Image with unknown copyright status removed Image line network.gif frame Image showing line network layout A linear bus topology is a network topology consisting of a main run of cable with a terminator at each end. All nodes file server, workstations, and peripherals are connected to the linear cable. Ethernet and LocalTalk networks use a linear bus topology. Advantages of a linear bus topology Easy to connect a computer or peripheral to a linear bus. Requires less cable length than a star topology . Disadvantages of a linear bus topology Entire network shuts down if there is a break in the main cable. Terminators are required at both ends of the backbone cable. Difficult to identify the problem if the entire network shuts down. Not meant to be used as a stand alone solution in a large building. External links http fcit.usf.edu network chap5 chap5.htm Category Network topology compu network stub id Topologi linier ... more details
A M bius strip , an object with only one surface and one edge. Such shapes are an object of study in topology. Topology from the Greek language Greek , place , and , study is a major area of mathematics ... . This later acquired the modern name of topology Specify . By the middle of the 20 sup th sup century, topology had become an important area of study within mathematics. The word topology is used both ... to define a topological space , a basic object of topology. Of particular importance are homeomorphism ... inverse . For instance, the function y x sup 3 sup is a homeomorphism of the real line . Topology includes many subfields. The most basic and traditional division within topology is General topology point set topology , which establishes the foundational aspects of topology and investigates concepts inherent to topological spaces basic examples include compactness and connectedness algebraic topology ... groups and homology mathematics homology and geometric topology , which primarily studies manifold ... dimensional topology and graph theory , do not fit neatly in this division. Image Trefoil knot arb.png ... Knot theory studies knot mathematics mathematical knot s. See also topology glossary for definitions of some of the terms used in topology and topological space for a more technical treatment of the subject ... problem solved by Euler. Topology began with the investigation of certain questions in geometry ... ref is regarded as one of the first academic treatises in modern topology. The term Topologie was introduced ... had used the word for ten years in correspondence before its first appearance in print. Topology ... chiefly are treated . The term topologist in the sense of a specialist in topology was used in 1905 ... corresponds exactly to the modern definition of topology. Modern topology depends strongly on the ideas ... part of algebraic topology . Maurice Fr chet , unifying the work on function spaces of Cantor, Vito ... 2010 For further developments, see point set topology and algebraic topology . Elementary introduction ... more details
Other uses Refimprove date December 2007 Distinguish Lineage The word linear comes from the Latin word linearis , which means created by lines . In mathematics , a linear map or function mathematics function ..., x is not necessarily a real number , but can in general be a member of any vector space . A linear function less restrictive definition of linear function, not coinciding with the definition of linear map, is used in elementary mathematics. The concept of linearity can be extended to linear Operator mathematics operator s. Important examples of linear operators include the derivative considered ... a differential equation can be expressed in linear form, it is particularly easy to solve by breaking .... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces also called linear spaces , linear transformations also called linear maps , and systems of linear equations. For a description of linear and nonlinear equations, see Linear equation . Nonlinear equations ... device s performance from ideal. Linear polynomials main linear equation In a different usage to the above, a polynomial of degree mathematics degree 1 is said to be linear, because the graph of a function of that form is a Line geometry line . Over the reals, a linear equation is one of the form ... of the term linear is not the same as the above, because linear polynomials over the real numbers ... . Boolean functions In Boolean algebra logic Boolean algebra , a linear function is a function ... in 0,1 . math A Boolean function is linear if A In every row of the truth table in which the value ... , exclusive or , tautology logic tautology , and contradiction are linear functions. Physics .... Electronics In electronics , the linear operating region of a transistor is where the collector ... to be used as an amplifier that preserves the High fidelity fidelity of analog signals. Linear ... line with arbitrary slope. Such linear electronic devices include linear filter , linear regulator ... more details
