On the semi-discretization of optimal control problems for networks of elastic strings: global optimality systems and domain decomposition

We consider a star-graph as an examplary network, with elastic strings stretched along the edges. The network is allowed to perform out-of-the plane displacements. We consider such networks as being controlled at its simple nodes via Dirichlet conditions. The objective is to steer given initial data to final target data in a given time T with minimal control costs. This problem is discussed in the continuous as well as in the discrete case. We discuss an iterative domain decomposition technique and its discrete analogue. We prove convergence and show some numerical results.     

Practical and flexible pattern matching over Ziv–Lempel compressed text

We address the problem of string matching on Ziv–Lempel compressed text. The goal is to search for a pattern in a text without uncompressing it. This is a highly relevant issue to keep compressed text databases where efficient searching is still possible. We develop a general technique for string matching when the text comes as a sequence of blocks. This abstracts the essential features of Ziv–Lempel compression. We then apply the scheme to each particular type of compression. We present an algorithm to find all the matches of a pattern in a text compressed using LZ77. When we apply our scheme to LZ78, we obtain a much more efficient search algorithm, which is faster than uncompressing the text and then searching it. Finally, we propose a new hybrid compression scheme which is between LZ77 and LZ78, being in practice as good to compress as LZ77 and as fast to search as LZ78. We show also how to search for some extended patterns on Ziv–Lempel compressed text, such as classes of characters and approximate string matching.     

Cascades of Hopf bifurcations from boundary delay

We consider a 1-dimensional reaction-diffusion equation with nonlinear boundary conditions of logistic type with delay. We deal with non-negative solutions and analyze the stability behavior of its unique positive equilibrium solution, which is given by the constant function u≡1. We show that if the delay is small, this equilibrium solution is asymptotically stable, similar as in the case without delay. We also show that, as the delay goes to infinity, this equilibrium becomes unstable and undergoes a cascade of Hopf bifurcations. The structure of this cascade will depend on the parameters appearing in the equation. This equation shows some dynamical behavior that differs from the case where the nonlinearity with delay is in the interior of the domain.     

Voronoi drawings of trees

We study the problem of characterizing sets of points whose Voronoi diagrams are trees and if so, what are the combinatorial properties of these trees. The second part of the problem can be naturally turned into the following graph drawing question: Given a tree T, can one represent T so that the resulting drawing is a Voronoi diagram of some set of points? We investigate the problem both in the Euclidean and in the Manhattan metric. The major contributions of this paper are as follows.

• We characterize those trees that can be drawn as Voronoi diagrams in the Euclidean metric.

• We characterize those sets of points whose Voronoi diagrams are trees in the Manhattan metric.

• We show that the maximum vertex degree of any tree that can be drawn as a Manhattan Voronoi diagram is at most five and prove that this bound is tight.

• We characterize those binary trees that can be drawn as Manhattan Voronoi diagrams.

Trajectories of lattice particles using the new approach to no integrability

We study two systems which lead to a lattice when an integration path is specified in “aesthetic field theory”. One of these cases involves nonsoliton type particles (magnitudes of maxima and minima oscillate in time). The other system is made up of soliton type particles. The two systems are intrinsically three-dimensional. We speak of the third dimension as “time”. In one of our solutions, the particles move on straight line trajectories, insofar as our numerical work indicates. In the other solution, the soliton type particles undergo what appears to be simple harmonic motion in both the x- and y-directions (loop motion). We then study these two systems using the new approach to integrability which involves a superposition principle and is characterized by a unique change function at each point. We still find multi maxima and minima. The systems are not as symmetric as the lattice. The soliton characteristic is preserved by the new method. We investigated the motion of lattice particles. We found evidence of maxima (minima) regions coalescing so that the location of the maxima (minima) became difficult to follow. The concept of location of particles may not even have a well-defined meaning here. We find examples of soliton particles appearing and disappearing. We conclude that the manner of integration in a no integrability theory can transform a system with well-defined trajectories into a system where particles can no longer be followed in time.     

