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Topic: Elliptic operator

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Timeline of mathematics

	Elliptic curves
		On the conjecture of Birch and Swinnerton-Dyer for elliptic curves, by Cristian D. Gonzalez-Aviles.
		Beppo Levi and the arithmetic of elliptic curves, by Norbert Schappacher.
		On an elliptic analogue of Zagier's conjecture, Jörg Wildeshaus.
	www.fermigier.com /fermigier/elliptic.html.en (746 words)

	Elliptic operator - Wikipedia, the free encyclopedia
		In mathematics, an elliptic operator is one of the major types of differential operator P.
		An important example of an elliptic operator is the Laplacian.
		Their solutions (harmonic functions of a general kind) tend to be smooth functions (if the coefficients in the operator are continuous).
	en.wikipedia.org /wiki/Elliptic_partial_differential_equation (245 words)

	Laplace operator - Wikipedia, the free encyclopedia
		The D'Alembert operator is often used to express the Klein-Gordon equation and the four-dimensional wave equation.
		It is a differential operator on the exterior algebra of a differentiable manifold.
		The discrete Laplace operator is an analog of the continuous Laplacian, defined on graphs and grids.
	en.wikipedia.org /wiki/Laplace_operator (1136 words)

	Atlas: Elliptic operators on manifolds with nonisolated singularities by Boris Sternin
		Atlas: Elliptic operators on manifolds with nonisolated singularities by Boris Sternin
		For elliptic operators on manifolds with edges, we consider the problem of determining the contributions of the strata to the index formula.
		For several classes of elliptic operators we obtain the contributions of the strata as homotopy invariant functionals of the corresponding symbols.
	atlas-conferences.com /cgi-bin/abstract/cakm-08 (713 words)

	[No title] (Site not responding. Last check: 2007-09-15)
		This approach is applicable to all periodic elliptic operators known to be of interest for math physics (including Maxwell), and in all these cases leads to the same model problem of complex analysis.
		Absence of singular continuous spectrum for periodic elliptic operators holds in a very general situation and is a rather straightforward consequence of Floquet theory (\cite{K}, \cite{RS}, \cite{Sj}%).
		This model operator happens to be of a generalized Cauchy-Riemann type \begin{equation} \frac \partial {\partial \overline{z}}+g(z) \label{model} \end{equation} on a complex plane, where the plane arises as a rational plane in $\bf{R}^d$ spanned by two integer vectors $l$ and $n$, and the function $g(z)$ is periodic.
	www.ma.utexas.edu /mp_arc/papers/98-630 (2259 words)

	Canisius College - Toshikazu Natsume
		In mathematical physics, elliptic operators appear as potential equations, parabolic ones appear as heat equations, and hyperbolic ones appear as wave equations.
		Most of the results for elliptic operators on bounded domains in Euclidean spaces also hold for elliptic operators on more general curved spaces, such as the surface of a doughnut, or of a pretzel.
		Elliptic operators on such closed surfaces were observed to have an especially nice property: to each such operator, an integer called the “index” of the operator can be assigned.
	www.canisius.edu /topos/natsume.asp (716 words)

	Harmonic function (Site not responding. Last check: 2007-09-15)
		The set of harmonic functions on a given open set U can be seen as the kernel of the Laplace operator Δ and is therefore a vector space over 'R' : sums, differences and scalar multiples of harmonic functions are again harmonic.
		This is a general fact about elliptic operator s, of which the Laplacian is a major example.
		The real and imaginary part of any holomorphic function yield harmonic functions on 'R' Conversely there is an operator taking a harmonic function u on a region in 'R' to its harmonic conjugate'' ''v, for which u+iv is a holomorphic function; here v is well-defined up to a real constant.
	www.serebella.com /encyclopedia/article-Harmonic_function.html (437 words)

	Elliptic operator: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-09-15)
		In mathematics and physics, the laplace operator or laplacian, denoted by δ, is a differential operator, specifically an important case of an...
		In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations...
		The newtonian potential is an operator in vector calculus that acts as the inverse to the negative laplacian, EHandler: no quick summary.
	www.absoluteastronomy.com /encyclopedia/e/el/elliptic_operator.htm (743 words)

