Linear multistep methods are numerical techniques used to solve ordinary differential equations (ODEs) by approximating the solutions at discrete points. Unlike single-step methods (like the Euler method or Runge-Kutta methods) that only use information from the current time step to compute the next step, linear multistep methods utilize information from multiple previous time steps.

Local convergence refers to the behavior of a sequence, series, or iterative method in relation to a specific point, usually in the context of numerical analysis, optimization, or iterative algorithms. It is an important concept in various fields such as mathematics, optimization, and numerical methods, especially when discussing convergence of sequences or functions.

A low-discrepancy sequence, also known as a quasi-random sequence, is a sequence of points in a multi-dimensional space that are designed to be more uniformly distributed than a purely random sequence. The goal of using a low-discrepancy sequence is to reduce the gaps between points and improve the uniformity of point distribution, which can lead to more efficient sampling and numerical integration, particularly in higher dimensions.

The modulus of smoothness is a concept used in functional analysis and approximation theory to measure the smoothness or regularity of a function. It provides a quantitative way to assess how "smooth" a function is by examining the variation of the function over a certain interval. The modulus of smoothness is often applied in the context of Banach spaces.

Numeric precision in Microsoft Excel refers to the level of detail and accuracy with which numbers are represented and calculated within the software. This includes considerations such as: 1. **Decimal Places**: The number of digits to the right of the decimal point that the software can display. Excel can handle a wide range of decimal places, but the display setting can affect how numbers appear.

The Overlap-Add method is a technique used in signal processing, particularly in the context of filtering and convolution. It is designed to efficiently compute the convolution of long signals with linear time-invariant (LTI) systems (filters) by breaking them into shorter segments. ### Key Concepts of the Overlap-Add Method: 1. **Segmentation**: The input signal is divided into overlapping segments.

Whitney's inequality is a result in the field of functional analysis and probability theory, particularly concerning the behavior of functions and measures. While the term may be used in different contexts, one common interpretation relates to bounds on stochastic processes or empirical measures. In one of its forms, Whitney's inequality gives a bound on the deviation of the empirical distribution from the true distribution.

The pseudo-spectral method is a numerical technique used for solving differential equations, particularly partial differential equations (PDEs). This method exploits the properties of orthogonal polynomial bases (such as Fourier series or Chebyshev polynomials) to transform the differential equations into a system of algebraic equations, making them more tractable for computation.

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.

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.

The Sterbenz lemma is a result in graph theory, particularly in the area of random graphs and percolation theory. It provides conditions under which a large connected component will exist in a random graph or a random structure. More specifically, the lemma is often discussed in the context of random graphs model \( G(n, p) \), where \( n \) is the number of vertices and \( p \) is the probability of an edge existing between any two vertices.

A surrogate model, often referred to as a meta-model or approximation model, is a mathematical model that approximates the behavior of a more complex, typically computationally expensive model or system. Surrogate models are commonly used in fields such as engineering, optimization, and machine learning to reduce the time and resources required to evaluate complex simulations or performances.

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.

A **transfer matrix** is a mathematical tool used in various fields, notably in physics, to analyze a system or process by relating the state of a system at one point to its state at another point. The concept is widely applied in statistical mechanics, condensed matter physics, quantum mechanics, and in the field of linear systems.

A geodesic grid is a type of coordinate system used primarily in geodesy, cartography, and various fields of mathematics and computer science to represent the surface of the Earth (or any spherical or spheroidal object) in a way that allows for accurate measurement and visualization.

Validated numerics is a computational technique used to ensure the accuracy and reliability of numerical results in scientific computing. It incorporates methods and frameworks to formally verify and validate the results of numerical computations, particularly when dealing with floating-point arithmetic, which can introduce errors due to its inherent limitations and approximations. Key aspects of validated numerics include: 1. **Bounding Enclosures**: Instead of producing a single numerical result, validated numerical methods often return an interval or bounding box that contains the true solution.

The Variational Multiscale Method (VMS) is a mathematical and computational technique used primarily in the field of fluid dynamics and continuum mechanics to effectively deal with the challenges of resolving various scales in turbulent flows. It is particularly useful for problems involving complex geometries and multi-physics interactions, where different physical phenomena occur at vastly different scales.

The National Weather Service (NWS) utilizes several numerical weather prediction models to forecast the weather. These models use complex mathematical equations to simulate the atmosphere's behavior based on current weather conditions, satellite data, and other observational data. The main functions of these models include: 1. **Data Assimilation**: The models take in vast amounts of observational data from various sources (e.g., satellites, radars, weather stations) to provide an accurate starting point for simulations.

Atmospheric reanalysis is a process that involves the integration of vast amounts of meteorological observations (such as temperature, pressure, wind, humidity, and precipitation) with sophisticated numerical weather models to produce a comprehensive, consistent, and high-quality representation of the Earth's atmosphere and its variability over time. This process typically covers a specific period, often spanning several decades, and generates datasets that are used for various research and practical applications, including climate studies, weather forecasting, and environmental monitoring.

The carbon cycle is the process through which carbon is exchanged between the Earth's atmosphere, land, oceans, and living organisms. It is a crucial component of the Earth's biosphere, facilitating the flow of carbon in various forms, such as carbon dioxide (CO2), organic compounds, and carbonates.