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# Topic: Linear operators

###### In the News (Sat 25 May 13)

 Linear - Wikipedia, the free encyclopedia Important examples of linear operators include the derivative considered as a differential operator, and many constructed from it, such as del and the Laplacian. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. In a slightly different usage to the above, a polynomial of degree 1 is said to be linear, because the graph of a function of that form is a line. en.wikipedia.org /wiki/Linear   (373 words)

 Learn more about Linear algebra in the online encyclopedia.   (Site not responding. Last check: 2007-10-22) Linear algebra is the branch of mathematics concerned with the study of vectorss, vector spaces (or linear spaces), linear transformations, and systems of linear equations. Linear algebra today has been extended to consider n-space, since most of the useful results from 2 and 3-space can be extended to n-dimensional space, but we may also use linear algebra to investigate infinite-dimensional spaces. Linear operators take elements from a linear space to another (or to itself), in a manner that is compatible with the addition and scalar multiplication given on the vector space(s). www.onlineencyclopedia.org /l/li/linear_algebra_1.html   (809 words)

 Linear Operators   (Site not responding. Last check: 2007-10-22) The simplest linear operator is the identity operator If the action of a linear operator on the basis vectors is known, then the action on any vector in the vector space is determined. The inverse operator of A, denoted by A electron6.phys.utk.edu /qm1/modules/m3/operators.htm   (286 words)

 Nonlinearity - Wikipedia, the free encyclopedia Examples of linear operators are matrices or linear combinations of powers of partial derivatives e.g. A map F(u) is a generalization of a linear operator. Equations involving maps include linear equations, and nonlinear equations as well as nonlinear systems (the last is a misnomer stemming from matrix equation 'systems', a nonlinear equation can be a scalar valued or matrix valued equation). en.wikipedia.org /wiki/Nonlinearity   (740 words)

 [No title]   (Site not responding. Last check: 2007-10-22) For instance, the integral yields a linear map from the space of all real-valued integrable functions on some interval to R, while differentiation is a linear transformation from the space of all differentiable functions to the space of all functions. In the finite dimensional case and if bases have been chosen, then the composition of linear maps corresponds to the multiplication of matrices, the addition of linear maps corresponds ot the addition of matrices, and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars. If V and W are finite dimensional, bases have been chosen and f is represented by the matrix A, then the rank of f is equal to the rank of the matrix A. www.informationgenius.com /encyclopedia/l/li/linear_transformation.html   (640 words)

 No Title   (Site not responding. Last check: 2007-10-22) Most of the investigations so far have been concerned with operators and relate principally to a study of operational quantities of operators, partial continuity, compactness, Tauberian theory, perturbation theory, index theory, and stability properties of the essential spectra. The results have applications to the theory of closed operators in Banach spaces, and in particular to Fredholm theory and differential operators. On the perturbation of unbounded linear operators with topologically complemented ranges, Journal of Functional Analysis 92 (1990), 468-473. www.mth.uct.ac.za /~webpages/cross   (263 words)

 Linear Operators Almost all operators encountered in quantum mechanics are linear operators. A linear operator is an operator which satisfies the following two conditions: The only other category of operators relevant to quantum mechanics is the set of antilinear operators, for which vergil.chemistry.gatech.edu /notes/quantrev/node14.html   (70 words)

 Abstract linear spaces His systèmes linéaires is a table of coefficients of a system of linear equations denoted by a single upper-case letter and Laguerre defines addition, subtraction and multiplication of of these linear sysyems. Peano defines a linear system to be any system of objects satisfying his four conditions. Peano defines linear operators on a linear space, shows that by using coordinates one obtains a matrix. www-groups.dcs.st-and.ac.uk /~history/HistTopics/Abstract_linear_spaces.html   (1865 words)

