This thesis is divided into two parts. The rst part studies homological algebra of quasihereditary algebras, with the underlying theme being to understand properties of the Yoneda algebra of standard modules. We will rst show how homological properties of a quasi-hereditary algebra are carried over to its tensor products and wreath products. We then determine the extgroups between indecomposable standard modules of a Cubist algebra of Chuang and Turner. We will also determine generators, hence the quiver, of the Yoneda algebra of standard modules for the rhombal algebras of Peach. We also obtain a higher multiplication vanishing condition for certain rhombal algebras. The second part of this thesis studies the notion of simple-minded systems, introduced by Koenig and Liu. Such systems were designed to generate the stable module categories of artinian algebras by extension, in the same way as the sets of simple modules. We classify simple-minded systems for representation- nite self-injective algebras, and establish connections of them to various notions in combinatorics and related derived categories. We also look at the notion of simple-minded systems de ned on triangulated categories, and obtain some classi cation results using a connection between the simple-minded systems of a triangulated category and of its orbit category.

In this dissertation, let B be a triangulated category and let D be an extension-closed subcategory of B. First, we give some new characterizations of an Auslander-Reiten triangle in D, which yields some necessary and sufficient conditions for D to have Auslander-Reiten triangles. Next, we study when an Auslander-Reiten triangle in B induces an Auslander-Reiten triangle in D. As an application, we study Auslander-Reiten triangles in a triangulated category with a t-structure. In case the t-structure has a t-hereditary heart, we establish the connection between the Auslander-Reiten triangles in B and the Auslander-Reiten sequences in the heart. Finally, we specialize to the bounded derived category of all modules of a noetherian algebra over a complete local noetherian commutative ring. Our result generalizes the corresponding result of Happel’s in the bounded derived category of finite dimensional modules of a finite dimensional algebra over an algebraically closed field.

Random triangulated surfaces are created by taking an even number, n, of triangles and arbitrarily ”gluing” together pairs of edges until every edge has been paired. The resulting surface can be described in terms of its number of boundary cycles, a random variable denoted by h. Building upon the work of Nicholas Pippenger and Kristin Schleich, and using a recent result from Alex Gamburd, we establish an improved approximation for the expectation of h for certain values of n. We use a computer simulation to exactly determine the distribution of h for small values of n, and present a method for calculating these probabilities. We also conduct an investigation into the related problem of creating one connected component out of n triangles.

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Imaging modalities such as CT, MRI, and SPECT have had a tremendous impact on diagnosis and treatment planning. These imaging techniques have given doctors the capability to visualize 3D anatomy structures of human body and soft tissues while being non-invasive. Unfortunately, the 3D images produced by these modalities often have boundaries between the organs and soft tissues that are difficult to delineate due to low signal to noise ratios and other factors. Image segmentation is employed as a method for differentiating Regions of Interest in these images by creating artificial contours or boundaries in the images. There are many different techniques for performing segmentation and automating these methods is an active area of research, but currently there are no generalized methods for automatic segmentation due to the complexity of the problem. Therefore hand-segmentation is still widely used in the medical community and is the €œGold standard€� by which all other segmentation methods are measured. However, existing manual segmentation techniques have several drawbacks such as being time consuming, introduce slice interpolation errors when segmenting slice-by-slice, and are generally not reproducible. In this thesis, we present a novel semi-automated method for 3D hand-segmentation that uses mesh extraction and surface parameterization to project several 3D meshes to 2D plane . We hypothesize that allowing the user to better view the relationships between neighboring voxels will aid in delineating Regions of Interest resulting in reduced segmentation time, alleviating slice interpolation artifacts, and be more reproducible.

This thesis consists of three papers about N-complexes and their uses in categorification. N-complexes are generalizations of chain complexes having a differential d satisfying dN = 0 rather than d2 = 0. Categorification is the process of finding a higher category analog of a given mathematical structure. Paper I: We study a set of homology functors indexed by positive integers a and b and their corresponding derived categories. We show that there is an optimal subcategory in the domain of every functor given by N-complexes with N = a + b. Paper II: In this paper we show that the lax nerve of the category of chain complexes is pointwise adjoint equivalent to the décalage of the simplicial category of N-complexes. This reveals additional simplicial structure on the lax nerve of the category of chain complexes which provides a categorfication of the triangulated homotopy category of chain complexes. We study this in general and present evidence that the axioms of triangulated categories have a simplicial origin. Paper III: Let n be a product of two distinct prime numbers. We construct a triangulated monoidal category having a Grothendieck ring isomorphic to the ring of n:th cyclotomic integers.

Applying engineering principles to tessellation origami-based designs enables the control of certain design properties, such as flexibility, bending stiffness, mechanical advantage, shape conformance, and deployment motion. The ability to control these and other properties will enable augmented design capabilities in environments which currently limit the design to specific materials, including space, medicine, harsh environments, and scaled environments (such as MEMS applications). Other applications will be able to achieve more complex motions or better satisfy design and performance requirements.This research demonstrates augmented design capabilities of origami tessellations in engineering design in rigid-foldable and non-rigid-foldable applications. First, a method to determine Poisson's ratio and mechanical advantage for deployable, rigid-foldable tessellations is presented. The results enable the selection and tailoring of patterns based on deployment motion of specific patterns.Secondly, a model that predicts the deployment stability of the non-rigid-foldable triangulated cylinder is presented. This model defines the geometry needed to obtain a maximum deployed height, always return to a closed position, or remain in either the open or closed configurations. The Stability Transition Ratio is the ratio of the inner to outer diameter that marks the point between monostable and bistable behavior in a triangulated cylinder and is dependent only on the number of sides.Lastly, this work presents methods to reduce sag in adult diapers by increasing shape conformance, promoting wicking capabilities, and improving the structure through the implementation of origami tessellations. Several basic fold patterns were evaluated and the results reported. Reducing sag increases comfort and decreases leaking.

Let S31(?) be the bivariate C1-cubic spline space over a triangulated quadrangulation ?. In this paper, an explicit representation of a locally supported basis of S31(?) is given using the interpolation conditions at vertices.

Geometrical modeling is a crucial aspect of simulations involving manufactured objects. Apart from the pure construction of curves and surfaces, the analysis of their quality is equally important in the design and manufacturing process. In computer-aided simulation, the original freeorm surfaces need to be tessellated into triangulated surfaces before the simulation procedure. To concurrently and interactively visualize the results from both simulation and surface interrogation in a virtual environment, I propose two novel surface interrogation algorithms for triangulated surfaces instead of the traditional freeorm surfaces. The novel algorithms are interactive and immersive versions of two well-established surface interrogation techniques ? reflection lines and generalized focal surfaces. These two algorithms have been designed to overcome some limitations of the traditional approaches and make them available for interactive and immersive applications. For reflection lines, the new algorithm maps the triangulated surface onto the light plane so that the computation of distance between reflection ray and light line in three dimensions can be reduced to computation of intersections between light lines and triangle edges in two dimensions, simplifying the computation. For generalized focal surfaces, the new algorithm estimates curvature by simple computation of the derivatives of a3rd degree triangular Bézier patch on each triangle and removes the requirement for a minimum number of neighbor points and implicit requirements on how the neighbor points are distributed. Proposed future work on real-time rendering of surface interrogation using a texture mapping technique is discussed.

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