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Topic: Discrete Laplace operator

	Discrete Laplace operator - Wikipedia, the free encyclopedia
		In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid.
		The discrete Laplace operator occurs in physics problems such as the Ising model and loop quantum gravity, as well as in the study of discrete dynamical systems.
		The Green's function of the discrete Schrödinger operator is given in the resolvent formalism by
	en.wikipedia.org /wiki/Discrete_Laplace_operator (404 words)

	Short Presentations
		For $k=d$, the star operator is a bijective mapping between $d$-forms (``densities'') and $0$-forms (``scalar functions''); a discrete analog is an operator which, given the integrals of a function over $n$ cells, reconstructs (approximately) the values of the function at $n$ nodes.
		The mimetic finite difference approach, a relatively new conservative discretization technique which attains second order approximations for the gradient and divergence operators in the whole computational domain (including the boundary), is applied to solve a set of one dimensional test problems associated to the steady diffusion equation, including both rough coefficients and grids.
		In the context of mimetic discretizations, Whitney’s interpolation formula is uniquely characterized as the mimetic discretization of “natural” differential operators on a manifold, which is compatible with the structure of a triangulation and boundary operators.
	www.sci.sdsu.edu /compscims/MIMETIC/Presentations.htm (3707 words)

	MISGAM Workshop Berlin 2005
		Knowing the Darboux transforation for this class of discrete surfaces allows to define discrete surfaces of constant mean curvature (cmc) and to derive some of their geometric properties.
		Boris Springborn: "A discrete Laplace-Beltrami operator for simplicial surfaces"
		Our Laplace operator is similar to the one defined by Pinkall and Polthier (the so called "cotan formula") except hat it is based on the intrinsic Delaunay triangulation of the simplicial surface.
	www.math.tu-berlin.de /geometrie/MISGAM/workshop2005/abstracts.html (1628 words)

	DIFFERENCE EQUATIONS \\ Applications and \\ Discrete Transforms Method
		The method of solving boundary value problems using the discrete Fourier transform is presented......The last chapter of the book, Chapter 7, is devoted to modeling various discrete problems with difference equations......There are a set of exercises at the end of each section and answers are given in the back of the book.
		As in the continuous case, discrete operational methods may not solve problems that are intractable by other methods, but they can facilitate the solution of a large class of discrete initial and boundary value problems.
		In the rest of the chapter we present some of the fundamental difference operators along with their basic properties and their inverses as ``sum'' operators, which are necessary for modeling difference equations as well as developing pairs for the basic discrete transforms.
	people.clarkson.edu /~jerria/book4.html (1647 words)

	Citations: Computing discrete minimal surfaces and their conjugates - Pinkall, Polthier (ResearchIndex)
		Since the mean curvature normal operator, also known as Laplace Beltrami operator, is a generalization of the Laplacian from flat spaces to manifolds [DHKW92] we first compute the Laplacian of the surface with respect to the conformal space parameters u and v.
		In this work, we rather have the goal to describ e discrete minimal surfaces as explicitly as possib#;J and thus we are limited to the more fundamental examples, for example the discrete minimal....
		Then the discrete divergence div h of v is given at each vertex p by div h v(p) Jc i 1 (7) where J denotes the rotation of a vector by 2 in each triangle,....
	citeseer.ist.psu.edu /context/497704/628950 (1380 words)

	Electric Charge and Potential: Spreadsheets for solving Laplace's equation
		They contain a formula that will be used to calculate their potential, in accordance with Laplace’s equation, subject to the specified boundary conditions.
		This means length in the Z direction; it is the charge per unit length of the object rooted in the given area and extending infinitely far perpendicular to the screen.
		Now the discrete approximation to the second derivative in the horizontal direction is b+c-2w, and in the vertical direction it is a+d-2w.
	www.av8n.com /physics/laplace.html (3593 words)

	Diffusion Wavelets
		In the paper “Diffusion Wavelets” we propose a construction of wavelets on discrete (or discretized continuous) graphs and spaces, that are adapted to the “geometry” of a given diffusion operator T, where the attribute “diffusion” is intended in a rather general sense.
		The motivation for starting with a given diffusion operator is that in many cases one is interested in studying functions on the graph/space, and hence it seems natural to start with a local operator generating local relationships between functions.
		We consider a circle as before, but now the diffusion operator is not translation invariant or homogeneous: the conductivity is non-constant and is depicted in the figure in the top-left position.
	www.math.yale.edu /~mmm82/diffusionwavelets.html (2317 words)

	RR-1958 : The Discrete Laplace operator in an octant (Site not responding. Last check: 2007-10-13)
		RR-1958 : The Discrete Laplace operator in an octant
		RR-1958 - The Discrete Laplace operator in an octant
		Abstract : Explicit integral formulas for the resolvent of the discrete Laplace operator in an ortant are obtained, thus providing an analytic continuation of the resolvent.
	www.inria.fr /rrrt/rr-1958.html (136 words)

