<p> Women are underrepresented in the STEM workforce. Trends are starting to change as more and more women are starting to choose majors that are related to STEM. However, the numbers decline sharply in engineering, physical sciences and computer sciences. This project was created as a resource to be used in schools to encourage the increase of women studying, and eventually working, in STEM fields. Research has shown that many women who choose STEM majors and careers were heavily influenced by informal STEM enrichment opportunities outside of the school day and female role models in STEM. This project is an NGSS aligned lesson that incorporates a historical vignette highlighting the work of a female scientist. The vignette can be used in a lesson or as a pull out in a different class or program.</p><p>

Rigidity theory deals in problems of the following form: given a structure defined by geometric constraints on a set of objects, what information about its geometric behavior is implied by the underlying combinatorial structure. The most well-studied class of structures is the bar-joint framework, which is made of fixed-length bars connected by universal joints with full rotational degrees of freedom; the allowed motions preserve the lengths and connectivity of the bars, and a framework is rigid if the only allowed motions are trivial motions of Euclidean space. A remarkable theorem of Maxwell-Laman says that rigidity of generic bar-joint frameworks depends only on the graph that has as its edges the bars and as its vertices the joints. We generalize the "degree of freedom counts" that appear in the Maxwell-Laman theorem to the very general setting of (k, ℓ)-sparse and (k, ℓ)-graded sparse hypergraphs. We characterize these in terms of their graph-graph theoretic and matroidal properties. For the fundamental algorithmic problems Decision, Extraction, Components, and Decomposition, we give efficient, implementable pebble game algorithms for all the (k, ℓ)-sparse and (k, ℓ)-graded-sparse families of hypergraphs we study. We then prove that all the matroids arising from (k , ℓ)-sparse are linearly representable by matrices with a certain "natural" structure that captures the incidence structure of the hypergraph and the sparsity parameters k and ℓ. Building on the combinatorial and linear theory discussed above, we introduce a new rigidity model: slider-pinning rigidity. This is an elaboration of the planar bar-joint model to include sliders, which constrain a vertex to move on a specific line. We prove the analogue of the Maxwell-Laman Theorem for slider pinning, using, as a lemma, a new proof of Whiteley's Parallel Redrawing Theorem. We conclude by studying the emergence of non-trivial rigid substructures in generic planar frameworks given by Erdos-Renyi random graphs. We prove that there is a sharp threshold for such substructures to emerge, and that, when they do, they are all linear size. This is consistent with experimental and simulation-based work done in the physics community on the formation of certain glasses.

The focus of this thesis is the study of deterministic and stochastic models that involve multiple interrelated scales. In the first part we study the stability of planar traveling waves for hyperbolic approximations of Hamilton-Jacobi equations. Such models were first introduced in the context of relaxation approximations for Hamilton-Jacobi equations in [6], while [4] treated the convergence of the relaxation approximation as the regularization parameter tends to zero. These convergence results are limited to smooth solutions, while it is well-known that solutions to Hamilton-Jacobi equations develop singularities in the gradient in finite time, even for smooth initial data. Here we extend the analysis to convergence and stability results in the case where singularities in the gradient are present. Related results on the stability and large time behavior of viscous approximations of hyperbolic conservation laws in one and several dimensions were obtained in [2, 3]. Furthermore, in [7], the stability of planar shocks was shown for relaxation approximations of scalar conservation laws. In these works an essential ingredient of the proof involves a derived integrated form of the conservation laws, reminiscent of a Hamilton-Jacobi equation, which necessitates the use of a shift function. Here the arguments simplify substantially since we already deal with a Hamilton-Jacobi equation and a shift function is not necessary. Furthermore, due the hyperbolic nature of the approximation, we obtain improved decay properties of the solution. In the second part of this thesis, we study mesoscopic models of particle diffusion in several interacting particle systems. These mesoscopic models are stochastic or deterministic integrodifferential equations and are derived through an exact coarse graining, directly from microscopic lattice models, and include detailed microscopic information on particle-particle interactions and particle dynamics. Previous results in [9, 14, 15] are limited to one particle species, however in many applications the diffusion mechanism involves several types of particles. In Chapter 2, we focus on deriving the mesoscopic theories for such complex multi-species dynamics. Starting from microscopic dynamics we derive mesoscopic models for both Metropolis and Arrhenius rules. Also we extend our results to the case where the external driving force is no longer a constant. In Chapter 3, we derive the mesoscopic theory for parabolic Arrhenius dynamics which typically models transport and diffusion in zeolites [26]. Finally, in Chapter 4, we discuss pattern formation in systems with both attractive and repulsive interactions, and determine how the competing microscopic interactions affect the overall morphology.

A large collection of digital images of natural scenes provides a database for analyzing and modeling small scene patches (e.g., 2 x 2) referred to as natural microimages. A pivotal finding is the stability of the empirical microimage distribution across scene samples and with respect to scaling. With a view toward potential applications (e.g. classification, clutter modeling, segmentation), we present a hierarchy of microimage probability models which capture essential local image statistics. Tools from information theory, algebraic geometry and of course statistical hypothesis testing are employed to assess the “match” between candidate models and the empirical distribution. Geometric symmetries play a key role in the model selection process. One central result is that the microimage distribution exhibits reflection and rotation symmetry and is well-represented by a Gibbs law with only pairwise interactions. However, the acceptance of the up-down reflection symmetry hypothesis is borderline and intensity inversion symmetry is rejected. Finally, possible extensions to larger patches via entropy maximization and to patch classification via vector quantization are briefly discussed.

