Kernel (mathematics)
Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the word kernel has several meanings. Kernel may mean a subset associated with a mapping
Map (mathematics)
In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...

:
  • The kernel of a mapping is the set of elements that map to the zero element
    Additive identity
    In mathematics the additive identity of a set which is equipped with the operation of addition is an element which, when added to any element x in the set, yields x...

     (such as zero or zero vector
    Null vector
    Null vector can refer to:* Null vector * A causal structure in Minkowski space...

    ), as in kernel of a linear operator and kernel of a matrix. In this context, kernel is often called nullspace.
  • More generally, the kernel in algebra
    Kernel (algebra)
    In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...

     is the set of elements that map to the neutral element. Here, the mapping is assumed to be a homomorphism
    Homomorphism
    In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...

    , that is, it preserves algebraic operations, and, in particular, maps neutral element to neutral element. The kernel is then the set of all elements that the mapping cannot distinguish from the neutral element.
  • The kernel in category theory
    Kernel (category theory)
    In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernels from algebra...

     is a generalization of this concept to morphism
    Morphism
    In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...

    s rather than mappings between sets.
  • In set theory, the kernel of a function is the set of all pairs of elements that the function cannot distinguish, that is, they map to the same value. This is a generalization of the kernel concept above to the case when there is no neutral element.
  • In set theory, the difference kernel or binary equalizer is the set of all elements where the values of two functions coincide.


Kernel may also mean a function of two variables, which is used to define a mapping:
  • In integral calculus, the kernel (also called integral kernel or kernel function) is a function of two variables that defines an integral transform, such as the function k in


  • In partial differential equation
    Partial differential equation
    In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

    s, when the solution of the equation for the right-hand side f can be written as Tf above, the kernel becomes the Green's function
    Green's function
    In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions...

    . The heat kernel
    Heat kernel
    In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a particular domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectrum of the Laplace operator, and is thus of some...

     is the Green's function of the heat equation.
  • In the case when the integral kernel depends only on the difference between its arguments, it becomes a convolution kernel, as in


  • In statistics
    Statistics
    Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

    , a kernel
    Kernel (statistics)
    A kernel is a weighting function used in non-parametric estimation techniques. Kernels are used in kernel density estimation to estimate random variables' density functions, or in kernel regression to estimate the conditional expectation of a random variable. Kernels are also used in time-series,...

     is a weighting function used in kernel density estimation
    Kernel density estimation
    In statistics, kernel density estimation is a non-parametric way of estimating the probability density function of a random variable. Kernel density estimation is a fundamental data smoothing problem where inferences about the population are made, based on a finite data sample...

     to estimate the probability density function
    Probability density function
    In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

     of a random variable
    Random variable
    In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

    .
  • In probability theory
    Probability theory
    Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

     and statistics
    Statistics
    Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

    , a stochastic kernel
    Stochastic kernel
    In statistics, a stochastic kernel estimate is an estimate of the transition function of a stochastic process. Often, this is an estimate of the conditional density function obtained using kernel density estimation...

     is the transition function of a stochastic process
    Stochastic process
    In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

    . In a discrete time process with continuous probability distributions, it is the same thing as the kernel of the integral operator that advances the probability density function
    Probability density function
    In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

    .
  • Kernel trick
    Kernel trick
    For machine learning algorithms, the kernel trick is a way of mapping observations from a general set S into an inner product space V , without ever having to compute the mapping explicitly, in the hope that the observations will gain meaningful linear structure in V...

     is a technique to write a nonlinear operator as a linear one in a space of higher dimension.
  • In operator theory, a positive definite kernel is a generalization of a positive matrix.
  • The kernel in a reproducing kernel Hilbert space
    Reproducing kernel Hilbert space
    In functional analysis , a reproducing kernel Hilbert space is a Hilbert space of functions in which pointwise evaluation is a continuous linear functional. Equivalently, they are spaces that can be defined by reproducing kernels...

    .
The source of this article is wikipedia, the free encyclopedia.  The text of this article is licensed under the GFDL.
 
x
OK