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Forensic footwear impression analysis is a valuable tool in criminal investigations. Extracting useful features from images of footwear impressions is a critical step in this process. However, the quality of these images can vary widely, making feature extraction challenging. In order to give a quality assessment rating to a footwear impression image, the image should first be analyzed to extract features from the impression. In this paper, we present a method to extract features from a 2D grayscale footwear impression image. A Hierarchical Grid Model implementation has been adapted from use on a 3D dataset to assist in finding features, referred to as lugs for footwear, of a 2D grayscale image of a footwear impression. Depending on the number of lugs found by the grid search, the image is assigned a quality rating to determine the likelihood of promising results from further analysis. Quality ratings range from high, good, moderate, low, and poor. We evaluated the performance of our method using a data set of footwear impression images with varying quality. Our results show that the proposed method is effective in extracting useful features from footwear impressions and provides quantitative results indicating agreement with ground truth. Our proposed method offers a solution for extracting features from footwear impression images with varying quality, providing an efficient and reliable tool for forensic investigations.

This dissertation introduces the DIverse MultiPLEx Generalized Dot Product Graph (DIMPLE-GDPG) network model where all layers of the network have the same collection of nodes and follow the Generalized Dot Product Graph (GDPG) model. In addition, all layers can be partitioned into groups such that the layers in the same group are embedded in the same ambient subspace but otherwise all matrices of connection probabilities can be different. In common particular cases, where layers of the network follow the Stochastic Block Model (SBM) and Degree Corrected Block Model (DCBM), this setting implies that the groups of layers have common community structures but all matrices of block connection probabilities can be different. For DCBM, each group can also equip with nodes' specific weights. We refer to this two versions as the DIMPLE model and the DIMPLE-DECOR model. While the DIMPLE-GDPG model generalizes the COmmon Subspace Independent Edge (COSIE) random graph model, the DIMPLE model generalizes a multitude of papers that study multilayer networks with the same community structures in all layers (which include the tensor block model, the checker-board model as well as the Mixture Multilayer Stochastic Block Model (MMLSBM) as particular cases). This dissertation introduces novel algorithms for the recovery of similar groups of layers, for the estimation of the ambient subspaces in the groups of layers in the DIMPLE-GDPG setting, and for the within-layer clustering in the case of the DIMPLE model. We also consider applications of the DIMPLE models to real-life data, and its comparison with the MMLSBM. And the DIMPLE model with its SBM-imposed structures provided better descriptions of the organization of layers than the ones obtained on the basis of the MMLSBM setting.

This dissertation consists of adaptations of two papers exploring the connections between tensor categories and localized excitations in (2+1)D topologically ordered systems. Topological orders have long been studied in terms of monoidal higher categories. In particular, topological orders in (2+1)D are obtained by string-net condensation from the data of unitary fusion categories, and topological domain walls and phase transitions between (2+1)D topological orders arise from anyon condensation, which can be understood in terms of the internal Morita theories of étale algebra objects in unitary modular tensor categories.Complementing existing models for domain walls, we introduce an extension of the Levin-Wen string-net model in which tuning a parameter implements anyon condensation. We then use tube algebra techniques to verify that the expected topological order results. We also describe the effects of the phase transition on anyons and string operators of the original topological order.We study compositions of parallel topological domain walls and their decompositions into superselection sectors. Our approach uses a description of particle mobility across domain walls in terms of tunneling operators. These are formalized in a 3-category of (2+1)D topological orders with a fixed anomaly described by a unitary modular tensor category A, algebraically characterized by the 3-category of A-enriched unitary fusion categories. We describe how A-enriched unitary fusion categories determine commuting projector string-net models for the corresponding anomalous (2+1)D topological orders on the boundary of (3+1)D Walker-Wang models with trivial topological order. We then give an explicit description in terms of étale algebra objects of the local operators which determine superselection sectors of parallel domain walls, characterizing the indecomposable domain wall in each sector.

Beyond its normative relevance, the question of how institutions obtain, maintain, utilize, and lose authority is an important historical and conceptual issue. Early American politics in the colonial and revolutionary eras provides fruitful ground for exploring how the authority of institutions affects politics, and how political developments affect institutional authority. Applying game theory to questions of historical political development allows us to clearly identify strategic incentives and generalize from particular cases. In three papers, this dissertation combines historical case studies and formal models to investigate different aspects of legal authority in American political development.In the first paper, I investigate how purely formal constraints on political actors are enforced through coordination among citizens, and conditions under which such coordination fails. Specifically, this paper argues that economic power is a crucial variable for understanding the rule of law. I make this argument in two ways. First, I present a formal model to show how economic inequality undermines the rule of law. In the model, coordination among citizens increases the expense that elites face to implement an unconstitutional policy. But this constraint only binds when elites’ material resources are sufficiently low relative to those of citizens. Second, I relate the model to a historical case. In the early American republic, elites repeatedly expressed concerns about the expense of suppressing armed challenges to the constitutional order, supporting compromise instead. Amid contemporary concerns about the rule of law, this argument indicates the value of renewed attention to economic power.In the second paper, I analyze contrasting political effects of different structures of legal authority. In the process of both European and American state development, legal institutions developed from many separate venues with overlapping jurisdictions to a hierarchical structure with authority concentrated in the lawmaking institutions of the state. How did this change in the structure of legal authority affect political competition among groups in society? This paper presents a formal model to show that concentrating legal authority increases incentives for competing groups to invest in political organization. In turn, this increases political inequality: as all groups invest more effort, groups with a resource advantage are better able to make use of their advantage. The model is consistent with a case comparison of 18th century Pennsylvania and Massachusetts, two British American colonies with contrasting legal structures, patterns of political organization, and legal outcomes. The third paper provides a strategic mechanism by which the boundaries of authority between institutions incrementally change over time. Fundamental political change often occurs gradually, and it is therefore important to understand the strategic underpinnings of gradual institutional change. This paper presents a game in which players are entirely strategic and share proposal power, while showing how one player can gradually accumulate authority over time at the expense of another. The model analyzes a bargaining problem between two actors. Over an infinite number of periods, each actor has a chance each period to propose a bundle of future authority plus present policy, which creates the potential for actors to trade authority for temporary policy concessions. With some degree of noise in the division of authority, a proposal that transfers authority reduces the reservation value of the respondent, creating an incentive for the proposer to trade present policy for future authority. The model shows how ongoing, everyday political interactions gradually rearrange authority relationships among political institutions.

