Academic Commons Search Results
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Academic Commons Search Resultsen-usBifurcation of localized eigenstates of perturbed periodic Schrödinger operators
http://academiccommons.columbia.edu/catalog/ac:182973
Vukicevic, Ivahttp://dx.doi.org/10.7916/D88C9V2ZThu, 05 Feb 2015 00:00:00 +0000A spatially localized initial condition for an energy-conserving wave equation with periodic coefficients disperses (spatially spreads) and decays as time advances. This dispersion is associated with the continuous spectrum of the underlying differential operator and the absence of discrete eigenvalues. The introduction of spatially localized perturbations in a periodic medium leads to ``defect modes'', states in which the wave is spatially localized and periodic in time. These modes are associated with eigenvalues which bifurcate from the continuous spectrum induced by the perturbation. This thesis investigates specific families of perturbations of one-dimensional periodic Schrödinger operators and studies the resulting bifurcating eigenvalues from the unperturbed continuous spectrum. For Q(x) a real-valued periodic function, the Schrödinger operator H_Q = -∂_x^2 + Q(x) has a continuous spectrum equal to the union of closed intervals, called spectral bands, separated by open spectral gaps. We find that upon the introduction of a bounded, ``small'', and sufficiently decaying perturbation W(x), the spectrum of H_{Q+W} has discrete eigenvalues (with corresponding eigenstates which are exponentially decaying in |x|) which lie in the open spectral gaps of H_Q. Our analysis covers two large classes of perturbations W(x): 1. W(x) = λ V(x), 0<λ ≪ 1, and V(x) sufficiently rapidly decaying as x → ± ∞; 2. W(x) = q(x, x/ε), 0<ε ≪ 1, where x ⟼ q(x,y) is spatially localized, q(x,y+1) = q(x,y) for x ∈ ℝ, and y ⟼ q(x,y) has mean zero. In Case 1. W(x) corresponds to a small and localized absolute change in the medium's material properties. In Case 2. W(x) corresponds to a high-contrast microstructure. Q(x) + W(x) may be pointwise very large, but on average it is a small perturbation of Q(x).Applied mathematics, Mathematicsiv2143Applied Physics and Applied MathematicsDissertationsEquivariant Gromov-Witten Theory of GKM Orbifolds
http://academiccommons.columbia.edu/catalog/ac:180940
Zong, Zhengyuhttp://dx.doi.org/10.7916/D8513WZCThu, 04 Dec 2014 00:00:00 +0000In this paper, we study the all genus Gromov-Witten theory for any GKM orbifold X. We generalize the Givental formula which is studied in the smooth case in [41] [42] [43] to the orbifold case. Specifically, we recover the higher genus Gromov-Witten invariants of a GKM orbifold X by its genus zero data. When X is toric, the genus zero Gromov-Witten invariants of X can be explicitly computed by the mirror theorem studied in [22] and our main theorem gives a closed formula for the all genus Gromov-Witten invariants of X. When X is a toric Calabi-Yau 3-orbifold, our formula leads to a proof of the remodeling conjecture in [38]. The remodeling conjecture can be viewed as an all genus mirror symmetry for toric Calabi-Yau 3-orbifolds. In this case, we apply our formula to the A-model higher genus potential and prove the remodeling conjecture by matching it to the B-model higher genus potential.MathematicsZhengyuMathematicsDissertationsA General Class of Heuristics for Minimum Weight Perfect Matching and Fast Special Cases with Doubly and Triply Logarithmic Errors
http://academiccommons.columbia.edu/catalog/ac:177681
Imielinska, Celina Z.; Kalantari, B.http://dx.doi.org/10.7916/D8SJ1J5XMon, 29 Sep 2014 00:00:00 +0000We give a class of heuristic algorithms for minimum weight perfect matching on a complete edgeweighted graph K(V) satisfying the triangle inequality, where V is a set of an even number, n, of vertices.This class is a generalization of the Onethird heuristics, the hypergreedy heuristic, and it possibly employs any given exact or approximate perfect matching algorithm as an auxiliary heuristic to an appropriate subgraph of K(V).Computer science, Mathematicsci42Biomedical InformaticsArticlesA Generalized Hypergreedy Algorithm for Weighted Perfect Matching
http://academiccommons.columbia.edu/catalog/ac:177678
Imielinska, Celina Z.; Kalantari, Bahmanhttp://dx.doi.org/10.7916/D8222SB3Mon, 29 Sep 2014 00:00:00 +0000We give a generalization of the hypergreedy algorithm for minimum weight perfect matching on a complete edge weighted graph whose weights satisfy the triangle inequality.Computer science, Mathematicsci42Biomedical InformaticsArticlesA Spacetime Alexandrov Theorem
http://academiccommons.columbia.edu/catalog/ac:175978
Wang, Ye-Kaihttp://dx.doi.org/10.7916/D8MG7MN2Mon, 07 Jul 2014 00:00:00 +0000Let Σ be an embedded spacelike codimension-2 submanifold in a spherically symmetric spacetime satisfying null convergence condition. Suppose Σ has constant null mean curvature and zero torsion. We prove that Σ must lie in a standard null cone. This generalizes the classical Alexandrov theorem which classifies embedded constant mean curvature hypersurfaces in Euclidean space. The proof follows the idea of Ros and Brendle. We first derive a spacetime Minkowski formula for spacelike codimension-2 submanifolds using conformal Killing-Yano 2-forms. The Minkowski formula is then combined with a Heintze-Karcher type geometric inequality to prove the main theorem. We also obtain several rigidity results for codimension-2 submanifolds in spherically symmetric spacetimes.MathematicsMathematicsDissertationsMultiple Dirichlet Series for Affine Weyl Groups
http://academiccommons.columbia.edu/catalog/ac:176818
Whitehead, Ianhttp://dx.doi.org/10.7916/D8BK19HTMon, 07 Jul 2014 00:00:00 +0000Let W be the Weyl group of a simply-laced affine Kac-Moody Lie group, excepting type A affine root systems of even rank. We construct a multiple Dirichlet series Z(x_1, ... x_n+1 meromorphic in a half-space, satisfying a group W of functional equations. This series is analogous to the multiple Dirichlet series for classical Weyl groups constructed by Brubaker-Bump-Friedberg, Chinta-Gunnells, and others. It is completely characterized by four natural axioms concerning its coefficients, axioms which come from the geometry of parameter spaces of hyperelliptic curves. The series constructed this way is optimal for computing moments of character sums and L-functions, including the fourth moment of quadratic L-functions at the central point via affine D4 and the second moment weighted by the number of divisors of the conductor via affine A_3. We also give evidence to suggest that this series appears as a first Fourier-Whittaker coefficient in an Eisenstein series on the twofold metaplectic cover of the relevant Kac-Moody group. The construction is limited to the rational function field, but it also describes the p-part of the multiple Dirichlet series over an arbitrary global field.MathematicsMathematicsDissertationsOn a triply-graded generalization of Khovanov homology
http://academiccommons.columbia.edu/catalog/ac:175996
Putyra, Krzysztofhttp://dx.doi.org/10.7916/D86971RXMon, 07 Jul 2014 00:00:00 +0000In this thesis we study a certain generalization of Khovanov homology that unifies both the original theory due to M. Khovanov, referred to as the even Khovanov homology, and the odd Khovanov homology introduced by P. Ozsv´ath, Z. Szab´o, and J. Rasmussen. The generalized Khovanov complex is a variant of the formal Khovanov bracket introduced by Bar Natan, constructed in a certain 2-categorical extension of cobordisms, in which the disjoint union is a cubical 2-functor, but not a strict one. This allows us to twist the usual relations between cobordisms with signs or, more generally, other invertible scalars. We prove the homotopy type of the complex is a link invariant, and we show how both even and odd Khovanov homology can be recovered. Then we analyze other link homology theories arising from this construction such as a unified theory over the ring Z_p :=Z[p]/(p²−1), and a variant of the algebra of dotted cobordisms, defined over k := Z[X,Y,Z^±1]/(X² = Y² = 1). The generalized chain complex is bigraded, but the new grading does not make it a stronger invariant. However, it controls up to some extend signs in the complex, the property we use to prove several properties of the generalized Khovanov complex such as multiplicativity with respect to disjoint unions and connected sums of links, and the duality between complexes for a link and its mirror image. In particular, it follows the odd Khovanov homology of anticheiral links is self-dual. Finally, we explore Bockstein-type homological operations, proving the unified theory is a finer invariant than the even and odd Khovanov homology taken together.Theoretical mathematics, MathematicsMathematicsDissertationsRational normal curves on complete intersections
http://academiccommons.columbia.edu/catalog/ac:175993
Pan, Xuanyuhttp://dx.doi.org/10.7916/D8KK98X0Mon, 07 Jul 2014 00:00:00 +0000We prove that the moduli space of rational normal curves on a low degree complete intersection passing several suitable points is a complete intersection.MathematicsMathematicsDissertationsDemazure-Lusztig Operators and Metaplectic Whittaker Functions on Covers of the General Linear Group
http://academiccommons.columbia.edu/catalog/ac:176190
Puskas, Annahttp://dx.doi.org/10.7916/D8J964J6Mon, 07 Jul 2014 00:00:00 +0000There are two different approaches to constructing Whittaker functions of metaplectic groups over non-archimedean local fields. One approach, due to Chinta and Offen for the general linear group and to McNamara in general, represents the spherical Whittaker function in terms of a sum over a Weyl group. The second approach, by Brubaker, Bump and Friedberg and separately by McNamara, expresses it as a sum over a highest weight crystal. This work builds a direct, combinatorial connection between the two approaches. This is done by exploring both in terms of Demazure and Demazure-Lusztig operators associated to the Weyl group of an irreducible root system. The relevance of Demazure and Demazure-Lusztig operators is indicated by results in the non-metaplectic setting: the Demazure character formula, Tokuyama's theorem and the work of Brubaker, Bump and Licata in describing Iwahori-Whittaker functions. The first set of results is joint work with Gautam Chinta and Paul E. Gunnells. We define metaplectic Demazure and Demazure-Lusztig operators for a root system of any type. We prove that they satisfy the same Braid relations and quadratic relations as their nonmetaplectic analogues. Then we prove two formulas for the long word in the Weyl group. One is a metaplectic generalization of Demazure's character formula, and the other connects the same expression to Demazure-Lusztig operators. Comparing the two results to McNamara's construction of metaplectic Whittaker functions results in a formula for the Whittaker functions in the spirit of the Demazure character formula. The second set of results relates to Tokuyama's theorem about the crystal description of type A characters. We prove a metaplectic generalization of this theorem. This establishes a combinatorial link between the two approaches to constructing Whittaker functions for metaplectic covers of any degree. The metaplectic version of Tokuyama's theorem is proved as a special case of a stronger result: a crystal description of polynomials produced by sums of Demazure-Lusztig operators acting on a monomial. These results make use of the Demazure and Demazure-Lusztig formulas above, and the branching structure of highest weight crystals of type A. The polynomials produced by sums of Demazure-Lusztig operators acting on a monomial are related to Iwahori fixed Whittaker functions in the nonmetaplectic setting.MathematicsMathematicsDissertationsTowards a definition of Shimura curves in positive characteristics
http://academiccommons.columbia.edu/catalog/ac:176077
Xia, Jiehttp://dx.doi.org/10.7916/D8ZP448CMon, 07 Jul 2014 00:00:00 +0000In the thesis, we present some answers to the question What is an appropriate definition of Shimura curves in positive characteristics ? The answer is obvious for Shimura curves of PEL type due to the moduli interpretation. Thus what is more interesting is the answer on Shimura curves of Hodge type. Inspired by an example constructed by David Mumford, we find conditions on a proper smooth curve over a field of positive characteristic which guarantee that it lifts to a Shimura curve of Hodge type over the complex numbers. These conditions are in terms of geometry mod p, such as Barsotti-Tate groups, Dieudonne isocrystals, crystalline Hodge cycles and l-adic monodromy. Thus one can take them as definitions of Shimura curves in positive characteristics. More generally, We define ``weak" Shimura curves in characteristic p. Along the way, we prove if a Barsotti-Tate group is versally deformed over a proper curve over an algebraically closed field of positive characteristic, then it admits a unique deformation to the corresponding Witt ring. This deformation result serves as one of the key ingredients in the proofs.Mathematicsjx2149MathematicsDissertationsProbabilistic Approaches to Partial Differential Equations with Large Random Potentials
