## Recursos de colección

#### Project Euclid (Hosted at Cornell University Library) (203.209 recursos)

Proceedings of the International Conference on Geometry, Integrability and Quantization

1. #### Star Product, Star Exponential and Applications

Yoshioka, Akira
We introduce star products for certain function space containing polynomials, and then we obtain an associative algebra of functions. In this algebra we can consider exponential elements, which are called star exponentials. Using star exponentials we can define star functions in the star product algebra. We explain several examples.

2. #### Fisher Metric for Diagonalizable Quadratic Hamiltonians and Application to Phase Transitions

We derive the extended entanglement entropy and the Fisher information metric in the case of quantum models, described by time-independent diagonal quadratic Hamiltonians. Our research is conducted within the framework of Thermo field dynamics. We also study the properties of the Fisher metric invariants to identify the phase structure of the quasi-particle systems in equilibrium.

3. #### Evidence of a ${\bf {6.12 \times 10^{18}}}$ GeV Particle: Detection and Mathematics

Smith, Paul T.
In a new approach the graviton is defined as the field particle of spacetime rather than the mediator of gravity. The unification equation is derived and used to predict that for a freely falling body, the energy of incident gravitons is $6.12\times 10^{18}$ GeV. Redshift and scattering of gravitons should produce diffraction patterns, galactic halos and expansion of the Universe. The energy of incident gravitons remains constant as the Universe evolves because of the Doppler shift as bodies fall towards redshifted gravitons. Complex space is used to represent gravitons and explain Young’s two-slit interference. The approach is corroborated by empirical...

4. #### Mathematical Models of Classical Electrodynamics

Savov, Sava
Four mathematical models of classical electrodynamics based on vector fields, tensor spaces, geometric algebras and differential forms are represented in parallel and compared.

5. #### Bifurcation of Closed Geodesics

Rýparová, Lenka; Mikeš, Josef
This paper is denoted to further study of geodesic bifurcation on surfaces of revolution. We demonstrate an example of bifurcation of closed geodesics on surfaces.

6. #### On a Class of Linear Weingarten Surfaces

We consider a class of linear Weingarten surfaces of revolution whose principal curvatures, meridional $k_{\mu}$ and parallel $k_{\pi}$, satisfy the relation $k_{\mu}=(n+1)k_{\pi}$, $n=0,\,1,\,2,\ldots\, .$ The first two members of this class of surfaces are the sphere $(n=0)$ and the Mylar balloon $(n=1)$. Elsewhere the Mylar balloon has been parameterized via the Jacobian and Weierstrassian elliptic functions and elliptic integrals. Here we derive six alternative parameterizations describing the third type of surfaces when $n=2$. The so obtained explicit formulas are applied for the derivation of the basic geometrical characteristics of this surface.

7. #### Geometric Aspects of Multiple Fourier Series Convergence on the System of Correctly Counted Sets

Olevskyi, Viktor; Olevska, Yuliia
For multiple Fourier series the convergence of partial sums essentially depends on the type of integer sets, to which the sequence numbers of their terms belong. The problem on the general form of such sets is studying in $u$-convergence theory ($u(K)$ - convergence) for multiple Fourier series. An alternative method of summation is based on the concept of the so-called correctly denumarable sets. In the paper some results describing the $u$-convergence relations and convergence on the system or correctly denumarable sets are presented. It is shown that the system of $U(K)$-sets containing a sphere of infinitely increasing radius for fixed...

8. #### Alternative Description of Rigid Body Kinematics and Quantum Mechanical Angular Momenta

In the present paper we investigate an alternative two-axes decomposition method for rotations that has been proposed in our earlier research. It is shown to provide a convenient parametrization for many important physical systems. As an example, the kinematics of a rotating rigid body is considered and a specific class of solutions to the Euler dynamical equations are obtained in the case of symmetric inertial ellipsoid. They turn out to be related to the Rabi oscillator in spin systems well known in quantum computation. The corresponding quantum mechanical angular momentum and Laplace operator are derived as well with the aid...

9. #### Construction of Symplectic-Haantjes Manifold of Certain Hamiltonian Systems

Hosokawa, Kiyonori; Takeuchi, Tsukasa; Yoshioka, Akira
Symplectic-Haantjes manifolds are constructed for several Hamiltonian systems following Tempesta-Tondo [5], which yields the complete integrability of systems.

10. #### On Conformal Mappings Onto Compact Einstein Manifolds

Hinterleitner, Irena; Guseva, Nadezda; Mikeš, Josef
In the present paper we prove non-existence theorems for conformal mappings of compact (pseudo-)Riemannian manifolds onto Einstein manifolds without boundary. We obtained certain conditions for which these mappings are only trivial.

