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Lyra geometry is a conformai geometry that originated from Weyl geometry. In this article, we derive the exterior field equation under a spherically symmetric gauge function [x.sup.0](r) and metric in Lyra geometry. When we impose a specific form of the gauge function [x.sup.0](r), the radial differential equation of the metric component [g.sub.00] will possess an irregular singular point (ISP) at r = 0. Moreover, we can apply the method of dominant balance to get the asymptotic behavior of the new space-time solution. The significance of this work is that we can use a series of smooth gauge functions [x.sup.0](r) to modulate the degree of divergence of the singularity at r =0, which will become a naked singularity under certain conditions. Furthermore, we investigate the physical meaning of this novel behavior of space-time in Lyra geometry and find out that no spaceship with finite integrated acceleration can arrive at this singularity at r =0. The physical meaning of the gauge function and integrability is also discussed. Key words: Lyra geometry, Weyl geometry, singularity, event horizon, dominant balance. La geometrie de Lyra est une geometrie conforme qui origine de la geometrie de Weyl. Dans ce travail, nous derivons l'equation du champ exterieur sous une fonction de jauge [x.sup.0](r) de symetrie spherique et une metrique en geometrie de Lyra. Quand nous imposons une forme specifique a la fonction de jauge [x.sup.0](r), l'equation differentielle radiale de la composante metrique [g.sub.00] possede alors un point singulier irregulier (PSI/ISP) a r = 0. De plus, nous appliquons la methode du comportement asymptotique dominant pour obtenir la forme asymptotique de notre nouvelle solution espace-temps. La signification de ce travail est que nous pourrions utiliser une serie de fonctions de jauge continues [x.sup.0](r) pour moduler le niveau de divergence de la singularite a r = 0, avec comme resultat que la singularite devient alors une singularite nue sous certaines conditions. De plus, nous analysons la signification physique de l'espace-temps dans la geometrie de Lyra et nous trouvons qu'aucun vaisseau spatial d'acceleration integree finie ne peut parvenir a la singularite a r =0. Nous discutons aussi la signification physique de la fonction de jauge et l'integrabilite. [Traduit par la Redaction] Mots-cles : geometrie de Lyra, singularite, horizon des evenements, comportement asymptotique dominant, geometrie de Weyl., 1. Introduction Weyl [1] created his version of generalized Riemannian geometry, Weyl geometry, to try to unify gravitation and electromagnetism. The structure of Weyl geometry is insightful because it has [...]

Distribution of the electric potential in a very long plate (for example, a long metal ruler) is determined. This is achieved by conformally mapping the plate into a plane, simplifying the geometry of the boundary conditions. Singularities of the potential are discussed as well as their regularization by the final size of electrical contacts. An analogy with renormalization is pointed out. Results are compared with previous studies. PACS Nos.: 01.55.+b, 03.50.De. Nous determinons la distribution du potentiel electrique dans une plaque tres longue (comme une regle metallique). Par transformation conforme, la regle devient un plan, simplifiant la geometrie des conditions limites. Nous analysons les singularites du potentiel, ainsi que leur regularisation suivant la dimension finie des contacts. Nous soulignons une analogie avec la renormalisation. Nous comparons nos resultats avec des etudes anterieures. [Traduit par la Redaction], 1. Introduction This paper is motivated by an intriguing question posed in an excellent collection of physics problems [1], based on an exam problem from the prestigious Moscow Institute of [...]