This dissertation is a compilation of four self-contained research papers that contribute to the literature of financial economics. All papers are co-authored by Professor Didier Sornette, my thesis supervisor, and further, depending on the project, by other authors. While three papers presented in this thesis were submitted to leading, peer-reviewed journals and are either awaiting review or are on revise and resubmit status, one paper is work in progress (the respective status is shown in the references as well as on the paper title page). The papers as presented in the thesis can include additional material that goes beyond the versions submitted. Understanding the formation and dynamics of asset prices is essential for many regulatory, policy and investment decisions. Chapter 2 and 3 of this dissertation aim to cast light into the dynamics of asset prices in the presence of two feedback effects, option markets and circuit breakers, using relative pricing models. Chapter 4 presents an absolute asset pricing model that provides a framework on how to correctly value firms in the presence of high clash flow growth. First, chapter 2 presents the discussion of a feedback effect between markets, namely from the options market onto the underlying’s market. We show that if the option market makers (OMM) hedge a larger share of their position compared to option market takers (OMT), this leads to a net delta hedge demand in the spot market. And due to liquidity effects (price impact), this mechanically leads to an increase or decrease in the underlying’s price volatility and a distortion of the underlying’s drift, depending on the positioning of the option market. We describe this dynamic with a relative pricing model, relative to the price without option hedging. Then we focus on the impact of option hedging on the spot markets volatility in the foreign exchange (FX) market. We use trade repository data to estimate the net positioning of the OMM in the FX options market and with it her delta hedge demand in the spot market. We find the OMM to be short gamma on average (implying she has a short option positioning on average) and thus the volatility to be increased by delta hedging of the OMM. Next, we an- alyze the impact of option hedging on the price stability. Using theoretical arguments as well as simulations, we show that the probability for drift bursts, i.e. explosive directional price moves, can be increased or decreased due to option positioning, depending on the option parameters and delta of the OMM. Using option positioning data for the GameStock stock, we document that there are indeed instances where explosive trends due to hedging were more likely in the first quarter of 2021, during which the stock showed exceptional price instability. Second, chapter 3 presents the discussion of a feedback effect that stems from the market architecture of exchanges, i.e. the price dynamics of so called circuit breakers (CB). A price-based CB for example is a trading halt imposed by an exchange after the price drops below (or rises above) a pre-specified value within a trading session. The rational behind a CB mechanism is that it gives traders time to calm down, interact with their counterparts and rethink their trading decision. Given that investors are continuously forming anticipations, their tactical positioning in the presence of the CB may lead to feedback of the CB on the price itself. We model the price dynamic under a CB using a relative pricing model, relative to the price without a CB, in a closed-form setup, in which the investors’ anticipation of the probability of a halt feedbacks on the price. In a general stochastic financial framework, this leads to coupled integral and stochastic differential equations. Our theory confirms two observations of this field for price-based CBs, namely an increased price volatility prior to the trigger and a negative effect of attracting the price to the CB level (known as the magnet effect). As our framework generalises to other CB mechanisms, we can then suggest a new CB design which minimises the adverse effects of a price-based CB. We e.g. show that a trading halt that is conditioned on a pre-specified volatility level, instead of a price level, mitigates the side effects. Third, chapter 4 contributes to the discipline of absolute asset pricing. We aim to refocus the discussion around what could be loosely referred to as an expected return factor. Motivated by economic intuition and empirical insights from the consumption-based capital asset pricing model we show that the expected return on a firm level has a linear relationship to the firm’s cash flow growth level. With that we argue that the discount rate should reflect this linear relationship and should thus have the same term structure as the assumed cash flow growth rate process. Based on that we first introduce a discounted cash flow (CDF) model with a time-varying discount rate matching the assumed cash flow growth rate process. This important feature greatly enhances the value of a DCF model, as the valuation is less sensitive to the cash flow growth rate process assumptions, as the discount rate is proportional to the assumed cash flow growth rate term structure, such that these two naturally complementary variables interact to massively reduce the range of reasonable valuation outcomes. Second, using the same time-varying discount rate in an internal cost of capital (ICC) model, we elaborate that the corresponding time-varying internal rate of return allows for a straightforward interpretation as investment return of a buy and hold strategy, compared to e.g. a constant ICC model, which by construction is a constant compound return. We show that our expected return proxy is superior in explaining realised returns compared to a constant internal rate of return, and even renders well known linear factors, which can be motivated from theory, insignificant. Last, the economic significance is confirmed in a trading strategy with yearly rebalancing.

In this paper we show a version of Johnson’s theorem for $\M(2,\R)$-sequences. In particular, motivated by the work of \cite{zhang2} and \cite{marx}, we show that one can identify the spectrum of Jacobi operator $J_{a,b}:\ell^2(\Z, \R)\rightarrow\ell^2(\Z,\R)$ by those energies $E \in \R$, whose cocycle map $B^{E} \in \M(2,\R)$ does not admit dominated splitting, i.e.$$\sigma(J_{a,b})=\{E\in\R:B^{E}\notin\CD\CS\}.$$ This result extends a well-known Johnson's theorem for Schr\"odinger operators, which identifies the spectrum of Schr\"odinger operator by all energy values $E \in \R$, such that the cocycle map $A^E \in \SL(2,\R)$ is not uniformly hyperbolic. Jacobi operator is a natural extension to Schr\"odinger operator, and dominated splitting generalizes the notion of uniform hyperbolicity. The biggest difference lies in the fact that for Schr\"odinger operator, the cocycle map takes value in $\SL(2,\R)$, whereas for Jacobi operator, the cocycle map takes value in $\M(2,\R)$. In particular, it is allowed to be singular, i.e. have zero determinant. This is one of the main challenges of this paper that we had to overcome. In this paper we also discuss the importance of Jacobi operators, develop definition of dominated splitting for $\M(2,\R)$-sequences, show stability theorem and finish this paper with the discussion of possibility of extending this result to dynamically defined Jacobi operators.

The following outlines my Ph.D thesis on Lagrange Inversion, Rogers-Ramanujan identities, and orthogonal polynomials. Sections $1-3$ serve as a summary of background material, much of which is based on various notes and papers either authored or co-authored by my advisor, Adriano Garsia. The subsequent sections contain related novel research. Specifically, section $4$ outlines the contents of a joint paper \cite{GG20} with Adriano Garsia, where we apply an expanded version of the combinatorial theory of orthogonal polynomials to produce new identities for the Fibonacci Polynomials. Section $5$ then details a purely combinatorial proof of $q-$Lagrange inversion, its application to Rogers-Ramanujan type identities, and more generalized Lagrange inversions involving an additional $t$ statistic.

