Distribution of the angle of rotation plane random walks

We study the probability distribution of the angle by which the tangent to the trajectory rotates in the course of a plane random walk. It is shown that the determination of this distribution function can be reduced to an integral equation, which can be rigorously transformed into a differential equation of Hill's type. We derive the asymptotic distribution for very long walks.

Path integral description of polymers using fractional Brownian walks

The statistical properties of fractional Brownian walks are used to construct a path integral representation of the conformations of polymers with different degrees of bond correlation. We specifically derive an expression for the distribution function of the chains’ end‐to‐end distance, and evaluate it by several independent methods, including direct evaluation of the discrete limit of the path integral, decomposition into normal modes, and solution of a partial differential equation. The distribution function is found to be Gaussian in the spatial coordinates of the monomer positions, as in the random walk description of the chain, but the contour variables, which specify the location of the monomer along the chain backbone, now depend on an index h, the degree of correlation of the fractional Brownian walk. The special case of h=1/2 corresponds to the random walk. In constructing the normal mode picture of the chain, we conjecture the existence of a theorem regarding the zeros of the Bessel function.

Dynamics of Fractional Brownian Walks

We investigate the dynamics of polymers whose solution configurations are represented by fractional Brownian walks. The calculation of the two dynamical quantities considered here, the longest relaxation time tau(r) and the intrinsic viscosity [eta], is formulated in terms of Langevin equations and is carried out within the continuum approach developed in an earlier paper. Our results for tau(r) and [eta] reproduce known scaling relations and provide reasonable numerical estimates of scaling amplitudes. The possible relevance of the work to the study of globular proteins and other compact polymeric phases is discussed.

Quantum Random Walks without Coin Toss

We construct a quantum random walk algorithm, based on the Dirac operator instead of the Laplacian. The algorithm explores multiple evolutionary branches by superposition of states, and does not require the coin toss instruction of classical randomised algorithms. We use this algorithm to search for a marked vertex on a hypercubic lattice in arbitrary dimensions. Our numerical and analytical results match the scaling behaviour of earlier algorithms that use a coin toss instruction.

Cascaded walks in protein sequence space: use of artificial sequences in remote homology detection between natural proteins

Over the past two decades, many ingenious efforts have been made in protein remote homology detection. Because homologous proteins often diversify extensively in sequence, it is challenging to demonstrate such relatedness through entirely sequence-driven searches. Here, we describe a computational method for the generation of `protein-like' sequences that serves to bridge gaps in protein sequence space. Sequence profile information, as embodied in a position-specific scoring matrix of multiply aligned sequences of bona fide family members, serves as the starting point in this algorithm. The observed amino acid propensity and the selection of a random number dictate the selection of a residue for each position in the sequence. In a systematic manner, and by applying a `roulette-wheel' selection approach at each position, we generate parent family-like sequences and thus facilitate an enlargement of sequence space around the family. When generated for a large number of families, we demonstrate that they expand the utility of natural intermediately related sequences in linking distant proteins. In 91% of the assessed examples, inclusion of designed sequences improved fold coverage by 5-10% over searches made in their absence. Furthermore, with several examples from proteins adopting folds such as TIM, globin, lipocalin and others, we demonstrate that the success of including designed sequences in a database positively sensitized methods such as PSI-BLAST and Cascade PSI-BLAST and is a promising opportunity for enormously improved remote homology recognition using sequence information alone.

Search on a hypercubic lattice using a quantum random walk. I. d > 2

Random walks describe diffusion processes, where movement at every time step is restricted to only the neighboring locations. We construct a quantum random walk algorithm, based on discretization of the Dirac evolution operator inspired by staggered lattice fermions. We use it to investigate the spatial search problem, that is, to find a marked vertex on a d-dimensional hypercubic lattice. The restriction on movement hardly matters for d > 2, and scaling behavior close to Grover's optimal algorithm (which has no restriction on movement) can be achieved. Using numerical simulations, we optimize the proportionality constants of the scaling behavior, and demonstrate the approach to that for Grover's algorithm (equivalent to the mean-field theory or the d -> infinity limit). In particular, the scaling behavior for d = 3 is only about 25% higher than the optimal d -> infinity value.

Determinant: Old algorithms, new insights (Extended Abstract)

In this paper we approach the problem of computing the characteristic polynomial of a matrix from the combinatorial viewpoint. We present several combinatorial characterizations of the coefficients of the characteristic polynomial, in terms of walks and closed walks of different kinds in the underlying graph. We develop algorithms based on these characterizations, and show that they tally with well-known algorithms arrived at independently from considerations in linear algebra.

Search on a fractal lattice using a quantum random walk

The spatial search problem on regular lattice structures in integer number of dimensions d >= 2 has been studied extensively, using both coined and coinless quantum walks. The relativistic Dirac operator has been a crucial ingredient in these studies. Here, we investigate the spatial search problem on fractals of noninteger dimensions. Although the Dirac operator cannot be defined on a fractal, we construct the quantum walk on a fractal using the flip-flop operator that incorporates a Klein-Gordon mode. We find that the scaling behavior of the spatial search is determined by the spectral (and not the fractal) dimension. Our numerical results have been obtained on the well-known Sierpinski gaskets in two and three dimensions.

