Journal of Geometry and Symmetry in Physics Articles (Project Euclid)
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The latest articles from Journal of Geometry and Symmetry in Physics on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2017 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Fri, 12 May 2017 12:16 EDTFri, 12 May 2017 12:16 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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On Matrix Representations of Geometric (Clifford) Algebras
http://projecteuclid.org/euclid.jgsp/1494605778
<strong>Ramon G. Calvet</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 43, 1--36.</p><p><strong>Abstract:</strong><br/>
Representations of geometric (Clifford) algebras with real square matrices are reviewed by
providing the general theorem as well as examples of lowest dimensions. New definitions for
isometry and norm are proposed. Direct and indirect isometries are identified respectively
with automorphisms and antiautomorphisms of the geometric algebra, while the norm of every
element is defined as the $n^\textit{th}$-root of the absolute value of the determinant of its
matrix representation of order $n$. It is deduced in which geometric algebras direct
isometries are inner automorphisms (similarity transformations of matrices). Indirect
isometries need reversion too. Finally, the most common isometries are reviewed in order to
write them in this way.
</p>projecteuclid.org/euclid.jgsp/1494605778_20170512121622Fri, 12 May 2017 12:16 EDTSymmetry and Solutions to the Helmholtz Equation Inside an Equilateral Triangle
http://projecteuclid.org/euclid.jgsp/1494605779
<strong>Nathaniel Stambaugh</strong>, <strong>Mark Semon</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 43, 37--45.</p><p><strong>Abstract:</strong><br/>
Solutions to the Helmholtz equation within an equilateral triangle which solve either the
Dirichlet or Neumann problem are investigated. This is done by introducing a pair of
differential operators, derived from symmetry considerations, which demonstrate interesting
relationships among these solutions. One of these operators preserves the boundary condition
while generating an orthogonal solution and the other leads to a bijection between solutions
of the Dirichlet and Neumann problems.
</p>projecteuclid.org/euclid.jgsp/1494605779_20170512121622Fri, 12 May 2017 12:16 EDTBranes on $G$-Manifolds
http://projecteuclid.org/euclid.jgsp/1494605780
<strong>André Viña</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 43, 47--71.</p><p><strong>Abstract:</strong><br/>
Let $X$ be Calabi-Yau manifold acted by a group $G$. We give a definition of
$G$-equivariance for branes on $X$, and assign to each equivariant brane an element of the
equivariant cohomology of $X$ that can be considered as a charge of the brane. We prove that
the spaces of strings stretching between equivariant branes support representations of $G$.
This fact allows us to give formulas for the dimension of some of such spaces, when $X$ is a
flag manifold of $G$.
</p>projecteuclid.org/euclid.jgsp/1494605780_20170512121622Fri, 12 May 2017 12:16 EDTClifford Algebra Implementations in Maxima
http://projecteuclid.org/euclid.jgsp/1494605781
<strong>Dimiter Prodanov</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 43, 73--105.</p><p><strong>Abstract:</strong><br/>
This tutorial focuses on the packages $\texttt{clifford}$ and $\texttt{cliffordan}$ for the
computer algebra system Maxima . Maxima is the open source descendant of the
first computer algebra system and features a rich functionality from a large number of shared
packages. The Maxima language is based on the ideas of functional programming, which is
particularly well suited for transformations of formal mathematical expressions. While
$\texttt{clifford}$ implements Clifford algebras $C\ell_{p,q,r}$ of arbitrary signatures and
order based on the elementary construction of Macdonald, $\texttt{cliffordan}$ features
geometric calculus functionality. Using $\texttt{clifford}$ expressions containing geometric,
outer and inner products can be simplified. Applications of $\texttt{clifford}$ and
$\texttt{cliffordan}$ in linear algebra and calculus are demonstrated.
</p>projecteuclid.org/euclid.jgsp/1494605781_20170512121622Fri, 12 May 2017 12:16 EDTOn Complex Homogeneous Space of Vectors with Constraintshttp://projecteuclid.org/euclid.jgsp/1499306418<strong>Emilija Celakoska</strong>, <strong>Elena Hadzieva</strong>, <strong>Vesna Celakoska-Jordanova</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 44, 1--11.</p><p><strong>Abstract:</strong><br/>
A homogeneous space $\mathcal{V}$ of complex constrained vectors in
$\mathbb{C}^3$, representing complex velocities is introduced. The corresponding
representation of the complex special orthogonal group of transformations acting on
$\mathcal{V}$ is also examined. The requirement for real vector magnitudes is addressed
by imposing orthogonality between the real and the imaginary parts of vectors and use
of the non-conjugate scalar product. We present the orthogonal transformations acting on
$\mathcal{V}$ in terms of the polar decomposition of complex orthogonal matrices.
