Search

In this paper we rigorously prove some statements on the symbolic dynamics for Bernoulli coupled map lattices studied by Kawasaki and Sasa. The advantage of our approach is that it is purely topological and it gives a simultaneous proof for the statements.

It has been conjectured that for ≥0 the entire spectrum of the non-Hermitian -symmetric Hamiltonian HN = p2 + x2(ix), where N = 2 + , is real. Strong evidence for this conjecture for the special case N = 3 was provided in a recent paper by Mezincescu (Mezincescu G A 2000 J. Phys. A: Math. Gen. 33 4911) in which the spectral zeta function Z3(1) for the Hamiltonian H3 = p2 + ix3 was calculated exactly. Here, the calculation of Mezincescu is generalized from the special case N = 3 to the region of all N≥2 (≥0) and the exact spectral zeta function ZN(1) for HN is obtained. Using ZN(1) it is shown that to extremely high precision (about three parts in 1018) the spectrum of HN for other values of N such as N = 4 is entirely real.

For original paper see Coveney et. al. ibid., vol.25, p.4947 (1992). The previous authors state, in the introductory section of their paper that they are looking only at a 'small part' of the Brussels formalism. The title of their paper does not reflect this restriction and refers to the general validity of the Brussels formalism. The criticisms are focused on a few papers published about twenty years ago. However, even restricted to these papers, their claims do not apply. The present authors point out two major misrepresentations.

For original paper see Coveney et. al., ibid., vol.25, p.4947 (1992). George and Prigogine (ibid., vol.26, p.3905, 1993) interpret the authors previous paper as destructive criticism of the Brussels formalism. However they argue that their aim was to discover conditions under which this method might be supported by rigorous analysis and that their results give a set of sufficient conditions, admittedly quite restrictive, for the validity of a part of the formalism. The Brussels school has never promulgated any agreed choice for the function space in which the formalism's operators are to act. In their work they chose a Hilbert space. This structure sufficed for the theorems they could prove, which concern only the P subspace. They agree that additional structure, such as that of a rigged Hilbert space, may be necessary to give effect to the Brussels formalism outside the P subspace, but since this additional structure would not have affected their results they do not see that its omission was a misrepresentation. Further comments are made.

For original paper see ibid. vol.20, p.915, (1987). The authors show that energy fluctuations, and thus the higher-order moments of energy, contain essential information and cannot be neglected even in the thermodynamical limit. In contrast with some of Jaworski's statements, they point out how all the thermodynamical properties depend in a crucial way on the fluctuations on the basis of realistic physical assumptions. Indeed, phenomenological thermodynamics allows one to conclude that the Gibbs canonical distribution is the only possible probability distribution which does not violate the second principle. It follows that the knowledge of the free energy as a function of temperature beta -1 is equivalent to that of the probability law governing fluctuations. This probability law is therefore a characteristic of a body which can be investigated by measuring either energy moments at fixed beta or the mean values of entropy and energy at varying beta .

This note discusses the possible use of perturbation methods in studying chaotic trajectories of ordinary differential equations, with particular focus on a recent paper on this topic by Rowlands (ibid., vol.16, p.585, 1983).

For original paper see Commun. Math. Phys., vol.104, p.457-70, 1986. The author links the recently proposed unification of the boson and fermion stochastic calculus with the general problem of boson-fermion equivalence (duality, reciprocity, etc.) for quantum fields. Even if via the Fock construction the common Fock space for bosons and fermions can be introduced, it still does not allow for the unrestricted boson-fermion equivalence for field theory models. All local fermion field theory models thus have boson equivalents (violating the weak local commutativity condition for space dimension three). The reverse statement is not valid: not all boson models admit a pure fermion reconstruction.

The geometry of crumpled paper balls is investigated. It is shown that these systems are fractals and their properties are studied.

It is pointed out that the different definitions of the fractional dimension of an arbitrary subset A of d, given by Naudts (1988) and by Barlow and Taylor (1989), are interrelated but obtained using completely different criteria.

We state and prove a theorem about the fact that B1-type approximants have reciprocals with different boundary structure from their own.

A detailed derivation of the energy lower bound model (SHRIMP model) for a translation-invariant many-fermion system has recently been proposed (Carr 1978). The author asserts that an N-particle shell model retaining antisymmetry in all N particles is obtained. The present author wishes to point out the error in this derivation which makes the SHRIMP model wrong leaving correct the related RIP and HIP models.

In this short paper we illustrate by few examples the special role played by higher-order Riccati equations in the construction of Backlund transformations for integrable systems.

In this short paper an algebraic expression for (lambda 0) x (3 0) reduction Wigner coefficients in the SU(3) superset of R(3) physical basis is obtained using a building-up process. The SU(3) U-functions are also given.

