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
Overconvergent global analytic geometry
Paugam, Frédéric
2014-01-01
We define a notion of global analytic space with overconvergent structure sheaf. This gives an analog on a general base Banach ring of Grosse-Kloenne's overconvergent p-adic spaces and of Bambozzi's generalized affinoid varieties over R. This also gives an affinoid version of Berkovich's and Poineau's global analytic spaces. This affinoid approach allows the introduction of a notion of strict global analytic space, that has some relations with the ideas of Arakelov geometry, since the base ex...
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.
Sharipov, Ruslan
2011-01-01
This book is a regular textbook of analytical geometry covering vector algebra and its applications to describing straight lines, planes, and quadrics in two and three dimensions. The stress is made on vector algebra by using skew-angular coordinates and by introducing some notations and prerequisites for understanding tensors. The book is addressed to students specializing in mathematics, physics, engineering, and technologies and to students of other specialities where educational standards require learning this subject.
Paugam, Frederic
2008-01-01
We define a new type of valuation of a ring that combines the notion of Krull valuation with that of multiplicative seminorm. This definition partially restores the broken symmetry between archimedean and non-archimedean valuations. This also allows us to define a notion of global analytic space that reconciles Berkovich's notion of analytic space of a (Banach) ring with Huber's notion of non-archimedean analytic space. After defining natural generalized valuation spectra and computing the sp...
Programming system for analytic geometry
After having outlined the characteristics of computing centres which do not comply with engineering tasks, notably the time required by all different tasks to be performed when developing a software (assembly, compilation, link edition, loading, run), and identified constraints specific to engineering, the author identifies the characteristics a programming system should have to suit engineering tasks. He discussed existing conversational systems and their programming language, and their main drawbacks. Then, he presents a system which aims at facilitating programming and addressing problems of analytic geometry and trigonometry
Some links between turtle geometry and analytic geometry
Rowe, Neil C.
1984-01-01
The computer language Logo facilitates the teaching of analytic geometry and calculus from the notion of curvature, through its turtle geometry facility. The author provides some theoretical basis for finding turtle geometry equivalents of familiar curves in analytic geometry, and vice versa, by some simple methods apparently previously unnoticed. In particular, he studied turtle geometry programs where the curvature of a line is a trigonometric function of its orientation. (Author)
Bruhat-Tits buildings and analytic geometry
Remy, Bertrand; Thuillier, Amaury; Werner, Annette
2012-01-01
This paper provides an overview of the theory of Bruhat-Tits buildings. Besides, we explain how Bruhat-Tits buildings can be realized inside Berkovich spaces. In this way, Berkovich analytic geometry canbe used to compactify buildings. We discuss in detail the example of the special linear group. Moreover, we give an intrinsic description of Bruhat-Tits buildings in the framework of non-Archimedean analytic geometry.
Recent topics in differential and analytic geometry
Ochiai, T
2014-01-01
Advanced Studies in Pure Mathematics, Volume 18-I: Recent Topics in Differential and Analytic Geometry presents the developments in the field of analytical and differential geometry. This book provides some generalities about bounded symmetric domains.Organized into two parts encompassing 12 chapters, this volume begins with an overview of harmonic mappings and holomorphic foliations. This text then discusses the global structures of a compact Kähler manifold that is locally decomposable as an isometric product of Ricci-positive, Ricci-negative, and Ricci-flat parts. Other chapters con
Analytic Coleman-de Luccia Geometries
Dong, Xi; /Stanford U., ITP /Stanford U., Phys. Dept. /SLAC; Harlow, Daniel; /Stanford U., ITP /Stanford U., Phys. Dept.
2012-02-16
We present the necessary and sufficient conditions for a Euclidean scale factor to be a solution of the Coleman-de Luccia equations for some analytic potential V ({psi}), with a Lorentzian continuation describing the growth of a bubble of lower-energy vacuum surrounded by higher-energy vacuum. We then give a set of explicit examples that satisfy the conditions and thus are closed-form analytic examples of Coleman-de Luccia geometries.
The analytic nodal method in cylindrical geometry
Nodal diffusion methods have been used extensively in nuclear reactor calculations, specifically for their performance advantage, but also for their superior accuracy. More specifically, the Analytic Nodal Method (ANM), utilising the transverse integration principle, has been applied to numerous reactor problems with much success. In this work, a nodal diffusion method is developed for cylindrical geometry. Application of this method to three-dimensional (3D) cylindrical geometry has never been satisfactorily addressed and we propose a solution which entails the use of conformal mapping. A set of 1D-equations with an adjusted, geometrically dependent, inhomogeneous source, is obtained. This work describes the development of the method and associated test code, as well as its application to realistic reactor problems. Numerical results are given for the PBMR-400 MW benchmark problem, as well as for a 'cylindrisized' version of the well-known 3D LWR IAEA benchmark. Results highlight the improved accuracy and performance over finite-difference core solutions and investigate the applicability of nodal methods to 3D PBMR type problems. Results indicate that cylindrical nodal methods definitely have a place within PBMR applications, yielding performance advantage factors of 10 and 20 for 2D and 3D calculations, respectively, and advantage factors of the order of 1000 in the case of the LWR problem
Gauge field geometry from complex and harmonic analyticities
The concept of preservation of harmonic analyticity is applied to find unconstrained prepotentials of hyper-Kehler geometry. The geometric meaning of prepotentials is revealed with introducing extra central charge coordinates. Finally, we establish the one-to one correspondence between hyper-Kehler geometry and off-shell d=4, N=2 supersymmetric σ-models. Their general Lagrangian is shown to be uniquely composed of hyper-Kehler prepotentials, with the analytic space coordinates replaced by analytic hypermultiplet superfields defined on the same set of harmonic variables
An analytical benchmark of MYRRHA ADS in cylindrical geometry
In this study, the steady and transient neutronic behaviour of MYRRHA ADS is investigated. For this purpose, a recently proposed analytical benchmark of the diffusion kinetics as 1D slab model of the MYRRHA ADS concept developed in Belgium has been extended to the cylindrical geometry which represents the system more realistically. Analytical calculations are performed using the Customized Solution Method and numerical Laplace inversion techniques such as Fixed-Talbot and Gaver-Wynn-Rho algorithms. Results are compared with the finite element program FLEXPDE registered and they are found to be in complete agreement. The necessity of modeling the MYRHHA reactor in cylindrical geometry rather than slab geometry to obtain more realistic benchmark results is demonstrated. (orig.)
Lagrangian and Hamiltonian Geometries. Applications to Analytical Mechanics
Miron, Radu
2012-01-01
The aim of the present text is twofold: to provide a compendium of Lagrangian and Hamiltonian geometries and to introduce and investigate new analytical Mechanics: Finslerian, Lagrangian and Hamiltonian. The fundamental equations (or evolution equations) of these Mechanics are derived from the variational calculus applied to the integral of action and these can be studied by using the methods of Lagrangian or Hamiltonian geometries. More general, the notions of higher order Lagrange or Hamilton spaces have been introduced and developed by the present author. The applications led to the notions of Lagrangian or Hamiltonian Analytical Mechanics of higher order. For short, in this text we aim to solve some difficult problems: The problem of geometrization of classical non conservative mechanical systems; The foundations of geometrical theory of new mechanics: Finslerian, Lagrangian and Hamiltonian;To determine the evolution equations of the classical mechanical systems for whose external forces depend on the hig...
Making and Understanding Analytic Geometry using the software Geogebra
Juliana Pereira Berti
2016-02-01
Full Text Available Mathematics Education has been undergoing transformations, especially with regard to the new requirements contained in legal documents. The various forms of expression language of mathematics and the use of technological resources are some of the suggestions of the National Curriculum Guidelines for Secondary Education. Thus, analytic geometry as a curricular component of mathematics also goes through these adjustments. In order to create alternatives to work that content and, thinking of the subject-object interaction, it developed a short course that uses the Dynamic Geometry through the GeoGebra software, as facilitator of the knowledge construction process. The methodology has seven activities. Are activities aimed at the line of study, including its equations and the relative position of the lines in the plan; the study of the circumference and the relative position of two circles in the plane; and the geometric interpretation of the ellipse, the parabola and hyperbole. We hope, therefore, provide educators strategies to reflect and rebuild their practices in math classes in high school.
Analytic hyperbolic geometry in N dimensions an introduction
Ungar, Abraham Albert
2014-01-01
The concept of the Euclidean simplex is important in the study of n-dimensional Euclidean geometry. This book introduces for the first time the concept of hyperbolic simplex as an important concept in n-dimensional hyperbolic geometry. Following the emergence of his gyroalgebra in 1988, the author crafted gyrolanguage, the algebraic language that sheds natural light on hyperbolic geometry and special relativity. Several authors have successfully employed the author's gyroalgebra in their exploration for novel results. Françoise Chatelin noted in her book, and elsewhere, that the computation la
In this paper we describe two analytical numerical methods applied to one-speed slab-geometry deep penetration transport problems. The linear discontinuous (LDN) equations are used to approximate the monoenergetic Boltzmann equation in slab geometry; they are obtained by considering a linear expansion of the angular flux inside each of the N elements of a uniform angular grid. The two analytical numerical methods are referred to as the spectral Green's function (SGF) nodal method and the Laplace transform (LTLDN) method. The SGF nodal method and the LTLDN method generate numerical solutions to the LDN equations that are completely free of spatial approximations, apart from finite arithmetic considerations. Numerical results to typical model problems and suggestions for future work are also presented. (orig.)
An analytical solution for the two-group kinetic neutron diffusion equation in cylindrical geometry
Recently the two-group Kinetic Neutron Diffusion Equation with six groups of delay neutron precursor in a rectangle was solved by the Laplace Transform Technique. In this work, we report on an analytical solution for this sort of problem but in cylindrical geometry, assuming a homogeneous and infinite height cylinder. The solution is obtained applying the Hankel Transform to the Kinetic Diffusion equation and solving the transformed problem by the same procedure used in the rectangle. We also present numerical simulations and comparisons against results available in literature. (author)
An analytical solution for the two-group kinetic neutron diffusion equation in cylindrical geometry
Fernandes, Julio Cesar L.; Vilhena, Marco Tullio, E-mail: julio.lombaldo@ufrgs.br, E-mail: vilhena@pq.cnpq.br [Programa de Pos Graduacao em Matematica Aplicada (DMPA/UFRGS), Universidade Federal do Rio Grande do Sul Porto Alegre, RS (Brazil); Bodmann, Bardo Ernst, E-mail: bardo.bodmann@ufrgs.br [Programa de Pos-Graduacao em Engenharia Mecanica (PROMEC/UFRGS), Universidade Federal do Rio Grande do Sul, Porto Alegre, RS (Brazil)
2011-07-01
Recently the two-group Kinetic Neutron Diffusion Equation with six groups of delay neutron precursor in a rectangle was solved by the Laplace Transform Technique. In this work, we report on an analytical solution for this sort of problem but in cylindrical geometry, assuming a homogeneous and infinite height cylinder. The solution is obtained applying the Hankel Transform to the Kinetic Diffusion equation and solving the transformed problem by the same procedure used in the rectangle. We also present numerical simulations and comparisons against results available in literature. (author)
In this work we report an analytical solution for the monoenergetic neutron diffusion kinetic equation in cartesian geometry. Bearing in mind that the equation for the delayed neutron precursor concentration is a first order linear differential equation in the time variable, to make possible the application of the GITT approach to the kinetic equation, we introduce a fictitious diffusion term multiplied by a positive small value ε. By this procedure, we are able to solve this set of equations. Indeed, applying the GITT technique to the modified diffusion kinetic equation, we come out with a matrix differential equation which has a well known analytical solution when ε goes to zero. We report numerical simulations as well study of numerical convergence of the results attained. (author)
Human eye analytical and mesh-geometry models for ophthalmic dosimetry using MCNP6
Eye tumors can be treated with brachytherapy using Co-60 plaques, I-125 seeds, among others materials. The human eye has regions particularly vulnerable to ionizing radiation (e.g. crystalline) and dosimetry for this region must be taken carefully. A mathematical model was proposed in the past [1] for the eye anatomy to be used in Monte Carlo simulations to account for dose distribution in ophthalmic brachytherapy. The model includes the description for internal structures of the eye that were not treated in previous works. The aim of this present work was to develop a new eye model based on the Mesh geometries of the MCNP6 code. The methodology utilized the ABAQUS/CAE (Simulia 3DS) software to build the Mesh geometry. For this work, an ophthalmic applicator containing up to 24 model Amersham 6711 I-125 seeds (Oncoseed) was used, positioned in contact with a generic tumor defined analytically inside the eye. The absorbed dose in eye structures like cornea, sclera, choroid, retina, vitreous body, lens, optical nerve and optical nerve wall were calculated using both models: analytical and MESH. (author)
Human eye analytical and mesh-geometry models for ophthalmic dosimetry using MCNP6
Angelocci, Lucas V.; Fonseca, Gabriel P.; Yoriyaz, Helio, E-mail: hyoriyaz@ipen.br [Instituto de Pesquisas Energeticas e Nucleares (IPEN/CNEN-SP), Sao Paulo, SP (Brazil)
2015-07-01
Eye tumors can be treated with brachytherapy using Co-60 plaques, I-125 seeds, among others materials. The human eye has regions particularly vulnerable to ionizing radiation (e.g. crystalline) and dosimetry for this region must be taken carefully. A mathematical model was proposed in the past [1] for the eye anatomy to be used in Monte Carlo simulations to account for dose distribution in ophthalmic brachytherapy. The model includes the description for internal structures of the eye that were not treated in previous works. The aim of this present work was to develop a new eye model based on the Mesh geometries of the MCNP6 code. The methodology utilized the ABAQUS/CAE (Simulia 3DS) software to build the Mesh geometry. For this work, an ophthalmic applicator containing up to 24 model Amersham 6711 I-125 seeds (Oncoseed) was used, positioned in contact with a generic tumor defined analytically inside the eye. The absorbed dose in eye structures like cornea, sclera, choroid, retina, vitreous body, lens, optical nerve and optical nerve wall were calculated using both models: analytical and MESH. (author)
Berrocal, Edouard; Churmakov, Dmitry Y; Romanov, Vadim P; Jermy, Mark C; Meglinski, Igor V
2005-05-01
Sprays and other industrially relevant turbid media can be quantitatively characterized by light scattering. However, current optical diagnostic techniques generate errors in the intermediate scattering regime where the average number of light scattering is too great for the single scattering to be assumed, but too few for the diffusion approximation to be applied. Within this transitional single-to-multiple scattering regime, we consider a novel crossed source-detector geometry that allows the intensity of single scattering to be measured separately from the higher scattering orders. We verify Monte Carlo calculations that include the imperfections of the experiment against analytical results. We show quantitatively the influence of the detector numerical aperture and the angle between the source and the detector on the relative intensity of the scattering orders in the intermediate single-to-multiple scattering regime. Monte Carlo and analytical calculations of double light-scattering intensity are made with small particles that exhibit isotropic scattering. The agreement between Monte Carlo and analytical techniques validates use of the Monte Carlo approach in the intermediate scattering regime. Monte Carlo calculations are then performed for typical parameters of sprays and aerosols with anisotropic (Mie) scattering in the intermediate single-to-multiple scattering regime. PMID:15881059
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
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...... matematik, geometri, og fysik til at forstå, hvad det er, der foregår....
Brunet, Edouard; Ajdari, Armand
2006-05-01
We set up an analytical framework that allows one to describe and compute streaming effects and electro-osmosis on an equal footing. This framework relies on the thin double layer approximation commonly used for description of electroosmotic flows, but rarely used for streaming problems. Using this framework we quantitatively assess the induction of bulk streaming current patterns by topographic or charge heterogeneities on surfaces. This too also permits analytical computation of all linear electrokinetic effects in complex microfluidic geometries, and we discuss a few immediate applications. PMID:16803036
Lozano Montero, Juan Andrés; García Herranz, Nuria; Ahnert Iglesias, Carolina; Aragonés Beltrán, José María
2008-01-01
In this work we address the development and implementation of the analytic coarse-mesh finite-difference (ACMFD) method in a nodal neutron diffusion solver called ANDES. The first version of the solver is implemented in any number of neutron energy groups, and in 3D Cartesian geometries; thus it mainly addresses PWR and BWR core simulations. The details about the generalization to multigroups and 3D, as well as the implementation of the method are given. The transverse integration procedure i...
