Holomorphic functional calculusIn mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. That is to say, given a holomorphic function f of a complex argument z and an operator T, the aim is to construct an operator, f(T), which naturally extends the function f from complex argument to operator argument. More precisely, the functional calculus defines a continuous algebra homomorphism from the holomorphic functions on a neighbourhood of the spectrum of T to the bounded operators.
Lyapunov stabilityVarious types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Aleksandr Lyapunov. In simple terms, if the solutions that start out near an equilibrium point stay near forever, then is Lyapunov stable. More strongly, if is Lyapunov stable and all solutions that start out near converge to , then is said to be asymptotically stable (see asymptotic analysis).
Continuous functional calculusIn mathematics, particularly in operator theory and C*-algebra theory, a continuous functional calculus is a functional calculus which allows the application of a continuous function to normal elements of a C*-algebra. Theorem. Let x be a normal element of a C*-algebra A with an identity element e. Let C be the C*-algebra of the bounded continuous functions on the spectrum σ(x) of x. Then there exists a unique mapping π : C → A, where π(f) is denoted f(x), such that π is a unit-preserving morphism of C*-algebras and π(1) = e and π(id) = x, where id denotes the function z → z on σ(x).
Functional programmingIn computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative programming paradigm in which function definitions are trees of expressions that map values to other values, rather than a sequence of imperative statements which update the running state of the program. In functional programming, functions are treated as first-class citizens, meaning that they can be bound to names (including local identifiers), passed as arguments, and returned from other functions, just as any other data type can.
Semigroup with involutionIn mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism, which—roughly speaking—brings it closer to a group because this involution, considered as unary operator, exhibits certain fundamental properties of the operation of taking the inverse in a group: uniqueness, double application "cancelling itself out", and the same interaction law with the binary operation as in the case of the group inverse.
Extensions of symmetric operatorsIn functional analysis, one is interested in extensions of symmetric operators acting on a Hilbert space. Of particular importance is the existence, and sometimes explicit constructions, of self-adjoint extensions. This problem arises, for example, when one needs to specify domains of self-adjointness for formal expressions of observables in quantum mechanics. Other applications of solutions to this problem can be seen in various moment problems. This article discusses a few related problems of this type.
Special classes of semigroupsIn mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups consists of all those semigroups in which the binary operation satisfies the commutativity property that ab = ba for all elements a and b in the semigroup. The class of finite semigroups consists of those semigroups for which the underlying set has finite cardinality.
Transformation semigroupIn algebra, a transformation semigroup (or composition semigroup) is a collection of transformations (functions from a set to itself) that is closed under function composition. If it includes the identity function, it is a monoid, called a transformation (or composition) monoid. This is the semigroup analogue of a permutation group. A transformation semigroup of a set has a tautological semigroup action on that set. Such actions are characterized by being faithful, i.e., if two elements of the semigroup have the same action, then they are equal.
Semigroup actionIn algebra and theoretical computer science, an action or act of a semigroup on a set is a rule which associates to each element of the semigroup a transformation of the set in such a way that the product of two elements of the semigroup (using the semigroup operation) is associated with the composite of the two corresponding transformations. The terminology conveys the idea that the elements of the semigroup are acting as transformations of the set.
Spectral densityThe power spectrum of a time series describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of a certain signal or sort of signal (including noise) as analyzed in terms of its frequency content, is called its spectrum.
