Weak topology | Trivial topology | Pointless topology | Initial topology | Grothendieck topology | differential topology | Boundary (topology) |
It follows from the Hahn–Banach separation theorem that the weak topology is Hausdorff, and that a norm-closed convex subset of a Banach space is also weakly closed.
for every set , having defined as the set of all precompact open subsets of with respect to the standard topology of finite dimensional vector spaces, and correspondingly the class of functions of locally bounded variation is defined as
For example: is the data wireframe, surface, or solid; is the topology (BREP) information required; must the face and edge identifications be preserved on subsequent modification; must the feature information and history be preserved between systems; and is PMI annotation to be transferred.
Carathéodory–Jacobi–Lie theorem, a generalization of Darboux's theorem in symplectic topology
Henri Cartan (1904-2008), French mathematician who worked in algebraic topology, son of the above
Coarse structure, a family of sets in geometry and topology to measure large-scale properties of a space
Trivial topology, the most coarse topology possible on a given set
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Initial topology, the most coarse topology in a certain category of topologies
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Weak topology, an example of topology coarser than the standard one
Countably generated space, a topological space in which the topology is determined by its countable subsets
The MTA-2 was an attempt to correct these problems while maintaining essentially the same processor architecture respun in one silicon ASIC, down from some 26 gallium arsenide ASICs in the original MTA; and while advancing the network design from a 4-D torus topology to a more scalable Cayley graph topology.
Beta- and gamma-crystallins (such as CRYGC) are similar in sequence, structure and domains topology, and thus have been grouped together as a protein superfamily called βγ-Crystallins.
Differential topology is the study of the (infinitesimal, local, and global) properties of structures on manifolds having no non-trivial local moduli, whereas differential geometry is the study of the (infinitesimal, local, and global) properties of structures on manifolds having non-trivial local moduli.
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Smooth manifolds are 'softer' than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology.
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Differential topology also deals with questions like these, which specifically pertain to the properties of differentiable mappings on Rn (for example the tangent bundle, jet bundles, the Whitney extension theorem, and so forth).
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For a list of differential topology topics, see the following reference: List of differential geometry topics.
The finite dimensional case differs from the general von Neumann algebras in that topology plays no role and they can be characterized using Wedderburn's theory of semisimple algebras.
In molecular biology, formylglycine-generating sulfatase enzyme is a protein domain which has a structure homologous to the complex alpha/beta topology found in sulfatase-modifying factors (SUMF1).
According to some sources, Peter Gabriel Bergmann brought Rainich's suggestion that algebraic topology (and knot theory in particular) should play a role in physics to the attention of John Archibald Wheeler, which shortly led to the Ph.D. thesis of Charles W. Misner.
Gordon Thomas Whyburn (7 January 1904 Lewisville, Texas – 8 September 1969 Charlottesville, Virginia) was an American mathematician who worked on topology.
Historically an area in which a large population of deer roamed around a mere, this topology gave rise to the school's name such that the school has always been associated with an area much larger than Eye.
In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology.
The idea of a Grothendieck topology (also known as a site) has been characterised by John Tate as a bold pun on the two senses of Riemann surface.
Homotopy lifting property, in algebraic topology, a technical condition on a continuous function from a topological space E to another one, B designed to support the picture of E "above" B
Subsequently, Berger popularized the subject in a series of articles and books, most recently in the march '08 issue of the Notices of the American Mathematical Society. A bibliography at the Website for systolic geometry and topology currently contains over 170 articles.
Nyberg received her Ph.D. in mathematics in 1980 from the University of Helsinki, with a dissertation in topology.
In topology, a branch of mathematics, the Knaster–Kuratowski fan (also known as Cantor's leaky tent or Cantor's teepee depending on the presence or absence of the apex) is a connected topological space with the property that the removal of a single point makes it totally disconnected.
In mathematics, Kuiper's theorem (after Nicolaas Kuiper) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space H.
