75 results on '"David B. Fairlie"'
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2. Interconnections among nonlinear field equations
- Author
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David B. Fairlie
- Subjects
Statistics and Probability ,Physics ,Integrable system ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,First order ,Nonlinear system ,symbols.namesake ,Nonlinear wave equation ,Modeling and Simulation ,symbols ,Field equation ,Mathematical Physics ,Lagrangian ,Mathematical physics - Abstract
Galileons are related to other implicitly integrable equations such as the first order nonlinear wave equation and the universal field equation, which is a Lagrangian for the Galileon.
- Published
- 2020
3. Geons of galileons
- Author
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David B. Fairlie and Thomas Curtright
- Subjects
High Energy Physics - Theory ,Physics ,Nuclear and High Energy Physics ,Cosmology and Nongalactic Astrophysics (astro-ph.CO) ,Spacetime ,010308 nuclear & particles physics ,Scalar (physics) ,FOS: Physical sciences ,Astrophysics - Astrophysics of Galaxies ,01 natural sciences ,Gravitation ,Theoretical physics ,General Relativity and Quantum Cosmology ,Classical mechanics ,High Energy Physics - Theory (hep-th) ,Astrophysics of Galaxies (astro-ph.GA) ,0103 physical sciences ,010306 general physics ,Astrophysics - Cosmology and Nongalactic Astrophysics - Abstract
We suggest that galileon theories should have an additional self-coupling of the fields to the trace of their own energy-momentum tensor. We explore the classical features of one such model, in flat 4D spacetime, with emphasis on solutions that are scalar analogues of gravitational geons. We discuss the stability of these scalar geons, and some of their possible signatures, including shock fronts., Comment: References added in v2
- Published
- 2012
- Full Text
- View/download PDF
4. Series Solution of the 1 + 2 Continuous Toda Chain
- Author
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R. Torres-Cordoba, David B. Fairlie, and A. N. Leznov
- Subjects
Series method ,General theory ,Series (mathematics) ,Chain (algebraic topology) ,Applied mathematics ,Symmetry (physics) ,Variable (mathematics) ,Mathematics - Abstract
The 1+2 dimensional continuous Toda chain presents a formidable challenge to the construction of solutions. Two variable reductions of the equation are known, but up till now nothing more is known. In this paper a way to solve the equation is presented, and some solutions are explicitly constructed. The method depends upon connecting series solutions of the symmetry equation of the Toda chain with the Toda solutions, as is guaranteed by general theory. Such solutions of the symmetry equation are obtainable a series method. An explicit solution is constructed by this method, and a general proce- dure is given to realize further solutions, which, however are given only in implicit form.
- Published
- 2012
5. Ternary Virasoro–Witt algebra
- Author
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David B. Fairlie, Cosmas K. Zachos, and Thomas Curtright
- Subjects
High Energy Physics - Theory ,Symmetric algebra ,Physics ,Nuclear and High Energy Physics ,010308 nuclear & particles physics ,Mathematics::Rings and Algebras ,FOS: Physical sciences ,Universal enveloping algebra ,Witt algebra ,Mathematical Physics (math-ph) ,01 natural sciences ,Filtered algebra ,Algebra ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,High Energy Physics - Theory (hep-th) ,Mathematics::Quantum Algebra ,Differential graded algebra ,0103 physical sciences ,Algebra representation ,Cellular algebra ,Virasoro algebra ,010306 general physics ,Mathematics::Symplectic Geometry ,Mathematical Physics - Abstract
A 3-bracket variant of the Virasoro-Witt algebra is constructed through the use of su(1,1) enveloping algebra techniques. The Leibniz rules for 3-brackets acting on other 3-brackets in the algebra are discussed and verified in various situations., Comment: 6 pages, LateX
- Published
- 2008
6. Lie algebras associated with one-dimensional aperiodic point sets
- Author
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Cosmas K. Zachos, Reidun Twarock, and David B. Fairlie
- Subjects
High Energy Physics - Theory ,Pure mathematics ,Multiplicative function ,FOS: Physical sciences ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Type (model theory) ,Set (abstract data type) ,High Energy Physics - Theory (hep-th) ,Aperiodic graph ,Product (mathematics) ,Lie algebra ,Point (geometry) ,Mathematical Physics ,Associative property ,Mathematics - Abstract
The set of points of a one-dimensional cut-and-project quasicrystal or model set, while not additive, is shown to be multiplicative for appropriate choices of acceptance windows. This leads to the definition of an associative additive graded composition law and permits the introduction of Lie algebras over such aperiodic point sets. These infinite dimensional Lie algebras are shown to be representatives of a new type of semi-direct product induced Lie algebras., Comment: 13 pages
- Published
- 2006
7. Morphing quantum mechanics and fluid dynamics
- Author
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David B. Fairlie and Thomas Curtright
- Subjects
High Energy Physics - Theory ,35Q35 ,76D99 ,FOS: Physical sciences ,General Physics and Astronomy ,01 natural sciences ,Schrödinger equation ,symbols.namesake ,Mathematics - Analysis of PDEs ,0103 physical sciences ,FOS: Mathematics ,Fluid dynamics ,010306 general physics ,Mathematical Physics ,Pressure gradient ,Physics ,Quantum Physics ,010308 nuclear & particles physics ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Morphing ,Formalism (philosophy of mathematics) ,Classical mechanics ,High Energy Physics - Theory (hep-th) ,symbols ,Quantum Physics (quant-ph) ,Hydrodynamic flow ,Analysis of PDEs (math.AP) - Abstract
We investigate the effects of given pressure gradients on hydrodynamic flow equations. We obtain results in terms of implicit solutions and also in the framework of an extra-dimensional formalism involving the diffusion/Schrodinger equation., Comment: Examples involving shocks and references added
- Published
- 2003
8. Extra dimensions and non-linear equations
- Author
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David B. Fairlie
- Subjects
Overdetermined system ,Extra dimensions ,Nonlinear system ,Underdetermined system ,Simultaneous equations ,Independent equation ,Quantum mechanics ,Mathematical analysis ,General Physics and Astronomy ,Space (mathematics) ,System of linear equations ,Mathematics - Abstract
It is shown how integrable non-linear equations of hydrodynamic type, which arise as continuations of Dirac-Born-Infeld equations to the case where the target space is of lower dimension than the base space, are associated with linear diffusion equations in a space of double the number of space co-ordinates.