Entropy of Semiclassical Measures of the Walsh-Quantized Baker’s Map

We study the baker’s map and its Walsh quantization, as a toy model of a quantized chaotic system. We focus on localization properties of eigenstates, in the semiclassical régime. Simple counterexamples show that quantum unique ergodicity fails for this model. We obtain, however, lower bounds on the entropies associated with semiclassical measures, as well as on the Wehrl entropies of eigenstates. The central tool of the proofs is an “entropic uncertainty principle”. Submitted: December 21, 2005; Accepted: March 1, 2006     

On countable connected locally connected almost regular Urysohn spaces

We construct connected, locally connected, almost regular, countable, Urysohn spaces. This answers a problem of G.X. Ritter. We show that there are 2^c such non-homeomorphic spaces. We also show that there are 2^c non-homeomorphic spaces which are further rigid. We discuss the group of homeomorphisms of such spaces.The following question was raised by G.X. Ritter: Does there exist a countable connected locally connected Urysohn space which is almost regular? We answer this question in the affirmative and in fact, show that not only are there as many as 2^c such spaces but that there are just as many rigid spaces with the same properties. Furthermore we show that every countable Urysohn space is a subspace of such a space. We also prove that every countable group is isomorphic to the group of autohomeomorphisms of some connected locally connected almost regular Urysohn space. Examples are given of groups of order c which can be represented in this manner.     

Mixed equilibrium problems with Z*-bifunctions and least element problems in Banach lattices

The purpose of this paper is to study the relations among a mixed equilibrium problem, a least element problem and a minimization problem in Banach lattices. We propose the concept of Z*-bifunctions as well as the concept of a feasible set for the mixed equilibrium problem. We prove that the feasible set of the mixed equilibrium problem is a sublattice provided that the associated bifunction is a strictly α-monotone Z*-bifunction. We establish the equivalence of the mixed equilibrium problem, the least element problem and the minimization problem under strict α-monotonicity and Z*-bifunction conditions.     

The core of zero-dimensional monomial ideals

The core of an ideal is the intersection of all its reductions. We describe the core of a zero-dimensional monomial ideal I as the largest monomial ideal contained in a general reduction of I. This provides a new interpretation of the core in the monomial case as well as an efficient algorithm for computing it. We relate the core to adjoints and first coefficient ideals, and in dimension two and three we give explicit formulas.     

Dynamics of a Discrete Brusselator Model: Escape to Infinity and Julia Set

We consider a discrete version of the Brusselator Model of the famous Belousov-Zhabotinsky reaction in chemistry. The original model is a reaction-diffusion equation and its discrete version is a coupled map lattice. We study the dynamics of the local map, which is a smooth map of the plane. We discuss the set of trajectories that escape to infinity as well as analyze the set of bounded trajectories – the Julia set of the system. The work was partially supported by National Science Foundation grant #DMS-0088971 and U.S.-Mexico Collaborative Research grant 0104675 The article is available online on SpringerLink (www.springerlink.com) using colors instead of greyscales in the pictures. Lecture held in the Seminario Matematico e Fisico on July 1, 2003 Received: March 2005     

Measures of serial extremal dependence and their estimation

The goal of this paper is two-fold: (1) We review classical and recent measures of serial extremal dependence in a strictly stationary time series as well as their estimation. (2) We discuss recent concepts of heavy-tailed time series, including regular variation and max-stable processes.     

Optimizing active ranges for consistent dynamic map labeling

Map labeling encounters unique issues in the context of dynamic maps with continuous zooming and panning—an application with increasing practical importance. In consistent dynamic map labeling, distracting behavior such as popping and jumping is avoided. We use a model for consistent dynamic labeling in which a label is represented by a 3d-solid, with scale as the third dimension. Each solid can be truncated to a single scale interval, called its active range, corresponding to the scales at which the label will be selected. The active range optimization (ARO) problem is to select active ranges so that no two truncated solids intersect and the sum of the heights of the active ranges is maximized. Simple ARO is a variant in which the active ranges are restricted so that a label is never deselected when zooming in. We investigate both the general and simple variants, for 1d- as well as 2d-maps.Different label shapes define different ARO variants. We show that 2d-ARO and general 1d-ARO are NP-complete, even for quite simple shapes. We solve simple 1d-ARO optimally with dynamic programming, and present a toolbox of algorithms that yield constant-factor approximations for a number of 1d- and 2d-variants.     

A new model for gas flow in pipe networks

We introduce a new model for gas dynamics in pipe networks by asymptotic analysis. The model is derived from the isothermal Euler equations. We present the derivation of the model as well as numerical results illustrating the validity and its properties. We compare the new model with existing models from the mathematical and engineering literature. We further give numerical results on a sample network. Copyright © 2009 John Wiley & Sons, Ltd.     