There is growing interest in developing pebble bed reactors (PBRs) as a candidate of very high temperature gas-cooled reactors (VHTRs). Until now, most existing methods of nuclear design analysis for this type of reactors are base on old finite-difference solvers or on statistical methods. But for realistic analysis of PBRs, there is strong desire of making available high fidelity nodal codes in three-dimensional (r,θ,z) cylindrical geometry. Recently, the Analytic Function Expansion Nodal (AFEN) method developed quite extensively in Cartesian (x,y,z) geometry and in hexagonal-z geometry was extended to two-group (r,z) cylindrical geometry, and gave very accurate results. In this thesis, we develop a method for the full three-dimensional cylindrical (r,θ,z) geometry and implement the method into a code named TOPS. The AFEN methodology in this geometry as in hexagonal geometry is 'robus' (e.g., no occurrence of singularity), due to the unique feature of the AFEN method that it does not use the transverse integration. The transverse integration in the usual nodal methods, however, leads to an impasse, that is, failure of the azimuthal term to be transverse-integrated over r-z surface. We use 13 nodal unknowns in an outer node and 7 nodal unknowns in an innermost node. The general solution of the node can be expressed in terms of that nodal unknowns, and can be updated using the nodal balance equation and the current continuity condition. For more realistic analysis of PBRs, we implemented em Marshak boundary condition to treat the incoming current zero boundary condition and the partial current translation (PCT) method to treat voids in the core. The TOPS code was verified in the various numerical tests derived from Dodds problem and PBMR-400 benchmark problem. The results of the TOPS code show high accuracy and fast computing time than the VENTURE code that is based on finite difference method (FDM)
A polynomial analytical method for one-group slab-geometry discrete ordinates heterogeneous problems
In this work we evaluate polynomial approximations to obtain the transfer functions that appear in SGF auxiliary equations (Green's Functions) for monoenergetic linearly anisotropic scattering SN equations in one-dimensional Cartesian geometry. For this task we use Lagrange Polynomials in order to compare the numerical results with the ones generated by the standard SGF method applied to SN problems in heterogeneous domains. This work is a preliminary investigation of a new proposal for handling the transverse leakage terms that appear in the transverse-integrated one-dimensional SN equations when we use the SGF - exponential nodal method (SGF-ExpN) in multidimensional rectangular geometry. (author)
During the last decade, the analytic function expansion nodal (AFEN) method has been developed and successfully applied to the static and kinetic problems in rectangular geometry and also applied to the static problems in hexagonal geometry. Although the results of two-dimensional hexagonal problems were very accurate, the accuracy becomes poor when the current hexagonal-z method is applied to the three-dimensional hexagonal problems. In this thesis, we develop a new method which improves the accuracy in three-dimensional hexagonal geometry and computerize the method into a new kinetics code. At first we add the edge fluxes in the upper and lower planes as additional nodal unknowns in axial direction to improve the accuracy. These nodal unknowns are updated through leakage balance equations by using a simple expansion of nodal fluxes at the vicinity of the edge fluxes. The relation of delayed neutron precursor densities between time steps is obtained analytically by using the transformed fluxes and assuming linear variations of the fission rates within a time step. The code developed for the steady state is verified in the cases of 2-D VVER-1000, 3-D SNR-300, and 3-D VVER-440 benchmark problems. The results of the static problems show higher accuracy than those of the original formulations in hexagonal geometry. Finally, a kinetics code is developed and tested by introducing step changes of nodal cross sections for the VVER-440 benchmark problem. The results appear to be accurate enough for this code to be useful for analyzing realistic three-dimensional hexagonal reactors
Analytical Model for Outdoor Millimeter Wave Channels using Geometry-Based Stochastic Approach
Muhammad, Nor Aishah; Wang, Peng; Li, Yonghui; Vucetic, Branka
2016-01-01
The severe bandwidth shortage in conventional microwave bands has spurred the exploration of the millimeter wave (MMW) spectrum for the next revolution in wireless communications. However, there is still lack of proper channel modeling for the MMW wireless propagation, especially in the case of outdoor environments. In this paper, we develop a geometry-based stochastic channel model to statistically characterize the effect of all the first-order reflection paths between the transmitter and re...
Prinsloo, Rian Hendrik
2006-01-01
Nodal diffusion methods have been used extensively in nuclear reactor calculations specifically for their performance advantage, but also their superior accuracy. In this work a nodal diffusion method is developed for three-dimensional cylindrical geometry. Recent developments in the Pebble Bed Modular Reactor (PBMR) project have sparked renewed interest in the application of different modelling methods to its design, and naturally included in this effort is a nodal method for ...
Modeling of cavities using the analytic modal method and an open geometry formalism
de Lasson, Jakob Rosenkrantz; Christensen, Thomas; Mørk, Jesper;
2012-01-01
We present an eigenmode expansion technique for calculating the properties of a dipole emitter inside a micropillar. We consider a solution domain of infinite extent, implying no outer boundary conditions for the electric field, and expand the field on analytic eigenmodes. In contrast to finite-s...
N. Hadadin
2011-07-01
Full Text Available The effects of basin hydrology on channel hydraulic variability for incised streams were investigated using available field data sets and models of watershed hydrology and channel hydraulics for Yazoo River Basin, USA. The study presents the hydraulic relations of bankfull discharge, channel width, mean depth, cross- sectional area, longitudinal slope, unit stream power, and runoff production as a function of drainage area using simple linear regression. The hydraulic geometry relations were developed for sixty one streams, twenty of them are classified as channel evaluation model (CEM Types IV and V and forty one of them are streams of CEM Types II and III. These relationships are invaluable to hydraulic and water resources engineers, hydrologists, and geomorphologists, involved in stream restoration and protection. These relations can be used to assist in field identification of bankfull stage and stream dimension in un-gauged watersheds as well as estimation of the comparative stability of a stream channel.
Results of this research show good fit of hydraulic geometry relationships in the Yazoo River Basin. The relations indicate that bankfull discharge, channel width, mean depth, cross-sectional area have stronger correlation to changes in drainage area than the longitudinal slope, unit stream power, and runoff production for streams CEM Types II and III. The hydraulic geometry relations show that runoff production, bankfull discharge, cross-sectional area, and unit stream power are much more responsive to changes in drainage area than are channel width, mean depth, and slope for streams of CEM Types IV and V. Also, the relations show that bankfull discharge and cross-sectional area are more responsive to changes in drainage area than are other hydraulic variables for streams of CEM Types II and III. The greater the regression slope, the more responsive to changes in drainage area will be.
Analytical modeling of turn-milling process geometry, kinematics and mechanics
Karagüzel, Umut; Karaguzel, Umut; Uysal, Emre; Budak, Erhan; Bakkal, Mustafa
2014-01-01
This paper presents an analytical approach for modeling of turn-milling which is a promising cutting process combining two conventional machining operations; turning and milling. This relatively new technology could be an alternative to turning for improved productivity in many applications but especially in cases involving hard-to-machine material or large work diameter. Intermittent nature of the process reduces forces on the workpiece, cutting temperatures and thus tool wear, and helps bre...
Stosic, Z.V.; Stevanovic, V.D. [Framatome ANP GmbH - NBTT, Erlangen (Germany)
2001-07-01
Nuclear fuel rod bundle thermal-hydraulics strongly depends on the presence of fuel rod spacers. High Reynolds number coolant flows around spacers of different geometry and position are numerically investigated in two-dimensions. Predicted is the influence of the spacer's geometry and its position within a flow channel, on the recirculation zones formation and corresponding reattachment lengths. The numerical procedure is based on the two-equation k-{epsilon} turbulence model and the control volume procedure. Numerical simulation and analyses of high Reynolds fluid flow over the bluff bodies of the nuclear fuel spacer shapes are performed. Two types of flow channels and conditions are considered: spacers mounted on the wall and spacers in a free fluid stream. The major findings are as follows: 1) In fluid flow over the spacer mounted on the wall, the large recirculation zone is formed behind the spacer's fin. The reattachment length and width of this large vortex are increasing with the spacer's fin inclination angle increase. The reattachment length is rapidly increasing for fin inclination angles greater than 45 degrees. The maximum value of the reattachment length is achieved with fin inclination angle of 75 degrees. For angles from 75 to 90 degrees this length is slightly decreasing (the result with the Re=1.6*10{sup 6}). 2) A small vortex is formed in front of the spacer's fin in case of the flow over wall mounted spacer. 3) In case of fluid flow around the spacer in a free stream, the vortexes are formed behind the fin. Vortex shedding is observed, which is contrary to the formation of a stable vortex behind the wall mounted spacers. No small recirculation zone exists in front of the fin, which is opposite to the flow structure in case of wall mounted spacer. (authors)
We construct a semi-analytic model for magnetohydrodynamic (MHD) flows in Kerr geometry that incorporates energy loading via neutrino annihilation on magnetic field lines threading the horizon. We compute the structure of the double-flow established in the magnetisphere for a wide range of energy injection rates and identify the different operation regimes. At low injection rates, the outflow is powered by the spinning black hole via the Blandford-Znajek mechanism, whereas at high injection rates, it is driven by the pressure of the plasma deposited on magnetic field lines. In the intermediate regime, both processes contribute to the outflow formation. The parameter that quantifies the load is the ratio of the net power injected below the stagnation radius and the maximum power that can be extracted magnetically from the black hole.
The Analytic Coarse-Mesh Finite-Difference method is developed in detail for multi-group and multi-dimensional diffusion calculations, including the general and particular modal solutions in the complex space for any number of groups. For rectangular multidimensional geometries, the Chao's generalized relations with transverse integration provide a high-order approximation of the ACMFD method, where all energy groups are coupled by matrix-vector FD relations and the errors are limited to the ones incurred by the interpolation of the transverse interface currents, in a non-linear iterative scheme. The implementation of the method in a multigroup 3D rectangular geometry nodal solver called ANDES is discussed, pointing out the encapsulation achieved for integration of the solver as an optional module within larger code systems. The performance of the ANDES solver in 3D rectangular (X-Y-Z) geometry and multi-groups is verified by its application to several 2D-3D model and international benchmarks (NEA-OECD), with given diffusion cross section sets in few-groups (2 to 8). The extensive verification, always required for new methods and codes, shows a quite fast convergence of ANDES in both the eigenvalue and transverse leakage iteration loops and with the nodal coarse-mesh size, allowing to reach the conclusion that quite high accuracy is achieved with rather large nodes, one node or four nodes per PWR fuel assembly, as compared with reference solutions obtained with fine-mesh finite-difference diffusion calculations using mesh sizes 64 to 128 times smaller than the ANDES nodes. (authors)
WANG YouYu; HUANG Tian; ZHAO XueMan; MEI JiangPing; Derek G CHETWYND
2008-01-01
Stiffness modeling is one of the most significant issues in the design of parallel kinematic machine (PKM).This paper presents a semi-analytical approach that enables the stiffness of PKM with complex machine frame geometry to be estimated effectively.This approach can be implemented by three steps:(i) decomposition of the entire system into two sub-systems associated with the parallel mechanism and the machine frame respectively;(ii) stiffness modeling of each sub-system using the analytical approach and the finite element analysis;and (iii) generation of the stiffness model of the entire system by means of linear superposition.In the modeling process of each sub-system,the virtual work princi-ple and overall deflection Jacobian are employed with special attention to the bending rigidity of the constrained passive limb and the interface stiffness of the machine frame.The stiffness distribution of a 5-DOF hybrid robot named TriVariant-B is investigated as an example to illustrate the effectivaness of this approach.The contributions of component rigidities to that of the system are evaluated using global indices.It shows that the results achieved by this approach have a good match to those obtained through finite element analysis and experiments.
Fernandes, Julio Cesar L.; Vilhena, Marco Tullio, E-mail: julio.lombaldo@ufrgs.b, E-mail: vilhena@pq.cnpq.b [Universidade Federal do Rio Grande do Sul (DMPA/UFRGS), Porto Alegre, RS (Brazil). Dept. de Matematica Pura e Aplicada. Programa de Pos Graduacao em Matematica Aplicada; Borges, Volnei; Bodmann, Bardo Ernest, E-mail: bardo.bodmann@ufrgs.b, E-mail: borges@ufrgs.b [Universidade Federal do Rio Grande do Sul (PROMEC/UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica
2011-07-01
The principal idea of this work, consist on formulate an analytical method to solved problems for diffusion of neutrons with isotropic scattering in one-dimensional cylindrical geometry. In this area were develop many works that study the same problem in different system of coordinates as well as cartesian system, nevertheless using numerical methods to solve the shielding problem. In view of good results in this works, we starting with the idea that we can represent a source in the origin of the cylindrical system by a Delta Dirac distribution, we describe the physical modeling and solved the neutron diffusion equation inside of cylinder of radius R. For the case of transport equation, the formulation of discrete ordinates S{sub N} consists in discretize the angular variables in N directions and in using a quadrature angular set for approximate the sources of scattering, where the Diffusion equation consist on S{sub 2} approximated transport equation in discrete ordinates. We solved the neutron diffusion equation with an analytical form by the finite Hankel transform. Was presented also the build-up factor for the case that we have neutron flux inside the cylinder. (author)
The objective of this work is to obtain an analytical solution of the neutron diffusion kinetic equation in one-dimensional cartesian geometry, to monoenergetic and multigroup problems. These equations are of the type stiff, due to large differences in the orders of magnitude of the time scales of the physical phenomena involved, which make them difficult to solve. The basic idea of the proposed method is applying the spectral expansion in the scalar flux and in the precursor concentration, taking moments and solving the resulting matrix problem by the Laplace transform technique. Bearing in mind that the equation for the precursor concentration is a first order linear differential equation in the time variable, to enable the application of the spectral method we introduce a fictitious diffusion term multiplied by a positive value which tends to zero. This procedure opened the possibility to find an analytical solution to the problem studied. We report numerical simulations and analysis of the results obtained with the precision controlled by the truncation order of the series. (author)
The analytic function expansion nodal (AFEN) method has been successfully applied to the rectangular and hexagonal geometries in the cartesian coordinates system. In this paper, we extended the AFEN method to the cylindrical geometry in the R-Z coordinates for the analysis of pebble bed modular reactors (PBMRs). To treat the mixed geometry of rectangular and triangular nodes appearing in the lower periphery of the reactors, we used half-interface averaged fluxes as nodal unknowns. Numerical results obtained attest to their accuracy and applicability to practical problems. (author)
Pablo Aguiar
2012-01-01
Full Text Available Positron emission mammography (PEM cameras are novel-dedicated PET systems optimized to image the breast. For these cameras it is essential to achieve an optimum trade-off between sensitivity and spatial resolution and therefore the main challenge for the novel cameras is to improve the sensitivity without degrading the spatial resolution. We carry out an analytical study of the effect of the different detector geometries on the photon sensitivity and the angle of incidence of the detected photons which is related to the DOI effect and therefore to the intrinsic spatial resolution. To this end, dual head detectors were compared to box and different polygon-detector configurations. Our results showed that higher sensitivity and uniformity were found for box and polygon-detector configurations compared to dual-head cameras. Thus, the optimal configuration in terms of sensitivity is a PEM scanner based on a polygon of twelve (dodecagon or more detectors. We have shown that this configuration is clearly superior to dual-head detectors and slightly higher than box, octagon, and hexagon detectors. Nevertheless, DOI effects are increased for this configuration compared to dual head and box scanners and therefore an accurate compensation for this effect is required.
Lefschetz, Solomon
2012-01-01
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.
