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Topic: Kernel algebra

Related Topics

Ring homomorphism

Kernel of a homomorphism

Fundamental theorem on homomorphisms

Monoid

Linear transformation

Identity element

Dimension

Associativity

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	Science Fair Projects - 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.
		Every Mal'cev algebra has a special neutral element (the zero vector in the case of vector spaces, the identity element in the case of groups, and the zero element in the case of rings or modules).
		The notion of ideal generalises to any Mal'cev algebra (as subspace in the case of vector spaces, normal subgroup in the case of groups, two-sided ring ideal in the case of rings, and submodule in the case of modules).
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Kernel_(algebra) (2023 words)

	Kernel (mathematics) - Wikipedia, the free encyclopedia
		Kernels in abstract algebra are general constructions which measure the failure of a homomorphism or function to be injective.
		For many algebraic structures, such as groups, rings, and vector spaces, there is a simpler definition of the kernel that is usually preferred.
		The kernel pair of a morphism f is defined as a pullback of f with itself.
	en.wikipedia.org /wiki/Kernel_(mathematics) (455 words)

	:::► Dictionary of Meaning www.dictionary-of-meaning.com ◄:::
		The dimension of the nullspace, called the nullity of M, is given by the number of columns of M minus the rank (matrix theory)rank of M, as a consequence of the rank-nullity theorem.
		Every Mal'cev algebra has a special neutral element (the zero vector in the case of vector spaces, the identity element in the case of group (mathematics)s, and the group (mathematics)s, and the zero element in the case of ring (mathematics)s or module (mathematics)s).
		The notion of ideal generalises to any Mal'cev algebra (as subspace (linear algebra)subspace in the case of vector spaces, normal subgroup in the case of groups, two-sided ring ideal in the case of rings, and submodule in the case of module (algebra)s).
	www.dictionary-of-meaning.com /kernel_(algebra).html (2117 words)

	Kernel (algebra) (Site not responding. Last check: 2007-10-21)
		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.
		Every Mal'cev algebra has a special neutral element (the zero vector in the case of vector spaces, the identity element in the case of groupss, and the zero element in the case of ringss or moduless).
		The notion of ideal generalises to any Mal'cev algebra (as subspace in the case of vector spaces, normal subgroup in the case of groups, two-sided ring ideal in the case of rings, and submodule in the case of moduless).
	www.sciencedaily.com /encyclopedia/kernel__algebra_ (1912 words)

	kernel (mathematics) (Site not responding. Last check: 2007-10-21)
		Unrelated to this, if f is any function in any context, then the kernel of f is a certain equivalence relation on the domain of f which is defined in terms of f.
		But in the case of Mal'cev algebras, it can be replaced by a simpler definition; the kernel of a homomorphism f is the preimage under f of the zero element of the codomain.
		Finally, for this last notion of kernel is generalised in a certain sense in category theory; the kernel of a morphism f is the difference kernel of f and the corresponding zero morphism (if this exists).
	www.yourencyclopedia.net /Kernel_(mathematics) (242 words)

	Kernel (algebra) - Info Voyager : Travel Guides : Information Portal (Site not responding. Last check: 2007-10-21)
		In mathematics, especially abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective.
		In many cases, the kernel of a homomorphism is a subset of the domain of the homomorphism (specifically, those elements which are mapped to the identity element in the codomain).
		The notion of kernel pair[?] is a further generalisation of the kernel as a congruence relation.
	www.infovoyager.com /info/ke/Kernel_of_a_homomorphism.html (793 words)

	Kernel (algebra) : QuicklyFind Info (Site not responding. Last check: 2007-10-21)
		Every Mal'cev algebra has a special neutral element (the zero vector in the case of vector spaces, the identity element in the case of groups, and the zero element in the case of rings or modules).
		The notion of ideal generalises to any Mal'cev algebra (as subspace in the case of vector spaces, normal subgroup in the case of groups, two-sided ring ideal in the case of rings, and submodule in the case of modules).
		The categorical generalisation of the kernel as a congruence relation is the kernel pair.
	www.quicklyfind.com /info/Kernel_(algebra).htm (1938 words)