We investigate f(R, T) gravity models (where R is the curvature scalar and T is the trace of the stress-energy tensor of ordinary matter) that are able to reproduce the four known types of future finite-time singularities. We choose a suitable expression for the Hubble parameter to realise the cosmic acceleration and we introduce two parameters, [alpha] and [H.sub.s], which characterise each type of singularity. We address the conformal anomaly and we observe that it cannot remove the sudden singularity or the Big Brake, but, for some values of [alpha], the Big Rip and the Big Freeze may be avoided. We also find that, even without taking into account the conformal anomaly, the Big Rip and the Big Freeze may be removed thanks to the presence of the T contribution of the f(R, T) theory. PACS Nos.: 04.50.Kd, 95.35.+d, 95.36.+X, 98.80.Qc. Nous analysons les modeles de gravite f(R, T), ou R est la courbure scalaire et T la trace du tenseur d'energieimpulsion, qui sont capables de reproduire les quatre types connus de singularites des temps futurs. Nous choisissons une expression raisonnable pour le parametre de Hubble, de facon a realiser l'acceleration cosmique et nous introduisons deux parametres, [alpha] et [H.sub.s] qui caracterisent chaque type de singularite. Nous nous penchons sur l'anomalie conforme et nous observons qu'elle ne peut eliminer la Singularite Brusque ou le Grand Gel, mais pour certaines valeurs de [alpha], on evite la Grande Dechirure et le Grand Gel. Nous trouvons aussi qu'il est possible d'eliminer la Grande Dechirure et le Grand Gel sans tenir compte de l'anomalie conforme, simplement par la contribution de T a la theorie f(R, T). [Traduit par la Redaction], 1. Introduction Observation indicates that the current expansion of the universe is accelerating [1, 2]. There are two large branches of cosmology attempting to explain this phenomenon [3-12] using two [...]

We investigate models of future finite-time singularities in f(T) theory, where T is the torsion scalar. The algebraic function f(T) is the teleparallel term, T, plus an arbitrary function, g(T). A suitable expression for the Hubble parameter is assumed and constraints are imposed to provide an expanding universe. Two parameters, [beta] and [H.sub.s], that appear in the Hubble parameter are relevant in specifying the types of singularities. Differential equations of g(T) are established and solved, leading to algebraic f(T) models for each type of future finite-time singularity. Moreover, we take into account the viscosity in the fluid and discuss three interesting cases: constant viscosity, viscosity proportional to [square root of (-T)], and the general one where the viscosity is proportional to [(-T).sup.n/2], where n is a natural number. We see that for the first and second cases, in general, the singularities are robust against the viscous fluid, while for the general case, the Big Rip and the Big Freeze can be avoided from the effects of the viscosity for some values of n. PACS Nos.: 95.36.+x, 04.50.Kd, 66.20.-d. Nous analysons la theorie des modeles de singularites de temps futur fini, f(T)ou T est le scalaire de torsion. La fonction algebrique est posee comme etant le terme teleparallele de T plus une fonction g(T). Nous supposons une expression adequate pour le parametre de Hubble et nous imposons des contraintes de facon a avoir un univers en expansion. Les deux parametres [beta] et [H.sub.s] qui apparaissent dans l'expression pour le parametre de Hubble sont pertinents pour specifier les types de singularite. Nous etablissons et solutionnons les equations differentielles pour g(T), menant aux modeles algebriques f(T) pour chaque type de singularite de temps futur fini. Nous tenons egalement compte de la viscosite du fluide et discutons trois cas interessants: viscosite constante, viscosite proportionnelle a [square root of (-T)] et un cas general ou la viscosite est proportionnelle a [(-T).sup.n/2], ou n est un entier naturel. Pour les deux premiers cas, generalement les singularites sont robustes devant la viscosite du fluide, alors que pour le cas general, la viscosite correspondant a certaines valeurs de n permet d'eviter la Grande Dechirure et le Grand Gel. [Traduit par la Redaction], 1. Introduction Recent observations of type Ia supernovae (SNIa) and Wilkinson microwave anisotropy probe (WMAP) [1, 2] indicate that our universe is currently undergoing an accelerating expansion, which confronts the [...]

