AutocorrelationAutocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable as a function of the time lag between them. The analysis of autocorrelation is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal obscured by noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies.
Nuclear magnetic resonance spectroscopy of proteinsNuclear magnetic resonance spectroscopy of proteins (usually abbreviated protein NMR) is a field of structural biology in which NMR spectroscopy is used to obtain information about the structure and dynamics of proteins, and also nucleic acids, and their complexes. The field was pioneered by Richard R. Ernst and Kurt Wüthrich at the ETH, and by Ad Bax, Marius Clore, Angela Gronenborn at the NIH, and Gerhard Wagner at Harvard University, among others.
Admittance parametersAdmittance parameters or Y-parameters (the elements of an admittance matrix or Y-matrix) are properties used in many areas of electrical engineering, such as power, electronics, and telecommunications. These parameters are used to describe the electrical behavior of linear electrical networks. They are also used to describe the small-signal (linearized) response of non-linear networks. Y parameters are also known as short circuited admittance parameters.
Cross-covarianceIn probability and statistics, given two stochastic processes and , the cross-covariance is a function that gives the covariance of one process with the other at pairs of time points. With the usual notation for the expectation operator, if the processes have the mean functions and , then the cross-covariance is given by Cross-covariance is related to the more commonly used cross-correlation of the processes in question.
Supersymmetric theory of stochastic dynamicsSupersymmetric theory of stochastic dynamics or stochastics (STS) is an exact theory of stochastic (partial) differential equations (SDEs), the class of mathematical models with the widest applicability covering, in particular, all continuous time dynamical systems, with and without noise. The main utility of the theory from the physical point of view is a rigorous theoretical explanation of the ubiquitous spontaneous long-range dynamical behavior that manifests itself across disciplines via such phenomena as 1/f, flicker, and crackling noises and the power-law statistics, or Zipf's law, of instantonic processes like earthquakes and neuroavalanches.
Stochastic differential equationA stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs have many applications throughout pure mathematics and are used to model various behaviours of stochastic models such as stock prices, random growth models or physical systems that are subjected to thermal fluctuations. SDEs have a random differential that is in the most basic case random white noise calculated as the derivative of a Brownian motion or more generally a semimartingale.
Simple harmonic motionIn mechanics and physics, simple harmonic motion (sometimes abbreviated ()) is a special type of periodic motion an object experiences due to a restoring force whose magnitude is directly proportional to the distance of the object from an equilibrium position and acts towards the equilibrium position. It results in an oscillation that is described by a sinusoid which continues indefinitely (if uninhibited by friction or any other dissipation of energy).
Limit (mathematics)In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals. The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to and direct limit in . In formulas, a limit of a function is usually written as (although a few authors use "Lt" instead of "lim") and is read as "the limit of f of x as x approaches c equals L".
Phase transitionIn chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of matter: solid, liquid, and gas, and in rare cases, plasma. A phase of a thermodynamic system and the states of matter have uniform physical properties. During a phase transition of a given medium, certain properties of the medium change as a result of the change of external conditions, such as temperature or pressure.
Limit inferior and limit superiorIn mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant.
