SPREADING OF N DIFFUSING SPECIES WITH DEATH AND BIRTH FEATURES

Fractals ◽  
1996 ◽  
Vol 04 (02) ◽  
pp. 161-168 ◽  
Author(s):  
S. HAVLIN ◽  
A. BUNDE ◽  
H. LARRALDE ◽  
Y. LEREAH ◽  
M. MEYER ◽  
...  
Keyword(s):  
Analytical Solution ◽  
Smooth Surface ◽  
Center Of Mass ◽  
Direct Measure ◽  
Dimensional Lattice ◽  
Random Walker ◽  
Random Walkers ◽  
Large N ◽  
Dimension 2 ◽  
The Mean

The number of distinct sites visited by a random walker after t steps is of great interest, as it provides a direct measure of the territory covered by a diffusing particle. We review the analytical solution to the problem of calculating SN(t), the mean number of distinct sites visited by N random walkers on a d-dimensional lattice, for d=1, 2, 3 in the limit of large N. There are three distinct time regimes for SN(t). A remarkable transition, for dimension ≥2, in the geometry of the set of visited sites is found. This set initially grows as a disk with a relatively smooth surface until it reaches a certain size, after which the surface becomes increasingly rough. We also review the results for a model for migration and spreading of populations and diseases. The model is based on N diffusing species, where each species has a probability α- of dying (or recovery from a disease) and a probability α+ to give birth (or to infect another species). It is found analytically that when α+ ≈ α- ≠ 0, after a crossover time t× ~ N/2α-, the territory covered by the population is localized around its center of mass while the center of mass diffuses regularly. When α+ > α-, the localization breaks down after a second crossover time and the species diffuse and spread around their center of mass. These results may explain the phenomena of migration and spreading of diseases and population appearing in nature.

scholarly journals Record statistics for random walks and Lévy flights with resetting

2021 ◽  
Author(s):  
Satya N Majumdar ◽  
Philippe Mounaix ◽  
Sanjib Sabhapandit ◽  
Gregory Schehr
Keyword(s):  
Time Series ◽  
Continuous Distribution ◽  
Strongly Correlated ◽  
Time Step ◽  
Random Walker ◽  
Large N ◽  
Starting Point ◽  
Exactly Solvable ◽  
Record Statistics ◽  
The Mean

Abstract We compute exactly the mean number of records $\langle R_N \rangle$ for a time-series of size $N$ whose entries represent the positions of a discrete time random walker on the line with resetting. At each time step, the walker jumps by a length $\eta$ drawn independently from a symmetric and continuous distribution $f(\eta)$ with probability $1-r$ (with $0\leq r < 1$) and with the complementary probability $r$ it resets to its starting point $x=0$. This is an exactly solvable example of a weakly correlated time-series that interpolates between a strongly correlated random walk series (for $r=0$) and an uncorrelated time-series (for $(1-r) \ll 1$). Remarkably, we found that for every fixed $r \in [0,1[$ and any $N$, the mean number of records $\langle R_N \rangle$ is completely universal, i.e., independent of the jump distribution $f(\eta)$. In particular, for large $N$, we show that $\langle R_N \rangle$ grows very slowly with increasing $N$ as $\langle R_N \rangle \approx (1/\sqrt{r})\, \ln N$ for $0<r <1$. We also computed the exact universal crossover scaling functions for $\langle R_N \rangle$ in the two limits $r \to 0$ and $r \to 1$. Our analytical predictions are in excellent agreement with numerical simulations.

Zur Lösung der BETHE-GOLDSTONE-Gleichung bei nicht-verschwindendem Gesamtimpuls I

1963 ◽  
Vol 18 (4) ◽  
pp. 531-538
Author(s):  
Dallas T. Hayes
Keyword(s):  
Approximate Solution ◽  
Analytical Solution ◽  
Nuclear Matter ◽  
Analytical Solutions ◽  
Projection Operator ◽  
Center Of Mass ◽  
Separable Potential ◽  
Localized Solutions ◽  
Non Local ◽  
Square Well

Localized solutions of the BETHE—GOLDSTONE equation for two nucleons in nuclear matter are examined as a function of the center-of-mass momentum (c. m. m.) of the two nucleons. The equation depends upon the c. m. m. as parameter due to the dependence upon the c. m. m. of the projection operator appearing in the equation. An analytical solution of the equation is obtained for a non-local but separable potential, whereby a numerical solution is also obtained. An approximate solution for small c. m. m. is calculated for a square-well potential. In the range of the approximation the two analytical solutions agree exactly.