An Algorithm for Global Optimization using the Taylor–Bernstein Form as Inclusion Function

We investigate the use of higher order inclusion functions in the Moore–Skelboe (MS) algorithm of interval analysis (IA) for unconstrained global optimization. We first propose an improvement of the Taylor–Bernstein (TB) form given in (Lin and Rokne (1996) 101) which has the property of higher order convergence. We make the improvement so that the TB form is more effective in practice. We then use the improved TB form as an inclusion function in a prototype MS algorithm and also modify the cut-off test and termination condition in the algorithm. We test and compare on several examples the performances of the proposed algorithm, the MS algorithm, and the MS algorithm with the Taylor model of Berz and Hoffstatter (1998; 97) as inclusion function. The results of these (preliminary) tests indicate that the proposed algorithm with the improved TB form as inclusion function is quite effective for low to medium dimension problems studied.     

Star Product for Second-Class Constraint Systems from a BRST Theory

We propose an explicit construction of the deformation quantization of a general second-class constraint system that is covariant with respect to local coordinates on the phase space. The approach is based on constructing the effective first-class constraint (gauge) system equivalent to the original second-class constraint system and can also be understood as a far-reaching generalization of the Fedosov quantization. The effective gauge system is quantized by the BFV–BRST procedure. The star product for the Dirac bracket is explicitly constructed as the quantum multiplication of BRST observables. We introduce and explicitly construct a Dirac bracket counterpart of the symplectic connection, called the Dirac connection. We identify a particular star product associated with the Dirac connection for which the constraints are in the center of the respective star-commutator algebra. It is shown that when reduced to the constraint surface, this star product is a Fedosov star product on the constraint surface considered as a symplectic manifold.     

Polygon Recognition and Symmetry Detection

We introduce an approach based on moving frames for polygon recognition and symmetry detection. We present detailed algorithms for the recognition of polygons in R² modulo the special Euclidean, Euclidean, equi-affine, skewed-affine, and similarity Lie groups. We also solve the case of polygons in the Poincar\'e half-plane under the action of SL(2) and explain a method applicable to Lie group actions in general. The time complexity of our algorithms is linear in the number of vertices and they are noise resistant. The signatures used allow the detection of partial, as well as approximate, equivalences.     

Holomorphic representation of constant mean curvature surfaces in Minkowski space: Consequences of non-compactness in loop group methods

We give an infinite dimensional generalized Weierstrass representation for spacelike constant mean curvature (CMC) surfaces in Minkowski 3-space R2,1. The formulation is analogous to that given by Dorfmeister, Pedit and Wu for CMC surfaces in Euclidean space, replacing the group SU₂ with SU1,1. The non-compactness of the latter group, however, means that the Iwasawa decomposition of the loop group, used to construct the surfaces, is not global. We prove that it is defined on an open dense subset, after doubling the size of the real form SU1,1, and prove several results concerning the behavior of the surface as the boundary of this open set is encountered. We then use the generalized Weierstrass representation to create and classify new examples of spacelike CMC surfaces in R2,1. In particular, we classify surfaces of revolution and surfaces with screw motion symmetry, as well as studying another class of surfaces for which the metric is rotationally invariant.     

Minimizing the probability of lifetime ruin under borrowing constraints

We determine the optimal investment strategy of an individual who targets a given rate of consumption and who seeks to minimize the probability of going bankrupt before she dies, also known as lifetime ruin. We impose two types of borrowing constraints: First, we do not allow the individual to borrow money to invest in the risky asset nor to sell the risky asset short. However, the latter is not a real restriction because in the unconstrained case, the individual does not sell the risky asset short. Second, we allow the individual to borrow money but only at a rate that is higher than the rate earned on the riskless asset.We consider two forms of the consumption function: (1) The individual consumes at a constant (real) dollar rate, and (2) the individual consumes a constant proportion of her wealth. The first is arguably more realistic, but the second is closely connected with Merton’s model of optimal consumption and investment under power utility. We demonstrate that connection in this paper, as well as include a numerical example to illustrate our results.     

Sinc-based computations of eigenvalues of Dirac systems

The celebrated classical sampling theorem is used to compute approximate values of the eigenvalues of Dirac systems with eigenvalue parameter in the boundary conditions. We deal with problems with an eigenparameter in one or two boundary conditions. The error analysis is established considering both truncation and amplitude errors associated with the sampling theorem. We indicate the role of the amplitude error as well as other parameters in the method via illustrative examples. AMS subject classification (2000)  34L16, 65L15, 94A20