Some classes of functions and Fourier coefficients with respect to general orthonormal systems

The a.e. convergence of an orthogonal series on [0, 1] depends strongly on the coefficients of this series. It is well known that a sufficient condition for the a.e. convergence of such a series is given by the Men’shov-Rademacher theorem. On the other hand, S. Banach proved that good differential properties of a function do not guarantee the a.e. convergence on [0, 1] of the Fourier series of this function with respect to general orthonormal systems (ONSs). In the present study, we find conditions on the functions of an ONS under which the Fourier coefficients of functions of some differential classes satisfy the hypothesis of the Men’shov-Rademacher theorem.     

Book Reviews

Book reviewed in this article: Man's Mark on the Land, by Arthur S. Gregor. Comets, Meteorites and Men, by Peter Lancaster Brown.     

An addition to Tandori’s summation theorem

We utilize the Bernstein-type almost monotone sequences instead of monotone sequences and verify that the classical Men?ov-condition \(\sum c_{n}^{2}\log \log^{2}n<\infty\) , which implies Abel- and (C,α>0)-summability of orthogonal series, is a necessary and sufficient condition for a larger class of sequences than it was proved by Tandori.     

BOOK REVIEWS

Book reviewed in this article: Electronics : Today and Tomorrow , by John Mills. General Physics , A Textbook for Colleges , by Oswald Blackwood, Ph.D. Essentials of Astronomy , by John Charles Duncan, Ph.D. Basic Air Navigation , by Eibert F. Blackburn. How to Pass Radio License Examinations , by Charles E. Drew. Introductory Astronomy , A Guide for Night Watchers , by J. B. Sidgwick. Plane and Spherical Trigonometry , by Donald H. Ballou, Ph.D. and Frederick, H. Steen, Ph.D. How to Pass a Written Examination , by Harry C. McKown. Military Applications of Mathematics , by Paul P. Hanson, M.A. Calculus Refresher for Technical Men , by Albert Klaf, B.S., M.E. Mathematics for Aircraft Engine Mechanics , by Harold Griffiths. Essentials of Business Arithmetic , Revised Edition, by Edward M. Kanzer. Mathematics for the Emergency , a text-workbook in essential mathematics. By C. J. Lapp. Liberal Education , by Mark Van Doren.     

Bessel sequences as projections of orthogonal systems

We prove generalizations of the Schur and Olevskii theorems on the continuation of systems of functions from an interval I to orthogonal systems on an interval J, I ? J. Only Bessel systems in L ²(I) are projections of orthogonal systems from the wider space L ²(J). This fact allows us to use a certain method for transferring the classical theorems on the almost everywhere convergence of orthogonal series (the Men’shov-Rademacher, Paley-Zygmund, and Garcia theorems) to series in Bessel systems. The projection of a complete orthogonal system from L ²(J) onto L ²)(I) is a tight frame, but not a basis.     

Book Vignettes

Thorarinsson , Sigurdur , Surtsey, the New Island in the North Atlantic Twitty , Victor Chandler , Of Scientists and Salamanders Chrestenson , H. E., Mappings of the Plane, With Applications to Trigonometry and Complex Numbers Fanning , A. E., Planets, Stars and Galaxies Schmidt , O. U., Abstract Theory of Groups Riley , Denis , and Young , Anthony , World Vegetation Edinger , James G., Watching for the Wind: The Seen and Unseen Influences on Local Weather Fehr , Howard F. and Hill , Thomas J., Contemporary Mathematics for Elementary Teachers Crowder, Norman A., Introduction to Genetics Kilmister , C. W., Men of Physics, Sir Arthur Eddington Ross , Herbert H., Understanding Evolution     

Book Vignettes

Lemons, Wayne, Learn Electronics Through Troubleshooting. Ore, Oyestein, Invitation to Number Theory. Salvadori, Mario, Mathematics in Architecture. Alter, Dinsmore (ed.), Lunar Atlas. Mayall, R. Newton and Mayall, Margaret W., Skyshooting—Photography for Amateur Astronomers. Brody, Boruch and Capaldi, Nicholas (eds.), Science: Men, Methods, Goals. Cissell, Robert and Cissell, Helen, Mathematics of Finance: Third Edition. Zwicky, Fritz, Discovery, Invention, Research Through the Morphological Approach. Killeffer, David H., How Did You Think of That? An Introduction to the Scientific Method. Warshofsky, Fred, The 21st Century: The New Age of Exploration. Krynowsky, William J.; Ramsey, William; Phillips, Clifford R., Jr. and Watenpaugh, Frank M., Foundations of Space Science.     

Subspaces and quotients of Banach spaces with shrinking unconditional bases

The main result is that a separable Banach space with the weak* unconditional tree property is isomorphic to a subspace as well as a quotient of a Banach space with a shrinking unconditional basis. A consequence of this is that a Banach space is isomorphic to a subspace of a space with a shrinking unconditional basis if and only if it is isomorphic to a quotient of a space with a shrinking unconditional basis, which solves a problem dating to the 1970s. The proof of the main result also yields that a uniformly convex space with the unconditional tree property is isomorphic to a subspace as well as a quotient of a uniformly convex space with an unconditional finite dimensional decomposition.     

