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A polyhedral graph G is called PCC if every vertex of G has strictly positive combinatorial curvature and the graph is not a prism or antiprism. In this thesis it is shown that the maximum order of a 3-regular PCC graph is 132 and the 3-regular PCC graphs which match that bound are enumerated. A new PCC graph with two 39-faces and 208 vertices is constructed, matching the number of vertices of the largest PCC graphs discovered by Nicholson and Sneddon. A conjecture that there are no PCC graphs with faces of size larger than 39 is made, along with a proof that if there are no faces of size larger than 122, then there is an upper bound of 244 on the order of PCC graphs. / Graduate

Counting lattice paths where the number of occurrences of a given pattern is monitored requires a careful analysis of the pattern. Not the length, but the characteristics of the pattern are responsible for the difficulties in finding explicit solutions. Certain features, like overlap and difference in number of ! and " steps determine the recursion formula. In the case of ballot paths, that is paths the stay weakly above the line y = x, the solutions to the recursions are typically polynomial sequences. The objects of Finite Operator Calculus are polynomial sequences, thus the theory can be used to solve the recursions. The theory of Finite Operator Calculus is strengthened and extended to the multivariate setting in order to obtain solutions, and to prepare for future applications. / by Shaun Sullivan. / Thesis (Ph.D.)--Florida Atlantic University, 2011. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2011. Mode of access: World Wide Web.

Thesis(M.S.)--Auburn University, 2005. / Abstract. Vita. Includes bibliographic references.

Testing is a crucial and necessary part of software development used to detect faults and make sure that the software works properly. One testing method proposed in research is named combinatorial testing, where a test suite covers all combinations of parameter values in a certain way. There is some evidence that indicated that most faults are triggered by a few number of parameter value combinations. There have been various tools proposed for using combinatorial testing in practice. ACTS is one of the most popular combinatorial research tools. Even if ACTS has been used in several industrial projects, few studies have shown how such a combinatorial testing tool can be used for modelling and test generation for industrial control software. In this study we apply modelling and test generation using ACTS on a Programmable Logic Control (PLC) software. This kind of software is used in the safety critical domain for controlling different computer devices, such as a train control management system, which is in charge of many of the critical and safety related procedures on a train. Testing this kind of software is very important because failures can contribute to the loss of lives and money.  We show how ACTS can be applied to PLC software. We evaluated ACTS in terms of applicability (i.e., how can ACTS can be applied directly on modelling the PLCs) and efficiency in terms of generation time and test suite size. We used 17 PLC programs provided by Bombardier Transportation. Based on the number of inputs in a program we divided the programs into three complexity categories: small, medium and large programs. An input space model was created for each program, where the needed information was obtained from both the programs and the engineers writing the programs. Each model was created as a system in the graphical interface of ACTS. The different algorithms and combinatorial techniques supported by ACTS were used to generate test suites by varying coverage of the parameter combinations (i.e. t-way) in the command line of ACTS. In this study we used a cut-off time of 1 hour for the test generation, as this is a realistic estimation based on discussion with industrial engineers. Our results showed that not all combinations of algorithms and combinatorial strengths could generate a test suite within the cut-off time. We argue that the results of the the modelling process and the results showing the efficiency of the test generation tool can be useful for practitioners considering to use combinatorial testing for PLC software.

We give several characterizations of partition lattices and projective geometries. Most of these characterizations use characteristic polynomials. A geometry is non—splitting if it cannot be expressed as the union of two of its proper flats. A geometry G is upper homogeneous if for all k, k = 1, 2, ... , r(G), and for every pair x, y of flats of rank k, the contraction G/x is isomorphic to the contraction G/y. Given a signed graph, we define a corresponding signed—graphic geometry. We give a characterization of supersolvable signed graphs. Finally, we give the following characterization of non—splitting supersolvable signed-graphic geometries : If a non-splitting supersolvable ternary geometry does not contain the Reid geometry as a subgeometry, then it is signed—graphic.

Combinatorial libraries are used in the search for ligands that bind to target proteins. Fmoc solid-phase peptide synthesis is routinely used to generate such libraries. Microwave-assisted peptide synthesis was employed here to decrease reaction times by 80-90%. Two One-Bead-One-Compound combinatorial libraries were synthesized on 130μm beads (one containing 750 members and the other 16, 807). The use of smaller solid supports would have many important practical advantages including; increased library diversity per unit mass, smaller quantities of library needed to generate hits, and screening could be conducted by using a standard flow cytometer. To this end, a miniaturized peptide library was synthesized on 20 μm beads to demonstrate proof of principle. A small sample from the 16,807-member library was screened against transferrin-AlexaFluro 647, a protein responsible for iron transport in vivo. A number of hits were identified and sequenced using techniques coupling nanomanipulation with nanoelectrospray mass spectrometry.

The development and identification of new biomaterials that can replace specific tissues and organs is desirable. In the presented PhD thesis polymer microarrays were applied for the screening of polyacrylates and polyurethanes and evaluation for material discovery for applications in the life sciences. In the first part of the thesis, the largest polymer microarray ever made with more than 7000 features was fabricated and subsequently used for the screening of polyacrylates that can control the fate of human embryonic stem cells. As stem cells have unique properties that offer the potential of replacing damaged or diseased tissue in future, the identification of cultivation substrates that can replace current biological and animal derived products was desirable. The water contact angle, roughness and cell doubling time of the cells on the identified polymers was determined and the stem cells characterised after 5 passages and compared to the currently most widely used animal derived substrate MatrigelTM. In the second part of the thesis, the development of a new polymer gradient microarray is presented. Initial studies involved the optimisation of printing parameters for the generation of linear polymer gradient lines and confirmed by XPS analysis. Cellular binding studies with the suspension cell line K562 and the adherent cell line HeLa were carried out and compared to previous binding studies to confirm the success of the concept. In further studies, the polymer gradients were functionalised with small molecules and proteins, allowing the generation of a protein gradient microarray with Semaphorin 3F. In binding studies with neuron cells it could be shown that the binding of the cells was concentration-dependent. The identification of polyacrylates for the effective and rapid activation and aggregation of platelets is described in the third part of the presented thesis. Here, polymer microarrays were applied for the binding of platelets in human blood samples. The amount of bound platelets as well as their activation state was compared to the natural agonist collagen by employing fluorescence intensity studies and scanning electron microscopy. In shear studies, the activation of the platelets by the polymers was evaluated under physiological conditions. The mechanism by which the polymer triggered the activation was further explored by protein binding studies. It was shown that the initial adsorption of fibrinogen and von Willebrand factor on the polymers lead to the adherence and aggregation of platelets. In the final part of the presented thesis, polymer microarrays were used to identify polymers that can sort and collect the precursor cells of platelets (megakaryocytes). For this purpose, the cell lines K562 and MEG-01 were used as cellular models. The identified polymers and the effect on the immobilised cells was further investigated by scanning electron microscopy, flow cytometry and miRNA studies. The adsorbed proteins on the different polymers were found to influence the cellular morphology on the different substrates.