In this paper we study a gravitational theory in which tachyons play the role of a scalar field. Also, nonminimal derivative coupling between tachyons and the Einstein tensor is considered in the model. We show that at the initial time, tachyon field behavior is exponential and its behavior at late times is liner in cosmic time. We investigate the asymptotic behavior of scale factor a(t) = [e.sup.α(t)] and show that in the limit t [right arrow]∞ to it is impossible to have [??] constant a and the model has an initial cosmological singularity (i.e., a [varies] [(t - [t.sub.i]).sup.2/β]) in the limit t - [t.sub.i]. PACS Nos: 95.36.+x, 98.80.-k, 04.50.kd Nous etudions ici une theorie de la gravitation dans laquelle les tachyons jouent le role de champ scalaire. Nous considerons aussi un couplage non minimal entre le tachyon et le tenseur d'Einstein. Nous montrons que le champ de tachyon a un comportement initialement exponentiel, devenant ulterieurement lineaire en temps cosmique. Nous analysons le comportement asymptotique du facteur d'echelle a(t) = [e.sup.α(t)] et montrons que dans la limite t →∞ to, il est impossible d'avoir un [??] constant et le modele a une singularite cosmologique initiale, c.-a-d., a [varia en proporcion con] (t - [t.sub.i])2/β dans la limite t - [t.sub.i]. [Traduit par la Redaction], 1. Introduction Scalar fields are important ingredients of physics, and in the last three decades their role in cosmology has been distinguished in the description of various sides of nature. [...]

Power law singularities and critical exponents in n-vector models are considered within a theoretical approach called GFD (grouping of Feynman diagrams) theory. It is discussed how possible values of the critical exponents can be related to specific n-vector models in this approach. A good agreement with the estimates of the perturbative renormalization group (RG) theory can be obtained. Predictions for corrections to scaling of the perturbative RG and GFD approaches are different. A nonperturbative proof is provided, supporting corrections to scaling of the GFD theory. Highly accurate experimental data very close to the l-transition point in liquid helium, as well as the Goldstone mode singularities in n-vector spin models, evaluated from Monte Carlo simulation results, are discussed with an aim to test the theoretical predictions. Our analysis shows that in both cases the data can be well interpreted within GFD theory. PACS Nos: 64.60.Fr, 64.60.Cn, 75.10.Hk, 05.50.+q, 05.10.Ln, 05.10.Cc Nous etudions les singularities en loi de puissance et les exposants critiques dans les modeles a n vecteurs, a l'interieur d'un cadre theorique appele le GDF (regroupement de diagrammes de Feynman). Nous analysons comment les valeurs possibles des exposants critiques peuvent etre reliees a des modeles a n vecteurs specifiques dans cette approche. Nous pouvons obtenir un bon accord avec les resultats de la theorie du groupe de renormalisation (RG) perturbative. Les predictions de correction d'echelle de la RG perturbative sont differentes de celles de l'approche GFD. Nous produisons une preuve non perturbative qui supporte la correction d'echelle de GFD. Afin de tester les predictions theoriques, nous analysons des donnees experimentales precises tres pres du point de transition λ dans l'helium liquide, ainsi les singularites du mode de Goldstone dans les modeles a n spins vectoriels, evaluees a partir de simulations Monte-Carlo. Nos resultats montrent dans les deux cas que les donnees peuvent etre bien interpretees par la theorie GDF. [Traduit par la Redaction], 1. Introduction Critical phenomena in interacting many-particle systems are associated with cooperative fluctuations of a large number of microscopic degrees of freedom. Various physical quantities typically exhibit power law singularities [...]

Toroidal shells are widely used in structural engineering. The governing equations of toroidal shells are very complicated because of its variable coefficients with singularity. To find their analytical solution, traditionally, the complex form governing equations were proposed and some useful solutions were obtained. Unfortunately, no any closed-form solution has even been obtained for either general or slender toroidal shells. This paper focus on a special case of toroidal shells, i.e., slender symmetrical toroidal shells. For the first time, the closed-form solution of this kind of shell has been successfully obtained from displacement form governing equations. The closed-form solution is demonstrated for the example of thermal compensation devices. The correction of well-known Dahl formula for slender toroidal shell has been proposed based on the solution obtained in this paper. DOI: 10.1061/(ASCE)EM.1943-7889.0000175 CE Database subject headings: Shells; Axisymmetry; Bending; Deformation; Elasticity; Closed form solutions. Author keywords: Toroidal shells; Axisymmetry; Revolution shells; Slender shells; Bending; Deformation; Elastic materials.

This paper describes the interval boundary element method treatment of the pointwise discretization error for systems with geometrically generated flux singularities. Worst-case interval bounds are provided for the local discretization error for all elements except for the element with a singular flux solution, for which the flux intensity factor is enclosed. An example is presented showing the behavior of the interval bounds on the local discretization error for systems with geometrically induced singularities. DOI: 10.1061/(ASCE)EM. 1943-7889.0000109 CE Database subject headings: Boundary element method; Errors. Author keywords: Boundary element method; Interval boundary element method; Discretization error; Flux singularity; Flux intensity factor; Discretization error bounds; Flux intensity factor bounds.

