A Pillai prime is a type of prime number characterized by its relationship to the factorial function. Specifically, a Pillai prime \( p \) is defined as a prime number for which there exists a positive integer \( n \) such that \( n! \equiv -1 \mod p \). This means that when \( n! \) (the factorial of \( n \)) is divided by the prime \( p \), it leaves a remainder of \( p - 1 \).

The Poisson binomial distribution is a generalization of the binomial distribution. It is used to model the number of successes in a sequence of independent Bernoulli trials, where each trial can have a different probability of success. In contrast, the binomial distribution assumes that each trial has the same probability of success. ### Key Characteristics: 1. **Independent Trials**: The trials are independent of each other.

The pseudogamma function is a mathematical function that generalizes the concept of the gamma function. While the traditional gamma function, denoted as \(\Gamma(z)\), is defined for complex numbers with a positive real part, the pseudogamma function can be used in a wider context, particularly in the field of number theory and special functions. One common interpretation of the pseudogamma function is based on the notion of providing alternatives or approximations to the gamma function.

The Sierpiński triangle, also known as the Sierpiński gasket or Sierpiński sieve, is a fractal and attractive fixed set with an overall shape that resembles an equilateral triangle. It is constructed through a recursive process that involves removing smaller triangles from a larger triangle. Here’s how it is usually created: 1. **Start with an equilateral triangle**: Begin with a solid equilateral triangle.

The Portable Document Format (PDF) is a file format developed by Adobe that allows documents to be presented in a manner independent of application software, hardware, and operating systems. Here’s a brief history of PDF: ### Early Development - **1980s**: The concept of a portable document format originated in the late 1980s. Adobe co-founder John Warnock initiated the idea to create electronic documents that could be easily shared across different systems and platforms.

Stirling numbers of the first kind, denoted by \(c(n, k)\), count the number of ways to express a permutation of \(n\) elements as a product of \(k\) disjoint cycles. In other words, they are used in combinatorial mathematics to determine how many different ways a set can be partitioned into cycles.

The Stirling numbers of the second kind, denoted as \( S(n, k) \), are a set of combinatorial numbers that count the ways to partition a set of \( n \) objects into \( k \) non-empty subsets. In other words, \( S(n, k) \) gives the number of different ways to group \( n \) distinct items into \( k \) groups, where groups can have different sizes but cannot be empty.

The Stirling transform is a mathematical technique used to convert sequences or series of numbers into a different form, often converting between combinatorial entities. It is particularly useful in the context of generating functions and combinatorial identities.

The Table of Newtonian series is a representation of polynomial expansions that can be used to express functions in terms of power series, particularly useful in numerical methods and approximation. Specifically, it refers to the series expansion and approximations that come from Newton's interpolation formula. Newton's interpolation formula is a method for estimating the value of a function at a given point based on known values of the function at discrete points.

Trinomial expansion refers to the process of expanding expressions that are raised to a power and involve three terms, typically represented in the form \((a + b + c)^n\), where \(a\), \(b\), and \(c\) are the terms and \(n\) is a non-negative integer. The formula for expanding a trinomial can be derived from the multinomial theorem, which generalizes the binomial theorem (the latter which deals only with two terms).

In measure theory, **content** is a concept used to generalize the idea of a measure for certain sets, particularly in the context of subsets of Euclidean spaces. While measures, such as Lebesgue measure, are defined for a broader class of sets and satisfy certain properties (like countable additivity), content is often used for more irregular sets that may not have a well-defined measure under the Lebesgue measure. **Key Aspects of Content:** 1.

In topology, a **cover** (or covering) of a topological space is a collection of subsets of that space whose union contains the entire space.

A Levi graph is a type of bipartite graph that provides a way to represent the relationships between points and lines (or more generally, between different types of geometric or combinatorial objects) in a projective geometry or other similar contexts. In the context of projective geometry: 1. **Vertices**: The vertices of a Levi graph can be divided into two disjoint sets, typically referred to as points and lines.

Cake-cutting refers to a problem and methodology in fair division, particularly in the context of allocating resources among multiple parties. It is often illustrated with the analogy of dividing a cake (or any divisible good) among several individuals in a way that each person believes they have received a fair share. The main goals of cake-cutting are to ensure fairness and avoid conflicts during the division process.

The Union-Closed Sets Conjecture is a problem in combinatorial set theory that deals with the properties of families of sets.

In set theory, a **universal set** is defined as the set that contains all possible elements within a particular context or discussion. It serves as a boundary for other sets being considered and encompasses all objects of interest relevant to a particular problem or situation.

Fairness criteria refer to a set of standards or principles used to evaluate and ensure equitable treatment and outcomes in various contexts, particularly in areas such as machine learning, data science, public policy, and social justice. The goal of applying fairness criteria is to mitigate bias, promote equity, and ensure that decisions and predictions are just and do not discriminate against any particular group based on characteristics such as race, gender, age, socioeconomic status, or other attributes.

The Dubins-Spanier theorems are results in the theory of stochastic processes, particularly concerning the behavior of certain classes of random walks, especially in relation to the concepts of ergodicity and recurrence. ### Key Concepts: 1. **Random Walks**: A type of stochastic process that describes a path consisting of a succession of random steps. Random walks can be one-dimensional or multidimensional and are foundational in probability theory.

Disjoint sets, also known as union-find or merge-find data structures, are a data structure that keeps track of a partition of a set into disjoint (non-overlapping) subsets. The main operations that can be performed on disjoint sets are: 1. **Find**: Determine which subset a particular element belongs to. This usually involves finding the "representative" or "root" of the set that contains the element.