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Parametrization (or parameterization) in an atmospheric model (either weather model or climate model) is a method of replacing processes that are too small-scale or complex to be physically represented in the model by a simplified process. This can be contrasted with other processes—e.g., large-scale flow of the atmosp... | Wikipedia/Parametrization_(atmospheric_modeling) |
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonli... | Wikipedia/Nonlinear_systems |
John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company that focuses on academic publishing and instructional materials. The company was founded in 1807 and produces books, journals, and encyclopedias, in print and electronically, as well as online products and services, tra... | Wikipedia/Interscience_Publishers |
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space has a natural vector space structure given by pointwise addition ... | Wikipedia/Space_of_functions |
Rod calculus or rod calculation was the mechanical method of algorithmic computation with counting rods in China from the Warring States to Ming dynasty before the counting rods were increasingly replaced by the more convenient and faster abacus. Rod calculus played a key role in the development of Chinese mathematics ... | Wikipedia/Rod_calculus |
The term "thermal energy" is often used ambiguously in physics and engineering. It can denote several different physical concepts, including:
Internal energy: The energy contained within a body of matter or radiation, excluding the potential energy of the whole system.
Heat: Energy in transfer between a system and its... | Wikipedia/Thermal_energy |
In mathematics, the composition operator
takes two functions,
f
{\displaystyle f}
and
g
{\displaystyle g}
, and returns a new function
h
(
x
)
:=
(
g
∘... | Wikipedia/Composite_function |
In mathematical logic, abstract algebraic logic is the study of the algebraization of deductive systems
arising as an abstraction of the well-known Lindenbaum–Tarski algebra, and how the resulting algebras are related to logical systems.
== History ==
The archetypal association of this kind, one fundamental to the hi... | Wikipedia/Abstract_algebraic_logic |
In propositional calculus, a propositional function or a predicate is a sentence expressed in a way that would assume the value of true or false, except that within the sentence there is a variable (x) that is not defined or specified (thus being a free variable), which leaves the statement undetermined. The sentence m... | Wikipedia/Propositional_function |
The Socratic method (also known as the method of Elenchus or Socratic debate) is a form of argumentative dialogue between individuals based on asking and answering questions. Socratic dialogues feature in many of the works of the ancient Greek philosopher Plato, where his teacher Socrates debates various philosophical ... | Wikipedia/Socratic_method |
In mathematical logic, the Lindenbaum–Tarski algebra (or Lindenbaum algebra) of a logical theory T consists of the equivalence classes of sentences of the theory (i.e., the quotient, under the equivalence relation ~ defined such that p ~ q exactly when p and q are provably equivalent in T). That is, two sentences are e... | Wikipedia/Lindenbaum–Tarski_algebra |
In abstract algebra, a monadic Boolean algebra is an algebraic structure A with signature
⟨·, +, ', 0, 1, ∃⟩ of type ⟨2,2,1,0,0,1⟩,
where ⟨A, ·, +, ', 0, 1⟩ is a Boolean algebra.
The monadic/unary operator ∃ denotes the existential quantifier, which satisfies the identities (using the received prefix notation for... | Wikipedia/Monadic_Boolean_algebra |
In mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory. In a Boolean-valued model, the truth values of propositions are not limited to "true" and "false", but instead take values in some fixed complete Boolean algebra.
Boolean-valued models were ... | Wikipedia/Boolean-valued_model |
In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation. The motivating example of a relation algebra is the algebra 2X 2 of all binary relations on a set X, that is, subsets of the cartesian square X2, with R•S interpreted a... | Wikipedia/Relation_algebra |
Metaphysics is the branch of philosophy that examines the basic structure of reality. It is traditionally seen as the study of mind-independent features of the world, but some theorists view it as an inquiry into the conceptual framework of human understanding. Some philosophers, including Aristotle, designate metaphys... | Wikipedia/Metaphysics |
In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and ordinary propositional logic. Interior algebras form a variety of modal alge... | Wikipedia/Interior_algebra |
In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment). A is a... | Wikipedia/Inclusion_(set_theory) |
In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first half of the 20th century. The theorem was first proved by Marshall H. Stone.... | Wikipedia/Representation_theorem_for_Boolean_algebras |
In abstract algebra, a branch of pure mathematics, an MV-algebra is an algebraic structure with a binary operation
⊕
{\displaystyle \oplus }
, a unary operation
¬
{\displaystyle \neg }
, and the constant
0
... | Wikipedia/MV-algebra |
In mathematics, a binary relation associates some elements of one set called the domain with some elements of another set called the codomain. Precisely, a binary relation over sets
X
{\displaystyle X}
and
Y
{\displaystyle Y}
is a ... | Wikipedia/Difunctional |
In mathematical logic, algebraic semantics is a formal semantics based on algebras studied as part of algebraic logic. For example, the modal logic S4 is characterized by the class of topological boolean algebras—that is, boolean algebras with an interior operator. Other modal logics are characterized by various other ... | Wikipedia/Algebraic_semantics_(mathematical_logic) |
