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Topic: Kernel mathematics

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	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.
		In set theory, the kernel of a function f : X → Y is an equivalence relation on X which is defined in terms of f.
		The kernel pair of a morphism f is defined as a pullback of f with itself.
	en.wikipedia.org /wiki/Kernel_(mathematics) (452 words)

	kernel (mathematics) (Site not responding. Last check: 2007-11-07)
		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 (Site not responding. Last check: 2007-11-07)
		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.
		Meta-Kernel The Meta Kernel is a kernel which enhances and accelerates many HP48 GX functions.
	www.serebella.com /encyclopedia/article-Kernel.html (325 words)

	Kernel (mathematics)
		This type of kernel is used in the kernel trick which has applications in several fields of applied mathematics.
		But in the case of Mal'cev algebra s, 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.nebulasearch.com /encyclopedia/article/Kernel_(mathematics).html (267 words)

	kernel -- Encyclopædia Britannica
		Normal kernels are replaced by smut “balls” containing powdery masses of brownish-fl spores having a dead-fish odour.
		The wheat kernel consists of a tiny plant called the embryo, or germ, which makes up about 3 percent of the weight of the kernel; the starchy endosperm; and the protective seed coat, or bran.
		A kernel of corn is wrapped in a tough, fibrous outer hull (bran).
	www.britannica.com /eb/article-9045160?tocId=9045160 (791 words)

	kernel (mathematics)
		In set theory, the kernel of a function f : X → Y is an equivalence relation on X which is defined in terms of f.
		The function f is injective iff the kernel is the diagonal in X × X.
		Unrelated to the meanings in algebra and category theory, there is the notion of a kernel of an integral operator.
	en.mcfly.org /kernel+%28mathematics%29 (396 words)

	HEGEL AND MATHEMATICS (Site not responding. Last check: 2007-11-07)
		He therefore, of course, also considered Hegel's philosophy of mathematics from the point of view of a criticism that distinguishes, which knew how to separate the positive kernel of the material and its faithful translation and transformation from the negative shell of the mystically-distorted ideal.
		Thus mathematical concepts and conformities to law are considered not as absolute, unchangeable, eternal truths, but as parts of the ideological superstructure of human society tied to the latter's fate.
		The class standpoint in mathematics must not, however, be interpreted in such a way that all previous mathematics is rejected as a whole and that in its place a mathematics constructed out of completely new elements must be set up according to totally new principles.
	www.marxists.org /reference/subject/philosophy/works/ru/kolman.htm (5559 words)

	Ecology and Society: The cost of restoration as a way of defining resilience: a viability approach applied to a model ...
		A mathematical formulation of resilience is built in the framework of the viability theory.
		Using the framework of the viability theory, we first propose a mathematical formulation of resilience that is consistent with the characteristics taken from the theoretical definitions.
		We have proposed a new mathematical formulation of resilience in the framework of the viability theory.
	www.ecologyandsociety.org /vol9/iss2/art8/main.html (8119 words)

	Kernel trick - InformationBlast
		The kernel trick was first published in the 1964 paper Theoretical foundations of the potential function method in pattern recognition learning.
		The kernel trick uses Mercer's condition, which states that any positive semi-definite kernel K(x, y) can be expressed as a dot product in a high-dimensional space.
		This is desirable, because the high-dimensional space may be infinite-dimensional (as is the case when the kernel is a Gaussian).
	www.informationblast.com /Kernel_trick.html (230 words)

	Digital Hermit - Kernel-Build-HOWTO
		This said, the newest Linux kernel (2.6.5 as of this writing) has noticeable improvements for the typical desktop user as a result of the improved scheduling system in the new kernel.
		Even for older kernels, rebuilding is often necessary for low memory situations, esoteric hardware, and in cases where every ounce of performance must be extracted.
		The Linux kernel is often likened to the conductor in an orchestra.
	www.digitalhermit.com /linux/Kernel-Build-HOWTO.html (5411 words)

	Earliest Known Uses of Some of the Words of Mathematics (K) (Site not responding. Last check: 2007-11-07)
		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).
		Kernel entered Statistics with the use of Fourier theory in describing estimates for spectral density and probability density functions.
		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 (1079 words)

