Prosthaphaeresis revisited

The trigonometric identities for the product of two sines and the product of two cosines were first published in 1588. The discovery of the two equations, however, clearly predates that publication. Credit for priority was debated at the end of the last century and tentatively assigned to Paul Wittich of Breslau. But at least some of the conclusions reached at that time were not completely sound, even with respect to the evidence then available; and in the meantime, new evidence has come to light. Reevaluation of the issue now suggests a very questionable role for Wittich in the discovery of either equation. The first one was certainly discovered by Johannes Werner in about 1510, and probably resurrected from his papers by Wittich, while the second one appears to have been discovered from a knowledge of the first by Joost Bürgi in about 1585. But if Wittich loses one aspect of his priority in this reevaluation, he gains in another aspect. For it is now clear that Wittich had developed the method of prosthaphaeresis—the idea of using a product formula to simplify calculations—well before he arrived on Hven in 1580.     

Napoleon revisited

Napoleon's Theorem can be neatly proved using a tessellation of the plane. The theorem can be generalized by using three similar triangles (instead of the three equilateral triangles) erected in different ways on the three sides of the triangle. Various interesting special cases occur.Dedicated to H. S. M. Coxeter on the occasion of his 80th birthday.     

Paracompactness revisited

New proofs of the characterization of paracompact frames as those frames which admit a complete uniformity, and of the existence of the paracompact regular reflection, based on the Samuel type completion, is presented.     

Forcing revisited

The purpose of this paper is to propose and explore a general framework within which a wide variety of model construction techniques from contemporary set theory can be subsumed. Taking our inspiration from presheaf constructions in category theory and Boolean ultrapowers, we will show that generic extensions, ultrapowers, extenders and generic ultrapowers can be construed as examples of a single model construction technique. In particular, we will show that Łoś's theorem can be construed as a specific case of Cohen's truth lemma, and we isolate the weakest conditions a filter must satisfy in order for the truth lemma to work.     

Transitivity revisited

A survey is given of research on transitivity.The debate about the assumption of transitivity turns upon the interpretation of certain real world experiences     

Glivenko theorems revisited

Glivenko-type theorems for substructural logics (over FL) are comprehensively studied in the paper [N. Galatos, H. Ono, Glivenko theorems for substructural logics over FL, Journal of Symbolic Logic 71 (2006) 1353-1384]. Arguments used there are fully algebraic, and based on the fact that all substructural logics are algebraizable (see [N. Galatos, H. Ono, Algebraization, parametrized local deduction theorem and interpolation for substructural logics over FL, Studia Logica 83 (2006) 279-308] and also [N. Galatos, P. Jipsen, T. Kowalski, H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, in: Studies in Logic and the Foundations of Mathematics, vol. 151, Elsevier, 2007] for the details).As a complementary work to the algebraic approach developed in [N. Galatos, H. Ono, Glivenko theorems for substructural logics over FL, Journal of Symbolic Logic 71 (2006) 1353-1384], we present here a concise, proof-theoretic approach to Glivenko theorems for substructural logics. This will show different features of these two approaches.     

Ramanujan expansions revisited

Dedicated to Wolfgang Schwarz on the occasion of his sixtieth birthday     

Health visitors revisited

Cambridgeshire Area Health Authority commissioned a team from the University of Aston Management Centre to carry out an operational research project in order to assist in planning the provision of health visitor services in Peterborough health District. Since Peterborough is a Development Area there has been a rapid increase in population which it is planned will continue for some years. This change in size and age-distribution of the population poses considerable problems for planning the provision of health visitor services.Health visitors are trained nurses who have highly independent roles in the National Health Service. They are largely concerned with preventive work, traditionally with children and increasingly with the elderly, in which it is difficult to set quantitative objectives and measure outputs.This paper will describe the research plan and the models developed. Considerable attention will be paid to the relationship of the research team to the client system, consisting of the Area and District managers and the individual health visitors. The implications of this type of study in extending the practice of O.R. will be considered.     

