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Topic: Differential operator

Related Topics

Operator theory

Operator algebra

Eigenfunction

Adjoint of an operator

Spectral theorem

Adjoint

Fractional calculus

Conjugate transpose

Laplace operator

Eigenvalue

Algebraic equation

Quantum field theory

Diagonal matrix

Pontryagin duality

Orthogonal functions

	NationMaster - Encyclopedia: Differential operator
		In mathematics, a differential operator is an operator defined as a function of the differentiation operator.
		The subring of operators that are polynomials in D with constant coefficients is, by contrast, commutative.
		In differential geometry and algebraic geometry it is often convenient to have a coordinate-independent description of differential operators between two vector bundles.
	www.nationmaster.com /encyclopedia/Differential-operator (2405 words)

	Differential operator - Wikipedia, the free encyclopedia
		In mathematics, a differential operator is an operator defined as a function of the differentiation operator.
		The subring of operators that are polynomials in D with constant coefficients is, by contrast, commutative.
		In differential geometry and algebraic geometry it is often convenient to have a coordinate-independent description of differential operators between two vector bundles.
	en.wikipedia.org /wiki/Differential_operator (800 words)

	PlanetMath: differential operator
		Roughly speaking, a differential operator is a mapping, typically understood to be linear, that transforms a function into another function by means of partial derivatives and multiplication by other functions.
		The notion of a differential operator can be generalized even further by allowing the operator to act on sections of a bundle.
		This is version 6 of differential operator, born on 2002-02-15, modified 2004-11-27.
	www.planetmath.org /encyclopedia/DifferentialOperator.html (312 words)

	Pseudo-differential operator - Wikipedia, the free encyclopedia
		In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator.
		Differential operators are local in the sense that one only needs the value of a function in a neighbourhood of a point to determine the effect of the operator.
		Differential algebra for a definition of pseudo-differential operators in the context of differential algebras and differential rings.
	en.wikipedia.org /wiki/Pseudo-differential_operator (655 words)

	PlanetMath: Green's function for differential operator
		Expression (1) is an example of initial value problem for an ordinary differential equation.
		Thus, function (3) is the Green's function for the operator equation (2) and then for the problem (1).
		This is version 4 of Green's function for differential operator, born on 2004-10-10, modified 2005-06-05.
	www.planetmath.org /encyclopedia/GreensFunctionForDifferentialOperator.html (142 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-06)
		The systematic study of self-adjoint differential operators of the second order on a finite interval dates from 1830 (the Sturm–Liouville problem) and was the subject of intensive study in the 19th century, in particular in connection with the theory of special functions.
		The theory of singular differential operators began in 1909–1910, when the spectral decomposition of a self-adjoint unbounded differential operator of the second order with an arbitrary spectral structure was discovered, and when, in principle, the concept of a deficiency index was introduced, and the first results in the theory of extensions were obtained.
		The systematic investigation of non-self-adjoint singular differential operators began in 1950, when the foundations of the theory of operator pencils were expounded and a method was found for proving the completeness of the system consisting of the eigenfunctions of a differential operator and of their associates.
	eom.springer.de /s/s086530.htm (2063 words)

	Amazon.com: "differential operator realizations": Key Phrase page (Site not responding. Last check: 2007-11-06)
		The Dirac ry matrices are shown to carry, as "operator states", the 4-dimensional vector representation of so(1).
		Differential operator realizations are obtained on these spaces,...
		The real generators of gl(N, Q)+ in the differential operator realizations are then...
	www.amazon.com /phrase/differential-operator-realizations (282 words)

	Differential Rings (New) [HB 67]
		The Galois theory of linear differential equations is the analogue for linear differential equations of the classical Galois theory of polynomial equations.
		Differential rings can be used to create differential operators and in a wider perspective to consider topics related to differential galois theory.
		These differential rings look like the ring they are created from and inherit all the functionality of that ring and similarly for their elements.
	magma.maths.usyd.edu.au /magma/ReleaseNotes/rel211/node29.htm (533 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-06)
		The classification adopted in the theory of linear differential operators refers mainly to linear differential operators that act in bundles of the same dimension, in fact to operators of the form (1) where the coefficients are square matrices.
		The formal theory of general linear differential operators is concerned with the concepts of formal integrability and the resolvent.
		In determining operators with constant coefficients such a representation is specified by an integral with respect to exponents (exponential representation), and for operators with periodic coefficients by an integral with respect to Floquet-generalized solutions.
	eom.springer.de /l/l059170.htm (1881 words)

