Geometry of the quantum projective plane
D'Andrea, Francesco
2009-01-01
We review some of the geometry of the quantum projective plane with emphasis on the construction of a differential calculus and of the Dirac operator (of a spin^c-structure). We also report on anti-self-dual connections on line bundles, the spectrum of the associated Laplacians, and the definition of classical and quantum characteristic classes.
The Casimir effect in the sphere-plane geometry
Canaguier-Durand, Antoine; Neto, Paulo A Maia; Lambrecht, Astrid; Reynaud, Serge
2012-01-01
We present calculations of the Casimir interaction between a sphere and a plane, using a multipolar expansion of the scattering formula. This configuration enables us to study the nontrivial dependence of the Casimir force on the geometry, and its correlations with the effects of imperfect reflection and temperature. The accuracy of the Proximity Force Approximation (PFA) is assessed, and is shown to be affected by imperfect reflexion. Our analytical and numerical results at ambient temperature show a rich variety of interplays between the effects of curvature, temperature, finite conductivity, and dissipation.
Noncommutative Geometry: Fuzzy Spaces, the Groenewold-Moyal Plane
Directory of Open Access Journals (Sweden)
Aiyalam P. Balachandran
2006-12-01
Full Text Available In this talk, we review the basics concepts of fuzzy physics and quantum field theory on the Groenewold-Moyal Plane as examples of noncommutative spaces in physics. We introduce the basic ideas, and discuss some important results in these fields. At the end we outline some recent developments in the field.
Multiple-View Geometry of the Refractive Plane
Chari, Visesh; Sturm, Peter
2009-01-01
International audience; Transparent refractive objects are one of the main problems in geometric vision that have been largely unexplored. The imaging and multi-view geometry of scenes with transparent or translucent objects with refractive properties is relatively less well understood than for opaque objects. The main objective of our work is to analyze the underlying multi-view relationships between cameras, when the scene being viewed contains a single refractive planar surface separating ...
The bifurcation diagram of drops in a sphere/plane geometry: influence of contact angle hysteresis
Ruiter, de Riëlle; Gorcum, van M.; Semprebon, C.; Duits, M.H.G.; Brinkmann, M.; Mugele, F.
2014-01-01
We study liquid drops that are present in a generic geometry, namely the gap in between a sphere and a plane. For the ideal system without contact angle hysteresis, the drop position is solely dependent on the contact angle, drop volume, and sphere/ plane separation distance. Performing a geometric
Plane Pendulum and Beyond by Phase Space Geometry
Klee, Bradley
2016-01-01
By careful analysis, the simple harmonic approximation leads to a wildly inaccurate prediction for the period of a simple plane pendulum. We make a perturbation ansatz for the phase space trajectory of a one-dimensional, anharmonic oscillator and apply conservation of energy to set undetermined functions. Iteration of the algorithm yields, to arbitrary precision, a solution to the equations of motion and the period of oscillation. Comparison with Jacobian elliptic functions leads to multidimensional applications such as the construction of approximate Seiffert spirals. Throughout we develop a quantum/classical analogy for the purpose of comparing time-independent perturbation theories.
The advanced geometry of plane curves and their applications
Zwikker, C
2005-01-01
""Of chief interest to mathematicians, but physicists and others will be fascinated ... and intrigued by the fruitful use of non-Cartesian methods. Students ... should find the book stimulating."" - British Journal of Applied PhysicsThis study of many important curves, their geometrical properties, and their applications features material not customarily treated in texts on synthetic or analytic Euclidean geometry. Its wide coverage, which includes both algebraic and transcendental curves, extends to unusual properties of familiar curves along with the nature of lesser known curves.Informativ
Geometry of the conics on the Minkowski plane
Aceff-Sanchez, F
2007-01-01
Conics in the Euclidean space have been known for their geometrical beauty and also for their power to model several phenomena in real life. It usually happens that when thinking about the conics in a semi-Riemannian manifold, the equations and the graphs that come to mind are those of the quadratic Euclidean equations. For example, a circle is always perceived like a closed curve. We study the geometry of the conics in the semi-Riemannian Minkowski spacetime, and interpret each equation with Euclidean eyes. By defining an extended geometric completeness for conics, we will show that the conic completeness of conics can be changed through a Euclidean mirror.
Plane geometry and convexity of polynomial stability regions
Henrion, Didier
2008-01-01
The set of controllers stabilizing a linear system is generally non-convex in the parameter space. In the case of two-parameter controller design (e.g. PI control or static output feedback with one input and two outputs), we observe however that quite often for benchmark problem instances, the set of stabilizing controllers seems to be convex. In this note we use elementary techniques from real algebraic geometry (resultants and Bezoutian matrices) to explain this phenomenon. As a byproduct, we derive a convex linear matrix inequality (LMI) formulation of two-parameter fixed-order controller design problem, when possible.
Geometry of state space in plane Couette flow
Cvitanović, P.; Gibson, J. F.
A large conceptual gap separates the theory of low-dimensional chaotic dynamics from the infinite-dimensional nonlinear dynamics of turbulence. Recent advances in experimental imaging, computational methods, and dynamical systems theory suggest a way to bridge this gap in our understanding of turbulence. Recent discoveries show that recurrent coherent structures observed in wall-bounded shear flows (such as pipes and plane Couette flow) result from close passes to weakly unstable invariant solutions of the Navier-Stokes equations. These 3D, fully nonlinear solutions (equilibria, traveling waves, and periodic orbits) structure the state space of turbulent flows and provide a skeleton for analyzing their dynamics. We calculate a hierarchy of invariant solutions for plane Couette, a canonical wall-bounded shear flow. These solutions reveal organization in the flow's turbulent dynamics and can be used to predict directly from the fundamental equations physical quantities such as bulk flow rate and mean wall drag. All results and the code that generates them are disseminated through through our group's open-source CFD software and solution database Channelflow.org and the collaborative e-book ChaosBook.org.
Spectral triplets, emergent geometry and entropy in Moyal plane
Chakraborty, B.; Scholtz, F. G.
2013-02-01
It was shown by Doplicher et.al. that the measurement of spacetime intervals of the order of Planck length scale is operationally impossible, as the process of measurement invariably gives rise to a black hole formation. This can be avoided by postulating non-vanishing commutation relations between the coordinates, which are now promoted to the level of operators. Formulation of quantum mechanics in these kinds of spaces through the introduction of Hilbert spaces of Hilbert-Schmidt operators is then shown to allow the construction of spectral triplets a la Connes naturally. The computation of spectral distance between pure and mixed states is then shown to exhibit a deep connection between entropy and geometry.
Koyuncu, Ilhan; Akyuz, Didem; Cakiroglu, Erdinc
2015-01-01
This study aims to investigate plane geometry problem-solving strategies of prospective mathematics teachers using dynamic geometry software (DGS) and paper-and-pencil (PPB) environments after receiving an instruction with GeoGebra (GGB). Four plane geometry problems were used in a multiple case study design to understand the solution strategies…
Geometry of magnetosonic shocks and plane-polarized waves: Coplanarity Variance Analysis (CVA)
Scudder, J. D.
2005-02-01
Minimum Variance Analysis (MVA) is frequently used for the geometrical organization of a time series of vectors. The Coplanarity Variance Analysis (CVA) developed in this paper reproduces the layer geometry involving coplanar magnetosonic shocks or plane-polarized wave trains (including normals and coplanarity directions) 300 times more precisely (CVA technique exploits the eigenvalue degeneracy of the covariance matrix present at planar structures to find a consistent normal to the coplanarity plane of the fluctuations. Although Tangential Discontinuities (TDs) have a coplanarity plane, the eigenvalues of their covariance matrix are usually not degenerate; accordingly, CVA does not misdiagnose TDs as shocks or plane-polarized waves. Together CVA and MVA may be used to sort between the hypotheses that the time series is caused by a one-dimensional current layer that has magnetic disturbances that are (1) coplanar, linearly polarized (shocks/plane waves), (2) intrinsically helical (rotational/tangential discontinuities), or (3) neither 1 nor 2.
Proofs with Coq of theorems in plane geometry using oriented angles
Guilhot, Frédérique
2002-01-01
Formalization of the theory of oriented angles of non zero vectors using Coq is reported. Using this theory, some classical plane geometry theorems are proved, among them : the theorem which gives a necessary and sufficient condition so that four points are cocyclic, the one which shows that the reflected points with respect to the sides of a triangle orthocenter are on its circumscribed circle, the Simson's theorem and the Napoleon's theorem. Elaboration of proofs using Coq that followed the...
Martinetti, Pierre; Tomassini, Luca
2013-10-01
We study the metric aspect of the Moyal plane from Connes' noncommutative geometry point of view. First, we compute Connes' spectral distance associated with the natural isometric action of on the algebra of the Moyal plane . We show that the distance between any state of and any of its translated states is precisely the amplitude of the translation. As a consequence, we obtain the spectral distance between coherent states of the quantum harmonic oscillator as the Euclidean distance on the plane. We investigate the classical limit, showing that the set of coherent states equipped with Connes' spectral distance tends towards the Euclidean plane as the parameter of deformation goes to zero. The extension of these results to the action of the symplectic group is also discussed, with particular emphasis on the orbits of coherent states under rotations. Second, we compute the spectral distance in the double Moyal plane, intended as the product of (the minimal unitization of) by . We show that on the set of states obtained by translation of an arbitrary state of , this distance is given by the Pythagoras theorem. On the way, we prove some Pythagoras inequalities for the product of arbitrary unital and non-degenerate spectral triples. Applied to the Doplicher- Fredenhagen-Roberts model of quantum spacetime [DFR], these two theorems show that Connes' spectral distance and the DFR quantum length coincide on the set of states of optimal localization.
Directory of Open Access Journals (Sweden)
Ivan H. Lenchuk
2014-02-01
Full Text Available Presented article concerns construction problems in plane geometry. Solved the problem of the formation of students' stereotypes efficient, economical in time visual representation of algorithms for solving problems on the modern computer screens. Used universal author’s method of fragmented typing tasks on the method of circles. Allocated rod-type problem with its subsequent filling with ingredients. Previously developed educational software (partially, GeoGebra ensure optimal realization of the construction. Their dynamic characteristics and constructive capabilities - quality visual- shaped stages of "evidence" and "research".
Tripoli, N K; Cohen, K L; Obla, P; Coggins, J M; Holmgren, D E
1996-06-01
To assess the accuracy with which the Keratron keratoscope (Optikon 2000, Rome, Italy) measured astigmatic test surfaces by a profile reconstruction algorithm within a plane geometry model and to discriminate between error caused by the model and error caused by other factors. Height was reported by the Keratron for eight surfaces with central astigmatism ranging from 4 to 16 diopters. A three-dimensional ray tracing simulation produced theoretic reflected ring patterns on which the Keratron's reconstruction algorithm was performed. The Keratron's measurements were compared with the surfaces' formulas and the ray-traced simulations. With a new mathematical filter for smoothing ring data, now part of the Keratron's software, maximum error was 0.47% of the total height and was usually less than 1% of local power for surfaces with 4 diopters of astigmatism. For surfaces with 16 diopters of astigmatism, maximum error was as high as 2.9% of total height and was usually less than 2.5% of local power. The reconstruction algorithm accounted for 40% and 70% of height error, respectively. The efficacy of keratoscopes cannot be assumed from their design theories but must be tested. Although plane geometry surface reconstruction contributed greatly to total height error, total error was so small that it is unlikely to affect clinical use.
Plane Geometry: From the Floor Plan of a House to the Quantity of Bricks
Directory of Open Access Journals (Sweden)
Cassiano Scott Puhl
2014-12-01
Full Text Available This work reports the application of a potentially meaningful teaching strategy for plane geometry which was applied in high school as an activity of identifying and reconstructing students’ previous knowledge on spatial geometry. The proposal aims to promote the meaningful learning, encourage students as active and more autonomous subjects and show the importance of studying this content. The study began with reading a story about the fire of Kiss Nightclub, where overcrowding was cited as one of the causes. To understand the mathematical sense of overcrowding, the students built the 1 m² and simulated the situation of the club on the night of the tragedy. Further, taking advantage of the interest and involvement of students, they were challenged to build the floor plan of a house. For this, chosen and measured bricks and ceramic would use in building the house. Finally, in joint work, all critically analyzed the work of the groups, plans and models, making a comparison between the various projects. The evaluation of this experience was very positive, because the students were motivated, learning together, and realizing the significance and meaning of the study of areas of plane figures.
RMT focal plane sensitivity to seismic network geometry and faulting style
Johnson, Kendra L.; Hayes, Gavin P.; Herrmann, Robert B.; Benz, Harley M.; McNamara, Dan E.; Bergman, Eric
2016-07-01
Modern tectonic studies often use regional moment tensors (RMTs) to interpret the seismotectonic framework of an earthquake or earthquake sequence; however, despite extensive use, little existing work addresses RMT parameter uncertainty. Here, we quantify how network geometry and faulting style affect RMT sensitivity. We examine how data-model fits change with fault plane geometry (strike and dip) for varying station configurations. We calculate the relative data fit for incrementally varying geometries about a best-fitting solution, applying our workflow to real and synthetic seismograms for both real and hypothetical station distributions and earthquakes. Initially, we conduct purely observational tests, computing RMTs from synthetic seismograms for hypothetical earthquakes and a series of well-behaved network geometries. We then incorporate real data and station distributions from the International Maule Aftershock Deployment (IMAD), which recorded aftershocks of the 2010 MW 8.8 Maule earthquake, and a set of regional stations capturing the ongoing earthquake sequence in Oklahoma and southern Kansas. We consider RMTs computed under three scenarios: (1) real seismic records selected for high data quality; (2) synthetic seismic records with noise computed for the observed source-station pairings and (3) synthetic seismic records with noise computed for all possible station-source pairings. To assess RMT sensitivity for each test, we observe the `fit falloff', which portrays how relative fit changes when strike or dip varies incrementally; we then derive the ranges of acceptable strikes and dips by identifying the span of solutions with relative fits larger than 90 per cent of the best fit. For the azimuthally incomplete IMAD network, Scenario 3 best constrains fault geometry, with average ranges of 45° and 31° for strike and dip, respectively. In Oklahoma, Scenario 3 best constrains fault dip with an average range of 46°; however, strike is best constrained by
Matrix Riccati equation formulation for radiative transfer in a plane-parallel geometry
Chang, Hung-Wen; Wu, Tso-Lun
1997-01-01
In this paper, we formulate the radiative transfer problem as an initial value problem via a pair of nonlinear matrix differential equations (matrix Riccati equations or MREs) which describe the reflection ( R) and transmission ( T) matrices of the specific intensities in a plane-parallel geometry. One first computes R and T matrices of some small but finite layer thickness (equivalent optical thickness 0959-7174/7/1/009/img1) and then repetitively applies the doubling method to the reflection and transmission matrices 0959-7174/7/1/009/img2 and 0959-7174/7/1/009/img3 until reaching the desired layer thickness. The initial matrices 0959-7174/7/1/009/img4 and 0959-7174/7/1/009/img5 can be computed quite accurately by either of the following methods: multiple-order, multiple-scattering approximation, iterative method or fourth-order Runge - Kutta techniques. In addition, the reflection coefficient matrix of a semi-infinite medium satisfies an algebraic matrix equation which can be solved repetitively by a matrix method. MREs offer an alternative way of solving plane-parallel radiative transport problems. This method requires only elementary matrix operations (addition, multiplication and inversion). For vector and/or beam-wave radiative transfer problems, large matrices are required to describe the physics adequately, and the MRE method provides a significant reduction in computer memory and computation time.
Kepler Commissioning Data for Measurement of the Pixel Response Function and Focal Plane Geometry
Bryson, Stephen T.
2017-01-01
This document describes the Kepler PRF/FPG data release. This data was taken on April 27-29, 2009, during Kepler's commissioning phase in order to measure the pixel response function (PRF) (Bryson et al., 2010a) and focal plane geometry (FPG) (Tenenbaum and Jenkins, 2010). 33,424 stellar targets were observed for 243 long cadences, each with a duration of 14.7 minutes (half the duration of a normal Kepler long cadence). During these 243 cadences the Kepler photometer was moved, pointing in a dither pattern to facilitate PRF measurement. Motion occurred during the even cadences (second, fourth, etc.), with the telescope in stable fine point at each pointing in the dither pattern during the odd cadences (first, third, etc.). The first and last cadences were at the center of the dither pattern. Motion cadences are included in this release, but they do not contain any data. For details on how this data was used to derive the Kepler PRF and FPG models, see Bryson et al. (2010a) and Tenenbaum and Jenkins (2010). Descriptions of the PRF and FPG models are found in Thompson et al. (2016), x2.3.5.17 and x2.3.5.16 respectively. The data in this release can be used to recompute the Kepler PRF and FPG. Such a reconstruction, however, would not reflect measured seasonal changes in the PRF described in Van Cleve et al. (2016b), x5.2.The dither pattern is shown in Figure 1. The crosses show the commanded pointings and the circles show the measured pointings. Measured pointings are different from the commanded pointings due to the early state of calibration of the fine guidance sensors during commissioning (Van Cleve et al., 2016a). The measured offsets from the center of the pattern are given in RADEC offsets and pixel offsets in Table 1. The order of the offsets was randomized during data collection to avoid time-dependent systematics.
Hayes, G.P.; Wald, D.J.
2009-01-01
A key step in many earthquake source inversions requires knowledge of the geometry of the fault surface on which the earthquake occurred. Our knowledge of this surface is often uncertain, however, and as a result fault geometry misinterpretation can map into significant error in the final temporal and spatial slip patterns of these inversions. Relying solely on an initial hypocentre and CMT mechanism can be problematic when establishing rupture characteristics needed for rapid tsunami and ground shaking estimates. Here, we attempt to improve the quality of fast finite-fault inversion results by combining several independent and complementary data sets to more accurately constrain the geometry of the seismic rupture plane of subducting slabs. Unlike previous analyses aimed at defining the general form of the plate interface, we require mechanisms and locations of the seismicity considered in our inversions to be consistent with their occurrence on the plate interface, by limiting events to those with well-constrained depths and with CMT solutions indicative of shallow-dip thrust faulting. We construct probability density functions about each location based on formal assumptions of their depth uncertainty and use these constraints to solve for the ‘most-likely’ fault plane. Examples are shown for the trench in the source region of the Mw 8.6 Southern Sumatra earthquake of March 2005, and for the Northern Chile Trench in the source region of the November 2007 Antofagasta earthquake. We also show examples using only the historic catalogues in regions without recent great earthquakes, such as the Japan and Kamchatka Trenches. In most cases, this method produces a fault plane that is more consistent with all of the data available than is the plane implied by the initial hypocentre and CMT mechanism. Using the aggregated data sets, we have developed an algorithm to rapidly determine more accurate initial fault plane geometries for source inversions of future
Institute of Scientific and Technical Information of China (English)
Song Falun; Cao Jinxiang; Wang Ge
2005-01-01
The purpose of the present work is to present a full-wave analysis of scattering from the weakly ionized plasma in the plane geometry. We have yielded an approximate solution in an analytic form to the electromagnetic wave scattering from the weakly ionizsd plasma. In the normal and oblique incidence, the analytic solution works well, as compared with the exact solution and the solution based on the Wenzell-Kramers-Brillouin-Jeffreys (WKBJ) approximation to the uniform density profile.
Banet, Matthias T.; Spencer, Mark F.; Raynor, Robert A.; Marker, Dan K.
2016-09-01
Digital holography in the pupil-plane recording geometry shows promise as a wavefront sensor for use in adaptive-optics systems. Because current wavefront sensors suffer from decreased performance in the presence of turbulence and thermal blooming, there is a need for a more robust wavefront sensor in such distributed-volume atmospheric conditions. Digital holography fulfills this roll by accurately estimating the wrapped phase of the complex optical field after propagation through the atmosphere to the pupil plane of an optical system. This paper examines wave-optics simulations of spherical-wave propagation through both turbulence and thermal blooming; it also quantifies the performance of digital holography as a wavefront sensor by generating field-estimated Strehl ratios as a function of the number of pixels in the detector array, the Rytov number, and the Fried coherence diameter. Altogether the results indicate that digital holography wavefront sensing in the pupil-plane recording geometry is a valid and accurate method for estimating the wrapped phase of the complex optical field in the presence of distributed-volume atmospheric aberrations.
A Comparison of Two Methods of Teaching Selected Topics in Plane Analytic Geometry.
Bundrick, Charles Michael
Reported are the results of a study to determine if there is a difference in learning as measured by an achievement test between high school students who study plane analytic geometric topics via a vector approach and those who study the same topics via a traditional approach. Secondary objectives concerned the transfer to further topics in solid…
Transport of Terrestrial gamma-Radiation in Plane Semi-Infinite Geometry
DEFF Research Database (Denmark)
Kirkegaard, Peter; Løvborg, Leif
1980-01-01
The plane one-dimensional photon transport equation is solved for the scattered γ-radiation flux in the case of two adjacent media. One medium represents a natural ground with uniformly distributed potassium, uranium, and thorium γ-ray emitters. The other medium is air with no radioactive...
Spencer, Mark F.; Raynor, Robert A.; Banet, Matthias T.; Marker, Dan K.
2017-03-01
This paper develops wave-optics simulations which explore the estimation accuracy of digital-holographic detection for wavefront sensing in the presence of distributed-volume or "deep" turbulence and detection noise. Specifically, the analysis models spherical-wave propagation through varying deep-turbulence conditions along a horizontal propagation path and formulates the field-estimated Strehl ratio as a function of the diffraction-limited sampling quotient and signal-to-noise ratio. Such results will allow the reader to assess the number of pixels, pixel field of view, pixel-well depth, and read-noise standard deviation needed from a focal-plane array when using digital-holographic detection in the off-axis image plane recording geometry for deep-turbulence wavefront sensing.
Julias, Margaret; Buettner, Helen M; Shreiber, David I.
2010-01-01
During traditional acupuncture, fine needles are inserted subcutaneously and rotated, which causes loose fascial tissue to wind around the needle. This coupling is stronger at acupuncture points, which tend to fall above intermuscular fascial planes, than control points, which lay above skeletal muscle. These different anatomical constraints may affect the mechanical coupling. Fascia at acupuncture points is bounded on two sides by skeletal muscle, but at control points is essentially unbound...
Julias, Margaret; Buettner, Helen M; Shreiber, David I
2011-02-01
During traditional acupuncture, fine needles are inserted subcutaneously and rotated, which causes loose fascial tissue to wind around the needle. This coupling is stronger at acupuncture points, which tend to fall above intermuscular fascial planes, than control points, which lay above skeletal muscle. These different anatomical constraints may affect the mechanical coupling. Fascia at acupuncture points is bounded on two sides by skeletal muscle, but at control points is essentially unbounded. These differences were approximated in simple in vitro models. To emulate the narrower boundary within the intermuscular plane, type I collagen was cast in circular gels of different radii. To model the channel-like nature of these planes, collagen was cast in elliptical gels with major and minor axes matching the large and small circular gels, respectively, and in planar gels constrained on two sides. Acupuncture needles were inserted into the gels and rotated via a computer-controlled motor while capturing the evolution of fiber alignment under cross-polarization. Small circular gels aligned faster, but failed earlier than large circular gels. Rotation in elliptical and planar gels generated more alignment-per-revolution than circular gels. Planar gels were particularly resistant to failure. Fiber alignment in circular gels was isotropic, but was stronger in the direction of the minor axis in elliptical and planar gels. In fibroblast-populated gels, cells followed the alignment of the collagen fibers, and also became denser in regions of stronger alignment. These results suggest that the anatomy at acupuncture points provides unique boundaries that accentuate the mechanical response to needle manipulation.
Numerical investigation of the formation of Trichel pulses in a needle-plane geometry
Dordizadeh, Peyman; Adamiak, Kazimierz; Castle, G. S. Peter
2015-10-01
This paper presents a numerical investigation of the formation of the Trichel pulses in negative dc corona discharge for a needle-plane configuration in atmospheric air. A 2D axisymmetric model of the problem considering three charged species and consisting of a hyperboloid needle with a tip radius of 35 μm and needle-plane spacing of 6 mm over the voltage range of -3.5 kV to -12 kV has been studied. A commercial finite element package COMSOL was used for simultaneously solving three convection-diffusion equations along with Poisson’s equation. In order to obtain a better understanding of the processes which lead to the pulse formation, a close look is taken at one of the pulses. The distributions of the major charged species and the timeline of peak-values of charged species densities and electric field are presented. Through tracing the evolution of the locations of the peak densities of the charged species some new insights have been provided. The configuration of the model was chosen so that the simulation results could be compared with the experimental data published by Lama and Gallo. The numerical results were in acceptable agreement with the experimental values. Some explanations are given for the discrepancies between experimental and simulation results. It is also shown, that as the frequency of the pulses increases with voltage, the transition from Trichel pulse discharge to glow discharge initiates and full glow discharge is reached at -12 kV.
Pedoe, Dan
1988-01-01
""A lucid and masterly survey."" - Mathematics Gazette Professor Pedoe is widely known as a fine teacher and a fine geometer. His abilities in both areas are clearly evident in this self-contained, well-written, and lucid introduction to the scope and methods of elementary geometry. It covers the geometry usually included in undergraduate courses in mathematics, except for the theory of convex sets. Based on a course given by the author for several years at the University of Minnesota, the main purpose of the book is to increase geometrical, and therefore mathematical, understanding and to he
Numerical solutions of matrix Riccati equations for radiative transfer in a plane-parallel geometry
Chang, Hung-Wen; Wu, Tso-Lun
1997-01-01
In this paper, we conduct numerical experiments with matrix Riccati equations (MREs) which describe the reflection ( R) and transmission ( T) matrices of the specific intensities in a layer containing randomly distributed scattering particles. The theoretical formulation of MREs is discussed in our previous paper where we show that R and T for a thick layer can be efficiently computed by successively doubling R and T matrices for a thin layer (with small optical thickness 0959-7174/7/1/010/img1). We can compute 0959-7174/7/1/010/img2 and 0959-7174/7/1/010/img3 very accurately using either a fourth-order Runge - Kutta scheme or the fourth-order iterative solution. The differences between these results and those computed by the eigenmode expansion technique (EMET) are very small (< 0.1%). Although the MRE formulation cannot be extended to handle the inhomogeneous term (source term) in the differential equation, we show that the force term can be reformulated as an equivalent boundary condition which is consistent with MRE methods. MRE methods offer an alternative way of solving plane-parallel radiative transport problems. For large problems that do not fit into computer memory, the MRE method provides a significant reduction in computer memory and computational time.
Modelling bacterial behaviour close to a no-slip plane boundary: the influence of bacterial geometry
Shum, H.
2010-01-13
We describe a boundary-element method used to model the hydrodynamics of a bacterium propelled by a single helical flagellum. Using this model, we optimize the power efficiency of swimming with respect to cell body and flagellum geometrical parameters, and find that optima for swimming in unbounded fluid and near a no-slip plane boundary are nearly indistinguishable. We also consider the novel optimization objective of torque efficiency and find a very different optimal shape. Excluding effects such as Brownian motion and electrostatic interactions, it is demonstrated that hydrodynamic forces may trap the bacterium in a stable, circular orbit near the boundary, leading to the empirically observable surface accumulation of bacteria. Furthermore, the details and even the existence of this stable orbit depend on geometrical parameters of the bacterium, as described in this article. These results shed some light on the phenomenon of surface accumulation of micro-organisms and offer hydrodynamic explanations as to why some bacteria may accumulate more readily than others based on morphology. © 2010 The Royal Society.
一个解决平面几何问题的有效方法%An Effective Solution to the Plane Geometry Problem
Institute of Scientific and Technical Information of China (English)
孙珍
2011-01-01
Euclidean geometry is a subsidiary geometry of projective geometry. It does not require too much skill and makes the problem easy to use projective geometry to consider some of the geometry problem. Establishment of projective coordinates is a solution to the question of plane geometry.%欧氏几何是射影几何的子几何,用射影的观点考虑一些几何问题,不需要太多技巧,并且在很大程度上使问题的解决变得容易,射影坐标的建立就是一个解决平面几何问题的有效方法.
Plane geometry drawing tutorial
Directory of Open Access Journals (Sweden)
Eduardo Gutiérrez de Ravé
2014-01-01
Full Text Available Se ha desarrollado un tutorial para facilitar la docencia del d ibujo geométrico. Con la idea de servir de apoyo a las explicac iones teóricas y prácticas de los conceptos correspondientes a los trazados ge ométricos planos necesarios en la ingeniería. Este tutorial es de fácil manejo y permite interactividad con el usuario, animaciones prá cticas, autoevaluaciones, explicaciones amplias del temario y l a enseñanza "paso a paso" de los conceptos gracias a los diferent es niveles de complejidad conceptual que incluye en su contenido.
Energy Technology Data Exchange (ETDEWEB)
Morice, J. [Bordeaux-1 Univ., Ecole Matmeca, 33 - Talence (France); Jaouen, St. [CEA Bruyeres-le-Chatel, Dept. Sciences de la Simulation et de l' Information, 91 (France)
2003-07-01
We derive the systems of equations satisfied by the linear Lagrangian perturbations of gas dynamics in planar, cylindrical and spherical geometries, using the canonical forms pointed out by B. Despres et al. (B. Despres, 2001 B. Despres and C. Mazeran, 2003). One of the interests of this approach is that it should be applied to more complex models (those which enter the B. Despres' formalism as 2T-hydrodynamics, MHD, reactive gas dynamics, etc.). Another one is that it is rather easy to derive entropic numerical schemes for the basic flow and their linearized versions for the perturbations. (authors)
Chang, Ling-Hua
2012-01-01
The angle between two compressed sparse vectors subject to the norm/distance constraints imposed by the restricted isometry property (RIP) of the sensing matrix plays a crucial role in the studies of many compressive sensing (CS) problems. Assuming that (i) u and v are two sparse vectors separated by an angle thetha, and (ii) the sensing matrix Phi satisfies RIP, this paper is aimed at analytically characterizing the achievable angles between Phi*u and Phi*v. Motivated by geometric interpretations of RIP and with the aid of the well-known law of cosines, we propose a plane geometry based formulation for the study of the considered problem. It is shown that all the RIP-induced norm/distance constraints on Phi*u and Phi*v can be jointly depicted via a simple geometric diagram in the two-dimensional plane. This allows for a joint analysis of all the considered algebraic constraints from a geometric perspective. By conducting plane geometry analyses based on the constructed diagram, closed-form formulae for the m...
Energy Technology Data Exchange (ETDEWEB)
Celestin, Sebastien; Bonaventura, Zdenek; Zeghondy, Barbar; Bourdon, Anne [Ecole Centrale Paris, EM2C, UPR CNRS 288, Grande voie des vignes, 92295 Chatenay-Malabry Cedex (France); Segur, Pierre, E-mail: sebastien.celestin@em2c.ecp.f, E-mail: anne.bourdon@em2c.ecp.f [Universite de Toulouse, LAPLACE, UMR CNRS 5213, INPT, UPS, 118 route de Narbonne, 31062 Toulouse Cedex 9 (France)
2009-03-21
This paper presents the application of the ghost fluid method (GFM) to solve Poisson's equation for streamer discharge simulations between electrodes of complex geometries. This approach allows one to use a simple rectilinear grid and nevertheless take into account the influence of the exact shape of the electrode on the calculation of the potential and the electric field. First, the validity of the GFM approach concerning the computation of the electric field is demonstrated by performing direct comparisons in a point-to-plane geometry of the Laplacian potential and electric field calculated with this method and given by the analytical solution. Second, the GFM is applied to the simulation of a positive streamer propagation in a hyperboloid-to-plane configuration studied by Kulikovsky (1998 Phys. Rev. E 57 7066-74). Very good agreement is obtained with the results of Kulikovsky (1998) on all positive streamer characteristics during its propagation in the interelectrode gap. Then the GFM is applied to simulate the discharge in preheated air at atmospheric pressure in point-to-point geometry. The propagation of positive and negative streamers from both point electrodes is observed. After the interaction of both discharges, the very rapid propagation of the positive streamer towards the cathode in the volume pre-ionized by the negative streamer is presented. This structure of the discharge is in qualitative agreement with the experiment.
Institute of Scientific and Technical Information of China (English)
Li Gu-Qiang
2005-01-01
The divergences at all levels for the statistical enytropy of a plane symmetry black hole arising from the massless Dirac field are considered using the brick-wall model. It is shown that if we ignore the usual contribution from the vacuum surrounding the system, then the statistical entropy consists of two paarts: one is the linearly divergent term which has the geometric character, the other consists of two logarithmically divergent terms which are not proportional to the surface area of the horizon. The entropy of the Dirac field on extremal plane symmetry spacetime background has higher divergence than usual.
Energy Technology Data Exchange (ETDEWEB)
Maruyama, R., E-mail: ryuji.maruyama@j-parc.jp [J-PARC Center, Japan Atomic Energy Agency, 2-4 Shirakata Shirane, Tokai, Ibaraki 319-1195 (Japan); Bigault, T.; Wildes, A.R.; Dewhurst, C.D. [Institut Laue Langevin, 71 avenue des Martyrs, 38042 Grenoble (France); Soyama, K. [J-PARC Center, Japan Atomic Energy Agency, 2-4 Shirakata Shirane, Tokai, Ibaraki 319-1195 (Japan); Courtois, P. [Institut Laue Langevin, 71 avenue des Martyrs, 38042 Grenoble (France)
2016-05-21
The in-plane magnetic structure of a layered system with a polycrystalline grain size less than the ferromagnetic exchange length was investigated using polarized neutron off-specular scattering and grazing incidence small angle scattering measurements to gain insight into the mechanism that controls the magnetic properties which are different from the bulk. These complementary measurements with different length scales and the data analysis based on the distorted wave Born approximation revealed the lateral correlation on a length scale of sub- μm due to the fluctuating orientation of the magnetization in the layer. The obtained in-plane magnetic structure is consistent with the random anisotropy model, i.e. competition between the exchange interactions between neighboring spins and the local magnetocrystalline anisotropy.
Dowthwaite, Jodi N.; Rosenbaum, Paula F.; Scerpella, Tamara A.
2011-01-01
Lumbar spine geometry, density and indices of bone strength were assessed relative to menarche status, using artistic gymnastics exposure during growth as a model of mechanical loading. Paired posteroanterior (PA) and supine lateral (LAT) DXA scans of L3 for 114 females (60 ex/gymnasts and 54 non-gymnasts) yielded output for comparison of paired (PALAT) versus standard PA and LAT outcomes. BMC, areal BMD, vertebral body dimensions, bone mineral apparent density (BMAD), axial compressive strength (IBS) and a fracture risk index were evaluated, modeling vertebral body geometry as an ellipsoid cylinder. Two-factor ANCOVA tested statistical effects of gymnastic exposure, menarche status and their interaction, adjusting for age and height as appropriate. Compared to non-gymnasts, ex/gymnasts exhibited greater PABMD, PABMC, PAWIDTH, PA CROSS-SECTIONAL AREA (CSA), PAVOLUME, LATBMD, LATBMAD, PALATCSA and PALATIBS (pgymnasts exhibited greater LATDEPTH/PAWIDTH, LATBMC/PABMC, LATVHEIGHT, LATAREA and Fracture Risk Index. Using ellipsoid vertebral geometric models, no significant differences were detected for PA or PALAT BMAD. In contrast, cuboid model results (Carter 1992) suggested erroneous ex/gymnast PABMAD advantages, resulting from invalid assumptions of proportional variation in linear skeletal dimensions. Gymnastic exposure was associated with shorter, wider vertebral bodies, yielding greater axial compressive strength and lower fracture risk, despite no BMAD advantage. Our results suggest the importance of plane-specific vertebral geometric adaptation to mechanical loading during growth. Paired scan output provides a more accurate assessment of this adaptation than PA or LAT plane scans alone. PMID:21839871
Kong, Dali; Zhang, Keke; Schubert, Gerald
2016-10-01
Unlike the even gravitational coefficients of Jupiter that are caused by both the rotational distortion and the equatorially symmetric zonal winds, the odd jovian gravitational coefficients are directly linked to the depth of the equatorially antisymmetric zonal winds. Accurate estimation of the wind-induced odd coefficients and comparison with measurements of those coefficients would be key to understanding the structure of the zonal winds in the deep interior of Jupiter. We consider two problems in connection with the jovian odd gravitational coefficients. In the first problem, we show, by solving the governing equations for the northern hemisphere of Jupiter subject to an appropriate condition at the equatorial plane, that the effect of non-spherical geometry makes an insignificant contribution to the lowermost-order odd gravitational coefficients. In the second problem, we investigate the effect of the equatorial smoothing used to avoid the discontinuity in the winds across the equatorial plane when the thermal wind equation is adopted to compute the odd gravitational coefficients. We reveal that, because of the dominant effect of the equatorial smoothing, the odd gravitational coefficients so obtained for deep zonal winds do not reflect physically realistic dynamics taking place in the deep interior of Jupiter.
Guggenheimer, Heinrich W
1977-01-01
This is a text of local differential geometry considered as an application of advanced calculus and linear algebra. The discussion is designed for advanced undergraduate or beginning graduate study, and presumes of readers only a fair knowledge of matrix algebra and of advanced calculus of functions of several real variables. The author, who is a Professor of Mathematics at the Polytechnic Institute of New York, begins with a discussion of plane geometry and then treats the local theory of Lie groups and transformation groups, solid differential geometry, and Riemannian geometry, leading to a
Robinson, Gilbert de B
2011-01-01
This brief undergraduate-level text by a prominent Cambridge-educated mathematician explores the relationship between algebra and geometry. An elementary course in plane geometry is the sole requirement for Gilbert de B. Robinson's text, which is the result of several years of teaching and learning the most effective methods from discussions with students. Topics include lines and planes, determinants and linear equations, matrices, groups and linear transformations, and vectors and vector spaces. Additional subjects range from conics and quadrics to homogeneous coordinates and projective geom
Introduction to projective geometry
Wylie, C R
2008-01-01
This lucid introductory text offers both an analytic and an axiomatic approach to plane projective geometry. The analytic treatment builds and expands upon students' familiarity with elementary plane analytic geometry and provides a well-motivated approach to projective geometry. Subsequent chapters explore Euclidean and non-Euclidean geometry as specializations of the projective plane, revealing the existence of an infinite number of geometries, each Euclidean in nature but characterized by a different set of distance- and angle-measurement formulas. Outstanding pedagogical features include w
Energy Technology Data Exchange (ETDEWEB)
Caldeira, A.D.; Dias, A.F.; Garcia, R.D.M. [Centro Tecnico Aeroespacial (CTA-IEAv), Sao Jose dos Campos, SP (Brazil). Inst. de Estudos Avancados
1997-12-01
A new version of the P{sub N} method for solving the two-group neutron transport equation in plane geometry without up scattering is reported. Contrasting with the conventional matrix approach, the developed version is based on a group-by-group solution of the transport equation which makes it specially well-suited for neutrons slowing-down calculations. (author). 14 refs., 2 tabs.
Euclidean Geometry via Programming.
Filimonov, Rossen; Kreith, Kurt
1992-01-01
Describes the Plane Geometry System computer software developed at the Educational Computer Systems laboratory in Sofia, Bulgaria. The system enables students to use the concept of "algorithm" to correspond to the process of "deductive proof" in the development of plane geometry. Provides an example of the software's capability and compares it to…
Introduction to finite geometries
Kárteszi, F
1976-01-01
North-Holland Texts in Advanced Mathematics: Introduction to Finite Geometries focuses on the advancements in finite geometries, including mapping and combinatorics. The manuscript first offers information on the basic concepts on finite geometries and Galois geometries. Discussions focus on linear mapping of a given quadrangle onto another given quadrangle; point configurations of order 2 on a Galois plane of even order; canonical equation of curves of the second order on the Galois planes of even order; and set of collineations mapping a Galois plane onto itself. The text then ponders on geo
Berger, Marcel
2010-01-01
Both classical geometry and modern differential geometry have been active subjects of research throughout the 20th century and lie at the heart of many recent advances in mathematics and physics. The underlying motivating concept for the present book is that it offers readers the elements of a modern geometric culture by means of a whole series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction. Starting with such natural, classical objects as lines, planes, circles, spheres, polygons, polyhedra, curves, surfaces,
Burdette, A C
1971-01-01
Analytic Geometry covers several fundamental aspects of analytic geometry needed for advanced subjects, including calculus.This book is composed of 12 chapters that review the principles, concepts, and analytic proofs of geometric theorems, families of lines, the normal equation of the line, and related matters. Other chapters highlight the application of graphing, foci, directrices, eccentricity, and conic-related topics. The remaining chapters deal with the concept polar and rectangular coordinates, surfaces and curves, and planes.This book will prove useful to undergraduate trigonometric st
Mahé, Louis; Roy, Marie-Françoise
1992-01-01
Ten years after the first Rennes international meeting on real algebraic geometry, the second one looked at the developments in the subject during the intervening decade - see the 6 survey papers listed below. Further contributions from the participants on recent research covered real algebra and geometry, topology of real algebraic varieties and 16thHilbert problem, classical algebraic geometry, techniques in real algebraic geometry, algorithms in real algebraic geometry, semialgebraic geometry, real analytic geometry. CONTENTS: Survey papers: M. Knebusch: Semialgebraic topology in the last ten years.- R. Parimala: Algebraic and topological invariants of real algebraic varieties.- Polotovskii, G.M.: On the classification of decomposing plane algebraic curves.- Scheiderer, C.: Real algebra and its applications to geometry in the last ten years: some major developments and results.- Shustin, E.L.: Topology of real plane algebraic curves.- Silhol, R.: Moduli problems in real algebraic geometry. Further contribu...
引导学生学习解析几何中的平面几何原型%Lead the students to study the plane geometric prototype of analytic geometry
Institute of Scientific and Technical Information of China (English)
王占君; 吕效国
2007-01-01
通过列举一些特定的例子说明,教师如何在数学教学过程中,有意识地引导学生学习解析几何中的平面几何原型,并结合解析几何中的习题将它们彼此之间联系起来,这有助于认识它们的一些新的特性.%Cited some specific examples and revealed how the teachers,in the teaching of mathematics, should intentionally lead the students to Study the plane geometric prototype of analytic geometry,so as to interrelate them and tap the resources of exercise in analytic geometry,thus realize their new properties.
Kulczycki, Stefan
2008-01-01
This accessible approach features two varieties of proofs: stereometric and planimetric, as well as elementary proofs that employ only the simplest properties of the plane. A short history of geometry precedes a systematic exposition of the principles of non-Euclidean geometry.Starting with fundamental assumptions, the author examines the theorems of Hjelmslev, mapping a plane into a circle, the angle of parallelism and area of a polygon, regular polygons, straight lines and planes in space, and the horosphere. Further development of the theory covers hyperbolic functions, the geometry of suff
Geometry from Information Geometry
Caticha, Ariel
2015-01-01
We use the method of maximum entropy to model physical space as a curved statistical manifold. It is then natural to use information geometry to explain the geometry of space. We find that the resultant information metric does not describe the full geometry of space but only its conformal geometry -- the geometry up to local changes of scale. Remarkably, this is precisely what is needed to model "physical" space in general relativity.
