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In mathematics, an upwards linked set A is a subset of a partially ordered set, P, in which any two of elements A have a common upper bound in P. Similarly, every pair of elements of a downwards linked set has a lower bound. Every centered set is linked, which includes, in particular, every directed set.
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https://en.wikipedia.org/wiki/Linked_set
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In mathematics, an η set (eta set) is a type of totally ordered set introduced by Hausdorff (1907, p. 126, 1914, chapter 6 section 8) that generalizes the order type η of the rational numbers.
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https://en.wikipedia.org/wiki/Eta_set
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In mathematics, an ω-bounded space is a topological space in which the closure of every countable subset is compact. More generally, if P is some property of subspaces, then a P-bounded space is one in which every subspace with property P has compact closure. Every compact space is ω-bounded, and every ω-bounded space is countably compact. The long line is ω-bounded but not compact. The bagpipe theorem describes the ω-bounded surfaces.
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https://en.wikipedia.org/wiki/Ω-bounded_space
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In mathematics, an ∞-topos is, roughly, an ∞-category such that its objects behave like sheaves of spaces with some choice of Grothendieck topology; in other words, it gives an intrinsic notion of sheaves without reference to an external space. The prototypical example of an ∞-topos is the ∞-category of sheaves of spaces on some topological space. But the notion is more flexible; for example, the ∞-category of étale sheaves on some scheme is not the ∞-category of sheaves on any topological space but it is still an ∞-topos. Precisely, in Lurie's Higher Topos Theory, an ∞-topos is defined as an ∞-category X such that there is a small ∞-category C and a left exact localization functor from the ∞-category of presheaves of spaces on C to X. A theorem of Lurie states that an ∞-category is an ∞-topos if and only if it satisfies an ∞-categorical version of Giraud's axioms in ordinary topos theory. A "topos" is a category behaving like the category of sheaves of sets on a topological space. In analogy, Lurie's definition and characterization theorem of an ∞-topos says that an ∞-topos is an ∞-category behaving like the category of sheaves of spaces.
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https://en.wikipedia.org/wiki/∞-topos
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In mathematics, analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three coordinate axes are given, each perpendicular to the other two at the origin, the point at which they cross. They are usually labeled x, y, and z. Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of real numbers, each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the plane determined by the other two axes.Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates and spherical coordinates, though there are an infinite number of possible methods. For more, see Euclidean space. Below are images of the above-mentioned systems.
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https://en.wikipedia.org/wiki/Three_dimensional_space
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In mathematics, analytic geometry (also called Cartesian geometry) describes every point in two-dimensional space by means of two coordinates. Two perpendicular coordinate axes are given which cross each other at the origin. They are usually labeled x and y. Relative to these axes, the position of any point in two-dimensional space is given by an ordered pair of real numbers, each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the other axis. Another widely used coordinate system is the polar coordinate system, which specifies a point in terms of its distance from the origin and its angle relative to a rightward reference ray.
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https://en.wikipedia.org/wiki/Plane_coordinates
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In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight.
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https://en.wikipedia.org/wiki/Coordinate_geometry
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It is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry. Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and circles, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space. As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometric shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom.
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https://en.wikipedia.org/wiki/Coordinate_geometry
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In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem).
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https://en.wikipedia.org/wiki/Analytic_Number_Theory
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In mathematics, ancient Egyptian multiplication (also known as Egyptian multiplication, Ethiopian multiplication, Russian multiplication, or peasant multiplication), one of two multiplication methods used by scribes, is a systematic method for multiplying two numbers that does not require the multiplication table, only the ability to multiply and divide by 2, and to add. It decomposes one of the multiplicands (preferably the smaller) into a set of numbers of powers of two and then creates a table of doublings of the second multiplicand by every value of the set which is summed up to give result of multiplication. This method may be called mediation and duplation, where mediation means halving one number and duplation means doubling the other number.
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https://en.wikipedia.org/wiki/Egyptian_multiplication
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It is still used in some areas.The second Egyptian multiplication and division technique was known from the hieratic Moscow and Rhind Mathematical Papyri written in the seventeenth century B.C. by the scribe Ahmes.Although in ancient Egypt the concept of base 2 did not exist, the algorithm is essentially the same algorithm as long multiplication after the multiplier and multiplicand are converted to binary. The method as interpreted by conversion to binary is therefore still in wide use today as implemented by binary multiplier circuits in modern computer processors.
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https://en.wikipedia.org/wiki/Egyptian_multiplication
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In mathematics, and disciplines in which mathematics plays a major role, hand-waving refers to either absence of formal proof or methods that do not meet mathematical rigor. In practice, it often involves the use of unrepresentative examples, unjustified assumptions, key omissions and faulty logic, and while these may be useful in expository papers and seminar presentations, they ultimately fall short of the standard of proof needed to establish a result. The mathematical profession tends to be receptive to informed critiques from any listener, and a claimant to a new result is expected to be able to answer any such question with a logical argument, up to a full proof. Should a speaker apparently fail to give such an answer, anyone in the audience who can supply the needed demonstration may sometimes upstage the speaker.
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https://en.wikipedia.org/wiki/Hand-waving
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The objector in such a case might receive some measure credit for the theorem the hand-waver presented. The opposite of hand-waving in mathematics (and related fields) is sometimes called nose-following, which refers to the unimaginative development of a narrow line of reasoning that—while correct—can also end up making the subject dry and uninteresting.The rationale for this culture of hyper-critical scrutiny is suggested by a quote of G. H. Hardy: " subject is the most curious of all—there is none in which truth plays such odd pranks. It has the most elaborate and the most fascinating technique, and gives unrivalled openings for the display of sheer professional skill."
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https://en.wikipedia.org/wiki/Hand-waving
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In mathematics, and especially affine differential geometry, the affine focal set of a smooth submanifold M embedded in a smooth manifold N is the caustic generated by the affine normal lines. It can be realised as the bifurcation set of a certain family of functions. The bifurcation set is the set of parameter values of the family which yield functions with degenerate singularities. This is not the same as the bifurcation diagram in dynamical systems.
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https://en.wikipedia.org/wiki/Affine_focal_set
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Assume that M is an n-dimensional smooth hypersurface in real (n+1)-space. Assume that M has no points where the second fundamental form is degenerate. From the article affine differential geometry, there exists a unique transverse vector field over M. This is the affine normal vector field, or the Blaschke normal field. A special (i.e. det = 1) affine transformation of real (n + 1)-space will carry the affine normal vector field of M onto the affine normal vector field of the image of M under the transformation.
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https://en.wikipedia.org/wiki/Affine_focal_set
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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 triangulated 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 called Π {\displaystyle \Pi } -stability and used to study BPS B-branes in string theory. This concept was made precise by Bridgeland, who phrased these stability conditions categorically, and initiated their study mathematically.
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https://en.wikipedia.org/wiki/Bridgeland_stability_condition
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In mathematics, and especially algebraic geometry, stability is a notion which characterises when a geometric object, for example a point, an algebraic variety, a vector bundle, or a sheaf, has some desirable properties for the purpose of classifying them. The exact characterisation of what it means to be stable depends on the type of geometric object, but all such examples share the property of having a minimal amount of internal symmetry, that is such stable objects have few automorphisms. This is related to the concept of simplicity in mathematics, which measures when some mathematical object has few subobjects inside it (see for example simple groups, which have no non-trivial normal subgroups). In addition to stability, some objects may be described with terms such as semi-stable (having a small but not minimal amount of symmetry), polystable (being made out of stable objects), or unstable (having too much symmetry, the opposite of stable).