In this paper, we present the analytical solution of the radial Schrodinger equation for the Hulthen potential within the framework of the asymptotic iteration method by using an approximation to the centrifugal potential for any l states. We obtain the energy eigenvalues and corresponding eigenfunctions for different screening parameters. The wavefunctions are physical and energy eigenvalues are in good agreement with the results obtained by other methods for different delta values. In order to demonstrate this, the results of the asymptotic iteration method are compared with the results of the supersymmetry, numerical integration, variational and shifted 1/N expansion methods., In this paper, we present the analytical solution of the radial Schr¨odingerequation for the Hulth´en potential within the framework of the asymptoticiteration method by using an approximation to the centrifugal potential for anyl states. We obtain the energy eigenvalues and corresponding eigenfunctionsfor different screening parameters. The wavefunctions are physical and energyeigenvalues are in good agreement with the results obtained by other methodsfor different δ values. In order to demonstrate this, the results of the asymptoticiterationmethod are compared with the results of the supersymmetry, numericalintegration, variational and shifted 1/N expansion methods.

An antilinear operator in complex vector spaces is an important operator in the study of modern quantum theory, quantum and semiclassical optics, quantum electronics and quantum chemistry. Consimilarity of complex matrices arises as a result of studying an antilinear operator referred to different bases in complex vector spaces, and the theory of consimilarity of complex matrices plays an important role in the study of quantum theory. This paper, by means of a real representation of a complex matrix, studies the relation between consimilarity and similarity of complex matrices, sets up an algebraic bridge between consimilarity and similarity and turns the theory of consimilarity into that of ordinary similarity. This paper also gives some applications of consimilarity of complex matrices.

In a paper published in Am. Math. Mon. (1966 73 1?23), Kac asked his famous question 'Can one hear the shape of a drum?'. Gordon et al answered this question negatively by constructing planar isospectral pairs in their paper published in Invent. Math. (1992 110 1?22). Only a finite number of pairs have been constructed till now. Further in J. Phys. A: Math. Gen. (2005 38 L477?83), Giraud showed that most of the known examples can be generated from solutions of a certain equation which involves certain involutions of an n-dimensional projective space over some finite field. He then generated all possible solutions when n = 2. In this letter we handle all dimensions, and show that no other examples arise.

In this paper we review some group theoretic techniques applied to discretizations of PDEs. Inspired by the recent years active research in Lie group- and exponential-time integrators for differential equations, we will in the first part of the paper present algorithms for computing matrix exponentials based on Fourier transforms on finite groups. As an example, we consider spherically symmetric PDEs, where the discretization preserves the 120 symmetries of the icosahedral group. This motivates the study of spectral element discretizations based on triangular subdivisions. In the second part of the paper, we introduce novel applications of multivariate non-separable Chebyshev polynomials in the construction of spectral element bases on triangular and simplicial sub-domains. These generalized Chebyshev polynomials are intimately connected to the theory of root systems and Weyl groups (used in the classification of semi-simple Lie algebras), and these polynomials share most of the remarkable properties of the classical Chebyshev polynomials, such as near-optimal Lebesgue constants for the interpolation error, the existence of FFT-based algorithms for computing interpolants and pseudo-spectral differentiation and existence of Gaussian integration rules. The two parts of the paper can be read independently.

In this paper, multisoliton solutions of the Hirota equation with variable coefficients are obtained by the Darboux transformation based on the Ablowitz–Kaup–Newell–Segur technology. As an example, we discuss the evolutional behaviour of a two-soliton solution in a soliton control fibre system. The results reveal that one may control the interaction between the pulses by choosing the third-order dispersion parameters d4 and h appropriately. Meanwhile, more generalized forms of bright soliton and dark soliton solutions of generalized higher order nonlinear Schrodinger equations (GHONLSE) with variable coefficients are obtained by the extended tanh-function method. Moreover, new bright and dark combined solitary wave, kink solitary wave and M-shaped solitary wave to GHONLSE with variable coefficients are firstly reported in this paper. Especially, the term proportional to α1 resulting from the group velocity decides the group velocity and the phase shift of these new solitary waves.

A usual loop algebra, not necessarily the matrix form of the loop algebra , is also made use of for constructing linear isospectral problems, whose compatibility conditions exhibit a zero-curvature equation from which integrable systems are derived. In order to look for the Hamiltonian structure of such integrable systems, a quadratic-form identity is created in the present paper whose special case is just the trace identity; that is, when taking the loop algebra , the quadratic-form identity presented in this paper is completely consistent with the trace identity.

