Children: Polynomials
An exponential polynomial is a type of mathematical expression that combines both polynomial terms and exponential terms.
An **external ray** is a concept in the field of complex dynamics, particularly in the study of Julia sets and the Mandelbrot set. It is used to describe a ray emanating from a point in the complex plane that enters or exits a fractal set. In more precise terms, external rays are typically defined in relation to a point on the boundary of a Julia set or the Mandelbrot set.
Fibonacci polynomials are a sequence of polynomials that are related to the Fibonacci numbers. They are defined recursively, similar to the Fibonacci numbers themselves. The \(n\)-th Fibonacci polynomial, denoted \(F_n(x)\), can be defined as follows: 1. \(F_0(x) = 0\), 2. \(F_1(x) = 1\), 3.
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.
A graph polynomial is a mathematical function associated with a graph that encodes information about the graph's structure and properties. There are various types of graph polynomials, each of which serves different purposes in combinatorics, algebra, and graph theory. Here are a few notable types: 1. **Chromatic Polynomial**: This polynomial counts the number of ways to color the vertices of a graph such that no two adjacent vertices share the same color.
The HOMFLY polynomial is a knot invariant, which means it is a mathematical object that can be used to distinguish different knots and links in three-dimensional space. It extends the concepts of the Alexander polynomial and the Jones polynomial, making it a more powerful tool in the study of knot theory. The HOMFLY polynomial was introduced by HOMFLY, which is an acronym for the initials of the authors: H. G. H. Kauffman, M. W. W. L.
Hermite polynomials are a set of orthogonal polynomials that arise in probability, combinatorics, and physics, particularly in the context of quantum mechanics and the study of harmonic oscillators. They are defined by a specific recurrence relation and can be generated using generating functions.
Hilbert's Nullstellensatz, or the "Zeroes Theorem," is a fundamental result in algebraic geometry that relates algebraic sets to ideals in polynomial rings. It essentially provides a bridge between geometric concepts and algebraic structures. There are two main forms of the Nullstellensatz, often referred to as the strong and weak versions.
Hilbert's thirteenth problem is one of the 23 problems proposed by the German mathematician David Hilbert in 1900. Specifically, the problem is concerned with the nature of continuous functions and their representations. Hilbert's thirteenth problem asks whether every continuous function of two variables can be represented as a composition of continuous functions of one variable.
The Hiptmair–Xu preconditioner is a mathematical tool used to improve the convergence of iterative methods for solving linear systems that arise from discretized partial differential equations (PDEs). It is particularly useful for problems governed by elliptic PDEs, including those that result from finite element discretizations. The preconditioner is named after its developers, who introduced it to address the challenges associated with solving large, sparse systems of equations.
A Hurwitz polynomial is a type of polynomial that has specific properties related to its roots, which are closely connected to stability in control theory and systems engineering. Specifically, a polynomial is called a Hurwitz polynomial if all of its roots have negative real parts, meaning they lie in the left half of the complex plane. This characteristic indicates that the system represented by the polynomial is stable.
An **integer-valued polynomial** is a polynomial function that takes integer values for all integer inputs.
An invariant polynomial is a polynomial function that remains unchanged under certain transformations or actions of a group, particularly in the context of algebraic structures or geometric spaces. Invariant polynomials often arise in representations of groups, algebraic geometry, and invariant theory. For instance, consider a group \( G \) acting on a vector space \( V \).
The Jacobian Conjecture is a long-standing open problem in the field of mathematics, specifically in algebraic geometry and polynomial functions. It was first proposed by the mathematician Ottheinrich Keller in 1939. The conjecture concerns polynomial mappings from \( \mathbb{C}^n \) (the n-dimensional complex space) to itself.
The Jones polynomial is an invariant of a knot or link, introduced by mathematician Vaughan Jones in 1984. It is a powerful tool in knot theory that provides a polynomial invariant, assigning to each oriented knot or link a polynomial with integer coefficients. The Jones polynomial \( V(L, t) \) is defined using a specific state-sum formula based on a diagram of the knot or link.
Kazhdan–Lusztig polynomials are a family of polynomial invariants associated with representation theory, algebraic geometry, and combinatorial mathematics. They were introduced by David Kazhdan and George Lusztig in the context of the representation theory of semisimple Lie algebras, the theory of Hecke algebras, and the study of algebraic varieties.
A knot polynomial is a mathematical invariant associated with knots and links in the field of knot theory, which is a branch of topology. Knot polynomials are used to distinguish between different knots and to study their properties. Some of the most well-known knot polynomials include: 1. **Alexander Polynomial**: This is one of the earliest knot polynomials, defined for a knot or link as a polynomial in one variable. It provides insights into the topology of the knot and can help distinguish between different knots.
The Lagrange polynomial is a form of polynomial interpolation used to find a polynomial that passes through a given set of points.
Laguerre polynomials are a sequence of orthogonal polynomials that arise in various areas of mathematics and physics, particularly in quantum mechanics and numerical analysis. They are named after the French mathematician Edmond Laguerre. There are two main types of Laguerre polynomials: the associated Laguerre polynomials and the simple Laguerre polynomials. ### 1.
A Laurent polynomial is a type of polynomial that allows for both positive and negative integer powers of the variable.