scholarly journals Addition Formula and Related Integral Equations for Heine—Stieltjes Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1231
Author(s):  
Hans Volkmer
Keyword(s):  
Integral Equations ◽  
Explicit Form ◽  
Spherical Harmonics ◽  
Reproducing Kernel ◽  
Addition Formula ◽  
Symmetric Products ◽  
Special Cases ◽  
Related Integral ◽  
Generalized Spherical Harmonics ◽  
The Relationship

It is shown that symmetric products of Heine–Stieltjes quasi-polynomials satisfy an addition formula. The formula follows from the relationship between Heine–Stieltjes quasi-polynomials and spaces of generalized spherical harmonics, and from the known explicit form of the reproducing kernel of these spaces. In special cases, the addition formula is written out explicitly and verified. As an application, integral equations for Heine–Stieltjes quasi-polynomials are found.

scholarly journals Surrogate Endpoint Evaluation: Principal Stratification Criteria and the Prentice Definition

2015 ◽  
Vol 3 (2) ◽  
pp. 157-175 ◽  
Author(s):  
Peter B. Gilbert ◽  
Erin E. Gabriel ◽  
Ying Huang ◽  
Ivan S.F. Chan
Keyword(s):  
Causal Effect ◽  
Surrogate Endpoint ◽  
Effect Modification ◽  
Principal Stratification ◽  
Clinical Endpoint ◽  
Efficacy Trial ◽  
Marker Selection ◽  
Special Cases ◽  
High Utility ◽  
The Relationship

AbstractA common problem of interest within a randomized clinical trial is the evaluation of an inexpensive response endpoint as a valid surrogate endpoint for a clinical endpoint, where a chief purpose of a valid surrogate is to provide a way to make correct inferences on clinical treatment effects in future studies without needing to collect the clinical endpoint data. Within the principal stratification framework for addressing this problem based on data from a single randomized clinical efficacy trial, a variety of definitions and criteria for a good surrogate endpoint have been proposed, all based on or closely related to the “principal effects” or “causal effect predictiveness (CEP)” surface. We discuss CEP-based criteria for a useful surrogate endpoint, including (1) the meaning and relative importance of proposed criteria including average causal necessity (ACN), average causal sufficiency (ACS), and large clinical effect modification; (2) the relationship between these criteria and the Prentice definition of a valid surrogate endpoint; and (3) the relationship between these criteria and the consistency criterion (i.e. assurance against the “surrogate paradox”). This includes the result that ACN plus a strong version of ACS generally do not imply the Prentice definition nor the consistency criterion, but they do have these implications in special cases. Moreover, the converse does not hold except in a special case with a binary candidate surrogate. The results highlight that assumptions about the treatment effect on the clinical endpoint before the candidate surrogate is measured are influential for the ability to draw conclusions about the Prentice definition or consistency. In addition, we emphasize that in some scenarios that occur commonly in practice, the principal strata subpopulations for inference are identifiable from the observable data, in which cases the principal stratification framework has relatively high utility for the purpose of effect modification analysis and is closely connected to the treatment marker selection problem. The results are illustrated with application to a vaccine efficacy trial, where ACN and ACS for an antibody marker are found to be consistent with the data and hence support the Prentice definition and consistency.

Geographical interpretation of measurements of average phase velocities of surface waves over great circular and great semi-circular paths

1964 ◽  
Vol 54 (2) ◽  
pp. 571-610
Author(s):  
George E. Backus
Keyword(s):  
Spherical Harmonics ◽  
Rayleigh Waves ◽  
Seismic Station ◽  
Great Circle ◽  
Explicit Formulas ◽  
Phase Velocities ◽  
Rapid Accumulation ◽  
Generalized Spherical Harmonics ◽  
Geographical Position ◽  
The Difference

ABSTRACT If the averages of the reciprocal phase velocity c−1 of a given Rayleigh or Love mode over various great circular or great semicircular paths are known, information can be extracted about how c−1 varies with geographical position. Assuming that geometrical optics is applicable, it is shown that if c−1 is isotropic its great circular averages determine only the sum of the values of c−1 at antipodal points and not their difference. The great semicircular averages determine the difference as well. If c−1 is anisotropic through any cause other than the earth's rotation, even great semicircular averages do not determine c−1 completely. Rotation has negligible effect on Love waves, and if it is the only anisotropy present its effect on Rayleigh waves can be measured and removed by comparing the averages of c−1 for the two directions of travel around any great circle not intersecting the poles of rotation. Only great circular and great semicircular paths are considered because every earthquake produces two averages of c−1 over such paths for each seismic station. No other paths permit such rapid accumulation of data when the azimuthal variations of the earthquakes' radiation patterns are unknown. Expansion of the data in generalized spherical harmonics circumvents the fact that the explicit formulas for c−1 in terms of its great circular or great semicircular integrals require differentiation of the data. Formulas are given for calculating the generalized spherical harmonics numerically.