Energy Technology Data Exchange (ETDEWEB)
Beaumont, A.; Leroy, J.; Crunteanu, A., E-mail: aurelian.crunteanu@xlim.fr [XLIM Research Institute UMR 7252, CNRS/University of Limoges, 123 avenue Albert Thomas, 87060 Limoges (France); Orlianges, J.-C. [SPCTS UMR 7513, CNRS/University of Limoges, 12 rue Atlantis, 87068 Limoges (France)
2014-04-21
Electrically activated metal-insulator transition (MIT) in vanadium dioxide (VO{sub 2}) is widely studied from both fundamental and practical points of view. It can give valuable insights on the currently controversial phase transition mechanism in this material and, at the same time, allows the development of original MIT-based electronic devices. Electrically triggered insulator-metal transitions are demonstrated in novel out-of-plane, metal-oxide-metal type devices integrating a VO{sub 2} thin film, upon applying moderate threshold voltages. It is shown that the current-voltage characteristics of such devices present clear negative differential resistance effects supporting the onset of continuous, current-driven phase oscillations across the vanadium dioxide material. The frequencies of these self-sustained oscillations are ranging from 90 to 300 kHz and they may be tuned by adjusting the injected current. A phenomenological model of the device and its command circuit is developed, and allows to extract the analytical expressions of the oscillation frequencies and to simulate the electrical oscillatory phenomena developed across the VO{sub 2} material. Such out-of-plane devices may further contribute to the general understanding of the driving mechanism in metal-insulator transition materials and devices, a prerequisite to promising applications in high speed/high frequency networks of oscillatory or resistive memories circuits.
Beaumont, A.; Leroy, J.; Orlianges, J.-C.; Crunteanu, A.
2014-04-01
Electrically activated metal-insulator transition (MIT) in vanadium dioxide (VO2) is widely studied from both fundamental and practical points of view. It can give valuable insights on the currently controversial phase transition mechanism in this material and, at the same time, allows the development of original MIT-based electronic devices. Electrically triggered insulator-metal transitions are demonstrated in novel out-of-plane, metal-oxide-metal type devices integrating a VO2 thin film, upon applying moderate threshold voltages. It is shown that the current-voltage characteristics of such devices present clear negative differential resistance effects supporting the onset of continuous, current-driven phase oscillations across the vanadium dioxide material. The frequencies of these self-sustained oscillations are ranging from 90 to 300 kHz and they may be tuned by adjusting the injected current. A phenomenological model of the device and its command circuit is developed, and allows to extract the analytical expressions of the oscillation frequencies and to simulate the electrical oscillatory phenomena developed across the VO2 material. Such out-of-plane devices may further contribute to the general understanding of the driving mechanism in metal-insulator transition materials and devices, a prerequisite to promising applications in high speed/high frequency networks of oscillatory or resistive memories circuits.
Kumaresan, S
2005-01-01
Including Affine and projective classification of Conics, 2 point homogeneity's of the planes, essential isometrics, non euclidean plan geometrics, in this book, the treatment of Geometry goes beyond the Kleinian views.
Analytic Geometry, A Tentative Guide.
Helwig, G. Alfred; And Others
This teacher's guide for a semester course in analytic geometry is based on the text "Analytic Geometry" by W. K. Morrill. Included is a daily schedule of suggested topics and homework assignments. Specific teaching hints are also given. The content of the course includes point and plane vectors, straight lines, point and space vectors, planes,…
Methods for euclidean geometry
Byer, Owen; Smeltzer, Deirdre L
2010-01-01
Euclidean plane geometry is one of the oldest and most beautiful topics in mathematics. Instead of carefully building geometries from axiom sets, this book uses a wealth of methods to solve problems in Euclidean geometry. Many of these methods arose where existing techniques proved inadequate. In several cases, the new ideas used in solving specific problems later developed into independent areas of mathematics. This book is primarily a geometry textbook, but studying geometry in this way will also develop students' appreciation of the subject and of mathematics as a whole. For instance, despite the fact that the analytic method has been part of mathematics for four centuries, it is rarely a tool a student considers using when faced with a geometry problem. Methods for Euclidean Geometry explores the application of a broad range of mathematical topics to the solution of Euclidean problems.
Introduction to differential geometry of plane curves
2015-01-01
A intenÃÃo desse trabalho serÃ de abordar de forma bÃsica e introdutÃria o estudo da Geometria Diferencial, que por sua vez tem seus estudos iniciados com as Curvas Planas. SerÃ necessÃrio um conhecimento de CÃlculo Diferencial, Integral e Geometria AnalÃtica para melhor compreensÃo desse trabalho, pois como seu prÃprio nome nos transparece Geometria Diferencial vem de uma junÃÃo do estudo da Geometria envolvendo CÃlculo. Assim abordaremos subtemas como curvas suaves, vetor tangente, co...
van den Broek, P.M.
1984-01-01
The aim of this paper is to give a detailed exposition of the relation between the geometry of twistor space and the geometry of Minkowski space. The paper has a didactical purpose; no use has been made of differential geometry and cohomology.
Introduction to non-Euclidean geometry
Wolfe, Harold E
2012-01-01
One of the first college-level texts for elementary courses in non-Euclidean geometry, this concise, readable volume is geared toward students familiar with calculus. A full treatment of the historical background explores the centuries-long efforts to prove Euclid's parallel postulate and their triumphant conclusion. Numerous original exercises form an integral part of the book.Topics include hyperbolic plane geometry and hyperbolic plane trigonometry, applications of calculus to the solutions of some problems in hyperbolic geometry, elliptic plane geometry and trigonometry, and the consistenc
Yale, Paul B
2012-01-01
This book is an introduction to the geometry of Euclidean, affine, and projective spaces with special emphasis on the important groups of symmetries of these spaces. The two major objectives of the text are to introduce the main ideas of affine and projective spaces and to develop facility in handling transformations and groups of transformations. Since there are many good texts on affine and projective planes, the author has concentrated on the n-dimensional cases.Designed to be used in advanced undergraduate mathematics or physics courses, the book focuses on ""practical geometry,"" emphasi
Arnold's Projective Plane and -Matrices
Directory of Open Access Journals (Sweden)
K. Uchino
2010-01-01
Full Text Available We will explain Arnold's 2-dimensional (shortly, 2D projective geometry (Arnold, 2005 by means of lattice theory. It will be shown that the projection of the set of nontrivial triangular -matrices is the pencil of tangent lines of a quadratic curve on Arnold's projective plane.
Linear connections on the quantum plane
Dubois-Violette, M; Masson, T; Mourad, J; Dubois-Violette, Michel; Madore, John; Masson, Thierry; Mourad, Jihad
1994-01-01
A general definition has been proposed recently of a linear connection and a metric in noncommutative geometry. It is shown that to within normalization there is a unique linear connection on the quantum plane and there is no metric.
Rodger, Alison
1995-01-01
Molecular Geometry discusses topics relevant to the arrangement of atoms. The book is comprised of seven chapters that tackle several areas of molecular geometry. Chapter 1 reviews the definition and determination of molecular geometry, while Chapter 2 discusses the unified view of stereochemistry and stereochemical changes. Chapter 3 covers the geometry of molecules of second row atoms, and Chapter 4 deals with the main group elements beyond the second row. The book also talks about the complexes of transition metals and f-block elements, and then covers the organometallic compounds and trans
An introduction to finite projective planes
Albert, Abraham Adrian
2015-01-01
Geared toward both beginning and advanced undergraduate and graduate students, this self-contained treatment offers an elementary approach to finite projective planes. Following a review of the basics of projective geometry, the text examines finite planes, field planes, and coordinates in an arbitrary plane. Additional topics include central collineations and the little Desargues' property, the fundamental theorem, and examples of finite non-Desarguesian planes.Virtually no knowledge or sophistication on the part of the student is assumed, and every algebraic system that arises is defined and
Pottmann, Helmut
2014-11-26
Around 2005 it became apparent in the geometry processing community that freeform architecture contains many problems of a geometric nature to be solved, and many opportunities for optimization which however require geometric understanding. This area of research, which has been called architectural geometry, meanwhile contains a great wealth of individual contributions which are relevant in various fields. For mathematicians, the relation to discrete differential geometry is significant, in particular the integrable system viewpoint. Besides, new application contexts have become available for quite some old-established concepts. Regarding graphics and geometry processing, architectural geometry yields interesting new questions but also new objects, e.g. replacing meshes by other combinatorial arrangements. Numerical optimization plays a major role but in itself would be powerless without geometric understanding. Summing up, architectural geometry has become a rewarding field of study. We here survey the main directions which have been pursued, we show real projects where geometric considerations have played a role, and we outline open problems which we think are significant for the future development of both theory and practice of architectural geometry.
Maor, Eli
2014-01-01
If you've ever thought that mathematics and art don't mix, this stunning visual history of geometry will change your mind. As much a work of art as a book about mathematics, Beautiful Geometry presents more than sixty exquisite color plates illustrating a wide range of geometric patterns and theorems, accompanied by brief accounts of the fascinating history and people behind each. With artwork by Swiss artist Eugen Jost and text by acclaimed math historian Eli Maor, this unique celebration of geometry covers numerous subjects, from straightedge-and-compass constructions to intriguing configur
Graustein, William C
2006-01-01
This first course in differential geometry presents the fundamentals of the metric differential geometry of curves and surfaces in a Euclidean space of three dimensions. Written by an outstanding teacher and mathematician, it explains the material in the most effective way, using vector notation and technique. It also provides an introduction to the study of Riemannian geometry.Suitable for advanced undergraduates and graduate students, the text presupposes a knowledge of calculus. The first nine chapters focus on the theory, treating the basic properties of curves and surfaces, the mapping of
Lefschetz, Solomon
2005-01-01
An introduction to algebraic geometry and a bridge between its analytical-topological and algebraical aspects, this text for advanced undergraduate students is particularly relevant to those more familiar with analysis than algebra. 1953 edition.
Ay, Nihat; Lê, Hông Vân; Schwachhöfer, Lorenz
2017-01-01
The book provides a comprehensive introduction and a novel mathematical foundation of the field of information geometry with complete proofs and detailed background material on measure theory, Riemannian geometry and Banach space theory. Parametrised measure models are defined as fundamental geometric objects, which can be both finite or infinite dimensional. Based on these models, canonical tensor fields are introduced and further studied, including the Fisher metric and the Amari-Chentsov tensor, and embeddings of statistical manifolds are investigated. This novel foundation then leads to application highlights, such as generalizations and extensions of the classical uniqueness result of Chentsov or the Cramér-Rao inequality. Additionally, several new application fields of information geometry are highlighted, for instance hierarchical and graphical models, complexity theory, population genetics, or Markov Chain Monte Carlo. The book will be of interest to mathematicians who are interested in geometry, inf...
Analogy and Dynamic Geometry System Used to Introduce Three-Dimensional Geometry
Mammana, M. F.; Micale, B.; Pennisi, M.
2012-01-01
We present a sequence of classroom activities on Euclidean geometry, both plane and space geometry, used to make three dimensional geometry more catchy and simple. The activity consists of a guided research activity that leads the students to discover unexpected properties of two apparently distant geometrical entities, quadrilaterals and…
Analogy and Dynamic Geometry System Used to Introduce Three-Dimensional Geometry
Mammana, M. F.; Micale, B.; Pennisi, M.
2012-01-01
We present a sequence of classroom activities on Euclidean geometry, both plane and space geometry, used to make three dimensional geometry more catchy and simple. The activity consists of a guided research activity that leads the students to discover unexpected properties of two apparently distant geometrical entities, quadrilaterals and…
Saleem, Zain Hamid
In this thesis we study a special class of black hole geometries called subtracted geometries. Subtracted geometry black holes are obtained when one omits certain terms from the warp factor of the metric of general charged rotating black holes. The omission of these terms allows one to write the wave equation of the black hole in a completely separable way and one can explicitly see that the wave equation of a massless scalar field in this slightly altered background of a general multi-charged rotating black hole acquires an SL(2, R) x SL(2, R) x SO(3) symmetry. The "subtracted limit" is considered an appropriate limit for studying the internal structure of the non-subtracted black holes because new 'subtracted' black holes have the same horizon area and periodicity of the angular and time coordinates in the near horizon regions as the original black hole geometry it was constructed from. The new geometry is asymptotically conical and is physically similar to that of a black hole in an asymptotically confining box. We use the different nice properties of these geometries to understand various classically and quantum mechanically important features of general charged rotating black holes.
Linear algebra, geometry and transformation
Solomon, Bruce
2014-01-01
Vectors, Mappings and Linearity Numeric Vectors Functions Mappings and Transformations Linearity The Matrix of a Linear Transformation Solving Linear Systems The Linear SystemThe Augmented Matrix and RRE Form Homogeneous Systems in RRE Form Inhomogeneous Systems in RRE Form The Gauss-Jordan Algorithm Two Mapping Answers Linear Geometry Geometric Vectors Geometric/Numeric Duality Dot-Product Geometry Lines, Planes, and Hyperplanes System Geometry and Row/Column Duality The Algebra of Matrices Matrix Operations Special Matrices Matrix Inversion A Logical Digression The Logic of the Inversion Alg
Energy Technology Data Exchange (ETDEWEB)
Caldeira, Alexandre David
1999-07-01
In this work P{sub N} solutions for the slowing-down and cell problems in slab geometry are developed. To highlight the main contributions of this development, one can mention: the new particular solution developed for the P{sub N} method applied to the slowing-down problem in the multigroup model, originating a new class of polynomials denominated Chandrasekhar generalized polynomials; the treatment of a specific situation, known as a degeneracy, arising from a particularity in the group constants and the first application of the P{sub N} method, for arbitrary N, in criticality calculations at the cell level reported in literature. (author)
Kendig, Keith
2015-01-01
Designed to make learning introductory algebraic geometry as easy as possible, this text is intended for advanced undergraduates and graduate students who have taken a one-year course in algebra and are familiar with complex analysis. This newly updated second edition enhances the original treatment's extensive use of concrete examples and exercises with numerous figures that have been specially redrawn in Adobe Illustrator. An introductory chapter that focuses on examples of curves is followed by a more rigorous and careful look at plane curves. Subsequent chapters explore commutative ring th
Akopyan, A V
2007-01-01
The book is devoted to the properties of conics (plane curves of second degree) that can be formulated and proved using only elementary geometry. Starting with the well-known optical properties of conics, the authors move to less trivial results, both classical and contemporary. In particular, the chapter on projective properties of conics contains a detailed analysis of the polar correspondence, pencils of conics, and the Poncelet theorem. In the chapter on metric properties of conics the authors discuss, in particular, inscribed conics, normals to conics, and the Poncelet theorem for confoca
REA, The Editors of
2012-01-01
REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Geometry I includes methods of proof, points, lines, planes, angles, congruent angles and line segments, triangles, parallelism, quadrilaterals, geometric inequalities, and geometric
Hartshorne, Robin
2000-01-01
In recent years, I have been teaching a junior-senior-level course on the classi cal geometries. This book has grown out of that teaching experience. I assume only high-school geometry and some abstract algebra. The course begins in Chapter 1 with a critical examination of Euclid's Elements. Students are expected to read concurrently Books I-IV of Euclid's text, which must be obtained sepa rately. The remainder of the book is an exploration of questions that arise natu rally from this reading, together with their modern answers. To shore up the foundations we use Hilbert's axioms. The Cartesian plane over a field provides an analytic model of the theory, and conversely, we see that one can introduce coordinates into an abstract geometry. The theory of area is analyzed by cutting figures into triangles. The algebra of field extensions provides a method for deciding which geometrical constructions are possible. The investigation of the parallel postulate leads to the various non-Euclidean geometries. And ...
Euclidean geometry and its subgeometries
Specht, Edward John; Calkins, Keith G; Rhoads, Donald H
2015-01-01
In this monograph, the authors present a modern development of Euclidean geometry from independent axioms, using up-to-date language and providing detailed proofs. The axioms for incidence, betweenness, and plane separation are close to those of Hilbert. This is the only axiomatic treatment of Euclidean geometry that uses axioms not involving metric notions and that explores congruence and isometries by means of reflection mappings. The authors present thirteen axioms in sequence, proving as many theorems as possible at each stage and, in the process, building up subgeometries, most notably the Pasch and neutral geometries. Standard topics such as the congruence theorems for triangles, embedding the real numbers in a line, and coordinatization of the plane are included, as well as theorems of Pythagoras, Desargues, Pappas, Menelaus, and Ceva. The final chapter covers consistency and independence of axioms, as well as independence of definition properties. There are over 300 exercises; solutions to many of the...
Pottmann, Helmut; Eigensatz, Michael; Vaxman, A.; Wallner, Johannes
2015-01-01
Around 2005 it became apparent in the geometry processing community that freeform architecture contains many problems of a geometric nature to be solved, and many opportunities for optimization which however require geometric understanding. This area of research, which has been called architectural
Pottmann, Helmut; Eigensatz, Michael; Vaxman, A.; Wallner, Johannes
2015-01-01
Around 2005 it became apparent in the geometry processing community that freeform architecture contains many problems of a geometric nature to be solved, and many opportunities for optimization which however require geometric understanding. This area of research, which has been called architectural
Petersen, Peter
2016-01-01
Intended for a one year course, this text serves as a single source, introducing readers to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialize in Riemannian geometry. This is one of the few Works to combine both the geometric parts of Riemannian geometry and the analytic aspects of the theory. The book will appeal to a readership that have a basic knowledge of standard manifold theory, including tensors, forms, and Lie groups. Important revisions to the third edition include: a substantial addition of unique and enriching exercises scattered throughout the text; inclusion of an increased number of coordinate calculations of connection and curvature; addition of general formulas for curvature on Lie Groups and submersions; integration of variational calculus into the text allowing for an early treatment of the Sphere theorem using a proof by Berger; incorporation of several recent results about manifolds with posit...
Kreyszig, Erwin
1991-01-01
An introductory textbook on the differential geometry of curves and surfaces in three-dimensional Euclidean space, presented in its simplest, most essential form, but with many explanatory details, figures and examples, and in a manner that conveys the theoretical and practical importance of the different concepts, methods and results involved. With problems at the end of each section, and solutions listed at the end of the book. Includes 99 illustrations.
General Geometry and Geometry of Electromagnetism
Shahverdiyev, Shervgi S.
2002-01-01
It is shown that Electromagnetism creates geometry different from Riemannian geometry. General geometry including Riemannian geometry as a special case is constructed. It is proven that the most simplest special case of General Geometry is geometry underlying Electromagnetism. Action for electromagnetic field and Maxwell equations are derived from curvature function of geometry underlying Electromagnetism. And it is shown that equation of motion for a particle interacting with electromagnetic...
"Improbable Feat": Plain Geometry, Easy-to-Learn Trig.
Stanley, Shirley; Mozier, Eugene
1983-01-01
Describes the college-level course offered at the State University of New York, which combines plane geometry and plane trigonometry. Explains lesson sequences and sample exercises to show how the course develops all of the major principles of geometry and trigonometry using three basic constructions: perpendicularity, angle congruency, and angle…
Berry, Edward A; Walker, F Ann
2008-05-01
Early investigation of the electron paramagnetic resonance spectra of bis-histidine-coordinated membrane-bound ferriheme proteins led to the description of a spectral signal that had only one resolved feature. These became known as "highly anisotropic low-spin" or "large g(max)" ferriheme centers. Extensive work with small-molecule model heme complexes showed that this spectroscopic signature occurs in bis-imidazole ferrihemes in which the planes of the imidazole ligands are nearly perpendicular, deltaphi = 57-90 degrees. In the last decade protein crystallographic studies have revealed the atomic structures of a number of examples of bis-histidine heme proteins. A frequent characteristic of these large g(max) ferrihemes in membrane-bound proteins is the occurrence of the heme within a four-helix bundle with a left-handed twist. The histidine ligands occur at the same level on two diametrically opposed helices of the bundle. These ligands have the same side-chain conformation and ligate heme iron on the bundle axis, resulting in a quasi-twofold symmetric structure. The two non-ligand-bearing helices also obey this symmetry, and have a conserved small residue, usually glycine, where the edge of the heme ring makes contact with the helix backbones. In many cases this small residue is preceded by a threonine or serine residue whose side-chain hydroxyl oxygen acts as a hydrogen-bond acceptor from the N(delta1) atom of the heme-ligating histidine. The deltaphi angle is thus determined by the common histidine side-chain conformation and the crossing angle of the ligand-bearing helices, in some cases constrained by hydrogen bonds to the serine/threonine residues on the non-ligand-bearing helices.
Horizons and plane waves: A review
Hubeny, V E; Hubeny, Veronika E.; Rangamani, Mukund
2003-01-01
We review the attempts to construct black hole/string solutions in asymptotically plane wave spacetimes. First, we demonstrate that geometries admitting a covariantly constant null Killing vector cannot admit event horizons, which implies that pp-waves can't describe black holes. However, relaxing the symmetry requirements allows us to generate solutions which do possess regular event horizons while retaining the requisite asymptotic properties. In particular, we present two solution generating techniques and use them to construct asymptotically plane wave black string/brane geometries.
Ciarlet, Philippe G
2007-01-01
This book gives the basic notions of differential geometry, such as the metric tensor, the Riemann curvature tensor, the fundamental forms of a surface, covariant derivatives, and the fundamental theorem of surface theory in a selfcontained and accessible manner. Although the field is often considered a classical one, it has recently been rejuvenated, thanks to the manifold applications where it plays an essential role. The book presents some important applications to shells, such as the theory of linearly and nonlinearly elastic shells, the implementation of numerical methods for shells, and
Algebraic geometry and theta functions
Coble, Arthur B
1929-01-01
This book is the result of extending and deepening all questions from algebraic geometry that are connected to the central problem of this book: the determination of the tritangent planes of a space curve of order six and genus four, which the author treated in his Colloquium Lecture in 1928 at Amherst. The first two chapters recall fundamental ideas of algebraic geometry and theta functions in such fashion as will be most helpful in later applications. In order to clearly present the state of the central problem, the author first presents the better-known cases of genus two (Chapter III) and
Directory of Open Access Journals (Sweden)
João Alberto da Silva
2009-12-01
Full Text Available O ensino da geometria plana nas séries finais do Ensino Fundamental é, muitas vezes, desprovido de sentido. Os professores optam por práticas pedagógicas que se fundamentam em algoritmos, sem preocuparem-se com os processos de pensamento que estão envolvidos na construção do pensamento geométrico. Essa pesquisa vale-se da Epistemologia Genética para investigar como adolescentes e adultos, que freqüentaram a escola e obtiveram êxito na aprendizagem de geometria, elaboram explicações a propósito de problemas que envolvem o cálculo da área e do perímetro de figuras planas. Os dados indicam que a totalidade dos entrevistados é capaz de realizar o cálculo através do algoritmo, mas muito poucos apresentam explicações elaboradas. Os modelos explicativos são os mais variados e dirigem-se de um pensamento baseado exclusivamente na percepção até a explicação lógico-matemática dos conceitos envolvidos. Palavras-chave: Ensino de Geometria. Modelos Explicativos. Jean Piaget. Epistemologia Genética.The teaching of plane geometry in elementary school is often lacking in meaning. Teachers choose teaching practices based on algorithms, without concern for the thinking processes involved in the construction of geometric thinking. This study is based on Genetic Epistemology to investigate how adolescents and adults who attended school, and were successful in learning geometry, construct explanations about problems involving the calculation of the area and the perimeter of plane figures. The data show that the interviewees are capable of doing the calculation with the algorithm, but very few show elaborated explanations. The explanatory models are the most varied, ranging from thinking based solely on perception to logical-mathematical explanations of the concepts involved. Keywords: Teaching of Geometry. Explanatory Models. Jean Piaget. Genetic Epistemology.
Directory of Open Access Journals (Sweden)
Poloni, Marinês Yole
2012-05-01
Full Text Available Este artigo tem por propósito discutir episódios da prática de duas professoras do Ensino Fundamental I que em um curso de formação continuada revisitaram alguns conceitos geométricos. O foco está na reconstrução dos conceitos dessas professoras, entretanto são explicitadas também decisões e estratégias metodológicas por elas tomadas a fim de mediar a aprendizagem dos alunos. A pesquisa de mestrado, que subsidia este texto, foi realizada ao longo do curso “Geometria em Ação”, o qual estava centrado no tema Figuras Planas e, nele, foi utilizado o software Cabri-Géomètre[1]. A fundamentação teórica foi construída a partir dos conceitos de reflexão de Schön, das vertentes do conhecimento didático de Ponte & Oliveira e da articulação entre teoria e prática de Tardif. A pesquisa de caráter qualitativo utilizou a metodologia de Design-Based Research. No artigo apresentamos reflexões tanto sobre a (reconstrução de conceitos geométricos, quanto sobre a prática docente. Concluímos, ao final do estudo, que ocorreram situações de reconstrução de conceitos geométricos por parte de ambas as professoras, particularmente quanto às definições e às propriedades de triângulos e quadriláteros. Em relação à prática docente, elas se conscientizaram das decisões tomadas tanto durante o planejamento de suas aulas quanto durante a aplicação das mesmas avaliando, posteriormente, suas decisões didáticas e pedagógicas. This paper discusses episodes of teaching practices of two primary school teachers whom, during a course of continuing education, have revisited some geometrical concepts. The focus is on the reconstruction of mathematical concepts of these teachers, however, we also present methodological strategies and decisions taken by them in order to support students' learning. The underlying research was carried out along the course "Geometria em Ação" (Geometry in Action, which was centered on the Planar
Spectral distance on the Moyal plane
Energy Technology Data Exchange (ETDEWEB)
Martinetti, Pierre [Universitaet Goettingen (Germany). Courant Centre
2010-07-01
We compute the spectral distance, defined in Connes' noncommutative geometry, in the Moyal plane. We find that the distance between the eigenstates m,m+1 of the quantum harmonic oscillator is proportional to m{sup -1/2}. We also show how to truncate the Moyal spectral triple in order to obtain quantum metric spaces in the sense of Rieffel.
A Relationship between Geometry and Algebra
Bejarano, Jose Ricardo Arteaga
2011-01-01
The three key documents for study geometry are: 1) "The Elements" of Euclid, 2) the lecture by B. Riemann at G\\"ottingen in 1854 entitled "\\"Uber die Hypothesen welche der Geometrie zu Grunde liegen" (On the hypotheses which underlie geometry) and 3) the "Erlangen Program", a document written by F. Klein (1872) on his income as professor at the Faculty of Philosophy and the Senate of the Erlangen University. The latter document F. Klein introduces the concept of group as a tool to study geometry. The concept of a group of transformations of space was known at the time. The purpose of this informative paper is to show a relationship between geometry and algebra through an example, the projective plane. Erlangen program until today continues being a guideline of how to study geometry.
Fixed Sagittal Plane Imbalance
Savage, Jason W.; Patel, Alpesh A.
2014-01-01
Study Design Literature review. Objective To discuss the evaluation and management of fixed sagittal plane imbalance. Methods A comprehensive literature review was performed on the preoperative evaluation of patients with sagittal plane malalignment, as well as the surgical strategies to address sagittal plane deformity. Results Sagittal plane imbalance is often caused by de novo scoliosis or iatrogenic flat back deformity. Understanding the etiology and magnitude of sagittal malalignment is ...
Effects of Geometry on the Steady Performance of Planing Hulls
DEFF Research Database (Denmark)
Wagner, M. K.; Andersen, Poul
2003-01-01
is applied to practical hull forms with chines spray rails and with varying deadrise over the length of the boat. The deadrise variation has a large influence on lift and drag. For a design situation, where the total lift and centre of effort is given, the influence on the total drag is less due to change...
Plane Transformations in a Complex Setting I: Homotheties-Translations
Dana-Picard, T.
2006-01-01
A previous note described how complex numbers can be used for elementary analytic geometry in the plane, describing lines, circles and their intersections using complex Cartesian equations. In the present note, a description of elementary plane transformations, namely homotheties and translations, their group structure and their operations on…
Plane Stratified Flow in a Room Ventilated by Displacement Ventilation
DEFF Research Database (Denmark)
Nielsen, Peter Vilhelm; Nickel, J.; Baron, D. J. G.
2004-01-01
The air movement in the occupied zone of a room ventilated by displacement ventilation exists as a stratified flow along the floor. This flow can be radial or plane according to the number of wall-mounted diffusers and the room geometry. The paper addresses the situations where plane flow...
Metrics and causality on Moyal planes
Franco, Nicolas
2015-01-01
Metrics structures stemming from the Connes distance promote Moyal planes to the status of quantum metric spaces. We discuss this aspect in the light of recent developments, emphasizing the role of Moyal planes as representative examples of a recently introduced notion of quantum (noncommutative) locally compact space. We move then to the framework of Lorentzian noncommutative geometry and we examine the possibility of defining a notion of causality on Moyal plane, which is somewhat controversial in the area of mathematical physics. We show the actual existence of causal relations between the elements of a particular class of pure (coherent) states on Moyal plane with related causal structure similar to the one of the usual Minkowski space, up to the notion of locality.
Integrable System and Motion of Curves in Projective and Similarity Geometries
Institute of Scientific and Technical Information of China (English)
HOU Yu-Qing
2006-01-01
Based on the naturalitame in the projective geometry, motions of curves in projective geometry are studied.It is shown that several integrable equations including Sawada-Kotera and KK equations arise from motion of plane curves in projective geometries. Motion of space curves described by acceleration feld and governed by endowing an extra space variable in similarity geometry p3 is also studied.
Translation planes foundations and construction principles
Knarr, Norbert
1995-01-01
The book discusses various construction principles for translation planes and spreads from a general and unifying point of view and relates them to the theory of kinematic spaces. The book is intended for people working in the field of incidence geometry and can be read by everyone who knows the basic facts about projective and affine planes. The methods developed work especially well for topological spreads of real and complex vector spaces. In particular, a complete classification of all semifield spreads of finite dimensional complex vector spaces is obtained.
Determination of grain boundary geometry using TEM
Jang, H.; Farkas, D.; Hosson, J.T.M. De
1992-01-01
An experimental method to obtain the grain boundary geometry using the transmission electron microscope is presented. The method allows Σ determination including grain boundary plane orientation. In order to determine the specialness of the grain boundary, three different criteria for maximum allowa
Determination of grain boundary geometry using TEM
Jang, H.; Farkas, D.; Hosson, J.T.M. De
An experimental method to obtain the grain boundary geometry using the transmission electron microscope is presented. The method allows Σ determination including grain boundary plane orientation. In order to determine the specialness of the grain boundary, three different criteria for maximum
Beyond core knowledge: Natural geometry
Spelke, Elizabeth; Lee, Sang Ah; Izard, Véronique
2010-01-01
For many centuries, philosophers and scientists have pondered the origins and nature of human intuitions about the properties of points, lines, and figures on the Euclidean plane, with most hypothesizing that a system of Euclidean concepts either is innate or is assembled by general learning processes. Recent research from cognitive and developmental psychology, cognitive anthropology, animal cognition, and cognitive neuroscience suggests a different view. Knowledge of geometry may be founded on at least two distinct, evolutionarily ancient, core cognitive systems for representing the shapes of large-scale, navigable surface layouts and of small-scale, movable forms and objects. Each of these systems applies to some but not all perceptible arrays and captures some but not all of the three fundamental Euclidean relationships of distance (or length), angle, and direction (or sense). Like natural number (Carey, 2009), Euclidean geometry may be constructed through the productive combination of representations from these core systems, through the use of uniquely human symbolic systems. PMID:20625445
Projective geometry and projective metrics
Busemann, Herbert
2005-01-01
The basic results and methods of projective and non-Euclidean geometry are indispensable for the geometer, and this book--different in content, methods, and point of view from traditional texts--attempts to emphasize that fact. Results of special theorems are discussed in detail only when they are needed to develop a feeling for the subject or when they illustrate a general method. On the other hand, an unusual amount of space is devoted to the discussion of the fundamental concepts of distance, motion, area, and perpendicularity.Topics include the projective plane, polarities and conic sectio
Bär, Christian; Schwarz, Matthias
2012-01-01
This volume contains a collection of well-written surveys provided by experts in Global Differential Geometry to give an overview over recent developments in Riemannian Geometry, Geometric Analysis and Symplectic Geometry. The papers are written for graduate students and researchers with a general interest in geometry, who want to get acquainted with the current trends in these central fields of modern mathematics.
Cukier, Mimi; Asdourian, Tony; Thakker, Anand
2012-01-01
Geometry provides a natural window into what it is like to do mathematics. In the world of geometry, playful experimentation is often more fruitful than following a procedure, and logic plus a few axioms can open new worlds. Nonetheless, teaching a geometry course in a way that combines both rigor and play can be difficult. Many geometry courses…
Barwick, Susan
2008-01-01
Unitals are key structures in projective planes, and have connections with other structures in algebra. This book presents a monograph on unitals embedded in finite projective planes. It offers a survey of the research literature on embedded unitals. It is suitable for graduate students and researchers who want to learn about this topic
Free string evolution across plane wave singularities
Craps, Ben; Evnin, Oleg
2009-01-01
In these proceedings, we summarize our studies of free string propagation in (near-)singular scale-invariant plane wave geometries. We analyze the singular limit of the evolution for the center-of-mass motion and all excited string modes. The requirement that the entire excitation energy of the string should be finite excludes consistent propagation across the singularity, in case no dimensionful scales are introduced at the singular locus (in an otherwise scale-invariant space-time).
Measuring Space-Time Geometry over the Ages
Stebbins, Albert
2012-01-01
Theorists are often told to express things in the "observational plane". One can do this for space-time geometry, considering "visual" observations of matter in our universe by a single observer over time, with no assumptions about isometries, initial conditions, nor any particular relation between matter and geometry, such as Einstein's equations. Using observables as coordinates naturally leads to a parametrization of space-time geometry in terms of other observables, which in turn prescrib...
Digital Differential Geometry Processing
Institute of Scientific and Technical Information of China (English)
Xin-Guo Liu; Hu-Jun Bao; Qun-Sheng Peng
2006-01-01
The theory and methods of digital geometry processing has been a hot research area in computer graphics, as geometric models serves as the core data for 3D graphics applications. The purpose of this paper is to introduce some recent advances in digital geometry processing, particularly mesh fairing, surface parameterization and mesh editing, that heavily use differential geometry quantities. Some related concepts from differential geometry, such as normal, curvature, gradient,Laplacian and their counterparts on digital geometry are also reviewed for understanding the strength and weakness of various digital geometry processing methods.
Fixed sagittal plane imbalance.
Savage, Jason W; Patel, Alpesh A
2014-12-01
Study Design Literature review. Objective To discuss the evaluation and management of fixed sagittal plane imbalance. Methods A comprehensive literature review was performed on the preoperative evaluation of patients with sagittal plane malalignment, as well as the surgical strategies to address sagittal plane deformity. Results Sagittal plane imbalance is often caused by de novo scoliosis or iatrogenic flat back deformity. Understanding the etiology and magnitude of sagittal malalignment is crucial in realignment planning. Objective parameters have been developed to guide surgeons in determining how much correction is needed to achieve favorable outcomes. Currently, the goals of surgery are to restore a sagittal vertical axis Sagittal plane malalignment is an increasingly recognized cause of pain and disability. Treatment of sagittal plane imbalance varies according to the etiology, location, and severity of the deformity. Fixed sagittal malalignment often requires complex reconstructive procedures that include osteotomy correction. Reestablishing harmonious spinopelvic alignment is associated with significant improvement in health-related quality-of-life outcome measures and patient satisfaction.
Meyer, Walter J
2006-01-01
Meyer''s Geometry and Its Applications, Second Edition, combines traditional geometry with current ideas to present a modern approach that is grounded in real-world applications. It balances the deductive approach with discovery learning, and introduces axiomatic, Euclidean geometry, non-Euclidean geometry, and transformational geometry. The text integrates applications and examples throughout and includes historical notes in many chapters. The Second Edition of Geometry and Its Applications is a significant text for any college or university that focuses on geometry''s usefulness in other disciplines. It is especially appropriate for engineering and science majors, as well as future mathematics teachers.* Realistic applications integrated throughout the text, including (but not limited to): - Symmetries of artistic patterns- Physics- Robotics- Computer vision- Computer graphics- Stability of architectural structures- Molecular biology- Medicine- Pattern recognition* Historical notes included in many chapters...
Institute of Scientific and Technical Information of China (English)
GUO Enli; MO Xiaohuan
2006-01-01
In this paper,a survey on Riemann-Finsler geometry is given.Non-trivial examples of Finsler metrics satisfying different curvature conditions are presented.Local and global results in Finsler geometry are analyzed.
Algebra and geometry of Hamilton's quaternions
Krishnaswami, Govind S
2016-01-01
Inspired by the relation between the algebra of complex numbers and plane geometry, William Rowan Hamilton sought an algebra of triples for application to three dimensional geometry. Unable to multiply and divide triples, he invented a non-commutative division algebra of quadruples, in what he considered his most significant work, generalizing the real and complex number systems. We give a motivated introduction to quaternions and discuss how they are related to Pauli matrices, rotations in three dimensions, the three sphere, the group SU(2) and the celebrated Hopf fibrations.
Geometry essentials for dummies
Ryan, Mark
2011-01-01
Just the critical concepts you need to score high in geometry This practical, friendly guide focuses on critical concepts taught in a typical geometry course, from the properties of triangles, parallelograms, circles, and cylinders, to the skills and strategies you need to write geometry proofs. Geometry Essentials For Dummies is perfect for cramming or doing homework, or as a reference for parents helping kids study for exams. Get down to the basics - get a handle on the basics of geometry, from lines, segments, and angles, to vertices, altitudes, and diagonals Conque
Affine and Projective Geometry
Bennett, M K
1995-01-01
An important new perspective on AFFINE AND PROJECTIVE GEOMETRY. This innovative book treats math majors and math education students to a fresh look at affine and projective geometry from algebraic, synthetic, and lattice theoretic points of view. Affine and Projective Geometry comes complete with ninety illustrations, and numerous examples and exercises, covering material for two semesters of upper-level undergraduate mathematics. The first part of the book deals with the correlation between synthetic geometry and linear algebra. In the second part, geometry is used to introduce lattice theory
Symplectic geometries on supermanifolds
Lavrov, P M
2007-01-01
Extension of symplectic geometry on manifolds to the supersymmetric case is considered. In the even case it leads to the even symplectic geometry (or, equivalently, to the geometry on supermanifolds endowed with an non-degenerate Poisson bracket) or to the geometry on an even Fedosov supermanifolds. It is proven that in the odd case there are two different scalar symplectic structures (namely, an odd closed differential 2-form and the antibracket) which can be used for construction of different symplectic geometries on supermanifolds.
Gualtieri, Marco
2010-01-01
Generalized Kahler geometry is the natural analogue of Kahler geometry, in the context of generalized complex geometry. Just as we may require a complex structure to be compatible with a Riemannian metric in a way which gives rise to a symplectic form, we may require a generalized complex structure to be compatible with a metric so that it defines a second generalized complex structure. We explore the fundamental aspects of this geometry, including its equivalence with the bi-Hermitian geometry on the target of a 2-dimensional sigma model with (2,2) supersymmetry, as well as the relation to holomorphic Dirac geometry and the resulting derived deformation theory. We also explore the analogy between pre-quantum line bundles and gerbes in the context of generalized Kahler geometry.
Energy Technology Data Exchange (ETDEWEB)
Lampton, Michael L.; Kim, A.; Akerlof, C.W.; Aldering, G.; Amanullah, R.; Astier, P.; Barrelet, E.; Bebek, C.; Bergstrom, L.; Berkovitz, J.; Bernstein, G.; Bester, M.; Bonissent, A.; Bower, C.; Carithers Jr., W.C.; Commins, E.D.; Day, C.; Deustua, S.E.; DiGennaro,R.; Ealet, A.; Ellis, R.S.; Eriksson, M.; Fruchter, A.; Genat, J.-F.; Goldhaber, G.; Goobar, A.; Groom, D.; Harris, S.E.; Harvey, P.R.; Heetderks, H.D.; Holland, S.E.; Huterer, D.; Karcher, A.; Kolbe, W.; Krieger, B.; Lafever, R.; Lamoureux, J.; Levi, M.E.; Levin, D.S.; Linder,E.V.; Loken, S.C.; Malina, R.; Massey, R.; McKay, T.; McKee, S.P.; Miquel, R.; Mortsell, E.; Mostek, N.; Mufson, S.; Musser, J.; Nugent, P.; Oluseyi, H.; Pain, R.; Palaio, N.; Pankow, D.; Perlmutter, S.; Pratt, R.; Prieto, E.; Refregier, A.; Rhodes, J.; Robinson, K.; Roe, N.; Sholl, M.; Schubnell, M.; Smadja, G.; Smoot, G.; Spadafora, A.; Tarle, G.; Tomasch,A.; von der Lippe, H.; Vincent, R.; Walder, J.-P.; Wang, G.
2002-07-29
The proposed SuperNova/Acceleration Probe (SNAP) mission will have a two-meter class telescope delivering diffraction-limited images to an instrumented 0.7 square-degree field sensitive in the visible and near-infrared wavelength regime. We describe the requirements for the instrument suite and the evolution of the focal plane design to the present concept in which all the instrumentation--visible and near-infrared imagers, spectrograph, and star guiders--share one common focal plane.
Blackfolds, Plane Waves and Minimal Surfaces
Armas, Jay
2015-01-01
Minimal surfaces in Euclidean space provide examples of possible non-compact horizon geometries and topologies in asymptotically flat space-time. On the other hand, the existence of limiting surfaces in the space-time provides a simple mechanism for making these configurations compact. Limiting surfaces appear naturally in a given space-time by making minimal surfaces rotate but they are also inherent to plane wave or de Sitter space-times in which case minimal surfaces can be static and compact. We use the blackfold approach in order to scan for possible black hole horizon geometries and topologies in asymptotically flat, plane wave and de Sitter space-times. In the process we uncover several new configurations, such as black helicoids and catenoids, some of which have an asymptotically flat counterpart. In particular, we find that the ultraspinning regime of singly-spinning Myers-Perry black holes, described in terms of the simplest minimal surface (the plane), can be obtained as a limit of a black helicoid...
Blackfolds, plane waves and minimal surfaces
Armas, Jay; Blau, Matthias
2015-07-01
Minimal surfaces in Euclidean space provide examples of possible non-compact horizon geometries and topologies in asymptotically flat space-time. On the other hand, the existence of limiting surfaces in the space-time provides a simple mechanism for making these configurations compact. Limiting surfaces appear naturally in a given space-time by making minimal surfaces rotate but they are also inherent to plane wave or de Sitter space-times in which case minimal surfaces can be static and compact. We use the blackfold approach in order to scan for possible black hole horizon geometries and topologies in asymptotically flat, plane wave and de Sitter space-times. In the process we uncover several new configurations, such as black helicoids and catenoids, some of which have an asymptotically flat counterpart. In particular, we find that the ultraspinning regime of singly-spinning Myers-Perry black holes, described in terms of the simplest minimal surface (the plane), can be obtained as a limit of a black helicoid, suggesting that these two families of black holes are connected. We also show that minimal surfaces embedded in spheres rather than Euclidean space can be used to construct static compact horizons in asymptotically de Sitter space-times.
Beamlet focal plane diagnostic
Energy Technology Data Exchange (ETDEWEB)
Caird, J.A.; Nielsen, N.D.; Patton, H.G.; Seppala, L.G.; Thompson, C.E.; Wegner, P.J.
1996-12-01
This paper describes the major optical and mechanical design features of the Beamlet Focal Plane Diagnostic system as well as measurements of the system performance, and typical data obtained to date. We also discuss the NIF requirements on the focal spot that we are interested in measuring, and some of our plans for future work using this system.