In this work we address the development and implementation of the analytic coarse-mesh finite-difference (ACMFD) method in a nodal neutron diffusion solver called ANDES. The first version of the solver is implemented in any number of neutron energy groups, and in 3D Cartesian geometries; thus it mainly addresses PWR and BWR core simulations. The details about the generalization to multigroups and 3D, as well as the implementation of the method are given. The transverse integration procedure is the scheme chosen to extend the ACMFD formulation to multidimensional problems. The role of the transverse leakage treatment in the accuracy of the nodal solutions is analyzed in detail: the involved assumptions, the limitations of the method in terms of nodal width, the alternative approaches to implement the transverse leakage terms in nodal methods - implicit or explicit -, and the error assessment due to transverse integration. A new approach for solving the control rod 'cusping' problem, based on the direct application of the ACMFD method, is also developed and implemented in ANDES. The solver architecture turns ANDES into an user-friendly, modular and easily linkable tool, as required to be integrated into common software platforms for multi-scale and multi-physics simulations. ANDES can be used either as a stand-alone nodal code or as a solver to accelerate the convergence of whole core pin-by-pin code systems. The verification and performance of the solver are demonstrated using both proof-of-principle test cases and well-referenced international benchmarks
In this work we report an analytical solution for the time dependent SN transport equation in a plane parallel geometry for unbounded domain, we mean for 0 ≤ x ≤ ∞. For such, we apply the Laplace transform technique in the time variable and the LTSN approach in the spatial variable. By this procedure we come out with an analytical solution for the angular flux in integral form applying the boundness of the angular flux at infinity. We present numerical simulations and also validation by the analysis of the asymptotic behavior of the scalar flux in a slab. (author)
Lozano Montero, Juan Andrés; Jiménez Escalante, Javier; García Herranz, Nuria; Aragonés Beltrán, José María
2010-01-01
In this paper the extension of the multigroup nodal diffusion code ANDES, based on the Analytic Coarse Mesh Finite Difference (ACMFD) method, from Cartesian to hexagonal geometry is presented, as well as its coupling with the thermal–hydraulic (TH) code COBRA-IIIc for hexagonal core analysis. In extending the ACMFD method to hexagonal assemblies, triangular-Z nodes are used. In the radial plane, a direct transverse integration procedure is applied along the three directions that are orthog...
Brown, Martha
1991-01-01
The purpose of this research was to investigate hypothesized relations of visuospatial and logical reasoning skills, and span of short-term memory to achievement in geometry. In addition, major subfactors of visuospatial ability (visualization, speeded rotations, spatial orientation, and disembedding) were assessed to determine which were significant predictors of geometry achievement. Vernon's (1965) model of intelligence and Baddeley's model of working memory provided the theoretical fra...
Delcey, Mickaël G. [Department of Chemistry – Ångström, The Theoretical Chemistry Programme, Uppsala University, Box 518, 751 20 Uppsala (Sweden); Freitag, Leon; González, Leticia, E-mail: leticia.gonzalez@univie.ac.at [Institut für Theoretische Chemie, Universität Wien, Währinger Straße 17, 1090 Vienna (Austria); Pedersen, Thomas Bondo [Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, 0315 Oslo (Norway); Aquilante, Francesco [Department of Chemistry – Ångström, The Theoretical Chemistry Programme, Uppsala University, Box 518, 751 20 Uppsala (Sweden); Dipartimento di Chimica “G. Ciamician,” Università di Bologna, V. F. Selmi 2, 40126 Bologna (Italy); Lindh, Roland, E-mail: roland.lindh@kemi.uu.se [Department of Chemistry – Ångström, The Theoretical Chemistry Programme, Uppsala University, Box 518, 751 20 Uppsala (Sweden); Uppsala Center for Computational Chemistry - UC3, Uppsala University, Box 518, 751 20 Uppsala (Sweden)
2014-05-07
We present a formulation of analytical energy gradients at the complete active space self-consistent field (CASSCF) level of theory employing density fitting (DF) techniques to enable efficient geometry optimizations of large systems. As an example, the ground and lowest triplet state geometries of a ruthenium nitrosyl complex are computed at the DF-CASSCF level of theory and compared with structures obtained from density functional theory (DFT) using the B3LYP, BP86, and M06L functionals. The average deviation of all bond lengths compared to the crystal structure is 0.042 Å at the DF-CASSCF level of theory, which is slightly larger but still comparable with the deviations obtained by the tested DFT functionals, e.g., 0.032 Å with M06L. Specifically, the root-mean-square deviation between the DF-CASSCF and best DFT coordinates, delivered by BP86, is only 0.08 Å for S{sub 0} and 0.11 Å for T{sub 1}, indicating that the geometries are very similar. While keeping the mean energy gradient errors below 0.25%, the DF technique results in a 13-fold speedup compared to the conventional CASSCF geometry optimization algorithm. Additionally, we assess the singlet-triplet energy vertical and adiabatic differences with multiconfigurational second-order perturbation theory (CASPT2) using the DF-CASSCF and DFT optimized geometries. It is found that the vertical CASPT2 energies are relatively similar regardless of the geometry employed whereas the adiabatic singlet-triplet gaps are more sensitive to the chosen triplet geometry.
We present a formulation of analytical energy gradients at the complete active space self-consistent field (CASSCF) level of theory employing density fitting (DF) techniques to enable efficient geometry optimizations of large systems. As an example, the ground and lowest triplet state geometries of a ruthenium nitrosyl complex are computed at the DF-CASSCF level of theory and compared with structures obtained from density functional theory (DFT) using the B3LYP, BP86, and M06L functionals. The average deviation of all bond lengths compared to the crystal structure is 0.042 Å at the DF-CASSCF level of theory, which is slightly larger but still comparable with the deviations obtained by the tested DFT functionals, e.g., 0.032 Å with M06L. Specifically, the root-mean-square deviation between the DF-CASSCF and best DFT coordinates, delivered by BP86, is only 0.08 Å for S0 and 0.11 Å for T1, indicating that the geometries are very similar. While keeping the mean energy gradient errors below 0.25%, the DF technique results in a 13-fold speedup compared to the conventional CASSCF geometry optimization algorithm. Additionally, we assess the singlet-triplet energy vertical and adiabatic differences with multiconfigurational second-order perturbation theory (CASPT2) using the DF-CASSCF and DFT optimized geometries. It is found that the vertical CASPT2 energies are relatively similar regardless of the geometry employed whereas the adiabatic singlet-triplet gaps are more sensitive to the chosen triplet geometry
Delcey, Mickaël G; Freitag, Leon; Pedersen, Thomas Bondo; Aquilante, Francesco; Lindh, Roland; González, Leticia
2014-05-01
We present a formulation of analytical energy gradients at the complete active space self-consistent field (CASSCF) level of theory employing density fitting (DF) techniques to enable efficient geometry optimizations of large systems. As an example, the ground and lowest triplet state geometries of a ruthenium nitrosyl complex are computed at the DF-CASSCF level of theory and compared with structures obtained from density functional theory (DFT) using the B3LYP, BP86, and M06L functionals. The average deviation of all bond lengths compared to the crystal structure is 0.042 Å at the DF-CASSCF level of theory, which is slightly larger but still comparable with the deviations obtained by the tested DFT functionals, e.g., 0.032 Å with M06L. Specifically, the root-mean-square deviation between the DF-CASSCF and best DFT coordinates, delivered by BP86, is only 0.08 Å for S0 and 0.11 Å for T1, indicating that the geometries are very similar. While keeping the mean energy gradient errors below 0.25%, the DF technique results in a 13-fold speedup compared to the conventional CASSCF geometry optimization algorithm. Additionally, we assess the singlet-triplet energy vertical and adiabatic differences with multiconfigurational second-order perturbation theory (CASPT2) using the DF-CASSCF and DFT optimized geometries. It is found that the vertical CASPT2 energies are relatively similar regardless of the geometry employed whereas the adiabatic singlet-triplet gaps are more sensitive to the chosen triplet geometry. PMID:24811621
Bruce, William J; Maxwell, E A; Sneddon, I N
1963-01-01
Analytic Trigonometry details the fundamental concepts and underlying principle of analytic geometry. The title aims to address the shortcomings in the instruction of trigonometry by considering basic theories of learning and pedagogy. The text first covers the essential elements from elementary algebra, plane geometry, and analytic geometry. Next, the selection tackles the trigonometric functions of angles in general, basic identities, and solutions of equations. The text also deals with the trigonometric functions of real numbers. The fifth chapter details the inverse trigonometric functions
An analytical description of magnetic islands is presented for the typical case of a single perturbation mode introduced to tokamak plasma equilibrium in the large aspect ratio approximation. Following the Hamiltonian structure directly in terms of toroidal coordinates, the well known integrability of this system is exploited, laying out a precise and practical way for determining the island topology features, as required in various applications, through an analytical and exact flux surface label
In this paper the extension of the multigroup nodal diffusion code ANDES, based on the Analytic Coarse Mesh Finite Difference (ACMFD) method, from Cartesian to hexagonal geometry is presented, as well as its coupling with the thermal-hydraulic (TH) code COBRA-IIIc for hexagonal core analysis. In extending the ACMFD method to hexagonal assemblies, triangular-Z nodes are used. In the radial plane, a direct transverse integration procedure is applied along the three directions that are orthogonal to the triangle interfaces. The triangular nodalization avoids the singularities, that appear when applying transverse integration to hexagonal nodes, and allows the advantage of the mesh subdivision capabilities implicit within that geometry. As for the thermal-hydraulics, the extension of the coupling scheme to hexagonal geometry has been performed with the capability to model the core using either assembly-wise channels (hexagonal mesh) or a higher refinement with six channels per fuel assembly (triangular mesh). Achieving this level of TH mesh refinement with COBRA-IIIc code provides a better estimation of the in-core 3D flow distribution, improving the TH core modelling. The neutronics and thermal-hydraulics coupled code, ANDES/COBRA-IIIc, previously verified in Cartesian geometry core analysis, can also be applied now to full three-dimensional VVER core problems, as well as to other thermal and fast hexagonal core designs. Verification results are provided, corresponding to the different cases of the OECD/NEA-NSC VVER-1000 Coolant Transient Benchmarks.
In the present paper a generalization is performed of a procedure to solve multigroup spherical harmonics equations, which has originally been proposed and developed for one-dimensional systems in cylindrical or spherical geometry, and later extended for a special case of a two-dimensional system in r-z geometry. The expressions are derived for the axial and the radial dependence of the group values of the neutron flux moments, in the P-3 approximation of the spherical harmonics method, in a cylindrically symmetrical system with an arbitrary number of material regions in both r- and z-directions. In the special case of an axially homogeneous system, these expressions reduce to the relations derived previously. (author)
A computer program has been developed which uses a technique of synthetic acceleration by diffusion by analytical schemes. Both in the diffusion equation as in that of transport, analytical schemes were used which allowed a substantial time saving in the number of iterations required by source iteration method to obtain the Keff. The program developed ASD (Synthetic Diffusion Acceleration) by diffusion was written in FORTRAN and can be executed on a personal computer with a hard disc and mathematical O-processor. The program is unlimited as to the number of regions and energy groups. The results obtained by the ASD program for Keff is nearly completely concordant with those of obtained utilizing the ANISN-PC code for different analytical type problems in this work. The ASD program allowed obtention of an approximate solution of the neutron transport equation with a relatively low number of internal reiterations with good precision. One of its applications would be in the direct determinations of axial distribution neutronic flow in a fuel assembly as well as in the obtention of the effective multiplication factor. (Author)
Tellgren, Erik I; Reine, Simen S; Helgaker, Trygve
2012-07-14
Analytical integral evaluation is a central task of modern quantum chemistry. Here we present a general method for evaluating differentiated integrals over standard Gaussian and mixed Gaussian/plane-wave hybrid orbitals. The main idea is to have a representation of basis sets that is flexible enough to enable differentiated integrals to be reinterpreted as standard integrals over modified basis functions. As an illustration of the method, we report a very simple implementation of Hartree-Fock level geometrical derivatives in finite magnetic fields for gauge-origin independent atomic orbitals, within the London program. As a quantum-chemical application, we optimize the structure of helium clusters and some well-known covalently bound molecules (water, ammonia and benzene) subject to strong magnetic fields. PMID:22653039
Fernandes, Julio C.L.; Vilhena, Marco T.; Bodmann, Bardo E.J., E-mail: julio.lombaldo@ufrgs.br, E-mail: mtmbvilhena@gmail.com, E-mail: bardo.bodmann@ufrgs.br [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Dept. de Matematica Pura e Aplicada; Dulla, Sandra; Ravetto, Piero, E-mail: sandra.dulla@polito.it, E-mail: piero.ravetto@polito.it [Dipartimento di Energia, Politecnico di Torino, Piemonte (Italy)
2015-07-01
In this work we generalize the solution of the one-dimensional neutron transport equation to a multi- group approach in planar geometry. The basic idea of this work consists in consider the hierarchical construction of a solution for a generic number G of energy groups, starting from a mono-energetic solution. The hierarchical method follows the reasoning of the decomposition method. More specifically, the additional terms from adding energy groups is incorporated into the recursive scheme as source terms. This procedure leads to an analytical representation for the solution with G energy groups. The recursion depth is related to the accuracy of the solution, that may be evaluated after each recursion step. The authors present a heuristic analysis of stability for the results. Numerical simulations for a specific example with four energy groups and a localized pulsed source. (author)
Jalili Bahabadi, Mohammad Hasan; Pazirandeh, Ali; Athari, Mitra [Islamic Azad Univ., Tehran (Iran, Islamic Republic of). Dept. of Nuclear Engineering, Science and Research Branch
2015-12-15
In this paper, we developed a new approach of analytic function expansion nodal (AFEN) method to solve the multi-group and multi-dimensional neutron diffusion equation in reactor cores with hexagonal fuel assembly. This method represents a multidimensional intra nodal flux distribution in terms of analytic basis functions at any points in the node. New types of boundary conditions have been considered that constrain the intranodal flux distributions in the hexagonal-z node, which include twelve radial surface-averaged partial currents and two axial surface-averaged partial currents. We utilized the coarse group rebalancing (CGR) method to increase the speed of code calculations. The computer code takes a few-groups cross sections produced by a lattice code and calculates the effective multiplication factor (k{sub eff}), flux in multi-group energy, reactivity, and the relative power density at each fuel assembly. Finally, the solution accuracy is tested for two well-known benchmark problems. The numerical results demonstrate that the new AFEN method is an accurate method for calculating k{sub eff} and power density distribution in hexagonal-z geometries.
In this paper, we developed a new approach of analytic function expansion nodal (AFEN) method to solve the multi-group and multi-dimensional neutron diffusion equation in reactor cores with hexagonal fuel assembly. This method represents a multidimensional intra nodal flux distribution in terms of analytic basis functions at any points in the node. New types of boundary conditions have been considered that constrain the intranodal flux distributions in the hexagonal-z node, which include twelve radial surface-averaged partial currents and two axial surface-averaged partial currents. We utilized the coarse group rebalancing (CGR) method to increase the speed of code calculations. The computer code takes a few-groups cross sections produced by a lattice code and calculates the effective multiplication factor (keff), flux in multi-group energy, reactivity, and the relative power density at each fuel assembly. Finally, the solution accuracy is tested for two well-known benchmark problems. The numerical results demonstrate that the new AFEN method is an accurate method for calculating keff and power density distribution in hexagonal-z geometries.