Usually in the study of singularity problems, only the most critical singular order is considered. For three-dimensional interface corner problems, if only the most critical singular order of stresses is considered, it is possible to lose the opportunity to compute the full modes of stress intensity factors. To fully understand the failure behavior of three-dimensional interface corners, a definition of the stress intensity factors for the lower singular orders is proposed in this paper based on that of the most critical singular order. Moreover, to compute the proposed multi-order stress intensity factors accurately and efficiently, a path-independent H-integral, which has been proven useful for the two-dimensional interface corners, is now modified into a domain-independent H-integral for the three-dimensional interface corner problems. Because the stress intensity factors characterize the fracture behavior focused on an arbitrary tip along the corner front, based on anisotropic elasticity the near tip solutions and complementary solutions of two-dimensional generalized plane strain problems are introduced and then utilized for computation of three-dimensional H-integral. To illustrate the validity of the present work, several three-dimensional numerical examples are analyzed and compared with the existing published solutions. Finally, two examples about the interface corners, which occur frequently in electric packages, are solved to show the feasibility and practicability of the proposed approach. [DOI: 10.1115/1.4000411] Keywords: stress intensity factors, three-dimensional interface corner, domain-independent H-integral

The particular solutions of a two-dimensional infinite wedge for various boundary conditions with lnr weak singularity have been investigated in this paper. The relations of the weak singularities and the discontinuities of the first kind of the boundary variables at a corner of a two-dimensional elastic body have been established. By using the relations, the singular behaviors of the unknown boundary variables at a corner of an elastic body can be obtained before solving the boundary value problem by using the boundary element method (BEM). Especially, if the boundary conditions at a corner are displacements prescribed, the values of the unknown tractions at the corner can be determined in advance. Thus, the difficulty related to the multivalued tractions at a corner in BEM analysis for problems with boundary, displacements prescribed has been overcome completely. In addition, more appropriate shape functions for the unknown boundary field variables of a corner element can be constructed, and the accuracy of the BEM may be greatly increased. [DOI: 10.1115/1.3168599] Keywords: corner problem, boundary, displacement and traction, discontinuity of the first kind, weak singularity, boundary element method

By coupling the Donnell-Mushtari shell equations to an analytical inviscid fluid solution, the linear dynamics of a rotating cylindrical shell with a corotating axial fluid flow is studied. Previously discovered mathematical singularities in the flow solution are explained here by the physical phenomenon of blocking. From a reference frame moving with the traveling waves in the shell wall the flow is identical to the flow in a rigid varicose tube. When the ratio of rotation rate to flow velocity approaches a critical value, the phenomenon of blocking creates a stagnation region between the humps of the wall. Since the linear model cannot account for this phenomenon, the solution blows up. [DOI: 10.1115/1.2998486]

Generally, the mixed boundary value problems in fracture and contact mechanics may be formulated in terms of integral equations. Through a careful asymptotic analysis of the kernels and by separating nonintegrable singular parts, the unique features of the unknown functions can then be recovered. In mechanics and potential theory, a characteristic feature of these singular kernels is the Cauchy singularity. In the absence of other nonintegrable kernels, Cauchy kernel would give a square-root or conventional singularity. On the other hand, if the kernels contain, in addition to a Cauchy singularity, other nonintegrable singular terms, the application of the complex function theory would show that the solution has a non-square-root or unconventional singularity. In this article, some typical examples from crack and contact mechanics demonstrating unique applications of such integral equations will be described. After some remarks on three-dimensional singularities, the key examples considered will include the generalized Cauchy kernels, membrane and sliding contact mechanics, coupled crack-contact problems, and crack and contact problems in graded materials. [DOI: 10.1115/1.2936241]

