Related papers: 3x+1 dynamics on rationals with fixed denominator
In the scalar-tensor theory of gravitation it seems nontrivial to establish if solutions of the cosmological equations in the presence of a cosmological constant behave as attractors independently of the initial values. We develop a general…
Results of a general study on the dynamics of cosmological scalar fields with arbitrary potentials are presented. Exact and approximate attractor solutions are found, with applications to quintessence, moduli stabilization and inflation.
We demonstrate that the number of cycles for two problems of the family of generalized 3x+1 mappings is possible finite.
We consider the problem of approaching real numbers with rational numbers with prime denominator and with a single numerator allowed for each denominator. We obtain basic results, both probabilistic and deterministic, draw connections to…
We develop some aspects of the theory of derivators, pointed derivators, and stable derivators. As a main result, we show that the values of a stable derivator can be canonically endowed with the structure of a triangulated category.…
A three dimensional generally covariant theory is described that has a 2+1 canonical decomposition in which the Hamiltonian constraint, which generates the dynamics, is absent. Physical observables for the theory are described and the…
The Collatz conjecture can be stated in terms of the reduced Collatz function R(x) = (3x+1)/2^m (where 2^m is the larger power of 2 that divides 3x+1). The conjecture is: Starting from any odd positive integer and repeating R(x) we…
Set out here are some fundamental theories that may be regarded as newly discovered metamathematics of the odd integers in relation to the Collatz conjecture (also called the 3x+1 problem). Originally motivated by the requirement to invent…
A four dimensional generally covariant field theory is presented which describes non-dynamical three geometries coupled to scalar fields. The theory has an infinite number of physical observables (or constants of the motion) which are…
In this paper, we convert Collatz map into a simple conjugate iterative maps defined in [0,1]. Such maps are more familiar to us and easier to deal with. Some new features of this map are observed by this method. An interesting heuristic…
We introduce a novel systematic construction for integrable (3+1)-dimensional dispersionless systems using nonisospectral Lax pairs that involve contact vector fields. In particular, we present new large classes of (3+1)-dimensional…
Exploring the Collatz Conjecture and changing the expression from 3n + 1 to 5n + 1, we found patterns in different sets of numbers. Some numbers reduce to one (as stated in the Collatz Conjecture), some might escape to infinity, and some…
We show that for any set of n distinct points in the complex plane, there exists a polynomial p of degree at most n+1 so that the corresponding Newton map, or even the relaxed Newton map, for p has the given points as a super-attracting…
A large class of real $3$-dimensional nilpotent polynomial vector fields of arbitrary degree is considered. The aim of this work is to present general properties of the discrete and continuous dynamical systems induced by these vector…
We determine the irreducible components of the space of 3x3 matrices of linear forms with vanishing determinant. We show that there are four irreducible components and we identify them concretely. In particular, under elementary row and…
Define $\theta(x)=(x-1)/3$ if $x\geq 1$, and $\theta(x)=2x/(1-x)$ if $x<1$. We conjecture that the orbit of every positive rational number ends in 0. In particular, there does not exist any positive rational fixed point for a map in the…
Numerical relativity is finally approaching a state where the evolution of rather general (3+1)-dimensional data sets can be computed in order to solve the Einstein equations. After a general introduction, three topics of current interest…
This paper presents four theorems that connect continuity postulates in mathematical economics to solvability axioms in mathematical psychology, and ranks them under alternative supplementary assumptions. Theorem 1 connects notions of…
Aspects of a generalized representation theory of current algebras in $3+1$ dimensions are discussed in terms of the Fock bundle method, the sesquilinear form approach (of Langmann and Ruijsenaars), and Hilbert space cocycles.
We prove that a triangulated category which is the underlying category of a stable derivator has a filtered enhancement, providing an affirmative answer to a conjecture in [3].