Homan, Ward; Decin, Leen; de Koter, Alex; van Marle, Allard Jan; Lombaert, Robin; Vlemmings, Wouter
2015-07-01
Context. Recent high-resolution observations have shown that stellar winds harbour complexities that strongly deviate from spherical symmetry, which generally is assumed as standard wind model. One such morphology is the Archimedean spiral, which is generally believed to be formed by binary interactions, as has been directly observed in multiple sources. Aims: We seek to investigate the manifestation in the observables of spiral structures embedded in the spherical outflows of cool stars. We aim to provide an intuitive bedrock with which upcoming ALMA data can be compared and interpreted. Methods: By means of an extended parameter study, we modelled rotational CO emission from the stellar outflow of asymptotic giant branch stars. To this end, we developed a simplified analytical parametrised description of a 3D spiral structure. This model is embedded into a spherical wind and fed into the 3D radiative transfer code LIME, which produces 3D intensity maps throughout velocity space. Subsequently, we investigated the spectral signature of rotational transitions of CO in the models, as well as the spatial aspect of this emission by means of wide-slit position-velocity (PV) diagrams. Additionally, we quantified the potential for misinterpreting the 3D data in a 1D context. Finally, we simulated ALMA observations to explore the effect of interferometric noise and artefacts on the emission signatures. Results: The spectral signatures of the CO rotational transition v = 0J = 3 - 2 are very efficient at concealing the dual nature of the outflow. Only a select few parameter combinations allow for the spectral lines to disclose the presence of the spiral structure. If the spiral cannot be distinguished from the spherical signal, this might result in an incorrect interpretation in a 1D context. Consequently, erroneous mass-loss rates would be calculated. The magnitude of these errors is mainly confined to a factor of a few, but in extreme cases can exceed an order of magnitude
A semi-analytical approach is presented to model the effects of complicated boundary conditions and rarefaction on the squeeze-film damping dependent quality factor in a double-gimballed MEMS torsion mirror. To compute squeeze-film damping in a rectangular torsion mirror with simple boundaries, compact models derived by solving the conventional Reynolds equation with zero pressure boundary conditions on the edges of the plate are generally used. These models are not applicable if the air-gap thickness is comparable to the length of the plate. To extend the validity of the existing models in devices with large air-gap thickness and complicated boundaries, we present a procedure that requires the computation of the effective length of the structure and uses this length for the computation of damping in all flow regimes using a modified effective viscosity model. The effective length is computed by comparing the damping obtained from a numerical solution of Navier–Stokes equations with that obtained from a Reynolds-equation-based compact model. To capture the effect of rarefaction in different flow regimes, we use two different approaches: the effective viscosity approach which is valid for continuum, slip, transition and molecular flow regimes, and an approach based on the free molecular model which is valid only in a molecular flow regime. We show that the effective length obtained for complicated structures in the continuum regime may still be used to capture the rarefaction effect in the slip, transition and molecular regimes. On comparing different empirical models based on the effective viscosity approach with experimental results, we find some anomaly in the region between the molecular regime and the intrinsic regime where non-fluid damping dominates. To improve modelling in the rarified regimes, we modify the best model among the existing models by minimizing error obtained with respect to the experimental results. We find that the proposed model captures
Spinning geometry = Twisted geometry
It is well known that the SU(2)-gauge invariant phase space of loop gravity can be represented in terms of twisted geometries. These are piecewise-linear-flat geometries obtained by gluing together polyhedra, but the resulting geometries are not continuous across the faces. Here we show that this phase space can also be represented by continuous, piecewise-flat three-geometries called spinning geometries. These are composed of metric-flat three-cells glued together consistently. The geometry of each cell and the manner in which they are glued is compatible with the choice of fluxes and holonomies. We first remark that the fluxes provide each edge with an angular momentum. By studying the piecewise-flat geometries which minimize edge lengths, we show that these angular momenta can be literally interpreted as the spin of the edges: the geometries of all edges are necessarily helices. We also show that the compatibility of the gluing maps with the holonomy data results in the same conclusion. This shows that a spinning geometry represents a way to glue together the three-cells of a twisted geometry to form a continuous geometry which represents a point in the loop gravity phase space. (paper)
Analytical geometry of three dimensions
McCrea, William H
2006-01-01
Brief but rigorous, this text is geared toward advanced undergraduates and graduate students. It covers the coordinate system, planes and lines, spheres, homogeneous coordinates, general equations of the second degree, quadric in Cartesian coordinates, and intersection of quadrics.Mathematician, physicist, and astronomer, William H. McCrea conducted research in many areas and is best known for his work on relativity and cosmology. McCrea studied and taught at universities around the world, and this book is based on a series of his lectures.
Technical calculus with analytic geometry
Gersting, Judith L
2010-01-01
This well-thought-out text, filled with many special features, is designed for a two-semester course in calculus for technology students with a background in college algebra and trigonometry. The author has taken special care to make the book appealing to students by providing motivating examples, facilitating an intuitive understanding of the underlying concepts involved, and by providing much opportunity to gain proficiency in techniques and skills.Initial chapters cover functions and graphs, straight lines and conic sections, new coordinate systems, the derivative, using the derivative, in
Modern calculus and analytic geometry
Silverman, Richard A
2012-01-01
A self-contained text for an introductory course, this volume places strong emphasis on physical applications. Key elements of differential equations and linear algebra are introduced early and are consistently referenced, all theorems are proved using elementary methods, and numerous worked-out examples appear throughout. The highly readable text approaches calculus from the student's viewpoint and points out potential stumbling blocks before they develop. A collection of more than 1,600 problems ranges from exercise material to exploration of new points of theory - many of the answers are fo
In this paper the X,Y-geometry SD-SGF-CN spectral nodal method, cf. spectral diamond-spectral Green's function-constant nodal, is used to determine the one-speed node-edge average angular fluxes in heterogeneous domains. This hybrid spectral nodal method uses the spectral diamond (SD) auxiliary equation for the multiplying regions and the spectral Green's function (SGF) auxiliary equation for the non-multiplying regions of the domain. Moreover, we consider constant approximations for the transverse-leakage terms in the transverse integrated SN nodal equations. We solve the SD-SGF-CN equations using the one-node block inversion (NBI) iterative scheme, which uses the most recent estimates available for the node-entering fluxes to evaluate the node-exiting fluxes in the directions that constitute the incoming fluxes for the adjacent node. Using these results, we offer an algorithm for analytical reconstruction of the coarse-mesh nodal solution within each spatial node, as localized numerical solutions are not generated by usual accurate nodal methods. Numerical results are presented to illustrate the accuracy of the present algorithm. (author)
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...
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...
Spain, Barry; Ulam, S; Stark, M
1960-01-01
Analytical Quadrics focuses on the analytical geometry of three dimensions. The book first discusses the theory of the plane, sphere, cone, cylinder, straight line, and central quadrics in their standard forms. The idea of the plane at infinity is introduced through the homogenous Cartesian coordinates and applied to the nature of the intersection of three planes and to the circular sections of quadrics. The text also focuses on paraboloid, including polar properties, center of a section, axes of plane section, and generators of hyperbolic paraboloid. The book also touches on homogenous coordi
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
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
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
Serdal Baltacı
2016-12-01
Full Text Available The potential of GeoGebra in teaching analytic geometry concepts was investigated in this paper. The study carried out with case study methodology and the participants were 6 pre-service mathematics teachers at 3rd grade of elementary mathematics education. All of the participants had the skill of well self-expression and they were volunteers for interview. Two participants were at high achievement levels, two participants were at medium achievement levels and two participants were low achievement levels. While carrying out each lesson, participants used worksheets which were prepared by the researchers. The data were obtained by semi-structured interviews which were carried out at the end of the courses and the data were analyzed with content analysis method. Research results showed that using dynamic mathematics software while studying on analytic geometry provides convenience for the participants and they felt more active while they were using software in the learning environment. [Bu çalışmada, analitik geometri kavramlarının öğretiminde GeoGebra’ nın potansiyeli incelenmiştir. Özel durum çalışması yöntemiyle yürütülen bu araştırmanın katılımcılarını, ilköğretim matematik öğretmenliği 3. sınıfa devam eden 6 öğretmen adayı oluşturmaktadır. Katılımcılar kendini ifade etme becerisi yüksek, mülakata gönüllü ve farklı başarı düzeyinde (yüksek, orta, düşük olan ikişer öğretmen adayından oluşmaktadır. Çalışmada analitik geometri dersleri, araştırmacılar tarafından geliştirilen çalışma yaprakları kullanılarak yürütülmüştür. Araştırmanın verileri derslerin sonunda yapılan yarı yapılandırılmış mülakatlarla toplanmıştır. Araştırmadan elde edilen veriler, içerik analizi yöntemi ile analiz edilmiştir. Araştırma sonuçları öğretmen adaylarının analitik geometri kavramlarını öğrenmede yazılımı kullanmalarının onlara kolaylık sa
Connes, Alain
1994-01-01
This English version of the path-breaking French book on this subject gives the definitive treatment of the revolutionary approach to measure theory, geometry, and mathematical physics developed by Alain Connes. Profusely illustrated and invitingly written, this book is ideal for anyone who wants to know what noncommutative geometry is, what it can do, or how it can be used in various areas of mathematics, quantization, and elementary particles and fields.Key Features* First full treatment of the subject and its applications* Written by the pioneer of this field* Broad applications in mathemat
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
Levenson, Esther; Tsamir, Pessia
2011-01-01
Recently the issue of early childhood mathematics has come to the fore and with it the importance of teaching geometrical concepts and reasoning from a young age. Geometry is a key domain mentioned in many national curricula and may also support the learning of other mathematical topics, such as number and patterns. This book is based on the rich experience (research and practice) of the authors and is devoted entirely to the learning and teaching of geometry in preschool. The first part of the book is dedicated to children's geometrical thinking, building concept images in line with concept d
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,
Desseyn, H. O.; And Others
1985-01-01
Compares linear-nonlinear and planar-nonplanar geometry through the valence-shell electron pairs repulsion (V.S.E.P.R.), Mulliken-Walsh, and electrostatic force theories. Indicates that although the V.S.E.P.R. theory has more advantages for elementary courses, an explanation of the best features of the different theories offers students a better…
Leonardo Paris
2012-06-01
Full Text Available Lo studio degli ingranaggi si basa sulle geometrie coniugate in cui due curve o due superfici si mantengono costantemente in contatto pur se in movimento reciproco. La teoria geometrica degli ingranaggi fino alla fine del XIX secolo era uno dei molteplici rami nelle applicazioni della Geometria Descrittiva. Lo studio si basa sulla conoscenza delle principali proprietà delle curve piane e gobbe e delle loro derivate. La specificità del tema è che queste geometrie nel momento in cui si devono relazionare con le loro coniugate, devono rispettare dei vincoli che altrimenti non avrebbero. Si vuole evidenziare attraverso casi concreti il ruolo della geometria descrittiva nel passaggio dal teorico al pratico riproponendo in chiave informatica, temi e procedure di indagine spesso passati in secondo piano se non addirittura dimenticati.
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 ...
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...
Gruber, Peter M
1987-01-01
This volume contains a fairly complete picture of the geometry of numbers, including relations to other branches of mathematics such as analytic number theory, diophantine approximation, coding and numerical analysis. It deals with convex or non-convex bodies and lattices in euclidean space, etc.This second edition was prepared jointly by P.M. Gruber and the author of the first edition. The authors have retained the existing text (with minor corrections) while adding to each chapter supplementary sections on the more recent developments. While this method may have drawbacks, it has the definit
Geometry The Language of Space and Form (Revised Edition)
Tabak, John
2011-01-01
Geometry, Revised Edition describes geometry in antiquity. Beginning with a brief description of some of the geometry that preceded the geometry of the Greeks, it takes up the story of geometry during the European Renaissance as well as the significant mathematical progress in other areas of the world. It also discusses the analytic geometry of Ren Descartes and Pierre Fermat, the alternative coordinate systems invented by Isaac Newton, and the solid geometry of Leonhard Euler. Also included is an overview of the geometry of one of the most successful mathematicians of the 19th century, Bernha
Leal, Andre Luiz do Carmo
2008-07-01
In this work we evaluate polynomial approximations to obtain the transfer functions that appear in SGF auxiliary equations (Green's Functions) for monoenergetic linearly anisotropic scattering SN equations in one-dimensional Cartesian geometry. For this task we use Lagrange Polynomials in order to compare the numerical results with the ones generated by the standard SGF method applied to SN problems in heterogeneous domains. This work is a preliminary investigation of a new proposal for handling the transverse leakage terms that appear in the transverse-integrated one-dimensional SN equations when we use the SGF - exponential nodal method (SGF-ExpN) in multidimensional rectangular geometry. (author)
Quantum Geometry and Quantum Gravity
Barbero González, Jesús Fernando
2008-01-01
The purpose of this contribution is to give an introduction to quantum geometry and loop quantum gravity for a wide audience of both physicists and mathematicians. From a physical point of view the emphasis will be on conceptual issues concerning the relationship of the formalism with other more traditional approaches inspired in the treatment of the fundamental interactions in the standard model. Mathematically I will pay special attention to functional analytic issues, the construction of t...
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.
Digital Differential Geometry Processing
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.
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...
Taylor, M
2006-01-01
Two charge BPS horizon free supergravity geometries are important in proposals for understanding black hole microstates. In this paper we construct a new class of geometries in the NS1-P system, corresponding to solitonic strings carrying fermionic as well as bosonic condensates. Such geometries are required to account for the full microscopic entropy of the NS1-P system. We then briefly discuss the properties of the corresponding geometries in the dual D1-D5 system.
Veronese geometry and the electroweak vacuum moduli space
We explain the origin of the Veronese surface in the vacuum moduli space geometry of the MSSM electroweak sector. While this result appeared many years ago using techniques of computational algebraic geometry, it has never been demonstrated analytically. Here, we present an analytical derivation of the vacuum geometry of the electroweak theory by understanding how the F- and D-term relations lead to the Veronese surface. We moreover give a detailed description of this geometry, realising an extra branch as a zero-dimensional point when quadratic Higgs lifting deformations are incorporated into the superpotential
Veronese geometry and the electroweak vacuum moduli space
He, Yang-Hui, E-mail: hey@maths.ox.ac.uk [Department of Mathematics, City University, London, Northampton Square, London EC1V 0HB (United Kingdom); School of Physics, NanKai University, Tianjin 300071 (China); Merton College, University of Oxford, Oxford OX1 4JD (United Kingdom); Jejjala, Vishnu, E-mail: vishnu@neo.phys.wits.ac.za [Centre for Theoretical Physics, NITheP, and School of Physics, University of the Witwatersrand, Johannesburg, WITS 2050 (South Africa); Matti, Cyril, E-mail: Cyril.Matti.1@city.ac.uk [Department of Mathematics, City University, London, Northampton Square, London EC1V 0HB (United Kingdom); Nelson, Brent D., E-mail: b.nelson@neu.edu [Department of Physics, Northeastern University, Boston, MA 02115 (United States); ICTP, Strada Costiera 11, Trieste 34014 (Italy)
2014-09-07
We explain the origin of the Veronese surface in the vacuum moduli space geometry of the MSSM electroweak sector. While this result appeared many years ago using techniques of computational algebraic geometry, it has never been demonstrated analytically. Here, we present an analytical derivation of the vacuum geometry of the electroweak theory by understanding how the F- and D-term relations lead to the Veronese surface. We moreover give a detailed description of this geometry, realising an extra branch as a zero-dimensional point when quadratic Higgs lifting deformations are incorporated into the superpotential.
Rossetto, V
2003-01-01
Motivated by recent experiments on DNA torsion-force-extension characteristics we consider the writhing geometry of open stiff molecules. We exhibit a cyclic motion which allows arbitrarily large twisting of the end of a molecule via an activated process. This process is suppressed for forces larger than femto-Newtons which allows us to show that experiments are sensitive to a generalization of the Calugareanu-White formula for the writhe. Using numerical methods we compare this formulation of the writhe with recent analytic calculations.
Thermodynamic geometry of holographic superconductors
Basak, Sayan; Nandi, Poulami; Sengupta, Gautam
2015-01-01
We obtain the thermodynamic geometry of a (2+1) dimensional strongly coupled quantum field theory at a finite temperature in a holographic set up through the gauge/gravity correspondence. The bulk dual gravitational theory is described by a 3+1 dimensional charged AdS black hole in the presence of a charged massive scalar field. The holographic free energy of the (2+1) dimensional strongly coupled boundary field theory is computed analytically through the bulk boundary correspondence. The thermodynamic metric and the corresponding scalar curvature is then obtained from the holographic free energy. The thermodynamic scalar curvature characterizes the superconducting phase transition of the boundary field theory.