- Published
- 2003
9. A Compact Formula for Rotations as Spin Matrix Polynomials
- Author
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David B. Fairlie, Thomas Curtright, and Cosmas K. Zachos
- Subjects
High Energy Physics - Theory ,Pure mathematics ,Pauli matrices ,FOS: Physical sciences ,01 natural sciences ,symbols.namesake ,0103 physical sciences ,0101 mathematics ,010306 general physics ,Mathematical Physics ,Special unitary group ,Mathematics ,Quantum Physics ,Group (mathematics) ,010102 general mathematics ,Mathematical Physics (math-ph) ,Rotation matrix ,Algebra ,Spin representation ,High Energy Physics - Theory (hep-th) ,Representation theory of SU ,symbols ,Geometry and Topology ,Quantum Physics (quant-ph) ,Analysis ,Group theory ,Rotation group SO - Abstract
Group elements of SU(2) are expressed in closed form as finite polynomials of the Lie algebra generators, for all definite spin representations of the rotation group. The simple explicit result exhibits connections between group theory, combinatorics, and Fourier analysis, especially in the large spin limit. Salient intuitive features of the formula are illustrated and discussed.
- Published
- 2014
10. The complex Bateman equation in a space of arbitrary dimension
- Author
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David B. Fairlie and A. N. Leznov
- Subjects
Partial differential equation ,Differential equation ,Weak solution ,Mathematical analysis ,First-order partial differential equation ,Characteristic equation ,Statistical and Nonlinear Physics ,Parabolic partial differential equation ,Physics::History of Physics ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Elliptic partial differential equation ,Hyperbolic partial differential equation ,Mathematical Physics ,Mathematics - Abstract
The general solution to the complex Bateman equation in a space of arbitrary dimensions is constructed.
- Published
- 2001
11. Dirac–Born–Infeld equations
- Author
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David B. Fairlie
- Subjects
High Energy Physics - Theory ,Physics ,Nuclear and High Energy Physics ,Dirac (software) ,Zero (complex analysis) ,FOS: Physical sciences ,Equations of motion ,General Relativity and Quantum Cosmology (gr-qc) ,Space (mathematics) ,String (physics) ,General Relativity and Quantum Cosmology ,High Energy Physics::Theory ,symbols.namesake ,High Energy Physics - Theory (hep-th) ,symbols ,Constant (mathematics) ,Mathematics::Symplectic Geometry ,Lagrangian ,Mathematical physics - Abstract
Properties of the Dirac-Born-Infeld Lagrangian analogous to those of the Nambu-Goto String are analysed. In particular the Lagrangian is shown to be constant or zero on the space of solutions of the equations of motion if the Lagrangian is taken to any power other than 1/2., Comment: 9 pages, LaTeX
- Published
- 1999
12. Moyal brackets, star products and the generalised Wigner function
- Author
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David B. Fairlie
- Subjects
High Energy Physics - Theory ,Quark ,Physics ,General Mathematics ,Applied Mathematics ,FOS: Physical sciences ,General Physics and Astronomy ,Bilinear interpolation ,Statistical and Nonlinear Physics ,Star (graph theory) ,Space (mathematics) ,Object (computer science) ,Convolution ,High Energy Physics - Theory (hep-th) ,Probability theory ,Wigner distribution function ,Mathematical physics - Abstract
The Wigner-Weyl- Moyal approach to Quantum Mechanics is recalled, and similarities to classical probability theory emphasised. The Wigner distribution function is generalised and viewed as a construction of a bosonic object, a target space co-ordinate, for example, in terms of a bilinear convolution of two fermionic objects, e.g. a quark antiquark pair. This construction is essentially non-local, generalising the idea of a local current. Such Wigner functions are shown to solve a BPS generalised Moyal-Nahm equation., 7 pages, LaTeX, to appear in special issue of the J. of Chaos, Solitons and Fractals
- Published
- 1999
13. [Untitled]
- Author
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David B. Fairlie and A. N. Leznov
- Subjects
Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Functional Relationship ,Complex system ,Applied mathematics ,Statistical and Nonlinear Physics ,Function (mathematics) ,Quantum field theory ,Special case ,Physics::History of Physics ,Mathematical Physics ,Mathematics ,Mathematical physics - Abstract
The general solution to the complex Bateman equation is constructed. It is given in implicit form in terms of a functional relationship for the unknown function. The known solution of the usual Bateman equation is recovered as a special case.
- Published
- 1999
14. Integrable top equations associated with projective geometry over
- Author
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Tatsuya Ueno and David B. Fairlie
- Subjects
Pure mathematics ,Series (mathematics) ,Integrable system ,Generalization ,Riemann surface ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Computer Science::Computers and Society ,Nonlinear Sciences::Chaotic Dynamics ,symbols.namesake ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Genus (mathematics) ,Physics::Atomic and Molecular Clusters ,Euler's formula ,symbols ,Mathematical Physics ,Mathematics ,Projective geometry - Abstract
We give a series of integrable top equations associated with the projective geometry over as a -dimensional generalization of the three-dimensional Euler top equations. The general solution of the -dimensional top is shown to be given by an integration over a Riemann surface with genus .
