Related papers: On a Central Transform of Integer Sequences
In this paper, we define Tribonacci and Tribonacci-Lucas matrix sequences and investigate their properties.
A method for computing the Hankel transform is proposed whereby the letter is reduced to a sum by representing the integrand as a smooth function times a Bessel function. The smooth function is replaced by its wavelet decomposition with a…
Integral transforms are invaluable mathematical tools to map functions into spaces where they are easier to characterize. We introduce the hyperdimensional transform as a new kind of integral transform. It converts square-integrable…
We present an algorithm to find invariant poynomial transformations of integer sequences, using the classical invariant theory approach.
We present a formal language with expressions denoting general symbol structures and queries which access information in those structures. A sequence-to-sequence network processing this language learns to encode symbol structures and query…
We study the settings where we are given a function of n variables defined in a given box of integers. We show that in many cases we can replace the given objective function by a new function with a much smaller domain. Our approach allows…
This paper presents a general expression for a number-theoretic Hilbert transform (NHT). The transformations preserve the circulant nature of the discrete Hilbert transform (DHT) matrix together with alternating values in each row being…
By a classical principle of probability theory, sufficiently thin subsequences of general sequences of random variables behave like i.i.d.\ sequences. This observation not only explains the remarkable properties of lacunary trigonometric…
Generalizations and variations of the fundamental lemma by Willems et al. are an active topic of recent research. In this note, we explore and formalize the links between kernel regression and some known nonlinear extensions of the…
We describe an efficient hierarchical method to compute attention in the Transformer architecture. The proposed attention mechanism exploits a matrix structure similar to the Hierarchical Matrix (H-Matrix) developed by the numerical…
Matrices are typically considered over fields or rings. Motivated by applications in parametric differential equations and data-driven modeling, we suggest to study matrices with entries from a Hilbert space and present an elementary theory…
In this paper we study the sets of integers which are $n$-th terms of Lucas sequences. We establish lower- and upper bounds for the size of these sets. These bounds are sharp for $n$ sufficiently large. We also develop bounds on the growth…
In the paper, some special linear combinations of the terms of rational cycles of generalized Collatz sequences are studied. It is proved that if the coefficients of the linear combinations satisfy some conditions then these linear…
In this paper, we propose a general framework that extends the theory of permutation patterns to higher dimensions and unifies several combinatorial objects studied in the literature. Our approach involves introducing the concept of a…
When a sequence of numbers is slowly converging, it can be transformed into a new sequence which, under some assumptions, could converge faster to the same limit. One of the most well--known sequence transformation is Shanks transformation…
We review a class of methods that can be collected under the name nonlinear transform coding (NTC), which over the past few years have become competitive with the best linear transform codecs for images, and have superseded them in terms of…
Inspired by the definition of modified ascent sequences, we introduce a new class of integer sequences called revised ascent sequences. These sequences are defined as Cayley permutations where each entry is a leftmost occurrence if and only…
The kernel matrix used in kernel methods encodes all the information required for solving complex nonlinear problems defined on data representations in the input space using simple, but implicitly defined, solutions. Spectral analysis on…
In this study, we apply "r" times the binomial transform to the Padovan and Perrin matrix sequences. Also, the Binet formulas, summations, generating functions of these transforms are found using recurrence relations. Finally, we give the…
Most machine learning algorithms, such as classification or regression, treat the individual data point as the object of interest. Here we consider extending machine learning algorithms to operate on groups of data points. We suggest…