Casimir interactions are interactions induced by quantum vacuum fluctuations and thermal fluctuations of the electromagnetic field. Using a path integral quantization for the gauge field, an effective Gaussian action will be derived which is the starting point to compute Casimir forces between macroscopic objects analytically and numerically. No assumptions about the independence of the material and shape dependent contributions to the interaction are made. We study the limit of flat surfaces in further detail and obtain a concise derivation of Lifshitz' theory of molecular forces. For the case of ideally conducting boundaries, the Gaussian action will be calculated explicitly. Both limiting cases are also discussed within the framework of a scalar field quantization approach, which is applicable for translationally invariant geometries. We develop a non-perturbative approach to calculate the Casimir interaction from the Gaussian action for periodically deformed and ideally conducting objects numerically. The obtained results reveal two different scaling regimes for the Casimir force as a function of the distance between the objects, their deformation wavelength and -amplitude. The results confirm that the interaction is non-additive, especially in the presence of strong geometric deformations. Furthermore, the numerical approach is extended to calculate lateral Casimir forces. The results are consistent with the results of the proximity-force approximation for large deformation wavelengths. A qualitatively different behaviour between the normal and lateral force is revealed. We also establish a relation between the boundary induced change of the of the density of states for the scalar Helmholtz equation and the Casimir interaction using the path integral method. For statically deformed boundaries, this relation can be expressed as a novel trace formula, which is formally similar to the so-called Krein-Friedel-Lloyd formula. While the latter formula describes the
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
Bendix: intuitive helix geometry analysis and abstraction
Dahl, Anna Caroline E.; Chavent, Matthieu; Sansom, Mark S P
2012-01-01
Summary: The flexibility of α-helices is important for membrane protein function and calls for better visualization and analysis. Software is presented that quantifies and projects the helix axis evolution over time, with the choice of uniform or analytic heatmap graphics according to the local geometry. Bendix supports static, molecular dynamics, atomistic and coarse-grained input.
Geometry of curves and surfaces with Maple
Rovenski, Vladimir
2000-01-01
This concise text on geometry with computer modeling presents some elementary methods for analytical modeling and visualization of curves and surfaces. The author systematically examines such powerful tools as 2-D and 3-D animation of geometric images, transformations, shadows, and colors, and then further studies more complex problems in differential geometry. Well-illustrated with more than 350 figures---reproducible using Maple programs in the book---the work is devoted to three main areas: curves, surfaces, and polyhedra. Pedagogical benefits can be found in the large number of Maple programs, some of which are analogous to C++ programs, including those for splines and fractals. To avoid tedious typing, readers will be able to download many of the programs from the Birkhauser web site. Aimed at a broad audience of students, instructors of mathematics, computer scientists, and engineers who have knowledge of analytical geometry, i.e., method of coordinates, this text will be an excellent classroom resource...
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.
Osborne, Ianna
2013-01-01
CMS faces real challenges with upgrade of the CMS detector through 2020. One of the challenges, from the software point of view, is managing upgrade simulations with the same software release as the 2013 scenario. We present the CMS geometry description software model, its integration with the CMS event setup and core software. The CMS geometry configuration and selection is implemented in Python. The tools collect the Python configuration fragments into a script used in CMS workflow. This flexible and automated geometry configuration allows choosing either transient or persistent version of the same scenario and specific version of the same scenario. We describe how the geometries are integrated and validated, how we define and handle different geometry scenarios in simulation and reconstruction. We discuss how to transparently manage multiple incompatible geometries in the same software release. Several examples are shown based on current implementation assuring consistent choice of scenario conditions. The...
Tame geometry with application in smooth analysis
Yomdin, Yosef
2004-01-01
The Morse-Sard theorem is a rather subtle result and the interplay between the high-order analytic structure of the mappings involved and their geometry rarely becomes apparent. The main reason is that the classical Morse-Sard theorem is basically qualitative. This volume gives a proof and also an "explanation" of the quantitative Morse-Sard theorem and related results, beginning with the study of polynomial (or tame) mappings. The quantitative questions, answered by a combination of the methods of real semialgebraic and tame geometry and integral geometry, turn out to be nontrivial and highly productive. The important advantage of this approach is that it allows the separation of the role of high differentiability and that of algebraic geometry in a smooth setting: all the geometrically relevant phenomena appear already for polynomial mappings. The geometric properties obtained are "stable with respect to approximation", and can be imposed on smooth functions via polynomial approximation.
Fundamental concepts of geometry
Meserve, Bruce E
2014-01-01
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.
Split Special Lagrangian Geometry
Harvey, F. Reese; Lawson Jr, H. Blaine
2010-01-01
One purpose of this article is to draw attention to the seminal work of J. Mealy in 1989 on calibrations in semi-riemannian geometry where split SLAG geometry was first introduced. The natural setting is provided by doing geometry with the complex numbers C replaced by the double numbers D, where i with i^2 = -1 is replaced by tau with tau^2 = 1. A rather surprising amount of complex geometry carries over, almost untouched, and this has been the subject of many papers. We briefly review this ...
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
Federal Laboratory Consortium — NETL’s analytical laboratories in Pittsburgh, PA, and Albany, OR, give researchers access to the equipment they need to thoroughly study the properties of materials...
Flannelly, W. G.; Fabunmi, J. A.; Nagy, E. J.
1981-01-01
Analytical methods for combining flight acceleration and strain data with shake test mobility data to predict the effects of structural changes on flight vibrations and strains are presented. This integration of structural dynamic analysis with flight performance is referred to as analytical testing. The objective of this methodology is to analytically estimate the results of flight testing contemplated structural changes with minimum flying and change trials. The category of changes to the aircraft includes mass, stiffness, absorbers, isolators, and active suppressors. Examples of applying the analytical testing methodology using flight test and shake test data measured on an AH-1G helicopter are included. The techniques and procedures for vibration testing and modal analysis are also described.
Geometry of multihadron production
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
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.
Discrete quantum geometries and their effective dimension
In several approaches towards a quantum theory of gravity, such as group field theory and loop quantum gravity, quantum states and histories of the geometric degrees of freedom turn out to be based on discrete spacetime. The most pressing issue is then how the smooth geometries of general relativity, expressed in terms of suitable geometric observables, arise from such discrete quantum geometries in some semiclassical and continuum limit. In this thesis I tackle the question of suitable observables focusing on the effective dimension of discrete quantum geometries. For this purpose I give a purely combinatorial description of the discrete structures which these geometries have support on. As a side topic, this allows to present an extension of group field theory to cover the combinatorially larger kinematical state space of loop quantum gravity. Moreover, I introduce a discrete calculus for fields on such fundamentally discrete geometries with a particular focus on the Laplacian. This permits to define the effective-dimension observables for quantum geometries. Analysing various classes of quantum geometries, I find as a general result that the spectral dimension is more sensitive to the underlying combinatorial structure than to the details of the additional geometric data thereon. Semiclassical states in loop quantum gravity approximate the classical geometries they are peaking on rather well and there are no indications for stronger quantum effects. On the other hand, in the context of a more general model of states which are superposition over a large number of complexes, based on analytic solutions, there is a flow of the spectral dimension from the topological dimension d on low energy scales to a real number between 0 and d on high energy scales. In the particular case of 1 these results allow to understand the quantum geometry as effectively fractal.
Discrete quantum geometries and their effective dimension
Thuerigen, Johannes
2015-07-02
In several approaches towards a quantum theory of gravity, such as group field theory and loop quantum gravity, quantum states and histories of the geometric degrees of freedom turn out to be based on discrete spacetime. The most pressing issue is then how the smooth geometries of general relativity, expressed in terms of suitable geometric observables, arise from such discrete quantum geometries in some semiclassical and continuum limit. In this thesis I tackle the question of suitable observables focusing on the effective dimension of discrete quantum geometries. For this purpose I give a purely combinatorial description of the discrete structures which these geometries have support on. As a side topic, this allows to present an extension of group field theory to cover the combinatorially larger kinematical state space of loop quantum gravity. Moreover, I introduce a discrete calculus for fields on such fundamentally discrete geometries with a particular focus on the Laplacian. This permits to define the effective-dimension observables for quantum geometries. Analysing various classes of quantum geometries, I find as a general result that the spectral dimension is more sensitive to the underlying combinatorial structure than to the details of the additional geometric data thereon. Semiclassical states in loop quantum gravity approximate the classical geometries they are peaking on rather well and there are no indications for stronger quantum effects. On the other hand, in the context of a more general model of states which are superposition over a large number of complexes, based on analytic solutions, there is a flow of the spectral dimension from the topological dimension d on low energy scales to a real number between 0 and d on high energy scales. In the particular case of 1 these results allow to understand the quantum geometry as effectively fractal.
Seif El-Nasr, Magy; Drachen, Anders; Canossa, Alessandro
2013-01-01
Game Analytics has gained a tremendous amount of attention in game development and game research in recent years. The widespread adoption of data-driven business intelligence practices at operational, tactical and strategic levels in the game industry, combined with the integration of quantitative...... measures in user-oriented game research, has caused a paradigm shift. Historically, game development has not been data-driven, but this is changing as the benefits of adopting and adapting analytics to inform decision making across all levels of the industry are becoming generally known and accepted....
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
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.
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
Viral nematics in confined geometries.
Manyuhina, O V; Lawlor, K B; Marchetti, M C; Bowick, M J
2015-08-14
Motivated by recent experiments on the rod-like virus bacteriophage fd, confined to circular and annular domains, we present a theoretical study of structural transitions in these geometries. Using the continuum theory of nematic liquid crystals, we examine the competition between bulk elasticity and surface anchoring, mediated by the formation of topological defects. We show analytically that bulk defects are unstable with respect to defects sitting at the boundary. In the case of an annulus, whose topology does not require the presence of topological defects, we find that nematic textures with boundary defects are stable compared to defect-free configurations when the anchoring is weak. Our simple approach, with no fitting parameters, suggests a possible symmetry breaking mechanism responsible for the formation of one-, two- and three-fold textures under annular confinement. PMID:26135676
Thermodynamic geometry of holographic superconductors
Sayan Basak
2016-02-01
Full Text Available We obtain the thermodynamic geometry of a (2+1 dimensional strongly coupled quantum field theory at a finite temperature in a holographic setup, through the gauge/gravity correspondence. The bulk dual gravitational theory is described by a (3+1 dimensional charged AdS black hole in the presence of a massive charged scalar field. The holographic free energy of the (2+1 dimensional strongly coupled boundary field theory is computed analytically through the bulk boundary correspondence. The thermodynamic metric and the corresponding scalar curvature are then obtained from the holographic free energy. The thermodynamic scalar curvature characterizes the superconducting phase transition of the boundary field theory.
Pappas, Marjorie L.
1995-01-01
Discusses analytical searching, a process that enables searchers of electronic resources to develop a planned strategy by combining words or phrases with Boolean operators. Defines simple and complex searching, and describes search strategies developed with Boolean logic and truncation. Provides guidelines for teaching students analytical…
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...
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)
Noncommutative Geometry and Physics
In this very short essay we shall describe a 'spectral' point of view on geometry which allows to start taking into account the lessons from both renormalization and of general relativity. We shall first do that for renormalization and explain in rough outline the content of our recent collaborations with Dirk Kreimer and Matilde Marcolli leading to the universal Galois symmetry of renormalizable quantum field theories provided by the renormalization group in its cosmic Galois group incarnation. As far as general relativity is concerned, since the functional integral cannot be treated in the traditional perturbative manner, it relies heavily as a 'sum over geometries' on the chosen paradigm of geometric space. This will give us the occasion to discuss, in the light of noncommutative geometry, the issue of 'observables' in gravity and our joint work with Ali Chamseddine on the spectral action, with a first attempt to write down a functional integral on the space of noncommutative geometries
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...
Confinement and related transport in Extrap geometry
The properties of the plasma equilibrium are investigated for the Extrap magnetic confinement geometry. An analytical solution for the profiles of the plasma parameters are found under the assumption that the energy is lost primarily in the radical direction by heat conduction and convection. An estimate of the radial particle confinement time is given, showing favorable scaling with plasma density and temperature. The conventional assumption of a uniform current density is shown to be unjustified in the case of an inhomogeneus electron temperature. An analytical expression is found for the pinch radius at different mechanisms of the heat transport. (author)
Implosions and hypertoric geometry
Dancer, A.; Kirwan, F.; Swann, A.
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....
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.
Ashtekar, Abhay
1999-01-01
Over the past six years, a detailed framework has been constructed to unravel the quantum nature of the Riemannian geometry of physical space. A review of these developments is presented at a level which should be accessible to graduate students in physics. As an illustrative application, I indicate how some of the detailed features of the micro-structure of geometry can be tested using black hole thermodynamics. Current and future directions of research in this area are discussed.
Compositional Geometry and Programming
Bantchev, Boyko
2013-01-01
Report published in the Proceedings of the National Conference on "Education and Research in the Information Society", Plovdiv, May, 2014 Functional, or compositional, geometry is a method of constructing complex pictures from simpler ones, apllying binding operations. The method is naturally related to functional programming and can be used as a tool for education or self-education in programming. The paper introduces to compositional geometry, bringing attention to its severa...
Vitório Pereira, Jorge
2015-01-01
This book takes an in-depth look at abelian relations of codimension one webs in the complex analytic setting. In its classical form, web geometry consists in the study of webs up to local diffeomorphisms. A significant part of the theory revolves around the concept of abelian relation, a particular kind of functional relation among the first integrals of the foliations of a web. Two main focuses of the book include how many abelian relations can a web carry and which webs are carrying the maximal possible number of abelian relations. The book offers complete proofs of both Chern’s bound and Trépreau’s algebraization theorem, including all the necessary prerequisites that go beyond elementary complex analysis or basic algebraic geometry. Most of the examples known up to date of non-algebraizable planar webs of maximal rank are discussed in detail. A historical account of the algebraization problem for maximal rank webs of codimension one is also presented.
Induced geometry from disformal transformation
Yuan, Fang-Fang, E-mail: ffyuan@nankai.edu.cn [School of Physics, Nankai University, Tianjin 300071 (China); Huang, Peng, E-mail: huangp46@mail.sysu.edu.cn [School of Astronomy and Space Science, Sun Yat-Sen University, Guangzhou 510275 (China)
2015-05-11
In this note, we use the disformal transformation to induce a geometry from the manifold which is originally Riemannian. The new geometry obtained here can be considered as a generalization of Weyl integrable geometry. Based on these results, we further propose a geometry which is naturally a generalization of Weyl geometry.
Software Geometry in Simulations
Alion, Tyler; Viren, Brett; Junk, Tom
2015-04-01
The Long Baseline Neutrino Experiment (LBNE) involves many detectors. The experiment's near detector (ND) facility, may ultimately involve several detectors. The far detector (FD) will be significantly larger than any other Liquid Argon (LAr) detector yet constructed; many prototype detectors are being constructed and studied to motivate a plethora of proposed FD designs. Whether it be a constructed prototype or a proposed ND/FD design, every design must be simulated and analyzed. This presents a considerable challenge to LBNE software experts; each detector geometry must be described to the simulation software in an efficient way which allows for multiple authors to easily collaborate. Furthermore, different geometry versions must be tracked throughout their use. We present a framework called General Geometry Description (GGD), written and developed by LBNE software collaborators for managing software to generate geometries. Though GGD is flexible enough to be used by any experiment working with detectors, we present it's first use in generating Geometry Description Markup Language (GDML) files to interface with LArSoft, a framework of detector simulations, event reconstruction, and data analyses written for all LAr technology users at Fermilab. Brett is the other of the framework discussed here, the General Geometry Description (GGD).
Introduction to combinatorial geometry
The combinatorial geometry package as used in many three-dimensional multimedia Monte Carlo radiation transport codes, such as HETC, MORSE, and EGS, is becoming the preferred way to describe simple and complicated systems. Just about any system can be modeled using the package with relatively few input statements. This can be contrasted against the older style geometry packages in which the required input statements could be large even for relatively simple systems. However, with advancements come some difficulties. The users of combinatorial geometry must be able to visualize more, and, in some instances, all of the system at a time. Errors can be introduced into the modeling which, though slight, and at times hard to detect, can have devastating effects on the calculated results. As with all modeling packages, the best way to learn the combinatorial geometry is to use it, first on a simple system then on more complicated systems. The basic technique for the description of the geometry consists of defining the location and shape of the various zones in terms of the intersections and unions of geometric bodies. The geometric bodies which are generally included in most combinatorial geometry packages are: (1) box, (2) right parallelepiped, (3) sphere, (4) right circular cylinder, (5) right elliptic cylinder, (6) ellipsoid, (7) truncated right cone, (8) right angle wedge, and (9) arbitrary polyhedron. The data necessary to describe each of these bodies are given. As can be easily noted, there are some subsets included for simplicity
CMS faces real challenges with upgrade of the CMS detector through 2020 and beyond. One of the challenges, from the software point of view, is managing upgrade simulations with the same software release as the 2013 scenario. We present the CMS geometry description software model, its integration with the CMS event setup and core software. The CMS geometry configuration and selection is implemented in Python. The tools collect the Python configuration fragments into a script used in CMS workflow. This flexible and automated geometry configuration allows choosing either transient or persistent version of the same scenario and specific version of the same scenario. We describe how the geometries are integrated and validated, and how we define and handle different geometry scenarios in simulation and reconstruction. We discuss how to transparently manage multiple incompatible geometries in the same software release. Several examples are shown based on current implementation assuring consistent choice of scenario conditions. The consequences and implications for multiple/different code algorithms are discussed.