The group link problem and the homogeneity of the space $\mathcal{V}$ are also discussed.
Finally, we briefly consider the convenience of the space $\mathcal{V}$ in theoretical
calculations.
</p>projecteuclid.org/euclid.jgsp/1499306418_20170705220031Wed, 05 Jul 2017 22:00 EDTCovering Maps and Ideal Embeddings of Compact
Homogeneous Spaceshttp://projecteuclid.org/euclid.jgsp/1499306419<strong>Bang-Yen Chen</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 44, 12--20.</p><p><strong>Abstract:</strong><br/>
The notion of ideal embeddings was introduced by the author at the Third Pacific Rim
Geometry Conference held at Seoul in 1996. Roughly speaking, an ideal embedding is an
isometrical embedding which receives the least possible amount of tension from the
surrounding space at each point.
In this article, we study ideal embeddings of irreducible compact homogenous spaces
in Euclidean spaces. Our main result states that if $\pi: M\to N$ is a covering map between
two irreducible compact homogeneous spaces with $\lambda_1(M)\ne \lambda_1(N)$,
then $N$ does not admit an ideal embedding in a Euclidean space, although $M$ could.
</p>projecteuclid.org/euclid.jgsp/1499306419_20170705220031Wed, 05 Jul 2017 22:00 EDTGeometric Quantization of Finite Toda Systems and Coherent Stateshttp://projecteuclid.org/euclid.jgsp/1499306420<strong>Rukmini Dey</strong>, <strong>Saibal Ganguli</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 44, 21--38.</p><p><strong>Abstract:</strong><br/>
Adler had showed that the Toda system can be given a coadjoint orbit description.
We quantize the Toda system by viewing it as a single orbit of a multiplicative group of
lower triangular matrices of determinant one with positive diagonal entries. We get a unitary
representation of the group with square integrable polarized sections of the quantization as
the module . We find the Rawnsley coherent states after completion of the above space of
sections. We also find non-unitary finite dimensional quantum Hilbert spaces for the system.
Finally we give an expression for the quantum Hamiltonian for the system.
</p>projecteuclid.org/euclid.jgsp/1499306420_20170705220031Wed, 05 Jul 2017 22:00 EDTTopological Quantum Numbers of Dyonic Fields Over
Taub-NUT and Taub-Bolt Spaceshttp://projecteuclid.org/euclid.jgsp/1499306421<strong>Daniel Flores-Alfonso</strong>, <strong>Hernando Quevedo</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 44, 39--54.</p><p><strong>Abstract:</strong><br/>
We calculate the Chern numbers of ${\rm SU}(2)$-homogeneous Einstein-Maxwell gravitational
instantons with boundary at infinity. By restating these numbers as Chern-Simons invariants
on the boundary apparent conflicting results emerge. We resolve this issue examining the
topological stability of the elf-gravitating Abelian fields. No quantization carrying physical
meaning is found when the background is a Taub-NUT space. However the magnetic charge
of dyons on Taub-Bolt spaces is found to be of topological quantum nature. In this framework
electric charge is quantized by a consistency condition.
</p>projecteuclid.org/euclid.jgsp/1499306421_20170705220031Wed, 05 Jul 2017 22:00 EDTMathematics in Caging of Roboticshttp://projecteuclid.org/euclid.jgsp/1499306422<strong>Hiroyasu Hamada</strong>, <strong>Satoshi Makita</strong>, <strong>Shigeki Matsutani</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 44, 55--66.</p><p><strong>Abstract:</strong><br/>
It is a crucial problem in robotics field
to cage an object using robots like multifingered hand.
However the problem what is the caging
for general geometrical objects and robots
has not been well-described in mathematics
though there were many rigorous studies on the methods how to
cage an object by certain robots.
In this article, we investigate the caging problem
more mathematically and describe the problem in terms of recursion of
the simple euclidean moves. Using this description,
we show that the caging has the degree of difficulty
which is closely related to a combinatorial
problem and a wire puzzle.
It implies that in order to capture an object by
caging, from a practical viewpoint the difficulty plays an important
role.
</p>projecteuclid.org/euclid.jgsp/1499306422_20170705220031Wed, 05 Jul 2017 22:00 EDTn -Characteristic Vector Fields of Contact Manifoldsshttp://projecteuclid.org/euclid.jgsp/1499306423<strong>Babak Hassanzadeh</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 44, 67--75.</p><p><strong>Abstract:</strong><br/>
In present paper we define and study $n$-characteristic vector fields. We present
definition of Tanaka-Webster connection, then use it for studying the behavior of
$n$-characteristic vector fields. Also we show some results about of these vector fields by
Tanaka-Webster connection.
