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On the trajectories of a particle in a translation invariant involutive field

Cristian Cobeli, Alexandru Zaharescu

Abstract

We introduce a double-folded operator that, upon iterative application, generates a dynamical system with two types of trajectories: a cyclic one and, another that grows endlessly on parabolas. These trajectories produce two distinct partitions of the set of lattice points in the plane. Our object is to analyze these trajectories and to point out a few special arithmetic properties of the integers they represent. We also introduce and study the parabolic-taxicab distance, which measures the fast traveling on the steps of the stairs defined by points on the parabolic trajectories whose coordinates are based on triangular numbers.

On the trajectories of a particle in a translation invariant involutive field

Abstract

We introduce a double-folded operator that, upon iterative application, generates a dynamical system with two types of trajectories: a cyclic one and, another that grows endlessly on parabolas. These trajectories produce two distinct partitions of the set of lattice points in the plane. Our object is to analyze these trajectories and to point out a few special arithmetic properties of the integers they represent. We also introduce and study the parabolic-taxicab distance, which measures the fast traveling on the steps of the stairs defined by points on the parabolic trajectories whose coordinates are based on triangular numbers.
Paper Structure (18 sections, 6 theorems, 59 equations, 11 figures, 2 tables)

This paper contains 18 sections, 6 theorems, 59 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Definitions and statements of the main results
  3. Preparatory Lemmas
  4. Arithmetic properties of K-generated trajectories
  5. The Cycles
  6. Cycles representing squares and cubes
  7. Cycles representing only primes
  8. An L2 average length of the steps of paths in K-generated cycles
  9. The average length of paths in K-generated cycles
  10. Proof of Theorem \ref{['TheoremMain']}
  11. One equivalence class
  12. The existence of min(R(a,b))
  13. The ladder on P(0,0)
  14. The vertex of the parabola passing through a given point
  15. The densities of represented numbers in mod p -- Proof of Theorem \ref{['TheoremLmodp']}
  16. ...and 3 more sections

Key Result

Theorem 1

Let $a$ and $b$ be integers. Then the following hold.

Figures (11)

  • Figure 1: The smallest $K$-generated cycles by $(a,b)\in[0,T]^2$, with $T=0,1,2,3,4$.
  • Figure 2: The smallest $K$-generated cycles by $(a,b)\in[-T,T]^2$, with $T=0,1,2,3,4$.
  • Figure 3: The lattice points in $\mathscr{P}_L(0,0)$, starting with the vertex $(0,0)$. The points on the upper branch have coordinates $(T_k,T_{k+1})$, while those on the lower branch have coordinates $(T_{k+1},T_k)$, where $T_k = k(k+1)/2$, for $k\ge 0$.
  • Figure 4: The balls $\mathscr{B}_{\mathrm{pc}}(O,r)$ centered at the origin $O=(0,0)$ and radii $r=10, 17$. The areas of the balls are $931$ and $4349$, respectively.
  • Figure 5: Two paths connecting $P=(5,2)$ and $Q = (15,23)$ through segments that combine taxicab steps with steps climbing the parabolic ladders shown in two zoom-levels. The first path, proving to be minimal, passes through points $(5, 2)$, $(5, 9)$, $(14, 9)$, $(14, 8)$, $(15, 8)$, $(15, 23)$, and thus $\operatorname{d_{pc}}(P,Q)=5$. The second path, shown dotted, has $9$ steps and passes through points $(5, 2)$, $(6, 2)$, $(7, 2)$, $(7, 13)$, $(8, 13)$, $(8, 12)$, $(17, 12)$, $(17, 23)$, $(16, 23)$, $(15, 23)$. Compare $\operatorname{d_{pc}}(P,Q)=5$ with the taxicab distance, which is equal to $(15-5)+(23-2)=31$, and with the Euclidean distance, which is $\sqrt{541}\approx 23.25941$.
  • ...and 6 more figures

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Conjecture 1
  • Theorem 3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 6.1
  • proof