Efficient data structures for masks on 2D grids

This article discusses various methods of representing and manipulating arbitrary coverage information in two dimensions, with a focus on space- and time-efficiency when processing such coverages, storing them on disk, and transmitting them between computers. While these considerations were originally motivated by the specific tasks of representing sky coverage and cross-matching catalogues of astronomical surveys, they can be profitably applied in many other situations as well.Comment: accepted by A&

Sixteen space-filling curves and traversals for d-dimensional cubes and simplices

This article describes sixteen different ways to traverse d-dimensional space recursively in a way that is well-defined for any number of dimensions. Each of these traversals has distinct properties that may be beneficial for certain applications. Some of the traversals are novel, some have been known in principle but had not been described adequately for any number of dimensions, some of the traversals have been known. This article is the first to present them all in a consistent notation system. Furthermore, with this article, tools are provided to enumerate points in a regular grid in the order in which they are visited by each traversal. In particular, we cover: five discontinuous traversals based on subdividing cubes into 2^d subcubes: Z-traversal (Morton indexing), U-traversal, Gray-code traversal, Double-Gray-code traversal, and Inside-out traversal; two discontinuous traversals based on subdividing simplices into 2^d subsimplices: the Hill-Z traversal and the Maehara-reflected traversal; five continuous traversals based on subdividing cubes into 2^d subcubes: the Base-camp Hilbert curve, the Harmonious Hilbert curve, the Alfa Hilbert curve, the Beta Hilbert curve, and the Butz-Hilbert curve; four continuous traversals based on subdividing cubes into 3^d subcubes: the Peano curve, the Coil curve, the Half-coil curve, and the Meurthe curve. All of these traversals are self-similar in the sense that the traversal in each of the subcubes or subsimplices of a cube or simplex, on any level of recursive subdivision, can be obtained by scaling, translating, rotating, reflecting and/or reversing the traversal of the complete unit cube or simplex.Comment: 28 pages, 12 figures. v2: fixed a confusing typo on page 12, line

The Critical Point of a Sigmoidal Curve: the Generalized Logistic Equation Example

Let $y(t)$ be a smooth sigmoidal curve, $y^{(n)}(t)$ be its $n$th derivative, $\{t_{m,i}\}$ and $\{t_{a,i}\}$, $i=1,2,\dots$ be the set of points where respectively the derivatives of odd and even order reach their extreme values. The "critical point of the sigmoidal curve" is defined to be the common limit of the sequences $\{t_{m,i}\}$ and $\{t_{a,i}\}$, provided that the limit exists. We prove that if $f(t)=\frac{dy}{dt}$ is an even function such that the magnitude of the analytic representation $|f_A(t)|=|f(t)+if_h(t)|$, where $f_h(t)$ is the Hilbert transform of $f(t)$, is monotone on $(0,\infty)$, then the point $t=0$ is the critical point in the sense above. For the general case, where $f(t)=\frac{dy}{dx}$ is not even, we prove that if $|f_A(t)|$ monotone on $(0,\infty)$ and if the phase of its Fourier transform $F(\omega)$ has a limit as $\omega\to \pm\infty$, then $t=0$ is still the critical point but as opposed to the previous case, the maximum of $f(t)$ is located away from $t=0$. We compute the Fourier transform of the generalized logistic growth functions and illustrate the notions above on these examples

A generalized Pancharatnam geometric phase formula for three level systems

We describe a generalisation of the well known Pancharatnam geometric phase formula for two level systems, to evolution of a three-level system along a geodesic triangle in state space. This is achieved by using a recently developed generalisation of the Poincare sphere method, to represent pure states of a three-level quantum system in a convenient geometrical manner. The construction depends on the properties of the group SU(3)\/ and its generators in the defining representation, and uses geometrical objects and operations in an eight dimensional real Euclidean space. Implications for an n-level system are also discussed.Comment: 12 pages, Revtex, one figure, epsf used for figure insertio

Modeling interest rate dynamics: an infinite-dimensional approach

We present a family of models for the term structure of interest rates which describe the interest rate curve as a stochastic process in a Hilbert space. We start by decomposing the deformations of the term structure into the variations of the short rate, the long rate and the fluctuations of the curve around its average shape. This fluctuation is then described as a solution of a stochastic evolution equation in an infinite dimensional space. In the case where deformations are local in maturity, this equation reduces to a stochastic PDE, of which we give the simplest example. We discuss the properties of the solutions and show that they capture in a parsimonious manner the essential features of yield curve dynamics: imperfect correlation between maturities, mean reversion of interest rates and the structure of principal components of term structure deformations. Finally, we discuss calibration issues and show that the model parameters have a natural interpretation in terms of empirically observed quantities.Comment: Keywords: interest rates, stochastic PDE, term structure models, stochastic processes in Hilbert space. Other related works may be retrieved on http://www.eleves.ens.fr:8080/home/cont/papers.htm