Search

Let a finite abelian group G act (linearly) on the space Rn and thus on its complexification Cn. Let W be the real part of the quotient Cn/G (in general W≠Rn/G). We give an algebraic formula for the radial index of a 1‐form on the real quotient W. It is shown that this index is equal to the signature of the restriction of the residue pairing to the G‐invariant part ΩωG of Ωω=ΩRn,0n/ω∧ΩRn,0n−1. For a G‐invariant function f, one has the so‐called quantum cohomology group defined in the quantum singularity theory (FJRW‐theory). We show that, for a real function f, the signature of the residue pairing on the real part of the quantum cohomology group is equal to the orbifold index of the 1‐form df on the preimage π−1(W) of W under the natural quotient map.

Topological singularity in a continuum theory of defects and a quantum field theory is studied from a viewpoint of differential geometry. The integrability conditions of singularity (Clairaut-Schwarz-Young theorem) are expressed by a torsion tensor and a curvature tensor when a Finslerian intrinsic parallelism holds for the multi-valued function. In the context of the quantum field theory, the singularity called an extended object is expressed by the torsion when the intrinsic parallelism is related to the spontaneous breakdown of symmetry. In the continuum theory of defects, the path-dependency of point and line defects within a crystal is interpreted by the non-vanishing condition of torsion tensor in a non-Riemannian space osculated from the Finsler space, and the domain is not simply connected. On the other hand, for the rotational singularity, an energy integral (J-integral) around a disclination field is path-independent when a nonlinear connection is single-valued. This means that the topological expression for the sole defect (Gauss-Bonnet theorem with genus g=1) is understood by the integrability of nonlinear connection.

In this paper, we study the following anisotropic problem with singularity: where is a bounded domain with smooth boundary. Using the critical point theory, we obtain the existence of weak solutions for the problem under suitable conditions. Copyright © 2015 John Wiley & Sons, Ltd.

This paper considers the one parameter families of extrinsic differential geometries of Lorentzian hypersurfaces on pseudo n-spheres. We give a continuous relationship among the three types Gauss indicatrices by one parameter map. Meanwhile, we also give the singularity analysis of the one parameter Gauss indicatrices of Lorentzian hypersurfaces on pseudo n-spheres by the Legendrian singularity theory. Copyright © 2014 John Wiley & Sons, Ltd.

A new numerical method for calculating the singularity orders of V-notches in Reissner's plate is proposed in this paper. By introducing the asymptotic expansion of the generalised displacement field at the notch tip into the equilibrium equations of a plate, a set of characteristic ordinary differential equations with respect to the singularity order are established. In addition, by adopting the variable substitution technique, the obtained non-linear characteristic equations are transformed into linear ones, which are solved by the interpolating matrix method. The singularity orders of moments and shear forces can be obtained simultaneously and can be distinguished from the corresponding characteristic angular functions conveniently. Four types of boundary conditions are proposed to investigate the influence of boundary conditions on the singularity order values. The effect of the Poisson's ratio on the singularity orders of the V-notch in Reissner's plate is discussed. The present method is versatile for the singularity analysis of single material V-notches and bi-material V-notches, and can be easily extended to multi-material V-notches.

In this paper, we give the explicit bounds for the data of objects involved in some basic theorems of singularity theory: the inverse, implicit and rank theorems for Lipschitz mappings, the splitting lemma and the Morse lemma, the density and openness of Morse functions. We expect that the results will make singularities more applicable and will be useful for numerical analysis and some fields of computing.

