Investigation of continuous-time quantum walk by using Krylov subspace-Lanczos algorithm

In papers\cite{js,jsa}, the amplitudes of continuous-time quantum walk on graphs possessing quantum decomposition (QD graphs) have been calculated by a new method based on spectral distribution associated to their adjacency matrix. Here in this paper, it is shown that the continuous-time quantum walk on any arbitrary graph can be investigated by spectral distribution method, simply by using Krylov subspace-Lanczos algorithm to generate orthonormal bases of Hilbert space of quantum walk isomorphic to orthogonal polynomials. Also new type of graphs possessing generalized quantum decomposition have been introduced, where this is achieved simply by relaxing some of the constrains imposed on QD graphs and it is shown that both in QD and GQD graphs, the unit vectors of strata are identical with the orthonormal basis produced by Lanczos algorithm. Moreover, it is shown that probability amplitude of observing walk at a given vertex is proportional to its coefficient in the corresponding unit vector of its stratum, and it can be written in terms of the amplitude of its stratum. Finally the capability of Lanczos-based algorithm for evaluation of walk on arbitrary graphs (GQD or non-QD types), has been tested by calculating the probability amplitudes of quantum walk on some interesting finite (infinite) graph of GQD type and finite (infinite) path graph of non-GQD type, where the asymptotic behavior of the probability amplitudes at infinite limit of number of vertices, are in agreement with those of central limit theorem of Ref.\cite{nko}.Comment: 29 pages, 4 figure

Discrete integrable systems generated by Hermite-Pad\'e approximants

We consider Hermite-Pad\'e approximants in the framework of discrete integrable systems defined on the lattice $\mathbb{Z}^2$. We show that the concept of multiple orthogonality is intimately related to the Lax representations for the entries of the nearest neighbor recurrence relations and it thus gives rise to a discrete integrable system. We show that the converse statement is also true. More precisely, given the discrete integrable system in question there exists a perfect system of two functions, i.e., a system for which the entire table of Hermite-Pad\'e approximants exists. In addition, we give a few algorithms to find solutions of the discrete system.Comment: 20 page

Applications of continued fractions in one and more variables

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Elementary properties of continued fractions are derived from sets of three-term recurrence relations and approximation methods are developed from this simple approach. First, a well-known method for numerical inversion of Laplace transforms is modified in two different ways to obtain exponential approximations. Differential-difference equations arising from certain Markov processes are solved by direct application of continued fractions and practical error estimates are obtained. Approximations of a slightly different form are then derived for certain generalised hypergeometric functions using those hypergeometric functions that satisfy three-term recurrence relations and have simple continued fraction expansions. Error estimates are also given in this case. The class of corresponding sequence algorithms is then described for the transformation of power series into continued fraction form. These algorithms are shown to have very general application and only break down if the required continued fraction does not exist. A continued fraction in two variables is then shown to exist and its correspondence with suitable double power series made feasible by the generalisation of the corresponding sequence method. A convergence theorem, due to Van Vleck, is adapted for use with this type of continued fraction and a comparison is made with Chisholm rational approximants in two variables. Some of these ideas are further generalised to the multivariate case. Such corresponding fractions are closely related to other fractions that may be used for point-wise bivariate or multivariate interpolation to function values known on a mesh of points. Interpolation algorithms are described and advantages and limitations discussed. The work presented forms a basis for a wide range of further research and some possible applications in numerical mathematics are indicated

Population size estimation via alternative parametrizations for Poisson mixture models

We exploit a suitable moment-based reparametrization of the Poisson mixtures distributions for developing classical and Bayesian inference for the unknown size of a finite population in the presence of count data. Here we put particular emphasis on suitable mappings between ordinary moments and recurrence coefficients that will allow us to implement standard maximization routines and MCMC routines in a more convenient parameter space. We assess the comparative performance of our approach in real data applications and in a simulation study

Matrix continued fractions associated with lattice paths, resolvents of difference operators, and random polynomials

We begin our analysis with the study of two collections of lattice paths in the plane, denoted $\mathcal{D}_{[n,i,j]}$ and $\mathcal{P}_{[n,i,j]}$. These paths consist of sequences of $n$ steps, where each step allows movement in three directions: upward (with a maximum displacement of $q$ units), rightward (exactly one unit), or downward (with a maximum displacement of $p$ units). The paths start from the point $(0,i)$ and end at the point $(n,j)$. In the collection $\mathcal{D}_{[n,i,j]}$, it is a crucial constraint that paths never go below the $x$-axis, while in the collection $\mathcal{P}_{[n,i,j]}$, paths have no such restriction. We assign weights to each path in both collections and introduce weight polynomials and generating series for them. Our main results demonstrate that certain matrices of size $q\times p$ associated with these generating series can be expressed as matrix continued fractions. These results extend the notable contributions previously made by P. Flajolet and G. Viennot in the scalar case $p=q=1$. The generating series can also be interpreted as resolvents of one-sided or two-sided difference operators of finite order. Additionally, we analyze a class of random banded matrices $H$, which have $p+q+1$ diagonals with entries that are independent and bounded random variables. These random variables have identical distributions along diagonals. We investigate the asymptotic behavior of the expected values of eigenvalue moments for the principal $n\times n$ truncation of $H$ as $n$ tends to infinity.Comment: 48 pages, 3 figure

Telescoping continued fractions for the error term in Stirling's formula

In this paper, we introduce telescoping continued fractions to find lower bounds for the error term $r_n$ in Stirling's approximation $\displaystyle n! = \sqrt{2\pi}n^{n+1/2}e^{-n}e^{r_n}.$ This improves lower bounds given earlier by Ces\`{a}ro (1922), Robbins (1955), Nanjundiah (1959), Maria (1965) and Popov (2017). The expression is in terms of a continued fraction, together with an algorithm to find successive terms of this continued fraction. The technique we introduce allows us to experimentally obtain upper and lower bounds for a sequence of convergents of a continued fraction in terms of a difference of two continued fractions.Comment: Comments invite