http://academiccommons.columbia.edu/catalog/ac:175891
Gu, Yuhttp://dx.doi.org/10.7916/D82R3PTDMon, 07 Jul 2014 00:00:00 +0000The thesis is devoted to an analysis of the heat equation with large random potentials in high dimensions. The size of the potential is chosen so that the large, highly oscillatory, random field is producing non-trivial effects in the asymptotic limit. We prove either homogenization, i.e., the random potential is replaced by some deterministic constant, or convergence to a stochastic partial differential equation, i.e., the random potential is replaced by some stochastic noise, depending on the correlation property. When the limit is deterministic, we provide estimates of the error between the heterogeneous and homogenized solutions when certain mixing assumption of the random potential is satisfied. We also prove a central limit type of result when the random potential is Gaussian or Poissonian. Lower dimensional and time-dependent cases are also treated. Most of the ingredients in the analysis are probabilistic, including a Feynman-Kac representation, a Brownian motion in random scenery, the Kipnis-Varadhan's method, and a quantitative martingale central limit theorem.Mathematicsyg2254Applied Physics and Applied MathematicsDissertations'Value Creation' Through Mathematical Modeling: Students' Mathematics Dispositions and Identities Developed in a Learning Community
http://academiccommons.columbia.edu/catalog/ac:176803
Park, Joo younghttp://dx.doi.org/10.7916/D87S7KXXMon, 07 Jul 2014 00:00:00 +0000This study examines how mathematical modeling activities within a collaborative group impact students' `value creation' through mathematics. Creating `value' in this study means to apply one's knowledge in a way that benefits the individual and society, and the notion of `value' was adopted from Makiguchi's theory of `value creation' (1930/1989). With a unified framework of Makiguchi's theory of `value', mathematical disposition, and identity, the study identified three aspects of value-beauty, gains, and social good-using observable evidence of mathematical disposition, identity, and sense of community. Sixty students who enrolled in a college algebra course participated in the study. The results showed significant changes in students' mathematics dispositions after engaging in the modeling activities. Analyses of students' written responses and interview data demonstrated that the modeling tasks associated with students' personal data and social interactions within a group contributed to students' developing their identity as doers of mathematics and creating social value. The instructional model aimed to balance the cognitive aspect and the affective skills of learning mathematics in a way that would allow students to connect mathematical concepts to their personal lives and social lives. As a result of the analysis of this study, there emerged a holistic view of the classroom as it reflects the Makiguchi's educational philosophy. Lastly, implications of this study for research and teaching are discussed.Mathematics education, MathematicsMathematics, Science, and Technology, Mathematics EducationDissertationsSequential Optimization in Changing Environments: Theory and Application to Online Content Recommendation Services
http://academiccommons.columbia.edu/catalog/ac:176086
Gur, Yonatanhttp://dx.doi.org/10.7916/D8639MWFMon, 07 Jul 2014 00:00:00 +0000Recent technological developments allow the online collection of valuable information that can be efficiently used to optimize decisions "on the fly" and at a low cost. These advances have greatly influenced the decision-making process in various areas of operations management, including pricing, inventory, and retail management. In this thesis we study methodological as well as practical aspects arising in online sequential optimization in the presence of such real-time information streams. On the methodological front, we study aspects of sequential optimization in the presence of temporal changes, such as designing decision making policies that adopt to temporal changes in the underlying environment (that drives performance) when only partial information about this changing environment is available, and quantifying the added complexity in sequential decision making problems when temporal changes are introduced. On the applied front, we study practical aspects associated with a class of online services that focus on creating customized recommendations (e.g., Amazon, Netflix). In particular, we focus on online content recommendations, a new class of online services that allows publishers to direct readers from articles they are currently reading to other web-based content they may be interested in, by means of links attached to said article. In the first part of the thesis we consider a non-stationary variant of a sequential stochastic optimization problem, where the underlying cost functions may change along the horizon. We propose a measure, termed {\it variation budget}, that controls the extent of said change, and study how restrictions on this budget impact achievable performance. As a yardstick to quantify performance in non-stationary settings we propose a regret measure relative to a dynamic oracle benchmark. We identify sharp conditions under which it is possible to achieve long-run-average optimality and more refined performance measures such as rate optimality that fully characterize the complexity of such problems. In doing so, we also establish a strong connection between two rather disparate strands of literature: adversarial online convex optimization; and the more traditional stochastic approximation paradigm (couched in a non-stationary setting). This connection is the key to deriving well performing policies in the latter, by leveraging structure of optimal policies in the former. Finally, tight bounds on the minimax regret allow us to quantify the "price of non-stationarity," which mathematically captures the added complexity embedded in a temporally changing environment versus a stationary one. In the second part of the thesis we consider another core stochastic optimization problem couched in a multi-armed bandit (MAB) setting. We develop a MAB formulation that allows for a broad range of temporal uncertainties in the rewards, characterize the (regret) complexity of this class of MAB problems by establishing a direct link between the extent of allowable reward "variation" and the minimal achievable worst-case regret, and provide an optimal policy that achieves that performance. Similarly to the first part of the thesis, our analysis draws concrete connections between two strands of literature: the adversarial and the stochastic MAB frameworks. The third part of the thesis studies applied optimization aspects arising in online content recommendations, that allow web-based publishers to direct readers from articles they are currently reading to other web-based content. We study the content recommendation problem and its unique dynamic features from both theoretical as well as practical perspectives. Using a large data set of browsing history at major media sites, we develop a representation of content along two key dimensions: clickability, the likelihood to click to an article when it is recommended; and engageability, the likelihood to click from an article when it hosts a recommendation. Based on this representation, we propose a class of user path-focused heuristics, whose purpose is to simultaneously ensure a high instantaneous probability of clicking recommended articles, while also optimizing engagement along the future path. We rigorously quantify the performance of these heuristics and validate their impact through a live experiment. The third part of the thesis is based on a collaboration with a leading provider of content recommendations to online publishers.Operations research, Business, MathematicsBusinessDissertationsConvex Optimization Algorithms and Recovery Theories for Sparse Models in Machine Learning
http://academiccommons.columbia.edu/catalog/ac:175385
Huang, Bohttp://dx.doi.org/10.7916/D8VM49DMMon, 07 Jul 2014 00:00:00 +0000Sparse modeling is a rapidly developing topic that arises frequently in areas such as machine learning, data analysis and signal processing. One important application of sparse modeling is the recovery of a high-dimensional object from relatively low number of noisy observations, which is the main focuses of the Compressed Sensing, Matrix Completion(MC) and Robust Principal Component Analysis (RPCA) . However, the power of sparse models is hampered by the unprecedented size of the data that has become more and more available in practice. Therefore, it has become increasingly important to better harnessing the convex optimization techniques to take advantage of any underlying "sparsity" structure in problems of extremely large size. This thesis focuses on two main aspects of sparse modeling. From the modeling perspective, it extends convex programming formulations for matrix completion and robust principal component analysis problems to the case of tensors, and derives theoretical guarantees for exact tensor recovery under a framework of strongly convex programming. On the optimization side, an efficient first-order algorithm with the optimal convergence rate has been proposed and studied for a wide range of problems of linearly constraint sparse modeling problems.Mathematics, Statistics, Operations researchIndustrial Engineering and Operations ResearchDissertationsThree-Manifold Mutations Detected by Heegaard Floer Homology
http://academiccommons.columbia.edu/catalog/ac:175403
Clarkson, Corrinhttp://dx.doi.org/10.7916/D8GF0RNGMon, 07 Jul 2014 00:00:00 +0000Given a self-diffeomorphism h of a closed, orientable surface S with genus greater than one and an embedding f of S into a three-manifold M, we construct a mutant manifold by cutting M along f(S) and regluing by h. We will consider whether there exist nontrivial gluings such that for any embedding, the manifold M and its mutant have isomorphic Heegaard Floer homology. In particular, we will demonstrate that if h is not isotopic to the identity map, then there exists an embedding of S into a three-manifold M such that the rank of the non-torsion summands of HF-hat of M differs from that of its mutant. We will also show that if the gluing map is isotopic to neither the identity nor the genus-two hyperelliptic involution, then there exists an embedding of S into a three-manifold M such that the total rank of HF-hat of M differs from that of its mutant.MathematicsMathematicsDissertationsBordered Heegaard Floer Homology and Graph Manifolds
http://academiccommons.columbia.edu/catalog/ac:175430
Hanselman, Jonathanhttp://dx.doi.org/10.7916/D8NZ85TFMon, 07 Jul 2014 00:00:00 +0000We use the techniques of bordered Heegaard Floer homology to investigate the Heegaard Floer homology of graph manifolds. Bordered Heegaard Floer homology allows us to split a graph manifold into pieces and perform computations for each piece separately. The resulting invariants can then be combined by a simple algebraic procedure to recover HFhat. Graph manifolds by definition decompose into pieces which are S¹-bundles over surfaces. This decomposition makes them particularly well suited to the divide-and-conquer techniques of bordered Heegaard Floer homology. In fact, the problem reduces to computing bordered Heegaard Floer invariants of just two pieces. The first invariant is the type D trimodule associated to the trivial S¹-bundle over the pair of pantsMathematicsMathematicsDissertationsConstant Scalar Curvature of Toric Fibrations
http://academiccommons.columbia.edu/catalog/ac:175513
Nyberg, Thomashttp://dx.doi.org/10.7916/D8TH8JVHMon, 07 Jul 2014 00:00:00 +0000We study the conditions under which a fibration of toric varieties, fibered over a flag variety, admits a constant scalar curvature Kähler metric. We first provide an introduction to toric varieties and toric fibrations and derive the scalar curvature equation. Next we derive interior a priori estimates of all orders and a global L^∞-estimate for the scalar curvature equation. Finally we extend the theory of K-Stability to this setting and construct test-configurations for these spaces.Mathematicstwn2103MathematicsDissertationsThe arithmetic and geometry of genus four curves
