Non-parametric efficiency, progress and regress measures for panel data: Methodological aspects

This purely methodological paper deals with the rôle of time in non-parametric efficiency analysis. Using both FDH and DEA technologies, it first shows how each observation in a panel can be characterized in efficiency terms vis-à-vis three different kinds of frontiers: (i) ‘contemporaneous’, (ii) ‘sequential’, and (iii) ‘intertemporal’. These are then compared with window analysis. Next, frontier shifts ‘outward’ and ‘inward’, interpreted as progress or regress are considered for the two kinds of technologies, and computational methods are described in detail for evaluating such shifts in either case. These are also contrasted with what is measured by the ‘Malmquist’ productivity index. Finally, an alternative way of identifying progress and regress, independent of the frontier notion and referring instead to some ‘benchmark’ notion, is extended here to panel data.     

Characterizing properties of approximate solutions for optimization problems

Approximate solutions for optimization problems become of interest if the ‘true’ optimum cannot be found: this may happen for the simple reason that an optimum does not exist or because of the ‘bounded rationality’ (or bounded accuracy) of the optimizer. This paper characterizes several approximate solutions by means of consistency and additional requirements. In particular we consider invariance properties. We prove that, where the domain contains optimization problems without maximum, there is no non-trivial consistent solution satisfying non-emptiness, translation and multiplication invariance. Moreover, we show that the class of ‘satisficing’ solutions is obtained, if the invariance axioms are replaced with Chernoff’s Choice Axiom.     

Fractal pattern growth simulation in electrodeposition and study of the shifting of center of mass

We presented simulation of fractal pattern in electrodeposition (Diffusion limited aggregation) using concept of off lattice walk.It is seen that the growth patterns are based on a parameter called ‘bias’. This parameter ‘bias’ controls the growth of patterns similar to that of electric field in electrodeposition technique. In present study the fractal patterns are grown for different values of ‘bias’. Dendritic patterns grown at lower value of ‘bias’ comprises open structure and show limited branching. As the bias is increased the growth tends to be dense and show more crowded branching. Box counting was implemented to calculate fractal dimension. The structural and textural complexities and are compared with the experimental observations.It was also noted that in the evolution of DLA patterns, the center of mass of the growth is shifted slightly. We tracked the position of the center of mass of simulated electro deposits under different electric field conditions. The center of mass exhibit random walk like patterns and it wanders around the origin or the starting point of the growth.     

On the structure of the optimal server control for fluid networks

This paper derives the optimal trajectories in a general fluid network with server control. The stationary optimal policy in the complete state space is constructed. The optimal policy is constant on polyhedral convex cones. An algorithm is derived that computes these cones and the optimal policy. Generalized Klimov indices are introduced, they are used for characterizing myopic and time-uniformly optimal policies.Received: November 2004 / Revised: February 2005The research of this author has been supported by the project ‘‘Stochastic Networks’’ of the Netherlands Organisation for Scientific Research NWO.     

Specialization, inequality and the social stability of economies with collective property rights

Many, if not most, economies are observed to exhibit some form of collective ownership with some goods. These economies may fail to be socially stable, in the sense that the economy has an empty core; some groups may have an incentive to ‘recontract out’. We show that, for a class of economies with collective ownership, sufficient ‘specialization’ in the endowment holdings of these economies gives rise to a non-empty core, so achieving social stability. It is shown, moreover, that reductions in income inequality are consistent with social stability, to the extent that these reductions preserve or increase ‘specialization’ in the economy. Finally, we show that our notion of specialization is not limited to the privatized sector of the economy. Even in economies in which there is no privately held property, sufficient specialization guarantees that the economy is socially stable.     

Exact equations for smoothed Wigner transforms and homogenization of wave propagation

The Wigner transform (WT) has been extensively used in the formulation of phase-space models for a variety of wave propagation problems including high-frequency limits, nonlinear and random waves. It is well known that the WT features counterintuitive ‘interference terms,’ which often make computation impractical. In this connection, we propose the use of the smoothed Wigner transform (SWT), and derive new, exact equations for it, covering a broad class of wave propagation problems. Equations for spectrograms are included as a special case. The ‘taming’ of the interference terms by the SWT is illustrated, and an asymptotic model for the Schrödinger equation is constructed and numerically verified.     

