Number Of Walks Research Articles

Multi-walker discrete time quantum walks on arbitrary graphs, their properties and their photonic implementation

Quantum walks have emerged as an interesting alternative to the usual circuit model for quantum computing. While still universal for quantum computing, the quantum walk model has very different physical requirements, which lends itself more naturally to some physical implementations, such as linear optics. Numerous authors have considered walks with one or two walkers, on one-dimensional graphs, and several experimental demonstrations have been performed. In this paper, we discuss generalizing the model of discrete time quantum walks to the case of an arbitrary number of walkers acting on arbitrary graph structures. We present a formalism that allows for the analysis of such situations, and several example scenarios for how our techniques can be applied. We consider the most important features of quantum walks—measurement, distinguishability, characterization and the distinction between classical and quantum interference. We also discuss the potential for physical implementation in the context of linear optics, which is of relevance to present-day experiments.

“Safety in Numbers” re-examined: Can we make valid or practical inferences from available evidence?

“Safety in Numbers”(SIN), a recent concept in transportation research, policy and planning, has emerged as a causal inference from the non-linear statistical association between estimates of the numbers of walkers or bicyclists in an area and the rate or number of traffic collisions experienced by pedestrians or cyclists. Proponents of SIN argue that greater numbers of walkers or cyclists modify the hazardous behaviors of motor vehicle drivers thus creating safer conditions. This paper critically examines the research on the non-linear association as an adequate empirical basis for this causal interpretation. Given the paucity of evidence supporting a specific mechanism for the SIN effect, alternative plausible explanations of the non-linear association behind SIN, and a potential for unintended consequences from its policy application, the authors call for caution in the use of SIN in transportation policy and planning dialogue and decision-making.

Model of inter-spine dynamics of PSD-95 molecules and its application to synaptic plasticity

We present a model of stochastic molecular transport in spiny dendrites. In this model the molecules perform a random walk between the spines that trap the walkers. If the molecules interact with each other inside the spines the trapping time in each spine depends on the number of molecules in the respective trap. The corresponding mathematical problem has non-trivial solutions even in the absence of external disorder due to self-organization phenomenon. We obtain the stationary distributions of the number of walkers in the traps for different kinds of on-site interactions between the walkers. We analyze how birth and death processes of the random walkers affect these distributions. We apply this model to describe the dynamics of the PSD-95 proteins in spiny dendrites. PSD-95 is the most abundant molecule in the post-synaptic density (PSD) located in the spines. It is observed that these molecules have high turnover rates and that neighboring spines are constantly exchanging individual molecules. We propose that the geometry of individual PSD-95 clusters determines the dependence of trapping times on the number of molecules inside the trap and thus can vary from spine to spine. Furthermore, we suggest that activity-dependent reorganization of the PSD-95 cluster can lead to synaptic plasticity in a form of long-term potentiation (LTP). In the model this is achieved by spine specific activity-dependent ubiquitinization of PSD-95 molecules, which transiently reduces the amount of PSD-95 in the spine but change the geometry of the PSD-95 cluster in such a way that self-organization process results in the overall increase of the number of PSD-95 molecules associated with LTP. We also show that such a dynamics of the PSD-95 molecules can set up the conditions for anomalous diffusive transport inside spiny dendrites and predict the distribution of the PSD sizes which has features of both exponential and Poisson distributions.

Maximal Fermi walk configurations on the directed square lattice and standard Young tableaux

We consider configurations of n walkers each of which starts at the origin of a directed square lattice and makes the same number t of steps from node to node along the edges of the lattice. Bose walkers are not allowed to cross, but can share edges. Fermi walk configurations must satisfy the additional constraint that no two walkers traverse the same path. Since, for given t, there are only a finite number of t-step paths, there is a limit nmax on the number of walkers allowed by the Fermi condition. The value of nmax is determined for six types of boundary conditions. The number of Fermi configurations of nmax walkers is also determined using a bijection to standard Young tableaux. In four cases there is no constraint on the endpoints of the walks and the relevant tableaux are shifted.

Communications: Survival of the fittest: Accelerating convergence in full configuration-interaction quantum Monte Carlo

We provide a very simple adaptation of our recently published quantum Monte Carlo algorithm in full configuration-interaction (Slater determinant) spaces which dramatically reduces the number of walkers required to achieve convergence. A survival criterion is imposed for newly spawned walkers. We define a set of initiator determinants such that progeny of walkers spawned from such determinants onto unoccupied determinants are able to survive, while the progeny of walkers not in this set can survive only if they are spawned onto determinants which are already occupied. The set of initiators is originally defined to be all determinants constructable from a subset of orbitals, in analogy with complete-active spaces. This set is dynamically updated so that if a noninitiator determinant reaches an occupation larger than a preset limit, it becomes an initiator. The new algorithm allows sign-coherent sampling of the FCI space to be achieved with relatively few walkers. Using the N(2) molecule as an illustration, we show that rather small initiator spaces and numbers of walkers can converge with submilliHartree accuracy to the known full configuration-interaction (FCI) energy (in the cc-pVDZ basis), in both the equilibrium geometry and the multiconfigurational stretched case. We use the same method to compute the energy with cc-pVTZ and cc-pVQZ basis sets, the latter having an FCI space of over 10(15) with very modest computational resources.