Bárány, Imre; Vilcu, Costin
2016-01-01
This volume presents easy-to-understand yet surprising properties obtained using topological, geometric and graph theoretic tools in the areas covered by the Geometry Conference that took place in Mulhouse, France from September 7–11, 2014 in honour of Tudor Zamfirescu on the occasion of his 70th anniversary. The contributions address subjects in convexity and discrete geometry, in distance geometry or with geometrical flavor in combinatorics, graph theory or non-linear analysis. Written by top experts, these papers highlight the close connections between these fields, as well as ties to other domains of geometry and their reciprocal influence. They offer an overview on recent developments in geometry and its border with discrete mathematics, and provide answers to several open questions. The volume addresses a large audience in mathematics, including researchers and graduate students interested in geometry and geometrical problems.
Algorithms in Algebraic Geometry
Dickenstein, Alicia; Sommese, Andrew J
2008-01-01
In the last decade, there has been a burgeoning of activity in the design and implementation of algorithms for algebraic geometric computation. Some of these algorithms were originally designed for abstract algebraic geometry, but now are of interest for use in applications and some of these algorithms were originally designed for applications, but now are of interest for use in abstract algebraic geometry. The workshop on Algorithms in Algebraic Geometry that was held in the framework of the IMA Annual Program Year in Applications of Algebraic Geometry by the Institute for Mathematics and Its
Fundamental concepts of geometry
Meserve, Bruce E
1983-01-01
Demonstrates relationships between different types of geometry. Provides excellent overview of the foundations and historical evolution of geometrical concepts. Exercises (no solutions). Includes 98 illustrations.
O'Leary, Michael
2010-01-01
Guides readers through the development of geometry and basic proof writing using a historical approach to the topic. In an effort to fully appreciate the logic and structure of geometric proofs, Revolutions of Geometry places proofs into the context of geometry's history, helping readers to understand that proof writing is crucial to the job of a mathematician. Written for students and educators of mathematics alike, the book guides readers through the rich history and influential works, from ancient times to the present, behind the development of geometry. As a result, readers are successfull
Euclidean geometry and transformations
Dodge, Clayton W
1972-01-01
This introduction to Euclidean geometry emphasizes transformations, particularly isometries and similarities. Suitable for undergraduate courses, it includes numerous examples, many with detailed answers. 1972 edition.
Determination of grain boundary geometry using TEM
Energy Technology Data Exchange (ETDEWEB)
Jang, H.; Farkas, D. (Department of Materials Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0237 (United States)); De Hosson, J.T.M. (Department of Applied Physics, University of Groningen, Nijenborgh 18, 9747 AG, Groningen (Netherlands))
1992-07-01
An experimental method to obtain the grain boundary geometry using the transmission electron microscope is presented. The method allows {Sigma} determination including grain boundary plane orientation. In order to determine the specialness of the grain boundary, three different criteria for maximum allowable deviations from exact CSL misorientations were examined. We tested these three criteria from a statistical distribution of grain boundary types in terms of {Sigma}. We compared grain boundary distributions from other studies in Ni{sub 3}Al and found discrepancies among them. It seems that the discrepancy came from the different criteria for special boundaries in {Sigma} determination and different experimental procedures they used. The statistical distribution of grain boundary plane orientations showed that low {Sigma} boundaries ({Sigma}{lt}11) were oriented to the plane of high density of coincident sites.
Ion distributions in plane and cylindrical chambers.
Rosen, R; George, E P
1975-11-01
The ion chamber equations of Thomson include both ion recombination and space-charge terms. Neglecting the space-charge term, an exact solution is obtained for the ion densities across a plane ionization chamber. The method is extended to the cylindrical chamber, and examples are given of the expected ion distributions in both geometries. Current-voltage relationships are derived for both chambers and compared with those of other workers. If the space-charge term is retained, the ion chamber equations for both geometries are not soluble in closed form. The cylindrical chamber is considered and a computer solution is obtained for the ion distributions and current. Comparison with the nonspace-charge solution shows that while there is only a small difference in the current-voltage relationship, a significant difference can occur in the ion concentrations.
Blackfolds, plane waves and minimal surfaces
Armas, Jay; Blau, Matthias
2015-01-01
Minimal surfaces in Euclidean space provide examples of possible non-compact horizon geometries and topologies in asymptotically flat space-time. On the other hand, the existence of limiting surfaces in the space-time provides a simple mechanism for making these configurations compact. Limiting surfaces appear naturally in a given space-time by making minimal surfaces rotate but they are also inherent to plane wave or de Sitter space-times in which case minimal surfaces can be static and comp...
Toric Geometry and String Theory
Bouchard, Vincent
2006-01-01
In this thesis we probe various interactions between toric geometry and string theory. First, the notion of a top was introduced by Candelas and Font as a useful tool to investigate string dualities. These objects torically encode the local geometry of a degeneration of an elliptic fibration. We classify all tops and give a prescription for assigning an affine, possibly twisted Kac-Moody algebra to any such top. Tops related to twisted Kac-Moody algebras can be used to construct string compactifications with reduced rank of the gauge group. Secondly, we compute all loop closed and open topological string amplitudes on orientifolds of toric Calabi-Yau threefolds, by using geometric transitions involving SO/Sp Chern-Simons theory, localization on the moduli space of holomorphic maps with involution, and the topological vertex. In particular, we count Klein bottles and projective planes with any number of handles in some Calabi-Yau orientifolds. We determine the BPS structure of the amplitudes, and illustrate ou...
Directory of Open Access Journals (Sweden)
Efim Khalimsky
1990-01-01
Full Text Available The importance of topological connectedness properties in processing digital pictures is well known. A natural way to begin a theory for this is to give a definition of connectedness for subsets of a digital plane which allows one to prove a Jordan curve theorem. The generally accepted approach to this has been a non-topological Jordan curve theorem which requires two different definitions, 4-connectedness, and 8-connectedness, one for the curve and the other for its complement.
Energy Technology Data Exchange (ETDEWEB)
Foda, Omar; Wheeler, Michael [Department of Mathematics and Statistics, University of Melbourne, Parkville, Victoria 3010 (Australia)
2007-01-15
Using BKP neutral fermions, we derive a product expression for the generating function of volume-weighted plane partitions that satisfy two conditions. If we call a set of adjacent equal height-h columns, h > 0, an h-path, then 1. Every h-path can assume one of two possible colours. 2. There is a unique way to move along an h-path from any column to another.
Modelling the Landing of a Plane in a Calculus Lab
Morante, Antonio; Vallejo, Jose A.
2012-01-01
We exhibit a simple model of a plane landing that involves only basic concepts of differential calculus, so it is suitable for a first-year calculus lab. We use the computer algebra system Maxima and the interactive geometry software GeoGebra to do the computations and graphics. (Contains 5 figures and 1 note.)
Modelling the Landing of a Plane in a Calculus Lab
Morante, Antonio; Vallejo, Jose A.
2012-01-01
We exhibit a simple model of a plane landing that involves only basic concepts of differential calculus, so it is suitable for a first-year calculus lab. We use the computer algebra system Maxima and the interactive geometry software GeoGebra to do the computations and graphics. (Contains 5 figures and 1 note.)
Supersymmetric Sigma Model Geometry
Ulf Lindström
2012-01-01
This is a review of how sigma models formulated in Superspace have become important tools for understanding geometry. Topics included are: The (hyper)k\\"ahler reduction; projective superspace; the generalized Legendre construction; generalized K\\"ahler geometry and constructions of hyperk\\"ahler metrics on Hermitean symmetric spaces.
Geometry of multihadron production
Energy Technology Data Exchange (ETDEWEB)
Bjorken, J.D.
1994-10-01
This summary talk only reviews a small sample of topics featured at this symposium: Introduction; The Geometry and Geography of Phase space; Space-Time Geometry and HBT; Multiplicities, Intermittency, Correlations; Disoriented Chiral Condensate; Deep Inelastic Scattering at HERA; and Other Contributions.
Supersymmetric Sigma Model geometry
Lindström, Ulf
2012-01-01
This is a review of how sigma models formulated in Superspace have become important tools for understanding geometry. Topics included are: The (hyper)k\\"ahler reduction; projective superspace; the generalized Legendre construction; generalized K\\"ahler geometry and constructions of hyperk\\"ahler metrics on Hermitean symmetric spaces.
1996-01-01
Designs and Finite Geometries brings together in one place important contributions and up-to-date research results in this important area of mathematics. Designs and Finite Geometries serves as an excellent reference, providing insight into some of the most important research issues in the field.
Foundations of algebraic geometry
Weil, A
1946-01-01
This classic is one of the cornerstones of modern algebraic geometry. At the same time, it is entirely self-contained, assuming no knowledge whatsoever of algebraic geometry, and no knowledge of modern algebra beyond the simplest facts about abstract fields and their extensions, and the bare rudiments of the theory of ideals.
Bergshoeff, Eric A.; Riccioni, Fabio; Alvarez-Gaumé, L.
2011-01-01
We probe doubled geometry with dual fundamental branes. i.e. solitons. Restricting ourselves first to solitonic branes with more than two transverse directions we find that the doubled geometry requires an effective wrapping rule for the solitonic branes which is dual to the wrapping rule for fundam
Energy Technology Data Exchange (ETDEWEB)
Grotz, Andreas
2011-10-07
In this thesis, a formulation of a Lorentzian quantum geometry based on the framework of causal fermion systems is proposed. After giving the general definition of causal fermion systems, we deduce space-time as a topological space with an underlying causal structure. Restricting attention to systems of spin dimension two, we derive the objects of our quantum geometry: the spin space, the tangent space endowed with a Lorentzian metric, connection and curvature. In order to get the correspondence to classical differential geometry, we construct examples of causal fermion systems by regularizing Dirac sea configurations in Minkowski space and on a globally hyperbolic Lorentzian manifold. When removing the regularization, the objects of our quantum geometry reduce to the common objects of spin geometry on Lorentzian manifolds, up to higher order curvature corrections.
The Role and Place of Fusionism in School Geometry Education
Directory of Open Access Journals (Sweden)
G. A. Klekovkin
2012-01-01
Full Text Available The paper deals with the issue of succession in school geometry education. By the analysis and synthesis of the integrative experience of the plane and spatial geometry teaching, it has being substantiated that the application of fusionism elements can provide the effective instrument for implementing the vertical content and process succession in geometry teaching. By means of the activity approach it is being proved that the fusion teaching of planimetry and stereometry in elementary school facilitates the development of children’s learning and cognitive skills. Using fusionism at the final years of Secondary School promotes the school leavers’ readiness for studying general spatial geometry in the system of higher education. The study is addressed to developers of the school mathematical educational standards, authors of school geometry manuals and geometry teachers.
Construction and decoding of a class of algebraic geometry codes
DEFF Research Database (Denmark)
Justesen, Jørn; Larsen, Knud J.; Jensen, Helge Elbrønd
1989-01-01
A class of codes derived from algebraic plane curves is constructed. The concepts and results from algebraic geometry that were used are explained in detail; no further knowledge of algebraic geometry is needed. Parameters, generator and parity-check matrices are given. The main result is a decod...... is a decoding algorithm which turns out to be a generalization of the Peterson algorithm for decoding BCH decoder codes......A class of codes derived from algebraic plane curves is constructed. The concepts and results from algebraic geometry that were used are explained in detail; no further knowledge of algebraic geometry is needed. Parameters, generator and parity-check matrices are given. The main result...
Riemann-Finsler Geometry with Applications to Information Geometry
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
Information geometry is a new branch in mathematics, originated from the applications of differential geometry to statistics. In this paper we briefly introduce RiemannFinsler geometry, by which we establish Information Geometry on a much broader base,so that the potential applications of Information Geometry will be beyond statistics.
Johnson, Aylmer
2004-01-01
Plane and Geodetic Surveying blends theory and practice, conventional techniques and GPS, to provide the ideal book for students of surveying.Detailed guidance is given on how and when the principle surveying instruments (theodolites, Total Stations, levels and GPS) should be used. Concepts and formulae needed to convert instrument readings into useful results are explained. Rigorous explanations of the theoretical aspects of surveying are given, while at the same time a wealth of useful advice about conducting a survey in practice is provided. An accompanying least squares adjustment program
Ariwahjoedi, Seramika; Kosasih, Jusak Sali; Rovelli, Carlo; Zen, Freddy Permana
2016-01-01
Following our earlier work, we construct statistical discrete geometry by applying statistical mechanics to discrete (Regge) gravity. We propose a coarse-graining method for discrete geometry under the assumptions of atomism and background independence. To maintain these assumptions, restrictions are given to the theory by introducing cut-offs, both in ultraviolet and infrared regime. Having a well-defined statistical picture of discrete Regge geometry, we take the infinite degrees of freedom (large n) limit. We argue that the correct limit consistent with the restrictions and the background independence concept is not the continuum limit of statistical mechanics, but the thermodynamical limit.
Bonola, Roberto
2010-01-01
This is an excellent historical and mathematical view by a renowned Italian geometer of the geometries that have risen from a rejection of Euclid's parallel postulate. Students, teachers and mathematicians will find here a ready reference source and guide to a field that has now become overwhelmingly important.Non-Euclidean Geometry first examines the various attempts to prove Euclid's parallel postulate-by the Greeks, Arabs, and mathematicians of the Renaissance. Then, ranging through the 17th, 18th and 19th centuries, it considers the forerunners and founders of non-Euclidean geometry, such
Santo, J
1999-01-01
The ALICE Geometry Database project consists of the development of a set of data structures to store the geometrical information of the ALICE Detector. This Database will be used in Simulation, Reconstruction and Visualisation and will interface with existing CAD systems and Geometrical Modellers.At the present time, we are able to read a complete GEANT3 geometry, to store it in our database and to visualise it. On disk, we store different geometry files in hierarchical fashion, and all the nodes, materials, shapes, configurations and transformations distributed in this tree structure. The present status of the prototype and its future evolution will be presented.
Discrete and computational geometry
Devadoss, Satyan L
2011-01-01
Discrete geometry is a relatively new development in pure mathematics, while computational geometry is an emerging area in applications-driven computer science. Their intermingling has yielded exciting advances in recent years, yet what has been lacking until now is an undergraduate textbook that bridges the gap between the two. Discrete and Computational Geometry offers a comprehensive yet accessible introduction to this cutting-edge frontier of mathematics and computer science. This book covers traditional topics such as convex hulls, triangulations, and Voronoi diagrams, as well a
Plane-wave least-squares reverse-time migration
Dai, Wei
2013-06-03
A plane-wave least-squares reverse-time migration (LSRTM) is formulated with a new parameterization, where the migration image of each shot gather is updated separately and an ensemble of prestack images is produced along with common image gathers. The merits of plane-wave prestack LSRTM are the following: (1) plane-wave prestack LSRTM can sometimes offer stable convergence even when the migration velocity has bulk errors of up to 5%; (2) to significantly reduce computation cost, linear phase-shift encoding is applied to hundreds of shot gathers to produce dozens of plane waves. Unlike phase-shift encoding with random time shifts applied to each shot gather, plane-wave encoding can be effectively applied to data with a marine streamer geometry. (3) Plane-wave prestack LSRTM can provide higher-quality images than standard reverse-time migration. Numerical tests on the Marmousi2 model and a marine field data set are performed to illustrate the benefits of plane-wave LSRTM. Empirical results show that LSRTM in the plane-wave domain, compared to standard reversetime migration, produces images efficiently with fewer artifacts and better spatial resolution. Moreover, the prestack image ensemble accommodates more unknowns to makes it more robust than conventional least-squares migration in the presence of migration velocity errors. © 2013 Society of Exploration Geophysicists.
Hackel, L.A.; Hermann, M.R.; Dane, C.B.; Tiszauer, D.H.
1995-12-12
A solid state laser is frequency tripled to 0.3 {micro}m. A small portion of the laser is split off and generates a Stokes seed in a low power oscillator. The low power output passes through a mask with the appropriate hole pattern. Meanwhile, the bulk of the laser output is focused into a larger stimulated Brillouin scattering (SBS) amplifier. The low power beam is directed through the same cell in the opposite direction. The majority of the amplification takes place at the focus which is the fourier transform plane of the mask image. The small holes occupy large area at the focus and thus are preferentially amplified. The amplified output is now imaged onto the multichip module where the holes are drilled. Because of the fourier plane amplifier, only about 1/10th the power of a competitive system is needed. This concept allows less expensive masks to be used in the process and requires much less laser power. 1 fig.
Zahm, A F
1924-01-01
This report gives the description and the use of a specially designed aerodynamic plane table. For the accurate and expeditious geometrical measurement of models in an aerodynamic laboratory, and for miscellaneous truing operations, there is frequent need for a specially equipped plan table. For example, one may have to measure truly to 0.001 inch the offsets of an airfoil at many parts of its surface. Or the offsets of a strut, airship hull, or other carefully formed figure may require exact calipering. Again, a complete airplane model may have to be adjusted for correct incidence at all parts of its surfaces or verified in those parts for conformance to specifications. Such work, if but occasional, may be done on a planing or milling machine; but if frequent, justifies the provision of a special table. For this reason it was found desirable in 1918 to make the table described in this report and to equip it with such gauges and measures as the work should require.
Cardone, V F; Diaferio, A; Tortora, C; Molinaro, R
2010-01-01
Modified Newtonian Dynamics (MOND) has been shown to be able to fit spiral galaxy rotation curves as well as giving a theoretical foundation for empirically determined scaling relations, such as the Tully - Fisher law, without the need for a dark matter halo. As a complementary analysis, one should investigate whether MOND can also reproduce the dynamics of early - type galaxies (ETGs) without dark matter. As a first step, we here show that MOND can indeed fit the observed central velocity dispersion $\\sigma_0$ of a large sample of ETGs assuming a simple MOND interpolating functions and constant anisotropy. We also show that, under some assumptions on the luminosity dependence of the Sersic n parameter and the stellar M/L ratio, MOND predicts a fundamental plane for ETGs : a log - linear relation among the effective radius $R_{eff}$, $\\sigma_0$ and the mean effective intensity $\\langle I_e \\rangle$. However, we predict a tilt between the observed and the MOND fundamental planes.
Hackel, Lloyd A.; Hermann, Mark R.; Dane, C. Brent; Tiszauer, Detlev H.
1995-01-01
A solid state laser is frequency tripled to 0.3 .mu.m. A small portion of the laser is split off and generates a Stokes seed in a low power oscillator. The low power output passes through a mask with the appropriate hole pattern. Meanwhile, the bulk of the laser output is focused into a larger stimulated Brillouin scattering (SBS) amplifier. The low power beam is directed through the same cell in the opposite direction. The majority of the amplification takes place at the focus which is the fourier transform plane of the mask image. The small holes occupy large area at the focus and thus are preferentially amplified. The amplified output is now imaged onto the multichip module where the holes are drilled. Because of the fourier plane amplifier, only .about.1/10th the power of a competitive system is needed. This concept allows less expensive masks to be used in the process and requires much less laser power.
Elementary differential geometry
Pressley, Andrew
2001-01-01
Curves and surfaces are objects that everyone can see, and many of the questions that can be asked about them are natural and easily understood Differential geometry is concerned with the precise mathematical formulation of some of these questions, and with trying to answer them using calculus techniques It is a subject that contains some of the most beautiful and profound results in mathematics yet many of these are accessible to higher-level undergraduates Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces while keeping the prerequisites to an absolute minimum Nothing more than first courses in linear algebra and multivariate calculus are required, and the most direct and straightforward approach is used at all times Numerous diagrams illustrate both the ideas in the text and the examples of curves and surfaces discussed there The book will provide an invaluable resource to all those taking a first course in differential geometry, for their lecture...
Chern, Shiing-Shen
1990-01-01
Discussed are the major historical developments of geometry. Euclid, Descartes, Klein's Erlanger Program, Gaus and Riemann, globalization, topology, Elie Cartan, and an application to molecular biology are included as topics. (KR)
Lectures on Symplectic Geometry
Silva, Ana Cannas
2001-01-01
The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and cl...
Emenaker, Charles E.
1999-01-01
Describes a sixth-grade interdisciplinary geometry unit based on Charles Dickens's "A Christmas Carol". Focuses on finding area, volume, and perimeter, and working with estimation, decimals, and fractions in the context of making gingerbread houses. (ASK)
Geometry of differential equations
Khovanskiĭ, A; Vassiliev, V
1998-01-01
This volume contains articles written by V. I. Arnold's colleagues on the occasion of his 60th birthday. The articles are mostly devoted to various aspects of geometry of differential equations and relations to global analysis and Hamiltonian mechanics.
Facilitating Understandings of Geometry.
Pappas, Christine C.; Bush, Sara
1989-01-01
Illustrates some learning encounters for facilitating first graders' understanding of geometry. Describes some of children's approaches using Cuisenaire rods and teacher's intervening. Presents six problems involving various combinations of Cuisenaire rods and cubes. (YP)
Introduction to tropical geometry
Maclagan, Diane
2015-01-01
Tropical geometry is a combinatorial shadow of algebraic geometry, offering new polyhedral tools to compute invariants of algebraic varieties. It is based on tropical algebra, where the sum of two numbers is their minimum and the product is their sum. This turns polynomials into piecewise-linear functions, and their zero sets into polyhedral complexes. These tropical varieties retain a surprising amount of information about their classical counterparts. Tropical geometry is a young subject that has undergone a rapid development since the beginning of the 21st century. While establishing itself as an area in its own right, deep connections have been made to many branches of pure and applied mathematics. This book offers a self-contained introduction to tropical geometry, suitable as a course text for beginning graduate students. Proofs are provided for the main results, such as the Fundamental Theorem and the Structure Theorem. Numerous examples and explicit computations illustrate the main concepts. Each of t...
Melzak, Z A
2008-01-01
Intended for students of many different backgrounds with only a modest knowledge of mathematics, this text features self-contained chapters that can be adapted to several types of geometry courses. 1983 edition.
Kollár, János
1997-01-01
This volume contains the lectures presented at the third Regional Geometry Institute at Park City in 1993. The lectures provide an introduction to the subject, complex algebraic geometry, making the book suitable as a text for second- and third-year graduate students. The book deals with topics in algebraic geometry where one can reach the level of current research while starting with the basics. Topics covered include the theory of surfaces from the viewpoint of recent higher-dimensional developments, providing an excellent introduction to more advanced topics such as the minimal model program. Also included is an introduction to Hodge theory and intersection homology based on the simple topological ideas of Lefschetz and an overview of the recent interactions between algebraic geometry and theoretical physics, which involve mirror symmetry and string theory.
Derived logarithmic geometry I
Steffen, Sagave; Timo, Schurg; Gabriele, Vezzosi
2016-01-01
In order to develop the foundations of logarithmic derived geometry, we introduce a model category of logarithmic simplicial rings and a notion of derived log \\'etale maps and use this to define derived log stacks.
Busemann, Herbert
2005-01-01
A comprehensive approach to qualitative problems in intrinsic differential geometry, this text examines Desarguesian spaces, perpendiculars and parallels, covering spaces, the influence of the sign of the curvature on geodesics, more. 1955 edition. Includes 66 figures.
DEFF Research Database (Denmark)
Kokkendorff, Simon Lyngby
2002-01-01
The subject of this Ph.D.-thesis is somewhere in between continuous and discrete geometry. Chapter 2 treats the geometry of finite point sets in semi-Riemannian hyperquadrics,using a matrix whose entries are a trigonometric function of relative distances in a given point set. The distance...... to the geometry of a simplex in a semi-Riemannian hyperquadric. In chapter 3 we study which finite metric spaces that are realizable in a hyperbolic space in the limit where curvature goes to -∞. We show that such spaces are the so called leaf spaces, the set of degree 1 vertices of weighted trees. We also...... establish results on the limiting geometry of such an isometrically realized leaf space simplex in hyperbolic space, when curvature goes to -∞. Chapter 4 discusses negative type of metric spaces. We give a measure theoretic treatment of this concept and related invariants. The theory developed...
Hohmann, Manuel
2014-01-01
From general relativity we have learned the principles of general covariance and local Lorentz invariance, which follow from the fact that we consider observables as tensors on a spacetime manifold whose geometry is modeled by a Lorentzian metric. Approaches to quantum gravity, however, hint towards a breaking of these symmetries and the possible existence of more general, non-tensorial geometric structures. Possible implications of these approaches are non-tensorial transformation laws between different observers and an observer-dependent notion of geometry. In this work we review two different frameworks for observer dependent geometries, which may provide hints towards a quantization of gravity and possible explanations for so far unexplained phenomena: Finsler spacetimes and Cartan geometry on observer space. We discuss their definitions, properties and applications to observers, field theories and gravity.
Implosions and hypertoric geometry
DEFF Research Database (Denmark)
Dancer, A.; Kirwan, F.; Swann, A.
2013-01-01
The geometry of the universal hyperkahler implosion for SU (n) is explored. In particular, we show that the universal hyperkahler implosion naturally contains a hypertoric variety described in terms of quivers. Furthermore, we discuss a gauge theoretic approach to hyperkahler implosion.......The geometry of the universal hyperkahler implosion for SU (n) is explored. In particular, we show that the universal hyperkahler implosion naturally contains a hypertoric variety described in terms of quivers. Furthermore, we discuss a gauge theoretic approach to hyperkahler implosion....
Intermediate algebra & analytic geometry
Gondin, William R
1967-01-01
Intermediate Algebra & Analytic Geometry Made Simple focuses on the principles, processes, calculations, and methodologies involved in intermediate algebra and analytic geometry. The publication first offers information on linear equations in two unknowns and variables, functions, and graphs. Discussions focus on graphic interpretations, explicit and implicit functions, first quadrant graphs, variables and functions, determinate and indeterminate systems, independent and dependent equations, and defective and redundant systems. The text then examines quadratic equations in one variable, system
Schreiber, Urs
2016-01-01
This is a survey of motivations, constructions and applications of higher prequantum geometry. In section 1 we highlight the open problem of prequantizing local field theory in a local and gauge invariant way, and we survey how a solution to this problem exists in higher differential geometry. In section 2 we survey examples and problems of interest. In section 3 we survey the abstract cohesive homotopy theory that serves to make all this precise and tractable.
Punzi, Raffaele; Wohlfarth, Mattias N R
2008-01-01
We reveal the non-metric geometry underlying omega-->0 Brans-Dicke theory by unifying the metric and scalar field into a single geometric structure. Taking this structure seriously as the geometry to which matter universally couples, we show that the theory is fully consistent with solar system tests. This is in striking constrast with the standard metric coupling, which grossly violates post-Newtonian experimental constraints.
Energy Technology Data Exchange (ETDEWEB)
Punzi, Raffaele [Zentrum fuer Mathematische Physik und II. Institut fuer Theoretische Physik, Universitaet Hamburg, Luruper Chaussee 149, 22761 Hamburg (Germany)], E-mail: raffaele.punzi@desy.de; Schuller, Frederic P. [Max Planck Institut fuer Gravitationsphysik, Albert Einstein Institut, Am Muehlenberg 1, 14467 Potsdam (Germany)], E-mail: fps@aei.mpg.de; Wohlfarth, Mattias N.R. [Zentrum fuer Mathematische Physik und II. Institut fuer Theoretische Physik, Universitaet Hamburg, Luruper Chaussee 149, 22761 Hamburg (Germany)], E-mail: mattias.wohlfarth@desy.de
2008-12-11
We reveal the non-metric geometry underlying {omega}{yields}0 Brans-Dicke theory by unifying the metric and scalar field into a single geometric structure. Taking this structure seriously as the geometry to which matter universally couples, we show that the theory is fully consistent with solar system tests. This is in striking contrast with the standard metric coupling, which grossly violates post-Newtonian experimental constraints.
McAteer, R. T. J.
2013-06-01
When Mandelbrot, the father of modern fractal geometry, made this seemingly obvious statement he was trying to show that we should move out of our comfortable Euclidean space and adopt a fractal approach to geometry. The concepts and mathematical tools of fractal geometry provides insight into natural physical systems that Euclidean tools cannot do. The benet from applying fractal geometry to studies of Self-Organized Criticality (SOC) are even greater. SOC and fractal geometry share concepts of dynamic n-body interactions, apparent non-predictability, self-similarity, and an approach to global statistics in space and time that make these two areas into naturally paired research techniques. Further, the iterative generation techniques used in both SOC models and in fractals mean they share common features and common problems. This chapter explores the strong historical connections between fractal geometry and SOC from both a mathematical and conceptual understanding, explores modern day interactions between these two topics, and discusses how this is likely to evolve into an even stronger link in the near future.
Ibrahim, Hazem
2016-09-19
The unrelenting increase in the population of mobile users and their traffic demands drive cellular network operators to densify their network infrastructure. Network densification shrinks the footprint of base stations (BSs) and reduces the number of users associated with each BS, leading to an improved spatial frequency reuse and spectral efficiency, and thus, higher network capacity. However, the densification gain comes at the expense of higher handover rates and network control overhead. Hence, user’s mobility can diminish or even nullifies the foreseen densification gain. In this context, splitting the control plane ( C -plane) and user plane ( U -plane) is proposed as a potential solution to harvest densification gain with reduced cost in terms of handover rate and network control overhead. In this paper, we use stochastic geometry to develop a tractable mobility-aware model for a two-tier downlink cellular network with ultra-dense small cells and C -plane/ U -plane split architecture. The developed model is then used to quantify the effect of mobility on the foreseen densification gain with and without C -plane/ U -plane split. To this end, we shed light on the handover problem in dense cellular environments, show scenarios where the network fails to support certain mobility profiles, and obtain network design insights.
3D blob dynamics in toroidal geometry
DEFF Research Database (Denmark)
Nielsen, Anders Henry; Reiser, Dirk
In this paper we study the simple case of the dynamics of a density perturbation localized in the edge region of a medium sized tokamak in a full 3D geometry. The 2D evolution of such a perturbation has been studied in details on the low-field side, where the gradient of the magnetic field always...... dynamics in a full 3D tokamak geometry including the edge and SOL region as well. Previous studies with the ATTEMPT code proved that density blobs appear for typical parameters in the TEXTOR tokamak. The code has been prepared for flux driven simulations with detailed control of the blob initial state....... The DIESEL code is an extension of the ESEL code [1]. It solves a simple interchange model in full 3D tokamak geometry, where the toroidal direction is divided into a number of drift planes. On each drift plane the equations are solved in a domain corresponding to the full 2D cross section of the tokamak...
Plane symmetric cosmological models
Yadav, Anil Kumar; Ray, Saibal; Mallick, A
2016-01-01
In this work, we perform the Lie symmetry analysis on the Einstein-Maxwell field equations in plane symmetric spacetime. Here Lie point symmetries and optimal system of one dimensional subalgebras are determined. The similarity reductions and exact solutions are obtained in connection to the evolution of universe. The present study deals with the electromagnetic energy of inhomogeneous universe where $F_{12}$ is the non-vanishing component of electromagnetic field tensor. To get a deterministic solution, it is assumed that the free gravitational field is Petrov type-II non-degenerate. The electromagnetic field tensor $F_{12}$ is found to be positive and increasing function of time. As a special case, to validate the solution set, we discuss some physical and geometric properties of a specific sub-model.
Duality and noncommutative planes
DEFF Research Database (Denmark)
Jøndrup, Søren
2015-01-01
We study extensions of simple modules over an associative ring A and we prove that for twosided ideals mm and nn with artinian factors the condition ExtA1(A/m,A/n)≠0 holds for the left A -modules A/mA/m and A/nA/n if and only if it holds for the right modules A/nA/n and A/mA/m. The methods pro...... proving this are applied to show that noncommutative models of the plane, i.e. algebras of the form k〈x,y〉/(f)k〈x,y〉/(f), where f∈([x,y])f∈([x,y]) are noetherian only in case (f)=([x,y])...
Sources of hyperbolic geometry
Stillwell, John
1996-01-01
This book presents, for the first time in English, the papers of Beltrami, Klein, and Poincaré that brought hyperbolic geometry into the mainstream of mathematics. A recognition of Beltrami comparable to that given the pioneering works of Bolyai and Lobachevsky seems long overdue-not only because Beltrami rescued hyperbolic geometry from oblivion by proving it to be logically consistent, but because he gave it a concrete meaning (a model) that made hyperbolic geometry part of ordinary mathematics. The models subsequently discovered by Klein and Poincaré brought hyperbolic geometry even further down to earth and paved the way for the current explosion of activity in low-dimensional geometry and topology. By placing the works of these three mathematicians side by side and providing commentaries, this book gives the student, historian, or professional geometer a bird's-eye view of one of the great episodes in mathematics. The unified setting and historical context reveal the insights of Beltrami, Klein, and Po...
Students Discovering Spherical Geometry Using Dynamic Geometry Software
Guven, Bulent; Karatas, Ilhan
2009-01-01
Dynamic geometry software (DGS) such as Cabri and Geometers' Sketchpad has been regularly used worldwide for teaching and learning Euclidean geometry for a long time. The DGS with its inductive nature allows students to learn Euclidean geometry via explorations. However, with respect to non-Euclidean geometries, do we need to introduce them to…
Students Discovering Spherical Geometry Using Dynamic Geometry Software
Guven, Bulent; Karatas, Ilhan
2009-01-01
Dynamic geometry software (DGS) such as Cabri and Geometers' Sketchpad has been regularly used worldwide for teaching and learning Euclidean geometry for a long time. The DGS with its inductive nature allows students to learn Euclidean geometry via explorations. However, with respect to non-Euclidean geometries, do we need to introduce them to…
Emergent Complex Network Geometry
Wu, Zhihao; Rahmede, Christoph; Bianconi, Ginestra
2014-01-01
Networks are mathematical structures that are universally used to describe a large variety of complex systems such as the brain or the Internet. Characterizing the geometrical properties of these networks has become increasingly relevant for routing problems, inference and data mining. In real growing networks, topological, structural and geometrical properties emerge spontaneously from their dynamical rules. Nevertheless we still miss a model in which networks develop an emergent complex geometry. Here we show that a single two parameter network model, the growing geometrical network, can generate complex network geometries with non-trivial distribution of curvatures, combining exponential growth and small-world properties with finite spectral dimensionality. In one limit, the non-equilibrium dynamical rules of these networks can generate scale-free networks with clustering and communities, in another limit planar random geometries with non-trivial modularity. Finally we find that these properties of the geo...
Computational synthetic geometry
Bokowski, Jürgen
1989-01-01
Computational synthetic geometry deals with methods for realizing abstract geometric objects in concrete vector spaces. This research monograph considers a large class of problems from convexity and discrete geometry including constructing convex polytopes from simplicial complexes, vector geometries from incidence structures and hyperplane arrangements from oriented matroids. It turns out that algorithms for these constructions exist if and only if arbitrary polynomial equations are decidable with respect to the underlying field. Besides such complexity theorems a variety of symbolic algorithms are discussed, and the methods are applied to obtain new mathematical results on convex polytopes, projective configurations and the combinatorics of Grassmann varieties. Finally algebraic varieties characterizing matroids and oriented matroids are introduced providing a new basis for applying computer algebra methods in this field. The necessary background knowledge is reviewed briefly. The text is accessible to stud...
Supersymmetry and noncommutative geometry
Beenakker, Wim; Suijlekom, Walter D van
2016-01-01
In this work the question whether noncommutative geometry allows for supersymmetric theories is addressed. Noncommutative geometry has seen remarkable applications in high energy physics, viz. the geometrical interpretation of the Standard Model, however such a question has not been answered in a conclusive way so far. The book starts with a systematic analysis of the possibilities for so-called almost-commutative geometries on a 4-dimensional, flat background to exhibit not only a particle content that is eligible for supersymmetry, but also have a supersymmetric action. An approach is proposed in which the basic `building blocks' of potentially supersymmetric theories and the demands for their action to be supersymmetric are identified. It is then described how a novel kind of soft supersymmetry breaking Lagrangian arises naturally from the spectral action. Finally, the above formalism is applied to explore the existence of a noncommutative version of the minimal supersymmetric Standard Model. This book is ...
Wetterich, C
2012-01-01
In models of emergent gravity the metric arises as the expectation value of some collective field. Usually, many different collective fields with appropriate tensor properties are candidates for a metric. Which collective field describes the "physical geometry"? We resolve this "metric ambiguity" by an investigation of the most general form of the quantum effective action for several metrics. In the long-distance limit the physical metric is universal and accounts for a massless graviton. Other degrees of freedom contained in the various metric candidates describe very massive scalars and symmetric second rank tensors. They only play a role at microscopic distances, typically around the Planck length. The universality of geometry at long distances extends to the vierbein and the connection. On the other hand, for distances and time intervals of Planck size geometry looses its universal meaning. Time is born with the big bang.
Cecil, Thomas E
2015-01-01
This exposition provides the state-of-the art on the differential geometry of hypersurfaces in real, complex, and quaternionic space forms. Special emphasis is placed on isoparametric and Dupin hypersurfaces in real space forms as well as Hopf hypersurfaces in complex space forms. The book is accessible to a reader who has completed a one-year graduate course in differential geometry. The text, including open problems and an extensive list of references, is an excellent resource for researchers in this area. Geometry of Hypersurfaces begins with the basic theory of submanifolds in real space forms. Topics include shape operators, principal curvatures and foliations, tubes and parallel hypersurfaces, curvature spheres and focal submanifolds. The focus then turns to the theory of isoparametric hypersurfaces in spheres. Important examples and classification results are given, including the construction of isoparametric hypersurfaces based on representations of Clifford algebras. An in-depth treatment of Dupin hy...
Zheng, Fangyang
2002-01-01
The theory of complex manifolds overlaps with several branches of mathematics, including differential geometry, algebraic geometry, several complex variables, global analysis, topology, algebraic number theory, and mathematical physics. Complex manifolds provide a rich class of geometric objects, for example the (common) zero locus of any generic set of complex polynomials is always a complex manifold. Yet complex manifolds behave differently than generic smooth manifolds; they are more coherent and fragile. The rich yet restrictive character of complex manifolds makes them a special and interesting object of study. This book is a self-contained graduate textbook that discusses the differential geometric aspects of complex manifolds. The first part contains standard materials from general topology, differentiable manifolds, and basic Riemannian geometry. The second part discusses complex manifolds and analytic varieties, sheaves and holomorphic vector bundles, and gives a brief account of the surface classifi...
Integral Geometry and Holography
Czech, Bartlomiej; McCandlish, Samuel; Sully, James
2015-01-01
We present a mathematical framework which underlies the connection between information theory and the bulk spacetime in the AdS$_3$/CFT$_2$ correspondence. A key concept is kinematic space: an auxiliary Lorentzian geometry whose metric is defined in terms of conditional mutual informations and which organizes the entanglement pattern of a CFT state. When the field theory has a holographic dual obeying the Ryu-Takayanagi proposal, kinematic space has a direct geometric meaning: it is the space of bulk geodesics studied in integral geometry. Lengths of bulk curves are computed by kinematic volumes, giving a precise entropic interpretation of the length of any bulk curve. We explain how basic geometric concepts -- points, distances and angles -- are reflected in kinematic space, allowing one to reconstruct a large class of spatial bulk geometries from boundary entanglement entropies. In this way, kinematic space translates between information theoretic and geometric descriptions of a CFT state. As an example, we...
2002-01-01
Discrete geometry investigates combinatorial properties of configurations of geometric objects. To a working mathematician or computer scientist, it offers sophisticated results and techniques of great diversity and it is a foundation for fields such as computational geometry or combinatorial optimization. This book is primarily a textbook introduction to various areas of discrete geometry. In each area, it explains several key results and methods, in an accessible and concrete manner. It also contains more advanced material in separate sections and thus it can serve as a collection of surveys in several narrower subfields. The main topics include: basics on convex sets, convex polytopes, and hyperplane arrangements; combinatorial complexity of geometric configurations; intersection patterns and transversals of convex sets; geometric Ramsey-type results; polyhedral combinatorics and high-dimensional convexity; and lastly, embeddings of finite metric spaces into normed spaces. Jiri Matousek is Professor of Com...
Ochiai, T.; Nacher, J. C.
2011-09-01
Recently, the application of geometry and conformal mappings to artificial materials (metamaterials) has attracted the attention in various research communities. These materials, characterized by a unique man-made structure, have unusual optical properties, which materials found in nature do not exhibit. By applying the geometry and conformal mappings theory to metamaterial science, it may be possible to realize so-called "Harry Potter cloaking device". Although such a device is still in the science fiction realm, several works have shown that by using such metamaterials it may be possible to control the direction of the electromagnetic field at will. We could then make an object hidden inside of a cloaking device. Here, we will explain how to design invisibility device using differential geometry and conformal mappings.
Cubic Curves, Finite Geometry and Cryptography
Bruen, A A; Wehlau, D L
2011-01-01
Some geometry on non-singular cubic curves, mainly over finite fields, is surveyed. Such a curve has 9,3,1 or 0 points of inflexion, and cubic curves are classified accordingly. The group structure and the possible numbers of rational points are also surveyed. A possible strengthening of the security of elliptic curve cryptography is proposed using a `shared secret' related to the group law. Cubic curves are also used in a new way to construct sets of points having various combinatorial and geometric properties that are of particular interest in finite Desarguesian planes.
Quantum Secret Sharing by applying Analytic Geometry
Liu, Ruilong
2010-01-01
In this paper, we investigate a novel $(2,2)$-threshold scheme and then generalize this to a $(n,n)$-threshold scheme for quantum secret sharing (QSS) which makes use of the fundamentals of Analytic Geometry. The dealer aptly selects GHZ states related to the coefficients which determine straight lines on a two-dimension plane. Then by computing each two of the lines intercept or not, we obtain a judging matrix whose rank can be used to determine the secret stored in entangled bits. Based on ...
Line geometry and electromagnetism II: wave motion
Delphenich, D H
2013-01-01
The fundamental role of line geometry in the study of wave motion is first introduced in the general context by way of the tangent planes to the instantaneous wave surfaces, in which it is first observed that the possible frequency-wave number 1-forms are typically constrained by a dispersion law that is derived from a constitutive law by way of the field equations. After a general review of the basic concepts that relate to quadratic line complexes, these geometric notions are applied to the study of electromagnetic waves, in particular.
The Geometry of Conventionality
Weatherall, James Owen
2013-01-01
Hans Reichenbach famously argued that the geometry of spacetime is conventional in relativity theory, in the sense that one can freely choose the spacetime metric so long as one is willing to postulate a "universal force field". Here we make precise a sense in which the field Reichenbach defines fails to be a "force". We then argue that there is an interesting and perhaps tenable sense in which geometry is conventional in classical spacetimes. We conclude with a no-go result showing that the variety of conventionalism available in classical spacetimes does not extend to relativistic spacetimes.
Bowyer, Adrian
1983-01-01
A Programmer's Geometry provides a guide in programming geometric shapes. The book presents formulas and examples of computer representation and coding of geometry. Each of the nine chapters of the text deals with the representation and solution of a specific geometrical problem, such as areas, vectors, and volumes. The last chapter provides a brief discussion on generating image through a computer. The codes presented in the book are written in FORTRAN 77. The text will be of great use to programmers who are working on projects that involve geometric calculations.