This paper provides an efficient analytical tool for solving the heat conduction equation in a multi-dimensional composite slab subject to generally time-dependent boundary conditions. A temporal Laplace transformation and novel separation of variables are applied to the heat equation. The time-dependent boundary conditions are approximated with Fourier series. Taking advantage of the periodic properties of Fourier series, the corresponding analytical solution is obtained and expressed explicitly through employing variable transformations. For such conduction problems, nearly all the published works necessitate numerical work such as computing residues or searching for eigenvalues even for a one-dimensional composite slab. In this paper, the proposed method involves no numerical iteration. The final closed form solution is straightforward; hence, the physical parameters are clearly shown in the formula. The accuracy of the developed analytical method is demonstrated by comparison with numerical calculations.

In this paper, we study the average inter-crossing number between two random walks and two random polygons in the three-dimensional space. The random walks and polygons in this paper are the so-called equilateral random walks and polygons in which each segment of the walk or polygon is of unit length. We show that the mean average inter-crossing number ICN between two equilateral random walks of the same length n is approximately linear in terms of n and we were able to determine the prefactor of the linear term, which is . In the case of two random polygons of length n, the mean average inter-crossing number ICN is also linear, but the prefactor of the linear term is different from that of the random walks. These approximations apply when the starting points of the random walks and polygons are of a distance ρ apart and ρ is small compared to n. We propose a fitting model that would capture the theoretical asymptotic behaviour of the mean average ICN for large values of ρ. Our simulation result shows that the model in fact works very well for the entire range of ρ. We also study the mean ICN between two equilateral random walks and polygons of different lengths. An interesting result is that even if one random walk (polygon) has a fixed length, the mean average ICN between the two random walks (polygons) would still approach infinity if the length of the other random walk (polygon) approached infinity. The data provided by our simulations match our theoretical predictions very well.

The present paper extends some results recently developed for the analysis of observability in nonlinear dynamical systems. The aim of the paper is to address the problem of embedding an attractor using more than one observable. A multivariate nonlinear observability matrix is proposed which includes the monovariable nonlinear and linear observability matrices as particular cases. Using the developed framework and a number of worked examples, it is shown that the choice of embedding coordinates is critical. Moreover, in some cases, to reconstruct the dynamics using more than one observable could be worse than to reconstruct using a scalar measurement. Finally, using the developed framework it is shown that increasing the embedding dimension, observability problems diminish and can even be eliminated. This seems to be a physically meaningful interpretation of the Takens embedding theorem.

We present three different linearizable extensions of the Gurevich–Zybin system. Their general solutions are found by reciprocal transformations. In this paper we rewrite the Gurevich–Zybin system as a Monge–Ampere equation. By application of reciprocal transformation this equation is linearized. Infinitely many local Hamiltonian structures, local Lagrangian representations, local conservation laws and local commuting flows are found. Moreover, all commuting flows can be written as Monge–Ampere equations similar to the Gurevich–Zybin system. The Gurevich–Zybin system describes the formation of large scale structure in the Universe. Second harmonic wave generation is known in nonlinear optics. In this paper we prove that the Gurevich–Zybin system is equivalent to a degenerate case of second harmonic generation. Thus, the Gurevich–Zybin system is recognized as a degenerate first negative flow of two-component Harry Dym hierarchy up to two Miura-type transformations. A reciprocal transformation between the Gurevich–Zybin system and degenerate case of the second harmonic generation system is found. A new solution for second harmonic generation is presented in implicit form.

A novel analytical approach has been developed for heat conduction in a multi-dimensional composite slab subject to time-dependent boundary changes of the first kind. Boundary temperatures are represented as Fourier series. Taking advantage of the periodic properties of boundary changes, the analytical solution is obtained and expressed explicitly. Nearly all the published works necessitate searching for associated eigenvalues in solving such a problem even for a one-dimensional composite slab. In this paper, the proposed method involves no iterative computation such as numerically searching for eigenvalues and no residue evaluation. The adopted method is simple which represents an extension of the novel analytical approach derived for the one-dimensional composite slab. Moreover, the method of 'separation of variables' employed in this paper is new. The mathematical formula for solutions is concise and straightforward. The physical parameters are clearly shown in the formula. Further comparison with numerical calculations is presented.

This paper is the last of a series of three articles presenting a classification of Vornoi and Delone tilings determined by point sets ?(?) ('quasicrystals'), built by the standard projection of the root lattice of type A4 to a two-dimensional plane spanned by the roots of the Coxeter group H2 (dihedral group of order 10). The acceptance window ? for ?(?) in the present paper is a regular decagon of any radius 0 < r < ?. There are 14 distinct VT sets of Voronoi tiles and 6 sets DT of Delone tiles, up to a uniform scaling by the factor and . The number of Voronoi tiles in different quasicrystal tilings varies between 3 and 12. Similarly, the number of Delone tiles is varying between 4 and 6. There are 7 VT sets of the 'generic' type and 7 of the 'singular' type. The latter occur for seven precise values of the radius of the acceptance window. Quasicrystals with acceptance windows with radii in between these values have constant VT sets, only the relative densities and arrangement of the tiles in the tilings change. Similarly, we distinguish singular and generic sets DT of Delone tiles.