The dynamical behavior of learning is known to be very slow for the multilayer perceptron, being often trapped in the 'plateau.' It has been recently understood that this is due to the singularity in the parameter space of perceptrons, in which trajectories of learning are drawn. The space is Riemannian from the point of view of information geometry and contains singular regions where the Riemannian metric or the Fisher information matrix degenerates. This paper analyzes the dynamics of learning in a neighborhood of the singular regions when the true teacher machine ties at the singularity. We give explicit asymptotic analytical solutions (trajectories) both for the standard gradient (SGD) and natural gradient (NGD) methods. It is clearly shown, in the case of the SGD method, that the plateau phenomenon appears in a neighborhood of the critical regions, where the dynamical behavior is extremely slow. The analysis of the NGD method is much more difficult, because the inverse of the Fisher information matrix diverges. We conquer the difficulty by introducing the 'blow-down' technique used in algebraic geometry. The NGD method works efficiently, and the state converges directly to the true parameters very quickly while it staggers in the case of the SGD method. The analytical results are compared with computer simulations, showing good agreement. The effects of singularities on learning are thus qualitatively clarified for both standard and NGD methods. Index Terms--Dynamics of learning, multilayer perceptrous, natural gradient (NGD) learning, singularity, standard gradient (SGD) learning.

1. Introduction Today a wide variety of crystal structures are solved using x-ray or neutron powder diffraction data in conjunction with Rietveld refinement techniques. The sequence of steps in the […]

Byline: A. G. Aleksandrov (1) Abstract: In the paper, the notion of generalized Fuchsian systems of differential equations with logarithmic singularities along a divisor D on a complex manifold is discussed. It is proved that such a system is characterized by the property of being regular singular along its singular locus in the classical sense. The proof is based on the main properties of logarithmic differential forms and vector fields it does not use the traditional technique of resolution of singularities by means of which this problem is usually reduced to the study of divisors with normal crossings. In the case where the system in question has singularities along a free Saito divisor, a purely algebraic method of computing the integrability condition in terms of the commutation relations on its coefficient matrices is described. Author Affiliation: (1) Institute of Control Sciences, Russian Academy of Sciences, Russia Article History: Registration Date: 11/01/2006 Article note: Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 15, Theory of Functions, 2004.

Byline: L. Yu. Kolotilina (1) Abstract: The paper considers the singularity/nonsingularity problem for matrices satisfying certain conditions of diagonal dominance. The conditions considered extend the classical diagonal dominance conditions and involve the directed graph of the matrix in question. Furthermore, in the case of the so-called mixed diagonal dominance, the corresponding conditions are allowed to involve both row and column sums for an arbitrary finite set of matrices diagonally conjugated to the original matrix. Conditions sufficient for the nonsingularity of quasi-irreducible matrices strictly diagonally dominant in certain senses are established, as well as necessary and sufficient conditions of singularity/nonsingularity for weakly diagonally dominant matrices in the irreducible case. The results obtained are used to describe inclusion regions for eigenvalues of arbitrary matrices. In particular, a direct extension of the Gerschgorin (r = 1) and Ostrowski-Brauer (r = 2) theorems to r aY= 3 is presented. Bibliography: 18 titles. Author Affiliation: (1) St. Petersburg Department, Steklov Mathematical Institute, St. Petersburg, Russia Article History: Registration Date: 22/11/2005 Received Date: 06/01/2004 Article note: Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 309, 2004, pp. 40--83.

Byline: V. M. Nezhinskij (1) Abstract: The objects studied are singular links of spheres of dimensions p.sub.1,..., p.sub.r, and p in the n-sphere. A theory of such singular links for the case where max{p.sub.1,..., p.sub.r} &lt min{2n/3 - 1, n - (p + 5)/3} is constructed. The theory generalizes (as far as it is possible) the theory of singular links of spheres of dimensions k,..., k, and p in the (2k + 1)-sphere, where k &gt 1, developed in the author's recent papers. Bibliography: 13 titles. Author Affiliation: (1) Russian State Pedagogical University, St. Petersburg, Russia Article History: Registration Date: 05/10/2005 Received Date: 10/01/2003 Article note: Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2003, pp. 287--294.

This paper is a handyman's manual for learning how to resolve the singularities of algebraic varieties defined over a field of characteristic zero by sequences of blowups.