Eisenhart, L P
1927-01-01
The use of the differential geometry of a Riemannian space in the mathematical formulation of physical theories led to important developments in the geometry of such spaces. The concept of parallelism of vectors, as introduced by Levi-Civita, gave rise to a theory of the affine properties of a Riemannian space. Covariant differentiation, as developed by Christoffel and Ricci, is a fundamental process in this theory. Various writers, notably Eddington, Einstein and Weyl, in their efforts to formulate a combined theory of gravitation and electromagnetism, proposed a simultaneous generalization o
Universal correlators from geometry
Energy Technology Data Exchange (ETDEWEB)
Dijkgraaf, Robbert [Institute for Theoretical Physics, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam (Netherlands); Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam (Netherlands); Temuerhan, Mine; Sinkovics, Annamaria [Institute for Theoretical Physics, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam (Netherlands)]. E-mail: sinkovic@science.uva.nl
2004-11-01
Matrix model correlators show universal behaviour at short distances. We provide a derivation for these universal correlators by inserting probe branes in the underlying effective geometry. We generalize these results to study correlators of branes and their universal behaviour in the Calabi-Yau crystals, where we find a role for a generalized brane insertion. (author)
Universal Correlators from Geometry
Dijkgraaf, Robbert; Sinkovics, Annamaria; Temurhan, Mine
2004-11-01
Matrix model correlators show universal behaviour at short distances. We provide a derivation for these universal correlators by inserting probe branes in the underlying effective geometry. We generalize these results to study correlators of branes and their universal behaviour in the Calabi-Yau crystals, where we find a role for a generalized brane insertion.
Universal Correlators from Geometry
Dijkgraaf, R; Temurhan, M; Dijkgraaf, Robbert; Sinkovics, Annamaria; Temurhan, Mine
2004-01-01
Matrix model correlators show universal behaviour at short distances. We provide a derivation for these universal correlators by inserting probe branes in the underlying effective geometry. We generalize these results to study correlators of branes and their universal behaviour in the Calabi-Yau crystals, where we find a role for a generalized brane insertion.
Atiyah, Michael; Dijkgraaf, Robbert; Hitchin, Nigel
2010-01-01
We review the remarkably fruitful interactions between mathematics and quantum physics in the past decades, pointing out some general trends and highlighting several examples, such as the counting of curves in algebraic geometry, invariants of knots and four-dimensional topology. PMID:20123740
Sliding vane geometry turbines
Sun, Harold Huimin; Zhang, Jizhong; Hu, Liangjun; Hanna, Dave R
2014-12-30
Various systems and methods are described for a variable geometry turbine. In one example, a turbine nozzle comprises a central axis and a nozzle vane. The nozzle vane includes a stationary vane and a sliding vane. The sliding vane is positioned to slide in a direction substantially tangent to an inner circumference of the turbine nozzle and in contact with the stationary vane.
Diophantine geometry an introduction
Hindry, Marc
2000-01-01
This is an introduction to diophantine geometry at the advanced graduate level. The book contains a proof of the Mordell conjecture which will make it quite attractive to graduate students and professional mathematicians. In each part of the book, the reader will find numerous exercises.
Towards relativistic quantum geometry
Energy Technology Data Exchange (ETDEWEB)
Ridao, Luis Santiago [Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR), Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Mar del Plata (Argentina); Bellini, Mauricio, E-mail: mbellini@mdp.edu.ar [Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350, C.P. 7600, Mar del Plata (Argentina); Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR), Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Mar del Plata (Argentina)
2015-12-17
We obtain a gauge-invariant relativistic quantum geometry by using a Weylian-like manifold with a geometric scalar field which provides a gauge-invariant relativistic quantum theory in which the algebra of the Weylian-like field depends on observers. An example for a Reissner–Nordström black-hole is studied.
Advanced geometries and regimes
Energy Technology Data Exchange (ETDEWEB)
Bulanov, S. S. [Univeristy of California, Berkeley, CA, 94720 (United States); Bulanov, S. V. [Kansai Photon Science Institute, JAEA, Kizugawa, Kyoto 619-0215 (Japan); Turchetti, G. [Dipartimento di Fisica, Università di Bologna and INFN Sezione di Bologna, Via Irnerio, 46-I-40126 Bologna (Italy); Limpouch, J.; Klimo, O.; Psikal, J. [Institute of Physics of the ASCR, ELI-Beamlines/HiLASE project, Na Slovance 2, 18221 Prague, Czech Republic and Czech Technical University in Prague, FNSPE, Brehova 7, 115 19 Prague (Czech Republic); Antici, P. [Dipartimento di Energetica ed INFM, Università di Roma, La Sapienza, 00165 Roma (Italy); Margarone, D.; Korn, G. [Institute of Physics of the ASCR, ELI-Beamlines/HiLASE project, Na Slovance 2, 18221 Prague (Czech Republic)
2013-07-26
We review and discuss different schemes of laser ion acceleration as well as advanced target geometries in connection with the development of the laser-driven proton source for hadron therapy of oncological diseases, which is a part of the ELIMED project.
Emergent Hyperbolic Network Geometry
Bianconi, Ginestra; Rahmede, Christoph
2017-02-01
A large variety of interacting complex systems are characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links, triangles, tetrahedra etc.) that have a natural geometric interpretation. As such simplicial complexes are widely used in quantum gravity approaches that involve a discretization of spacetime. Here, by extending our knowledge of growing complex networks to growing simplicial complexes we investigate the nature of the emergent geometry of complex networks and explore whether this geometry is hyperbolic. Specifically we show that an hyperbolic network geometry emerges spontaneously from models of growing simplicial complexes that are purely combinatorial. The statistical and geometrical properties of the growing simplicial complexes strongly depend on their dimensionality and display the major universal properties of real complex networks (scale-free degree distribution, small-world and communities) at the same time. Interestingly, when the network dynamics includes an heterogeneous fitness of the faces, the growing simplicial complex can undergo phase transitions that are reflected by relevant changes in the network geometry.
Spacetime and Euclidean Geometry
Brill, D R; Brill, Dieter; Jacobson, Ted
2004-01-01
Using only the principle of relativity and Euclidean geometry we show in this pedagogical article that the square of proper time or length in a two-dimensional spacetime diagram is proportional to the Euclidean area of the corresponding causal domain. We use this relation to derive the Minkowski line element by two geometric proofs of the "spacetime Pythagoras theorem".
Spacetime and Euclidean geometry
Brill, Dieter; Jacobson, Ted
2006-04-01
Using only the principle of relativity and Euclidean geometry we show in this pedagogical article that the square of proper time or length in a two-dimensional spacetime diagram is proportional to the Euclidean area of the corresponding causal domain. We use this relation to derive the Minkowski line element by two geometric proofs of the spacetime Pythagoras theorem.
DEFF Research Database (Denmark)
Booss-Bavnbek, Bernhelm
2011-01-01
This paper applies I.M. Gelfand's distinction between adequate and non-adequate use of mathematical language in different contexts to the newly opened window of model-based measurements of intracellular dynamics. The specifics of geometry and dynamics on the mesoscale of cell physiology are elabo...
Wares, Arsalan; Elstak, Iwan
2017-01-01
The purpose of this paper is to describe the mathematics that emanates from the construction of an origami box. We first construct a simple origami box from a rectangular sheet and then discuss some of the mathematical questions that arise in the context of geometry and algebra. The activity can be used as a context for illustrating how algebra…
Boyer, Carl B
2012-01-01
Designed as an integrated survey of the development of analytic geometry, this study presents the concepts and contributions from before the Alexandrian Age through the eras of the great French mathematicians Fermat and Descartes, and on through Newton and Euler to the "Golden Age," from 1789 to 1850.
Wares, Arsalan; Elstak, Iwan
2017-01-01
The purpose of this paper is to describe the mathematics that emanates from the construction of an origami box. We first construct a simple origami box from a rectangular sheet and then discuss some of the mathematical questions that arise in the context of geometry and algebra. The activity can be used as a context for illustrating how algebra…
Barkin, Yu. V.; Ferrandiz, J. M.
2009-04-01
theory of Mercury librations in longitude by using three characteristics of Mercury rotation determined in the paper [3]. Two from these parameters are values of angle of librations in longitude and angular velocity in moment of passage of perihelion of Mercury orbit on 17 April 2002: (^g)0 = 0007 ± 0001, (^?? )0 = (2.10± 0.06)? ars/d. Third parameter determined in [3] is a dynamical coefficient: K = (B -A)(4Cm ) = (5.08± 0.30) × 10-5. B > A are principal moment of inertia, corresponding to equatorial axes of inertia; Cm is a polar moment of inertia of the mantle of Mercury. 1 Analytical theory of plane Mercury librations. This theory describes forced and free librations of Mercury in longitude in the frame of plane problem about resonant librations of Mercury considered or as non-spherical rigid body, or as system of rigid non-spherical mantle and liquid ellipsoidal core. Saving the main terms for the perturbations of angle of librations ^g and angular velocity ^? in both mentioned cases we will have formulae [6]: ^g = K(E sin M + E sin2M + E sin 3M + E sin4M + E sin5M ) 1 2 3 4 5+K0 sin(E KM- - φ) (A)
An introduction to Minkowski geometries
Farnsworth, David L.
2016-07-01
The fundamental ideas of Minkowski geometries are presented. Learning about Minkowski geometries can sharpen our students' understanding of concepts such as distance measurement. Many of its ideas are important and accessible to undergraduate students. Following a brief overview, distance and orthogonality in Minkowski geometries are thoroughly discussed and many illustrative examples and applications are supplied. Suggestions for further study of these geometries are given. Indeed, Minkowski geometries are an excellent source of topics for undergraduate research and independent study.
Gravitational Couplings for Gop-Planes and y-Op-Planes
Ospina-Giraldo, J F
2000-01-01
The Wess-Zumino actions for generalized orientifold planes (GOp-planes) and y-deformed orientifold planes (yOp-planes) are presented and two series power expantions are realized from whiches processes that involves GOp-planes,yOp-planes, RR-forms, gravitons and gaugeons, are obtained. Finally non-standard GOp-planes and y-Op-planes are showed.
Reeve, W. D., Ed.
There are a number of recurring topics in the articles that comprise this yearbook, such as the nature of both informal and demonstrative geometry, the reasons for teaching both, and the extent of such courses. Other emphasized topics are use of the analytic method, whether to combine plane and solid geometry, the place of algebra and trigonometry…
Detecting Translation Errors in CAD Surfaces and Preparing Geometries for Mesh Generation
Energy Technology Data Exchange (ETDEWEB)
Petersson, N Anders; Chand, K K
2001-08-27
The authors have developed tools for the efficient preparation of CAD geometries for mesh generation. Geometries are read from IGES files and then maintained in a boundary-representation consisting of a patchwork of trimmed and untrimmed surfaces. Gross errors in the geometry can be identified and removed automatically while a user interface is provided for manipulating the geometry (such as correcting invalid trimming curves or removing unwanted details). Modifying the geometry by adding or deleting surfaces and/or sectioning it by arbitrary planes (e.g. symmetry planes) is also supported. These tools are used for robust and accurate geometry models for initial mesh generation and will be applied to in situ mesh generation requirements of moving and adaptive grid simulations.
Evolutes of Hyperbolic Plane Curves
Institute of Scientific and Technical Information of China (English)
Shyuichi IZUMIYA; Dong He PEI; Takashi SANO; Erika TORII
2004-01-01
We define the notion of evolutes of curves in a hyperbolic plane and establish the relationships between singularities of these subjects and geometric invariants of curves under the action of the Lorentz group. We also describe how we can draw the picture of an evolute of a hyperbolic plane curve in the Poincar(e) disk.
Conceptual Design of Wave Plane
DEFF Research Database (Denmark)
Frigaard, Peter; Trewers, Andrew; Kofoed, Jens Peter;
The Wave Plane is a patented Wave Energy device of the overtopping type, designed to capture potential as well as kinetic energy. This is as such different to other overtopping devices, who usually only focus on potential energy. If Wave Plane A/S can deliver the turbine technology to utilize both...
3D blob dynamics in toroidal geometry
DEFF Research Database (Denmark)
Nielsen, Anders Henry; Reiser, Dirk
. The DIESEL code is an extension of the ESEL code [1]. It solves a simple interchange model in full 3D tokamak geometry, where the toroidal direction is divided into a number of drift planes. On each drift plane the equations are solved in a domain corresponding to the full 2D cross section of the tokamak......In this paper we study the simple case of the dynamics of a density perturbation localized in the edge region of a medium sized tokamak in a full 3D geometry. The 2D evolution of such a perturbation has been studied in details on the low-field side, where the gradient of the magnetic field always...... point radial inward, see e.g. [1-2]. Here, the initial condition is implemented in two very different 3D numerical codes, ATTEMPT [3], and a new developed code, DIESEL (Disk version of ESEL), and the results are compared and discussed in detail. The ATTEMPT code has been employed to study the blob...
3D plane-wave least-squares Kirchhoff migration
Wang, Xin
2014-08-05
A three dimensional least-squares Kirchhoff migration (LSM) is developed in the prestack plane-wave domain to increase the quality of migration images and the computational efficiency. Due to the limitation of current 3D marine acquisition geometries, a cylindrical-wave encoding is adopted for the narrow azimuth streamer data. To account for the mispositioning of reflectors due to errors in the velocity model, a regularized LSM is devised so that each plane-wave or cylindrical-wave gather gives rise to an individual migration image, and a regularization term is included to encourage the similarities between the migration images of similar encoding schemes. Both synthetic and field results show that: 1) plane-wave or cylindrical-wave encoding LSM can achieve both computational and IO saving, compared to shot-domain LSM, however, plane-wave LSM is still about 5 times more expensive than plane-wave migration; 2) the regularized LSM is more robust compared to LSM with one reflectivity model common for all the plane-wave or cylindrical-wave gathers.
Cylindrical geometry hall thruster
Raitses, Yevgeny; Fisch, Nathaniel J.
2002-01-01
An apparatus and method for thrusting plasma, utilizing a Hall thruster with a cylindrical geometry, wherein ions are accelerated in substantially the axial direction. The apparatus is suitable for operation at low power. It employs small size thruster components, including a ceramic channel, with the center pole piece of the conventional annular design thruster eliminated or greatly reduced. Efficient operation is accomplished through magnetic fields with a substantial radial component. The propellant gas is ionized at an optimal location in the thruster. A further improvement is accomplished by segmented electrodes, which produce localized voltage drops within the thruster at optimally prescribed locations. The apparatus differs from a conventional Hall thruster, which has an annular geometry, not well suited to scaling to small size, because the small size for an annular design has a great deal of surface area relative to the volume.
Lee, Jeongseog; Safdi, Benjamin R
2014-01-01
Entanglement entropy in even dimensional conformal field theories (CFTs) contains well-known universal terms arising from the conformal anomaly. Renyi entropies are natural generalizations of the entanglement entropy that are much less understood. Above two spacetime dimensions, the universal terms in the Renyi entropies are unknown for general entangling geometries. We conjecture a new structure in the dependence of the four-dimensional Renyi entropies on the intrinsic and extrinsic geometry of the entangling surface. We provide evidence for this conjecture by direct numerical computations in the free scalar and fermion field theories. The computation involves relating the four-dimensional free massless Renyi entropies across cylindrical entangling surfaces to corresponding three-dimensional massive Renyi entropies across circular entangling surfaces. Our numerical technique also allows us to directly probe other interesting aspects of three-dimensional Renyi entropy, including the massless renormalized Reny...
Spectral Geometry and Causality
Kopf, T
1996-01-01
For a physical interpretation of a theory of quantum gravity, it is necessary to recover classical spacetime, at least approximately. However, quantum gravity may eventually provide classical spacetimes by giving spectral data similar to those appearing in noncommutative geometry, rather than by giving directly a spacetime manifold. It is shown that a globally hyperbolic Lorentzian manifold can be given by spectral data. A new phenomenon in the context of spectral geometry is observed: causal relationships. The employment of the causal relationships of spectral data is shown to lead to a highly efficient description of Lorentzian manifolds, indicating the possible usefulness of this approach. Connections to free quantum field theory are discussed for both motivation and physical interpretation. It is conjectured that the necessary spectral data can be generically obtained from an effective field theory having the fundamental structures of generalized quantum mechanics: a decoherence functional and a choice of...
Multivariate calculus and geometry
Dineen, Seán
2014-01-01
Multivariate calculus can be understood best by combining geometric insight, intuitive arguments, detailed explanations and mathematical reasoning. This textbook has successfully followed this programme. It additionally provides a solid description of the basic concepts, via familiar examples, which are then tested in technically demanding situations. In this new edition the introductory chapter and two of the chapters on the geometry of surfaces have been revised. Some exercises have been replaced and others provided with expanded solutions. Familiarity with partial derivatives and a course in linear algebra are essential prerequisites for readers of this book. Multivariate Calculus and Geometry is aimed primarily at higher level undergraduates in the mathematical sciences. The inclusion of many practical examples involving problems of several variables will appeal to mathematics, science and engineering students.
Algebra, Arithmetic, and Geometry
Tschinkel, Yuri
2009-01-01
The two volumes of "Algebra, Arithmetic, and Geometry: In Honor of Y.I. Manin" are composed of invited expository articles and extensions detailing Manin's contributions to the subjects, and are in celebration of his 70th birthday. The well-respected and distinguished contributors include: Behrend, Berkovich, Bost, Bressler, Calaque, Carlson, Chambert-Loir, Colombo, Connes, Consani, Dabrowski, Deninger, Dolgachev, Donaldson, Ekedahl, Elsenhans, Enriques, Etingof, Fock, Friedlander, Geemen, Getzler, Goncharov, Harris, Iskovskikh, Jahnel, Kaledin, Kapranov, Katz, Kaufmann, Kollar, Kont
Bengtsson, Ingemar; Zyczkowski, Karol
2007-12-01
Preface; 1. Convexity, colours and statistics; 2. Geometry of probability distributions; 3. Much ado about spheres; 4. Complex projective spaces; 5. Outline of quantum mechanics; 6. Coherent states and group actions; 7. The stellar representation; 8. The space of density matrices; 9. Purification of mixed quantum states; 10. Quantum operations; 11. Duality: maps versus states; 12. Density matrices and entropies; 13. Distinguishability measures; 14. Monotone metrics and measures; 15. Quantum entanglement; Epilogue; Appendices; References; Index.
Differential geometry and thermodynamics
Quevedo, H
2003-01-01
In this work we present the first steps of a new approach to the study of thermodynamics in the context of differential geometry. We introduce a fundamental differential 1-form and a metric on a pseudo-Euclidean manifold coordinatized by means of the extensive thermodynamic variables. The study of the connection and the curvature of these objects is initialized in this work by using Cartan structure equations. (Author)
Krauss, L M; Krauss, Lawrence M.; Turner, Michael S.
1999-01-01
The recognition that the cosmological constant may be non-zero forces us to re-evaluate standard notions about the connection between geometry and the fate of our Universe. An open Universe can recollapse, and a closed Universe can expand forever. As a corollary, we point out that there is no set of cosmological observations we can perform that will unambiguously allow us to determine what the ultimate destiny of the Universe will be.
Inflation from quantum geometry.
Bojowald, Martin
2002-12-23
Quantum geometry predicts that a universe evolves through an inflationary phase at small volume before exiting gracefully into a standard Friedmann phase. This does not require the introduction of additional matter fields with ad hoc potentials; rather, it occurs because of a quantum gravity modification of the kinetic part of ordinary matter Hamiltonians. An application of the same mechanism can explain why the present day cosmological acceleration is so tiny.
DEFF Research Database (Denmark)
Tamke, Martin; Ramsgaard Thomsen, Mette; Riiber Nielsen, Jacob
2009-01-01
The versatility of wood constructions and traditional wood joints for the production of non standard elements was in focus of a design based research. Herein we established a seamless process from digital design to fabrication. A first research phase centered on the development of a robust parame...... parametric model and a generic design language a later explored the possibilities to construct complex shaped geometries with self registering joints on modern wood crafting machines. The research was carried out as collaboration with industrial partners....
Integral geometry and holography
Czech, Bartłomiej; Lamprou, Lampros; McCandlish, Samuel; Sully, James
2015-10-01
We present a mathematical framework which underlies the connection between information theory and the bulk spacetime in the AdS3/CFT2 correspondence. A key concept is kinematic space: an auxiliary Lorentzian geometry whose metric is defined in terms of conditional mutual informations and which organizes the entanglement pattern of a CFT state. When the field theory has a holographic dual obeying the Ryu-Takayanagi proposal, kinematic space has a direct geometric meaning: it is the space of bulk geodesics studied in integral geometry. Lengths of bulk curves are computed by kinematic volumes, giving a precise entropic interpretation of the length of any bulk curve. We explain how basic geometric concepts — points, distances and angles — are reflected in kinematic space, allowing one to reconstruct a large class of spatial bulk geometries from boundary entanglement entropies. In this way, kinematic space translates between information theoretic and geometric descriptions of a CFT state. As an example, we discuss in detail the static slice of AdS3 whose kinematic space is two-dimensional de Sitter space.
Hilbert, completeness and geometry
Directory of Open Access Journals (Sweden)
Giorgio Venturi
2011-11-01
Full Text Available This paper aims to show how the mathematical content of Hilbert's Axiom of Completeness consists in an attempt to solve the more general problem of the relationship between intuition and formalization. Hilbert found the accordance between these two sides of mathematical knowledge at a logical level, clarifying the necessary and sufficient conditions for a good formalization of geometry. We will tackle the problem of what is, for Hilbert, the definition of geometry. The solution of this problem will bring out how Hilbert's conception of mathematics is not as innovative as his conception of the axiomatic method. The role that the demonstrative tools play in Hilbert's foundational reflections will also drive us to deal with the problem of the purity of methods, explicitly addressed by Hilbert. In this respect Hilbert's position is very innovative and deeply linked to his modern conception of the axiomatic method. In the end we will show that the role played by the Axiom of Completeness for geometry is the same as the Axiom of Induction for arithmetic and of Church-Turing thesis for computability theory. We end this paper arguing that set theory is the right context in which applying the axiomatic method to mathematics and we postpone to a sequel of this work the attempt to offer a solution similar to Hilbert's for the completeness of set theory.
Integral geometry and valuations
Solanes, Gil
2014-01-01
Valuations are finitely additive functionals on the space of convex bodies. Their study has become a central subject in convexity theory, with fundamental applications to integral geometry. In the last years there has been significant progress in the theory of valuations, which in turn has led to important achievements in integral geometry. This book originated from two courses delivered by the authors at the CRM and provides a self-contained introduction to these topics, covering most of the recent advances. The first part, by Semyon Alesker, is devoted to the theory of convex valuations, with emphasis on the latest developments. A special focus is put on the new fundamental structures of the space of valuations discovered after Alesker's irreducibility theorem. Moreover, the author describes the newly developed theory of valuations on manifolds. In the second part, Joseph H. G. Fu gives a modern introduction to integral geometry in the sense of Blaschke and Santaló, based on the notions and tools presented...
Generalised Geometry and Flux Vacua
Larfors, Magdalena
2015-01-01
This note discusses the connection between generalised geometry and flux compactifications of string theory. Firstly, we explain in a pedestrian manner how the supersymmetry constraints of type II ${\\mathcal{N}}=1$ flux compactifications can be restated as integrability constraints on certain generalised complex structures. This reformulation uses generalised complex geometry, a mathematical framework that geometrizes the B-field. Secondly, we discuss how exceptional generalised geometry may provide a similar geometrization of the RR fields. Thirdly, we examine the connection between generalised geometry and non-geometry, and finally we present recent developments where generalised geometry is used to construct explicit examples of flux compactifications to flat space.
Geometry of solar coronal rays
Filippov, B. P.; Martsenyuk, O. V.; Platov, Yu. V.; Den, O. E.
2016-02-01
Coronal helmet streamers are the most prominent large-scale elements of the solar corona observed in white light during total solar eclipses. The base of the streamer is an arcade of loops located above a global polarity inversion line. At an altitude of 1-2 solar radii above the limb, the apices of the arches sharpen, forming cusp structures, above which narrow coronal rays are observed. Lyot coronagraphs, especially those on-board spacecrafts flying beyond the Earth's atmosphere, enable us to observe the corona continuously and at large distances. At distances of several solar radii, the streamers take the form of fairly narrow spokes that diverge radially from the Sun. This radial direction displays a continuous expansion of the corona into the surrounding space, and the formation of the solar wind. However, the solar magnetic field and solar rotation complicate the situation. The rotation curves radial streams into spiral ones, similar to water streams flowing from rotating tubes. The influence of the magnetic field is more complex and multifarious. A thorough study of coronal ray geometries shows that rays are frequently not radial and not straight. Coronal streamers frequently display a curvature whose direction in the meridional plane depends on the phase of the solar cycle. It is evident that this curvature is related to the geometry of the global solar magnetic field, which depends on the cycle phase. Equatorward deviations of coronal streamers at solar minima and poleward deviations at solar maxima can be interpreted as the effects of changes in the general topology of the global solar magnetic field. There are sporadic temporal changes in the coronal rays shape caused by remote coronal mass ejections (CMEs) propagating through the corona. This is also a manifestation of the influence of the magnetic field on plasma flows. The motion of a large-scale flux rope associated with a CME away from the Sun creates changes in the structure of surrounding field
Foundation of Euclidean and non-Euclidean geometries according to F. Klein
Redei, L; Stark, M
1968-01-01
Foundation of Euclidean and Non-Euclidean Geometries according to F. Klein aims to remedy the deficiency in geometry so that the ideas of F. Klein obtain the place they merit in the literature of mathematics.This book discusses the axioms of betweenness, lattice of linear subspaces, generalization of the notion of space, and coplanar Desargues configurations. The central collineations of the plane, fundamental theorem of projective geometry, and lines perpendicular to a proper plane are also elaborated. This text likewise covers the axioms of motion, basic projective configurations, properties
Introductory non-Euclidean geometry
Manning, Henry Parker
1963-01-01
This fine and versatile introduction begins with the theorems common to Euclidean and non-Euclidean geometry, and then it addresses the specific differences that constitute elliptic and hyperbolic geometry. 1901 edition.
Geometry for the Secondary School
Moalem, D.
1977-01-01
A sequential but non-axiomatic high school geometry course which includes Euclidean, transformation, and analytic geometry and vectors and matrices, and emphasizes the invariance property of transformations, is outlined. Sample problems, solutions, and comments are included. (MN)
Linear connections on matrix geometries
Madore, J; Mourad, J; Madore, John; Masson, Thierry; Mourad, Jihad
1994-01-01
A general definition of a linear connection in noncommutative geometry has been recently proposed. Two examples are given of linear connections in noncommutative geometries which are based on matrix algebras. They both possess a unique metric connection.
Editors, LearningExpress
2010-01-01
Whether you're new to geometry or just looking for a refresher, this completely revised and updated third edition of Geometry Success in 20 Minutes a Day offers a 20-step lesson plan that provides quick and thorough instruction in practical, critical skills. Stripped of unnecessary math jargon but bursting with geometry essentials, Geometry Success in 20 Minutes a Day is an invaluable resource for both students and adults.
Teaching of Geometry in Bulgaria
Bankov, Kiril
2013-01-01
Geometry plays an important role in the school mathematics curriculum all around the world. Teaching of geometry varies a lot (Hoyls, Foxman, & Kuchemann, 2001). Many countries revise the objectives, the content, and the approaches to the geometry in school. Studies of the processes show that there are not common trends of these changes…
2015-01-01
This stimulating volume offers a broad collection of the principles of geometry and trigonometry and contains colorful diagrams to bring mathematical principles to life. Subjects are enriched by references to famous mathematicians and their ideas, and the stories are presented in a very comprehensible way. Readers investigate the relationships of points, lines, surfaces, and solids. They study construction methods for drawing figures, a wealth of facts about these figures, and above all, methods to prove the facts. They learn about triangle measure for circular motion, sine and cosine, tangent
Abhyankar, Shreeram Shankar
1964-01-01
This book provides, for use in a graduate course or for self-study by graduate students, a well-motivated treatment of several topics, especially the following: (1) algebraic treatment of several complex variables; (2) geometric approach to algebraic geometry via analytic sets; (3) survey of local algebra; (4) survey of sheaf theory. The book has been written in the spirit of Weierstrass. Power series play the dominant role. The treatment, being algebraic, is not restricted to complex numbers, but remains valid over any complete-valued field. This makes it applicable to situations arising from
Dooner, David B
2012-01-01
Building on the first edition published in 1995 this new edition of Kinematic Geometry of Gearing has been extensively revised and updated with new and original material. This includes the methodology for general tooth forms, radius of torsure', cylinder of osculation, and cylindroid of torsure; the author has also completely reworked the '3 laws of gearing', the first law re-written to better parallel the existing 'Law of Gearing" as pioneered by Leonard Euler, expanded from Euler's original law to encompass non-circular gears and hypoid gears, the 2nd law of gearing describing a unique relat
Flegg, H Graham
2001-01-01
This excellent introduction to topology eases first-year math students and general readers into the subject by surveying its concepts in a descriptive and intuitive way, attempting to build a bridge from the familiar concepts of geometry to the formalized study of topology. The first three chapters focus on congruence classes defined by transformations in real Euclidean space. As the number of permitted transformations increases, these classes become larger, and their common topological properties become intuitively clear. Chapters 4-12 give a largely intuitive presentation of selected topics.
Ionization coefficient approach to modeling breakdown in nonuniform geometries.
Energy Technology Data Exchange (ETDEWEB)
Warne, Larry Kevin; Jorgenson, Roy Eberhardt; Nicolaysen, Scott D.
2003-11-01
This report summarizes the work on breakdown modeling in nonuniform geometries by the ionization coefficient approach. Included are: (1) fits to primary and secondary ionization coefficients used in the modeling; (2) analytical test cases for sphere-to-sphere, wire-to-wire, corner, coaxial, and rod-to-plane geometries; a compilation of experimental data with source references; comparisons between code results, test case results, and experimental data. A simple criterion is proposed to differentiate between corona and spark. The effect of a dielectric surface on avalanche growth is examined by means of Monte Carlo simulations. The presence of a clean dry surface does not appear to enhance growth.
Topology, ergodic theory, real algebraic geometry Rokhlin's memorial
Turaev, V
2001-01-01
This book is dedicated to the memory of the outstanding Russian mathematician, V. A. Rokhlin (1919-1984). It is a collection of research papers written by his former students and followers, who are now experts in their fields. The topics in this volume include topology (the Morse-Novikov theory, spin bordisms in dimension 6, and skein modules of links), real algebraic geometry (real algebraic curves, plane algebraic surfaces, algebraic links, and complex orientations), dynamics (ergodicity, amenability, and random bundle transformations), geometry of Riemannian manifolds, theory of Teichmüller
Transmission through metallic chains: Role of distortions and contact geometry
Energy Technology Data Exchange (ETDEWEB)
Wunderlich, Thomas; Akgenc, Berna; Schuster, Cosima; Eckern, Ulrich [Institut fuer Physik, Universitaet Augsburg, 86135 (Germany)
2010-07-01
We present results of electronic structure and transport calculations for metallic chains, based on density functional theory and scattering theory combined with the the non-equilibrium Green's function technique. Starting from a simple model system of monovalent metallic chains we investigate the influence of distortions on the electronic structure and the transport properties of H and Li chains. Furthermore we calculate the electronic structure of Au chains which are contacted to leads via different geometries, and study the influence of the contact geometry on the transmission coefficient. In particular, we compare chains, pyramides and planes in the contact region. A comparison with analytical results is given.
Evaluation of a cone beam computed tomography geometry for image guided small animal irradiation.
Yang, Yidong; Armour, Michael; Wang, Ken Kang-Hsin; Gandhi, Nishant; Iordachita, Iulian; Siewerdsen, Jeffrey; Wong, John
2015-07-01
The conventional imaging geometry for small animal cone beam computed tomography (CBCT) is that a detector panel rotates around the head-to-tail axis of an imaged animal ('tubular' geometry). Another unusual but possible imaging geometry is that the detector panel rotates around the anterior-to-posterior axis of the animal ('pancake' geometry). The small animal radiation research platform developed at Johns Hopkins University employs the pancake geometry where a prone-positioned animal is rotated horizontally between an x-ray source and detector panel. This study is to assess the CBCT image quality in the pancake geometry and investigate potential methods for improvement. We compared CBCT images acquired in the pancake geometry with those acquired in the tubular geometry when the phantom/animal was placed upright simulating the conventional CBCT geometry. Results showed signal-to-noise and contrast-to-noise ratios in the pancake geometry were reduced in comparison to the tubular geometry at the same dose level. But the overall spatial resolution within the transverse plane of the imaged cylinder/animal was better in the pancake geometry. A modest exposure increase to two folds in the pancake geometry can improve image quality to a level close to the tubular geometry. Image quality can also be improved by inclining the animal, which reduces streak artifacts caused by bony structures. The major factor resulting in the inferior image quality in the pancake geometry is the elevated beam attenuation along the long axis of the phantom/animal and consequently increased scatter-to-primary ratio in that orientation. Not withstanding, the image quality in the pancake-geometry CBCT is adequate to support image guided animal positioning, while providing unique advantages of non-coplanar and multiple mice irradiation. This study also provides useful knowledge about the image quality in the two very different imaging geometries, i.e. pancake and tubular geometry, respectively.
Discrete Plane Segmentation and Estimation from a Point Cloud Using Local Geometric Patterns
Institute of Scientific and Technical Information of China (English)
Yukiko Kenmochi; Lilian Buzer; Akihiro Sugimoto; Ikuko Shimizu
2008-01-01
This paper presents a method for segmenting a 3D point cloud into planar surfaces using recently obtained discrete-geometry results. In discrete geometry, a discrete plane is defined as a set of grid points lying between two parallel planes with a small distance, called thickness. In contrast to the continuous case, there exist a finite number of local geometric patterns (LGPs) appearing on discrete planes. Moreover, such an LGP does not possess the unique normal vector but a set of normal vectors. By using those LGP properties, we first reject non-linear points from a point cloud, and then classify non-rejected points whose LGPs have common normal vectors into a planar-surface-point set. From each segmented point set, we also estimate the values of parameters of a discrete plane by minimizing its thickness.
Liu, Siqi; Macquart, J-P; Brisken, Walter; Deller, Adam
2015-01-01
Our analysis of archival VLBI data of PSR 0834+06 revealed that its scintillation properties can be precisely modelled using the inclined sheet model (Pen & Levin 2014), resulting in two distinct lens planes. These data strongly favour the grazing sheet model over turbulence as the primary source of pulsar scattering. This model can reproduce the parameters of the observed diffractive scintillation with an accuracy at the percent level. Comparison with new VLBI proper motion results in a direct measure of the ionized ISM screen transverse velocity. The results are consistent with ISM velocities local to the PSR 0834+06 sight-line (through the Galaxy). The simple 1D structure of the lenses opens up the possibility of using interstellar lenses as precision probes for pulsar lens mapping, precision transverse motions in the ISM, and new opportunities for removing scattering to improve pulsar timing. We describe the parameters and observables of this double screen system. While relative screen distances can i...
Byron, S.
1985-03-01
The low pressure gas-filled thyratron is scalable in the long dimension. Internally the tube is formed as a tetrode, with an auxiliary grid placed between the cathode and the control grid. A dc or pulsed power source drives the auxiliary grid both to insure uniform cathode emission and to provide a grid-cathode plasma prior to commutation. The high voltage holdoff structure consists of the anode, the control grid and its electrostatic shielding baffles, and a main quartz insulator. A small gas flow supply and exhaust system is used that eliminates the need for a hydrogen reservoir and permits other gases, such as helium, to be used. The thyratron provides a low inductance, high current, long lifetime switch configuration: useful for switch-on applications involving large scale lasers and other similar loads that are distributed in a linear geometry.
Critique of information geometry
Energy Technology Data Exchange (ETDEWEB)
Skilling, John, E-mail: skilling@eircom.net [Maximum Entropy Data Consultants Ltd, Kenmare (Ireland)
2014-12-05
As applied to probability, information geometry fails because probability distributions do not form a metric space. Probability theory rests on a compelling foundation of elementary symmetries, which also support information (aka minus entropy, Kullback-Leibler) H(p;q) as the unique measure of divergence from source probability distribution q to destination p. Because the only compatible connective H is from≠to asymmetric, H(p;q)≠H(q;p), there can be no compatible geometrical distance (which would necessarily be from=to symmetric). Hence there is no distance relationship compatible with the structure of probability theory. Metrics g and densities sqrt(det(g)) interpreted as prior probabilities follow from the definition of distance, and must fail likewise. Various metrics and corresponding priors have been proposed, Fisher's being the most popular, but all must behave unacceptably. This is illustrated with simple counter-examples.
Bochnak, Jacek; Roy, Marie-Françoise
1998-01-01
This book is a systematic treatment of real algebraic geometry, a subject that has strong interrelation with other areas of mathematics: singularity theory, differential topology, quadratic forms, commutative algebra, model theory, complexity theory etc. The careful and clearly written account covers both basic concepts and up-to-date research topics. It may be used as text for a graduate course. The present edition is a substantially revised and expanded English version of the book "Géometrie algébrique réelle" originally published in French, in 1987, as Volume 12 of ERGEBNISSE. Since the publication of the French version the theory has made advances in several directions. Many of these are included in this English version. Thus the English book may be regarded as a completely new treatment of the subject.
Covariant Macroscopic Quantum Geometry
Hogan, Craig J
2012-01-01
A covariant noncommutative algebra of position operators is presented, and interpreted as the macroscopic limit of a geometry that describes a collective quantum behavior of the positions of massive bodies in a flat emergent space-time. The commutator defines a quantum-geometrical relationship between world lines that depends on their separation and relative velocity, but on no other property of the bodies, and leads to a transverse uncertainty of the geometrical wave function that increases with separation. The number of geometrical degrees of freedom in a space-time volume scales holographically, as the surface area in Planck units. Ongoing branching of the wave function causes fluctuations in transverse position, shared coherently among bodies with similar trajectories. The theory can be tested using appropriately configured Michelson interferometers.
Advanced geometries and regimes
Bulanov, S. S.; Bulanov, S. V.; Turchetti, G.; Limpouch, J.; Klimo, O.; Psikal, J.; Stockem, A.; Fiuza, F.; Silva, L. O.; Antici, P.; Margarone, D.; Korn, G.
2013-08-01
We review and discuss different schemes of laser ion acceleration as well as advanced target geometries in connection with the development of the laser-driven proton source for hadron therapy of oncological diseases, which is a part of the ELIMED project. At the request of the Proceedings Editors and Dr. Stepan Bulanov, University of California, Berkeley, the above article has been updated to include three additional authors: A. Stockem, F. Fiuza, and L. O. Silva. All additional authors have consented to their name being added to the paper. Furthermore, the updated article PDF contains amendments to a number of references as detailed within the pages attached to the end of the updated article PDF file. The updated article was re-published on 8 August 2013.
Magnetism in curved geometries
Streubel, Robert; Fischer, Peter; Kronast, Florian; Kravchuk, Volodymyr P.; Sheka, Denis D.; Gaididei, Yuri; Schmidt, Oliver G.; Makarov, Denys
2016-09-01
Extending planar two-dimensional structures into the three-dimensional space has become a general trend in multiple disciplines, including electronics, photonics, plasmonics and magnetics. This approach provides means to modify conventional or to launch novel functionalities by tailoring the geometry of an object, e.g. its local curvature. In a generic electronic system, curvature results in the appearance of scalar and vector geometric potentials inducing anisotropic and chiral effects. In the specific case of magnetism, even in the simplest case of a curved anisotropic Heisenberg magnet, the curvilinear geometry manifests two exchange-driven interactions, namely effective anisotropy and antisymmetric exchange, i.e. Dzyaloshinskii-Moriya-like interaction. As a consequence, a family of novel curvature-driven effects emerges, which includes magnetochiral effects and topologically induced magnetization patterning, resulting in theoretically predicted unlimited domain wall velocities, chirality symmetry breaking and Cherenkov-like effects for magnons. The broad range of altered physical properties makes these curved architectures appealing in view of fundamental research on e.g. skyrmionic systems, magnonic crystals or exotic spin configurations. In addition to these rich physics, the application potential of three-dimensionally shaped objects is currently being explored as magnetic field sensorics for magnetofluidic applications, spin-wave filters, advanced magneto-encephalography devices for diagnosis of epilepsy or for energy-efficient racetrack memory devices. These recent developments ranging from theoretical predictions over fabrication of three-dimensionally curved magnetic thin films, hollow cylinders or wires, to their characterization using integral means as well as the development of advanced tomography approaches are in the focus of this review.
Klein geometries, parabolic geometries and differential equations of finite type
Abadoglu, Ender
2009-01-01
We define the infinitesimal and geometric orders of an effective Klein geometry G/H. Using these concepts, we prove i) For any integer m>1, there exists an effective Klein geometry G/H of infinitesimal order m such that G/H is a projective variety (Corollary 9). ii) An effective Klein geometry G/H of geometric order M defines a differential equation of order M+1 on G/H whose global solution space is G (Proposition 18).
Higher geometry an introduction to advanced methods in analytic geometry
Woods, Frederick S
2005-01-01
For students of mathematics with a sound background in analytic geometry and some knowledge of determinants, this volume has long been among the best available expositions of advanced work on projective and algebraic geometry. Developed from Professor Woods' lectures at the Massachusetts Institute of Technology, it bridges the gap between intermediate studies in the field and highly specialized works.With exceptional thoroughness, it presents the most important general concepts and methods of advanced algebraic geometry (as distinguished from differential geometry). It offers a thorough study
Cell division plane orientation based on tensile stress in Arabidopsis thaliana.
Louveaux, Marion; Julien, Jean-Daniel; Mirabet, Vincent; Boudaoud, Arezki; Hamant, Olivier
2016-07-26
Cell geometry has long been proposed to play a key role in the orientation of symmetric cell division planes. In particular, the recently proposed Besson-Dumais rule generalizes Errera's rule and predicts that cells divide along one of the local minima of plane area. However, this rule has been tested only on tissues with rather local spherical shape and homogeneous growth. Here, we tested the application of the Besson-Dumais rule to the divisions occurring in the Arabidopsis shoot apex, which contains domains with anisotropic curvature and differential growth. We found that the Besson-Dumais rule works well in the central part of the apex, but fails to account for cell division planes in the saddle-shaped boundary region. Because curvature anisotropy and differential growth prescribe directional tensile stress in that region, we tested the putative contribution of anisotropic stress fields to cell division plane orientation at the shoot apex. To do so, we compared two division rules: geometrical (new plane along the shortest path) and mechanical (new plane along maximal tension). The mechanical division rule reproduced the enrichment of long planes observed in the boundary region. Experimental perturbation of mechanical stress pattern further supported a contribution of anisotropic tensile stress in division plane orientation. Importantly, simulations of tissues growing in an isotropic stress field, and dividing along maximal tension, provided division plane distributions comparable to those obtained with the geometrical rule. We thus propose that division plane orientation by tensile stress offers a general rule for symmetric cell division in plants.
Numerical shadow and geometry of quantum states
Energy Technology Data Exchange (ETDEWEB)
Dunkl, Charles F [Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137 (United States); Gawron, Piotr; Miszczak, Jaroslaw A; Puchala, Zbigniew [Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, Baltycka 5, 44-100 Gliwice (Poland); Holbrook, John A [Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario N1G 2W1 (Canada); Zyczkowski, Karol, E-mail: cfd5z@virginia.edu, E-mail: gawron@iitis.pl, E-mail: jholbroo@uoguelph.ca, E-mail: miszczak@iitis.pl, E-mail: z.puchala@iitis.pl, E-mail: karol@tatry.if.uj.edu.pl [Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Krakow (Poland)
2011-08-19
The totality of normalized density matrices of dimension N forms a convex set Q{sub N} in R{sup N2-1}. Working with the flat geometry induced by the Hilbert-Schmidt distance, we consider images of orthogonal projections of Q{sub N} onto a two-plane and show that they are similar to the numerical ranges of matrices of dimension N. For a matrix A of dimension N, one defines its numerical shadow as a probability distribution supported on its numerical range W(A), induced by the unitarily invariant Fubini-Study measure on the complex projective manifold CP{sup N-1}. We define generalized, mixed-state shadows of A and demonstrate their usefulness to analyse the structure of the set of quantum states and unitary dynamics therein.
