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Embedded Contact Homology
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer introduced the first version of Floer homology, now called Lagrangian Floer homology, in his proof of the Arnold conjecture in symplectic geometry. Floer also developed a closely related theory for Lagrangian submanifolds of a symplectic manifold. A third construction, also due to Floer, associates homology groups to closed three-dimensional manifolds using the Yang–Mills functional. These constructions and their descendants play a fundamental role in current investigations into the topology of symplectic and contact manifolds as well as (smooth) three- and four-dimensional manifolds. Floer homology is typically defined by associating to the object of interest an infinite-dimensional manifold and a real valued function on it. In the sy ... [...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] |
Abelian Group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation \cdot that combines any two elements a and b of A to form another element of A, denoted a \cdot b. The symbo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed Point (mathematics)
A fixed point (sometimes shortened to fixpoint, also known as an invariant point) is a value that does not change under a given transformation. Specifically, in mathematics, a fixed point of a function is an element that is mapped to itself by the function. In physics, the term fixed point can refer to a temperature that can be used as a reproducible reference point, usually defined by a phase change or triple point. Fixed point of a function Formally, is a fixed point of a function if belongs to both the domain and the codomain of , and . For example, if is defined on the real numbers by f(x) = x^2 - 3 x + 4, then 2 is a fixed point of , because . Not all functions have fixed points: for example, , has no fixed points, since is never equal to for any real number. In graphical terms, a fixed point means the point is on the line , or in other words the graph of has a point in common with that line. Fixed-point iteration In numerical analysis, ''fixed-point iter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Isotopy
In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the symplectic structure of phase space, and is called a canonical transformation. Formal definition A diffeomorphism between two symplectic manifolds f: (M,\omega) \rightarrow (N,\omega') is called a symplectomorphism if :f^*\omega'=\omega, where f^* is the pullback of f. The symplectic diffeomorphisms from M to M are a (pseudo-)group, called the symplectomorphism group (see below). The infinitesimal version of symplectomorphisms gives the symplectic vector fields. A vector field X \in \Gamma^(TM) is called symplectic if :\mathcal_X\omega=0. Also, X is symplectic iff the flow \phi_t: M\rightarrow M of X is a symplectomorphism for every t. These vector fields build a Lie subalgebra of \Gamma^(TM). Here, \Gamma^(TM) is the set of smooth vector ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Loop Space
"Free Loop (One Night Stand)" (titled as "Free Loop" on ''Daniel Powter'') is a song written by Canadian singer Daniel Powter. It was his second single and the follow-up to his successful song, " Bad Day". In the United Kingdom, WEA failed to realize the single was deemed ineligible to chart since "Bad Day" was included as B-side whilst still being in the UK top 40 at the time. In the US, it did not chart on the ''Billboard'' Hot 100, but it did reach the top 30 of the ''Billboard'' Adult Contemporary chart. The song was featured in Cheetah Mobile's 2016 iOS/Android app Piano Tiles 2. It was also used in a 2007 advertisement showcasing the Ford I-Max in Taiwan. Music video The video, directed by Marc Webb, who also directed the music videos for " Bad Day" and "Lie to Me," shows a young Daniel Powter starting at a piano in a music store and puts his fingers on the window and magically plays the piano. The scene then cuts to an adult version of Powter, who is now able to buy the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Cover
A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete space D and for every x \in X an open neighborhood U \subset X, such that \pi^(U)= \displaystyle \bigsqcup_ V_d and \pi, _:V_d \rightarrow U is a homeomorphism for every d \in D . Often, the notion of a covering is used for the covering space E as well as for the map \pi : E \rightarrow X. The open sets V_ are called sheets, which are uniquely determined up to a homeomorphism if U is connected. For each x \in X the discrete subset \pi^(x) is called the fiber of x. The degree of a covering is the cardinality of the space D. If E is path-connected, then the covering \pi : E \rightarrow X is denoted as a path-connected covering. Examples * For every topological space X there exists the covering \pi:X \rightarrow X with \pi(x)=x, which is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Action
