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Bridgeland Stability Condition
In mathematics, and especially algebraic geometry, a Bridgeland stability condition, defined by Tom Bridgeland, is an algebro-geometric stability condition defined on elements of a triangulated category. The case of original interest and particular importance is when this derived category is the derived category of coherent sheaves on a Calabi–Yau manifold, and this situation has fundamental links to string theory and the study of D-branes. Such stability conditions were introduced in a rudimentary form by Michael Douglas Michael Kirk Douglas (born September 25, 1944) is an American actor and film producer. He has received numerous accolades, including two Academy Awards, five Golden Globe Awards, a Primetime Emmy Award, the Cecil B. DeMille Award, and the A ... called \Pi-stability and used to study BPS B-branes in string theory.Douglas, M.R., Fiol, B. and Römelsberger, C., 2005. Stability and BPS branes. Journal of High Energy Physics, 2005(09), p. 006. This concep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subcategory
In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, a subcategory of ''C'' is a category obtained from ''C'' by "removing" some of its objects and arrows. Formal definition Let ''C'' be a category. A subcategory ''S'' of ''C'' is given by *a subcollection of objects of ''C'', denoted ob(''S''), *a subcollection of morphisms of ''C'', denoted hom(''S''). such that *for every ''X'' in ob(''S''), the identity morphism id''X'' is in hom(''S''), *for every morphism ''f'' : ''X'' → ''Y'' in hom(''S''), both the source ''X'' and the target ''Y'' are in ob(''S''), *for every pair of morphisms ''f'' and ''g'' in hom(''S'') the composite ''f'' o ''g'' is in hom(''S'') whenever it is defined. These conditions ensure that ''S'' is a category in its own right: its collection of objects is ob(''S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
0905
''M83'' is the debut studio album by French electropop band M83, released on 18 April 2001 on Gooom. The album was reissued on 6 September 2005 on Mute Records for its North American release, and is thus sometimes referred to as ''0905''. Produced by both M83 and Morgan Daguenet, the album is predominantly instrumental, with dialogue samples from various films and television programmes appearing across the songs. The track titles, if read sequentially, form a short story. Background and recording Recorded as a duo by founding members, Anthony Gonzalez and Nicolas Fromageau, the album was recorded at Echotone, in the Autumn of 2000, with co-producer Morgan Daguenet. The track "Slowly", included on the album's 2005 reissue, was recorded two years later at the same location. Track listing Notes * The track length of "Night" is 4:44 on the 2005 reissue, bringing the total length of the album to 70:23. * The first few seconds of "Violet Tree" features a sample of audio from epis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
T-structure
In the branch of mathematics called homological algebra, a ''t''-structure is a way to axiomatize the properties of an abelian subcategory of a derived category. A ''t''-structure on \mathcal consists of two subcategories (\mathcal^, \mathcal^) of a triangulated category or stable infinity category which abstract the idea of complexes whose cohomology vanishes in positive, respectively negative, degrees. There can be many distinct ''t''-structures on the same category, and the interplay between these structures has implications for algebra and geometry. The notion of a ''t''-structure arose in the work of Beilinson, Bernstein, Deligne, and Gabber on perverse sheaves. Definition Fix a triangulated category \mathcal with translation functor /math>. A ''t''-structure on \mathcal is a pair (\mathcal^, \mathcal^) of full subcategories, each of which is stable under isomorphism, which satisfy the following three axioms. # If ''X'' is an object of \mathcal^ and ''Y'' is an object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. '' Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grothendieck Group
In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a homomorphic image of will also contain a homomorphic image of the Grothendieck group of . The Grothendieck group construction takes its name from a specific case in category theory, introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory. This specific case is the monoid of isomorphism classes of objects of an abelian category, with the direct sum as its operation. Grothendieck group of a commutative monoid Motivation Given a commutative monoid , "the most general" abelian group that arises from is to be constructed by introducing inverse elements to all elements of . Such an abelian group always exists; it is called the Grothendieck group of . It is character ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Vector Bundle
In mathematics, a stable vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may be built from stable ones using Harder–Narasimhan filtration. Stable bundles were defined by David Mumford in and later built upon by David Gieseker, Fedor Bogomolov, Thomas Bridgeland and many others. Motivation One of the motivations for analyzing stable vector bundles is their nice behavior in families. In fact, Moduli spaces of stable vector bundles can be constructed using the Quot scheme in many cases, whereas the stack of vector bundles \mathbfGL_n is an Artin stack whose underlying set is a single point. Here's an example of a family of vector bundles which degenerate poorly. If we tensor the Euler sequence of \mathbb^1 by \mathcal(1) there is an exact sequence0 \to \mathcal(-1) \to \mathcal\oplus \mathcal \to \mathcal(1) \to 0which represents a non-zero element in v \in \text^1(\mathcal(1),\m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
HN Filtration In Triangulated Category
HN or Hn may refer to: Arts and entertainment * Hn., musical score notation for French horn * A numbering system for Royal Doulton Figurines, e..g. HN211 Businesses and organizations *Heavylift Cargo Airlines (IATA airline designator) * Hutchinson and Northern Railway, Kansas, US, reporting mark Media * HN, former alternate name for Headline News * Hacker News Places * Heilbronn, Germany, vehicle registration * Herceg Novi, a town in Montenegro * Honduras (ISO 3166-1 country code) ** .hn, the Internet country code top-level domain (ccTLD) for Honduras * Hunan, a province of China * Thesprotia, regional unit of Greece (vehicle plate code ⟨ΗΝ⟩, for capital Igoumenitsa ) Other uses * Hospitalman, a United States Navy Hospital corpsman rate * , a two-letter combination used in some languages ** Reduction of /hn/ to /n/ in Old/Middle English * Harmonic numbers See also * English horn, "E. hn." in music scores * Saxhorn The saxhorn is a family of valved b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bogomol'nyi–Prasad–Sommerfield State
In theoretical physics, massive representations of an extended supersymmetry algebra called BPS states have mass equal to the supersymmetry central charge ''Z''. Quantum mechanically, if the supersymmetry remains unbroken, exact equality to the modulus of ''Z'' exists. Their importance arises as the supermultiplets shorten for generic massive representations, with stability and mass formula exact. ''d'' = 4 ''N'' = 2 The generators for the odd part of the superalgebra have relations: : \begin \ & = 2 \sigma_^m P_m \delta^A_B\\ \ & = 2 \epsilon_ \epsilon^ \bar\\ \ & = -2 \epsilon_ \epsilon_ Z\\ \end where: \alpha \dot are the Lorentz group indices, A and B are R-symmetry indices. Take linear combinations of the above generators as follows: : \begin R_\alpha^A & = \xi^ Q_\alpha^A + \xi \sigma_^0 \bar^\\ T_\alpha^A & = \xi^ Q_\alpha^A - \xi \sigma_^0 \bar^\\ \end Consider a state ψ which has 4 momentum (M,0,0,0). Applying the following operator to this state gives: : \begin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