- Published
- 1998
15. MOYAL BRACKETS IN M-THEORY
- Author
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David B. Fairlie
- Subjects
High Energy Physics - Theory ,M-theory ,Physics ,Nuclear and High Energy Physics ,Spinor ,Existential quantification ,FOS: Physical sciences ,General Physics and Astronomy ,Astronomy and Astrophysics ,Space (mathematics) ,High Energy Physics - Theory (hep-th) ,Phase space ,Limit (mathematics) ,Representation (mathematics) ,Mathematical physics - Abstract
The infinite limit of Matrix Theory in 4 and 10 dimensions is described in terms of Moyal Brackets. In those dimensions there exists a Bogomol'nyi bound to the Euclideanized version of these equations, which guarantees that solutions of the first order equations also solve the second order Matrix Theory equations. A general construction of such solutions in terms of a representation of the target space co-ordinates as non-local spinor bilinears, which are generalisations of the standard Wigner functions on phase space, is given., Comment: 10 pages, Latex, no figures. References altered, typos corrected
- Published
- 1998
16. Equations with an infinite number of explicit conservation laws
- Author
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David B. Fairlie
- Subjects
Conservation law ,Class (set theory) ,Infinite set ,Polynomial ,Differential equation ,General Mathematics ,Multivariable calculus ,Applied mathematics ,Charge (physics) ,Generating function (physics) ,Mathematics - Abstract
A large class of first-order partial nonlinear differential equations in two independent variables which possess an infinite set of polynomial conservation laws derived from an explicit generating function is constructed. The conserved charge densities are all homogeneous polynomials in the unknown functions which satisfy the differential equations in question. The simplest member of the class of equations is related to the Born–Infeld Equation in two dimensions. It is observed that some members of this class possess identical charge densities. This enables the construction of a set of multivariable equations with an infinite number of conservation laws.
- Published
- 1997
17. The algebraic and Hamiltonian structure of the dispersionless Benney and Toda hierarchies
- Author
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Ian A. B. Strachan and David B. Fairlie
- Subjects
High Energy Physics - Theory ,Physics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Logarithm ,Applied Mathematics ,FOS: Physical sciences ,Computer Science Applications ,Theoretical Computer Science ,symbols.namesake ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Corollary ,High Energy Physics - Theory (hep-th) ,Hamiltonian structure ,Signal Processing ,symbols ,Exactly Solvable and Integrable Systems (nlin.SI) ,Algebraic number ,Hypergeometric function ,Hamiltonian (quantum mechanics) ,Legendre polynomials ,Mathematical Physics ,Mathematical physics - Abstract
The algebraic and Hamiltonian structures of the multicomponent dispersionless Benney and Toda hierarchies are studied. This is achieved by using a modified set of variables for which there is a symmetry between the basic fields. This symmetry enables formulae normally given implicitly in terms of residues, such as conserved charges and fluxes, to be calculated explicitly. As a corollary of these results the equivalence of the Benney and Toda hierarchies is established. It is further shown that such quantities may be expressed in terms of generalized hypergeometric functions, the simplest example involving Legendre polynomials. These results are then extended to systems derived from a rational Lax function and a logarithmic function. Various reductions are also studied., Comment: 29 pages, LaTeX
- Published
- 1996
18. The integrable mapping as the discrete group of inner symmetry of integrable systems
- Author
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A. N. Leznov and David B. Fairlie
- Subjects
Physics ,Integrable system ,Discrete group ,General Physics and Astronomy ,Locally integrable function ,Invariant (mathematics) ,Mathematical physics - Abstract
It is shown that each integrable mapping is connected with a hierarchical completely integrable sytem of equations of evolution type which are invariant with respect to the transformation described by this mapping.
- Published
- 1995
19. A fresh look at generalized Veneziano amplitudes
- Author
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David B. Fairlie and Jean Nuyts
- Subjects
High Energy Physics - Theory ,Physics ,Nuclear and High Energy Physics ,Dual resonance model ,Logarithm ,FOS: Physical sciences ,Duality (optimization) ,String theory ,Veneziano amplitude ,Scattering amplitude ,High Energy Physics::Theory ,Amplitude ,High Energy Physics - Theory (hep-th) ,Finite set ,Mathematical physics - Abstract
The dual resonance model, which was a precursor of string theory was based upon the idea that two-particle scattering amplitudes should be expressible equivalently as a sum of contributions of an infinite number of $s$ channel poles each corresponding to a finite number of particles with definite spin, or as a similar sum of $t$ channel poles. The famous example of Veneziano \cite{ven} satisfies all these requirements, and is additionally ghost free.We recall other trajectories which provide solutions to the duality constraints, e.g. the general Mobi\"us trajectories and the logarithmic trajectories, which were thought to be lacking this last feature. We however demonstrate, partly empirically, the existence of a regime within a particular deformation of the Veneziano amplitude for logarithmic trajectories for which the amplitude remains ghost free., Comment: 18 pages, LaTeX ,DTP/94/19
- Published
- 1995
20. Integrable Systems in Higher Dimensions
- Author
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David B. Fairlie
- Subjects
Physics ,Physics and Astronomy (miscellaneous) ,Integrable system ,Mathematical physics - Published
- 1995
21. On certain quantum deformations of gl(N,R)
- Author
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David B. Fairlie and Jean Nuyts
- Subjects
High Energy Physics - Theory ,Physics ,Pure mathematics ,FOS: Physical sciences ,Creation and annihilation operators ,Lie group ,Statistical and Nonlinear Physics ,General linear group ,Type (model theory) ,High Energy Physics - Theory (hep-th) ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Limit (mathematics) ,Quantum ,Mathematical Physics - Abstract