Measurement Based Impromptu Deployment of a Multi-Hop Wireless Relay Network

We study the problem of optimal sequential (''as-you-go'') deployment of wireless relay nodes, as a person walks along a line of random length (with a known distribution). The objective is to create an impromptu multihop wireless network for connecting a packet source to be placed at the end of the line with a sink node located at the starting point, to operate in the light traffic regime. In walking from the sink towards the source, at every step, measurements yield the transmit powers required to establish links to one or more previously placed nodes. Based on these measurements, at every step, a decision is made to place a relay node, the overall system objective being to minimize a linear combination of the expected sum power (or the expected maximum power) required to deliver a packet from the source to the sink node and the expected number of relay nodes deployed. For each of these two objectives, two different relay selection strategies are considered: (i) each relay communicates with the sink via its immediate previous relay, (ii) the communication path can skip some of the deployed relays. With appropriate modeling assumptions, we formulate each of these problems as a Markov decision process (MDP). We provide the optimal policy structures for all these cases, and provide illustrations of the policies and their performance, via numerical results, for some typical parameters.

Optimal sequential wireless relay placement on a random lattice path

Our work is motivated by impromptu (or ``as-you-go'') deployment of wireless relay nodes along a path, a need that arises in many situations. In this paper, the path is modeled as starting at the origin (where there is the data sink, e.g., the control center), and evolving randomly over a lattice in the positive quadrant. A person walks along the path deploying relay nodes as he goes. At each step, the path can, randomly, either continue in the same direction or take a turn, or come to an end, at which point a data source (e.g., a sensor) has to be placed, that will send packets to the data sink. A decision has to be made at each step whether or not to place a wireless relay node. Assuming that the packet generation rate by the source is very low, and simple link-by-link scheduling, we consider the problem of sequential relay placement so as to minimize the expectation of an end-to-end cost metric (a linear combination of the sum of convex hop costs and the number of relays placed). This impromptu relay placement problem is formulated as a total cost Markov decision process. First, we derive the optimal policy in terms of an optimal placement set and show that this set is characterized by a boundary (with respect to the position of the last placed relay) beyond which it is optimal to place the next relay. Next, based on a simpler one-step-look-ahead characterization of the optimal policy, we propose an algorithm which is proved to converge to the optimal placement set in a finite number of steps and which is faster than value iteration. We show by simulations that the distance threshold based heuristic, usually assumed in the literature, is close to the optimal, provided that the threshold distance is carefully chosen. (C) 2014 Elsevier B.V. All rights reserved.

Distribution, Demography, and Conservation of Lion-tailed Macaques (Macaca silenus) in the Anamalai Hills Landscape, Western Ghats, India

The status of the endemic and endangered lion-tailed macaque (Macaca silenus) has not been properly assessed in several regions of the Western Ghats of southern India. We conducted a study in Parambikulam Forest Reserve in the state of Kerala to determine the distribution, demography, and status of lion-tailed macaques. We laid 5km(2) grid cells on the map of the study area (644km(2)) and made four replicated walks in each grid cell using GPS. We gathered data on lion-tailed macaque group locations, demography, and site covariates including trail length, duration of walk, proportion of evergreen forest, height of tallest trees, and human disturbance index. We also performed occupancy modeling using PRESENCE ver. 3.0. We estimated a minimum of 17 groups of macaques in these hills. Low detection and occupancy probabilities indicated a low density of lion-tailed macaques in the study area. Height of the tallest trees correlated positively whereas human disturbance and proportion of evergreen forest correlated negatively with occupancy in grid cells. We also used data from earlier studies carried out in the surrounding Anamalai Tiger Reserve and Nelliyampathy Hills to discuss the conservation status in the large Anamalai Hills Landscape. This landscape harbors an estimated population of 1108 individuals of lion-tailed macaques, which is about one third of the entire estimated wild population of this species. A conservation plan for this landscape could be used as a model for conservation in other regions of the Western Ghats.

Markov chains, R-trivial monoids and representation theory

We develop a general theory of Markov chains realizable as random walks on R-trivial monoids. It provides explicit and simple formulas for the eigenvalues of the transition matrix, for multiplicities of the eigenvalues via Mobius inversion along a lattice, a condition for diagonalizability of the transition matrix and some techniques for bounding the mixing time. In addition, we discuss several examples, such as Toom-Tsetlin models, an exchange walk for finite Coxeter groups, as well as examples previously studied by the authors, such as nonabelian sandpile models and the promotion Markov chain on posets. Many of these examples can be viewed as random walks on quotients of free tree monoids, a new class of monoids whose combinatorics we develop.

As-You-Go Deployment of a 2-Connected Wireless Relay Network for Sensor-Sink Interconnection

A person walks along a line (which could be an idealisation of a forest trail, for example), placing relays as he walks, in order to create a multihop network for connecting a sensor at a point along the line to a sink at the start of the line. The potential placement points are equally spaced along the line, and at each such location the decision to place or not to place a relay is based on link quality measurements to the previously placed relays. The location of the sensor is unknown apriori, and is discovered as the deployment agent walks. In this paper, we extend our earlier work on this class of problems to include the objective of achieving a 2-connected multihop network. We propose a network cost objective that is additive over the deployed relays, and accounts for possible alternate routing over the multiple available paths. As in our earlier work, the problem is formulated as a Markov decision process. Placement algorithms are obtained for two source location models, which yield a discounted cost MDP and an average cost MDP. In each case we obtain structural results for an optimal policy, and perform a numerical study that provides insights into the advantages and disadvantages of multi-connectivity. We validate the results obtained from numerical study experimentally in a forest-like environment.