The purpose of this study was to determine the effects of dynamic geometry software (DGS) on secondary students’ understanding of geometry concepts. This paper examines how dynamic geometry software (DGS) is used in the secondary mathematics classroom and its effects on student understanding and achievement through a review of the literature. The use of technology in mathematics education is well documented and its effects are noted within this paper. The focus of the literature review in Chapter 2 includes: 1) how the computer technology is used in the classroom; 2) the use of DGS to promote student understanding of inductive and deductive reasoning; 3) creation of proof and the discovery of geometric theorems; and 4). the role of the software on student performance. In Chapter 3 I conclude by reviewing the research findings and giving recommendations.

An esteemed 2016 paper set a new lower bound on the clause density required to strongly refute a k-XOR formula in polynomial time. This paper used the sum of squares algorithm in conjunction with an improved bound for the injective tensor norm to reach its result, which is limited to formulas with all clauses the same size. We consider how this technique could be expanded to formulas with mixed clause sizes. We specifically focus our efforts on what we view as the simplest combination of clause sizes: an XOR formula with clauses of sizes k and 2k with k even. While this thesis does not establish a new refutation bound for this mixed size formula, it does give a prospective structure for the proof, and shows how to expand the 2016 paper’s techniques to mixed size problems. Additionally, this thesis gives an overview of the 2016 paper for new researchers.

Debiased Sinkhorn divergence (DS divergence) is a distance function of regularized optimal transport that measures the dissimilarity between two probability measures of optimal transport. This thesis analyzes the advantages of using DS divergence when compared to the more computationally expensive Wasserstein distance as well as the classical Euclidean norm. Specifically, theory and numerical experiments are used to show that Debiased Sinkhorn divergence has geometrically desirable properties such as maintained convexity after data normalization. Data normalization is often needed to calculate Sinkhorn divergence as well as Wasserstein distance, as these formulas only accept probability distributions as inputs and do not directly apply to signed data such as time signals and seismic waves; however, in doing so one may lose or distort information about the original signal. The investigations in this paper show that for high frequency signal inputs, Wasserstein distance may need a much more dramatic normalization compared to Debiased Sinkhorn in order to preserve convexity, leading to a loss of information about the original signal, the amplification of noise, and possibly machine overload, thus posing the desirability of the Debiased Sinkhorn divergence method.

In today’s world, deep learning models are widely used in a variety of fields. Audio applications include speech recognition, audio classification, and music information retrieval. In this paper, we will focus on the classification of music genres using an artificial neural network. The development of audio machine learning techniques has created an independence from traditional, more time-consuming signal processing techniques. Starting with raw audio data, we will gain an understanding of what audio is and its digital representation. Then, the focus will be on obtaining frequency information from audio signals through the use of spectrograms. Transforming the spectrograms into the perceptually relevant mel scale allows us to eventually extract mel frequency cepstral coefficients (MFCC) from audio files. We will then make use of our network architecture to process the MFCC’s. A convolutional neural network, our network of choice here, is trained to classify audio files into one of nine musical genres with an accuracy of 89.1% using the GTZAN dataset, which is only about 4 percentage points below the state-of-the-art performance for this dataset.

This paper provides a brief overview of some of the existing literature concerningannihilating random walks and provides a toy example for the reader. Two novelannihilating random walk models are additionally considered. The first examines thecase where a + b → ∅. The second examines a case where there is both annihilatingand coalescence between a and b particles, where a + a → a, and a + b → ∅.For both problems, computational results from simulations are provided, with keeninterest on the decay of a particles. Possibilities for future research are also examined.

In this thesis we study extensions to higher dimensions of several versions of percolation within a torus. We include the contents of two papers, one covering independent percolation and the other covering a well-known dependent model.Percolation traditionally studies the appearance of infinite components in random subgraphs of lattices. The canonical subgraphs studied this way are obtained by taking a constant fraction of the vertices or edges independently at random. One can build higher dimensional random complexes in this way, but it is not clear what the equivalent of the infinite component should be. Bobrowski and Skraba offered a partial answer in finite volume complexes called homological percolation. In the torus T^d, homological percolation in dimension i means that the subcomplex contains a representative of a nontrivial element of H_i(T^d). In section 2 we consider analogues of both bond and site percolation in the torus and show that such percolation has a sharp threshold function in all dimensions. We also show that percolation in half the dimension of the torus occurs at p=1/2, analogous to the classical Harris-Kesten theorem.Another percolation model of interest is the random-cluster model, which weights configurations of independent percolation according to the number of connected components. This is a particularly interesting model because it can be coupled with the Ising and Potts models of magnetism. In section 3 we study a higher dimensional version of this model introduced by Hiraoka and Shirai, which also admits a coupling to a higher dimensional Potts model. It is worth noting that the two dimensional random-cluster model is associated to a Potts lattice gauge theory, which is related to interesting questions from physics. We prove similar results about sharp thresholds for homological percolation in the random-cluster model, which sheds some light on the Wilson loops in the associated lattice gauge theory.

Jacobi’s Four Squares Theorem is a celebrated result of number theory that provides a formula for the number of ways a positive integer n can be written as a sum of four integral squares. In this paper, we prove this theorem using the theory of modular forms.

This dissertation explores how computer science became a discipline in the United States and how this process of discipline formation and institutionalization shaped the content and culture of computer science. Like other fields of knowledge, computer science in the United States emerged out of established disciplines; however, unlike many other scientific or engineering fields, computer science developed simultaneously out of multiple other disciplines and multiple different interdisciplinary arrangements. This dissertation explores how each of these disciplines in specific university contexts contributed to the construction of the character of computer science and the definition of the identity of the computer scientist and how different interdisciplinary relationships provided opportunities for the development of distinct visions of computer science. Drawing on archival research from university and department records as well as the papers of university administrators, computer scientists, mathematicians, electrical engineers, and others, this work reveals how the processes of discipline formation shaped everything from the intellectual foundation and internal culture of computer science to the modern American university’s institutional structure and approach to interdisciplinary knowledge-making.

