Geometric Explanation Research Articles

Numerical Integration of Periodic Functions: A Few Examples

The constant K has the same meaning as above. 1CAS = Computer Algebra System. 2The interval [0, 2π] is for convenience only. Everything we say can easily be extended to an arbitrary interval [a, b]. 3One notices that the error bound for the midpoint rule is one half that of the trapezoidal rule; compare (2) with (3). For a pretty geometrical explanation of why one can expect the midpoint rule to be better by about a factor of two, the reader is referred to Stewart [13, p. 460].

Some remarks on Chow varieties and Euler–Chow series

In this article we give a geometric explanation of the fact that the Betti numbers of the d-fold symmetric product of the proyective space of dimension n are the same as those of the Grassmanian of d-planes in the complex vector space of dimension n+ d. In fact, we give a correspondence which is the graph of a rational morphism which induces an isomorphism, and whose matrix is the identity. We also prove some properties of Euler–Chow series and state some open problems related to this series.

Dynamic of the organisation and structural inertia

– The story of organisation is often related as a successive period of stability and crisis. This article is based on the hypothesis that it is the intra-organisational inertia (i.e.: the organisation lack of adaptation to any modifications of the environment) the cause of usual severe ruptures in the evolution of firms. Indeed if we consider as theorists of organisations that the interrelation settled between the organisation and its environment is structural, the questions of representation, the evolution and adaptation become essential to understand the dynamic organisation. Starting by a geometric construction a model describing the organisation dynamics will be introduced, with elements enabling us to understand what an organisational change is (construction of a taxonomy) by the forces that generate it (construction of a dynamic). It will be more precisely question of trying to identify the notion of force compared with the notion of efficiency. And taking into account “this idea of crisis” which characterise all organisational transformation a geometric explanation based on the rupture of the organisation trajectory and singularity will be suggested. The interpretation of the first results thanks to a mere example of a model application in dynamic organisation brings to the fore, the importance of its history (for instance training in this example) in its adaptation capacity proving that it may become the cause of inertia, and organisational inefficiency. Classification Codes: L21.

Polarized Submillimeter Emission from Filamentary Molecular Clouds

We model the submillimeter patterns that are expected for filamentary clouds that are threaded by helical magnetic fields. We previously developed a three-parameter model of such clouds, which are described by a concentration parameter C and two flux-to-mass ratios Γz and Γ to specify the mass loading of the poloidal and toroidal field lines, respectively. Our models provide a simple and purely geometric explanation for the well-known polarization hole effect, in which the submillimeter percentage decreases toward the regions of peak intensity. This occurs because of a cancellation between contributions to the from the backbone of poloidal flux along the filament's axis and its surrounding envelope, which is dominated by the toroidal field component. A systematic exploration of our parameter space allows us to classify the patterns due to filaments aligned approximately perpendicular to the plane of the sky into three basic types. The vectors are parallel to filaments when Bz,S/B,S 0.1, where Bz,S and B,S are, respectively, the poloidal and toroidal magnetic field components at the outer surface of the filament. The vectors are perpendicular to filaments when Bz,S/B,S 0.37. Intermediate cases result in patterns that contain 90° flips in the orientation of the vectors. The flips are symmetric about the central axis for filaments oriented parallel to the plane of the sky, but more complicated asymmetric patters result from filaments that are inclined at some angle.

Streaked speckle inCu3Aucoherent x-ray diffraction

Coherent x-ray diffraction from the binary alloy Cu{sub 3}Au has speckled superstructure Bragg reflections due to its antiphase domain structure. For nonspecular superstructure reflections, it is possible to vary the diffraction geometry through a range of incidence and exit angles. Under grazing-exit conditions on a prepared Cu{sub 3}Au(111) thin-film sample, the superstructure speckles are found to become highly elongated and rotated on the detector. A detailed geometrical explanation of this behavior is developed to explain the effect.

About the Mixing and CP Violation in Neutrino System

Suppose that the geometrical explanation to the weak CP phase in quark sector is also valid for neutrinos, the mixing and CP violation in neutrino system are discussed. We find that a JCp larger than implies the large-mixing solution for solar neutrino problem. In the case of bi-maximal mixing, we predict relative large CP violation with JCp larger than 10-3 in neutrino system, except the third mixing angle approaching to 0 or very closely.

A Note on Mascarenhas' Counterexample about Global Convergence of the Affine Scaling Algorithm

Mascarenhas gave an instance of linear programming problems to show that the long-step affine scaling algorithm can fail to converge to an optimal solution with the step-size λ=0.999 . In this note, we give a simple and clear geometrical explanation for this phenomenon in terms of the Newton barrier flow induced by projecting the homogeneous affine scaling vector field conically onto a hyperplane where the objective function is constant. Based on this interpretation, we show that the algorithm can fail for "any" λ greater than about 0.91 (a more precise value is 0.91071), which is considerably shorter than λ = 0.95 and 0.99 recommended for efficient implementations.

