A genetic algorithm for the generalised assignment problem

A new algorithm for the generalised assignment problem is described in this paper. The algorithm is adapted from a genetic algorithm which has been successfully used on set covering problems, but instead of genetically improving a set of feasible solutions it tries to genetically restore feasibility to a set of near-optimal ones. Thus it may be regarded as operating in a dual sense to the more familiar genetic approach. The algorithm has been tested on generalised assignment problems of substantial size and compared to an exact integer programming approach and a well-established heuristic approach.     

A simple dual algorithm for the generalised assignment problem

A new algorithm for the generalised assignment problem is described in this paper. The dual-type algorithm uses a simple heuristic derived from a relaxation of the problem. The algorithm has been tested on generalised assignment problems of substantial size and compared to an exact integer programming approach and a well-established heuristic approach. Computational results look promising in terms of speed and solution quality.     

A generalised ratio identity

An identity is derived relating the distribution of a continuous time, skip-free process with two impenetrable barriers to the distribution of a process with one barrier. It is indicated how the methods used here can also be used to give results for related processes with different types of barriers. The identity is then used to derive explicit results giving the distribution of the process with two impenetrable barriers.     

A remark on generalised Putnam-Fuglede theorems

Given , the algebra of operators on a Hilbert space , define and by and . Let and be two classes of operators strictly larger than the class of normal operators. Define (resp., if (resp., for all and . This note shows that the equivalence holds for a number of the commonly considered classes of operators.

     

A constructive generalised Goursat normal form

We provide necessary and sufficient conditions on the derived type of a vector field distribution V in order that it be locally equivalent to a partial prolongation of the contact distribution , on the 1st order jet bundle of maps from R to Rq, q?1. This result fully generalises the Goursat normal form from the theory of exterior differential systems. Our proof is constructive: it is proven that if V is locally equivalent to a partial prolongation of then the explicit construction of contact coordinates algorithmically depends upon the integration of a sequence of geometrically defined and algorithmically determined integrable Pfaffian systems on M. Though the tools required are rather different, our main theorem may be regarded as a generalisation of the work of R.B. Gardner and W.F. Shadwick on the feedback linearisation of autonomous nonlinear control systems.     

Extending the multivariate generalised and generalised distributions

The GGH family of multivariate distributions is obtained by scale mixing on the Exponential Power distribution using the Extended Generalised Inverse Gaussian distribution. The resulting GGH family encompasses the multivariate generalised hyperbolic (GH), which itself contains the multivariate t and multivariate Variance-Gamma (VG) distributions as special cases. It also contains the generalised multivariate t distribution [O. Arslan, Family of multivariate generalised t distribution, Journal of Multivariate Analysis 89 (2004) 329–337] and a new generalisation of the VG as special cases. Our approach unifies into a single GH-type family the hitherto separately treated t-type [O. Arslan, A new class of multivariate distribution: Scale mixture of Kotz-type distributions, Statistics and Probability Letters 75 (2005) 18–28; O. Arslan, Variance–mean mixture of Kotz-type distributions, Communications in Statistics-Theory and Methods 38 (2009) 272–284] and VG-type cases. The GGH distribution is dual to the distribution obtained by analogous mixing on the scale parameter of a spherically symmetric stable distribution. Duality between the multivariate t and multivariate VG [S.W. Harrar, E. Seneta, A.K. Gupta, Duality between matrix variate t and matrix variate V.G. distributions, Journal of Multivariate Analysis 97 (2006) 1467–1475] does however extend in one sense to their generalisations.     

A generalised life-expectancy model for a population

In a recent paper, it was shown that the life-expectancy behaviour for homogeneous and non-homogeneous population differs markedly for moderately high and low ages. One therefore has two curves for each of these regions and the problem is how to construct a single life-expectancy curve which blends these two curves together in a satisfactory way. This note offers a simple mathematical and analytical model which is useful in estimating an acceptable life expectancy under all severities of life situation. A simple numerical example is given to illustrate the application of the method.     