	MATHEON Workshop
		It is known that the standard finite element method produces upper approximations of the eigenvalues.
		In the second part of the talk the hierarchical matrix technique for non-local operators is applied to decompose the sparse Galerkin stiffness matrix of a second order elliptic operator into a product of triangular matrices.
		Our results are readily applicable to the classical Ritz method for compact symmetric integral operators and to finite element method eigenvalue approximation for symmetric positive definite differential operators.
	www-iam.mathematik.hu-berlin.de /matheon (576 words)

	Hodge theory - Wikipedia, the free encyclopedia
		In mathematics, Hodge theory is one aspect of the study of the algebraic topology of a smooth manifold M.
		One major consequence of this is that the de Rham cohomology groups on a compact manifold are finite-dimensional.
		This follows since the operators Δ are elliptic, and the kernel of an elliptic operator on a compact manifold is always a finite-dimensional vector space.
	en.wikipedia.org /wiki/Hodge_theory (682 words)

	Differential and Integral Operators
		Elliptic boundary value problems for general elliptic systems in complete scales of Banach spaces, by I. Roitberg, studies elliptic BVPs for general elliptic systems in the case where the boundary conditions contain both the function from the systems and the additional functions defined at the boundary of the domain.
		Green's formula for elliptic operators with a shift and its applications, by Z. Sheftel, introduces the notion of normal boundary conditions with a shift, and deduces the Green’s formula for such boundary conditions and partial differential equations of even order.
		The contributions to different aspects of operator theory and its applications contained in this volume are of interest for the research workers in the domain.
	www.ici.ro /ici/revista/sic1999_2/art09.html (1098 words)

	[No title] (Site not responding. Last check: 2007-09-15)
		The proof of the corresponding equivariant index theorem, namely that this topological index is equal to G. Kasparov's equivariant analytic index for the elliptic operator, requires the use of equivariant asymptotic homomorphisms of C*-algebras and the associated equivariant version of the E-theory groups of A. Connes and N. Higson.
		The proposed research deals with the index theory of elliptic differential operators on manifolds which commute with the action of a locally compact transformation group, i.e., studying the spaces of solutions of differential equations which exhibit certain degrees of internal symmetry.
		Being able to compute the indices of these elliptic operators should lead to a clearer understanding of a variety of problems in geometry and representation theory, which are important to physics.
	www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9706767.txt (302 words)

	No Title
		By the end of the sixties this had been justified in full generality and it was believed that the same is true for equations and variational problems with higher derivatives.
		In particular, in 1984 he and S. Nazarov found an example of a strongly elliptic fourth order equation with constant coefficients for which the vertex of a cone is not regular in the Wiener sense.
		The classic oblique derivative problem is an important example of the class of operators investigated by Maz'ya and Paneah, provided the vector field is tangent to the boundary along a submanifold of codimension 1 and transversal to this submanifold.
	www.mai.liu.se /~vlmaz/60birthday/60birthday.html (2803 words)

	Definition of Atiyah-Singer index theorem
		In the mathematics of manifolds and differential operators, the Atiyah-Singer index theorem is an important unifying result that connects topology and analysis.
		It deals with elliptic differential operators (such as the Laplacian) on compact manifolds.
		Here E is a differential operator acting on smooth sections of the vector bundle.
	www.wordiq.com /definition/Atiyah-Singer_index_theorem (738 words)

	Publications of Robert Denk (Site not responding. Last check: 2007-09-15)
		Elliptic boundary value problems with large parameter for mixed order systems (with L. Volevich).
		The spectrum of a parametrized partial differential operator occurring in hydrodynamics (with M. Möller, C. Tretter).
		On elliptic operator pencils with general boundary conditions.
	www.uni-regensburg.de /Fakultaeten/nat_Fak_I/Denk/publ.html (354 words)

