Series Of Complex Numbers Research Articles

(e)-Convergence for double sequences

We define the notion of (e)-summability of double sequences and series of complex numbers. We also obtain a criteria for this summability method with regards to Berezin symbols of an diagonal operator, and show regularity of (e)-summability method for double sequences.

Random generation of 2D geometry-controlled particles via the epicycle series

The discrete element method (DEM) represents a classical and popular research approach to exploring the particle shape effects of granular materials. For many modellers, the acquisition of irregular particle outlines is a problem of considerable practical importance. To mitigate this problem, a novel method based on epicycle series is introduced in this study, and it can produce geometry-controlled particles from two major shape indexes. First, the three key steps of the proposed method are explained: (1) a series of complex numbers, namely, the epicycle series, is used to represent the discrete outline of a particle; (2) the magnitudes of the epicycle descriptors are correlated with the particle shape indexes; and (3) the particle shape is controlled by predicting and adjusting the epicycle descriptors. Then, sensitivity analysis and a case study are conducted to test the proposed method's capacity for reproducing particle shape features. The results suggest that the proposed framework can generate granular particles with accurately controlled shape indexes and realistic outlines, which can facilitate the DEM modelling of irregular particles.

SSAW: A new sequence similarity analysis method based on the stationary discrete wavelet transform

BackgroundAlignment-free sequence similarity analysis methods often lead to significant savings in computational time over alignment-based counterparts.ResultsA new alignment-free sequence similarity analysis method, called SSAW is proposed. SSAW stands for Sequence Similarity Analysis using the Stationary Discrete Wavelet Transform (SDWT). It extracts k-mers from a sequence, then maps each k-mer to a complex number field. Then, the series of complex numbers formed are transformed into feature vectors using the stationary discrete wavelet transform. After these steps, the original sequence is turned into a feature vector with numeric values, which can then be used for clustering and/or classification.ConclusionsUsing two different types of applications, namely, clustering and classification, we compared SSAW against the the-state-of-the-art alignment free sequence analysis methods. SSAW demonstrates competitive or superior performance in terms of standard indicators, such as accuracy, F-score, precision, and recall. The running time was significantly better in most cases. These make SSAW a suitable method for sequence analysis, especially, given the rapidly increasing volumes of sequence data required by most modern applications.

Berezin symbols and Borel summability

We prove in terms of so-called Berezin symbols some theorems for Borel summability method for sequences and series of complex numbers. Namely, we characterize the Borel convergent sequences and series; prove regularity of Borel summability method, and prove a new Tauberian type theorem for Borel summability.

Statistical convergence of sequences and series of complex numbers with applications in Fourier Analysis and Summability

This is a survey paper on recent results indicated in the title. In contrast to the famous examples of Kolmogorov and Fejer on the pointwise divergence of Fourier series, the statistical convergence of the Fourier series of any integrable function takes place at almost every point; and the statistical convergencr of the Fourier series of any continuous function is uniform. Furthermore, Tauberian conditions are also presented, under which ordinary convergence of any sequence of real or complex numbers follows from its statistical summability.

( e)-convergence and related problem

We answer negatively to a question of Zorboska (2003) [13], which is concerned to the boundary behavior of Berezin symbols of Bergman space operators. We also introduce the notions of ( e)-summability of sequences and series of complex numbers, and study some of their properties. As a corollary, we obtain the classical Abel theorems of summability theory.

A counter-example in summability theory

Let Σ denote the set of all seriesof complex numbers. By a ‘summability method’, say A, we mean a function from some subset (the set of ‘A -summable series’) of Σ into the set of complex numbers. We will use the language generally associated with this definition, and will take for granted the case in which A is (C, 1). A summability method A will be called linear if, whenever a, b are A -summable, then so is λa + μb (where λ, μ are any complex constants) and if the. A -sums of a, b, λa + μb are then related in the natural way. We call A regular if, whenever a converges to σ, it is A -summable to σ. If A is a regular summability method, then any condition P on the series (1) will be called a Tauberian condition for A if any A -summable series which satisfies P is convergent.