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# Topic: Boundary topology

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 Boundary (topology) - Wikipedia, the free encyclopedia In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. In discussing boundaries of manifolds or simplexes and their simplicial complexes, one often meets the assertion that the boundary of the boundary is always empty. For example, the topological boundary of a closed disk viewed as a topological space is empty, while its boundary in the sense of manifolds is the circle surrounding the disk. en.wikipedia.org /wiki/Boundary_(topology)   (750 words)

 [No title] Topology Glossary Mainly extracted from (a) UC Davis Math:Profile Glossary (http://www.math.ucdavis.edu/profiles/glossary.html) by Greg Kuperberg (http://www.math.ucdavis.edu/profiles/kuperberg.html), and (b) Topology Atlas Glossary (http://www.achilles.net/~mtalbot/TopoGloss.html). boundary of Manifold The boundary Bd M of a manifold M is the set of points of M that have boundary patches. An early result in topology states that every closed 3-manifold (closed meaning that the manifold is finite and connected but has no boundary) has a Heegaard splitting and a resulting description in terms of a Heegaard diagram, which describes how the two handlebodies are glued together. www.ornl.gov /sci/ortep/topology/defs.txt   (5717 words)

 WIND User's Guide - Boundary Conditions The use of implicit boundary conditions with an implicit solver is known to improve the stability of the method and lead to faster convergence at the expense of greater computational effort and complexity. boundary condition imposes a no-slip condition of the flow, a zero pressure gradient, and the appropriate heat transfer condition (adiabatic or constant temperature) at the zone boundary (wall surface). boundary condition may be used for internal flows at boundaries where subsonic flow is leaving the computational domain, such as at the exit plane of an inlet, diffuser, or auxiliary flow duct. www.grc.nasa.gov /WWW/winddocs/wind5.0/user/bc.html   (2963 words)

 PlanetMath: boundary / frontier From the definition, it follows that the boundary of any set is a closed set. The term boundary' (but not frontier') is used in a different sense for topological manifolds: the boundary This is version 14 of boundary / frontier, born on 2003-04-22, modified 2006-08-27. planetmath.org /encyclopedia/Boundary.html   (123 words)

 Storage and Topology in the Redesigned MAF/TIGER A distinct advantage of using topology derives from the relationships that are maintained between the "topological primitives", i.e., nodes, edges, and faces. Topology is utilized to a significant degree by many of the applications that comprise the Census Bureau's Geographic Support System. Alternatively, if topology is used to organize spatial data, the coordinates for a section of road that also bounds geographic areas are only stored once, and the correction becomes more straightforward, although it is still subject to business rules that govern feature consistency. www.census.gov /geo/mtep_obj2/topo_and_data_stor.html   (4867 words)

 Topology   (Site not responding. Last check: 2007-10-13) Topology is a branch of pure mathematics that deals with the abstract relationships found in geometry and analysis. The study of topology requires a solid background in calculus and a facility with logic and proofs. Some have described topology as seeking the most general kind of space in which continuity can be defined. www.stetson.edu /~mhale/topology/index.htm   (462 words)

 Cartan's Corner : Point Set Topology Remarks: In the definition of a topology when the number of elements of the set in not finite, the logical intersection of open sets is restricted to any pair, and the logical union of closed sets to restricted to any pair. Relative to the topology T4(open), the interior of (ab) is the singleton, (a): In all cases, note that the union of the interior and the boundary is equal to the union of the set and its limit points. www22.pair.com /csdc/car/carfre64.htm   (2727 words)

 Euclid's Elements, Book I, Definitions 13 and 14 Plane figures are not solid figures since they are not contained by any boundaries in space. A circle is defined in I.Def.15 as is a plane figure and has its circumference as its boundary. A sphere is defined in XI.Def.14 as a solid figure and has a spherical surface as its boundary. aleph0.clarku.edu /~djoyce/java/elements/bookI/defI13.html   (489 words)

 PhD Abstracts The numerical implementation of the boundary integral equation, referred to as the boundary element method, uses isoparametric linear and/or quadratic polynomial approximation of the boundary geometry, displacements and tractions. The boundary element formulation for the analysis of linear elastic fatigue crack propagation is given. The boundary integral equation is one of several equivalent forms of governing equations that can be used to compute approximate solutions to boundary value problems in elasticity and potential flow analvsis. www.cfg.cornell.edu /publications/phd_abstracts.htm   (5583 words)

 Understanding Topology and Shapefiles Today, topology in GIS is generally defined as the spatial relationships between adjacent or neighboring features. The typical digitizing procedure is to digitize all lines, build topology, and label polygons and then clean up slivers, dangles, and under- and overshoots and build topology again, repeating the clean and build phases as many times as necessary. The first step in enforcing planar topology in a shapefile is to remove twisted or self-intersecting polygon rings and to ensure that the "inside" of the polygon is on the correct side of the polygon boundary. www.esri.com /news/arcuser/0401/topo.html   (1912 words)

