Quadratic transformations of the sixth Painlevé equation with application to algebraic solutions

In 1991, one of the authors showed the existence of quadratic transformations between the Painlevé VI equations with local monodromy differences (1/2, a, b, ±1/2) and (a, a, b, b). In the present paper we give concise forms of these transformations. They are related to the quadratic transformations obtained by Manin and Ramani–Grammaticos–Tamizhmani via Okamoto transformations. To avoid cumbersome expressions with differentiation, we use contiguous relations instead of the Okamoto transformations. The 1991 transformation is particularly important as it can be realized as a quadratic‐pull back transformation of isomonodromic Fuchsian equations. The new formulas are illustrated by derivation of explicit expressions for several complicated algebraic Painlevé VI functions. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

COSEARCH: A Parallel Cooperative Metaheuristic

In order to design a well-balanced metaheuristic for robustness, we propose the COSEARCH approach which manages the cooperation of complementary heuristic methods via an adaptive memory which contains a history of the search already done. In this paper, we present the idiosyncrasies of the COSEARCH approach and its application for solving large scale instances of the quadratic assignment problem (QAP). We propose an original design of the adaptive memory in order to focus on high quality regions of the search and avoid attractive but deceptive areas. For the QAP, we have hybridized three heuristic agents of complementary behaviours: a Tabu Search is used as the main search algorithm, a Genetic Algorithm is in charge of the diversification and a Kick Operator is applied to intensify the search. The evaluations have been executed on large scale network of workstations via a parallel environment which supports fault tolerance and adaptive dynamic scheduling of tasks.     

SDP diagonalizations and perspective cuts for a class of nonseparable MIQP

We present a new approach, requiring the solution of a SemiDefinite Program, for decomposing the Hessian of a nonseparable mixed-integer quadratic problem to permit using perspective cuts to improve its continuous relaxation bound. The new method favorably compares with a previously proposed one requiring a minimum eigenvalue computation.     

Equality and inequality constrained optimization algorithms with convergent stepsizes

In this paper, we consider two algorithms for nonlinear equality and inequality constrained optimization. Both algorithms utilize stepsize strategies based on differentiable penalty functions and quadratic programming subproblems. The essential difference between the algorithms is in the stepsize strategies used. The objective function in the quadratic subproblem includes a linear term that is dependent on the penalty functions. The quadratic objective function utilizes an approximate Hessian of the Lagrangian augmented by the penalty functions. In this approximation, it is possible to ignore the second-derivative terms arising from the constraints in the penalty functions.The penalty parameter is determined using a strategy, slightly different for each algorithm, that ensures boundedness as well as a descent property. In particular, the boundedness follows as the strategy is always satisfied for finite values of the parameter.These properties are utilized to establish global convergence and the condition under which unit stepsizes are achieved. There is also a compatibility between the quadratic objective function and the stepsize strategy to ensure the consistency of the properties for unit steps and subsequent convergence rates.This research was funded by SERC and ESRC research contracts. The author is grateful to Professors Laurence Dixon and David Mayne for their comments. The numerical results in the paper were obtained using a program written by Mr. Robin Becker.     

An analysis of reduced Hessian methods for constrained optimization

We study the convergence properties of reduced Hessian successive quadratic programming for equality constrained optimization. The method uses a backtracking line search, and updates an approximation to the reduced Hessian of the Lagrangian by means of the BFGS formula. Two merit functions are considered for the line search: the ₁ function and the Fletcher exact penalty function. We give conditions under which local and superlinear convergence is obtained, and also prove a global convergence result. The analysis allows the initial reduced Hessian approximation to be any positive definite matrix, and does not assume that the iterates converge, or that the matrices are bounded. The effects of a second order correction step, a watchdog procedure and of the choice of null space basis are considered. This work can be seen as an extension to reduced Hessian methods of the well known results of Powell (1976) for unconstrained optimization.This author was supported, in part, by National Science Foundation grant CCR-8702403, Air Force Office of Scientific Research grant AFOSR-85-0251, and Army Research Office contract DAAL03-88-K-0086.This author was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contracts W-31-109-Eng-38 and DE FG02-87ER25047, and by National Science Foundation Grant No. DCR-86-02071.     

On random walks arising in queueing systems: ergodicity and transience via quadratic forms as lyapounov functions — Part I

A simple and quite general approach is proposed to derive criteria for transience and ergodicity of a certain class of irreducibleN-dimensional Markov chains in ₊ ^N assuming a boundedness condition on the second moment of the jumps. The method consists in constructing convenient smooth supermartingales outside some compact set. The Lyapounov functions introduced belong to the set of quadratic forms in ₊ ^N and do not always have a definite sign. Existence and construction of these forms is shown to be basically equivalent to finding vectors satisfying a system of linear inequalities.Part I is restricted toN=2, in which case a complete characterization is obtained for the type of random walks analyzed by Malyshev and Mensikov, thus relaxing their condition of boundedness of the jumps. The motivation for this work is partly from a large class of queueing systems that give rise to random walks in ₊ ^N      

