Geometric Notions Research Articles

A Functional From Geometry with Applications to Discrepancy Estimates and the Radon Transform

Estimates of discrepancy, or irregularities of distribution, are obtained for measures without atoms. Two estimators are used, the half-space, or separation, discrepancy ${D_S}$ and a geometric functional ${I^\alpha }$. A representation formula for the generalized energy integral ${I^\alpha }$ is developed. Norm inequalities for the Radon transform are obtained as an application of the continuous discrepancy results. Integral geometric notions play a prominent role.

On the integrability of second-order G-structures with applications to continuous theories of dislocations

The material structure of a uniform elastic body whose mechanical response is sensitive to both the first and second gradients of the deformation is shown to be closely mirrored by the geometric notion of second-order G-structures. The presence of inhomogeneities, such as dislocations and disclinations, manifests itself through the lack of integrability of the associated G-structure. A measure of second-order inhomogeneity is introduced and conditions of integrability are derived for important particular structure groups.

Uncertainty in the weighted gap metric: A geometric approach

The stability of control systems is studied in the context of weighted input-output signal spaces. Necessary and sufficient conditions for a controller to stabilize a plant are given in terms of geometric notions. These geometric quantities can be calculated by solving H ∞ optimization problems. Maximally stabilizing controllers in a weighted signal space are introduced and characterized in terms of Nehari extensions. The robustness properties of maximally stabilizing controllers are analysed in terms of weighted coprime factor uncertainty. Necessary and sufficient conditions are established for a controller of a given plant to be the maximally stabilizing controller of the plant with respect to a weight. An upper bound for the mixed-sensitivity of a control system is given where the controller is the maximally stabilizing controller of the plant.

On parabolic geometry and irreversible macroscopic time

A theory of 2-dimensional parabolic geometry, which is a special degenerate case of the Banach-Minkowski space or the Kropina type Finsler space, is presented and developed. Then it is shown how this mathematical model can be applied to irreversible non-equilibrium thermodynamics. In particular, the concept of macroscopic irreversibility of time is explicated by means of geometrical notions of parabolic geometry.

Leibniz and Topological Equivalence

Did Leibniz invent or, if you prefer, discover topology with his analysis situs? Yes, urge Nicholas Rescher (1978, p. 70), George MacDonald Ross (1984, p. 29) and Ian Hacking (1984, p. 213). No, urge Hans Freudenthal (1954/1972), Benson Mates (1986, p. 240) and Michael Otte (1989, p. 24). James Alexander (1932/1967, p. 249), drawing a distinction between point set and combinatorial methods, cautiously remarked that combinatorial topology “is more nearly a development of Leibniz's original idea.” Less cautiously, Morris Kline (1972, p. 1163) remarked that “to the extent that he was at all clear, Leibniz envisioned what we now call combinatorial topology.” Louis Couturat (1961, p. 429), Rudolf Carnap (1922, p. 81) and Ernst Cassirer (1950, p. 49) proposed projective geometry as the realization of Leibniz's project. Dennis Martin (1983, p. 5) sees topology as a development from analysis situs. Javier Echeverria (1988, p. 218), reporting on his archival research, argues that Leibniz “successfully introduced very general geometrical notions that boil down to what is known today as topology.” And a good many others, for and against, might be cited.

Mathematical notions in preliterate societies

Many preliterate societies show considerable mathematical development. It is not unusual to encounter numeral systems going up to the hundreds of thousands or higher and to encounter some arithmetic ability. Many geometric notions are present, including simple forms like circles, lines, rectangles and parallelograms, and triangles, Other shapes and curves occur with less frequency. Some simpler geometric constructions are carried out, and perhaps some properties of similar right triangles are known. Logical reasoning shows up in kinship relationships and in some games of strategy. People seem to develop basic mathematical notions largely in response to what they consider important aspects of life. Large numbers come up when people need to count higher, whether counting shells used as money or yams harvested and stored or animals. More precise geometric constructions also enter in only where the society values precision. Part of determining exactly what a society has accomplished mathematically involves finding those aspects of life where the development of mathematical notions was important to the people.

Method for evaluating the fractal dimension of curves using convex hulls.

A local analysis for simple curves is performed using the geometric notions of size and deviation of an arc. For typical parametrized curves, the rate of decrease of these averaged measurements is related to the value of the fractal dimension. A method for calculating this dimension is derived, generalizing the variation method used for graphs of continuous functions to a large family of curves. Our method, the constant-deviation variable-step method, is tested on various types of curves (fractal Brownian motion, Gosper curve, and geographical coastline). A comparison is made with other methods such as the divider-step method, variation method, and diameter method. We show that our method offers the same level of performance while exhibiting much more general applications.