	Differential operator
		In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator.
		An operator P is said to be a k-th order differential operator if it factors through the jet bundle.
		In abstract algebra the concept of derivation means that differential operators may still be defined, in the absence of calculus concepts based on geometry.
	www.xasa.com /wiki/en/wikipedia/d/di/differential_operator.html (540 words)

	[No title]
		In the functions for differential operators in the DEtools package, the names Dx and x (other names can be used as well) can be specified either by an entry called domain, or by setting _Envdiffopdomain to [Dx,x].
		The output of this procedure is this operator R. Like the gcd for polynomials, this R is a combination R=af+bg for some operators a and b.
		The output of this procedure is a linear differential operator M of minimal order such that for all solutions y1..
	www.math.fsu.edu /~hoeij/daisy/help_pages (4796 words)

	Element Operations on Differential Operators
		iff the differential operator L is the unity element of its parent.
		Returns the order of the differential operator L. In the case that L is identically 0, the order is defined to be -1.
		Given a differential operator L and a ring element f, return the ring element obtained after applying L to f, as an element of the base ring of L. The element f must be coercible into the base ring of L. Example
	www.math.lsu.edu /magma/text904.htm (763 words)

	Prof. Hans Wilhelm Alt: Script Analysis IV -- Linear differential operators
		The operator L is called linear differential operator with constant coefficients (resp.
		This differential equation in fact is a system consisting of M scalar differential equations for N unknown functions.
		We prove, that the necessary condition derived in sect:1-2-(1) for solving the differential equation ∇u=f indeed is a sufficient condition.
	www.iam.uni-bonn.de /~alt/ss2002/EN/analysis4-hyp_2.html (999 words)

	Language and Classification (Site not responding. Last check: 2007-11-06)
		In thinking of partial differential equations, it is a common practice to carry over the language that has been used for matrix or ordinary differential equations in as far as possible.
		The techniques of studying partial differential operators and the properties of these operators change depending on the "type" of operator.
		Before comparing the similarity in procedures for changing the partial differential equation to standard form with the preceeding arithmetic, we pause to emphasize the differences in geometry for the types: elliptic, hyperbolic, and parabolic.
	www.math.gatech.edu /~harrell/pde/herod/jvh10.html (1342 words)

	Tutorial for Green's Functions, Materials Reliability Division, N.I.S.T
		For a discussion of the concept of self-adjoint and non self-adjoint differential operators please refer, for example, to the text by Morse and Feshbach.
		Since L is a differential operator, it is reasonable to expect its inverse to be an integral operator.
		Note that we have used the linearity of the differential and inverse operators in addition to equations (4), (5), and (6) to arrive at the final answer.
	www.boulder.nist.gov /div853/greenfn/tutorial.html?.html (1037 words)

	Partial Differential Equations and Bivariate Orthogonal Polynomials -- from Mathematica Information Center
		One approach, due to Krall and Sheffer in 1967 and pursued by others is to determine which linear partial differential operators have orthogonal polynomial solutions with all the polynomials in the family of the same degree sharing the same eigenvalue.
		In fact, such an operator only determines a multi-dimensional eigenspace associated with each eigenvalue; it does not determine the individual polynomials, even up to a multiplicative constant.
		In contrast, our approach is to seek pairs of linear differential operators which, have joint eigenfunctions that are comprises of family of bivariate orthogonal polynomials.
	library.wolfram.com /infocenter/Articles/1816 (193 words)

	Partial differential operators.
		The operator is called an elliptic operator if the eigenvalues of A are non-zero and have the same algebraic sign.
		If one has a second order partial differential equation with constant coefficients that is not in the standard form, there is a method to change it into this form.
		For ordinary differential equations boundary value problems, the dot product came with the problem in a sense: it was an integral over an appropriate interval on which the functions were defined.
	www.mathphysics.com /pde/green/g16.html (4669 words)

	Optimal Edge-Based Shape Detection
		In this work, we derive a one-dimensional smoothing operator for a step function using a different criterion: minimizing the sum of the noise power and the mean squared error between input and output.
		Since this operator suppresses noise while preserving the step shape in an optimal way, the derivative of the response function is less noisy and close to an impulse function, thus achieving very accurate detection and localization of the step edge.
		This operation is repeated for all possible positions, and the maximum response is chosen as indicating the presence of the specified shape.
	www.cfar.umd.edu /~hankyu/shape_html (2034 words)