Numerical shadow and geometry of quantum states
Dunkl, Charles F; Holbrook, John A; Miszczak, Jarosław A; Puchała, Zbigniew; Życzkowski, Karol
2011-01-01
The totality of normalised density matrices of order N forms a convex set Q_N in R^(N^2-1). Working with the flat geometry induced by the Hilbert-Schmidt distance we consider images of orthogonal projections of Q_N onto a two-plane and show that they are similar to the numerical ranges of matrices of order N. For a matrix A of a order N one defines its numerical shadow as a probability distribution supported on its numerical range W(A), induced by the unitarily invariant Fubini-Study measure on the complex projective manifold CP^(N-1). We define generalized, mixed-states shadows of A and demonstrate their usefulness to analyse the structure of the set of quantum states and unitary dynamics therein.
Semidefinite geometry of the numerical range
Henrion, Didier
2008-01-01
The numerical range of a matrix is studied geometrically via the cone of positive semidefinite matrices (or semidefinite cone for short). In particular it is shown that the feasible set of a two-dimensional linear matrix inequality (LMI), an affine section of the semidefinite cone, is always dual to the numerical range of a matrix, which is therefore an affine projection of the semidefinite cone. Both primal and dual sets can also be viewed as convex hulls of explicit algebraic plane curve components. Several numerical examples illustrate this interplay between algebra, geometry and semidefinite programming duality. Finally, these techniques are used to revisit a theorem in statistics on the independence of quadratic forms in a normally distributed vector.
Semidefinite geometry of the numerical range
Henrion, Didier
2010-01-01
The numerical range of a matrix is studied geometrically via the cone of positive semidefinite matrices (or semidefinite cone for short). In particular it is shown that the feasible set of a two-dimensional linear matrix inequality (LMI), an affine section of the semidefinite cone, is always dual to the numerical range of a matrix, which is therefore an affine projection of the semidefinite cone. Both primal and dual sets can also be viewed as convex hulls of explicit algebraic plane curve components. Several numerical examples illustrate this interplay between algebra, geometry and semidefinite programming duality. Finally, these techniques are used to revisit a theorem in statistics on the independence of quadratic forms in a normally distributed vector.
Geometry and mechanics of thin growing bilayers.
Pezzulla, Matteo; Smith, Gabriel P; Nardinocchi, Paola; Holmes, Douglas P
2016-05-11
We investigate how thin sheets of arbitrary shapes morph under the isotropic in-plane expansion of their top surface, which may represent several stimuli such as nonuniform heating, local swelling and differential growth. Inspired by geometry, an analytical model is presented that rationalizes how the shape of the disk influences morphing, from the initial spherical bending to the final isometric limit. We introduce a new measure of slenderness that describes a sheet in terms of both thickness and plate shape. We find that the mean curvature of the isometric state is three fourths the natural curvature, which we verify by numerics and experiments. We finally investigate the emergence of a preferred direction of bending in the isometric state, guided by numerical analyses. The scalability of our model suggests that it is suitable to describe the morphing of sheets spanning several orders of magnitude.
An introduction to incidence geometry
De Bruyn, Bart
2016-01-01
This book gives an introduction to the field of Incidence Geometry by discussing the basic families of point-line geometries and introducing some of the mathematical techniques that are essential for their study. The families of geometries covered in this book include among others the generalized polygons, near polygons, polar spaces, dual polar spaces and designs. Also the various relationships between these geometries are investigated. Ovals and ovoids of projective spaces are studied and some applications to particular geometries will be given. A separate chapter introduces the necessary mathematical tools and techniques from graph theory. This chapter itself can be regarded as a self-contained introduction to strongly regular and distance-regular graphs. This book is essentially self-contained, only assuming the knowledge of basic notions from (linear) algebra and projective and affine geometry. Almost all theorems are accompanied with proofs and a list of exercises with full solutions is given at the end...
Guide to Computational Geometry Processing
DEFF Research Database (Denmark)
Bærentzen, Jakob Andreas; Gravesen, Jens; Anton, François;
be processed before it is useful. This Guide to Computational Geometry Processing reviews the algorithms for processing geometric data, with a practical focus on important techniques not covered by traditional courses on computer vision and computer graphics. This is balanced with an introduction......, metric space, affine spaces, differential geometry, and finite difference methods for derivatives and differential equations Reviews geometry representations, including polygonal meshes, splines, and subdivision surfaces Examines techniques for computing curvature from polygonal meshes Describes...
Linear algebra and projective geometry
Baer, Reinhold
2005-01-01
Geared toward upper-level undergraduates and graduate students, this text establishes that projective geometry and linear algebra are essentially identical. The supporting evidence consists of theorems offering an algebraic demonstration of certain geometric concepts. These focus on the representation of projective geometries by linear manifolds, of projectivities by semilinear transformations, of collineations by linear transformations, and of dualities by semilinear forms. These theorems lead to a reconstruction of the geometry that constituted the discussion's starting point, within algebra
Finding Proofs in Tarskian Geometry
Beeson, Michael; Wos, Larry
2016-01-01
We report on a project to use a theorem prover to find proofs of the theorems in Tarskian geometry. These theorems start with fundamental properties of betweenness, proceed through the derivations of several famous theorems due to Gupta and end with the derivation from Tarski's axioms of Hilbert's 1899 axioms for geometry. They include the four challenge problems left unsolved by Quaife, who two decades ago found some \\Otter proofs in Tarskian geometry (solving challenges issued in Wos's 1998...
Phase structures in fuzzy geometries
Govindarajan, T R; Gupta, K S; Martin, X
2012-01-01
We study phase structures of quantum field theories in fuzzy geometries. Several examples of fuzzy geometries as well as QFT's on such geometries are considered. They are fuzzy spheres and beyond as well as noncommutative deformations of BTZ blackholes. Analysis is done analytically and through simulations. Several features like novel stripe phases as well as spontaneous symmetry breaking avoiding Colemen, Mermin, Wagner theorem are brought out. Also we establish that these phases are stable due to topological obstructions.
Thermodynamics of Asymptotically Conical Geometries.
Cvetič, Mirjam; Gibbons, Gary W; Saleem, Zain H
2015-06-12
We study the thermodynamical properties of a class of asymptotically conical geometries known as "subtracted geometries." We derive the mass and angular momentum from the regulated Komar integral and the Hawking-Horowitz prescription and show that they are equivalent. By deriving the asymptotic charges, we show that the Smarr formula and the first law of thermodynamics hold. We also propose an analog of Christodulou-Ruffini inequality. The analysis can be generalized to other asymptotically conical geometries.
Planetary Image Geometry Library
Deen, Robert C.; Pariser, Oleg
2010-01-01
The Planetary Image Geometry (PIG) library is a multi-mission library used for projecting images (EDRs, or Experiment Data Records) and managing their geometry for in-situ missions. A collection of models describes cameras and their articulation, allowing application programs such as mosaickers, terrain generators, and pointing correction tools to be written in a multi-mission manner, without any knowledge of parameters specific to the supported missions. Camera model objects allow transformation of image coordinates to and from view vectors in XYZ space. Pointing models, specific to each mission, describe how to orient the camera models based on telemetry or other information. Surface models describe the surface in general terms. Coordinate system objects manage the various coordinate systems involved in most missions. File objects manage access to metadata (labels, including telemetry information) in the input EDRs and RDRs (Reduced Data Records). Label models manage metadata information in output files. Site objects keep track of different locations where the spacecraft might be at a given time. Radiometry models allow correction of radiometry for an image. Mission objects contain basic mission parameters. Pointing adjustment ("nav") files allow pointing to be corrected. The object-oriented structure (C++) makes it easy to subclass just the pieces of the library that are truly mission-specific. Typically, this involves just the pointing model and coordinate systems, and parts of the file model. Once the library was developed (initially for Mars Polar Lander, MPL), adding new missions ranged from two days to a few months, resulting in significant cost savings as compared to rewriting all the application programs for each mission. Currently supported missions include Mars Pathfinder (MPF), MPL, Mars Exploration Rover (MER), Phoenix, and Mars Science Lab (MSL). Applications based on this library create the majority of operational image RDRs for those missions. A
Digital geometry in image processing
Mukhopadhyay, Jayanta
2013-01-01
Exploring theories and applications developed during the last 30 years, Digital Geometry in Image Processing presents a mathematical treatment of the properties of digital metric spaces and their relevance in analyzing shapes in two and three dimensions. Unlike similar books, this one connects the two areas of image processing and digital geometry, highlighting important results of digital geometry that are currently used in image analysis and processing. The book discusses different digital geometries in multi-dimensional integral coordinate spaces. It also describes interesting properties of
Initiation to global Finslerian geometry
Akbar-Zadeh, Hassan
2006-01-01
After a brief description of the evolution of thinking on Finslerian geometry starting from Riemann, Finsler, Berwald and Elie Cartan, the book gives a clear and precise treatment of this geometry. The first three chapters develop the basic notions and methods, introduced by the author, to reach the global problems in Finslerian Geometry. The next five chapters are independent of each other, and deal with among others the geometry of generalized Einstein manifolds, the classification of Finslerian manifolds of constant sectional curvatures. They also give a treatment of isometric, affine, p
Affine Contractions on the Plane
Celik, D.; Ozdemir, Y.; Ureyen, M.
2007-01-01
Contractions play a considerable role in the theory of fractals. However, it is not easy to find contractions which are not similitudes. In this study, it is shown by counter examples that an affine transformation of the plane carrying a given triangle onto another triangle may not be a contraction even if it contracts edges, heights or medians.…
Distributed storage in the plane
Altman, Eitan; Avrachenkov, Konstatin; Goseling, Jasper
2013-01-01
We consider storage devices located in the plane according to a general point process and specialize the results for the homogeneous Poisson process. A large data file is stored at the storage devices, which have limited storage capabilities. Hence, they can only store parts of the data. Clients can
Distributed storage in the plane
Altman, Eitan; Avrachenkov, Konstatin; Goseling, Jasper
2014-01-01
We consider storage devices located in the plane according to a general point process and specialize the results for the homogeneous Poisson process. A large data file is stored at the storage devices, which have limited storage capabilities. Hence, they can only store parts of the data. Clients can
Plane and parabolic solar panels
Sales, J H O
2009-01-01
We present a plane and parabolic collector that absorbs radiant energy and transforms it in heat. Therefore we have a panel to heat water. We study how to increment this capture of solar beams onto the panel in order to increase its efficiency in heating water.
Distributed storage in the plane
Altman, Eitan; Avrachenkov, Konstatin; Goseling, Jasper
2013-01-01
We consider storage devices located in the plane according to a general point process and specialize the results for the homogeneous Poisson process. A large data file is stored at the storage devices, which have limited storage capabilities. Hence, they can only store parts of the data. Clients can
Distributed storage in the plane
Altman, Eitan; Avrachenkov, Konstatin; Goseling, Jasper
2014-01-01
We consider storage devices located in the plane according to a general point process and specialize the results for the homogeneous Poisson process. A large data file is stored at the storage devices, which have limited storage capabilities. Hence, they can only store parts of the data. Clients can
Seesaw geometry and leptogenesis
Di Bari, P
2005-01-01
The representation of the seesaw orthogonal matrix in the complex plane establishes a graphical correspondence between neutrino mass models and geometrical configurations, particularly useful to study relevant aspects of leptogenesis. We first derive the CP asymmetry bound for hierarchical heavy neutrinos and then an expression for the effective leptogenesis phase, determining the conditions for maximal phase and placing a lower bound on the phase suppression for generic models. Reconsidering the lower bounds on the lightest right-handed (RH) neutrino mass M_1 and on the reheating temperature T_{reh}, we find that models where the lightest neutrino mass m_1 is dominated by one of the two heavier right-handed (RH) neutrinos, typically arising from connections with quark masses, undergo both phase suppression and strong wash-out such that M_1 (T_{reh})\\gtrsim 1.5\\times 10^{11} (2x10^{10}) GeV. The window 10^9 GeV \\lesssim M_1,T_{reh}\\lesssim 10^{10} GeV is accessible only for a class of models where m_1 is domi...
Speziale, Simone
2013-01-01
We define and investigate a quantisation of null hypersurfaces in the context of loop quantum gravity on a fixed graph. The main tool we use is the parametrisation of the theory in terms of twistors, which has already proved useful in discussing the interpretation of spin networks as the quantization of twisted geometries. The classical formalism can be extended in a natural way to null hypersurfaces, with the Euclidean polyhedra replaced by null polyhedra with space-like faces, and SU(2) by the little group ISO(2). The main difference is that the simplicity constraints present in the formalims are all first class, and the symplectic reduction selects only the helicity subgroup of the little group. As a consequence, information on the shapes of the polyhedra is lost, and the result is a much simpler, abelian geometric picture. It can be described by an Euclidean singular structure on the 2-dimensional space-like surface defined by a foliation of space-time by null hypersurfaces. This geometric structure is na...
Bhatia, Rajendra
2013-01-01
This book is an outcome of the Indo-French Workshop on Matrix Information Geometries (MIG): Applications in Sensor and Cognitive Systems Engineering, which was held in Ecole Polytechnique and Thales Research and Technology Center, Palaiseau, France, in February 23-25, 2011. The workshop was generously funded by the Indo-French Centre for the Promotion of Advanced Research (IFCPAR). During the event, 22 renowned invited french or indian speakers gave lectures on their areas of expertise within the field of matrix analysis or processing. From these talks, a total of 17 original contribution or state-of-the-art chapters have been assembled in this volume. All articles were thoroughly peer-reviewed and improved, according to the suggestions of the international referees. The 17 contributions presented are organized in three parts: (1) State-of-the-art surveys & original matrix theory work, (2) Advanced matrix theory for radar processing, and (3) Matrix-based signal processing applications.
Functional integration over geometries
Mottola, E
1995-01-01
The geometric construction of the functional integral over coset spaces {\\cal M}/{\\cal G} is reviewed. The inner product on the cotangent space of infinitesimal deformations of \\cal M defines an invariant distance and volume form, or functional integration measure on the full configuration space. Then, by a simple change of coordinates parameterizing the gauge fiber \\cal G, the functional measure on the coset space {\\cal M}/{\\cal G} is deduced. This change of integration variables leads to a Jacobian which is entirely equivalent to the Faddeev-Popov determinant of the more traditional gauge fixed approach in non-abelian gauge theory. If the general construction is applied to the case where \\cal G is the group of coordinate reparametrizations of spacetime, the continuum functional integral over geometries, {\\it i.e.} metrics modulo coordinate reparameterizations may be defined. The invariant functional integration measure is used to derive the trace anomaly and effective action for the conformal part of the me...
Itin, Yakov
2007-01-01
The possible extensions of GR for description of fermions on a curved space, for supergravity and for loop quantum gravity require a richer set of 16 independent variables. These variables can be assembled in a coframe field, i.e., a local set of four linearly independent 1-forms. In the ordinary formulation, the coframe gravity does not have any connection to a specific geometry even being constructed from the geometrical meaningful objects. A geometrization of the coframe gravity is an aim of this paper. We construct a complete class of the coframe connections which are linear in the first order derivatives of the coframe field on an $n$ dimensional manifolds with and without a metric. The subclasses of the torsion-free, metric-compatible and flat connections are derived. We also study the behavior of the geometrical structures under local transformations of the coframe. The remarkable fact is an existence of a subclass of connections which are invariant when the infinitesimal transformations satisfy the Ma...
Directory of Open Access Journals (Sweden)
Šárka Nedomová
2013-01-01
Full Text Available Precise quantification of the profile of egg can provide a powerful tool for the analysis of egg shape for various biological problems. A new approach to the geometry of a Ostrich’s egg profile is presented here using an analysing the egg’s digital photo by edge detection techniques. The obtained points on the eggshell counter are fitted by the Fourier series. The obtained equations describing an egg profile have been used to calculate radii of curvature. The radii of the curvature at the important point of the egg profile (sharp end, blunt end and maximum thickness are independent on the egg shape index. The exact values of the egg surface and the egg volume have been obtained. These quantities are also independent on the egg shape index. These quantities can be successively estimated on the basis of simplified equations which are expressed in terms of the egg length, L¸ and its width, B. The surface area of the eggshells also exhibits good correlation with the egg long circumference length. Some limitations of the most used procedures have been also shown.
Geometries from field theories
Aoki, Sinya; Kikuchi, Kengo; Onogi, Tetsuya
2015-10-01
We propose a method to define a d+1-dimensional geometry from a d-dimensional quantum field theory in the 1/N expansion. We first construct a d+1-dimensional field theory from the d-dimensional one via the gradient-flow equation, whose flow time t represents the energy scale of the system such that trArr 0 corresponds to the ultraviolet and trArr infty to the infrared. We then define the induced metric from d+1-dimensional field operators. We show that the metric defined in this way becomes classical in the large-N limit, in the sense that quantum fluctuations of the metric are suppressed as 1/N due to the large-N factorization property. As a concrete example, we apply our method to the O(N) nonlinear σ model in two dimensions. We calculate the 3D induced metric, which is shown to describe an anti-de Sitter space in the massless limit. Finally, we discuss several open issues for future studies.
Optimum Stirling engine geometry
Energy Technology Data Exchange (ETDEWEB)
Senft, J.R. [University of Wisconsin, River Walls, WI (United States). Mathematics Dept.
2002-07-01
This paper combines the author's work on mechanical efficiency of reciprocating engines with the classic Schmidt thermodynamic model for Stirling engines and revisits the problem of identifying optimal engine geometry. All previous optimizations using the Schmidt theory focused on obtaining a maximal specific indicated cyclic work. This does not necessarily produce the highest shaft output. Indeed, some optima based upon indicated work would yield engines that cannot run at all due to excessive intrinsic mechanical losses. The analysis presented in this paper shows how to optimize for shaft or brake work output. Specifically, it presents solutions to the problem of finding the piston-to-displacer swept volume ratio and phase angle which will give the maximum brake output for a given total swept volume, given temperature extremes, a given mean operating pressure, and a given engine mechanism effectiveness. The paper covers the split-cylinder or gamma-type Stirling in detail, serving as a model for similar analysis of the other Stirling engine configurations. (author)
On quaternions and octonions their geometry, arithmetic, and symmetry
AUTHOR|(CDS)2067326
2003-01-01
This book investigates the geometry of quaternion and octonion algebras. Following a comprehensive historical introduction, the book illuminates the special properties of 3- and 4-dimensional Euclidean spaces using quaternions, leading to enumerations of the corresponding finite groups of symmetries. The second half of the book discusses the less familiar octonion algebra, concentrating on its remarkable "triality symmetry" after an appropriate study of Moufang loops. The authors also describe the arithmetics of the quaternions and octonions. The book concludes with a new theory of octonion factorization. Topics covered include the geometry of complex numbers, quaternions and 3-dimensional groups, quaternions and 4-dimensional groups, Hurwitz integral quaternions, composition algebras, Moufang loops, octonions and 8-dimensional geometry, integral octonions, and the octonion projective plane.
Interactions between Digital Geometry and Combinatorics on Words
Brlek, Srečko
2011-01-01
We review some recent results in digital geometry obtained by using a combinatorics on words approach to discrete geometry. Motivated on the one hand by the well-known theory of Sturmian words which model conveniently discrete lines in the plane, and on the other hand by the development of digital geometry, this study reveals strong links between the two fields. Discrete figures are identified with polyominoes encoded by words. The combinatorial tools lead to elegant descriptions of geometrical features and efficient algorithms. Among these, radix-trees are useful for efficiently detecting path intersection, Lyndon and Christoffel words appear as the main tools for describing digital convexity; equations on words allow to better understand tilings by translations.
Numerical Prediction of Induced Pressure and Lift of the Planing Surfaces
Institute of Scientific and Technical Information of China (English)
Hassan GHASSEMI; Ahmad Reza KOHANSAL; Mahmoud GHIASSI
2009-01-01
This paper discusses the numerical prediction of the induced pressure and lift of the planing surfaces in a steady motion based on the potential flow solver as well as the spray drag by use of the practical method.The numerical method for computation of the induced pressure and lift is potential-based boundary element method.Special technique is identified to present upwash geometry and to determine the spray drag.Numerical results of a planing flat plate and planing craft model 4666 are presented.It is shown that the method is robust and efficient and the results agree well with the experimental measurements with various Froude humors.
Gravitational Couplings for y-Gop-Planes
Ospina-Giraldo, J F
2000-01-01
The Wess-Zumino action for y deformed and generalized orientifold planes (yGOp-planes) is presented and one power expantion is realized from which processes that involves yGOp-planes, RR-forms, gravitons and gaugeons, are obtained. Finally non-standard yGOp-planes are showed.
Linear Instability of the Plane Couette and Plane Poiseuille Flows
Chefranov, Sergey G
2015-01-01
We show possibility of the Plane Couette (PC) flow instability for Reynolds number Re>Reth=140. This new result of the linear hydrodynamic stability theory is obtained on the base of refusal from the traditionally used assumption on longitudinal periodicity of the disturbances along the direction of the fluid flow. We found that earlier existing understanding on the linear stability of this flow for any arbitrary large Reynolds number is directly related with an assumption on the separation of the variables of the spatial variability for the disturbance field and their periodicity in linear theory of stability. By the refusal from the pointed assumptions also for the Plane Poiseuille (PP) flow, we get a new threshold Reynolds value Reth=1040 that with 4% accuracy agrees with the experiment contrary to more than 500% discrepancy for the earlier known estimate Reth=5772 obtained in the frame of the linear theory but when using the "normal" disturbance form (S. A. Orszag, 1971).
Interactive visualizations of blowups of the plane.
Schenzel, Peter; Stussak, Christian
2013-06-01
Blowups are an important technique in algebraic geometry that permit the smoothing of singular algebraic varieties. It is a challenge to visualize this process even in the case of blowups of points X in the affine plane AA(IR)(2). First results were obtained by Brodmann with the aid of the so-called toroidal blowup, a compact embedding of the blowup into affine 3-space. In fact, Brodmann provides a rational parametrization of the toroidal blowup, but its visualization fails in the neighborhood of X because the parametrization tends to indefinite terms of the form 0/0. Our approach is based on implicitization of the parametric form. By methods from commutative algebra we are able to reduce the implicitization to the computation of a single, fairly simple resultant. This provides an algebraic equation of the implicit surface of the toroidal blowup including the so-called exceptional fiber associated with X. Surprisingly, the degree of the equation grows only linearly with the degree of the parametrization. By applying additional clipping techniques to the implicit surface we are able to visualize the toroidal blowup as well as its deformations by several parameters interactively in real time using GPU-based ray casting techniques. The methods of the paper provide insights in the structure of blowups of points, even if the points are interactively moved or tend to degenerations.
Reilingh, M.L.; Tuijthof, G.J.M.; Van Dijk, C.N.; Blankevoort, L.
2011-01-01
Background: Malalignment of the hindfoot can be corrected with a calcaneal osteotomy (CO). A well-selected osteotomy angle in the sagittal plane will reduce the shear force in the osteotomy plane while walking. The purpose was to determine the presence of a relationship between the foot geometry and
On the topology of real algebraic plane curves
DEFF Research Database (Denmark)
Cheng, Jinsan; Lazard, Sylvain; Peñaranda, Luis
2010-01-01
We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given...... coordinate system even if the curve is not in generic position. Previous methods based on the cylindrical algebraic decomposition use sub-resultant sequences and computations with polynomials with algebraic coefficients. A novelty of our approach is to replace these tools by Gröbner basis computations...... and isolation with rational univariate representations. This has the advantage of avoiding computations with polynomials with algebraic coefficients, even in non-generic positions. Our algorithm isolates critical points in boxes and computes a decomposition of the plane by rectangular boxes. This decomposition...
Gravitational scattering of zero-rest-mass plane waves
De Logi, W. K.; Kovacs, S. J., Jr.
1977-01-01
The Feyman-diagram technique is used to calculate the differential cross sections for the scattering of zero-rest-mass plane waves of spin 0, 1, and 2 by linearized Schwarzschild and Kerr geometries in the long-wavelength weak-field limit. It is found that the polarization of right (or left) circularly polarized electromagnetic waves is unaffected by the scattering process (i.e., helicity is conserved) and that the two helicity (polarization) states of the photon are scattered differently by the Kerr geometry. This coupling between the photon helicity and the angular momentum of the scatterer also leads to a partial polarization of unpolarized incident light. For gravitational waves, on the other hand, there is neither helicity conservation nor helicity-dependent scattering; the angular momentum of the scatterer has no polarizing effect on incident unpolarized gravitational waves.
Characteristics of stereo images from detectors in focal plane array.
Son, Jung-Young; Yeom, Seokwon; Chun, Joo-Hwan; Guschin, Vladmir P; Lee, Dong-Su
2011-07-01
The equivalent ray geometry of two horizontally aligned detectors at the focal plane of the main antenna in a millimeter wave imaging system is analyzed to reveal the reason why the images from the detectors are fused as an image with a depth sense. Scanning the main antenna in both horizontal and vertical directions makes each detector perform as a camera, and the two detectors can work like a stereo camera in the millimeter wave range. However, the stereo camera geometry is different from that of the stereo camera used in the visual spectral range because the detectors' viewing directions are diverging to each other and they are a certain distance apart. The depth sense is mainly induced by the distance between detectors. The images obtained from the detectors in the millimeter imaging system are perceived with a good depth sense. The disparities responsible for the depth sense are identified in the images.
Geometric structure of pseudo-plane quadratic flows
Sun, Che
2017-03-01
Quadratic flows have the unique property of uniform strain and are commonly used in turbulence modeling and hydrodynamic analysis. While previous applications focused on two-dimensional homogeneous fluid, this study examines the geometric structure of three-dimensional quadratic flows in stratified fluid by solving a steady-state pseudo-plane flow model. The complete set of exact solutions reveals that steady quadratic flows have an invariant conic type in the non-rotating frame and a non-rotatory vertical structure in the rotating frame. Three baroclinic solutions with vertically non-aligned formulation disprove an earlier conjecture. All elliptic and hyperbolic solutions, except for the inertial ones, exhibit vertical concentricity. The rich geometry of quadratic flows stands in contrast to the depleted geometry of high-degree polynomial flows. A paradox in the steady solutions of shallow-water reduced-gravity models is also explained.
Surrogate Modeling for Geometry Optimization
DEFF Research Database (Denmark)
Rojas Larrazabal, Marielba de la Caridad; Abraham, Yonas; Holzwarth, Natalie;
2009-01-01
A new approach for optimizing the nuclear geometry of an atomic system is described. Instead of the original expensive objective function (energy functional), a small number of simpler surrogates is used.......A new approach for optimizing the nuclear geometry of an atomic system is described. Instead of the original expensive objective function (energy functional), a small number of simpler surrogates is used....
On the number of singular points of plane curves
Orevkov, S Y
1995-01-01
This is an extended, renovated and updated report on a joint work which the second named author presented at the Conference on Algebraic Geometry held at Saitama University, 15-17 of March, 1995. The main result is an inequality for the numerical type of singularities of a plane curve, which involves the degree of the curve, the multiplicities and the Milnor numbers of its singular points. It is a corollary of the logarithmic Bogomolov-Miyaoka-Yau's type inequality due to Miyaoka. It was first proven by F. Sakai at 1990 and rediscovered by the authors independently in the particular case of an irreducible cuspidal curve at 1992. Our proof is based on the localization, the local Zariski--Fujita decomposition and uses a graph discriminant calculus. The key point is a local analog of the BMY-inequality for a plane curve germ. As a corollary, a boundedness criterium for a family of plane curves has been obtained. Another application of our methods is the following fact: a rigid rational cuspidal plane curve canno...
Leibniz operad on symplectic plane and cohomological vector fields
Uchino, K
2011-01-01
By using help of algebraic operad theory, Leibniz algebra theory and symplectic geometry are connected. We introduce the notion of cohomological vector field defined on nongraded symplectic plane. It will be proved that the cohomological vector fields induce the finite dimensional Leibniz algebras by the derived bracket construction. This proposition is a Leibniz analogue of the cohomological field theory in the category of Lie algebras. The basic properties of the cohomological fields will be studied, in particular, we discuss a factorization problem with the cohomological fields and introduce the notion of double-algebra in the category of Leibniz algebras.
Classification of flat slant surfaces in complex Euclidean plane
Chen, Bang-Yen
2002-01-01
It is well-known that the classification of flat surfaces in Euclidean 3space is one of the most basic results in differential geometry. For surfaces in the complex Euclidean plane $C^{2}$ endowed with almost complex structure $J$ , flat surfaces are the simplest ones from intrinsic point of views. On the other hand, from $J$ -action point of views, the most natural surfaces in $C^{2}$ are slant surfaces, i.e., surfaces with constant Wintinger angle. In this paper the author completely classi...
Shafarevich, Igor Rostislavovich
1994-01-01
Shafarevich Basic Algebraic Geometry 2 The second edition of Shafarevich's introduction to algebraic geometry is in two volumes. The second volume covers schemes and complex manifolds, generalisations in two different directions of the affine and projective varieties that form the material of the first volume. Two notable additions in this second edition are the section on moduli spaces and representable functors, motivated by a discussion of the Hilbert scheme, and the section on Kähler geometry. The book ends with a historical sketch discussing the origins of algebraic geometry. From the Zentralblatt review of this volume: "... one can only respectfully repeat what has been said about the first part of the book (...): a great textbook, written by one of the leading algebraic geometers and teachers himself, has been reworked and updated. As a result the author's standard textbook on algebraic geometry has become even more important and valuable. Students, teachers, and active researchers using methods of al...
Index Theorems on Torsional Geometries
Kimura, Tetsuji
2007-01-01
We study various topological invariants on a differential geometry in the presence of a totally anti-symmetric torsion H under the closed condition dH=0. By using the identification between the Clifford algebra on a geometry and the canonical quantization condition of fermion in the quantum mechanics, we construct the N=1 quantum mechanical sigma model in the Hamiltonian formalism and extend this model to N=2 system, equipped with the totally anti-symmetric tensor associated with the torsion on the target space geometry. Next we construct transition elements in the Lagrangian path integral formalism and apply them to the analyses of the Witten indices in supersymmetric systems. We improve the formulation of the Dirac index on the torsional geometry which has already been studied. We also formulate the Euler characteristic and the Hirzebruch signature on the torsional geometry.
Directory of Open Access Journals (Sweden)
Otto Bachmann
1984-01-01
Full Text Available In this paper we continue the study of projective planes which admit collineation groups of low rank (Kallaher [1] and Bachmann [2,3]. A rank 5 collineation group of a projective plane ℙ of order n≠3 is proved to be flag-transitive. As in the rank 3 and rank 4 case this implies that is ℙ not desarguesian and that n is (a prime power of the form m4 if m is odd and n=m2 with m≡0mod4 if n is even. Our proof relies on the classification of all doubly transitive groups of finite degree (which follows from the classification of all finite simple groups.
Hydrodynamics of planing monohull watercraft
Vorus, William S
2017-01-01
This book addresses the principles involved in the design and engineering of planing monohull power boats, with an emphasis on the theoretical fundamentals that readers need in order to be fully functional in marine design and engineering. Author William Vorus focuses on three topics: boat resistance, seaway response, and propulsion and explains the physical principles, mathematical details, and theoretical details that support physical understanding. In particular, he explains the approximations and simplifications in mathematics that lead to success in the applications of planing craft design engineering, and begins with the simplest configuration that embodies the basic physics. He leads readers, step-by-step, through the physical complications that occur, leading to a useful working knowledge of marine design and engineering. Included in the book are a wealth of examples that exemplify some of the most important naval architecture and marine engineering problems that challenge many of today’s engineers.
Forgács, Péter; Romańczukiewicz, Tomasz
2013-01-01
It is shown that in a large class of systems plane waves can act as tractor beams: i.e., an incident plane wave can exert a pulling force on the scatterer. The underlying physical mechanism for the pulling force is due to the sufficiently strong scattering of the incoming wave into another mode having a larger wave number, in which case excess momentum is created behind the scatterer. Such a tractor beam or negative radiation pressure effect arises naturally in systems where the coupling between the scattering channels is due to Aharonov-Bohm (AB) gauge potentials. It is demonstrated that this effect is also present if the AB potential is an induced, ("artificial") gauge potential such as the one found in J. March-Russell, J. Preskill, F. Wilczek, Phys. Rev. Lett. 58 2567 (1992).
Eight plane IPND mechanical testing.
Energy Technology Data Exchange (ETDEWEB)
Zhao, A.; Guarino, V.; Wood, K.; Nephew, T.; Ayres, D.; Lee, A.; High Energy Physics; FNAL
2008-03-18
A mechanical test of an 8 plane IPND mechanical prototype, which was constructed using extrusions from the testing/tryout of the 16 cell prototype extrusion die in Argonne National Laboratory, was conducted. There were 4 vertical and 4 horizontal planes in this 8 plane IPND prototype. Each vertical plane had four 16 cell extrusions, while each horizontal plane had six 16 cell extrusions. Each plane was glued together using the formulation of Devcon adhesive, Devcon 60. The vertical extrusions used in the vertical planes shares the same dimensions as the horizontal extrusions in the horizontal planes with the average web thickness of 2.1 mm and the average wall thickness of 3.1 mm. This mechanical prototype was constructed with end-seals on the both ends of the vertical extrusions. The gaps were filled with epoxy between extrusions and end-seals. The overall dimension of IPND is 154.8 by 103.1 by 21.7 inches with the weight of approximately 1200 kg, as shown in a figure. Two similar mechanical tests of 3 layer and 11 layer prototypes have been done in order to evaluate the strength of the adhesive joint between extrusions in the NOvA detector. The test showed that the IPND prototype was able to sustain under the loading of weight of itself and scintillator. Two FEA models were built to verify the measurement data from the test. The prediction from FEA slice model seems correlated reasonably well to the test result, even under a 'rough' estimated condition for the wall thickness (from an untuned die) and an unknown property of 'garage type' extrusion. A full size of FEA 3-D model also agrees very well with the test data from strain gage readings. It is worthy to point out that the stress distribution of the structure is predominantly determined by the internal pressure, while the buckling stability relies more on the loading weight from the extrusions themselves and scintillate. Results of conducted internal pressure tests, including 3- cell, 11
DEFF Research Database (Denmark)
Rathkjen, Arne
A state of plane stress is illustrated by means of two families of curves, each family representing constant values of a derivative of Airy's stress function. The two families of curves form a map giving in the first place an overall picture of regions of high and low stress, and in the second...... place, the map comprises a complete graphic representation of the stress at any point....
The Geometry Description Markup Language
Institute of Scientific and Technical Information of China (English)
RadovanChytracek
2001-01-01
Currently,a lot of effort is being put on designing complex detectors.A number of simulation and reconstruction frameworks and applications have been developed with the aim to make this job easier.A very important role in this activity is played by the geometry description of the detector apparatus layout and its working environment.However,no real common approach to represent geometry data is available and such data can be found in various forms starting from custom semi-structured text files,source code (C/C++/FORTRAN),to XML and database solutions.The XML(Extensible Markup Language)has proven to provide an interesting approach for describing detector geometries,with several different but incompatible XML-based solutions existing.Therefore,interoperability and geometry data exchange among different frameworks is not possible at present.This article introduces a markup language for geometry descriptions.Its aim is to define a common approach for sharing and exchanging of geometry description data.Its requirements and design have been driven by experience and user feedback from existing projects which have their geometry description in XML.
Hamilton geometry: Phase space geometry from modified dispersion relations
Barcaroli, Leonardo; Gubitosi, Giulia; Loret, Niccoló; Pfeifer, Christian
2015-01-01
We describe the Hamilton geometry of the phase space of particles whose motion is characterised by general dispersion relations. In this framework spacetime and momentum space are naturally curved and intertwined, allowing for a simultaneous description of both spacetime curvature and non-trivial momentum space geometry. We consider as explicit examples two models for Planck-scale modified dispersion relations, inspired from the $q$-de Sitter and $\\kappa$-Poincar\\'e quantum groups. In the first case we find the expressions for the momentum and position dependent curvature of spacetime and momentum space, while for the second case the manifold is flat and only the momentum space possesses a nonzero, momentum dependent curvature. In contrast, for a dispersion relation that is induced by a spacetime metric, as in General Relativity, the Hamilton geometry yields a flat momentum space and the usual curved spacetime geometry with only position dependent geometric objects.
Hull, C. M.
1993-01-01
The higher-spin geometries of $W_\\infty$-gravity and $W_N$-gravity are analysed and used to derive the complete non-linear structure of the coupling to matter and its symmetries. The symmetry group is a subgroup of the symplectic diffeomorphisms of the cotangent bundle of the world-sheet, and the $W_N$ geometry is obtained from a non-linear truncation of the $W_\\infty$ geometry. Quantum W-gravity is briefly discussed. (Talk given at {\\it Pathways to Fundamental Interactions}, the 16th John Ho...
Walsh, Edward T
2014-01-01
This introductory text is designed to help undergraduate students develop a solid foundation in geometry. Early chapters progress slowly, cultivating the necessary understanding and self-confidence for the more rapid development that follows. The extensive treatment can be easily adapted to accommodate shorter courses. Starting with the language of mathematics as expressed in the algebra of logic and sets, the text covers geometric sets of points, separation and angles, triangles, parallel lines, similarity, polygons and area, circles, space geometry, and coordinate geometry. Each chapter incl
Fallow), Stray
2009-01-01
Having trouble with geometry? Do Pi, The Pythagorean Theorem, and angle calculations just make your head spin? Relax. With Head First 2D Geometry, you'll master everything from triangles, quads and polygons to the time-saving secrets of similar and congruent angles -- and it'll be quick, painless, and fun. Through entertaining stories and practical examples from the world around you, this book takes you beyond boring problems. You'll actually use what you learn to make real-life decisions, like using angles and parallel lines to crack a mysterious CSI case. Put geometry to work for you, and
Adler, Irving
1967-01-01
This richly detailed overview surveys the development and evolution of geometrical ideas and concepts from ancient times to the present. In addition to the relationship between physical and mathematical spaces, it examines the interactions of geometry, algebra, and calculus. The text proves many significant theorems and employs several important techniques. Chapters on non- Euclidean geometry and projective geometry form brief, self-contained treatments.More than 100 exercises with answers and 200 diagrams illuminate the text. Teachers, students (particularly those majoring in mathematics educa
Belogurov, S.; Berchun, Yu; Chernogorov, A.; Malzacher, P.; Ovcharenko, E.; Semennikov, A.
2011-12-01
Due to conceptual difference between geometry descriptions in Computer-Aided Design (CAD) systems and particle transport Monte Carlo (MC) codes direct conversion of detector geometry in either direction is not feasible. An original set of tools has been developed for building a GEANT4/ROOT compatible geometry in the CATIA CAD system and exchanging it with mentioned MC packages using GDML file format. A Special structure of a CATIA product tree, a wide range of primitives, different types of multiple volume instantiation, and supporting macros have been implemented.
Quantum Consequences of Parameterizing Geometry
Wanas, M. I.
2002-12-01
The marriage between geometrization and quantization is not successful, so far. It is well known that quantization of gravity , using known quantization schemes, is not satisfactory. It may be of interest to look for another approach to this problem. Recently, it is shown that geometries with torsion admit quantum paths. Such geometries should be parameterizied in order to preserve the quantum properties appeared in the paths. The present work explores the consequences of parameterizing such geometry. It is shown that quantum properties, appeared in the path equations, are transferred to other geometric entities.
Hull, C M
1993-01-01
The higher-spin geometries of $W_\\infty$-gravity and $W_N$-gravity are analysed and used to derive the complete non-linear structure of the coupling to matter and its symmetries. The symmetry group is a subgroup of the symplectic diffeomorphisms of the cotangent bundle of the world-sheet, and the $W_N$ geometry is obtained from a non-linear truncation of the $W_\\infty$ geometry. Quantum W-gravity is briefly discussed. (Talk given at {\\it Pathways to Fundamental Interactions}, the 16th John Hopkins Workshop on Current Problems in Particle Theory, Gothenborg, 1992.)
Thermal Phase in Bubbling Geometries
Institute of Scientific and Technical Information of China (English)
LIU Chang-Yong
2008-01-01
We use matrix model to study thermal phase in bubbling half-BPS type IIB geometries with SO(4)×SO(4) symmetry.Near the horizon limit,we find that thermal vacua of bubbling geometries have disjoint parts,and each part is one kind of phase of the thermal system.We connect the thermal dynamics of bubbling geometries with one-dimensional fermions thermal system.Finally,we try to give a new possible way to resolve information loss puzzle.
An improved combinatorial geometry model for arbitrary geometry in DSMC
Kargaran, H.; Minuchehr, A.; Zolfaghari, A.
2017-03-01
This paper focuses on a new direct simulation Monte Carlo (DSMC) code based on combinatorial geometry (CG) for simulation of any rarefied gas flow. The developed code, called DgSMC-A, has been supplied with an improved CG modeling able to significantly optimize the particle-tracking process, resulting in a highly reduced runtime compared to the conventional codes. The improved algorithm inserts a grid over the geometry and saves those grid elements containing some part of the geometry border. Since only a small part of a grid is engaged with the geometry border, significant time can be saved using the proposed algorithm. Embedding the modified algorithm in the DgSMC-A resulted in a fast, robust and self-governing code needless to any mesh generator. The code completely handles complex geometries created with first-and second-order surfaces. In addition, we developed a new surface area calculator in the CG methodology for complex geometries based on the Monte Carlo method with acceptable accuracy. Several well-known test cases are examined to indicate the code ability to deal with a wide range of realistic problems. Results are also found to be in good agreement with references and experimental data.
Natural Language as a Tool for Analyzing the Proving Process: The Case of Plane Geometry Proof
Robotti, Elisabetta
2012-01-01
In the field of human cognition, language plays a special role that is connected directly to thinking and mental development (e.g., Vygotsky, "1938"). Thanks to "verbal thought", language allows humans to go beyond the limits of immediately perceived information, to form concepts and solve complex problems (Luria, "1975"). So, it appears language…
Li, B; Li, Baowen; Robnik, Marko
1995-01-01
In this work we study the geometrical properties of the high-lying eigenfunctions (200,000 and above) which are deep in the semiclassical regime. The system we are analyzing is the billiard system inside the region defined by the quadratic (complex) conformal map w = z + \\lambda z^{2} of the unit disk |z| \\le 1 as introduced by Robnik (1983), with the shape parameter value \\lambda = 0.15, so that the billiard is still convex and has KAM-type classical dynamics, where regular and irregular regions of classical motion coexist in the classical phase space. By inspecting 100 and by showing 36 consecutive numerically calculated eigenfunctions we reach the following conclusions: (1) Percival's (1973) conjectured classification in regular and irregular states works well: the mixed type states "living" on regular {\\em and} irregular regions disappear in the semiclassical limit. (2) The irregular (chaotic) states can be strongly localized due to the slow classical diffusion, but become fully extended in the semiclassi...
Reprocessing of X-rays in AGN. I. Plane parallel geometry -- test of pressure equilibrium
Dumont, A M; Collin, S; Zycki, P T
2002-01-01
We present a model of the vertical stratification and the spectra of an irradiated medium under the assumption of constant pressure. Such a solution has properties intermediate between constant density models and hydrostatic equilibrium models, and it may represent a flattened configuration of gas clumps accreting onto the central black hole. Such a medium develops a hot skin, thicker than hydrostatic models, but thinner than constant density models, under comparable irradiation. The range of theoretical values of the alpha_ox index is comparable to those from hydrostatic models and both are close to the observed values for Seyfert galaxies but lower than in quasars. The amount of X-ray Compton reflection is consistent with the observed range. The characteristic property of the model is a frequently multicomponent iron K alpha line.
