Related papers: A Two-Operator Calculus for Arithmetic-Progression…
Using $p$-adic numbers, we partially categorize the cycles of a sizable class of polynomial dynamical systems. In turn, we prove a few results related to the non-trivial cycles of the $\textit{Collatz map}$ $\text{Col} : \mathbb{Z}_+ \to…
We describe a new Maple package for treating boundary problems for linear ordinary differential equations, allowing two-/multipoint as well as Stieltjes boundary conditions. For expressing differential operators, boundary conditions, and…
It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential…
This paper is an overview and survey of work on the 3x+1 problem, also called the Collatz problem, and generalizations of it. It gives a history of the problem. It addresses two questions: (1) What can mathematics currently say about this…
In this paper, we study the dynamics of an operator $\mathcal T$ naturally associated to the so-called Collatz map, which maps an integer $n \geq 0$ to $n / 2$ if $n$ is even and $3n + 1$ if $n$ is odd. This operator $\mathcal T$ is defined…
Several applications of slicing require a program to be sliced with respect to more than one slicing criterion. Program specialization, parallelization and cohesion measurement are examples of such applications. These applications can…
We consider the problem of enumerating Dyck paths staying weakly above the x-axis with a limit to the number of consecutive up steps, or a limit to the number of consecutive down steps. We use Finite Operator Calculus to obtain formulas for…
The \textit{Collatz's conjecture} is an unsolved problem in mathematics. It is named after Lothar Collatz in 1973. The conjecture also known as Syrucuse conjecture or problem. Take any positive integer $ n $. If $ n $ is even then divide it…
In this paper we propose two proximal gradient algorithms for fractional programming problems in real Hilbert spaces, where the numerator is a proper, convex and lower semicontinuous function and the denominator is a smooth function, either…
In this paper, we consider the problem of minimizing the sum of two convex functions subject to linear linking constraints. The classical alternating direction type methods usually assume that the two convex functions have relatively easy…
We establish an equivalent condition to the validity of the Collatz conjecture, using elementary methods. We derive some conclusions and show several examples of our results. We also offer a variety of exercises, problems and conjectures.
An operational approach to the Collatz Conjecture is presented. Scenarios are defined as strings of characters "s" (for "spike") and "d" (for "down") which symbolize the Collatz operations (3m+1)/2 and m/2 in a Collatz Series connecting two…
We present efficient algorithms to calculate trajectories for periodic Lorentz gases consisting of square lattices of circular obstacles in two dimensions, and simple cubic lattices of spheres in three dimensions; these become increasingly…
We present a methodology for parallel acceleration of learning in the presence of matrix orthogonality and unitarity constraints of interest in several branches of machine learning. We show how an apparently sequential elementary rotation…
The Path Contraction and Cycle Contraction problems take as input an undirected graph $G$ with $n$ vertices, $m$ edges and an integer $k$ and determine whether one can obtain a path or a cycle, respectively, by performing at most $k$ edge…
Two simple Cellular Automata, which mimic the Collatz-Ulam iterated map (3x+1 map), are introduced. These Cellular Automata allow to test efficiently the Collatz conjecture for very large numbers.
We consider a class of linear programs on graphs with total variation regularization and a budgetary constraint. For these programs, we give a characterization of basic solutions in terms of rooted spanning forests with orientation on the…
Just as the $\lambda$-calculus uses three primitives (abstraction, application, variable) as the foundation of functional programming, inheritance-calculus uses three primitives (record, definition, inheritance) as the foundation of…
This course, intended for undergraduates familiar with elementary calculus and linear algebra, introduces the extension of differential calculus to functions on more general vector spaces, such as functions that take as input a matrix and…
Derivatives and integration operators are well-studied examples of linear operators that commute with scaling up to a fixed multiplicative factor; i.e., they are scale-invariant. Fractional order derivatives (integration operators) also…