Form Of Power Series Expansion Research Articles

Noncommutative QCD, first-order-in-θ-deformed instantons and 't Hooft vertices

For commutative Euclidean time, we study the existence of field configurations that {\it a)} are formal power series expansions in $h\theta^{\m\n}$, {\it b)} go to ordinary (anti-)instantons as $h\theta^{\m\n}\to 0$, and {\it c)} render stationary the classical action of Euclidean noncommutative SU(3) Yang-Mills theory. We show that the noncommutative (anti-)self-duality equations have no solutions of this type at any order in $h\theta^{\m\n}$. However, we obtain all the deformations --called first-order-in-$\theta$-deformed instantons-- of the ordinary instanton that, at first order in $h\theta^{\m\n}$, satisfy the equations of motion of Euclidean noncommutative SU(3) Yang-Mills theory. We analyze the quantum effects that these field configurations give rise to in noncommutative SU(3) with one, two and three nearly massless flavours and compute the corresponding 't Hooft vertices, also, at first order in $h\theta^{\m\n}$. Other issues analyzed in this paper are the existence at higher orders in $h\theta^{\m\n}$ of topologically nontrivial solutions of the type mentioned above and the classification of the classical vacua of noncommutative SU(N) Yang-Mills theory that are power series in $h\theta^{\m\n}$.

Algebraic expansions for curvature-coupled scalar field models

A late time asymptotic perturbative analysis of curvature-coupled complex scalar field models with accelerated cosmological expansion is carried out on the level of formal power series expansions. For this, algebraic analogues of the Einstein scalar field equations in Gaussian coordinates for spacetime dimensions greater than two are postulated and formal solutions are constructed inductively and shown to be unique. The results obtained this way are found to be consistent with already known facts on the asymptotics of such models. In addition, the algebraic expansions are used to provide a prospect of the large time behaviour that might be expected of the considered models.

Limitation of application of the center manifold reduction in aeroelasticity

In this paper, the method of center manifold reduction is applied to the limit cycle calculations of a three-dimensional thin airfoil placed in an incompressible flow. Limit cycle oscillations are caused by a cubic structural restoring force corresponding to the aileron rotation. The equation of motion is written as an integro-differential equation and also as an approximate set of ordinary differential equations. Two different implementations of the method of center manifold reduction for these two cases are briefly outlined. It is emphasized, that the formal power series expansions used in the method of center manifold reduction typically diverge and cause the method not to give satisfactory results. An example is presented, when the method of center manifold reduction cannot even qualitatively predict the occurrence of a stable limit cycle of a small amplitude.

Clifford-algebraic random walks on the hypercube

The n-dimensional hypercube is a simple graph on 2n vertices labeled by binary strings, or words, of length n. Pairs of vertices are adjacent if and only if they differ in exactly one position as binary words; i.e., the Hamming distance between the words is one. A discrete-time random walk is easily defined on the hypercube by “flipping” a randomly selected digit from 0 to 1 or vice-versa at each time step. By associating the words as blades in a Clifford algebra of particular signature, combinatorial properties of the geometric product can be used to represent this random walk as a sequence within the algebra. A closed-form formula is revealed which yields probability distributions on the vertices of the hypercube at any time k ≥ 0 by a formal power series expansion of elements in the algebra. Furthermore, by inducing a walk on a larger Clifford algebra, probabilities of self-avoiding walks and expected first hitting times of specific vertices are recovered. Moreover, because the Clifford algebras used in the current work are canonically isomorphic to fermion algebras, everything appearing here can be rewritten using fermion creation/annihilation operators, making the discussion relevant to quantum mechanics and/or quantum computing.

