Support Points of the Unit Ball of H p (1 p ∞)

The problem of constructing Q-optimal experimental designs for polynomial regression on the interval [–1, 1] is considered. It is shown that well-known Malyutov – Fedorov designs using D-optimal designs (so-called Legendre spectrum) are other than Q-optimal designs. This statement is a direct consequence of Shabados remark which disproved the Erdős hypothesis that the spectrum (support points) of saturated D-optimal designs for polynomial regression on a segment appeared to be support points of saturated Q-optimal designs. We present a saturated exact Q-optimal design for polynomial regression with s = 3 which proves the Shabados notion and then extend this statement to approximate designs. It is shown that when s = 3, 4 the Malyutov – Fedorov theorem on approximate Q-optimal design is also incorrect, though it still stands for s = 1, 2. The Malyutov – Fedorov designs with Legendre spectrum are considered from the standpoint of their proximity to Q-optimal designs. Case studies revealed that they are close enough for small degrees s of polynomial regression. A universal expression for Q-optimal distribution of the weights pi for support points xi for an arbitrary spectrum is derived. The expression is used to tabulate the distribution of weights for Malyutov – Fedorov designs at s = 3, ..., 6. The general character of the obtained expression is noted for Q-optimal weights with A-optimal weight distribution (Pukelsheim distribution) for the same problem statement. In conclusion a brief recommendation on the numerical construction of Q-optimal designs is given. It is noted that in this case in addition to conventional numerical methods some software systems of symbolic computations using methods of resultants and elimination theory can be successfully applied. The examples of Q-optimal designs considered in the paper are constructed using precisely these methods.

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Holomorphically embedded discs with rapidly growing area

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500021399 ◽

1999 ◽

Vol 129 (2) ◽

pp. 343-349

Author(s):

Josip Globevnik

Keyword(s):

Unit Ball ◽

Open Unit ◽

Open Unit Ball

It is shown that if V is a closed submanifold of the open unit ball of ℂ2 biholomorphically equivalent to a disc, then the area of V ∩ r can grow arbitrarily rapidly as r ↗ 1. It is also shown that if V is a closed submanifold of ℂ2 biholomorphically equivalent to a disc, then the area of V ∩ r can grow arbitrarily rapidly as r ↗ ∞.

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Polynomial extreme elements of the closed unit ball ofH∞(B)

Complex Variables Theory and Application An International Journal ◽

10.1080/17476939408814730 ◽

1994 ◽

Vol 25 (1) ◽

pp. 49-56 ◽

Cited By ~ 1

Author(s):

John N. McDonald

Keyword(s):

Unit Ball ◽

Closed Unit Ball

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Homogeneously Polyanalytic Kernels on the Unit Ball and the Siegel Domain

Complex Analysis and Operator Theory ◽

10.1007/s11785-021-01145-z ◽

2021 ◽

Vol 15 (6) ◽

Author(s):

Christian Rene Leal-Pacheco ◽

Egor A. Maximenko ◽

Gerardo Ramos-Vazquez

Keyword(s):

Unit Ball ◽

Siegel Domain

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An Algorithm for Nonparametric Estimation of a Multivariate Mixing Distribution with Applications to Population Pharmacokinetics

Pharmaceutics ◽

10.3390/pharmaceutics13010042 ◽

2020 ◽

Vol 13 (1) ◽

pp. 42

Author(s):

Walter M. Yamada ◽

Michael N. Neely ◽

Jay Bartroff ◽

David S. Bayard ◽

James V. Burke ◽

...

Keyword(s):

Drug Development ◽

Population Pharmacokinetics ◽

Empirical Bayes ◽

Applied Mathematics ◽

Grid Method ◽

Bayes Estimation ◽

Population Pharmacokinetic ◽

Support Points ◽

Primal Dual ◽

Log Normal

Population pharmacokinetic (PK) modeling has become a cornerstone of drug development and optimal patient dosing. This approach offers great benefits for datasets with sparse sampling, such as in pediatric patients, and can describe between-patient variability. While most current algorithms assume normal or log-normal distributions for PK parameters, we present a mathematically consistent nonparametric maximum likelihood (NPML) method for estimating multivariate mixing distributions without any assumption about the shape of the distribution. This approach can handle distributions with any shape for all PK parameters. It is shown in convexity theory that the NPML estimator is discrete, meaning that it has finite number of points with nonzero probability. In fact, there are at most N points where N is the number of observed subjects. The original infinite NPML problem then becomes the finite dimensional problem of finding the location and probability of the support points. In the simplest case, each point essentially represents the set of PK parameters for one patient. The probability of the points is found by a primal-dual interior-point method; the location of the support points is found by an adaptive grid method. Our method is able to handle high-dimensional and complex multivariate mixture models. An important application is discussed for the problem of population pharmacokinetics and a nontrivial example is treated. Our algorithm has been successfully applied in hundreds of published pharmacometric studies. In addition to population pharmacokinetics, this research also applies to empirical Bayes estimation and many other areas of applied mathematics. Thereby, this approach presents an important addition to the pharmacometric toolbox for drug development and optimal patient dosing.

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