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.     

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.     

Interpolation of characteristic classes of singular hypersurfaces

We show that the Chern–Schwartz–MacPherson class of a hypersurface X in a nonsingular variety M ‘interpolates’ between two other notions of characteristic classes for singular varieties, provided that the singular locus of X is smooth and that certain numerical invariants of X are constant along this locus. This allows us to define a lift of the Chern–Schwartz–MacPherson class of such ‘nice’ hypersurfaces to intersection homology. As another application, the interpolation result leads to an explicit formula for the Chern–Schwartz–MacPherson class of X in terms of its polar classes.     

On dynamically non-trivial three-valued logics: oscillatory and bifurcatory species

We interpret three-valued logical systems as pools of abstract chemical species, logical variables, where chemical reactions are catalyzed by logical connectives; then we study global- and space-time dynamics of these ‘logical’ chemical reactors. We develop a family of combinatorial systems {T,F,*},, where T, F and * are truth-values, is commutative binary operator, acts as a Boolean conjunction on {T,F};**=*, T*=a and F*=b, a,b{T,F,*}. We consider nine combinatorial systems of the family, specified by values of a and b, and derive from each member ab of the family an artificial chemical system, where interactions between reactants T, F and * are governed by . Computational experiments with well-stirred reactors show that all systems but T* and *T exhibit a dull behaviour and converge to their only stable points. In reactor of the system T*, catalyzed by , concentrations of reactants oscillate while reactor of the system *T finishes its evolution in one of two stable points; thus we call the models of T* and *T oscillatory and bifurcatory systems. We enrich the systems with negation connective, define additional connectives via ¬ and and undertake a detailed study of integral dynamic of the artificial stirred chemical reactors and of space-time dynamic of thin-layer non-stirred chemical reactors derived from the logical connectives. The thin-layer reactors show rich space-time dynamic ranging from breathing patterns and mobile localizations to fractal structures. A primitive hierarchy of connectives’ phenomenological complexity is constructed.     

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.     

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.     

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.     

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.     

Component-by-Component Construction of Good Lattice Rules with a Composite Number of Points

We develop algorithms to construct rank-1 lattice rules in weighted Korobov spaces of periodic functions and shifted rank-1 lattice rules in weighted Sobolev spaces of non-periodic functions. Analyses are given which show that the rules so constructed achieve strong QMC tractability error bounds. Unlike earlier analyses, there is no assumption that n, the number of quadrature points, be a prime number. However, we do assume that there is an upper bound on the number of distinct prime factors of n. The generating vectors and shifts characterizing the rules are constructed ‘component-by-component,’ that is, the (d+1)th components of the generating vectors and shifts are obtained using one-dimensional searches, with the previous d components kept unchanged.     

Quantum repulsive Nonlinear Schrödinger models and their ‘Superconductivity’

The fundamental role played by the quantum repulsive Nonlinear Schrödinger (NLS) equation in the evolution of our understanding of the phenomenon of superconductivity in appropriate metals at very low temperatures is surveyed. The first major work was that in 1947 by N. N. Bogoliubov, who studied the very physical 3-space-dimensions problem and super fluidity; and the survey takes the form of an actual dedication to that outstanding scientist who died four years ago. The 3-space-dimensions NLS equation is not integrable either classically or quantum mechanically. But a number of recently discovered closely related lattices in one space dimension (one space plus one time dimension) are integrable as both classical lattices and quantum lattices while their continuum limits are the now well-known fundamental and integrable system the quantum ‘Bose gas’. These models are all examined in this paper in a physical application of recent so-called ‘quantum groups’ theory, itself fundamental to integrability theory. The ‘superfluid’ phase transitions shown by these lattices, as well as by the bose gas, all at zero temperature in 1 + 1 dimensions, are analysed in terms of the behaviour of certain lattice correlation functions which are either quantum or, in the case of the so-called XY-model, classical correlation functions. Although the repulsive NLS models in 1 + 1 are integrable, they do not have actual soliton solutions. Nevertheless the material as surveyed here is a fundamental application of soliton-theory in the broader context of integrability or near-integrability which has had profound effects in the evolution of current understandings in all of modern theoretical physics.     