A REDUCE program for the normalization of polynomial Hamiltonians

The program LINA01 is proposed for the direct and the inverse normalization of Hamiltonian systems and for the calculation of formal integrals of motion of them. The calculations required in LINA01 are made on the basis of Lie canonical transformation method. The program package of LINA01 is written on REDUCE. Program summary Title of program:LINA01 Catalogue identifier:ADUV Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADUV Program obtainable from:CPC Program Library, Queen's University of Belfast, N. Ireland Computer: IBM PC PENTIUM 4/2.40 GHz 512 Mb Operating systems under which the program has been tested: Windows XP Programming language used: REDUCE vs. 3.7 No. of lines in distributed program, including test data, etc.:485 No. of bytes in distributed program, including test data, etc.:4320 Distribution format:tar.gz Nature of physical problem. The transformation bringing a given Hamiltonian function into the normal form (namely, the normalization) is one of the conventional methods for non-linear Hamiltonian systems [A.J. Lichtenberg, M.A. Lieberman, Regular and Stochastic Motion, Springer-Verlag, Berlin, 1983; G.D. Birkhoff, Dynamical Systems, A.M.S. Colloquium Publications, New York, 1927; F. Gustavson, Astron. J. 71 (1966) 670; G.I. Hori, Astron. Soc. Japan 18 (1966) 287; A. Deprit, Cel. Mech. 1 (1969) 12; A.A. Kamel, Cel. Mech. 3 (1970) 90]. Recently, beyond classical mechanics, the normal form method has been applied to quantization of chaotic Hamiltonian systems with the aim of finding quantum signature of chaos [L.E. Reichl, The Transition to Chaos. Conservative Classical Systems: Quantum Manifestations, Springer, New York, 1992]. Besides those utilities, the normalization requires quite cumbersome algebraic calculations of polynomials, so that the computer algebraic approach is worth studying to promote further investigations around the normalization together with the ones around the inverse normalization. Method of solution. The canonical transformation proceeding the normalization is expressed in terms of the Lie transformation power series, which is also referred to as the Hori–Deprit transformation. After (formal) power series expansion as above, the fundamental equation of the normalization is solved for the normal form together with the generating function of transformation recursively from degree-3 to the degree desired to be normalized. The generating function thus obtained is applied to the calculation of (formal) integrals of motion. Restrictions due to the complexity of the problem. The computation time rises in a combinatorial manner as the desired degree of normalization does. Especially, such a combinatorial growth of computation is more significant in the inverse normalization than in the direct one. The hardware (processor and memory, for example) available for the computation may restrict either the degree of normalization or the computation time.

On inner constraints in plane circular arches

A one-dimensional model of plane circular arches with rigid sections is introduced. Suitable strain measures are defined as deviations from rigid displacements. If the arch is thin, constitutive arguments make the shearing strain negligible. Hence, the shearing indeformability will be assumed as inner constraint. By means of a formal power series expansion of the exact measures of deformation it is shown that the shearing indeformability implies some constraints on the axial strain. In particular, the first-order axial strain must vanish in the case of infinitesimal displacements. The same procedure is applied to pure flexible arches, in order to compare the two sets of results. It is shown that the hypothesis of finite pure flexibility is not compatible with small deformations of the arch. An example is provided to evaluate the effects of the two constraints at the first non-linear step of the perturbation expansions.

Renormalization group flows and continual Lie algebras

We study the renormalization group flows of two-dimensional metrics in sigma models and demonstrate that they provide a continual analogue of the Toda field equations based on the infinite dimensional algebra G(d/dt;1). The resulting Toda field equation is a non-linear generalization of the heat equation, which is integrable in target space and shares the same dissipative properties in time. We provide the general solution of the renormalization group flows in terms of free fields, via Backlund transformations, and present some simple examples that illustrate the validity of their formal power series expansion in terms of algebraic data. We study in detail the sausage model that arises as geometric deformation of the O(3) sigma model, and give a new interpretation to its ultra-violet limit by gluing together two copies of Witten's two-dimensional black hole in the asymptotic region. We also provide some new solutions that describe the renormalization group flow of negatively curved spaces in different patches, which look like a cane in the infra-red region. Finally, we revisit the transition of a flat cone C/Z_n to the plane, as another special solution, and note that tachyon condensation in closed string theory exhibits a hidden relation to the infinite dimensional algebra G(d/dt;1) in the regime of gravity. Its exponential growth holds the key for the construction of conserved currents and their systematic interpretation in string theory, but they still remain unknown.

Application of the Floquet theory to multiple quantum NMR of dipolar-coupled multi-spin systems under magic angle spinning

A new analytical Liouville-space representation of the time-propagator under magic angle spinning (MAS) is introduced using the formalized quantum Floquet theory. This approach has the advantage that it is applicable to the analysis of any type of NMR experiment where MAS is combined with multiple-pulse excitation. General relationships describing the spectral parameters in multiple-quantum (MQ) MAS spectra are derived in this representation. Their use is illustrated with an application to double-quantum (DQ) NMR spectra of dipolar-coupled multi-spin systems. Corresponding to the separation of the MAS time-propagator into a rotor modulated and a dephasing component, two distinct mechanisms for DQ excitation are identified. One of them exploits the rotor-modulated component to excite DQ coherences through dipolar-recoupling techniques, which are familiar for spin pairs. Analytical expressions of the integral intensities and linewidths in the resulting DQ sideband pattern are derived in the form of power series expansions of the inverse rotor frequency, of which coefficients depend on structural parameters. In a multi-spin system they can most reliably be extracted in the fast spinning regime. The other mechanism exploits the dephasing component, which is characteristic to multi-spin systems only. This is shown to give rise to DQ coherences by free evolution at full rotor periods. The possibility to exploit it for selective excitation of higher order MQ coherences is discussed. In either case, the dephasing component also leads to residual broadening. The main results of the theoretical developments are demonstrated experimentally on adamantane.

