相关论文: Some Observations on the 3x+1 Problem
We show that for most choices of an initial seed $x_0$, the sequence of the first $N$ iterates of $x_0$ under the $3x+1$ map approximately satisfies Benford's law.
By creating some new concepts and methods: checking tree, long unit path, direct contradiction unit pair, indirect contradiction unit pair, additional contradiction unit pair, 2-unit layer and 3-unit layer, redundant units, and destroying…
The present paper aims to survey known results and to point out the wealth of rather important open problems that are out there.
This article seeks to encourage a mathematical dialog regarding a possible solution to Beals Conjecture. It breaks down one of the worlds most difficult math problems into laymans terms and encourages people to question some of the most…
In this paper, we provide a comprehensive solution to the open problem regarding the existence of a recurrence formula for computing fixed points of the Josephus function precisely when the reduction constant is three. Incorporating this…
The Collatz conjecture, which posits that any positive integer will eventually reach 1 through a specific iterative process, is a classic unsolved problem in mathematics. This research focuses on designing an efficient algorithm to compute…
We consider multi-dimensional assignment problems in a probabilistic setting. Our main results are: (i) A new efficient algorithm for the 3-dimensional planar problem, based on enumerating and selecting from a set of "alternating-path…
The main objective of this paper is to investigate the explicit form, stability character and global behavior of solutions of the following two systems of rational difference equations x_{n+1}=((+(-)1)/(y_{n}(x_{n-1}+(-)1)+1)),…
Here we briefly discuss how negative numbers, or "negative probabilities", can naturally arise in probabilistic expressions and be given an operational interpretation. Like the use of negative numbers in arithmetical expressions, the use of…
Testing hypotheses is an issue of primary importance in the scientific research, as well as in many other human activities. Much clarification about it can be achieved if the process of learning from data is framed in a stochastic model of…
Data values in a dataset can be missing or anomalous due to mishandling or human error. Analysing data with missing values can create bias and affect the inferences. Several analysis methods, such as principle components analysis or…
A mapping conjugate to the Collatz mapping seems to imply that $\N=\{1,2,3,\ldots\}$ is partitioned in a trivial loop $\{1\}$ and `strings' that are ordered subsets of $\{\N \setminus 1\}$ that run from an element of $\{2+3\0\}$ to an…
Constrained counting is important in domains ranging from artificial intelligence to software analysis. There are already a few approaches for counting models over various types of constraints. Recently, hashing-based approaches achieve…
The research area of algorithms with predictions has seen recent success showing how to incorporate machine learning into algorithm design to improve performance when the predictions are correct, while retaining worst-case guarantees when…
To determine whether a number is congruent or not is an old and difficult topic and progress is slow. The paper presents a new theorem when a prime number is a congruent number or not. The proof is not necessarily any simpler or shorter…
We give an endorsement for Cornacchia's famous algorithm. Thus we do not claim anything new but an approach which is supposed to be simpler than those of previous works written with the same aim.
Consider convex optimization problems subject to a large number of constraints. We focus on stochastic problems in which the objective takes the form of expected values and the feasible set is the intersection of a large number of convex…
The $3x+k$ function $T_{k}(n)$ sends $n$ to $(3n+k)/2$ resp. $n/2,$ according as $n$ is odd, resp. even, where $k \equiv \pm 1~(\bmod \, 6)$. The map $T_k(\cdot)$ sends integers to integers, and for $m \ge 1$ let $n \rightarrow m$ mean that…
The problem of sequential change diagnosis is considered, where observations are obtained on-line, an abrupt change occurs in their distribution, and the goal is to quickly detect the change and accurately identify the post-change…
An algorithm is discussed for converting a class of recursive processes to a parallel system. It is argued that this algorithm can be superior to certain methods currently found in the literature for an important subset of problems. The…