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https://en.wikipedia.org/wiki/Stability_(algebraic_geometry)
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In mathematics, and especially complex geometry, the Mabuchi functional or K-energy functional is a functional on the space of Kähler potentials of a compact Kähler manifold whose critical points are constant scalar curvature Kähler metrics. The Mabuchi functional was introduced by Toshiki Mabuchi in 1985 as a functional which integrates the Futaki invariant, which is an obstruction to the existence of a Kähler–Einstein metric on a Fano manifold.The Mabuchi functional is an analogy of the log-norm functional of the moment map in geometric invariant theory and symplectic reduction. The Mabuchi functional appears in the theory of K-stability as an analytical functional which characterises the existence of constant scalar curvature Kähler metrics. The slope at infinity of the Mabuchi functional along any geodesic ray in the space of Kähler potentials is given by the Donaldson–Futaki invariant of a corresponding test configuration. Due to the variational techniques of Berman–Boucksom–Jonsson in the study of Kähler–Einstein metrics on Fano varieties, the Mabuchi functional and various generalisations of it have become critically important in the study of K-stability of Fano varieties, particularly in settings with singularities.
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https://en.wikipedia.org/wiki/K-energy_functional
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In mathematics, and especially complex geometry, the holomorphic tangent bundle of a complex manifold M {\displaystyle M} is the holomorphic analogue of the tangent bundle of a smooth manifold. The fibre of the holomorphic tangent bundle over a point is the holomorphic tangent space, which is the tangent space of the underlying smooth manifold, given the structure of a complex vector space via the almost complex structure J {\displaystyle J} of the complex manifold M {\displaystyle M} .
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https://en.wikipedia.org/wiki/Holomorphic_tangent_bundle
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In mathematics, and especially differential and algebraic geometry, K-stability is an algebro-geometric stability condition, for complex manifolds and complex algebraic varieties. The notion of K-stability was first introduced by Gang Tian and reformulated more algebraically later by Simon Donaldson. The definition was inspired by a comparison to geometric invariant theory (GIT) stability. In the special case of Fano varieties, K-stability precisely characterises the existence of Kähler–Einstein metrics. More generally, on any compact complex manifold, K-stability is conjectured to be equivalent to the existence of constant scalar curvature Kähler metrics (cscK metrics).
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https://en.wikipedia.org/wiki/K-stability
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In mathematics, and especially differential geometry and algebraic geometry, a stable principal bundle is a generalisation of the notion of a stable vector bundle to the setting of principal bundles. The concept of stability for principal bundles was introduced by Annamalai Ramanathan for the purpose of defining the moduli space of G-principal bundles over a Riemann surface, a generalisation of earlier work by David Mumford and others on the moduli spaces of vector bundles.Many statements about the stability of vector bundles can be translated into the language of stable principal bundles. For example, the analogue of the Kobayashi–Hitchin correspondence for principal bundles, that a holomorphic principal bundle over a compact Kähler manifold admits a Hermite–Einstein connection if and only if it is polystable, was shown to be true in the case of projective manifolds by Subramanian and Ramanathan, and for arbitrary compact Kähler manifolds by Anchouche and Biswas.
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https://en.wikipedia.org/wiki/Stable_principal_bundle
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In mathematics, and especially differential geometry and gauge theory, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. A principal G-connection on a principal G-bundle P over a smooth manifold M is a particular type of connection which is compatible with the action of the group G. A principal connection can be viewed as a special case of the notion of an Ehresmann connection, and is sometimes called a principal Ehresmann connection. It gives rise to (Ehresmann) connections on any fiber bundle associated to P via the associated bundle construction. In particular, on any associated vector bundle the principal connection induces a covariant derivative, an operator that can differentiate sections of that bundle along tangent directions in the base manifold. Principal connections generalize to arbitrary principal bundles the concept of a linear connection on the frame bundle of a smooth manifold.
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https://en.wikipedia.org/wiki/Connection_(principal_bundle)
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In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a linear connection on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a covariant derivative, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on the tangent bundle of a pseudo-Riemannian manifold, which gives a standard way to differentiate vector fields.
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https://en.wikipedia.org/wiki/Connection_on_a_vector_bundle
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Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear. Linear connections are also called Koszul connections after Jean-Louis Koszul, who gave an algebraic framework for describing them (Koszul 1950). This article defines the connection on a vector bundle using a common mathematical notation which de-emphasizes coordinates. However, other notations are also regularly used: in general relativity, vector bundle computations are usually written using indexed tensors; in gauge theory, the endomorphisms of the vector space fibers are emphasized. The different notations are equivalent, as discussed in the article on metric connections (the comments made there apply to all vector bundles).
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https://en.wikipedia.org/wiki/Connection_on_a_vector_bundle
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In mathematics, and especially differential geometry and mathematical physics, gauge theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused with the closely related concept of a gauge theory in physics, which is a field theory which admits gauge symmetry. In mathematics theory means a mathematical theory, encapsulating the general study of a collection of concepts or phenomena, whereas in the physical sense a gauge theory is a mathematical model of some natural phenomenon. Gauge theory in mathematics is typically concerned with the study of gauge-theoretic equations.
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https://en.wikipedia.org/wiki/Gauge_theory_(mathematics)
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These are differential equations involving connections on vector bundles or principal bundles, or involving sections of vector bundles, and so there are strong links between gauge theory and geometric analysis. These equations are often physically meaningful, corresponding to important concepts in quantum field theory or string theory, but also have important mathematical significance. For example, the Yang–Mills equations are a system of partial differential equations for a connection on a principal bundle, and in physics solutions to these equations correspond to vacuum solutions to the equations of motion for a classical field theory, particles known as instantons. Gauge theory has found uses in constructing new invariants of smooth manifolds, the construction of exotic geometric structures such as hyperkähler manifolds, as well as giving alternative descriptions of important structures in algebraic geometry such as moduli spaces of vector bundles and coherent sheaves.
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https://en.wikipedia.org/wiki/Gauge_theory_(mathematics)
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In mathematics, and especially differential geometry, an affine sphere is a hypersurface for which the affine normals all intersect in a single point. The term affine sphere is used because they play an analogous role in affine differential geometry to that of ordinary spheres in Euclidean differential geometry. An affine sphere is called improper if all of the affine normals are constant. In that case, the intersection point mentioned above lies on the hyperplane at infinity. Affine spheres have been the subject of much investigation, with many hundreds of research articles devoted to their study.
