Arrangements on Parametric Surfaces I: General Framework and Infrastructure

We introduce a framework for the construction, maintenance, and manipulation of arrangements of curves embedded on certain two-dimensional orientable parametric surfaces in three-dimensional space. The framework applies to planes, cylinders, spheres, tori, and surfaces homeomorphic to them. We reduce the effort needed to generalize existing algorithms, such as the sweep line and zone traversal algorithms, originally designed for arrangements of bounded curves in the plane, by extensive reuse of code. We have realized our approach as the Cgal package Arrangement_on_surface_2. We define a compact and modular interface for our framework; for a given application a required small subset of the interface can be identified. Then, only this subset must be implemented. A companion paper describes concretizations for several types of surfaces and curves embedded on them, and applications. This is the first implementation of a generic algorithm that can handle arrangements on a large class of parametric surfaces.     

Improved algorithms for finding low-weight polynomial multiples in \mathbb {F}_{2}^{}[x] and some cryptographic applications

In this paper we present an improved algorithm for finding low-weight multiples of polynomials over the binary field using coding theoretic methods. The associated code defined by the given polynomial has a cyclic structure, allowing an algorithm to search for shifts of the sought minimum-weight codeword. Therefore, a code with higher dimension is constructed, having a larger number of low-weight codewords and through some additional processing also reduced minimum distance. Applying an algorithm for finding low-weight codewords in the constructed code yields a lower complexity for finding low-weight polynomial multiples compared to previous approaches. As an application, we show a key-recovery attack against  that has a lower complexity than the chosen security level indicate. Using similar ideas we also present a new probabilistic algorithm for finding a multiple of weight 4, which is faster than previous approaches. For example, this is relevant in correlation attacks on stream ciphers.     

A Group-Theoretic Approach to Parameter Identifiability of PDE Systems

The contribution is devoted to the parameter identifiability problem of (nonlinear) PDE systems. Especially, we discuss the (local) identifiability of parameters along a trajectory. The analysis relies on a coordinate-free formulation for systems, including boundary conditions, and we motivate an approach by (Lie) transformation groups, whose success for PDE systems depends on a consequent extent to the accompanying boundary conditions. The (non-)identifiability of parameters is related to the (non-)existence of group generators, wherewith (local) conditions can be derived. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Riccati equations and Lie series

Lie series and a special matrix notation for first-order differential operators are used to show that the Lie group properties of matrix Riccati equations arise in a natural way. The Lie series notation makes it evident that the solutions of a matrix Riccati equation are curves in a group of nonlinear transformations that is a generalization of the linear fractional transformations familiar from the classical complex analysis. It is easy to obtain a linear representation of the Lie algebra of the nonlinear group of transformations and then this linearization leads directly to the standard linearization of the matrix Riccati equations. We note that the matrix Riccati equations considered here are of the general rectangular type.     

From Gumbel to Tracy-Widom

The Tracy-Widom distribution that has been much studied in recent years can be thought of as an extreme value distribution. We discuss interpolation between the classical extreme value distribution exp( ? exp( ? x)), the Gumbel distribution, and the Tracy-Widom distribution. There is a family of determinantal processes whose edge behaviour interpolates between a Poisson process with density exp( ? x) and the Airy kernel point process. This process can be obtained as a scaling limit of a grand canonical version of a random matrix model introduced by Moshe, Neuberger and Shapiro. We also consider the deformed GUE ensemble, $M=M_0+\sqrt{2S} V$ , with M ₀ diagonal with independent elements and V from GUE. Here we do not see a transition from Tracy-Widom to Gumbel, but rather a transition from Tracy-Widom to Gaussian.     

Control and stability analysis of a turbocharged Diesel engine using singular perturbation methods

The control of the air path of a diesel engine with a variable geometry turbocharger and exhaust gas recirculation can be improved significantly by nonlinear methods. The control objective is to supply an amount of fresh air by the compressor and to achieve a desired pressure in the intake manifold in the neighborhood of an operating point of the engine. The mathematical model of the air path is split into a slow and a fast part. For the proposed controller, asymptotic stability of the closed loop system in the sense of Lyapunov is guaranteed by the use of singular perturbation methods. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Supercritical Spatially Homogeneous Branching in Admits no Equilibria

At the beginning of investigations in spatially homogeneous branching processes in Euclidean space (Liemant [1]) it seemed to be obvious that the existence of equilibria implies criticality of branching. This prejudice was disproved by the example [2] of a subcritical homogeneous branching equilibrium in dimension one. We prove that supercritical homogeneous branching processes in Euclidean space and, more general, in a broad class of topological groups have no (non – void, homogeneous) equilibria.     

Investigations of shear free turbulent diffusion in a rotating frame

We reconsider the problem of shear free turbulent diffusion in a rotating frame, rotating about x₁. Shear free turbulence is generated at a vibrating grid in the x₂–x₃ plane and diffuses away from the grid in x₁ direction. An important property of this flow case is that there is no mean flow‐velocity. With the help of Lie‐group methods Reynolds‐stress transport models can be analyzed for this kind of flow in a rotating frame. From the analysis it can be found, that the turbulent diffusion only influences a finite domain. Implicating this solution in the model equations shows that even fully nonlinear Reynolds‐stress transport models (non‐linear in the Reynolds‐stresses for the pressure‐strain model) are insensitive to rotation for this type of flow. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A sequential perspective on searching for static targets

We present a sequential approach to detect static targets with imperfect sensors, which range from tower-mounted cameras to satellites. The scenario is operationally relevant to many military, homeland security, search and rescue, environmental engineering, counter-narcotics, and law enforcement applications. The idea is to stop the search as soon as there is enough probabilistic evidence about the targets’ locations, given an operator-prescribed error tolerance, knowledge of the sensors’ parameters, and a sequence of detection signals from the sensors. By stopping the search as soon as possible, we promote efficiency by freeing up sensors and operators to perform other tasks. The model we develop has the added benefits of decreasing operator workload and providing negative information as a search progresses.