Quillen-Suslin theory revisited

We partially answer a question of T.Y. Lam by obtaining partial generalized Local Global and Monic Inversion Principles for the action of the elementary subgroup on the set of unimodular rows over a commutative ring.     

Kolmogorov Theorem revisited

Kolmogorov Theorem on the persistence of invariant tori of real analytic Hamiltonian systems is revisited. In this paper we are mainly concerned with the lower bound on the constant of the Diophantine condition required by the theorem. From the existing proofs in the literature, this lower bound turns to be of O(ε1/4), where ε is the size of the perturbation. In this paper, by means of careful estimates on Kolmogorov's method, we show that this lower bound can be weakened to be of O(ε1/2). This condition coincides with the optimal one of KAM Theorem. Moreover, we also obtain optimal estimates for the distance between the actions of the perturbed and unperturbed tori. We believe that some ideas contained in this paper may be used for improving several estimates in the general KAM context.     

Local cuts revisited

We present a variant of the local cut generation procedure by Applegate, Bixby, Chvátal and Cook. Unlike the original procedure, our method immediately yields a facet of the projected polytope as the solution of a single LP, without the need for the time-consuming tilting step. Moreover, our facets have big volume in general.     

Stabilized SQP revisited

The stabilized version of the sequential quadratic programming algorithm (sSQP) had been developed in order to achieve superlinear convergence in situations when the Lagrange multipliers associated to a solution are not unique. Within the framework of Fischer (Math Program 94:91–124, 2002), the key to local superlinear convergence of sSQP are the following two properties: upper Lipschitzian behavior of solutions of the Karush-Kuhn-Tucker (KKT) system under canonical perturbations and local solvability of sSQP subproblems with the associated primal-dual step being of the order of the distance from the current iterate to the solution set of the unperturbed KKT system. According to Fernández and Solodov (Math Program 125:47–73, 2010), both of these properties are ensured by the second-order sufficient optimality condition (SOSC) without any constraint qualification assumptions. In this paper, we state precise relationships between the upper Lipschitzian property of solutions of KKT systems, error bounds for KKT systems, the notion of critical Lagrange multipliers (a subclass of multipliers that violate SOSC in a very special way), the second-order necessary condition for optimality, and solvability of sSQP subproblems. Moreover, for the problem with equality constraints only, we prove superlinear convergence of sSQP under the assumption that the dual starting point is close to a noncritical multiplier. Since noncritical multipliers include all those satisfying SOSC but are not limited to them, we believe this gives the first superlinear convergence result for any Newtonian method for constrained optimization under assumptions that do not include any constraint qualifications and are weaker than SOSC. In the general case when inequality constraints are present, we show that such a relaxation of assumptions is not possible. We also consider applying sSQP to the problem where inequality constraints are reformulated into equalities using slack variables, and discuss the assumptions needed for convergence in this approach. We conclude with consequences for local regularization methods proposed in (Izmailov and Solodov SIAM J Optim 16:210–228, 2004; Wright SIAM J. Optim. 15:673–676, 2005). In particular, we show that these methods are still locally superlinearly convergent under the noncritical multiplier assumption, weaker than SOSC employed originally.     

Autoregressive-output-analysis methods revisited

We revisit and update the autoregressive-output-analysis method for constructing a confidence interval for the steady-state mean of a simulated process by using Rissanen's predictive least-squares criterion to estimate the autoregressive order of the process. This order estimator is strongly consistent when the output is autoregressive. The order estimator is combined with the standard autoregressive-output-analysis method to form a confidence-interval procedure. Alternatives for estimating the degrees of freedom for the procedure are investigated. The main result is an asymptotically valid confidence-interval procedure that, empirically, has good small-sample properties.     

Least elements revisited

Least element theory is extended, for linear problems, to general multiple-objective problems, and the pre-Leontief matrix and left-hand matrix inverse notions are generalized.This work was completed while the author was visiting the Center for Advanced Studies and the Department of Systems Engineering at the University of Virginia. The author is indebted to the referees for their helpful comments.