http://academiccommons.columbia.edu/catalog/ac:175469
Xue, Hanghttp://dx.doi.org/10.7916/D87P8WHMMon, 07 Jul 2014 00:00:00 +0000We construct a point in the Jacobian of a non-hyperelliptic genus four curve which is defined over a quadratic extension of the base field. We attempt to answer two questions: 1. Is this point torsion? 2. If not, does it generate the Mordell--Weil group of the Jacobian? We show that this point generates the Mordell--Weil group of the Jacobian of the universal genus four curve. We construct some families of genus four curves over the function field of $\bP^1$ over a finite field and prove that half of the Jacobians in this family are generated by this point via the other half are not. We then turn to the case where the base field is a number field or a function field. We compute the Neron--Tate height of this point in terms of the self-intersection of the relative dualizing sheaf of (the stable model of) the curve and some local invariants depending on the completion of the curve at the places where this curve has bad or smooth hyperelliptic reduction. In the case where the reduction satisfies some certain conditions, we compute these local invariants explicitly.Mathematicshx2119MathematicsDissertationsSelf-duality and singularities in the Yang-Mills flow
http://academiccommons.columbia.edu/catalog/ac:175717
Waldron, Alexhttp://dx.doi.org/10.7916/D81V5C3RMon, 07 Jul 2014 00:00:00 +0000We investigate the long-time behavior and smooth convergence properties of the Yang-Mills flow in dimension four. Two chapters are devoted to equivariant solutions and their precise blowup asymptotics at infinite time. The last chapter contains general results. We show that a singularity of pure + or - charge cannot form within finite time, in contrast to the analogous situation of harmonic maps between Riemann surfaces. This implies long-time existence given low initial self-dual energy. In this case we study convergence of the flow at infinite time: if a global weak Uhlenbeck limit is anti-self-dual and has vanishing self-dual second cohomology, then the limit exists smoothly and exponential convergence holds. We also recover the classical grafting theorem, and derive asymptotic stability of this class of instantons in the appropriate sense.MathematicsMathematicsDissertationsPro-p-Iwahori-Hecke Algebras in the mod-p Local Langlands Program
http://academiccommons.columbia.edu/catalog/ac:175711
Koziol, Karolhttp://dx.doi.org/10.7916/D89C6VKVMon, 07 Jul 2014 00:00:00 +0000Let p be a prime number, and F a nonarchimedean local field of residual characteristic p. This thesis is dedicated to the study of the pro-p-Iwahori-Hecke algebra H_{F_p}(G, I(1)) in the mod-p Local Langlands Program, where G is the group of F-points of a connected, reductive group, and I(1) is a pro-p-Iwahori subgroup of G. When G = U(2,1)(E/F) is an unramified unitary group in three variables, we first describe the structure and simple modules of the algebra H_{F_p}(G, I(1)). We then adapt methods of Schneider-Stuhler and Paskunas to construct, for each supersingular H_{F_p}(G, I(1))-module, a supersingular representation of G. These are exactly the representations which are expected to correspond to irreducible Galois parameters. When G = U(1,1)(Q_{p^2} /Q_p) is an unramified unitary group in two variables, we use the pro-p-Iwahori-Hecke algebra H_{F_p}(G_S , I_S(1)) of the derived subgroup G_S to classify the supersingular representations of G. Combining this with previous results, we obtain a classification of all irreducible representations of G, and then construct a correspondence between representations of G and Galois parameters. Finally, when G = GL_n(F) and G_S = SL_n(F), we show how to relate the two algebras H_{F_p}(G, I(1)) and H_{F_p}(G_S, I_S(1)). Using this interplay, we prove a numerical correspondence between L-packets of supersingular H_{F_p}(G_S , I_S(1))-modules and irreducible projective n-dimensional Galois representations, and prove that this correspondence is induced by a functor when F = Q_p.MathematicsMathematicsDissertationsCanonical Metrics in Sasakian Geometry
http://academiccommons.columbia.edu/catalog/ac:175504
Collins, Tristanhttp://dx.doi.org/10.7916/D86Q1VCSMon, 07 Jul 2014 00:00:00 +0000The aim of this thesis is to study the existence problem for canonical Sasakian metrics, primarily Sasaki-Einstein metrics. We are interested in providing both necessary conditions, as well as sufficient conditions for the existence of such metrics. We establish several sufficient conditions for the existence of Sasaki-Einstein metrics by studying the Sasaki-Ricci flow. In the process, we extend some fundamental results from the study of the Kahler-Ricci flow to the Sasakian setting. This includes finding Sasakian analogues of Perelman's energy and entropy functionals which are monotonic along the Sasaki-Ricci flow. Using these functionals we extend Perelman's deep estimates for the Kahler-Ricci flow to the Sasaki-Ricci flow. Namely, we prove uniform scalar curvature, diameter and non-collapsing estimates along the Sasaki-Ricci flow. We show that these estimates imply a uniform transverse Sobolev inequality. Furthermore, we introduce the sheaf of transverse foliate vector fields, and show that it has a natural, transverse complex structure. We show that the convergence of the flow is intimately related to the space of global transversely holomorphic sections of this sheaf. We introduce an algebraic obstruction to the existence of constant scalar curvature Sasakian metrics, extending the notion of K-stability for projective varieties. Finally, we show that, for regular Sasakian manifolds whose quotients are Kahler-Einstein Fano manifolds, the Sasaki-Ricci flow, or equivalently, the Kahler-Ricci flow, converges exponentially fast to a (transversely) Kahler-Einstein metric.Mathematicstcc2119MathematicsDissertationsA Characterization of Markov Equivalence Classes for Acyclic Digraphs
http://academiccommons.columbia.edu/catalog/ac:173896
Andersson, Steen A.; Madigan, David B.; Perlman, Michael D.http://dx.doi.org/10.7916/D8FX77J3Thu, 15 May 2014 00:00:00 +0000Undirected graphs and acyclic digraphs (ADG's), as well as their mutual extension to chain graphs, are widely used to describe dependencies among variables in multiviarate distributions. In particular, the likelihood functions of ADG models admit convenient recursive factorizations that often allow explicit maximum likelihood estimates and that are well suited to building Bayesian networks for expert systems. Whereas the undirected graph associated with a dependence model is uniquely determined, there may be many ADG's that determine the same dependence (i.e., Markov) model. Thus, the family of all ADG's with a given set of vertices is naturally partitioned into Markov-equivalence classes, each class being associated with a unique statistical model. Statistical procedures, such as model selection of model averaging, that fail to take into account these equivalence classes may incur substantial computational or other inefficiences. Here it is show that each Markov-equivalence class is uniquely determined by a single chain graph, the essential graph, that is itself simultaneously Markov equivalent to all ADG's in the equivalence class. Essential graphs are characterized, a polynomial-time algorithm for their construction is given, and their applications to model selection and other statistical questions are described.Mathematics, Statistics, Theoretical mathematicsdm2418StatisticsArticlesA One-Pass Sequential Monte Carlo Method for Bayesian Analysis of Massive Datasets
http://academiccommons.columbia.edu/catalog/ac:173899
Balakrishnan, Suhrid; Madigan, David B.http://dx.doi.org/10.7916/D8B56GTPThu, 15 May 2014 00:00:00 +0000For Bayesian analysis of massive data, Markov chain Monte Carlo (MCMC) techniques often prove infeasible due to computational resource constraints. Standard MCMC methods generally require a complete scan of the dataset for each iteration. Ridgeway and Madigan (2002) and Chopin (2002b) recently presented importance sampling algorithms that combined simulations from a posterior distribution conditioned on a small portion of the dataset with a reweighting of those simulations to condition on the remainder of the dataset. While these algorithms drastically reduce the number of data accesses as compared to traditional MCMC, they still require substantially more than a single pass over the dataset. In this paper, we present "1PFS," an efficient, one-pass algorithm. The algorithm employs a simple modification of the Ridgeway and Madigan (2002) particle filtering algorithm that replaces the MCMC based "rejuvenation" step with a more efficient "shrinkage" kernel smoothing based step. To show proof-of-concept and to enable a direct comparison, we demonstrate 1PFS on the same examples presented in Ridgeway and Madigan (2002), namely a mixture model for Markov chains and Bayesian logistic regression. Our results indicate the proposed scheme delivers accurate parameter estimates while employing only a single pass through the data.Mathematics, Statisticsdm2418StatisticsArticlesCorrection: Separation and completeness properties for AMP chain graph Markov models
http://academiccommons.columbia.edu/catalog/ac:173887
Madigan, David B.; Levitz, Michael; Perlman, Michael D.http://dx.doi.org/10.7916/D8QF8R05Wed, 14 May 2014 00:00:00 +0000Correction of table 2 on page 1757 of 'Separation and completeness properties for AMP chain graph Markov models', Annals of Statistics, volume 29 (2001).Mathematics, Statisticsdm2418StatisticsArticlesA Flexible Bayesian Generalized Linear Model for Dichotomous Response Data with an Application to Text Categorization
http://academiccommons.columbia.edu/catalog/ac:173817
Eyheramendy, Susana; Madigan, David B.http://dx.doi.org/10.7916/D86M34ZFTue, 13 May 2014 00:00:00 +0000We present a class of sparse generalized linear models that include probit and logistic regression as special cases and offer some extra flexibility. We provide an EM algorithm for learning the parameters of these models from data. We apply our method in text classification and in simulated data and show that our method outperforms the logistic and probit models and also the elastic net, in general by a substantial margin.Mathematics, Statistics, Theoretical mathematicsdm2418StatisticsBook chaptersLocation Estimation in Wireless Networks: A Bayesian Approach
http://academiccommons.columbia.edu/catalog/ac:173820
Madigan, David B.; Ju, Wen-Hua; Krishnan, P.; Krishnakumar, A. S. ; Zorych, Ivanhttp://dx.doi.org/10.7916/D82V2D74Tue, 13 May 2014 00:00:00 +0000We present a Bayesian hierarchical model for indoor location estimation in wireless networks. We demonstrate that out model achieves accuracy that is similar to other published models and algorithms. By harnessing prior knowledge, our model drastically reduces the requirement for training data as compared with existing approaches.Mathematics, Statistics, Applied mathematicsdm2418StatisticsArticlesA Hierarchical Model for Association Rule Mining of Sequential Events: An Approach to Automated Medical Symptom Prediction
http://academiccommons.columbia.edu/catalog/ac:173838
McCormick, Tyler H.; Rudin, Cynthia; Madigan, David B.http://dx.doi.org/10.7916/D89C6VJDTue, 13 May 2014 00:00:00 +0000In many healthcare settings, patients visit healthcare professionals periodically and report multiple medical conditions, or symptoms, at each encounter. We propose a statistical modeling technique, called the Hierarchical Association Rule Model (HARM), that predicts a patient’s possible future symptoms given the patient’s current and past history of reported symptoms. The core of our technique is a Bayesian hierarchical model for selecting predictive association rules (such as “symptom 1 and symptom 2 → symptom 3 ”) from a large set of candidate rules. Because this method “borrows strength” using the symptoms of many similar patients, it is able to provide predictions specialized to any given patient, even when little information about the patient’s history of symptoms is available.Mathematics, Statistics, Medicinedm2418StatisticsArticles[Least Angle Regression]: Discussion
http://academiccommons.columbia.edu/catalog/ac:173841