Separation axioms are a useful property in general topology. Important among them is the concept of normality. Normality can be used to guarantee the existence of various continuous functions.Three theorems towards that goal are Urysohn’s Lemma, the Katetov-Tong Insertion Lemma and Tietze Extension Theorem. Much work has been done on extending these theorems into the realm of lattice valued topological spaces. This paper compiles much of the existent work on the topic, clearly and concisely elaborating on the proofs of these theorems in the current literature.

In [Bau06], Baues computed the secondary Steenrod algebra, the algebra of all secondary cohomology operations. Together with Jibladze, they showed that this gives an algorithm that computes all Adams d_2 differentials for the sphere [BJ04]. The goal of this paper is to reinterpret their results in the language of synthetic spectra in order to achieve stronger computational results. Using this, we obtain an algorithm that computes hidden extensions on the E_3 page that jump by one filtration, in addition to the d_2 differentials of Baues–Jibladze. We then implement and run this algorithm for the sphere up to the 140th stem. Combined with a generalized version of the Leibniz rule, these hidden extensions allow us to compute many longer differentials with ease. In particular, we resolve all remaining unknown d_2, d_3, d_4 and d_5 differentials of the sphere up to the 95th stem.

In this paper, I examine magic squares of squares (MSS) of order 5 over Zp where p is a prime number. The first approach to the problem is to find how many distinct elements an MSS may have (called the degree of the MSS). In the next step, I study the relationship between the magic sum and the center entry of any MSS. In order to develop construction methods and configurations for magic squares of squares of order 5 with desired degrees, I study Pythagorean triples and sequences of consecutive quadratic residues modulo p. Properties of these sequences are provided and applied to construct desired magic squares of squares. This research focuses on magic squares of squares of order 5 in which the center 3 x 3 square is a magic square of squares of order 3. I claim that the magic sum of such an MSS M is 5c, where c is the center element of M and the degree of M must be odd when p > 5. The main results of the thesis include several configurations for the construction of MSS of a given degree and the existence ofMSSs of all possible odd degrees over Zp for infinitely many primes p. Chapter 1 presents an overview of modular arithmetic as well as some important definitions. Chapter 2 gives the results about the magic sum and degrees. In Chapter 3, I investigate special sequences of quadratic residues and describe properties of them. In Chapter 4, by applying special sequences of quadratic residues, several configurations are developed and they are used to construct MSSs of a given degree. The main results of this thesis are provided in Chapter 4 as well.

One of the most investigated topics in operations research is the Traveling Salesman Problem (TSP) and the algorithms that can be used to solve it. Despite its relatively simple formulation, its computational difficulty keeps it and potential solution methods at the forefront of current research. This paper defines and analyzes numerous proposed solutions to the TSP in order to facilitate understanding of the problem. Additionally, the efficiencies of different heuristics are studied and compared to the aforementioned algorithms’ accuracy, as a quick algorithm is often formulated at the expense of an exact solution.

In their 1976 paper, Nagel and Rudin characterize the closed unitarily and Möbius invariant spaces of continuous and Lp functions on a sphere, for 1≤p∞ functions on a sphere is not so easily solved. In this paper we provide an analogous solution by considering L∞ functions on a sphere with the weak*-topology. We further investigate the weak*-closed unitarily and Möbius invariant algebras of L∞ functions on a sphere.Each set of pairs of nonnegative integers induces a weak*-closed unitarily invariant space, and conversely each weak*-closed unitarily invariant space is induced by such a set. For algebras, we show that a set of pairs of nonnegative integers induces a weak*-closed unitarily invariant algebra of L∞ functions on a sphere if and only if the set induces a closed unitarily invariant algebra of continuous functions on a sphere. Thus, the same criterion which Rudin formulates in his 1979 paper to characterize the sets which induce closed unitarily invariant algebras of continuous functions on a sphere for dimension at least 3 also serves to characterize the sets which induce weak*-closed unitarily invariant algebras of L∞ functions on a sphere for dimension at least 3. Further, the same exceptions which arise in dimension 2 for continuous functions also arise in dimension 2 for L∞ functions.Finally, we determine the weak*-closed Möbius invariant spaces of L∞ functions on a sphere, of which there are six: the null space, the constant functions, the space of almost everywhere radial limits of holomorphic functions on the ball, the space of conjugates of members of the previous space, the weak*-closure of the algebraic sum of the previous two spaces, and the space of L∞ functions. Of these spaces, all are algebras except the fifth, which is not an algebra if and only if the dimension of the sphere is greater than 1.

In this doctoral dissertation, we consider the Cauchy problem for the 3D incompressible Navier-Stokes equations. Here, we are interested in a smooth periodic solution of the problem which happens to be a special case of a paper by Otto Kreiss and Jens Lorenz. More precisely, we will look into a special case of their paper by two approaches. In the first approach, we will try to follow the similar techniques as in the original paper for smooth periodic solution. Because of the involvement of the Fourier expansion in the process, we encounter with some intriguing factors in the periodic case which are absolutely not a part in the original paper. While in the second approach, we decompose our solution space using the Helmholtz-Weyl decomposition and introduce a new tool \say{the Leray projector} to eliminate the pressure term from the Navier-Stokes equations and go from there. This approach is completely different than the technique of dealing with the pressure term in the paper by Otto Kreiss and Jens Lorenz.

The purpose of this thesis is to examine how the exact penalty function provides solutions for nonsmooth optimization problems. It is known that under some conditions, an optimal solution of nonlinear program is also an optimal solution of the exact penalty function θp for sufficiently large parameter value. Therefore, if we find an optimal solution of the exact penalty function, this point may also be an optimal solution of the nonlinear program. We review two papers on the strict local minima of order m and the weak sharp minima of order m, an important class of nonisolated minima of the standard exact penalty function. This study presents the review of the two papers, an example to show the limitation of Corollary 3.1 in [1], and sufficient condition to ensure the optimal solution of the given example.