In this paper all deformations of the general linear group, subject to certain restrictions which in particular ensure a smooth passage to the Lie group limit, are obtained. Representations are given in terms of certains sets of creation and annihilation operators. These creation and annihilation operators may belong to a generalisation of the $q$-quark type or $q$-hadronic type, of $q$-boson or $q$-fermion type. We are also led to a natural definition of $q$-direct sums of q-algebras., Comment: 15 pages
- Published
- 1994
22. Comments on Galileons
- Author
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David B. Fairlie
- Subjects
High Energy Physics - Theory ,Statistics and Probability ,Kaluza–Klein theory ,Second order equation ,General Physics and Astronomy ,Equations of motion ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,General Relativity and Quantum Cosmology (gr-qc) ,Invariant (physics) ,Galilean transformation ,First order ,General Relativity and Quantum Cosmology ,symbols.namesake ,High Energy Physics - Theory (hep-th) ,Modeling and Simulation ,Fluid dynamics ,symbols ,Field equation ,Mathematical Physics ,Mathematics ,Mathematical physics - Abstract
The recent progress in the study of Galileons, i.e. equations of second order with an action invariant under a Galilean transformation is related to work on `Universal Field Equations' \cite{dbfgov} which are second order equations arising by an iterative procedure from arbitrary Lagrangians of weight one in their first derivatives. It is pointed out that the Galileon is simply a Kaluza-Klein reduction of a Universal Field Equation. An implicit solution to the equation of motion is presented, and a class of explicit solutions pointed out. The multi-field extensions of both types of equations are derived from a first order formalism, which is simply the substantive derivative of fluid dynamics., Comment: 10 pages, no figures
- Published
- 2011
- Full Text
- View/download PDF
23. Euler hierarchies and universal equations
- Author
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David B. Fairlie and Jan Govaerts
- Subjects
High Energy Physics - Theory ,Physics ,Pure mathematics ,Field (physics) ,Integrable system ,FOS: Physical sciences ,Equations of motion ,Duality (optimization) ,Statistical and Nonlinear Physics ,String (physics) ,symbols.namesake ,High Energy Physics - Theory (hep-th) ,Euler's formula ,symbols ,Field theory (psychology) ,Korteweg–de Vries equation ,Mathematical Physics - Abstract
Finite Euler hierarchies of field theory Lagrangians leading to universal equations of motion for new types of string and membrane theories and for {\it classical} topological field theories are constructed. The analysis uses two main ingredients. On the one hand, there exists a generic finite Euler hierarchy for one field leading to a universal equation which generalises the Plebanski equation of self-dual four dimensional gravity. On the other hand, specific maps are introduced between field theories which provide a ``triangular duality'' between certain classes of arbitrary field theories, classical topological field theories and generalised string and membrane theories. The universal equations, which derive from an infinity of inequivalent Lagrangians, are generalisations of certain reductions of the Plebanski and KdV equations, and could possibly define new integrable systems, thus in particular integrable membrane theories. Some classes of solutions are constructed in the general case. The general solution to some of the universal equations is given in the simplest cases., 42 p., Plain TeX, replaces previous unprintable version corrupted by mailer
- Published
- 1992
24. Neutral and charged quommutators with and without symmetries
- Author
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Jean Nuyts and David B. Fairlie
- Subjects
Theoretical physics ,Pure mathematics ,Physics and Astronomy (miscellaneous) ,Homogeneous space ,Creation and annihilation operators ,Commutation ,Operator theory ,Engineering (miscellaneous) ,Symmetry (physics) ,Mathematics - Abstract
We present two sets of qualgebras involving operators which generalize creation and annihilation operators. These two groups of operators satisfy separately quommutation relations rather than commutation or anticommutation relations. The quommutators of the creation and annihilation operators generate new “neutral operators” which themselves are subjected to quommutation relations. Two solutions are presented. In the second one, some new symmetry relations are added to the system. In a certain sense these extra relations, rather than imposing new constraints on the parameters, increase their freedom.
- Published
- 1992
25. The algebra of Weyl symmetrised polynomials and its quantum extension
- Author
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David B. Fairlie and I. M. Gelfand
- Subjects
Algebra ,Filtered algebra ,Symmetric algebra ,Algebra representation ,Division algebra ,Current algebra ,Cellular algebra ,Statistical and Nonlinear Physics ,Universal enveloping algebra ,Mathematical Physics ,Moyal bracket ,Mathematics - Abstract
The Algebra of Weyl symmetrised polynomials in powers of Hamiltonian operatorsP andQ which satisfy canonical commutation relations is constructed. This algebra is shown to encompass all recent infinite dimensional algebras acting on two-dimensional phase space. In particular the Moyal bracket algebra and the Poisson bracket algebra, of which the Moyal is the unique one parameter deformation are shown to be different aspects of this infinite algebra. We propose the introduction of a second deformation, by the replacement of the Heisenberg algebra forP, Q with aq-deformed commutator, and construct algebras ofq-symmetrised Polynomials.
- Published
- 1991
26. Multiparameter associative generalizations of canonical commutation relations and quantized planes
- Author
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David B. Fairlie and Cosmas K. Zachos
- Subjects
Physics ,Nuclear and High Energy Physics ,Quantization (physics) ,Hamiltonian formalism ,Quantum harmonic oscillator ,Quantum mechanics ,Quantum algebra ,Quantum ,Associative property ,Mathematical physics - Abstract
We explicitly deform classical oscillators to multiparameter quantum oscillators which do not commute with each other. These “anyonic” oscillators lead to the construction of a consistent quantum algebra GLq(N) with 1 2 (N−1)(N−2) dependent parameters, some of whose features and applications we discuss.