Osborne, I.; Brownson, E.; Eulisse, G.; Jones, C. D.; Lange, D. J.; Sexton-Kennedy, E.
2014-06-01
CMS faces real challenges with upgrade of the CMS detector through 2020 and beyond. One of the challenges, from the software point of view, is managing upgrade simulations with the same software release as the 2013 scenario. We present the CMS geometry description software model, its integration with the CMS event setup and core software. The CMS geometry configuration and selection is implemented in Python. The tools collect the Python configuration fragments into a script used in CMS workflow. This flexible and automated geometry configuration allows choosing either transient or persistent version of the same scenario and specific version of the same scenario. We describe how the geometries are integrated and validated, and how we define and handle different geometry scenarios in simulation and reconstruction. We discuss how to transparently manage multiple incompatible geometries in the same software release. Several examples are shown based on current implementation assuring consistent choice of scenario conditions. The consequences and implications for multiple/different code algorithms are discussed.
Wobbling geometry in simple triaxial rotor
Shi, W X
2014-01-01
The spectroscopy properties and angular momentum geometry for the wobbling motion of a simple triaxial rotor are investigated within the triaxial rotor model up to spin $I=40\\hbar$. The obtained exact solutions of energy spectra and reduced quadrupole transition probabilities are compared to the approximate analytic solutions by harmonic approximation formula and Holstein-Primakoff formula. It is found that the low lying wobbling bands can be well described by the analytic formulas. The evolution of the angular momentum geometry as well as the $K$-distribution with respect to the rotation and the wobbling phonon excitation are studied in detail. It is demonstrated that with the increasing of wobbling phonon number, the triaxial rotor changes its wobbling motions along the axis with the largest moment of inertia to the axis with the smallest moment of inertia. In this process, a specific evolutionary track that can be used to depict the motion of a triaxial rotating nuclei is proposed.
Compaction of granular material inside confined geometries
Benjy eMarks
2015-06-01
Full Text Available In both nature and the laboratory, loosely packed granular materials are often compacted inside confined geometries. Here, we explore such behaviour in a quasi-two dimensional geometry, where parallel rigid walls provide the confinement. We use the discrete element method to investigate the stress distribution developed within the granular packing as a result of compaction due to the displacement of a rigid piston. We observe that the stress within the packing increases exponentially with the length of accumulated grains, and show an extension to current analytic models which fits the measured stress. The micromechanical behaviour is studied for a range of system parameters, and the limitations of existing analytic models are described. In particular, we show the smallest sized systems which can be treated using existing models. Additionally, the effects of increasing piston rate, and variations of the initial packing fraction, are described.
Update on Scroll Compressor Chamber Geometry
Bell, Ian; Groll, Eckhard; Braun, James; King, Galen
2010-01-01
The geometry of the scroll compressor determines the efficiency of the scroll compressor and controls all elements of its operation. It is therefore critical to be able to accurately model the volumes of the compressor over the course of a revolution. This paper proposes a novel quasi-analytic formulation of the suction, compression and discharge chambers based on a change of variables from involute angle to polar integration angle. This solution has been compared against a reference polyg...
This book is comprised of nineteen chapters, which describes introduction of analytical chemistry, experimental error and statistics, chemistry equilibrium and solubility, gravimetric analysis with mechanism of precipitation, range and calculation of the result, volume analysis on general principle, sedimentation method on types and titration curve, acid base balance, acid base titration curve, complex and firing reaction, introduction of chemical electro analysis, acid-base titration curve, electrode and potentiometry, electrolysis and conductometry, voltammetry and polarographic spectrophotometry, atomic spectrometry, solvent extraction, chromatograph and experiments.
The division for Analytical Chemistry continued to try and develope an accurate method for the separation of trace amounts from mixtures which, contain various other elements. Ion exchange chromatography is of special importance in this regard. New separation techniques were tried on certain trace amounts in South African standard rock materials and special ceramics. Methods were also tested for the separation of carrier-free radioisotopes from irradiated cyclotron discs
Geometry - Construction - Architecture
Riccardo Migliari
2012-06-01
Full Text Available Geometry, in Drawing, is suffering from a profound crisis, more than other disciplines of the area, and this crisis has confined it, according a few, within a field of study of top-grade specialization, that only few scholars are interested in. The reason for this crisis can be found in the changes that were induced, during the last thirty years, by the advent of the information technology, but not only there. It can also be found, in my opinion, in the rightful ambition of all of us architects to carry out research, an ambition that the traditional Geometry, both descriptive and more in general graphical, could not satisfy. In fact, that part of descriptive geometry, which deals with the graphical representation on a plane of three-dimensional objects, has been outclassed by 3D modelling, both in simplicity of execution and in quality of the results. Whilst the graphical geometry has been neglected even by the mathematicians, during the axiomatic turning-point of the 1920s, a part from some exceptions - Coxeter - and a few recent reconsiderations. Yet, geometry continues to play a leading role in drawing, in general, and in architecture, in particular, and I refer to the historical studies, to the studies of surfaces, to the relation with art and, last but not least important, to teaching.
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...
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...
Quantum Geometry and Interferometry
Hogan, Craig
2012-01-01
All existing experimental results are currently interpreted using classical geometry. However, there are theoretical reasons to suspect that at a deeper level, geometry emerges as an approximate macroscopic behavior of a quantum system at the Planck scale. If directions in emergent quantum geometry do not commute, new quantum-geometrical degrees of freedom can produce detectable macroscopic deviations from classicality: spatially coherent, transverse position indeterminacy between any pair of world lines, with a displacement amplitude much larger than the Planck length. Positions of separate bodies are entangled with each other, and undergo quantum-geometrical fluctuations that are not describable as metric fluctuations or gravitational waves. These fluctuations can either be cleanly identified or ruled out using interferometers. A Planck-precision test of the classical coherence of space-time on a laboratory scale is now underway at Fermilab.
Strings on Bubbling Geometries
Lin, Hai; Shock, Jonathan P
2010-01-01
We study gauge theory operators which take the form of a product of a trace with a Schur polynomial, and their string theory duals. These states represent strings excited on bubbling AdS geometries which are dual to the Schur polynomials. These geometries generically take the form of multiple annuli in the phase space plane. We study the coherent state wavefunction of the lattice, which labels the trace part of the operator, for a general Young tableau and their dual description on the droplet plane with a general concentric ring pattern. In addition we identify a density matrix over the coherent states on all the geometries within a fixed constraint. This density matrix may be used to calculate the entropy of a given ensemble of operators. We finally recover the BMN string spectrum along the geodesic near any circle from the ansatz of the coherent state wavefunction.
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...
Chern, Shiing-Shen
2005-01-01
Riemann-Finsler geometry is a subject that concerns manifolds with Finsler metrics, including Riemannian metrics. It has applications in many fields of the natural sciences. Curvature is the central concept in Riemann-Finsler geometry. This invaluable textbook presents detailed discussions on important curvatures such as the Cartan torsion, the S-curvature, the Landsberg curvature and the Riemann curvature. It also deals with Finsler metrics with special curvature or geodesic properties, such as projectively flat Finsler metrics, Berwald metrics, Finsler metrics of scalar flag curvature or iso
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
Elementary differential geometry
O'Neill, Barrett
2006-01-01
Written primarily for students who have completed the standard first courses in calculus and linear algebra, ELEMENTARY DIFFERENTIAL GEOMETRY, REVISED SECOND EDITION, provides an introduction to the geometry of curves and surfaces. The Second Edition maintained the accessibility of the first, while providing an introduction to the use of computers and expanding discussion on certain topics. Further emphasis was placed on topological properties, properties of geodesics, singularities of vector fields, and the theorems of Bonnet and Hadamard. This revision of the Second Edition p
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.
Cooper, Brett D.; Barger, Rita
2009-01-01
The many connections between music and mathematics are well known. The length of a plucked string determines its tone, the time signature of a piece of music is a ratio, and note durations are measured in fractions. One connection commonly overlooked is that between music and geometry--specifically, geometric transformations, including…
Towards relativistic quantum geometry
Ridao, Luis Santiago; Bellini, Mauricio
2015-12-01
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.
Metrics for Probabilistic Geometries
Tosi, Alessandra; Hauberg, Søren; Vellido, Alfredo; Lawrence, Neil D.
We investigate the geometrical structure of probabilistic generative dimensionality reduction models using the tools of Riemannian geometry. We explicitly define a distribution over the natural metric given by the models. We provide the necessary algorithms to compute expected metric tensors where...
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...
Hsü, K J; Hsü, A J
1990-01-01
Music critics have compared Bach's music to the precision of mathematics. What "mathematics" and what "precision" are the questions for a curious scientist. The purpose of this short note is to suggest that the mathematics is, at least in part, Mandelbrot's fractal geometry and the precision is the deviation from a log-log linear plot.
Towards relativistic quantum geometry
Luis Santiago Ridao
2015-12-01
Full Text Available 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.
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.
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 elabo...
The lectures given at the ''5th Symposium of Mathematics in Abidjan: Differential Geometry and Mechanics'' are presented. They are divided into four chapters: Riemannian metric on a differential manifold, curvature tensor fields on a Riemannian manifold, some classical functionals on Riemannian manifolds and questions. 11 refs
The basic ideas of description methods of physical fields and elementary particle interactions are discussed. One of such ideas is the conception of space-time geometry. In this connection experimental measurement methods are analyzed. It is shown that measure procedures are the origin of geometrical axioms. The connection between space symmetry properties and the conservation laws is considered
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.
The paper concerns the physical principles behind the analytical techniques employing high energy ion microbeams, with special attention to features that affect their use with microbeams. Particle-induced x-ray emission (PIXIE) is discussed with respect to X-ray production, thick-target PIXIE, a microbeam PIXIE system, sensitivity, and microbeam PIXIE applications. An explanation of nuclear reaction analysis (NRA) is given for NRA with charged particle detection, NRA with neutron detection and NRA with gamma detection. The essentials of Rutherford back scattering (RBS) are given, along with the elastic recoil detection analysis, which has very close connections with RBS but was introduced much more recently. Finally a comparison of the microbeam's capability with those of its main competitors is presented. (UK)
Kinetics of binding and geometry of cells on molecular biochips
Chechetkin, V.R.
2011-01-01
We examine how the shape of cells and the geometry of experiment affect the reaction-diffusion kinetics at the binding between target and probe molecules on molecular biochips. In particular, we compare the binding kinetics for the probes immobilized on surface of the semispherical and flat circular cells, the limit of thin slab of analyte solution over probe cell as well as hemispherical gel pads and cells printed in gel slab over a substrate. It is shown that hemispherical geometry provides...
Polarizability of nanowires at surfaces: Exact solution for general geometry
Jung, Jesper; Pedersen, Thomas G.
2012-01-01
The polarizability of a nanostructure is an important parameter that determines the optical properties. An exact semi-analytical solution of the electrostatic polarizability of a general geometry consisting of two segments forming a cylinder that can be arbitrarily buried in a substrate is derived using bipolar coordinates, cosine-, and sine-transformations. Based on the presented expressions, we analyze the polarizability of several metal nanowire geometries that are important within plasmon...
Novel geometry gradient coils for MRI designed by genetic algorithm
Williams, Guy Barnett
2001-01-01
This thesis concerns the design of gradient coils for magnetic resonance imaging systems. The method of design by genetic algorithm optimisation is applied to novel gradient geometries both by use of conventional computer facilities, and, by parallelisation of the design algorithm, on a supercomputer architecture. Geometries and regions of interests which are inaccessible to analytic solution are considered, and the criteria which are difficult to include in such algorithms, such as the robus...
Smania, Daniel
2007-07-01
We describe a new and robust method to prove rigidity results in complex dynamics. The new ingredient is the geometry of the critical puzzle pieces: under control of geometry and ``complex bounds'', two generalized polynomial-like maps which admit a topological conjugacy, quasiconformal outside the filled-in Julia set, are indeed quasiconformally conjugate. The proof uses a new abstract removability-type result for quasiconformal maps, following ideas of Heinonen and Koskela and of Kallunki and Koskela, optimized for applications in complex dynamics. We prove, as the first application of this new method, that, for even criticalities distinct from two, the period two cycle of the Fibonacci renormalization operator is hyperbolic with 1 -dimensional unstable manifold.
Chamseddine, Ali H; Mukhanov, Viatcheslav
2014-01-01
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 manifolds with large quantized volume are then obtained as solutions. When this condition is adopted in the gravitational action it leads to the quantization of the four volume with the cosmological constant obtained as an integration constant. Restricting the condition to a three dimensional hypersurface implies quantization of the three volume and the possible appearance of mimetic dark matter. When restricting to a two dimensional hypersurface, under appropriate bounda...
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.
Zupan, Karmen
2013-01-01
Observing is an important process in learning geometry. In the first part of the thesis observing is considered from a psychological perspective: the Gestalt theory, its history, the distribution of gestalt qualities, as well as various studies and theories of templates. The observing process is considered also from the didactic point of view by means of the van Hiele’s theory of levels of geometric reasoning. Since colours also influence observing, a list of advices for teachers about using ...
Tysver, Joseph Bryce
1982-01-01
This report presents the potential use of 3-D data at NUWES on trial runs to provide information on the geometry of two vehicles in the vicinity of intercept. Smoothing on data segments provides velocity components as well as smoothed estimates or vehicle locations. Analysis of this smoothed data can be analyzed to establish (1) distance between vehicles (2) vehicular heading directional angles (3) look angle for attack vehicle, (4) attack angle (5) projected intercept point and time, (6) p...
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....
Noncommutative Geometry Year 2000
Connes, Alain
2000-01-01
We describe basic concepts of noncommutative geometry and a general construction extending the familiar duality between ordinary spaces and commutative algebras to a duality between Quotient spaces and Noncommutative algebras. Basic tools of the theory, K-theory, Cyclic cohomology, Morita equivalence, Operator theoretic index theorems, Hopf algebra symmetry are reviewed. They cover the global aspects of noncommutative spaces, such as the transformation $\\theta \\to 1/\\theta$ for the NC torus $...
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.
Relativistic quantum equation in nonstationary space-time geometry
Gaffour, L, E-mail: lgaffour.usba@yahoo.fr [Department of Physics, Laboratory of Elaboration and Caracterisation of Materials, Faculty of Sciences, Sidi-Bel-Abbes university, 22000 (Algeria)
2011-07-08
An analytical solution for the 1-1 D relativistic quantum equation in dependent space-time geometry is given when the imaginary part of the particle mass is negligible. The solution is accomplished through covariant transformations of wave equation. The solutions have a modal structure and the modes are dynamics. The relativistic redshift turns out to be a manifestation of the space-time geometry motion. As illustrations, two space-time geometry motions corresponding to nonlinear mappings and the linear Lorentz transformations are discussed.
Geometry Optimization Of Marinelli Sample In Environmental Radioactivity Measurement
The problem of geometry optimization in environmental radioactivity determination has been studied by many scientists in the world. However, up to now, there have been not any published articles which studied optimum sample geometry in any given volumes. In this work, the simulation program MCNP was used to build the detection efficiency analytical formulas which can be used to calculate optimized geometries of Marinelli beaker in environmental radioactivity measurement. The geometry optimization was carried out with the sample volumes from 10 ml to 450 ml and the gamma-ray energy from 60 keV to 2 MeV. These optimized geometries give the highest detection efficiencies corresponding to given volumes. The goal of our study is to improve the limitation in low-level radioactivity measurement of environmental samples. (author)
Two lectures on D-geometry and noncommutative geometry
This is a write-up of lectures given at the 1998 Spring School at the Abdus Salam ICTP. We give a conceptual introduction to D-geometry, the study of geometry as seen by D-branes in string theory, and to noncommutative geometry as it has appeared in D-brane and Matrix theory physics. (author)
Holographic free energy and thermodynamic geometry
Ghorai, Debabrata
2016-01-01
We analytically obtain the free energy and thermodynamic geometry of holographic superconductors in $2+1$-dimensions. The gravitational theory in the bulk dual to this $2+1$-dimensional strongly coupled theory lives in the $3+1$-dimensions and is that of a charged $AdS$ black hole together with a massive charged scalar field. The matching method is applied to obtain the nature of the fields near the horizon using which the holographic free energy is computed through the gauge/gravity duality. The critical temperature is obtained for a set of values of the matching point of the near horizon and the boundary behaviour of the fields. The thermodynamic geometry is then computed from the free energy of the boundary theory. From the divergence of the thermodynamic scalar curvature, the critical temperature is obtained once again. We then compare this result for the critical temperature with that obtained from the matching method.