Infobox Writing system name Linear A type Undeciphered typedesc likely Syllabic and Ideographic languages Eteocretan language Eteocretan unknown time Possibly from MM IB to LM IIIA differentiated to Linear B and Linear Cypriot iso15924 Lina Image Linear A tablets filt.jpg thumb 237px right Linear A incised on tablets found in Akrotiri, Santorini . Linear A is one of two scripts used in ancient Crete before Mycenaean Greek language Mycenaean Greek Linear B , the second being Cretan hieroglyphs . In Minoan Civilization Minoan times, before the Mycenaean Greek dominion, Linear A was the official script ... discovered and named by Arthur Evans . In 1952, Michael Ventris discovered that Linear B was being used ... used this information to achieve a significant and now well accepted decipherment of Linear B, although many points remain to be elucidated. A failure to discover the language of Linear A has prevented the same sort of progress being made in its decipherment. Though the two scripts Linear A and B share some of the same symbols, using the syllable s associated with Linear B in Linear A writings ... around 1450 BC. Linear A seems to have been used as a complete syllabary around 1900 1800 BC, although several signs appear as mason marks earlier. It is possible that the Trojan script Trojan Linear ... BC, which is the period of the construction of the first palaces. Theories of decipherment Image Linear A vase filt.jpg thumb right 180px Linear A incised on a vase, also found in Akrotiri. As the Minoan ... is correct. The simplest approach to decipherment may be to presume that the values of Linear A match more or less the values given to the fully transliterated Linear B script, used for Mycenean Greek ... has a comprehensive list of known texts written in Linear A. ref This point of view has been of great ... Linear A and B therefore, 12 signs have the same values in both syllabaries DA, I, JA, KI, PA ... state of understanding of the language of Linear A the known elements are too scarce to build a safe ... more details
Unreferenced date December 2009 In functional analysis and related areas of mathematics a dual topology is a locally convex topology on a dual pair , two vector space s with a bilinear form defined on them, so that one vector space becomes the continuous dual of the other space. The different dual topologies for a given dual pair are characterized by the Mackey Arens theorem. All locally convex topologies with their continuous dual are trivially a dual pair and the locally convex topology is a dual topology. Several topological properties depend only on the dual pair and not on the chosen dual topology and thus it is often possible to substitute a complicated dual topology by a simpler one. Definition Given a dual pair math X, Y, langle , rangle math , a dual topology on math X math is a locally convex topology math tau math so that math X, tau simeq Y. math That is the continuous dual of math X, tau math is equal to math Y math up to linear isomorphism . Properties Theorem by George Mackey Mackey Given a dual pair, the bounded set topological vector space bounded set s under any dual topology are identical. Under any dual topology the same sets are barrelled set barrelled . Characterization of dual topologies The Mackey Arens theorem , named after George Mackey and Richard Friedrich Arens Richard Arens , characterizes all possible dual topologies on a locally convex space s. The theorem shows that the coarser topology coarsest dual topology is the weak topology , the topology of uniform convergence on all finite subsets of math X math , and the finer topology finest topology is the Mackey topology , the topology of uniform convergence on all weakly compact subsets of math ... and math X math its continuous dual then math tau math is a dual topology on math X math if and only if it is a topology of uniform convergence on a family of absolutely convex and weak topology weakly compact subsets of math X math DEFAULTSORT Dual Topology Category Topology of function spaces ... more details
of the predual B H sub sub . By definition, the continuous linear functionals in the norm topology ...In the mathematics mathematical field of functional analysis there are several standard topology topologies which are given to the algebra B H of bounded linear operator s on a Hilbert space H . Introduction Let T sub n sub be a sequence of linear operators on the Hilbert space H . Consider the statement ... in the uniform operator topology . If math T n x to Tx math for all x in H , then we say math T n to T math in the strong operator topology . Finally, suppose math T n x to Tx math in the weak topology of H . This means that math F T n x to F T x math for all linear functionals F on H . In this case we say that math T n to T math in the weak operator topology . All of these notions make sense ... are all locally convex, which implies that they are defined by a family of seminorm s. In analysis, a topology ... modes of convergence are, respectively, strong and weak. In topology proper, these terms ... , x sup sup x sup 1 2 sup . If B is a vector space of linear maps on the vector space A , then A , B is defined to be the weakest topology on A such that all elements of B are continuous. The norm topology or uniform topology or uniform operator topology is defined by the usual norm x on B H . It is stronger than all the other topologies below. The weak topology weak Banach space topology is B H , B H sup sup , in other words the weakest topology such that all elements of the dual B H sup sup are continuous. It is the weak topology on the Banach space B H . It is stronger than the ultraweak and weak operator topologies. Warning the weak Banach space topology and the weak operator topology and the ultraweak topology are all sometimes called the weak topology, but they are different. The Mackey topology or Arens Mackey topology is the strongest locally convex topology on B H such that the dual is B H sub sub , and is also the uniform convergence topology on B H sub sub , B H compact ... more details