The production of polymer is a very important sector in the chemical processing industry as demand for polymeric materials increases, and these polymerization processes exhibit nonlinear dynamics posing difficulties on process operation and control. It is desirable to understand the relationship between open-loop controllability and process conditions. The phase behavior that is an open-loop indicator of the controllability for the methyl methacrylate and propylene polymerization reactor over the entire feasible operating region is determined, and the influences of design/operating parameters and model uncertainties on this inherent characteristic at the design stage are analyzed. Based on zero dynamics and singularity theory, directly in the nonlinear setting, a methodology for the preliminary analysis of the phase behavior over the whole feasible operating region is presented and applied into the above two reactors. The above research results demonstrate that to modify certain parameters in certain direction would improve the controllability and are expected to provide helpful guidelines for improving plant and control system design. This is desirable because it allows identification of the cause of the limitation to give an indication where to modify the design. © 2010 American Institute of Chemical Engineers AIChE J, 57: 2795–2807, 2011

This paper is a computation of the homotopy type of K, the space of long knots in R^3, the same space of knots studied by Vassiliev via singularity theory. Each component of K corresponds to an isotopy class of long knot, and we `enumerate' the components via the companionship trees associated to the knot. The knots with the simplest companionship trees are: the unknot, torus knots, and hyperbolic knots. The homotopy-type of these components of K were computed by Hatcher. In the case the companionship tree has height, we give a fibre-bundle description of those components of K, recursively, in terms of the homotopy types of `simpler' components of K, in the sense that they correspond to knots with shorter companionship trees. The primary case studied in this paper is the case of a knot which has a hyperbolic manifold contained in the JSJ-decomposition of its complement.

We determine relations between the Euler obstruction of a map f and the Chern obstruction of a collection of 1-forms associated to f . We obtain explicit applications in singularity theory. On one hand, in [10], Grulha defines a generalization of the Euler obstruction of a holomorphic function to the case of a map-germ f : (V, 0) → (C, 0) where (V, 0) is the germ of a reduced equidimensional complex analytic variety of dimension d ≥ p. The main result in that paper is a formula that computes the Euler obstruction of f in terms of the Euler obstruction for p-frames as defined by Brasselet, Seade and Suwa in [3]. On the other hand, Ebeling and Gusein-Zade introduced in [4] the notion of Chern obstruction for singular spaces using collections of differential 1-forms. This number is well defined for any germ of reduced equidimensional complex analytic space, and the authors provide a formula for the Chern obstruction in terms of intersection number. One of the main results of the present paper is that the Euler obstruction of a map can be written as a Chern obstruction, therefore as an intersection multiplicity.

Singularity theory is applied for the study of constrained tensegrity systems. Previous studies have already been performed on non-constrained systems; however, the present one allows for general non-symmetric equilibrium configurations. A modified T3 tensegrity model comprising seven rigid bars, three elastic cables and one rotational spring is considered. The stability of this model is examined performing singularity theory for deformations under conservative quasistatic loading. Critical conditions for branching of the equilibrium paths are defined and their post-critical behavior is discussed. Classification of the simple and compound singularities of the total potential energy function is effected. The theory is implemented into the fold catastrophe of the asymmetric configuration of the modified T3 tensegrity model and an elliptic umbilic singularity for the case of compound branching. It is pointed out that singularity studies with constraints demand a quite different mathematical approach than those without constraints.

In this paper, the classification of hypersurfaces in with an isolated singularity are studied. If the singularity is of type A k , under certain restrictions of the degree of the hypersurfaces, a classification up to homeomorphism, which is a diffeomorphism on the nonsingular part, is obtained. Examples of cubic hypersurfaces with an A 5 -singularity are constructed.

The definition of Thom polynomials for Lagrange, Legendre and critical point function singularities is given. The approach is based on the notion of classifying space of singularities. This approach provides a universal method for computing Thom polynomials. Characteristic classes of complex Lagrange and Legendre singularities of small codimension are computed. Two independent methods for these computations are presented. The first is the direct one based on the resolution of singularities, the Gysin homomorphism, and the notion of adjacency exponents. The second uses the existence theorem for Thom polynomials and is based on Rimanyi's idea of using symmetries. The expressions for the resulting polynomials reduced modulo 2 agree with those obtained by Vassiliev for the real case. 2000 Mathematics Subject Classification 32S20.

