Children: Polynomials
The ring of symmetric functions is a mathematical structure in the field of algebra, particularly in combinatorics and representation theory. It consists of symmetric polynomials, which are polynomials that remain unchanged when any of their variables are permuted. This ring serves as a fundamental object of study due to its rich structure and various applications.
Romanovski polynomials are a class of orthogonal polynomials that generalize classical orthogonal polynomials such as Hermite, Laguerre, and Legendre polynomials. They are named after the Russian mathematician A. V. Romanovski, who studied these polynomials in the context of certain orthogonal polynomial systems. These polynomials can be characterized by their orthogonality properties with respect to specific weight functions on defined intervals, and they satisfy certain recurrence relations.
The Rook polynomial is a combinatorial polynomial used in the study of permutations and combinatorial objects on a chessboard-like grid, specifically related to the placement of rooks on a chessboard. The Rook polynomial encodes information about the number of ways to place a certain number of non-attacking rooks on a chessboard of specified dimensions.
In mathematics, particularly in complex analysis and algebra, a root of unity is a complex number that, when raised to a certain positive integer power \( n \), equals 1.
The Rosenbrock function, often referred to as the Rosenbrock's valley or Rosenbrock's banana function, is a non-convex function used as a performance test problem for optimization algorithms. It is defined in two dimensions as: \[ f(x, y) = (a - x)^2 + b(y - x^2)^2 \] where \(a\) and \(b\) are constants.
The Routh–Hurwitz stability criterion is a mathematical test used in control theory to determine the stability of a linear time-invariant (LTI) system based on the coefficients of its characteristic polynomial. Specifically, it helps assess whether all poles of the system's transfer function have negative real parts, which is a necessary condition for the system to be stable.
Ruffini's rule is a mathematical technique used for dividing polynomials, especially when dividing a polynomial by a linear divisor of the form \( (x - c) \). This method provides a systematic way to find the quotient and remainder of polynomial division without performing long division.
Shapiro polynomials, also known as Shapiro's polynomials or Shapiro's equations, are a specific sequence of polynomials that arise in the study of certain mathematical problems, particularly in the context of probability and combinatorics. These polynomials are associated with various mathematical constructs, such as generating functions and interpolation. The Shapiro polynomials are defined recursively, and they exhibit properties related to roots and symmetry, making them useful in various theoretical frameworks.
The Sheffer sequence refers to a specific type of sequence of polynomials that can be used in the context of combinatorics and algebra. In particular, it is associated with generating functions and is useful in the study of combinatorial structures. More formally, the Sheffer sequence is a sequence of polynomials \( \{ P_n(x) \} \) such that there is an exponential generating function associated with it.
The Sister Beiter conjecture is a conjecture in the field of number theory, specifically relating to the distribution of prime numbers. It was proposed by the mathematician Sister Mary Beiter, who is known for her work in this area. The conjecture suggests that there is a certain predictable pattern or behavior in the distribution of prime numbers, particularly regarding their spacing and density within the set of natural numbers.
The stability radius is a concept used in control theory and systems analysis to measure the robustness of a control system with respect to changes in its parameters or structure. Specifically, it quantifies the maximum amount of perturbation (or change) that can be introduced to a system before it becomes unstable. ### Key points related to stability radius: 1. **Perturbation**: This refers to any changes in the system dynamics, such as alterations in system parameters, modeling errors, or external disturbances.
A stable polynomial is a concept primarily used in control theory and mathematics, particularly in the study of dynamical systems. A polynomial is defined as stable if all of its roots (or zeros) lie in the left half of the complex plane.
Stanley symmetric functions are a family of symmetric functions that arise in combinatorics, particularly in the study of partitions, representation theory, and algebraic geometry. They were introduced by Richard Stanley in the context of the theory of symmetric functions and are particularly important in the study of stable combinatorial structures.
Stirling polynomials are a family of polynomials related to Stirling numbers, which arise in combinatorics, particularly in the context of partitioning sets and distributions of objects. There are two main types of Stirling numbers: the "Stirling numbers of the first kind" \( S(n, k) \) and the "Stirling numbers of the second kind" \( \left\{ n \atop k \right\} \).
A **symmetric polynomial** is a polynomial in several variables that remains unchanged (symmetric) under any permutation of its variables.
The Theory of Equations is a branch of mathematics that deals with the study of equations and their properties, solutions, and relationships. It primarily focuses on polynomial equations, which are equations in which the unknown variable is raised to a power and combined with constants. Here are some key concepts within the Theory of Equations: 1. **Polynomial Equations**: These are equations of the form \( P(x) = 0 \), where \( P(x) \) is a polynomial.
Thomae's formula is a mathematical result in the theory of functions of several complex variables, particularly concerning the computation of certain types of integrals in the context of complex analysis. More specifically, Thomae's formula provides a way to express a certain type of integrals related to the complex form of the elliptic functions.
Touchard polynomials, named after the French mathematician Jacques Touchard, are a sequence of polynomials that arise in the study of combinatorial structures, particularly in connection with the enumeration of permutations and other combinatorial configurations. These polynomials can be defined using the generating function approach for certain combinatorial objects, such as exponential generating functions for permutations with specific properties. Touchard polynomials can be expressed in several equivalent ways, including through a recursive formula or by explicit polynomial forms.