</p>projecteuclid.org/euclid.jgsp/1499306423_20170705220031Wed, 05 Jul 2017 22:00 EDTOrthogonal Spheres, Light Cones and Causality in Minkowski Spacehttp://projecteuclid.org/euclid.jgsp/1499306424<strong>Robert J. Low</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 44, 77--81.</p><p><strong>Abstract:</strong><br/>
We describe a curious relationship between orthogonal spheres and causal
relationships in Minkowski space.
</p>projecteuclid.org/euclid.jgsp/1499306424_20170705220031Wed, 05 Jul 2017 22:00 EDTTwo Types of Lorentz Transformations for Massless Fieldshttp://projecteuclid.org/euclid.jgsp/1499306425<strong>Victor L. Mironov</strong>, <strong>Sergey V. Mironov</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 44, 83--96.</p><p><strong>Abstract:</strong><br/>
We describe a curious relationship between orthogonal spheres and causal
relationships in Minkowski space.
</p>projecteuclid.org/euclid.jgsp/1499306425_20170705220031Wed, 05 Jul 2017 22:00 EDTElastic Bending Energy: A Variational Approachhttps://projecteuclid.org/euclid.jgsp/1512442901<strong>Riccardo Capovilla</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 45, 1--45.</p><p><strong>Abstract:</strong><br/>
Geometric continuum models for fluid lipid membranes are considered using classical field theory, within a covariant variational approach. The approach is cast as a higher-derivative Lagrangian formulation of continuum classical field theory, and it can be seen as a covariant version of the field theoretical variational approach that uses the height representation. This novel Lagrangian formulation is presented first for a generic reparametrization invariant geometric model, deriving its equilibrium equation, or shape equation, and its linear and angular stress tensors, using the classical Canham-Helfrich elastic bending energy for illustration. The robustness of the formulation is established by extending it to the presence of external forces, and to the case of heterogenous lipid membranes, breaking reparametrization invariance. In addition, a useful and compact general expression for the second variation of the free energy is obtained within the Lagrangian formulation, as a first step towards the study of the stability of membrane configurations. The simple structure of the expressions derived for the basic entities that appear in the mechanics of a lipid membrane is a direct consequence of the well established power of a Lagrangian variational approach. The paper is self-contained, and it is meant to provide, besides a new framework, also a convenient introduction to the mechanics of lipid membranes.
</p>projecteuclid.org/euclid.jgsp/1512442901_20171204220145Mon, 04 Dec 2017 22:01 ESTGeneralized Seiberg-Witten Equations on a Riemann Surfacehttps://projecteuclid.org/euclid.jgsp/1512442902<strong>Rukmini Dey</strong>, <strong>Varun Thakre</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 45, 47--66.</p><p><strong>Abstract:</strong><br/>
In this paper we consider twice-dimensionally reduced, generalized Seiberg-Witten (S-W) equations, defined on a compact Riemann surface. A novel feature of the reduction technique is that the resulting equations produce an extra “Higgs field”. Under suitable regularity assumptions, we show that the moduli space of gauge-equivalent classes of solutions to the reduced equations, is a smooth Kähler manifold and construct a pre-quantum line bundle over the moduli space of solutions.
</p>projecteuclid.org/euclid.jgsp/1512442902_20171204220145Mon, 04 Dec 2017 22:01 ESTKähler Dynamics for the Universal Multi-Robot Fleethttps://projecteuclid.org/euclid.jgsp/1512442903<strong>Vladimir G. Ivancevic Ivancevic</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 45, 67--85.</p><p><strong>Abstract:</strong><br/>
A general model is formulated for a universal fleet of all unmanned vehicles, including Aerial Vehicles (UAVs), Ground Vehicles (UGVs), Sea Vehicles (USVs) and Underwater Vehicles (UUVs), as a geometric Kähler dynamics and control system. Based on the Newton-Euler dynamics of each vehicle, a control system for the universal autonomous fleet is designed as a combined Lagrangian and Hamiltonian form. The associated continuous system representing a very large universal fleet is given in Appendix in the form of the Kähler-Ricci flow
</p>projecteuclid.org/euclid.jgsp/1512442903_20171204220145Mon, 04 Dec 2017 22:01 ESTGeodesics on Rotational Surfaces in Pseudo-Galilean Spacehttps://projecteuclid.org/euclid.jgsp/1512442904<strong>Dae Won Yoon</strong>, <strong>Murat Kemal Karacan</strong>, <strong>Bahaddin Bukcu</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 45, 87--94.</p><p><strong>Abstract:</strong><br/>
In this paper, we study rotational surfaces in the pseudo-Galilean three-space $\mathbb G_3^1$ with pseudo-Euclidean rotations and isotropic rotations. In particular, we investigate properties of geodesics on rotational surfaces in $\mathbb G_3^1$ and give some examples.
