Bogomol'nyi–Prasad–Sommerfield Bound

	Bogomol'nyi–Prasad–Sommerfield Bound The Bogomol'nyi–Prasad–Sommerfield bound (named after Evgeny Bogomolny, M.K. Prasad, and Charles Sommerfield) is a series of inequalities for solutions of partial differential equations depending on the homotopy class of the solution at infinity. This set of inequalities is very useful for solving soliton equations. Often, by insisting that the bound be satisfied (called "saturated"), one can come up with a simpler set of partial differential equations to solve the Bogomolny equations. Solutions saturating the bound are called "BPS states" and play an important role in field theory and string theory. Example In a theory of non-abelian Yang–Mills–Higgs, the energy at a given time ''t'' is given by :E=\int d^3x\, \left frac\pi^T \pi + V(\varphi) + \frac\operatorname\left[\vec\cdot\vec+\vec\cdot\vec\rightright] where \pi is the covariant derivative of the Higgs field and ''V'' is the potential. If we assume that ''V'' is nonnegative and is zero only for the Higgs vacu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Evgeny Bogomolny Yevgeni, Yevgeny, Yevgenii or Yevgeniy (russian: Евгений), also transliterated as Evgeni, Evgeny, Evgenii or Evgeniy, is the Russian form of the masculine given name Eugene. People with the name include: :''Note: Occasionally, a person may be in more than one section.'' Arts and entertainment * Yevgeny Aryeh (1947–2022), Israeli theater director, playwright, scriptwriter and set designer Yevgeni Bauer (1865–1917), Russian film director and screenwriter Yevgeni Grishkovetz (born 1967), Russian writer, dramatist, stage director and actor Evgeny Kissin (born 1971), Russian pianist Yevgeny Leonov (1926–1994), Soviet and Russian actor Yevgeni Mokhorev (born 1967), Russian photographer Evgeny Mravinsky (1903–1988), Russian conductor Evgeny Svetlanov (1928–2002), Russian conductor Yevgeni Urbansky (1932–1965), Soviet Russian actor Yevgeniy Yevstigneyev (1926–1992), Soviet and Russian actor Yevgeny Yevtushenko (1933–2017), Soviet and Russian poet Yevgeny ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]
	Covariant Derivative In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component (dependent on the embedding) and the intrinsic covariant derivative component. The name is motivated by the importance of changes of coordinate in physics: the covariant derivative transforms covariantly under a general coordinate transformation, that is, linearly via the Jacobia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Partial Differential Equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the many open questions are the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Topological Charge In physics, a topological quantum number (also called topological charge) is any quantity, in a physical theory, that takes on only one of a discrete set of values, due to topology, topological considerations. Most commonly, topological quantum numbers are topological invariants associated with topological defects or soliton-type solutions of some set of differential equations modeling a physical system, as the solitons themselves owe their stability to topological considerations. The specific "topological considerations" are usually due to the appearance of the fundamental group or a higher-dimensional homotopy group in the description of the problem, quite often because the boundary, on which the boundary conditions are specified, has a non-trivial homotopy group that is preserved by the differential equations. The topological quantum number of a solution is sometimes called the winding number of the solution, or, more precisely, it is the degree of a continuous mapping. Recent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Group Extension In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence :1\to N\;\overset\;G\;\overset\;Q \to 1. If G is an extension of Q by N, then G is a group, \iota(N) is a normal subgroup of G and the quotient group G/\iota(N) is isomorphic to the group Q. Group extensions arise in the context of the extension problem, where the groups Q and N are known and the properties of G are to be determined. Note that the phrasing "G is an extension of N by Q" is also used by some. Since any finite group G possesses a maximal normal subgroup N with simple factor group G/N, all finite groups may be constructed as a series of extensions with finite simple groups. This fact was a motivation for completing the classification of finite simple groups. An extension is called a central extension if the subgroup N lies in the center o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Magnetic Flux In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B over that surface. It is usually denoted or . The SI unit of magnetic flux is the weber (Wb; in derived units, volt–seconds), and the CGS unit is the maxwell. Magnetic flux is usually measured with a fluxmeter, which contains measuring coils and electronics, that evaluates the change of voltage in the measuring coils to calculate the measurement of magnetic flux. Description The magnetic interaction is described in terms of a vector field, where each point in space is associated with a vector that determines what force a moving charge would experience at that point (see Lorentz force). Since a vector field is quite difficult to visualize at first, in elementary physics one may instead visualize this field with field lines. The magnetic flux through some surface, in this simplified picture, is proportional to the