Projector Monte Carlo method based on Slater determinants: Test application to singlet excited states of H 2O and LiF

A projector Monte Carlo method based on Slater determinants (PMC-SD) is expanded to excited state calculations. Target excited states are calculated state-by-state by eliminating the components of the lower states from the imaginary-time propagator. Test calculations are performed for the singlet excited states of H 2O and LiF. As the calculations of H 2O show, the accuracy of the PMC-SD method is improved systematically by increasing the number of walkers. The full-CI energies are obtainable as a limit for a given basis set. The avoided crossing of covalent and ionic states in the dissociation of LiF is well reproduced using the PMC-SD method.

Exact encounter times for many random walkers on regular and complex networks

The exact mean time between encounters of a given particle in a system consisting of many particles undergoing random walks in discrete time is calculated, on both regular and complex networks. Analytical results are obtained both for independent walkers, where any number of walkers can occupy the same site, and for walkers with an exclusion interaction, when no site can contain more than one walker. These analytical results are then compared with numerical simulations, showing very good agreement.

Unintended consequences of expense ratio guidelines: The Avon breast cancer walks

This case study examines how donor and rating agency focus on percentage-based expense ratios exacerbated pressures on the Avon Product Foundation’s breast cancer walks. Beginning in 2002, Avon changed its business and accounting practices in ways that eventually helped it report better compliance with charity monitor guidelines. However, the number of walkers and amounts of funds raised dropped; the new accounting practices are less transparent and of questionable conformity with GAAP.

Characterizing the human mobility pattern in a large street network

Previous studies demonstrated empirically that human mobility exhibits Lévy flight behavior. However, our knowledge of the mechanisms governing this Lévy flight behavior remains limited. Here we analyze over 72,000 people's moving trajectories, obtained from 50 taxicabs during a six-month period in a large street network, and illustrate that the human mobility pattern, or the Lévy flight behavior, is mainly attributed to the underlying street network. In other words, the goal-directed nature of human movement has little effect on the overall traffic distribution. We further simulate the mobility of a large number of random walkers and find that (1) the simulated random walkers can reproduce the same human mobility pattern, and (2) the simulated mobility rate of the random walkers correlates pretty well (an R square up to 0.87) with the observed human mobility rate.

Fermion Monte Carlo without fixed nodes: A game of life, death, and annihilation in Slater determinant space

We have developed a new quantum Monte Carlo method for the simulation of correlated many-electron systems in full configuration-interaction (Slater determinant) spaces. The new method is a population dynamics of a set of walkers, and is designed to simulate the underlying imaginary-time Schrödinger equation of the interacting Hamiltonian. The walkers (which carry a positive or negative sign) inhabit Slater determinant space, and evolve according to a simple set of rules which include spawning, death and annihilation processes. We show that this method is capable of converging onto the full configuration-interaction (FCI) energy and wave function of the problem, without any a priori information regarding the nodal structure of the wave function being provided. Walker annihilation is shown to play a key role. The pattern of walker growth exhibits a characteristic plateau once a critical (system-dependent) number of walkers has been reached. At this point, the correlation energy can be measured using two independent methods--a projection formula and a energy shift; agreement between these provides a strong measure of confidence in the accuracy of the computed correlation energies. We have verified the method by performing calculations on systems for which FCI calculations already exist. In addition, we report on a number of new systems, including CO, O(2), CH(4), and NaH--with FCI spaces ranging from 10(9) to 10(14), whose FCI energies we compute using modest computational resources.

조명환경구역 구분을 위한 도심상업지역 보행자 도로의 조도분포 실태조사

본 연구의 목적은 도심상업지역의 조도실태를 조사하여, CIE 조명환경구역 규정이 국내에 적용 가능한가를 검토하는 것이다. 현장조사를 통하여 도심상업지역 보행자 도로의 수평면 조도와 수직면 조도를 측정하였고, 조명환경에 대한 보행자의 만족도를 조사하기 위하여 설문을 실시하였다. 조사가 이루어진 장소는 모두 4곳으로 보행자 도로의 조명상태와 야간의 보행자 수가 각각 다른 곳이었다. 조도측정 결과 인구유동 및 조명상태에 따라 조도값이 차이를 보이고 있었으며, 설문조사 결과 보행자도로의 조도가 보행자의 조명환경 만족도에 영향을 미치는 것으로 나타났다. 마지막으로 CIE 조명환경구역의 조도 규정은 국내 도심상업지역에 적용할 수 있는 것으로 나타났다.