</p>projecteuclid.org/euclid.jgsp/1512442904_20171204220145Mon, 04 Dec 2017 22:01 EST2+2 Moulton Configurationhttps://projecteuclid.org/euclid.jgsp/1512442905<strong>Naoko Yoshimi</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 45, 95--110.</p><p><strong>Abstract:</strong><br/>
We pose a new problem of collinear central configurations in Newtonian $n$-body problem. It is known that the configuration of two bodies moving along the Newtonian force is always a collinear central configuration. Can we add new two bodies on the straight line of initial two bodies without changing the move of the initial two bodies and the configuration of the four bodies is central, too? We call it 2+2 Moulton configuration. We find three special solutions to this problem and find each mass of new two bodies is zero.
</p>projecteuclid.org/euclid.jgsp/1512442905_20171204220145Mon, 04 Dec 2017 22:01 ESTIs Spacetime as Physical as Is Space?https://projecteuclid.org/euclid.jgsp/1518577290<strong>Mayeul Arminjon</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 46, 1--24.</p><p><strong>Abstract:</strong><br/>
Two questions are investigated by looking successively at classical mechanics, special relativity, and relativistic gravity: first, how is space related with spacetime? The proposed answer is that each given reference fluid, that is a congruence of reference trajectories, defines a physical space. The points of that space are formally defined to be the world lines of the congruence. That space can be endowed with a natural structure of 3-D differentiable manifold, thus giving rise to a simple notion of spatial tensor -- namely, a tensor on the space manifold. The second question is: does the geometric structure of the spacetime determine the physics, in particular, does it determine its relativistic or preferred-frame character? We find that it does not, for different physics (either relativistic or not) may be defined on the same spacetime structure - and also, the same physics can be implemented on different spacetime structures.
</p>projecteuclid.org/euclid.jgsp/1518577290_20180213220145Tue, 13 Feb 2018 22:01 ESTTwistor Spaces and Compact Manifolds Admitting Both Kähler and Non-Kähler Structureshttps://projecteuclid.org/euclid.jgsp/1518577291<strong>Ljudmila Kamenova</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 46, 25--35.</p><p><strong>Abstract:</strong><br/>
In this expository paper we review some twistor techniques and recall the problem of finding compact differentiable manifolds that can carry both Kähler and non-Kähler complex structures. Such examples were constructed independently by Atiyah, Blanchard and Calabi in the 1950’s. In the 1980’s Tsanov gave an example of a simply connected manifold that admits both Kähler and non-Kähler complex structures - the twistor space of a $K3$ surface. Here we show that the quaternion twistor space of a hyperkähler manifold has the same property.
</p>projecteuclid.org/euclid.jgsp/1518577291_20180213220145Tue, 13 Feb 2018 22:01 ESTCommuting Pairs of Generalized Contact Metric Structureshttps://projecteuclid.org/euclid.jgsp/1518577292<strong>Janet Talvacchia</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 46, 37--50.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove a theorem that gives a simple criterion for generating commuting pairs of generalized almost complex structures on spaces that are the product of two generalized almost contact metric spaces. We examine the implications of this theorem with regard to the definitions of generalized Sasakian and generalized co-Kähler geometry.
</p>projecteuclid.org/euclid.jgsp/1518577292_20180213220145Tue, 13 Feb 2018 22:01 ESTA Note on the Class of Surfaces with Constant Skew Curvatureshttps://projecteuclid.org/euclid.jgsp/1518577293<strong>Magdalena Toda</strong>, <strong>Alexander Pigazzini</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 46, 51--58.</p><p><strong>Abstract:</strong><br/>
The goal of this paper is to analyze surfaces with constant skew curvature (CSkC), and show that the class of CSkC surfaces with non-constant principal curvatures does not contain any Bonnet surfaces.
</p>projecteuclid.org/euclid.jgsp/1518577293_20180213220145Tue, 13 Feb 2018 22:01 ESTComposition Algebras, Exceptional Jordan Algebra and Related Groupshttps://projecteuclid.org/euclid.jgsp/1518577294<strong>Ivan Todorov</strong>, <strong>Svetla Drenska</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 46, 59--93.</p><p><strong>Abstract:</strong><br/>
Normed division rings are reviewed in the more general framework of composition algebras that include the split (indefinite metric) case. The Jordan - von Neumann - Wigner classification of finite dimensional Jordan algebras is outlined with special attention to the 27 dimensional exceptional Jordan algebra $\frak{J}$. The automorphism group $\rm{F}_4$ of $\frak{J}$ and its maximal Borel-de~Siebenthal subgroups $\frac{\rm SU(3)\times \rm SU(3)}{\mathbb{Z}_3}$ and ${\rm Spin}(9)$ are studied in some detail with an eye to possible applications to the fundamental fermions in the Standard Model of particle physics.