 History of Operator Theory   (Site not responding. Last check: 2007-10-22) In the first textbook on operator theory, Théorie des Opérations Linéaires, published in Warsaw 1932, Stefan Banach states that the subject of the book is the study of functions on spaces of infinite dimension, especially those he coyly refers to as spaces of type B, otherwise Banach spaces (definition). Since all these subjects predated operator theory as such by a century or two, it is no surprise that some of the earliest antecedents of operator theory are to be found in them. For example, infinite-dimensional operators can have continuous spectrum, as became evident (though not in that language) when George Hill presented the theory of periodic Sturm-Liouville equations in order to study the stability of the lunar orbit. www.mathphysics.com /opthy/OpHistory.html   (2635 words)

 Functional analysis   (Site not responding. Last check: 2007-10-22) Linear continuous functionals in Hilbert spaces (the Riesz representation theorem). The relation between the kernel and the image of an operator and its adjoint. Operator inequalities and relations to the spectrum and the functional calculus. www.ma.ic.ac.uk /~grigor/fa.htm   (296 words)

 DEOS Boundary Value ProblemPage   (Site not responding. Last check: 2007-10-22) The measurable quantities are (at the level of spherical and constant radius approximation) linear functionals of the fundamental unknown, the earth's gravity (disturbance) potential. With the spherical harmonics as system of eigenfunctions the eigenvalue representation of the linear operators is derived, and consequently the spectral connection among all (measurable) linear gravity functionals. The operators connecting the disturbing potential and its radial derivatives and the operators to get the disturbing potential at a different height, have the spherical harmonics as eigenfunctions and are fully determined by their eigenvalues. www.deos.tudelft.nl /gravi/info/theory/bvp.shtml   (630 words)

 Theory of Linear Operators in Hilbert Space   (Site not responding. Last check: 2007-10-22) This includes the selfadjoint operators which represent observables in quantum physics, and the more interesting ones are unbounded. But there is a mathematical distinction between formally selfadjoint operators (also called symmetric operators) and the selfadjoint ones. For every theorem relating to a bounded linear operator on Hilbert space, replace the operator by a matrix on Euclidean n-space.. www.hallscience.com /store/books_0486677486_Theory-of-Linear-Operators-in-Hilbert-Space.html   (394 words)

 Logical/Linear Operators   (Site not responding. Last check: 2007-10-22) A family of these operators appropriate for measuring the low-order differential structure of image curves is developed. These L/L operators are derived by decomposing a linear model into logical components to ensure that certain structural preconditions for the existence of an image curve are upheld. By thus reducing the incidence of false-positive responses, these operators are a substantial improvement over (thresholded) linear operators which attempt to resolve the same class of features. www.ai.sri.com /%7Eleei/loglin.html   (216 words)

 Bounded Operators form a Vector Space   (Site not responding. Last check: 2007-10-22) If a and b are vector spaces then the linear operators from a into b form another vector space. Linear functions from a into b are sometimes denoted hom(a,b). Now let a and be be normed spaces, and note that the bounded operators from a into b form a vector space. www.mathreference.com /top-ban,bhom.html   (387 words)

 Theory of Linear Operators in Hilbert Space This classic textbook by two mathematicians from the U.S.S.R.’s prestigious Kharkov Mathematics Institute introduces linear operators in Hilbert space, and presents in detail the geometry of Hilbert space and the spectral theory of unitary and self-adjoint operators. Concise treatment, written for students and researchers interested in the interaction of function theory and operator theory, focuses on theory of shift operators, Toeplitz operators and Hardy classes of vector- and operator-valued functions. Topics include normal operators, analytic functions of operators, shift operators, invariant subspace lattices, compact operators, invariant and hyperinvariant subspaces, von Neumann algebras, transitive operator algebras, and algebras associated with invariant subspaces. www.doverdirect.com /0486677486.html   (205 words)

 Askaroo : Linear Operators   (Site not responding. Last check: 2007-10-22) In the language of abstract algebra, a linear transformation is a homomorphism Linear Operators and Foundations of Quantum Mechanics - Von Neumann centennial conference. His research interests are in the area of semi groups of linear operators and other aspects of the underlying theory of partial differential equations. www.askaroo.com /info/Linear_operators.html   (1223 words)

