacement of alld-cycles by elementary faces, and a bijective approach via slice decomposition, which consists in cutting the maps along shortest paths. Both lead to explicit expressions for the generating functions ofd-irreducible maps with controlled face degrees, summarized in some elegant ‘pointing formula’. We provide an equivalent description ofd-irreducible slices in terms of so-calledd-oriented trees. We finally show that irreducible maps give rise to a hierarchy of discrete integrable equations which include equations encountered previously in the context of naturally embedded trees.

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