	Eitan Grinspun's home page
		These models are derived from an axiomatic treatment of discrete Laplace operators, using these operators to obtain linear models for discrete mean curvature from which bending energies are assembled.
		On the one hand, it is similar in its simplicity to some of the discrete curvature operators commonly used in graphics; on the other hand, it passes a number of important convergence tests and produces consistent results for different types of meshes and mesh refinement
		In this paper we introduce a discrete shell model describing the behavior of thin flexible structures, such as hats, leaves, and aluminum cans, which are characterized by a curved undeformed configuration.
	www.cs.columbia.edu /~eitan (1710 words)

	References
		On the dynamics of a general unitary operator.
		Operators with singular continuous spectrum: I. General operators.
		Operators with singular continuous spectrum VI, Graph Laplacians and Laplace-Beltrami operators.
	www.math.harvard.edu /~knill/oldinterests/discrete/node8.html (275 words)

	geometric analysis : SFB 647
		The associated geometric flow and discrete Willmore surfaces are studied.
		Our Laplace operator is similar to the one defined by Pinkall and Polthier (the so called "cotan formula'') except that it is based on the intrinsic Delaunay triangulations of the simplicial surfaces.
		Mirror symmetry on CY manifolds exchanges the symplectic structure on M, actually a complexified Kähler structure, with the complex structure on a mirror dual CY manifold W. The deformation theory of each of these structures can be described by a topological string theory called the topological A- and the B-model respectively.
	geometricanalysis.mi.fu-berlin.de /sfb.htm (1167 words)

	DSP notation (Site not responding. Last check: 2007-10-13)
		It is recommended that f Hz be used for the continuous variable of frequency and that v cycles/sample be used as a discrete variable.
		Further, frequency may be further limited in the discrete case to taking a limited number of values (as in the case of the discrete Fourier transform).
		The unilateral Laplace transform, or one-side Laplace, is defined exactly the same but the lowlimit of the integral is 0 rather than −∞.
	cnx.org /content/m10161/latest (1508 words)

	Citebase - A discrete Laplace-Beltrami operator for simplicial surfaces (Site not responding. Last check: 2007-10-13)
		Authors: Bobenko, Alexander I. Springborn, Boris A. We define a discrete Laplace-Beltrami operator for simplicial surfaces.
		Our Laplace operator is similar to the well known finite-elements Laplacian (the so called ``cotan formula'') except that it is based on the intrinsic Delaunay triangulation of the simplicial surface.
		This leads to new definitions of discrete harmonic functions, discrete mean curvature, and discrete minimal surfaces.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0503219 (239 words)

	Discrete Math (Site not responding. Last check: 2007-10-13)
		The text is geared to both computer science discrete math and math majors, but its emphasis on understanding logic discrete math and proof provides for an effective alternative to calculus for non-majors as well.
		Discrete modelling - Discrete modelling is the discrete analogue of continuous modelling.
		In discrete modelling, discrete formulae are fit to data.
	so98.iasoft.org /discretemath.html (1114 words)

	Discrete Surface Parametrizations
		The aim of discrete differential geometry is the discretization of classical differential geometry, that is, to find proper discrete analogs of differential geometric notions and to develop at the discrete level a corresponding theory.
		Linear and nonlinear theories of discrete analytic functions.
		Hexagonal circle patterns with constant intersection angles and discrete Painleve and Riccati equations.
	www.math.tu-berlin.de /geometrie/ddg/index.shtml (230 words)

	Discrete Wireless (Site not responding. Last check: 2007-10-13)
		These restrictions may be applicable to orders placed by the same Wal-Mart account, the same credit card, discrete wireless and also to orders that use the same billing and/or shipping address.
		Wireless mobility management provides an "alerting" function for call completion to a wireless terminal, monitors wireless link performance to determine when an automatic link transfer is required, and coordinates link transfers between wireless access interfaces.
		Discrete Laplace operator - In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid.
	so98.iasoft.org /discretewireless.html (829 words)

	An Efficient Iterative Method for the Generalized Stokes Problem
		, A behaves like the mass matrix M at one extreme and the Laplace operator T at the other, causing problems for the common iterative methods employed to solve this system.
		The generalized Stokes problem is one of the most time-consuming steps in the large scale simulation of incompressible fluid flows.
		is analogous to a Laplace operator on the domain.
	www.aem.umn.edu /Solid-Liquid_Flows/papers/Sarin/eff_iter_ab.html (1210 words)

	Mathematics of Sampled Data Systems
		Starting in the Laplace domain, the Laplace operator ‘s’ is replaced by a first order difference operator, i.e.
		The index indicates how many integer multiples of the sampling interval is involved in the time shift, with the sign of the index denoting whether it is a forward (plus sign) or backward (minus sign) shift.
		The sampling interval must be sufficiently small for accurate conversion from the continuous time domain to the discrete time domain.
	lorien.ncl.ac.uk /ming/digicont/digimath/sampled2.htm (333 words)