 Tractability of tensor product linear operators   (Site not responding. Last check: 2007-10-22) The problem of approximating linear functionals is fully analyzed in the first two classes of information. We prove that tractability of linear functionals depends on the given space of functions. For some spaces all nontrivial normed linear functionals are intractable, whereas for other spaces all linear functionals are tractable. www.minet.uni-jena.de /~novak/noslwo.html   (169 words)

 linear operators | TutorGig.co.uk Dictionary   (Site not responding. Last check: 2007-10-22) The Analysis of Linear Partial Differential Operators: Distribution Th.. Linear Operators and Linear Systems: An Analytical Approach to Control.. Semigroups of Linear Operators and Applications to Partial Differentia.. www.tutorgig.co.uk /dict.jsp?keywords=linear+operators   (171 words)

 Amazon.ca: Books: Linear Operators, Part 1, General Theory   (Site not responding. Last check: 2007-10-22) Functional analysis and operator theory is a deep and useful subject, and is still an active area of research. Weakly compact operators, so important in the theory of integral equations are discussed, along with the compact operators, so ubiquitous now in operator theory. The finite dimensional case is reviewed first, and the spectral theory of compact operators, a generalization of the Fredholm theory of linear integral equations, is discussed in great detail. www.amazon.ca /exec/obidos/ASIN/0471608483/geometrynet-20/ref=nosim   (1354 words)

 Lie algebras of bounded operators   (Site not responding. Last check: 2007-10-22) But the structure of the linear maps on such a space is too little understood; for these linear maps one cannot speak about something like the Jordan canonical structure of the matrices. We mean the bounded linear operators on a complex Banach space. Wonderfully, the influence between the operator theory and the Lie algebra theory turned out to be in both senses. www.imar.ro /~dbeltita/laobo.html   (373 words)

 Remarks on the Cauchy problem for multi-valued linear operators - Knuckles, Neubrander (ResearchIndex) Abstract: Introduction Multi-valued linear operators appear, for example, as adjoints and closures of single-valued linear operators. Whereas these operators are always closed (see Section 2), degenerate Cauchy problems Bu 0 (t) = Au(t); u(0) = x lead to possibly nonclosed multi-valued operators A = B \Gamma1 A. 2 Multivalued linear operators and degenerate evolution equati.. citeseer.ist.psu.edu /knuckles94remarks.html   (466 words)

 Citations: Triangular and Jordan Representations of Linear Operators - Brodskii (ResearchIndex) Citations: Triangular and Jordan Representations of Linear Operators - Brodskii (ResearchIndex) Brodskii, Triangular and Jordan Representations on Linear operators, Transl. Brodskii, Triangular and Jordan Representations of Linear Operators, Amer. citeseer.ist.psu.edu /context/373614/0   (247 words)

 47: Operator theory Operator theory studies transformations between the vector spaces studied in Functional Analysis, such as differential operators or self-adjoint operators. The analysis might study the spectrum of an individual operator or the semigroup structure of a collection of them. Parameterizations of unitary operators on a Hilbert space (and thus parameterization of the unitary and orthogonal groups). www.math.niu.edu /~rusin/known-math/index/47-XX.html   (240 words)

 Linear Operators on Matrices: Preserving Spectrum and Displacement Structure Linear Operators on Matrices: Preserving Spectrum and Displacement Structure:SIAM Journal on Matrix Analysis and Applications Vol. In this paper we characterize those linear operators on general matrices that preserve singular values and displacement rank. We also characterize those linear operators on Hermitian matrices that preserve eigenvalues and displacement inertia. epubs.siam.org /sam-bin/dbq/article/23444   (85 words)

 Linear Operators Professionally built for the professional installer Linear Operators Professionally built for the professional installer LinearĀ®, the garage door industry’s premier source for radio controls, launched a new line of professionally installed, chain driven, garage door operators. The head unit’s built in courtesy light goes on automatically when the operator is triggered. www.linear-access-controls.com /ligadoop.html   (268 words)

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