Given a sequence of non-negative real numbers 0 ; 1 ; : : : which sum to 1, we consider random graphs having approximately i n vertices of degree i. Essentially, we show that if P i(i \Gamma 2) i ? 0 then such graphs almost surely have a giant component, while if P i(i \Gamma 2) i ! 0 then almost surely all components in such graphs are small. We can apply these results to G n;p ; G n;M , and other well-known models of random graphs. There are also applications related to the chromatic number of sparse random graphs.

functions near the liquid-liquid

study on the field-induced quantum

the complexity of finding first-order

Many people have inspired, guided, and helped me during my tenure as a doctoral student, and I would like to thank them all for a great graduate school experience. First and foremost I would like to thank my advisor, Dr. John Hegseth, for his supervision as my advisor throughout the time it took me to complete this research and write the dissertation. He challenged my thinking, writing and research skills and served as a teaching mentor and a role model. Without his insightful direction, many of the results presented here would not have been possible. My dissertation committee deserves recognition for their time, valuable suggestions and support: Dr. Juliette Ioup for her continuous support and her willingness to read and make constructive suggestions during my research on this dissertation. I am grateful to her for that, for all the hours of consultation, suggestions enthusiasm and professionalism; Dr. Dimitrios Charalampidis for generously giving his time and expertise to better my work; Dr. Joseph Murphy for his insight and sound advise through this entire research; Dr. Jinke Tang for his kind

. We construct a sequence of eigenfunctions on T 2 with a bounded number of critical points. S. T. Yau raised a question about the number and distribution of critical points of eigenfunctions of the Laplacian on a Riemannian manifold ([4, # 76], [5, # 43]). In [6] he investigated this problem

point called the “critical ” point, all noise traders may place the same order (sell) at the same time, thus causing a crash. The interplay between the progressive strengthening of imitation and the ubiquity of noise is characterized by the hazard rate, i.e. the probability per unit time that the crash

, and optimization to ensure the triangles are well shaped. For added e#ciency, piecewise polynomial metaballs are often used, but introduce special problems in #nding the critical points necessary to determine topological type. There are numerous details important to implementing such techniques on piecewise

attenuation of the sound mode around the critical point

head model with large noise and small hair textures, and the corresponding mean curvature visualization. (b) The mean curvature function yields 7,629 critical points due to the curvature function’s sensitivity to noise. (c) The corresponding mesh saliency. (d) Salient critical points with lower