Remainders in Numerical Integration Formulae Involving Derivatives

The remainders which correspond to the numerical integrationformulae considered in Cicenia (1976) are derived in a simpleand convenient form for feasible cases. Then they are appliedto some of the functions used in Cicenia (1976) for testingquadrature formulae there, and comparisons are made betweenerror estimates obtained from them and the actual errors tabulatedin Cicenia (1976).     

Estimating the covariance matrix: a new approach

In this paper, we consider the problem of estimating the covariance matrix and the generalized variance when the observations follow a nonsingular multivariate normal distribution with unknown mean. A new method is presented to obtain a truncated estimator that utilizes the information available in the sample mean matrix and dominates the James-Stein minimax estimator. Several scale equivariant minimax estimators are also given. This method is then applied to obtain new truncated and improved estimators of the generalized variance; it also provides a new proof to the results of Shorrock and Zidek (Ann. Statist. 4 (1976) 629) and Sinha (J. Multivariate Anal. 6 (1976) 617).     

An examiniation of criticisms of ridge regression simulation methodology

Page1 (1981) claimed that there is a serious design flaw in some of the recent simulation studies of ridge estimators, in particular those in Hoerl, Kennard and Baldwin (1976) and Lawless and Wang (1976). Farebrother (1983) argued that the major criticism in Page1 (1981) is unsubstantiated. In this paper we obtain a series expansion for the mean squared error of the ordinary ridge estimator, use it to prove that Pagel's claim is incorrect and reinforce Farebrother's comments.     

A note on the missing value principle and the EM-Algorithm for estimation and prediction in sampling from finite populations with a multinormal superpopulation model

It is shown that both the missing value principle (MVP) of Orchard and Woodbury (1972) and the EM-Algorithm of Dempster, Laird and Rubin (1972) yield a unique predictor of the population total, under a superpopulation multinormal model. The predictor obtained is the maximum likelihood predictor introduced by Royal (1976).     

Resonance sets of the feedback control equation

Summary The concept of a resonance set is used to state and prove sufficient conditions for the existence of a periodic solution of the nonlinear vector differential equation f(D)x + + BMg(D)x=p, in which D=d/dt and the functions M, p are periodic in the t variable. Entrata in Redazione il 19 giugno 1976.     

Rate of convergence in the strong law of large numbers for martingales

Summary The rate of convergence in the random strong law of large numbers for martingale differences is established. The results obtained generalize theorems given by R. Chen (1976) and I.A. Ahmad (1980).     

An Application of Forecasting in the Alcoholic Drinks Industry

The sales of cider in the U.K. experienced an upsurge in demand during the good summer of 1975 and the record drought of 1976. Conventional linear growth seasonal forecasting systems are unable to isolate out the effects of this exceptional weather and would produce a high forecast for the (average/poor) year of 1977. Furthermore at the end of 1976 the Government imposed an Excise Duty which depressed demand still further. A model capable of isolating and measuring these (and other) effects was required and this was achieved within the framework of the Dynamic Linear Model proposed by Harrison and Stevens¹ usually referred to as the Bayesian methods.The model finally included growth and seasonality plus the effect of exceptional weather together with price, inflation and the transfer effects of price changes. The forecasts have been very satisfactory and enable future alternative pricing strategies to be investigated.     

Stabilized approximate solutions of the inverse time problem for a parabolic evolution equation

The set of all solutions of a composite fuzzy relation equation of Sanchez (Inform. and Control30 (1976)), defined on finite spaces, is studied by determining and characterizing all the lower solutions of such an equation.     

Nonnegative definite and positive definite solutions to the matrix equation AXA* = B

The general nonegative definite solution to the matrix equation AXA^* = B is established in a form which can be viewed as advantageous over that derived by Khatri and Mitra (1976). The problem of determining an existence criterion and a representation of a positive definite to this equation is considered.     

A characterization of the Poisson distribution

In 1976 the author of this note provided a characterization of the Poisson distribution in the set of infinitely divisible distributions. We give another characterization of the Poisson distribution in the set of infinitely divisible distributions.     

