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The $3x+1$ problem, also called the Collatz conjecture, is a very interesting unsolved mathematical problem related to computer science. This paper generalized this problem by relaxing the constraints, i.e., generalizing this deterministic…

In this paper it will be shown that the Standard Model in 3+1 dimensions is a gauge fixed version of a 2T-physics field theory in 4+2 dimensions, thus establishing that 2T-physics provides a correct description of Nature from the point of…

This study analyzes the Collatz map through nonlinear dynamics. By embedding integers in Sharkovsky's ordering, we show that odd initial values suffice for full dynamical characterization. We introduce ``direction phases'' to partition…

This paper gives a simple proof of the Wirsching-Goodwin representation of integers connected to 1 in the $3x+1$ problem (see \cite{Wirsching} and \cite{Goodwin}). This representation permits to compute all the ascending Collatz sequences…

Let $T \colon \mathbb{N} \to \mathbb{N}$ denote the $3x+1$ function, where $T(n)=n/2$ if $n$ is even, $T(n)=(3n+1)/2$ if $n$ is odd. As an accelerated version of $T$, we define a jump at $n \ge 1$ by jp$(n) = T^{(\ell)}(n)$, where $\ell$ is…

We study the complexity of constraint satisfaction problems for templates $\Gamma$ that are first-order definable in $(\Bbb Z; succ)$, the integers with the successor relation. Assuming a widely believed conjecture from finite domain…

We consider deterministic homogenization for discrete-time fast-slow systems of the form $$ X_{k+1} = X_k + n^{-1}a_n(X_k,Y_k) + n^{-1/2}b_n(X_k,Y_k)\;, \quad Y_{k+1} = T_nY_k\;$$ and give conditions under which the dynamics of the slow…

We study synchronization for linearly coupled temporal networks of heterogeneous time-dependent nonlinear agents via the convergence of attracting trajectories of each node. The results are obtained by constructing and studying the…

We analyze the coherent dynamics of excitons in three dimensional topologically disordered networks with traps. If the interactions between the nodes of the network are long ranged, i.e., algebraically decaying as a function of the distance…

Join-preserving maps on the discrete time scale $\omega^+$, referred to as time warps, have been proposed as graded modalities that can be used to quantify the growth of information in the course of program execution. The set of time warps…

In this paper, we consider the one-to-one correspondence between a 2-adic integer and its parity sequence under iteration of the so-called "3x+1" map. First, we prove a new formula for the inverse transform. Next, we briefly review what is…

In this paper we propose some continuation theorems for the periodic problem \begin{equation*} \begin{cases} \, x_{i}' = g_{i}(t,x_{i+1}), &i=1,\ldots,n-1, \\ \, x_{n}' = h(t,x_{1},\ldots,x_{n}), \\ \, x_{i}(0)=x_{i}(T), &i=1,\ldots,n,…

We prove that deterministic motion in dissipative systems emerges as a strict geometric attractor of contact flow, not a statistical approximation. Building on the contact geometry of stochastic vector bundles, we develop time-dependent…

We study nondeterministic and probabilistic versions of a discrete dynamical system (due to T. Antal, P. L. Krapivsky, and S. Redner) inspired by Heider's social balance theory. We investigate the convergence time of this dynamics on…

We present a scenario in $1 + 1$ and $3 + 1$ dimensional space time which is paradoxical in the presence of a time machine. We show that the paradox cannot be resolved and the scenario has {\em no} consistent classical solution. Since the…

In this paper, we study abstract dynamical systems with discrete phase spaces. One example of such a system is induced by the $3 x{+}1$-map on the set of all natural numbers, also known as the Collatz map. Our main focus is on dynamical…

Our contribution is a bounded cubic compilation theorem. For each fixed resource parameter $k$, syntactic proof checking at resource level $k$ is faithfully represented by a finite bounded-domain system of cubic polynomial equations. Every…

The three gap theorem, also known as the Steinhaus conjecture or three distance theorem, states that the gaps in the fractional parts of $\alpha,2\alpha,\ldots, N\alpha$ take at most three distinct values. Motivated by a question of…

We derive a canonical form for 2-group gauge theory in 3+1D which shows they are either equivalent to Dijkgraaf-Witten theory or to the so-called "EF1" topological order of Lan-Wen. According to that classification, recently argued from a…

In this work, we introduce another extension U of the 3n+1 function to the real line. We propose a conjecture about the U-trajectories that generalizes the famous 3n+1 (or Collatz) conjecture. We then prove our main result about the…