In functional analysis and related areas of mathematics , the Mackey topology , named after George Mackey , is the finer topology finest topology for a topological vector space which still preserves the continuous dual . In other words the Mackey topology does not make linear functions continuous which were discontinuous in the default topology. The Mackey topology is the opposite of the weak topology , which is the coarser topology coarsest topology on a topological vector space which preserves the continuity of all linear functions in the continuous dual. The Mackey Arens theorem states that all possible dual topology dual topologies are finer than the weak topology and coarser than the Mackey topology. Definition Given a dual pair math X,X math with math X math a topological vector space and math X math its continuous dual the Mackey topology math tau X,X math is a polar topology defined on math X math by using the set of all absolutely convex and weak topology weakly compact sets in math X math . Examples Every metrisable locally convex space math X, tau math with continuous dual math X math carries the Mackey topology, that is math tau tau X, X math , or to put it more succinctly every Mackey space carries the Mackey topology Every Fr chet space math X, tau math carries the Mackey topology and the topology coincides with the strong topology , that is math tau tau X, X beta X, X math See also polar topology weak topology strong topology References springer id M m062080 title Mackey topology author A.I. Shtern cite journal last Mackey first G.W. authorlink George Mackey title On convex topological linear spaces journal Trans. Amer. Math. Soc. volume 60 year 1946 pages 519 537 doi 10.2307 1990352 url http jstor.org stable 1990352 issue 3 publisher Transactions of the American Mathematical Society, Vol. 60, No. 3 cite book last Bourbaki first Nicolas authorlink Nicolas Bourbaki title Topological vector spaces series Elements of mathematics publisher Addison Wesley year ... more details
center Bridge topology is an important topology with many uses in both linear and non linear applications ...The topology of an electronic circuit is the form taken by the Network analysis electrical circuits network ... are regarded as being the same topology. Strictly speaking, replacing a component with one of an entirely different type is still the same topology. In some contexts, however, these can loosely ... and low pass topologies even though the network topology is identical. A more correct term for these classes ... value is Prototype filter prototype network . Mathematical topology Electronic network topology is related to topology mathematical topology , in particular, for networks which contain only two terminal devices, circuit topology can be viewed as an application of graph theory . In a Network analysis ... are the edge graph theory edges of graph theory. Two networks of this kind have the same topology ... branches in both circuits. Topology names Many topology names relate to their appearance ... topology equivalents.svg thumb 700px center All these topologies are identical. Series topology is a general ... is a common name for the topology in filter design. For a network with three branches there are four ... with three branches Note that the parallel series topology is another representation of the Delta topology discussed below. Series and parallel topologies can continue to be constructed with greater ... 375px center Y and topologies Y and are important topologies in linear network analysis due to these being the simplest possible three terminal networks. A Y transform is available for linear circuits ... rules. The Y topology is also called star topology. However, star topology may also refer to the more ... main Electronic filter topology Image Filter topologies.svg 425px left The topologies shown opposite ... section is identical topology to the potential divider topology. The T section is identical topology to the Y topology. The section is identical topology to the topology. All these topologies can ... more details
incomplete date August 2009 In mathematics , general topology or point set topology is the branch of topology ... from other branches of topology in that the topological spaces may be very general, and do not have to be at all similar to manifold s. Definition A topology is a pair X , consisting of a set mathematics ... intersection of open sets is an open set. X and the empty set are open sets. History General topology ... line once known as the topology of point sets , this usage is now obsolete the introduction of the manifold concept the study of metric space s, esp. normed linear space s, in the early days of functional analysis . General topology assumed its present form around 1940. It captures, one might ... topology that basic notions are defined and theorems about them proved. This includes the following open set open and closed set s interior topology interior and closure topology closure neighbourhood topology neighbourhood and closeness topology closeness compact space compactness and connected space connectedness continuous function topology continuous function mathematics function s limit of a sequence .... Set theoretic topology examines such questions