Natural Language as a Tool for Analyzing the Proving Process: The Case of Plane Geometry Proof
Robotti, Elisabetta
2012-01-01
In the field of human cognition, language plays a special role that is connected directly to thinking and mental development (e.g., Vygotsky, "1938"). Thanks to "verbal thought", language allows humans to go beyond the limits of immediately perceived information, to form concepts and solve complex problems (Luria, "1975"). So, it appears language…
Hyperbolic Metamaterials with Complex Geometry
DEFF Research Database (Denmark)
Lavrinenko, Andrei; Andryieuski, Andrei; Zhukovsky, Sergei
2016-01-01
We investigate new geometries of hyperbolic metamaterialssuch as highly corrugated structures, nanoparticle monolayer assemblies, super-structured or vertically arranged multilayersand nanopillars. All structures retain basic propertiesof hyperbolic metamaterials, but have functionality improved...
Instability of supersymmetric microstate geometries
Eperon, Felicity C; Santos, Jorge E
2016-01-01
We investigate the classical stability of supersymmetric, asymptotically flat, microstate geometries with five non-compact dimensions. Such geometries admit an "evanescent ergosurface": a timelike hypersurface of infinite redshift. On such a surface, there are null geodesics with zero energy relative to infinity. These geodesics are stably trapped in the potential well near the ergosurface. We present a heuristic argument indicating that this feature is likely to lead to a nonlinear instability of these solutions. We argue that the precursor of such an instability can be seen in the behaviour of linear perturbations: nonlinear stability would require that all linear perturbations decay sufficiently rapidly but the stable trapping implies that some linear perturbation decay very slowly. We study this in detail for the most symmetric microstate geometries. By constructing quasinormal modes of these geometries we show that generic linear perturbations decay slower than any inverse power of time.
Molecular motion in restricted geometries
Indian Academy of Sciences (India)
Siddharth Gautam; S Mitra; R Mukhopadhyay
2008-10-01
Molecular dynamics in restricted geometries is known to exhibit anomalous behaviour. Diffusion, translational or rotational, of molecules is altered significantly on confinement in restricted geometries. Quasielastic neutron scattering (QENS) offers a unique possibility of studying molecular motion in such systems. Both time scales involved in the motion and the geometry of motion can be studied using QENS. Molecular dynamics (MD) simulation not only provides insight into the details of the different types of motion possible but also does not suffer limitations of the experimental set-up. Here we report the effect of confinement on molecular dynamics in various restricted geometries as studied by QENS and MD simulations: An example where the QENS technique provided direct evidence of phase transition associated with change in the dynamical behaviour of the molecules is also discussed.
Moment methods in extremal geometry
De Laat, D.
2016-01-01
In this thesis we develop techniques for solving problems in extremal geometry. We give an infinite dimensional generalization of moment techniques from polynomial optimization. We use this to construct semidefinite programming hierarchies for approximating optimal packing densities and ground state
An introduction to differential geometry
Willmore, T J
2012-01-01
This text employs vector methods to explore the classical theory of curves and surfaces. Topics include basic theory of tensor algebra, tensor calculus, calculus of differential forms, and elements of Riemannian geometry. 1959 edition.
Instability of supersymmetric microstate geometries
Energy Technology Data Exchange (ETDEWEB)
Eperon, Felicity C.; Reall, Harvey S.; Santos, Jorge E. [Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA (United Kingdom)
2016-10-07
We investigate the classical stability of supersymmetric, asymptotically flat, microstate geometries with five non-compact dimensions. Such geometries admit an “evanescent ergosurface”: a timelike hypersurface of infinite redshift. On such a surface, there are null geodesics with zero energy relative to infinity. These geodesics are stably trapped in the potential well near the ergosurface. We present a heuristic argument indicating that this feature is likely to lead to a nonlinear instability of these solutions. We argue that the precursor of such an instability can be seen in the behaviour of linear perturbations: nonlinear stability would require that all linear perturbations decay sufficiently rapidly but the stable trapping implies that some linear perturbation decay very slowly. We study this in detail for the most symmetric microstate geometries. By constructing quasinormal modes of these geometries we show that generic linear perturbations decay slower than any inverse power of time.
Moment methods in extremal geometry
De Laat, D.
2016-01-01
In this thesis we develop techniques for solving problems in extremal geometry. We give an infinite dimensional generalization of moment techniques from polynomial optimization. We use this to construct semidefinite programming hierarchies for approximating optimal packing densities and ground state
Guide to Computational Geometry Processing
DEFF Research Database (Denmark)
Bærentzen, Jakob Andreas; Gravesen, Jens; Anton, François
be processed before it is useful. This Guide to Computational Geometry Processing reviews the algorithms for processing geometric data, with a practical focus on important techniques not covered by traditional courses on computer vision and computer graphics. This is balanced with an introduction......Optical scanning is rapidly becoming ubiquitous. From industrial laser scanners to medical CT, MR and 3D ultrasound scanners, numerous organizations now have easy access to optical acquisition devices that provide huge volumes of image data. However, the raw geometry data acquired must first......, metric space, affine spaces, differential geometry, and finite difference methods for derivatives and differential equations Reviews geometry representations, including polygonal meshes, splines, and subdivision surfaces Examines techniques for computing curvature from polygonal meshes Describes...
Advances in discrete differential geometry
2016-01-01
This is one of the first books on a newly emerging field of discrete differential geometry and an excellent way to access this exciting area. It surveys the fascinating connections between discrete models in differential geometry and complex analysis, integrable systems and applications in computer graphics. The authors take a closer look at discrete models in differential geometry and dynamical systems. Their curves are polygonal, surfaces are made from triangles and quadrilaterals, and time is discrete. Nevertheless, the difference between the corresponding smooth curves, surfaces and classical dynamical systems with continuous time can hardly be seen. This is the paradigm of structure-preserving discretizations. Current advances in this field are stimulated to a large extent by its relevance for computer graphics and mathematical physics. This book is written by specialists working together on a common research project. It is about differential geometry and dynamical systems, smooth and discrete theories, ...
SNAP Satellite Focal Plane Development
Energy Technology Data Exchange (ETDEWEB)
Bebek, C.; Akerlof, C.; Aldering, G.; Amanullah, R.; Astier, P.; Baltay, C.; Barrelet, E.; Basa, S.; Bercovitz, J.; Bergstrom, L.; Berstein, G.P.; Bester, M.; Bohlin, R.; Bonissent, A.; Bower, C.; Campbell, M.; Carithers, W.; Commins, E.; Day, C.; Deustua, S.; DiGennaro, R.; Ealet, A.; Ellis, R.; Emmett, W.; Eriksson, M.; Fouchez,D.; Fruchter, A.; Genat, J-F.; Goldhaber, G.; Goobar, A.; Groom, D.; Heetderks, H.; Holland, S.; Huterer, D.; Johnson, W.; Kadel, R.; Karcher,A.; Kim, A.; Kolbe, W.; Lafever, R.; Lamoureaux, J.; Lampton, M.; Lefevre, O.; Levi, M.; Levin, D.; Linder, E.; Loken, S.; Malina, R.; Mazure, A.; McKay, T.; McKee, S.; Miquel, R.; Morgan, N.; Mortsell, E.; Mostek, N.; Mufson, S.; Musser, J.; Roe, N.; Nugent, P.; Oluseyi, H.; Pain, R.; Palaio, N.; Pankow, D.; Perlmutter, S.; Prieto, E.; Rabinowitz,D.; Refregier, A.; Rhodes, J.; Schubnell, M.; Sholl, M.; Smadja, G.; Smith, R.; Smoot, G.; Snyder, J.; Spadafora, A.; Szymkowiak, A.; Tarle,G.; Taylor, K.; Tilquin, A.; Tomasch, A.; Vincent, D.; von der Lippe, H.; Walder, J-P.; Wang, G.
2003-07-07
The proposed SuperNova/Acceleration Probe (SNAP) mission will have a two-meter class telescope delivering diffraction-limited images to an instrumented 0.7 square degree field in the visible and near-infrared wavelength regime. The requirements for the instrument suite and the present configuration of the focal plane concept are presented. A two year R&D phase, largely supported by the Department of Energy, is just beginning. We describe the development activities that are taking place to advance our preparedness for mission proposal in the areas of detectors and electronics.
Paper Interfaces for Learning Geometry
Bonnard, Quentin; Verma, Himanshu; Kaplan, Frédéric; Dillenbourg, Pierre
2012-01-01
Paper interfaces offer tremendous possibilities for geometry education in primary schools. Existing computer interfaces designed to learn geometry do not consider the integration of conventional school tools, which form the part of the curriculum. Moreover, most of computer tools are designed specifically for individual learning, some propose group activities, but most disregard classroom-level learning, thus impeding their adoption. We present an augmented reality based tabletop system with ...
Courant Algebroids in Parabolic Geometry
Armstrong, Stuart
2011-01-01
To a smooth manifold $M$, a parabolic geometry associates a principal bundle, which has a parabolic subgroup of a semisimple Lie group as its structure group, and a Cartan connection. We show that the adjoint tractor bundle of a regular normal parabolic geometry can be endowed with the structure of a Courant algebroid. This gives a class of examples of transitive Courant algebroids that are not exact.
Topology and geometry for physicists
Nash, Charles
2011-01-01
Differential geometry and topology are essential tools for many theoretical physicists, particularly in the study of condensed matter physics, gravity, and particle physics. Written by physicists for physics students, this text introduces geometrical and topological methods in theoretical physics and applied mathematics. It assumes no detailed background in topology or geometry, and it emphasizes physical motivations, enabling students to apply the techniques to their physics formulas and research. ""Thoroughly recommended"" by The Physics Bulletin, this volume's physics applications range fr
Higgs mass in noncommutative geometry
Energy Technology Data Exchange (ETDEWEB)
Devastato, A.; Martinetti, P. [Dipartimento di Fisica, Universita di Napoli Federico II, Via Cintia, 80126 Napoli (Italy); INFN, Sezione di Napoli, Via Cintia, 80126 Napoli (Italy); Lizzi, F. [Dipartimento di Fisica, Universita di Napoli Federico II, Via Cintia, 80126 Napoli (Italy); INFN, Sezione di Napoli, Via Cintia, 80126 Napoli (Italy); Departament de Estructura i Constituents de la Materia, Universitat de Barcelona, Marti y Franques, Barcelona, Catalonia (Spain)
2014-09-11
In the noncommutative geometry approach to the standard model, an extra scalar field σ - initially suggested by particle physicist to stabilize the electroweak vacuum - makes the computation of the Higgs mass compatible with the 126 GeV experimental value. We give a brief account on how to generate this field from the Majorana mass of the neutrino, following the principles of noncommutative geometry. (Copyright copyright 2014 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)
Traveling hairpin-shaped fluid vortices in plane Couette flow.
Deguchi, K; Nagata, M
2010-11-01
Traveling-wave solutions are discovered in plane Couette flow. They are obtained when the so-called steady hairpin vortex state found recently by Gibson [J. Fluid Mech. 638, 243 (2009)] and Itano and Generalis [Phys. Rev. Lett. 102, 114501 (2009)] is continued to sliding Couette flow geometry between two concentric cylinders by using the radius ratio as a homotopy parameter. It turns out that in the plane Couette flow geometry two traveling waves having the phase velocities with opposite signs are associated with their appearance from the steady hairpin vortex state, where the amplitude of the phase velocities increases gradually from zero as the Reynolds number is increased. The solutions obviously inherit the streaky structure of the hairpin vortex state, but shape preserving flow patterns propagate in the streamwise direction. Other striking features of the solution are asymmetric mean flow profiles and strong quasistreamwise vortices which occupy the vicinity of only the top or bottom moving boundary, depending on the sign of the phase velocity. Furthermore, we find that the pitchfork bifurcation associated with the appearance of the solution becomes imperfect when the flow is perturbed by a Poiseuille flow component.
The Common Geometry Module (CGM).
Energy Technology Data Exchange (ETDEWEB)
Tautges, Timothy James
2004-12-01
The Common Geometry Module (CGM) is a code library which provides geometry functionality used for mesh generation and other applications. This functionality includes that commonly found in solid modeling engines, like geometry creation, query and modification; CGM also includes capabilities not commonly found in solid modeling engines, like geometry decomposition tools and support for shared material interfaces. CGM is built upon the ACIS solid modeling engine, but also includes geometry capability developed beside and on top of ACIS. CGM can be used as-is to provide geometry functionality for codes needing this capability. However, CGM can also be extended using derived classes in C++, allowing the geometric model to serve as the basis for other applications, for example mesh generation. CGM is supported on Sun Solaris, SGI, HP, IBM, DEC, Linux and Windows NT platforms. CGM also includes support for loading ACIS models on parallel computers, using MPI-based communication. Future plans for CGM are to port it to different solid modeling engines, including Pro/Engineer or SolidWorks. CGM is being released into the public domain under an LGPL license; the ACIS-based engine is available to ACIS licensees on request.
On free fermions and plane partitions
Foda, O; Zuparic, M
2008-01-01
We use free fermion methods to re-derive a result of Okounkov and Reshetikhin relating charged fermions to random plane partitions, and to extend it to relate neutral fermions to strict plane partitions.
Image plane sweep volume illumination.
Sundén, Erik; Ynnerman, Anders; Ropinski, Timo
2011-12-01
In recent years, many volumetric illumination models have been proposed, which have the potential to simulate advanced lighting effects and thus support improved image comprehension. Although volume ray-casting is widely accepted as the volume rendering technique which achieves the highest image quality, so far no volumetric illumination algorithm has been designed to be directly incorporated into the ray-casting process. In this paper we propose image plane sweep volume illumination (IPSVI), which allows the integration of advanced illumination effects into a GPU-based volume ray-caster by exploiting the plane sweep paradigm. Thus, we are able to reduce the problem complexity and achieve interactive frame rates, while supporting scattering as well as shadowing. Since all illumination computations are performed directly within a single rendering pass, IPSVI does not require any preprocessing nor does it need to store intermediate results within an illumination volume. It therefore has a significantly lower memory footprint than other techniques. This makes IPSVI directly applicable to large data sets. Furthermore, the integration into a GPU-based ray-caster allows for high image quality as well as improved rendering performance by exploiting early ray termination. This paper discusses the theory behind IPSVI, describes its implementation, demonstrates its visual results and provides performance measurements.
The INTEGRAL Galactic Plane Scanning
Fiocchi, Mariateresa
2013-01-01
After the first nine years of INTEGRAL operational life, the discovery of new sources and source types, a large fraction of which are highly transient or highly absorbed, is certainly one of the most compelling results and legacies of INTEGRAL. Frequent monitoring of the Galactic Plane in AO8 and AO9 campaigns allowed us to detect transient sources, both known and new, confirming that the gamma-ray sky is dominated by the extreme variability of different classes of objects. Regular scans of the Galactic Plane by INTEGRAL provide the most sensitive hard X-ray wide survey to date of our Galaxy, with flux limits of the order of 0.3 mCrab for an exposure time of ~2Ms. Many transient sources have been detected on a wide range of time scales (~hours to months) and identified by triggered followup observations, mainly by Swift/XRT and optical/infrared telescopes. These discoveries are very important to characterize the X-ray binary population in our Galaxy, that is necessary input for evolution studies. The transien...
The HAWC Galactic Plane Survey
Hui, Michelle
2016-03-01
The High Altitude Water Cherenkov (HAWC) Observatory is an all-sky surveying instrument that covers 2/3 of the sky in 24 hours. It is designed with an emphasis on continuous sky coverage for transient events, and on the measurement of extended and large-scale structures. The array is located in Sierra Negra, Mexico at an elevation of 4,100 m and was inaugurated in March 2015. The HAWC array consists of 300 water Cherenkov detectors and is sensitive to extensive air showers triggered by cosmic rays and gamma rays from 100 GeV to >100 TeV. Thanks to its modular design, data taking began in Summer 2013 with 1/3 of the array. Analysis of the first year of data with the partial array shows detections that are coincident with known TeV supernova remnants and pulsar wind nebulae along the Galactic plane. Spectral and morphological analyses are ongoing to study the particle population and acceleration mechanism of these objects. With a growing data set taken with the completed array, source searches are underway for both point-like and extended emission along the Galactic plane, which contain many objects such as pulsar wind nebulae, young star clusters, and binaries.
Lines, Circles, Planes and Spheres
Purdy, George B
2009-01-01
Let $S$ be a set of $n$ points in $\\mathbb{R}^3$, no three collinear and not all coplanar. If at most $n-k$ are coplanar and $n$ is sufficiently large, the total number of planes determined is at least $1 + k \\binom{n-k}{2}-\\binom{k}{2}(\\frac{n-k}{2})$. For similar conditions and sufficiently large $n$, (inspired by the work of P. D. T. A. Elliott in \\cite{Ell67}) we also show that the number of spheres determined by $n$ points is at least $1+\\binom{n-1}{3}-t_3^{orchard}(n-1)$, and this bound is best possible under its hypothesis. (By $t_3^{orchard}(n)$, we are denoting the maximum number of three-point lines attainable by a configuration of $n$ points, no four collinear, in the plane, i.e., the classic Orchard Problem.) New lower bounds are also given for both lines and circles.
Singularities from colliding plane gravitational waves
Tipler, Frank J.
1980-12-01
A simple geometrical argument is given which shows that a collision between two plane gravitational waves must result in singularities. The argument suggests that these singularities are a peculiar feature of plane waves, because singularities are also a consequence of a collision between self-gravitating plane waves of other fields with arbitrarily small energy density.
Singularities from colliding plane gravitational waves
Energy Technology Data Exchange (ETDEWEB)
Tipler, F.J.
1980-12-15
A simple geometrical argument is given which shows that a collision between two plane gravitational waves must result in singularities. The argument suggests that these singularities are a peculiar feature of plane waves, because singularities are also a consequence of a collision between self-gravitating plane waves of other fields with arbitrarily small energy density.
Evolved stars in galactic plane surveys
Verbeek, K.
2013-01-01
For the first time in history the entire Galactic Plane is digitally mapped from La Palma and Chile by the European Galactic Plane surveys EGAPS (UVEX, IPHAS and VPHAS+, see http://www.uvexsurvey.org http://www.iphas.org and http://www.vphasplus.org). The complete Galactic plane (3600 square degrees
Homogeneity and plane-wave limits
Figueroa-O'Farrill, J M; Philip, S; Farrill, Jos\\'e Figueroa-O'; Meessen, Patrick; Philip, Simon
2005-01-01
We explore the plane-wave limit of homogeneous spacetimes. For plane-wave limits along homogeneous geodesics the limit is known to be homogeneous and we exhibit the limiting metric in terms of Lie algebraic data. This simplifies many calculations and we illustrate this with several examples. We also investigate the behaviour of (reductive) homogeneous structures under the plane-wave limit.
Holomorphic Cartan geometries and rational curves
Biswas, Indranil
2010-01-01
We prove that any compact K\\"ahler manifold bearing a holomorphic Cartan geometry contains a rational curve just when the Cartan geometry is inherited from a holomorphic Cartan geometry on a lower dimensional compact K\\"ahler manifold.
Can nanotechnology experimentally solve the plane-plane challenge
Siria, Alessandro; Auvert, Geoffroy; Comin, Fabio; Chevrier, Joel
2010-01-01
Non-contact interaction between two parallel flat surfaces is a central paradigm in sciences. This situation is the starting point for a wealth of different models: the capacitor description in electrostatics, hydrodynamic flow, thermal exchange, the Casimir force, direct contact study, third body confinement such as liquids or films of soft condensed matter. The control of parallelism is so demanding that no versatile single force machine in this geometry has been proposed so far. Using a combination of nanopositioning based on inertial motors and microcrystal shaping with Focused Ion Beams (FIB) we propose here an experimental set up that should enable one to measure interactions between movable surfaces separated by gaps in the micrometer and the nanometer ranges.
The Application of Axial Plane Beam%有轴平面束定理的应用
Institute of Scientific and Technical Information of China (English)
周明; 徐井海
2015-01-01
平面束定理是解析几何中关于空间直线和平面部分的重要内容,有轴平面束定理对判定直线与平面的位置关系、两直线共面定理、直纹曲面的性质和平面方程的求法起着重要作用.%The plane beam theorem is an important content in plane analytic geometry on a straight line of space and plane part,by using the plane beam theorem ,it plays an important role that we can determine the location of the straight line and plane relationship, two straight line coplanar theorem, the method to get the nature of the ruled surface and plane equation.
Symmetry Broken Exact Coherent Structures in Plane Couette Flow
Gopalaswamy, Varchas; Borrero-Echeverry, Daniel
2015-11-01
Invariant solutions of the fully resolved Navier-Stokes equation, known as exact coherent structures (ECS) are an exciting and potentially revolutionary method for understanding turbulent dynamics. The geometry of plane Couette flow leads to the existence of ECS with a high degree of symmetry. However, turbulent flows do not display a high degree of symmetry, so it is unclear whether these symmetric ECS can truly capture the turbulent dynamics. We report the discovery of four new periodic orbits - P85 and P60 which are fully symmetric, and P32 and P8, which have partially broken symmetry. Projections of these periodic orbits in the dissipation-energy input plane reveal that P32, P60 and P85 lie in the turbulent region of the state space, whereas P8 lies very far away from this region. Parametric continuation in the spanwise periodic cell length Lz suggests that P8 undergoes two bifurcations, which are verified by analysis of various properties of P8 in the dissipation-energy input plane, and by observations of changes in the stability of eigenvectors that are consistent with bifurcations.
Numerical simulations of aerodynamic contribution of flows about a space-plane-type configuration
Matsushima, Kisa; Takanashi, Susume; Fujii, Kozo; Obayashi, Shigeru
1987-01-01
The slightly supersonic viscous flow about the space-plane under development at the National Aerospace Laboratory (NAL) in Japan was simulated numerically using the LU-ADI algorithm. The wind-tunnel testing for the same plane also was conducted with the computations in parallel. The main purpose of the simulation is to capture the phenomena which have a great deal of influence to the aerodynamic force and efficiency but is difficult to capture by experiments. It includes more accurate representation of vortical flows with high angles of attack of an aircraft. The space-plane shape geometry simulated is the simplified model of the real space-plane, which is a combination of a flat and slender body and a double-delta type wing. The comparison between experimental results and numerical ones will be done in the near future. It could be said that numerical results show the qualitatively reliable phenomena.
Geometry of inferior endplates of the cervical spine.
Lou, Jigang; Liu, Hao; Rong, Xin; Li, Huibo; Wang, Beiyu; Gong, Quan
2016-03-01
Device subsidence is a well-known complication following cervical disc arthroplasty. Its occurrence has been closely tied with the endplate-implant contact interface. But current literature on the geometry of cervical endplate is very scarce. The aim of this anatomical investigation was to analyze geometry of inferior endplates of the cervical vertebrae, thereby identifying the common endplate shape patterns and providing morphological reference values consummating the design of the implant. Reformatted CT scans of 85 individuals were analyzed and endplate concave depth, endplate concave apex location, sagittal diameter of endplate, coronal concave angle, as well as transverse diameter of endplate were measured in mid-sagittal plane and specified coronal plane. According to the endplate concave apex location, the inferior endplates in mid-sagittal plane were classified into 3 types: type I with posteriorly positioned apex, type II with middle situated concave apex and type III with anteriorly positioned apex. Moreover, the inferior endplates in specified coronal plane were also classified into three types: concave, flat and irregular. Based on visual assessment, for the mid-sagittal plane, type I endplate accounted for 26.9% of all the 510 endplates of 85 individuals, while the proportion of type II and type III endplates were 53.9 and 19.2% respectively. For the specified coronal plane, 68.6% of all the 510 endplates were evaluated as concave, 26.9% as flat and the remaining 4.5% as irregular. Among all measured segments, C3 had the largest endplate concave depth values in mid-sagittal plane, while C7 the least; C5 and C6 had the largest sagittal endplate diameter values, while C2 the least. For each level, the sagittal endplate concave depth and endplate diameter of females were significantly smaller than those of males (P0.05). Increasing from C2 to C7, the endplate transverse diameters of females were significantly smaller than those of males (Psagittal and
Thermodynamics of black plane solution
Rodrigues, Manuel E; Houndjo, Stéphane J M
2012-01-01
We obtain a new phantom black plane solution in 4D of the Einstein-Maxwell theory coupled with a cosmological constant. We analyse their basic properties and obtain the extensive and intensive thermodynamic variables, as well as the specific heat and the first law. Through the specific heat and the so-called geometric methods, we analyse in detail their thermodynamic properties, the extreme and phase transition limits, as well as the local and global stabilities of the system. The normal case is shown with an extreme limit and the phantom one with a phase transition only for null mass. The systems present local and global stabilities for certain values of the entropy density with respect to the electric charge, for the canonical and grand canonical ensembles.
Thermodynamics of black plane solution
Rodrigues, Manuel E.; Jardim, Deborah F.; Houndjo, Stéphane J. M.; Myrzakulov, Ratbay
2013-11-01
We obtain a new phantom black plane solution in D of the Einstein-Maxwell theory coupled with a cosmological constant. We analyse their basic properties, as well as its causal structure, and obtain the extensive and intensive thermodynamic variables, as well as the specific heat and the first law. Through the specific heat and the so-called geometric methods, we analyse in detail their thermodynamic properties, the extreme and phase transition limits, as well as the local and global stabilities of the system. The normal case is shown with an extreme limit and the phantom one with a phase transition only for null mass, which is physically inaccessible. The systems present local and global stabilities for certain values of the entropy density with respect to the electric charge, for the canonical and grand canonical ensembles.
On plane submerged laminar jets
Coenen, Wilfried; Sanchez, Antonio L.
2016-11-01
We address the laminar flow generated when a developed stream of liquid of kinematic viscosity ν flowing along channel of width 2 h discharges into an open space bounded by two symmetric plane walls departing from the channel rim with an angle α 1 . Attention is focused on values of the jet volume flux 2 Q such that the associated Reynolds number Re = Qh / ν is of order unity. The formulation requires specification of the boundary conditions far from the channel exit. If the flow is driven by the volume flux, then the far-field solution corresponds to Jeffery-Hamel self-similar flow. However, as noted by Fraenkel (1962), such solutions exist only for α potential flow driven by the jet entrainment, and a Falkner-Skan near-wall boundary layer. Numerical integrations of the Navier-Stokes equations are used to ascertain the existence of these different solutions.
Calculation of the electrical of induction heating coils in two dimensional axissymmetric geometry
Energy Technology Data Exchange (ETDEWEB)
Nerg, J.; Partanen, J. [Lappeenranta University of Technology (Finland). Department of Energy Technology, Laboratory of Electrical Engineering
1997-12-31
The effect of the workpiece temperature on the electrical parameters of a plane, spiral inductor is discussed. The effect of workpiece temperature on the electrical efficiency, power transfer to the workpiece and electromagnetic distortion are also presented. Calculation is performed in two dimensional axissymmetric geometry using a FEM program. (orig.) 5 refs.
Pupil geometry and pupil re-imaging in telescope arrays
Traub, Wesley A.
1990-01-01
This paper considers the issues of lateral and longitudinal pupil geometry in ground-based telescope arrays, such as IOTA. In particular, it is considered whether or not pupil re-imaging is required before beam combination. By considering the paths of rays through the system, an expression is derived for the optical path errors in the combined wavefront as a function of array dimensions, telescope magnification factor, viewing angle, and field-of-view. By examining this expression for the two cases of pupil-plane and image-plane combination, operational limits can be found for any array. As a particular example, it is shown that for IOTA no pupil re-imaging optics will be needed.
General Construction of Tubular Geometry
Mukhopadhyay, Partha
2016-01-01
We consider the problem of locally describing tubular geometry around a submanifold embedded in a (pseudo)Riemannian manifold in its general form. Given the geometry of ambient space in an arbitrary coordinate system and equations determining the submanifold in the same system, we compute the tubular expansion coefficients in terms of this {\\it a priori data}. This is done by using an indirect method that crucially applies the tubular expansion theorem for vielbein previously derived. With an explicit construction involving the relevant coordinate and non-coordinate frames we verify consistency of the whole method up to quadratic order in vielbein expansion. Furthermore, we perform certain (long and tedious) higher order computation which verifies the first non-trivial spin connection term in the expansion for the first time. Earlier a similar method was used to compute tubular geometry in loop space. We explain this work in the light of our general construction.
Quantum geometry and gravitational entropy
Energy Technology Data Exchange (ETDEWEB)
Simon, Joan; Balasubramanian, Vijay; Czech, Bart Iomiej; Larjo, Klaus; Marolf, Donald; Simon, Joan
2007-05-29
Most quantum states have wavefunctions that are widely spread over the accessible Hilbert space and hence do not have a good description in terms of a single classical geometry. In order to understand when geometric descriptions are possible, we exploit the AdS/CFT correspondence in the half-BPS sector of asymptotically AdS_5 x S5 universes. In this sector we devise a"coarse-grained metric operator" whose eigenstates are well described by a single spacetime topology and geometry. We show that such half-BPS universes have a non-vanishing entropy if and only if the metric is singular, and that the entropy arises from coarse-graining the geometry. Finally, we use our entropy formula to find the most entropic spacetimes with fixed asymptotic moments beyond the global charges.
Conventionalism and integrable Weyl geometry
Pucheu, M. L.
2015-03-01
Since the appearance of Einstein's general relativity, gravitation has been associated to the space-time curvature. This theory introduced a geometrodynamic language which became a convenient tool to predict matter behaviour. However, the properties of space-time itself cannot be measurable by experiments. Taking Poincaré idea that the geometry of space-time is merely a convention, we show that the general theory of relativity can be completely reformulated in a more general setting, a generalization of Riemannian geometry, namely, the Weyl integrable geometry. The choice of this new mathematical language implies, among other things, that the path of particles and light rays should now correspond to Weylian geodesies. Such modification in the dynamic of bodies brings a new perception of physical phenomena that we will explore.
Wave propagation on microstate geometries
Keir, Joseph
2016-01-01
Supersymmetric microstate geometries were recently conjectured to be nonlinearly unstable due to numerical and heuristic evidence, based on the existence of very slowly decaying solutions to the linear wave equation on these backgrounds. In this paper, we give a thorough mathematical treatment of the linear wave equation on both two and three charge supersymmetric microstate geometries, finding a number of surprising results. In both cases we prove that solutions to the wave equation have uniformly bounded local energy, despite the fact that three charge microstates possess an ergoregion; these geometries therefore avoid Friedman's "ergosphere instability". In fact, in the three charge case we are able to construct solutions to the wave equation with local energy that neither grows nor decays, although this data must have nontrivial dependence on the Kaluza-Klein coordinate. In the two charge case we construct quasimodes and use these to bound the uniform decay rate, showing that the only possible uniform dec...
Geometry of black hole spacetimes
Andersson, Lars; Blue, Pieter
2016-01-01
These notes, based on lectures given at the summer school on Asymptotic Analysis in General Relativity, collect material on the Einstein equations, the geometry of black hole spacetimes, and the analysis of fields on black hole backgrounds. The Kerr model of a rotating black hole in vacuum is expected to be unique and stable. The problem of proving these fundamental facts provides the background for the material presented in these notes. Among the many topics which are relevant for the uniqueness and stability problems are the theory of fields on black hole spacetimes, in particular for gravitational perturbations of the Kerr black hole, and more generally, the study of nonlinear field equations in the presence of trapping. The study of these questions requires tools from several different fields, including Lorentzian geometry, hyperbolic differential equations and spin geometry, which are all relevant to the black hole stability problem.
Effect of pelvic tilt on lumbar spine geometry.
Delisle, A; Gagnon, M; Sicard, C
1997-12-01
The purpose of this study was to use a noninvasive method to determine the effect of pelvic tilt on the lumbar spine geometry in the sagittal plane. Five healthy male subjects were instructed in performing active forward and backward pelvic tilt manoeuvres in the standing position. The lumbar spine geometry (severity of lordosis, pelvis and lumbar vertebrae orientations) was estimated with a lumbar spine geometric model. The voluntary backward pelvic tilt succeeded in reducing the depth of the lumbar spine curvature, but the forward tilt did not change it. Both pelvic tilt manoeuvres influenced the absolute orientations of the lower lumbar vertebrae and the relative orientations of some lumbar vertebrae. Interestingly, the L5/S1 joint showed was little affected by the pelvic tilt manoeuvres.
Geometric Transformations in Engineering Geometry
Directory of Open Access Journals (Sweden)
I. F. Borovikov
2015-01-01
Full Text Available Recently, for business purposes, in view of current trends and world experience in training engineers, research and faculty staff there has been a need to transform traditional courses of descriptive geometry into the course of engineering geometry in which the geometrical transformations have to become its main section. On the basis of critical analysis the paper gives suggestions to improve a presentation technique of this section both in the classroom and in academic literature, extend an application scope of geometrical transformations to solve the position and metric tasks and simulation of surfaces, as well as to design complex engineering configurations, which meet a number of pre-specified conditions.The article offers to make a number of considerable amendments to the terms and definitions used in the existing courses of descriptive geometry. It draws some conclusions and makes the appropriate proposals on feasibility of coordination in teaching the movement transformation in the courses of analytical and descriptive geometry. This will provide interdisciplinary team teaching and allow students to be convinced that a combination of analytical and graphic ways to solve geometric tasks is useful and reasonable.The traditional sections of learning courses need to be added with a theory of projective and bi-rational transformations. In terms of application simplicity and convenience it is enough to consider the central transformations when solving the applied tasks. These transformations contain a beam of sub-invariant (low-invariant straight lines on which the invariant curve induces non-involution and involution projectivities. The expediency of nonlinear transformations application is shown in the article by a specific example of geometric modeling of the interfacing surface "spar-blade".Implementation of these suggestions will contribute to a real transformation of a traditional course of descriptive geometry to the engineering geometry
Dominant fault plane orientations of intermediate-depth earthquakes beneath South America
Warren, Linda M.
2014-07-01
The South American subduction zone exhibits considerable variation: the subduction angle alternates between flat and steep; the subducting plate has complex structures; and arc volcanism in the overlying plate has gaps. I investigate the effect of these differences in incoming plate structure and slab geometry on intermediate-depth earthquakes, specifically their fault orientations and rupture characteristics, and find that slab geometry has the largest impact on fault orientation. I use rupture directivity to estimate rupture direction and rupture velocity and to distinguish the fault plane from the auxiliary plane of the focal mechanism. From analysis of 163 large (Mw≥5.7) intermediate-depth (60-360 km depth) earthquakes from along the length of South America, estimated rupture azimuths and plunges show no trends, appearing to be randomly distributed on the determined population of fault plane orientations, and a majority of earthquakes are made up of multiple subevents. As seen in other subduction zones, subduction segments descending at normal angles have predominantly subhorizontal faults. Flat slab segments also have a dominant fault orientation, but those earthquakes slip along the conjugate nodal plane of the focal mechanism. In strongly curved slab segments, such as at the downdip edge of flat segments where the slab resubducts, earthquakes may slip along either nodal plane orientation. While both fault orientations could be consistent with the reactivation of fossil outer rise faults, the fault orientations are also consistent with expectations for newly created faults in agreement with the ambient stress field. Fault reactivation alone does not explain why different fault orientations are active in segments with different geometries, so the preferred explanation for having regionally consistent fault orientations is that they minimize the total work of the system. The previously observed predominance of subhorizontal faults appears to be a consequence
A molecular dynamics study of freezing in a confined geometry
Ma, Wen-Jong; Banavar, Jayanth R.; Koplik, Joel
1992-01-01
The dynamics of freezing of a Lennard-Jones liquid in narrow channels bounded by molecular walls is studied by computer simulation. The time development of ordering is quantified and a novel freezing mechanism is observed. The liquid forms layers and subsequent in-plane ordering within a layer is accompanied by a sharpening of the layer in the transverse direction. The effects of channel size, the methods of quench, the liquid-wall interaction and the roughness of walls on the freezing mechanism are elucidated. Comparison with recent experiments on freezing in confined geometries is presented.
RCS Analysis of Plate Geometries, parts 1 and 2
Balanis, Constantine A.; Polka, Lesley A.; Polycarpou, Anastasis C.
1993-01-01
High-frequency techniques for Radar Cross Section (RCS) prediction of plate geometries and a physical optics/equivalent currents model for the RCS of trihedral corner reflectors are addressed. In part 1, a Uniform Theory of Diffraction (UTD) model for the principal-plane radar cross section (RCS) of a perfectly conducting, rectangular plate coated on one side with an electrically thin, lossy dielectric is presented. In part 2, the scattering in the interior regions of both square and triangular trihedral corner reflectors are examined.
A molecular dynamics study of freezing in a confined geometry
Ma, Wen-Jong; Banavar, Jayanth R.; Koplik, Joel
1992-01-01
The dynamics of freezing of a Lennard-Jones liquid in narrow channels bounded by molecular walls is studied by computer simulation. The time development of ordering is quantified and a novel freezing mechanism is observed. The liquid forms layers and subsequent in-plane ordering within a layer is accompanied by a sharpening of the layer in the transverse direction. The effects of channel size, the methods of quench, the liquid-wall interaction and the roughness of walls on the freezing mechanism are elucidated. Comparison with recent experiments on freezing in confined geometries is presented.
KEMAJUAN BELAJAR SISWA PADA GEOMETRI TRANSFORMASI MENGGUNAKAN AKTIVITAS REFLEKSI GEOMETRI
Directory of Open Access Journals (Sweden)
Irkham Ulil Albab
2014-10-01
Full Text Available Abstrak: Penelitian ini bertujuan untuk mendeskripsikan kemajuan belajar siswa pada materi geometri transformasi yang didukung dengan serangkaian aktivitas belajar berdasarkan Pendidikan Matematika Realistik Indonesia. Penelitian didesain melalui tiga tahap, yaitu tahapan perancangan desain awal, pengujian desain melalui pembelajaran awal dan pembelajaran eksperimental, dan tahap analisis retrospektif. Dalam penelitian ini, Hypothetical Learning Trajectory, HLT (HLT berperan penting sebagai desain pembelajaran sekaligus instrumen penelitian. HLT diujikan terhadap 26 siswa kelas VII. Data dikumpulkan dengan teknik wawancara, pengamatan, dan catatan lapangan. Hasil penelitian menunjukkan bahwa desain pembelajaran ini mampu menstimulasi siswa untuk memberikan karakteristik refleksi dan transformasi geometri lainnya secara informal, mengklasifikasikannya dalam transformasi isometri pada level kedua, dan menemukan garis bantuan refleksi pada level yang lebih formal. Selain itu, garis bantuan refleksi digunakan oleh siswa untuk menggambar bayangan refleksi dan pola pencerminan serta memahami bentuk rotasi dan translasi sebagai kombinasi refleksi adalah level tertinggi. Keyword: transformasi geometri, kombinasi refleksi, rotasi, translasi, design research, HLT STUDENTS’ LEARNING PROGRESS ON TRANSFORMATION GEOMETRY USING THE GEOMETRY REFLECTION ACTIVITIES Abstract: This study was aimed at describing the students’ learning progress on transformation geometry supported by a set of learning activities based on Indonesian Realistic Mathematics Education. The study was designed into three stages, that is, the preliminary design stage, the design testing through initial instruction and experiment, and the restrospective analysis stage. In this study, Hypothetical Learning Trajectory (HLT played an important role as an instructional design and a research instrument. HLT was tested to 26 seventh grade students. The data were collected through interviews
Stochastic geometry and its applications
Chiu, Sung Nok; Kendall, Wilfrid S; Mecke, Joseph
2013-01-01
An extensive update to a classic text Stochastic geometry and spatial statistics play a fundamental role in many modern branches of physics, materials sciences, engineering, biology and environmental sciences. They offer successful models for the description of random two- and three-dimensional micro and macro structures and statistical methods for their analysis. The previous edition of this book has served as the key reference in its field for over 18 years and is regarded as the best treatment of the subject of stochastic geometry, both as a subject with vital a
Geometry Design of Wooden Barrels
Directory of Open Access Journals (Sweden)
Ivan CISMARU
2010-12-01
Full Text Available The aim of this paper is to present a design methodology of the wooden barrel geometry, as an algorithm of successive calculations. Thus, starting from the required elements (volume, length, shape, maximum height of storage space the user will be able to define the geometry which must be obtained by processing. Based on these calculations, one can define the structure, size and shape of the staves in order to establish the processing technology of both components and subassemblies (jacket and bottoms which are to form the final product by their assembling using metal circles.
Geometry, topology, and string theory
Energy Technology Data Exchange (ETDEWEB)
Varadarajan, Uday
2003-07-10
A variety of scenarios are considered which shed light upon the uses and limitations of classical geometric and topological notions in string theory. The primary focus is on situations in which D-brane or string probes of a given classical space-time see the geometry quite differently than one might naively expect. In particular, situations in which extra dimensions, non-commutative geometries as well as other non-local structures emerge are explored in detail. Further, a preliminary exploration of such issues in Lorentzian space-times with non-trivial causal structures within string theory is initiated.
Spatial geometry and special relativity
DEFF Research Database (Denmark)
Kneubil, Fabiana Botelho
2016-01-01
In this work, it is shown the interplay of relative and absolute entities, which are present in both spatial geometry and special relativity. In order to strengthen the understanding of special relativity, we discuss firstly an instance of geometry and the existence of both frame......-dependent and frame-independent entities. We depart from a subject well known by students, which is the three-dimensional geometric space in order to compare, afterwards, with the treatment of four-dimensional space in the special relativity. The differences and similarities between these two subjects are also...
Introduction to topology and geometry
Stahl, Saul
2014-01-01
An easily accessible introduction to over three centuries of innovations in geometry Praise for the First Edition ". . . a welcome alternative to compartmentalized treatments bound to the old thinking. This clearly written, well-illustrated book supplies sufficient background to be self-contained." -CHOICE This fully revised new edition offers the most comprehensive coverage of modern geometry currently available at an introductory level. The book strikes a welcome balance between academic rigor and accessibility, providing a complete and cohesive picture of the science with an unparallele
Lectures on classical differential geometry
Struik, Dirk J
1988-01-01
Elementary, yet authoritative and scholarly, this book offers an excellent brief introduction to the classical theory of differential geometry. It is aimed at advanced undergraduate and graduate students who will find it not only highly readable but replete with illustrations carefully selected to help stimulate the student's visual understanding of geometry. The text features an abundance of problems, most of which are simple enough for class use, and often convey an interesting geometrical fact. A selection of more difficult problems has been included to challenge the ambitious student.Writ
Gauging Geometry: A Didactic Lecture
Kannenberg, L
2016-01-01
Local inertial frame invariance is taken as the fundamental principle of physical geometry, where a local inertial frame is represented by a verbein. Invariance of the vierbein with respect to local Lorentz transformations then expresses local inertial frame invariance. The dynamics of physical geometry develops as a gauge theory of the verbein that is closely analogous to the Yang-Mills field provided the verbein connection and curvature correspond to the geometric potential and field respectively. The resulting theory is shown to be equivalent to Einstein's tensor form of relativistic gravitation.
