Some new transformations for Bailey pairs and WP-Bailey pairs

We derive several new transformations relating WP-Bailey pairs. We also consider the corresponding transformations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions which can also be expressed as certain types of Lambert series.     

A finite element method for the Tricomi problem

Summary We consider the numerical solution of the Tricomi problem. Using a weak formulation based on different spaces of test and trial functions, we construct a new Galerkin procedure for the Tricomi problem. Existence, uniqueness, and uniform stability of the approximate solution is proven, and a priori error bounds are given.Research supported in part by the Department of Energy under contract DOE E(40-1)3443     

On the zeros and poles of Padé approximants toe z . III

Summary In this paper, we continue our study of the location of the zeros and poles of general Padé approximants toe ^z . We state and prove here new results for the asymptotic location of the normalized zeros and poles for sequences of Padé approximants toe ^z , and for the asymptotic location of the normalized zeros for the associated Padé remainders toe ^z . In so doing, we obtain new results for nontrivial zeros of Whittaker functions, and also generalize earlier results of Szegö and Olver.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2688Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2729, and by the Energy Research and Development Administration (ERDA) under Grant EY-76-S-02-2075     

On optimal triangular meshes for minimizing the gradient error

Summary Construction of optimal triangular meshes for controlling the errors in gradient estimation for piecewise linear interpolation of data functions in the plane is discussed. Using an appropriate linear coordinate transformation, rigorously optimal meshes for controlling the error in quadratic data functions are constructed. It is shown that the transformation can be generated as a curvilinear coordinate transformation for anyC data function with nonsingular Hessian matrix. Using this transformation, a construction of nearly optimal meshes for general data functions is described and the error equilibration properties of these meshes discussed. In particular, it is shown that equilibration of errors is not a sufficient condition for optimality. A comparison of meshes generated under several different criteria is made, and their equilibrating properties illustrated.This work was supported by the Natural Sciences and Engineering Research Council of Canada, by the Information Technology Research Centre, which is funded by the Province of Ontario, by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy under contract DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc., and through an appointment to the U.S. Department of Energy Postgraduate Research Program administered by Oak Ridge Associated Universities     

Partitioned quasi-Newton methods for nonlinear equality constrained optimization

We derive new quasi-Newton updates for the (nonlinear) equality constrained minimization problem. The new updates satisfy a quasi-Newton equation, maintain positive definiteness on the null space of the active constraint matrix, and satisfy a minimum change condition. The application of the updates is not restricted to a small neighbourhood of the solution. In addition to derivation and motivational remarks, we discuss various numerical subtleties and provide results of numerical experiments.Research partially supported by the Applied Mathematical Sciences Research Program (KC-04-02) of the Office of Energy Research of the US Department of Energy under grant DE-FG02-86ER25013.A000, and by the US Army Research Office through the Mathematical Sciences Institute, Cornell University.     

The number of self-dual types in finite-valued logics

We obtain the cycle indices of two groups of self-dual transformations of functions of an arbitrary finite-valued logic, and from them we derive explicit formulas for the number of types of functions relative to these groups.     

Composite sequence transformations

Summary The aim of this work is to introduce the new concept of composite sequence transformations and to show, by very simple examples and theorems, that it can be useful in accelerating the convergence of sequences. Generalizations of classical transformations and results are obtained.Work performed under the Nato Research Grant 027.81.Presented at the International Conference on Numerical Analysis, Munich, March 19–21, 1984     

LU decompositions of generalized diagonally dominant matrices

Summary Using the simple vehicle ofM-matrices, the existence and stability ofLU decompositions of matricesA which can be scaled to diagonally dominant (possibly singular) matrices are investigated. Bounds on the growth factor for Gaussian elimination onA are derived. Motivation for this study is provided in part by applications to solving homogeneous systems of linear equationsAx=0, arising in Markov queuing networks, input-output models in economics and compartmental systems, whereA or –A is an irreducible, singularM-matrix.This paper extends earlier work by Funderlic and Plemmons and by Varga and Cai.Research sponsored by the Applied Mathematical Sciences Research Program, Office of Energy Research, U.S. Department of Energy under contract W-7405-eng-26 with the Union Carbide CorporationResearch supported in part by the National Science Foundation under Grant No. MCS 8102114Research supported in part by the U.S. Army Research Office under contract no. DAAG 29-81-k-0132     

A family of Bell transformations

We introduce a family of sequence transformations, defined via partial Bell polynomials, that may be used for a systematic study of a wide variety of problems in enumerative combinatorics. This family includes some of the transformations listed in the paper by Bernstein & Sloane, now seen as transformations under the umbrella of partial Bell polynomials. Our goal is to describe these transformations from the algebraic and combinatorial points of view. We provide functional equations satisfied by the generating functions, derive inverse relations, and give a convolution formula. While the full range of applications remains unexplored, in this paper we show a glimpse of the versatility of Bell transformations by discussing the enumeration of several combinatorial configurations, including rational Dyck paths, rooted planar maps, and certain classes of permutations.     