</p>projecteuclid.org/euclid.jgsp/1518577294_20180213220145Tue, 13 Feb 2018 22:01 ESTFlat Affine and Symplectic Geometries on Lie Groupshttps://projecteuclid.org/euclid.jgsp/1518577295<strong>Andrés Villabón</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 46, 95--121.</p><p><strong>Abstract:</strong><br/>
In this paper we exhibit a family of flat left invariant affine structures on the double Lie group of the oscillator Lie group of dimension 4, associated to each solution of classical Yang-Baxter equation given by Boucetta and Medina. On the other hand, using Koszul's method, we prove the existence of an immersion of Lie groups between the group of affine transformations of a flat affine and simply connected manifold and the classical group of affine transformations of $\mathbb{R}^n$. In the last section, for each flat left invariant affine symplectic connection on the group of affine transformations of the real line, describe by Medina-Saldarriaga-Giraldo, we determine the affine symplectomorphisms. Finally we exhibit the Hess connection, associated to a Lagrangian bi-foliation, which is flat left invariant affine.
</p>projecteuclid.org/euclid.jgsp/1518577295_20180213220145Tue, 13 Feb 2018 22:01 ESTIndefinite Eisenstein Lattices: A Modern Ball-Rendevous with Poincarè, Picard, Hecke, Shimura, Mumford, Deligne and Hirzebruchhttps://projecteuclid.org/euclid.jgsp/1525939245<strong>Rolf-Peter Holzapfel</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 47, 1--40.</p><p><strong>Abstract:</strong><br/>
In \cite{Ho02} we have counted indefinite metrics (two-dimensional, integrally defined, over Gauss numbers) with a fixed norm (discriminant). We would like to call them also \emph{indefinite class numbers}. In this article we change from Gauss to Eisenstein numbers. We have to work on the complex two-dimensional unit ball, an Eisenstein lattice on it and the quotient surface. It turns out that the compactified quotient is the complex plane $\IP^2$. In the first part we present a new proof of this fact. In the second part we construct explicitly a Heegner series with the help of Legendre-symbol coefficients. They can be interpreted as ``indefinite class numbers'' we look for. Geometrically they appear also as number of plane curves with (normed) Eisenstein disc uniformization.
</p>projecteuclid.org/euclid.jgsp/1525939245_20180510040050Thu, 10 May 2018 04:00 EDTCassini Ovals in Harmonic Motion Orbitshttps://projecteuclid.org/euclid.jgsp/1525939246<strong>Khristo N. Boyadzhiev</strong>, <strong>Irina A. Boyadzhiev</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 47, 41--49.</p><p><strong>Abstract:</strong><br/>
We discover the appearance of interesting Cassinian ovals in the motion of a two-dimensional harmonic oscillator. The trajectories of the oscillating points are ellipses depending on a parameter. The locus of the foci of these ellipses is a Cassini oval. The form of this oval depends on the magnitude of the initial velocity.
</p>projecteuclid.org/euclid.jgsp/1525939246_20180510040050Thu, 10 May 2018 04:00 EDTNatural Coordinate System in Curved Space-Timehttps://projecteuclid.org/euclid.jgsp/1525939247<strong>Ying-Qiu Gu</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 47, 51--62.</p><p><strong>Abstract:</strong><br/>
In this paper we establish a generally and globally valid coordinate system in curved space-time with the simultaneous hypersurface orthogonal to the time coordinate. The time coordinate can be presented according to practical evolving process and keep synchronous with the evolution of the realistic world. In this coordinate system, it is convenient to express the physical laws and to calculate physical variables with clear geometrical meaning. We call it ``natural coordinate system''. The constructing method for the natural coordinate system is concretely provided, and its physical and geometrical meanings are discussed in detail. In natural coordinate system, we make classical approximation of spinor equation to get Newtonian mechanics, and then make weak field approximation of Einstein's equation and low speed approximation of particles moving in the space-time. From the analysis and examples we find it is helpful to understand the nature of space-time.
</p>projecteuclid.org/euclid.jgsp/1525939247_20180510040050Thu, 10 May 2018 04:00 EDTDeformations of Symplectic Structures by Moment Mapshttps://projecteuclid.org/euclid.jgsp/1525939248<strong>Tomoya Nakamura</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 47, 63--84.</p><p><strong>Abstract:</strong><br/>
We study deformations of symplectic structures on a smooth manifold $M$ via the quasi-Poisson theory. We can deform a given symplectic structure $\omega $ with a Hamiltonian $G$-action to a new symplectic structure $\omega ^t$ parametrized by some element $t$ in $\Lambda^2\mathfrak{g}$. We can obtain concrete examples for the deformations of symplectic structures on the complex projective space and the complex Grassmannian. Moreover applying the deformation method to any symplectic toric manifold, we show that manifolds before and after deformations are isomorphic as a symplectic toric manifold.
