λ-Deformation: A Canonical Framework for Statistical Manifolds of Constant Curvature

This paper systematically presents the λ-deformation as the canonical framework of deformation to the dually flat (Hessian) geometry, which has been well established in information geometry. We show that, based on deforming the Legendre duality, all objects in the Hessian case have their correspondence in the λ-deformed case: λ-convexity, λ-conjugation, λ-biorthogonality, λ-logarithmic divergence, λ-exponential and λ-mixture families, etc. In particular, λ-deformation unifies Tsallis and Rényi deformations by relating them to two manifestations of an identical λ-exponential family, under subtractive or divisive probability normalization, respectively. Unlike the different Hessian geometries of the exponential and mixture families, the λ-exponential family, in turn, coincides with the λ-mixture family after a change of random variables. The resulting statistical manifolds, while still carrying a dualistic structure, replace the Hessian metric and a pair of dually flat conjugate affine connections with a conformal Hessian metric and a pair of projectively flat connections carrying constant (nonzero) curvature. Thus, λ-deformation is a canonical framework in generalizing the well-known dually flat Hessian structure of information geometry.     

Hyperbolic Billiards on Surfaces of Constant Curvature

We establish sufficient conditions for the hyperbolicity of the billiard dynamics on surfaces of constant curvature. This extends known results for planar billiards. Using these conditions, we construct large classes of billiard tables with positive Lyapunov exponents on the sphere and on the hyperbolic plane. Received: 26 January 1999 / Accepted: 17 May 1999     

Curvature Properties of Lorentzian Manifolds with Large Isometry Groups

The curvature of Lorentzian manifolds (M ⁿ ,g), admitting a group of isometries of dimension at least 1/2n(n − 1) + 1, is completely described. Interesting behaviours are found, in particular as concerns local symmetry, local homogeneity and conformal flatness. Second and third authors supported by funds of MURST and the University of Salento.     

The Dynamics of Pendulums on Surfaces of Constant Curvature

We define the notion of a pendulum on a surface of constant curvature and study the motion of a mass at a fixed distance from a pivot. We consider some special cases: first a pivot that moves with constant speed along a geodesic, and then a pivot that undergoes acceleration along a fixed geodesic.      

Space-Like Submanifolds with Constant Scalar Curvature

Let M ⁿ be a space-like submanifold in a de Sitter space M _p ^n+p (c) with constant scalar curvature. This Letter gives intrinsic conditions for M ⁿ to be totally umbilical.     

Parasupersymmetry of Non-Linear and Isotropic Oscillator on Constant Curvature

In this paper we discuss the one-dimensional non-linear harmonic oscillator and isotropic oscillator in positive curvature. We will see that in the special case two different oscillator have a same behavior. Finally, by using the parasupersymmetry algebra we obtain the partner potential and superpotential. These superpotential lead us to obtain the corresponding superchares.     

Two-Dimensional Coulomb Systems on a Surface of Constant Negative Curvature

We study the equilibrium statistical mechanics of classical two-dimensional Coulomb systems living on a pseudosphere (an infinite surface of constant negative curvature). The Coulomb potential created by one point charge exists and goes to zero at infinity. The pressure can be expanded as a series in integer powers of the density (the virial expansion). The correlation functions have a thermodynamic limit, and remarkably that limit is the same one for the Coulomb interaction and some other interaction law. However, special care is needed for defining a thermodynamic limit of the free energy density. There are sum rules expressing the property of perfect screening. These generic properties can be checked on the Debye–Hückel approximation, and on two exactly solvable models, the one-component plasma and the two-component plasma, at some special temperature.     

Casorati Inequalities for Statistical Submanifolds in Kenmotsu Statistical Manifolds of Constant ϕ-Sectional Curvature with Semi-Symmetric Metric Connection

In this paper, we prove some inequalities between intrinsic and extrinsic curvature invariants, namely the normalized δ-Casorati curvatures and the scalar curvature of statistical submanifolds in Kenmotsu statistical manifolds of constant ϕ-sectional curvature that are endowed with semi-symmetric metric connection. Furthermore, we investigate the equality cases of these inequalities. We also describe an illustrative example.     

Intertwining Symmetry Algebras of Quantum Superintegrable Systems on Constant Curvature Spaces

A class of quantum superintegrable Hamiltonians defined on a hypersurface in a n+1 dimensional ambient space with signature (p,q) is considered and a set of intertwining operators connecting them are determined. It is shown that the intertwining operators can be chosen such that they generate the su(p,q) and so(2p,2q) Lie algebras and lead to the Hamiltonians through Casimir operators. The physical states corresponding to the discrete spectrum of bound states as well as the degeneration are characterized in terms of some particular unitary representations.     

The Generalized Uncertainty Principle

The uncertainty principle lies at the heart of quantum physics, and is widely thought of as a fundamental limit of the measurement precision of incompatible observables. Here it is shown that the traditional uncertainty relation in fact belongs to the leading order approximation of a generalized uncertainty relation. That is, the leading order linear dependence of observables gives the Heisenberg type of uncertainty relations, while higher order nonlinear dependence may reveal more different and interesting correlation properties. Applications of the generalized uncertainty relation and the high order nonlinear dependence between observables in quantum information science are also discussed.     

Quantum Mechanics on Hilbert Manifolds: The Principle of Functional Relativity

Quantum mechanics is formulated as a geometric theory on a Hilbert manifold. Images of charts on the manifold are allowed to belong to arbitrary Hilbert spaces of functions including spaces of generalized functions. Tensor equations in this setting, also called functional tensor equations, describe families of functional equations on various Hilbert spaces of functions. The principle of functional relativity is introduced which states that quantum theory (QT) is indeed a functional tensor theory, i.e., it can be described by functional tensor equations. The main equations of QT are shown to be compatible with the principle of functional relativity. By accepting the principle as a hypothesis, we then explain the origin of physical dimensions, provide a geometric interpretation of Planck’s constant, and find a simple model of the two-slit experiment and the process of measurement.     

Three-Dimensional Metrics as Deformations of a Constant Curvature Metric

Any three-dimensional metric g may be locally obtained from a constant curvature metric, h, by a deformation like

Ti∶LiNbO_3二维渐变折射率光波导传播常数的计算

从Ti∶LiNbO3波导截面折射率分布的特点出发，得出一套计算传播常数简单而精确的方法。文中对该法进行了分析，并应用该方法计算了钛扩散波导的色散关系。该方法为钛扩散光波导集成光学器件的设计提供了一种较可靠的理论手段。     

影响弹簧扭转常数不确定度的因素分析

运用不确定度理论,推导了扭摆法测弹簧扭转常数的不确定度传递公式。通过应用塑料圆柱、金属圆筒和金属细杆三种不同形状的标准件来测量弹簧扭转常数,分析了影响测量弹簧扭转常数不确定度的几个因素,给出了减小实验误差的有效途径。     

Two-Dimensional Krall–Sheffer Polynomials andQuantum Systems on Spaces with Constant Curvature

Krall and Sheffer found in 1967 that there exists at most nine different types of two-dimensional orthogonal polynomials which are eigensolutions of a second-order linear differential operator with polynomial coefficients. We show that, for all these types, there correspond quantum mechanical systems on a Euclidean (pseudo-Eeuclidean) plane, two-dimensional sphere, or hyperboloid.     

Generalized Uncertainty Principle from QuantumGeometry

The generalized uncertainty principle of string theory is derived in the frameworkof quantum geometry by taking into account the existence of an upper limit onthe acceleration of massive particles.