The paper presents a skeleton-based approach for robust detection of perceptually salient shape features. Given ashape approximated by a polygonal surface, its skeleton is extracted using a three-dimensional Voronoi diagramtechnique proposed recently by Amenta et al. [3]. Shape creases, ridges and ravines, are detected as curvescorresponding to skeletal edges. Salient shape regions are extracted via skeleton decomposition into patches.The approach explores the singularity theory for ridge and ravine detection, combines several filtering methodsfor skeleton denoising and for selecting perceptually important ridges and ravines, and uses a topological analysisof the skeleton for detection of salient shape regions. ACM CSS: I.3.5 Computational Geometry and Object Modeling

The stability characteristics of a class of unstructured models of continuous bioreactors are analyzed using elementary concepts of singularity theory and continuation techniques. The class consists of models for which the non-biomass product formation rate is linearly proportional to the utilization rate of limiting substrate. The kinetics expressions of cell growth and product synthesis are allowed to assume general forms of substrate and product. Global analytical conditions are derived that allow the construction of a practical picture in the multidimensional parameter space delineating the different static behavior these models can predict, including unique steady states, coexistence of non-trivial steady states with wash-out conditions, and multistability resulting from hysteresis. These general results are applied to specific examples of bioprocesses and allow the study of the effect of kinetic and operating parameters on the stability characteristics of these models.

The stability characteristics of a bioreactor with cell recycle involving the competition between microbial cultures are investigated. The unstructured model, based on Andrew's inhibitory kinetics, involves the pure and simple competition between two microorganisms for a single pollutant. The singularity theory used for this study allows an in-depth analysis of both the static and dynamic bifurcation mechanisms occurring in the system. The hysteresis with five solutions is the highest singularity the system can exhibit. With inhibitory kinetic expressions, the model can also predict self-sustained oscillations for a wide range of parameters. The analysis of clean feed conditions shows that the model cannot exhibit periodic behavior regardless of the growth kinetics model. Analytical criteria are also derived for the coexistence of the competing cultures and for the prevention of wash-out conditions. The stability characteristics for Monod kinetics, derived as a limiting case of the inhibitory kinetic expressions, are incorporated in the general framework offered by the singularity theory.

A Maple package which performs the symbolic algebra central to problems in local singularity theory is described. This is a generalisation of previous projects, which dealt only with problems in elementary catastrophe theory. Applications to specific problems are described, and a survey given of the powerful techniques from singularity theory that are used by the package. A description of the underlying algorithm is given, and some of the more important computational aspects discussed. The package, user manual and installation instructions are available in the appendices to this article.

We set up a singularity-theoretic framework for classifying one-parameter steady-state bifurcations with hidden symmetries. This framework also permits a non-trivial linearization at the bifurcation point. Many problems can be reduced to this situation; for instance, the bifurcation of steady or periodic solutions to certain elliptic partial differential equations with Neumann or Dirichlet boundary conditions. We formulate an appropriate equivalence relation with its associated tangent spaces, so that the usual methods of singularity theory become applicable. We also present an alternative method for computing those matrix-valued germs that appear in the equivalence relations employed in the classification of equivariant bifurcation problems. This result is motivated by hidden symmetries appearing in a class of partial differential equations defined on an N-dimensional rectangle under Neumann boundary conditions.

The biodegradation of pollutants in continuous operation when the microbial population exhibits wall attachment is studied. The proposed model for wall attachment assumes two morphological forms of the microbial cell connected by metamorphosis reactions with first order exchange kinetics. An analysis of the stability of the bioreactor, carried out using elementary principles of the singularity theory and continuation techniques, allows for classification in the multidimensional parameter space of the various stability behaviors exhibited by the reactor model, for both substrate-inhibited and Monod kinetics. The analysis also shows the enhanced stability behavior of the bioreactor due to wall attachment.