	Presentations of Ivan Avramidi (Site not responding. Last check: 2007-11-07)
		The criterion of strong ellipticity is discussed and the first non-trivial coefficient of the asymptotic expansion of the trace of the heat kernel is computed.
		The boundary-value problem for Laplace-type operators acting on smooth sections of a vector bundle over a compact Riemannian manifold with nonstandard singular boundary conditions, which include Dirichlet conditions on one part of the boundary and Neumann ones on another part of the boundary, is studied.
		The resolvent kernel and the heat kernel in the leading approximation are explicitly constructed and the leading heat-kernel asymptotics are computed.
	infohost.nmt.edu /~iavramid/talks/naples01abs.html (415 words)

	Mathematics (Site not responding. Last check: 2007-11-07)
		These benefits are designed to stimulate interest in mathematics by providing expository books and articles on contemporary mathematics and on recent developments at the frontiers of mathematical research, and by exchanging information about important events in the mathematical world.
		A major emphasis of the MAA is the teaching of mathematics at the collegiate level, but anyone who is interested in mathematics is welcome to join.
		The Geometry Center is a mathematics research and education center at the University of Minnesota.
	www.cox-internet.com /vlopez11/math.html (836 words)

	Citations: Kernel estimation of regression function - Gasser, Muller (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		Though boundary kernel methods and re#ection methods have been proposed for correcting these boundary e#ects, it is widely recognized that local polynomial methods are more attractive in terms of computational implementation and optimal minimax properties.
		The parameters may be estimated by an approximate maximum likelihood method proposed by Beran (1995) A data driven algorithm for estimating 1 SEMIFAR models, which is a mixture of these two approaches, was proposed by Beran (1997) This paper summarizes recent results on the estimation of....
		The behavior of these various kernel regression estimators is quite similar in practice, however, since they are all based on locally weighted averages.
	citeseer.ist.psu.edu /context/187758/0 (3458 words)

	IBM Research \| IBM Research \| Mathematical Sciences
		We are a center for basic and applied research in the mathematical sciences and related areas.
		Mathematical Sciences affects all areas, ranging from more efficient arithmetic calculations on chips to the invention of the Trackpoint for the Thinkpad to the creation of sophisticated systems for the evaluation and management of risk.
		The focus of this research is two-fold: (i) to provide a library of reusable kernel optimization functions that are customized for steel manufacturing, and (ii) generalize these to make them usable for a broader set of production processes in other industries.
	domino.research.ibm.com /comm/research.nsf/pages/d.math.html (327 words)

	Kernel (Site not responding. Last check: 2007-11-07)
		The kernel of a seed is all that is within the coat of the seed as the edible contained in the shell of a nut ; hence anything included in a shell or integument.
		In mathematics kernel has several different somewhat unrelated meanings; kernel (mathematics) or go directly to kernel of a function kernel (algebra) or kernel (category theory).
		This is the first real-time kernel I use and with the help of the explanations and examples in the book it's very easy to use.
	www.freeglossary.com /Kernel (585 words)

	Kernel
		A kernel is the inner part of a seed.
		In mathematics, kernel has different, unrelated meanings, in algebra and analysis: see kernel (mathematics).
		The kernel of a homomorphism is the set of all elements that the homomorphism sends to the identity element.
	www.fastload.org /ke/Kernel.html (145 words)

	Presentations of Ivan Avramidi (Site not responding. Last check: 2007-11-07)
		The heat kernel diagonal of a second-order Laplace type partial differential operator acting on sections of a vector bundle over a Riemannian manifold is studied.
		An algebraic method for the calculation of the heat kernel coefficients is developed.
		It is shown that the heat semigroup for the scalar Laplacian on symmetric spaces may be presented in form of an average over the Lie group of isometries.
	infohost.nmt.edu /~iavramid/talks/tgtc02abs.html (197 words)

	ADDRESS GIVEN BY DAVID HILBERT, 1930 (Site not responding. Last check: 2007-11-07)
		Mathematics builds the connecting bridges and is constantly enhancing their capabilities.
		This language however is mathematics, and these signs are the figures of mathematics.
		Mathematics owes its prestige, to the extent that it has any among the general public, to these sciences along with their numerous broader applications.
	math.ucsd.edu /~williams/motiv/hilbert.html (534 words)