The universal instability in general geometry
Helander, P.; Plunk, G. G. [Max-Planck-Institut für Plasmaphysik, Wendelsteinstr. 1, 17491 Greifswald (Germany)
2015-09-15
The “universal” instability has recently been revived by Landreman et al. [Phys. Rev. Lett. 114, 095003 (2015)], who showed that it indeed exists in plasma geometries with straight (but sheared) magnetic field lines. Here, it is demonstrated analytically that this instability can be presented in more general sheared and toroidal geometries. In a torus, the universal instability is shown to be closely related to the trapped-electron mode, although the trapped-electron drive is usually dominant. However, this drive can be weakened or eliminated, as in the case in stellarators with the maximum-J property, leaving the parallel Landau resonance to drive a residual mode, which is identified as the universal instability.
The universal instability in general geometry
The “universal” instability has recently been revived by Landreman et al. [Phys. Rev. Lett. 114, 095003 (2015)], who showed that it indeed exists in plasma geometries with straight (but sheared) magnetic field lines. Here, it is demonstrated analytically that this instability can be presented in more general sheared and toroidal geometries. In a torus, the universal instability is shown to be closely related to the trapped-electron mode, although the trapped-electron drive is usually dominant. However, this drive can be weakened or eliminated, as in the case in stellarators with the maximum-J property, leaving the parallel Landau resonance to drive a residual mode, which is identified as the universal instability
LEARNING GEOMETRY THROUGH MIMESIS AND DIGITAL CONSTRUCT
Maria Mion POP
2015-12-01
Full Text Available The theme proposed by us is useful to teachers and students for mathematics in the compulsory school cycle. The issues faced by school teachers/parents are the difficulty with which students read and understand the lessons/examples/synthesis in order to assimilate technical terms. The echoic and iconic memory facilitates the learning of the specific curriculum of linear, spatial and analytical geometry by the students using digital platform designed by us; it facilitates the acquiring of the theoretical elements of applied geometry by encoding-decoding, so that the teacher's role becomes the one of the advisor and not only a person who transmits the information. The utility of the program extends from mainstream schools to special schools.
A geometric framework for quantum mechanics arose during the mid 1970s when authors such as Cantoni explored the notion of generalized transition probabilities, and Kibble promoted the idea that the space of pure quantum states provides a natural quantum mechanical analogue for classical phase space. This central idea can be seen easily since the projection of Schroedinger's equation from a Hilbert space into the space of pure spaces is a set of Hamilton's equations. Over the intervening years considerable work has been carried out by a variety of authors and a mature description of quantum mechanics in geometric terms has emerged with many applications. This current offering would seem ideally placed to review the last thirty years of progress and relate this to the most recent work in quantum entanglement. Bengtsson and Zyczkowski's beautifully illustrated volume, Geometry of Quantum States (referred to as GQS from now on) attempts to cover considerable ground in its 466 pages. Its topics range from colour theory in Chapter 1 to quantum entanglement in Chapter 15-to say that this is a whirlwind tour is, perhaps, no understatement. The use of the work 'introduction' in the subtitle of GQS, might suggest to the reader that this work be viewed as a textbook and I think that this interpretation would be incorrect. The authors have chosen to present a survey of different topics with the specific aim to introduce entanglement in geometric terms-the book is not intended as a pedagogical introduction to the geometric approach to quantum mechanics. Each of the fifteen chapters is a short, and mostly self-contained, essay on a particular aspect or application of geometry in the context of quantum mechanics with entanglement being addressed specifically in the final chapter. The chapters fall into three classifications: those concerned with the mathematical background, those which discuss quantum theory and the foundational aspects of the geometric framework, and
Introductory non-Euclidean geometry
Manning, Henry Parker
2013-01-01
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.
Reflexive functors in Algebraic Geometry
Sancho, Pedro
2015-01-01
Reflexive functors of modules naturally appear in Algebraic Geometry. In this paper we define a wide and elementary family of reflexive functors of modules, closed by tensor products and homomorphisms, in which Algebraic Geometry can be developed.
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
Kokkendorff, Simon Lyngby
2002-01-01
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 is then...... applied to show, that hyperbolic spaces are of strictly negative type. We also give an application to maximal distributions of subharmonic kernels. The most important application is probably the discussion of closed geodesics and negative type. Among other things we show, that a compact Riemannian...
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
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
Duality and asymptotic geometries
Boonstra, H J; Skenderis, K
1997-01-01
We consider a series of duality transformations that leads to a constant shift in the harmonic functions appearing in the description of a configuration of branes. This way, for several intersections of branes, we can relate the original brane configuration which is asymptotically flat to a geometry of the type $adS_k \\xx E^l \\xx S^m$. The implications of our results for supersymmetry enhancement, M(atrix) theory at finite N, and for supergravity theories in diverse dimensions are discussed.
The geometry of thermodynamics
Quevedo, Hernando
2007-01-01
We present a review of the main aspects of geometrothermodynamics, an approach which allows us to associate a specific Riemannian structure to any classical thermodynamic system. In the space of equilibrium states, we consider a Legendre invariant metric, which is given in terms of the fundamental equation of the corresponding thermodynamic system, and analyze its geometric properties in the case of the van der Waals gas, and black holes. We conclude that the geometry of this particular metric reproduces the thermodynamic behavior of the van der Waals gas, and the Reissner-Nordstr\\"om black hole, but it is not adequate for the thermodynamic description of Kerr black holes.
Developing thinking in geometry
Johnston-Wilder, Sue
2005-01-01
'Geometry is often given less time in the teaching timetable than other aspects of mathematics. This book encourages practitioners to think about and raise its profile, indeed achieving what its title suggest' - Primary Practice `This creative, innovative and fascinating book/CD package is one you ""MUST BUY"". All prospective, new and experienced teachers of mathematics can use it to transform their teaching. All readers can use it to reignite their fascination with mathematics' - Professor Sylvia Johnson, Sheffield Hallam University 'This book exudes activity and interactivity. Moreov
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
Isaev, Alexander
2007-01-01
Kobayashi-hyperbolic manifolds are an object of active research in complex geometry. In this monograph the author presents a coherent exposition of recent results on complete characterization of Kobayashi-hyperbolic manifolds with high-dimensional groups of holomorphic automorphisms. These classification results can be viewed as complex-geometric analogues of those known for Riemannian manifolds with high-dimensional isotropy groups, that were extensively studied in the 1950s-70s. The common feature of the Kobayashi-hyperbolic and Riemannian cases is the properness of the actions of the holomorphic automorphism group and the isometry group on respective manifolds.
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.
Nonlinear connections and spinor geometry
Nadejda A. Vicol
2004-05-01
Full Text Available We present an introduction to the geometry of higher-order vector and covector bundles (including higher-order generalizations of the Finsler geometry and Kaluza-Klein gravity and review the basic results on Clifford and spinor structures on spaces with generic local anisotropy modeled by anholonomic frames with associated nonlinear connection structures. We emphasize strong arguments for application of Finsler-like geometries in modern string and gravity theory, noncommutative geometry and noncommutative field theory, and gravity.
Stability and mix in spherical geometry
We consider a spherical system composed of N concentric fluid shells having perturbations of amplitude ηi at interface i, i=1,2,...,N-1. For arbitrary implosion-explosion histories Ri(t), we present the set of N-1 second-order differential equations describing the time evolution of the ηi which are coupled to the two adjacent ηi±1. We report analytical solutions for the N=2 and N=3 cases. We also present a model to describe the evolution of a turbulent mixing layer in spherical geometry when the interface between two fluids undergoes a constant acceleration or a shock
The Geometry of Walker Manifolds
Gilkey, Peter
2009-01-01
This book, which focuses on the study of curvature, is an introduction to various aspects of pseudo-Riemannian geometry. We shall use Walker manifolds (pseudo-Riemannian manifolds which admit a non-trivial parallel null plane field) to exemplify some of the main differences between the geometry of Riemannian manifolds and the geometry of pseudo-Riemannian manifolds and thereby illustrate phenomena in pseudo-Riemannian geometry that are quite different from those which occur in Riemannian geometry, i.e. for indefinite as opposed to positive definite metrics. Indefinite metrics are important in
The authors address the problem of following the trajectory of a particle in simulations. It is necessary to follow the motion of the particle, and to determine its intersection with different geometric surfaces in the problem, in order to relate the stepping of the particle trajectory into real motion through the physical problem at hand. The distance a particle moves before encountering a surface is needed to compare with the actual transport distance that is about to be used in the simulation. Basic mathematical expressions are developed for the intersections of particle trajectories with plane and conic surfaces. The authors show how these are used in the EGS4 code system, which should be typical of the general problem. They also review geometry packages currently being used in electron-photon Monte Carlo programs
Emergent geometry of membranes
de Badyn, Mathias Hudoba; Karczmarek, Joanna L.; Sabella-Garnier, Philippe; Yeh, Ken Huai-Che
2015-11-01
In work [1], a surface embedded in flat ℝ 3 is associated to any three hermitian matrices. We study this emergent surface when the matrices are large, by constructing coherent states corresponding to points in the emergent geometry. We find the original matrices determine not only shape of the emergent surface, but also a unique Poisson structure. We prove that commutators of matrix operators correspond to Poisson brackets. Through our construction, we can realize arbitrary noncommutative membranes: for example, we examine a round sphere with a non-spherically symmetric Poisson structure. We also give a natural construction for a noncommutative torus embedded in ℝ 3. Finally, we make remarks about area and find matrix equations for minimal area surfaces.
Optically defined mechanical geometry
Barasheed, Abeer Z.; Müller, Tina; Sankey, Jack C.
2016-05-01
In the field of optomechanics, radiation forces have provided a particularly high level of control over the frequency and dissipation of mechanical elements. Here we propose a class of optomechanical systems in which light exerts a similarly profound influence over two other fundamental parameters: geometry and mass. By applying an optical trap to one lattice site of an extended phononic crystal, we show it is possible to create a tunable, localized mechanical mode. Owing to light's simultaneous and constructive coupling with the structure's continuum of modes, we estimate that a trap power at the level of a single intracavity photon should be capable of producing a significant effect within a realistic, chip-scale device.
Critique of information geometry
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
Geometry of Periodic Monopoles
Maldonado, Rafael
2013-01-01
BPS monopoles on $\\mathbb{R}^2\\timesS^1$ correspond, via the generalized Nahm transform, to certain solutions of the Hitchin equations on the cylinder $\\mathbb{R}\\times S^1$. The moduli space M of two monopoles with their centre-of-mass fixed is a 4-dimensional manifold with a natural hyperk\\"ahler metric, and its geodesics correspond to slow-motion monopole scattering. The purpose of this paper is to study the geometry of M in terms of the Nahm/Hitchin data, i.e. in terms of structures on $\\mathbb{R}\\times S^1$. In particular, we identify the moduli, derive the asymptotic metric on M, and discuss several geodesic surfaces and geodesics on M. The latter include novel examples of monopole dynamics.
Geometry, Renormalization, And Supersymmetry
Berg, G M
2001-01-01
This thesis is about understanding, applying and improving quantum field theory. We compute renormalization group flows as the evolution of a “coarse-graining” operator without the need for a Euclidean formulation. Renormalization is cast in the form of a Lie algebra of (in general infinite) matrices that generate, by exponentiation, counterterms for diagrams with subdivergences. These results may shed light on noncommutative geometry. We check our results in a scalar three-loop example. Then, we consider the renormalization of a certain supersymmetric gauge theory, the low-energy limit of a string model. We compare results to those computed directly in the string model and find agreement. Finally, we discuss the possibility of detecting quantum-mechanical phases distinguishing the two Pin groups, double covers of the full Lorentz group. Majorana fermions, if detected, would provide an important testing ground; such particles can restrict the choice of Pin group.
Principles of algebraic geometry
Griffiths, Phillip A
1994-01-01
A comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applications through the study of interesting examples and the development of computational tools. Coverage ranges from analytic to geometric. Treats basic techniques and results of complex manifold theory, focusing on results applicable to projective varieties, and includes discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex as well as special top
A new approach to two-charge fuzzball geometries
Rui-Yan Yu
2009-04-01
A few years ago, Mathur proposed a `fuzzball' conjecture to give a microscopic description of black hole entropy. In the fuzzball scenario, the entropy in a two-charge black hole corresponds to microstates of a two-charge string (brane) system, e.g., a winding fundamental string with momentum modes. The geometry of such a two-charge system is fuzzy near the horizon, and is very difficult to get analytically in general. In this paper, we show a new method to get geometries of two-charge fuzzball. Our method is based on the multipole expansion. We find that the method is powerful enough to get a clean analytic form of metric of the fuzzball with one-momentum mode. It is expected to get multi-mode geometries using this method in the near future.
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
On ''conformal spinor geometry'': An attempt to ''understand'' internal symmetry
The natural homomorphism of pure spinors corresponding to a given Clifford algebra Csub(2n) to polarized isotropic n-planes of complex Euclidean space Esub(2n)sup(c) is taken as a starting point for the construction of a geometry called spinor geometry where pure spinors are the only elements out of which all tensors have to be constructed (analytically as bilinear polynomia of the components of a pure spinor). C4 and C6 spinor geometry are analyzed but it seems that C8 spinor geometry is necessary to construct Minkowski space Msup(3,1). C6 spinor field equations give rise in Minkowski space to a pair of Dirac equations (for conformal semispinors) presenting an SU(2) internal symmetry algebra. Mass is generated by spontaneously breaking the original O(4,2) symmetry of the spinor equation. (author)
Euclidean distance geometry and applications
Liberti, Leo; Lavor, Carlile; Maculan, Nelson; Mucherino, Antonio
2012-01-01
Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of the most important applications: molecular conformation, localization of sensor networks and statics.
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
Universal Hyperbolic Geometry I: Trigonometry
Wildberger, N J
2009-01-01
Hyperbolic geometry is developed in a purely algebraic fashion from first principles, without a prior development of differential geometry. The natural connection with the geometry of Lorentz, Einstein and Minkowski comes from a projective point of view, with trigonometric laws that extend to `points at infinity', here called `null points', and beyond to `ideal points' associated to a hyperboloid of one sheet. The theory works over a general field not of characteristic two, and the main laws ...
Gravity, Geometry and the Quantum
Ashtekar, Abhay
2006-01-01
After a brief introduction, basic ideas of the quantum Riemannian geometry underlying loop quantum gravity are summarized. To illustrate physical ramifications of quantum geometry, the framework is then applied to homogeneous isotropic cosmology. Quantum geometry effects are shown to replace the big bang by a big bounce. Thus, quantum physics does not stop at the big-bang singularity. Rather there is a pre-big-bang branch joined to the current post-big-bang branch by a `quantum bridge'. Furth...
The Geometry of Homological Triangles
Smarandache, Florentin
2012-01-01
This book is addressed to students, professors and researchers of geometry, who will find herein many interesting and original results. The originality of the book The Geometry of Homological Triangles consists in using the homology of triangles as a "filter" through which remarkable notions and theorems from the geometry of the triangle are unitarily passed. Our research is structured in seven chapters, the first four are dedicated to the homology of the triangles, while the last ones to their applications.