The purpose of this note is to generalise, and give a more illuminating proof, of a theorem of [ 13 ] (Theorem 1.1 below). Before stating it, we provide some introductory information. Consider the following two sequences of pictures: in each we see a 1-parameter family X ℝ, t of real algebraic hypersurfaces, which undergoes a bifurcation when the parameter t is equal to 0. Note that in Figure 1, both (i) (a) and (i) (b), and in (ii) (b), the surface X ℝ, t has a purely 1-dimensional part, which we have indicated with a dotted line, and that in (i) (b) we have drawn a curve vertically along the middle of the surface to make clearer the way it passes through itself. The reader will observe that in (a) the surface X ℝ, t is homotopically a 2-sphere when t >0 and a 0-sphere when t X ℝ, t is a homotopy 1-sphere both for t 0. Such sequences are typical in singularity theory; each is in fact the family of algebraic closures of images of a versal deformation of a codimension 1 singularity of mapping. Now suppose that the complexification X [Copf ], t is a homotopy n -sphere. In [ 13 ] the second author pointed out that it follows that X ℝ, t is a homotopy sphere for t ≠0 (allowing the empty set as a −1-sphere). Indeed, in the local situation, or globally in the weighted homogeneous case, there are well-defined integers k + and k − between −1 and n such that X ℝ, t ≃ S k + for t >0 and X ℝ, t ≃ S k − for t We describe X ℝ, t for t ∈ℝ−0 as ‘good’ if the homotopy dimension of X ℝ, t is equal to n . In this case the inclusion X ℝ, t [rarrhk ] X t is a homotopy equivalence [ 13 , 1.1].

In this paper singularity theory, large activation energy asymptotic, and numerical methods are used to present a comprehensive study of the steady-state multiplicity features of three classical adiabatic autothermal reactor models: tubular reactor with internal heat exchange, tubular reactor with external heat exchange, and the CSTR with external heat exchange. Specifically, the authors derive the exact uniqueness-multiplicity boundary, determine typical cross-sections of the bifurcation set, and classify the different types of bifurcation diagrams of conversion vs. residence time. Asymptotic (limiting) models are used to determine analytical expressions for the uniqueness boundary and the ignition and extinction points. The analytical results are used to present simple, explicit and accurate expressions defining the boundary of the region of autothermal operation in the physical parameter space.

Singularity theory is combined with asymptotic analysis to determine the exact uniqueness-multiplicity boundary and ignition and extinction locus for the non-adiabatic, autothermal tubular reactor model. It is found that the steady-state behavior of the nonadiabatic reactor is described by the two limiting cases of adiabatic and strongly cooled models. The adiabatic case has been examined in a previous study. Here, we develop limiting models to describe the strongly cooled asymptotes. We also classify the different types of bifurcation diagrams of conversion vs. residence time using the results of singularity theory with a distinguished parameter. Analytical criteria are developed for predicting the conditions under which autothermal operation is feasible when heat losses are significant.

This paper has dealt with biophysically accurate, or plausible, excitation systems. These are obtained from experiments, and so are complicated, often of high order, and are continually being updated by new experimental results. This is especially true for the excitation equations that represent cardiac tissue. Biophysically relevant problems require quantitatively accurate answers: What happens at what values of what parameter? Thus, numerical methods and results are important, and there is a strong tendency for both the model and the method of analysis to be given as computer algorithms; a current model for cardiac membrane dynamics is a computer package, and the use of path tracking procedures such as AUTO or PATH is becoming common. Perhaps the best way to proceed in the investigation of complicated and exotic dynamics in models of cardiac tissue would be to combine model formulation from voltage clamp data (the derivation of the excitation equations) with bifurcation, and even perhaps singularity theory analysis of dynamical systems, into an expert system.

The ball-on-plate example in the above paper by the authors [3] contains a mistake in the vector field. This note presents the singularity result of the corrected vector field.