In mathematics, the notion of cylindric algebra, developed by Alfred Tarski, arises naturally in the algebraization of first-order logic with equality. This is comparable to the role Boolean algebras play for propositional logic. Cylindric algebras are Boolean algebras equipped with additional cylindrification operatio... | Wikipedia/Cylindric_algebra |
In algebra and logic, a modal algebra is a structure
⟨
A
,
∧
,
∨
,
−
,
0
,
1
,
◻
⟩
{\displaystyle \langle A,\land ,\lor ,-,0,1,\Box \rangle }
such that
... | Wikipedia/Modal_algebra |
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field of rational numbers. Intuitively, it consists of ratios between integral do... | Wikipedia/Field_of_rational_functions |
In mathematics, the modular lambda function λ(τ) is a highly symmetric holomorphic function on the complex upper half-plane. It is invariant under the fractional linear action of the congruence group Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the modular curve X(... | Wikipedia/Modular_lambda_function |
In mathematics, a quintic function is a function of the form
g
(
x
)
=
a
x
5
+
b
x
4
+
c
... | Wikipedia/General_quintic_equation |
Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, "statistical properties" refers to properties which are expressed through the behavior of time averages of various functions along trajectories of dynamical s... | Wikipedia/Ergodic_theory |
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ... | Wikipedia/Numerical_methods |
In mathematics, the power series method is used to seek a power series solution to certain differential equations. In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients.
== Method =... | Wikipedia/Power_series_solution_of_differential_equations |
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differ... | Wikipedia/First-order_differential_equation |
In mathematical systems theory, a multidimensional system or m-D system is a system in which not only one independent variable exists (like time), but there are several independent variables.
Important problems such as factorization and stability of m-D systems (m > 1) have recently attracted the interest of many resea... | Wikipedia/Multidimensional_systems |
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differ... | Wikipedia/Non-homogeneous_differential_equation |
In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory, e.g. in mathematical models that include ultrametric pseudo-differential equation... | Wikipedia/Pseudo-differential_operators |
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applications include many problems in a wide variety of fields such as biology, neuro... | Wikipedia/Statistical_mechanics |
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator
D
{\displaystyle D}
D
f
(
x
... | Wikipedia/Fractional_differential_equations |
In mathematical analysis, particularly numerical analysis, the rate of convergence and order of convergence of a sequence that converges to a limit are any of several characterizations of how quickly that sequence approaches its limit. These are broadly divided into rates and orders of convergence that describe how qui... | Wikipedia/Rate_of_convergence |
The Petrov–Galerkin method is a mathematical method used to approximate solutions of partial differential equations which contain terms with odd order and where the test function and solution function belong to different function spaces. It can be viewed as an extension of Bubnov-Galerkin method where the bases of test... | Wikipedia/Petrov–Galerkin_method |
In mathematics, the method of characteristics is a technique for solving particular partial differential equations. Typically, it applies to first-order equations, though in general characteristic curves can also be found for hyperbolic and parabolic partial differential equation. The method is to reduce a partial di... | Wikipedia/Method_of_characteristics |
A functional differential equation is a differential equation with deviating argument. That is, a functional differential equation is an equation that contains a function and some of its derivatives evaluated at different argument values.
Functional differential equations find use in mathematical models that assume a s... | Wikipedia/Functional_differential_equation |
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonli... | Wikipedia/Non-linear_differential_equation |
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differ... | Wikipedia/Ordinary_differential_equations |
Stochastic 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 mode... | Wikipedia/Stochastic_partial_differential_equation |
In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying hybrid systems. It has applications in any field that requires simultane... | Wikipedia/Time_scale_calculus |
In mathematics, an abstract differential equation is a differential equation in which the unknown function and its derivatives take values in some generic abstract space (a Hilbert space, a Banach space, etc.). Equations of this kind arise e.g. in the study of partial differential equations: if to one of the variables ... | Wikipedia/Abstract_differential_equation |
The infinite element method is a numerical method for solving problems of engineering and mathematical physics. It is a modification of finite element method. The method divides the domain concerned into sections of infinite length. In contrast with a finite element which is approximated by polynomial expressions on a ... | Wikipedia/Infinite_element_method |
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonli... | Wikipedia/Non-linear_differential_equations |
In mathematical analysis, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form:
f
(
x
1
,... | Wikipedia/Integral_equations |
In mathematics, a differential-algebraic system of equations (DAE) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to such a system.