Guijosa, A
1999-01-01
This thesis explores some aspects of the recently uncovered connection between gauge theories and gravity, known as the AdS/CFT, or bulk-boundary, correspondence. This is a remarkable statement of equivalence between string or M-theory on certain backgrounds and field theories living on the boundaries of the corresponding spacetimes. Under the duality between four-dimensional N = 4 SU(N) superYang-Mills (SYM) and Type IIB string theory on AdS5 × S5, a baryon is mapped onto N fundamental strings terminating on a wrapped D5-brane. We examine the structure and energetics of this system from the vantage point of the fivebrane worldvolume action, making use of the Born-Infeld string approach. We construct supersymmetric fivebrane embeddings which correspond to gauge theory configurations with n external quarks, 0 ≤ n ≤ N. The extension of these solutions to the full asymptotically flat geometry of N D3-branes provides a detailed description of the creation of strings as the fivebrane is...
Raugas, M V
2001-01-01
In this work, we investigate different features of the quantum geometry associated to string theory. In chapter one, we use the language of quiver theory to determine the toric part of the classical moduli space of D-branes on a nonabelian Calabi-Yau threefold singularity and see that it admits topology-changing transitions. In chapter two, we determine the worldvolume field theories of D3-branes transverse to Calabi-Yau cones over singular Fano surfaces, which can be realized as partial resolutions of an orbifold of complex three-space by a discrete group. A mismatch between the isometries of the complex line bundle over the Fano surfaces and the discrete symmetries of the effective superpotential is found. In chapter three, we investigate the quantum volume of D-branes wrapped around cycles of various dimension in Calabi-Yau fourfolds and fivefolds. We consider the cases of the sextic and heptic hypersurface Calabi-Yau manifolds, as well as one example in weighted projective space, and find expressions for...
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.
Optimum Stirling engine geometry
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)
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
Quantum roots in geometry : II
The present work is a review of a series of papers, published in the last ten. years, comprising an attempt to find a suitable avenue from geometry to quantum. It shows clearly that, any non-symmetric geometry admits some built-in quantum features. These features disappear completely once the geometry becomes symmetric (torsion-less). It is shown that, torsion of space-time plays an important role in both geometry and physics. It interacts with the spin of the moving particle and with its charge. The first interaction, Spin-Torsion Interaction, has been used to overcome the discrepancy in the results of the COW-experiment. The second interaction, Charge-Torsion Interaction, is similar to the Aharonov-Bohm effect. As a byproduct, a new version of Absolute Parallelism (AP) geometry, the Parameterized Absolute Parallelism (PAP) geometry, has been established and developed. This version can be used to construct field theories that admit some quantum features. Riemannian geometry and conventional AP-geometry are special cases of PAP-geometry
Virtual Monopole Geometry and Confinement
La, H S
1999-01-01
Generalizing the geometry of the gauge covariant variables in Yang-Mills theory proposed by Johnson and Haagensen, the 4-d geometry associated with a monopole is defined for SU(2). There are three relevant geometries: AdS$_2\\times S^2$, $R^2\\times S^2$ and $H_+\\times S^2$, depending on the asymptotic behavior of the torsion. Using this geometry, the Wilson loop average is computed {\\it à la} Nambu-Goto action. In case of AdS$_2\\times S^2$, it satisfies the area law.
In this paper, results related to a limited scope assessment of the geometry-distortion-induced effects on key reactor physics parameters of a CANDU reactor are discussed. These results were generated by simulations using refined analytical methods and detailed modeling of CANDU reactor core with aged lattice cell geometry. (authors)
Casimir effects for classical and quantum liquids in slab geometry: A brief review
We analytically explore Casimir effects for confinement of classical and quantum fluctuations in slab (film) geometry (i) for classical (critical) fluctuations over 4He liquid around the λ point, and (ii) for quantum (phonon) fluctuations of Bogoliubov excitations over an interacting Bose-Einstein condensate. We also briefly review Casimir effects for confinement of quantum vacuum fluctuations confined to two plates of different geometries
Diophantine and tropical geometry, and uniformity of rational points on curves
Katz, Eric; Rabinoff, Joseph; Zureick-Brown, David
2016-01-01
We describe recent work connecting combinatorics and tropical/non-Archimedean geometry to Diophantine geometry, particularly the uniformity conjectures for rational points on curves and for torsion packets of curves. The method of Chabauty--Coleman lies at the heart of this connection, and we emphasize the clarification that tropical geometry affords throughout the theory of $p$-adic integration, especially to the comparison of analytic continuations of $p$-adic integrals and to the analysis ...
Identification of Plasmonic Modes in Parabolic Cylinder Geometry by Quasi-Separation of Variables
KURIHARA, Kazuyoshi; Otomo, Akira; Yamamoto, Kazuhiro; TAKAHARA, Junichi; Tani, Masahiko; Kuwashima, Fumiyoshi
2014-01-01
This paper describes the plasmonic modes in the parabolic cylinder geometry as a theoretical complement to the previous paper (J Phys A 42:185401) that considered the modes in the circular paraboloidal geometry. In order to identify the plasmonic modes in the parabolic cylinder geometry, analytic solutions for surface plasmon polaritons are examined by solving the wave equation for the magnetic field in parabolic cylindrical coordinates using quasi-separation of variables in combination with ...
Surrogate Modeling for Geometry Optimization
Rojas Larrazabal, Marielba de la Caridad; Abraham, Yonas; Holzwarth, Natalie; Plemmons, Robert
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....
GPS: Geometry, Probability, and Statistics
Field, Mike
2012-01-01
It might be said that for most occupations there is now less of a need for mathematics than there was say fifty years ago. But, the author argues, geometry, probability, and statistics constitute essential knowledge for everyone. Maybe not the geometry of Euclid, but certainly geometrical ways of thinking that might enable us to describe the world…
Limits of downstream hydraulic geometry
Wohl, Ellen
2004-10-01
Adjustments to flow width, depth, and velocity in response to changes in discharge are commonly characterized by using downstream hydraulic geometry relationships. The spatial limits of these relationships within a drainage basin have not been systematically quantified. Where the erosional resistance of the channel substrate is sufficiently large, hydraulic driving forces presumably will be unable to adjust channel form. Data sets from 10 mountain rivers in the United States, Panama, Nepal, and New Zealand are used in this study to explore the limits of downstream hydraulic geometry relationships. Where the ratio of stream power to sediment size (Ω/D84) exceeds 10,000 kg/s3, downstream hydraulic geometry is well developed; where the ratio falls below 10,000 kg/s3, downstream hydraulic geometry relationships are poorly developed. These limitations on downstream hydraulic geometry have important implications for channel engineering and simulations of landscape change.
Lobachevsky's Geometry and Research of Geometry of the Universe
Brylevskaya, L. I.
2008-10-01
For the first time N. I. Lobachevsky gave a talk on the new geometry in 1826; three years after he had published a work "On the fundamentals of geometry", containing all fundamental theorems and methods of non-Euclidean geometry. A small part of the article was devoted to the study of geometry of the Universe. The interpretation of geometrical concepts in pure empirical way was typical for mathematicians at the beginning of the XIX century; in this connection it was important for scientists to find application of his geometry. Having the purpose to determine experimentally the properties of real physical Space, Lobachevsky decided to calculate the sum of angles in a huge triangle with two vertexes in opposite points of the terrestrial orbit and the third -- on the remote star. Investigating the possibilities of solution of the set task, Lobachevsky faced the difficulties of theoretical, technical and methodological character. More detailed research of different aspects of the problem led Lobachevsky to the comprehension of impossibility to obtain the values required for the goal achievement, and he called his geometry an imaginary geometry.
MacNeill, Sheila; Campbell, Lorna M.; Hawksey, Martin
2014-01-01
This article presents an overview of the development and use of analytics in the context of education. Using Buckingham Shum's three levels of analytics, the authors present a critical analysis of current developments in the domain of learning analytics, and contrast the potential value of analytics research and development with real world…
Controlling electromagnetic fields at boundaries of arbitrary geometries
Teo, Jonathon Yi Han; Wong, Liang Jie; Molardi, Carlo; Genevet, Patrice
2016-08-01
Rapid developments in the emerging field of stretchable and conformable photonics necessitate analytical expressions for boundary conditions at metasurfaces of arbitrary geometries. Here, we introduce the concept of conformal boundary optics: a design theory that determines the optical response for designer input and output fields at such interfaces. Given any object, we can realize coatings to achieve exotic effects like optical illusions and anomalous diffraction behavior. This approach is relevant to a broad range of applications from conventional refractive optics to the design of the next-generation of wearable optical components. This concept can be generalized to other fields of research where designer interfaces with nontrivial geometries are encountered.
Controlling Electromagnetic Fields at Boundaries of Arbitrary Geometries
Teo, Jonathon Yi Han; Molardi, Carlo; Genevet, Patrice
2015-01-01
Rapid developments in the emerging field of stretchable and conformable photonics necessitate analytical expressions for boundary conditions at metasurfaces of arbitrary geometries. Here, we introduce the concept of conformal boundary optics: a design theory that determines the optical response for designer input and output fields at such interfaces. Given any object, we can realise coatings to achieve exotic effects like optical illusions and anomalous diffraction behaviour. This approach is relevant to a broad range of applications from conventional refractive optics to the design of the next-generation of wearable optical components. This concept can be generalized to other fields of research where designer interfaces with nontrivial geometries are encountered.
The geometry description markup language
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. The author 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
The Geometry Description Markup Language
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.
Wanas, M I
2006-01-01
The present work is a review of a series of papers, published in the last ten years, comprising an attempt to find a suitable avenue from geometry to quantum. It shows clearly that, any non-symmetric geometry admits some built-in quantum features. These features disappear completely once the geometry becomes symmetric (torsion-less). It is shown that, torsion of space-time plays an important role in both geometry and physics. It interacts with the spin of the moving particle and with its charge. The first interaction, {\\bf{Spin-Torsion Interaction}}, has been used to overcome the discrepancy in the results of the COW-experiment. The second interaction, {\\bf{Charge-Torsion Interaction}}, is similar to the Aharonov-Bohm effect. As a byproduct, a new version of Absolute Parallelism (AP) geometry, the Parameterized Absolute Parallelism (PAP) geometry, has been established and developed. This version can be used to construct field theories that admit some quantum features. Riemannian geometry and conventional AP-g...
Riemannian statistics geometry: A counterpart approach of inference geometry
Velazquez, L
2011-01-01
Riemannian statistics geometry is proposed in this work as a counterpart approach of inference geometry. This geometry framework is inspired on the existence of a notable analogy between the general theorems of inference theory and the the general fluctuation theorems associated with a parametric family of distribution functions $dp(I|\\theta)$, which describes the stochastic behavior of a set of continuous stochastic variables driven by a set of control parameters $\\theta$. In this approach, statistical properties are rephrased as purely geometric notions derived from the Riemannian structure on the manifold $\\mathcal{M}_{\\theta}$ of stochastic variables $I$. Consequently, this theory arises as an alternative framework for applying the powerful methods of differential geometry for the statistical analysis.
Fluctuation geometry: A counterpart approach of inference geometry
Velazquez, L.
2011-01-01
Starting from an axiomatic perspective, \\emph{fluctuation geometry} is developed as a counterpart approach of inference geometry. This approach is inspired on the existence of a notable analogy between the general theorems of \\emph{inference theory} and the the \\emph{general fluctuation theorems} associated with a parametric family of distribution functions $dp(I|\\theta)=\\rho(I|\\theta)dI$, which describes the behavior of a set of \\emph{continuous stochastic variables} driven by a set of contr...
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
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
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
A proposal of an open PET geometry
Yamaya, Taiga [Molecular Imaging Center, National Institute of Radiological Sciences, 4-9-1 Anagawa, Inage-ku, Chiba, 263-8555 (Japan); Inaniwa, Taku [Research Center for Charged Particle Therapy, National Institute of Radiological Sciences, 4-9-1 Anagawa, Inage-ku, Chiba 263-8555 (Japan); Minohara, Shinichi [Research Center for Charged Particle Therapy, National Institute of Radiological Sciences, 4-9-1 Anagawa, Inage-ku, Chiba 263-8555 (Japan); Yoshida, Eiji [Molecular Imaging Center, National Institute of Radiological Sciences, 4-9-1 Anagawa, Inage-ku, Chiba, 263-8555 (Japan); Inadama, Naoko [Molecular Imaging Center, National Institute of Radiological Sciences, 4-9-1 Anagawa, Inage-ku, Chiba, 263-8555 (Japan); Nishikido, Fumihiko [Molecular Imaging Center, National Institute of Radiological Sciences, 4-9-1 Anagawa, Inage-ku, Chiba, 263-8555 (Japan); Shibuya, Kengo [Molecular Imaging Center, National Institute of Radiological Sciences, 4-9-1 Anagawa, Inage-ku, Chiba, 263-8555 (Japan); Lam, Chih Fung [Molecular Imaging Center, National Institute of Radiological Sciences, 4-9-1 Anagawa, Inage-ku, Chiba, 263-8555 (Japan); Murayama, Hideo [Molecular Imaging Center, National Institute of Radiological Sciences, 4-9-1 Anagawa, Inage-ku, Chiba, 263-8555 (Japan)
2008-02-07
The long patient port of a PET scanner tends to put stress on patients, especially patients with claustrophobia. It also prevents doctors and technicians from taking care of patients during scanning. In this paper, we proposed an 'open PET' geometry, which consists of two axially separated detector rings. A long and continuous field-of-view (FOV) including a 360 deg. opened gap between two detector rings can be imaged enabling a fully 3D image reconstruction of all the possible lines-of-response. The open PET will become practical if iterative image reconstruction methods are applied even though image reconstruction of the open PET is analytically an incomplete problem. First we implemented a 'masked' 3D ordered subset expectation maximization (OS-EM) in which the system matrix was obtained from a long 'gapless' scanner by applying a mask to detectors corresponding to the open space. Next, in order to evaluate imaging performance of the proposed open PET geometry, we simulated a dual HR+ scanner (ring diameter of D = 827 mm, axial length of W = 154 mm x 2) separated by a variable gap. The gap W was the maximum limit to have axially continuous FOV of 3W though the maximum diameter of FOV at the central slice was limited to D/2. Artifacts, observed on both sides of the open space when the gap exceeded W, were effectively reduced by inserting detectors partially into unnecessary open spaces. We also tested the open PET geometry using experimental data obtained by the jPET-D4. The jPET-D4 is a prototype brain scanner, which has 5 rings of 24 detector blocks. We simulated the open jPET-D4 with a gap of 66 mm by eliminating 1 block-ring from experimental data. Although some artifacts were seen at both ends of the opened gap, very similar images were obtained with and without the gap. The proposed open PET geometry is expected to lead to realization of in-beam PET, which is a method for an in situ monitoring of charged particle therapy, by
Highlights of Noncommutative Spectral Geometry
Sakellariadou, Mairi
2012-01-01
A summary of noncommutative spectral geometry as an approach to unification is presented. The role of the doubling of the algebra, the seeds of quantization and some cosmological implications are briefly discussed.
Molecular motion in restricted geometries
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.
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.
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, ...
Guide to Computational Geometry Processing
Bærentzen, Jakob Andreas; Gravesen, Jens; Anton, François;
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 to the......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 be......, 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...
When to carry out analytic continuation?
Zuo, J M
1998-01-01
This paper discusses the analytic continuation in the thermal field theory by using the theory of $\\eta-\\xi$ spacetime. Taking a simple model as example, the $2\\times 2$ matrix real-time propagator is solved from the equation obtained through continuation of the equation for the imaginary-time propagator. The geometry of the $\\eta-\\xi$ spacetime plays important role in the discussion.
Higgs mass in noncommutative geometry
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)
Duality principle and braided geometry
Majid, S
1994-01-01
We give an overview of a new kind symmetry in physics which exists between observables and states and which is made possible by the language of Hopf algebras and quantum geometry. It has been proposed by the author as a feature of Planck scale physics. More recent work includes corresponding results at the semiclassical level of Poisson-Lie groups and at the level of braided groups and braided geometry.
Higgs mass in noncommutative geometry
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)
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.
Geometry of the quantum universe
A quantum universe with the global shape of a (Euclidean) de Sitter spacetime appears as dynamically generated background geometry in the causal dynamical triangulation (CDT) regularisation of quantum gravity. We investigate the micro- and macro-geometry of this universe, using geodesic shell decompositions of spacetime. More specifically, we focus on evidence of fractality and global anisotropy, and on how they depend on the bare coupling constants of the theory.
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