Related papers: On a Central Transform of Integer Sequences

We discuss the integer sequence transform $a \mapsto b$ where $b_n$ is the number of real roots of the polynomial $a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$. It is shown that several sequences $a$ give the trivial sequence $b = (0,1,0,1,…

This work proposes kernel transform learning. The idea of dictionary learning is well known; it is a synthesis formulation where a basis is learnt along with the coefficients so as to generate or synthesize the data. Transform learning is…

We look at thin interpolating sequences and the role they play in uniform algebras, Hardy spaces, and model spaces.

Matrices and more generally multidimensional arrays, form the backbone of computational studies. In this paper we demonstrate increases in computational efficiency by performing partial-tracing/tensor-contractions on sparse-arrays. It was…

In this paper, we develop a new technique which we call representation theory of the real hyperrectangle, which describes how to compute the eigenvectors and eigenvalues of certain matrices arising from hyperrectangles. We show that these…

This article is written with the hope to draw attention to a method that uses integral transforms to find exact values for a large class of convergent series (and, in particular, series of rational terms). We apply the method to some series…

Transformer is a powerful architecture that achieves superior performance on various sequence learning tasks, including neural machine translation, language understanding, and sequence prediction. At the core of the Transformer is the…

Although dominant in natural language processing, transformer-based models remain challenged by the task of long-sequence processing, because the computational cost of self-attention operations in transformers swells quadratically with the…

In a previous paper, we have given an algebraic model to the set of intervals. Here, we apply this model in a linear frame. We define a notion of diagonalization of square matrices whose coefficients are intervals. But in this case, with…

Nowadays, hyperspectral image classification widely copes with spatial information to improve accuracy. One of the most popular way to integrate such information is to extract hierarchical features from a multiscale segmentation. In the…

We propose a simple technique that, if combined with algorithms for computing functions of triangular matrices, can make them more efficient. Basically, such a technique consists in a specific scaling similarity transformation that reduces…

Let $A$ be a proper Riordan array with general element $a_{n,k}$. We study the one parameter family of matrices whose general elements are given by $a_{2n+r, n+k+r}$. We show that each such matrix can be factored into a product of a Riordan…

Kendall transformation is a conversion of an ordered feature into a vector of pairwise order relations between individual values. This way, it preserves ranking of observations and represents it in a categorical form. Such transformation…

In this article we discuss the presentation of a random binary matrix using sequence of whole nonnegative numbers. We examine some advantages and disadvantages of this presentation as an alternative of the standard presentation using…

A central limit theorem for binary tree is numerically examined. Two types of central limit theorem for higher-order branches are formulated. A topological structure of a binary tree is expressed by a binary sequence, and the…

Many models for chaotic systems consist of joining two integrable systems with incompatible constants of motion. The quantum counterparts of such models have a propagator which factorizes into two integrable parts. Each part can be…

We define a class of "algebraic" random matrix channels for which one can generically compute the limiting Shannon transform using numerical techniques and often enumerate the low SNR series expansion coefficients in closed form. We…

Number sequences defined by a linear recursion relation are studied by means of generating functions. Indices of the terms in the recursion relation have arbitrary differenses. In addition to formulas for the nth term an algorithm is…

This article discusses the notion of convergence of sequences of iterated function systems. The technique of iterated function systems is one of the several methods to construct objects with fractal nature, and the fractals obtained with…

We describe how the reversion of a series is related to convolutional recurrence relations for the series, and we place this relationship in the context of Riordan arrays. As an example of the approach, we give new recurrence relations for…