	HlbrtKE.htm
		The tool that serves as intermediary between theory and practice, between thought and observation, is mathematics.
		Kant pronounced: "I assert that, in every particular natural science, one encounters intrinsically scientific substance only to the extent that mathematics is present." Indeed, we do not master a scientific theory until we have shelled and completely pried free its mathematical kernel.
		Without mathematics, the astronomy and physics of today would be impossible.
	topo.math.u-psud.fr /~lcs/Hilbert/HlbrtKE.htm (595 words)

	Kernel (Site not responding. Last check: 2007-11-07)
		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 term kernel can also mean a single seed or grain, as "a kernel of corn".
		This is a disambiguation page; that is, one that points to various pages that might otherwise have the same name.
	www.theezine.net /k/kernel.html (138 words)

	Index of /~jcorneli/d/essay (Site not responding. Last check: 2007-11-07)
		Human-generated hypertext is not meant to be excluded, but given the quantity of extant mathematics, and the fact that much of the text falls under copyright, and the need for some standards for the database entries, automation of many parts of the process will be critical.
		If they choose the route of simply digitizing the portion of mathematical writing that lies in the public domain, then their product will be free in the sense of freedom, but not comprehensive in any concrete sense.
		It is not our ambition to put all publishers of mathematics out of business any more than it is our ambition to disenfranchise teachers or producers of mathematics; on the contrary, the HDM will create new opportunities in all of these areas, even as it transforms them.
	www.ma.utexas.edu /~jcorneli/d/essay (2845 words)

	Kernel - TheBestLinks.com - Cereal, Corn, Linux kernel, Mathematics, ... (Site not responding. Last check: 2007-11-07)
		Kernel - TheBestLinks.com - Cereal, Corn, Linux kernel, Mathematics,...
		Kernel, Cereal, Corn, Linux kernel, Mathematics, Operating system, Seed, Kernel...
		This is a disambiguation page, i.e., a navigational aid which lists other pages that might otherwise share the same title.
	www.thebestlinks.com /Kernel.html (212 words)

	Amazon.com: Books: Kernel Methods for Pattern Analysis (Site not responding. Last check: 2007-11-07)
		This book provides professionals with a large selection of algorithms, kernels and solutions ready for implementation and suitable for standard pattern discovery problems in fields such as bioinformatics, text analysis and image analysis.
		The unifying framework of kernel methods has proven state of the art results and the community has been waiting for a book like this to make both theory and practice of kernel methods accesssible for readers of all different backgrounds (researchers, students, practioners from both academia and industry,...).
		The book covers all the relevant topics in the state of the art of kernel methods, a field of research in which the authors have been a driving force since the beginning.
	www.amazon.com /exec/obidos/tg/detail/-/0521813972?v=glance (1166 words)

	A Comparative Study of Kernel Functions for Primal-Dual Interior-Point Algorithms in Linear Optimization (Site not responding. Last check: 2007-11-07)
		In both the algorithm and its analysis we use a single neighborhood of the central path; the neighborhood naturally depends on the kernel function.
		An important conclusion is that inverse functions of suitable restrictions of the kernel function and its first derivative more or less determine the behavior of the corresponding IPMs.
		It is shown that small-update methods based on the new kernel functions all have the same complexity as the classical primal-dual IPM, namely, $O(\sqrt{n}\log\frac{n}{\e})$.
	epubs.siam.org /sam-bin/dbq/article/42311 (334 words)

	Citations: Kernel on random graphs - Vega (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		Since existence of a kernel is a 1 1 property, it follows that the countable random digraph has a kernel, too.
		double kernel if it is a kernel and can be split in two non empty sub kernels A 1 and A 2 such that F j= 8x(x 62 A (9y 2 A 1 (Rxy) 9y 2 A 2 (Rxy) The notion of a kernel comes from the theory of digraphs, i.e.
		....we do not have a proof that existence of a single kernel in a random frame is not almost surely valid, but we have a very strong evidence for that.
	citeseer.ist.psu.edu /context/1127082/0 (911 words)

	Newsletter.oct95 Mathematica 2.2 on UNIX and OpenVMS Systems (Site not responding. Last check: 2007-11-07)
		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)

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