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Topic: Initial topology

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Topology glossary

product topology

closure topology

completeness topology

Zariski topology

base topology

continuity topology

differential topology

weak topology

general topology

neighborhood topology

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	NationMaster - Encyclopedia: Continuous function (topology)
		In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense.
		In topology and related areas of mathematics, the initial topology (projective topology or weak topology) on a set, with respect to a family of functions on, is the coarsest topology on X which makes those functions continuous.
		In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology (or the relative topology, or the induced topology).
	www.nationmaster.com /encyclopedia/Continuous-function-%28topology%29 (2819 words)

	Initial topology (Site not responding. Last check: 2007-10-08)
		The initial topology on (with respect to the family of functions) is the weakest topology such that every is continuous.
		The topology is generated by sets of the form, where is an open set in.
		For example, the subspace topology for is the initial topology with respect to the inclusion map.
	www.xasa.com /wiki/en/wikipedia/i/in/initial_topology.html (140 words)

	Initial topology - Wikipedia, the free encyclopedia
		In topology and related areas of mathematics, the initial topology (projective topology or weak topology) on a set X, with respect to a family of functions on X, is the coarsest topology on X which makes those functions continuous.
		The subspace topology is the initial topology on the subspace with respect to the inclusion map.
		The weak topology on a locally convex space is the initial topology with respect to the continuous linear forms of its dual space.
	www.wikipedia.org /wiki/Initial_topology (501 words)

	PlanetMath: initial topology
		The initial topology is sometimes called topology generated by a family of mappings [2], weak topology [4] or projective topology.
		From the viewpoint of category theory, the initial topology is an initial source.
		This is version 6 of initial topology, born on 2005-09-10, modified 2006-07-07.
	www.planetmath.org /encyclopedia/InitialTopology.html (202 words)

	NationMaster - Encyclopedia: Tychonoff space
		In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces.
		In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider.
		In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way.
	www.nationmaster.com /encyclopedia/Tychonoff-space (2001 words)

	Weak topology - Wikipedia, the free encyclopedia
		In mathematics, weak topology is an alternative term for initial topology.
		The term is most commonly used for the initial topology of a normed vector space with respect to its (continuous) dual.
		If X is equipped with the weak topology, then addition and scalar multiplication remain continuous operations, and X is a locally convex topological vector space.
	en.wikipedia.org /wiki/Weak_topology (369 words)

	1-1 Media and Topologies \| StudyNotes.net
		In bus topologies, all computers are connected to a single cable or "trunk or backbone", by a transceiver either directly or by using a short drop cable.
		In a ring topology network computers are connected by a single loop of cable, the data signals travel around the loop in one direction, passing through each computer.
		Ring topology is an active topology because each computer repeats (boosts) the signal before passing it on to the next computer.
	studynotes.net /net1.htm (1445 words)

	wiki/Category:Topology Definition / wiki/Category:Topology Research (Site not responding. Last check: 2007-10-08)
		Topology is concerned with the study of the so-called topological properties of figures, that is to say properties that do not change under bicontinuous one-to-one transformations (called homeomorphisms).
		Discrete spaceIn topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.
		Weak topology In mathematics, the weak topology on a set, with respect to a collection of functions from that set into topological spaces, is the weakest (that is, smallest or coarsest) topology on the set which makes all the functions continuous.
	www.elresearch.com /wiki/Category:Topology (2445 words)

	Initial topology -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		The initial topology on (with respect to the family of functions) is the (Click link for more info and facts about weakest topology) weakest topology such that every is continuous.
		The topology is generated by sets of the form, where is an (Click link for more info and facts about open set) open set in.
		The (Click link for more info and facts about subspace topology) subspace topology is the initial topology on the subspace with respect to the (Click link for more info and facts about inclusion map) inclusion map.
	www.absoluteastronomy.com /encyclopedia/i/in/initial_topology.htm (188 words)

	Initial topology - Wikipedia, the free encyclopedia - TESTVERSION (Site not responding. Last check: 2007-10-08)
		The initial topology on $X (with respect to the family of functions) is the weakest$ topology such that every $f_i is continuous .$
		The topology is generated by sets of the form $f_i^{-1}(U), where U is an open set in Y_i .$
		For example, the subspace topology for $X \subset Y is the$ initial topology with respect to the inclusion map.
	www.wissen-im-web.net /wiki/Initial_topology (166 words)