Dynamical system analysis for the Einstein–Yang–Mills equations

Local solutions of the static, spherically symmetric Einstein–Yang–Mills (EYM) equations with SU(2) gauge group are studied using dynamical systems methods. This approach enables us to classify EYM solutions in a neighborhood of the origin, to prove the existence of solutions with an oscillating metric as well as the existence of local solutions for all known formal power series expansions, to study the extendibility of solutions, and to find some new local singular solutions.

An asymptotic theory of thin micropolar plates

The asymptotic expansion technique is used to obtain the two-dimensional dynamic equations of thin micropolar elastic plates from the three-dimensional dynamic equations of micropolar elasticity theory. To this end, all the field variables are scaled via an appropriate thickness parameter such that it reflects the expected behavior of the plate. A formal power series expansion of the three-dimensional solution is used by considering the thickness parameter as a small parameter. Without any a priori assumption on the form of the field variables, it is shown that the zeroth-order approximation simultaneously includes both the plate equations previously presented in the literature and the standard assumptions on the specific forms of the field variables. Some aspects of the present asymptotic plate equations are also discussed.

Stress gradient calculations at notches

Calculations of the relative stress gradient ahead of notches are presented. Attention has been focused on finite width plates under tension weakened either by a central hole or by two semicircular lateral notches. In this paper stress field solutions given by Neuber as well as analytical solutions in the form of power series expansions, given by Howland (central hole) and Ling (semi-circular notches), have been employed in the derivation of the stress gradient. Comparisons have been made with gradient calculations derived from approximate stress field formulas developed by Xu et al. (Practical stress expressions for stress concentration regions. Fatigue Fract Eng Mater Struct 1995;18(7/8)885–95) and by Glinka and Newport (Universal features of elastic notch-tip stress fields. Int J Fatigue 1987;9(3):143–50). It is shown that the effect of the finite dimensions of the specimen on the stress gradient is correctly captured only if exact analytical solutions are employed. The results presented in this paper are useful for calculating the stress intensity factors for cracks growing in notches and to assess the fatigue strength of notched components.

Approach towardsN-nucleon effective generators of the Poincaré group derived from a field theory

It is shown that the ten Hermitian generators of the Poincar\'e group derived from standard Hermitian Lagrangians which describe interacting fields can be block diagonalized by one and the same unitary transformation such that the space of a fixed number of nucleons is separated from the rest of the space. The existence proof is carried through using a formal power series expansion in the coupling constant to all orders. In this manner one arrives at effective Hermitian generators of the Poincar\'e group which act in the two subspaces separately.

A series solution for nonlinear differential equations using delta operators

In this paper, we shall generalize our previous results [1] to the case of series expansion in powers of several polynomials. For this, we shall extend the ideas of delta operators and their basic polynomial sequences, introduced in conjunction with the algebra (over a field of characteristic zero) of all polynomials in one variable [2] to the algebra (over a field of characteristic zero) of all polynomials in n indeterminates. We apply this technique to derive the formal power series expansion of the input-output map describing a nonlinear system with polynomial inputs.

The motion of a fast spinning disc which comes out from the limiting case γ″ 0 ≈ 0

In this paper, the problem of motion of a rigid body about a fixed point in a central Newtonian force field is studied for a singular value of the natural frequency (ω = 1) being excluded from the limiting case γ″ 0 ≈ 0 [1]. Poincaré's small parameter method [2–4] is used to investigate periodic solutions of a quasilinear autonomous system in the form of power series expansions containing assumed small parameter. Such a motion is analysed geometrically using Euler's angles to describe the orientation of the body at any instant of time. A Programm is worked out to represent the obtained analytical solutions graphically. On the other hand, the quasilinear autonomous system is solved numerically using the fourth-order Runge-Kutta method [5] and the graphical representations of such solutions are obtained through other programm. At the end part of this paper, the analytical and the numerical solutions are compared.