</p>projecteuclid.org/euclid.jgsp/1525939248_20180510040050Thu, 10 May 2018 04:00 EDTSpanning Class in the Category of Braneshttps://projecteuclid.org/euclid.jgsp/1525939249<strong>Andrés Viña</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 47, 85--104.</p><p><strong>Abstract:</strong><br/>
Normed division rings are reviewed in the more general framework of composition algebras that include the split (indefinite metric) case. The Jordan - von Neumann - Wigner classification of finite dimensional Jordan algebras is outlined with special attention to the 27 dimensional exceptional Jordan algebra $\frak{J}$. The automorphism group $\rm{F}_4$ of $\frak{J}$ and its maximal Borel-de~Siebenthal subgroups $\frac{\rm SU(3)\times \rm SU(3)}{\mathbb{Z}_3}$ and ${\rm Spin}(9)$ are studied in some detail with an eye to possible applications to the fundamental fermions in the Standard Model of particle physics.
</p>projecteuclid.org/euclid.jgsp/1525939249_20180510040050Thu, 10 May 2018 04:00 EDTUnsteady Roto-Translational Viscous Flow: Analytical Solution to Navier-Stokes Equations in Cylindrical Geometryhttps://projecteuclid.org/euclid.jgsp/1527127300<strong>Alessio Bocci</strong>, <strong>Giovanni Mingari Scarpello</strong>, <strong>Daniele Ritelli</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 48, 1--21.</p><p><strong>Abstract:</strong><br/>
We study the unsteady viscous flow of an incompressible, isothermal (Newtonian) fluid whose motion is induced by the sudden swirling of a cylindrical wall and is also starting with an axial velocity component. Basic physical assumptions are that the pressure axial gradient keeps its hydrostatic value and the radial velocity is zero. In such a way the Navier-Stokes PDEs become uncoupled and can be solved separately. Accordingly, we provide analytic solutions to the unsteady speed components, i.e., the axial $v_z(r,t)$ and the circumferential $v_\theta(r,t)$, by means of expansions of Fourier-Bessel type under time damping. We also find: the surfaces of dynamical equilibrium, the wall shear stress during time and the Stokes streamlines.
</p>projecteuclid.org/euclid.jgsp/1527127300_20180523220143Wed, 23 May 2018 22:01 EDTCentralizer of Reeb vector field in Contact Lie Groupshttps://projecteuclid.org/euclid.jgsp/1527127301<strong>Babak Hassanzadeh</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 48, 23--31.</p><p><strong>Abstract:</strong><br/>
We consider the centralizer of Reeb vector field of a contact Lie group with a left invariant Riemannian metric while contact structure is left invariant. Then we decompose the Lie algebra of this Lie groups to centralizer of Reeb vector field and its orthogonal complement and using this decomposition the contact Lie group is investigated. Furthermore, in last section a special automorphism is defined and studied which it keeps the contact form.
</p>projecteuclid.org/euclid.jgsp/1527127301_20180523220143Wed, 23 May 2018 22:01 EDTAn Examination of Perpendicular Intersections of BFRS and MFRS in $E^3$https://projecteuclid.org/euclid.jgsp/1527127302<strong>Şeyda Kılıçoğlu</strong>, <strong>Süleyman Şenyurt</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 48, 33--45.</p><p><strong>Abstract:</strong><br/>
We already have defined and found the parametric equations of Frenet ruled surfaces which are called Bertrandian Frenet Ruled Surfaces (BFRS) and Mannheim Frenet Ruled Surfaces (MFRS) of a curve $\alpha ,$ in terms of the Frenet apparatus. In this paper, we find a matrix which gives us all sixteen positions of normal vector fields of eight BFRS and MFRS in terms of the Frenet apparatus. Further using the orthogonality conditions of the eight normal vector fields, we give perpendicular intersection curves of the eight BFRS and MFRS.