Comparison theorems in Riemannian geometry
Cheeger, Jeff
2008-01-01
The central theme of this book is the interaction between the curvature of a complete Riemannian manifold and its topology and global geometry. The first five chapters are preparatory in nature. They begin with a very concise introduction to Riemannian geometry, followed by an exposition of Toponogov's theorem-the first such treatment in a book in English. Next comes a detailed presentation of homogeneous spaces in which the main goal is to find formulas for their curvature. A quick chapter of Morse theory is followed by one on the injectivity radius. Chapters 6-9 deal with many of the most re
Lectures on Algebraic Geometry I
Harder, Gunter
2012-01-01
This book and the following second volume is an introduction into modern algebraic geometry. In the first volume the methods of homological algebra, theory of sheaves, and sheaf cohomology are developed. These methods are indispensable for modern algebraic geometry, but they are also fundamental for other branches of mathematics and of great interest in their own. In the last chapter of volume I these concepts are applied to the theory of compact Riemann surfaces. In this chapter the author makes clear how influential the ideas of Abel, Riemann and Jacobi were and that many of the modern metho
Kang, Junmo; Jariwala, Deep; Ryder, Christopher R.; Wells, Spencer A.; Choi, Yongsuk; Hwang, Euyheon; Cho, Jeong Ho; Marks, Tobin J.; Hersam, Mark C.
2016-01-01
Black phosphorus (BP) has recently emerged as a promising narrow band gap layered semiconductor with optoelectronic properties that bridge the gap between semi-metallic graphene and wide band gap transition metal dichalcogenides such as MoS2. To date, BP field-effect transistors have utilized a lateral geometry with in-plane transport dominating device characteristics. In contrast, we present here a vertical field-effect transistor geometry based on a graphene/BP van der Waals heterostructure...
Generalized plane gravitational waves of non-symmetric unified field theories in plane symmetry
Directory of Open Access Journals (Sweden)
Sanjiv R. Bhoyar
2012-12-01
Full Text Available In this paper we investigated the plane wave solutions of both the weak and strong non-symmetric unified field equations of Einstein and Bonner in a generalized plane symmetric space-time in the sense of Taub [Ann. Math. 53, 472 (1951] for plane gravitational waves. We show that the plane wave solutions of Einstein and Bonner field equations exist in plane symmetry.
Measuring Space-Time Geometry over the Ages
Energy Technology Data Exchange (ETDEWEB)
Stebbins, Albert; /Fermilab
2012-05-01
Theorists are often told to express things in the 'observational plane'. One can do this for space-time geometry, considering 'visual' observations of matter in our universe by a single observer over time, with no assumptions about isometries, initial conditions, nor any particular relation between matter and geometry, such as Einstein's equations. Using observables as coordinates naturally leads to a parametrization of space-time geometry in terms of other observables, which in turn prescribes an observational program to measure the geometry. Under the assumption of vorticity-free matter flow we describe this observational program, which includes measurements of gravitational lensing, proper motion, and redshift drift. Only 15% of the curvature information can be extracted without long time baseline observations, and this increases to 35% with observations that will take decades. The rest would likely require centuries of observations. The formalism developed is exact, non-perturbative, and more general than the usual cosmological analysis.
KAMPUNG SENI ISLAM DI MAKASSAR DENGAN PENDEKATAN ARSITEKTUR ISLAM GEOMETRI
Directory of Open Access Journals (Sweden)
Yaumil Maghfirah Asaf
2015-06-01
work produced. So it can be recognized by both local and international. The approach used in the building of Islamic Art Village is Islamic architecture geometry. Geometry is a branch of mathematics that studies of point, line, plane and space objects along with their properties, the measurements, and the relationship between each other. Islamic architectural design more use patterns in the form of lines, circles and other geometric patterns arranged to form a unity which implies spiritual and aesthetic value or beauty of a high level. Islamic art looks association complex geometry, between forms, ornaments, and façade Key Word: Village Islamic Art, Islamic architecture geometry
Differential geometry of groups in string theory
Energy Technology Data Exchange (ETDEWEB)
Schmidke, W.B. Jr.
1990-09-01
Techniques from differential geometry and group theory are applied to two topics from string theory. The first topic studied is quantum groups, with the example of GL (1{vert bar}1). The quantum group GL{sub q}(1{vert bar}1) is introduced, and an exponential description is derived. The algebra and coproduct are determined using the invariant differential calculus method introduced by Woronowicz and generalized by Wess and Zumino. An invariant calculus is also introduced on the quantum superplane, and a representation of the algebra of GL{sub q}(1{vert bar}1) in terms of the super-plane coordinates is constructed. The second topic follows the approach to string theory introduced by Bowick and Rajeev. Here the ghost contribution to the anomaly of the energy-momentum tensor is calculated as the Ricci curvature of the Kaehler quotient space Diff(S{sup 1})/S{sup 1}. We discuss general Kaehler quotient spaces and derive an expression for their Ricci curvatures. Application is made to the string and superstring diffeomorphism groups, considering all possible choices of subgroup. The formalism is extended to associated holomorphic vector bundles, where the Ricci curvature corresponds to the anomaly for different ghost sea levels. 26 refs.
Quantum Entanglement and Projective Ring Geometry
Directory of Open Access Journals (Sweden)
Michel Planat
2006-08-01
Full Text Available The paper explores the basic geometrical properties of the observables characterizing two-qubit systems by employing a novel projective ring geometric approach. After introducing the basic facts about quantum complementarity and maximal quantum entanglement in such systems, we demonstrate that the 15 × 15 multiplication table of the associated four-dimensional matrices exhibits a so-far-unnoticed geometrical structure that can be regarded as three pencils of lines in the projective plane of order two. In one of the pencils, which we call the kernel, the observables on two lines share a base of Bell states. In the complement of the kernel, the eight vertices/observables are joined by twelve lines which form the edges of a cube. A substantial part of the paper is devoted to showing that the nature of this geometry has much to do with the structure of the projective lines defined over the rings that are the direct product of n copies of the Galois field GF(2, with n = 2, 3 and 4.
Energy Technology Data Exchange (ETDEWEB)
Hallez, Y
2007-12-15
The present work based on Direct Numerical Simulations is devoted to the study of mixing between two miscible fluids of different densities. The movement of these fluids is induced by buoyancy. Three geometries are considered: a cylindrical tube, a square channel and a plane two-dimensional flow. For cylindrical tubes, the results of numerical simulations fully confirm previous experimental findings by Seon et al., especially regarding the existence of three different flow regimes, depending on the tilt angle. The comparison of the various geometries shows that tridimensional flows in tubes or channels are similar, whereas the two-dimensional model fails to give reliable information about real 3D flows, either from a quantitative point of view or for a phenomenological understanding. A peculiar attention is put on a joint analysis of the concentration and vorticity fields and allows us to explain several subtle aspects of the mixing dynamics. (author)
A Machine Learning Approach to Recovery of Scene Geometry from Images
Trinh, Hoang
2010-01-01
Recovering the 3D structure of the scene from images yields useful information for tasks such as shape and scene recognition, object detection, or motion planning and object grasping in robotics. In this thesis, we introduce a general machine learning approach called unsupervised CRF learning based on maximizing the conditional likelihood. We apply our approach to computer vision systems that recover the 3-D scene geometry from images. We focus on recovering 3D geometry from single images, stereo pairs and video sequences. Building these systems requires algorithms for doing inference as well as learning the parameters of conditional Markov random fields (MRF). Our system is trained unsupervisedly without using ground-truth labeled data. We employ a slanted-plane stereo vision model in which we use a fixed over-segmentation to segment the left image into coherent regions called superpixels, then assign a disparity plane for each superpixel. Plane parameters are estimated by solving an MRF labelling problem, t...
Teaching Activity-Based Taxicab Geometry
Ada, Tuba
2013-01-01
This study aimed on the process of teaching taxicab geometry, a non-Euclidean geometry that is easy to understand and similar to Euclidean geometry with its axiomatic structure. In this regard, several teaching activities were designed such as measuring taxicab distance, defining a taxicab circle, finding a geometric locus in taxicab geometry, and…
Math 1813 (PIPI): Analytic Geometry.
Oklahoma State Univ., Stillwater. Coll. of Engineering.
This study guide, designed for use at Oklahoma State University, contains lists of activities for students to perform based on the "mastery of learning" concept. The activities include readings, problems, self evaluations, and assessment tasks. The units included are: Lines in a Plane, Conics, Transformations, Polar Coordinates,…
Learners engaging with transformation geometry
African Journals Online (AJOL)
The article concludes by recommending that opportunities need to ... the plane around a centre point, the students in her study expected the shape to slide to the ..... Ken: “... you have got to translate these coordinates to the rules they have ...
Cell division plane orientation based on tensile stress in Arabidopsis thaliana
Louveaux, Marion; Julien, Jean-Daniel; Mirabet, Vincent; Boudaoud, Arezki; Hamant, Olivier
2016-01-01
Cell geometry has long been proposed to play a key role in the orientation of symmetric cell division planes. In particular, the recently proposed Besson–Dumais rule generalizes Errera’s rule and predicts that cells divide along one of the local minima of plane area. However, this rule has been tested only on tissues with rather local spherical shape and homogeneous growth. Here, we tested the application of the Besson–Dumais rule to the divisions occurring in the Arabidopsis shoot apex, which contains domains with anisotropic curvature and differential growth. We found that the Besson–Dumais rule works well in the central part of the apex, but fails to account for cell division planes in the saddle-shaped boundary region. Because curvature anisotropy and differential growth prescribe directional tensile stress in that region, we tested the putative contribution of anisotropic stress fields to cell division plane orientation at the shoot apex. To do so, we compared two division rules: geometrical (new plane along the shortest path) and mechanical (new plane along maximal tension). The mechanical division rule reproduced the enrichment of long planes observed in the boundary region. Experimental perturbation of mechanical stress pattern further supported a contribution of anisotropic tensile stress in division plane orientation. Importantly, simulations of tissues growing in an isotropic stress field, and dividing along maximal tension, provided division plane distributions comparable to those obtained with the geometrical rule. We thus propose that division plane orientation by tensile stress offers a general rule for symmetric cell division in plants. PMID:27436908
Pechereau, Francois; Jansky, Jaroslav; Bourdon, Anne
2012-10-01
In recent years, experimental studies on flue gas treatment have demonstrated the efficiency of plasma assisted catalysis for the treatment of a wide range of pollutants at a low energetic cost. In plasma reactors, usual catalyst supports are pellets, monoliths or porous media, and then atmospheric pressure discharges have to interact with many obstacles and to propagate in microcavities and pores. As a first step to better understand atmospheric pressure discharge dynamics in these complex geometries, in this work, we have carried out numerical simulations using a 2D-axisymmetric fluid model for a point-to-plane discharge with a dielectric plane obstacle placed in the path of the discharge. First, we have simulated the discharge ignition at the point electrode, its propagation in the gap and its impact and expansion on the dielectric plane. Depending on the applied voltage, the dielectric plane geometry and permittivity, we have identified conditions for the reignition of a second discharge behind the plane obstacle. These conditions will be discussed and compared with recent experimental results on the same configuration.
Plane-polarized Raman continuum in the insulating and superconducting layered cuprates
Reznik, D.; Cooper, S. L.; Klein, M. V.; Lee, W. C.; Ginsberg, D. M.; Maksimov, A. A.; Puchkov, A. V.; Tartakovskii, I. I.; Cheong, S.-W.
1993-09-01
Electronic properties of copper oxygen planes (and chains in Y-Ba-Cu-O) were studied with Raman spectroscopy of plane-polarized photons. The electronic continuum was found to be independent of doping in 2:1:4 and 1:2:3 materials at energies above ~1000 cm-1. Temperature dependence at low energies differs significantly in undoped, lightly doped, and fully doped YBa2Cu3O6+x. A feature consistent with the superconducting gap was observed below Tc in YBa2Cu3O6.9 in all scattering geometries. However, the gaplike redistribution was not complete, with 40-60 % of states not shifted to higher energies at temperatures well below Tc. Above Tc the temperature dependence strongly depends on scattering geometry: the continuum is temperature independent (marginal-Fermi-liquid-like) in XX (x2) and X'X' (x2+y2) geometry; it has a Bose-factor temperature dependence in X'Y' (x2-y2) geometry, and a weak temperature dependence somewhat smaller than the Bose factor in YY (y2) geometry. A two-boson-like temperature dependence of the low-energy continuum is found in YBa2Cu3O6.1 and Sm2CuO4. It becomes one-particle-like in Y-Ba-Cu-O once small doping levels are introduced. Constraints these results place on theoretical models are discussed.
Differential geometry meets the cell.
Marshall, Wallace F
2013-07-18
A new study by Terasaki et al. highlights the role of physical forces in biological form by showing that connections between stacked endoplasmic reticulum cisternae have a shape well known in classical differential geometry, the helicoid, and that this shape is a predictable consequence of membrane physics.
Energy Technology Data Exchange (ETDEWEB)
Byrd, M.
1997-10-01
The group SU(3) is parameterized in terms of generalized {open_quotes}Euler angles{close_quotes}. The differential operators of SU(3) corresponding to the Lie Algebra elements are obtained, the invariant forms are found, the group invariant volume element is found, and some relevant comments about the geometry of the group manifold are made.
Fractal geometry and stochastics IV
Bandt, Christoph
2010-01-01
Over the years fractal geometry has established itself as a substantial mathematical theory in its own right. This book collects survey articles covering many of the developments, like Schramm-Loewner evolution, fractal scaling limits, exceptional sets for percolation, and heat kernels on fractals.
Signature geometry and quantum engineering
Samociuk, Stefan
2013-09-01
As the operating frequency of electromagnetic based devices increase, physical design geometry is playing an ever more important role. Evidence is considered in support of a relationship between the dimensionality of primitive geometric forms, such as transistors, and corresponding electromagnetic coupling efficiency. The industry of electronics is defined as the construction of devices by the patterning of primitive forms to physical materials. Examples are given to show the evolution of these primitives, down to nano scales, are requiring exacting geometry and three dimensional content. Consideration of microwave monolithic integrated circuits,(MMIC), photonics and metamaterials,(MM), support this trend and also add new requirements of strict geometric periodicity and multiplicity. Signature geometries,(SG), are characterized by distinctive attributes and examples are given. The transcendent form transcode algorithm, (TTA) is introduced as a multi dimensional SG and its use in designing photonic integrated circuits and metamaterials is discussed . A creative commons licensed research database, TRANSFORM, containing TTA geometries in OASIS file formats is described. An experimental methodology for using the database is given. Multidimensional SG and extraction of three dimensional cross sections as primitive forms is discussed as a foundation for quantum engineering and the exploitation of phenomena other than the electromagnetic.
Instructional Identities of Geometry Students
Aaron, Wendy Rose; Herbst, Patricio
2012-01-01
We inspect the hypothesis that geometry students may be oriented toward how they expect that the teacher will evaluate them as students or otherwise oriented to how they expect that their work will give them opportunities to do mathematics. The results reported here are based on a mixed-methods analysis of twenty-two interviews with high school…
Loop groups and noncommutative geometry
Carpi, Sebastiano
2015-01-01
We describe the representation theory of loop groups in terms of K-theory and noncommutative geometry. This is done by constructing suitable spectral triples associated with the level l projective unitary positive-energy representations of any given loop group LG. The construction is based on certain supersymmetric conformal field theory models associated with LG.
3DHZETRN: Inhomogeneous Geometry Issues
Wilson, John W.; Slaba, Tony C.; Badavi, Francis F.
2017-01-01
Historical methods for assessing radiation exposure inside complicated geometries for space applications were limited by computational constraints and lack of knowledge associated with nuclear processes occurring over a broad range of particles and energies. Various methods were developed and utilized to simplify geometric representations and enable coupling with simplified but efficient particle transport codes. Recent transport code development efforts, leading to 3DHZETRN, now enable such approximate methods to be carefully assessed to determine if past exposure analyses and validation efforts based on those approximate methods need to be revisited. In this work, historical methods of representing inhomogeneous spacecraft geometry for radiation protection analysis are first reviewed. Two inhomogeneous geometry cases, previously studied with 3DHZETRN and Monte Carlo codes, are considered with various levels of geometric approximation. Fluence, dose, and dose equivalent values are computed in all cases and compared. It is found that although these historical geometry approximations can induce large errors in neutron fluences up to 100 MeV, errors on dose and dose equivalent are modest (<10%) for the cases studied here.
The Basics of Information Geometry
Caticha, Ariel
2014-01-01
To what extent can we distinguish one probability distribution from another? Are there quantitative measures of distinguishability? The goal of this tutorial is to approach such questions by introducing the notion of the "distance" between two probability distributions and exploring some basic ideas of such an "information geometry".
College geometry a unified development
Kay, David C
2011-01-01
""The book is a comprehensive textbook on basic geometry. … Key features of the book include numerous figures and many problems, more than half of which come with hints or even complete solutions. Frequent historical comments add to making the reading a pleasant one.""-Michael Joswig, Zentralblatt MATH 1273
Foucault pendulum through basic geometry
von Bergmann, Jens; von Bergmann, HsingChi
2007-10-01
We provide a thorough explanation of the Foucault pendulum that utilizes its underlying geometry on a level suitable for science students not necessarily familiar with calculus. We also explain how the geometrically understood Foucault pendulum can serve as a prototype for more advanced phenomena in physics known as Berry's phase or geometric phases.
Analogical Reasoning in Geometry Education
Magdas, Ioana
2015-01-01
The analogical reasoning isn't used only in mathematics but also in everyday life. In this article we approach the analogical reasoning in Geometry Education. The novelty of this article is a classification of geometrical analogies by reasoning type and their exemplification. Our classification includes: analogies for understanding and setting a…
Open problems in algebraic geometry
Edixhoven, S.J.; Moonen, B.J.J.; Oort, F.
2000-01-01
The open problems presented here were collected on the occasion of a workshop on Arithmetic Geometry at the University ofUtrecht, 26{30 June, 2000. This workshop was organized by the editors of the present article, and was made possible by support of: | NWO, the Netherlands Organization for
Data Imprecision in Computational Geometry
Löffler, M.
2009-01-01
The field of computational geometry is concerned with the design and analysis of geometric algorithms. For such algorithms, correctness and efficiency proofs are constructed, or problems are proven to be hard when no correct and efficient algorithms exist. In order to be able to do this, several ass
GEOMETRY AND COMPLEXITY IN ARCHITECTURE
Directory of Open Access Journals (Sweden)
RUSU Maria Ana
2015-06-01
Full Text Available As Constantin Brancuși (1876-1956 said „Simplicity is complexity itself“, simplicity and regularity through the use of basic geometric forms has always played a central role in architectural design, during the 20th century. A diachronic perspective, shows as the use of geometry and mathematics to describe built form provided a common basis for communication between the processes of design, fabrication and stability. Classic ways of representing geometry, based on descriptive methods, favor precise language of bidimensionality easy to represent in a rectangular coordinate system. In recent years, the importance of geometry has been re-emphasized by significant advances in the digital age, where computers are increasingly used in design, fabrication and construction to explore the art of the possible. Contemporary architecture transcend the limitations of Euclidean geometry and create new forms that are emerging through the convergence of complex systems, computational design and robotic fabrication devices, but which can also achieve higher levels of performance. Freeform architectural shapes and structures play an increasingly important role in 21st century architectural design. Through a series of examples, the paper relates to contemporary architectural explorations of complex, curvilinear surfaces in the digital age and discusses how it has required rethinking the mode in which we traditionally operate as architects. The analysis creates the possibility of comparisons between original and current design.
Dynamics of Crowd Behaviors: From Complex Plane to Quantum Random Fields
Ivancevic, Vladimir G.; Reid, Darryn J.
2015-11-01
The following sections are included: * Complex Plane Dynamics of Crowds and Groups * Introduction * Complex-Valued Dynamics of Crowd and Group Behaviors * Kähler Geometry of Crowd and Group Dynamics * Computer Simulations of Crowds and Croups Dynamics * Braids of Agents' Behaviors in the Complex Plane * Hilbert-Space Control of Crowds and Groups Dynamics * Quantum Random Fields: A Unique Framework for Simulation, Optimization, Control and Learning * Introduction * Adaptive Quantum Oscillator * Optimization and Learning on Banach and Hilbert Spaces * Appendix * Complex-Valued Image Processing * Linear Integral Equations * Riemann-Liouville Fractional Calculus * Rigorous Geometric Quantization * Supervised Machine-Learning Methods * First-Order Logic and Quantum Random Fields
Focal Plane Instrumentation of VERITAS
Nagai, T; Sleege, G; Petry, D
2007-01-01
VERITAS is a new atmospheric Cherenkov imaging telescope array to detect very high energy gamma rays above 100 GeV. The array is located in southern Arizona, USA, at an altitude of 1268m above sea level. The array consists of four 12-m telescopes of Davies-Cotton design and structurally resembling the Whipple 10-m telescope. The four focal plane instruments are equipped with high-resolution (499 pixels) fast photo-multiplier-tube (PMT) cameras covering a 3.5 degree field of view with 0.15 degree pixel separation. Light concentrators reduce the dead-space between PMTs to 25% and shield the PMTs from ambient light. The PMTs are connected to high-speed preamplifiers allowing operation at modest anode current and giving good single photoelectron peaks in situ. Electronics in the focus box provides real-time monitoring of the anode currents for each pixel and ambient environmental conditions. A charge injection subsystem installed in the focus box allows daytime testing of the trigger and data acquisition system b...
Radioactivity in the galactic plane
Walraven, G. D.; Haymes, R. C.
1976-01-01
The paper reports the detection of a large concentration of interstellar radioactivity during balloon-altitude measurements of gamma-ray energy spectra in the band between 0.02 and 12.27 MeV from galactic and extragalactic sources. Enhanced counting rates were observed in three directions towards the plane of the Galaxy; a power-law energy spectrum is computed for one of these directions (designated B 10). A large statistical deviation from the power law in a 1.0-FWHM interval centered near 1.16 MeV is discussed, and the existence of a nuclear gamma-ray line at 1.15 MeV in B 10 is postulated. It is suggested that Ca-44, which emits gamma radiation at 1.156 MeV following the decay of radioactive Sc-44, is a likely candidate for this line, noting that Sc-44 arises from Ti-44 according to explosive models of supernova nucleosynthesis. The 1.16-MeV line flux inferred from the present data is shown to equal the predicted flux for a supernova at a distance of approximately 3 kpc and an age not exceeding about 100 years.
A Collaborative Knowledge Plane for Autonomic Networks
Mbaye, Maïssa; Krief, Francine
Autonomic networking aims to give network components self-managing capabilities. Several autonomic architectures have been proposed. Each of these architectures includes sort of a knowledge plane which is very important to mimic an autonomic behavior. Knowledge plane has a central role for self-functions by providing suitable knowledge to equipment and needs to learn new strategies for more accuracy.However, defining knowledge plane's architecture is still a challenge for researchers. Specially, defining the way cognitive supports interact each other in knowledge plane and implementing them. Decision making process depends on these interactions between reasoning and learning parts of knowledge plane. In this paper we propose a knowledge plane's architecture based on machine learning (inductive logic programming) paradigm and situated view to deal with distributed environment. This architecture is focused on two self-functions that include all other self-functions: self-adaptation and self-organization. Study cases are given and implemented.
DEFF Research Database (Denmark)
Byg din egen boomerang, kast den, se den flyve, forstå hvorfor og hvordan den vender tilbage, og grib den. Det handler om opdriften på vingerne når du flyver, men det handler også og allermest om den mærkværdige gyroskop-effekt, du bruger til at holde balancen, når du kører på cykel. Vi vil bruge...
High-order exact solutions for pseudo-plane ideal flows
Sun, Che
2016-08-01
A steady pseudo-plane ideal flow (PIF) model is derived from the 3D Euler equations under Boussinesq approximation. The model is solved analytically to yield high-degree polynomial exact solutions. Unlike quadratic flows, the cubic and quartic solutions display reduced geometry in the form of straightline jet, circular vortex, and multipolar strain field. The high-order circular-vortex solutions are vertically aligned and even the non-aligned multipolar strain-field solutions display vertical concentricity. Such geometry reduction is explained by an analytical theorem stating that only straightline jet and circular vortex have functional solutions to the PIF model.
From Hilbert's Axioms to Circle-squaring in the Hyperbolic Plane
Hasvoldseter, Jin
2011-01-01
This thesis is based on M. J. Greenberg's article "Old and New Results in the Foundation of Elementary Plane Euclidean and Non-Euclidean Geometries" (American Mathematical Monthly , Vol 117, No 3 pp. 198-219). The aim of this thesis is to give a more complete description of some of the interesting topics in this article. We will start with Hilbert's axioms and Euclid's propositions, and then focus on hyperbolic geometry. We will proceed to give a complete proof of the uniformity theorem by us...
RF/Optical Demonstration: Focal Plane Assembly
Hoppe, D. J.; Chung, S.; Kovalik, J.; Gama, E.; Fernandez, M. M.
2016-11-01
In this article, we describe the second-generation focal plane optical assembly employed in the RF/optical demonstration at DSS-13. This assembly receives reflected light from the two mirror segments mounted on the RF primary. The focal plane assembly contains a fast steering mirror (FSM) to stabilize the focal plane spot, a pupil camera to aid in aligning the two segments, and several additional cameras for receiving the optical signal prior to as well as after the FSM loop.
Large Format Uncooled Focal Plane Array Project
National Aeronautics and Space Administration — Black Forest Engineering has identified innovative modifications in uncooled focal plane array (UFPA) architecture and processing that allows development of large...
3D geometry analysis of the medial meniscus--a statistical shape modeling approach.
Vrancken, A C T; Crijns, S P M; Ploegmakers, M J M; O'Kane, C; van Tienen, T G; Janssen, D; Buma, P; Verdonschot, N
2014-10-01
The geometry-dependent functioning of the meniscus indicates that detailed knowledge on 3D meniscus geometry and its inter-subject variation is essential to design well functioning anatomically shaped meniscus replacements. Therefore, the aim of this study was to quantify 3D meniscus geometry and to determine whether variation in medial meniscus geometry is size- or shape-driven. Also we performed a cluster analysis to identify distinct morphological groups of medial menisci and assessed whether meniscal geometry is gender-dependent. A statistical shape model was created, containing the meniscus geometries of 35 subjects (20 females, 15 males) that were obtained from MR images. A principal component analysis was performed to determine the most important modes of geometry variation and the characteristic changes per principal component were evaluated. Each meniscus from the original dataset was then reconstructed as a linear combination of principal components. This allowed the comparison of male and female menisci, and a cluster analysis to determine distinct morphological meniscus groups. Of the variation in medial meniscus geometry, 53.8% was found to be due to primarily size-related differences and 29.6% due to shape differences. Shape changes were most prominent in the cross-sectional plane, rather than in the transverse plane. Significant differences between male and female menisci were only found for principal component 1, which predominantly reflected size differences. The cluster analysis resulted in four clusters, yet these clusters represented two statistically different meniscal shapes, as differences between cluster 1, 2 and 4 were only present for principal component 1. This study illustrates that differences in meniscal geometry cannot be explained by scaling only, but that different meniscal shapes can be distinguished. Functional analysis, e.g. through finite element modeling, is required to assess whether these distinct shapes actually influence
3D geometry analysis of the medial meniscus – a statistical shape modeling approach
Vrancken, A C T; Crijns, S P M; Ploegmakers, M J M; O'Kane, C; van Tienen, T G; Janssen, D; Buma, P; Verdonschot, N
2014-01-01
The geometry-dependent functioning of the meniscus indicates that detailed knowledge on 3D meniscus geometry and its inter-subject variation is essential to design well functioning anatomically shaped meniscus replacements. Therefore, the aim of this study was to quantify 3D meniscus geometry and to determine whether variation in medial meniscus geometry is size- or shape-driven. Also we performed a cluster analysis to identify distinct morphological groups of medial menisci and assessed whether meniscal geometry is gender-dependent. A statistical shape model was created, containing the meniscus geometries of 35 subjects (20 females, 15 males) that were obtained from MR images. A principal component analysis was performed to determine the most important modes of geometry variation and the characteristic changes per principal component were evaluated. Each meniscus from the original dataset was then reconstructed as a linear combination of principal components. This allowed the comparison of male and female menisci, and a cluster analysis to determine distinct morphological meniscus groups. Of the variation in medial meniscus geometry, 53.8% was found to be due to primarily size-related differences and 29.6% due to shape differences. Shape changes were most prominent in the cross-sectional plane, rather than in the transverse plane. Significant differences between male and female menisci were only found for principal component 1, which predominantly reflected size differences. The cluster analysis resulted in four clusters, yet these clusters represented two statistically different meniscal shapes, as differences between cluster 1, 2 and 4 were only present for principal component 1. This study illustrates that differences in meniscal geometry cannot be explained by scaling only, but that different meniscal shapes can be distinguished. Functional analysis, e.g. through finite element modeling, is required to assess whether these distinct shapes actually influence
On the C*-algebras of foliations in the plane
Wang, Xiaolu
1987-01-01
The main result of this original research monograph is the classification of C*-algebras of ordinary foliations of the plane in terms of a class of -trees. It reveals a close connection between some most recent developments in modern analysis and low-dimensional topology. It introduces noncommutative CW-complexes (as the global fibred products of C*-algebras), among other things, which adds a new aspect to the fast-growing field of noncommutative topology and geometry. The reader is only required to know basic functional analysis. However, some knowledge of topology and dynamical systems will be helpful. The book addresses graduate students and experts in the area of analysis, dynamical systems and topology.
Fast decoding of codes from algebraic plane curves
DEFF Research Database (Denmark)
Justesen, Jørn; Larsen, Knud J.; Jensen, Helge Elbrønd
1992-01-01
Improvement to an earlier decoding algorithm for codes from algebraic geometry is presented. For codes from an arbitrary regular plane curve the authors correct up to d*/2-m2 /8+m/4-9/8 errors, where d* is the designed distance of the code and m is the degree of the curve. The complexity of finding...... the error locator is O(n7/3 ), where n is the length of the code. For codes from Hermitian curves the complexity of finding the error values, given the error locator, is O(n2), and the same complexity can be obtained in the general case if only d*/2-m2/2 errors are corrected...
Casimir attraction in multilayered plane parallel magnetodielectric systems
Ellingsen, S A
2006-01-01
A powerful procedure is presented for calculating the Casimir attraction between plane parallel multilayers made up of homogeneous regions with arbitrary magnetic and dielectric properties by use of the Minkowski energy-momentum tensor. The theory is applied to numerous geometries and shown to reproduce a number of results obtained by other authors. Although the various pieces of theory drawn upon are well known, the relative ease with which the Casimir force density in even complex planar structures may be calculated, appears not to be widely appreciated, and no single paper to the author's knowledge renders explicitly the procedure demonstrated herein. Results may be seen as an important building block in the settling of issues of fundamental interest, such as the long-standing dispute over the thermal behaviour of the Casimir force or the question of what is the correct stress tensor to apply, a discussion re-quickened by the newly suggested alternative theory due to Raabe and Welsch.
Electron states in quantum rings with structural distortions under axial or in-plane magnetic fields
Energy Technology Data Exchange (ETDEWEB)
Planelles, J [Departament de Quimica Fisica i Analitica, Universitat Jaume I, Box 224, E-12080 Castello (Spain); Rajadell, F [Departament de Quimica Fisica i Analitica, Universitat Jaume I, Box 224, E-12080 Castello (Spain); Climente, J I [Departament de Quimica Fisica i Analitica, Universitat Jaume I, Box 224, E-12080 Castello (Spain)
2007-09-19
A comprehensive study of anisotropic quantum rings, QRs, subject to axial and in-plane magnetic field, both aligned and transverse to the anisotropy direction, is carried out. Elliptical QRs for a wide range of eccentricity values and also perfectly circular QRs including one or more barriers disturbing the QR current are considered. These models mimic anisotropic geometry deformations and mass diffusion occurring in the QR fabrication process. Symmetry considerations and simplified analytical models supply physical insight into the obtained numerical results. Our study demonstrates that, except for unusual extremely large eccentricities, QR geometry deformations only appreciably influence a few low-lying states, while the effect of barriers disturbing the QR current is stronger and affects all studied states to a similar extent. We also show that the response of the electron states to in-plane magnetic fields provides accurate information on the structural anisotropy.
Geometry of Membrane Sigma Models
Vysoky, Jan
2015-01-01
String theory still remains one of the promising candidates for a unification of the theory of gravity and quantum field theory. One of its essential parts is relativistic description of moving multi-dimensional objects called membranes (or p-branes) in a curved spacetime. On the classical field theory level, they are described by an action functional extremalising the volume of a manifold swept by a propagating membrane. This and related field theories are collectively called membrane sigma models. Differential geometry is an important mathematical tool in the study of string theory. It turns out that string and membrane backgrounds can be conveniently described using objects defined on a direct sum of tangent and cotangent bundles of the spacetime manifold. Mathematical field studying such object is called generalized geometry. Its integral part is the theory of Leibniz algebroids, vector bundles with a Leibniz algebra bracket on its module of smooth sections. Special cases of Leibniz algebroids are better ...
Differential geometry and mathematical physics
Rudolph, Gerd
Starting from an undergraduate level, this book systematically develops the basics of • Calculus on manifolds, vector bundles, vector fields and differential forms, • Lie groups and Lie group actions, • Linear symplectic algebra and symplectic geometry, • Hamiltonian systems, symmetries and reduction, integrable systems and Hamilton-Jacobi theory. The topics listed under the first item are relevant for virtually all areas of mathematical physics. The second and third items constitute the link between abstract calculus and the theory of Hamiltonian systems. The last item provides an introduction to various aspects of this theory, including Morse families, the Maslov class and caustics. The book guides the reader from elementary differential geometry to advanced topics in the theory of Hamiltonian systems with the aim of making current research literature accessible. The style is that of a mathematical textbook,with full proofs given in the text or as exercises. The material is illustrated by numerous d...
Fractal geometry and computer graphics
Sakas, Georgios; Peitgen, Heinz-Otto; Englert, Gabriele
1992-01-01
Fractal geometry has become popular in the last 15 years, its applications can be found in technology, science, or even arts. Fractal methods and formalism are seen today as a general, abstract, but nevertheless practical instrument for the description of nature in a wide sense. But it was Computer Graphics which made possible the increasing popularity of fractals several years ago, and long after their mathematical formulation. The two disciplines are tightly linked. The book contains the scientificcontributions presented in an international workshop in the "Computer Graphics Center" in Darmstadt, Germany. The target of the workshop was to present the wide spectrum of interrelationships and interactions between Fractal Geometry and Computer Graphics. The topics vary from fundamentals and new theoretical results to various applications and systems development. All contributions are original, unpublished papers.The presentations have been discussed in two working groups; the discussion results, together with a...
The geometry of surfaces contact
Directory of Open Access Journals (Sweden)
Siegl J.
2007-11-01
Full Text Available This contribution deals with a geometrical exact description of contact between two given surfaces which are defined by the vector functions. These surfaces are substituted at a contact point by approximate surfaces of the second order in accordance with the Taylor series and consequently there is derived a differential surface of these second order surfaces. Knowledge of principal normal curvatures, their directions and the tensor (Dupin indicatrix of this differential surface are necessary for description of contact of these surfaces. For description of surface geometry the first and the second surface fundamental tensor and a further methods of the differential geometry are used. A geometrical visualisation of obtained results of this analysis is made. Method and results of this study will be applied to contact analysis of tooth screw surfaces of screw machines.
Grassmannian geometry of scattering amplitudes
Arkani-Hamed, Nima; Cachazo, Freddy; Goncharov, Alexander; Postnikov, Alexander; Trnka, Jaroslav
2016-01-01
Outlining a revolutionary reformulation of the foundations of perturbative quantum field theory, this book is a self-contained and authoritative analysis of the application of this new formulation to the case of planar, maximally supersymmetric Yang–Mills theory. The book begins by deriving connections between scattering amplitudes and Grassmannian geometry from first principles before introducing novel physical and mathematical ideas in a systematic manner accessible to both physicists and mathematicians. The principle players in this process are on-shell functions which are closely related to certain sub-strata of Grassmannian manifolds called positroids - in terms of which the classification of on-shell functions and their relations becomes combinatorially manifest. This is an essential introduction to the geometry and combinatorics of the positroid stratification of the Grassmannian and an ideal text for advanced students and researchers working in the areas of field theory, high energy physics, and the...
Foliation theory in algebraic geometry
McKernan, James; Pereira, Jorge
2016-01-01
Featuring a blend of original research papers and comprehensive surveys from an international team of leading researchers in the thriving fields of foliation theory, holomorphic foliations, and birational geometry, this book presents the proceedings of the conference "Foliation Theory in Algebraic Geometry," hosted by the Simons Foundation in New York City in September 2013. Topics covered include: Fano and del Pezzo foliations; the cone theorem and rank one foliations; the structure of symmetric differentials on a smooth complex surface and a local structure theorem for closed symmetric differentials of rank two; an overview of lifting symmetric differentials from varieties with canonical singularities and the applications to the classification of AT bundles on singular varieties; an overview of the powerful theory of the variety of minimal rational tangents introduced by Hwang and Mok; recent examples of varieties which are hyperbolic and yet the Green-Griffiths locus is the whole of X; and a classificati...
Euclidean distance geometry an introduction
Liberti, Leo
2017-01-01
This textbook, the first of its kind, presents the fundamentals of distance geometry: theory, useful methodologies for obtaining solutions, and real world applications. Concise proofs are given and step-by-step algorithms for solving fundamental problems efficiently and precisely are presented in Mathematica®, enabling the reader to experiment with concepts and methods as they are introduced. Descriptive graphics, examples, and problems, accompany the real gems of the text, namely the applications in visualization of graphs, localization of sensor networks, protein conformation from distance data, clock synchronization protocols, robotics, and control of unmanned underwater vehicles, to name several. Aimed at intermediate undergraduates, beginning graduate students, researchers, and practitioners, the reader with a basic knowledge of linear algebra will gain an understanding of the basic theories of distance geometry and why they work in real life.
Groups and Geometries : Siena Conference
Kantor, William; Lunardon, Guglielmo; Pasini, Antonio; Tamburini, Maria
1998-01-01
On September 1-7, 1996 a conference on Groups and Geometries took place in lovely Siena, Italy. It brought together experts and interested mathematicians from numerous countries. The scientific program centered around invited exposi tory lectures; there also were shorter research announcements, including talks by younger researchers. The conference concerned a broad range of topics in group theory and geometry, with emphasis on recent results and open problems. Special attention was drawn to the interplay between group-theoretic methods and geometric and combinatorial ones. Expanded versions of many of the talks appear in these Proceedings. This volume is intended to provide a stimulating collection of themes for a broad range of algebraists and geometers. Among those themes, represented within the conference or these Proceedings, are aspects of the following: 1. the classification of finite simple groups, 2. the structure and properties of groups of Lie type over finite and algebraically closed fields of f...
Gear geometry of cycloid drives
Institute of Scientific and Technical Information of China (English)
CHEN BingKui; FANG TingTing; LI ChaoYang; WANG ShuYan
2008-01-01
According to differential geometry and gear geometry,the equation of meshing for small teeth difference planetary gearing and a universal equation of conjugated profile are established based on cylindrical pin tooth and given motion.The correct meshing condition,contact line,contact ratio,calculating method for pin tooth's maximum contact point are developed.Investigation on the theory of conjugated meshing is carried out when the tooth difference numbers between pin wheel and cycloidal gear are 1,2,3 and -1,respectively.A general method called enveloping method to generate hypocycloid and epicycloid is put forward.The correct mesh-ing condition for cycloid pin wheel gearing is provided,and the contact line and the contact ratio are also discussed.
Geometry of polycrystals and microstructure
Directory of Open Access Journals (Sweden)
Ball John M.
2015-01-01
Full Text Available We investigate the geometry of polycrystals, showing that for polycrystals formed of convex grains the interior grains are polyhedral, while for polycrystals with general grain geometry the set of triple points is small. Then we investigate possible martensitic morphologies resulting from intergrain contact. For cubic-totetragonal transformations we show that homogeneous zero-energy microstructures matching a pure dilatation on a grain boundary necessarily involve more than four deformation gradients. We discuss the relevance of this result for observations of microstructures involving second and third-order laminates in various materials. Finally we consider the more specialized situation of bicrystals formed from materials having two martensitic energy wells (such as for orthorhombic to monoclinic transformations, but without any restrictions on the possible microstructure, showing how a generalization of the Hadamard jump condition can be applied at the intergrain boundary to show that a pure phase in either grain is impossible at minimum energy.
Gear geometry of cycloid drives
Institute of Scientific and Technical Information of China (English)
2008-01-01
According to differential geometry and gear geometry, the equation of meshing for small teeth difference planetary gearing and a universal equation of conjugated profile are established based on cylindrical pin tooth and given motion. The correct meshing condition, contact line, contact ratio, calculating method for pin tooth’s maximum contact point are developed. Investigation on the theory of conjugated meshing is carried out when the tooth difference numbers between pin wheel and cycloidal gear are 1, 2, 3 and ?1, respectively. A general method called enveloping method to generate hypocycloid and epicycloid is put forward. The correct meshing condition for cycloid pin wheel gearing is provided, and the contact line and the contact ratio are also discussed.
The geometry of celestial mechanics
Geiges, Hansjörg
2016-01-01
Celestial mechanics is the branch of mathematical astronomy devoted to studying the motions of celestial bodies subject to the Newtonian law of gravitation. This mathematical introductory textbook reveals that even the most basic question in celestial mechanics, the Kepler problem, leads to a cornucopia of geometric concepts: conformal and projective transformations, spherical and hyperbolic geometry, notions of curvature, and the topology of geodesic flows. For advanced undergraduate and beginning graduate students, this book explores the geometric concepts underlying celestial mechanics and is an ideal companion for introductory courses. The focus on the history of geometric ideas makes it perfect supplementary reading for students in elementary geometry and topology. Numerous exercises, historical notes and an extensive bibliography provide all the contextual information required to gain a solid grounding in celestial mechanics.
Core foundations of abstract geometry.
Dillon, Moira R; Huang, Yi; Spelke, Elizabeth S
2013-08-27
Human adults from diverse cultures share intuitions about the points, lines, and figures of Euclidean geometry. Do children develop these intuitions by drawing on phylogenetically ancient and developmentally precocious geometric representations that guide their navigation and their analysis of object shape? In what way might these early-arising representations support later-developing Euclidean intuitions? To approach these questions, we investigated the relations among young children's use of geometry in tasks assessing: navigation; visual form analysis; and the interpretation of symbolic, purely geometric maps. Children's navigation depended on the distance and directional relations of the surface layout and predicted their use of a symbolic map with targets designated by surface distances. In contrast, children's analysis of visual forms depended on the size-invariant shape relations of objects and predicted their use of the same map but with targets designated by corner angles. Even though the two map tasks used identical instructions and map displays, children's performance on these tasks showed no evidence of integrated representations of distance and angle. Instead, young children flexibly recruited geometric representations of either navigable layouts or objects to interpret the same spatial symbols. These findings reveal a link between the early-arising geometric representations that humans share with diverse animals and the flexible geometric intuitions that give rise to human knowledge at its highest reaches. Although young children do not appear to integrate core geometric representations, children's use of the abstract geometry in spatial symbols such as maps may provide the earliest clues to the later construction of Euclidean geometry.
Topics in modern differential geometry
Verstraelen, Leopold
2017-01-01
A variety of introductory articles is provided on a wide range of topics, including variational problems on curves and surfaces with anisotropic curvature. Experts in the fields of Riemannian, Lorentzian and contact geometry present state-of-the-art reviews of their topics. The contributions are written on a graduate level and contain extended bibliographies. The ten chapters are the result of various doctoral courses which were held in 2009 and 2010 at universities in Leuven, Serbia, Romania and Spain.
