Combinatorial methods and recurrence relations with two indices

In this paper, we solve several recurrence relations with two indices by using combinatorial methods. Moreover, we present several recurrence relations with two indices related to Dyck paths and Schröder paths.     

Semilocal Convergence of Steffensen-Type Algorithms for Solving Nonlinear Equations

In this article, we provide a semilocal analysis for the Steffensen-type method (STTM) for solving nonlinear equations in a Banach space setting using recurrence relations. Numerical examples to validate our main results are also provided in this study to show that STTM is faster than other methods ([7 I. K. Argyros , J. Ezquerro , J. M. Gutiérrez , M. Hernández , and S. Hilout ( 2011 ). On the semilocal convergence of efficient Chebyshev-Secant-type methods . J. Comput. Appl. Math. 235 : 3195 – 3206 .[Crossref], [Web of Science ®] , [Google Scholar], 13 J. A. Ezquerro and M. A. Hernández ( 2009 ). An optimization of Chebyshev's method . J. Complexity 25 : 343 – 361 .[Crossref], [Web of Science ®] , [Google Scholar]]) using similar convergence conditions.     

On the conditional increments of degradation processes

Recently, Wang and Xu (2010) developed an efficient EM algorithm for the semiparametric inference of the inverse Gaussian (IG) process. In the presence of missing degradation data, the algorithm needs to compute the mean of the IG increment during some time interval [_s1,_s2]$[s_1,s_2]$ conditional on that the process is tied down on two points before _s1$s_1$ and after _s2$s_2$, respectively. This study derives the conditional distribution of this increment and gives the expectation of the increment and of the reciprocal of the increment. The results simplify the implementation of the EM algorithm for the IG process. In a similar vein, we further derive distributions for the conditional increments of the Wiener process and the Gamma process, respectively.     

On the principle of maximum entropy and the risk analysis of disaster loss

Disasters that occur everywhere in the most disordered way indicate that disaster entropy has reached the maximum value. Under given constraint conditions, when disaster entropy is the maximum value, the disaster loss series should follow P-III distribution. The occurrence interval of disaster loss refers to the average time interval that disaster loss of certain degree happens in the future. We could, according to the field disaster data and using P-III distribution function, calculate the value of future disaster loss with certain recurrence interval. Explicit in concept and easy to use, such a method has significant meaning in practice.     

Sums of products of Cauchy numbers

In this paper, we consider a kind of sums involving Cauchy numbers, which have not been studied in the literature. By means of the method of coefficients, we give some properties of the sums. We further derive some recurrence relations and establish a series of identities involving the sums, Stirling numbers, generalized Bernoulli numbers, generalized Euler numbers, Lah numbers, and harmonic numbers. In particular, we generalize some relations between two kinds of Cauchy numbers and some identities for Cauchy numbers and Stirling numbers.     

Construction and calculation of reproducing kernel determined by various linear differential operators

We presented a method to construct and calculate the reproducing kernel for the linear differential operator with constant coefficients and a single latent root; further, we gave the formula for calculation. Additionally, by studying the recurrence relation of the reproducing kernel with arithmetic latent roots, we found that the reproducing kernel with a multi-knots interpolation constraint can be concisely represented by one with an initial-value constraint.     

Sharp Error Terms for Return Time Statistics under Mixing Conditions

We describe the statistics of repetition times of a string of symbols in a stochastic process. Denote by τ _A the time elapsed until the process spells a finite string A and by S _A the number of consecutive repetitions of A. We prove that, if the length of the string grows unboundedly, (1) the distribution of τ _A , when the process starts with A, is well approximated by a certain mixture of the point measure at the origin and an exponential law, and (2) S _A is approximately geometrically distributed. We provide sharp error terms for each of these approximations. The errors we obtain are point-wise and also allow us to get approximations for all the moments of τ _A and S _A . To obtain (1) we assume that the process is φ-mixing, while to obtain (2) we assume the convergence of certain conditional probabilities.      

A Mathematica package for q-holonomic sequences and power series

We describe a Mathematica package for dealing with q-holonomic sequences and power series. The package is intended as a q-analogue of the Maple package gfun and the Mathematica package GeneratingFunctions. It provides commands for addition, multiplication, and substitution of these objects, for converting between various representations (q-differential equations, q-recurrence equations, q-shift equations), for computing sequence terms and power series coefficients, and for guessing recurrence equations given initial terms of a sequence. C. Koutschan partially supported by the Austrian Science Foundation (FWF) grants SFB F1305.     

Multidimensional random walk with reflections

Reflected random walk in higher dimension arises from an ordinary random walk (sum of i.i.d. random variables): whenever one of the reflecting coordinates becomes negative, its sign is changed, and the process continues from that modified position. One-dimensional reflected random walk is quite well understood from work in 7 decades, but the multidimensional model presents several new difficulties. Here we investigate recurrence questions.