when they have substantial relations to set theory , as is often the case. Other main branches of topology are algebraic topology , geometric topology , and differential topology . As the name implies, general topology provides the common foundation for these areas. An important variant of general topology is pointless topology , which, rather ... also Glossary of general topology for detailed definitions List of general topology topics for related articles Category of topological spaces References Some standard books on general topology include Bourbaki cite Topologie G n rale cite cite General Topology cite ISBN 0 387 19374 X John L. Kelley cite General Topology cite ISBN 0 387 90125 6 James Munkres cite Topology cite ISBN 0 13 181629 2 Ryszard Engelking cite General Topology cite ISBN 3 88538 006 4 Citation last1 Steen first1 Lynn Arthur ... more details
different for other spaces of linear maps see below. The weak topology weak convergence in normed ... with the topology of pointwise convergence of linear functionals. Other properties By definition, the weak ...dablink This article discusses the weak topology on a normed vector space. For the weak topology induced by a family of maps see initial topology . For the weak topology generated by a cover of a space see coherent topology . 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 or topological vector ... respectively, compact, etc. with respect to the weak topology. Likewise, functions are sometimes called ..., derivative differentiable , analytic function analytic , etc. with respect to the weak topology ... space carrying a topology as part of its definition. For example, a normed vector space X is, by using the norm to measure distances, also a topological vector space. This topology is also called the strong topology on X . The weak topology on X is defined using the continuous dual space X sup sup . This dual space consists of all linear operator linear functions from X into the base field R or C which are continuous function topology continuous with respect to the strong topology. The weak topology on X is the weakest topology the topology with the fewest open sets such that all elements of X sup sup remain continuous. Explicitly, a subbase for the weak topology is the collection ... field R or C . In other words, a subset of X is open in the weak topology if and only if it can be written ... sets of the form &phi sup 1 sup U . More generally, if X is a vector space and F is any family of linear functionals on X in the algebraic dual space , then the initial topology of X with respect to the family F , denoted by &sigma X , F , is sometimes also called the weak topology with respect to F . If F X is the continuous dual space of X , then the weak topology with respect to F coincides with the weak ... more details
In functional analysis and related areas of mathematics a polar topology , topology of math mathcal A math convergence or topology of uniform convergence on the sets of math mathcal A math is a method to define locally convex topology locally convex topologies on the vector space s of a dual pair . Definition Given a dual pair math X,Y, langle , rangle math and a family math mathcal A math of Set mathematics sets in math X math such that for all math A math in math mathcal A math the polar set math A 0 math is an absorbent set absorbent subset of math Y math , the polar topology on math Y math is defined by a family of semi norm s math p A A in mathcal A math . For each math A math in math mathcal A math we define math p A y sup vert langle x , y rangle vert x in A math . The semi norm math p A y math is the gauge mathematics gauge of the polar set math A 0 math . Examples a dual topology is a polar topology the converse is not necessarily true a locally convex topology is the polar topology defined by the family of equicontinuous sets of the dual space , that is the sets of all continuous linear form s which are equicontinuous Using the family of all finite sets in math X math we get the coarsest polar topology math sigma Y,X math on math Y math . math sigma Y,X math is identical to the weak topology . Using the family of all sets in math X math where the polar set is absorbent, we get the finest polar topology math beta Y,X math on math Y math Notes A polar topology is sometimes called topology of uniform convergence on the sets of math mathcal A math because given a dual pair math X,Y, langle , rangle math and a polar topology math tau math on math Y math defined by the gauges of the polar sets math A 0 math , a sequence math y n math in math Y, tau math converges to math y math if and only if for all semi norms math p A math math lim n to infty p A y n y lim n to infty ... with respect to math x in A math . Unreferenced date March 2008 Category Topology of function ... more details