Geometry for the accelerating universe
Punzi, R; Wohlfarth, M N R; Punzi, Raffaele; Schuller, Frederic P.; Wohlfarth, Mattias N.R.
2006-01-01
The Lorentzian spacetime metric is replaced by an area metric which naturally emerges as a generalized geometry in quantum string and gauge theory. Employing the area metric curvature scalar, the gravitational Einstein-Hilbert action is re-interpreted as dynamics for an area metric. Without the need for dark energy or fine-tuning, area metric cosmology explains the observed small acceleration of the late Universe.
Number theory III Diophantine geometry
1991-01-01
From the reviews of the first printing of this book, published as Volume 60 of the Encyclopaedia of Mathematical Sciences: "Between number theory and geometry there have been several stimulating influences, and this book records of these enterprises. This author, who has been at the centre of such research for many years, is one of the best guides a reader can hope for. The book is full of beautiful results, open questions, stimulating conjectures and suggestions where to look for future developments. This volume bears witness of the broad scope of knowledge of the author, and the influence of several people who have commented on the manuscript before publication ... Although in the series of number theory, this volume is on diophantine geometry, and the reader will notice that algebraic geometry is present in every chapter. ... The style of the book is clear. Ideas are well explained, and the author helps the reader to pass by several technicalities. Reading and rereading this book I noticed that the topics ...
Black Holes as Effective Geometries
Balasubramanian, Vijay; El-Showk, Sheer; Messamah, Ilies
2008-01-01
Gravitational entropy arises in string theory via coarse graining over an underlying space of microstates. In this review we would like to address the question of how the classical black hole geometry itself arises as an effective or approximate description of a pure state, in a closed string theory, which semiclassical observers are unable to distinguish from the "naive" geometry. In cases with enough supersymmetry it has been possible to explicitly construct these microstates in spacetime, and understand how coarse-graining of non-singular, horizon-free objects can lead to an effective description as an extremal black hole. We discuss how these results arise for examples in Type II string theory on AdS_5 x S^5 and on AdS_3 x S^3 x T^4 that preserve 16 and 8 supercharges respectively. For such a picture of black holes as effective geometries to extend to cases with finite horizon area the scale of quantum effects in gravity would have to extend well beyond the vicinity of the singularities in the effective t...
Introduction to geometry and relativity
2013-01-01
This book provides a lucid introduction to both modern differential geometry and relativity for advanced undergraduates and first-year graduate students of applied mathematics and physical sciences. This book meets an overwhelming need for a book on modern differential geometry and relativity that is student-friendly, and which is also suitable for self-study. The book presumes a minimal level of mathematical maturity so that any student who has completed the standard Calculus sequence should be able to read and understand the book. The key features of the book are: Detailed solutions are provided to the Exercises in each chapter; Many of the missing steps that are often omitted from standard mathematical derivations have been provided to make the book easier to read and understand; A detailed introduction to Electrodynamics is provided so that the book is accessible to students who have not had a formal course in this area; In its treatment of modern differential geometry, the book employs both a modern, c...
Slipping and Rolling on an Inclined Plane
Aghamohammadi, Cina; Aghamohammadi, Amir
2011-01-01
In the first part of the paper, using a direct calculation two-dimensional motion of a particle sliding on an inclined plane is investigated for general values of friction coefficient ([mu]). A parametric equation for the trajectory of the particle is also obtained. In the second part of the paper, the motion of a sphere on the inclined plane is…
The European Galactic Plane Surveys: EGAPS
Groot, P.J.; Drew, J.; Greimel, R.; Gaensicke, B.; Knigge, C.; Irwin, M.; Mampaso, A.; Augusteijn, T.; Morales-Rueda, L.; Barlow, M.; Iphas, C.; Uvex, C.; Vphas, C.
2006-01-01
Introduction: The European Galactic Plane Surveys (EGAPS) will for the first time ever map the complete galactic plane (10x360 degrees) down to 21st magnitude in u', g', r', i' and H-alpha and partly in He I 5875. It will complete a database of ~1 billion objects. The aim of EGAPS is to study popula
Fast & Furious focal-plane wavefront sensing
Korkiakoski, V.A.; Keller, C.U.; Doelman, N.; Kenworthy, M.; Otten, G.; Verhaegen, M.H.G.
2014-01-01
We present two complementary algorithms suitable for using focal-plane measurements to control a wavefront corrector with an extremely high-spatial resolution. The algorithms use linear approximations to iteratively minimize the aberrations seen by the focal-plane camera. The first algorithm, Fast &
Focal-plane sensor-processor chips
Zarándy, Ákos
2011-01-01
Focal-Plane Sensor-Processor Chips explores both the implementation and application of state-of-the-art vision chips. Presenting an overview of focal plane chip technology, the text discusses smart imagers and cellular wave computers, along with numerous examples of current vision chips.
Candela, Anna Maria; Sánchez, Miguel
2013-01-01
Recently, classical results on completeness of trajectories of Hamiltonian systems obtained at the beginning of the seventies, have been revisited, improved and applied to Lorentzian Geometry. Our aim here is threefold: to give explicit proofs of some technicalities in the background of the specialists, to show that the introduced tools allow to obtain more results for the completeness of the trajectories, and to apply these results to the completeness of spacetimes that generalize classical plane and pp-waves.
Geometry success in 20 minutes a day
LLC, LearningExpress
2014-01-01
Whether you're new to geometry or just looking for a refresher, Geometry Success in 20 Minutes a Day offers a 20-step lesson plan that provides quick and thorough instruction in practical, critical skills. Stripped of unnecessary math jargon but bursting with geometry essentials, Geometry Success in 20 Minutes a Day: Covers all vital geometry skills, from the basic building blocks of geometry to ratio, proportion, and similarity to trigonometry and beyond Provides hundreds of practice exercises in test format Applies geometr
Algebraic Geometry and Number Theory Summer School
Sarıoğlu, Celal; Soulé, Christophe; Zeytin, Ayberk
2017-01-01
This lecture notes volume presents significant contributions from the “Algebraic Geometry and Number Theory” Summer School, held at Galatasaray University, Istanbul, June 2-13, 2014. It addresses subjects ranging from Arakelov geometry and Iwasawa theory to classical projective geometry, birational geometry and equivariant cohomology. Its main aim is to introduce these contemporary research topics to graduate students who plan to specialize in the area of algebraic geometry and/or number theory. All contributions combine main concepts and techniques with motivating examples and illustrative problems for the covered subjects. Naturally, the book will also be of interest to researchers working in algebraic geometry, number theory and related fields.
Local geometry of electromagnetic fields and its role in molecular multipole transitions
Yang, Nan
2010-01-01
Electromagnetic fields with complex spatial variation routinely arise in Nature. We study the response of a small molecule to monochromatic fields of arbitrary three-dimensional geometry. First, we consider the allowed configurations of the fields and field gradients at a single point in space. Many configurations cannot be generated from a single plane wave, regardless of polarization, but any allowed configuration can be generated by superposition of multiple plane waves. There is no local configuration of the fields and gradients that requires near-field effects. Second, we derive a set of local electromagnetic quantities, where each couples to a particular multipole transition. These quantities are small or zero in plane waves, but can be large in regions of certain superpositions of plane waves. Our findings provide a systematic framework for designing far-field and near-field experiments to drive multipole transitions. The proposed experiments provide information on molecular structure that is inaccessi...
Slipping and rolling on an inclined plane
Energy Technology Data Exchange (ETDEWEB)
Aghamohammadi, Cina [Department of Electrical Engineering, Sharif University of Technology, PO Box 11365-11155, Tehran (Iran, Islamic Republic of); Aghamohammadi, Amir, E-mail: mohamadi@alzahra.ac.ir [Department of Physics, Alzahra University, Tehran 19938-91176 (Iran, Islamic Republic of)
2011-07-15
In the first part of the paper, using a direct calculation two-dimensional motion of a particle sliding on an inclined plane is investigated for general values of friction coefficient ({mu}). A parametric equation for the trajectory of the particle is also obtained. In the second part of the paper, the motion of a sphere on the inclined plane is studied. It is shown that the evolution equation for the contact point of a sliding sphere is similar to that of a point particle sliding on an inclined plane whose friction coefficient is 7/2 {mu}. If {mu} > 2/7 tan {theta}, for any arbitrary initial velocity and angular velocity, the sphere will roll on the inclined plane after some finite time. In other cases, it will slip on the inclined plane. In the case of rolling, the centre of the sphere moves on a parabola. Finally the velocity and angular velocity of the sphere are exactly computed.
Slipping and Rolling on an Inclined Plane
Aghamohammadi, Cina; 10.1088/0143-0807/32/4/017
2011-01-01
In the first part of the article using a direct calculation two-dimensional motion of a particle sliding on an inclined plane is investigated for general values of friction coefficient ($\\mu$). A parametric equation for the trajectory of the particle is also obtained. In the second part of the article the motion of a sphere on the inclined plane is studied. It is shown that the evolution equation for the contact point of a sliding sphere is similar to that of a point particle sliding on an inclined plane whose friction coefficient is $2/7}\\ \\mu$. If $\\mu> 2/7 \\tan\\theta$, for any arbitrary initial velocity and angular velocity the sphere will roll on the inclined plane after some finite time. In other cases, it will slip on the inclined plane. In the case of rolling center of the sphere moves on a parabola. Finally the velocity and angular velocity of the sphere are exactly computed.
Confocal X-ray fluorescence micro-spectroscopy experiment in tilted geometry
Energy Technology Data Exchange (ETDEWEB)
Czyzycki, Mateusz, E-mail: Mateusz.Czyzycki@desy.de [DESY Photon Science, Notkestr. 85, D-22607 Hamburg (Germany); AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, Al. A. Mickiewicza 30, 30-059 Krakow (Poland); Wrobel, Pawel; Lankosz, Marek [AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, Al. A. Mickiewicza 30, 30-059 Krakow (Poland)
2014-07-01
This paper provides a generalized mathematical model to describe the intensity of primary X-ray fluorescence radiation collected in the tilted confocal geometry mode, where the collimating optics is rotated over an angle relative to a horizontal plane. The influence of newly introduced terms, which take into account the tilted geometry mode, is discussed. The model is verified with a multi-layer test sample scanned in depth. It is proved that for low-Z matrices, the rotation of the detection channel does not induce any significant differences in a reconstruction of the thickness and chemical composition of layers, so that it may safely be ignored. - Highlights: • A mathematical model for confocal XRF spectroscopy in tilted geometry was derived. • Tilted geometry influenced the analytical capabilities of XRF instrument slightly. • Thickness and the chemical composition of multi-layers were determined.
Network geometry with flavor: From complexity to quantum geometry
Bianconi, Ginestra; Rahmede, Christoph
2016-03-01
Network geometry is attracting increasing attention because it has a wide range of applications, ranging from data mining to routing protocols in the Internet. At the same time advances in the understanding of the geometrical properties of networks are essential for further progress in quantum gravity. In network geometry, simplicial complexes describing the interaction between two or more nodes play a special role. In fact these structures can be used to discretize a geometrical d -dimensional space, and for this reason they have already been widely used in quantum gravity. Here we introduce the network geometry with flavor s =-1 ,0 ,1 (NGF) describing simplicial complexes defined in arbitrary dimension d and evolving by a nonequilibrium dynamics. The NGF can generate discrete geometries of different natures, ranging from chains and higher-dimensional manifolds to scale-free networks with small-world properties, scale-free degree distribution, and nontrivial community structure. The NGF admits as limiting cases both the Bianconi-Barabási models for complex networks, the stochastic Apollonian network, and the recently introduced model for complex quantum network manifolds. The thermodynamic properties of NGF reveal that NGF obeys a generalized area law opening a new scenario for formulating its coarse-grained limit. The structure of NGF is strongly dependent on the dimensionality d . In d =1 NGFs grow complex networks for which the preferential attachment mechanism is necessary in order to obtain a scale-free degree distribution. Instead, for NGF with dimension d >1 it is not necessary to have an explicit preferential attachment rule to generate scale-free topologies. We also show that NGF admits a quantum mechanical description in terms of associated quantum network states. Quantum network states evolve by a Markovian dynamics and a quantum network state at time t encodes all possible NGF evolutions up to time t . Interestingly the NGF remains fully classical but
Towards a Nano Geometry? Geometry and Dynamics on Nano Scale
Booss-Bavnbek, Bernhelm
2012-01-01
This paper applies I.M. Gelfand's distinction between adequate and non-adequate use of mathematical language in different contexts to the newly opened window of model-based measurements of intracellular dynamics. The specifics of geometry and dynamics on the mesoscale of cell physiology are elaborated - in contrast to the familiar Newtonian mechanics and the more recent, but by now also rather well established quantum field theories. Examples are given originating from the systems biology of insulin secreting pancreatic beta-cells and the mathematical challenges of an envisioned non-invasive control of magnetic nanoparticles.
Geometric plane shapes for computer-generated holographic engraving codes
Augier, Ángel G.; Rabal, Héctor; Sánchez, Raúl B.
2017-04-01
We report a new theoretical and experimental study on hologravures, as holographic computer-generated laser-engravings. A geometric theory of images based on the general principles of light ray behaviour is shown. The models used are also applicable for similar engravings obtained by any non-laser method, and the solutions allow for the analysis of particular situations, not only in the case of light reflection mode, but also in transmission mode geometry. This approach is a novel perspective allowing the three-dimensional (3D) design of engraved images for specific ends. We prove theoretically that plane curves of very general geometric shapes can be used to encode image information onto a two-dimensional (2D) engraving, showing notable influence on the behaviour of reconstructed images that appears as an exciting investigation topic, extending its applications. Several cases of code using particular curvilinear shapes are experimentally studied. The computer-generated objects are coded by using the chosen curve type, and engraved by a laser on a plane surface of suitable material. All images are recovered optically by adequate illumination. The pseudoscopic or orthoscopic character of these images is considered, and an appropriate interpretation is presented.
Calculation of a static potential created by plane fractal cluster
Nigmatullin, Raoul R.; Alekhin, Alexander P.
2011-12-01
In this paper we demonstrate new approach that can help in calculation of electrostatic potential of a fractal (self-similar) cluster that is created by a system of charged particles. For this purpose we used the simplified model of a plane dendrite cluster [1] that is generated by a system of the concentric charged rings located in some horizontal plane (see Fig. 2). The radiuses and charges of the system of concentric rings satisfy correspondingly to relationships: rn = r0ξn and en = e0bn, where n determines the number of a current ring. The self-similar structure of the system considered allows to reduce the problem to consideration of the functional equation that similar to the conventional scaling equation. Its solution represents itself the sum of power-low terms of integer order and non-integer power-law term multiplied to a log-periodic function [5,6]. The appearance of this term was confirmed numerically for internal region of the self-similar cluster ( r0 ≪ r ≪ rN-1 ), where r0, rN-1 determine the smallest and the largest radiuses of the limiting rings correspondingly. The results were obtained for homogeneously ( b > 0) and heterogeneously ( b < 0) charged rings. We expect that this approach allows to consider more complex self-similar structures with different geometries of charge distributions.
A Whirlwind Tour of Computational Geometry.
Graham, Ron; Yao, Frances
1990-01-01
Described is computational geometry which used concepts and results from classical geometry, topology, combinatorics, as well as standard algorithmic techniques such as sorting and searching, graph manipulations, and linear programing. Also included are special techniques and paradigms. (KR)
Integral Transport Theory in One-dimensional Geometries
Energy Technology Data Exchange (ETDEWEB)
Carlvik, I.
1966-06-15
A method called DIT (Discrete Integral Transport) has been developed for the numerical solution of the transport equation in one-dimensional systems. The characteristic features of the method are Gaussian integration over the coordinate as described by Kobayashi and Nishihara, and a particular scheme for the calculation of matrix elements in annular and spherical geometry that has been used for collision probabilities in earlier Flurig programmes. The paper gives a general theory including such things as anisotropic scattering and multi-pole fluxes, and it gives a brief description of the Flurig scheme. Annular geometry is treated in some detail, and corresponding formulae are given for spherical and plane geometry. There are many similarities between DIT and the method of collision probabilities. DIT is in many cases faster, because for a certain accuracy in the fluxes DIT often needs fewer space points than the method of collision probabilities needs regions. Several computer codes using DIT, both one-group and multigroup, have been written. It is anticipated that experience gained in calculations with these codes will be reported in another paper.
Quantum fluctuations of lightcone in 4-dimensional spacetime with parallel plane boundaries
Yu, H; Yu, Hongwei; Wu, Pu-Xun
2003-01-01
Quantum fluctuations of lightcone are examined in a 4-dimensional spacetime with two parallel planes. Both the Dirichlet and the Neumann boundary conditions are considered. In all the cases we have studied, quantum lightcone fluctuations are greater where the Neumann boundary conditions are imposed, suggesting that quantum lightcone fluctuations depend not only on the geometry and topology of the spacetime as has been argued elsewhere but also on boundary conditions. Our results also show that quantum lightcone fluctuations are larger here than that in the case of a single plane. Therefore, the confinement of gravitons in a smaller region by the presence of a second plane reinforces the quantum fluctuations and this can be understood as a consequence of the uncertainty principle.
Cutting performance orthogonal test of single plane puncture biopsy needle based on puncture force
Xu, Yingqiang; Zhang, Qinhe; Liu, Guowei
2017-04-01
Needle biopsy is a method to extract the cells from the patient's body with a needle for tissue pathological examination. Many factors affect the cutting process of soft tissue, including the geometry of the biopsy needle, the mechanical properties of the soft tissue, the parameters of the puncture process and the interaction between them. This paper conducted orthogonal experiment of main cutting parameters based on single plane puncture biopsy needle, and obtained the cutting force curve of single plane puncture biopsy needle by studying the influence of the inclination angle, diameter and velocity of the single plane puncture biopsy needle on the puncture force of the biopsy needle. Stage analysis of the cutting process of biopsy needle puncture was made to determine the main influencing factors of puncture force during the cutting process, which provides a certain theoretical support for the design of new type of puncture biopsy needle and the operation of puncture biopsy.
Two- and three-dimensional computation of solitary wave runup on non-plane beach
Directory of Open Access Journals (Sweden)
B. H. Choi
2008-06-01
Full Text Available Solitary wave runup on a non-plane beach is studied analytically and numerically. For the theoretical approach, nonlinear shallow-water theory is applied to obtain the analytical solution for the simplified bottom geometry, such as an inclined channel whose cross-slope shape is parabolic. It generalizes Carrier-Greenspan approach for long wave runup on the inclined plane beach that is currently used now. For the numerical study, the Reynolds Averaged Navier-Stokes (RANS system is applied to study soliton runup on an inclined beach and the detailed characteristics of the wave processes (water displacement, velocity field, turbulent kinetic energy, energy dissipation are analyzed. In this study, it is theoretically and numerically proved that the existence of a parabolic cross-slope channel on the plane beach causes runup intensification, which is often observed in post-tsunami field surveys.
Slip patterns and preferred dislocation boundary planes
DEFF Research Database (Denmark)
Winther, G.
2003-01-01
The planes of deformation induced extended planar dislocation boundaries are analysed in two different co-ordinate systems, namely the macroscopic system defined by the deformation axes and the crystallographic system given by the crystallographic lattice. The analysis covers single and polycryst......The planes of deformation induced extended planar dislocation boundaries are analysed in two different co-ordinate systems, namely the macroscopic system defined by the deformation axes and the crystallographic system given by the crystallographic lattice. The analysis covers single...... and polycrystals of fcc metals in three deformation modes (rolling, tension and torsion). In the macroscopic system, boundaries lie close to the macroscopically most stressed planes. In the crystallographic system, the boundary plane depends on the grain/crystal orientation. The boundary planes in both co......-ordinate systems are rationalised based on the slip. The more the slip is concentrated on a slip plane, the closer the boundaries lie to this. The macroscopic preference arises from the macroscopic directionality of the slip. The established relations are applied to (a) prediction of boundary planes from slip...
Generalised Complex Geometry in Thermodynamical Fluctuation Theory
Directory of Open Access Journals (Sweden)
P. Fernández de Córdoba
2015-08-01
Full Text Available We present a brief overview of some key concepts in the theory of generalized complex manifolds. This new geometry interpolates, so to speak, between symplectic geometry and complex geometry. As such it provides an ideal framework to analyze thermodynamical fluctuation theory in the presence of gravitational fields. To illustrate the usefulness of generalized complex geometry, we examine a simplified version of the Unruh effect: the thermalising effect of gravitational fields on the Schroedinger wavefunction.
Safe domain and elementary geometry
Richard, J M
2004-01-01
A classical problem of mechanics involves a projectile fired from a given point with a given velocity whose direction is varied. This results in a family of trajectories whose envelope defines the border of a 'safe' domain. In the simple cases of a constant force, harmonic potential and Kepler or Coulomb motion, the trajectories are conic curves whose envelope in a plane is another conic section which can be derived either by simple calculus or by geometrical considerations. The case of harmonic forces reveals a subtle property of the maximal sum of distances within an ellipse.
Laminar simulation of intersubchannel mixing in a triangular nuclear fuel bundle geometry
Energy Technology Data Exchange (ETDEWEB)
Zaretsky, A.; Lightstone, M.F., E-mail: lightsm@mcmaster.ca; Tullis, S.
2015-12-15
Highlights: • Quasi-periodic flow was observed through rod-to-wall gaps. • Triangular subchannel flows were fundamentally irregular. • Cross-gap flow was influenced both by local and adjacent cross-gap intensity. • Phase-linking between gaps induced cross-plane peripheral circulation through rod–wall gaps. • Cross-gap flow structure was dependent on subchannel geometry. - Abstract: Predicting temperature distributions in fuel rod bundles is an important component of nuclear reactor safety analysis. Intersubchannel mixing acts to homogenize coolant temperatures thus reducing the likelihood of localized regions of high fuel temperature. Previous research has shown that intersubchannel mixing in nuclear fuel rod bundles is enhanced by a large-scale quasi-periodic energetic fluid motion, which transports fluid on the cross-plane between the narrow gaps connecting subchannels. This phenomenon has also been observed in laminar flows. Unsteady laminar flow simulations were performed in a simplified bundle of three rods with a pipe. Three similar geometries of varying gap width were examined, and a thermal trace was implemented on the first geometry. Thermal mixing was driven by the advection of energy between subchannels by the cross-plane flow. Flow through the rod-to-wall gaps in the wall subchannels alternated with a dominant frequency, particularly when rod-to-wall gaps were smaller than rod-to-rod gaps. Significant phase-linking between rod-to-wall gaps was also observed such that a peripheral circulation occurred through each gap simultaneously. Cross-plane flow through the rod-to-rod gaps in the triangular subchannel was irregular in each case. This was due to the fundamental irregularity of the triangular subchannel geometry. Vortices were continually broken up by cross-plane flow from other gaps due to the odd number of fluid pathways within the central subchannel. Cross-plane flow in subchannel geometries is highly interconnected between gaps. The
Yosano, Akira; Katakura, Akira; Takaki, Takashi; Shibahara, Takahiko
2009-05-01
In this study, we investigated how method of mandibular fixation influenced longterm postoperative stability of the maxilla in Class III cases. In particular, we investigated change in the maxillary occlusal plane after Occlusal Plane Alteration. Therefore, we focused on change in the palatal plane to evaluate stability of the maxillary occlusal plane, as the position of the palatal plane affects the maxillary occlusal plane. This study included 16 patients diagnosed with mandibular protrusion. Alteration of the occlusal plane was achieved by clockwise rotation of the maxilla by Le Fort I osteotomy and mandibular setback was performed by bilateral sagittal split ramus osteotomy. We analyzed and examined lateral cephalometric radiographs taken at 1 month, 3 months, 6 months, and 1 year after surgery. Stability achieved by two methods of mandibular fixation was compared. In one group of patients (group S) titanium screws were used, and in the other group (group P) titanium-locking mini-plates were used. No significant displacement was recognized in group S, whereas an approximately 0.7mm upward vertical displacement was recognized in the anterior nasal spine in group P. As a result, not only the angle of the palatal plane and S-N plane, but also occlusal plane angle in group P showed a greater decrease than that in group S. The results suggest that fixing the mandible with screws yielded greater stability of the maxilla and maxillary occlusal plane than fixing the mandible with titanium plates.
Geometry in the Early Years: A Commentary
Dindyal, Jaguthsing
2015-01-01
The primary goal of this paper is to provide a commentary on the teaching and learning of geometry in the early years of schooling with the set of papers in this issue as a guiding factor. It is structured around issues about geometry education of young learners, such as: what should we teach in geometry and why; representation of geometrical…
Students' Misconceptions and Errors in Transformation Geometry
Ada, Tuba; Kurtulus, Aytac
2010-01-01
This study analyses the students' performances in two-dimensional transformation geometry and explores the mistakes made by the students taking the analytic geometry course given by researchers. An examination was given to students of Education Faculties who have taken the analytic geometry course at Eskisehir Osmangazi University in Turkey. The…
Global continuation for distance geometry problems
Energy Technology Data Exchange (ETDEWEB)
More, J.J.; Wu, Zhijun
1995-03-01
Distance geometry problems arise in the interpretation of NMR data and in the determination of protein structure. The authors formulate the distance geometry problem as a global minimization problem with special structure, and show the global smoothing techniques and a continuation approach for global optimization can be used to determine solutions of distance geometry problems with a nearly 100% probability of success.
Geometry in the Early Years: A Commentary
Dindyal, Jaguthsing
2015-01-01
The primary goal of this paper is to provide a commentary on the teaching and learning of geometry in the early years of schooling with the set of papers in this issue as a guiding factor. It is structured around issues about geometry education of young learners, such as: what should we teach in geometry and why; representation of geometrical…
Number Theory, Analysis and Geometry
Goldfeld, Dorian; Jones, Peter
2012-01-01
Serge Lang was an iconic figure in mathematics, both for his own important work and for the indelible impact he left on the field of mathematics, on his students, and on his colleagues. Over the course of his career, Lang traversed a tremendous amount of mathematical ground. As he moved from subject to subject, he found analogies that led to important questions in such areas as number theory, arithmetic geometry, and the theory of negatively curved spaces. Lang's conjectures will keep many mathematicians occupied far into the future. In the spirit of Lang's vast contribution to mathematics, th
Integral geometry and representation theory
Gel'fand, I M; Vilenkin, N Ya
1966-01-01
Generalized Functions, Volume 5: Integral Geometry and Representation Theory is devoted to the theory of representations, focusing on the group of two-dimensional complex matrices of determinant one.This book emphasizes that the theory of representations is a good example of the use of algebraic and geometric methods in functional analysis, in which transformations are performed not on the points of a space, but on the functions defined on it. The topics discussed include Radon transform on a real affine space, integral transforms in the complex domain, and representations of the group of comp
Porous media geometry and transports
Adler, Pierre
1992-01-01
The goal of ""Porous Media: Geometry and Transports"" is to provide the basis of a rational and modern approach to porous media. This book emphasizes several geometrical structures (spatially periodic, fractal, and random to reconstructed) and the three major single-phase transports (diffusion, convection, and Taylor dispersion).""Porous Media"" serves various purposes. For students it introduces basic information on structure and transports. Engineers will find this book useful as a readily accessible assemblage of al the major experimental results pertaining to single-phase tr
Stochastic geometry for image analysis
Descombes, Xavier
2013-01-01
This book develops the stochastic geometry framework for image analysis purpose. Two main frameworks are described: marked point process and random closed sets models. We derive the main issues for defining an appropriate model. The algorithms for sampling and optimizing the models as well as for estimating parameters are reviewed. Numerous applications, covering remote sensing images, biological and medical imaging, are detailed. This book provides all the necessary tools for developing an image analysis application based on modern stochastic modeling.
Adding momentum to supersymmetric geometries
Energy Technology Data Exchange (ETDEWEB)
Lunin, Oleg, E-mail: olunin@albany.edu [Department of Physics, University at Albany (SUNY), Albany, NY 12222 (United States); Mathur, Samir D., E-mail: mathur.16@osu.edu [Department of Physics, Ohio State University, Columbus, OH 43210 (United States); Turton, David, E-mail: turton.7@osu.edu [Department of Physics, Ohio State University, Columbus, OH 43210 (United States)
2013-03-11
We consider general supersymmetric solutions to minimal supergravity in six dimensions, trivially lifted to IIB supergravity. To any such solution we add a traveling wave deformation involving the additional directions. The deformed solution is given in terms of a function which is harmonic in the background geometry. We also present a family of explicit examples describing microstates of the D1-D5 system on T{sup 4}. In the case where the background contains a large AdS region, the deformation is identified as corresponding to an action of a U(1) current of the D1-D5 orbifold CFT on a given state.
Adding momentum to supersymmetric geometries
Lunin, Oleg; Turton, David
2012-01-01
We consider general supersymmetric solutions to minimal supergravity in six dimensions, trivially lifted to IIB supergravity. To any such solution we add a travelling-wave deformation involving the additional directions. The deformed solution is given in terms of a function which is harmonic in the background geometry. We also present a family of explicit examples describing microstates of the D1-D5 system on T^4. In the case where the background contains a large AdS region, the deformation is identified as corresponding to an action of a U(1) current of the D1-D5 orbifold CFT on a given state.
Generalised geometry for string corrections
Coimbra, André; Triendl, Hagen; Waldram, Daniel
2014-01-01
We present a general formalism for incorporating the string corrections in generalised geometry, which necessitates the extension of the generalised tangent bundle. Not only are such extensions obstructed, string symmetries and the existence of a well-defined effective action require a precise choice of the (generalised) connection. The action takes a universal form given by a generalised Lichnerowitz--Bismut theorem. As examples of this construction we discuss the corrections linear in $\\alpha'$ in heterotic strings and the absence of such corrections for type II theories.
Loop quantum geometry: a primer
Energy Technology Data Exchange (ETDEWEB)
Corichi, Alejandro [Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, A. Postal 70-543, Mexico D.F. 04510 (Mexico)
2005-01-15
This is the written version of a lecture given at the 'VI Mexican School of Gravitation and Mathematical Physics' (Nov 21-27, 2004, Playa del Carmen, Mexico), introducing the basics of Loop Quantum Geometry. The purpose of the written contribution is to provide a Primer version, that is, a first entry into Loop Quantum Gravity and to present at the same time a friendly guide to the existing pedagogical literature on the subject. This account is geared towards graduate students and non-experts interested in learning the basics of the subject.
Exceptional geometry and Borcherds superalgebras
Palmkvist, Jakob
2015-01-01
We study generalized diffeomorphisms in exceptional geometry with U-duality group E_{n(n)} from an algebraic point of view. By extending the Lie algebra e_n to an infinite-dimensional Borcherds superalgebra, involving also the extension to e_{n+1}, the generalized Lie derivatives can be expressed in a simple way, and the expressions take the same form for any n less than 8. The closure of the transformations then follows from the Jacobi identity and the grading of e_{n+1} with respect to e_n.
Bondi accretion in trumpet geometries
Miller, August J.; Baumgarte, Thomas W.
2017-02-01
The Bondi solution, which describes the radial inflow of a gas onto a non-rotating black hole, provides a powerful test for numerical relativistic codes. However, the Bondi solution is usually derived in Schwarzschild coordinates, which are not well suited for dynamical spacetime evolutions. Instead, many current numerical relativistic codes adopt moving-puncture coordinates, which render black holes in trumpet geometries. Here we transform the Bondi solution into trumpet coordinates, which result in regular expressions for the fluid flow extending into the black-hole interior. We also evolve these solutions numerically and demonstrate their usefulness for testing and calibrating numerical codes.
Complexity and Shock Wave Geometries
Stanford, Douglas
2014-01-01
In this paper we refine a conjecture relating the time-dependent size of an Einstein-Rosen bridge to the computational complexity of the of the dual quantum state. Our refinement states that the complexity is proportional to the spatial volume of the ERB. More precisely, up to an ambiguous numerical coefficient, we propose that the complexity is the regularized volume of the largest codimension one surface crossing the bridge, divided by $G_N l_{AdS}$. We test this conjecture against a wide variety of spherically symmetric shock wave geometries in different dimensions. We find detailed agreement.
Loop Quantum Geometry: A primer
Corichi, Alejandro
2005-01-01
This is the written version of a lecture given at the ``VI Mexican School of Gravitation and Mathematical Physics" (Nov 21-27, 2004, Playa del Carmen, Mexico), introducing the basics of Loop Quantum Geometry. The purpose of the written contribution is to provide a Primer version, that is, a first entry into Loop Quantum Gravity and to present at the same time a friendly guide to the existing pedagogical literature on the subject. This account is geared towards graduate students and non-expert...
Isosurfaces geometry, topology, and algorithms
Wenger, Rephael
2013-01-01
Ever since Lorensen and Cline published their paper on the Marching Cubes algorithm, isosurfaces have been a standard technique for the visualization of 3D volumetric data. Yet there is no book exclusively devoted to isosurfaces. Isosurfaces: Geometry, Topology, and Algorithms represents the first book to focus on basic algorithms for isosurface construction. It also gives a rigorous mathematical perspective on some of the algorithms and results. In color throughout, the book covers the Marching Cubes algorithm and variants, dual contouring algorithms, multilinear interpolation, multiresolutio
Quanta of geometry and unification
Chamseddine, Ali H.
2016-11-01
This is a tribute to Abdus Salam’s memory whose insight and creative thinking set for me a role model to follow. In this contribution I show that the simple requirement of volume quantization in spacetime (with Euclidean signature) uniquely determines the geometry to be that of a noncommutative space whose finite part is based on an algebra that leads to Pati-Salam grand unified models. The Standard Model corresponds to a special case where a mathematical constraint (order one condition) is satisfied. This provides evidence that Salam was a visionary who was generations ahead of his time.
Quanta of Geometry and Unification
Chamseddine, Ali H
2016-01-01
This is a tribute to Abdus Salam's memory whose insight and creative thinking set for me a role model to follow. In this contribution I show that the simple requirement of volume quantization in space-time (with Euclidean signature) uniquely determines the geometry to be that of a noncommutative space whose finite part is based on an algebra that leads to Pati-Salam grand unified models. The Standard Model corresponds to a special case where a mathematical constraint (order one condition) is satisfied. This provides evidence that Salam was a visionary who was generations ahead of his time.
Streptococcus anginosus infections: crossing tissue planes.
Sunwoo, Bernie Y; Miller, Wallace T
2014-10-01
Streptococcus anginosus has long been recognized to cause invasive pyogenic infections. This holds true for thoracic infections where S. anginosus has a propensity for abscess and empyema formation. Early diagnosis is important given the significant morbidity and mortality associated with thoracic S. anginosus infections. Yet, distinguishing thoracic S. anginosus clinically is difficult. We present three cases of thoracic S. anginosus that demonstrated radiographic extension across tissue planes, including the interlobar fissure, diaphragm, and chest wall. Few infectious etiologies are known to cross tissue planes. Accordingly, we propose S. anginosus be considered among the differential diagnosis of potential infectious etiologies causing radiographic extension across tissue planes.
Single-view geometric calibration for C-arm inverse geometry CT.
Slagowski, Jordan M; Dunkerley, David A P; Hatt, Charles R; Speidel, Michael A
2017-01-01
Accurate and artifact-free reconstruction of tomographic images requires precise knowledge of the imaging system geometry. A projection matrix-based calibration method to enable C-arm inverse geometry CT (IGCT) is proposed. The method is evaluated for scanning-beam digital x-ray (SBDX), a C-arm mounted inverse geometry fluoroscopic technology. A helical configuration of fiducials is imaged at each gantry angle in a rotational acquisition. For each gantry angle, digital tomosynthesis is performed at multiple planes and a composite image analogous to a cone-beam projection is generated from the plane stack. The geometry of the C-arm, source array, and detector array is determined at each angle by constructing a parameterized three-dimensional-to-two-dimensional projection matrix that minimizes the sum-of-squared deviations between measured and projected fiducial coordinates. Simulations were used to evaluate calibration performance with translations and rotations of the source and detector. The relative root-mean-square error in a reconstruction of a numerical thorax phantom was 0.4% using the calibration method versus 7.7% without calibration. In phantom studies, reconstruction of SBDX projections using the proposed method eliminated artifacts present in noncalibrated reconstructions. The proposed IGCT calibration method reduces image artifacts when uncertainties exist in system geometry.
Foundations of arithmetic differential geometry
Buium, Alexandru
2017-01-01
The aim of this book is to introduce and develop an arithmetic analogue of classical differential geometry. In this new geometry the ring of integers plays the role of a ring of functions on an infinite dimensional manifold. The role of coordinate functions on this manifold is played by the prime numbers. The role of partial derivatives of functions with respect to the coordinates is played by the Fermat quotients of integers with respect to the primes. The role of metrics is played by symmetric matrices with integer coefficients. The role of connections (respectively curvature) attached to metrics is played by certain adelic (respectively global) objects attached to the corresponding matrices. One of the main conclusions of the theory is that the spectrum of the integers is "intrinsically curved"; the study of this curvature is then the main task of the theory. The book follows, and builds upon, a series of recent research papers. A significant part of the material has never been published before.
Quanta of Geometry: Noncommutative Aspects
Chamseddine, Ali H.; Connes, Alain; Mukhanov, Viatcheslav
2015-03-01
In the construction of spectral manifolds in noncommutative geometry, a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of real scalar fields naturally appears and implies, by equality with the index formula, the quantization of the volume. We first show that this condition implies that the manifold decomposes into disconnected spheres, which will represent quanta of geometry. We then refine the condition by involving the real structure and two types of geometric quanta, and show that connected spin manifolds with large quantized volume are then obtained as solutions. The two algebras M2(H ) and M4(C ) are obtained, which are the exact constituents of the standard model. Using the two maps from M4 to S4 the four-manifold is built out of a very large number of the two kinds of spheres of Planckian volume. We give several physical applications of this scheme such as quantization of the cosmological constant, mimetic dark matter, and area quantization of black holes.
Warped Geometry of Brane Worlds
Felder, G; Kofman, L A; Felder, Gary; Frolov, Andrei; Kofman, Lev
2002-01-01
We study the dynamical equations for a warp factor and a bulk scalar in 5d brane world scenarios. These equations are similar to those for the time dependence of the scale factor and a scalar field in 4d cosmology, but with the sign of the scalar field potential reversed. Based on this analogy, we introduce two novel methods for studying the warped geometry. First, we construct the full phase portraits of the warp factor/scalar system for several examples of the bulk potential. This allows us to view the global properties of the warped geometry. For flat branes, the phase portrait is two dimensional. Moving along typical phase trajectories, the warp factor is initially increasing and finally decreasing. All trajectories have timelike gradient-dominated singularities at one or both of their ends, which are reachable in a finite distance and must be screened by the branes. For curved branes, the phase portrait is three dimensional. However, as the warp factor increases the phase trajectories tend towards the tw...
Weyl gravity and Cartan geometry
Attard, J.; François, J.; Lazzarini, S.
2016-04-01
We point out that the Cartan geometry known as the second-order conformal structure provides a natural differential geometric framework underlying gauge theories of conformal gravity. We are concerned with two theories: the first one is the associated Yang-Mills-like Lagrangian, while the second, inspired by [1], is a slightly more general one that relaxes the conformal Cartan geometry. The corresponding gauge symmetry is treated within the Becchi-Rouet-Stora-Tyutin language. We show that the Weyl gauge potential is a spurious degree of freedom, analogous to a Stueckelberg field, that can be eliminated through the dressing field method. We derive sets of field equations for both the studied Lagrangians. For the second one, they constrain the gauge field to be the "normal conformal Cartan connection.''Finally, we provide in a Lagrangian framework a justification of the identification, in dimension 4, of the Bach tensor with the Yang-Mills current of the normal conformal Cartan connection, as proved in [2].
Geometry and the Quantum: Basics
Chamseddine, Ali H; Mukhanov, Viatcheslav
2014-01-01
Motivated by the construction of spectral manifolds in noncommutative geometry, we introduce a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of scalar fields. This commutation relation appears in two versions, one sided and two sided. It implies the quantization of the volume. In the one-sided case it implies that the manifold decomposes into a disconnected sum of spheres which will represent quanta of geometry. The two sided version in dimension 4 predicts the two algebras M_2(H) and M_4(C) which are the algebraic constituents of the Standard Model of particle physics. This taken together with the non-commutative algebra of functions allows one to reconstruct, using the spectral action, the Lagrangian of gravity coupled with the Standard Model. We show that any connected Riemannian Spin 4-manifold with quantized volume >4 (in suitable units) appears as an irreducible representation of the two-sided commutation relations in dimension 4 and that these represen...
Fuzzy Logic for Incidence Geometry.
Tserkovny, Alex
The paper presents a mathematical framework for approximate geometric reasoning with extended objects in the context of Geography, in which all entities and their relationships are described by human language. These entities could be labelled by commonly used names of landmarks, water areas, and so forth. Unlike single points that are given in Cartesian coordinates, these geographic entities are extended in space and often loosely defined, but people easily perform spatial reasoning with extended geographic objects "as if they were points." Unfortunately, up to date, geographic information systems (GIS) miss the capability of geometric reasoning with extended objects. The aim of the paper is to present a mathematical apparatus for approximate geometric reasoning with extended objects that is usable in GIS. In the paper we discuss the fuzzy logic (Aliev and Tserkovny, 2011) as a reasoning system for geometry of extended objects, as well as a basis for fuzzification of the axioms of incidence geometry. The same fuzzy logic was used for fuzzification of Euclid's first postulate. Fuzzy equivalence relation "extended lines sameness" is introduced. For its approximation we also utilize a fuzzy conditional inference, which is based on proposed fuzzy "degree of indiscernibility" and "discernibility measure" of extended points.
Ring polymers in confined geometries
Usatenko, Z; Kuterba, P
2016-01-01
The investigation of a dilute solution of phantom ideal ring polymers and ring polymers with excluded volume interactions (EVI) in a good solvent confined in a slit geometry of two parallel repulsive walls and in a solution of colloidal particles of big size were performed. Taking into account the correspondence between the field theoretical $\\phi^4$ $O(n)$-vector model in the limit $n\\to 0$ and the behavior of long-flexible polymer chains in a good solvent the correspondent depletion interaction potentials, depletion forces and the forces which exert phantom ideal ring and ring polymer chains with EVI on the walls were obtained in the framework of the massive field theory approach at fixed space dimensions d=3 up to one-loop order. Additionally, the investigation of a dilute solution of phantom ideal ring polymers in a slit geometry of two inert walls and mixed walls with one repulsive and other one inert wall were performed and correspondent depletion interaction potentials and the depletion forces were cal...
Fuzzy Logic for Incidence Geometry
2016-01-01
The paper presents a mathematical framework for approximate geometric reasoning with extended objects in the context of Geography, in which all entities and their relationships are described by human language. These entities could be labelled by commonly used names of landmarks, water areas, and so forth. Unlike single points that are given in Cartesian coordinates, these geographic entities are extended in space and often loosely defined, but people easily perform spatial reasoning with extended geographic objects “as if they were points.” Unfortunately, up to date, geographic information systems (GIS) miss the capability of geometric reasoning with extended objects. The aim of the paper is to present a mathematical apparatus for approximate geometric reasoning with extended objects that is usable in GIS. In the paper we discuss the fuzzy logic (Aliev and Tserkovny, 2011) as a reasoning system for geometry of extended objects, as well as a basis for fuzzification of the axioms of incidence geometry. The same fuzzy logic was used for fuzzification of Euclid's first postulate. Fuzzy equivalence relation “extended lines sameness” is introduced. For its approximation we also utilize a fuzzy conditional inference, which is based on proposed fuzzy “degree of indiscernibility” and “discernibility measure” of extended points. PMID:27689133