 Computing Hierarchical Curve-Skeletons of 3D Objects We place electrical point charges on the object's boundary, considering each boundary voxel to be a point charge (a boundary voxel is an object voxel which has a background neighbor). The repulsive force at an interior object voxel due to a nearby point charge is defined as a force pushing the voxel away from the charge with a strength that is inverse proportional to the distance between the voxel and the charge, raised at a certain power m which we call "the field strength". A disadvantage of boundary seeding based on the curvature is that it is affected even by small amounts of noise present on the object’s boundary. www.caip.rutgers.edu /~cornea/Skeletonization/index.html   (1596 words)

 Untitled Document A good summary, this could also be a kind of manifesto for the extension of topology (a "boundary language") into all domains where treatment involves both technological and social/cultural/spiritual considerations. In the nursing perspective, this boundary is invariant and marks the position of the patient as a biological entity. In common parlance, the boundary of the nursing perspective is equivalent to the topological surface of the individual, that is, at first approximation, to the skin. art3idea.psu.edu /hoath/hoath_4.html   (1170 words)

 Manifold Summary In dimension 2, there is a complete classification of the 'closed' manifolds, surfaces that have no boundary curves and that fit in a finite region. The non-orientable compact 2-manifolds are the projective plane (a manifold formed from a flat circular paper by gluing opposite points on the boundary circle), the Klein bottle, which is obtained from the projective plane by attaching a handle, and surfaces formed from the projective plane by adding 2 handles, 3 handles, and so forth. To fully understand the mathematics behind manifolds, it is necessary to know elementary concepts regarding sets and functions, and a working knowledge of calculus and topology would be helpful. www.bookrags.com /Manifold   (6881 words)

 genetic-programming.com-Home-Page The topology of a circuit, in turn, includes several aspects of the circuit, For example, the genetically evolved parameterized topology has a total of 9 components. The boundaries of the passband and the stopband may vary over the range between 1,000 Hz and 100,000 Hz. The programâ€™s inputs are the boundary of the passband (F1) and the boundary of the stopband (F2). www.genetic-programming.com /parameterized.html   (1618 words)

 Wind-US User's Guide - Boundary Conditions Wind-US imposes the boundary conditions explicitly after the interior solution has been computed for each zone after each iteration, by default. Implicit boundary conditions are available in Wind-US in a limited manner, primarily for wall boundary conditions. They are treated as normal coupled boundaries, with the connection data stored in the common grid (.cgd) file when setting boundary conditions with GMAN. www.grc.nasa.gov /WWW/winddocs/user/bc.html   (3003 words)

 Topology   (Site not responding. Last check: 2007-10-13) If the unit disk contains half of its boundary, or a line segment contains one end point and not the other, it is neither open nor closed. A topology is a set of points, and a collection of subsets that are designated as "open". To be a proper topology, the empty set and the entire set must be open, and open sets remain open under finite intersection and arbitrary union. www.mathreference.com /top,intro.html   (430 words)

 Glossary of Topology Terms boundary of A is the union of points for which every open set around them intersects both A and its complement OR the image of a differential map in a complex. the boundary Bd M of a manifold M is the set of points of M that have boundary patches. The metric d gives rise to a topology on P; a basis for this topology is the collection of sets { q: d(p,q)0. cage.rug.ac.be /~hvernaev/TopoGloss.html   (2536 words)

 Cisco - Understanding Multiple Spanning Tree Protocol (802.1s) In the topology described in the first diagram, there are only two different final logical topologies, so only two spanning tree instances are really necessary. In this diagram, the port on B1 is at the boundary of region A, whereas the ports on B2 and B3 are internal to region B: MST Instances The topology is determined by the instance, regardless of the VLANs mapped to it. www.cisco.com /warp/public/473/147.html   (4028 words)

 Rehren Duality | Musings The bottom line is that the “AQFT,” that Rehren would like to construct as the “dual” of a CFT on the boundary, bears little resemblance to a bona fide quantum field theory. Well-known arguments lead to the conclusion that the partition function (as a function of the boundary-value data) satisfies the requisite properties to be the generating functional of a (conformal) QFT on the boundary. Jacques rightly insists in the CONCEPTUAL difference between “restriction” (boundary fields are bulk fields evaluated at z=0) and “duality” (prescribed boundary values of bulk fields are sources for operators in the boundary theory). golem.ph.utexas.edu /~distler/blog/archives/000987.html   (5742 words)

 key terms In contrast to 'projective' ways of representing and knowing, topology is a 'method of touch' or, to use the literary term, 'metonymy.' Topology is capable of describing the contingent circumstances of experience and often contrasts with our imagined pictures of experience, which use framing and thematization to leave out key details. Topology stands for many things: a means of 'journaling' experience, a means of accounting for self-reference and recursion, and the calculus of George Spencer Brown, a one-symbol logic that improves on Boolean logic's either-or mandate. The connection between inside and outside, violating rules of boundaries and transitive space, is the most important for architecture and the arts because of the 'inside-out' effect. art3idea.psu.edu /boundaries/documents/key_terms.html   (2768 words)