	Atiyah-Singer index theorem (Site not responding. Last check: 2007-09-15)
		In the mathematics of manifolds and differential operators the Atiyah-Singer index theorem is an important unifying result that topology and analysis.
		Such a Fredholm operator has an index defined as the difference between the of the kernel of E (solutions of Ef = 0 the harmonic functions in a general sense) and the of the cokernel of E (the constraints on the right-hand-side of inhomogeneous equation like Ef = g).
		Given the examples from Hodge theory Cauchy-Riemann operators in several variables and the topologists' work on the Theorem at the time the required concepts perhaps all 'up in the air' by 1960.
	www.freeglossary.com /Atiyah-Singer_theorem (891 words)

	[No title] (Site not responding. Last check: 2007-09-15)
		The results apply to elliptic equations and systems (including overdetermined ones) on abelian coverings of compact Riemannian manifolds, as well as to holomorphic functions on abelian coverings of compact complex manifolds, and to periodic equations on abelian coverings of combinatorial and quantum graphs.
		The ellipticity is understood in the sense of the nonvanishing of the principal symbol of the operator $P$ on the cotangent bundle (with the zero section removed) $T^*X\setminus (X\times \{0\})$.
		In particular, the bottom of the spectrum of the operator $H$ is at $\Lambda_{0}$.
	www.ma.utexas.edu /mp_arc/html/papers/05-91 (9553 words)

	[No title]
		By an EBVP we mean an elliptic differential operator over a compact manifold with boundary, endowed with a global (injectively elliptic) boundary condition.
		We have (with $\Ss := \Ss_0$): %Theorem 1 \begin{theorem} For $\z$-comparable operators $A_1,A_2$ \begin{equation}\label{e:prop1} {\rm det}_{\z,\myth}(A_1,A_2) = {\rm det}_F \Ss\,.
		Associated to $\D$ we have the first-order elliptic operator acting on sections of $E^0 \oplus E^1$ \begin{equation} \wD = \begin{pmatrix} 0 & D^ \\ D & -I \ \end{pmatrix} : H^1 (X, E^0 \oplus E^1)\to L^2 (X, E^0 \oplus E^1) \.
	www.maths.tcd.ie /EMIS/journals/ERA-AMS/2001-01-003/2001-01-003.tex.html (2872 words)

	Bulletin of the American Mathematical Society (Site not responding. Last check: 2007-09-15)
		Abstract: In its original form, the Atiyah-Singer Index Theorem equates two global quantities of a closed manifold, one analytic (the index of an elliptic operator) and one topological (a characteristic number).
		For operators naturally associated to a Riemannian metric on a closed manifold, the topological side of the Index Theorem can often be expressed as the integral of local (i.e.
		In §§2,3, we discuss further developments in index theory which lead to spectral invariants, the eta invariant and the determinant of an elliptic operator, that are definitely nonlocal.
	www.ams.org /bull/1997-34-04/S0273-0979-97-00731-3/home.html (884 words)

	VNTL Publishers
		In this paper we consider the variational parabolic inequality for the operator $u_{tt}+A_{3}u+A_{4}u_{t}+g(u_{t})$ in unbounded domain, where $A_{3}$ is a linear elliptic operator of the fourth order and $A_{4}$ is a linear elliptic operator of the second order.
		Unbounded operators on intermediate spaces of Banach spaces generated by the method of real interpolation are investigated.
		Two nonlinear majorants for nonlinear operator are constructed depending on the conditions imposed on it.
	vd.litech.net /en/?journal=28 (817 words)

	A robust multilevel approach for minimizing H(div)-dominated functionals in an H1-conforming finite element space
		In this paper, we address a system of equations that is generated from a perturbation of the non-elliptic operator [I - ∇ ∇ · ] by a negative ε Δ.
		For ε near to one, this operator is elliptic, but as ε approaches zero, the operator becomes non-elliptic as it is dominated by its non-elliptic part.
		The robustness of the multigrid algorithm depends on a relaxation operator that yields a smooth error.
	math.lanl.gov /~austint/Publications/austin-2004-robust.shtml (372 words)

