Given graphs G,H, is it possible to find a matching which, when deleted from G, destroys all copies of H? The answer is obvious for some inputs—notably, when G is a large complete graph the answer is “no”—but in general this can be a very difficult question. In this thesis, we study this decision problem when H is a fixed tree or cycle; our aim is to identify those H for which it can be solved efficiently. The H-transverse matching problem, TM(H) for short, asks whether an input graph admits a matching M such that no subgraph of G − M is isomorphic to H. The main goal of this thesis is the following dichotomy. When H is a triangle or one of a few small-diameter trees, there is a polynomial-time algorithm to find an H-transverse matching if one exists. However, TM(H) is NP-complete when H is any longer cycle or a tree of diameter ≥ 4. In addition, we study the restriction of these problems to structured graph classes. / Graduate

The final Chapter concerns a problem of partitioning graphs subject to certain restrictions. We prove that several subproblems are NP-complete.

High-Frequency Surface Wave Radar (HFSWR) is a radar technology that offers numerous advantages for surveillance of coastal waters beyond the exclusive economic zone. However, target detection and tracking is primarily limited by ionospheric interference. Ionospheric clutter is characterized by a high degree of nonhomogeneity and nonstationarity, which makes its suppression difficult using conventional processing techniques. Space-time adaptive processing techniques have enjoyed great success in airborne radar, but have not yet been investigated in the context of HFSWR. This thesis is primarily concerned with the evaluation of existing STAP techniques in the HFSWR scenario and the development of a new multistage adaptive processing approach, dubbed the Fast Fully Adaptive (FFA) scheme, which was developed with the particular constraints of the HFSWR interference environment in mind. Three different spatio-temporal partitioning schemes are introduced and a thorough investigation of the performance of the FFA is conducted.

High-Frequency Surface Wave Radar (HFSWR) is a radar technology that offers numerous advantages for surveillance of coastal waters beyond the exclusive economic zone. However, target detection and tracking is primarily limited by ionospheric interference. Ionospheric clutter is characterized by a high degree of nonhomogeneity and nonstationarity, which makes its suppression difficult using conventional processing techniques. Space-time adaptive processing techniques have enjoyed great success in airborne radar, but have not yet been investigated in the context of HFSWR. This thesis is primarily concerned with the evaluation of existing STAP techniques in the HFSWR scenario and the development of a new multistage adaptive processing approach, dubbed the Fast Fully Adaptive (FFA) scheme, which was developed with the particular constraints of the HFSWR interference environment in mind. Three different spatio-temporal partitioning schemes are introduced and a thorough investigation of the performance of the FFA is conducted.

We study the geometry of immanants, which are polynomials on n^2 variables that are defined by irreducible representations of the symmetric group Sn. We compute stabilizers of immanants in most cases by computing Lie algebras of stabilizers of immanants. We also study tensor network states, which are special tensors defined by contractions. We answer a question about tensor network states asked by Grasedyck. Both immanants and tensor network states are related to the Geometric Complexity Theory program, in which one attempts to use representation theory and algebraic geometry to solve an algebraic analogue of the P versus N P problem. We introduce the Geometric Complexity Theory (GCT) program in Section one and we introduce the background for the study of immanants and tensor network states. We also explain the relation between the study of immanants and tensor network states and the GCT program. Mathematical preliminaries for this dissertation are in Section two, including multilinear algebra, representation theory, and complex algebraic geometry. In Section three, we first give a description of immanants as trivial (SL(E) x SL(F )) ><| delta(Sn)-modules contained in the space S^n(E X F ) of polynomials of degree n on the vector space E X F , where E and F are n dimensional complex vectorspaces equipped with fixed bases and the action of Sn on E (resp. F ) is induced by permuting elements in the basis of E (resp. F ). Then we prove that the stabilizer of an immanant for any non-symmetric partition is T (GL(E) x GL(F )) ><| delta(Sn) ><| Z2, where T (GL(E) x GL(F )) is the group of pairs of n x n diagonal matrices with the product of determinants equal to 1, delta(Sn) is the diagonal subgroup of Sn x Sn. We also prove that the identity component of the stabilizer of any immanant is T (GL(E) x GL(F )). In Section four, we prove that the set of tensor network states associated to a triangle is not Zariski closed and we give two reductions of tensor network states from complicated cases to simple cases. In Section five, we calculate the dimension of the tangent space and weight zero subspace of the second osculating space of GL_(n^2) .[perm_n] at the point [perm_n] and determine the Sn x Sn-module structure of this space. We also determine some lines on the hyper-surface determined by the permanent polynomial. In Section six, we give a summary of this dissertation.

Biodiversity is a concept that plays a key role in both scientific theories such as the species-area law and conservation politics. Currently, however, little agreement exists on how biodiversity should be defined, let alone measured. This has led to suggestions that biodiversity is not a metaphysically robust concept, with major implications for its usefulness in formulating scientific theories and making conservation decisions. A general discussion of biodiversity is presented, highlighting its application both in scientific and conservation contexts, its relationship with environmental ethics, and existing approaches to its measurement. To overcome the limitations of existing biodiversity concepts, a new concept of biocomplexity is proposed. This concept equates the biodiversity of any biological system with its effective complexity. Biocomplexity is shown to be the only feasible measure of biodiversity that captures the essential features desired of a general biodiversity concept. In particular, it is a well-defined, measurable and strongly intrinsic property of any biological system. Finally, the practical application of biocomplexity is discussed.

Thesis (Ph. D.)--University of Texas at El Paso, 2007. / Title from title screen. Vita. CD-ROM. Includes bibliographical references. Also available online.

Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 78-80).

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

Thesis (Ph. D.)--Carleton University, 2003. / Includes bibliographical references (p. 226-237). Also available in electronic format on the Internet.