In a series of papers Calogero and Graffi (2003 Phys. Lett. A 313 356–62) and Calogero (Phys. Lett. A (submitted); J. Nonlinear Math. Phys. (to appear)) treated the quantization of several one-degree-of-freedom Hamiltonians containing a parameter, c, which plays no role in the classical motion, but is critical to the value of the eigenvalue of the ground state. In this paper we examine the classical and quantum problems from the point of view of their Noether and Lie point symmetries respectively and demonstrate the construction of the quantal wavefunctions from the Lie point symmetries of the Schrodinger equation.

After expanding the distribution function about an anisotropic Planck function, the new moment closure method of Banach and Larecki applied to the Boltzmann–Peierls equation for the phonon gas dynamics leads to a whole hierarchy of closed systems of moment equations. The system of equations for the energy density and the heat flux is the first, non-perturbative member of this hierarchy of closures. In our previous paper (2004 J. Phys. A: Math. Gen. 37 9805), emphasis was placed on deriving the next member, the 9-moment anisotropic closure that involves the flux of the heat flux as an extra gas-state variable. Here, as a first step in effectively analysing this system, we present a study of the one-dimensional, rotationally symmetric reduction of these equations. Under the assumption of Callaway's model, a systematic procedure is derived which shows that the obtained system of three evolution equations for three nonvanishing gas-state variables can be cast into a symmetric hyperbolic form. For the sake of completeness, we describe explicitly the region of symmetric hyperbolicity in parameter space (the space defined by the gas-state variables). The evolution system is symmetric hyperbolic for significant ranges of physical conditions, i.e., there are effectively no unphysical limitations on the magnitude of the energy density and the heat flux. This paper also deals with the eigenvalue problem and calculates approximately the characteristic speeds.

In a recent paper, the canonical forms of a new multi-parameter class of Abel differential equations, so-called AIR, all of whose members can be mapped into Riccati equations, were shown to be related to the differential equations for the hypergeometric 2F1, 1F1 and 0F1 functions. In this paper, a connection between the AIR canonical forms and the Heun General (GHE), Confluent (CHE) and Biconfluent (BHE) equations is presented. This connection fixes the value of one of the Heun parameters, expresses another one in terms of those remaining, and provides closed form solutions in terms of pFq functions for the resulting GHE, CHE and BHE, respectively depending on four, three and two irreducible parameters. This connection also turns evident what is the relation between the Heun parameters such that the solutions admit Liouvillian form, and suggests a mechanism for relating linear equations with N and N-1 singularities through the canonical forms of a non-linear equation of one order less., Original version submitted to Journal of Physics A: 16 pages, related to math.GM/0002059 and math-ph/0402040. Revised version according to referee's comments: 23 pages. Sign corrected (June/17) in formula (79). Second revised version (July/25): 25 pages. See also http://lie.uwaterloo.ca/odetools.htm

In this paper, we study the two-dimensional Burgers–Korteweg–de Vries (2D-BKdV) equation by analysing an equivalent two-dimensional autonomous system, which indicates that under some particular conditions, the 2D-BKdV equation has a unique bounded travelling wave solution. Then by using a direct method, a travelling solitary wave solution to the 2D-BKdV equation is expressed explicitly, which appears to be more efficient than the existing methods proposed in the literature. At the end of the paper, the asymptotic behaviour of the proper solutions of the 2D-BKdV equation is established by applying the qualitative theory of differential equations.

An algebraic method is devised to uniformly construct a series of exact solutions for general integrable and nonintegrable nonlinear evolution equations. Compared with most existing tanh methods, the Jacobi function expansion method or other sophisticated methods, the proposed method not only gives new and more general solutions, but also provides a guideline to classify the various types of the solutions according to the values of some parameters. The solutions obtained in this paper include (a) polynomial solutions, (b) exponential solutions, (c) rational solutions, (d) triangular periodic wave solutions, (e) hyperbolic and solitary wave solutions and (f) Jacobi and Weierstrass doubly periodic wave solutions. The efficiency of the method can be demonstrated on a large variety of nonlinear equations such as those considered in this paper, new (2 + 1)-dimensional Calogero–KdV equation, (3 + 1)-dimensional Jimbo–Miwa equation, symmetric regular long wave equation, Drinfel'd–Sokolov–Wilson equation, (2 + 1)-dimensional generalized dispersive long wave equation, double sine-Gordon equation, Calogero–Degasperis–Fokas equation and coupled Schrodinger–Boussinesq equation. In addition, the links among our proposed method, the tanh method, the extended method and the Jacobi function expansion method are also clarified generally.

