Related papers: How Expressive Are Friendly School Partitions?
We study the simultaneous embeddability of a pair of partitions of the same underlying set into disjoint blocks. Each element of the set is mapped to a point in the plane and each block of either of the two partitions is mapped to a region…
The number of partitions of n into parts divisible by a or b equals the number of partitions of n in which each part and each difference of two parts is expressible as a non-negative integer combination of a or b. This generalizes…
Rounding has proven to be a fundamental tool in theoretical computer science. By observing that rounding and partitioning of $\mathbb{R}^d$ are equivalent, we introduce the following natural partition problem which we call the {\em secluded…
Recent advances in distilling pretrained language models have discovered that, besides the expressiveness of knowledge, the student-friendliness should be taken into consideration to realize a truly knowledgable teacher. Based on a pilot…
Recently, Andrews introduced separable integer partition classes and studied some well-known theorems. In this atricle, we will investigate six types of partitions from the view of the point of separable integer partition classes.
This paper considers three different partition relations from partition calculus, two of which are pair relations and one of which is a triple relation. An examination of the first partition relation and the ramification argument used to…
For three natural classes of dynamic decision problems; 1. additively separable problems, 2. discounted problems, and 3. discounted problems for a fixed discount factor; we provide necessary and sufficient conditions for one sequential…
We conduct a computational analysis of fair and optimal partitions in additively separable hedonic games. We show that, for strict preferences, a Pareto optimal partition can be found in polynomial time while verifying whether a given…
Let $B$ be an infinite subset of $\mathbf{N}$. When we consider partitions of natural numbers into elements of $B$, a partition number without a restriction of the number of equal parts can be expressed by partition numbers with a…
The set of tritangent planes to smooth tropical space sextic curves has 15 connected components, recording continuous displacements of planes preserving the tritangency condition. These 15 tritangent classes are polyhedral complexes in…
Whether teaching in a classroom or a Massive Online Open Course it is crucial to present the material in a way that benefits the audience as a whole. We identify two important tasks to solve towards this objective, 1 group students so that…
We consider the popular matching problem in a roommates instance with strict preference lists. While popular matchings always exist in a bipartite instance, they need not exist in a roommates instance. The complexity of the popular matching…
Partitioning a set of elements into an unknown number of mutually exclusive subsets is essential in many machine learning problems. However, assigning elements, such as samples in a dataset or neurons in a network layer, to an unknown and…
We introduce a new concept, which we call partition expanders. The basic idea is to study quantitative properties of graphs in a slightly different way than it is in the standard definition of expanders. While in the definition of expanders…
Separation is a classical problem in mathematics and computer science. It asks whether, given two sets belonging to some class, it is possible to separate them by another set of a smaller class. We present and discuss the separation problem…
In this study, we consider the real-world problem of assigning students to classes, where each student has a preference list, ranking a subset of classes in order of preference. Though we use existing approaches to include the daily class…
In this article, we first investigate the partitions whose parts are congruent to $a$ or $b$ modulo $k$ with the aid of separable integer partition classes with modulus $k$ introduced by Andrews. Then, we introduce the…
Iterative peer grading activities may keep students engaged during in-class project presentations. Effective methods for collecting and aggregating peer assessment data are essential. Students tend to grade projects favorably. So, while…
Alon, Bohman, Holzman and Kleitman proved that any partition of a $d$-dimensional discrete box into proper sub-boxes must consist of at least $2^d$ sub-boxes. Recently, Leader, Mili\'{c}evi\'{c} and Tan considered the question of how many…
10 is the smallest positive integer which is whether solitary or friendly is still an open question in mathematics. In this paper, we provide upper bounds for each of the prime divisors of a friend of 10. This paper is precisely a…