Gateaux derivativeIn mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, a French mathematician who died at age 25 in World War I, it is defined for functions between locally convex topological vector spaces such as Banach spaces. Like the Fréchet derivative on a Banach space, the Gateaux differential is often used to formalize the functional derivative commonly used in the calculus of variations and physics.
Wh-movementIn linguistics, wh-movement (also known as wh-fronting, wh-extraction, or wh-raising) is the formation of syntactic dependencies involving interrogative words. An example in English is the dependency formed between what and the object position of doing in "What are you doing?" Interrogative forms are sometimes known within English linguistics as wh-words, such as what, when, where, who, and why, but also include other interrogative words, such as how.
DerivativeIn mathematics, the derivative shows the sensitivity of change of a function's output with respect to the input. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
Fundamental solutionIn mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not address boundary conditions). In terms of the Dirac delta "function" δ(x), a fundamental solution F is a solution of the inhomogeneous equation Here F is a priori only assumed to be a distribution. This concept has long been utilized for the Laplacian in two and three dimensions.
Stochastic partial differential equationStochastic partial differential equations (SPDEs) generalize partial differential equations via random force terms and coefficients, in the same way ordinary stochastic differential equations generalize ordinary differential equations. They have relevance to quantum field theory, statistical mechanics, and spatial modeling. One of the most studied SPDEs is the stochastic heat equation, which may formally be written as where is the Laplacian and denotes space-time white noise.
Generalizations of the derivativeIn mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, geometry, etc. The Fréchet derivative defines the derivative for general normed vector spaces . Briefly, a function , an open subset of , is called Fréchet differentiable at if there exists a bounded linear operator such that Functions are defined as being differentiable in some open neighbourhood of , rather than at individual points, as not doing so tends to lead to many pathological counterexamples.
Interrogative wordAn interrogative word or question word is a function word used to ask a question, such as what, which, when, where, who, whom, whose, why, whether and how. They are sometimes called wh-words, because in English most of them start with wh- (compare Five Ws). They may be used in both direct questions (Where is he going?) and in indirect questions (I wonder where he is going). In English and various other languages the same forms are also used as relative pronouns in certain relative clauses (The country where he was born) and certain adverb clauses (I go where he goes).
Numerical methods for partial differential equationsNumerical methods for partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs). In principle, specialized methods for hyperbolic, parabolic or elliptic partial differential equations exist. Finite difference method In this method, functions are represented by their values at certain grid points and derivatives are approximated through differences in these values.
Pronunciation of English ⟨wh⟩The pronunciation of the digraph in English has changed over time, and still varies today between different regions and accents. It is now most commonly pronounced /w/, the same as a plain initial , although some dialects, particularly those of Scotland, Ireland, and the Southern United States, retain the traditional pronunciation /hw/, generally realized as ʍ, a voiceless "w" sound. The process by which the historical /hw/ has become /w/ in most modern varieties of English is called the wine–whine merger.
Gödel's incompleteness theoremsGödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.
Mathematical constantA mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Constants arise in many areas of mathematics, with constants such as e and pi occurring in such diverse contexts as geometry, number theory, statistics, and calculus. Some constants arise naturally by a fundamental principle or intrinsic property, such as the ratio between the circumference and diameter of a circle (pi).
ClauseIn language, a clause is a constituent that comprises a semantic predicand (expressed or not) and a semantic predicate. A typical clause consists of a subject and a syntactic predicate, the latter typically a verb phrase composed of a verb with any objects and other modifiers. However, the subject is sometimes unvoiced if it is retrievable from context, especially in null-subject language but also in other languages, including English instances of the imperative mood. A complete simple sentence contains a single clause with a finite verb.
