Knots and Classical 3-Geometries

It has been conjectured by Rovelli that there is a correspondence between the space of link classes of a Riemannian 3-manifold and the space of 3-geometries (on the same manifold). An exact statement of his conjecture will be established and then verified for the case when the 3-manifold is compact, orientable and closed.Comment: 14p. type-set in AmS-TeX version 2.

Symmetries of the Einstein Equations

Generalized symmetries of the Einstein equations are infinitesimal transformations of the spacetime metric that formally map solutions of the Einstein equations to other solutions. The infinitesimal generators of these symmetries are assumed to be local, \ie at a given spacetime point they are functions of the metric and an arbitrary but finite number of derivatives of the metric at the point. We classify all generalized symmetries of the vacuum Einstein equations in four spacetime dimensions and find that the only generalized symmetry transformations consist of: (i) constant scalings of the metric (ii) the infinitesimal action of generalized spacetime diffeomorphisms. Our results rule out a large class of possible ``observables'' for the gravitational field, and suggest that the vacuum Einstein equations are not integrable.Comment: 15 pages, FTG-114-USU, Plain Te

Physics of the Pseudogap State: Spin-Charge Locking

The properties of the pseudogap phase above Tc of the high-Tc cuprate superconductors are described by showing that the Anderson-Nambu SU(2) spinors of an RVB spin gap 'lock' to those of the electron charge system because of the resulting improvement of kinetic energy. This enormously extends the range of the vortex liquid state in these materials. As a result it is not clear that the spinons are ever truly deconfined. A heuristic description of the electrodynamics of this pseudogap-vortex liquid state is proposed.Comment: Submitted to Phys Rev Letter

Perturbation analysis of the limit cycle of the free van der Pol equation

A power series expansion in the damping parameter, epsilon, of the limit cycle of the free van der Pol equation is constructed and analyzed. Coefficients in the expansion are computed in exact rational arithmetic using the symbolic manipulation system MACSYMA and using a FORTRAN program. The series is analyzed using Pade approximants. The convergence of the series for the maximum amplitude of the limit cycle is limited by two pair of complex conjugate singularities in the complex epsilon-plane. A new expansion parameter is introduced which maps these singularities to infinity and leads to a new expansion for the amplitude which converges for all real values of epsilon. Amplitudes computed from this transformed series agree very well with reported numerical and asymptotic results. For the limit cycle itself, convergence of the series expansion is limited by three pair of complex conjugate branch point singularities. Two pair remain fixed throughout the cycle, and correspond to the singularities found in the maximum amplitude series, while the third pair moves in the epsilon-plane as a function of t from one of the fixed pairs to the other. The limit cycle series is transformed using a new expansion parameter, which leads to a new series that converges for larger values of epsilon

New Symbolic Tools for Differential Geometry, Gravitation, and Field Theory

DifferentialGeometry is a Maple software package which symbolically performs fundamental operations of calculus on manifolds, differential geometry, tensor calculus, Lie algebras, Lie groups, transformation groups, jet spaces, and the variational calculus. These capabilities, combined with dramatic recent improvements in symbolic approaches to solving algebraic and differential equations, have allowed for development of powerful new tools for solving research problems in gravitation and field theory. The purpose of this paper is to describe some of these new tools and present some advanced applications involving: Killing vector fields and isometry groups, Killing tensors and other tensorial invariants, algebraic classification of curvature, and symmetry reduction of field equations.Comment: 42 page

Transthoracic three-dimensional echocardiography for the assessment of straddling tricuspid or mitral valves

Background The advent of 3D echocardiography has provided a technique which, potentially, could afford significant additional information over conventional cross-sectional echocardiography in the assessment of patients with straddling atrioventricular valves prior to surgical correction. Methods Eight patients, aged from 1 month to 9˙2 years, were examined with 3D echocardiography. All but three had discordant ventriculoarterial connections or double outlet right ventricle. Data suitable for reconstruction was acquired with transthoracic scanning. Right and left ventricular volumes were calculated in the 3D dataset. Results 3D echocardiography proved capable of defining the exact degree of straddling by imaging theproportion of tension apparatus attached to either side of the ventricular septum. It was able also to display the atrioventricular junction “en face”, thus permitting identification of the precise site of insertion of the muscular ventricular septum relative to the atrioventricular junction. This made it possiblefirst, to calculate the degree of valvar override, and second, to predict the location of the penetrating atrioventricular bundle. End-diastolic volume of the right ventricle in those with straddling tricuspid valves was 73 (61–83)% of normal, and, of the left ventricle in those with mitral valvar straddling 71 (40‐97)% of normal. Conclusions 3D echocardiography can aid in planning the optimal surgical procedure in patients with straddling or overrriding atrioventricular valves, as it provides diagnostic information superiorto standard crosssectional techniques. It also allows for exact measurement of the volumes of the respective ventricles

A computerized symbolic integration technique for development of triangular and quadrilateral composite shallow-shell finite elements

Computerized symbolic integration was used in conjunction with group-theoretic techniques to obtain analytic expressions for the stiffness, geometric stiffness, consistent mass, and consistent load matrices of composite shallow shell structural elements. The elements are shear flexible and have variable curvature. A stiffness (displacement) formulation was used with the fundamental unknowns consisting of both the displacement and rotation components of the reference surface of the shell. The triangular elements have six and ten nodes; the quadrilateral elements have four and eight nodes and can have internal degrees of freedom associated with displacement modes which vanish along the edges of the element (bubble modes). The stiffness, geometric stiffness, consistent mass, and consistent load coefficients are expressed as linear combinations of integrals (over the element domain) whose integrands are products of shape functions and their derivatives. The evaluation of the elemental matrices is divided into two separate problems - determination of the coefficients in the linear combination and evaluation of the integrals. The integrals are performed symbolically by using the symbolic-and-algebraic-manipulation language MACSYMA. The efficiency of using symbolic integration in the element development is demonstrated by comparing the number of floating-point arithmetic operations required in this approach with those required by a commonly used numerical quadrature technique

Incomplete markets with no Hart points

We provide a geometric test of whether a general equilibrium incomplete markets (GEI) economy has Hart points---points at which the rank of the securities payoff matrix drops. Condition (H) says that, at each nonterminal node, there is an affine set (of appropriate dimension) that intersects all of a well-specified set of convex polyhedra. If the economy has Hart points, then Condition (H) is satisfied; consequently, if condition (H) fails, the economy has no Hart points. The shapes of the convex polyhedra are determined by the number of physical goods and the dividends of the securities, but are independent of the endowments and preferences of the agents. Condition (H) fails, and thus there are no Hart points, in interesting classes of economies with only short-lived securities, including economies obtained by discretizing an economy with a continuum of states and sufficiently diverse payoffs.Incomplete Markets, GEI model, Hart points