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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.

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

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.

The multi-field generalisation of the Bateman equation arises from considerations of the continuation of String and Brane equations to the case where the base space is of higher dimension than the target space. The complex extension of this equation possesses a remarkably large invariance group, and admits a very simple implicit form for its general solution, in addition to the special case of holomorphic and anti-holomorphic explicit solutions. A class of inequivalent Lagrangians for this equation is discovered.

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.

A relation between the classical Faulhaber and Bernoulli polynomials and the theory of the Korteweg–de Vries equation is established.

The general solution to the complex Bateman equation in a space of arbitrary dimensions is constructed.

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.

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 .

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.

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

Deformations of the canonical commutation relations which have the effect of altering the spectrum of a standard Hamiltonian, bilinear in creation and annihilation operators are described. The problem of going over from an eigenvalue situation, as is the case in the vast majority of papers in the literature, to a theory with time evolution is discussed, and a special example with deformation parameter an N th root of unity is constructed which possesses a consistent time evolution. This work is an account of some recent studies of associative deformations of the Heisenberg algebra of several creation and annihilation algebras, with Jean Nuyts of the University of Mons, Hainaut, together with some observations of my own concerning the difficulty of implementing time evolution in a quantum group context. It builds on earlier work with Cosmas Zachos (Argonne National Laboratory, USA), which in turn is re;ated to work of Manin, and Wess, Zumino and collaborators. The main idea is that, if quantum groups have any role in physics, then they must manifest themselves at the level of the basic rules of quantisation.

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.

The authors take a standard Hamiltonian, quadratic in a set of creation and annihilation operators, and demonstrate an alternative quantization by quommutators; i.e. a set of rules for transposition of these operators, different from the those of the usual canonical commutation relations which yields a new spectrum, of a type reminiscent of logarithmic trajectories.

The formalism of the Wigner distribution function is reviewed. In addition to the Liouville equation, which expresses the time rate of change of this function in terms of its Moyal bracket with the Hamiltonian, and its expression as a projection operator, a third equation is proposed with the aid of an auxiliary variable s, to which a formal solution is constructed in terms of known quantum-mechanical eigenfunctions and eigenvalues. In addition, an ab initio solution to the three equations in terms of an error function is found for the free particle in one dimension. Two views are advanced: the orthodox, that this new equation is merely a consistency requirement, and the speculative, that the measurement process has something to do with the choice of s.

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.

SynopsisSome disparate ideas in the literature are drawn together. The work of P. J. Olver and his associates on Lagrangians which vanish for arbitrary variations, the so-called null Lagrangians, is viewed as a parallel development to Witten's study of topological field theories. A theorem of Olver, that all hyperjacobians are expressible as divergences, and are thus candidates for the construction of null Lagrangians, is shown to follow directly from the observation that such entities appear in a power series development of the general associative product, and this technique facilitates the construction of multi-dimensional examples.

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.

This paper explores features of the infinite‐dimensional algebras that have been previously introduced. In particular, it is shown that the classical simple Lie algebras (AN, BN, CN, DN) may be expressed in an ‘‘egalitarian’’ basis with trigonometric structure constants. The transformation to the standard Cartan–Weyl basis, and the particularly transparent N→∞ limit that this formulation allows is provided.

Some features of Manin’s construction of quantum groups are developed and extended to supergroups.

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

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)

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

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

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

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.

The Yang equations for all self dual solutions of SU(r,s) gauge theory are exhibited in simple form. Algebraic and Backlund transformations of the solutions of these equations are derived. The Backlund transformations change an SU(r,s) solution into an SU(r−1, s+1) solution.

Linear relations among the fields of a gauge theory in eight dimensions derived from a secular equation invariant under a SO(7) subgroup are particularly interesting. The authors present here the spherical solutions corresponding to this case when the gauge group is SO(7) or SO(8). In the former case the solution can be made regular everywhere.

Ignoring the problem of sources and singularities, explicit expressions are constructed for the ansatze of Atiyah and Ward. These take an especially simple form in theR gauge of Yang. Some non-linear transformation properties of the self-duality equations in this gauge provide an inductive proof of the ansatze. There is a six-parameter family of these Backlund transformations. They take real SU(2) gauge fields into real SU(1, 1) gauge fields and vice versa.

It is shown that the entire Virasoro, Ramond and Neveu-Schwarz algebras can each be constructed from a finite number of well-chosen generators satisfying a small number of conditions. The most economical sets consist of just two starting generators in all cases, subject to eight conditions for the Virasoro case, five conditions for the Ramond case, and nine conditions for the Neveu-Schwarz case.

In a finite difference scheme for numerical approximation the usual notion of product is replaced by a convolution to circumvent the failure of the product rule for differentiation. The solution displays a complementarity: the more localised is the product the more extended is the approximation to the derivative and vice versa.

Nontrivial examples of Krichever–Novikov algebras are constructed. The construction involves parameters that satisfy eigenvalue conditions. In the linear case, for an r‐term algebra, the eigenvalues are 0,1,...,r−3.

The scattering amplitude for a potential expressible as a sum of n separable potentials is obtained as the solution of a set of n simultaneous linear equations. As n→∞ the Fredholm solution for the integral form of the Schrödinger equation is recovered.

The infinite sequence of conserved quantities which were recently discovered in free-field theories by Kibble, are shown to be a consequence of translation invariance and the quadratic nature of the Lagrangian. This observation leads to a natural extension of Kibble’s work by exploiting the full Lorentz invariance of the free-field equations, leading to the construction of further new conservation laws. The question whether the existence of a conservation law implies an invariance of the Lagrangian is raised and answered in the negative.

A proof by means of graphical techniques is given of an inequality conjectured by Nakanishi concerning the Feynman integrand for any term in the perturbation expansion of the scattering amplitude.

Some properties of crossing matrices are deduced which are independent of the particular symmetry group from which the crossing matrices are derived. In particular, a, factorization of any elastic crossing matrix, analogous to the factorization of a rotation matrix in n dimensions is found, and the connection between, crossing and unitary matrices elucidated. In the special case of the crossing matrix for elastic scattering of particles which transform as representations of SU2, as well as the usual consistency requirement that each row should sum to unity, a new consistency requirement on the elements in a given column is proved. As a byproduct of this work, a possibly new quadratic identity for Racah coefficients is exhibited.

The model due to Ruijgrok and Van Hove is studied for a particular choice of coupling constants. The resulting theory displays an exact criterion for the existence of ghost states. In the limit as the number of intermediates states increases to simulate a realistic field theory the ghost behaviour disappears.

Generalized self‐duality type relations for gauge fields in higher‐dimensional spaces are discussed from the point of view of orbits, strata, and stability groups. It is explained how to obtain them as integrability conditions for first‐order gauge‐invariant linear differential equations.