Madigan, David B.; Ridgeway, Greghttp://dx.doi.org/10.7916/D81V5C29Tue, 13 May 2014 00:00:00 +0000Algorithms for simultaneous shrinkage and selection in regression and classification provide attractive solutions to knotty old statistical challenges. Nevertheless, as far as we can tell, Tibshirani's Lasso algorithm has had little impact on statistical practice. Two particular reasons for this may be the relative inefficiency of the original Lasso algorithm and the relative complexity of more recent Lasso algorithms [e.g., Osborne, Presnell and Turlach (2000)]. Efron, Hastie, Johnstone and Tibshirani have provided an efficient, simple algorithm for the Lasso as well as algorithms for stagewise regression and the new least angle regression. As such this paper is an important contribution to statistical computing.Mathematics, Statisticsdm2418StatisticsArticlesA Note on Equivalence Classes of Directed Acyclic Independence Graphs
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Madigan, David B.http://dx.doi.org/10.7916/D8TB150CTue, 13 May 2014 00:00:00 +0000Directed acyclic independence graphs (DAIGs) play an important role in recent developments in probabilistic expert systems and influence diagrams (Chyu [1]). The purpose of this note is to show that DAIGs can usefully be grouped into equivalence classes where the members of a single class share identical Markov properties. These equivalence classes can be identified via a simple graphical criterion. This result is particularly relevant to model selection procedures for DAIGs (see, e.g., Cooper and Herskovits [2] and Madigan and Raftery [4]) because it reduces the problem of searching among possible orientations of a given graph to that of searching among the equivalence classes.Mathematics, Statisticsdm2418StatisticsArticlesSeparation and Completeness Properties for Amp Chain Graph Markov Models
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Levitz, Michael; Perlman, Michael D.; Madigan, David B.http://dx.doi.org/10.7916/D8X34VJGTue, 13 May 2014 00:00:00 +0000Pearl’s well-known d-separation criterion for an acyclic directed graph (ADG) is a pathwise separation criterion that can be used to efficiently identify all valid conditional independence relations in the Markov model determined by the graph. This paper introduces p-separation, a pathwise separation criterion that efficiently identifies all valid conditional independences under the Andersson–Madigan–Perlman (AMP) alternative Markov property for chain graphs (= adicyclic graphs), which include both ADGs and undirected graphs as special cases. The equivalence of p-separation to the augmentation criterion occurring in the AMP global Markov property is established, and p-separation is applied to prove completeness of the global Markov property for AMP chain graph models. Strong completeness of the AMP Markov property is established, that is, the existence of Markov perfect distributions that satisfy those and only those conditional independences implied by the AMP property(equivalently, by p-separation). A linear-time algorithm for determining p-separation is presented.Mathematics, Statistics, Theoretical mathematicsdm2418StatisticsArticles[Bayesian Analysis in Expert Systems]: Comment: What's Next?
http://academiccommons.columbia.edu/catalog/ac:173856
Madigan, David B.http://dx.doi.org/10.7916/D8W37TFJTue, 13 May 2014 00:00:00 +0000"These papers represent two of the many different graphical modeling camps that have emerged from a flurry of activity in the past decade. The paper by Cox and Wermuth falls within the statistical graphical modeling camp and provides a useful generalization of that body of work. There is, of course, a price to be paid for this generality, namely that the interpretation of the graphs is more complex...The paper by Spiegelhalter, Dawid, Lauritzen and Cowell falls within the probabilistic expert system camp. This is a tour de force by researchers responsible for much of the astonishing progress in this area. Ten years ago, probabilistic models were shunned by the artificial intelligence community. That they are now widely accepted and used is due in large measure to the insights and efforts of these authors, along with other pioneers such as Judea Pearl and Peter Cheeseman..." -- page 261Mathematics, Statisticsdm2418StatisticsArticlesSurface wave phase velocities of the Western United States from a two-station method
http://academiccommons.columbia.edu/catalog/ac:172091
Foster, Anna; Ekström, Göran; Nettles, Meredith K.http://dx.doi.org/10.7916/D87W6979Thu, 13 Mar 2014 00:00:00 +0000We calculate two-station phase measurements using single-station measurements made on USArray Transportable Array data for surface waves at periods from 25 to 100 s. The phase measurements are inverted for baseline Love and Rayleigh wave phase velocity maps on a 0.5° × 0.5° grid. We make estimates of the arrival angle for each event at each station using a mini array method similar to beamforming, and apply this information to correct the geometry of the two-station measurements. These corrected measurements are inverted for an additional set of phase velocity maps. Arrival angles range from 0° to ±15°, and the associated corrections result in local changes of up to 4 per cent in the final phase velocity maps. We select our preferred models on the basis of the internal consistency of the measurements, finding that the arrival-angle corrections improve the two-station phase measurements, but that Love wave arrival-angle estimates may be contaminated by overtone interference. Our preferred models compare favourably with recent studies of the phase velocity of the Western United States. The corrected Rayleigh wave models achieve greater variance reduction than the baseline Rayleigh wave models, and the baseline Love wave models, which are more difficult to obtain, are robust and could be used in conjunction with the Rayleigh wave models to constrain radially anisotropic earth structure.Geophysics, Mathematicsaef2127, ge21, mn2237Lamont-Doherty Earth Observatory, Earth and Environmental SciencesArticlesArrival-angle anomalies across the USArray Transportable Array
http://academiccommons.columbia.edu/catalog/ac:172088
Foster, Anna; Ekström, Göran; Hjorleifsdottir, Valahttp://dx.doi.org/10.7916/D8CJ8BJ9Thu, 13 Mar 2014 00:00:00 +0000We construct composite maps of surface-wave arrival-angle anomalies using clustered earthquakes and an array method for measuring wave-front geometry. This results in observations of arrival angles covering the entire footprint of the USArray Transportable Array during 2006–2010. Bands of arrival-angle deviations in the propagation direction indicate the presence of heterogeneous velocity structure both inside and outside of the array. We compare the observed patterns to arrival angles predicted using two global tomographic models, the mantle model S362ANI and the surface-wave-dispersion model GDM52. We use both ray-theory-based prediction methods and measurements on synthetic data calculated using a spectral-element method. Both models and all prediction methods produce similar mean arrival angles and long-wavelength patterns of anomalies which are similar to the observations. Predicted short-wavelength features generally do not agree with the observations. The spectral-element method produces some complexity that is not obtained using the ray-theory-based methods; this predicted complexity is similar in character to the observed patterns, but does not match them.Geophysics, Mathematicsaef2127, ge21Lamont-Doherty Earth Observatory, Earth and Environmental SciencesArticlesInvisible Mathematics in Italo Calvino's Le città invisibili
http://academiccommons.columbia.edu/catalog/ac:166630
Moreno-Viqueira, Ileanahttp://hdl.handle.net/10022/AC:P:22021Fri, 18 Oct 2013 00:00:00 +0000This dissertation examines the use of mathematical concepts as an essential structural and thematic element in Italo Calvino's Le città invisibili. The author`s conception of literature as a combinatorial art, intrinsically mathematical itself, is the point of departure. Focal to the study is Calvino's interest in that which is an essential part of the combinatorial game and the key to Gödel Incompleteness Theory, namely, the elements of surprise and the unexpected - the exceptions to the rule. Other critical approaches to Calvino's work, like semiotic, structuralism and scientific are interrelated to Mathematics, but what this study proposes is a strictly mathematical approach to complement that which has already been pointed out. A mathematical perspective based on an understanding of Mathematics as more than just numbers encompasses the whole analysis. Mathematics is given its proper place as a humanistic discipline. It is an interdisciplinary proposal of literature and science, pertinent to Calvino's writing. The purpose is to unveil a "hidden math" which from the perspective of this study is an intrinsic tool in Calvino's writing process of Città. As a versatile writer, Calvino manages to use mathematics in such subtle ways that it may not be perceptible at first sight. Most importantly, within these mathematical concepts and images lies, in part, the potential character of literature for which the author aims: that latent yet invisible possibility, that search for new forms (like the cities). These ideas, particularly related to potential literature, are also analyzed from his interest and involvement in Oulipo (Ouvroir de Littérature Potentielle). The study begins by unfolding what aspects of combinatorial mathematics are present in Le città invisibili; how these concepts as well as other images are used in the construction and design of the cities and the book; and to find out why Calvino finds recourse to mathematics as a narrative and creative strategy. Calvino's use of mathematical concepts are studied as a "visual instrument" in the organization and construction of his imaginative writing and, furthermore, as a means to achieve "lightness" structurally and thematically through the abstract, aesthetic and, at times, even humorous nature of mathematics. In their own way mathematics and literature attempt to make visible what is invisible, and they both struggle to remove weight from their own "systems" of expression. In conclusion, the investigation intends to demonstrate through Calvino's Le città invisibili, how mathematics and literature complement each other in the search for new forms, new ideas, new stories.Literature, MathematicsItalianDissertationsAssociated Polynomials and Uniform Methods for the Solution of Linear Problems
http://academiccommons.columbia.edu/catalog/ac:166451
Traub, Joseph F.http://hdl.handle.net/10022/AC:P:21984Thu, 10 Oct 2013 00:00:00 +0000To every polynomial P of degree n we associate a sequence of n-1 polynomials of increasing degree which we call the associated polynomials of P. The associated polynomials depend in a particularly simple way on the coefficients of P. These polynomials have appeared in many guises in the literature, usually related to some particular application and most often going unrecognized. They have been called Horner polynomials and Laguerre polynomials. Often what occurs is not an associated polynomial itself but a number which is an associated polynomial evaluated at a zero of P. The properties of associated polynomials have never been investigated in themselves. We shall try to demonstrate that associated polynomials provide a useful unifying concept. Although many of the results of this paper are new, we shall also present known results in our framework.Mathematicsjft2Computer ScienceArticlesComputational Complexity of Iterative Processes
http://academiccommons.columbia.edu/catalog/ac:166442
Traub, Joseph F.http://hdl.handle.net/10022/AC:P:21981Thu, 10 Oct 2013 00:00:00 +0000The theory of optimal algorithmic processes is part of computational complexity. This paper deals with analytic computational complexity. The relation between the goodness of an iteration algorithm and its new function evaluation and memory requirements are analyzed. A new conjecture is stated.Computer science, Mathematicsjft2Computer ScienceArticlesA Class of Globally Convergent Iterations for the Solution of Polynomial Equations
http://academiccommons.columbia.edu/catalog/ac:166448
Traub, Joseph F.http://hdl.handle.net/10022/AC:P:21983Thu, 10 Oct 2013 00:00:00 +0000We introduce a class of new iteration functions which are ratios of polynomials of the same degree and hence defined at infinity. The poles of these rational functions occur at points which cause no difficulty. The classical iteration functions are given as explicit functions of P and its derivatives. The new iteration functions are constructed according to a certain algorithm. This construction requires only simple polynomial manipulation which may be performed on a computer. We shall treat here only the important case that the zeros of P are distinct and that the dominant zero is real. The extension to multiple zeros, dominant complex zeros, and sub-dominant zeros will be given in another paper. We shall restrict ourselves to questions relevant to the calculation of zeros. Certain aspects of our investigations which are of broader interest will be reported elsewhere.Mathematics, Computer sciencejft2Computer ScienceArticlesA Three-State Algorithm for Real Polynomials Using Quadratic Iteration
