A new product construction for partial difference sets

Relatively few constructions are known of negative Latin square type Partial Difference Sets (PDSs), and most of the known constructions are in elementary abelian groups. We present a product construction that produces negative Latin square type PDSs, and we apply this product construction to generate examples in p-groups of exponent bigger than p.     

Characteristics of Good Meta-Heuristic Algorithms for the Frequency Assignment Problem

Most writers on frequency assignment algorithms have described the details of a single algorithm, and evaluated the algorithm on selected data sets. There has been relatively little emphasis on describing the common features that are important if an algorithm is to have good performance. This paper describes the key features, with particular emphasis on algorithms for weighted fixed spectrum problems. The use of algorithms handling weighted constraints has become increasingly common in recent years. The advantages and disadvantages of weighting constraints are demonstrated.     

Implementing pure adaptive search for global optimization using Markov chain sampling

The Pure Adaptive Search (PAS) algorithm for global optimization yields a sequence of points, each of which is uniformly distributed in the level set corresponding to its predecessor. This algorithm has the highly desirable property of solving a large class of global optimization problems using a number of iterations that increases at most linearly in the dimension of the problem. Unfortunately, PAS has remained of mostly theoretical interest due to the difficulty of generating, in each iteration, a point uniformly distributed in the improving feasible region. In this article, we derive a coupling equivalence between generating an approximately uniformly distributed point using Markov chain sampling, and generating an exactly uniformly distributed point with a certain probability. This result is used to characterize the complexity of a PAS-implementation as a function of (a) the number of iterations required by PAS to achieve a certain solution quality guarantee, and (b) the complexity of the sampling algorithm used. As an application, we use this equivalence to show that PAS, using the so-called Random ball walk Markov chain sampling method for generating nearly uniform points in a convex region, can be used to solve most convex programming problems in polynomial time.     

A tits alternative for groups that are residually of bounded rank

In this short note a Tits alternative for certain kinds of groups which are residually of rank at mostr is obtained. The main theorem states that ifG is a group that is residually (locally (soluble-by-finite) of rankr), then eitherG is locally (soluble-by-finite) orG contains a non-abelian free subgroup. The third author would like to thank the School of Mathematics at the University of Wales, Cardiff for its hospitality whild part of this work was being done.     

Determining angular frequency from images of rotating objects via a generalized fast Fourier transform

In certain real-world applications, one needs to estimate the angular frequency of a spinning object. We consider the image processing problem of estimating this rate of rotation from a video of the object taken by a camera aligned with the axis of rotation. For many types of spinning objects, this problem can be addressed with existing techniques: simply register two consecutive video frames. We focus, however, on objects whose shape and intensity changes greatly from frame to frame, such as spinning plumes of plasma that emerge from a certain type of spacecraft thruster. To estimate the angular frequency of such objects, we introduce the Geometric Sum Transform (GST), a new rotation-based generalization of the discrete Fourier transform (DFT). Taking the GST of a given video produces a new sequence of images, the most coherent of which corresponds to the object’s true rate of rotation. After formally demonstrating this fact, we provide a fast algorithm for computing the GST which generalizes the decimation-in-frequency approach for performing a Fast Fourier Transform (FFT). We further show that computing a GST is, in fact, mathematically equivalent to computing a system of DFTs, provided one can decompose each video frame in terms of an eigenbasis of a rotation operator. We conclude with numerical experimentation.