Note on the equivalence of Kuhn-Tucker complementarity conditions to an equation

This note presents a more general and simple proof with geometric interpretations of the equivalence of the complementarity problem to an equation (or a system of equations), given by Mangasarian in 1976. Although this fact has been used by the author and others in a different context, it is believed that it should be presented to a more general audience of optimization specialists.     

On the facets of the lift-and-project relaxations of graph subdivisions

We study the behavior of lift-and-project procedures for solving combinatorial optimization problems as described by Lovász and Schrijver (1991), in the context of the stable set problem on graphs. Following the work of Wolsey (1976), we investigate how to generate facets of the relaxations obtained by these procedures from facets of the relaxations of the original graph, after applying fundamental graph operations. We show our findings for the odd subdivision of an edge and its generalization, the stretching of a vertex operation.     

On the stable global convergence of particular quasi-newton-methods

In the paper the classical BFGS-method with a Goldstein-Armijo step length rule is considered in the case when the first order information about the function is inexact. There are given accuracy restrictions for which the perturbed method is globally convergent and additional conditions for q-superlinear convergence. For the proofs there were essentially used results of Powell (1976) and of Dennis and Walker (1984, 1985)     

Nonnegative definite and positive definite solutions to the matrix equation AXA * = B

The general nonegative definite solution to the matrix equation AXA^* = B is established in a form which can be viewed as advantageous over that derived by Khatri and Mitra (1976). The problem of determining an existence criterion and a representation of a positive definite to this equation is considered.     

Equational theory of algebras with a majority polynomial

This note gives a simple proof of a result of R. N. McKenzie that any finitely based variety of algebras which admit a majority polynomial and is definable by absorption identities is one-based. In the case of lattices, for example, this yields a “short” identity with only seven variables. Presented by R. McKenzie. Presented in the Special Session on Varieties of Algebras at the 735th Meeting of the AMS held at Reno, Nevada in the spring of 1976 (abstract 735-A14, Notices, AMS 23 (1976)). This research was supported by the grant #A8215 from the National Research Council of Canada.     

A Crank‐Nicolson and ADI Galerkin method with quadrature for hyperbolic problems

We propose, analyze, and implement fully discrete two‐time level Crank‐Nicolson methods with quadrature for solving second‐order hyperbolic initial boundary value problems. Our algorithms include a practical version of the ADI scheme of Fernandes and Fairweather [SIAM J Numer Anal 28 (1991), 1265–1281] and also generalize the methods and analyzes of Baker [SIAM J Numer Anal 13 (1976), 564–576] and Baker and Dougalis [SIAM J Numer Anal 13 (1976), 577–598]. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Modulus of continuity of the solution to the Dirichlet problem for a second-order parabolic equation

The modulus of continuity of the solution to the Dirichlet problem is investigated for a second-order parabolic equation at a regular boundary point. A bound for the modulus of continuity is obtained in terms of the capacity. The coefficients of the equation are required to satisfy a Dini condition (uniformly).Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 587–593, April, 1976.     

Mathematical statement and investigation of the problem of heat transfer between fluid flow in a channel and the environment

Translated from Issledovaniya po Prikladnoi Matematike, No. 5, pp. 111–123, 1976.     

A note on the cross-covariance operator and on congruence relations for Hilbert space valued stochastic processes

Explicit formulas are derived for the congruence mappings that connect three Hilbert spaces associated with a second-order stochastic process. In particular, an insightful expression is obtained for the mapping that connects a process to its corresponding reproducing kernel Hilbert space. In addition, a useful infinite dimensional extension of a result from Khatri (1976) which pertains to cross-covariance operators is provided.     

The Schur group of an abelian number field

We characterize the maximum r-local index of a Schur algebra over an abelian number field K in terms of global information determined by the field K for an arbitrary rational prime, r. This completes and unifies previous results of Janusz in [G.J. Janusz, The Schur group of an algebraic number field, Ann. of Math. (2) 103 (1976) 253-281] and Pendergrass in [J.W. Pendergrass, The 2-part of the Schur group, J. Algebra 41 (1976) 422-438].