Pseudo memory effects, majorization and entropy in quantum random walks

A quantum random walk on the integers exhibits pseudo memory effects, in that its probability distribution after N steps is determined by reshuffling the first N distributions that arise in a classical random walk with the same initial distribution. In a classical walk, entropy increase can be regarded as a consequence of the majorization ordering of successive distributions. The Lorenz curves of successive distributions for a symmetric quantum walk reveal no majorization ordering in general. Nevertheless, entropy can increase, and computer experiments show that it does so on average. Varying the stages at which the quantum coin system is traced out leads to new quantum walks, including a symmetric walk for which majorization ordering is valid but the spreading rate exceeds that of the usual symmetric quantum walk.

Teleonomic entropy: measuring the phase-space of end-directed systems

We introduce a novel way of measuring the entropy of a set of values undergoing changes. Such a measure becomes useful when analyzing the temporal development of an algorithm designed to numerically update a collection of values such as artificial neural network weights undergoing adjustments during learning. We measure the entropy as a function of the phase-space of the values, i.e. their magnitude and velocity of change, using a method based on the abstract measure of entropy introduced by the philosopher Rudolf Carnap. By constructing a time-dynamic two-dimensional Voronoi diagram using Voronoi cell generators with coordinates of value- and value-velocity (change of magnitude), the entropy becomes a function of the cell areas. We term this measure teleonomic entropy since it can be used to describe changes in any end-directed (teleonomic) system. The usefulness of the method is illustrated when comparing the different approaches of two search algorithms, a learning artificial neural network and a population of discovering agents. (C) 2004 Elsevier Inc. All rights reserved.

A tutorial on the cross-entropy method

The cross-entropy (CE) method is a new generic approach to combinatorial and multi-extremal optimization and rare event simulation. The purpose of this tutorial is to give a gentle introduction to the CE method. We present the CE methodology, the basic algorithm and its modifications, and discuss applications in combinatorial optimization and machine learning. combinatorial optimization

The cross-entropy method for network reliability estimation

Consider a network of unreliable links, modelling for example a communication network. Estimating the reliability of the network-expressed as the probability that certain nodes in the network are connected-is a computationally difficult task. In this paper we study how the Cross-Entropy method can be used to obtain more efficient network reliability estimation procedures. Three techniques of estimation are considered: Crude Monte Carlo and the more sophisticated Permutation Monte Carlo and Merge Process. We show that the Cross-Entropy method yields a speed-up over all three techniques.

Application of the cross-entropy method to the buffer allocation problem in a simulation-based environment

The buffer allocation problem (BAP) is a well-known difficult problem in the design of production lines. We present a stochastic algorithm for solving the BAP, based on the cross-entropy method, a new paradigm for stochastic optimization. The algorithm involves the following iterative steps: (a) the generation of buffer allocations according to a certain random mechanism, followed by (b) the modification of this mechanism on the basis of cross-entropy minimization. Through various numerical experiments we demonstrate the efficiency of the proposed algorithm and show that the method can quickly generate (near-)optimal buffer allocations for fairly large production lines.

Heavy tails, importance sampling and cross-entropy

We consider the problem of estimating P(Yi + (...) + Y-n > x) by importance sampling when the Yi are i.i.d. and heavy-tailed. The idea is to exploit the cross-entropy method as a toot for choosing good parameters in the importance sampling distribution; in doing so, we use the asymptotic description that given P(Y-1 + (...) + Y-n > x), n - 1 of the Yi have distribution F and one the conditional distribution of Y given Y > x. We show in some specific parametric examples (Pareto and Weibull) how this leads to precise answers which, as demonstrated numerically, are close to being variance minimal within the parametric class under consideration. Related problems for M/G/l and GI/G/l queues are also discussed.

Various topological excitations in the SO(4) gauge field in higher dimensions

Following the original analysis Of Zhang and Hu for the 4-dimensional generalization of Quantum Hall effect, there has been much work from different viewpoints on the higher dimensional condensed matter systems. In this paper, we discuss three kinds of topological excitations in the SO(4) gauge field of condensed matter systems in 4-dimension-the instantons and anti-instantons, the 't Hooft-Polyakov monopoles, and the 2-membranes. Using the phi-mapping topological theory, it is revealed that there are 4-, 3-, and 2-dimensional topological currents inhering in the SO (4) gauge field, and the above three kinds of excitations can be directly and explicitly derived from these three kinds of currents, respectively. Moreover, it is shown that the topological charges of these excitations are characterized by the Hopf indices and Brouwer degrees of phi-mapping. (c) 2005 Elsevier Inc. All rights reserved.