	Kernel (algebra) -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		One can define kernels for (Similarity of form) homomorphisms between (A self-contained component (unit or item) that is used in combination with other components) modules over a (Jewelry consisting of a circlet of precious metal (often set with jewels) worn on the finger) ring in an analogous manner.
		The first isomorphism theorem in general universal algebra states that this quotient algebra is (Click link for more info and facts about naturally isomorphic) naturally isomorphic to the (An iconic mental representation) image of f (which is a (Click link for more info and facts about subalgebra) subalgebra of B).
		Note that the definition of kernel here (as in the monoid example) doesn't depend on the algebraic structure; it is a purely (A group of things of the same kind that belong together and are so used) set-theoretic concept.
	www.absoluteastronomy.com /encyclopedia/k/ke/kernel_(algebra).htm (2163 words)

	wiki/Kernel (algebra) Definition / wiki/Kernel (algebra) Research (Site not responding. Last check: 2007-10-21)
		The term "abstract algebra" is used to distinguish the field from "elementary algebra" or "high school algebra" which teaches the correct rules for manipulating formulas and algebraic expressions involving real and complex numbers....
		The fundamental theorem on homomorphismsIn abstract algebra, for a number of algebraic structures, the fundamental theorem on homomorphisms relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism....
		Let M and N be monoids and let f be a monoid homomorphismIn abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element.
	www.elresearch.com /wiki/Kernel_(algebra) (3168 words)

	Earliest Known Uses of Some of the Words of Mathematics (K)
		Kernel occurs in English in 1909 in M. Bôcher's Introduction to the Study of Integral Equations: "K is called the kernel of these equations." (OED2).
		A JSTOR search found the "Fejér kernel" and "Dirichlet kernel" in Charles N. Moore's "On the Application of Borel's Method to the Summation of Fourier's Series" (Proceedings of the National Academy, 11, (1925), 284-287) but it is unlikely that this was the first published use of these terms.
		The use of kernel in algebra appears to be unrelated to its use in integral equations and Fourier analysis.
	members.aol.com /jeff570/k.html (1368 words)

	[No title]
		We prove a formula for the composition of relative convolutions and deduce from it that {\em an algebra of relative convolutions induced by a Lie algebra \algebra{g} is a representation of the algebra of group convolutions on the Lie group $\object{Exp}\algebra{g}$}.
		The algebra of relative convolutions defined by the last frame is the algebra of operators generated by two-sided convolutions on \Heisen{n} and operators of multiplication by functions, which form a meta-Heisenberg group~\cite{Folland94}.
		There is our conclusion\footnote{This is an answer to the reasonable question of E.~Stein:``Does the algebra of two-sided convolutions contain at least one interesting operator?''}: \begin{prop} The algebra of the Toeplitz operators with PDO pre-symbols is naturally imbedded into the algebra of relative convolutions generated by two-sided convolutions on \Heisen{n} and operators of multiplication by functions.
	www.ma.utexas.edu /mp_arc/papers/94-305 (9590 words)

	[No title]
		From a $3$-graded Lie algebra ${\mathfrak g}$ one obtains a KKT algebra ${\mathfrak g}^{\#}$, defining \begin{equation}\label{E:3G-KKT} {\mathfrak g}^{\#} ={\mathfrak g'}\s /({\mathfrak g'_0} \cap Z_{\mathfrak g'}), \end{equation} where ${\mathfrak g'}$ is the 3-graded Lie subalgebra of ${\mathfrak g}$ spanned by ${\mathfrak g_{-1}}$ and ${\mathfrak g_1}$, and $Z_{\mathfrak g'}$ denotes its center.
		The relation between the canonical kernel function and the Bergman kernel of $\myD$ is the following (\cite{SA}): \begin{Prop}\label{P:berker} The Bergman kernel $K(z,w\s)$ of $\myD $ is given by \begin{equation*}\label{E:berg} K(z,w\s) = \dfrac{1\s }{ {\rm vol}\, \myD } \bl \det \Ad^{-1}_{\pp}\Ke (z,w\s).
		In (\cite{DF}) and later, independently in (\cite{DO1}), the reproducing kernel of a general vector-valued holomorphic discrete series representation is shown to be equal to the representation calculated at the canonical kernel function, up to a multiplicative factor.
	www.math.psu.edu /era-mirror/2003-01-018/2003-01-018.tex.html (2337 words)

	Kernel (Site not responding. Last check: 2007-10-21)
		The kernel of a seed is all that is within the outer coat of the seed, as the edible substance contained in the shell of a nut; hence, anything included in a shell, husk, or integument.
		The kernel of an operating system is its essential component, such as the Linux kernel.
		In mathematics, kernel has several different, somewhat unrelated meanings; see kernel (mathematics), or go directly to kernel of a function, kernel (algebra), null space (also called the kernel, in linear algebra), or kernel (category theory).
	www.yotor.com /wiki/en/ke/Kernel.htm (129 words)

