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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…
We introduce an infinite set of integer mappings that generalize the well-known Collatz-Ulam mapping and we conjecture that an infinite subset of these mappings feature the remarkable property of the Collatz conjecture, namely that they…
The Collatz conjecture is a famous math problem that was introduced by Lothar Collatz in 1937, and nobody has yet succeeded in proving or disproving it. In this article, I will analyze this problem with a new approach and I will discuss my…
We survey recent work that relates Pitman's transformation to a variety of classical integrable systems, including the box-ball system, the ultra-discrete and discrete KdV equations, and the ultra-discrete and discrete Toda lattice…
In 20th century mathematics, the field of topology, which concerns the properties of geometric objects under continuous transformation, has proved surprisingly useful in application to the study of discrete mathematics, such as…
We propose Reduced Collatz Conjecture that is equivalent to Collatz Conjecture, which states that every positive integer can return to an integer less than it, instead of 1. Reduced Collatz Conjecture is easier to explore because certain…
Efficient simulation of SDEs is essential in many applications, particularly for ergodic systems that demand efficient simulation of both short-time dynamics and large-time statistics. However, locally Lipschitz SDEs often require special…
Many engineering problems involve learning hidden dynamics from indirect observations, where the physical processes are described by systems of partial differential equations (PDE). Gradient-based optimization methods are considered…
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…
Let $q$ be an odd prime, and let $T_{q}:\mathbb{Z}\rightarrow\mathbb{Z}$ be the Shortened $qx+1$ map, defined by $T_{q}\left(n\right)=n/2$ if $n$ is even and $T_{q}\left(n\right)=\left(qn+1\right)/2$ if $n$ is odd. The study of the dynamics…
Repeatedly adding or subtracting the digital reversal to or from an integer, depending on which one is larger, can be treated as a dynamical system. On one hand, a three-digit version of this map running only two steps is the 1089…
A structured approach for the Collatz conjecture is presented using just the odd integers that are, in turn, divided into categories based on the roles they play such as Starter, Intermediary and Terminal. The expression 4x+1 is used as a…
This paper mainly investigates the circular open dimension problem (CODP), which consists of packing a set of circles of known radii into a strip of fixed width and unlimited length without overlapping. The objective is to minimize the…
Decisions made by machine learning systems have increasing influence on the world, yet it is common for machine learning algorithms to assume that no such influence exists. An example is the use of the i.i.d. assumption in content…
In this paper we investigate how the subgradients of the value function of a discrete-time convex Bolza problem evolve over time. In particular, we develop a discrete-time version of the characteristic method introduced by Rockafellar and…
Determining physical properties inside an object without access to direct measurements of target regions can be formulated as a specific type of \textit{inverse problem}. One of such problems is applied in \textit{Electrical Impedance…
The capacity to transfer knowledge across scientific domains relies on shared organizational principles. However, existing transfer-learning methodologies often fail to bridge radically heterogeneous systems, particularly under severe data…
Solving partial differential equations (PDEs) by learning the solution operators has emerged as an attractive alternative to traditional numerical methods. However, implementing such architectures presents two main challenges: flexibility…
Value Iteration is a widely used algorithm for solving Markov Decision Processes (MDPs). While previous studies have extensively analyzed its convergence properties, they primarily focus on convergence with respect to the infinity norm. In…
The aim of this research is to introduce the notion of multi-transitivity in non-autonomous discrete dynamical systems(NDDS) with respect to a vector. Necessary and sufficient conditions are obtained under which a NDDS is multitransitive…