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Jenkins, M.A.; Traub, Joseph F.http://hdl.handle.net/10022/AC:P:21982Thu, 10 Oct 2013 00:00:00 +0000We introduce a new three-stage process for calculating the zeros of a polynomial with real coefficients. The algorithm finds either a linear or quadratic factor, working completely in real arithmetic. In the third stage the algorithm uses one of two variable-shift iterations corresponding to the linear or quadratic case. The iteration for a linear factor is a real arithmetic version of the third stage of the algorithm for complex polynomials which we studied in an earlier paper. A new variable-shift iteration is introduced in this paper which is suitable for quadratic factors. If the complex algorithm and the new real algorithm are applied to the same real polynomial, then the real algorithm is about four times as fast. We prove that the mathematical algorithm always converges and show that the rate of convergence of the third stage is faster than second order. The problem and algorithm may be recast into matrix form. The third stage is a quadratic form of shifted inverse powering and a quadratic form of generalized Rayleigh iteration. The results of extensive testing are summarized. For an ALGOL W program run on an IBM 360/67 we found that for polynomials ranging in degree from 20 to 50, the time required to calculate all zeros averaged 2n² milliseconds. An ALGOL 60 implementation of the algorithm and a program which calculates a posteriors bounds on the zeros may be found in Jenkins’ 1969 Stanford dissertation.Mathematics, Computer sciencejft2Computer ScienceArticlesGeneralized Sequences with Applications to the Discrete Calculus
http://academiccommons.columbia.edu/catalog/ac:166457
Traub, Joseph F.http://hdl.handle.net/10022/AC:P:21986Thu, 10 Oct 2013 00:00:00 +0000Mikusinski [17] has introduced a theory of generalized functions which is algebraic in nature. Generalized functions are introduced in a way which is analogous to the extension of the concept of number from integers to rationals. In this paper, an analogous theory of "generalized sequences" is constructed for the discrete calculus. This theory serves a dual purpose. It provides a rigorous foundation for an operational calculus and provides a powerful formalism for the solution of discrete problems.Mathematicsjft2Computer ScienceArticlesOptimal Order and Efficiency for Iterations with Two Evaluations
http://academiccommons.columbia.edu/catalog/ac:166436
Kung, H.T.; Traub, Josephe F.http://hdl.handle.net/10022/AC:P:21979Thu, 10 Oct 2013 00:00:00 +0000The problem is to calculate a simple zero of a nonlinear function f. We consider rational iterations without memory which use two evaluations of f or its derivatives. It is shown that the optimal order is 2. This settles a conjecture of Kung and Traub that an iteration using n evaluations without memory is of order at most 2ⁿ⁻¹, for the case n=2. Furthermore we show that any rational two-evaluation iteration of optimal order must use either two evaluations of f or one evaluation of f and one of f'. From this result we completely settle the question of the optimal efficiency, in our efficiency measure, for any two-evaluation iteration without memory. Depending on the relative cost of evaluating f and f', the optimal efficiency is achieved by either Newton iteration or the iteration ᴪ.Mathematicsjft2Computer ScienceArticlesThe Algebraic Theory of Matrix Polynomials
http://academiccommons.columbia.edu/catalog/ac:166433
Dennis Jr., J.E.; Traub, Joseph F.; Weber, R.P.http://hdl.handle.net/10022/AC:P:21978Thu, 10 Oct 2013 00:00:00 +0000A matrix S is a solvent of the matrix polynomial M(X)=A₀Xᵐ +...+ Am if M(S)=O where A, X, and S are square matrices. In this paper we develop the algebraic theory of matrix polynomials and solvents. We define division and interpolation, investigate the properties of block Vandermonde matrices, and define and study the existence of a complete set of solvents. We study the relation between the matrix polynomial problem and the lambda-matrix problem, which is to find a scalar A₀λᵐ + A₁λᵐ⁻¹ +...+ Am is singular. In a future paper we extend Traub’s algorithm for calculating zeros of scalar polynomials to matrix polynomials and establish global convergence properties of this algorithm for a class of matrix polynomials.Mathematicsjft2Computer ScienceArticlesAlgorithms for Solvents of Matrix Polynomials
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Dennis Jr., J.E.; Traub, Joseph F.; Weber, R.P.http://hdl.handle.net/10022/AC:P:21980Thu, 10 Oct 2013 00:00:00 +0000In an earlier paper we developed the algebraic theory of matrix polynomials. Here we introduce two algorithms for computing "dominant" solvents. Global convergence of the algorithms under certain conditions is established.Mathematicsjft2Computer ScienceArticlesConstruction of Globally Convergent Iteration Functions for the Solution of Polynomial Equations
http://academiccommons.columbia.edu/catalog/ac:166454
Traub, Joseph F.http://hdl.handle.net/10022/AC:P:21985Thu, 10 Oct 2013 00:00:00 +0000Iteration functions for the approximation of zeros of a polynomial P are usually given as explicit functions of P and its derivatives. We introduce a class of iteration functions which are themselves constructed according to a certain algorithm given below. The construction of the iteration functions requires only simple polynomial manipulation which may be performed on a computer.Mathematicsjft2Computer ScienceArticlesOn Lagrange-Hermite Interpolation
http://academiccommons.columbia.edu/catalog/ac:166460
Traub, Joseph F.http://hdl.handle.net/10022/AC:P:21987Thu, 10 Oct 2013 00:00:00 +0000Mathematics, Applied mathematicsjft2Computer ScienceArticlesThe Cognitive and Demographic Variables that Underlie Notetaking and Review in Mathematics: Does Quality of Notes Predict Test Performance in Mathematics?
http://academiccommons.columbia.edu/catalog/ac:163324
Belanfante, Elizabeth Andreahttp://hdl.handle.net/10022/AC:P:21089Tue, 16 Jul 2013 00:00:00 +0000Taking and reviewing lecture notes is an effective and prevalent method of studying employed by students at the post-secondary level (Armbruster, 2000; Armbruster, 2009; Dunkel and Davy, 1989; Peverly et al., 2009). However, few studies have examined the cognitive variables that underlie this skill. In addition, these studies have focused on more verbally based domains, such as history and psychology. The current study examined the practical utility of notes in actual class settings. It is the first study that has attempted to examine the outcomes and cognitive skills associated with note-taking and review in any area of mathematics. It also set out to establish the importance of quality of notes and quality of review sheets to test performance in graduate level probability and statistics courses. Finally, this dissertation sought to explore the extent to which variables besides notes also contribute to test performance in this domain. Participants included 74 graduate students enrolled in introductory probability and statistics courses at a private graduate teacher education college in a large city in the Northeast United States. Participants took notes during class and provided the researcher with a copy of their notes for several lectures. Participants were also required to write down additional information on the back of two formula sheets that were used as an aid on the midterm exam. The independent variables included handwriting speed, gender, spatial visualization ability, background knowledge, verbal ability, and working memory. The dependent variables were quality of lecture notes, quality of supplemental review sheets, and midterm performance. All measures were group administered. Results revealed that gender was the only predictor of quality of lecture notes. Quality of lecture notes was the only significant predictor of quality of supplemental review sheets. Neither quality of lecture notes nor quality of supplemental review sheets predicted overall test performance. Instead, background knowledge and instructor significantly predicted overall test performance. Handwriting speed was a marginally significant predictor of overall test performance. Future research aimed at replicating these findings and expanding the results to include other mathematical domains and educational levels is recommended.Mathematics, Statistics, Educationeab2111Health and Behavior Studies, School PsychologyDissertationsApproximate dynamic programming for large scale systems
http://academiccommons.columbia.edu/catalog/ac:169790
Desai, Vijay V.http://hdl.handle.net/10022/AC:P:20875Fri, 28 Jun 2013 00:00:00 +0000Sequential decision making under uncertainty is at the heart of a wide variety of practical problems. These problems can be cast as dynamic programs and the optimal value function can be computed by solving Bellman's equation. However, this approach is limited in its applicability. As the number of state variables increases, the state space size grows exponentially, a phenomenon known as the curse of dimensionality, rendering the standard dynamic programming approach impractical. An effective way of addressing curse of dimensionality is through parameterized value function approximation. Such an approximation is determined by relatively small number of parameters and serves as an estimate of the optimal value function. But in order for this approach to be effective, we need Approximate Dynamic Programming (ADP) algorithms that can deliver `good' approximation to the optimal value function and such an approximation can then be used to derive policies for effective decision-making. From a practical standpoint, in order to assess the effectiveness of such an approximation, there is also a need for methods that give a sense for the suboptimality of a policy. This thesis is an attempt to address both these issues. First, we introduce a new ADP algorithm based on linear programming, to compute value function approximations. LP approaches to approximate DP have typically relied on a natural `projection' of a well studied linear program for exact dynamic programming. Such programs restrict attention to approximations that are lower bounds to the optimal cost-to-go function. Our program -- the `smoothed approximate linear program' -- is distinct from such approaches and relaxes the restriction to lower bounding approximations in an appropriate fashion while remaining computationally tractable. The resulting program enjoys strong approximation guarantees and is shown to perform well in numerical experiments with the game of Tetris and queueing network control problem. Next, we consider optimal stopping problems with applications to pricing of high-dimensional American options. We introduce the pathwise optimization (PO) method: a new convex optimization procedure to produce upper and lower bounds on the optimal value (the `price') of high-dimensional optimal stopping problems. The PO method builds on a dual characterization of optimal stopping problems as optimization problems over the space of martingales, which we dub the martingale duality approach. We demonstrate via numerical experiments that the PO method produces upper bounds and lower bounds (via suboptimal exercise policies) of a quality comparable with state-of-the-art approaches. Further, we develop an approximation theory relevant to martingale duality approaches in general and the PO method in particular. Finally, we consider a broad class of MDPs and introduce a new tractable method for computing bounds by consider information relaxation and introducing penalty. The method delivers tight bounds by identifying the best penalty function among a parameterized class of penalty functions. We implement our method on a high-dimensional financial application, namely, optimal execution and demonstrate the practical value of the method vis-a-vis competing methods available in the literature. In addition, we provide theory to show that bounds generated by our method are provably tighter than some of the other available approaches.Operations research, Mathematicsvvd2101Industrial Engineering and Operations Research, BusinessDissertationsProperties of Hamiltonian Torus Actions on Closed Symplectic Manifolds
