Oka manifolds: From Oka to Stein and back

Oka theory has its roots in the classical Oka-Grauert principle whose main result is Grauert's classification of principal holomorphic fiber bundles over Stein spaces. Modern Oka theory concerns holomorphic maps from Stein manifolds and Stein spaces to Oka manifolds. It has emerged as a subfield of complex geometry in its own right since the appearance of a seminal paper of M. Gromov in 1989. In this expository paper we discuss Oka manifolds and Oka maps. We describe equivalent characterizations of Oka manifolds, the functorial properties of this class, and geometric sufficient conditions for being Oka, the most important of which is Gromov's ellipticity. We survey the current status of the theory in terms of known examples of Oka manifolds, mention open problems and outline the proofs of the main results. In the appendix by F. Larusson it is explained how Oka manifolds and Oka maps, along with Stein manifolds, fit into an abstract homotopy-theoretic framework. The article is an expanded version of the lectures given by the author at the Winter School KAWA-4 in Toulouse, France, in January 2013. A more comprehensive exposition of Oka theory is available in the monograph F. Forstneric, Stein Manifolds and Holomorphic Mappings (The Homotopy Principle in Complex Analysis), Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 56, Springer-Verlag, Berlin-Heidelberg (2011).Comment: With an appendix by Finnur Larusson. To appear in Ann. Fac. Sci. Toulouse Math. (6), vol. 22, no. 4. This version is identical with the published tex

The Power Domination Toolbox

Phasor Measurement Units (PMUs) are placed at strategic nodes in an electrical power network to directly monitor nearby transmission lines and monitor further parts of the network through conservation of energy laws.Efficient placement of PMUs is modeled by the graph theoretic process called Power Domination (PD). This paper describes a Power Domination Toolbox (PDT) that efficiently identifies potential PMU locations. The PDT leverages the graph theoretic literature to reduce the complexity of determining optimal PMU placements by: reducing the size of the network (contraction), identification of preferred nodes, elimination of redundant nodes, assignment of a qualitative score to the remaining nodes, and parallel processing techniques. After pre-processing steps to reduce network size, current state-of-the-art PD techniques based on the minimum rank sage library (MRZG) are used to analyze the network. The PDT is an extension of MRZG in Python and maintains the compatibility of MRZG with SageMath. The PDT can identify minimum PMU placements for networks with hundreds of nodes on personal computers and can analyze larger networks on high performance computers. The PDT affords users the ability to investigate power domination on networks previously considered infeasible due to the number of nodes resulting in a prohibitively long run-time.Comment: 12 pages, 9 figure

Automated Image Analysis of Offshore Infrastructure Marine Biofouling

Supplementary Materials: The following are available online at www.mdpi.com/2077-1312/6/1/2/s1 Acknowledgments: This project was funded by the Natural Environmental Research Council (NERC) project No.: NE/N019865/1. The authors would like to thank Melanie Netherway and Don Orr, from our project partner (company requested to remain anonymous) for the provision of survey footage and for supporting the project. In addition, many thanks to Oscar Beijbom, University California Berkley for providing guidance and support to the project. Additional thanks to Calum Reay, Bibby Offshore; George Gair, Subsea 7; and Alan Buchan, Wood Group Kenny for help with footage collection and for allowing us to host workshops with them and their teams, their feedback and insights were very much appreciated.Peer reviewedPublisher PD

Department of Applied Mathematics Academic Program Review, Self Study / June 2010

The Department of Applied Mathematics has a multi-faceted mission to provide an exceptional mathematical education focused on the unique needs of NPS students, to conduct relevant research, and to provide service to the broader community. A strong and vibrant Department of Applied Mathematics is essential to the university's goal of becoming a premiere research university. Because research in mathematics often impacts science and engineering in surprising ways, the department encourages mathematical explorations in a broad range of areas in applied mathematics with specific thrust areas that support the mission of the school

Stratification and domination in graphs.

Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2006.In a recent manuscript (Stratification and domination in graphs. Discrete Math. 272 (2003), 171-185) a new mathematical framework for studying domination is presented. It is shown that the domination number and many domination related parameters can be interpreted as restricted 2-stratifications or 2-colorings. This framework places the domination number in a new perspective and suggests many other parameters of a graph which are related in some way to the domination number. In this thesis, we continue this study of domination and stratification in graphs. Let F be a 2-stratified graph with one fixed blue vertex v specified. We say that F is rooted at the blue vertex v. An F-coloring of a graph G is a red-blue coloring of the vertices of G such that every blue vertex v of G belongs to a copy of F (not necessarily induced in G) rooted at v. The F-domination number yF(GQ of G is the minimum number of red vertices of G in an F-coloring of G. Chapter 1 is an introduction to the chapters that follow. In Chapter 2, we investigate the X-domination number of prisms when X is a 2-stratified 4-cycle rooted at a blue vertex where a prism is the cartesian product Cn x K2, n > 3, of a cycle Cn and a K2. In Chapter 3 we investigate the F-domination number when (i) F is a 2-stratified path P3 on three vertices rooted at a blue vertex which is an end-vertex of the F3 and is adjacent to a blue vertex and with the remaining vertex colored red. In particular, we show that for a tree of diameter at least three this parameter is at most two-thirds its order and we characterize the trees attaining this bound. (ii) We also investigate the F-domination number when F is a 2-stratified K3 rooted at a blue vertex and with exactly one red vertex. We show that if G is a connected graph of order n in which every edge is in a triangle, then for n sufficiently large this parameter is at most (n — /n)/2 and this bound is sharp. In Chapter 4, we further investigate the F-domination number when F is a 2- stratified path P3 on three vertices rooted at a blue vertex which is an end-vertex of the P3 and is adjacent to a blue vertex with the remaining vertex colored red. We show that for a connected graph of order n with minimum degree at least two this parameter is bounded above by (n —1)/2 with the exception of five graphs (one each of orders four, five and six and two of order eight). For n > 9, we characterize those graphs that achieve the upper bound of (n — l)/2. In Chapter 5, we define an f-coloring of a graph to be a red-blue coloring of the vertices such that every blue vertex is adjacent to a blue vertex and to a red vertex, with the red vertex itself adjacent to some other red vertex. The f-domination number yz{G) of a graph G is the minimum number of red vertices of G in an f-coloring of G. Let G be a connected graph of order n > 4 with minimum degree at least 2. We prove that (i) if G has maximum degree A where A 4 with maximum degree A where A 5 with maximum degree A where