In geometry, a **facet** refers to a flat surface that forms part of the boundary of a higher-dimensional geometric object. The term is most commonly used in the context of polyhedra and polytopes. 1. **In Polyhedra**: For a three-dimensional polyhedron, a facet typically refers to one of the polygonal faces of the polyhedron. For example, a cube has six facets, all of which are square faces.

A Gale diagram, also known as a Gale's diagram or Gale's bipartite representation, is a graphical representation used in combinatorial optimization, particularly in the context of matching problems. In essence, a Gale diagram illustrates the relationships between two sets of items, typically referred to as agents and tasks, in a bipartite graph format. It facilitates visualization of the possible pairings between the two sets, often highlighting preferences or weights associated with each potential pairing.

Kalai's 3-dimensional conjecture, proposed by Gil Kalai, pertains to the geometry of convex polytopes. The conjecture specifically addresses the conditions under which a simplicial complex can be realized as the nerve of a covering by open sets in a topological space. More concretely, it asserts that any simplicial complex that has a specific homotopy type will have a realization in a three-dimensional space.

The Bombieri norm is a concept encountered in the study of number theory, particularly in the context of the distribution of prime numbers and analytic number theory. Named after mathematician Enrico Bombieri, the Bombieri norm is often defined in the context of bounding sums or integrals that involve characters or exponential sums, playing a role in various results related to prime number distributions, especially in the understanding of the Riemann zeta function and L-functions.

In mathematics, a "bring radical" refers to a specific type of radical expression used to solve equations involving higher-degree polynomials, especially the general quintic equation. The bring radical is derived from the "Bring-Jerrard form" of a cubic polynomial. In essence, the Bring radical is often studied in the context of finding roots of polynomials that do not have explicit formulas involving only radicals for degrees five and higher.

The Coefficient Diagram Method (CDM) is a technique used in the field of control systems and engineering, specifically for the design and analysis of robust and high-performance control systems. It provides a systematic way to create control laws by using polynomial representations of system dynamics and control objectives. ### Key Aspects of the Coefficient Diagram Method 1.

A polynomial sequence is a sequence of numbers or terms that can be defined by a polynomial function. Specifically, a sequence \( a_n \) is said to be a polynomial sequence if there exists a polynomial \( P(x) \) of degree \( d \) such that: \[ a_n = P(n) \] for all integers \( n \) where \( n \geq 0 \) (or sometimes for \( n \geq 1 \)).

Tensiomyography (TMG) is a non-invasive diagnostic tool used to assess muscle function and determine muscle contractile properties. It measures the muscle's response to electrical stimulation, providing data on muscle tone, contraction time, relaxation time, and other parameters. In a typical TMG procedure, a small electrical impulse is delivered to the muscle, and specialized sensors placed on the skin record the muscle's contraction and relaxation patterns.

Steinitz's theorem, named after mathematician Georg Cantor Steinitz, is a fundamental result in convex geometry and linear algebra related to the representation of convex polyhedra. The theorem states that a finite set of points in a Euclidean space forms the vertices of a convex polyhedron if and only if: 1. The points can be represented as the convex hull of those points.

"Unique sink orientation" is not a widely recognized term in literature or specific fields of study as of my last knowledge update in October 2023, and it may refer to a concept in a specialized area such as ecology, hydrology, or perhaps even computer science or data structures where "sink" (referring to a point where items are collected or processed) is a relevant term.

Generating functions are a powerful mathematical tool used in combinatorics, probability, and other areas of mathematics to encode sequences of numbers into a formal power series. Essentially, a generating function provides a way to express an infinite sequence as a single entity, allowing for easier manipulation and analysis.

Bernoulli polynomials are a sequence of classical orthogonal polynomials that arise in various areas of mathematics, particularly in number theory, combinatorics, and approximation theory. They are defined using the following generating function: \[ \frac{t}{e^t - 1} = \sum_{n=0}^{\infty} B_n \frac{t^n}{n!

The geometrical properties of polynomial roots involve understanding how the roots (or solutions) of a polynomial equation are distributed in the complex plane, as well as their relationship to the coefficients of the polynomial. Here are some key geometrical concepts and properties related to the roots of polynomials: ### 1. **Complex Roots and the Complex Plane**: - Roots of polynomials can be real or complex.

An algebraic equation is a mathematical statement that expresses the equality between two algebraic expressions. It involves variables (often represented by letters such as \(x\), \(y\), etc.), constants, and arithmetic operations, such as addition, subtraction, multiplication, and division.

An algebraic function is a type of mathematical function that can be defined as the root of a polynomial equation.

Angelescu polynomials are a class of orthogonal polynomials that arise in certain contexts in mathematics, particularly in algebra and analysis. They are typically defined via specific recurrence relations or differential equations. While they are not as widely known as classical families like Legendre, Hermite, or Chebyshev polynomials, they do have special properties and applications in various areas, including numerical analysis and approximation theory. The properties and definitions of Angelescu polynomials often depend on the context in which they arise.

Bernoulli polynomials of the second kind, denoted by \( B_n^{(2)}(x) \), are a sequence of polynomials that are closely related to the traditional Bernoulli polynomials. They are defined through specific properties and relationships with other mathematical functions.

The term "binomial type" can refer to a few different concepts depending on the context, especially in mathematics and statistics. Here are a few interpretations: 1. **Binomial Distribution**: In statistics, a binomial type often refers to the binomial distribution, which models the number of successes in a fixed number of independent Bernoulli trials (experiments with two possible outcomes: success or failure).

The Bollobás–Riordan polynomial is a polynomial invariant associated with a graph-like structure called a "graph with a surface". It generalizes several concepts in graph theory, including the Tutte polynomial for planar graphs and other types of polynomials related to graph embeddings. The Bollobás–Riordan polynomial is primarily used in the study of graphs embedded in surfaces, particularly in the context of `k`-edge-connected graphs and their combinatorial properties.