http://academiccommons.columbia.edu/catalog/ac:161552
Fanoe, Andrew L.http://hdl.handle.net/10022/AC:P:20455Fri, 24 May 2013 00:00:00 +0000In this thesis, we will study the properties of certain Hamiltonian torus actions on closed symplectic manifolds. First, we will consider counting Hamiltonian torus actions on closed, symplectic manifolds M with 2-dimensional second cohomology. In particular, all such manifolds are bundles with fiber and base equal to projective spaces. We use cohomological techniques to show that there is a unique toric structure if the fiber has a smaller dimension than the base. Furthermore, if the fiber and base are both at least 2-dimensional projective spaces, we show that there is a finite number of toric structures on M that are compatible with some symplectic structure on M. Additionally, we show there is uniqueness in certain other cases, such as the case where M is a monotone symplectic manifold. Finally, we will be interested in the existence of symplectic, non-Hamiltonian circle actions on closed symplectic 6-manifolds. In particular, we will use J-holomorphic curve techniques to show that there are no such actions that satisfy certain fixed point conditions. This lends support to the conjecture that there are no such actions with a non-empty set of isolated fixed points.Mathematicsalf2140MathematicsDissertationsSingular theta lifts and near-central special values of Rankin-Selberg L-functions
http://academiccommons.columbia.edu/catalog/ac:161464
Garcia, Luis Emiliohttp://hdl.handle.net/10022/AC:P:20421Thu, 23 May 2013 00:00:00 +0000In this thesis we study integrals of a product of two automorphic forms of weight 2 on a Shimura curve over Q against a function on the curve with logarithmic singularities at CM points obtained as a Borcherds lift. We prove a formula relating periods of this type to a near-central special value of a Rankin-Selberg L-function. The results provide evidence for Beilinson's conjectures on special values of L-functions.Mathematicslg2440MathematicsDissertationsDel Pezzo surfaces with irregularity and intersection numbers on quotients in geometric invariant theory
http://academiccommons.columbia.edu/catalog/ac:161405
Maddock, Zachary Alexanderhttp://hdl.handle.net/10022/AC:P:20379Mon, 20 May 2013 00:00:00 +0000This thesis comprises two parts covering distinct topics in algebraic geometry. In Part I, we construct the first examples of regular del Pezzo surfaces for which the first cohomology group of the structure sheaf is nonzero. Such surfaces, which only exist over imperfect fields, arise as generic fibres of fibrations of singular del Pezzo surfaces in positive characteristic whose total spaces are smooth, and their study is motivated by the minimal model program. We also find a restriction on the integer pairs that are possible as the irregularity (that is, the dimension of the first cohomology group of the structure sheaf) and anti-canonical degree of regular del Pezzo surfaces with positive irregularity. In Part II, we consider a connected reductive group acting linearly on a projective variety over an arbitrary field. We prove a formula that compares intersection numbers on the geometric invariant theory quotient of the variety by the reductive group with intersection numbers on the geometric invariant theory quotient of the variety by a maximal torus, in the case where all semi-stable points are properly stable. These latter intersection numbers involve the top equivariant Chern class of the maximal torus representation given by the quotient of the adjoint representation on the Lie algebra of the reductive group by that of the maximal torus. We provide a purely algebraic proof of the formula when the root system decomposes into irreducible root systems of type A. We are able to remove this restriction on root systems by applying a related result of Shaun Martin from symplectic geometry.Mathematicszam2104MathematicsDissertationsFinancial Portfolio Risk Management: Model Risk, Robustness and Rebalancing Error
http://academiccommons.columbia.edu/catalog/ac:161415
Xu, Xingbohttp://hdl.handle.net/10022/AC:P:20382Mon, 20 May 2013 00:00:00 +0000Risk management has always been in key component of portfolio management. While more and more complicated models are proposed and implemented as research advances, they all inevitably rely on imperfect assumptions and estimates. This dissertation aims to investigate the gap between complicated theoretical modelling and practice. We mainly focus on two directions: model risk and reblancing error. In the first part of the thesis, we develop a framework for quantifying the impact of model error and for measuring and minimizing risk in a way that is robust to model error. This robust approach starts from a baseline model and finds the worst-case error in risk measurement that would be incurred through a deviation from the baseline model, given a precise constraint on the plausibility of the deviation. Using relative entropy to constrain model distance leads to an explicit characterization of worst-case model errors; this characterization lends itself to Monte Carlo simulation, allowing straightforward calculation of bounds on model error with very little computational effort beyond that required to evaluate performance under the baseline nominal model. This approach goes well beyond the effect of errors in parameter estimates to consider errors in the underlying stochastic assumptions of the model and to characterize the greatest vulnerabilities to error in a model. We apply this approach to problems of portfolio risk measurement, credit risk, delta hedging, and counterparty risk measured through credit valuation adjustment. In the second part, we apply this robust approach to a dynamic portfolio control problem. The sources of model error include the evolution of market factors and the influence of these factors on asset returns. We analyze both finite- and infinite-horizon problems in a model in which returns are driven by factors that evolve stochastically. The model incorporates transaction costs and leads to simple and tractable optimal robust controls for multiple assets. We illustrate the performance of the controls on historical data. Robustness does improve performance in out-of-sample tests in which the model is estimated on a rolling window of data and then applied over a subsequent time period. By acknowledging uncertainty in the estimated model, the robust rules lead to less aggressive trading and are less sensitive to sharp moves in underlying prices. In the last part, we analyze the error between a discretely rebalanced portfolio and its continuously rebalanced counterpart in the presence of jumps or mean-reversion in the underlying asset dynamics. With discrete rebalancing, the portfolio's composition is restored to a set of fixed target weights at discrete intervals; with continuous rebalancing, the target weights are maintained at all times. We examine the difference between the two portfolios as the number of discrete rebalancing dates increases. We derive the limiting variance of the relative error between the two portfolios for both the mean-reverting and jump-diffusion cases. For both cases, we derive ``volatility adjustments'' to improve the approximation of the discretely rebalanced portfolio by the continuously rebalanced portfolio, based on on the limiting covariance between the relative rebalancing error and the level of the continuously rebalanced portfolio. These results are based on strong approximation results for jump-diffusion processes.Operations research, Finance, Mathematicsxx2126Industrial Engineering and Operations Research, BusinessDissertationsPurity of the stratification by Newton polygons and Frobenius-periodic vector bundles
http://academiccommons.columbia.edu/catalog/ac:161158
Yang, Yanhonghttp://hdl.handle.net/10022/AC:P:20334Wed, 15 May 2013 00:00:00 +0000This thesis includes two parts. In the first part, we show a purity theorem for stratifications by Newton polygons coming from crystalline cohomology, which says that the family of Newton polygons over a noetherian scheme have a common break point if this is true outside a subscheme of codimension bigger than 1. The proof is similar to the proof of [dJO99, Theorem 4.1]. In the second part, we prove that for every ordinary genus-2 curve X over a finite field k of characteristic 2 with automorphism group Z/2Z × S_3, there exist SL(2,k[[s]])-representations of π_1(X) such that the image of π_1(X^-) is infinite. This result produces a family of examples similar to Laszlo's counterexample [Las01] to a question regarding the finiteness of the geometric monodromy of representations of the fundamental group [dJ01].Mathematicsyy2244MathematicsDissertationsLocal Regularity of the Complex Monge-Ampere Equation
http://academiccommons.columbia.edu/catalog/ac:161155
Wang, Yuhttp://hdl.handle.net/10022/AC:P:20333Wed, 15 May 2013 00:00:00 +0000In this thesis, we present a self-contained account of the current development in the local regularity theory of the complex Monge-Ampere equation through the modern fully-nonlinear PDE point of view. We have apply the modern elliptic techniques to establish new local regularity results. These includes: regularity of small perturbed solutions, Holder regularity of the Hessian of the W^{2,p} solutions and a Liouville-type theorem.Mathematicsyw2340MathematicsDissertationsOdd symmetric functions and categorification
http://academiccommons.columbia.edu/catalog/ac:161123
Ellis, Alexander Palenhttp://hdl.handle.net/10022/AC:P:20307Tue, 14 May 2013 00:00:00 +0000We introduce q- and signed analogues of several constructions in and around the theory of symmetric functions. The most basic of these is the Hopf superalgebra of odd symmetric functions. This algebra is neither (super-)commutative nor (super-)cocommutative, yet its combinatorics still exhibit many of the striking integrality and positivity properties of the usual symmetric functions. In particular, we give odd analogues of Schur functions, Kostka numbers, and Littlewood-Richardson coefficients. Using an odd analogue of the nilHecke algebra, we give a categorification of the integral divided powers form of U_q^+(sl_2) inequivalent to the one due to Khovanov-Lauda. Along the way, we develop a graphical calculus for indecomposable modules for the odd nilHecke algebra.Mathematicsape2104MathematicsDissertationsExamining the Effects of Gender, Poverty, Attendance, and Ethnicity on Algebra, Geometry, and Trigonometry Performance in a Public High School
http://academiccommons.columbia.edu/catalog/ac:160486
Shafiq, Hasanhttp://hdl.handle.net/10022/AC:P:20075Wed, 01 May 2013 00:00:00 +0000Over the last few decades school accountability for student performance has become an issue at the forefront of education. The federal No Child Left Behind Act of 2001 (NCLB) and various regulations by individual states have set standards for student performance at both the district and individual public and charter school levels, and certain consequences apply if the performance of students in an institution is deemed unsatisfactory. Conversely, rewards come to districts or schools that perform especially well or make a certain degree of improvement over their earlier results. Albeit with certain conditions, the federal government makes additional education money available to the states under NCLB. While testing is nothing new in American public education, the concept of district/school accountability for performance is at least relatively so. In New York City, where New York State Regents Examinations (NYSRE) have been a measure of student performance for many years, scores on these tests are low, often preventing students from receiving course credit, which in turn results in failure to graduate on schedule. In addition, rates of graduation from public high schools are low. The city and state have kept data on student performance broken out by a number of factors including socioeconomic status, ethnicity, attendance, and gender which point to an achievement gap among different groups. This study investigates a series of those factors associated with the mastery of high school Algebra, Geometry, and Trigonometry. This study concerns itself specifically with the effect that gender, socioeconomic status, attendance, and ethnicity may have on student achievement in a mathematics course and on standardized tests, specifically the NYSRE, an annual rite of passage for students in grades 9 through 11. This research considered and ran tests on data gathered from a single large New York City high school. In this study, a 12 two-way (between-groups) univariate