Related papers: Asymptotic Analysis of Regular Sequences

We give an asymptotic for the number of prime solutions to $Q(x_1,\dots, x_8) = N$, subject to a mild non-degeneracy condition on the homogeneous quadratic form $Q$. The argument initially proceeds via the circle method, but this does not…

Let $f_1,\ldots,f_k : \mathbb{N} \rightarrow \mathbb{C}$ be multiplicative functions taking values in the closed unit disc. Using an analytic approach in the spirit of Hal\'{a}sz' mean value theorem, we compute multidimensional averages of…

This is a discussion of miscellaneous summation, integration and transformation formulas obtained using Fourier analysis. The topics covered are: Series of the form $\sum_{n\in\mathbb{Z}} c_ne^{\pi i \gamma n^2}$; Fusion of integrals, and…

In a previous work [arXiv:1009.4363], we have studied the evolution of a scalar field with a quartic coupling, driven by a classical source that initializes it to a non-perturbatively large value. At leading order in the coupling, the…

We consider a coupled system of two singularly perturbed reaction-diffusion equations, with two small parameters $0< \epsilon \le \mu \le 1$, each multiplying the highest derivative in the equations. The presence of these parameters causes…

In the paper we develop an approach to asymptotic normality through factorial cumulants. Factorial cumulants arise in the same manner from factorial moments, as do (ordinary) cumulants from (ordinary) moments. Another tool we exploit is a…

We establish asymptotic formulas for sums of reciprocals of primes in arithmetic progressions, generalizing recent results on multiple Mertens evaluations by Tenenbaum, Qi, and Hu. Specifically, for any fixed constant $K>0$, we derive…

We study the arithmetic function sopfr$(n)$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$. In particular we obtain the asymptotic formula $$ \sum_{n \leq x} \rm{sopfr}(n) \sim \frac{\pi^2}{12}…

This paper studies properties of binary runlength-limited sequences with additional constraints on their Hamming weight and/or their number of runs of identical symbols. An algebraic and a probabilistic (entropic) characterization of the…

A \emph{sprout sequence} is a sequence $\frakr=(R_0=1,R_1,R_2,\dots)$ of symmetric functions in the variables $\bmx=(x_1,x_2,\dots)$ over a field $K$ generated from a power series $F(t)=1+a_1t+a_2t^2+\cdots$ by the rule $\sum_{n\geq…

We study the existence, strong consistency and asymptotic normality of estimators obtained from estimating functions, that are p-dimensional martingale transforms. The problem is motivated by the analysis of evolutionary clustered data,…

The coefficient sequences of multivariate rational functions appear in many areas of combinatorics. Their diagonal coefficient sequences enjoy nice arithmetic and asymptotic properties, and the field of analytic combinatorics in several…

We focus on a data sequence produced by repetitive quantum measurement on an internal hidden quantum system, and call it a hidden Markovian process. Using a quantum version of the Perron-Frobenius theorem, we derive novel upper and lower…

We describe the utility of integral representations for sums of basic hypergeometric functions. In particular we use these to derive an infinite sequence of transformations for symmetrizations over certain variables which the functions…

Calculations in field theory are usually accomplished by employing some variants of perturbation theory, for instance using loop expansions. These calculations result in asymptotic series in powers of small coupling parameters, which as a…

Datasets from the fields of bioinformatics, chemometrics, and face recognition are typically characterized by small samples of high-dimensional data. Among the many variants of linear discriminant analysis that have been proposed in order…

The Gaussian entire function is a random entire function, characterised by a certain invariance with respect to isometries of the plane. We study the fluctuations of the increment of the argument of the Gaussian entire function along planar…

We establish asymptotic expansions for factorial moments of following distributions: number of cycles in a random permutation, number of inversions in a random permutation, and number of comparisons used by the randomized quick sort…

This paper develops further and systematically the asymptotic expansion theory that was initiated by Foias and Saut in [11]. We study the long-time dynamics of a large class of dissipative systems of nonlinear ordinary differential…

We consider the semi-parametric estimation of a scale parameter of a one-dimensional Gaussian process with known smoothness. We suggest an estimator based on quadratic variations and on the moment method. We provide asymptotic…