Linear homeomorphisms of non-classical Hilbert spaces

Let K be a complete infinite rank valued field. In [4] we studied Norm Hilbert Spaces (NHS) over K i.e. K-Banach spaces for which closed subspaces admit projections of norm ≤ 1. In this paper we prove the following striking properties of continuous linear operators on NHS. Surjective endomorphisms are bijective, no NHS is linearly homeomorphic to a proper subspace (Theorem 3.7), each operator can be approximated, uniformly on bounded sets, by finite rank operators (Theorem 3.8). These properties together — in real or complex theory shared only by finite-dimensional spaces — show that NHS are more ‘rigid’ than classical Hilbert spaces.     

Developing a platform for comparison of hospital admission systems: An illustration

There is an increasing need to develop a platform for comparing hospital admission planning systems due to a shift in the service paradigm in the health sector. The current service concept of hospital admission planning aims at optimising the use of scarce hospital resources without paying much attention to the level of service offered to patients. As patients nowadays do not accept long waiting times for hospital admission, it becomes necessary to consider alternative admission service concepts. Waiting lists have also become a political issue, and alternative concepts have been advocated such as giving all patients an appointment for admission. A simulation model was built to examine the impacts of extreme admission service concepts in a simplified hospital setting. The alternative concepts considered are based on the ‘zero waiting time’ principle (immediate treatment), and the ‘booked admissions’ principle (using an appointment for admission). The results of these admission service concepts are compared with the results of the current concept, based on the ‘maximising resource use’ principle. The paper deals with the development of a framework and tool that allows evaluating different, somehow conflicting, hospital admission planning concepts and the usefulness of such framework and tool for more refined/real-life approaches to hospital admission planning.     

A correction to the definition of local center

It is known that in order to solve the minimax facility location problem on a graph with a finite set of demand points, only a finite set of possible location points, called ‘local centers’ must be considered.It has been shown that the continuous m-center problem on a graph can be solved by using a series of set covering problems in which each local center covers the demand points at a distance not greater than a corresponding number called ‘the range’ of the local center.However, all points which are at the same distance from more than two demand points, and from which there is no direction where all these distances are decreasing, must also be considered as local centers. This paper proves that, in some special cases, it is not sufficient to consider only the points where this occurs with respect to pairs of demand points. The definition of local center is corrected and the corresponding results and algorithms are revised.     

An arithmetic proof of John’s ellipsoid theorem

Using an idea of Voronoi in the geometric theory of positive definite quadratic forms, we give a transparent proof of John’s characterization of the unique ellipsoid of maximum volume contained in a convex body. The same idea applies to the ‘hard part’ of a generalization of John’s theorem and shows the difficulties of the corresponding ‘easy part’.Received: 8 October 2004     

Some existence and construction results of polygonal designs

This paper revisits the existence and construction problems for polygonal designs (a special class of partially balanced incomplete block designs associated with regular polygons). We present new polygonal designs with various parameter sets by explicit construction. In doing so we employ several construction methods — some conventional and some new. We also establish a link between a class of polygonal designs of block size 3 and the cyclically generated ‘λ-fold triple systems’. Finally, we show that the existence question for a certain class of polygonal designs is equivalent to the existence question for ‘perfect grouping systems’ which we introduce.     

Estimation of a mean in the presence of an extreme observation

The performance of Anscombe, semi-Winsorization and Winsorization (A, S and W) rules for dealing with extreme observations are investigated for observations from N(μ, σ²) and the simple case where it is assumed that at most one observation in the sample may be biased, arising from N(μ + aσ, σ²) and the primary objective is to estimate μ when σ is unknown. Each of these rules is separately treated in terms of the estimated standard deviation, range and interquartile range. A Monte Carlo method is used to evaluate certain expectation integrals that arise in the computations. We give the results for sample sizes n = 6, 8, 10, 12, 14, 16, 20, 30, 40, 50, 60, 80, 100 of determining the constants necessary to give ‘premiums’ of 0.01 and 0.05 for each of the rules. The performance of the rules is measured in terms of ‘protection’. Features of the resulting tables are discussed.     

Classification of a family of symmetric graphs with complete 2-arc-transitive quotients

In this paper we give a classification of a family of symmetric graphs with complete 2-arc-transitive quotients. Of particular interest are two subfamilies of graphs which admit an arc-transitive action of a projective linear group. The graphs in these subfamilies can be defined in terms of the cross ratio of certain 4-tuples of elements of a finite projective line, and thus may be called the second type ‘cross ratio graphs’, which are different from the ‘cross ratio graphs’ studied in [A. Gardiner, C. E. Praeger, S. Zhou, Cross-ratio graphs, J. London Math. Soc. (2) 64 (2001), 257–272]. We also give a combinatorial characterisation of such second type cross ratio graphs.     