	G2V2: Geometry, Graphics, Vision, Visualization Seminar
		We rely heavily on the discrete Laplace- Beltrami operator and its eigenfunctions in this construction, and discrete Morse theory provides the linkage between our scalar fields and the topology of the surface.
		Many geometric computations can be reduced to algebraic operations, so much exact geometric computation boils down to performing exact algebraic operations.
		This talk will include an overview of these operations, along with the current best solutions, and some of the recent work that holds promise for significant advances in the future.
	www.cise.ufl.edu /~ungor/seminars/G2V2_Fall05 (1193 words)

	CAARMS ABSTRACTS (Site not responding. Last check: 2007-10-13)
		One can define a discrete Laplace operator, which in the limit (when also suitably) defined is the continuous analogue.
		Then, we may ask the same questions in the discrete setting.
		We will then discuss the analogous problems in the discrete setting (joint research with F.R.K. Chung).
	www.wam.umd.edu /~rlj/Oden.html (220 words)

	Dr. Dobb's \| Loose Ends \| August 9, 2006
		In practice, the operators s and D are indistinguishable.
		It's because the Laplace transform gives them access to a huge body of knowledge and techniques such as Nyquist diagrams, Bode plots, root locus plots, the Routh criterion, and more.
		There are two multiplication operators for vectors, no division operator, and a handful of miscellaneous operations, so the package might be smaller than you think.
	www.ddj.com /191901764 (3216 words)

	Markov Processes and Related Filelds
		The Laplace Operator in an Orthant (Algebraic Geometry Approach) pp.
		79-90 Explicit integral formulas for the resolvent of the discrete Laplace operator in an orthant are obtained, thus providing an analytic continuation of the resolvent.
		We develop the analysis of the fluid dynamics in [H. Chen and A. Mandelbaum, Discrete flow networks: bottleneck analysis and fluid approximations, Math.
	www.math.msu.su /~malyshev/abs95.htm (2139 words)

	Documentation for Concepts 2.0
		Discrete equivalent of the Laplacian in 2D for linear and quadratic FEM.
		This bilinear form computes the stiffness matrix resulting from the discretization of the Laplacian (with integration by parts):
		If this method is not reimplemented in a derived class, the default behaviour is to call the application operator without
	www.math.ethz.ch /~concepts/doxygen/html/classLaplace.html (304 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		A few interesting examples will be given including torsion groups of intermediate growth, Basilica group and the lamplighter group.
		Then we will focus on the last example and describe the spectrum of discrete Laplace operator on Cayley grap of this group.
		The information about spectrum will be used to solve one question of M.Atiayh on range of L^2-Betti numbers.
	www-math.mit.edu /~combin/abstracts/nov03/gri.html (114 words)

	Re: Why do Fermions form a Cube?
		the real questions lay directly underneath the figure: (A) Why is the charge operator equal to the discrete Laplace-Beltrami operator (apart from sign) in the complementary cube space?
		(C) The particle quantum numbers essentially coincide with the [eigenvalues of the] isometric operators in the complementary space (of which there are 4 independent ones).
		Why are the particle states linked directly to the symmetry group in the complementary space?
	www.lns.cornell.edu /spr/1999-07/msg0017240.html (313 words)

	APPLICATIONS OF MATHEMATICS, Vol. 46, No. 3, pp. 231-239, 2001 (Site not responding. Last check: 2007-10-13)
		Eigenvalues and eigenfunctions of the Laplace operator on an equilateral triangle
		Abstract: A discretized boundary value problem for the Laplace equation with the Dirichlet and Neumann boundary conditions on an equilateral triangle with a triangular mesh is transformed into a problem of the same type on a rectangle.
		Explicit formulae for all eigenvalues and all eigenfunctions are given.
	am.math.cas.cz /am46-3/4.html (155 words)

	Grigorchuk Abstract (Site not responding. Last check: 2007-10-13)
		After a quick introduction to the theory of groups generated by finite automata we will concentrate in more details on the so-called "lamplighter" group L which will be realized as a group generated by a 2-state automaton.
		This will allow us to completely compute the spectrum of the discrete Laplace operator on L (that turns out to be a pure point spectrum).
		We will then use a trick of G. Baumslag to embed L into a finitely presented group.
	www.math.uiuc.edu /Colloquia/03FA/grigorchuk_oct30-03.html (118 words)

	Homework 4
		Set up a system of linear equations corresponding to the discrete Laplace problem with boundary conditions indicated in the picture below and write down the systems of equations corresponding to Jacobi and Gauss-Seidel iterative methods (see textbook pages 99-102).
		X-tra credit: The Poisson kernel is given in polar coordinates as
		Check by direct computation that it is harmonic, that is that Laplace’s operator
	www.mcs.drexel.edu /~knowak/pde/Homework_4.htm (86 words)

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