On the spectral triple of a noncommutative manifold $(\mathcal{A},H,D)$, despite the absence of underlying space of points, one can still consider its scalar curvature in terms of spectral information of the Dirac operator $D$, for example using short-time asymptotic expansion of the heat kernel $e^{-tD^2}$. In the recent decade, the conformal theory on a noncommutative two tori was firstly started by Connes and Tretkoff(nee Cohen), and later greatly developed by Connes, Moscovici and many others. Noncommutative conformal geometry on a noncommutative torus $\mathcal{A}_\theta$ is the study of quantized Gaussian curvature under noncommutative conformal change of ``metric'' by a positive operator-valued Weyl factor $k=e^h,h^*=h$. In this dissertation, by using Lesch and Moscovici's extension of Connes' pseudo-differential calculus to the Heisenberg modules, we will calculate the scalar curvature of a non-conformal change of metric by means of two commuting positive operator-valued factors $k_1,k_2$.\\ The first part of this paper, inspired by work by L.Dabrowski and S.Andrzej, contains extension of the rearrangement lemma that was systematized by M.Lesch, to non-conformal operators, by which we mean the elliptic operators with principal symbol $\sum_j k_j^2\xi_j^2$ with distinct $k_1,...,k_m$. By adapting the technique used by Y.Liu, we interpret the result of rearrangement as generalized hyper-geometric functions on Grassmannians, generalizing the conformal results of Y.Liu , namely when $k_1=k_2$. Second part of this paper consists of calculation of scalar curvature density associated to a non-conformal Laplacian operator $\Delta_{k_1,...,k_m}$ on a $m$-torus $\mathcal{A}_\Theta^m$. Third part is calculation of index density of a non-conformal Dirac operator $D_{k_1,k_2;\mathcal{E}(g,\theta)}$ on the Heisenberg module $\mathcal{E}(g,\theta)$. In appendix A, we will justify our terminology ``non-conformal''. We show that such a non-conformal Dirac operator on Heisenberg module amounts to a conformal change on the endomorphism algebra $\mathrm{End}_{\mathcal{A}^o_\theta}(\mathcal{T}^*(\mathcal{A}_\theta))$ of the cotangent bundle together with a change of complex structure. In appendix B, we put the elementary but crucial lemma of Gaussian averages, which will be used in the proof of extended rearrangement lemma. In appendix C, we list the definitions of hyper-geometric functions and their generalization, and propositions that will be needed in the statement of the rearrangement lemma.

This dissertation is comprised of two papers studying the topology of certain spaces of symplectic embeddings. The first paper shows how given two 4-dimensional ellipsoids whose symplectic sizes satisfy a specified inequality, a certain loop of symplectic embeddings between the two ellipsoids is noncontractible. The statement about symplectic ellipsoids is a particular case of a more general result which claims that given two convex toric domains whose first and second ECH capacities satisfy a specified inequality, one can prove that a certain loop of symplectic embeddings between the two convex toric domains is noncontractible.The second paper proves how given two 2n-dimensional symplectic ellipsoids whose symplectic sizes satisfy certain inequalities, a certain map from the n-torus to the space of symplectic embeddings from one ellipsoid to the other induces an injective map at the level of homology with mod 2 coefficients.

This work is divided into two parts which are concerned, respectively, with the combinatorics of the cardinals $\aleph_1$ and $\aleph_2$. The first part of the thesis contains the result due to the author and his advisor, Itay Neeman, that the Abraham-Rubin-Shelah Open Coloring Axiom is consistent with a large continuum; this answers a long-standing open question in forcing. Most of Part 1 appears in our submitted manuscript \cite{GN}. After surveying the relevant background in the first chapter, we proceed in the second chapter to define the notion of a Partition Product. This is a type of iteration built out of smaller ones in specific ways, roughly with memory conditions on the names and with isomorphism and coherence conditions on the various ``memories." We will prove a number of useful facts about partition products in Chapter 2. In Chapter 3, we show how to construct so-called Preassignments of Colors in the context of partition products; this forms the technical heart of Part 1. And finally, in Chapter 4, we show how to construct partition products in $L$; in particular, we construct the partition product which yields the model witnessing our theorem.Part 2 of the thesis addresses questions about a variety of combinatorial principles on $\aleph_2$. In each chapter in Part 2, we will be concerned with showing that some amount of Stationary Reflection holds at $\omega_2$, more specifically showing that various amounts of stationary reflection are compatible with other principles of wide interest. In Chapter 5, we provide an overview of these combinatorial principles and some of their history; we also spend some time collecting standard facts about Mitchell-type forcings which we will use in the subsequent chapters. Chapter 5 concludes with a proof that various Mitchell-type posets and their quotients are proper, a result which we assume is known, but which we have not encountered ourselves elsewhere.Chapters 6 and 7 address questions arising from the recent paper \emph{The Eightfold Way} by Cummings, Friedman, Magidor, Rinot, and Sinapova (see \cite{8fold}). In Chapter 6, we answer an open question asked at the end of that paper by showing that it is consistent, from a Mahlo cardinal, that the Tree Property $(\TP)$ and Approachability $(\AP)$ both fail at $\om_2$, while stationary reflection ($\RP$) holds at $\omega_2$. The authors of \cite{8fold} obtained the consistency of this same configuration from a weakly compact cardinal; our result proves the consistency of this configuration from optimal assumptions. We remark here that we present the original proof discovered by the author of this thesis. Later, the author, working with John Krueger, provided a more streamlined proof of this same result; this proof will appear in the forthcoming paper \cite{GiltonKrueger8}. Chapter 6 also includes an unrelated Easton-style lemma for preserving stationary subsets of countable cofinality; this result is due to the author and Omer Ben-Neria.In Chapter 7, we show that for any Boolean combination, $\Phi$, of $\TP$ and $\AP$, $\Phi$ is consistent with a strong form of simultaneous stationary reflection on $\omega_2$, namely that every stationary $S\seq\om_2\cap\cof(\om)$ reflects almost everywhere. This strengthens some of the results from \cite{8fold}. In Chapter 8, we return to the model from \cite{GiltonKruegerHS}, making good on a promise from the postscript therein. In \cite{GiltonKruegerHS}, the author and John Krueger originally sought to show that stationary reflection on $\om_2$ is consistent with a large continuum, and we built an involved mixed-support iteration to achieve such a model. However, we later learned from I. Neeman that such a model can be constructed by simply adding Cohen reals over the original Harrington-Shelah model (\cite{HS}). In Chapter 8 we will show that after a modification of our original preparatory iteration, we may obtain a model in which $\RP$ and $\AP$ both hold, in which $2^\om>\om_2$, and in which there are neither special Aronszajn trees on $\om_2$ nor weak Kurepa trees on $\omega_1$. This is a configuration which cannot be obtained simply by adding Cohen reals over the original Harrington-Shelah model nor by the methods of disjoint stationary sequences from \cite{GiltonKrueger8}. We hope that this demonstrates the usefulness of such a mixed support iteration.In the final chapter, we provide a list of open questions which we would like to address in future work.