	[No title]
		While the theorem applies to any compact mfld with an elliptic operator, the cohomology theory which plays a vital role is K-theory, which is complex-oriented.
		The versions of elliptic cohom constructed by Hopkins and Miller, which are now being called rings of topological modular forms and one of which is the proper target for the Witten genus, follow this pattern.
		According to a friend who studies elliptic operators on manifolds, the KO-theory orientation class is, in an appropriate setting, represented by the Dirac operator.
	www.math.niu.edu /~rusin/known-math/00_incoming/ellip_coho (735 words)

	[No title] (Site not responding. Last check: 2007-09-15)
		The 2004 Abel Prize in mathematics is announced to be awarded to Michael F. Atiyah and Isadore M. Singer for their index theorem.
		In mathematics, in particular in partial differential equations and differential geometry, an elliptic complex generalizes the notion of an elliptic operator to sequences.
		Elliptic complexes isolate those features common to the de Rham complex and the Dolbeault complex which are essential for performing
	www.humpdaytimes.co.uk /htl_dfn.php (364 words)

	Atiyah-Singer index theorem - All About All (Site not responding. Last check: 2007-09-15)
		In the mathematics of manifolds and differential operators, the Atiyah–Singer index theorem is an important unifying result that connects topology and analysis.
		The first research announcement was a 1963 paper (Atiyah-Singer), and the sequence of major papers began with The index of elliptic operators: I in the Annals of Mathematics (1968).
		on elliptic operators when i is the inclusion map of a submanifold, that preserves the index.
	www.answers-zone.com /article/Atiyah-Singer_index_theorem (956 words)

	publications
		Maz'ya, V. The selfadjointness of the Laplace operator.
		-capacity, imbedding theorems and the spectrum of a selfadjoint elliptic operator.
		Gel'man, I. V.; Maz'ya, V. Estimates on the boundary for differential operators with constant coefficients in a half-space.
	www.mai.liu.se /~vlmaz/publications/publications.html (8463 words)

	R
		The theory of elliptic boundary value problems is by now well understood, having been developed to a high degree of sophistication.
		The class of operators I am working with have symplectic characteristic set lying over the boundary and are hyppoelliptic with loss of one derivative.
		After transferring the boundary value problem to a problem on the boundary only, there should be a close connection between the characteristic set of the pseudodifferential operators obtained this way (the trace of the original partial differential operators on the boundary) and the characteristic set of the pseudodifferential operators giving the boundary conditions.
	www.math.temple.edu /~cristi/professional/rp.html (607 words)

	[No title]
		The following table of approximate minimum c values of CCON was constructed from solution of the c elliptic equation, in a unit square domain with the c source and diffusion functions given by a uniformly c distributed random number, on an Alliant FX/40.
		This table c is intended as a guide to the user; the actual minimum c attainable value of CCON depends on the particular problem c and on the computer used.
		The flag ICFLAG determines the operation of the c routine: c c If the routine is called with ICFLAG=0, only c preprocessing is done, the arrays C and IPC c are determined, and CAPCN immediately returns to c the calling program without solving.
	netlib2.cs.utk.edu /toms/732 (2320 words)

	cm conference abstract: Justin Wan (Site not responding. Last check: 2007-09-15)
		Multigrid for solving elliptic partial differential equations (PDEs) with smooth coefficients has been proven, both numerically and theoretically, to be a successful and powerful techniques.
		Since the discretization matrix is nonsymmetric in general, the energy norm induced by the symmetric positive definite matrix in the case of elliptic PDEs no longer applicable.
		Our idea is to construct two sets of basis functions, one by minimizing the energy norm of the elliptic operator, and the other set by minimizing the L2 norm, subject to appropriate constraints.
	www.mgnet.org /mgnet/Conferences/CopperMtn03/abs/wan.html (246 words)

	Department of Mathematics - University of Sussex (Site not responding. Last check: 2007-09-15)
		The 'typical' elliptic operator is the Laplace operator and so the course begins with such properties of solutions of Laplace's equation as mean-value theorems and various maximum principles.
		Higher-order elliptic equations are also examined, and the existence of solutions of the corresponding Dirichlet boundary-value problem is established.
		The main technical tool is the Fredholm theory which requires the knowledge of the properties of compact operators.
	www.maths.sussex.ac.uk /GRC/Degrees/MSc/elliptic.html (111 words)

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