Mostrando recursos 1 - 20 de 233

  1. Deformation Quantization in the Teaching of Lie Group Representations

    Balsomo, Alexander J.; Nable, Job A.
    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...

  2. Symmetry Groups of Systems of Entangled Particles

    Ungar, Abraham A.
    A Lorentz transformation of signature $(m,n)$, $m,n \in {\mathbb N}$, 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...

  3. An Examination of Perpendicular Intersections of BFRS and MFRS in $E^3$

    Kılıçoğlu, Şeyda; Şenyurt, Süleyman
    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.

  4. Centralizer of Reeb Vector Field in Contact Lie Groups

    Hassanzadeh, Babak
    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.

  5. Unsteady Roto-Translational Viscous Flow: Analytical Solution to Navier-Stokes Equations in Cylindrical Geometry

    Bocci, Alessio; Scarpello, Giovanni Mingari; Ritelli, Daniele
    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...

  6. Spanning Class in the Category of Branes

    Viña, Andrés
    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.

  7. Deformations of Symplectic Structures by Moment Maps

    Nakamura, Tomoya
    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.

  8. Natural Coordinate System in Curved Space-Time

    Gu, Ying-Qiu
    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...

  9. Cassini Ovals in Harmonic Motion Orbits

    Boyadzhiev, Khristo N.; Boyadzhiev, Irina A.
    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.

  10. Indefinite Eisenstein Lattices: A Modern Ball-Rendevous with Poincarè, Picard, Hecke, Shimura, Mumford, Deligne and Hirzebruch

    Holzapfel, Rolf-Peter
    In [21] we have counted indefinite metrics (two-dimensional, integrally defined, over Gauss numbers) with a fixed norm (discriminant). We would like to call them also 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 ${\mathbb P}^2$. 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...

  11. Flat Affine and Symplectic Geometries on Lie Groups

    Villabón, Andrés
    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...

  12. Composition Algebras, Exceptional Jordan Algebra and Related Groups

    Todorov, Ivan; Drenska, Svetla
    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.

  13. A Note on the Class of Surfaces with Constant Skew Curvatures

    Toda, Magdalena; Pigazzini, Alexander
    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.

  14. Commuting Pairs of Generalized Contact Metric Structures

    Talvacchia, Janet
    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.

  15. Twistor Spaces and Compact Manifolds Admitting Both Kähler and Non-Kähler Structures

    Kamenova, Ljudmila
    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.

  16. Is Spacetime as Physical as Is Space?

    Arminjon, Mayeul
    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...

  17. 2+2 Moulton Configuration

    Yoshimi, Naoko
    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.

  18. Geodesics on Rotational Surfaces in Pseudo-Galilean Space

    Yoon, Dae Won; Karacan, Murat Kemal; Bukcu, Bahaddin
    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.

  19. Kähler Dynamics for the Universal Multi-Robot Fleet

    Ivancevic, Vladimir G.
    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.

  20. Generalized Seiberg-Witten Equations on a Riemann Surface

    Dey, Rukmini; Thakre, Varun
    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.

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