Singularity Analysis Method Research Articles

2A1-I06 マルチ駆動リニアモータを用いたパラレルリンク機構に関する研究 : 第1報:機構の形態と力学モデル(パラレルロボット・メカニズム)

Parallel link mechanisms with multi drive linear motors (MDLMs) are proposed. The multi drive is a control method of linear motors that multi moving parts are individually driven on a stator part. Various configurations of parallel link mechanisms are constructed on the MDLMs are proposed. The proposed mechanisms have additional features of wide moving area with original characteristics of the parallel mechanisms of rigid mechanisms, high precision and high speed. More over, they are suitable for force control because of low friction direct drive actuators. In this paper, kinematics and dynamics of 2-DOF (xy) with 2MDLMs, and 3-DOF (xyθ) with 3MDLMs (non-redundant) and 4MDLMs (redundant) planer parallel link mechanisms are investigated. Singularity analysis and internal force control method of the 3-DOF with 4MDLMs redundant mechanism are derived and confirmed by numerical simulations. Condition of a dynamical decoupling and constant inertia design of the 2-DOF with 2MDLMs and 3-DOF with 4MDLMs are derived.

Effective matrix formalism for singularity analysis of differential equations and new intergrable system in nonlinear elasticity

AbstractWe introduce a simple matrix formalism for Taylor series and generalized Laurent series that can be used for implementing the Taylor method for nonlinear ODEs and singularity analysis of differential equations. Advantages of this approach over conventional techniques are shown on model examples. Surprisingly, the same formalism can be used for proving C‐integrability of a 3D model in nonlinear elasticity. An alternative proof is obtained by using similarity between the model in nonlinear elasticity and the classic Pohlmeier‐Lund‐Regge model from high energy physics. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Generating Functions and the Solutions of Full History Recurrence Equations

Abstract We study a class of full history recurrence equations using generating functions. The generating functions are shown to satisfy a first order linear differential equation. Singularity analysis and saddle-point methods can then be employed to derive asymptotic approximation for the coefficients of the generating functions.

Multifractal imaging filtering and decomposition methods in space, Fourier frequency, and eigen domains

Abstract. The patterns shown on two-dimensional images (fields) used in geosciences reflect the end products of geo-processes that occurred on the surface and in the subsurface of the Earth. Anisotropy of these types of patterns can provide information useful for interpretation of geo-processes and identification of features in the mapped area. Quantification of the anisotropy property is therefore essential for image processing and interpretation. This paper introduces several techniques newly developed on the basis of multifractal modeling in space, Fourier frequency, and eigen domains, respectively. A singularity analysis method implemented in the space domain can be used to quantify the intensity and anisotropy of local singularities. The second method, called S-A, characterizes the generalized scale invariance property of a field in the Fourier frequency domain. The third method characterizes the field using a power-law model on the basis of eigenvalues and eigenvectors of the field. The applications of these methods are demonstrated with a case study of Environment Scan Electric Microscope (ESEM) microimages for identification of sphalerite (ZnS) ore minerals from the Jinding Pb/Zn/Ag mineral deposit in Shangjiang District, Yunnan Province, China.

An asymptotic expansion for the q-binomial series using singularity analysis for generating functions

For each k, 1≤k≤n, let denote the Gaussian (or q-binomial) Coefficients, where . Then the q-binomial series, or Rogers-Szegö polynomials, are defined as for fixed 0<q<1. The q-binomial series is one of a great variety of series that generalize corresponding hypergeometric series by replacing a number a by (1−q a )/(1−q), or basic number, recovering the hypergeometric series when q→∞. There exists a vast literature on hypergeometric and basic hyper geometric series as well as their applications in many disciplines. Our purpose in this paper is twofold. We first present a review of the concept of a basic analogue of a hypergeometric function and some applications of the q-binomial series and the Gauss coefficients. Then, we outline an important asymptotic analysis method called singularity analysis that can be applied to generating functions of combinatorial enumeration sequences in order to obtain asymptotic expressions of their coefficients (i.e. the sequence). Finally, we use this method in order to derive an asymptotic expression for the q-binomial series. Therefore, we present another example of the power of the singularity analysis method, giving accurate estimates for a frequently occurring hypergeometric expression.

On Asymptotics for the Signless Noncentral q‐Stirling Numbers of the First Kind

In this paper, we derive asymptotic formulas for the signless noncentral q‐Stirling numbers of the first kind and for the corresponding series. The signless noncentral q‐Stirling numbers of the first kind appear as coefficients of a polynomial of q‐number [t]q, expressing the noncentral ascending q‐factorial of t of order m and noncentrality parameter k. In this paper, we have two main purposes. The first is to give an expression by which we obtain the asymptotic behavior of these coefficients, using the saddle point method. The second main purpose is to derive an asymptotic expression for the signless noncentral q‐Stirling of the first kind series by using the singularity analysis method. We then apply our first formula to provide asymptotic expressions for probability functions of the number of successes in m trials and of the number of trials until the occurrence of the nth success in sequences of Bernoulli trials with varying success probability which are both written in terms of the signless noncentral q‐Stirling numbers of the first kind. In addition, we present some numerical calculations using the computer program MAPLE indicating that our expressions are close to the actual values of the signless noncentral q‐Stirling numbers of the first kind and of the corresponding series even for moderate values of m.