scholarly journals The truncated Hyper-Poisson queues: Hk/Ma,b/C/N with balking, reneging and general bulk service rule

2008 ◽  
Vol 18 (1) ◽  
pp. 23-36 ◽  
Author(s):  
A.I. Shawky ◽  
M.S. El-Paoumy
Keyword(s):  
Steady State ◽  
Analytical Solution ◽  
Explicit Form ◽  
Recurrence Relations ◽  
Expected Number ◽  
Special Cases ◽  
Queue Discipline ◽  
Bulk Service ◽  
General Bulk Service Rule ◽  
Steady State Probabilities

The aim of this paper is to derive the analytical solution of the queue: Hk/Ma,b/C/N with balking and reneging in which (I) units arrive according to a hyper-Poisson distribution with k independent branches, (II) the queue discipline is FIFO; and (III) the units are served in batches according to a general bulk service rule. The steady-state probabilities, recurrence relations connecting various probabilities introduced are found and the expected number of units in the queue is derived in an explicit form. Also, some special cases are obtained. .

scholarly journals Asymptotic Behavior of Solutions of Integral Equations with Homogeneous Kernels

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 180
Author(s):  
Oleg Avsyankin
Keyword(s):  
Integral Equation ◽  
Asymptotic Behavior ◽  
Integral Equations ◽  
Spherical Harmonics ◽  
Special Class ◽  
Continuous Functions ◽  
Asymptotic Behavior Of Solutions ◽  
Free Term ◽  
Research Technique ◽  
Multidimensional Integral

The multidimensional integral equation of second kind with a homogeneous of degree (−n) kernel is considered. The special class of continuous functions with a given asymptotic behavior in the neighborhood of zero is defined. It is proved that, if the free term of the integral equation belongs to this class and the equation itself is solvable, then its solution also belongs to this class. To solve this problem, a special research technique is used. The above-mentioned technique is based on the decomposition of both the solution and the free term in spherical harmonics.

scholarly journals On dual integral equations with Hankel kernel and an arbitrary weight function

1986 ◽  
Vol 9 (2) ◽  
pp. 293-300 ◽  
Author(s):  
C. Nasim
Keyword(s):  
Weight Function ◽  
Integral Equations ◽  
Single Equation ◽  
Dual Integral Equations ◽  
Fredholm Type ◽  
Operational Procedure ◽  
Mellin Transforms ◽  
Special Cases ◽  
General Order ◽  
Arbitrary Weight

In this paper we deal with dual integral equations with an arbitrary weight function and Hankel kernels of distinct and general order. We propose an operational procedure, which depends on exploiting the properties of the Mellin transforms, and readily reduces the dual equations to a single equation. This then can be inverted by the Hankel inversion to give us an equation of Fredholm type, involving the unknown function. Most of the known results are then derived as special cases of our general result.

scholarly journals Non-linear integral equations for Heun functions

1969 ◽  
Vol 16 (4) ◽  
pp. 281-289 ◽  
Author(s):  
B. D. Sleeman
Keyword(s):  
Integral Equations ◽  
Heun Equation ◽  
Mathieu Functions ◽  
Special Cases ◽  
Mathieu’S Equation ◽  
Heun Functions ◽  
Non Linear ◽  
Linear Integral ◽  
Image Position ◽  
Linear Integral Equations

Some years ago Lambe and Ward (1) and Erdélyi (2) obtained integral equations for Heun polynomials and Heun functions. The integral equations discussed by these authors were of the formFurther, as is well known, the Heun equation includes, among its special cases, Lamé's equation and Mathieu's equation and so (1.1) may be considered a generalisation of the integral equations satisfied by Lamé polynomials and Mathieu functions. However, integral equations of the type (1.1) are not the only ones satisfied by Lamé polynomials; Arscott (3) discussed a class of non- linear integral equations associated with these functions. This paper then is concerned with discussing the existence of non-linear integral equations satisfied by solutions of Heun's equation.

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