This paper presents the first motion planning methodology applicable to articulated, nonpoint nonholonomic robots with guaranteed collision avoidance and convergence properties. It is based on a new class of nonsmooth Lyapunov functions and a novel extension of the navigation function method to account for nonpoint articulated robots. The dipolar inverse Lyapunov functions introduced are appropriate for nonholonomic control and offer superior performance characteristics compared to existing tools. The new potential field technique uses diffeomorphic transformations and exploits the resulting point-world topology. The combined approach is applied to the problem of handling deformable material by multiple nonholonomic mobile manipulators in an obstacle environment to yield a centralized coordinating control law. Simulation results verify asymptotic convergence of the robots, obstacle avoidance, boundedness of object deformations, and singularity avoidance for the manipulators. Index Terms--Cooperative mobile manipulators, inverse Lyapunov functions, nonholonomic motion planning, potential fields.

Abstract: The paper deals with the system [rho] u.sub.tt - u.sub.xx + Vu = 0, x > 0, t > 0 \\ u| .sub.t = 0 = u.sub.t |.sub.t = 0 = 0 \\ u| .sub.x = 0 = f, . \\ where [rho] = [rho] ( x ) and V = V( x ) are 2 2 -matrix functions [rho] = diag\ [rho] .sub.1 ,[rho] .sub.2 \,[rho] .sub.[alpha] > 0 \ f is a boundary control u = u( x,t ) is the solution. The singularities of the fundamental solution corresponding to the controls ( *20c [delta] \\ 0 \\ )\ and\ ( *20c 0 \\ [delta] \\ ) ( [delta] = [delta] ( t ) is the Dirac [delta] -function) are under investigation. In the case of [rho] .sub.1 ( x ) [rho] .sub.2 ( x ) , the singularities of the fundamental solution are described in terms of the standard scale [delta] ,\ [delta] ,\ [delta] ,.... . In the presence of points x = x.sub.* :[rho] .sub.1 ( x.sub.* ) = [rho] .sub.2 ( x.sub.* ) an interesting effect occurs: singularities of intermediate (fractional) orders appear. Bibliography: 1 title. Article History: Registration Date: 21/10/2004

Byline: R. F. Sayakhova (1) Abstract: A homotopy classification of three-component singular links with components of dimensions 1, 1, and m &gt 1 in the three-sphere S.sup.3 is obtained. It is shown that there are links of such type that are pseudo-homotopic but not homotopic. Bibliography: 8 titles. Author Affiliation: (1) St.Petersburg State University, Russia Article History: Registration Date: 22/10/2004

It is known that any scattered wave field carrying energy into infinity must have source singularity centers within a bounded space. Otherwise, the scattered field should be identically equal to zero everywhere [1]. In this paper, attention is paid to localization of these singularities under the assumption that every scattered wave is determined uniquely by its own singularities. Investigations have shown that these singularities are distributed as 'bright centers' and the distance between them depends on frequency. To determine the position (localization) of the scattered wave field singularities, the functions describing converging and diverging waves are used. Based on these concepts and the method of auxiliary sources, an efficient numerical method to reconstruct a field up to its singularities is suggested. The localization of singularities is used for partial representation of the scattered fields, which reduces significantly the number of unknowns in describing the scattering process and leading into optimized inverse scattering problem solutions. Index Terms--Auxiliary absorbers, auxiliary sources, scattered field singularities.

Research is presented concerning the development of a novel derivation for doubly connected domains within the Schwarz-Christofell transformation. The construction of the pre-Schwarzian by reflection of the singularities is discussed.

Byline: Yu. Z. Povstenko (1) Abstract: We investigate the stress field in the vicinity of a point defect in a nonlocal elastic solid. Unlike the classical solution, the stress field has no singularities. We estimate the values of the elastic energy of a defect and maximum stresses. Author Affiliation: (1) Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv Article History: Registration Date: 11/10/2004

When analyzing field problems with the finite element method, the numerical solution will converge against the physical solution. This paper deals with the influence of singularities on the convergence behaviour. Index terms - Convergence of numerical methods, finite element methods, electrostatic analysis, magnetostatics

The kinematic analysis and the determination of the singularity loci of spatial four-degree-of-freedom parallel manipulators with prismatic or revolute actuators are discussed in this article. A new method for the derivation of the velocity equations and the corresponding Jacobian matrices is presented. The numerical determination of the workspace boundaries is then briefly discussed. Finally, the determination of the singularity loci is performed using the velocity equations and examples are given to illustrate the results obtained. Spatial four-degree-of-freedom parallel manipulators can be used in several robotic applications as well as in flight simulators. The determination of their singularity loci is a very important design issue.