A minimum average risk approach to shrinkage estimators of the normal mean

For the problem of estimating the normal mean based on a random sample X ₁,...,X _n when a prior value ₀ is available, a class of shrinkage estimators % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-qqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xHapdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaubeaeqaje% aWbaGaamOBaaWcbeqdbaGafqiVd0MbaKaaaaGccaqGGaGaaiikaiaa% dUgacaGGPaGaeyypa0Jaam4AaiaacIcadaqfqaqabKqaahaacaqGUb% aaleqaneaacaqGubaaaOGaaiykaiaabccadaqfqaqabKqaahaacaWG% UbaaleqaneaaceqGybGbaebaaaGccaqGGaGaey4kaSIaaeiiaiaacI% cacaaIXaGaaeiiaiabgkHiTiaabccacaWGRbGaaiikamaavababeqc% baCaaiaab6gaaSqab0qaaiaabsfaaaGccaGGPaGaaiykamaavababe% qcbaCaaiaad6gaaSqab0qaaiabeY7aTbaaaaa!5615!\[\mathop {\hat \mu }\nolimits_n {\rm{ }}(k) = k(\mathop {\rm{T}}\nolimits_{\rm{n}} ){\rm{ }}\mathop {{\rm{\bar X}}}\nolimits_n {\rm{ }} + {\rm{ }}(1{\rm{ }} - {\rm{ }}k(\mathop {\rm{T}}\nolimits_{\rm{n}} ))\mathop \mu \nolimits_n \] is considered, where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-qqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xHapdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaubeaeqaje% aWbaGaamOBaaWcbeqcdawaaiaadsfaaaGccaqGGaGaaeypaiaabcca% caWGUbWaaWbaaSqabeaacaaIXaGaai4laiaaikdaaaGccaGGOaWaa0% aaaeaacaWGybaaamaaBaaajeaWbaGaamOBaaWcbeaakiaabccacqGH% sislcaqGGaWaaubeaeqajeaWbaGaaGimaaWcbeqdbaGaaeiVdaaaki% aacMcacaqGGaGaae4laiabeccaGiabeo8aZbaa!4C33!\[\mathop T\nolimits_n {\rm{ = }}n^{1/2} (\overline X _n {\rm{ }} - {\rm{ }}\mathop {\rm{\mu }}\nolimits_0 ){\rm{ /}} \sigma \] and k is a weight function. For certain choices of k, % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-qqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xHapdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaubeaeqaje% aWbaGaamOBaaWcbeqdbaGafqiVd0MbaKaaaaGccaqGGaGaaiikaiaa% dUgacaGGPaaaaa!3CEE!\[\mathop {\hat \mu }\nolimits_n {\rm{ }}(k)\] coincides with previously studied preliminary test and shrinkage estimators. We consider choosing k from a natural non-parametric family of weight functions so as to minimize average risk relative to a specified prior p. We study how, by varying p, the MSE efficiency (relative to \-X) properties of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-qqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xHapdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaubeaeqaje% aWbaGaamOBaaWcbeqdbaGafqiVd0MbaKaaaaGccaqGGaGaaiikaiaa% dUgacaGGPaaaaa!3CEE!\[\mathop {\hat \mu }\nolimits_n {\rm{ }}(k)\] can be controlled. In the process, a certain robustness property of the usual family of posterior mean estimators, corresponding to the conjugate normal priors, is observed.     

Investigations on 1-Azabicyclic Compounds. 25. Stereochemistry, Hydrogen Bonds, and Gas-Liquid Chromatography of Pyrrolizidine Alcohols

Data have been summarized of several studies on the separation of isomeric pyrrolizidine alcohols by GLC using liquid polar stationary phases. It was shown that the order of emergence of isomers from the chromatographic column is determined to a significant extent by competition of intermolecular hydrogen bonds formed in the sorbate-sorbent systems and intramolecular hydrogen bonds in the molecules of the same pyrrolizidine alcohols. The preference for one or other type of hydrogen bond depends on the stereochemistry of the pyrrolizidine alcohols. Analysis of the geometric conditions for the formation of intramolecular hydrogen bonds in the investigated compounds in conjunction with chromatographic resolution data enables their configurations to be assigned. The anomalously short retention times of highly strained 5-hydroxyalkyl-3-methylpyrrolizidines are explained by the existence in them of a bicyclic conformation predominantly with a trans linkage and with f avora ble geometric conditions for forming intramolecular hydrogen bonds in them.     

Quadratic Padé approximants and the intruder state problem of multireference perturbation methods

Simple and quadratic Padé resummation methods are applied to high‐order series from multireference many‐body perturbation theory (MR‐MBPT) calculations using various partitioning schemes (Møller–Plesset, Epstein–Nesbet, and forced degeneracy) to determine their efficacy in resumming slowly convergent or divergent series. The calculations are performed for the ground and low‐lying excited states of (i) CH₂, (ii) BeH₂ at three geometries, and (iii) Be, for which full configuration interaction (CI) calculations are available for comparison. The 49 perturbation series that are analyzed include those with oscillatory and monotonic divergence and convergence, including divergences that arise from either frontdoor or backdoor intruder states. Both the simple and quadratic Padé approximations are found to speed the convergence of slowly convergent or divergent series. However, the quadratic Padé method generally outperforms the simple Padé resummation. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005