Modelling ship operational reliability over a mission under regular inspections

A ship is required to operate for a fixed mission period. Should a critical item of equipment fail at sea, the ship is subject to a costly event with potentially high risk to ship and crew. Given warning of a pending defect, the ship can try to return to port under its own power and thus attempt to avoid an at sea failure. Defects which lead to a failure are detected by inspection, and the task is to select the appropriate frequency of inspection to balance the number of occasions that a ship fails at sea and the number of preventive inspection based returns to port during a mission to correct a defect. The modelling entails using the delay time concept. Expressions are established for the expected number of preventive and failure returns over a mission, and an example given of a cost based balance to select an optimal inspection period. Although addressing ship reliability, the model has relevance to the mission reliability of any repairable equipment with remote main repair facilities.     

On Martingale Approximation of Adapted Processes

We show that the existence of a martingale approximation of a stationary process depends on the choice of the filtration. There exists a stationary linear process which has a martingale approximation with respect to the natural filtration, but no approximation with respect to a larger filtration with respect to which it is adapted and regular. There exists a stationary process adapted, regular, and having a martingale approximation with respect to a given filtration but not (regular and having a martingale approximation) with respect to the natural filtration.     

Price coordination in distribution channels: A dynamic perspective

In this study, we investigate two important questions related to dynamic pricing in distribution channels: (i) Are coordinated pricing decisions efficient in a context where prices have carry-over effects on demand? (ii) Should firms practice a skimming or a penetration strategy if they choose to coordinate or to decentralize their activities? To answer these questions, we consider a differential game that takes place in a bilateral monopoly where the past retail prices paid by consumers contribute to the building of a reference price. The latter is used by consumers as a benchmark to evaluate the value of the product, and by firms to decide whether to adopt a skimming or a penetration strategy.     

Lifting Property of Direct Sums of Hollow Modules

We give necessary and sufficient conditions for a module M with a decomposition that complements maximal direct summands to be a lifting module. We apply our results to get a necessary and sufficient condition for a direct sum of modules to be a quasi-discrete (discrete) module. We also give some sufficient conditions for a lifting module to be a direct sum of indecomposables.     

On the properties of orthorecursive expansions with respect to subspaces

In a Hilbert space, for orthorecursive expansions with respect to closed subspaces, we establish a criterion for expansions of elements of a certain finite-dimensional subspace with respect to a finite sequence of subspaces to coincide with the expanded elements. This implies a criterion for an element to be equal to its orthorecursive expansion with respect to a finite sequence of subspaces. We also obtain a number of results related to the best approximations of elements by partial sums of their orthorecursive expansions with respect to a sequence of finite-dimensional subspaces.     

Linear facility location — Solving extensions of the basic problem

The basic problem is to locate a linear facility to minimize the sume of weighted shortest Euclidean distances from demand points to the facility. We extend the analysis to locating a constrained linear facility, a radial facility, a linear facility where distances are rectangular and a linear facility under the minimax criterion. Each case is shown to admit a simple solution technique.     

A parallel domain reduction method

We relate a particular version of a parallel multigrid method to a domain decomposition method, showing that the parallel multigrid method reduces computation to a small portion of the domain and then extends the solution to the entire domain using the correct reflections to get the exact solution. We extend a particular example to double the parallelism in a nonobvious manner. While the techniques of this paper are applied to twodimensional problems, they can be applied to higher dimensional problems in an obvious manner.     

Characterization of isolated homogeneous hypersurface singularities in C~4 Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

Let V be a hypersurface with an isolated singularity at the origin in Cn 1. It is a natural question to ask when V is defined by weighted homogeneous polynomial or homogeneous polynomial up to biholomorphic change of coordinates. In 1971, a beautiful theorem of Saito gives a necessary and sufficient condition for V to be defined by a weighted homogeneous polynomial. For a two-dimensional isolated hypersurface singularity V, Xu and Yau found a coordinate free characterization for V to be defined by a homogeneous polynomial. Recently Lin and Yau gave necessary and sufficient conditions for a 3-dimensional isolated hypersurface singularity with geometric genus bigger than zero to be defined by a homogeneous polynomial. The purpose of this paper is to prove that Lin-Yau's theorem remains true for singularities with geometric genus equal to zero.     

Koszul Bicomplexes and Generalized Koszul Complexes in Projective Dimension One

We describe Koszul type complexes associated with a linear map from any module to a free module, and vice versa with a linear map from a free module to an arbitrary module, generalizing the classical Koszul complexes. Given a short complex of finite free modules, we assemble these complexes to what we call Koszul bicomplexes. They are used in order to investigate the homology of the Koszul complexes in projective dimension one. As in the case of the classical Koszul complexes, this homology turns out to be grade sensitive. In a special setup, we obtain necessary conditions for a map of free modules to be lengthened to a short complex of free modules.     

Simd language design using prescriptive semantics

The goal of this article is to isolate a set of primitives necessary for the construction of SIMD programs and to give a denotational semantics for these primitives. The intent is to devise a language with a simple semantics rather than to propose a language which may be conveniently implemented. The approach taken results in the addition of a synchronous parallel assignment statement and a synchronous communication statement to the familiar sequential programming language control structures of composition, projection and iteration.     

Perturbation singulière d'un problème de frottement sec non monotone

In this Note we deal with a singularly perturbed system constituted by a differential inclusion which has a unique solution for each value of the perturbation parameter. The associated degenerated problem, that corresponds to a dynamic dry friction problem, has many solutions. We show that perturbed problem solutions converge to a particular solution of the degenerated problem when the perturbation parameter goes to zero. The singular perturbation approach allows an analysis of a criterion used to select a solution of the degenerated problem, and suggests a method to study more elaborated dry friction problems.