</p>projecteuclid.org/euclid.jgsp/1527127302_20180523220143Wed, 23 May 2018 22:01 EDTSymmetry Groups of Systems of Entangled Particleshttps://projecteuclid.org/euclid.jgsp/1527127303<strong>Abraham A. Ungar</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 48, 47--77.</p><p><strong>Abstract:</strong><br/>
A Lorentz transformation of signature $(m,n)$, $m,n\in\Nb$, is a pseudo-rotation in a pseudo-Euclidean space of signature $(m,n)$. Accordingly, the Lorentz transformation of signature $(1,3)$ is the common Lorentz transformation of special relativity theory. It is known that entangled particles involve Lorentz symmetry violation. Hence, the aim of this article is to expose and illustrate the symmetry groups of systems of entangled particles uncovered in [44] It turns out that the Lorentz transformations of signature $(m,n)$ form the symmetry group by which systems of $m$ $n$-dimensional entangled particles can be understood, just as the common Lorentz group of signature $(1,3)$ forms the symmetry group by which Einstein's special theory of relativity can be understood. Consequently, it is useful to extend special relativity theory by incorporating Lorentz transformation groups of signature $(m,3)$ for all $m\geq 2$. The resulting extended special relativity theory, then, provides not only the symmetry group of the $(1+3)$-dimensional spacetime of particles, but also the symmetry group of the $(m+3)$-dimensional spacetime of systems of $m$ entangled three-dimensional particles, for each $m\geq 2$.
</p>projecteuclid.org/euclid.jgsp/1527127303_20180523220143Wed, 23 May 2018 22:01 EDTDeformation Quantization in the Teaching of Lie Group Representationshttps://projecteuclid.org/euclid.jgsp/1527127304<strong>Alexander J. Balsomo</strong>, <strong>Job A. Nable</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 48, 79--100.</p><p><strong>Abstract:</strong><br/>
We present straightforward and concrete computations of the unitary irreducible representations of the Euclidean motion group ${\rm M}(2)$ employing the methods of deformation quantization. Deformation quantization is a quantization method of classical mechanics and is an autonomous approach to quantum mechanics, arising from the Wigner quasiprobability distributions and Weyl correspondence. We advertise the utility and power of deformation theory in Lie group representations. In implementing this idea, many aspects of the method of orbits are also learned, thus further adding to the mathematical toolkit of the beginning graduate student of physics. Furthermore, the essential unity of many topics in mathematics and physics (such as Lie theory, quantization, functional analysis and symplectic geometry) is witnessed, an aspect seldom encountered in textbooks, in an elementary way.
</p>projecteuclid.org/euclid.jgsp/1527127304_20180523220143Wed, 23 May 2018 22:01 EDTThe Solution to the Three-Body Problem and Some Applicationshttps://projecteuclid.org/euclid.jgsp/1538705200<strong>Ramon Gonzàlez Calvet</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 49, 1--61.</p><p><strong>Abstract:</strong><br/>
Here we provide and explain the coordinate transformation according to which every weighted quadratic form of the absolute Cartesian coordinates or velocities of three particles is separable into quadratic terms of the relative and centre-of-mass coordinates or velocities. This solution is applied to define a new set of weighted colour coordinates $YJK$ in the colour space, and also to solve the dynamical system Sun-Earth-Moon. The weighted Laplacian and hence the quantum Hamiltonian operator for a system of three particles are also given in relative coordinates, and applied to calculate the vibrational energy levels of carbon dioxide and the electronic energy of the ground state of the hydrogen-molecule-ion and two-electron atomic systems like the helium atom..
</p>projecteuclid.org/euclid.jgsp/1538705200_20181004220650Thu, 04 Oct 2018 22:06 EDTConformal Changes of Odd-Dimensional Generalized Structureshttps://projecteuclid.org/euclid.jgsp/1538705201<strong>Fereshteh Malek</strong>, <strong>Niloufar H. Kashani</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 49, 63--76.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider the integrability of generalized almost contact and contact manifolds after conformal changes. We also study conditions under which the generalized almost contact and normal generalized contact structures, be normal after conformal changes.
</p>projecteuclid.org/euclid.jgsp/1538705201_20181004220650Thu, 04 Oct 2018 22:06 EDTThe Mechanics and Mathematics of Biological Growth by Alain Gorielyhttps://projecteuclid.org/euclid.jgsp/1538705202<strong>Jean-François Ganghoffer</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 49, 77--79.</p>projecteuclid.org/euclid.jgsp/1538705202_20181004220650Thu, 04 Oct 2018 22:06 EDTVasil Valdemarov Tsanovhttps://projecteuclid.org/euclid.jgsp/1538705203<p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 49, 81--106.</p>projecteuclid.org/euclid.jgsp/1538705203_20181004220650Thu, 04 Oct 2018 22:06 EDTLorentz-Invariant Second-Order Tensors and an Irreducible Set of Matriceshttps://projecteuclid.org/euclid.jgsp/1548471825<strong>Mayeul Arminjon</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 50, 1--10.</p><p><strong>Abstract:</strong><br/>
We prove that, up to multiplication by a scalar, the Minkowski metric tensor is the only second-order tensor that is Lorentz-invariant. To prove this, we show that a specific set of three $4\times 4$ matrices, made of two rotation matrices plus a Lorentz boost, is irreducible.