Theoretical study of mean-field Boltzmann machine learning by information geometry

Mean-field Boltzmann machine learning is recognized as a practical method to circumvent the difficulty that Boltzmann machine learning is very time-consuming. However, its theoretical meaning is still not clear. In this paper, based on information geometry, we give an information-theoretic interpretation of mean-field Boltzmann machine learning and a clear geometrical explanation of the approximation used there. Based on this interpretation, computer simulations for evaluating the effectiveness of mean-field Boltzmann machine learning are carried out for two-unit Boltzmann machines. The necessity of geometrical analysis in demonstrating the effectiveness of mean-field Boltzmann machine learning is discussed. © 1999 Scripta Technica, Electron Comm Jpn Pt 3, 82(8): 30–39, 1999

Theoretical study of mean‐field Boltzmann machine learning by information geometry

Mean-field Boltzmann machine learning is recognized as a practical method to circumvent the difficulty that Boltzmann machine learning is very time-consuming. However, its theoretical meaning is still not clear. In this paper, based on information geometry, we give an information-theoretic interpretation of mean-field Boltzmann machine learning and a clear geometrical explanation of the approximation used there. Based on this interpretation, computer simulations for evaluating the effectiveness of mean-field Boltzmann machine learning are carried out for two-unit Boltzmann machines. The necessity of geometrical analysis in demonstrating the effectiveness of mean-field Boltzmann machine learning is discussed. © 1999 Scripta Technica, Electron Comm Jpn Pt 3, 82(8): 30–39, 1999

The Cramer-Rao bound for position and amplitude estimation of multiple pulses in Gaussian noise

Estimating the position or amplitude of a pulse is a common problem in radar and seismic signal processing. In this paper an approximation to the Cramer-Rao bound for estimation of positions and amplitudes of multiple pulses is derived. It is shown that a vector can be associated with each parameter and the effect of a parameter on the estimation of another parameter is expressed as the normalized cross correlation of the vectors associated with these two parameters. We give examples to show that this approximation is good. These results provide a simple, geometric explanation for the effect of one pulse on the estimation of another.

The quantum tetrahedron in 3 and 4 dimensions

Recent work on state sum models of quantum gravity in 3 and 4 dimensions has led to interest in the `quantum tetrahedron'. Starting with a classical phase space whose points correspond to geometries of the tetrahedron in R^3, we use geometric quantization to obtain a Hilbert space of states. This Hilbert space has a basis of states labeled by the areas of the faces of the tetrahedron together with one more quantum number, e.g. the area of one of the parallelograms formed by midpoints of the tetrahedron's edges. Repeating the procedure for the tetrahedron in R^4, we obtain a Hilbert space with a basis labelled solely by the areas of the tetrahedron's faces. An analysis of this result yields a geometrical explanation of the otherwise puzzling fact that the quantum tetrahedron has more degrees of freedom in 3 dimensions than in 4 dimensions.

Linearizations of ordinary differential equations by area preserving maps

We clarify the class of second and third order ordinary differential equations which can be tranformed to the simplest equations Y″ = 0 and Y‴ = 0. The coordinate changes employed to transform the equations are respectively area preserving maps for second order equations and contact form preserving maps for third order equations. A geometric explanation of the results is also given by using connections and associated covariant differentials both on tangent and cotangent spaces.

A note on the NLS and the Schrödinger flow of maps

We prove, by finding nice Lax pairs, that the NLS −: i ψ t + ψ xx − 2| ψ| 2 ψ = 0 is gauge equivalent to the Schrödinger flow of maps from R 1 into the hyperbolic 2-space. This gives a geometric explanation for the NLS −.

A Geometrical Explanation for the Deflection of Light

A Geometrical Explanation for the Deflection of Light

Integral equation methods in plane-strain elasticity with boundary reinforcement

In this paper, a linear theory of elastic boundary reinforcement of an elastic solid is developed for plane–strain deformations. The reinforcement consists of a thin elastic coating bonded to part of the boundary of the solid. The elastic properties of the coating incorporate both extensibility and bending rigidity. Interior and exterior mixed–boundary problems are formulated and solved using integral equation methods. The boundary value problems are reduced to systems of singular integro–differential equations to which Noether–type theorems are shown to apply. The case corresponding to a coating which has only extensibility properties and no bending rigidity is particularly interesting from a mathematical point of view and is given special attention. Finally, existence and uniqueness results are presented for both interior and exterior reinforcement problems of plane–strain.

