Related papers: Ultra-recursive sequences
We extend the close interplay between continued fractions, orthogonal polynomials, and Gaussian quadrature rules to several variables in a special but natural setting which we characterize in terms of moment sequences. The crucial condition…
A novel notion of unpredictable strings is revealed and utilized to define deterministic unpredictable sequences on a finite number of symbols. We prove the first law of large strings for random processes in discrete time, which confirms…
Let $ \prod_{i=1}^d (X-\alpha_i Y) \in{\mathbb C}[X,Y]$ be a binary form and let $\epsilon_1,\dots,\epsilon_d$ be nonzero complex numbers. We consider the family of binary forms $ \prod_{i=1}^d (X-\alpha_i \epsilon_i^aY)$, $a\in {\mathbb…
Recurrence analysis is a well settled method allowing to discern chaos from order, and determinism from noise. We apply this tool to study time series representing geodesic and inspiraling motion of a test particle in a deformed Kerr…
We use fast-growing finite and infinite sequences of natural numbers and more complicated constructs to define models of hypercomputation and interpret non-arithmetic predicates, with the strongest extensions reaching full second order…
We study the number of factorizations of a positive integer, where the parts of the factorization are of l different colors (or kinds). Recursive or explicit formulas are derived for the case of unordered and ordered, distinct and…
It is natural for probabilistic programs to use conditionals to express alternative substructures in models, and loops (recursion) to express repeated substructures in models. Thus, probabilistic programs with conditionals and recursion…
A sequence is a fractal sequence if it contains itself as a proper subsequence. (The self-containment property resembles that of visual fractals) A doubly fractal sequence of integers is defined by operations called upper trimming and lower…
We take causality and uniqueness of events observation as our driving forces. They are built in in the way we define distinct observers, which then require a finite time to communicate between each other. This unavoidably leads to the…
Over the long history of machine learning, which dates back several decades, recurrent neural networks (RNNs) have been used mainly for sequential data and time series and generally with 1D information. Even in some rare studies on 2D…
We propose a method to search for signs of causal structure in spatiotemporal data making minimal a priori assumptions about the underlying dynamics. To this end, we generalize the elementary concept of recurrence for a point process in…
We characterize a family of number triangles whose production matrices are closely related to the original number triangle. We study a number of such triangles that are of combinatorial significance. For a specific subfamily, these…
In this note we study the convergence of recursively defined infinite series. We explore the role of the derivative of the defining function at the origin (if it exists), and develop a comparison test for such series which can be used even…
Time-series data exists in every corner of real-world systems and services, ranging from satellites in the sky to wearable devices on human bodies. Learning representations by extracting and inferring valuable information from these time…
To approximate a simple root of an equation we construct families of iterative maps of higher order of convergence. These maps are based on model functions which can be written as an inner product. The main family of maps discussed is…
A sequence is called $C$-finite if it satisfies a linear recurrence with constant coefficients. We study sequences which satisfy a linear recurrence with $C$-finite coefficients. Recently, it was shown that such $C^2$-finite sequences…
We extend previous work on anti-recurrence sequences of Kimberling and Moses, Zaslavsky, and Bosma et al. Kimberling and Moses have formulated several questions on these sequences, which can be combined into the meta-conjecture that…
We recast homogeneous linear recurrence sequences with fixed coefficients in terms of partial Bell polynomials, and use their properties to obtain various combinatorial identities and multifold convolution formulas. Our approach relies on a…
In this paper we consider a problem of searching a space of predictive models for a given training data set. We propose an iterative procedure for deriving a sequence of improving models and a corresponding sequence of sets of non-linear…
We prove that the difference between the $n$-th hyperfibonacci number of $r$-th generation and its two consecutive predecessors is the $n$-th regular $(r-1)$-topic number. Using this fact we provide an equivalent recursive definition of…