A generalised Skolem-Mahler-Lech theorem for affine varieties

We give a corrected and strengthened statement and proof ofthe ‘p-adic analytic arc lemma’ in a paper of theauthor (J. London Math. Soc. (2) 73 (2006) 367–379). Weshow that the analytic arc is guaranteed to exist for p 5 andgive a counterexample showing that this sometimes cannot bedone when p = 2. Footnotes 2000 Mathematics Subject Classification 11D45 (primary), 14R10,11D88 (secondary). Received September 16, 2007; revised January 22, 2008; published online March 30, 2008.     

The {\cal {D Q R}} algorithm, basic theory, convergence, and conditional stability

Summary. We describe a fast matrix eigenvalue algorithm that uses a matrix factorization and reverse order multiply technique involving three factors and that is based on the symmetric matrix factorization as well as on –orthogonal reduction techniques where is computed from the given matrix . It operates on a similarity reduction of a real matrix to general tridiagonal form and computes all of 's eigenvalues in operations, where the part of the operations is possibly performed over , instead of the 7–8 real flops required by the eigenvalue algorithm. Potential breakdo wn of the algorithm can occur in the reduction to tridiagonal form and in the –orthogonal reductions. Both, however, can be monitored during the computations. The former occurs rather rarely for dimensions and can essentially be bypassed, while the latter is extremely rare and can be bypassed as well in our conditionally stable implementation of the steps. We prove an implicit theorem which allows implicit shifts, give a convergence proof for the algorithm and show that is conditionally stable for general balanced tridiagonal matrices . Received April 25, 1995 / Revised version received February 9, 1996     

A hemisystem of a nonclassical generalised quadrangle

The concept of a hemisystem of a generalised quadrangle has its roots in the work of B. Segre, and this term is used here to denote a set of points such that every line ℓ meets in half of the points of ℓ. If one takes the point-line geometry on the points of the hemisystem, then one obtains a partial quadrangle and hence a strongly regular point graph. The only previously known hemisystems of generalised quadrangles of order (q, q ²) were those of the elliptic quadric , q odd. We show in this paper that there exists a hemisystem of the Fisher–Thas–Walker–Kantor generalised quadrangle of order (5, 5²), which leads to a new partial quadrangle. Moreover, we can construct from our hemisystem the 3· A ₇-hemisystem of , first constructed by Cossidente and Penttila.      

Factorisation of generalised theta functions. I

We prove a version of factorisation, relating the space of sections of theta bundles on the moduli spaces of (parabolic, rank 2) vector bundles on curves of genusg andg–1.Oblatum 23-I-1992 & 21-VI-1993     

A generalised inductive approach to the lace expansion

 The lace expansion is a powerful tool for analysing the critical behaviour of self-avoiding walks and percolation. It gives rise to a recursion relation which we abstract and study using an adaptation of the inductive method introduced by den Hollander and the authors. We give conditions under which the solution to the recursion relation behaves as a Gaussian, both in Fourier space and in terms of a local central limit theorem. These conditions are shown elsewhere to hold for sufficiently spread-out models of networks of self-avoiding walks in dimensions d > 4, and for sufficiently spread-out models of critical oriented percolation in dimensions d + 1 > 5, providing a unified approach and an essential ingredient for a detailed analysis of the branching behaviour of these models. Received: 13 September 2000 / Revised version: 16 May 2001 / Published online: 20 December 2001     

A sum labelling for the generalised friendship graph

We provide an optimal sum labelling scheme for the generalised friendship graph, also known as the flower (a symmetric collection of cycles meeting at a common vertex) and show that its sum number is 2.     

A note on Schrage's generalised variable upper bounds

An improved procedure for implementing pivoting, based upon ideas from generalised upper bounds, is suggested for Schrage's generalised variable upper bounds.