The set of the solutions of such a system is a differential algebraic variety, and corresponds to an ideal in a differential... | Wikipedia/Differential_algebraic_equation |
In mathematics, the method of undetermined coefficients is an approach to finding a particular solution to certain nonhomogeneous ordinary differential equations and recurrence relations. It is closely related to the annihilator method, but instead of using a particular kind of differential operator (the annihilator) i... | Wikipedia/Method_of_undetermined_coefficients |
In mathematics, and more specifically in analysis, a holonomic function is a smooth function of several variables that is a solution of a system of linear homogeneous differential equations with polynomial coefficients and satisfies a suitable dimension condition in terms of D-modules theory. More precisely, a holonom... | Wikipedia/Holonomic_function |
In mathematical analysis, the Dirac delta function (or δ distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. Thus it can be represented heuristically as
... | Wikipedia/Dirac_delta_function |
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties in 1786. This is often written as
∇
2
f
=
0
... | Wikipedia/Laplace_equation |
In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time domain (if applicable) are discretized, or broken into a finite number of intervals, and the values of the ... | Wikipedia/Finite_difference_method |
The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations.
In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. ... | Wikipedia/Finite_volume_method |
In mathematics, a recurrence relation is an equation according to which the
n
{\displaystyle n}
th term of a sequence of numbers is equal to some combination of the previous terms. Often, only
k
{\displaystyle k}
previous terms of th... | Wikipedia/Difference_equations |
In mathematics, a system of differential equations is a finite set of differential equations. Such a system can be either linear or non-linear. Also, such a system can be either a system of ordinary differential equations or a system of partial differential equations.
== Linear systems of differential equations ==
A... | Wikipedia/System_of_differential_equations |
In mathematics, an ordinary differential equation is called a Bernoulli differential equation if it is of the form
y
′
+
P
(
x
)
y
=
Q
(
x
)
y
... | Wikipedia/Bernoulli_differential_equation |
Natural science or empirical science is one of the branches of science concerned with the description, understanding and prediction of natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer review and reproducibility of findings are used to try to ensure the validit... | Wikipedia/Natural_science |
In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. When the variable is time, they are also called time-invariant systems.
Many laws in physics, where the independent variable is usually as... | Wikipedia/Autonomous_differential_equation |
In mathematics, in the area of numerical analysis, Galerkin methods are a family of methods for converting a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete problem by applying linear constraints determined by finite sets of basis functions. They are named aft... | Wikipedia/Galerkin_method |
In numerical analysis, the Runge–Kutta methods (English: RUUNG-ə-KUUT-tah) are a family of implicit and explicit iterative methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations. These methods were developed around 1900 by the German ... | Wikipedia/Runge–Kutta_methods |
In mathematics, the Korteweg–De Vries (KdV) equation is a partial differential equation (PDE) which serves as a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an integrable PDE, exhibiting typical behaviors such as a large number of explicit solutions, i... | Wikipedia/Korteweg–De_Vries_equation |
In numerical analysis, the Crank–Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations. It is a second-order method in time. It is implicit in time, can be written as an implicit Runge–Kutta method, and it is numerically stable. The metho... | Wikipedia/Crank–Nicolson_method |
In mathematics, an exact differential equation or total differential equation is a certain kind of ordinary differential equation which is widely used in physics and engineering.
== Definition ==
Given a simply connected and open subset D of
R
... | Wikipedia/Exact_differential_equation |
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer scien... | Wikipedia/Differential_operators |
Computer simulation is the running of a mathematical model on a computer, the model being designed to represent the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be determined by comparing their results to the real-world outcomes they aim to predict. C... | Wikipedia/Computer_model |
In fluid mechanics, the thin-film equation is a partial differential equation that approximately predicts the time evolution of the thickness h of a liquid film that lies on a surface. The equation is derived via lubrication theory which is based on the assumption that the length-scales in the surface directions are si... | Wikipedia/Thin-film_equation |
A mathematical model is an abstract description of a concrete system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in applied mathematics and in the natural sciences (such as physics, biology, earth science, chemis... | Wikipedia/Mathematical_modelling |
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathematical functions in terms of dynamic variables. These variables are usually ... | Wikipedia/Equations_of_motion |
In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interactions of atoms and molecules. Electromagnetism can be thought of as a combinatio... | Wikipedia/Electrodynamics |
Finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. Computers ar... | Wikipedia/Finite_element_method |
Stochastic 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 mode... | Wikipedia/Stochastic_partial_differential_equations |
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The development of the computer algebra systems in the second half of the 20th c... | Wikipedia/Computer_algebra_system |
In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if
L
{\displaystyle L}
is a linear differential operat... | Wikipedia/Green's_function |
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematicia... | Wikipedia/Euler–Lagrange_equation |
In mathematics, a differential-algebraic system of equations (DAE) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to such a system.