In mathematics , particularly in the area of functional analysis and topological vector space s, the vague topology is an example of the weak topology weak topology which arises in the study of measure theory measures on locally compact Hausdorff space s. Let X be a locally compact Hausdorff space . Let M X be the space of complex numbers complex Radon measure s on X , and C sub 0 sub X sup sup denote the dual of C sub 0 sub X , the Banach space of complex continuous function s on X vanish at infinity vanishing at infinity equipped with the uniform norm . By the Riesz representation theorem M X is isometry isometric to C sub 0 sub X sup sup . The isometry maps a measure &mu to a linear functional math I mu f int X f , d mu. math The vague topology is the Weak topology weak topology on C sub 0 sub X sup sup . The corresponding topology on M X induced by the isometry from C sub 0 sub X sup sup is also called the vague topology on M X . Thus, in particular, one may refer to vague convergence of measure &mu sub n sub &rarr &mu . One application of this is to probability theory for example, the central limit theorem is essentially a statement that if &mu sub n sub are the probability measure s for certain sums of independent random variables , then &mu sub n sub converge weakly to a normal distribution , i.e. the measure &mu sub n sub is approximately normal for large n . References citation author Dieudonn , Jean authorlink Jean Dieudonn chapter 13.4. The vague topology title Treatise on analysis volume II publisher Academic Press year 1970 . G.B. Folland, Real Analysis Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999. planetmath title Weak topology of the space of Radon measures id 7212 Category Real analysis Category Topology of function spaces ... more details
In mathematics , the Chabauty topology is a certain topological structure introduced in 1950 by Claude Chabauty , on the set of all closed subgroup s of a locally compact group G . The intuitive idea may be seen in the case of the set of all lattice group lattices in a Euclidean space E . There these are only certain of the closed subgroups others can be found by in a sense taking limiting cases or degenerating a certain sequence of lattices. One can find linear subspaces or discrete groups that are lattices in a subspace, depending on how one takes a limit. This phenomenon suggests that the set of all closed subgroups carries a useful topology. This topology can be derived from the Vietoris topology construction, a topological structure on all non empty subsets of a space. More precisely, it is an adaptation of the Fell topology construction, which itself derives from the Vietoris topology concept. References Claude Chabauty, Limite d ensembles et g om trie des nombres . Bulletin de la Soci t Math matique de France, 78 1950 , p. 143 151 Category Topological groups ... more details
In general topology and related areas of mathematics , the initial topology projective topology , or projective limit topology , or weak topology on a Set mathematics set math X math , with respect to a family of functions on math X math , is the coarsest topology on X which makes those functions continuous function topology continuous . The subspace topology and product topology constructions are both special cases of initial topologies. Indeed, the initial topology construction can be viewed as a generalization of these. The duality mathematics dual construction is called the final topology . Definition ... math f i X to Y i math the initial topology &tau on math X math is the coarsest topology on X such that each math f i X, tau to Y i math is continuous function topology continuous . Explicitly, the initial topology may be described as the topology subbase generated by sets of the form ... cases of the initial topology. The subspace topology is the initial topology on the subspace with respect to the inclusion map . The product topology is the initial topology with respect to the family ... theoretic inverse limit together with the initial topology determined by the canonical morphisms. The weak topology on a locally convex space is the initial topology with respect to the continuous linear form s of its dual space . Given a family mathematics family of topologies &tau sub i sub on a fixed set X the initial topology on X with respect to the functions id sub X sub X &rarr X , &tau ... on X . That is, the initial topology &tau is the topology generated by the union set theory union of the topologies ... topology with respect to its family of bounded function bounded real valued continuous functions. Every topological space X has the initial topology with respect to the family of continuous functions from X to the Sierpi ski space . Properties Characteristic property The initial topology on X ... &isin I . Image InitialTopology 01.png center Characteristic property of the initial topology Evaluation ... more details
In functional analysis and related areas of mathematics the strong topology is the finer topology finest polar topology , the topology with the most open set s, on a dual pair . The coarser topology coarsest polar topology is called weak topology polar topology weak topology . Definition Given a dual pair math X,Y, langle , rangle math the strong topology math beta Y, X math on math Y math is the polar topology defined by using the family of all sets in math X math where the polar set in math Y math is Absorption law absorbent . Examples Given a normed vector space math X math and its continuous dual math X math then math beta X , X math topology on math X math is identical to the topology induced by the operator norm . Conversely math beta X, X math topology on math X math is identical to the topology induced by the norm mathematics norm . Properties In barrelled space s the strong topology is identical to the Mackey topology . mathanalysis stub Category Topology of function spaces ... more details