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https://en.wikipedia.org/wiki/Affine_sphere
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In mathematics, and especially differential geometry, the Quillen metric is a metric on the determinant line bundle of a family of operators. It was introduced by Daniel Quillen for certain elliptic operators over a Riemann surface, and generalized to higher-dimensional manifolds by Jean-Michel Bismut and Dan Freed.The Quillen metric was used by Quillen to give a differential-geometric interpretation of the ample line bundle over the moduli space of vector bundles on a compact Riemann surface, known as the Quillen determinant line bundle. It can be seen as defining the Chern–Weil representative of the first Chern class of this ample line bundle. The Quillen metric construction and its generalizations were used by Bismut and Freed to compute the holonomy of certain determinant line bundles of Dirac operators, and this holonomy is associated to certain anomaly cancellations in Chern–Simons theory predicted by Edward Witten.The Quillen metric was also used by Simon Donaldson in 1987 in a new inductive proof of the Hitchin–Kobayashi correspondence for projective algebraic manifolds, published one year after the resolution of the correspondence by Shing-Tung Yau and Karen Uhlenbeck for arbitrary compact Kähler manifolds.
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https://en.wikipedia.org/wiki/Quillen_metric
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In mathematics, and especially differential topology and gauge theory, Donaldson's theorem states that a definite intersection form of a compact, oriented, smooth manifold of dimension 4 is diagonalisable. If the intersection form is positive (negative) definite, it can be diagonalized to the identity matrix (negative identity matrix) over the integers. The original version of the theorem required the manifold to be simply connected, but it was later improved to apply to 4-manifolds with any fundamental group.
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https://en.wikipedia.org/wiki/Donaldson_theorem
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In mathematics, and especially differential topology, functional analysis and singularity theory, the Whitney topologies are a countably infinite family of topologies defined on the set of smooth mappings between two smooth manifolds. They are named after the American mathematician Hassler Whitney.
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https://en.wikipedia.org/wiki/Whitney_topology
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In mathematics, and especially gauge theory, Donaldson theory is the study of the topology of smooth 4-manifolds using moduli spaces of anti-self-dual instantons. It was started by Simon Donaldson (1983) who proved Donaldson's theorem restricting the possible quadratic forms on the second cohomology group of a compact simply connected 4-manifold. Important consequences of this theorem include the existence of an Exotic R4 and the failure of the smooth h-cobordism theorem in 4 dimensions. The results of Donaldson theory depend therefore on the manifold having a differential structure, and are largely false for topological 4-manifolds. Many of the theorems in Donaldson theory can now be proved more easily using Seiberg–Witten theory, though there are a number of open problems remaining in Donaldson theory, such as the Witten conjecture and the Atiyah–Floer conjecture.
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https://en.wikipedia.org/wiki/Donaldson_invariant
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In mathematics, and especially gauge theory, Seiberg–Witten invariants are invariants of compact smooth oriented 4-manifolds introduced by Edward Witten (1994), using the Seiberg–Witten theory studied by Nathan Seiberg and Witten (1994a, 1994b) during their investigations of Seiberg–Witten gauge theory. Seiberg–Witten invariants are similar to Donaldson invariants and can be used to prove similar (but sometimes slightly stronger) results about smooth 4-manifolds. They are technically much easier to work with than Donaldson invariants; for example, the moduli spaces of solutions of the Seiberg–Witten equations tends to be compact, so one avoids the hard problems involved in compactifying the moduli spaces in Donaldson theory.
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https://en.wikipedia.org/wiki/Seiberg–Witten_equations
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For detailed descriptions of Seiberg–Witten invariants see (Donaldson 1996), (Moore 2001), (Morgan 1996), (Nicolaescu 2000), (Scorpan 2005, Chapter 10). For the relation to symplectic manifolds and Gromov–Witten invariants see (Taubes 2000). For the early history see (Jackson 1995).
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https://en.wikipedia.org/wiki/Seiberg–Witten_equations
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In mathematics, and especially gauge theory, the Bogomolny equation for magnetic monopoles is the equation F A = ⋆ d A Φ , {\displaystyle F_{A}=\star d_{A}\Phi ,} where F A {\displaystyle F_{A}} is the curvature of a connection A {\displaystyle A} on a principal G {\displaystyle G} -bundle over a 3-manifold M {\displaystyle M} , Φ {\displaystyle \Phi } is a section of the corresponding adjoint bundle, d A {\displaystyle d_{A}} is the exterior covariant derivative induced by A {\displaystyle A} on the adjoint bundle, and ⋆ {\displaystyle \star } is the Hodge star operator on M {\displaystyle M} . These equations are named after E. B. Bogomolny and were studied extensively by Michael Atiyah and Nigel Hitchin.The equations are a dimensional reduction of the self-dual Yang–Mills equations from four dimensions to three dimensions, and correspond to global minima of the appropriate action. If M {\displaystyle M} is closed, there are only trivial (i.e. flat) solutions.
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https://en.wikipedia.org/wiki/Bogomolny_equation
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In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n {\displaystyle n} -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} by the Euclidean metric.
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https://en.wikipedia.org/wiki/Euclidean_topology
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In mathematics, and especially general topology, the interlocking interval topology is an example of a topology on the set S := R+ \ Z+, i.e. the set of all positive real numbers that are not positive whole numbers. To give the set S a topology means to say which subsets of S are "open", and to do so in a way that the following axioms are met: The union of open sets is an open set. The finite intersection of open sets is an open set. S and the empty set ∅ are open sets.
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https://en.wikipedia.org/wiki/Interlocking_interval_topology
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In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for tangency. One needs a definition of intersection number in order to state results like Bézout's theorem. The intersection number is obvious in certain cases, such as the intersection of the x- and y-axes in a plane, which should be one.
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https://en.wikipedia.org/wiki/Intersection_multiplicity
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The complexity enters when calculating intersections at points of tangency, and intersections which are not just points, but have higher dimension. For example, if a plane is tangent to a surface along a line, the intersection number along the line should be at least two. These questions are discussed systematically in intersection theory.
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https://en.wikipedia.org/wiki/Intersection_multiplicity
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In mathematics, and especially in category theory, a commutative diagram is a diagram of objects, also known as vertices, and morphisms, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition. Commutative diagrams play the role in category theory that equations play in algebra.
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https://en.wikipedia.org/wiki/Mathematical_diagram
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In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result. It is said that commutative diagrams play the role in category theory that equations play in algebra.
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https://en.wikipedia.org/wiki/Commutative_diagrams
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In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the icosahedron) and the rhombic triacontahedron. Every polyhedron with icosahedral symmetry has 60 rotational (or orientation-preserving) symmetries and 60 orientation-reversing symmetries (that combine a rotation and a reflection), for a total symmetry order of 120. The full symmetry group is the Coxeter group of type H3. It may be represented by Coxeter notation and Coxeter diagram . The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group A5 on 5 letters.