analyses of variance (ANOVAs) were conducted to assess whether there were differences in students' mathematics achievement scores by gender, ethnicity, attendance, and family socio-economic status (SES). In addition, three Pearson correlation analyses were conducted to determine whether there was a correlation among Integrated Algebra, Geometry, and Algebra II/Trigonometry unit examination scores and Regents scores. Nine Pearson correlation analyses were conducted to determine whether there was a correlation between Regents scores and mathematics achievement unit examination scores. A correlation was run between each mathematics achievement score with the Regents score from each subject. Six two-way (between-groups) ANOVA were also conducted to assess whether there were difference in students' mathematics achievement among Black males, Black females, Hispanic males, and Hispanic females. Data were gathered, merged, and transferred into a Statistical Package for the Social Sciences (SPSS) 19.0 (IBM, 2010) for analysis. The findings indicate that attendance and family SES have a meaningful relationship to mathematics achievement in the New York City public high school which was the subject of this investigation. On the other hand, gender and ethnicity showed no relationship to students' mathematics achievement. As an implication of this research, school policies must focus more on the achievement gap of students from low-SES families and must encourage students to maintain good attendance. Students should have access to different forms of academic interventions that go beyond after-school or Saturday tutoring; academic intervention services; community counseling or mediation; or peer intervention or peer counseling through which students learn basic mathematics skills from each other to achieve college readiness.Mathematics education, Mathematicsmhs2143Mathematics, Science, and Technology, Mathematics EducationDissertationsp-adic Heights of Heegner points on Shimura curves
http://academiccommons.columbia.edu/catalog/ac:160525
Disegni, Danielhttp://hdl.handle.net/10022/AC:P:20097Wed, 01 May 2013 00:00:00 +0000Let f be a primitive Hilbert modular form of weight 2 and level N for the totally real field F, and let p be an odd rational prime such that f is ordinary at all primes dividing p. When E is a CM extension of F of relative discriminant prime to Np, we give an explicit construction of the p-adic Rankin-Selberg L-function L_p(f_E,-) and prove that when the sign of its functional equation is -1, its central derivative is given by the p-adic height of a Heegner point on the abelian variety A associated to f. This p-adic Gross-Zagier formula generalises the result obtained by Perrin-Riou when F=Q and N satisfies the so-called Heegner condition. We deduce applications to both the p-adic and the classical Birch and Swinnerton-Dyer conjectures for A.Mathematicsdd2438MathematicsDissertationsLocalization and Heegaard Floer Homology
http://academiccommons.columbia.edu/catalog/ac:160806
Hendricks, Kristenhttp://hdl.handle.net/10022/AC:P:20103Wed, 01 May 2013 00:00:00 +0000In this thesis we use Seidel-Smith localization for Lagrangian Floer cohomology to study invariants of cyclic branched covers of three-manifolds and symmetry groups of knots by constructing localization spectral sequences in Heegaard Floer homology.Mathematicskeh2141MathematicsDissertationsTetrahedra and Their Nets: Mathematical and Pedagogical Implications
http://academiccommons.columbia.edu/catalog/ac:160274
Mussa, Deregehttp://hdl.handle.net/10022/AC:P:20037Tue, 30 Apr 2013 00:00:00 +0000If one has three sticks (lengths), when can you make a triangle with the sticks? As long as any two of the lengths sum to a value strictly larger than the third length one can make a triangle. Perhaps surprisingly, if one is given 6 sticks (lengths) there is no simple way of telling if one can build a tetrahedron with the sticks. In fact, even though one can make a triangle with any triple of three lengths selected from the six, one still may not be able to build a tetrahedron. At the other extreme, if one can make a tetrahedron with the six lengths, there may be as many 30 different (incongruent) tetrahedra with the six lengths. Although tetrahedra have been studied in many cultures (Greece, India, China, etc.) Over thousands of years, there are surprisingly many simple questions about them that still have not been answered. This thesis answers some new questions about tetrahedra, as well raising many more new questions for researchers, teachers, and students. It also shows in an appendix how tetrahedra can be used to illustrate ideas about arithmetic, algebra, number theory, geometry, and combinatorics that appear in the Common Cores State Standards for Mathematics (CCSS -M). In particular it addresses representing three-dimensional polyhedra in the plane. Specific topics addressed are a new classification system for tetrahedra based on partitions of an integer n, existence of tetrahedra with different edge lengths, unfolding tetrahedra by cutting edges of tetrahedra, and other combinatorial aspects of tetrahedra.Mathematics education, Mathematicsdhm2114Mathematics, Science, and Technology, Mathematics EducationDissertationsSato-Tate Problem for GL(3)
http://academiccommons.columbia.edu/catalog/ac:160461
Zhou, Fanhttp://hdl.handle.net/10022/AC:P:20048Tue, 30 Apr 2013 00:00:00 +0000Based upon the work of Goldfeld and Kontorovich on the Kuznetsov trace formula of Maass forms for SL(3,Z), we prove a weighted vertical equidistribution theorem (with respect to the generalized Sato-Tate measure) for the p-th Hecke eigenvalue of Maass forms, with the rate of convergence. With a conjectured orthogonality relation between the Fourier coefficients of Maass forms for SL(N,Z) for N≥4, we generalize the above equidistribution theorem to N≥4.Mathematicsfz2133MathematicsDissertationsTournaments With Forbidden Substructures and the Erdos-Hajnal Conjecture
http://academiccommons.columbia.edu/catalog/ac:160247
Choromanski, Krzysztofhttp://hdl.handle.net/10022/AC:P:20024Mon, 29 Apr 2013 00:00:00 +0000A celebrated Conjecture of Erdos and Hajnal states that for every undirected graph H there exists ɛ(H)>0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or a stable set of size at least n^{ɛ(H)}. In 2001 Alon, Pach and Solymosi proved that the conjecture has an equivalent directed version, where undirected graphs are replaced by tournaments and cliques and stable sets by transitive subtournaments. This dissertation addresses the directed version of the conjecture and some problems in the directed setting that are closely related to it. For a long time the conjecture was known to be true only for very specific small graphs and graphs obtained from them by the so-called substitution procedure proposed by Alon, Pach and Solymosi. All the graphs that are an outcome of this procedure have nontrivial homogeneous sets. Tournaments without nontrivial homogeneous sets are called prime. They play a central role here since if the conjecture is not true then the smallest counterexample is prime. We remark that for a long time the conjecture was known to be true only for some prime graphs of order at most 5. There exist 5-vertex graphs for which the conjecture is still open, however one of the corollaries of the results presented in the thesis states that all tournaments on at most 5 vertices satisfy the conjecture. In the first part of the thesis we will establish the conjecture for new infinite classes of tournaments containing infinitely many prime tournaments. We will first prove the conjecture for so-called constellations. It turns out that almost all tournaments on at most 5 vertices are either constellations or are obtained from constellations by substitutions. The only 5-vertex tournament for which this is not the case is a tournament in which every vertex has outdegree 2. We call this the tournament C_{5}. Another result of this thesis is the proof of the conjecture for this tournament. We also present here the structural characterization of the tournaments satisfying the conjecture in almost linear sense. In the second part of the thesis we focus on the upper bounds on coefficients epsilon(H) for several classes of tournaments. In particular we analyze how they depend on the structure of the tournament. We prove that for almost all h-vertex tournaments ɛ(H) ≤ 4/h(1+o(1)). As a byproduct of the methods we use here, we get upper bounds for ɛ(H) of undirected graphs. We also present upper bounds on ɛ(H) of tournaments with small nontrivial homogeneous sets, in particular prime tournaments. Finally we analyze tournaments with big ɛ(H) and explore some of their structural properties.Mathematicskmc2178Industrial Engineering and Operations ResearchDissertationsGood Mathematics Teaching: Perspectives of Beginning Secondary Teachers
http://academiccommons.columbia.edu/catalog/ac:159494
Leong, Kwan Euhttp://hdl.handle.net/10022/AC:P:19322Mon, 11 Mar 2013 00:00:00 +0000What is good mathematics teaching? The answer depends on whom you are asking. Teachers, researchers, policymakers, administrators, and parents usually provide their own view on what they consider is good mathematics teaching and what is not. The purpose of this study was to determine how beginning teachers define good mathematics teaching and what they report as being the most important attributes at the secondary level. This research explored whether there was a relationship between the demographics of the participants and the attributes of good teaching. In addition, factors that influence the understanding of good mathematics teaching were explored. A mixed methodology was used to gather information from the research participants regarding their beliefs and classroom practices of good mathematics teaching. The two research instruments used in this study were the survey questionnaire and a semi-structured interview. Thirty-three respondents who had one to two years of classroom experience comprised the study sample. They had graduated from a school of education in an eastern state and had obtained their teacher certification upon completing their studies. The beginning mathematics teachers selected these four definitions of good teaching as their top choices: 1) have High Expectations that all students are capable of learning; 2) have strong content knowledge (Subject Matter Knowledge); 3) create a Learning Environment that fosters the development of mathematical power; and 4) bring Enthusiasm and excitement to classroom. The three most important attributes in good teaching were: Classroom Management, Motivation, and Strong in Content Knowledge. One interesting finding was the discovery of four groups of beginning teachers and how they were associated with specific attributes of good mathematics teaching according to their demographics. Beginning teachers selected Immediate Classroom Situation, Mathematical Beliefs, Pedagogical Content Knowledge, and Colleagues as the top four factors from the survey analysis that influenced their understanding of good mathematics teaching. The study's results have implications for informing the types of mathematical knowledge required for pre-service teachers that can be incorporated into teacher education programs and define important attributes of good mathematics teaching during practicum.Mathematics education, Mathematics, Teacher educationMathematics, Science, and Technology, Mathematics EducationDissertationsProof and Reasoning in Secondary School Algebra Textbooks
http://academiccommons.columbia.edu/catalog/ac:156775