In mathematics , a base or basis B for a topological space X with topological space topology T is a collection ... . We say that the base generates the topology T . Bases are useful because many properties of topologies can be reduced to statements about a base generating that topology, and because many topologies ... properties of bases are The base elements cover topology cover X . Let B sub 1 sub , B sub ... either of these, then it is not a base for any topology on X . It is a subbase , however, as is any ... topology on X for which B is a base it is called the topology generated by B . This topology ... common way of defining topologies. A sufficient but not necessary condition for B to generate a topology ..., the collection of all open interval s in the real line forms a base for a topology on the real ... they are a base for the standard topology on the real number s. However, a base is not unique. Many bases, even of different sizes, may generate the same topology. For example, the open intervals with rational endpoints are also a base for the standard real topology, as are the open intervals with irrational ... of all open intervals. In contrast to a basis linear algebra basis of a vector space in linear algebra , a base need not be maximal indeed, the only maximal base is the topology itself. In fact ... the topology. The smallest possible cardinality of a base is called the weight of the topological space ... of the forms , a and a , , where a is a real number. Then S is not a base for any topology on R . To show this, suppose it were. Then, for example, , 1 and 0, would be in the topology generated ... property fails, since no base element can fit inside this intersection. Given a base for a topology ... The order topology is usually defined as the topology generated by a collection of open interval like sets. The metric topology is usually defined as the topology generated by a collection of open ball s. A second countable space is one that has a countable base. The discrete topology has the Singleton ... more details
of an ordinal indexed sequence. See also lower limit topology long line topologyLinear ...In mathematics , an order topology is a certain topology that can be defined on any totally ordered set . It is a natural generalization of the topology of the real numbers to arbitrary totally ordered ... counter examples If X is a totally ordered set, the order topology on X is generated by the subbase ... form a base topology base for the order topology. The open sets in X are the sets that are a union set theory union of possibly infinitely many such open intervals and rays. The order topology makes ... , Q , and N are the order topologies. Induced order topology If Y is a subset of X , then Y inherits a total order from X . Y therefore has an order topology, the induced order topology . As a subset of X , Y also has a subspace topology . The subspace topology is always at least as finer topology fine as the induced order topology, but they are not in general the same. For example, consider the subset Y 1 &cup 1 n sub n &isin N sub in the rational numbers rationals . Under the subspace topology, the singleton set 1 is open in Y , but under the induced order topology, any open set containing ... space whose topology is not an order topology Though the subspace topology of Y 1 &cup 1 n sub ... an order topology on Y indeed, in the subspace topology every point is isolated i.e., singleton y is open in Y for every y in Y , so the subspace topology is the discrete topology on Y the topology in which every subset of Y is an open set , and the discrete topology on any set is an order topology. To define a total order on Y that generates the discrete topology on Y , simply modify the induced ... n &isin N . Then, in the order topology on Y generated by sub 1 sub , every point of Y is isolated ... order on Z generates the subspace topology on Z , so that the subspace topology will not be an order topology even though it is the subspace topology of a space whose topology is an order topology ... more details
Digital topology deals with properties and features of two dimensional 2D or Three dimensional space three dimensional 3D digital images that correspond to topological properties e.g., connectedness or topological features e.g., Boundary topology boundaries of objects. Concepts and results of digital topology are used to specify and justify important low level image analysis algorithms, including algorithms .... History Digital topology was first studied in the late 1960 s by the computer image analysis researcher ... and developing the field. The term digital topology was itself invented by Rosenfeld, who used it in a 1973 publication for the first time. A related work called the grid cell topology appeared ... topology . Rosenfeld et al proposed digital connectivity such as 4 connectivity and 8 connectivity ... grid cell topology to 3D and high dimensions. In early 1980s, digital surface s were studied. Morgenthaler ... A basic early result in digital topology says that 2D binary images require the alternative use ... in the 2D grid cell topology , and the result generalizes to 3D the alternative use of 6 or 26 adjacency corresponds to open or closed sets in the 3D grid cell topology . Grid cell topology also applies ... values and applying a maximum label rule see book by Klette and Rosenfeld, 2004 . Digital topology is highly related to combinatorial topology . The main differences between them are 1 digital topology mainly studies digital objects that is formed by grid cells, and 2 digital topology also deals with non .... It usually means a piecewise linear manifold made by simplicial complexes . A digital manifold ... . See also Digital geometry Combinatorial topology Computational geometry Computational topology Topological data analysis Topology Discrete mathematics References cite book author Herman, G.T. title ... Surfaces and Manifolds A Theory of Digital Discrete Geometry and Topology publisher SP Computing ... topologytopology stub fa ... more details