ASYNCHRONOUS UPDATING RESTORES THE LAW OF LARGE NUMBERS IN GLOBALLY COUPLED SYSTEMS

2002 ◽  
Vol 12 (03) ◽  
pp. 663-669 ◽  
Author(s):  
SUDESHNA SINHA
Keyword(s):  
Law Of Large Numbers ◽  
Mean Field ◽  
Coupled Systems ◽  
Large N ◽  
Large Numbers ◽  
Statistical Behavior ◽  
Remarkable Result ◽  
Asynchronous Updating ◽  
The Mean ◽  
The Law

It was observed in earlier studies, that the mean field of globally coupled maps evolving under synchronous updating rules violated the law of large numbers, and this remarkable result generated widespread research interest. In this work we demonstrate that incorporating increasing degrees of asynchronicity in the updating rules rapidly restores the statistical behavior of the mean field. This is clear from the decay of the mean square deviation of the mean field with respect to lattice size N, for varying degrees of asynchronicity, which shows 1/N behavior upto very large N even when the updating is far from fully asynchronous. This is also evidenced through increasing 1/f2 behavior regimes in the power spectrum of the mean field under increasing asynchronicity.

scholarly journals A Reaction-Diffusion Model for Interference in Meiotic Crossing Over

Genetics ◽  
2002 ◽  
Vol 161 (1) ◽  
pp. 365-372 ◽  
Author(s):  
Youhei Fujitani ◽  
Shintaro Mori ◽  
Ichizo Kobayashi
Keyword(s):  
Genetic Distance ◽  
Diffusion Model ◽  
Genetic Model ◽  
Initial Density ◽  
Reaction Diffusion ◽  
Crossover Point ◽  
Random Walker ◽  
Random Walkers ◽  
Reaction Diffusion Model ◽  
Two Parameters

Abstract One crossover point between a pair of homologous chromosomes in meiosis appears to interfere with occurrence of another in the neighborhood. It has been revealed that Drosophila and Neurospora, in spite of their large difference in the frequency of crossover points, show very similar plots of coincidence—a measure of the interference—against the genetic distance of the interval, defined as one-half the average number of crossover points within the interval. We here propose a simple reaction-diffusion model, where a “randomly walking” precursor becomes immobilized and matures into a crossover point. The interference is caused by pair-annihilation of the random walkers due to their collision and by annihilation of a random walker due to its collision with an immobilized point. This model has two parameters—the initial density of the random walkers and the rate of its processing into a crossover point. We show numerically that, as the former increases and/or the latter decreases, plotted curves of the coincidence vs. the genetic distance converge on a unique curve. Thus, our model explains the similarity between Drosophila and Neurospora without parameter values adjusted finely, although it is not a “genetic model” but is a “physical model,” specifying explicitly what happens physically.

scholarly journals Entanglement cost of generalised measurements

2003 ◽  
Vol 3 (5) ◽  
pp. 405-422
Author(s):  
R. Jozsa ◽  
M. Koashi ◽  
N. Linden ◽  
S. Popescu ◽  
S. Presnell ◽  
...  
Keyword(s):  
Quantum Measurements ◽  
Great Circle ◽  
Bloch Sphere ◽  
Numerical Techniques ◽  
Bipartite Entanglement ◽  
Von Neumann ◽  
Tensor Product Space ◽  
Large N ◽  
Dimension 2 ◽  
The Cost

Bipartite entanglement is one of the fundamental quantifiable resources of quantum information theory. We propose a new application of this resource to the theory of quantum measurements. According to Naimark's theorem any rank 1 generalised measurement (POVM) M may be represented as a von Neumann measurement in an extended (tensor product) space of the system plus ancilla. By considering a suitable average of the entanglements of these measurement directions and minimising over all Naimark extensions, we define a notion of entanglement cost E_{\min}(M) of M. We give a constructive means of characterising all Naimark extensions of a given POVM. We identify various classes of POVMs with zero and non-zero cost and explicitly characterise all POVMs in 2 dimensions having zero cost. We prove a constant upper bound on the entanglement cost of any POVM in any dimension. Hence the asymptotic entanglement cost (i.e. the large n limit of the cost of n applications of M, divided by n) is zero for all POVMs. The trine measurement is defined by three rank 1 elements, with directions symmetrically placed around a great circle on the Bloch sphere. We give an analytic expression for its entanglement cost. Defining a normalised cost of any $d$-dimensional POVM by E_{\min} (M)/\log_2 d, we show (using a combination of analytic and numerical techniques) that the trine measurement is more costly than any other POVM with d>2, or with d=2 and ancilla dimension 2. This strongly suggests that the trine measurement is the most costly of all POVMs.