- Published
- 1991
27. A Coding of Real Null Four-Momenta into World-Sheet Co-ordinates
- Author
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David B. Fairlie
- Subjects
Physics ,High Energy Physics - Theory ,Article Subject ,QC1-999 ,Applied Mathematics ,Mathematical analysis ,FOS: Physical sciences ,General Physics and Astronomy ,ENCODE ,law.invention ,Ordinate ,High Energy Physics - Theory (hep-th) ,law ,Cartesian coordinate system ,Coding (social sciences) - Abstract
The results of minimizing the action for string-like systems on a simply-connected world sheet are shown to encode the Cartesian components of real null momentum four-vectors into co-ordinates on the world sheet. This identification arises consistently from different approaches to the problem., 9 pages, Latex2e
- Published
- 2008
28. Deformations and renormalisations ofW ∞
- Author
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J. Nuyts and David B. Fairlie
- Subjects
Subalgebra ,Closure (topology) ,Statistical and Nonlinear Physics ,Conformal map ,Extension (predicate logic) ,Physics::History of Physics ,Algebra ,Renormalization ,Limit (mathematics) ,Mathematical Physics ,Moyal bracket ,Mathematical physics ,Mathematics ,Spin-½ - Abstract
Deformations of the infiniteN limit of the ZamolodchikovWN algebra are discussed. A recent one, due to Pope, Romans and Shen with non-zero central extensions for every conformal spin is shown to be formally renormalisable to one representable in Moyal bracket form. Another deformation is discovered which, like the algebra of Pope et al. possesses automatic closure, but has non-zero central extension only in the Virasoro subalgebra.
- Published
- 1990
29. Matrix membranes and integrability
- Author
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David B. Fairlie, Cosmas K. Zachos, and Thomas Curtright
- Subjects
Physics ,Matrix (mathematics) ,symbols.namesake ,Commutator ,Poisson bracket ,Integrable system ,Lax pair ,Homogeneous space ,Euler's formula ,symbols ,Moyal bracket ,Mathematical physics - Abstract
This is a pedagogical digest of results reported in Phys Lett B405 (1997) 37, and an explicit implementation of Euler's construction for the solution of the Poisson Bracket dual Nahm equation. But it does not cover 9 and 10-dimensional systems, and subsequent progress on them [hep-th/9707190]. Cubic interactions are considered in 3 and 7 space dimensions, respectively, for bosonic membranes in Poisson Bracket form. Their symmetries and vacuum configurations are explored. Their associated first order equations are transformed to Nahm's equations, and are hence seen to be integrable, for the 3-dimensional case, by virtue of the explicit Lax pair provided. Most constructions introduced also apply to matrix commutator or Moyal Bracket analogs.
- Published
- 2007
30. An Atavistic Lie Algebra
- Author
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David B. Fairlie and Cosmas K. Zachos
- Subjects
High Energy Physics - Theory ,Physics ,Nuclear and High Energy Physics ,Pure mathematics ,Mathematics::Operator Algebras ,Simple Lie group ,Adjoint representation ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Killing form ,Affine Lie algebra ,Lie conformal algebra ,Graded Lie algebra ,Adjoint representation of a Lie algebra ,Representation of a Lie group ,High Energy Physics - Theory (hep-th) ,Mathematics::Quantum Algebra ,Mathematical Physics - Abstract
An infinite-dimensional Lie Algebra is proposed which includes, in its subalgebras and limits, most Lie Algebras routinely utilized in physics. It relies on the finite oscillator Lie group, and appears applicable to twisted noncommutative QFT and CFT., Comment: 9 pages, LateX2e
- Published
- 2006
- Full Text
- View/download PDF
31. Vertex Ring-Indexed Lie Algebras
- Author
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Cosmas K. Zachos and David B. Fairlie
- Subjects
Physics ,High Energy Physics - Theory ,Nuclear and High Energy Physics ,Pure mathematics ,Ring (mathematics) ,Non-associative algebra ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Affine Lie algebra ,Lie conformal algebra ,Graded Lie algebra ,Adjoint representation of a Lie algebra ,High Energy Physics - Theory (hep-th) ,Lie algebra ,Nest algebra ,Mathematical Physics - Abstract
Infinite-dimensional Lie algebras are introduced, which are only partially graded, and are specified by indices lying on cyclotomic rings. They may be thought of as generalizations of the Onsager algebra, but unlike it, or its sl(n) generalizations, they are not subalgebras of the loop algebras associated with sl(n). In a particular interesting case associated with sl(3), their indices lie on the Eisenstein integer triangular lattice, and these algebras are expected to underlie vertex operator combinations in CFT, brane physics, and graphite monolayers., Plain LaTex2e, 7 pages
- Published
- 2005
32. Fock Space Representations for Non-Hermitian Hamiltonians
- Author
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Jean Nuyts and David B. Fairlie
- Subjects
Physics ,High Energy Physics - Theory ,Sesquilinear form ,General Physics and Astronomy ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Hermitian matrix ,Fock space ,symbols.namesake ,Operator (computer programming) ,High Energy Physics - Theory (hep-th) ,symbols ,Hamiltonian (quantum mechanics) ,Quantum ,Mathematical Physics ,Harmonic oscillator ,Eigenvalues and eigenvectors ,Mathematical physics - Abstract
The requirement of Hermiticity of a Quantum Mechanical Hamiltonian, for the description of physical processes with real eigenvalues which has been challenged notably by Carl Bender, is examined for the case of a Fock space Hamilitonian which is bilinear in two creation and destruction operators. An interpretation of this model as a Schr\"odinger operator leads to an identification of the Hermitian form of the Hamiltonian as the Landau model of a charged particle in a plane, interacting with a constant magnetic field at right angles to the plane. When the parameters of the Hamiltonian are suitably adjusted to make it non-Hermitian, the model represents two harmonic oscillators at right angles interacting with a constant magnetic field in the third direction, but with a pure imaginary coupling, and real energy eigenvalues. It is now ${\cal PT}$ symmetric. Multiparticle states are investigated., Comment: LaTeX2e, 18 pages
