Pythagorean addition refers to a mathematical concept that arises from the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Quantification of Margins and Uncertainties (QMU) is a systematic approach used typically in engineering, particularly in the fields of aerospace, nuclear, and other complex systems, to assess and manage the uncertainties and margins in performance predictions of a system. The objective of QMU is to provide a comprehensive understanding of how uncertainties in various inputs and parameters affect the performance and reliability of a system.
A Radial Basis Function (RBF) is a real-valued function whose value depends only on the distance from a center point, typically in a multi-dimensional space. RBFs are used in various applications, including interpolation, function approximation, and machine learning, particularly in radial basis function networks and support vector machines. ### Key Characteristics: 1. **Distance-Based**: The function typically measures the distance from a point in space to a center point (also called a basis center).
Radial Basis Function (RBF) interpolation is a method used in numerical analysis and computational mathematics to interpolate scattered data points in multidimensional space. It is particularly effective for problems where the data is irregularly spaced, as it can approximate values at unmeasured points based on the values of known points. ### Key Concepts: 1. **Radial Basis Function**: An RBF is a real-valued function whose value depends only on the distance from a center point.
The rate of convergence refers to the speed at which a sequence approaches its limit or a solution in mathematical analysis, numerical methods, and optimization. Specifically, it quantifies how quickly the terms of a sequence get closer to a given value as the number of iterations or the index of the sequence increases.
Regge calculus is a mathematical formulation used in the field of general relativity and quantum gravity that provides a way to discretize spacetime. Developed by Tullio Regge in the 1960s, this approach allows for the study of Einstein's equations and gravitational dynamics in a non-continuous, piecewise linear manner.
The Regularized Meshless Method (RMM) is a numerical approach used to solve partial differential equations (PDEs) and other related problems in computational mechanics and engineering. It is part of the broader category of meshless methods, which are techniques for approximating solutions to differential equations without relying on a structured grid or mesh. This can be particularly useful for problems involving complex geometries, moving boundaries, or other situations where traditional mesh techniques may struggle.
Relative change and absolute change (often referred to simply as "difference") are two ways to express changes in a value, and they serve different purposes in analysis. ### Absolute Change (Difference) - **Definition**: Absolute change refers to the straightforward difference between two values.
In numerical analysis, the term "residual" refers to the difference between a computed solution and the exact solution of a mathematical problem. It quantifies the error or discrepancy in a numerical approximation.
Richardson extrapolation is a mathematical technique used to improve the accuracy of an approximation of a quantity by combining estimates obtained with different step sizes. It is commonly utilized in numerical analysis, especially when dealing with methods for solving differential equations, approximating integrals, or performing other numerical calculations.
A Riemann solver is a numerical method used to solve hyperbolic partial differential equations (PDEs) that arise in various applications, such as fluid dynamics, gas dynamics, and traffic flow. The term "Riemann problem" refers to an initial value problem for a conservation law which consists of a hyperbolic PDE with piecewise constant initial data — typically defined by two constant states separated by a discontinuity.
Ross' π lemma is a result in the field of measure theory, particularly concerning the integration of functions and properties related to measurability. The lemma is often used in situations involving the interchange of limits and integrals. Although it may not be universally recognized by all mathematicians under the name "Ross' π lemma," it is primarily attributed to the work of mathematician A. Ross. In essence, the lemma establishes conditions under which one can exchange limits and integrals for sequences of measurable functions.
The Ross–Fahroo Lemma is a result in the field of optimization, specifically in the context of optimal control and differential inclusions. It provides conditions under which the solution of an optimal control problem can be related to a particular type of differential equation or inclusions. While the lemma itself involves technical mathematical concepts, its application typically involves deriving necessary conditions for optimality and exploring the structure of control problems, particularly where the control may be subject to various constraints.
The Ross–Fahroo pseudospectral method is a numerical approach used in optimal control and trajectory optimization problems. It combines the concepts of pseudospectral methods with optimization techniques to solve nonlinear optimal control problems effectively. ### Key Features: 1. **Pseudospectral Methods**: These methods involve the use of polynomial approximations based on a set of collocation points (often Chebyshev or Legendre nodes) to approximate the state and control variables.
Round-off error, also known as rounding error, refers to the difference between the true value of a number and its approximated value due to the limitations of numerical representation in computers or mathematical calculations. This type of error occurs when a number cannot be represented exactly in a finite number of digits, leading to rounding during calculations.
Runge–Kutta methods are a family of iterative techniques used for solving ordinary differential equations (ODEs). These methods are employed to find numerical approximations to the solutions of initial value problems, where the goal is to compute the future values of a function given its current state and the rate of change defined by the differential equation. The most commonly used member of this family is the classical fourth-order Runge-Kutta method, often abbreviated as RK4.
The Runge–Kutta–Fehlberg method is a numerical technique used to solve ordinary differential equations (ODEs). It is an adaptive step size method, which is an extension of the classical Runge-Kutta methods. The method is primarily designed to achieve a balance between accuracy and computational efficiency, allowing for the use of variable step sizes based on the estimated error.
A Scale Co-occurrence Matrix (SCM) is often used in fields such as natural language processing, image analysis, and various data analysis tasks where the relationships between different entities or features are important. While the specific use and definition of a Scale Co-occurrence Matrix may vary depending on the context, here’s a general understanding: ### Definition: - **Co-occurrence Matrix**: A general co-occurrence matrix is a table that displays how often different items or features occur together across a dataset.
Semi-infinite programming (SIP) is a type of optimization problem that involves a finite number of variables but an infinite number of constraints.
Series acceleration refers to a set of mathematical techniques used to accelerate the convergence of an infinite series, making it converge more quickly or improving the accuracy of its sum. This is particularly useful when dealing with series that converge slowly, as it allows for more efficient computations and can help achieve a desired level of accuracy with fewer terms. Some common methods of series acceleration include: 1. **Euler's Transformation**: This is used primarily for alternating series to improve their convergence.