The graceful tree conjecture was first introduced over 50 years ago, and to this day it remains largely unresolved. Ideas for how to label arbitrary trees have been sparse, and so most work in this area focuses on demonstrating that particular classes of trees are graceful. In my research, I continue this effort and establish the gracefulness of some new tree types using previously developed techniques for constructing graceful trees. Meanwhile, little work has been done on developing computational methods for obtaining graceful labelings, as direct approaches are computationally infeasible for even moderately large trees. With this in mind, I have designed a new computational approach for constructing a graceful labeling for trees with sufficiently many leaves. This approach leverages information about the local structures present in a given tree in order to construct a suitable labeling. It has been shown to work for many small cases and thoughts on how to extend this approach for larger trees are put forth. / McAnulty College and Graduate School of Liberal Arts; / Computational Mathematics / MS; / Thesis;

Cascading failures involve a failure in one system component that triggers failures in successive system components, potentially leading to system wide failures. While frequently used fault tolerant techniques can reduce the severity and the frequency of such failures, they continue to occur in practice. To better understand how failures cascade, we have conducted a qualitative analysis of 55 cascading failures, described in 26 publicly available incident reports. Through this analysis we have identified 16 types of cascading mechanisms (organized into eight categories) that capture the nature of the system interactions that contribute to cascading failures. We also discuss three themes based on the observation that the cascading failures we have analyzed occurred in one of three ways: a component being unable to tolerate a failure in another component, through the actions of support or automation systems as they respond to an initial failure, or during system recovery. We believe that the 16 cascading mechanisms we present and the three themes we discuss, provide important insights into some of the challenges associated with engineering a truly resilient and well-supported system.

優美圖是圖論中較有趣的研究課題。在本文中，我們將對C_n⊕S_m的圖形提供一套優美的標法。 / Graceful graph is a more interesting research problem in the graph theory. In this paper, we will give a graceful labeling to the gragh C_n⊕S_m.

樹是一個沒有迴路的連接圖。而4-caterpillar是一種樹，它擁有單一路徑連接到數個長度為3的路徑的端點。一個有n個邊的無向圖G的優美標法是一個從G的點到{0,1,2,...,n}的一對一函數，使得每一個邊的標號都不一樣，其中，邊的標號是兩個相鄰的點的編號差的絕對值。在這篇論文當中，我們最主要的目的是使用一個演算法來完成4-caterpillars的優美標法。 / A tree is connected acyclic graph. A 4-caterpillar is a tree with a single path only incident to the end-vertices of paths of length 3. A graceful labelling of an undirected graph G with n edges is a one-to-one function from the set of vertices of G to the set {0,1,2,...,n} such that the induced edge labels are all distinct, where the edge label is the difference between two endvertex labels. In this thesis, our main purpose is to use an algorithm to yield graceful labellings of 4-caterpillars.

Gaits have become an integral part of the design method of robots heading to complex terrains. But research into optimal ways to transition between different gaits is still lacking, and is the primary motivation behind this research. An essential characteristic of gaits is periodicity, and considering that a novel notion of graceful transition is proposed: a graceful transition is one that has maximally persisting periodicity. This particular notion of persistence in the characteristic behavior can be generalized. Therefore, a comprehensive framework for the general problem of connecting any two trajectories of a dynamical system, with an underlying characteristic behavior, over a finite time interval and in a manner that the behavior persists maximally during the transition, is developed and presented. This transition is called the Gluskabi Raccordation, and the characteristic behavior is defined by a kernel representation. Along with establishing this framework, the kernel representations for some interesting characteristic behaviors are also identified. The problem of finding the Gluskabi Raccordations is then solved for different combinations of characteristic behaviors and dynamical systems, and compact widely applicable results are obtained. Lastly, the problem of finding graceful gait transitions is treated within this newly established broader framework, and these graceful gait transitions are obtained for the case of a two-piece worm model.

For k∈{Z}+ and G a simple connected graph, a k-radio labeling f:VG→Z+ of G requires all pairs of distinct vertices u and v to satisfy |f(u)-f(v)|≥ k+1-d(u,v). When k=1, this requirement gives rise to the familiar labeling known as vertex coloring for which each vertex of a graph is labeled so that adjacent vertices have different "colors". We consider k-radio labelings of G when k=diam(G). In this setting, no two vertices can have the same label, so graphs that have radio labelings of consecutive integers are one extreme on the spectrum of possibilities; graphs that can be labeled with such a labeling are called radio graceful. In this thesis, we give four main results on the existence of radio graceful graphs, which focus on Hamming graphs (Cartesian products of complete graphs) and a generalization of the Petersen graph. In particular, we prove the existence of radio graceful graphs of arbitrary diameter, a result previously unknown. Two of these main results show that, under certain conditions, the tth Cartesian power Gt of a radio graceful graph G is also radio graceful. We will also speak to occasions when Gt is not radio graceful despite G being so, as well as some partial results about necessary and sufficient conditions for a graph G so that Gt is radio graceful.

A decomposition D of a graph H by a graph G is a partition of the edge set of H such that the subgraph induced by the edges in each part of the partition is isomorphic to G. A mixed graph on V vertices is an ordered pair (V,C), where V is a set of vertices, |V| = v, and C is a set of ordered and unordered pairs, denoted (x, y) and [x, y] respectively, of elements of V [8]. An ordered pair (x, y) ∈ C is called an arc of (V,C) and an unordered pair [x, y] ∈ C is called an edge of graph (V,C). A path on n vertices is denoted as Pn. A partial orientation on G is obtained by replacing each edge [x, y] ∈ E(G) with either (x, y), (y, x), or [x, y] in such a way that there are twice as many arcs as edges. The complete mixed graph on v vertices, denoted Mv, is the mixed graph (V,C) where for every pair of distinct vertices v1, v2 ∈ V , we have {(v1, v2), (v2, v1), [v1, v2]} ⊂ C. The goal of this thesis is to establish necessary and sufficient conditions for decomposition of Mv by all possible partial orientations of P4.

In this thesis, we consider the decomposition of the complete mixed graph on v vertices denoted Mv, into every possible mixed graph on three vertices which has (like Mv) twice as many arcs as edges. Direct constructions are given in most cases. Decompositions of theλ-fold complete mixed graph λMv, are also studied.

The purpose of this study was to explore teaching for meaningfulness in a school setting. I feel that a better understanding of teaching for meaningfulness can support the practice of teachers who desire to teach for meaningfulness. I elected to do a case study using qualitative methods while focusing for five months on a teacher's literacy program. Through this look at the everyday life of one public school teacher and her class a picture of an exemplary example of teaching for meaningfulness could be painted. Interviews, fieldnotes, journals, observations, various site artifacts, and a brief teacher autobiography comprised the primary data collection sources. Informants included a third grade teacher, eighteen third grade students, and a school language arts coordinator. / Ph. D.

International Telemetering Conference Proceedings / October 17-20, 1994 / Town & Country Hotel and Conference Center, San Diego, California / Use of top-down design principles and standard interface techniques provides the basis for a global telemetry data collection, analysis, and satellite control network with a high degree of survivability via use of distributed architecture. Use of Commercial Off-The-Shelf (COTS) hardware and software minimizes costs and provides for easy expansion and adaption to new satellite constellations. Adaptive techniques and low cost multiplexers provide for graceful system wide degradation and flexible data distribution.