- Published
- 2004
33. Implicit Solutions of PDE's
- Author
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David B. Fairlie
- Subjects
Variables ,Partial differential equation ,Differential equation ,media_common.quotation_subject ,FOS: Physical sciences ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Lorentz covariance ,Nonlinear system ,53C26 ,Quadratic equation ,Discriminant ,Order (group theory) ,Applied mathematics ,Mathematical Physics ,media_common ,Mathematics - Abstract
Further investigations of implicit solutions to non-linear partial differential equations are pursued. Of particular interest are the equations which are Lorentz invariant. The question of which differential equations of second order for a single unknown $\phi$ are solved by the imposition of an inhomogeneous quadratic relationship among the independent variables, whose coefficients are functions of $\phi$ is discussed, and it is shown that if the discriminant of the quadratic vanishes, then an implicit solution of the so-called Universal Field Equation is obtained. The relation to the general solution is discussed., Comment: 11 pages LaTeX2e
- Published
- 2004
34. Implicit solutions to some Lorentz invariant non-linear equations revisited
- Author
-
David B. Fairlie
- Subjects
Hessian matrix ,Partial differential equation ,Order (ring theory) ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Lorentz covariance ,symbols.namesake ,Nonlinear system ,symbols ,Field equation ,Linear wave equation ,Mathematical Physics ,Mathematical physics ,Ansatz ,Mathematics - Abstract
An implicit solution to the vanishing of the so-called Universal Field Equation, or Bordered Hessian, which dates at least as far back as 1935 \cite{chaundy} is revived, and derived from a much later form of the solution. A linear ansatz for an implicit solution of second order partial differential equations, previously shown to have wide applicability \cite{fai} is at the heart of the Chaundy solution, and is shown to yield solutions even to the linear wave equation., Comment: Accepted for publication in J Nonlinear Math. Phys. (Vol. 12, 2005)
- Published
- 2004
- Full Text
- View/download PDF
35. Extra Dimensions and Nonlinear Equations
- Author
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David B. Fairlie and Thomas Curtright
- Subjects
35Q35 ,76D99 ,High Energy Physics - Theory ,Boundary (topology) ,FOS: Physical sciences ,01 natural sciences ,Mathematics - Analysis of PDEs ,Simple (abstract algebra) ,0103 physical sciences ,FOS: Mathematics ,010306 general physics ,Mathematical Physics ,Mathematics ,Partial differential equation ,010308 nuclear & particles physics ,Mathematical analysis ,Fluid Dynamics (physics.flu-dyn) ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Physics - Fluid Dynamics ,Nonlinear system ,Extra dimensions ,High Energy Physics - Theory (hep-th) ,Feature (computer vision) ,Boundary theory ,Analysis of PDEs (math.AP) - Abstract
Solutions of nonlinear multi-component Euler-Monge partial differential equations are constructed in n spatial dimensions by dimension-doubling, a method that completely linearizes the problem. Nonlocal structures are an essential feature of the method. The Euler-Monge equations may be interpreted as a boundary theory arising from a linearized bulk system such that all boundary solutions follow from simple limits of those for the bulk., Scientific Workplace LATEX
- Published
- 2002
36. A universal solution
- Author
-
David B. Fairlie
- Subjects
High Energy Physics - Theory ,Class (set theory) ,Partial differential equation ,Implicit function ,Field (physics) ,Order (ring theory) ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Lorentz covariance ,High Energy Physics - Theory (hep-th) ,Large set (Ramsey theory) ,Variational principle ,Mathematical Physics ,Mathematical physics ,Mathematics - Abstract
The phenomenon of an implicit function which solves a large set of second order partial differential equations obtainable from a variational principle is explicated by the introduction of a class of universal solutions to the equations derivable from an arbitrary Lagrangian which is homogeneous of weight one in the field derivatives. This result is extended to many fields. The imposition of Lorentz invariance makes such Lagrangians unique, and equivalent to the Companion Lagrangians introduced in [baker]., Comment: arxiv version is already official
- Published
- 2001
- Full Text
- View/download PDF
37. Lagrange Brackets and U(1) fields
- Author
-
David B. Fairlie
- Subjects
Physics ,High Energy Physics - Theory ,Nuclear and High Energy Physics ,Field (physics) ,Formal structure ,FOS: Physical sciences ,Poisson bracket ,symbols.namesake ,High Energy Physics::Theory ,High Energy Physics - Theory (hep-th) ,symbols ,Covariant transformation ,Tensor ,Brane ,U-1 ,Mathematics::Symplectic Geometry ,Lagrangian ,Mathematical physics - Abstract
The idea of a companion Lagrangian associated with $p$-Branes is extended to include the presence of U(1) fields. The Brane Lagrangians are constructed with $F_{ij}$ represented in terms of Lagrange Brackets, which make manifest the reparametrisation invariance of the theory; these are replaced by Poisson Brackets in the companion Lagrangian, which is now covariant under field redefinition. The ensuing Lagrangians possess a similar formal structure to those in the absence of an anti-symmetric field tensor., 7 pages, LaTeX, reference corrected
- Published
- 2000
38. Hamilton-Jacobi equations and Brane associated Lagrangians
- Author
-
David B. Fairlie and Linda Baker
- Subjects
Physics ,High Energy Physics - Theory ,Nuclear and High Energy Physics ,Equations of motion ,FOS: Physical sciences ,Hamilton–Jacobi equation ,symbols.namesake ,High Energy Physics::Theory ,Square root ,High Energy Physics - Theory (hep-th) ,Brane cosmology ,symbols ,Covariant transformation ,Brane ,Special case ,Hamiltonian (quantum mechanics) ,Mathematics::Symplectic Geometry ,Mathematical physics - Abstract
