Geometry of Linear Subspace Arrangements with Connections to Matroid Theory

This dissertation is devoted to the study of the geometric properties of subspace configurations, with an emphasis on configurations of points. One distinguishing feature is the widespread use of techniques from Matroid Theory and Combinatorial Optimization. In part we generalize a theorem of Edmond\u27s about partitions of matroids in independent subsets. We then apply this to establish a conjectured bound on the Castelnuovo-Mumford regularity of a set of fat points. We then study how the dimension of an ideal of point changes when intersected with a generic fat subspace. In particular we introduce the concept of a ``very unexpected hypersurface\u27\u27 passing through a fixed set of points Z. We show in certain cases these can be characterized via combinatorial data and geometric data from the Hyperplane Arrangement dual to Z. This generalizes earlier results on unexpected curves in the plane due to Faenzi, Valles, Cook, Harbourne, Migliore and Nagel

Structure Analysis of Some Generalizations of Matchings and Matroids under Algorithmic Aspects of Matchings and Matroids Under Algorithmic Aspects

Combinatorial optimization problems whose underlying structures are matchings or matroids are well-known to be solvable with efficient algorithms. Matroids can even be characterized by a simple greedy algorithm. In the first part of this thesis, some generalizations of matroids which allow the ground set to be partially ordered are considered. In particular, it will be shown that a special type of lattice polyhedra, for which Dietrich and Hoffman recently established a dual greedy algorithm, can be reduced to ordinary polymatroids. Moreover, strong exchange structures, Gauss greedoids and Delta-matroids will be extended from Boolean lattices to general distributive lattices, and the resulting structures will be characterized by certain greedy-type algorithms. While a matching of maximal size can be determined by a polynomial algorithm, the dual problem of finding a vertex cover of minimal size in general graphs is one of the hardest problems in combinatorial optimization. However, in case the graph belongs to the class of K\"onig-Egerv\'ary graphs, a maximum matching can be used to construct a minimum vertex cover. Lovasz and Korach characterized König-Egervary graphs by the exclusion of forbidden subgraphs. In the second part of this dissertation, the structure of König-Egervary graphs and the more general Red/Blue-split graphs will be analyzed. Red/Blue-split graphs have red and blue colored edges and the vertices of which can be split into two stable sets with respect to the red and blue edges, respectively. An algorithm that either determines a feasible partition of the vertices, or returns a red-blue colored subgraph (called ``flower'') characterizing non-Red/Blue-split graphs will be presented. This characterization allows the deduction of Lovasz and Korach's characterizations of König-Egerv\'ary graphs in case the red edges of the flower form a maximum matching. Furthermore, weighted Red/Blue-split graphs which model integrally solvable simple systems are introduced. A simple system is an inequality system where the sum of absolute values in each row of the integral matrix does not exceed the value two. A shortest-path algorithm and the presented Red/Blue-split algorithm will be used to find an integral solution of a simple system. These two algorithms lead to a characterization of weighted Red/Blue-split graphs by forbidden weighted subgraphs

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Real Algebraic Geometry With a View Toward Moment Problems and Optimization

Continuing the tradition initiated in MFO workshop held in 2014, the aim of this workshop was to foster the interaction between real algebraic geometry, operator theory, optimization, and algorithms for systems control. A particular emphasis was given to moment problems through an interesting dialogue between researchers working on these problems in finite and infinite dimensional settings, from which emerged new challenges and interdisciplinary applications

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

The use of artificial neural networks in classifying lung scintigrams

An introduction to nuclear medical imaging and artificial neural networks (ANNs) is first given. Lung scintigrams are classified using ANNs in this study. Initial experiments using raw data are first reported. These networks did not produce suitable outputs, and a data compression method was next employed to present an orthogonal data input set containing the largest amount of information possible. This gave some encouraging results, but was neither sensitive nor accurate enough for clinical use. A set of experiments was performed to give local information on small windows of scintigram images. By this method areas of abnormality could be sent into a subsequent classification network to diagnose the cause of the defect. This automatic method of detecting potential defects did not work, though the networks explored were found to act as smoothing filters and edge detectors. Network design was investigated using genetic algorithms (GAs). The networks evolved had low connectivity but reduced error and faster convergence than fully connected networks. Subsequent simulations showed that randomly partially connected networks performed as well as GA designed ones. Dynamic parameter tuning was explored in an attempt to produce faster convergence, but the previous good results of other workers could not be replicated. Classification of scintigrams using manually delineated regions of interest was explored as inputs to ANNs, both in raw state and as principal components (PCs). Neither representation was shown to be effective on test data

Fault propagation, detection and analysis in process systems

Process systems are often complicated and liable to experience faults and their effects. Faults can adversely affect the safety of the plant, its environmental impact and economic operation. As such, fault diagnosis in process systems is an active area of research and development in both academia and industry. The work reported in this thesis contributes to fault diagnosis by exploring the modelling and analysis of fault propagation and detection in process systems. This is done by posing and answering three research questions. What are the necessary ingredients of a fault diagnosis model? What information should a fault diagnosis model yield? Finally, what types of model are appropriate to fault diagnosis? To answer these questions , the assumption of the research is that the behaviour of a process system arises from the causal structure of the process system. On this basis, the research presented in this thesis develops a two-level approach to fault diagnosis based on detailed process information, and modelling and analysis techniques for representing causality. In the first instance, a qualitative approach is developed called a level 1 fusion. The level 1 fusion models the detailed causality of the system using digraphs. The level 1 fusion is a causal map of the process. Such causal maps can be searched to discover and analyse fault propagation paths through the process. By directly building on the level 1 fusion, a quantitative level 2 fusion is developed which uses a type of digraph called a Bayesian network. By associating process variables with fault variables, and using conditional probability theory, it is shown how measured effects can be used to calculate and rank the probability of candidate causes. The novel contributions are the development of a systematic approach to fault diagnosis based on modelling the chemistry, physics, and architecture of the process. It is also shown how the control and instrumentation system constrains the casualty of the process. By demonstrating how digraph models can be reversed, it is shown how both cause-to-effect and effect-to-cause analysis can be carried out. In answering the three research questions, this research shows that it is feasible to gain detailed insights into fault propagation by qualitatively modelling the physical causality of the process system. It is also shown that a qualitative fault diagnosis model can be used as the basis for a quantitative fault diagnosis modelOpen Acces