Broadly speaking, this thesis investigates mathematical questions motivated by computer science. The involved topics include communication avoiding algorithms, classical analysis, convex geometry, and low-rank matrix approximation. In total, the thesis consists of four self-contained sections, each adapted from papers the author has been a part of.The first two sections are both motivated by the Brascamp-Lieb inequalities, which are also often referred to as Hölder-Brascamp-Lieb inequalities. These inequalities have featured prominently in recent theoretical computer science work, due to connections to geometric complexity theory, harmonic analysis, communication-avoidance, and many other areas. Moreover, work generalizing the inequalities in various ways, such as to nonlinear versions, has been impactful to the study of differential equations.Section 1 studies the application of Hölder-Brascamp-Lieb (HBL) inequalities to the design of communication optimal algorithms. In particular, it describes optimal tiling (blocking) strategies for nested loops that lack data dependencies and exhibit affine memory access patterns. The problem roughly amounts to maximizing the volume of an object provided some of its linear images have bounded volume. The methods used are algorithmic.Another reason for the interest in these inequalities is because they are an interesting test case for non-convex optimization techniques. The optimal constant for a particular instance of the inequality is given by solving a non-convex optimization problem that is still highly structured. Of particular relevance to this thesis is that it can be formulated as a geodesically-convex problem, considered in the context of the manifold of positive definite matrices of determinant $1$. Even using the methods of Section 1, the procedure is not necessarily polynomial time, and this motivates further study of geodesic convexity.This lead to the work of Section 2, which discusses a notion of halfspace for Hadamard manifolds that is natural in the context of convex optimization. For this notion of halfspace, we generalize a classic result of Grunbaum, which itself is a corollary of Helly's theorem. Namely, given a probability distribution on the manifold, there is a point for which all halfspaces based at this point have at least 1/(n+1) of the mass, n being the dimension of the manifold. As an application, the gradient oracle complexity of geodesic convex optimization is polynomial in the parameters defining the problem. In particular it is polynomial in -log(epsilon), where epsilon is the desired error. This is a step toward the open question of whether such an algorithm exists.The remaining two sections of the paper present a different research direction, randomized numerical linear algebra. Numerical linear algebra has long been an important part of scientific computing. Due to the current trend of increasing matrix sizes and growing importance of fast, approximate solutions in industry, randomized methods are quickly increasing in popularity. Sections 3 and 4 in this thesis aim to show that randomized low-rank approximation algorithms satisfy many of the properties of classical rank-revealing factorizations.Section 3 introduces a Generalized Randomized QR-decomposition (RURV) that may be applied to arbitrary products of matrices and their inverses, without needing to explicitly compute the products or inverses. This factorization is a critical part of a communication-optimal spectral divide-and-conquer algorithm for the nonsymmetric eigenvalue problem. In this paper, we establish that this randomized QR-factorization satisfies the strong rank-revealing properties. We also formally prove its stability, making it suitable in applications. Finally, we present numerical experiments which demonstrate that our theoretical bounds capture the empirical behavior of the factorization.Section 4 concerns a Generalized LU-Factorization (GLU) for low-rank matrix approximation. We relate this to past approaches and extensively analyze its approximation properties. The established deterministic guarantees are combined with sketching ensembles satisfying Johnson-Lindenstrauss properties to present complete bounds. Particularly good performance is shown for the sub-sampled randomized Hadamard transform (SRHT) ensemble. Moreover, the factorization is shown to unify and generalize many past algorithms. It also helps to explain the effect of sketching on the growth factor during Gaussian Elimination.

The studies selected for this paper includes traditional and technological methods used in classrooms. The headings for these in the paper include: Math Strategies, Technology Use Strategies, and Cooperative Math and Technology Use. The reasoning behind this is to show a comparison between the different results given in all formats.

SRB measures present a useful class of invariant measures that help understand the chaotic nature of random dynamical systems. Inspired by a paper from Alex Blumenthal and Lai-Sang Young from 2018 titled ”Equivalence of physical and SRB measures in random dynamical systems” this BSc project discusses the existence of SRB measures in chaotic random motions. We look at an innovative geometric construction of SRB measures via a graph transform approach that uses adapted one- sided charts. By pushing forward the stationary measures and approximating the unstable manifolds with pushed forward leaves we show that the sample measures are SRB.

In recent work published by Biswal, Chari, Shereen, and Wand the authors defined a family of symmetric polynomials indexed by pairs of dominant integral weights, G_{\nu, \lambda}(z,q) where z=(z_1, \cdots. z_{n+1})\in\C^{n+1}, and determined that G_{0, \lambda}(z,q) is the graded character of a level two Demazure module for sl_{n+1}[t]. The aim of this thesis is to construct analogues of these polynomials for the generalized Demazure modules for so_{2n}[t] as they are presented by Chari, Davis, and Moruzzi. We do this by constructing modules which interpolate from the presentation provided in that paper and local Weyl modules. We then create short exact sequences between them to relate their graded characters. This allows us to identify coefficients in the corresponding graded characters with the coefficients in G_{\nu,\lambda}(z,q).

Motivated by the Calabi-Yau/Landau-Ginzburg correspondence, we study BCOV-type torsion for Landau-Ginzburg B-models. To pave the way for understanding the Landau-Ginzburg model from the index theoretic point of view, we study Witten deformation, which was introduced in one of Witten's extremely influential papers in 1987, on noncompact manifolds. In Part I, we explore the analysis of Witten deformation on noncompact manifolds: we show the asymptotic growth of eigenvalues and the decay of eigenfunctions near infinity as well as the expansion and the estimate of the heat kernel for Schr\"odinger-type operators on noncompact manifolds. With heat kernel expansions of Sch\"ordinger-type operators, we define the Ray-Singer metric (analytic torsion) for the Witten deformation associated with some flat vector bundle and explore several nice properties of it (independence of metrics, Cheeger-Muller/Bismut-Zhang theorem, e.t.c.). Next, we move on to study Landau Ginzburg B-models in Part II. In chapter 6, we investigate the genus-zero theory for Landau-Ginzburg B-models, and establish Calabi-Yau/Landau-Ginzburg correspondence for the $tt^*$ structure and the Weil-Peterson type metric.