	Coarsest Topology Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-10-08)
		All possible polar topologies on a dual pair are finer than the weak topology and coarser than the strong topology.
		The join, however, is not generally the union of those topologies (the union of two topologies need not be a topology) but rather the topology generated by the union.
		In the case of topologies, the greatest element is the discrete topology and the least element is the trivial topology.
	www.karr.net /encyclopedia/Coarsest_topology (633 words)

	Topology Correction of Digital Images.html
		In an initial work (MICCAI 2003), a multiple region growing process, concurrently acting on the foreground and the background, divides the segmentation into connected components and successive minimum cost decisions guarantee convergence to correct spherical topology.
		The initial segmentation had 3 topological defects (3 handles: euler characteristic of the tessellation X = -4); the final surface has the correct spherical topology.
		Correction of the topology of the cortical surface is a much more challenging task: its highly convoluted nature often causes numerous topological defects, which interact with each other, and are difficult to precisely locate and correct.
	www.ai.mit.edu /projects/medical-vision/DigitalTopologyCorrection/DigitalTopologyCorrection.html (1055 words)

	Colloquia and Seminars - UNL - Department of Mathematics (Site not responding. Last check: 2007-10-08)
		The problem is classical in (finite) group theory but requires quite different techniques in monoid theory.
		In view of the recursive definition of regular languages (and of some powerful theorems!), the problem then reduces to computing the closure of a finitely generated subgroup of a free group F in the profinite topology.
		As this is known to be computable, the original problem regarding finite monoids is decidable.
	www.math.unl.edu /pi/colloquia/abstract-20040325.txt (229 words)

	cage
		Topologies can be bounded (where they have an edge, and any addresses off that edge are taken as having some background state), or unbounded (usually where they wrap around on themselves, such as for the surface of a sphere).
		Topologies have the ability to normalize addresses, and additionally can make morphologically identical clones of the same size and shape (not necessarily their actual cellular states) for synchronous automata that need to maintain multiple topologies for the sake of simultaneous update.
		An initializer is used to set the initial states of an automaton map to the desired settings before the automaton begins.
	www.alcyone.com /software/cage (1457 words)

	Hidden Markov Modeling (Site not responding. Last check: 2007-10-08)
		The initial topology for an HMM can be determined by estimating how many different states are involved in specifying a sign.
		In this case, an initial topology of 5 states was considered sufficient for the most complex sign.
		However, after testing several different topologies, a four state HMM with one skip transition was determined to be appropriate for this task (Figure 1).
	web.media.mit.edu /~testarne/asl/asl-tr466/node5.html (161 words)

	Initial consonant mutation Definition / Initial consonant mutation Research
		The prototypical example of consonant mutation is the initial consonant mutation of all modern Celtic languages Celtic languages are a branch of the Indo-European languages.
		Initial consonant mutation is located additionally in Southern Paiute and in several West African languages such as FulaThe Fula is an ethnic group of people spread over many countries in West Africa, from Mauritania in the northwest to Cameroon in the east.
		Initial consonant mutation is the phenomenon in which the first consonant of a word is changed according to a certain grammatical environment.
	www.elresearch.com /Initial_consonant_mutation (498 words)

	Generalized Unstructured Decimation
		If the initial local boundary loop is non-convex, some of the n-simplices which result will intersect valid n-simplices which are external to the local boundary loop.
		The initial image, there were 15314 vertices, of which 76 percent were removed in the final image.
		The deletion of an interior vertex from a general surface topology requires that the size of the Valid stack be exactly two less than the original number of incident triangles.
	www.cs.wpi.edu /~matt/courses/cs563/talks/powwie/p2/decimate.htm (3731 words)

	PlanetMath: product topology
		is defined to be the initial topology with respect to the projection maps; that is,
		The main reason for this can be expressed in terms of category theory: the product topology is the topology of the direct categorical product in the category Top (see Theorem 1 below).
		This is version 30 of product topology, born on 2002-06-12, modified 2006-10-01.
	www.planetmath.org /encyclopedia/Product2.html (181 words)

	Aquaporin 1
		Similarly, it is unknown whether this change in topology is driven by the energy of helical packing other aspects of protein folding.
		Similar reorientation events have been reported for engineered chimeras and for Sec61 alpha, which suggests that modification of initial topology may be a relatively common feature of polytopic protein biogenesis.
		Topology of each TM segment is established independently and no reorientation of topology occurs.
	www.ohsu.edu /skachlab/web_animations/cotranslat_and_aqp1_biogen.html (736 words)

