The resistance of randomly grown trees

Copyright @ 2011 IOP Publishing Ltd. This is a preprint version of the published article which can be accessed from the link below.An electrical network with the structure of a random tree is considered: starting from a root vertex, in one iteration each leaf (a vertex with zero or one adjacent edges) of the tree is extended by either a single edge with probability p or two edges with probability 1 − p. With each edge having a resistance equal to 1 omega, the total resistance Rn between the root vertex and a busbar connecting all the vertices at the nth level is considered. A dynamical system is presented which approximates Rn, it is shown that the mean value (Rn) for this system approaches (1 + p)/(1 − p) as n → ∞, the distribution of Rn at large n is also examined. Additionally, a random sequence construction akin to a random Fibonacci sequence is used to approximate Rn; this sequence is shown to be related to the Legendre polynomials and its mean is shown to converge with |(Rn) − (1 + p)/(1 − p)| ∼ n−1/2.Engineering and Physical Sciences Research Council (EPSRC

Reliable Computation of the Zeros of Solutions of Second Order Linear ODEs Using a Fourth Order Method

A fourth order fixed point method to compute the zeros of solutions of second order homogeneous linear ODEs is obtained from the approximate integration of the Riccati equation associated with the ODE. The method requires the evaluation of the logarithmic derivative of the function and also uses the coefficients of the ODE. An algorithm to compute with certainty all the zeros in an interval is given which provides a fast, reliable, and accurate method of computation. The method is illustrated by the computation of the zeros of Gauss hypergeometric functions (including Jacobi polynomials) and confluent hypergeometric functions (Laguerre polynomials, Hermite polynomials, and Bessel functions included) among others. The examples show that typically 4 or 5 iterations per root are enough to provide more than 100 digits of accuracy, without requiring a priori estimations of the roots

Random Geometric Series

Integer sequences where each element is determined by a previous randomly chosen element are investigated analytically. In particular, the random geometric series x_n=2x_p with 0<=p<=n-1 is studied. At large n, the moments grow algebraically, n^beta(s) with beta(s)=2^s-1, while the typical behavior is x_n n^ln 2. The probability distribution is obtained explicitly in terms of the Stirling numbers of the first kind and it approaches a log-normal distribution asymptotically.Comment: 6 pages, 2 figure

Canadians Should Travel Randomly

We study online algorithms for the Canadian Traveller Problem (CTP) introduced by Papadimitriou and Yannakakis in 1991. In this problem, a traveller knows the entire road network in advance, and wishes to travel as quickly as possible from a source vertex s to a destination vertex t, but discovers online that some roads are blocked (e.g., by snow) once reaching them. It is PSPACE-complete to achieve a bounded competitive ratio for this problem. Furthermore, if at most k roads can be blocked, then the optimal competitive ratio for a deterministic online algorithm is 2k + 1, while the only randomized result known is a lower bound of k + 1. In this paper, we show for the first time that a polynomial time randomized algorithm can beat the best deterministic algorithms, surpassing the 2k + 1 lower bound by an o(1) factor. Moreover, we prove the randomized algorithm achieving a competitive ratio of (1 + [√2 over 2])k + 1 in pseudo-polynomial time. The proposed techniques can also be applied to implicitly represent multiple near-shortest s-t paths.NSC Grant 102-2221-E-007-075-MY3Japan Society for the Promotion of Science (KAKENHI 23240002

Greedy Solution of Ill-Posed Problems: Error Bounds and Exact Inversion

The orthogonal matching pursuit (OMP) is an algorithm to solve sparse approximation problems. Sufficient conditions for exact recovery are known with and without noise. In this paper we investigate the applicability of the OMP for the solution of ill-posed inverse problems in general and in particular for two deconvolution examples from mass spectrometry and digital holography respectively. In sparse approximation problems one often has to deal with the problem of redundancy of a dictionary, i.e. the atoms are not linearly independent. However, one expects them to be approximatively orthogonal and this is quantified by the so-called incoherence. This idea cannot be transfered to ill-posed inverse problems since here the atoms are typically far from orthogonal: The ill-posedness of the operator causes that the correlation of two distinct atoms probably gets huge, i.e. that two atoms can look much alike. Therefore one needs conditions which take the structure of the problem into account and work without the concept of coherence. In this paper we develop results for exact recovery of the support of noisy signals. In the two examples in mass spectrometry and digital holography we show that our results lead to practically relevant estimates such that one may check a priori if the experimental setup guarantees exact deconvolution with OMP. Especially in the example from digital holography our analysis may be regarded as a first step to calculate the resolution power of droplet holography

ИССЛЕДОВАНИЕ ПУТЕЙ ОПТИМИЗАЦИИ МАТЕРИАЛОВ ДЛЯ СОЛНЕЧНЫХ БАТАРЕЙ ВТОРОГО ПОКОЛЕНИЯ МЕТОДОМ МНОГОУРОВНЕВОГО МОДЕЛИРОВАНИЯ

Theoretical analysis of optimization options for the properties of CdTe absorber layer is an important task for increasing the efficiency of CdTe/CdS heterojunction based thin−film solar cells. Properties of the materials (e.g. the density of free carriers) often depend essentially on the parameters of the deposition process and subsequent treatment which determine the defect composition of the material. In this work a model based on the lattice kinetic Monte−Carlo method is developed to describe the process of CdTe deposition as a function of temperature and Cd and Te fluxes. To determine the effect of the treatment conditions on CdTe conductivity, we developed a quasichemical model based on the electrical neutrality equation for point defects concentrations that are described by defects formation reaction constants. Parameter obtained from the first−principles density functional calculations were used when developing the models. The developed deposition model correctly describes the transition from evaporation to precipitation as well as the increased evaporation rates in excess of Cd. To explain the observed electrical properties of CdTe after Cl−treatment, we complemented the quasichemical defect model by a deep acceptor complex defect that allowed us to describe both the high−temperature dependence of conductivity on the Cd pressure and the dependence of resistivity on Cl concentration at room temperature. Теоретическое исследование путей оптимизации свойств слоя поглотителя CdTe является важной задачей на пути повышения эффективности тонкопленочных солнечных элементов на основе гетероперехода CdTe/CdS. Свойства получаемых материалов, такие как плотность свободных носителей заряда, часто сильно зависят от параметров процессов осаждения слоев и их последующей обработки, в существенной степени определяющих дефектный состав получаемых материалов. Для описания зависимости скорости процесса осаждения CdTe от температуры и потоков Cd и Te разработана модель на основе решеточного кинетического метода Монте–Карло. Для определения влияния условий обработки CdTe на проводимость разработана квазихимическая модель, основанная на уравнении электронейтральности для концентраций точечных дефектов, описываемых константами реакций образования дефектов. При разработке моделей использованы значения параметров дефектов и реакций, полученных из первопринципных расчетов с помощью теории функционала плотности. Разработанная модель осаждения корректно описывает переход от испарения к осаждению, а также повышение скорости испарения при избытке Cd. Для объяснения наблюдаемых электрических свойств CdTe после обработки Cl квазихимическая модель дефектов дополнена глубоким акцепторным комплексным дефектом, что позволило описать как высокотемпературную зависимость проводимости от давления кадмия, так и зависимость сопротивления от концентрации хлора при комнатной температуре.