Symbolic Model Checking is a technique for checking certain properties of a finite state model of a computer system. The most widely used symbolic representation is based on Ordered Binary Decision Diagrams. In [4], G. Avrunin showed how computational geometry and Gröbner basis techniques may be used for symbolic model checking. The present work investigates the details of this approach, and its practicality, using theoretical means and experiments based on a prototype implementation.

This work analyzes two perspectives, Atomism and Infinite Divisibility, in the light of modern mathematical knowledge and recent developments in computer graphics. A developmental perspective is taken which relates ideas leading to atomism and infinite divisibility. A detailed analysis of and a new resolution for Zeno's paradoxes are presented. Aristotle's arguments are analyzed. The arguments of some other philosophers are also presented and discussed. All arguments purporting to prove one position over the other are shown to be faulty, mostly by question begging. Included is a sketch of the consistency of infinite divisibility and a development of the atomic perspective modeled on computer graphics screen displays. The Pythagorean theorem is shown to depend upon the assumption of infinite divisibility. The work concludes that Atomism and infinite divisibility are independantly consistent, though mutually incompatible, not unlike the wave/particle distinction in physics.

No description available.

<p>Network analysis provides tools for addressing fundamental applications in graphs such as webpage ranking, protein-function prediction, and product categorization and recommendation. As real-world networks grow to have millions of nodes and billions of edges, the scalability of network analysis algorithms becomes increasingly important. Whereas many standard graph algorithms rely on matrix-vector operations that require exploring the entire graph, this thesis is concerned with graph algorithms that are local (that explore only the graph region near the nodes of interest) as well as the localized behavior of global algorithms. We prove that two well-studied matrix functions for graph analysis, PageRank and the matrix exponential, stay localized on networks that have a skewed degree sequence related to the power-law degree distribution common to many real-world networks. Our results give the first theoretical explanation of a localization phenomenon that has long been observed in real-world networks. We prove our novel method for the matrix exponential converges in sublinear work on graphs with the specified degree sequence, and we adapt our method to produce the first deterministic algorithm for computing the related heat kernel diffusion in constant-time. Finally, we generalize this framework to compute any graph diffusion in constant time. </p>

<p> Information complexity enables the use of information-theoretic tools in communication complexity theory. Prior to the results presented in this thesis, information complexity was mainly used for proving lower bounds and direct-sum theorems in the setting of communication complexity. We present three results that demonstrate new connections between information complexity and communication complexity.</p><p> In the first contribution we thoroughly study the information complexity of the smallest nontrivial two-party function: the AND function. While computing the communication complexity of AND is trivial, computing its exact information complexity presents a major technical challenge. In overcoming this challenge, we reveal that information complexity gives rise to rich geometrical structures. Our analysis of information complexity relies on new analytic techniques and new characterizations of communication protocols. We also uncover a connection of information complexity to the theory of elliptic partial differential equations. Once we compute the exact information complexity of AND, we can compute <i> exact communication complexity</i> of several related functions on <i> n</i>-bit inputs with some additional technical work. Previous combinatorial and algebraic techniques could only prove bounds of the form &Theta;(<i> n</i>). Interestingly, this level of precision is typical in the area of information theory, so our result demonstrates that this meta-property of precise bounds carries over to information complexity and in certain cases even to communication complexity. Our result does not only strengthen the lower bound on communication complexity of disjointness by making it more exact, but it also shows that information complexity provides the exact upper bound on communication complexity. In fact, this result is more general and applies to a whole class of communication problems.</p><p> In the second contribution, we use self-reduction methods to prove strong lower bounds on the information complexity of two of the most studied functions in the communication complexity literature: Gap Hamming Distance (GHD) and Inner Product mod 2 (IP). In our first result we affirm the conjecture that the information complexity of GHD is linear even under the uniform distribution. This strengthens the &Omega;(<i>n</i>) bound shown by Kerenidis et al. (2012) and answers an open problem by Chakrabarti et al. (2012). We also prove that the information complexity of IP is arbitrarily close to the trivial upper bound <i>n</i> as the permitted error tends to zero, again strengthening the &Omega;(<i>n</i>) lower bound proved by Braverman and Weinstein (2011). More importantly, our proofs demonstrate that self-reducibility makes the connection between information complexity and communication complexity lower bounds a two-way connection. Whereas numerous results in the past used information complexity techniques to derive new communication complexity lower bounds, we explore a generic way, in which communication complexity lower bounds imply information complexity lower bounds <i> in a black-box manner</i>.</p><p> In the third contribution we consider the roles that private and public randomness play in the definition of information complexity. In communication complexity, private randomness can be trivially simulated by public randomness. Moreover, the communication cost of simulating public randomness with private randomness is well understood due to Newman's theorem (1991). In information complexity, the roles of public and private randomness are reversed: public randomness can be trivially simulated by private randomness. However, the information cost of simulating private randomness with public randomness is not understood. We show that protocols that use only public randomness admit a rather strong compression. In particular, efficient simulation of private randomness by public randomness would imply a version of a direct sum theorem in the setting of communication complexity. This establishes a yet another connection between the two areas. (Abstract shortened by UMI.)</p>

<p> From the early days of AI, computers have been programmed to play games against human players. Most of the AI work has sought to build world-champion programs to play turn-based games such as Chess and Checkers, however computer games increasingly provide for entertaining real-time play. In this dissertation, we present an extension of recursive game theory, which can be used to analyze games involving simultaneous movement. We include an algorithm which can be used to practically solve recursive games, and present a proof of its correctness. We also define a game theory of lowered expectations to deal with situations where game theory currently fails to give players a definitive strategy, and demonstrate its applicability using several example games.</p>