Singularity analysis of fine-tuning Stewart platform for large radio telescope using genetic algorithm

A new singularity analysis method for general six degree-of-freedom (DOF) Stewart platform using genetic algorithm (GA) is proposed in this paper. The Jacobian matrix of Stewart platform is first deduced, then the square of determinant of the Jacobian matrix is selected as the objective function, and the minimal of this objective function is searched in the workspace of Stewart platform by the GA. The singularity of Stewart platform depends on this minimal objective function: if this value is zero, the singularity of Stewart platform will take place, otherwise, the Stewart platform is singularity-free. The effectiveness of this new genetic singularity analysis method is validated by the singularity analysis of a six-DOF fine-tuning Stewart platform for the next generation large radio telescope. The results have shown that the fine-tuning Stewart platform is singularity-free, which has laid a solid base for the requirement of high precision trajectory tracking for the next generation large radio telescope.

High-order differential approximants

We introduce a new form of differential approximant for the summation of power series. The method is a special type of Padé–Hermite approximant. It consists of a high-order linear differential equation with polynomial coefficients that is satisfied approximately by the partial sum of the power series. This method is able to reproduce the polylogarithmic functions exactly. Numerical evidence suggests that this is currently one of the best methods of singularity analysis for many problems.

AdS box graphs, unitarity and operator product expansions

We develop a method of singularity analysis for conformal graphs which, in particular, is applicable to the holographic image of AdS supergravity theory. It can be used to determine the critical exponents for any such graph in a given channel. These exponents determine the towers of conformal blocks that are exchanged in this channel. We analyze the scalar AdS box graph and show that it has the same critical exponents as the corresponding CFT box graph. Thus pairs of external fields couple to the same exchanged conformal blocks in both theories. This is looked upon as a general structural argument supporting the Maldacena hypothesis.

Bäcklund transformation of partial differential equations from the Painlevé-Gambier classification. II. Tzitzéica equation

From the existing methods of singularity analysis only, we derive the two equations which define the Bäcklund transformation of the Tzitzéica equation. This is achieved by defining a truncation in the spirit of the approach of Weiss et al., so as to preserve the Lorentz invariance of the Tzitzéica equation. If one assumes a third-order scattering problem, this truncation admits a unique solution, thus leading to a matrix Lax pair and a Darboux transformation. In order to obtain the Bäcklund transformation (BT), which is the main new result of this paper, one represents the Lax pair by an equivalent two-component Riccati pseudopotential. This yields two different BTs; the first one is a BT for the Hirota–Satsuma equation, while the second one is a BT for the Tzitzéica equation. One of the two equations defining the BT is the fifth ordinary differential equation of Gambier.

A centro-based characterization of singularities in the workspace of planar closed-loop manipulators

The paper presents a centro-based characterization of singularities in planar multi-DOF mechanisms with arbitrary kinematic structure. A multi-DOF mechanism is treated as an assembly of several single DOF link sets and hence the terms like affected link set, active link set and active centro have been defined. Criteria for end-effector dependent and end-effector independent singularities of a closed-loop manipulator with given output link have been established in the paper in terms of centroes. Based on the relative disposition of the centroes, the singularities have been classified into five types and their physical implications have been explained. End-effector dependent singularities are shown to lie on the coupler curve of a larger single DOF mechanism. A simple linkage transformation has been proposed for the study of manipulators with physical limits on joint motion under the general framework of close-loop manipulators. Several examples have been worked out to demonstrate the centro-based method of singularity analysis.

Singularity Analysis and Boundary Integral Equation Method for Frictional Crack Problems in Two-Dimensional Elasticity

This study investigates the stress singularities in the neighborhood of the tip of a sliding crack with Coulomb-type frictional contact surfaces, and applies the boundary integral equation method to solve some frictional crack problems in plane elasticity. A universal approach to the determination of the complex order of stress singularity is established analytically by using the series expansion of the complex stress functions. When the cracks are open, or when no friction exists between the upper and lower crack faces, our results agree with those given by Williams. When displacement and traction are prescribed on the upper and lower crack surfaces (or vice versa), our result agrees with those by Muskhelishvili. For the case of a closed crack with frictional contact, the only nonzero stress intensity factor is that for pure shear or sliding mode. By using the boundary integral equation method, we derive analytically that the stress intensity factor due to the interaction of two colinear frictional cracks under far field biaxial compression can be expressed in terms of E(k) and K(k) (the complete elliptic integrals of the first and second kinds), where k=[1-(a/b)2]1/2 with 2a the distance between the two inner crack tips and b- a the length of the cracks. For the case of an infinite periodic colinear crack array under remote biaxial compression, the mode II stress intensity factor is found to be proportional to [2b tan(π a/2b)]1/2 where 2a and 2b are the crack length and period of the crack array.

Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations

The integrability of systems of ordinary differential equations with polynomial vector fields is investigated by using the singularity analysis methods. Three types of results are obtained. First, a general relationship between the degrees of first integrals and the so-called Kowalevskaya exponents is derived. Second, it is shown that all solutions of algebraically integrable systems can be expanded in Puiseux series. Third, a new method to study partially integrable systems is studied. These different aspects allow us to study algorithmically the integrability, partial integrability, and nonintegrability of differential systems.

On Ramanujan's Q-function

This study provides a detailed analysis of a function which Knuth discovered to play a central rôle in the analysis of hashing with linear probing. The function, named after Knuth Q( n), is related to several of Ramanujan's investigations. It surfaces in the analysis of a variety of algorithms and discrete probability problems including hashing, the birthday paradox, random mapping statistics, the “rho” method for integer factorization, union-find algorithms, optimum caching, and the study of memory conflicts. A process related to the complex asymptotic methods of singularity analysis and saddle point integrals permits to precisely quantify the behaviour of the Q( n) function. In this way, tight bounds are derived. They answer a question of Knuth ( The Art of Computer Programming, Vol. 1, 1968, [Ex. 1.2.11.3.13]), itself a rephrasing of earlier questions of Ramanujan in 1911–1913.

New method for confluent singularity analysis of power series

A new method for calculating critical parameters from power series expansions, recently developed by the authors, is modified to take into account confluent singularities. The new procedure allows one to obtain wholly unbiased approaches to all physically meaningful critical parameters. The high- temperature, zero-field magnetic susceptibility series for the spin-1/2 Ising model for the face-centered cubic lattice is discussed as an illustrative example. Present results compare favorably with previously reported ones and agree closely with bow from renormalkation group theory.

Soliton structure of the Drinfel’d–Sokolov–Wilson equation

An integrable equation due to Drinfel’d and Sokolov [Sov. Math. Dokl. 23, 457 (1981)] and Wilson [Phys. Lett. A 89, 332 (1982)] (DSW) is studied in detail. It is shown how this system can be obtained as a six-reduction of the Kadomtsev–Petviashvili hierarchy. This equation presents a novel type of solutions called static solitons: they are static solutions that interact with moving solitons without deformations. Examples of such solutions are given, together with a general procedure for their construction. Finally the Painlevé analysis of the DSW equation is performed directly on the bilinear form, which constitutes a new application of the singularity analysis method.

On the complete and partial integrability of non-Hamiltonian systems

The methods of singularity analysis are applied to several third order non-Hamiltonian systems of physical significance including the Lotka-Volterra equations, the three-wave interaction and the Rikitake dynamo model. Complete integrability is defined and new completely integrable systems are discovered by means of the Painlevé property. In all these cases we obtain integrals, which reduce the equations either to a final quadrature or to an irreducible second order ordinary differential equation (ODE) solved by Painlevé transcendents. Relaxing the Painlevé property we find many partially integrable cases whose movable singularities are poles at leading order, with In( t - t 0 ) terms entering at higher orders. In an N th order, generalized Rössler model a precise relation is established between the partial fulfillment of the Painlevé conditions and the existence of N - 2 integrals of the motion.

Hyperscaling in the three-dimensional Ising model

We analyze the 21st-order series of Nickel for the susceptibility and the correlation length of the spin-$s$ Ising model for the bcc lattice with the use of an unbiased method of confluent singularity analysis tailored to the loose-packed lattices. This modified five-fit method of analysis assumes a parametrization which includes one confluent correction, e.g., for the susceptibility $\ensuremath{\chi}={\mathrm{At}}^{\ensuremath{-}\ensuremath{\gamma}}(1+{\mathrm{Bt}}^{{\ensuremath{\Delta}}_{1}})$, where $t=1\ensuremath{-}\frac{{T}_{c}}{T}$. The method determines sequences for these five parameters, e.g., ${\mathrm{lim}}_{n\ensuremath{\rightarrow}\ensuremath{\infty}}{\ensuremath{\gamma}}_{n}=\ensuremath{\gamma}$, by equating five consecutive even or odd coefficients in the perturbation series with the five corresponding coefficients in the expansion of the assumed parametrization and by solving the five equations for the five parameters numerically. Our analysis of the spin-$s$ Ising-model series shows that the indices $\ensuremath{\gamma}$ and $2\ensuremath{\nu}$ are smaller than indicated by early analysis on shorter series and that the confluent correction is apparently not significant for spin-\textonehalf{}, validating the use of ratio methods for the spin-\textonehalf{} series. Considering the results of both the confluent singularity analysis and the standard analysis of the spin-\textonehalf{} series, we estimate $\ensuremath{\gamma}=1.242 (+0.003, \ensuremath{-}0.005)$ and $2\ensuremath{\nu}=1.268(+0.006, \ensuremath{-}0.008)$, in basic agreement with renormalization-group calculations, hyperscaling, and the biased analysis of Nickel's series. Test-function analysis demonstrates the reliability of the method. Estimates for the leading index represent a significant improvement over older methods, e.g., the ratio method. Estimates for the correction exponent and amplitude are consistently high, presumably because the method must try to have one correction mimic all the corrections to the leading singularity.