The extended Jacobian method is a popular approach for controlling a kinematically redundant arm which allows one to resolve redundancy by locally optimizing an objective function and to gain repeatability for a cyclic end effector trajectory. It is a special case of a family of methods called constraint function methods. It has been found that the occurrence of algorithmic singularities can cause severe difficulties and the advantages of the methods such as repeatability might no longer exist. The purpose of this paper is to study the characteristics of algorithmic singularities, especially those of corank 1. A result of the authors on kinematic singularities is used to obtain a sufficient condition for the existence of solution paths at algorithmic singularities of the constrained function method. The phenomenon of branch repeatability is shown to occur at an algorithmic singularity. We also show that the extended Jacobian method cannot successfully optimize the objective function beyond the singularity without loss of continuity of the joint derivative. Examples are given to demonstrate the use of our theoretical results. Index Terms - Algorithmic singularity, extended Jacobian method, redundancy resolution.

Upon solving the inverse kinematics problem of robot manipulators, the inherent singularity problem should always be considered. When a manipulator is approaching a singular configuration, a certain degree of freedom will be lost such that there are no feasible solutions of the manipulator to move into this singular direction. In this paper, the singularities of a 6-DOF PUMA manipulator are analyzed in detail and all the corresponding singular directions in task space are clearly identified. In order to resolve this singularity problem, an approach denoted Singularity Isolation Plus Compact QP (SICQP) method is proposed. The SICQP method decomposes the work space into achievable and unachievable (i.e., singular) directions. Then, the exactness in the singular directions are released such that extra redundancy is provided to the achievable directions. Finally, the Compact QP method is applied to maintain the exactness in the achievable directions, and to minimize the tracking errors in the singular directions under the condition that feasible joint solutions must be obtained. In the end, some simulation results for PUMA manipulator are given to demonstrate the effectiveness of the SICQP method.

This paper is devoted to the description of the set of all singular configurations of serial robot-manipulators. For 6 degrees of freedom serial robot-manipulators we have developed a theory which allows to describe higher order singularities. By using Lie algebra properties of the screw space we give an algorithm, which determines the degree of a singularity from the knowledge of the actual configuration of axes of the robot-manipulator only. The local shape of the singular set in a neighbourhood of a singular configuration can be determined as well. We also solve the problem of escapement from a singular configuration. For serial robot-manipulators with the number of degrees of freedom different from six we show that up to certain exceptions singular configurations can be avoided by a small change of the motion of the end-effector. We also give an algorithm which allows to determine equations of the singular set for any serial robot-manipulator. We discuss some special cases and give examples of singular sets including PUMA 560.

The Eshelby-Stroh formalism for anisotropic elasticity is applied to evaluate stress singularities at grain triple junctions. The analysis defined different asymmetric grain boundary configurations and random orientations for both cubic and orthotropic grains. Both grain types showed super singularities for two-dimensional polycrystals in the fully symmetric case.

Characteristic equations are established to define stress singularity orders at the tip of a crack ending at an interface between two orthotropic media. Variations in alignment and angular relations to the interface are then examined for two general anisotropic materials. A general solution to the problem shows roots as gamma functions. Extension of the problem to include Krenk's parameters shows generalized Dundurs' constants as vital factors in analysis.

Arbitrary normal and shearing processes based on elasticity theory are applied to the three-phase bonded structure problem, which are defined by two interfaces and two free surfaces. Bonded wedge geometries displayed a direct relationship between material pairing and stress singularity order. Material bonding order also conditioned the singularity order for the given bonded structure.