	High-performance linear algebra algorithms using new generalized data structures for matrices
		By “kernel” routine, we mean a routine that performs matrix-multiply-type operations on matrix operands that are contiguous and of a form and size that permits optimal use of the L1 cache.
		Also, the combination of using the NDS with kernel routines is general, and for matrix factorization it helps to overcome the current performance problems introduced by having a nonuniform, deep memory hierarchy.
		A kernel routine for a level 3 BLAS or for a factorization routine is the piece of code that performs the floating-point operations.
	www.research.ibm.com /journal/rd/471/gustavson.html (13138 words)

	Linear Algebra -- Kernel and Range
		Determine the kernel and range of the following linear transformation from R^3 into R^3.
		Linear Algebra -- Kernel and Range - I don't understand this kernel and range stuff.
		Linear Algebra: Linear transformation - Is there a linear transformation T in R^3 -> R^3 for which: T[2 1 3] = [4 1 9] T[3 1 0] = [9 1 0] T[3 2 3] = [9 4 9]
	www.brainmass.com /homework-help/math/other/3059 (222 words)

	Kernel (algebra) (Site not responding. Last check: 2007-10-21)
		Then the kernel of f is the subset of the directproduct A × A consisting of all those ordered pairs ofelements of A whose components are both mapped by f to the same element in B.
		The first isomorphism theorem in general universal algebra states that this quotient algebra is naturally isomorphic to the image of f (which is a subalgebra ofB).
		The notion of ideal generalises to any Mal'cev algebra (as subspace in the case of vector spaces, normal subgroup in the case of groups, two-sided ringideal in the case of rings, and submodule in the case of modules).
	www.therfcc.org /kernel-algebra--210931.html (1786 words)

	`Yacas`: A do-it-yourself symbolic algebra environment (Site not responding. Last check: 2007-10-21)
		The kernel does not contain any definitions of symbolic mathematical operations and tries to be as general and free as possible of predefined notions or policies in the domain of symbolic computation.
		To the kernel, this operator is on the same footing as any other function defined by the user and can be redefined.
		Kernel primitives are available for arbitrary-precision arithmetic, string manipulation, array and list access and manipulation, for basic control flow, for assigning variables (atoms) and for defining rules for functions (atoms with a function syntax).
	yacas.sourceforge.net /essayschapter1.html (3645 words)

	kernel - OneLook Dictionary Search
		KERNEL : 1911 edition of the Encyclopedia Britannica [home, info]
		Phrases that include kernel: apricot kernel oil, bergman kernel, kernel user interface package, palm kernel, andorra kernel language, more...
		Words similar to kernel: center, core, essence, gist, heart, inwardness, kerneling, marrow, meat, nitty-gritty, nub, pith, substance, sum, grain, nutmeat, seed, more...
	www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=kernel (335 words)

	Newsletter.oct95 Mathematica 2.2 on UNIX and OpenVMS Systems (Site not responding. Last check: 2007-10-21)
		The kernel is the computational engine of Mathematica, in which core functional features are implemented.
		Kernel mathematics functionality includes definition of all elementary functions and many special functions, algebraic manipulation, solving sets of linear and nonlinear equations, series, sums, limits, differential and integral calculus, Fourier transforms, ordinary differential equations, and linear algebra.
		Kernel graphics functionality includes all common types of 2D and 3D plots, contour plots, density plots, parametric plots, 2D and 3D animation, and numerous options to finely control the appearance of plots, from line width and tick mark spacing to color, lighting, and shading models.
	www.utexas.edu /acits/newsletter/oct95/mathematica.html (1144 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Intel® Math Kernel Library (Intel® MKL) offers highly optimized, extensively threaded math routines for scientific, engineering, and financial applications that require maximum performance.
		In this release of Intel Math Kernel Library (Intel MKL), we have focused on providing optimized multi-threaded performance for the new Quad-Core Intel Xeon processor 5300 series and its close relative the Dual-Core Intel® Xeon® processor 5100 series systems.
		Get more information on performance tests and on the performance of Intel products.
	www.intel.com /software/products/mkl (1734 words)