</p>projecteuclid.org/euclid.jgsp/1548471825_20190125220351Fri, 25 Jan 2019 22:03 ESTThird-Order Jacobsthal Generalized Quaternionshttps://projecteuclid.org/euclid.jgsp/1548471826<strong>Gamaliel Cerda-Morales</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 50, 11--27.</p><p><strong>Abstract:</strong><br/>
In this paper, the third-order Jacobsthal generalized quaternions are introduced. We use the well-known identities related to the third-order Jacobsthal and third-order Jacobsthal-Lucas numbers to obtain the relations regarding these quaternions. Furthermore, the third-order Jacobsthal generalized quaternions are classified by considering the special cases of quaternionic units. We derive the relations between third-order Jacobsthal and third-order Jacobsthal-Lucas generalized quaternions.
</p>projecteuclid.org/euclid.jgsp/1548471826_20190125220351Fri, 25 Jan 2019 22:03 ESTSpaces Realized and Non-Realized as Dold-Lashof Classifying Spaceshttps://projecteuclid.org/euclid.jgsp/1548471827<strong>Mohamed Rachid Hilali</strong>, <strong>Abdelhadi Zaim</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 50, 29--56.</p><p><strong>Abstract:</strong><br/>
Let $X$ be a simply connected CW-complex of finite type. Denote by ${\rm Baut}_{1}(X)$ the Dold-Lashof classifying space of fibrations with fiber $X$. \ This paper is a survey about the problem of realizing Dold-Lashof classifying spaces. We will also present some new results: we show that not all rank-two rational $H$-spaces can be realized as ${\rm Baut}_{1}(X)$ for simply connected, rational elliptic space $X$. Moreover, we construct an infinite family of rational spaces $X,$ such that ${\rm Baut}_{1}(X)$ is rationally a finite $H$-space of rank-two (up to rational homotopy type).
</p>projecteuclid.org/euclid.jgsp/1548471827_20190125220351Fri, 25 Jan 2019 22:03 ESTMathematical Model of Elastic Closed Flexible Shells with Nonlocal Shape Deviationshttps://projecteuclid.org/euclid.jgsp/1548471828<strong>Viktor Olevskyi</strong>, <strong>Yuliia Olevska</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 50, 57--69.</p><p><strong>Abstract:</strong><br/>
A model of deformation and mechanical stability of a thin-walled shell with geometric deviations, which is close to a circular cylindrical shell, under the action of axial compression and normal pressure is developed. The model uses the scheme of a flexible shell of zero Gaussian curvature with a perturbed edge, which makes it possible to apply the methods of the geometrically nonlinear theory of torso shells.
</p>projecteuclid.org/euclid.jgsp/1548471828_20190125220351Fri, 25 Jan 2019 22:03 ESTExamples of Automorphism Groups of Ind-Varieties of Generalized Flagshttps://projecteuclid.org/euclid.jgsp/1548471829<strong>Ivan Penkov</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 50, 71--77.</p><p><strong>Abstract:</strong><br/>
We compute the automorphism groups of finite and cofinite ind-grassmannians, as well as of the ind-variety of the maximal flags indexed by ${\mathbb Z}_{>0}$. We pay special attention to differences with the case of ordinary flag varieties.
</p>projecteuclid.org/euclid.jgsp/1548471829_20190125220351Fri, 25 Jan 2019 22:03 EST2+1-Moulton Configurationhttps://projecteuclid.org/euclid.jgsp/1548471830<strong>Naoko Yoshimi</strong>, <strong>Akira Yoshioka</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 50, 79--95.</p><p><strong>Abstract:</strong><br/>
We pose a new problem of collinear central configuration in Newtonian $n$-body problem. For a given two-body, we ask whether we can add a new body in a way such that i) the configuration of the total three-body is also collinear central with the configuration of the initial two-body being fixed and further ii) the initial two-body keeps its motion without any change during the process. We find three solutions to the above problem. We also consider a similar problem such that while the condition i) is satisfied but by modifying the condition ii) the motion of the initial two-body is not necessarily equal to the original one. We also find explicit solutions to the second problem.
</p>projecteuclid.org/euclid.jgsp/1548471830_20190125220351Fri, 25 Jan 2019 22:03 ESTThe Restricted Three-Body Problem and Holomorphic Curves by Urs Frauenfelder and Otto van Koerthttps://projecteuclid.org/euclid.jgsp/1548471831<strong>Ramon González Calvet</strong>. <p><strong>Source: </strong>Journal of Geometry and Symmetry in Physics, Volume 50, 97--100.</p>projecteuclid.org/euclid.jgsp/1548471831_20190125220351Fri, 25 Jan 2019 22:03 EST