Partial Boolean algebras, first studied by Kochen and Specker in the 1960s, provide the structure for Bell–Kochen–Specker theorems which deny the existence of non-contextual hidden variable theories. In this paper, we study partial Boolean algebras which are 'algebraic' in the sense that their elements have coordinates in an algebraic number field. Several of these algebras have been discussed recently in a debate on the validity of Bell–Kochen–Specker theorems in the context of finite precision measurements. The main result of this paper is that every algebraic finitely-generated partial Boolean algebra B(T) is finite when the underlying space is three-dimensional, answering a question of Kochen and showing that Conway and Kochen's infinite algebraic partial Boolean algebra has minimum dimension. This result contrasts the existence of an infinite (non-algebraic) B(T) generated by eight elements in an abstract orthomodular lattice of height 3. We then initiate a study of higher-dimensional algebraic partial Boolean algebras. First, we describe a restriction on the determinants of the elements of B(T) that are generated by a given set T. We then show that when the generating set T consists of the rays spanning the minimal vectors in a real irreducible root lattice, B(T) is infinite just if that root lattice has an A5 sublattice. Finally, we characterize the rays of B(T) when T consists of the rays spanning the minimal vectors of the root lattice E8.

A new general method is presented which allows one to find all distinct Voronoi and Delone tiles in any quasicrystal from a large family. This includes the tiles which may be present with arbitrarily low density >0. At all stages, the method requires only consideration of a (possibly large) finite number of cases. Our method is applicable, in principle, to quasicrystals in any dimension and with any irrationality. This is the first of three papers where the Voronoi and Delone tilings are studied. Two-dimensional point sets, 'quasicrystals', arising from the A4-root lattice by means of the standard projection to a two-dimensional plane with the irrationality τ = 1/2(1 + √5), are considered. In general, we require that the acceptance window be bounded with non-empty interior. Specific results are provided here for rhombic acceptance windows of any size oriented along the direction of simple roots of the Coxeter group H2. Within one quasicrystal the tiles are distinguished by their shape, size and orientation. The rhombic window case is indispensable for subsequent classification of Voronoi and Delone tiles in quasicrystals with general shape of the acceptance window. Voronoi and Delone tiles of quasicrystals with circular and decagonal windows of any size are given in subsequent papers. Let VT denote the set of distinct Voronoi tiles making up a quasicrystal with a given acceptance window. There are three VT sets of the 'generic' type and three of the 'singular' type. The latter occur for one precise value of the size of the acceptance window. Any other VT set is a uniform scaling of the tiles listed here. Similar results, differing in detail, are provided for the sets of distinct Delone tiles DT. Altogether there are four different sets DT of Delone tiles.

In a previous paper (Yehia H M 1986 J. Mec. Theor. Appl. 5 55–71) we have introduced a method for constructing integrable conservative two-dimensional mechanical systems whose second integral of motion is polynomial in the velocities. This method has proved successful in constructing a multitude of irreversible systems (involving gyroscopic forces) with a second quadratic integral (Yehia H M 1992 J. Phys. A: Math. Gen. 25 197–221). The objective of this paper is to apply the same method for the systematic construction of mechanical systems with a cubic integral. As in our previous works, the configuration space is not assumed to be a Euclidean plane. This widens the range of applicability of the results to diverse mechanical systems to include such problems as rigid body dynamics. Several new reversible and irreversible integrable systems are obtained. Some of these systems generalize previously known ones by introducing additional parameters which may change either or both of the configuration manifold and the potential of the forces acting on the system. Other systems are completely new. An application is given to problems of rigid body dynamics. The famous classical integrable case due to Goriachev and Chaplygin and all its subsequent generalizations by several authors are further generalized to include certain variable gyroscopic forces that preserve a cubic integral. On the other hand, the above case and another less famous case of rigid body dynamics due to Goriachev and the Hall–Toda case of particle mechanics are all obtained as special cases, corresponding to different choices of certain parameters, from one more general system unifying them all.

We consider the Schr?dinger operator H = (i? + A)2 in the space L2(2) with a magnetic potential A(x) = a()(?x2, x1) |x|?2, where a is an arbitrary function on the unit circle. Our goal is to study spectral properties of the corresponding scattering matrix S(?), ? > 0. We obtain its stationary representation and show that its singular part (up to compact terms) is a pseudodifferential operator of zero order whose symbol is an explicit function of a. We deduce from this result that the essential spectrum of S(?) does not depend on ? and consists of two complex conjugated and perhaps overlapping closed intervals of the unit circle. Finally, we calculate the diagonal singularity of the scattering amplitude (kernel of S(?) considered as an integral operator). In particular, we show that for all these properties only the behaviour of a potential at infinity is essential. The preceding papers on this subject treated the case a() = const and used the separation of variables in the Schr?dinger equation in the polar coordinates. This technique does not, of course, work for arbitrary a. From an analytical point of view, our paper relies on some modern tools of scattering theory and well-known properties of pseudodifferential operators.

