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Recurrence quantification analysis is a method for measuring the complexity of dynamical systems. Recurrence determinism is a fundamental characteristic of it, closely related to correlation sum. In this paper, we study asymptotic behavior…
We show that it is consistent relative to ZF, that there is no well-ordering of $\mathbb{R}$ while a wide class of special sets of reals such as Hamel bases, transcendence bases, Vitali sets or Bernstein sets exists. To be more precise, we…
In the present paper we show that there are infinitely many classes of term functions in the free-void generated diagonalizable algebra, which are precomplete with respect to parametrical expressibility of functions.
For a big tt-category, we give a characterization of the Telescope Conjecture (TC) in terms of definable tensor-ideals generated by homological residue fields. We formulate a stalk-locality property of (TC) and prove that it holds in the…
We propose a simple experimental procedure based on the Elitzur-Vaidman scheme to test the persistence of macroscopic superpositions. We conjecture that its implementation will reveal the persistence of superpositions of macroscopic objects…
Windowed recurrences are sliding window calculations where a function is applied iteratively across the window of data, and are ubiquitous throughout the natural, social, and computational sciences. In this monograph we explore the…
We study generalized sums of linear orders. These are binary operations that, given linear orders $A$ and $B$, return an order $A \oplus B$ that can be decomposed as an isomorphic copy of $A$ interleaved with a copy of $B$. We show that…
Zeckendorf's theorem states every positive integer has a unique decomposition as a sum of non-adjacent Fibonacci numbers. This result has been generalized to many sequences $\{a_n\}$ arising from an integer positive linear recurrence, each…
The pursue of what are properties that can be identified to permit an automated reasoning program to generate and find new and interesting theorems is an interesting research goal (pun intended). The automatic discovery of new theorems is a…
We consider the problem of symbolic integration of $\int G(x,y(x)) dx$ where $G$ is rational and $y(x)$ is a non algebraic solution of a differential equation $y'(x)=F(x,y(x))$ with $F$ rational. As $y$ is transcendental, the Galois action…
This paper continues the author's previous work on a limit-free algebraic-geometric construction of the derivative in the class of polynomial functions and extends the proposed framework to elementary functions. Derivatives of rational…
We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such…
The first part of the paper is a brief overview of Hindman's finite sums theorem, its prehistory and a few of its further generalizations, and a modern technique used in proving these and similar results, which is based on idempotent…
The celebrated Zeilberger algorithm which finds holonomic recurrence equations for definite sums of hypergeometric terms $F(n,k)$ is extended to certain nonhypergeometric terms. An expression $F(n,k)$ is called a hypergeometric term if both…
We give an effective procedure that produces a natural number in its output from any natural number in its input, that is, it computes a total function. The elementary operations of the procedure are Turing-computable. The procedure has a…
This paper aims at reviewing and analysing the method of reflections. The latter is an iterative procedure designed to linear boundary value problems set in multiply connected domains. Being based on a decomposition of the domain boundary,…
Zeckendorf's Theorem states that any positive integer can be uniquely decomposed into a sum of distinct, non-adjacent Fibonacci numbers. There are many generalizations, including results on existence of decompositions using only even…
The notion of geometric construction is introduced. This notion allows to compare incidence configurations in the algebraic and tropical plane. We provide an algorithm such that, given a tropical instance of a geometric construction, it…
We consider the following problem: Given a nested sum expression, find a sum representation such that the nested depth is minimal. We obtain a symbolic summation framework that solves this problem for sums defined, e.g., over…
Non-wellfounded proof theory results from allowing proofs of infinite height in proof theory. To guarantee that there is no vicious infinite reasoning, it is usual to add a constraint to the possible infinite paths appearing in a proof.…