	Structure Operations on Differential Operator Rings (Site not responding. Last check: 2007-11-06)
		As outlined in the introduction, a differential operator ring R is of the form F[D], for a differential ring F. The ring F is called the base ring or coefficient ring of R. BaseRing(R) : RngDiffOp -> Rng
		By construction the variable D of a differential operator ring F[D] is related to the derivation delta_F.
		The differential belonging to the derivation of the differential operator ring R. The derivation must have been constructed in such a way that it is defined by a differential.
	wwwmaths.anu.edu.au /research.groups/aat/htmlhelp/text873.htm (246 words)

	Differential Operator Rings (Site not responding. Last check: 2007-11-06)
		Given a differential operator ring R with n indeterminates and a sequence S of n strings, assign the elements of S to the names of the variables of R. This procedure only changes the names used in the printing of the elements of R. Creation of Differential Operators
		The easiest way to create an element in a given ring is to use the angle bracket construction to attach a name to the indeterminate of the differential operator ring.
		The i-th indeterminate of the differential ring R, where i must be 1.
	www.math.lsu.edu /magma/text902.htm (190 words)

	Differential Equation Theory (Site not responding. Last check: 2007-11-06)
		However, the real role of a differential equation lies in relating conditions in one region to those in another, through a recursive process which allows us to progressively work out the solution fiom one place to another, a little bit at a time.
		The framework is prescribed by the differential equation, but the information which is to be relayed from one place to another is contained in the boundary conditions.
		Oddly enough, when we come to regard a differential equation as serving only to define a mapping from the solution space to the boundary value space, it is possible to avoid an excessive preoccupation with the actual boundary values themselves.
	delta.cs.cinvestav.mx /~mcintosh/comun/quant/node4.html (1451 words)

	unified_field.nb (Site not responding. Last check: 2007-11-06)
		The unified force field proposed is modeled on a simplification of the electromagnetic field strength tensor, being formed by a quaternion differential operator acting on a potential, Box* A*.
		In the standard approach to generating the Maxwell equations, a differential operator acts on the electromagnetic field strength tensor.
		An operator that adds nothing to the trace but distorts the lengths of the scalar and 3-vector with the constraint that the difference in lengths is constant will also suffice.
	world.std.com /~sweetser/quaternions/gravity/unified_field/s.html (7009 words)

	The vector differential operator in three dimensions
		In the course of the development of the science of physics, three intimately related differential operators emerged with rôles pivotal to the abstract formalisation of the laws of physics as they were understood before the ramifications of electromagnetism displaced the three-dimensional model of space which is the home of these three.
		It turns out that grad and div are (albeit somewhat disguised by use of the metric) the first and last in a unified chain of differential operators; the chain's length is equal to the dimension of the vector space and curl does indeed correspond to the middle item in the chain for three dimensions.
		Thus differentiating a scalar field should yield a co-vector field; differentiating one of those should yield a second rank tensor of type dual(V)⊗dual(V); differentiating one of those should yield a third rank tensor of rank dual(V)⊗dual(V)⊗dual(V) and so on.
	www.chaos.org.uk /~eddy/math/nabla.html (967 words)

	Publications up to 2000 (Site not responding. Last check: 2007-11-06)
		Differentiable dependence of eigenvalues of operators in Banach spaces, J.
		On the essential spectrum of a differentially rotationg star in the axisymmetric case, Proc.
		The essential spectrum of a system of singular ordinary differential operators of mixed order; Part I: The general problem and an almost regular case, Math.
	sunsite.wits.ac.za /mmoller/node2.html (663 words)

	5.2 Stress-energy bi-tensor operator and noise kernel
		The fundamental problem of defining a quantum operator for the stress tensor is immediately visible: The field operator appears quadratically.
		All field operator products present in the first expectation value that could be divergent, are canceled by similar products in the second term.
		This will allow us to express the noise kernel in terms of a pair of differential operators acting on a combination of four and two point functions.
	relativity.livingreviews.org /Articles/lrr-2004-3/articlesu8.html (1395 words)

	XVIII. Partial differential operators.
		In thinking of partial differential equations, we shall carry over the language that we used for matrix or ordinary differential equations as far as possible.
		Just as in ordinary differential equations, in partial differential equations some boundary conditions will be needed to solve the equations.
		Before we pursue the idea of rescaling and translating in second order partial differential equations in order to come up with canonical forms, we need to recall that there is also the troublesome need to rotate the axis in order to get some quadratic forms into the standard one.
	www.mathphysics.com /pde/ch18wr.html (4173 words)

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