Unreferenced date December 2009 In functional analysis and related areas of mathematics the weak topology is the coarser topology coarsest polar topology , the topology with the fewest open set s, on a dual pair . The finer topology finest polar topology is called strong topology polar topology strong topology . Under the weak topology the Bounded set topological vector space bounded set s coincide with the relatively compact set s which leads to the important Bourbaki Alaoglu theorem . Definition Given a dual pair math X,Y, langle , rangle math the weak topology math sigma X,Y math is the weakest polar topology on math X math so that math X, sigma X,Y simeq Y math . That is the continuous dual of math X, sigma X,Y math is equal to math Y math up to isomorphism . The weak topology is constructed as follows For every math y math in math Y math on math X math we define a semi norm on math X math math p y X to mathbb R math with math p y x vert langle x , y rangle vert qquad x in X math This family of semi norms defines a locally convex topology on math X math . Examples Given a normed vector space math X math and its continuous dual math X math , math sigma X, X math is called the weak topology on math X math and math sigma X , X math the weak star topology weak topology on math X math DEFAULTSORT Weak Topology Polar Topology Category Topology of function spaces ... more details
functions in exactly the same fashion as the physical linear bus topology i.e., all nodes share ... . 2. The physical linear bus topology is sometimes considered to be a special case of the physical ... factor of 1 would be classified as a physical lineartopology. 3. The branching factor, f ... chained network can take two basic forms linear and ring. A lineartopology puts a two way link ... useful. This is similar in some ways to a grid network , where a linear or ring topology is used ... topologies. Network topology is the layout pattern of interconnections of the various elements Data ...?id 3516 title network topology author ATIS committee PRQC publisher Alliance for Telecommunications ... may be physical or logical. Physical topology means the physical design of a network including the devices, location and cable installation. Logical topology refers to how data is actually transferred in a network as opposed to its physical design. In general physical topology relates to a core network whereas logical topology relates to basic network. Topology can be considered as a virtual shape ... not necessarily mean that it represents a ring topology. Any particular network topology is determined ... nodes. The study of network topology uses graph theory . Distances between nodes, physical interconnections ... topology and a logical topology. Any given node in the LAN has one or more links to one or more ... that may be used to describe the physical topology of the network. Likewise, the mapping of the data flow between the nodes in the network determines the logical topology of the network. The physical and logical topologies may or may not be identical in any particular network. Basic topology types The study of network topology recognizes seven basic topologies ref name Bicsi, B. 2002 Bicsi, B., 2002 ... topology Bus point to multipoint topology Star topology Ring topology Tree topology Mesh topology Hybrid topology This classification is based on the interconnection between computers be it physical ... more details
Algorithmic topology , or computational topology , is a subfield of topology with an overlap with areas of computer science , in particular computational geometry and computational complexity theory . A primary concern of algorithmic topology, as its name suggests, is to develop efficient algorithm s for solving topological problems, or using topological methods to solve algorithmic problems from other fields. Major algorithms by subject area Algorithmic 3 manifold theory A large family of algorithms concerning 3 manifold s revolve around normal surface theory, which is a phrase that encompasses several techniques to turn problems in 3 manifold theory into integer linear programming problems. Rubinstein s 3 sphere recognition algorithm . This is an algorithm that takes as input a triangulated 3 manifold and determines whether or not the manifold is homeomorphic to the 3 sphere . It has exponential .... Introducing Regina, the 3 manifold topology software, Experimental Mathematics 13 2004 , 267 272 ... structures on 3 manifolds with solvable word problem, Geometry and Topology 6 2002 1 26 ref At present ..., Algorithmic topology and the classification of 3 manifolds, Springer Verlag 2003 ref Conversion ... triangulation. This algorithm has a roughly linear run time in the number of crossings in the diagram ..., D.Thurston. 3 manifolds efficiently bound 4 manifolds. Journal of Topology 2008 1 3 703 745 ref ... geometry Digital topology Topological data analysis Spatial temporal reasoning Experimental ... of Topology in Science and Engineering http comptopfs.stanford.edu Computational Topology at Stanford ... Topology for Computing publisher Cambridge year 2005 isbn 0 521 83666 2 http books.google.com books?id MDXa6gFRZuIC Computational Topology An Introduction , Herbert Edelsbrunner, John L. Harer, AMS Bookstore, 2010, ISBN 978 0 8218 4925 5 DEFAULTSORT Computational Topology Category Computational topology ... science Topology ... more details