Mixed hyperbolic-elliptic systems in self-similar flows

From the observation that self-similar solutions of conservation laws in two space dimensions change type, it follows that for systems of more than two equations, such as the equations of gas dynamics, the reduced systems will be of mixed hyperbolic-elliptic type, in some regions of space. In this paper, we derive mixed systems for the isentropic and adiabatic equations of compressible gas dynamics. We show that the mixed systems which arise exhibit complicated nonlinear dependence. In a prototype system, the nonlinear wave system, this behavior is much simplified, and we outline the solution to some typical Riemann problems.Dedicated to Constantine Dafermos on his 60^th birthdayResearch supported by the National Science Foundation, grant DMS-9970310.Research supported by the Department of Energy, grant DE-FG-03-94-ER25222 and by the National Science Foundation, grant DMS-9973475 (POWRE).Research supported by the Department of Energy, grant DE-FG-03-94-ER25222 and by the National Science Foundation, grant DMS-0103823.     

A distance on curves modulo rigid transformations

We propose a geometric method for quantifying the difference between parametrized curves in Euclidean space by introducing a distance function on the space of parametrized curves up to rigid transformations (rotations and translations). Given two curves, the distance between them is defined as the infimum of an energy functional which, roughly speaking, measures the extent to which the jet field of the first curve needs to be rotated to match up with the jet field of the second curve. We show that this energy functional attains a global minimum on the appropriate function space, and we derive a set of first-order ODEs for the minimizer.     

The equivalence of strict convexity and injectivity of the gradient in bounded level sets

It is shown that Lipschitzian functions are strictly convex if and only if their generalized gradients are disjoint at distinct interior points of a given bounded level set.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, US Department of Energy, under Contract W-31-109-Eng-38.     

Computing theCS decomposition of a partitioned orthonormal matrix

Summary This paper describes an algorithm for simultaneously diagonalizing by orthogonal transformations the blocks of a partitioned matrix having orthonormal columns.This work was supported by the Air Force Office of Scientific Research under Contract No. AFOSR-82-0078     

Direct and inverse error estimates for finite elements with mesh refinements

The finite element method is used to solve a second order elliptic boundary value problem on a polygonal domain. Mesh refinements and weighted Besov spaces are used to obtain optimal error estimates and inverse theorems.Research performed while at the University of Maryland under a Fulbright fellowshipResearch supported in part by the Department of Energy under the contract E(40-1)3443Research supported in part by the National Institutes of Health under the grant 5R01-AM-20373     

Bounding separable recourse functions with limited distribution information

The recourse function in a stochastic program with recourse can be approximated by separable functions of the original random variables or linear transformations of them. The resulting bound then involves summing simple integrals. These integrals may themselves be difficult to compute or may require more information about the random variables than is available. In this paper, we show that a special class of functions has an easily computable bound that achieves the best upper bound when only first and second moment constraints are available.This research has been partially supported by the National Science Foundation under Grants ECS-8304065 and ECS-8815101, by the Office of Naval Research Grant N00014-86-K-0628 and by the National Research Council under a Research Associateship at the Naval Postgraduate School, Monterey, California.     

On Some Integral Transformations and Their Application to the Solution of Boundary-Value Problems in Mathematical Physics

We obtain a formula for the expansion of an arbitrary function in a series in the eigenfunctions of the Sturm–Liouville boundary-value problem for the differential equation of cone functions. On the basis of this result, we derive a series of integral transformations (including well-known ones) and inversion formulas for them. We apply these formulas to the solution of initial boundary-value problems in the theory of heat conduction for circular hollow cones truncated by spherical surfaces.     

The Euler Maclaurin expansion for the Cauchy Principal Value integral

Summary The classical Euler Maclaurin Summation Formula expresses the difference between a definite integral over [0, 1] and its approximation using the trapezoidal rule with step lengthh=1/m as an asymptotic expansion in powers ofh together with a remainder term. Many variants of this exist some of which form the basis of extrapolation methods such as Romberg Integration. in this paper a variant in which the integral is a Cauchy Principal Value integral is derived. The corresponding variant of the Fourier Coefficient Asymptotic Expansion is also derived. The possible role of the former in numerical quadrature is discussed.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contract W-31-109-Eng-38     

Coordinate Transformations and Derivation of Open-Loop Nash Equilibria

Leitmann (Ref. 1) introduced coordinate transformations to derive global optima of a class of dynamic optimization problems. We present applications of this method to derive open-loop Nash equilibria for finite-time horizon differential games. The method of coordinate transformations is especially useful in cases where the original game does not satisfy the global curvature conditions normally imposed in sufficient optimality conditions.     

Families of multisums as mock theta functions

In view of the Bailey lemma and the relations between Hecke-type sums and Appell–Lerch sums given by Hickerson and Mortenson, we find that many Bailey pairs given by Slater can be used to deduce mock theta functions. Therefore, by constructing generalized Bailey pairs with more parameters, we derive some new families of mock theta functions. Meanwhile, some identities between new mock theta functions and classical ones are established. Furthermore, based on the proofs of the main theorems, many q-hypergeometric transformations are obtained.     

On the rate of convergence of infinite horizon discounted optimal value functions

In this paper we investigate the rate of convergence of the optimal value function of an infinite horizon discounted optimal control problem as the discount rate tends to zero. Using the Integration Theorem for Laplace transformations we provide conditions on averaged functionals along suitable trajectories yielding quadratic pointwise convergence. From this we derive under appropriate controllability conditions criteria for linear uniform convergence of the value functions on control sets. Applications of these results are given and an example is discussed in which both linear and slower rates of convergence occur depending on the cost functional.