This article seeks to relate a recent proposal for the association of a covariant Field Theory with a string or brane Lagrangian to the Hamilton-Jacobi formalism for strings and branes. It turns out that since in this special case, the Hamiltonian depends only upon the momenta of the Jacobi fields and not the fields themselves, it is the same as a Lagrangian, subject to a constancy constraint. We find that the associated Lagrangians for strings or branes have a covariant description in terms of the square root of the same Lagrangian. If the Hamilton-Jacobi function is zero, rather than a constant, then it is in in one dimension lower, reminiscent of the `holographic' idea. In the second part of the paper, we discuss properties of these Lagrangians, which lead to what we have called `Universal Field Equations', characteristic of covariant equations of motion., Comment: 23 pages,LaTeX2e, clarified text, generalised proof in appendix
- Published
- 2000
- Full Text
- View/download PDF
39. Moyal Nahm Equations
- Author
-
David B. Fairlie and Linda Baker
- Subjects
High Energy Physics - Theory ,Large class ,Physics ,Pure mathematics ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Set (abstract data type) ,Matrix (mathematics) ,High Energy Physics - Theory (hep-th) ,Simple (abstract algebra) ,Nahm equations ,Wigner distribution function ,Limit (mathematics) ,Algebra over a field ,Mathematical Physics - Abstract
Various aspects of the Nahm equations in 3 and 7 dimensions are investigated. The residues of the variables at simple poles in the 7-dimensional case form an algebra. A large class of matrix representations of this algebra is constructed. The large $N$ limit of these equations is taken by replacing the commutators by Moyal Brackets, and a set of non trivial solutions in a generalised form of Wigner distribution functions is obtained., Comment: 15 pages, latex, no figures
- Published
- 1999
- Full Text
- View/download PDF
40. Higher-dimensional Generalisations of the Euler Top Equations
- Author
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Tatsuya Ueno and David B. Fairlie
- Subjects
High Energy Physics - Theory ,Physics ,Conservation law ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,General Physics and Astronomy ,FOS: Physical sciences ,String theory ,symbols.namesake ,High Energy Physics - Theory (hep-th) ,Euler's formula ,symbols ,Nahm equations ,Physics::Atomic and Molecular Clusters ,Exactly Solvable and Integrable Systems (nlin.SI) ,Mathematical physics - Abstract
Generalisations of the familiar Euler top equations in three dimensions are proposed which admit a sufficiently large number of conservation laws to permit integrability by quadratures. The usual top is a classical analogue of the Nahm equations. One of the examples discussed here is a seven-dimensional Euler top, which arises as a classical counterpart to the eight-dimensional self-dual equations which are currently believed to play a role in new developments in string theory., Comment: 8 pages, Latex, 1 eps.file. Minor corrections, eq.(22) added, version to be published in Phys. Lett. A
- Published
- 1997
- Full Text
- View/download PDF
41. The Reversed q-Exponential Functional Relation
- Author
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Ming-Yuan Wu and David B. Fairlie
- Subjects
Condensed Matter::Quantum Gases ,Pure mathematics ,Condensed Matter::Other ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Condensed Matter::Mesoscopic Systems and Quantum Hall Effect ,Functional relation ,Mathematics - Quantum Algebra ,FOS: Mathematics ,q-exponential ,Order (group theory) ,Quantum Algebra (math.QA) ,Multiplication ,Mathematical Physics ,Mathematics - Abstract
After obtaining some useful identities, we prove an additional functional relation for $q$ exponentials with reversed order of multiplication, as well as the well known direct one in a completely rigorous manner., Comment: 6 pages, LaTeX, no figures
- Published
- 1997
- Full Text
- View/download PDF
42. Two-index generalisations of Superconformal Algebras
- Author
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Jean Nuyts and David B. Fairlie
- Subjects
Physics ,High Energy Physics - Theory ,Pure mathematics ,Index (economics) ,Structure (category theory) ,FOS: Physical sciences ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,High Energy Physics::Theory ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,High Energy Physics - Theory (hep-th) ,Algebra over a field ,Mathematical Physics ,Moyal bracket - Abstract
The superconformal algebras of Ademollo et al are generalised to a multi-index form. The structure obtained is similar to the Moyal Bracket analogue of the Neveu-Schwarz Algebra., 8 pages, LaTeX, no figures
- Published
- 1996
43. The Hamiltonian structure of the dispersionless Toda hierarchy
- Author
-
Ian A. B. Strachan and David B. Fairlie
- Subjects
Physics ,Laplace's equation ,Conservation law ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Condensed Matter Physics ,symbols.namesake ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Hamiltonian structure ,symbols ,Exactly Solvable and Integrable Systems (nlin.SI) ,Hamiltonian (quantum mechanics) ,Axial symmetry ,Mathematical physics - Abstract
The Hamiltonian structure of the two-dimensional dispersionless Toda hierarchy is studied, this being a particular example of a system of hydrodynamic type. The polynomial conservation laws for the system turn out, after a change of variable, to be associated with the axially symmetric solutions of the 3-dimensional Laplace equation and this enables a generating function for the Hamiltonian densities to be derived in closed form., 12 pages, latex, no figures
- Published
- 1995
44. General solution of the universal equation in n-dimensional space
- Author
-
A. N. Leznov and David B. Fairlie
- Subjects
Pure mathematics ,N dimensional ,Integrable system ,Differential equation ,Mathematical analysis ,Statistical and Nonlinear Physics ,Space (mathematics) ,Legendre's equation ,Legendre transformation ,symbols.namesake ,symbols ,Legendre polynomials ,Mathematical Physics ,General Theoretical Physics ,Mathematics - Abstract
It is shown that general solution of the Universal Equation in n-dimensional space is closely connected with an exactly (but only implicitly) integrable system @ @xn + n 1 X =1 @ @x = 0. (0.01) Using the explicit form of the solution of this system it is possible to construct the general solution of the Universal Equation which was found before by the Legendre transformation.