In a previous paper (Popov D 2001 J. Phys. A: Math. Gen. 34 5283) we have constructed the coherent states (CSs) of the pseudoharmonic oscillator (PHO). These states are of the Barut–Girardello type. In the present paper, we have constructed and investigated some properties of the photon-added Barut–Girardello coherent states (PA-BGCSs) of the PHO. These states are not orthogonal but are normalized and satisfy a unity resolution relation (the completeness relation). We have found the analytical form for the positive weight function in the resolution of unity. By using these states, we have calculated some expectation values (the first two powers of the number operator N), which have evinced some non-classical properties. In order to examine the statistical properties of a PHO gas, which obeys the quantum canonical distribution, the diagonal P-representation of the density operator ρ(m) is constructed and the thermal expectation values are calculated. All these quantities are expressed in terms of Meijer G-functions, and so the PA-BGCSs are a new field of application for these functions. Also, the time dependence of the PA-BGCSs is examined. If in the obtained results we put m = 0, we recover all the results from our previous paper.

A new direct and unified algebraic method for constructing multiple travelling wave solutions of general nonlinear evolution equations is presented and implemented in a computer algebraic system. Compared with most of the existing tanh methods, the Jacobi elliptic function method or other sophisticated methods, the proposed method not only gives new and more general solutions, but also provides a guideline to classify the various types of the travelling wave solutions according to the values of some parameters. The solutions obtained in this paper include (a) kink-shaped and bell-shaped soliton solutions, (b) rational solutions, (c) triangular periodic solutions and (d) Jacobi and Weierstrass doubly periodic wave solutions. Among them, the Jacobi elliptic periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition. The efficiency of the method can be demonstrated on a large variety of nonlinear evolution equations such as those considered in this paper, KdV–MKdV, Ito's fifth MKdV, Hirota, Nizhnik–Novikov–Veselov, Broer–Kaup, generalized coupled Hirota–Satsuma, coupled Schrodinger–KdV, (2 + 1)-dimensional dispersive long wave, (2 + 1)-dimensional Davey–Stewartson equations. In addition, as an illustrative sample, the properties of the soliton solutions and Jacobi doubly periodic solutions for the Hirota equation are shown by some figures. The links among our proposed method, the tanh method, extended tanh method and the Jacobi elliptic function method are clarified generally.

The aim of this work is to nd numerical solutions for dierential equations by expanding the unknown function in terms of Chebyshev polynomials and solving a system of linear equations directly for the values of the function at the extrema (or zeros) of the Chebyshev polynomial of order N. The solutions are exact at these points, apart from round-o com- puter errors and the convergence of other numerical methods used in connection to solving the linear system of equations. Applications to initial-value problems in time-dependent quantum eld theory, and second order boundary-value problems in fluid dynamics are pre- sented. of the functions rather than the Chebyshev coecients. The two approaches are formally equivalent in the sense that if we have the values of the function, the Chebyshev coecients can be calculated. In this paper we use the discrete orthogonality relation- ships of the Chebyshev polynomials to exactly discretize various continuous equations by reducing the study of the solutions to the Hilbert space of functions dened on the set of (N+1) extrema of TN(x), spanned by a dis- crete (N+1) term Chebyshev polynomial basis. In our approach we follow closely the procedures outlined by El-gendy (6) for the calculation of integrals, but extend his work to the calculation of derivatives. We also show that similar procedures can be applied for a second grid given by the zeros of TN(x). The paper is organized as follows: In Section II we re- view the basic properties of the Chebyshev polynomial and derive the general theoretical ingredients that allow us to discretize the various equations. The key element is the calculation of derivatives and integrals without ex- plicitly calculating the Chebyshev expansion coecients. In Sections III and IV we apply the formalism to obtain numerical solutions of initial-value and boundary-value problems, respectively. We accompany the general pre- sentation with examples, and compare the solution ob- tained using the proposed Chebyshev method with the numerical solution obtained using the nite-dierences method. Our conclusions are presented in Section V.

