A **Happy Number** is defined as a number that eventually reaches 1 when replaced repeatedly by the sum of the squares of its digits. If it does not reach 1, it will enter a cycle that does not include 1, and it is then considered an unhappy number. The process for determining if a number is happy can be described as follows: 1. Take the number and replace it with the sum of the squares of its digits. 2. Repeat this process.

A **Feedback Arc Set** (FAS) is a concept in graph theory that refers to a specific type of subset of edges in a directed graph (digraph). The purpose of a feedback arc set is to eliminate cycles in the graph. More formally, a feedback arc set of a directed graph is a set of edges such that, when these edges are removed, the resulting graph becomes acyclic (i.e., it contains no cycles).

Masayoshi Tomizuka is a prominent figure in the field of engineering, particularly known for his work in control systems and robotics. He is a professor at the University of California, Berkeley, and has made significant contributions to areas such as adaptive control, estimation theory, and the development of algorithms for robotics and automation. His research often focuses on improving the performance of dynamic systems and addressing challenges in real-time control.

Once upon a time, when Ciro Santilli had a job, he had a programming problem.

A senior developer came over, and rather than trying to run and modify the code like an idiot, which is what Ciro Santilli usually does (see also experimentalism remarks at Section "Ciro Santilli's bad old event memory"), he just stared at the code for about 10 minutes.

We knew that the problem was likely in a particular function, but it was really hard to see why things were going wrong.

After the 10 minutes of examining every line in minute detail, he said:and truly, that was the cause.

And so, Ciro was enlightened.

A **Connected Dominating Set (CDS)** is a concept from graph theory, particularly in the study of network design and communication networks. It consists of a subset of vertices (nodes) in a graph that satisfies two main properties: 1. **Dominating Set**: The subset of vertices \( S \) is a dominating set, which means that every vertex not in \( S \) is adjacent to at least one vertex in \( S \).