Visibility of the higher-dimensional central projection into the projective sphere

We can describe higher-dimensional classical spaces by analytical projective geometry, if we embed the d-dimensional real space onto a d + 1-dimensional real projective metric vector space. This method allows an approach to Euclidean, hyperbolic, spherical and other geometries uniformly [8]. To visualize d-dimensional solids, it is customary to make axonometric projection of them. In our opinion the central projection gives more information about these objects, and it contains the axonometric projection as well, if the central figure is an ideal point or an s-dimensional subspace at infinity. We suggest a general method which can project solids into any picture plane (space) from any central figure, complementary to the projection plane (space). Opposite to most of the other algorithms in the literature, our algorithm projects higher-dimensional solids directly into the two-dimensional picture plane (especially into the computer screen), it does not use the three-dimensional space for intermediate step. Our algorithm provides a general, so-called lexicographic visibility criterion in Definition and Theorem 3.4, so it determines an extended visibility of the d-dimensional solids by describing the edge framework of the two-dimensional surface in front of us. In addition we can move the central figure and the image plane of the projection, so we can simulate the moving position of the observer at fixed objects on the computer screen (see first our figures in reverse order). Supported by DAAD 2008 Multimedia Technology for Mathematics and Computer Science Education.     

因子von Neumann代数上导子的等价刻画

设R是含非平凡幂等元P的素环,C∈R,C=PC.本文证明可加映射△:R→R在C可导,即△(AB)=△(A)B+A△(B),A,B∈R,AB=C当且仅当存在导子δ:R→R,使得△(A)=δ(A)+△(I)A,A∈R.没有I_1型中心直和项的von Neumann代数上的可导映射也有类似结论.利用该结论证明了,若非零算子C∈B(X),使得ran(C)或ker(C)在X中可补,则可加映射△:B(X)→B(X)在C可导当且仅当它是导子.特别地,证明了因子von Neumann代数上的可加映射在任意但固定的非零算子可导当且仅当它是导子.     

A Silicon-Based Arrayed Waveguide Grating Operating in Two Communication Channels

Abstract

This article proposes a silicon-based arrayed waveguide grating operating in two wavelength bands simultaneously with central wavelengths of 1,550.12 nm and 1,310.12 nm. If input light to an arrayed waveguide grating consists of wavelengths around 1,550.12 nm, the proposed system will act as a four-channel demultiplexer with channel spacing of 0.8 nm. On the other hand, when input wavelengths are distributed around 1,310.12 nm, the same arrayed waveguide grating will divide the input to seven channels with channel spacing of 0.33 nm.     

McKean-Vlasov limit for interacting random processes in random media

We apply large-deviation theory to particle systems with a random mean-field interaction in the McKean-Vlasov limit. In particular, we describe large deviations and normal fluctuations around the McKean-Vlasov equation. Due to the randomness in the interaction, the McKean-Vlasov equation is a collection of coupled PDEs indexed by the state space of the single components in the medium. As a result, the study of its solution and of the finite-size fluctuation around this solution requires some new ingredient as compared to existing techniques for nonrandom interaction.     

On the structure of random plane-oriented recursive trees and their branches

This paper is an investigation of the structural properties of random plane-oriented recursive trees and their branches. We begin by an enumeration of these trees and some general properties related to the outdegrees of nodes. Using generalized Pólya urn models we study the exact and limiting distributions of the size and the number of leaves in the branches of the tree. The exact distribution for the leaves in the branches is given by formulas involving second-order Eulerian numbers. A martingale central limit theorem for a linear combination of the number of leaves and the number of internal nodes is derived. The distribution of that linear combination is a mixture of normals with a beta distribution as its mixing density. The martingale central limit theorem allows easy determination of the limit laws governing the leaves in the branches. Furthermore, the asymptotic joint distribution of the number of nodes of outdegree 0, 1 and 2 is shown to be trivariate normal. © 1993 John Wiley & Sons, Inc.     

Multivariate Variational Inequalities and the Central Limit Theorem

Multivariate variational inequalities are obtained in terms of thew-functions and the trace of a Fisher-type information matrix. In consequence of these inequalities, the multivariate central limit theorem arises in the sense of the total variation.     

Hölderian Invariance Principle for Triangular Arrays of Random Variables

In this paper, we extend the Hölderian invariance principle of Lamperti [6] to the case of partial-sum processes based on a triangular array of row-wise independent random variables. As an application, we obtain necessary and sufficient conditions for the almost sure (resp. in probability) weak Hölder convergence of partial-sum processes based on bootstrapped samples.     

正四面体套中心构型的存在唯一性

Two cases of the nested configurations in R^3 consisting of two regular quadrilaterals are discussed. One case of them do not form central configuration, the other case can be central configuration. In the second case the existence and uniqueness of the central configuration are studied. If the configuration is a central configuration, then all masses of outside layer are equivalent, similar to the masses of inside layer. At the same time the following relation between r(the ratio of the sizes) and mass ratio b = m/m must be satisfied b=24(3的立方根)（3r^2 2r 3）^-3/2-8（1-r）|1-r|^-3-3(6r的立方根)/24(3的立方根)（3 r）（3r^2 2r 3）^-3/2-8r（1-r）|1-r|^-3-3(6r^-2的立方根)in which the masses at outside layer are not less than the masses at inside layer, and the solution of this kind of central configuration is unique for the given ratio (b) of masses.     

Mann-Whitney test for associated sequences

Let {X ₁, ...,X _m } and {Y ₁, ...,Y _n } be two samples independent of each other, but the random variables within each sample are stationary associated with one dimensional marginal distribution functionsF andG, respectively. We study the properties of the classical Wilcoxon-Mann-Whitney statistic for testing for stochastic dominance in the above set up.     

Asymptotic Behavior of Heat Kernels on Spheres of Large Dimensions

Forn2, let (μ^x_τ, n)_τ0be the distributions of the Brownian motion on the unit sphereS^{n n+1}starting in some pointxSⁿ. This paper supplements results of Saloff-Coste concerning the rate of convergence ofμ^x_τ, nto the uniform distributionU_nonSⁿforτ→∞ depending on the dimensionn. We show that,[formula]forτ_n:=(ln n+2s)/(2n), where erf denotes the error function. Our proof depends on approximations of the measuresμ^x_τ, nby measures which are known explicitly via Poisson kernels onSⁿ, and which tend, after suitable projections and dilatations, to normal distributions on forn→∞. The above result as well as some further related limit results will be derived in this paper in the slightly more general context of Jacobi-type hypergroups.