Related papers: Evolution patterns in Collatz problem
We consider the evolution by crystalline curvature of a planar set in a stratified medium, modeled by a periodic forcing term. We characterize the limit evolution law as the period of the oscillations tends to zero. Even if the model is…
Evolutionary algorithms have been frequently applied to constrained continuous optimisation problems. We carry out feature based comparisons of different types of evolutionary algorithms such as evolution strategies, differential evolution…
In this paper we consider traces at initial times for functions with mixed time-space smoothness. Such results are often needed in the theory of evolution equations. Our result extends and unifies many previous results. Our main improvement…
Design patterns are generalized solutions to frequently recurring problems. They were initially developed by architects and computer scientists to create a higher level of abstraction for their designs. Here, we extend these concepts to…
The Collatz process is defined on natural numbers by iterating the map $T(x) = T_0(x) = x/2$ when $x\in\mathbb{N}$ is even and $T(x)=T_1(x) =(3x+1)/2$ when $x$ is odd. In an effort to understand its dynamics, and since Generalised Collatz…
We present a general formalism which allows us to derive the evolution equations describing one-dimensional (1D) and isotropic 2D interfacelike systems, that is based on symmetries, conservation laws, multiple scale arguments, and exploits…
Model evolution is widely considered as a subject under research. Despite its role in research, common purpose concepts, approaches, solutions, and methodologies are missing. Limiting the scope to model libraries makes model evolution and…
We focus on evolution equations on co-evolving, infinite, graphs and establish a rigorous link with a class of nonlinear continuity equations, whose vector fields depend on the graphs considered. More precisely, weak solutions of the…
This paper presents a high-level conceptual framework to help orient the discussion and implementation of open-endedness in evolutionary systems. Drawing upon earlier work by Banzhaf et al., three different kinds of open-endedness are…
An evolution-type differential equation encodes the intersection theory of tautological classes on the Hilbert scheme of a family of nodal curves.
An involution is usually defined as a mapping that is its own inverse. In this paper, we study quaternion involutions that have the additional properties of distribution over addition and multiplication. We review formal axioms for such…
We study pattern densities in binary sequences, finding optimal limit sequences with fixed pattern densities.
Any permutation has a disjoint cycle decomposition and concept generates an equivalence class on the symmetry group called the cycle-type. The main focus of this work is on permutations of restricted cycle-types, with particular emphasis on…
The ancestral sequence reconstruction problem is the inference, back in time, of the properties of common sequence ancestors from measured properties of contemporary populations. Standard algorithms for this problem assume independent…
A brief presentation of basic definitions and notation used in permutation patterns research.
The Collatz conjecture is explored using polynomials based on a binary numeral system. It is shown that the degree of the polynomials, on average, decreases after a finite number of steps of the Collatz operation, which provides a weak…
We introduce kernels and resolvents on preordered sets and derive sharp resolvent inequalities that entail Gronwall inequalities for functions of several variables. In this way, we can prove a fixed point result for operators on topological…
This paper formalizes a latent variable inference problem we call {\em supervised pattern discovery}, the goal of which is to find sets of observations that belong to a single ``pattern.'' We discuss two versions of the problem and prove…
A new delay equation is introduced to describe the punctuated evolution of complex nonlinear systems. A detailed analytical and numerical investigation provides the classification of all possible types of solutions for the dynamics of a…
This paper explores special conditions on the starting value of a Collatz sequence which imply that the Collatz conjecture is true. This is the result of the collaboration of a retired mathematics professor (Koelzer) and a retired physics…