We present a Jacobi–Davidson like correction formula for left and right eigenvector approximations for non-Hermitian nonlinear eigenvalue problems. It exploits techniques from singularity theory for characterizing singular points of nonlinear equations. Unlike standard nonlinear Jacobi-Davidson, the correction formula does not contain derivative information and works with orthogonal projectors only. Moreover, the basic method is modified in that the new eigenvalue approximation is taken as a nonlinear Rayleigh functional obtained as root of a certain scalar nonlinear equation the existence of which – as well as a first order perturbation expansion – is shown. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Phase transformation plays an important role in thermodynamics and materials science. Based on the theory of singularities, a new method to construct phase diagrams is presented. Analysing singularities on base of root sequences, see Tamaschke [16], will help to develop singularity graphs, where workings by H. Whitney, R. Thom, and V. I. Arnold provide fundamentals. The generated singularity graphs build the starting point for singularity phase diagrams. A powerful characteristic of such singularity graphs is that higher-dimensional surfaces can be transformed to a two-dimensional diagram. The attained singularity diagram can be used in materials science for analytical models of temperature-concentrated diagrams. As tools from algebra and analysis build a sound basis for singularity diagrams, it is possible to evolve computer software generating these phase diagrams. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The complete static behavior of a large class of unstructured models of continuous bioprocesses is classified using elementary concepts of the singularity theory and continuation techniques. The class consists of models for which the cell growth rate is proportional to the rate of utilization of limiting substrate while the kinetics of cell growth, utilization of limiting substrate and synthesis of the desired non-biomass product are allowed to assume general forms of substrate and product. This class of models was used extensively in the literature to model fermentation processes. Global analytical conditions are derived that allow the construction of a practical picture in the multidimensional parameter space delineating the different static behavior these models can predict, including unique steady states, coexistence of wash-out conditions with non-trivial steady states and multistability resulting from hysteresis. These general results are applied to a number of experimentally validated models of fermentation processes, and allow the study of the effect of kinetic and operating parameters on the stability characteristics of these models. Practical criteria are also derived for the safe operation of the bioprocesses.

In this article we outline an application of the Thorn-Mather singularity theory of C" mappings to a problem in bifurcation theory. A detailed description of the results stated here may be found in Reference 1. Unless otherwise stated all mappings are assumed to be germs of C a functions near the origin. map G: IR" x IR - IR"'

This paper applies the theory of singularities of differentiable mappings - specifically the unfolding theorem - to study the effect of imperfections in a system subject to bifurcation. In a number of special cases we have classified (up to a suitable equivalence) all the possible perturbations of the bifurcation equations by a finite number of imperfection parameters. These cases include both bifurcation from a double eigenvalue and from a simple eigenvalue degenerate in the sense of Crandall-Rabinowitz.

This paper has two purposes: to study from the singularity theory point of view certain bifurcation problems which commute with the five-dimensional irreducible representation of the orthogonal group O(3) and to give some implications of this study for the Benard problem in spherical geometry. We shall now describe in general terms our results and compare them with the work of Chossat [3] whose paper motivated our own interest in the problem. The Benard problem is concerned with convection in a viscous fluid when it is heated from below. The fluid is assumed to be confined in a spherical shell of outer radius R, and inner radius qRo, where q is near 0.3. This choice of q is partially motivated by considering convection within the molten layer of the core of the earth (see Busse [l] for further discussion). We consider the Binard problem in the Boussinesq approximation. In this model there is a trivial solution representing pure heat conduction radially outward. As the temperature on the inner sphere (that is, the Rayleigh number R) is increased, this trivial solution loses stability, say at R = R'. The Benard problem is the study of the resulting bifurcation. Chossat studies this problem using the Lyapunov-Schmidt reduction. In the Boussinesq approximation the fluid is driven from the pure conduction state by a term in the momentum equation involving the gravity vector g(r) and a term in the temperature equation involving the gradient of the conduction temperature VT,. Assuming that the production of heat and the density are uniform throughout the shell, these vectors have the form