This dissertation investigates the origins of the completeness theorem for first-order predicate logic in the algebraic logic work of L\"{o}wenheim and Skolem. When G\"{o}del proved the completeness theorem in 1929, he was unaware that all the components of a completeness proof were already contained in earlier papers by L\"{o}wenheim and Skolem in which they prove the model-theoretic result known as the L\"{o}wenheim-Skolem theorem. This is not, however, a question of G\"{o}del’s completeness proof having been preempted. For neither were L\"{o}wenheim or Skolem show recognition of the result that G\"{o}del would later make explicit. When the similarity between the proofs was noticed in the 1950s, the fact that Skolem in particular had not put the pieces together to prove completeness before G\"{o}del seemed a puzzling oversight. G\"{o}del offered his own answer to the puzzle, appealing to alleged prejudices Skolem had against transfinite methods of reasoning. Chapter One shows how the puzzle emerged and how G\"{o}del purported to explain it. Chapters Two and Three give reconstructions of the original proofs of L\"{o}wenheim and G\"{o}del. I analyze the meaning of G\"{o}del’s claim that ``finitary prejudices’’ were at the heart of the failure to recognize completeness, and assess the evidence for this claim in L\"{o}wenheim and Skolem. Chapter Four reconstructs Skolem’s proof of L\"{o}wenheim’s theorem and establishes the technical background to understand the relation it bears to the completeness theorem. In Chapter 5, I argue that G\"{o}del’s own answer to the puzzle rests on a false premise. When certain contextual features are accounted for, Skolem’s failure to recognize a completeness theorem in his own work is not the oversight it now seems. I investigate Skolem’s search for a decidability proof for first-order logic and the role this played in leading Skolem away from the discovery of completeness.

This thesis aims to develop a comprehensive picture of large deviations of affine processes. Affine processes have been studied so extensively, that it would not be foolish to admit that their theory is somewhat complete. Though some questions still remain, the thesis of [cuchiero2011] establishes a comprehensive understanding of affine processes on general convex state spaces. Starting with the basic definition of an affine process as a time-homogeneous Markov process with conditional marginal distributions having a characteristic function with log-affine dependence on the initial state, this paper establishes numerous useful properties of the process. Among these properties include that affine processes have cadlag modifications (paths have left-limits and are right-continuous), and such modifications are themselves jump-diffusions with respect to their natural filtration. This result allowed [keller2015] to prove the transform formula for the real and complex moments, which establishes existence of the process and its finite moments, so long as a system of generalized Riccati differential equations have some solution (which need not be unique).Despite this very comprehensive understanding, large deviation asymptotics of families of affine processes do not have a similar general approach. Seeing as the class of affine processes is quite general, surely there are large deviation principles for certain families of affine processes. The simplest of which include families of state-independent processes like the Brownian motion or Poisson process, whose large deviations are understood asymptotically ε → 0 as time and space are scaled linearly with respective factors 1/ε and ε; this is a simple consequence of the classical Mogulskii's theorem (see [dembo2010]). More general results indeed exist, like [kang2014] which proves a large deviation principle for continuous affine processes on the so-called canonical state space. We are concerned with large deviation principles of similar asymptotics---those which could be interpreted as regularizing the noise of a dynamical system---but on the level of generality that affine processes are already understood.Though our original intent was to prove a large deviation principle for the general class of affine processes on convex state spaces, we discovered other remarkable facts along the way. The following two are of key interest. Firstly, the measure-change techniques one often uses in large deviations are typically split into those of a finite-dimensional (twisting/tilting) and infinite-dimensional (exponential martingale) flavor, but for affine processes one may in fact parameterize the finite-dimensional measure-densities in terms of the infinite-dimensional ones. This was not a fact seen in [kang2014], which---as they mentioned---forced them to choose a proof using exponential martingales. We show how their original finite-dimensional approach was sufficient in establishing an integral form for their rate function. Secondly, seeing as no large deviations result has been made on the general level of affine processes, there has been no attempt at understanding the structure of the associated rate functions. In our work of developing a proof for the principle, we saw the techniques mentioned in [duffy2004], which ultimately inspired us to develop a calculus associated with our rate functions. These results remarkably have no dependence on the affine assumption, and so if a future large deviation principle is to be proved for asymptotic families of jump-diffusions of the same regularization, this thesis will have already established semi-closed forms for the associated rate function.All this said, we do still prove a large deviation principle for the general class of affine processes on convex state spaces. While we require usual light-tail assumptions, we feel the generality is comparable to the treatment of affine processes in [cuchiero2011] and [keller2015]. In fact, we even choose to present affine processes with similar generality up until the point of us needing the assumption. At this juncture, we are able to represent the affine processes as special jump-diffusions, which have representations that are far more familiar to the larger audience that have studied Ito processes. For those that have only seen continuous diffusions, we also offer an Appendix which we feel is a great reference to the calculus associated with jump-diffusions.It goes without saying that our proof is not without the incredible theory of large deviations that already exists. We in particular discuss it at the level of understanding one has from reading [dembo2010], but we for the most part cite [feng2006] and its large deviation results on the Skorokhod space. We also use [puhalskii2001] to use a density argument in developing an integral form for our rate function.

The purpose of this paper is to analyze how to predict a business’s operation status during the spread of COVID-19 between the years 2020 to 2022. Exploratory Data Analysis techniques including barplot, geographical visualization, and word cloud are utilized. Three models are chosen to fit and predict business operation status: Logistic Regression, Support Vector Machine, and Bidirectional Long Short Term Memory. After tuning the probability cutoff threshold to 0.0510 with the F1 score, the logistic regression using the solver Newton's Conjugate Gradient generates an accuracy score of around 0.8120. Support Vector Machine algorithm leads to an accuracy score of around 0.9763. By using GloVe for word embedding, bidirectional LSTM results in a loss of 0.1113 and an accuracy score of 0.9758 after 2 epochs. From the performance of the three models, SVM seems to be the most effective and efficient in predicting business operation status through user reviews.