PORTFOLIO MODEL UNDER FRACTAL MARKET BASED ON MEAN-DCCA

Fractals ◽  
2020 ◽  
Vol 28 (07) ◽  
pp. 2050142
Author(s):  
WEIDE CHUN ◽  
HESEN LI ◽  
XU WU
Keyword(s):  
Analytical Solution ◽  
Correlation Analysis ◽  
Capital Market ◽  
Cross Correlation ◽  
Investment Portfolio ◽  
Cross Correlation Analysis ◽  
Risk Criterion ◽  
The Mean ◽  
Mean Variance Portfolio ◽  
Mean Variance

Under the realistic background that the capital market nowadays is a fractal market, this paper embeds the detrended cross-correlation analysis (DCCA) into the return-risk criterion to construct a Mean-DCCA portfolio model, and gives an analytical solution. Based on this, the validity of Mean-DCCA portfolio model is verified by empirical analysis. Compared to the mean-variance portfolio model, the Mean-DCCA portfolio model is more conducive for investors to build a sophisticated investment portfolio under multi-time-scale, improve the performance of portfolios, and overcome the defect that the mean-variance portfolio model has not considered the existence of fractal correlation characteristics between assets.

scholarly journals Investigating the Influence of Shaft Balance Point on Clubhead Speed: A Simulation Study

Proceedings ◽  
2020 ◽  
Vol 49 (1) ◽  
pp. 156
Author(s):  
William McNally ◽  
John McPhee
Keyword(s):  
Total Mass ◽  
Center Of Mass ◽  
Golf Club ◽  
Physical Interaction ◽  
Balance Point ◽  
Mass Distributions ◽  
Biomechanical Models ◽  
Significant Difference ◽  
The Mean ◽  
Simulation Results

In this study, a dynamic golfer model was used to investigate the influence of the golf shaft’s balance point (i.e., center of mass) on the generation of clubhead speed. Three hypothetical shaft designs having different mass distributions, but the same total mass and stiffness, were proposed. The golfer model was then stochastically optimized 100 times using each shaft. A statistically significant difference was found between the mean clubhead speeds at impact (p < 0.001), where the clubhead speed increased as the balance point moved closer to the grip. When comparing the two shafts with the largest difference in balance point, a 1.6% increase in mean clubhead speed was observed for a change in balance point of 18.8 cm. The simulation results have implications for shaft design and demonstrate the usefulness of biomechanical models for capturing the complex physical interaction between the golfer and golf club.

scholarly journals Estimation of leg stiffness using an approximation to the planar spring–mass system in high-speed running

2020 ◽  
Vol 17 (1) ◽  
pp. 172988141989071
Author(s):  
Wei Guo ◽  
Changrong Cai ◽  
Mantian Li ◽  
Fusheng Zha ◽  
Pengfei Wang ◽  
...  
Keyword(s):  
Analytical Solution ◽  
High Speed ◽  
Stance Phase ◽  
Center Of Mass ◽  
Critical Role ◽  
Line Center ◽  
Motion Model ◽  
Mass System ◽  
Leg Stiffness ◽  
Wide Range

Leg stiffness plays a critical role in legged robots’ speed regulation. However, the analytic solutions to the differential equations of the stance phase do not exist, of course not for the exact analytical solution of stiffness. In view of the challenge in dealing with every circumstance by numerical methods, which have been adopted to tabulate approximate answers, the “harmonic motion model” was used as approximation of the stance phase. However, the wide range leg sweep angles and small fluctuations of the “center of mass” in fast movement were overlooked. In this article, we raise a “triangle motion model” with uniform forward speed, symmetric movement, and straight-line center of mass trajectory. The characters are then shifted to a quadratic equation by Taylor expansion and obtain an approximate analytical solution. Both the numerical simulation and ADAMS-Matlab co-simulation of the control system show the accuracy of the triangle motion model method in predicting leg stiffness even in the ultra-high-speed case, and it is also adaptable to low-speed cases. The study illuminates the relationship between leg stiffness and speed, and the approximation model of the planar spring–mass system may serve as an analytical tool for leg stiffness estimation in high-speed locomotion.

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