 Boundaries in topological space However, she also make reference to the "discrete" topology in which all sets are open and all sets are closed. The boundary of any set A is cl(A)- int(A)= A- A= {}, not the complement of A. In the "indiscrete" topology, where the only open sets are X and {}, the only closed sets are also X and {}. If there is a topology T on X, and A is a proper subset of X, then there is an induced topology on A, I'll call it T(A), the open sets of which are formed from the intersection of A with open sets in X. Yes. www.physicsforums.com /showthread.php?t=96411   (2001 words)

 Amazon.com: Algebraic Topology: Books: Allen Hatcher   (Site not responding. Last check: 2007-10-13) Topology from the Differentiable Viewpoint by John Milnor Its richness in examples (in both the main text and the problems) exposes a beginner to the underlying mechanisms of geometry in algebraic topology; its choice and arrangement of topics strike a perfect balance between accesibility and substantiveness; its lively and motivating exposition makes a student reluctant to attend the often boring topology classes. There are many really good textbooks on algebraic topology and each has its own merit: Bredon for his effort in explaining everything that can be dealt without using spectral equences, Fomenko and Novikov for their effort in unifying differential geometry and algebraic/differential topology. www.amazon.com /Algebraic-Topology-Allen-Hatcher/dp/0521795400   (1975 words)

 APG Test Consultants, Inc. Our Programming staff are specialists in many of the advanced HP 3070 features such as Interconnect Plus boundary scan, high performance analog/digital test techniques and serial telecommunications test. This module reports the simulation time of every failing event, provides the failing pin/port and boundary scan cell information and displays the expected TAP state and instruction information for each failure. Industry leaders in boundary scan such as Panasonic, ATT and StorageTek all use closed loop design procedures to develop and verify IEEE-1149.1 compliant devices. www.apgtest.com /boundary_scan.html   (686 words)

 Geometry Class Reference   (Site not responding. Last check: 2007-10-13) By default all top level Grids, which is to say, all Grids which are spawned from a "Region" take the same Geometry as the Region, itself. All Regions within a dataset have the same Geometry but not necessarily the same topology (boundary conditions). A Geometry is defined by the the dimensionality (an int) the metric type (cartesian, spherical, etc), the boundary type (periodic, or cartesian), the space type (real, fourier, etc.) and the limits of the coordinates... www.physics.drexel.edu /BASIN/docs/classGeometry.html   (242 words)

 ANSYS ICEM CFD - Unigraphics Interface   (Site not responding. Last check: 2007-10-13) In MULCAD, the boundary conditions are defined on subfaces using the "Unspec. For more details on how to define boundary conditions in MULCAD or in the DDN-Mesher Interface, please refer to the General Remarks section. The boundary condition types must be assigned on the block faces, except for singular faces where they are assigned to edges. www.ansys.com /products/icemcfd-interfaces/cfl3d.html   (375 words)

 [Abstract] Reconstruction of Topology Valid Boundary of Discrete Object from 3D Range Images However, in a great number of applications as immersion systems for surgery simulation for example, validity of the resulting surface avoiding crack pathing and triangle degeneracies is primordial. In the present paper, a topology based approach is proposed in order to ensure the validity of the boundary reconstruc tion surface. The geometric embedment of the underlying data structure is also evaluated over different surface recon struction and representation resolutions. www.actapress.com /Abstract.aspx?paperId=21703   (168 words)

 School of Theoretical Physics - Seminars The normalization factors for these probabilities involve, in general, correlators for manifolds of non-trivial topology which are obtained by gluing simpler four-manifolds. The boundary between the matrix phase which has no commutative analogue and the the fuzzy sphere phase is demarcated by the ${\bf S}^2_N-$to-matrix critical line. More specifically, we extract from this framework a set of boundary conditions, to be imposed on an inner sphere which represents a spatial slice of the fl hole horizon. www.stp.dias.ie /TPPG/seminars.shtml   (2068 words)

 ANSYS ICEM CFD - ACUSOLVE Interface The translator writes the ACUSOLVE files using an unstructured domain file, a boundary conditions file and a parameters file (acusolve.par). To set boundary conditions, click on the icon "Boundary conditions" from the Output menu. When all desired boundary conditions have been specified, select "Accept" at the bottom of the panel. www.ansys.com /products/icemcfd-interfaces/acusolve.htm   (481 words)

 ICEM CFD/CAE OUTPUT INTERFACES: CFX   (Site not responding. Last check: 2007-10-13) Note: It is still possible to create the old topology file, but the boundary conditions described in this document are not fully supported for the old topology file anymore. The input for the translator are structured domain file(s), the boundary condition file (boco) and the topology file (topo_mulcad_out). In P-CUBE, use the block boundary condition icons to define the 3D boundary condition type and the face boundary condition icons to define the 2D patch types. www-berkeley.ansys.com /interfaces/cfx4.html   (490 words)

 [No title] Implementation of a topological data structure similar to the WINGED EDGE data structure, representing the boundary of a polyhedral solid. The boundary (surface) of a solid is represented by means of a set of topological entities. It could as well be used to represent the topology of a (restricted kind of) computer network, in which e.g. breplibrary.sourceforge.net   (1034 words)

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