In this dissertation, we determine best-to-date upper bounds on the resolvent degree of solving general polynomials. Chapters 1 and 2 provide the necessary background information and Chapters 3 and 4 establish new results.In Chapter 1, we provide a high-level introduction of the dissertation and provide a history of the literature on resolvent degree. We also establish standard mathematical notation and terminology.In Chapter 2, Section 1, we recall the definitions of essential dimension and resolvent degree, as well as some standard results. In Section 2, we give an introduction to the theory of Tschirnhaus transformations and explain how we will use this theory to obtain upper bounds on resolvent degree by determining special points on Tschirnhaus complete intersections.In Chapter 3, Section 1, we recover the classical notion of the polars of a hypersurface at a point, so that we can introduce the polar cone of a hypersurface at a point and the connection between the polar cone at a point and lines on the hypersurface through that point. We then extend these notions to intersections of hypersurfaces and introduce iterated polar cones and their connections to k-planes in Section 2. Finally, we recover the obliteration algorithm of Sylvester and present it in a modern geometric context in Section 3.In Chapter 4, we establish our bounds on resolvent degree and Sections 1, 2, and 3 each highlight different constructions. In Section 4.4, we indicate obstructions to further bounds on resolvent degree via iterated polar cone constructions. We then proceed to establish approximations via elementary functions and compare the bounds we obtain with previous bounds in Sections 5 and 6. Finally, we posit several questions for future research in Section 7.Appendix A contains numerical information regarding our bounding function G'(m) and the previous best bounding function F(m), as well as implementations of the geometric obliteration algorithm. Appendix B contains three translations [Sut2019, Sut2021A, Sut2021B] of papers in the classical resolvent degree literature [Che1954, Kle1905, Wim1927].

A set is called Totally Multicolored (TMC) if no elements in the set are colored the same. For all natural numbers t, m, let R(t, m) be the least natural number n such that for every t-coloring of the set {1, 2, 3, ..., R(t, m)} there exist a solution set {x_1, x_2, . . ., x_m} to L(m), x_1 + x_2 + x_3 + ... + x_(m−1) = x_m such that x_i does not equal x_j for all i that does not equal j, that avoids being Totally Multicolored. This paper shows a function to find R(t,m) for any t greater than or equal to 1 and m greater than or equal to 3. See the abstract in the text for the function that gives R(t,m).

In order for an electrical network to be printed on a flat surface without changing the network's input or output, it is important to consider if any wires will cross and if this problem can be avoided. If a circular network can be printed so that no wires cross, the network is said to be circular planar. In this paper, we identify a number of wiring patterns that make circular planarity impossible. We find exactly 3 wiring patterns using circular pairs with sets of two nodes, and we find exactly 78 wiring patterns using circular pairs with sets of three nodes.

This dissertation is a compilation of five chapters, three of which are standalone papers. Chapter one is an introduction to video prompting, and I present the rationale for the following chapters. Chapter two is a systematic literature review on using video-based technology to teach academic skills to learners with moderate to profound intellectual disabilities. In chapter three, I describe a research study in which I used video prompting to teach the use of a number line to learners with moderate and severe intellectual disability. Chapter four is a practitioner paper focused on video prompting and prompt fading in the classroom and provides suggestions for planning the fading of introduced video prompts. Finally, in chapter five I present a review of my research during this doctoral program, including my dissertation, and future directions based on the current line, as well as potential related lines of study.

In 1932, Esther Klein made the observation that among any 5 points in general position in the plane, one can always find 4 points that form a convex quadrilateral. This problem came to be known as the Happy Ending Problem. She then posed the following generalization of her observation: how many points in the plane are needed to guarantee a convex n-gon, if such a number of points exists at all? Two of the mathematicians Klein was working with at the time, Paul Erdős and George Szekeres, published their progress on Klein’s generalized problem in 1935. Their paper proved that for any natural number n ≥ 3, there exists a minimal number ES(n) such that any configuration of ES(n) points in general position in the plane is guaranteed to contain a convex n-gon. They also attempted to find a formula for ES(n), and while they did not actually prove that ES(n) could be written as a function of n, they did conjecture that ES(n) = 2n-2 + 1. This has come to be known as the Erdős–Szekeres Conjecture. Despite the work of many mathematicians since, this conjecture still remains unproven. This paper details the history of the Happy Ending Problem and the Erdős–Szekeres Conjecture, provides proofs for values of ES(n) for some specific n, and discusses the progress that has been made toward proving the Erdős–Szekeres Conjecture.

This paper will be researching special properties of Rouquier blocks of partitions of integers. This problem is motivated by ongoing work in representation theory of the symmetric group. For each choice of integer parameters w and p with p = 2 and w = 1, there is an object called a Rouquier block; this block can be visualized as a collection of points in a plane, each corresponding to a partition. In this group of points, we say a pair of points is “connected” if certain conditions on the partitions are met. In this paper, we will be finding a formula that will restrict the diameter of this graph. We want to minimize the distance between the two points that are the furthest away from each other. A formula to give the most efficient path is either impossible to find or is too complicated to be useful; rather, we will set a ceiling on this distance, so that, given any two blocks, we will know the largest “most efficient” path length possible. Once this is done, we will also extend our problem to find the largest degree in a graph.

In their undergraduate studies, new calculus students learn how to integrate a function using the Riemann integral. The Riemann integral is used in Mathematics and Engineering and is a fairly simple concept to understand. However, the Riemann integral has some drawbacks and limitations. These limitations include improper integrals, application of the Fundamental theorem of Calculus, and others that will be discussed. By using the generalized Riemann integral, we can eliminate these limitations by only slightly changing our definition of the Riemann integral. However, what has been written on it isn’t particularly easy to understand. Because of this, the theory is only taught at the higher undergraduate or graduate level. This paper will take a look at the generalized Riemann integral. Being a more general version of the standard Riemann integral, it is not subject to these restrictions. The focus of this paper will be in one and two dimensions, but the theory from there can be easily generalized into higher dimensions. In this paper we will look at the generalized Riemann integral and will show all the ways all in which it is more powerful than the standard Riemann integral taught in calculus courses. While showing that the generalized Riemann is superior, we will also break the theory down for both the one and two-dimensional cases in a more understandable way.

Imagine two players are playing a strategy board game. With one player holding black stones and the other player holding white stones, they take turns to place their pieces. After a while, they start to recognize some patterns. With three colors in front of them, black, white, and the color of the game board, they are curious about if they can form a recognizable image. Instead of playing the game to win, they start to collaborate with each other to form a mosaic. Later, they invite more players and provide them their own stones in different shades of gray. The goal of this project is to use mathematical optimization techniques to design mosaics that simultaneously resemble well-known images and look like the current state of a board game like Gomoku or something that could be played on a Go board or chessboard. The work described here builds upon techniques devised by Professor Robert Bosch and some of his students. The paper will be structured as follows: First, we will discuss the data, variables, objective function, and universal constraints. Then we will explore constraints for two specific games, Gomoku and chess. Next, we will present several examples, sharing both images and statistics, and we will talk about methods we have devised for improving the results. We will conclude with a discussion of future directions for our research.