- Published
- 1994
45. General solutions of the Monge-Amp\'{e}re equation in $n$-dimensional space
- Author
-
A. N. Leznov and David B. Fairlie
- Subjects
Physics ,High Energy Physics - Theory ,Pure mathematics ,Integrable system ,N dimensional ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Mathematics::Complex Variables ,Mathematics::Analysis of PDEs ,General Physics and Astronomy ,Space (mathematics) ,Mathematics - Analysis of PDEs ,Homogeneous ,Geometry and Topology ,Mathematical Physics - Abstract
It is shown that the general solution of a homogeneous Monge-Amp\`{e}re equation in $n$-dimensional space is closely connected with the exactly (but only implicitly) integrable system \frac {\partial \xi_{j}}{\partial x_0}+\sum_{k=1}^{n-1} \xi_{k} \frac {\partial \xi_{j}}{\partial x_{k}}=0 \label{1} Using the explicit form of solution of this system it is possible to construct the general solution of the Monge-Amp\`{e}re equation., Comment: 8 pages
- Published
- 1994
46. Study of Quommutators of Quantum Variables and Generalized Derivatives
- Author
-
Jean Nuyts and David B. Fairlie
- Subjects
Physics ,High Energy Physics - Theory ,Transposition (telecommunications) ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Deformation (meteorology) ,High Energy Physics - Theory (hep-th) ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Coherent states ,Quantum Algebra (math.QA) ,Algebra over a field ,Quantum ,Mathematical Physics ,Mathematical physics - Abstract
A general deformation of the Heisenberg algebra is introduced with two deformed operators instead of just one. This is generalised to many variables, and permits the simultaneous existence of coherent states, and the transposition of creation operators., 17 pages (Previous version was truncated in transmission)
- Published
- 1993
47. Necessary conditions for ternary algebras
- Author
-
David B. Fairlie and Jean Nuyts
- Subjects
High Energy Physics - Theory ,Statistics and Probability ,Physics ,Pure mathematics ,High Energy Physics - Theory (hep-th) ,Modeling and Simulation ,Canonical normal form ,FOS: Physical sciences ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Ternary operation ,Mathematical Physics - Abstract
Ternary algebras, constructed from ternary commutators, or as we call them, ternutators, defined as the alternating sum of products of three operators, have been shown to satisfy cubic identities as necessary conditions for their existence. Here we examine the situation where we permit identities not solely constructed from ternutators or nested ternutators and we find that in general, these impose additional restrictions; for example, the anti-commutators or commutators of the operators must obey some linear relations among themselves., Comment: 10 pages
- Published
- 2010
48. Linearisation of Universal Field Equations
- Author
-
David B. Fairlie and Jan Govaerts
- Subjects
Large class ,Physics ,High Energy Physics - Theory ,Conjecture ,Dynamical systems theory ,Integrable system ,media_common.quotation_subject ,Existential quantification ,General Physics and Astronomy ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Infinity ,Legendre transformation ,symbols.namesake ,High Energy Physics - Theory (hep-th) ,symbols ,Field equation ,Mathematical Physics ,media_common ,Mathematical physics - Abstract
The Universal Field Equations, recently constructed as examples of higher dimensional dynamical systems which admit an infinity of inequivalent Lagrangians are shown to be linearised by a Legendre transformation. This establishes the conjecture that these equations describe integrable systems. While this construction is implicit in general, there exists a large class of solutions for which an explicit form may be written., 11pp., DTP-92/47, NI-92/011
- Published
- 1992
49. Universal Field Equations with Reparametrisation Invariance
- Author
-
David B. Fairlie and Jan Govaerts
- Subjects
High Energy Physics - Theory ,Physics ,Nuclear and High Energy Physics ,Lorentz transformation ,Equations of motion ,FOS: Physical sciences ,Invariant (physics) ,symbols.namesake ,High Energy Physics::Theory ,High Energy Physics - Theory (hep-th) ,Euclidean geometry ,symbols ,Covariant transformation ,Inverse function ,Field equation ,Scalar field ,Mathematical physics - Abstract
New reparametrisation invariant field equations are constructed which describe $d$-brane models in a space of $d+1$ dimensions. These equations, like the recently discovered scalar field equations in $d+1$ dimensions, are universal, in the sense that they can be derived from an infinity of inequivalent Lagrangians, but are nonetheless Lorentz (Euclidean) invariant. Moreover, they admit a hierarchical structure, in which they can be derived by a sequence of iterations from an arbitrary reparametrisation covariant Lagrangian, homogeneous of weight one. None of the equations of motion which appear in the hierarchy of iterations have derivatives of the fields higher than the second. The new sequence of Universal equations is related to the previous one by an inverse function transformation. The particular case of $d=2$, giving a new reparametrisation invariant string equation in 3 dimensions is solved., Comment: 9pages
- Published
- 1992
- Full Text
- View/download PDF
50. Ternutator identities
- Author
-
David B. Fairlie, Gregor Weingart, Chandrashekar Devchand, and Jean Nuyts
- Subjects
High Energy Physics - Theory ,Statistics and Probability ,Jacobi identity ,Physics ,Pure mathematics ,Institut für Mathematik ,Structure (category theory) ,FOS: Physical sciences ,General Physics and Astronomy ,Commutator (electric) ,Statistical and Nonlinear Physics ,law.invention ,symbols.namesake ,Quadratic equation ,High Energy Physics - Theory (hep-th) ,law ,Modeling and Simulation ,Product (mathematics) ,symbols ,Ternary operation ,Mathematical Physics - Abstract
The ternary commutator or ternutator, defined as the alternating sum of the product of three operators, has recently drawn much attention as an interesting structure generalising the commutator. The ternutator satisfies cubic identities analogous to the quadratic Jacobi identity for the commutator. We present various forms of these identities and discuss the possibility of using them to define ternary algebras., 12 pages, citation added
- Published
- 2009
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