Norm-conserving diffusion Monte Carlo method and diagrammatic expansion of interacting Drude oscillators: Application to solid xenon

The quantum Drude oscillator (QDO) model, which allows many-body polarization and dispersion to be treated both on an equal footing and beyond the dipole limit, is investigated using two approaches to the linear scaling diffusion Monte Carlo (DMC) technique. The first is a general purpose norm-conserving DMC (NC-DMC) method wherein the number of walkers, $N$, remains strictly constant thereby avoiding the sudden death or explosive growth of walker populations with an error that vanishes as $\mathcal{O}({N}^{\ensuremath{-}1})$ in the absence of weights. As NC-DMC satisfies detailed balance, a phase space can be defined that permits both an exact trajectory weighting and a fast mean-field trajectory weighting technique to be constructed which can eliminate or reduce the population bias, respectively. The second is a many-body diagrammatic expansion for trial wave functions in systems dominated by strong on-site harmonic coupling and a dense matrix of bilinear coupling constants such as the QDO in the dipole limit; an approximate trial function is introduced to treat two-body interactions outside the dipole limit. Using these approaches, high accuracy is achieved in studies of the fcc-solid phase of the highly polarizable atom, xenon, within the QDO model. It is found that 200 walkers suffice to generate converged results for systems as large as 500 atoms. The quality of QDO predictions compared to experiment and the ability to generate these predictions efficiently demonstrate the feasibility of employing the QDO approach to model long-range forces.

Form factors, plane partitions, and random walks

An exactly solvable boson model, the so-called “phase model,” is considered. A relation between certain transition matrix elements of this model and boxed plane partitions, three-dimensional Young diagrams placed into a box of finite size, is established, It is shown that the natural model describing the behavior of friendly walkers, ones that can share the same lattice sites, is the “phase model.” An expression for the number of all admissible nests of lattice paths made by a fixed number of friendly walkers for a certain number of steps is obtained. Bibliography 35 titles.

Low dimensional description of pedestrian-induced oscillation of the Millennium Bridge

When it opened to pedestrian traffic in the year 2000, London’s Millennium Bridge exhibited an unwanted, large side-to-side oscillation, which was apparently due to a resonance between the stepping frequency of walkers and one of the bridge modes. Models for this event, and similar events on other bridges, have been proposed. The model most directly addressing the synchronization mechanism of individual walkers and the resulting global response of the bridge-pedestrian system is the one developed by Eckhardt et al. [Phys. Rev. E 75, 021110 (2007)]. This model treats individual walkers with a phase oscillator description and is inherently high dimensional with system dimensionality (N+2), where N is the number of walkers. In the present work we use a method proposed by Ott and Antonsen [Chaos 18, 037113 (2008)] to reduce the model of Eckhardt et al. to a low dimensional dynamical system, and we employ this reduced description to study the global dynamics of the bridge/pedestrian interaction. More generally, this treatment serves as an interesting example of the possibility of low dimensional macroscopic behavior in large systems of coupled oscillators.

Sistema robotizado para la reeducación de la marcha en pacientes con lesión medular incompleta

A spinal cord injury involves the loss or alteration of motor patterns in walking, the recovery of which depends partly on the rearrangement of the preserved neural circuits. AIM. To evaluate the changes that take place in the gait of patients with incomplete spinal cord injuries who were treated with a robotic walking system in association with conventional therapy. The study conducted was an open-label, prospective, descriptive trial with statistical inference in patients with C2-L3 spinal cord injuries that were classified as degrees C and D according to the American Spinal Injury Association (ASIA) scale. The variables that were analysed on the first and the last day of the study were: number of walkers, 10-m gait test, the Walking Index for Spinal Cord Injury scale revision, technical aids, muscle balance in the lower limbs, locomotor subscale of the measure of functional independence, modified Ashworth scale for spasticity and the visual analogue scale for pain. At the end, data were recorded from the impression of change scale. The analysis was conducted by means of Student's t, chi squared and Pearson's correlation; p < or = 0.05. Forty-five patients, with a mean age of 44 +/- 14.3 years, finished the study; 76% were males, injury was caused by trauma in 58% of cases, and the time of progression was 139 +/- 70 days. Statistically significant increases were observed in the number of subjects capable of walking, walking speed, less need for technical aids, strength in the lower limbs and independence in activities of daily living. Treatment using the robotic system in association with conventional therapy improves walking capacity in patients with incomplete spinal cord injuries.