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https://en.wikipedia.org/wiki/Icosahedral_group
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In mathematics, and especially in homotopy theory, a crossed module consists of groups G {\displaystyle G} and H {\displaystyle H} , where G {\displaystyle G} acts on H {\displaystyle H} by automorphisms (which we will write on the left, ( g , h ) ↦ g ⋅ h {\displaystyle (g,h)\mapsto g\cdot h} , and a homomorphism of groups d: H ⟶ G , {\displaystyle d\colon H\longrightarrow G,} that is equivariant with respect to the conjugation action of G {\displaystyle G} on itself: d ( g ⋅ h ) = g d ( h ) g − 1 {\displaystyle d(g\cdot h)=gd(h)g^{-1}} and also satisfies the so-called Peiffer identity: d ( h 1 ) ⋅ h 2 = h 1 h 2 h 1 − 1 {\displaystyle d(h_{1})\cdot h_{2}=h_{1}h_{2}h_{1}^{-1}}
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https://en.wikipedia.org/wiki/Crossed_module
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In mathematics, and especially in order theory, a nucleus is a function F {\displaystyle F} on a meet-semilattice A {\displaystyle {\mathfrak {A}}} such that (for every p {\displaystyle p} in A {\displaystyle {\mathfrak {A}}} ): p ≤ F ( p ) {\displaystyle p\leq F(p)} F ( F ( p ) ) = F ( p ) {\displaystyle F(F(p))=F(p)} F ( p ∧ q ) = F ( p ) ∧ F ( q ) {\displaystyle F(p\wedge q)=F(p)\wedge F(q)} Every nucleus is evidently a monotone function.
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https://en.wikipedia.org/wiki/Nucleus_(order_theory)
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In mathematics, and especially symplectic geometry, the Thomas–Yau conjecture asks for the existence of a stability condition, similar to those which appear in algebraic geometry, which guarantees the existence of a solution to the special Lagrangian equation inside a Hamiltonian isotopy class of Lagrangian submanifolds. In particular the conjecture contains two difficulties: first it asks what a suitable stability condition might be, and secondly if one can prove stability of an isotopy class if and only if it contains a special Lagrangian representative. The Thomas–Yau conjecture was proposed by Richard Thomas and Shing-Tung Yau in 2001, and was motivated by similar theorems in algebraic geometry relating existence of solutions to geometric partial differential equations and stability conditions, especially the Kobayashi–Hitchin correspondence relating slope stable vector bundles to Hermitian Yang–Mills metrics.
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https://en.wikipedia.org/wiki/Thomas–Yau_conjecture
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The conjecture is intimately related to mirror symmetry, a conjecture in string theory and mathematical physics which predicts that mirror to a symplectic manifold (which is a Calabi–Yau manifold) there should be another Calabi–Yau manifold for which the symplectic structure is interchanged with the complex structure. In particular mirror symmetry predicts that special Lagrangians, which are the Type IIA string theory model of BPS D-branes, should be interchanged with the same structures in the Type IIB model, which are given either by stable vector bundles or vector bundles admitting Hermitian Yang–Mills or possibly deformed Hermitian Yang–Mills metrics. Motivated by this, Dominic Joyce rephrased the Thomas–Yau conjecture in 2014, predicting that the stability condition may be understood using the theory of Bridgeland stability conditions defined on the Fukaya category of the Calabi–Yau manifold, which is a triangulated category appearing in Kontsevich's homological mirror symmetry conjecture.
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https://en.wikipedia.org/wiki/Thomas–Yau_conjecture
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In mathematics, and especially the discipline of representation theory, the Schur indicator, named after Issai Schur, or Frobenius–Schur indicator describes what invariant bilinear forms a given irreducible representation of a compact group on a complex vector space has. It can be used to classify the irreducible representations of compact groups on real vector spaces.
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https://en.wikipedia.org/wiki/Schur_indicator
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In mathematics, and especially topology and differential geometry, a pinched torus (or croissant surface) is a kind of two-dimensional surface. It gets its name from its resemblance to a torus that has been pinched at a single point. A pinched torus is an example of an orientable, compact 2-dimensional pseudomanifold.
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https://en.wikipedia.org/wiki/Pinched_torus
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In mathematics, and especially topology, a Poincaré complex (named after the mathematician Henri Poincaré) is an abstraction of the singular chain complex of a closed, orientable manifold. The singular homology and cohomology groups of a closed, orientable manifold are related by Poincaré duality. Poincaré duality is an isomorphism between homology and cohomology groups. A chain complex is called a Poincaré complex if its homology groups and cohomology groups have the abstract properties of Poincaré duality.A Poincaré space is a topological space whose singular chain complex is a Poincaré complex. These are used in surgery theory to analyze manifold algebraically.
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https://en.wikipedia.org/wiki/Poincaré_complex
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In mathematics, and especially topology, a Pytkeev space is a topological space that satisfies qualities more subtle than a convergence of a sequence. They are named after E. G. Pytkeev, who proved in 1983 that sequential spaces have this property.
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https://en.wikipedia.org/wiki/Pytkeev_space
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In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a variable may be said to be either free or bound. The terms are opposites. A free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not a parameter of this or any container expression.
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https://en.wikipedia.org/wiki/Bound_variable
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Some older books use the terms real variable and apparent variable for free variable and bound variable, respectively. The idea is related to a placeholder (a symbol that will later be replaced by some value), or a wildcard character that stands for an unspecified symbol. In computer programming, the term free variable refers to variables used in a function that are neither local variables nor parameters of that function.
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https://en.wikipedia.org/wiki/Bound_variable
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The term non-local variable is often a synonym in this context. An instance of a variable symbol is bound, in contrast, if the value of that variable symbol has been bound to a specific value or range of values in the domain of discourse or universe. This may be achieved through the use of logical quantifiers, variable-binding operators, or an explicit statement of allowed values for the variable (such as, "...where n {\displaystyle n} is a positive integer".)
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https://en.wikipedia.org/wiki/Bound_variable
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A variable symbol overall is bound if at least one occurrence of it is bound.pp.142--143 Since the same variable symbol may appear in multiple places in an expression, some occurrences of the variable symbol may be free while others are bound,p.78 hence "free" and "bound" are at first defined for occurrences and then generalized over all occurrences of said variable symbol in the expression. However it is done, the variable ceases to be an independent variable on which the value of the expression depends, whether that value be a truth value or the numerical result of a calculation, or, more generally, an element of an image set of a function. While the domain of discourse in many contexts is understood, when an explicit range of values for the bound variable has not been given, it may be necessary to specify the domain in order to properly evaluate the expression.
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https://en.wikipedia.org/wiki/Bound_variable
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For example, consider the following expression in which both variables are bound by logical quantifiers: ∀ y ∃ x ( x = y ) . {\displaystyle \forall y\,\exists x\,\left(x={\sqrt {y}}\right).} This expression evaluates to false if the domain of x {\displaystyle x} and y {\displaystyle y} is the real numbers, but true if the domain is the complex numbers. The term "dummy variable" is also sometimes used for a bound variable (more commonly in general mathematics than in computer science), but this should not be confused with the identically named but unrelated concept of dummy variable as used in statistics, most commonly in regression analysis.
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https://en.wikipedia.org/wiki/Bound_variable
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In mathematics, and in particular algebraic geometry, K-stability is an algebro-geometric stability condition for projective algebraic varieties and complex manifolds. K-stability is of particular importance for the case of Fano varieties, where it is the correct stability condition to allow the formation of moduli spaces, and where it precisely characterises the existence of Kähler–Einstein metrics. K-stability was first defined for Fano manifolds by Gang Tian in 1997 in response to a conjecture of Shing-Tung Yau from 1993 that there should exist a stability condition which characterises the existence of a Kähler–Einstein metric on a Fano manifold. It was defined in reference to the K-energy functional previously introduced by Toshiki Mabuchi.
