Image Manifolds which are Isometric to Euclidean Space

Recently, the Isomap procedure [10] was proposed as a new way to recover a low-dimensional parametrization of data lying on a low-dimensional submanifold in high-dimensional space. The method assumes that the submanifold, viewed as a Riemannian submanifold of the ambient high-dimensional space, is isometric to a convex subset of Euclidean space. This naturally raises the question: what datasets can reasonably be modeled by this condition? In this paper, we consider a special kind of image data: families of images generated by articulation of one or several objects in a scene—for example, images of a black disk on a white background with center placed at a range of locations. The collection of all images in such an articulation family, as the parameters of the articulation vary, makes up an articulation manifold, a submanifold of L ². We study the properties of such articulation manifolds, in particular, their lack of differentiability when the images have edges. Under these conditions, we show that there exists a natural renormalization of geodesic distance which yields a well-defined metric. We exhibit a list of articulation models where the corresponding manifold equipped with this new metric is indeed isometric to a convex subset of Euclidean space. Examples include translations of a symmetric object, rotations of a closed set, articulations of a horizon, and expressions of a cartoon face. The theoretical predictions from our study are borne out by empirical experiments with published Isomap code. We also note that in the case where several components of the image articulate independently, isometry may fail; for example, with several disks in an image avoiding contact, the underlying Riemannian manifold is locally isometric to an open, connected, but not convex subset of Euclidean space. Such a situation matches the assumptions of our recently-proposed Hessian Eigenmaps procedure, but not the original Isomap procedure.     

Number-theoretic functions which are equivalent to number of divisors

Let d(n) denote the number of positive integral divisors of n. In this paper we show that the Möbius function, μ(N), can be computed by a single call to an oracle for d(n). We also show that any function that depends solely on the exponents in the prime factorization of N can be computed by at most log₂ N calls to an oracle for d(N).     

Improved lower bounds on the number of multiplications/divisions which are necessary to evaluate polynomials

We improve some lower bounds which have been obtained by Strassen and Lipton. In particular there exist polynomials of degree n with 0–1 coefficients that cannot be evaluated with less than $\text{n}\text{/(4}\text{log}\text{n)}$ nonscalar multiplications/divisions. The evaluation of p(x) $\text{∑}\text{δ=0}\text{n}\text{e}{}^{\mathrm{<mtext>2\pi i</mtext><mtext>2\delta </mtext>}}\text{x}{}^{\mathrm{\delta }}$ requires at least $\text{n}\text{(12}\text{log}\text{n)}$ multiplications/divisions and at least $\text{n/(8}\text{log}\text{n)}$ nonscalar multiplications/divisions. We specify polynomials with algebraic coefficients that require $\text{1}\text{2}\text{n}$ multiplications/divisions.     

对高校教材建设的几点看法——由《网页设计与制作教程》一书热销引发的思考

本文通过分析目前高校教材建设存在的一些问题以及笔者编著的《网页设计与制作教程》一书热销的原因,给出了笔者对高校教材建设的一些个人看法,供大家借鉴和思考。     

Cybercrimes Of The 21st Century: This series examines cybercrime across the globe,the way in which governments are responding to its challenge,and how we can minimize the risks it poses to business

Over 50% of all frauds committed in the first half of 2000 were ‘cybercrimes’. Not a day passes where you cannot pick up a paper and read of some new Internet fraud scam. Internet crime appears to be growing faster than the Net itself. For every new online service, another one has been hacked into, either deliberately, or as a result of some security flaw. That this is a fact has been reinforced by the release of new figures in December of last year by the International Chamber of Commerce. Its Commercial Crime Services division has reported that, “Web-based fraud is rising dramatically, accounting for two thirds of all the fraud cases it investigated for its business members in 2000.”