Topological side-chain classification of beta-turns: Ideal motifs for peptidomimetic development

beta-turns are important topological motifs for biological recognition of proteins and peptides. Organic molecules that sample the side chain positions of beta-turns have shown broad binding capacity to multiple different receptors, for example benzodiazepines. beta-turns have traditionally been classified into various types based on the backbone dihedral angles (phi 2, psi 2, phi 3 and psi 3). Indeed, 57-68% of beta-turns are currently classified into 8 different backbone families (Type I, Type II, Type I', Type II', Type VIII, Type VIa1, Type VIa2 and Type VIb and Type IV which represents unclassified beta-turns). Although this classification of beta-turns has been useful, the resulting beta-turn types are not ideal for the design of beta-turn mimetics as they do not reflect topological features of the recognition elements, the side chains. To overcome this, we have extracted beta-turns from a data set of non-homologous and high-resolution protein crystal structures. The side chain positions, as defined by C-alpha-C-beta vectors, of these turns have been clustered using the kth nearest neighbor clustering and filtered nearest centroid sorting algorithms. Nine clusters were obtained that cluster 90% of the data, and the average intra-cluster RMSD of the four C-alpha-C-beta vectors is 0.36. The nine clusters therefore represent the topology of the side chain scaffold architecture of the vast majority of beta-turns. The mean structures of the nine clusters are useful for the development of beta-turn mimetics and as biological descriptors for focusing combinatorial chemistry towards biologically relevant topological space.

Cyclic MrIA: A stable and potent cyclic conotoxin with a novel topological fold that targets the norepinephrine transporter

Conotoxins, disulfide-rich peptides from the venom of cone snails, have created much excitement over recent years due to their potency and specificity for ion channels and their therapeutic potential. One recently identified conotoxin, MrIA, a 13-residue member of the chi-conotoxin family, inhibits the human norepinephrine transporter (NET) and has potential applications in the treatment of pain. In the current study, we show that the, beta-hairpin structure of native MrIA is retained in a synthetic cyclic version, as is biological activity at the NET. Furthermore, the cyclic version has increased resistance to trypsin digestion relative to the native peptide, an intriguing result because the cleavage site for the trypsin is not close to the cyclization site. The use of peptides as drugs is generally hampered by susceptibility to proteolysis, and so, the increase in enzymatic stability against trypsin observed in the current study may be useful in improving the therapeutic potential of MrIA. Furthermore, the structure reported here for cyclic MrIA represents a new topology among a growing number of circular disulfide-rich peptides.

Summarizing level-two topological relations in large spatial datasets

Summarizing topological relations is fundamental to many spatial applications including spatial query optimization. In this article, we present several novel techniques to effectively construct cell density based spatial histograms for range (window) summarizations restricted to the four most important level-two topological relations: contains, contained, overlap, and disjoint. We first present a novel framework to construct a multiscale Euler histogram in 2D space with the guarantee of the exact summarization results for aligned windows in constant time. To minimize the storage space in such a multiscale Euler histogram, an approximate algorithm with the approximate ratio 19/12 is presented, while the problem is shown NP-hard generally. To conform to a limited storage space where a multiscale histogram may be allowed to have only k Euler histograms, an effective algorithm is presented to construct multiscale histograms to achieve high accuracy in approximately summarizing aligned windows. Then, we present a new approximate algorithm to query an Euler histogram that cannot guarantee the exact answers; it runs in constant time. We also investigate the problem of nonaligned windows and the problem of effectively partitioning the data space to support nonaligned window queries. Finally, we extend our techniques to 3D space. Our extensive experiments against both synthetic and real world datasets demonstrate that the approximate multiscale histogram techniques may improve the accuracy of the existing techniques by several orders of magnitude while retaining the cost efficiency, and the exact multiscale histogram technique requires only a storage space linearly proportional to the number of cells for many popular real datasets.

The cross-entropy method for continuous multi-extremal optimization

In recent years, the cross-entropy method has been successfully applied to a wide range of discrete optimization tasks. In this paper we consider the cross-entropy method in the context of continuous optimization. We demonstrate the effectiveness of the cross-entropy method for solving difficult continuous multi-extremal optimization problems, including those with non-linear constraints.

Temperature of a trapped unitary Fermi gas at finite entropy

We present theoretical predictions for the equation of state of a harmonically trapped Fermi gas in the unitary limit. Our calculations compare Monte Carlo results with the equation of state of a uniform gas using three distinct perturbation schemes. We show that in experiments the temperature can be usefully calibrated by making use of the entropy, which is invariant during an adiabatic conversion into the weakly interacting limit of molecular BEC. We predict the entropy dependence of the equation of state.

Entropy generation for microscale forced convection: effects of different thermal boundary conditions, velocity slip, temperature jump, viscous dissipation, and duct geometry

This work presents closed form solutions for fully developed temperature distribution and entropy generation due to forced convection in microelectromechanical systems (MEMS) in the Slip-flow regime, for which the Knudsen number lies within the range 0.001