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https://en.wikipedia.org/wiki/K-stability_of_Fano_varieties
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Tian's definition of K-stability was reformulated by Simon Donaldson in 2001 in a purely algebro-geometric way.K-stability has become an important notion in the study and classification of Fano varieties. In 2012 Xiuxiong Chen, Donaldson, and Song Sun and independently Gang Tian proved that a smooth Fano manifold is K-polystable if and only if it admits a Kähler–Einstein metric. This was later generalised to singular K-polystable Fano varieties due to the work of Berman–Boucksom–Jonsson and others.
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https://en.wikipedia.org/wiki/K-stability_of_Fano_varieties
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K-stability is important in constructing moduli spaces of Fano varieties, where observations going back to the original development of geometric invariant theory show that it is necessary to restrict to a class of stable objects to form good moduli. It is now known through the work of Chenyang Xu and others that there exists a projective coarse moduli space of K-polystable Fano varieties of finite type. This work relies on Caucher Birkar's proof of boundedness of Fano varieties, for which he was awarded the 2018 Fields medal. Due to the reformulations of the K-stability condition by Fujita–Li and Odaka, the K-stability of Fano varieties may be explicitly computed in practice. Which Fano varieties are K-stable is well understood in dimension one, two, and three.
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https://en.wikipedia.org/wiki/K-stability_of_Fano_varieties
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In mathematics, and in particular differential geometry and complex geometry, a complex analytic variety or complex analytic space is a generalization of a complex manifold which allows the presence of singularities. Complex analytic varieties are locally ringed spaces which are locally isomorphic to local model spaces, where a local model space is an open subset of the vanishing locus of a finite set of holomorphic functions.
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https://en.wikipedia.org/wiki/Analytic_variety
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In mathematics, and in particular differential geometry and gauge theory, Hitchin's equations are a system of partial differential equations for a connection and Higgs field on a vector bundle or principal bundle over a Riemann surface, written down by Nigel Hitchin in 1987. Hitchin's equations are locally equivalent to the harmonic map equation for a surface into the symmetric space dual to the structure group. They also appear as a dimensional reduction of the self-dual Yang–Mills equations from four dimensions to two dimensions, and solutions to Hitchin's equations give examples of Higgs bundles and of holomorphic connections. The existence of solutions to Hitchin's equations on a compact Riemann surface follows from the stability of the corresponding Higgs bundle or the corresponding holomorphic connection, and this is the simplest form of the Nonabelian Hodge correspondence.
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https://en.wikipedia.org/wiki/Hitchin's_equations
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The moduli space of solutions to Hitchin's equations was constructed by Hitchin in the rank two case on a compact Riemann surface and was one of the first examples of a hyperkähler manifold constructed. The nonabelian Hodge correspondence shows it is isomorphic to the Higgs bundle moduli space, and to the moduli space of holomorphic connections. Using the metric structure on the Higgs bundle moduli space afforded by its description in terms of Hitchin's equations, Hitchin constructed the Hitchin system, a completely integrable system whose twisted generalization over a finite field was used by Ngô Bảo Châu in his proof of the fundamental lemma in the Langlands program, for which he was afforded the 2010 Fields medal.
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https://en.wikipedia.org/wiki/Hitchin's_equations
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In mathematics, and in particular functional analysis, the shift operator, also known as the translation operator, is an operator that takes a function x ↦ f(x) to its translation x ↦ f(x + a). In time series analysis, the shift operator is called the lag operator. Shift operators are examples of linear operators, important for their simplicity and natural occurrence.
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https://en.wikipedia.org/wiki/Unilateral_shift
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The shift operator action on functions of a real variable plays an important role in harmonic analysis, for example, it appears in the definitions of almost periodic functions, positive-definite functions, derivatives, and convolution. Shifts of sequences (functions of an integer variable) appear in diverse areas such as Hardy spaces, the theory of abelian varieties, and the theory of symbolic dynamics, for which the baker's map is an explicit representation. Triangulated category is a categorified analogue of the shift operator.
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https://en.wikipedia.org/wiki/Unilateral_shift
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In mathematics, and in particular functional analysis, the tensor product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert spaces is another Hilbert space. Roughly speaking, the tensor product is the metric space completion of the ordinary tensor product. This is an example of a topological tensor product. The tensor product allows Hilbert spaces to be collected into a symmetric monoidal category.
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https://en.wikipedia.org/wiki/Tensor_product_of_Hilbert_spaces
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In mathematics, and in particular game theory, Sion's minimax theorem is a generalization of John von Neumann's minimax theorem, named after Maurice Sion. It states: Let X {\displaystyle X} be a compact convex subset of a linear topological space and Y {\displaystyle Y} a convex subset of a linear topological space. If f {\displaystyle f} is a real-valued function on X × Y {\displaystyle X\times Y} with f ( x , ⋅ ) {\displaystyle f(x,\cdot )} upper semicontinuous and quasi-concave on Y {\displaystyle Y} , ∀ x ∈ X {\displaystyle \forall x\in X} , and f ( ⋅ , y ) {\displaystyle f(\cdot ,y)} lower semicontinuous and quasi-convex on X {\displaystyle X} , ∀ y ∈ Y {\displaystyle \forall y\in Y} then, min x ∈ X sup y ∈ Y f ( x , y ) = sup y ∈ Y min x ∈ X f ( x , y ) . {\displaystyle \min _{x\in X}\sup _{y\in Y}f(x,y)=\sup _{y\in Y}\min _{x\in X}f(x,y).}
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https://en.wikipedia.org/wiki/Sion's_minimax_theorem
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In mathematics, and in particular gauge theory and complex geometry, a Hermitian Yang–Mills connection (or Hermite-Einstein connection) is a Chern connection associated to an inner product on a holomorphic vector bundle over a Kähler manifold that satisfies an analogue of Einstein's equations: namely, the contraction of the curvature 2-form of the connection with the Kähler form is required to be a constant times the identity transformation. Hermitian Yang–Mills connections are special examples of Yang–Mills connections, and are often called instantons. The Kobayashi–Hitchin correspondence proved by Donaldson, Uhlenbeck and Yau asserts that a holomorphic vector bundle over a compact Kähler manifold admits a Hermitian Yang–Mills connection if and only if it is slope polystable.
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https://en.wikipedia.org/wiki/Hermitian_Yang–Mills_connection
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In mathematics, and in particular homotopy theory, a hypercovering (or hypercover) is a simplicial object that generalises the Čech nerve of a cover. For the Čech nerve of an open cover U → X {\displaystyle {\mathcal {U}}\to X} , one can show that if the space X {\displaystyle X} is compact and if every intersection of open sets in the cover is contractible, then one can contract these sets and get a simplicial set that is weakly equivalent to X {\displaystyle X} in a natural way. For the étale topology and other sites, these conditions fail. The idea of a hypercover is to instead of only working with n {\displaystyle n} -fold intersections of the sets of the given open cover U {\displaystyle {\mathcal {U}}} , to allow the pairwise intersections of the sets in U = U 0 {\displaystyle {\mathcal {U}}={\mathcal {U}}_{0}} to be covered by an open cover U 1 {\displaystyle {\mathcal {U}}_{1}} , and to let the triple intersections of this cover to be covered by yet another open cover U 2 {\displaystyle {\mathcal {U}}_{2}} , and so on, iteratively. Hypercoverings have a central role in étale homotopy and other areas where homotopy theory is applied to algebraic geometry, such as motivic homotopy theory.
