Message from the Editor-in-Chief

Chapter 15 - Algebraic Methods for Computer Aided Geometric Design

Contributors

Optimal Control for Systems Described by Difference Equations

Polyhedral and Algebraic Methods in Computational Geometry provides a thorough introduction into algorithmic geometry and its applications. It presents its primary topics from the viewpoints of discrete, convex and elementary algebraic geometry. The first part of the book studies classical problems and techniques that refer to polyhedral structures. The authors include a study on algorithms for computing convex hulls as well as the construction of Voronoi diagrams and Delone triangulations. The second part of the book develops the primary concepts of (non-linear) computational algebraic geometry. Here, the book looks at Grbner bases and solving systems of polynomial equations. The theory is illustrated by applications in computer graphics, curve reconstruction and robotics. Throughout the book, interconnections between computational geometry and other disciplines (such as algebraic geometry, optimization and numerical mathematics) are established. Polyhedral and Algebraic Methods in Computational Geometry is directed towards advanced undergraduates in mathematics and computer science, as well as towards engineering students who are interested in the applications of computational geometry.

Shape analysis plays an important role in many applications. In particular, in molecular biology, analyzing molecular shapes is essential to the fundamental problem of understanding how molecules interact. This project aims at developing efficient and effective algorithms to characterize and analyze molecular structures using geometric and topological methods. Two main components of this project are (1) developing novel molecular shape descriptors; and (2) identifying and representing meaningful features based on those descriptors. The project also produces accompanying (visualization) software. Results from this project (09/2006-10/2009) include the following publications. We have also set up web-servers for the software developed in this period, so that our new methods are accessible to a broader scientific community. The web sites are given below as well. In this final technical report, we first list publications and software resulted from this project. We then briefly explain the research conducted and main accomplishments during the period of this project.

Preface: Volume 81

<i>Journal of Ecology</i> news

Algebraic, Geometric, and Topological Methods in Optimization

Computational algebraic geometry has experienced significant advancements in recent years, driven by both theoretical breakthroughs and practical applications comprehensive review of the latest techniques in computational algebraic geometry, highlighting their development and impact on various domains. We begin by discussing foundational methods, including Gröbner bases, resultants, and elimination theory, which have been pivotal in solving polynomial systems and understanding algebraic varieties emerging algorithms that leverage improvements in computational efficiency, such as homotropy continuation methods and numerical algebraic geometry techniques. the applications of these advancements across different fields. In robotics and computer vision, for example, algebraic geometry methods are applied to solve problems related to motion planning and object recognition. In cryptography, recent developments in computational algebraic geometry offer new approaches to designing secure cryptographic systems resistant to quantum attacks. We also discuss the integration of algebraic geometry with machine learning and data science, emphasizing its role in improving predictive modeling and pattern recognition.

Updates for the Journal of Pediatric Nursing Readership

Editorial: Twenty‐Five Years Later…

The bandwidth of a n-vertex graph G is the smallest integer b such that there exists a bijective function f : V(G) → {1,...,n}, called a layout of G, such that for every edge uv ∈ E(G), |f(u) − f(v)| ≤ b. In the Bandwidth problem we are given as input a graph G and integer b, and asked whether the bandwidth of G is at most b. We present two results concerning the parameterized complexity of the Bandwidth problem on trees.First we show that an algorithm for Bandwidth with running time f(b)n o(b) would violate the Exponential Time Hypothesis, even if the input graphs are restricted to be trees of pathwidth at most two. Our lower bound shows that the classical 2O(b) n b + 1 time algorithm by Saxe [SIAM Journal on Algebraic and Discrete Methods, 1980] is essentially optimal.Our second result is a polynomial time algorithm that given a tree T and integer b, either correctly concludes that the bandwidth of T is more than b or finds a layout of T of bandwidth at most b O(b). This is the first parameterized approximation algorithm for the bandwidth of trees.

