Search results
Results from the WOW.Com Content Network
The classical finite-difference approximations for numerical differentiation are ill-conditioned. However, if is a holomorphic function, real-valued on the real line, which can be evaluated at points in the complex plane near , then there are stable methods.
Given a differentiable manifold [a] of dimension , a divergence on is a -function : [,) satisfying: [1] [2] (,) for all , (non-negativity),(,) = if and only if ...
The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale (also known as a time-set), which may be an arbitrary closed subset of the reals. In this way, results apply not only to the set of real numbers or set of integers but to more general time scales such as a Cantor set.
Let : be a continuously-differentiable, strictly convex function defined on a convex set. The Bregman distance associated with F for points p , q ∈ Ω {\displaystyle p,q\in \Omega } is the difference between the value of F at point p and the value of the first-order Taylor expansion of F around point q evaluated at point p :
A function of a real variable is differentiable at a point of its domain, if its domain contains an open interval containing , and the limit = (+) exists. [2] This means that, for every positive real number , there exists a positive real number such that, for every such that | | < and then (+) is defined, and | (+) | <, where the vertical bars denote the absolute value.
Lemma 1. ′ =, where ′ is the differential of . This equation means that the differential of , evaluated at the identity matrix, is equal to the trace.The differential ′ is a linear operator that maps an n × n matrix to a real number.
Let be a function in the Lebesgue space ([,]).We say that in ([,]) is a weak derivative of if ′ = ()for all infinitely differentiable functions with () = =.. Generalizing to dimensions, if and are in the space () of locally integrable functions for some open set, and if is a multi-index, we say that is the -weak derivative of if
Discrete differential calculus is the study of the definition, properties, and applications of the difference quotient of a function. The process of finding the difference quotient is called differentiation.