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https://en.wikipedia.org/wiki/Hypercovering
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In mathematics, and in particular in arithmetic combinatorics, a Salem-Spencer set is a set of numbers no three of which form an arithmetic progression. Salem–Spencer sets are also called 3-AP-free sequences or progression-free sets. They have also been called non-averaging sets, but this term has also been used to denote a set of integers none of which can be obtained as the average of any subset of the other numbers. Salem-Spencer sets are named after Raphaël Salem and Donald C. Spencer, who showed in 1942 that Salem–Spencer sets can have nearly-linear size. However a later theorem of Klaus Roth shows that the size is always less than linear.
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https://en.wikipedia.org/wiki/Salem–Spencer_set
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In mathematics, and in particular in combinatorics, the combinatorial number system of degree k (for some positive integer k), also referred to as combinadics, or the Macaulay representation of an integer, is a correspondence between natural numbers (taken to include 0) N and k-combinations. The combinations are represented as strictly decreasing sequences ck > ... > c2 > c1 ≥ 0 where each ci corresponds to the index of a chosen element in a given k-combination. Distinct numbers correspond to distinct k-combinations, and produce them in lexicographic order.
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https://en.wikipedia.org/wiki/Combinatorial_number_system
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The numbers less than ( n k ) {\displaystyle {\tbinom {n}{k}}} correspond to all k-combinations of {0, 1, ..., n − 1}. The correspondence does not depend on the size n of the set that the k-combinations are taken from, so it can be interpreted as a map from N to the k-combinations taken from N; in this view the correspondence is a bijection. The number N corresponding to (ck, ..., c2, c1) is given by N = ( c k k ) + ⋯ + ( c 2 2 ) + ( c 1 1 ) {\displaystyle N={\binom {c_{k}}{k}}+\cdots +{\binom {c_{2}}{2}}+{\binom {c_{1}}{1}}} .The fact that a unique sequence corresponds to any non-negative number N was first observed by D. H. Lehmer.
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https://en.wikipedia.org/wiki/Combinatorial_number_system
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Indeed, a greedy algorithm finds the k-combination corresponding to N: take ck maximal with ( c k k ) ≤ N {\displaystyle {\tbinom {c_{k}}{k}}\leq N} , then take ck−1 maximal with ( c k − 1 k − 1 ) ≤ N − ( c k k ) {\displaystyle {\tbinom {c_{k-1}}{k-1}}\leq N-{\tbinom {c_{k}}{k}}} , and so forth. Finding the number N, using the formula above, from the k-combination (ck, ..., c2, c1) is also known as "ranking", and the opposite operation (given by the greedy algorithm) as "unranking"; the operations are known by these names in most computer algebra systems, and in computational mathematics.The originally used term "combinatorial representation of integers" was shortened to "combinatorial number system" by Knuth, who also gives a much older reference; the term "combinadic" is introduced by James McCaffrey (without reference to previous terminology or work). Unlike the factorial number system, the combinatorial number system of degree k is not a mixed radix system: the part ( c i i ) {\displaystyle {\tbinom {c_{i}}{i}}} of the number N represented by a "digit" ci is not obtained from it by simply multiplying by a place value. The main application of the combinatorial number system is that it allows rapid computation of the k-combination that is at a given position in the lexicographic ordering, without having to explicitly list the k-combinations preceding it; this allows for instance random generation of k-combinations of a given set. Enumeration of k-combinations has many applications, among which are software testing, sampling, quality control, and the analysis of lottery games.
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https://en.wikipedia.org/wiki/Combinatorial_number_system
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In mathematics, and in particular in group theory, a cyclic permutation is a permutation consisting of a single cycle. In some cases, cyclic permutations are referred to as cycles; if a cyclic permutation has k elements, it may be called a k-cycle. Some authors widen this definition to include permutations with fixed points in addition to at most one non-trivial cycle. In cycle notation, cyclic permutations are denoted by the list of their elements enclosed with parentheses, in the order to which they are permuted.
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https://en.wikipedia.org/wiki/Adjacent_transposition
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For example, the permutation (1 3 2 4) that sends 1 to 3, 3 to 2, 2 to 4 and 4 to 1 is a 4-cycle, and the permutation (1 3 2)(4) that sends 1 to 3, 3 to 2, 2 to 1 and 4 to 4 is considered a 3-cycle by some authors. On the other hand, the permutation (1 3)(2 4) that sends 1 to 3, 3 to 1, 2 to 4 and 4 to 2 is not a cyclic permutation because it separately permutes the pairs {1, 3} and {2, 4}. The set of elements that are not fixed by a cyclic permutation is called the orbit of the cyclic permutation. Every permutation on finitely many elements can be decomposed into cyclic permutations on disjoint orbits. The individual cyclic parts of a permutation are also called cycles, thus the second example is composed of a 3-cycle and a 1-cycle (or fixed point) and the third is composed of two 2-cycles.
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https://en.wikipedia.org/wiki/Adjacent_transposition
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In mathematics, and in particular in mathematical analysis, the Gagliardo–Nirenberg interpolation inequality is a result in the theory of Sobolev spaces that relates the L p {\displaystyle L^{p}} -norms of different weak derivatives of a function through an interpolation inequality. The theorem is of particular importance in the framework of elliptic partial differential equations and was originally formulated by Emilio Gagliardo and Louis Nirenberg in 1958. The Gagliardo-Nirenberg inequality has found numerous applications in the investigation of nonlinear partial differential equations, and has been generalized to fractional Sobolev spaces by Haim Brezis and Petru Mironescu in the late 2010s.
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https://en.wikipedia.org/wiki/Gagliardo–Nirenberg_interpolation_inequality
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In mathematics, and in particular in the field of algebra, a Hilbert–Poincaré series (also known under the name Hilbert series), named after David Hilbert and Henri Poincaré, is an adaptation of the notion of dimension to the context of graded algebraic structures (where the dimension of the entire structure is often infinite). It is a formal power series in one indeterminate, say t {\displaystyle t} , where the coefficient of t n {\displaystyle t^{n}} gives the dimension (or rank) of the sub-structure of elements homogeneous of degree n {\displaystyle n} . It is closely related to the Hilbert polynomial in cases when the latter exists; however, the Hilbert–Poincaré series describes the rank in every degree, while the Hilbert polynomial describes it only in all but finitely many degrees, and therefore provides less information. In particular the Hilbert–Poincaré series cannot be deduced from the Hilbert polynomial even if the latter exists. In good cases, the Hilbert–Poincaré series can be expressed as a rational function of its argument t {\displaystyle t} .
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https://en.wikipedia.org/wiki/Hilbert–Poincaré_series
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In mathematics, and in particular in the mathematical background of string theory, the Goddard–Thorn theorem (also called the no-ghost theorem) is a theorem describing properties of a functor that quantizes bosonic strings. It is named after Peter Goddard and Charles Thorn. The name "no-ghost theorem" stems from the fact that in the original statement of the theorem, the natural inner product induced on the output vector space is positive definite. Thus, there were no so-called ghosts (Pauli–Villars ghosts), or vectors of negative norm. The name "no-ghost theorem" is also a word play on the no-go theorem of quantum mechanics.