We study the classical Bandwidth problem from the viewpoint of parameterized algorithms. In the Bandwidth problem we are given a graph G = (V,E) together with a positive integer k, and asked whether there is an bijective function β: {1, ..., n} →V such that for every edge uv ∈ E, |β − 1(u) − β − 1(v)| ≤ k. The problem is notoriously hard, and it is known to be NP-complete even on very restricted subclasses of trees. The best known algorithm for Bandwidth for small values of k is the celebrated algorithm by Saxe [SIAM Journal on Algebraic and Discrete Methods, 1980 ], which runs in time \(2^{{\mathcal{O}}(k)}n^{k+1}\). In a seminal paper, Bodlaender, Fellows and Hallet [STOC 1994 ] ruled out the existence of an algorithm with running time of the form \(f(k)n^{{\mathcal{O}}(1)}\) for any function f even for trees, unless the entire W-hierarchy collapses.We initiate the search for classes of graphs where Bandwidth is fixed parameter tractable (FPT), that is, solvable in time \(f(k)n^{{\mathcal{O}}(1)}\) for some function f. In this paper we present an algorithm with running time \(2^{{\mathcal O}(k \log k)} n^2\) for Bandwidth on AT-free graphs, a well-studied graph class that contains interval, permutation, and cocomparability graphs. Our result is the first non-trivial FPT algorithm for Bandwidth on a graph class where the problem remains NP-complete.KeywordsInterval GraphGraph ClassHair LengthPermutation GraphVertex PairThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This year sees an important landmark reached for Strain and the British Society for Strain Measurement, namely 40 years of continuous support for industrial and academic members. The importance of the interchange of research and development information has never been of greater importance, with the rapidly advancing technologies threatening to leave the uninformed behind. The society encourages cooperation at all levels and between widely different branches of engineering. Strain now has an editorial board with members from 12 countries, showing the wide appeal and influence of the journal. Much has been written recently about the developments of the publication of Strain. We now have a very high-quality journal that is read worldwide. This issue of Strain contains two full papers, two technical notes and a tutorial. Two of the papers deal with sensors in a generic fashion. The first by Sodano et al., suggests a novel means of energy harvesting using the piezoelectric effect. At present, the idea of using wireless technology in conjunction with sensors to eliminate problems brought about by ‘wiring’ is gaining much attention. The advantages of such devices are most greatly exploited in a structural health monitoring or machine condition monitoring context. Applications such as embedded sensors in composite structures, inaccessible locations such as in large civil and offshore structures and in particular large rotating machinery such as wind turbines come to mind as suitable candidates for wireless sensors. The wireless sensors can be remotely interrogated with minimal effect on the structure. The only difficulty is powering the sensor. Batteries are bulky, have a finite lifespan and also require wiring; therefore it is desirable to define a means of ‘self-powering’. The paper defines a means of obtaining power by the conversion of mechanical energy from the motion of the structure to electrical energy to power the sensor. A model of the energy that can be produced was developed in the paper and validated using a non-homogeneous cantilever beam. The experimental study showed that the power-harvesting exercise produced damping in the beam where the energy was stored rather than dissipated into the structure. The second paper by Overton and Worden deals with sensor placement to optimise fault detection in a structure. The paper uses an ant colony metaphor to determine optimum sensor placement for the location and classification of impacts on a composite panel using a neural network. The ant colony metaphor is based on the idea of ants foraging for food using the shortest path by tracking a pheromone trail deposited by other ants. The composite panel used in the work simulated an aircraft skin and was instrumented with 17 piezoceramic strain sensors, which were used initially to train the neural network. The ant colony technique was used to optimise the positioning of three sensors and then six sensors. In both cases, the ant colony approach outperformed a genetic algorithm. The work in this paper will be useful in reducing the amount of computational effort used in structural health monitoring, as optimisation reduces the number of sensors necessary for accurate fault location and hence reduces costs associated with online monitoring. The technical note by Guntaka et al. deals with a new recording medium in the full-field analysis area of holographic interferometry. The new material is self-developing and has the potential to facilitate real-time holographic inspections. The technique was validated on a thick PVC pipe subjected to internal pressure. The technical note by Noroozi et al. deals with the use of neural networks in conjunction with photoelastic data. FEA is used to train the neural network and then the technique is applied in an inverse fashion to derive the relative retardation obtained from photoelastic data and hence the value of the applied load. This work provides an important first step in combining FEA and photoelasticity to produce a hybrid technique for stress analysis. The tutorial by Truman will be of value to all stress analysts and deals with the application of tensor elasticity to isotropic bodies. Two well-known problems in plane elasticity are cited as examples and the reader is shown how the solutions for these problems are derived. The tutorial follows on from the first in the series presented in the November 2003 issue of Strain. To conclude, the journal Strain and the Society have offered much information and support through the hard work of many dedicated members and thanks goes to all that have contributed so much over the last 40 years. Some have been active throughout virtually all of that time, which is a tremendous show of dedication and enthusiasm for their chosen field.

Abstract. Geometric and topological methods are applied to significance testing in the wavelet domain. A geometric test was developed for assigning significance to pointwise significance patches in local wavelet spectra, i.e., contiguous regions of significant wavelet power coefficients with respect to some noise model. This geometric significance test was found to produce results similar to an existing areawise significance test while being more computationally flexible and efficient. The geometric significance test can be readily applied to pointwise significance patches at various pointwise significance levels in wavelet power and coherence spectra. The geometric test determined that features in wavelet power of the North Atlantic Oscillation (NAO) are indistinguishable from a red-noise background, suggesting that the NAO is a stochastic, unpredictable process, which could render difficult the future projections of the NAO under a changing global system. The geometric test did, however, identify features in the wavelet power spectrum of an El Niño index (Niño 3.4) as distinguishable from a red-noise background. A topological analysis of pointwise significance patches determined that holes, deficits in pointwise significance embedded in significance patches, are capable of identifying important structures, some of which are undetected by the geometric and areawise tests. The application of the topological methods to ideal time series and to the time series of the Niño 3.4 and NAO indices showed that the areawise and geometric tests perform similarly in ideal and geophysical settings, while the topological methods showed that the Niño 3.4 time series contains numerous phase-coherent oscillations that could be interacting nonlinearly.