A geometric explanation of the effects of mild streamline curvature on the turbulence anisotropy

A new explanation of the well documented effects of mild streamline curvature on the anisotropy of sheared turbulence has been developed. Its main underlying assumption is that the mean streamline curvature has no direct effect on the production and dynamics of each individual turbulent eddy, which is produced with the structure of purely sheared turbulence, and is subsequently convected downstream while retaining its initial anisotropy relative to fixed inertial coordinates. The local Reynolds stress anisotropy accumulates the contributions of all surviving eddies produced upstream, which, because the mean shear keeps changing direction, have different anisotropies, when viewed in terms of the local curvilinear coordinates; thus, the local anisotropy is influenced by the history of flow curvature only indirectly. A model developed to demonstrate the validity of the hypothesis requires only the specification of the turbulence anisotropies in a fully developed, rectilinear, reference flow (e.g., a rectilinear uniformly sheared flow), the geometrical features of the flow under study, and a dimensionless mean eddy lifetime. It predicts accurately the observed asymptotic turbulence structure of uniformly sheared flow subjected to prolonged, constant curvature and the exponential adjustment of this structure to stepwise changes in flow curvature. Predictions of the shear stress anisotropy in a curved mixing layer are also in good agreement with published data. In all these cases, the present model makes better predictions than two popular Reynolds stress models and a rapid distortion model.

Moon-Earth-Sun: The oldest three-body problem

The daily motion of the Moon through the sky has many unusual features that a careful observer can discover without the help of instruments. The three different frequencies for the three degrees of freedom have been known very accurately for 3000 years, and the geometric explanation of the Greek astronomers was basically correct. Whereas Kepler's laws are sufficient for describing the motion of the planets around the Sun, even the most obvious facts about the lunar motion cannot be understood without the gravitational attraction of both the Earth and the Sun. Newton discussed this problem at great length, and with mixed success; it was the only testing ground for his Universal Gravitation. This background for today's many-body theory is discussed in some detail because all the guiding principles for our understanding can be traced to the earliest developments of astronomy. They are the oldest results of scientific inquiry, and they were the first ones to be confirmed by the great physicist-mathematicians of the 18th century. By a variety of methods, Laplace was able to claim complete agreement of celestial mechanics with the astronomical observations. Lagrange initiated a new trend wherein the mathematical problems of mechanics could all be solved by the same uniform process; canonical transformations eventually won the field. They were used for the first time on a large scale by Delaunay to find the ultimate solution of the lunar problem by perturbing the solution of the two-body Earth-Moon problem. Hill then treated the lunar trajectory as a displacement from a periodic orbit that is an exact solution of a restricted three-body problem. Newton's difficultly in explaining the motion of the lunar perigee was finally resolved, and the Moon's orbit was computed by a new method that became the universal standard until after WW II. Poincar\'e opened the 20th century with his analysis of trajectories in phase space, his insistence on investigating periodic orbits even in ergodic systems, and his critique of perturbation theory, particularly in the case of the Moon's motion. Space exploration, astrophysics, and the landing of the astronauts on the Moon led to a new flowering of celestial mechanics. Lunar theory now has to confront many new data beyond the simple three-body problem in order to improve its accuracy below the precision of 1 arcsecond; the computer dominates all the theoretical advances. This review is intended as a case study of the many stages that characterize the slow development of a problem in physics from simple observations through many forms of explanation to a high-precision fit with the data.

Geometry of Mixed-Mode Oscillations in the 3-D Autocatalator

We present a geometric explanation of a basic mechanism generating mixed-mode oscillations in a prototypical simple model of a chemical oscillator. Our approach is based on geometric singular perturbation theory and canard solutions. We explain how the small oscillations are generated near a special point, which is classified as a folded saddle-node for the reduced problem. The canard solution passing through this point separates small oscillations from large relaxation type oscillations. This allows to define a one-dimensional return map in a natural way. This bimodal map is capable of explaining the observed bifurcation sequence convincingly.

Smooth trajectory tracking of three-link robot: a self-organizing CMAC approach

A neuro fuzzy system which is embedded in the conventional control theory is proposed to tackle physical learning control problems. The control scheme is composed of two elements. The first element, the fuzzy sliding mode controller (FSMC), is used to drive the state variables to a specific switching hyperplane or a desired trajectory. The second one is developed based on the concept of the self organizing fuzzy cerebellar model articulation controller (FCMAC) and adaptive heuristic critic (AHC). Both compose a forward compensator to reduce the chattering effect or cancel the influence of system uncertainties. A geometrical explanation on how the FCMAC algorithm works is provided and some refined procedures of the AHC are presented as well. Simulations on smooth motion of a three-link robot is given to illustrate the performance and applicability of the proposed control scheme.

Kinematics of rigid bodies in general spatial motion: second-order motion properties

This work treats the kinematics of rigid bodies in general spatial motion using vector algebra. New results as well as other known properties are derived and expressed in simple form with geometrical reasoning and explanation. A special emphasis goes to the second-order motion properties. A simple definition and a geometrical approach for the investigation of the existence of the acceleration center is considered. The existence of a unique acceleration center or a single infinity of acceleration centers forming a line of acceleration centers is investigated in detail. The concept of the acceleration center is used to simply determine the acceleration of any point in the moving body in matrix form. Furthermore the distribution of the acceleration field corresponding to the acceleration center is studied in a systematic and easy way. It is believed that this is the first time this subject has been tackled from a general perspective, considering completely general the spatial motion of rigid bodies, in a simple and straightforward approach. Planar and spherical kinematics are stemmed as special cases of the kinematics of bodies in spatial motion. This study is intended to clear away the subject of second-order motion properties and lead to a general understanding.