The set of the solutions of such a system is a differential algebraic variety, and corresponds to an ideal in a differential... | Wikipedia/Differential-algebraic_system_of_equations |
The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynam... | Wikipedia/Wave_equation |
In mathematics, delay differential equations (DDEs) are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times.
DDEs are also called time-delay systems, systems with aftereffect or dead-time, hereditary systems... | Wikipedia/Delay_differential_equation |
Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals.
Many differential equations cannot be... | Wikipedia/Numerical_ordinary_differential_equations |
Electrostatics is a branch of physics that studies slow-moving or stationary electric charges.
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word ḗlektron (ἤλεκτρον), meaning 'amber', was thus the root of the word electricity. Electro... | Wikipedia/Electrostatics |
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits.
L... | Wikipedia/Limit_(category_theory) |
In mathematics, specifically in order theory and functional analysis, if
C
{\displaystyle C}
is a cone at the origin in a topological vector space
X
{\displaystyle X}
such that
0
∈
C
... | Wikipedia/Normal_cone_(functional_analysis) |
In mathematics, the upper topology on a partially ordered set X is the coarsest topology in which the closure of a singleton
{
a
}
{\displaystyle \{a\}}
is the order section
a
]
=
{
x
≤
a... | Wikipedia/Upper_topology |
In graph theory and order theory, a comparability graph is an undirected graph that connects pairs of elements that are comparable to each other in a partial order. Comparability graphs have also been called transitively orientable graphs, partially orderable graphs, containment graphs, and divisor graphs.
An incompara... | Wikipedia/Comparability_graph |
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least one maximal element.
The lemma was proved (assuming the axiom of choice) by Kazimier... | Wikipedia/Zorn's_Lemma |
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.
== In calculus and analysis ==
In calculus, a function
... | Wikipedia/Monotonic_function |
In mathematics, given two partially ordered sets P and Q, a function f: P → Q between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema. That is, for every directed subset D of P with supremum in P, its image has a supremum in Q, and that supremum is the image of t... | Wikipedia/Scott_topology |
In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a totally ordered set.
There are several variants and generalizations of the ... | Wikipedia/Lexicographic_order |
In mathematics and theoretical computer science, the Lawson topology, named after Jimmie D. Lawson, is a topology on partially ordered sets (posets) used in the study of domain theory. The lower topology on a poset P is generated by the subbasis consisting of all complements of principal filters on P. The Lawson topolo... | Wikipedia/Lawson_topology |
In the mathematical area of order theory, there are various notions of the common concept of distributivity, applied to the formation of suprema and infima. Most of these apply to partially ordered sets that are at least lattices, but the concept can in fact reasonably be generalized to semilattices as well.
== Distr... | Wikipedia/Distributivity_(order_theory) |
In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set (poset). The most familiar example is the completeness of the real numbers. A special use of the term refers to complete partial orders or complete lattices. However, many... | Wikipedia/Completeness_(order_theory) |
Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where it is used to specify denotational semantics, especially ... | Wikipedia/Domain_theory |
In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the same set, but with the inverse order, i.e. x ≤ y holds in Pop if and only if y ≤ x holds in P. It is easy to see ... | Wikipedia/Duality_(order_theory) |
In general topology, an Alexandrov topology is a topology in which the intersection of an arbitrary family of open sets is open (while the definition of a topology only requires this for a finite family). Equivalently, an Alexandrov topology is one whose open sets are the upper sets for some preorder on the space.
Spac... | Wikipedia/Alexandrov_topology |
In mathematics, specifically in order theory and functional analysis, the order topology of an ordered vector space
(
X
,
≤
)
{\displaystyle (X,\leq )}
is the finest locally convex topological vector space (TVS) topology on
... | Wikipedia/Order_topology_(functional_analysis) |
In mathematics, particularly in functional analysis, the closed graph theorem is a result connecting the continuity of a linear operator to a topological property of their graph. Precisely, the theorem states that a linear operator between two Banach spaces is continuous if and only if the graph of the operator is clo... | Wikipedia/Closed_graph_theorem_(functional_analysis) |
In mathematics, specifically functional analysis, a Banach space is said to have the approximation property (AP), if every compact operator is a limit of finite-rank operators. The converse is always true.
Every Hilbert space has this property. There are, however, Banach spaces which do not; Per Enflo published the fir... | Wikipedia/Approximation_property |
In mathematics, Choquet theory, named after Gustave Choquet, is an area of functional analysis and convex analysis concerned with measures which have support on the extreme points of a convex set C. Roughly speaking, every vector of C should appear as a weighted average of extreme points, a concept made more precise by... | Wikipedia/Choquet_theory |
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.
A special ca... | Wikipedia/Open_mapping_theorem_(functional_analysis) |
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra.
Von Neumann algebras were originally introduced by John von Neumann, motivated by his study of s... | Wikipedia/Von_Neumann_algebra |
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