num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Adjoint Representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL(n, \mathbb), the Lie group of real ''n''-by-''n'' invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible ''n''-by-''n'' matrix g to an endomorphism of the vector space of all linear transformations of \mathbb^n defined by: x \mapsto g x g^ . For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action of ''G'' on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields. Definition Let ''G'' be a Lie group, and let :\Psi: G \to \operatorname(G) be the mapping , with Aut(''G'') the automorphism group of ''G'' and given by the inner automorphism (conjugation) :\Psi_g(h)= ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Higgs Field The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the quantum excitation of the Higgs field, one of the fields in particle physics theory. In the Standard Model, the Higgs particle is a massive scalar boson with zero spin, even (positive) parity, no electric charge, and no colour charge, that couples to (interacts with) mass. It is also very unstable, decaying into other particles almost immediately. The Higgs field is a scalar field, with two neutral and two electrically charged components that form a complex doublet of the weak isospin SU(2) symmetry. Its " Mexican hat-shaped" potential leads it to take a nonzero value ''everywhere'' (including otherwise empty space), which breaks the weak isospin symmetry of the electroweak interaction, and via the Higgs mechanism gives mass to many particles. Both the field and the boson are named after physicist Peter Higgs, who in 1964, alo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Yang–Mills–Higgs Equations In mathematics, the Yang–Mills–Higgs equations are a set of non-linear partial differential equations for a Yang–Mills field, given by a connection, and a Higgs field, given by a section of a vector bundle (specifically, the adjoint bundle). These equations are :\begin D_AF_A + Phi, D_A\Phi&= 0, \\ D_AD_A\Phi &= 0 \end with a boundary condition :\lim_, \Phi, (x) = 1 where : ''A'' is a connection on a vector bundle, : ''D'' is the exterior covariant derivative, : ''F'' is the curvature of that connection, : Φ is a section of that vector bundle, : ∗ is the Hodge star, and : �,·is the natural, graded bracket. These equations are named after Chen Ning Yang, Robert Mills, and Peter Higgs. They are very closely related to the Ginzburg–Landau equations, when these are expressed in a general geometric setting. M.V. Goganov and L.V. Kapitanskii have shown that the Cauchy problem for hyperbolic Yang–Mills–Higgs equations in Hamiltonian gauge on 4-dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Charles M Charles is a masculine given name predominantly found in English and French speaking countries. It is from the French form ''Charles'' of the Proto-Germanic name (in runic alphabet) or ''karilaz'' (in Latin alphabet), whose meaning was "free man". The Old English descendant of this word was '' Ċearl'' or ''Ċeorl'', as the name of King Cearl of Mercia, that disappeared after the Norman conquest of England. The name was notably borne by Charlemagne (Charles the Great), and was at the time Latinized as ''Karolus'' (as in ''Vita Karoli Magni''), later also as '' Carolus''. Some Germanic languages, for example Dutch and German, have retained the word in two separate senses. In the particular case of Dutch, ''Karel'' refers to the given name, whereas the noun ''kerel'' means "a bloke, fellow, man". Etymology The name's etymology is a Common Germanic noun ''karilaz'' meaning "free man", which survives in English as churl (< Old English ''ċeorl''), which developed its dep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	String Theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its mass, charge, and other properties determined by the vibrational state of the string. In string theory, one of the many vibrational states of the string corresponds to the graviton, a quantum mechanical particle that carries the gravitational force. Thus, string theory is a theory of quantum gravity. String theory is a broad and varied subject that attempts to address a number of deep questions of fundamental physics. String theory has contributed a number of advances to mathematical physics, which have been applied to a variety of problems in black hole physics, early universe cosmology, nuclear physics, and conde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	BPS State BPS, Bps or bps may refer to: Science and mathematics Plural of bp, base pair, a measure of length of DNA Plural of bp, basis point, one one-hundredth of a percentage point - ‱ Battered person syndrome, a physical and psychological condition found in victims of abuse Best practice statement, a qualification of a method used in guidelines documents Bisphenol S, an organic chemical compound Bladder pain syndrome, a disorder characterised by pain associated with urination Bogomol'nyi–Prasad–Sommerfield bound, a mathematical concept in field and string theory Bogomol'nyi–Prasad–Sommerfield state, solutions saturating the BPS bound * BPS domain, a protein domain * Bronchopulmonary sequestration, where a section of lung tissue has a decreased blood supply Bovine papular stomatitis, a zoonotic farmyard pox Computing IBM Basic Programming Support, BPS/360 Bits per second (bps), a data rate unit Bytes per second (Bps), a data rate unit Bits per sample (bps), referri ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]