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     

Families of m-convex polygons:

Polygons are described as almost-convex if their perimeter differs from the perimeter of their minimum bounding rectangle by twice their ‘concavity index’, m. Such polygons are called m-convex polygons. We first use the inclusion–exclusion principle to rederive the known generating function for 1-convex self-avoiding polygons (SAPs). We then use our results to derive the exact anisotropic generating functions for osculating and neighbour-avoiding 1-convex SAPs, their isotropic form having recently been conjectured.     

Conveniently Hölder Homomorphisms are Smooth in the Convenient Sense

We show that every ‘conveniently Hölder’ homomorphism between Lie groups in the sense of convenient differential calculus is smooth (in the convenient sense). In particular, every ℓip⁰-homomorphism is smooth.     

Multi-segmental representations and approximation of set-valued functions with 1D images

In this work univariate set-valued functions (SVFs, multifunctions) with 1D compact sets as images are considered. For such a continuous SFV of bounded variation (CBV multifunction), we show that the boundaries of its graph are continuous, and inherit the continuity properties of the SVF. Based on these results we introduce a special class of representations of CBV multifunctions with a finite number of ‘holes’ in their graphs. Each such representation is a finite union of SVFs with compact convex images having boundaries with continuity properties as those of the represented SVF. With the help of these representations, positive linear operators are adapted to SVFs. For specific positive approximation operators error estimates are obtained in terms of the continuity properties of the approximated multifunction.     

Chaos control and synchronization for a special generalized Lorenz canonical system – The SM system

This paper presents some simple feedback control laws to study global stabilization and global synchronization for a special chaotic system described in the generalized Lorenz canonical form (GLCF) when τ = −1 (which, for convenience, we call Shimizu–Morioka system, or simply SM system). For an arbitrarily given equilibrium point, a simple feedback controller is designed to globally, exponentially stabilize the system, and reach globally exponent synchronization for two such systems. Based on the system’s coefficients and the structure of the system, simple feedback control laws and corresponding Lyapunov functions are constructed. Because all conditions are obtained explicitly in terms of algebraic expressions, they are easy to be implemented and applied to real problems. Numerical simulation results are presented to verify the theoretical predictions.     

Monotone second-best optimal contracts

We review the two stage nonlinear programming approach to the principal-agent problem proposed by Grossman and Hart (G-H) in 1983. The objective of the principal is to determine an optimal payment schedule to an agent that depends only on observable outcomes when (random) outcomes are influenced by unobservable actions of the agent. We establish that these ‘second-best’ optimal contracts are monotonically relted to observable outputs under fairly mild conditions. Two methods of proof are employed to establish that the monotone likelihood ratio condition (MLRC) on the probabilities of outcomes is sufficient for monotonicity of second-best optimal contracts in a three-state model. The first uses ideas related to the variation-diminishing property of totally positive matrices. The second approach uses a theorem of the alternative from linear programming theory to establish a condition that must hold under MLRC. The paper concludes with a counter-example that is a perturbed version of an example in G-H that shows that MLRC is not sufficient for monotonicity of second-best optimal contracts when there are four or more states.     

Linguistic modelling and information coarsening based on prototype theory and label semantics

The theory of prototypes provides a new semantic interpretation of vague concepts. In particular, the calculus derived from this interpretation results in the same calculus as label semantics proposed by Lawry. In the theory of prototypes, each basic linguistic label L has the form ‘about P’, where P is a set of prototypes of L and the neighborhood size of the underlying concept is described by the word ‘about’ which represents a probability density function δ on [0,+∞). In this paper we propose an approach to vague information coarsening based on the theory of prototypes. Moreover, we propose a framework for linguistic modelling within the theory of prototypes, in which the rules are concise and transparent. We then present a linguistic rule induction method from training data based on information coarsening and data clustering. Finally, we apply this linguistic modelling method to some benchmark time series prediction problems, which show that our linguistic modelling and information coarsening methods are potentially powerful tools for linguistic modelling and uncertain reasoning.     

Point Evaluation and Hardy Space on a Homogeneous Tree

We consider stationary multiscale systems as defined by Basseville, Benveniste, Nikoukhah and Willsky. We show that there are deep analogies with the discrete time non stationary setting as developed by the first author, Dewilde and Dym. Following these analogies we define a point evaluation with values in a C*–algebra and the corresponding “Hardy space” in which Cauchy’s formula holds. This point evaluation is used to define in this context the counterpart of classical notions such as Blaschke factors.     

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.