11. #### Quantization of Locally Symmetric Kähler Manifolds

Hara, Kentaro; Sako, Akifumi
We introduce noncommutative deformations of locally symmetric Kähler manifolds. A Kähler manifold $M$ is said to be a locally symmetric Kähler manifold if the covariant derivative of the curvature tensor is vanishing . An algebraic derivation process to construct a locally symmetric Kähler manifold is given. As examples, star products for noncommutative Riemann surfaces and noncommutative $\mathbb {CP}^N$ are constructed.

12. #### On Holomorphically Projective Mappings of Equidistant Parabolic Kähler Spaces

Chudá, Hana; Mikeš, Josef; Peška, Patrik; Shiha, Mohsen
In this paper we construct holomorphically projective mappings of equidistant parabolic Kähler spaces. We discus fundamental equations of these mappings as well.

13. #### The Elasticity of Quantum Spacetime Fabric

Cartas, Viorel Laurentiu
The present paper aims to emphasize the geometrical features of the quantum spacetime, considering gravity as an emergent feature similar to the elasticity of the solid state. A small scale structure is needed to explain the emergent gravity and how spacetime atoms are continuously created in the process of the expansion of the universe. A simple geometrical model has been introduced.

14. #### Projective Bivector Parametrization of Isometries in Low Dimensions

Brezov, Danail S.
The paper provides a pedagogical study on vectorial parameterizations first proposed by O. Rodrigues for the rotation group in $\mathbb{R}^3$ by means of the so-called Rodrigue’s vector. Although his technique yields significant advantages in both theoretical and applied context, the vectorial interpretation is easily seen to be completely wrong and in order to benefit most from this otherwise fruitful approach, we put it in the proper perspective, namely, that of Clifford’s geometric algebras, spin groups and projective geometry. This allows for a natural generalization and straightforward implementations in various physical models, some of which are pointed out below in the...

15. #### Kauffman Bracket on Rational Tangles and Rational Knots

Bataineh, Khaled
Computing Kauffman bracket grows exponentially with the number of crossings in the knot diagram. In this paper we illustrate how Kauffman bracket for rational tangles and rational knots can be computed so that it involves a low number of terms. Kauffman bracket and Jones polynomial are known to have connections with statistical mechanics, quantum theory and quantum field theory.

16. #### On Hyper Generalized Weakly Symmetric Manifolds

Baishya, Kanak K.; Zengin, Füsun; Mikeš, Josef
This paper aims to introduce the notion of hyper generalized weakly symmetric manifolds with a non-trivial example.

17. #### On Charge Conservation in a Gravitational Field

Arminjon, Mayeul
According to the “gravitationally-modified” Maxwell equations that were proposed for an alternative scalar theory with an “ether”, electric charge would not be conserved in time-dependent gravitational field. We define an asymptotic expansion scheme for the electromagnetic field in a weak gravitational field. This allows us to assess the amounts of charge production or destruction which are thus predicted. These amounts seem high enough to discard that version of the gravitationally-modified Maxwell equations. We show that this failure is due to the former assumption of additivity of the energy tensors: an “interaction energy tensor” has to be added. Then the standard...

18. #### Clifford Algebras and Their Applications to Lie Groups and Spinors

Shirokov, Dmitry
We discuss some well-known facts about Clifford algebras: matrix representations, Cartan’s periodicity of 8, double coverings of orthogonal groups by spin groups, Dirac equation in different formalisms, spinors in $n$ dimensions, etc. We also present our point of view on some problems. Namely, we discuss the generalization of the Pauli theorem, the basic ideas of the method of averaging in Clifford algebras, the notion of quaternion type of Clifford algebra elements, the classification of Lie subalgebras of specific type in Clifford algebra, etc.

19. #### On Auto and Hetero Bäcklund Transformations for the Hénon-Heiles Systems

Tsiganov, Andrey; Sozonov, Alexey
We consider a canonical transformation of parabolic coordinates on the plain associated with integrable Hénon-Heiles systems and suppose that this transformation together with some additional relations may be considered as a counterpart of the auto and hetero Bäcklund transformations.

20. #### Deformation Quantization of Kähler Manifolds and Their Twisted Fock Representation

Sako, Akifumi; Umetsu, Hiroshi
We introduce the notion of twisted Fock representations of noncommutative Kähler manifolds and give their explicit expressions. The so-called twisted Fock representation is a representation of the Heisenberg like algebra whose states are constructed by acting creation operators on a vacuum state. “Twisted” means that creation operators are not Hermitian conjugate of annihilation operators. In deformation quantization of Kähler manifolds with separation of variables formulated by Karabegov, local complex coordinates and partial derivatives of the Kähler potential with respect to coordinates satisfy the commutation relations between the creation and annihilation operators. Based on these relations, the twisted Fock representation of...

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