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https://en.wikipedia.org/wiki/No-ghost_theorem
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In mathematics, and in particular in the theory of solitons, the Dym equation (HD) is the third-order partial differential equation u t = u 3 u x x x . {\displaystyle u_{t}=u^{3}u_{xxx}.\,} It is often written in the equivalent form for some function v of one space variable and time v t = ( v − 1 / 2 ) x x x . {\displaystyle v_{t}=(v^{-1/2})_{xxx}.\,} The Dym equation first appeared in Kruskal and is attributed to an unpublished paper by Harry Dym. The Dym equation represents a system in which dispersion and nonlinearity are coupled together.
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https://en.wikipedia.org/wiki/Dym_equation
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HD is a completely integrable nonlinear evolution equation that may be solved by means of the inverse scattering transform. It obeys an infinite number of conservation laws; it does not possess the Painlevé property. The Dym equation has strong links to the Korteweg–de Vries equation.
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https://en.wikipedia.org/wiki/Dym_equation
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C.S. Gardner, J.M.
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https://en.wikipedia.org/wiki/Dym_equation
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Greene, Kruskal and R.M. Miura applied to the solution of corresponding problem in Korteweg–de Vries equation. The Lax pair of the Harry Dym equation is associated with the Sturm–Liouville operator.
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https://en.wikipedia.org/wiki/Dym_equation
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The Liouville transformation transforms this operator isospectrally into the Schrödinger operator. Thus by the inverse Liouville transformation solutions of the Korteweg–de Vries equation are transformed into solutions of the Dym equation. An explicit solution of the Dym equation, valid in a finite interval, is found by an auto-Bäcklund transform u ( t , x ) = 2 / 3 . {\displaystyle u(t,x)=\left^{2/3}.}
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https://en.wikipedia.org/wiki/Dym_equation
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In mathematics, and in particular linear algebra, the Moore–Penrose inverse A + {\displaystyle A^{+}} of a matrix A {\displaystyle A} is the most widely known generalization of the inverse matrix. It was independently described by E. H. Moore in 1920, Arne Bjerhammar in 1951, and Roger Penrose in 1955. Earlier, Erik Ivar Fredholm had introduced the concept of a pseudoinverse of integral operators in 1903. When referring to a matrix, the term pseudoinverse, without further specification, is often used to indicate the Moore–Penrose inverse.
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https://en.wikipedia.org/wiki/Moore-Penrose_inverse
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The term generalized inverse is sometimes used as a synonym for pseudoinverse. A common use of the pseudoinverse is to compute a "best fit" (least squares) solution to a system of linear equations that lacks a solution (see below under § Applications). Another use is to find the minimum (Euclidean) norm solution to a system of linear equations with multiple solutions.
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https://en.wikipedia.org/wiki/Moore-Penrose_inverse
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The pseudoinverse facilitates the statement and proof of results in linear algebra. The pseudoinverse is defined and unique for all matrices whose entries are real or complex numbers. It can be computed using the singular value decomposition. In the special case where A {\displaystyle A} is a normal matrix (for example, a Hermitian matrix), the pseudoinverse A + {\displaystyle A^{+}} annihilates the kernel of A {\displaystyle A} and acts as a traditional inverse of A {\displaystyle A} on the subspace orthogonal to the kernel.
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https://en.wikipedia.org/wiki/Moore-Penrose_inverse
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In mathematics, and in particular model theory, a prime model is a model that is as simple as possible. Specifically, a model P {\displaystyle P} is prime if it admits an elementary embedding into any model M {\displaystyle M} to which it is elementarily equivalent (that is, into any model M {\displaystyle M} satisfying the same complete theory as P {\displaystyle P} ).
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https://en.wikipedia.org/wiki/Prime_model
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In mathematics, and in particular modular representation theory, a decomposition matrix is a matrix that results from writing the irreducible ordinary characters in terms of the irreducible modular characters, where the entries of the two sets of characters are taken to be over all conjugacy classes of elements of order coprime to the characteristic of the field. All such entries in the matrix are non-negative integers. The decomposition matrix, multiplied by its transpose, forms the Cartan matrix, listing the composition factors of the projective modules.
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https://en.wikipedia.org/wiki/Decomposition_matrix
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In mathematics, and in particular ordinary differential equations, a Green's matrix helps to determine a particular solution to a first-order inhomogeneous linear system of ODEs. The concept is named after George Green. For instance, consider x ′ = A ( t ) x + g ( t ) {\displaystyle x'=A(t)x+g(t)\,} where x {\displaystyle x\,} is a vector and A ( t ) {\displaystyle A(t)\,} is an n × n {\displaystyle n\times n\,} matrix function of t {\displaystyle t\,} , which is continuous for t ∈ I , a ≤ t ≤ b {\displaystyle t\in I,a\leq t\leq b\,} , where I {\displaystyle I\,} is some interval. Now let x 1 ( t ) , … , x n ( t ) {\displaystyle x^{1}(t),\ldots ,x^{n}(t)\,} be n {\displaystyle n\,} linearly independent solutions to the homogeneous equation x ′ = A ( t ) x {\displaystyle x'=A(t)x\,} and arrange them in columns to form a fundamental matrix: X ( t ) = .
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https://en.wikipedia.org/wiki/Green's_matrix
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{\displaystyle X(t)=\left.\,} Now X ( t ) {\displaystyle X(t)\,} is an n × n {\displaystyle n\times n\,} matrix solution of X ′ = A X {\displaystyle X'=AX\,} . This fundamental matrix will provide the homogeneous solution, and if added to a particular solution will give the general solution to the inhomogeneous equation. Let x = X y {\displaystyle x=Xy\,} be the general solution.
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https://en.wikipedia.org/wiki/Green's_matrix
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Now, x ′ = X ′ y + X y ′ = A X y + X y ′ = A x + X y ′ . {\displaystyle {\begin{aligned}x'&=X'y+Xy'\\&=AXy+Xy'\\&=Ax+Xy'.\end{aligned}}} This implies X y ′ = g {\displaystyle Xy'=g\,} or y = c + ∫ a t X − 1 ( s ) g ( s ) d s {\displaystyle y=c+\int _{a}^{t}X^{-1}(s)g(s)\,ds\,} where c {\displaystyle c\,} is an arbitrary constant vector. Now the general solution is x = X ( t ) c + X ( t ) ∫ a t X − 1 ( s ) g ( s ) d s .
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https://en.wikipedia.org/wiki/Green's_matrix
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{\displaystyle x=X(t)c+X(t)\int _{a}^{t}X^{-1}(s)g(s)\,ds.\,} The first term is the homogeneous solution and the second term is the particular solution. Now define the Green's matrix G 0 ( t , s ) = { 0 t ≤ s ≤ b X ( t ) X − 1 ( s ) a ≤ s < t . {\displaystyle G_{0}(t,s)={\begin{cases}0&t\leq s\leq b\\X(t)X^{-1}(s)&a\leq s
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https://en.wikipedia.org/wiki/Green's_matrix
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In mathematics, and in particular representation theory, Frobenius reciprocity is a theorem expressing a duality between the process of restricting and inducting. It can be used to leverage knowledge about representations of a subgroup to find and classify representations of "large" groups that contain them. It is named for Ferdinand Georg Frobenius, the inventor of the representation theory of finite groups.