	Reorientation of Aquaporin-1 Topology during Maturation in the Endoplasmic Reticulum -- Lu et al. 11 (9): 2973 -- ...
		The topologies of myc and P reporters are derived from the accessibility of epitopes based on patterns of protected fragments.
		Potential topologies of epitopes indicated beneath autoradiograms are deduced from the protease accessibility of myc and P reporters.
		At the times indicated after RNA injection (A) or the initiation of in vitro translation (B-E), samples were digested with PK and immunoprecipitated as in Figures 3 and 4.
	www.molbiolcell.org /cgi/content/full/11/9/2973 (7884 words)

	Highly designable phenotypes and mutational buffers emerge from a systematic mapping between network topology and ...
		Blue line, distribution of designabilities for the 5,688 5-cycles arising from four-node networks under rule one; red line, Poisson distribution constructed with mean 35.6, corresponding to the average designability for the 5-cycle distribution.
		The continuous dynamics demonstrate that the Boolean model not only exhibits the correct oscillatory phenomena but also precisely captures the three-to-one frequency ratio of the buffer to the other nodes (refer to the dynamical phenotype in Fig.
		We define the "hit rate" as the number of parameter sets that yield the dynamics of interest divided by the total number of parameter sets screened.
	www.pnas.org /cgi/content/full/0507032103/DC1 (555 words)

	Hawley and Krolik: Discussion
		The contrast between the initially-poloidal and initially-toroidal cases arises from the fact that the inflow time at any given radius of the disk proper is longer in the toroidal simulation than in the poloidal due to the weaker magnetic stress.
		The linear analysis of the MRI shows that the fastest growing wavelengths of vertical fields are axisymmetric and have larger vertical wavelengths, while the toroidal field instability is nonaxisymmetric, favors small wavelengths, and grows transiently as time-dependent radial wavenumbers swing from leading to trailing.
		This does not mean, however, that the initial purely toroidal field evolution is the best model; it is in many ways a singular state, and there are sharp differences between it and models that have even a small amount of local net poloidal field.
	www.astro.virginia.edu /VITA/papers/plunge/section5.html (2648 words)

	3.11 Euclidean path integrals and quantum cosmology
		In the Hartle-Hawking approach to quantum cosmology, the initial wave function of the universe is described by a path integral for a compact manifold
		This approach finesses the question of initial conditions for the universe by simply omitting an initial boundary, and it postpones the question of the nature of time in quantum gravity: Information about time is hidden in the boundary geometry
		This result involves a clever representation of a spacetime triangulation as a Feynman graph in a field theory on a group manifold, allowing the sum over topologies to be reexpressed as a sum of field theory Feynman diagrams.
	www.emis.de /journals/LRG/Articles/lrr-2005-1/articlesu19.html (1026 words)

	PlanetMath: initial source
		has the initial topology with respect to the family
		Cross-references: subcategories, characterization, Tychonoff space, line, real, continuous maps, completely regular, initial topology, topological spaces, category, composite, source, concrete category
		This is version 3 of initial source, born on 2006-06-30, modified 2007-02-06.
	planetmath.org /encyclopedia/InitialMonosource.html (89 words)

	[No title]
		An initial intended usage of routing policy systems such as the RIPE database had been in an advisory capacity, documenting the intended routing policies for the purpose of debugging.
		The root repository must be initially populated with the allocation of an inetnum covering the prefix 0/0, indicating that some address allocation authority exists.
		We then examine the operational cases where either initial topology is not tree structured or cases where topology changes.
	www.arin.net /reference/rfc/rfc2725.txt (12963 words)

	Development of a Wireless Sensor Platform for Autonomous Topology
		Initial tests were done to determine the angular error associated with a full circle turn and subsequent repeated circles would measure the drift associated with circular motion.
		The ideal end result of this testing phase is the realization of a near perfect homogeneity between the units, this does not indicate that the presence of error is not anticipated rather that the presence of error should be somewhat static and predictable throughout each of the units.
		Using the initial topology software developed the robots will run the gambit of operations and form the proper topology.
	designserver.rit.edu /Archives/P04043/final3.html (271 words)

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