	Parallelism in ALDOR -- the communication library Piit for parallel, distributed computation (Site not responding. Last check: 2007-10-21)
		During the last decade, an important thread of research in the area of computer algebra has addressed questions of parallelism: parallel analysis of the complexity of problems, development of parallel algorithms and the design/implementation of parallel systems.
		Even if main challenges (data management, tasks scheduling and interface design) in the implementation of a computer algebra system have solutions, most of these systems have been depreciated due to the high costs of development necessary to ensure portability of sources with good efficient programs at runtime.
		One key point in implementing a parallel computer algebra system is the ability to reuse a sequential computer algebra kernel that will be plugged in to a library supporting parallelism (thread kernel, data communication).
	www.inf.ethz.ch /personal/mannhart/publications/europar99 (3226 words)

	Dense Linear Algebra Kernels on Heterogeneous Platforms: Redistribution Issues - Beaumont, Legrand, Rastello, Robert ...
		Abstract: In this paper, we deal with redistribution issues for dense linear algebra kernels on heterogeneous platforms.
		In this context, processors speeds may well vary during the execution of a large kernel, which requires ecient strategies for redistributing the data along the computations.
		The strategy that we propose is to redistribute data after some well identi ed static phases and therefore, it is neither fully static nor fully dynamic.
	citeseer.ist.psu.edu /beaumont00dense.html (489 words)

	CONCERT - Coordination Programming
		Parallelism may be expressed by several forms of annotations; the compiler generates C code with PACLIB task creation and synchronization statements.
		PACLIB [HSN+92] [SH93b] [SH93a] [SH93c] [Sch94b] [HNS95] is a shared memory parallel variant of the runtime kernel of the computer algebra library SACLIB [BCE+92] for symbolic and algebraic computation.
		A variety of computer algebra applications have been developed in PACLIB on a Sequent Symmetry system with 20 processors [HL94] [LP94].
	www.risc.uni-linz.ac.at /people/schreine/papers/ipa97/index_3.html (453 words)

	Kernel (algebra) : Kernel (universal algebra)
		terms defined : Kernel (algebra) : Kernel (universal algebra)
		Under the for his proficiency in science, and the Sultan showered favors upon of the eight learned men employed to do it; the result was the Jalali computation of time,' says Gibbon, 'which surpasses the Julian, and author of some astronomical tables, entitled 'Ziji-Malikshahi,' and of his on Algebra.
		Though all these, like our Smiths, Archers, Millers, Fletchers, to the close; it is told in the anonymous preface which is sometimes.
	www.termsdefined.net /ke/kernel-(universal-algebra).html (1008 words)

	Linear Algebra (Site not responding. Last check: 2007-10-21)
		The Range and Kernel of a Linear Transformation
		Let A be a linear transformation of the vector space U into the vector space V.
		The collection of all those vectors x in U such that Ax = 0 is called the kernel of A and is denoted by ker(A) Let A be a linear transformation of the vector space U into the vector space V.
	lagrange.la.asu.edu /VirtualClass/Algebra/RangeKern.html (206 words)

	Linear Algebra Toolkit
		This Linear Algebra Toolkit is comprised of the modules listed below.
		Each module is designed to help a linear algebra student learn and practice a basic linear algebra procedure, such as Gauss-Jordan reduction, calculating the determinant, or checking for linear independence.
		Every effort has been made to make it compatible with a broad range of browsers, however, no guarantee can be made that every browser will properly handle this application.
	www.math.odu.edu /~bogacki/cgi-bin/lat.cgi (488 words)

	ABSTRACT ALGEBRA ON LINE: Contents
		It is intended for undergraduate students taking an abstract algebra class at the junior/senior level, as well as for students taking their first graduate algebra course.
		It is based on the books Abstract Algebra, by John A. Beachy and William D. Blair, and Abstract Algebra II, by John A. Beachy.
		An algebraic extension of an algebraic extension is algebraic(6.2.10)
	www.math.niu.edu /~beachy/aaol/contents.html (401 words)

	Debian -- Software Packages in "testing", devel section
		Kernel module source, reporting changes on files to /dev/fwatch
		Linux 2.4.27 kernel headers for AMD K7 kernel-headers-2.4.27-2-k7-smp (2.4.27-11)
		Linux kernel headers 2.6.8 on Itanium II kernel-headers-2.6.8-2-mckinley-smp (2.6.8-14)
	packages.debian.org /testing/devel (3421 words)

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