In this paper we discuss singular configurations of serial robot-manipulators with respect to their removability. A removable singularity is a singularity which can be removed from a motion of the end-effector by a small change of the motion. The most interesting situation appears for robot-manipulators with 5 degrees of freedom, because the case of 4 degrees of freedom is easy and singular configurations of robot-manipulators with 6 degrees of freedom are nonremovable. In the paper we give the complete list of all 5R robot-manipulators which have nonremovable singularities. The image of the singular set in the parameter space for such manipulators can be a plane, a quadric, a cylinder or an algebraic surface of degree 3 or 7. All of them are explicitly given.

Using known techniques of desingularization for singular plane projective curves over finite fields [F.sub.q], q = [2.sup.m], we found several new binary plane projective curves of genus 3, 4, and 5 with the maximal number of [F.sub.q]-rational points (q = [2.sup.m], m = 3, 4, 5, 6, 7, 8, and 9) on their smooth projective models, which are close to or meet Serre's upper bound.

A geometric application of screw theory is used to develop a recursive algorithm for computing all singular configurations of revolute-jointed manipulators with arbitrary link geometries and an arbitrary number of joints. The depth of the recursion is linear in the number of joints, n, while the computational burden is proportional to [2.sup.n-2]. This method does not require explicit construction of the Jacobian matrix elements or a determinant operation. The method is also robust with respect to the bifurcations that occur for industrial robot geometries. Further, the screw axis of the singular motion is determined at no additional cost.

The still-unsolved Opaque Square Problem has been extended to various shapes, polygons and plane regions. An analysis illustrates three possible solutions to the Opaque Cube Problem, with the best one achieving an optimal surface area of 4.2324. The Opaque Sphere Problem, on the other hand, is shown to possibly have no solution because the surface or object that blocks sight may have to be widened, thus complicating what defines opaqueness. Topics for future research are also recommended.

A fractional slope on the log-log Bode plot has been observed in characterizing a certain type of physical phenomena and has been called the fractal system or the fractional power pole. In order to represent and study its dynamical behavior, a method of singularity function is presented which consists of cascaded branches of a number of pole-zero (negative real) pairs or simple RC section. The distribution spectrum of the system can also be easily calculated and its accuracy depends on a prescribed error specified in the beginning. The method is then extended to a multiple-fractal system which consists of a number of fractional power poles and the result can be simulated by a combination of singularity functions, each representing a single-fractal system.

In this paper, the stability robustness of linear discrete-time singular systems with structured parameter perturbations is investigated. A new sufficient criterion is presented for ensuring that the linear discrete-time singular system remains regular, impulse-free, and asymptotically stable under structured parameter perturbations. The presented criterion is proved to be less conservative than the existing one reported recently.

Correct association of equivalent current and magnetic charge sheets in adjacent elements over the surface of non-magnetic conducting media removes the logarithmic singularities which otherwise occur at the edges and corners. This not only improves the accuracy of computed fields near corners but is also shown to allow magnetic flux to enter the medium naturally, and to remove the need to modify the surface impedance near corners in eddy-current problems.

This paper introduces a novel extension to a scalar two-dimensional polynomial finite-element basis to better cope with wedge singularities in waveguide problems. An error estimate for the computed cutoff frequencies of the waveguide shows that the relative [H.sup.1] error of the modal solution is critical. We demonstrate that the present extension significantly improves the approximation properties of a polynomial basis, especially in the [H.sup.1] norm. Numerical examples show that the present extension compares well with other recent techniques. Combining variable order elements with singular basis extension provides further significant reduction of the computational burden. Index Terms--High-order FEM methods, wedge singularities.

This paper presents two kinds of robust controllers for stabilizing singularly perturbed discrete bilinear systems. The first one is an [Epsilon]-dependent controller that stabilizes the closed-loop system for all [Epsilon] [element of] ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the prespecified upper bound of the singular perturbation parameter. The second one is an [Epsilon]-independent controller, which is able to stabilize the system in the entire state space for all [Epsilon] [element] (0, [Epsilon]*, where [Epsilon]*] is the exact upper [Epsilon]-bound. The [Epsilon]* can be calculated by the critical stability criterion once the robust controller is determined. An example is presented to illustrate the proposed schemes. Index Terms--Lyapunov equation, robust controller, singularly perturbed discrete bilinear systems, singular perturbation parameter.