A ?geometric? view of the dynamics of trajectories of computer programs

This paper applies mathematical ideas which arise from differential geometry and control theory to computer programming theory. For the sake of simplicity, most questions of logic and language are put to the side. A computer program is regarded as a collection of data, considered as a collection of cross-sections of a fiber space whose base is a discrete space. These distinguished cross-sections are called the trajectories of the computer program system. For the simplest von Neumann type of program, the base space is the nonnegative integers: The trajectories may then often be thought of as the orbits of a discretetime dynamical system. Feedback control theory enters when one attempts to provide a ‘rational’ way of generating such dynamical systems, satisfying certain conditions, such as termination and correctness. The greatest common divisor and bubble sort programs — standard examples in computer science textbooks — are treated as models for the development of a general theory. Geometrically, we will also consider them as the discrete analogue of what are called path systems in differential geometry. These examples can also be treated as the analogue of closed-loop feedback control systems — themselves a sort of path system — that are ‘driven’ to the final answer by a process that is similar to the way that Lyapunov theory is used in the linear regulator problem of control theory. This use of the Lyapunov ideas of dynamical system theory gives a technique for proving the correctness and termination of some programs. In order to provide a mathematical structure to a programming theory that is, in some sense, analogous to the group-theoretic structure of fiber bundle differential geometry, an algebraic structure of a category which acts on the fiber space is associated with such programs. The computation of the binomial coefficients will also be treated as an example of a system of partial difference equations, leading to the description of some features of concurrent programs. Prolongations and correspondences are defined for programs, based on the analogous ideas for differential systems. Finally, some of the geometric background of modal logic is developed, particularly as it is used in following the temporal development of computer programs. The approach is related to the classical geometric notion of correspondence, and to the Jonsson-Tarski theory of operators on Boolean algebras.

Sur la convexité holomorphe. Théorie locale

We define a geometric convexity notion for certain open subsets of C n . We prove some results about local cohomology expliciting the topology of the last non zero cohomology group ; the cohomology here considered is the Dolbeault’s cohomology of differential forms.

Geometric theory and feedback invariants of generalized linear systems: A matrix pencil approach

Generalized linear systems are classified according to their algebraic structure and their regularizability and normalizability properties under generalized state feedback are investigated. New feedback invariants are introduced in terms of the input-space restriction pencil and the Plucker matrix of the system. It is shown that the classification of subspaces of the state space of generalized linear systems is reduced to an equivalent problem of classifying the subspacesV of the domain of an ordered pair (F, G), F, Geℝm×n. The set of strict equivalence invariants of the restriction pencil (F, G)/V provides a complete, basis-free algebraic characterization and leads to the definition of notions of geometric invariance. The key geometric notions that emerge are those of (F, G)-, (G, F)-, complete-(F, G)-, partitioned(F, G)-, cyclic-, and semicyclic-invariant subspaces. A complete set of geometric algorithms leading to the computation of the above families is also given. The above families of invariant subspaces are characterized in terms of the invariants of (F, G)/V, and this provides the links with their dynamic characterization. These results provide an “open-loop” or “feedback-free” unifying treatment of spaces of generalized systems, which for the case of proper systems is reduced to the standard geometric theory results.

Some Families of Isoparametric Hypersurfaces and Rigidity in a Complex Hyperbolic Space

The geometric notion of equivalence for submanifolds in a chosen ambient space is that of congruence. In this study, a certain type of isoparametric hypersurface of a complex hyperbolic space form is shown to have a rigid immersion by utilizing the congruences of a Lorentzian hyperbolic space form that lies as an ${S^1}$-fiber bundle over the complex hyperbolic space. Several families of isoparametric hypersurfaces (namely tubes and horospheres) are constructed whose immersions are rigid.

Some families of isoparametric hypersurfaces and rigidity in a complex hyperbolic space

The geometric notion of equivalence for submanifolds in a chosen ambient space is that of congruence. In this study, a certain type of isoparametric hypersurface of a complex hyperbolic space form is shown to have a rigid immersion by utilizing the congruences of a Lorentzian hyperbolic space form that lies as an S 1 {S^1} -fiber bundle over the complex hyperbolic space. Several families of isoparametric hypersurfaces (namely tubes and horospheres) are constructed whose immersions are rigid.