Higher order differentiability properties of the composition and of the inversion operator

This paper contains theorems of r-th order Fréchet differentiability, with r≥1, for the autonomous composition operator and for the inversion operator in Schauder spaces. The optimality of the differentiability theorems for the composition is indicated by means of an ‘inverse result’. A main point of this paper is that (higher order) ‘sharp’ differentiability theorems for the composition operator can be proved by approximating the operator by composition operators whose superposing function is a polynomial, an idea which may be employed in other function space settings.     

Cyclic permutations and evolutionary trees

Given a tree with leaf set X, there are certain ways of arranging the elements of X in a circular order so that can be embedded in the plane and ‘preserve’ this ordering. We investigate some new combinatorial properties of these ‘circular orderings.’ We then use these properties to establish two results concerning dissimilarity maps on X that are induced by edge-weighted trees with leaf set X.     

Binary relations as single primitive notions for hyperbolic three-space and the inversive plane

By interpreting J.A. Lester's [9] result on inversive-distance-preserving mappings as an axiomatizability statement, and by using the Liebmann isomorphism between the inversive plane and hyperbolic three-space, we point out that hyperbolic three-spaces (and inversive geometry) coordinatized by Euclidean fields can be axiomatized with planes (or circles) as variables, by using only the plane-orthogonality (or circle-orthogonality) predicate _p (or _c), or by using only the predicate δ′ (or δ), where δ′(p,p′) (or δ(A, B)) is interpreted as ‘the distance between the planes p and p′ is equal to the length of the segment s whose angle of parallelism is (i. e. II(s) = )’ (or as ‘the numerical distance between the disjoint circles A and B has the value , which corresponds to s via Liebmann's isomorphism’).     

Generalized p-Center problems: Complexity results and approximation algorithms

In an earlier paper, two alternative p-Center problems, where the centers serving costumers must be chosen so that exactly one node from each of p prespecified disjoint pairs of nodes is selected, were shown to be NP-complete. This paper considers a generalized version of these problems, in which the nodes from which the p servers are to be selected are partitioned into k sets and the number of servers selected from each set must be within a prespecified range. We refer to these problems as the ‘Set’ p-Center problems. We establish that the triangle inequality (Δ-inequality) versions of these problems, in which the edge weights are assumed to satisfy the triangle inequality, are also NP-complete. We also provide a polynomial time approximation algorithm for the two Δ-inequality Set p-Center problems that is optimal for one of the problems in the sense that no algorithm with polynomial running time can provide a better constant factor performance guarantee, unless P = NP. For the special case ‘alternative’ p-Center problems, which we refer to as the ‘Pair’ p-Center problems, we extend the previous results in several ways. For example, the results mentioned above for the Set p-Center problems also apply to the Pair p-Center problems. Furthermore, we establish and exploit a correspondence between satisfiability and the dominating set type of problems that naturally arise when considering the decision versions of the Pair p-Center problems.     

Path-dependent processes and the emergence of macro-structure

Path-dependent systems of the ‘autocatalytic’ or self-reinforcing type typically possess a multiplicity of possible asymptotic outcomes or structures, with early random fluctuations determining which structure is ‘selected’.We explore a wide class of such systems, which we call non-linear Polya systems, where increments to proportions or concentrations occur with probabilities that are non-linear functions of present proportions or concentrations. We show that such systems converge to outcomes (proportions or concentrations) that are represented by the stable fixed points of these functions. These limit theorems are strong laws of large numbers for path-dependent increments, and as such they generalize the standard Borel strong law for independent increments. They are powerful and easy to use.We show applications in chemical kinetics, industrial location theory and in the emergence of technological structure in the economy.     

Projective Completions of Jordan Pairs,Part II: Manifold Structures and Symmetric Spaces

We define symmetric spaces in arbitrary dimension and over arbitrary non-discrete topological fields , and we construct manifolds and symmetric spaces associated to topological continuous quasi-inverse Jordan pairs and -triple systems. This class of spaces, called smooth generalized projective geometries, generalizes the well-known (finite or infinite-dimensional) bounded symmetric domains as well as their ‘compact-like’ duals. An interpretation of such geometries as models of Quantum Mechanics is proposed, and particular attention is paid to geometries that might be considered as ‘standard models’ – they are associated to associative continuous inverse algebras and to Jordan algebras of hermitian elements in such an algebra.Mathematics Subject Classiffications (2000). primary: 17C36, 46H70, 17C65; secondary: 17C30, 17C90