In physics, action is a scalar quantity describing how a physical system has dynamics (physics), changed over time. Action is significant because the equations of motion of the system can be derived through the principle of stationary action. In the simple case of a single particle moving with a constant velocity (uniform linear motion), the action is the momentum of the particle times the distance it moves, integral (mathematics), added up along its path; equivalently, action is twice the particle's kinetic energy times the duration for which it has that amount of energy. For more complicated systems, all such quantities are combined. More formally, action is a functional (mathematics), mathematical functional which takes the trajectory (also called path or history) of the system as its argument and has a real number as its result. Generally, the action takes different values for different paths. Action has dimensional analysis, dimensions of energy × time or momentu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectomorphism
In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the symplectic structure of phase space, and is called a canonical transformation. Formal definition A diffeomorphism between two symplectic manifolds f: (M,\omega) \rightarrow (N,\omega') is called a symplectomorphism if :f^*\omega'=\omega, where f^* is the pullback of f. The symplectic diffeomorphisms from M to M are a (pseudo-)group, called the symplectomorphism group (see below). The infinitesimal version of symplectomorphisms gives the symplectic vector fields. A vector field X \in \Gamma^(TM) is called symplectic if :\mathcal_X\omega=0. Also, X is symplectic iff the flow \phi_t: M\rightarrow M of X is a symplectomorphism for every t. These vector fields build a Lie subalgebra of \Gamma^(TM). Here, \Gamma^(TM) is the set of smooth vector ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gromov's Compactness Theorem (topology)
In the Mathematics, mathematical field of symplectic topology, Gromov's compactness theorem states that a sequence of pseudoholomorphic curve, pseudoholomorphic curves in an almost complex manifold with a uniform energy bound must have a subsequence which limits to a pseudoholomorphic curve which may have nodes or (a finite tree of) "bubbles". A bubble is a holomorphic sphere which has a transverse intersection with the rest of the curve. This theorem, and its generalizations to punctured pseudoholomorphic curves, underlies the compactness results for flow lines in Floer homology and symplectic field theory. If the complex structures on the curves in the sequence do not vary, only bubbles can occur; nodes can occur only if the complex structures on the domain are allowed to vary. Usually, the energy bound is achieved by considering a symplectic manifold with compatible almost-complex structure as the target, and assuming that curves to lie in a fixed homology class in the target ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudoholomorphic Curves
In mathematics, specifically in topology and geometry, a pseudoholomorphic curve (or ''J''-holomorphic curve) is a smooth map from a Riemann surface into an almost complex manifold that satisfies the Cauchy–Riemann equation. Introduced in 1985 by Mikhail Gromov, pseudoholomorphic curves have since revolutionized the study of symplectic manifolds. In particular, they lead to the Gromov–Witten invariants and Floer homology, and play a prominent role in string theory. Definition Let X be an almost complex manifold with almost complex structure J. Let C be a smooth Riemann surface (also called a complex curve) with complex structure j. A pseudoholomorphic curve in X is a map f : C \to X that satisfies the Cauchy–Riemann equation :\bar \partial_ f := \frac(df + J \circ df \circ j) = 0. Since J^2 = -1, this condition is equivalent to :J \circ df = df \circ j, which simply means that the differential df is complex-linear, that is, J maps each tangent space :T_xf(C)\subseteq T_xX to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flow Lines
Flow may refer to: Science and technology * Fluid flow, the motion of a gas or liquid * Flow (geomorphology), a type of mass wasting or slope movement in geomorphology * Flow (mathematics), a group action of the real numbers on a set * Flow (psychology), a mental state of being fully immersed and focused * Flow, a spacecraft of NASA's GRAIL program Computing * Flow network, graph-theoretic version of a mathematical flow * Flow analysis * Calligra Flow, free diagramming software * Dataflow, a broad concept in computer systems with many different meanings * Microsoft Flow (renamed to Power Automate in 2019), a workflow toolkit in Microsoft Dynamics * Neos Flow, a free and open source web application framework written in PHP * webMethods Flow, a graphical programming language * FLOW (programming language), an educational programming language from the 1970s * Flow (web browser), a web browser with a proprietary rendering engine Arts, entertainment and media * ''Flow'' (journal), an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
