Hypersurface

About: Hypersurface is a research topic. Over the lifetime, 8816 publications have been published within this topic receiving 127191 citations.

Polynomial Theory of Complex Systems

Abstract: A complex multidimensional decision hypersurface can be approximated by a set of polynomials in the input signals (properties) which contain information about the hypersurface of interest. The hypersurface is usually described by a number of experimental (vector) points and simple functions of their coordinates. The approach taken in this paper to approximating the decision hypersurface, and hence the input-output relationship of a complex system, is to fit a high-degree multinomial to the input properties using a multilayered perceptronlike network structure. Thresholds are employed at each layer in the network to identify those polynomials which best fit into the desired hypersurface. Only the best combinations of the input properties are allowed to pass to succeeding layers, where more complex combinations are formed. Each element in each layer in the network implements a nonlinear function of two inputs. The coefficients of each element are determined by a regression technique which enables each element to approximate the true outputs with minimum mean-square error. The experimental data base is divided into a training and testing set. The training set is used to obtain the element coefficients, and the testing set is used to determine the utility of a given element in the network and to control overfitting of the experimental data. This latter feature is termed "decision regularization.

On the proof of the positive mass conjecture in general relativity

Abstract: LetM be a space-time whose local mass density is non-negative everywhere. Then we prove that the total mass ofM as viewed from spatial infinity (the ADM mass) must be positive unlessM is the flat Minkowski space-time. (So far we are making the reasonable assumption of the existence of a maximal spacelike hypersurface. We will treat this topic separately.) We can generalize our result to admit wormholes in the initial-data set. In fact, we show that the total mass associated with each asymptotic regime is non-negative with equality only if the space-time is flat.

Parallel coordinates: a tool for visualizing multi-dimensional geometry

Abstract: A methodology for visualizing analytic and synthetic geometry in RN is presented. It is based on a system of parallel coordinates which induces a non-projective mapping between N-Dimensional and 2-Dimensional sets. Hypersurfaces are represented by their planar images which have some geometrical properties analogous to the properties of the hypersurface that they represent. A point ← → line duality when N = 2 generalizes to lines and hyperplanes enabling the representation of polyhedra in RN. The representation of a class of convex and non-convex hypersurfaces is discussed together with an algorithm for constructing and displaying any interior point. The display shows some local properties of the hypersurface and provides information on the point's proximity to the boundary. Applications to Air Traffic Control, Robotics, Computer Vision, Computational Geometry, Statistics, Instrumentation and other areas are discussed.

A covariant entropy conjecture

Abstract: We conjecture the following entropy bound to be valid in all space-times admitted by Einstein's equation: let A be the area of any two-dimensional surface. Let L be a hypersurface generated by surface-orthogonal null geodesics with non-positive expansion. Let S be the entropy on L. Then S ≤ A/4. We present evidence that the bound can be saturated, but not exceeded, in cosmological solutions and in the interior of black holes. For systems with limited self-gravity it reduces to Bekenstein's bound. Because the conjecture is manifestly time reversal invariant, its origin cannot be thermodynamic, but must be statistical. It thus places a fundamental limit on the number of degrees of freedom in nature.

Zero rest-mass fields including gravitation: asymptotic behaviour

Abstract: A zero rest-mass field of arbitrary spin s determines, at each event in space-time, a set of 2 s principal null directions which are related to the radiative behaviour of the field. These directions exhibit the characteristic ‘peeling-off9 behaviour of Sachs, namely that to order r - k -1 ( k = 0, . . . , 2 s ), 2 s - k of them coincide radially, r being a linear parameter in any advanced or retarded radial direction. This result is obtained in part I for fields of any spin in special relativity, by means of an inductive spinor argument which depends ultimately on the appropriate asymptotic behaviour of a very simple Hertz-type complex scalar potential. Spin ( s - ½) fields are used as potentials for spin s fields, etc. Several examples are given to illustrate this, In particular, the method is used to obtain physically sensible singularity-free waves for each spin which can be of any desired algebraic type. In part II, a general technique is described, for discussing asymptotic properties of fields in curved space-times which is applicable to all asymptotically flat or asymptotically de Sitter space-times. This involves the introduction of ‘points at infinity’ in a consistent way. These points constitute a hypersurface boundary I to a manifold whose interior is conformally identical with the original space-time. Zero rest-mass fields exhibit an essential conformal invariance, so their behaviour at ‘infinity’ can be studied at this hypersurface. Continuity at I for the transformed field implies that the ‘peeling-off’ property holds. Furthermore, if the Einstein empty-space equations hold near I then continuity at I for the transformed gravitational field is a consequence. This leads to generalizations of results due to Bondi and Sachs. The case when the Einstein-Maxwell equations hold near I is also similarly treated here. The hypersurface I is space-like, time-like or null according as the cosmological constant is positive, negative or absent. The technique affords a covariant approach to the definition of radiation fields in general relativity. If I is not null, however, the radiation field concept emerges as necessarily origin dependent. Further applications of the technique are also indicated.