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kernel

(kûr'nəl)

A grain or seed, as of a cereal grass, enclosed in a husk.
The inner, usually edible seed of a nut or fruit stone.
The most material and central part; the core: “that hard kernel of gaiety that never breaks” (Evelyn Waugh).

[Middle English, from Old English cyrnel.]

kerneled ker'neled adj.

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Computer Desktop Encyclopedia: kernel

The nucleus of an operating system. It is the closest part to the machine level and may activate the hardware directly or interface to another software layer that drives the hardware. The kernel orchestrates the entire operation of the computer by slicing time for each system function and each application as well as managing all the computer's resources. It typically resides in memory at all times. See microkernel, monolithic kernel, kernel space and kernel panic.

Thesaurus: kernel

noun

A fertilized plant ovule capable of germinating: pip, pit², seed. See start/end.
A source of further growth and development: bud¹, embryo, germ, nucleus, seed, spark¹. See start/end.
The most central and material part: core, essence, gist, heart, marrow, meat, nub, pith, quintessence, root¹, soul, spirit, stuff, substance. Law gravamen. See be.

Veterinary Dictionary: kernel

Whole grains and the meats of nuts and stone fruit pips or pits.

Word Tutor: kernel

IN BRIEF: A seed.

Can you put the kernel back in the nut or the broken egg in its shell? — zaadz.com.

Tutor's tip: The officers nicknamed the "colonel" (military rank between lieutenant colonel and brigadier general) "kernel" (part of an ear of corn) for his penchant to eat corn at every meal.

Wikipedia: kernel (mathematics)

In mathematics, the word kernel has several meanings. In many cases it refers to a general construction which measures the failure of a function or homomorphism to be injective.

In set theory

Main article: Kernel (set theory)

In set theory, the kernel of a function $f : X \to Y$ is an equivalence relation on $X$ which is defined in terms of $f$ :

$\ker\left(f\right) = \{\left(x_1,x_2\right) \in X \times X : f\left(x_1\right) = f\left(x_2\right)\}.$

The function f is injective if and only if the kernel is the diagonal in $X \times X$ .

In abstract algebra

Main article: Kernel (algebra)

Let $f$ be a homomorphism. The equivalence relation $\ker\left(f\right)$ defined in the previous section becomes a congruence relation on $X$ (i.e. the equivalence relation is compatible with the algebraic structure). For many algebraic structures, such as groups, rings, and vector spaces, there is a simpler definition of the kernel that is usually preferred; in these cases the equivalence relation is entirely determined by the equivalence class of the neutral element, and the kernel is defined as the preimage of the neutral element in $Y$ :

$\ker\left(f\right) = \{x \in X : f\left(x\right) = 0\}.$

The congruence relation is replaced with the notion of a normal subgroup, in the case of groups, or an ideal, in the case of rings. For linear operators between vector spaces, the kernel is also known as the null space.

In linear algebra and functional analysis

Main article: Kernel (linear operator)

The same definition is used in linear algebra as in abstract algebra: the kernel or nullspace of a linear operator T is the set of solutions to the equation Tx = 0.

Of a matrix

Main article: Kernel (matrix)

The kernel, or nullspace, of a matrix A is the set of vectors that, when multiplied by A, give the zero vector.

In category theory

Main article: Kernel (category theory)

There exist several notions in category theory which seek to generalize the concept of a kernel in algebra. In categories with zero morphisms, the kernel of a morphism $f$ is defined as the equalizer of $f$ and the parallel zero morphism. Additionally, the kernel pair of a morphism $f$ (similar to a congruence relation in algebra) is defined as the pullback of $f$ with itself. In the category of sets this is simply the kernel of a function.

A difference kernel is another name for a binary equalizer. The name comes from preadditive categories, where one can define the equalizer of $f$ and $g$ as the kernel of the difference:

$\mathrm{eq}\left(f, g\right) = \ker\left(f - g\right).$

Difference kernels, however, make sense in arbitrary categories and are often used in conjunction with kernel pairs.

In integral calculus

In reference to a series, the kernel conveys the idea of the generating function. Similarly, in integral calculus, the kernel is the part of the integrand that defines the integral transform; specifically, the kernel of the operator $T k$ defined by

$(T_k f)(x) = \int_X k(x, x') f(x') \, dx'.$

is the function k. k is also called a kernel function.

In probability theory and statistics

Main article: Kernel (statistics)

A stochastic kernel is the transition function of a stochastic process (usually discrete). 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.

		Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2007. Published by Houghton Mifflin Company. All rights reserved. Read more
		Computer Desktop Encyclopedia. THIS COPYRIGHTED DEFINITION IS FOR PERSONAL USE ONLY. All other reproduction is strictly prohibited without permission from the publisher. © 1981-2008 Computer Language Company Inc. All rights reserved. Read more
		Thesaurus. Roget's II: The New Thesaurus, Third Edition by the Editors of the American Heritage® Dictionary Copyright © 1995 by Houghton Mifflin Company. Published by Houghton Mifflin Company. All rights reserved. Read more
		Veterinary Dictionary. Saunders Comprehensive Veterinary Dictionary 3rd Edition. Copyright © 2007 by D.C. Blood, V.P. Studdert and C.C. Gay, Elsevier. All rights reserved. Read more
		Word Tutor. Copyright © 2004-present by eSpindle Learning, a 501(c) nonprofit organization. All rights reserved. eSpindle provides personalized spelling and vocabulary tutoring online; free trial. Read more
		Wikipedia. This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Kernel (mathematics)". Read more
		Translations. Copyright © 2007, WizCom Technologies Ltd. All rights reserved. Read more

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