This paper discusses the connections between numerical methods for ordinary differential equations, fixed point iterations and sequence transformations. Books on chaos and fractals are concerned, on the one hand, with continuous dynamical systems, and, on the other hand, with the dynamics of iterations in discrete dynamical systems. There exists a strong connection between numerical methods for the integration of ordinary differential equations (ODEs) and fixed point iterations (FPIs). This connection has already been studied (see [14, chapter 5, pp 197–343] or [25, chapter 6, pp 165–201] for an introduction, and [30] for an extended review), but the dynamics of the iterations generated by a numerical method for ODEs was mainly used for understanding the behaviour of the solution of the differential equation. In this paper, we are mostly interested in FPIs. We will see that FPIs can be considered as coming from methods for the numerical integration of ODEs. Reciprocally, numerical methods for ODEs can be used in the solution of fixed point problems. So, FPIs can lead to new methods for the numerical integration of ODEs, and conversely. The link with sequence transformations, used in numerical analysis to accelerate the convergence, will also be discussed.

One of the central problems in the representation theory of compact groups concerns multiplicity, wherein an irreducible representation occurs more than once in the decomposition of the n-fold tensor product of irreducible representations. The problem is that there are no operators arising from the group itself whose eigenvalues can be used to label the equivalent representations occurring in the decomposition. In this paper we use invariant theory along with so-called generalized Casimir operators to show how to resolve the multiplicity problem for the U(N) groups. The starting point is to augment the n-fold tensor product space with the contragredient representation of interest and construct a subspace of U(N) invariants. The setting for this construction is a polynomial space embedded in a Fock space of complex variables which carries all the irreducible representations of U(N) (or GLN ). The dimension of the invariant subspace is equal to the multiplicity occurring in the tensor product decomposition. Generalized Casimir operators are operators from the universal enveloping algebra of outer product U(N) groups that commute with the diagonal U(N) action and whose eigenvalues can be used to label the multiplicity. Using the notion of dual representations we show how to rewrite the generalized Casimir operators and prove that they act invariantly on the invariant subspace. A complete set of commuting generalized Casimir operators can therefore be used to construct eigenvectors that form an orthonormal basis in the invariant subspace. Different sets of generalized commuting Casimir operators generate different orthonormal bases in the invariant subspace; the overlaps between the eigenvectors of different commuting sets of generalized Casimir operators are called invariant coefficients. We show that Racah coefficients are special cases of invariant coefficients in which the generalized Casimir operators have been chosen with respect to a definite coupling scheme in the tensor product. The paper concludes with an example of the threefold tensor product of the eight-dimensional irreducible representation of U(3) in which the multiplicity of the chosen irreducible representation is 6. Eigenvectors in the six-dimensional invariant subspace are computed for different sets of generalized Casimir operators and invariant coefficients, including Racah coefficients.

In this letter we continue the investigation of finite XXZ spin chains with periodic boundary conditions and odd number of sites, initiated in paper \cite{S}. As it turned out, for a special value of the asymmetry parameter $\Delta=-1/2$ the Hamiltonian of the system has an eigenvalue, which is exactly proportional to the number of sites $E=-3N/2$. Using {\sc Mathematica} we have found explicitly the corresponding eigenvectors for $N \le 17$. The obtained results support the conjecture of paper \cite{S} that this special eigenvalue corresponds to the ground state vector. We make a lot of conjectures concerning the correlations of the model. Many remarkable relations between the wave function components are noticed. It is turned out, for example, that the ratio of the largest component to the least one is equal to the number of the alternating sing matrices., Comment: Latex2e, 6 pages

In a previous paper (Lou S-y 1995 J. Phys. A: Math. Gen. 28 7227), a generalized dromion structure was revealed for the (2+1)-dimensional KdV equation, which was first derived by Boiti et al (Boiti M, Leon J J P, Manna M and Pempinelli F 1986 Inverse Problems 2 271) using the idea of the weak Lax pair. In this paper, using a Backlund transformation and the variable separation approach, we find there exist much more abundant localized structures for the (2+1)-dimensional KdV equation. The abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution. Some special types of dromion solution, lumps, breathers, instantons and the ring type of soliton, are discussed by selecting the arbitrary functions appropriately. The dromion solutions can be driven by sets of straight-line and curved-line ghost solitons. The dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves. The breathers may breathe both in amplitude and in shape.

This paper explores the connection between wavelet methods and an efficient computational algorithm - the discrete singular convolution (DSC). Many new DSC kernels are constructed and they are identified as wavelet scaling functions. Two approaches are proposed to generate wavelets from DSC kernels. Two well known examples, the Canny filter and the Mexican hat wavelet, are found to be special cases of the present DSC kernel-generated wavelets. A family of wavelet generators proposed in this paper are found to form an infinite-dimensional Lie group which has an invariant subgroup of translation and dilation. If DSC kernels form an orthogonal system, they are found to span a wavelet subspace in a multiresolution analysis.