In mathematics , a strong topology is a topology which is stronger than some other default topology. This term is used to describe different topologies depending on context, and it may refer to the final topology on the disjoint union topology disjoint union the topology arising from a normed vector space norm the strong operator topology the strong topology polar topology , which subsumes all topologies above. Note that a topology is stronger than a topology is a Comparison of topologies finer topology if contains all the open sets of . In algebraic geometry , it usually means the topology of an algebraic variety as complex manifold or subspace of complex projective space , as opposed to the Zariski topology which is rarely even a Hausdorff space . See also Weak topology mathdab Category Topology ... more details
For other uses of triangulation in mathematics Triangulation disambiguation In mathematics , topology ... homology and cohomology theories. Piecewise linear structures For topological manifold s, there is a slightly stronger notion of triangulation a piecewise linear triangulation sometimes just called ... linear sphere. The link of a simplex s in a simplicial complex K is a subcomplex of K consisting ... simplex in K . For instance, in a two dimensional piecewise linear manifold formed by a set ... 3 fold Suspension topology suspension of the Poincar sphere is a topological manifold homeomorphic to the n sphere with a triangulation that is not piecewise linear it has a simplex whose ... manifold that is not homeomorphic to a sphere. The question of which manifolds have piecewise linear triangulations has led to much research in topology. Differentiable manifold s Stewart Cairns, harvs ... and Robert Hardt admit a piecewise linear triangulation. Topological manifold s of dimensions 2 and 3 are always triangulable by an Hauptvermutung essentially unique triangulation up to piecewise linear ... have an infinite number of triangulations, all piecewise linear inequivalent. In dimension greater ... it is known that some do not have piecewise linear manifold piecewise linear triangulations see ... Skeleton topology skeletons of Whitney triangulations are exactly the Neighbourhood graph theory ... Topology publisher American Mathematical Society year 2007 isbn 0821842307 citation last .... title A History of Algebraic and Differential Topology, 1900 1960 publisher Birkh user year 1989 isbn ... 1997 isbn 3 540 53334 6 citation authorlink Edwin E. Moise last Moise first E. title Geometric Topology ... last Munkres first J. authorlink James Munkres title Elementary Differential Topology, revised ... and Topology, Vol. I publisher Princeton University Press year 1997 isbn 0 691 08304 5 citation ... 2 year 1992 pages 147 164 doi 10.1016 0095 8956 92 90015 P Category Topology Category Algebraic ... more details
In mathematics , the uniform topology on a space has several different meanings depending on the context In functional analysis, it sometimes refers to a polar topology on a topological vector space. In general topology, it is the topology carried by a uniform space . In real analysis, it is the topology of uniform convergence . Disambig ... more details
Unreferenced date December 2009 In functional analysis , a branch of mathematics , the ultraweak topology , also called the weak topology , or weak operator topology or weak topology , on the set B H of bounded operator s on a Hilbert space is the weak topology weak topology obtained from the predual B sub sub H of B H , the trace class operators on H . In other words it is the weakest topology such that all elements of the predual are continuous when considered as functions on B H . Relation with the weak operator topology The ultraweak topology is similar to the weak operator topology. For example, on any norm bounded set the weak operator and ultraweak topologies are the same, and in particular the unit ball is compact in both topologies. The ultraweak topology is stronger than the weak operator topology. One problem with the weak operator topology is that the dual of B H with the weak operator topology is too small . The ultraweak topology fixes this problem the dual is the full predual B sub sub H of all trace class operators. In general the ultraweak topology is more useful than the weak operator topology, but it is more complicated to define, and the weak operator topology is often more apparently convenient. The ultraweak topology can be obtained from the weak operator topology as follows. If H sub 1 sub is a separable infinite dimensional Hilbert space then B H can be embedded in B H H sub 1 sub by tensoring with the identity map on H sub 1 sub . Then the restriction of the weak operator topology on B H H sub 1 sub is the ultraweak topology of B H . See also Topologies on the set of operators on a Hilbert space ultrastrong topology weak operator topology DEFAULTSORT Ultraweak Topology Category Topology of function spaces Category Von Neumann algebras ... more details
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