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https://en.wikipedia.org/wiki/Frobenius_reciprocity
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In mathematics, and in particular singularity theory, an Ak singularity, where k ≥ 0 is an integer, describes a level of degeneracy of a function. The notation was introduced by V. I. Arnold. Let f: R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } be a smooth function. We denote by Ω ( R n , R ) {\displaystyle \Omega (\mathbb {R} ^{n},\mathbb {R} )} the infinite-dimensional space of all such functions.
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https://en.wikipedia.org/wiki/Ak_singularity
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Let diff ( R n ) {\displaystyle \operatorname {diff} (\mathbb {R} ^{n})} denote the infinite-dimensional Lie group of diffeomorphisms R n → R n , {\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{n},} and diff ( R ) {\displaystyle \operatorname {diff} (\mathbb {R} )} the infinite-dimensional Lie group of diffeomorphisms R → R . {\displaystyle \mathbb {R} \to \mathbb {R} .} The product group diff ( R n ) × diff ( R ) {\displaystyle \operatorname {diff} (\mathbb {R} ^{n})\times \operatorname {diff} (\mathbb {R} )} acts on Ω ( R n , R ) {\displaystyle \Omega (\mathbb {R} ^{n},\mathbb {R} )} in the following way: let φ: R n → R n {\displaystyle \varphi :\mathbb {R} ^{n}\to \mathbb {R} ^{n}} and ψ: R → R {\displaystyle \psi :\mathbb {R} \to \mathbb {R} } be diffeomorphisms and f: R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } any smooth function.
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https://en.wikipedia.org/wiki/Ak_singularity
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We define the group action as follows: ( φ , ψ ) ⋅ f := ψ ∘ f ∘ φ − 1 {\displaystyle (\varphi ,\psi )\cdot f:=\psi \circ f\circ \varphi ^{-1}} The orbit of f , denoted orb(f), of this group action is given by orb ( f ) = { ψ ∘ f ∘ φ − 1: φ ∈ diff ( R n ) , ψ ∈ diff ( R ) } . {\displaystyle {\mbox{orb}}(f)=\{\psi \circ f\circ \varphi ^{-1}:\varphi \in {\mbox{diff}}(\mathbb {R} ^{n}),\psi \in {\mbox{diff}}(\mathbb {R} )\}\ .} The members of a given orbit of this action have the following fact in common: we can find a diffeomorphic change of coordinate in R n {\displaystyle \mathbb {R} ^{n}} and a diffeomorphic change of coordinate in R {\displaystyle \mathbb {R} } such that one member of the orbit is carried to any other.
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https://en.wikipedia.org/wiki/Ak_singularity
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A function f is said to have a type Ak-singularity if it lies in the orbit of f ( x 1 , … , x n ) = 1 + ε 1 x 1 2 + ⋯ + ε n − 1 x n − 1 2 ± x n k + 1 {\displaystyle f(x_{1},\ldots ,x_{n})=1+\varepsilon _{1}x_{1}^{2}+\cdots +\varepsilon _{n-1}x_{n-1}^{2}\pm x_{n}^{k+1}} where ε i = ± 1 {\displaystyle \varepsilon _{i}=\pm 1} and k ≥ 0 is an integer. By a normal form we mean a particularly simple representative of any given orbit. The above expressions for f give normal forms for the type Ak-singularities. The type Ak-singularities are special because they are amongst the simple singularities, this means that there are only a finite number of other orbits in a sufficiently small neighbourhood of the orbit of f. This idea extends over the complex numbers where the normal forms are much simpler; for example: there is no need to distinguish εi = +1 from εi = −1.
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https://en.wikipedia.org/wiki/Ak_singularity
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In mathematics, and in particular the necklace splitting problem, the Hobby–Rice theorem is a result that is useful in establishing the existence of certain solutions. It was proved in 1965 by Charles R. Hobby and John R. Rice; a simplified proof was given in 1976 by A. Pinkus.
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https://en.wikipedia.org/wiki/Hobby–Rice_theorem
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In mathematics, and in particular the study of Hilbert spaces, a crinkled arc is a type of continuous curve. The concept is usually credited to Paul Halmos. Specifically, consider f: → X , {\displaystyle f\colon \to X,} where X {\displaystyle X} is a Hilbert space with inner product ⟨ ⋅ , ⋅ ⟩ . {\displaystyle \langle \cdot ,\cdot \rangle .} We say that f {\displaystyle f} is a crinkled arc if it is continuous and possesses the crinkly property: if 0 ≤ a < b ≤ c < d ≤ 1 {\displaystyle 0\leq a
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https://en.wikipedia.org/wiki/Crinkled_arc
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In mathematics, and in particular the study of Weierstrass elliptic functions, the equianharmonic case occurs when the Weierstrass invariants satisfy g2 = 0 and g3 = 1. This page follows the terminology of Abramowitz and Stegun; see also the lemniscatic case. (These are special examples of complex multiplication.) In the equianharmonic case, the minimal half period ω2 is real and equal to Γ 3 ( 1 / 3 ) 4 π {\displaystyle {\frac {\Gamma ^{3}(1/3)}{4\pi }}} where Γ {\displaystyle \Gamma } is the Gamma function.
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https://en.wikipedia.org/wiki/Equianharmonic
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The half period is ω 1 = 1 2 ( − 1 + 3 i ) ω 2 . {\displaystyle \omega _{1}={\tfrac {1}{2}}(-1+{\sqrt {3}}i)\omega _{2}.} Here the period lattice is a real multiple of the Eisenstein integers.
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https://en.wikipedia.org/wiki/Equianharmonic
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The constants e1, e2 and e3 are given by e 1 = 4 − 1 / 3 e ( 2 / 3 ) π i , e 2 = 4 − 1 / 3 , e 3 = 4 − 1 / 3 e − ( 2 / 3 ) π i . {\displaystyle e_{1}=4^{-1/3}e^{(2/3)\pi i},\qquad e_{2}=4^{-1/3},\qquad e_{3}=4^{-1/3}e^{-(2/3)\pi i}.} The case g2 = 0, g3 = a may be handled by a scaling transformation.
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https://en.wikipedia.org/wiki/Equianharmonic
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In mathematics, and in particular the study of algebra, an Akivis algebra is a nonassociative algebra equipped with a binary operator, the commutator {\displaystyle } and a ternary operator, the associator {\displaystyle } that satisfy a particular relationship known as the Akivis identity. They are named in honour of Russian mathematician Maks A. Akivis. Formally, if A {\displaystyle A} is a vector space over a field F {\displaystyle \mathbb {F} } of characteristic zero, we say A {\displaystyle A} is an Akivis algebra if the operation ( x , y ) ↦ {\displaystyle \left(x,y\right)\mapsto \left} is bilinear and anticommutative; and the trilinear operator ( x , y , z ) ↦ {\displaystyle \left(x,y,z\right)\mapsto \left} satisfies the Akivis identity: , z ] + , x ] + , y ] = + + − − − .
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https://en.wikipedia.org/wiki/Akivis_algebra
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