This paper completes a series devoted to explicit constructions of finite-dimensional irreducible representations of the classical Lie algebras. Here the case of odd orthogonal Lie algebras (of type B) is considered (two previous papers dealt with C and D types). A weight basis for each representation of the Lie algebra o(2n+1) is constructed. The basis vectors are parametrized by Gelfand--Tsetlin-type patterns. Explicit formulas for the matrix elements of generators of o(2n+1) in this basis are given. The construction is based on the representation theory of the Yangians. A similar approach is applied to the A type case where the well-known formulas due to Gelfand and Tsetlin are reproduced., Comment: 29 pages, Latex

Time delays can occur naturally or as transport lags in many physico-chemical as well as biological systems. Incorporating them into a lumped parameter system results in a system of first-order ordinary delay-differential equations (DDEs). In this paper, we develop two-parameter periodic solutions near a Hopf point in such systems using the general reductive perturbation theory and apply the results to a nonisothermal chemical reactor with delayed feedback. The paper suggests that the two-parameter result can be generalized to multiple time delays and other parameters. Results of this work can be useful in constructing plane wave solutions, rotating waves, phase singularity and other interesting phenomena for temporal kinetic systems with time delays.

In an earlier paper a q-Schrodinger equation was obtained based on a particular quantization procedure, called Borel quantization, and a related q-deformation of the Witt algebra. This q-deformation is a deformation in the category of Lie algebras and hence the corresponding q-Witt algebra has a trivial Hopf algebra structure. In this paper, we extend the above algebra by the addition of a set of shift-type generators, which appear in the expression for the quantum mechanical position operator and hence lead to a new type of quantum kinematics. The latter gives rise to a new kind of evolution equation and it is shown that in the limit q1 a specific class of Schrodinger equations is obtained from it. This specification of a particular class is a new phenomenon, because in earlier references, where a different q-deformation has been implemented or no deformation has been used at all, such a class could not be determined uniquely. The extended algebra used here has a nontrivial Hopf structure. The appearance of the shift-type generator in the q-deformed picture hence leads to a selection of a particular type of dynamics and delivers in the limit q1 new information for the characterization of the corresponding dynamics in the undeformed situation.

In this paper we compute new classes of symmetry reduction and associated exact solutions of a generalized nonlinear Schrodinger equation (GNLS), the generalized terms modelling dispersion and scattering. Several authors have obtained symmetry reductions of one-, two- and three-dimensional nonlinear Schrodinger equations; in all cases to date reductions have been based on a real new independent variable. In this paper we compute reductions in which the new independent variable is complex. We seek new reductions from a two-dimensional GNLS to a PDE in two independent variables and also reductions to ODEs. Five new classes of reduction are found.

In this paper we answer the question, whether any Delone set , invariant under quasiaddition of Berman and Moody, can be identified with a cut and project quasicrystal. For any such set , we find an acceptance window , which is bounded but has only convex interior. The cut and project quasicrystal is then identified with an affine image of . Constructive methods used in the paper, allow one, in principle, to put bounds on from a given fragment of a Delone set.

Bifurcation diagrams and plots of Lyapunov exponents in the r- plane for Duffing-type oscillators exhibit a regular pattern of repeating self-similar `tongues' with complex internal structure. We demonstrate here how this behaviour is easily understood qualitatively and quantitatively from a Poincare map of the system in action-angle variables in the limit of large driving force or, equivalently, small driving frequency. This map approaches the one-dimensional form as derived in paper I. This second paper describes our approach to calculating the various constants and functions introduced in paper I. It gives numerical applications of the theory and tests its range of validity by comparison with results from the numerical integration of Duffing-type equations. Finally we show how to extend the range in the parameter space where the map is applicable.

This paper is the first of two papers devoted to the study of the Clebsch-Gordan (CG) problem for the three-dimensional Lorentz group in an elliptic (or SO(2)) basis. Here we describe the reduction of the tensor product of two unitary irreducible representations (UIRs) of the continuous series, i.e. belonging to either the principal or complementary series. The corresponding CG coefficients are defined as matrix elements of an intertwining operator between the tensor product representation and the irreducible component appearing in the decomposition. We then obtain an expression for CG coefficients in terms of a single function, namely in terms of the bilateral series with unit argument defined in the complex space of the variable . In the general case the functions are expressed in terms of two hypergeometric functions with unit argument; however, it reduces to the single function if at least one of the coupling UIRs belong to a discrete series. We derive a completeness relation for CG coefficients for all the cases under consideration.

In this paper we continue with the study of linear statistics of random Hermitian matrix ensembles. We generalize the result of a previous paper on the probability density function of linear statistics to the finite N ensembles where the interval of support of the eigenvalue spectrum is a single interval. Combining the Hankel determinant formula for orthogonal polynomials which are associated with Hermitian matrix ensembles and the linear statistics theorem for finite N, we obtain the strong or oscillatory asymptotics for the polynomials orthogonal with respect to weight functions supported on the real axis.