Analysis of pedestrian dynamics in counter flow via an extended lattice gas model

The modeling of human behavior is an important approach to reproduce realistic phenomena for pedestrian flow. In this paper, an extended lattice gas model is proposed to simulate pedestrian counter flow under the open boundary conditions by considering the human subconscious behavior and different maximum velocities. The simulation results show that the presented model can capture some essential features of pedestrian counter flows, such as lane formation, segregation effect, and phase separation at higher densities. In particular, an interesting feature that the faster walkers overtake the slower ones and then form a narrow-sparse walkway near the central partition line is discovered. The phase diagram comparison and analysis show that the subconscious behavior plays a key role in reducing the occurrence of jam cluster. The effects of the symmetrical and asymmetrical injection rate, different partition lines, and different combinations of maximum velocities on pedestrian flow are investigated. An important conclusion is that it is needless to separate faster and slower pedestrians in the same direction by a partition line. Furthermore, the increase of the number of faster walkers does not always benefit the counter flow in all situations. It depends on the magnitude and asymmetry of injection rate. And at larger maximum velocity, the obtained critical transition point corresponding to the maximum flow rate of the fundamental diagram is in good agreement with the empirical results.

Projector Monte Carlo method based on configuration state functions. Test applications to the H 4 system and dissociation of LiH

A projector Monte Carlo method based on configuration state functions (PMC-CSF) is developed, which does not require the nodes of trial wave functions. The accuracy of the PMC-CSF method is systematically improved by increasing the number of walkers. As demonstrated by test applications to the H 4 and LiH systems, full-CI energies are obtainable as a limit for a given basis set. In addition, the PMC-CSF method is suitable for parallel calculations.

Analysis models for unguided search in unstructured P2P networks

Random walk and flooding are basic mechanisms for searching unstructured overlays. This paper shows that node coverage is an important metric for query performance in random graph based P2P networks. We then present two analytical models: the algebraic model and the combinatorial model. These models are useful in setting query parameters and evaluating search efficiency. We evaluate these two models in terms of various performance metrics against simulation results. We also study the impact of the setup parameters, such as node degree, hop count, and number of walkers, on the query performance, and on the precision of the models.

Reef walking on Red Sea reef flats – Quantifying impacts and identifying motives

Reef walking on the shallow reef flats of fringing reefs is an activity commonly observed on Red Sea coral reefs. Of the 137 in-water observations of reef walkers made at Ras Um Sidd, Sharm El Sheikh, South Sinai, 12.4% caused coral damage. Eighteen incidents of coral damage were recorded, 44.4% being breakages of branching corals, 39.8% abrasions of various growth forms and 16.7% detachments of soft coral colonies. Sediment was stirred-up on 31.4% of observations. Tramplers broke 0.1 branches per minute and raised 1.5 sediment clouds, thereby causing more damage than reported from divers. Damage caused depended on coral community structure and past trampling history, with communities dominated by fragile branching corals being most susceptible to mechanical damage and communities heavily trampled in the past being less susceptible. Given that main motives for reef walking included crossing the reef flat, adjusting gear or communicating, management interventions, such as the construction of a walkway combined with a public awareness campaign, could reduce the numbers of reef walkers by up to 39%.

The Capacitance of Pristine Ice Crystals and Aggregate Snowflakes

Abstract A new method of accurately calculating the capacitance of realistic ice particles is described: such values are key to accurate estimates of deposition and evaporation (sublimation) rates in numerical weather models. The trajectories of diffusing water molecules are directly sampled, using random “walkers.” By counting how many of these trajectories intersect the surface of the ice particle (which may be any shape) and how many escape outside a spherical boundary far from the particle, the capacitances of a number of model ice particle habits have been estimated, including hexagonal columns and plates, “scalene” columns and plates, bullets, bullet rosettes, dendrites, and realistic aggregate snowflakes. For ice particles with sharp edges and corners this method is an efficient and straightforward way of solving Laplace’s equation for the capacitance. Provided that a large enough number of random walkers are used to sample the particle geometry (∼104) the authors expect the calculated capacitances to be accurate to within ∼1%. The capacitance for the modeled aggregate snowflakes (C/Dmax = 0.25, normalized by the maximum dimension Dmax) is shown to be in close agreement with recent aircraft measurements of snowflake sublimation rates. This result shows that the capacitance of a sphere (C/Dmax = 0.5), which is commonly used in numerical models, overestimates the evaporation rate of snowflakes by a factor of 2. The effect of vapor “screening” by crystals growing in the vicinity of one another has also been investigated. The results clearly show that neighboring crystals growing on a filament in cloud chamber experiments can strongly constrict the vapor supply to one another, and the resulting growth rate measurements may severely underestimate the rate for a single crystal in isolation (by a factor of 3 in this model setup).