The forward problem in electrocardiology refers to the challenge of predicting the electric potentials on the body surface generated by the heart's electrical activity. In simpler terms, it involves modeling how the electrical signals produced by the heart propagate through the body and how those signals can be observed on the skin surface. ### Key Aspects of the Forward Problem: 1. **Electrical Activity of the Heart**: The heart generates electrical signals during each heartbeat, primarily through actions of specialized cardiac cells.
Gal’s accurate tables refer to a set of mathematical tables created by the Danish astronomer and mathematician, Niels Bohr Gal, in the early 20th century. These tables are specifically designed for accurate calculations in celestial mechanics, such as determining the positions of celestial objects or calculating the orbits of planets and moons.
The Galerkin method is a numerical technique for solving differential equations, particularly those arising in boundary value problems. It belongs to a family of methods known as weighted residual methods, which are used to approximate solutions to various mathematical problems, including partial differential equations (PDEs) and ordinary differential equations (ODEs). ### Key Concepts: 1. **Weak Formulation**: The Galerkin method begins by reformulating a differential equation into its weak (or variational) form.
The Generalized-strain mesh-free formulation refers to a numerical method used in the field of computational mechanics, particularly in the context of finite element analysis (FEA) and computational continuum mechanics. This approach is part of a broader category of mesh-free methods, which are designed to overcome some of the limitations associated with traditional mesh-based methods, such as the Finite Element Method (FEM).
The Generalized Gauss–Newton (GGN) method is an extension of the standard Gauss–Newton algorithm used for solving nonlinear least squares problems. The Gauss–Newton method is a nonlinear optimization technique that provides a way to find the minimum of a sum of squares of nonlinear functions. It is particularly useful when dealing with problems where the objective function can be expressed as a sum of squared residuals.
GetFEM++ is an open-source software library designed for the finite element method (FEM) in the numerical simulation of partial differential equations. It provides a flexible and extensible framework for solving problems in various fields such as engineering, physics, and applied mathematics.
Gradient Discretisation Method (GDM) is a numerical method used in the context of solving partial differential equations (PDEs), particularly those arising in fluid dynamics and other fields of continuum mechanics. The GDM is designed to achieve a balance between accuracy and computational efficiency, especially when dealing with the advection-dominated problems that are common in these fields.
A **guard digit** is a concept used in numerical computation and arithmetic to improve the accuracy of calculations, particularly in floating-point arithmetic. It refers to an extra digit that is added to the significant part (or mantissa) of a number during calculations to help minimize errors that can arise from rounding. When performing arithmetic operations, such as addition or multiplication, intermediate results can lose precision due to the limited number of digits that can be represented (the precision limit of the floating-point representation).
The Hermes Project is a research initiative focused on the development of a high-performance, open-source JavaScript engine designed for running JavaScript applications on mobile devices. The primary aim of the project is to optimize JavaScript execution for React Native, a popular framework for building mobile applications using JavaScript and React. Key features of the Hermes Project include: 1. **Performance Optimization**: Hermes is designed to improve the start-up time and overall performance of applications.
The "Hundred-dollar, Hundred-digit Challenge" is an educational activity designed to engage students in mathematical problem-solving and creative thinking. The challenge typically involves creating a series of problems or exercises that utilize exactly one hundred digits to make a total of one hundred dollars. Participants are often encouraged to use various mathematical operations and creative strategies to form their solutions.
INTLAB is a software package designed for the rigorous and verified numerical computation of mathematical problems. It is specifically aimed at interval arithmetic, a technique used to handle uncertainties and errors that arise in numerical calculations. By using intervals to represent ranges of values, INTLAB allows for more reliable results compared to traditional floating-point arithmetic.
Interval arithmetic is a mathematical technique used to handle and represent ranges of values, rather than single precise numbers. In interval arithmetic, numbers are represented as intervals, which consist of a lower bound and an upper bound. For example, an interval \([a, b]\) represents all real numbers \(x\) such that \(a \leq x \leq b\).
An **Interval Contractor** is a concept primarily used in mathematical optimization and interval analysis. It refers to a technique or method that manages and works with intervals, which are ranges of values rather than specific points. This approach is especially useful in dealing with uncertainties and variables that can take on a range of values. In optimization problems, interval arithmetic is employed to identify feasible solutions that satisfy various constraints, even when those constraints contain uncertainties.
Interval propagation is a numerical method used primarily in the field of computer science, engineering, and mathematics to efficiently manage and analyze uncertainty in computations, particularly in the context of systems that involve constraints or nonlinear relationships. The main idea behind interval propagation is to work with ranges (or intervals) of possible values rather than with single point estimates.
Isotonic regression is a non-parametric regression technique used to find a best-fit line or curve that preserves the order of the data points. The objective of isotonic regression is to find a piecewise constant function that minimizes the sum of squared deviations from the observed values while ensuring that the fitted values are non-decreasing (i.e., they maintain the order of the independent variable).
An iterative method is a mathematical or computational technique that generates a sequence of approximations to a solution of a problem, with each iteration building upon the previous one. This approach is often used when direct methods are difficult to apply or when a solution cannot be expressed explicitly. ### Key Characteristics of Iterative Methods: 1. **Initial Guess**: An initial approximation, called the guess or starting point, is required. The success of the method can depend heavily on the choice of this initial value.
The Iterative Rational Krylov Algorithm (IRKA) is a numerical method used primarily for model order reduction of linear dynamical systems. It is particularly useful in control theory and numerical linear algebra for reducing the complexity of systems while preserving their essential dynamical properties. Here's a brief overview of the concepts and methodology involved in IRKA: ### Background 1. **Model Order Reduction (MOR)**: In many applications, high-dimensional systems (e.g.
The Jenkins–Traub algorithm is a numerical method used for finding the roots of polynomials. It is particularly effective for finding all the roots, including both real and complex roots, of a polynomial with real coefficients. The algorithm is notable for its efficiency and robustness. ### Key Features of Jenkins–Traub Algorithm: 1. **Root-Finding**: It finds all the roots of a polynomial in a systematic manner, starting from an initial guess and refining this guess iteratively.
The Kahan summation algorithm, also known as compensated summation, is a numerical technique used to improve the precision of the summation of a sequence of floating-point numbers. It mitigates the error that can occur when small numbers are added to large numbers, a common issue in floating-point arithmetic due to limited precision. ### How it Works The algorithm maintains an extra variable (often called `c`, for "compensation") that keeps track of small error terms.
The Kantorovich Theorem is a result in the field of mathematics, particularly in functional analysis and optimal transport theory. Named after the Soviet mathematician Leonid Kantorovich, the theorem provides conditions under which certain optimization problems can be solved effectively. One of the most significant applications of the Kantorovich Theorem is in the context of the optimal transport problem, which involves finding the most efficient way to transport goods from suppliers to consumers while minimizing costs.