Zeros and poles of linear periodic multivariable discrete-time systems

In this paper the notions of transmission zero, invariant zero, and structural zero (that is the geometric notion of zero) are extended to linear periodic discrete-time systems, as well as the notion of pole. Their meaning is clarified and their relationships stressed, thus extending the time-invariant theory. In particular, the invariant zeros and the structural zeros are shown to coincide, together with their multiplicities, and the former are shown to be independent of a linear state feedback. Moreover, the nonzero transmission zeros, the nonzero invariant zeros, and the nonzero poles are shown to be independent of time, together with their multiplicities. Some of these results are obtained with the help of the notions of reachability subspace and inner controllable subspace, which constitute a further development of the geometric theory for this class of systems.

Second Order Directional Derivatives in Parametric Optimization Problems

We characterize the second order directional derivative in the sense of Ben-Tal and Zowe of a marginal function Ψ of the following general form: Ψ(x) − Inf{f(x, y) ∣ y ∈ Ω(x)}. This requires us to introduce some geometrical notions as, for example, the first and second order tangent directions to a multifunction. The main results are then applied to some particular types of marginal functions.

Operator Theoretical Realization of some Geometric Notions

Operator Theoretical Realization of some Geometric Notions

Electromagnetic chirality and its applications

Chirality is a geometric notion which refers to the handedness of an object. A chiral object is, by definition, a body that cannot be brought into congruence with its mirror image by translation and rotation. In other words, such a body lacks bilateral symmetry, and cannot be superposed on its mirror image. An object of this sort has the property of handedness and is said to be either right-handed or left-handed. An object that is not chiral is achiral. Some chiral objects occur in two versions relatedto each other as a chiral object and its mirror image. Objects so related are said to be enanriomorphs of each other. If a chiral object is found to be lefthanded, its enantiomorph is right-handed, and vice versa. Two examples of chiral objects, the Mbbius strip and the irregular tetrahedron, and their enantiomorphs are shown in Fig. 1. Note that these objects and their mirror images are incongruent

Operator theoretical realization of some geometric notions

This paper studies the realization of certain geometric constructions in Cowen-Douglas operator class. Through this realization, some operator theoretical phenomena are easily seen from the corresponding geometric phenomena. In particular, we use this technique to solve the first-order equivalence problem and introduce a new operation among certain operators.

The climate attractor over short timescales

Recent work has highlighted the possibilities of using certain ideas from the theory of dynamical systems for the study of global climate. These ideas include the geometrical notion of correlation or scaling dimension, first used to analyse attractors arising in mathematical and laboratory systems1–4 and later applied in a geophysical context5,6. The conclusion of the original geophysical work5 has been criticized in the light of a reanalysis7 which questions the existence of a low-dimensional climate attractor. Here results of a similar analysis conducted on daily meteorological observations from 1946 to 1982, are announced. This new work overcomes limitations of previous analyses, and supports the existence of such an attractor.

Conformally flat Riemannian metrics, Schrödinger operators, and semiclassical approximation

A relationship between Laplace-Beltrami and Schrödinger operators on Euclidean domains is analyzed and exploited for several purposes: We use the Schrödinger equation to analyze the spectra of Laplace-Beltrami operators with periodic metrics on R v , and use geometric notions and nonlinear differential equations to bound spectra and Green functions of Schrödinger operators in various ways. We also have a new, more operator-theoretic analysis of the semiclassical limit and the Liouville-Green (or JWKB) approximation in one dimension.

On the Segré, Weyr characteristics of right (left) regular matrix pencils

The classical Segré, Weyr characteristic theory of the standard eigenvalue-eigenvector problem defined on A ϵ ℝn×n is extended to the case of right (left) regular pencils sF — G, F, G ϵ ℝm×n. The notions of α-Toeplitz matrices and of the right (left) α. — (F, G) sequences Jα r(F,G) (Jα 1(F,G)) respectively are introduced. For right (left) regular pairs it is shown that Jα r(F,G) (Jα 1(F,G)) are piecewise arithmetic progression sequences (PAPS); the singular points of those PAPS define the possible degrees of the elementary divisors (e.d.) at α, whereas the deviations, or gaps from the arithmetic progression sequence, define the multiplicity of the corresponding degree e.d.s. The links between the algebraic notion of the Segré characteristic and the geometric notion of the Weyr characteristic are established for such families of pencils. Two different procedures for the computation of the Segré characteristic are given; the first is based on the analysis of singular points of PAPS and the second is based on the conjugate partitioning property of the algebraic multiplicity of a zero of sF - G at α by the Weyr and Segré characteristics at a. In the latter case a Ferrers-type diagram is suggested as the tool for the computation of the Segré characteristic from the Weyr characteristic. The results provide a procedure for the computation of the Weierstrass part of the Kronecker form of right (left) regular pencils by rank tests on α-Toeplitz matrices.