His works have been performed by groups such as The California Ear Unit, Topology, Clocked Out, Ensemble Scintilla Divina, the MATA Ensemble, The Collective and artists such as Michael Kieran Harvey, Ross Bolleter and Hiroshi Chu Okubo.
The resulting topological space, sometimes written Rl and called the Sorgenfrey line after Robert Sorgenfrey, often serves as a useful counterexample in general topology, like the Cantor set and the long line.
In descriptive set theory and general topology, Luzin space or Luzin set, a hypothetical uncountable topological T1 space without isolated points in which every nowhere-dense subset is at most countable
Helpful here is J. K. Gibson-Graham’s complex topology of the diversity of contemporary market economies describing different types of transactions, labour, and economic agents.
Edgar Heilbronner had described twisted annulenes which had Möbius topology, but in including the twist of these systems, he concluded that Möbius systems could never be lower in energy than the Hückel counterparts.
In comparison, other network topology discovery techniques using SNMP or Route analytics aim for greater accuracy with less emphasis on overhead reduction.
Star network topology can be used to lower complexity and cost of peripherals, and also simplifies encryption key management.
Examples of topologies include the Zariski topology in algebraic geometry that reflects the algebraic nature of varieties, and the topology on a differential manifold in differential topology where each point within the space is contained in an open set that is homeomorphic to an open ball in a finite-dimensional Euclidean space.
Pointless topology, an approach to topology that avoids mentioning points
grompp, a preprocessor for simulation input files for GROMACS (a fast, free, open-source code for some problems in computational chemistry) which calls the system C preprocessor (or other preprocessor as determined by the simulation input file) to parse the topology, using mostly the #define and #include mechanisms to determine the effective topology at grompp run time.
In its original form, it was applied to an unramified double covering of a Riemann surface, and was used by F. Schottky and H. W. E. Jung in relation with the Schottky problem, as it now called, of characterising Jacobian varieties among abelian varieties.
Quadrics QsNetII interconnect like its predecessor QsNet uses a 'fat tree' topology, QsNetII scales up to 4096 nodes, each node might have multiple CPUs so that systems of >10,000 CPUs can be constructed.
One of the first papers in topology was the demonstration, by Leonhard Euler, that it was impossible to find a route through the town of Königsberg (now Kaliningrad) that would cross each of its seven bridges exactly once.
Sylvain Edward Cappell (born 1946), a Belgian American mathematician and former student of William Browder at Princeton University, is a topologist who has spent most of his career at the Courant Institute of Mathematical Sciences at NYU, where he is now the Silver Professor of Mathematics.
The first paper in 1995 deals with Donaldson's polynomial invariants and introduced Kronheimer–Mrowka basic class, which have been used to prove a variety of results about the topology and geometry of 4-manifolds, and partly motivated Witten's introduction of the Seiberg–Witten invariants.
Bourbaki, Nicolas; Elements of Mathematics: General Topology, Addison-Wesley (1966).
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Bredon, Glen E., Topology and Geometry (Graduate Texts in Mathematics), Springer; 1st edition (October 17, 1997).
the Ultra-Linear output stage (a tapped push-pull output transformer providing power intermediate between triodes and pentodes, at lower distortion than either) was originated by Alan Blumlein in 1937 in the UK, but popularised following publication of a paper by David Hafler and Keroes in the USA in 1951, and became the dominant topology during the post war recovery of consumer products
As with unit disk graphs, the Vietoris–Rips complex has been applied in computer science to model the topology of ad hoc wireless communication networks.
He also substantially contributed to the development of the theory of capacities, nonlinear potential theory, the asymptotic and qualitative theory of arbitrary order elliptic equations, the theory of ill-posed problems, the theory of boundary value problems in domains with piecewise smooth boundary.
XIO is usually used in a star topology, using a router ASIC called Crossbow (Xbow) to connect up to eight fully symmetrical devices in a system (one of them is usually the memory controller / CPU bridge, called HEART in Octane or Hub in Origin).
It occurs in the proofs of several theorems of crucial importance, for instance the Hahn–Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that every nonzero ring has a maximal ideal and that every field has an algebraic closure.