Related papers: Generalized 3x + 1 Mappings : searching for cycles
In this paper we study a certain class of polycyclic groups. We outline a method for constructing a poly-$\mathbb{Z}$ group $G_n$ by describing a process for selecting maps that are used to extend $G_i$ to $G_{i+1}$ for $1 \leq i \leq n-1$…
We discuss integrable discretizations of 3-dimensional cyclic systems, that is, orthogonal coordinate systems with one family of circular coordinate lines. In particular, the underlying circle congruences are investigated in detail, and…
The 3x+ 1 problem concerns iteration of the map on the integers given by T(n) = (3n+1)/2 if n is odd; T(n) = n/2 if n is even. The 3x+1 Conjecture asserts that for every positive integer n > 1 the forward orbit of n under iteration by T…
The work in this article is concerned with two different types of families of finite sets: separating families and splitting families (they are also called "systems"). These families have applications in combinatorial search, coding theory,…
Mapping network flows provides insight into the organization of networks, but even though many real-networks are bipartite, no method for mapping flows takes advantage of the bipartite structure. What do we miss by discarding this…
We study learning of indexed families from positive data where a learner can freely choose a hypothesis space (with uniformly decidable membership) comprising at least the languages to be learned. This abstracts a very universal learning…
Metric search commonly involves finding objects similar to a given sample object. We explore a generalization, where the desired result is a fair tradeoff between multiple query objects. This builds on previous results on complex queries,…
A nonlinear cyclic system with delay and the overall negative feedback is considered. The characteristic equation of the linearized system is studied in detail. Sufficient conditions for the oscillation of all solutions and for the…
Group synchronization plays a crucial role in global pipelines for Structure from Motion (SfM). Its formulation is nonconvex and it is faced with highly corrupted measurements. Cycle consistency has been effective in addressing these…
Networks are widely used in the biological, physical, and social sciences as a concise mathematical representation of the topology of systems of interacting components. Understanding the structure of these networks is one of the outstanding…
In this work we consider families of smooth vector fields having a persistent polycycle with $n$ hyperbolic saddles. We derive the asymptotic expansion of the return map associated to the polycycle, determining explicitly its leading terms.…
We study the dynamics of a family of replicator maps, depending on two parameters. Such studies are motivated by the analysis of the dynamics of evolutionary games under selections. From the dynamics viewpoint, we prove the existence of…
This paper considers generalizations of open mappings, closed mappings, pseudo-open mappings, and quotient mappings from topological spaces to generalized topological spaces. Characterizations of these classes of mappings are obtained and…
Given a triangle-free planar graph G and a 9-cycle C in G, we characterize situations where a 3-coloring of C does not extend to a proper 3-coloring of G. This extends previous results when C is a cycle of length at most 8.
To better understand the overlapping modular organization of large networks with respect to flow, here we introduce the map equation for overlapping modules. In this information-theoretic framework, we use the correspondence between…
We prove new lower bounds on the crossing number of a complete graphs assuming that it is drawn in such a way that it contains a Hamiltonian cycle with no crossings.
Given a plane graph $G$ (i.e., a planar graph with a fixed planar embedding) and a simple cycle $C$ in $G$ whose vertices are mapped to a convex polygon, we consider the question whether this drawing can be extended to a planar…
We construct a collection of higher Chow cycles on certain surfaces which degenerate to an arrangement of planes in general position. When its degree is 4, this construction gives a new explicit proof of the Hodge-D-Conjecture for a certain…
Taking a new approach towards analyzing the Collatz Problem, or, 3x+1 conjecture. Introducing some new functions, the Collatz-2 and Collatz-3 sequences, as well as deducing results related to Collatz-2 and Collatz-3 sequences.
A method to predict the emergence of different kinds of ordered collective behaviors in systems of globally coupled chaotic maps is proposed. The method is based on the analogy between globally coupled maps and a map subjected to an…