Propagating beam analysis of fiber wavelength filters with a depressed-index section

A wavelength filter consisting of single-mode and few-mode fibers is investigated numerically. A simple finite-difference beam-propagation method, in which a transparent boundary condition can be imposed, is developed for circularly symmetric waveguides. After confirming the validity of the numerical method by the mode-mismatch loss, we calculate the propagating field in the fiber wavelength filter, in which interference between LP₀₁, and LP₀₂ modes occurs. To improve the filtering operation, a depressed-index fiber is employed for the few-mode fiber. The effects of the radius and refractive index of the depressed section on the transmission power are revealed and discussed. Power is suppressed to less than 0.1% at 1.3 μm, while maintaining power transmission of more than 85% at 1.55 μm. It is also found that the filtering operation shifts to higher wavelengths as the input power is increased when we choose a self-focusing nonlinear material in the depressed section.     

The diameter of a fuzzy set

In pattern recognition one often wants to measure geometric properties of imprecisely defined subsets of an image. This paper proposes definitions of intrinsic and extrinsic diameter for fuzzy subsets which reduce to the ordinary definitions when the subsets are crisp. We also define height and width for a fuzzy subset and show how they relate to the area (i.e., integral of membership). For convex fuzzy subsets the intrinsic diameter cannot exceed the extrinsic diameter, but it can be smaller. Finally, for piecewise constant convex fuzzy subsets the intrinsic diameter cannot exceed half the fuzzy perimeter, but this need not be true in the nonconvex case.     

Information,Quantum Mechanics,and Gravity

This is a basically expository article, with some new observations, tracing connections of the quantum potential to Fisher information, to Kähler geometry of the projective Hilbert space of a quantum system, and to the Weyl-Ricci scalar curvature of a Riemannian flat spacetime with quantum matter.Á Denise     

Common complements of two subspaces of a Hilbert space

In this paper we find a necessary and sufficient condition for two closed subspaces, and , of a Hilbert space to have a common complement, i.e. a subspace having trivial intersection with and and such that .Unlike the finite-dimensional case the condition is significantly more subtle than simple equalities of dimensions and codimensions, and non-trivial examples of subspaces without a common complement are possible.     

Sub-Riemannian constant mean curvature surfaces in the Heisenberg group as limits

The 3-dimensional Heisenberg group H together with its standard sub-Riemannian metric g₀ is viewed as the limit of a family of Riemannian manifolds, (H,g_u), u>0. For each u>0, we consider some invariant surfaces with constant mean curvature in (H,g_u). These surfaces of (H,g_u) have very nice limits as u0. We then define the mean curvature of a hypersurface in (H,g₀) to be the limit of its mean curvature in (H,g_u). We show that in a more general case, this definition is appropriate. Mathematics Subject Classification (2000) Principal 53C17; Secondary 22E25     

Conformally invariant powers of the Laplacian -- A complete nonexistence theorem

We show that on conformal manifolds of even dimension there is no conformally invariant natural differential operator between density bundles with leading part a power of the Laplacian for n/2$">. This shows that a large class of invariant operators on conformally flat manifolds do not generalise to arbitrarily curved manifolds and that the theorem of Graham, Jenne, Mason and Sparling, asserting the existence of curved version of for , is sharp.

     

Orbits of Positive Operators from a Differentiable Viewpoint

Let be a unital C^*-algebra and G the group of units of . A geometrical study of the action of G over the set ⁺ of all positive elements of is presented. The orbits of elements with closed range by this action are provided with a structure of differentiable homogeneous space with a natural connection. The orbits are partitioned in 'components' which also have a rich geometrical structure.     

Para-Hopf Algebroids and their Cyclic Cohomology

We introduce the concept of para-Hopf algebroid and define their cyclic cohomology in the spirit of Connes–Moscovici cyclic cohomology for Hopf algebras. Para-Hopf algebroids are closely related to, but different from, Hopf algebroids. Their definition is motivated by attempting to define a cyclic cohomology theory for Hopf algebroids in general. We show that many of Hopf algebraic structures, including the Connes–Moscovici algebra , are para-Hopf algebroids     

Letter: Epicyclic Orbital Oscillations in Newton's and Einstein's Gravity from the Geodesic Deviation Equation

In a recent paper Abramowicz and Kluniak [1] have discussed the problem of epicyclic oscillations in Newton's and Einstein's dynamics and have shown that Newton's dynamics in a properly curved three-dimensional space is identical to test-body dynamics in the three-dimensional optical geometry of Schwarzschild space-time. One of the main results of this paper was the proof that different behaviour of radial epicyclic frequency and Keplerian frequency in Newtonian and General Relativistic regimes had purely geometric origin contrary to claims that nonlinearity of Einstein's theory was responsible for this effect. In this paper we obtain the same result from another perspective: by representing these two distinct problems (Newtonian and Einstein's test body motion in central gravitational field) in a uniform way — as a geodesic motion. The solution of geodesic deviation equation reproduces the well known results concerning epicyclic frequencies and clearly demonstrates geometric origin of the difference between Newtonian and Einstein's problems.