Analytical series expressions for Hantush's M and S functions

M. S. Hantush established relationships between the dynamics of groundwater mounding beneath recharge zones and two integral functions, M and S. Exact algebraic expressions for these functions are developed in terms of a formal power series expansion. This expansion may be reordered to provide two independent analytical partial summations involving elementary functions. The convergence characteristics of these two formulae are discussed and compared with numerical quadratures of M and, hence, S. The algebraic expressions are used to generate identities for related integrals. Compact algebraic approximations to M and S can be deduced from the series expansions with essentially arbitrary accuracy, while retaining valuable functional information. For example, a simple two‐term truncated sum yields a reasonable approximation to M over a useful range of arguments. The results are amenable for use in further theoretical studies of groundwater percolation and mounding where numerical quadratures may be undesirable.

Nonlinear ripples of Kelvin–Helmholtz type which arise from an interfacial mode interaction

An analysis is made of the small-amplitude capillary–gravity waves which occur on the interface of two fluids and which arise out of the interaction between the Mth and Nth harmonics of the fundamental mode. The method employed is that of multiple scales in both space and time and a pair of coupled nonlinear partial differential equations for the slowly varying wave amplitudes is derived. These equations describe, correct up to third order, the progression of a wavetrain and are generalizations of the nonlinear Schrödinger-type equations used by many authors to model wave propagation. The equations are solved and formal power series expansions of the corresponding wave profiles obtained. Many different wave configurations can arise, some symmetric others asymmetric. It is found that an important influence on the type of waves which can occur is whether the ratio of the interacting wave modes is greater or less than two. Finally, an examination of the stability of the waves to plane wave perturbations is carried out.

The motion of fast spinning rigid body about a fixed point with definite natural frequency

In this paper, the problem of motion of a rigid body about a fixed point under the action of a Newtonian force field is studied when the natural frequency value w = 1 3 . This singularity appears in [1] and deals with different bodies being classified according to the moments of inertia. Using Poincaré's small parameter method [2], periodic solutions — with zero basic amplitudes — of the quasilinear autonomous system are obtained in the form of power series expansions, up to the third approximation, containing assumed small parameter. Also, Lagrange's gyroscope and Euler's one are derived as special cases from our solutions. At the end, the geometric interpretation of motion using Euler's angles is considered to show that the resulted motion is of regular precession type which depends on four arbitrary constants.

Gevrey series and dynamic bifurcations for analytic slow-fast mappings

The surprising effects of slow sweep of a parameter in a family of dynamical systems exhibiting bifurcations are known as dynamic bifurcations. In this paper we study dynamic bifurcations in the context of analytic slow-fast mappings of the form Fv(x, lambda )=(f(x, lambda ), lambda +v) for small v in the neighbourhood of an analytic curve x=y( lambda ) of fixed points of f(., lambda ). Firstly, we construct a formal power series expansion U in powers of v for invariant curves lambda to U( lambda ,v) close to the curve of fixed points, and prove it is Gevrey-1. Secondly we show how the behaviour of the orbits, and in particular the existence of a delay for the dynamic bifurcation, can be derived as a direct consequence of the analytical properties of this formal series.

Power Spectral Density of Nonlinear System Response: The Recursion Method

Spectral densities of the response of nonlinear systems to white noise excitation are considered. By using a formal solution of the associated Fokker-Planck-Kolmogorov equation, response spectral densities are represented by formal power series expansion for large frequencies. The coefficients of the series, known as the spectral moments, are determined in terms of first-order response statistics. Alternatively, a J-fraction representation of spectral densities can be achieved by using a generalization of the Lanczos algorithm for matrix tridiagonalization, known as the “recursion method.” Sequences of rational approximations of increasing order are obtained. They are used for numerical calculations regarding the single-well and double-well Duffing oscillators, and Van der Pol type oscillators. Digital simulations demonstrate that the proposed approach can be quite reliable over large variations of the system parameters. Further, it is quite versatile as it can be used for the determination of the spectrum of the response of a broad class of randomly excited nonlinear oscillators, with the sole prerequisite being the availability, in exact or approximate form, of the stationary probability density of the response.

Expansion of a ratio of hypergeometric functions of two variables in branching continued fractions

We obtain two-dimensional analogs of the continued fractions of Gauss which are an expansion of the ratio of the Appel hypergeometric functions. It is proved that these fractions are those corresponding to the formal power series expansion of the given ratio. Convergence criteria are established for the branching continued fractions under consideration.