Dituri, Philip Charleshttp://hdl.handle.net/10022/AC:P:19092Fri, 15 Feb 2013 00:00:00 +0000The purpose of this study was to determine the extent to which the modeling of deductive reasoning and proof-type thinking occurs in a mathematics course in which students are not explicitly preparing to write formal mathematical proofs. Algebra was chosen because it is the course that typically directly precedes a student's first formal introduction to proof in geometry in the United States. The lens through which this study aimed to examine the intended curriculum was by identifying and reviewing the modeling of proof and deductive reasoning in the most popular and widely circulated algebra textbooks throughout the United States. Textbooks have a major impact on mathematics classrooms, playing a significant role in determining a teacher's classroom practices as well as student activities. A rubric was developed to analyze the presence of reasoning and proof in algebra textbooks, and an analysis of the coverage of various topics was performed. The findings indicate that, roughly speaking, students are only exposed to justification of mathematical claims and proof-type thinking in 38% of all sections analyzed. Furthermore, only 6% of coded sections contained an actual proof or justification that offered the same ideas or reasoning as a proof. It was found that when there was some justification or proof present, the most prevalent means of convincing the reader of the truth of a concept, theorem, or procedure was through the use of specific examples. Textbooks attempting to give a series of examples to justify or convince the reader of the truth of a concept, theorem, or procedure often fell short of offering a mathematical proof because they lacked generality and/or, in some cases, the inductive step. While many textbooks stated a general rule at some point, most only used deductive reasoning within a specific example if at all. Textbooks rarely expose students to the kinds of reasoning required by mathematical proof in that they rarely expose students to reasoning about mathematics with generality. This study found a lack of sufficient evidence of instruction or modeling of proof and reasoning in secondary school algebra textbooks. This could indicate that, overall, algebra textbooks may not fulfill the proof and reasoning guidelines set forth by the NCTM Principles and Standards and the Common Core State Standards. Thus, the enacted curriculum in mathematics classrooms may also fail to address the recommendations of these influential and policy defining organizations.Mathematics education, Mathematics, Educationpcd2102Mathematics, Science, and Technology, Mathematics EducationDissertationsRare Events in Stochastic Systems: Modeling, Simulation Design and Algorithm Analysis
http://academiccommons.columbia.edu/catalog/ac:156733
Shi, Yixihttp://hdl.handle.net/10022/AC:P:19034Wed, 13 Feb 2013 00:00:00 +0000This dissertation explores a few topics in the study of rare events in stochastic systems, with a particular emphasis on the simulation aspect. This line of research has been receiving a substantial amount of interest in recent years, mainly motivated by scientific and industrial applications in which system performance is frequently measured in terms of events with very small probabilities.The topics mainly break down into the following themes: Algorithm Analysis: Chapters 2, 3, 4 and 5. Simulation Design: Chapters 3, 4 and 5. Modeling: Chapter 5. The titles of the main chapters are detailed as follows: Chapter 2: Analysis of a Splitting Estimator for Rare Event Probabilities in Jackson Networks Chapter 3: Splitting for Heavy-tailed Systems: An Exploration with Two Algorithms Chapter 4: State Dependent Importance Sampling with Cross Entropy for Heavy-tailed Systems Chapter 5: Stochastic Insurance-Reinsurance Networks: Modeling, Analysis and Efficient Monte CarloEngineering, Mathematicsys2347Industrial Engineering and Operations ResearchDissertationsPerrault's watch and Beltrami's pseudosphere: A story without a moral
http://academiccommons.columbia.edu/catalog/ac:153401
Bartocci, Claudiohttp://hdl.handle.net/10022/AC:P:14920Fri, 12 Oct 2012 00:00:00 +0000If a specter haunted 19th century mathematics, it was the specter of the pseudosphere, i.e. the two-dimensional space with constant negative curvature. Already toward the end of the previous century, Johann Heinrich Lambert (1728 1777), in his original investigation about Euclid’s fifth postulate,1 hinted that the angles of a triangle could sum to less than two right angles in the case where the triangle lies on an "imaginary sphere" (imaginäre Kugelfläche) [Lambert 1895, p. 203]. Lambert’s highly speculative remark was suggested by the «analogie entre le cercle et l’hyperbole» he had already exploited in his work on the irrationality of pi [Lambert 1768]; however, it would have been impossible to express that insight in a mathematically more definite form simply because, at the time, essential geometrical notions – first of all, that of curvature – were still to be developed.MathematicsItalian AcademyWorking papersAnalysis of Mathematical Fiction with Geometric Themes
http://academiccommons.columbia.edu/catalog/ac:153198
Shloming, Jennifer Rebeccahttp://hdl.handle.net/10022/AC:P:14870Wed, 10 Oct 2012 00:00:00 +0000Analysis of mathematical fiction with geometric themes is a study that connects the genre of mathematical fiction with informal learning. This study provides an analysis of 26 sources that include novels and short stories of mathematical fiction with regard to plot, geometric theme, cultural theme, and presentation. The authors' mathematical backgrounds are presented as they relate to both geometric and cultural themes. These backgrounds range from having little mathematical training to advance graduate work culminating in a Ph.D. in mathematics. This thesis demonstrated that regardless of background, the authors could write a mathematical fiction novel or short story with a dominant geometric theme. The authors' pedagogical approaches to delivering the geometric themes are also discussed. Applications from this study involve a pedagogical component that can be used in a classroom setting. All the sources analyzed in this study are fictional, but the geometric content is factual. Six categories of geometric topics were analyzed: plane geometry, solid geometry, projective geometry, axiomatics, topology, and the historical foundations of geometry. Geometry textbooks aligned with these categories were discussed with regard to mathematical fiction and formal learning. Cultural patterns were also analyzed for each source of mathematical fiction. There were also an analysis of the integration of cultural and geometric themes in the 26 sources of mathematical fiction; some of the cultural patterns discussed are gender bias, art, music, academia, mysticism, and social issues. On the basis of this discussion, recommendations for future studies involving the use of mathematical fiction were made.Mathematics education, Mathematicsjrs2137Mathematics, Science, and Technology, Mathematics EducationDissertationsA Meshless Method for Magnetohydrodynamics and Applications to Protoplanetary Disks
http://academiccommons.columbia.edu/catalog/ac:151776
McNally, Colin Powellhttp://hdl.handle.net/10022/AC:P:14425Fri, 17 Aug 2012 00:00:00 +0000This thesis presents an algorithm for simulating the equations of ideal magnetohydrodynamics and other systems of differential equations on an unstructured set of points represented by sample particles. Local, third-order, least-squares, polynomial interpolations (Moving Least Squares interpolations) are calculated from the field values of neighboring particles to obtain field values and spatial derivatives at the particle position. Field values and particle positions are advanced in time with a second order predictor-corrector scheme. The particles move with the fluid, so the time step is not limited by the Eulerian Courant-Friedrichs-Lewy condition. Full spatial adaptivity is implemented to ensure the particles fill the computational volume, which gives the algorithm substantial flexibility and power. A target resolution is specified for each point in space, with particles being added and deleted as needed to meet this target. Particle addition and deletion is based on a local void and clump detection algorithm. Dynamic artificial viscosity fields provide stability to the integration. The resulting algorithm provides a robust solution for modeling flows that require Lagrangian or adaptive discretizations to resolve. The code has been parallelized by adapting the framework provided by Gadget-2. A set of standard test problems, including one part in a million amplitude linear MHD waves, magnetized shock tubes, and Kelvin-Helmholtz instabilities are presented. Finally we demonstrate good agreement with analytic predictions of linear growth rates for magnetorotational instability in a cylindrical geometry. We provide a rigorous methodology for verifying a numerical method on two dimensional Kelvin-Helmholtz instability. The test problem was run in the Pencil Code, Athena, Enzo, NDSPHMHD, and Phurbas. A strict comparison, judgment, or ranking, between codes is beyond the scope of this work, although this work provides the mathematical framework needed for such a study. Nonetheless, how the test is posed circumvents the issues raised by tests starting from a sharp contact discontinuity yet it still shows the poor performance of Smoothed Particle Hydrodynamics. We then comment on the connection between this behavior and the underlying lack of zeroth-order consistency in Smoothed Particle Hydrodynamics interpolation. In astrophysical magnetohydrodynamics (MHD) and electrodynamics simulations, numerically enforcing the divergence free constraint on the magnetic field has been difficult. We observe that for point-based discretization, as used in finite-difference type and pseudo-spectral methods, the divergence free constraint can be satisfied entirely by a choice of interpolation used to define the derivatives of the magnetic field. As an example we demonstrate a new class of finite-difference type derivative operators on a regular grid which has the divergence free property. This principle clarifies the nature of magnetic monopole errors. The principles and techniques demonstrated in this chapter are particularly useful for the magnetic field, but can be applied to any vector field. Finally, we examine global zoom-in simulations of turbulent magnetorotationally unstable flow. We extract and analyze the high-current regions produced in the turbulent flow. Basic parameters of these regions are abstracted, and we build one dimensional models including non-ideal MHD, and radiative transfer. For sufficiently high temperatures, an instability resulting from the temperature dependence of the Ohmic resistivity is found. This instability concentrates current sheets, resulting in the possibility of rapid heating from temperatures on the order of 600 Kelvin to 2000 Kelvin in magnetorotationally turbulent regions of protoplanetary disks. This is a possible local mechanism for the melting of chondrules and the formation of other high-temperature materials in protoplanetary disks.Astronomy, Astrophysics, Mathematicscpm2118Astronomy and AstrophysicsDissertationsStatistical inference in two non-standard regression problems
http://academiccommons.columbia.edu/catalog/ac:151460
Seijo, Emilio Franciscohttp://hdl.handle.net/10022/AC:P:14317Wed, 08 Aug 2012 00:00:00 +0000This thesis analyzes two regression models in which their respective least squares estimators have nonstandard asymptotics. It is divided in an introduction and two parts. The introduction motivates the study of nonstandard problems and presents an outline of the contents of the remaining chapters. In part I, the least squares estimator of a multivariate convex regression function is studied in great detail. The main contribution here is a proof of the consistency of the aforementioned estimator in a completely nonparametric setting. Model misspecification, local rates of convergence and multidimensional regression models mixing convexity and componentwise monotonicity constraints will also be considered. Part II deals with change-point regression models and the issues that might arise when applying the bootstrap to these problems. The classical bootstrap is shown to be inconsistent on a simple change-point regression model, and an alternative (smoothed) bootstrap procedure is proposed and proved to be consistent. The superiority of the alternative method is also illustrated through a simulation study. In addition, a version of the continuous mapping theorem specially suited for change-point estimators is proved and used to derive the results concerning the bootstrap.Statistics, Applied mathematics, Mathematicsefs2113StatisticsDissertationsTaming unstable inverse problems: Mathematical routes toward high-resolution medical imaging modalities
http://academiccommons.columbia.edu/catalog/ac:146713
Monard, Francoishttp://hdl.handle.net/10022/AC:P:13162Mon, 07 May 2012 00:00:00 +0000This thesis explores two mathematical routes that make the transition from some severely ill-posed parameter reconstruction problems to better-posed versions of them. The general introduction starts by defining what we mean by an inverse problem and its theoretical analysis. We then provide motivations that come from the field of medical imaging. The first part consists in the analysis of an inverse problem involving the Boltzmann transport equation, with applications in Optical Tomography. There we investigate the reconstruction of the spatially-dependent part of the scattering kernel, from knowledge of angularly averaged outgoing traces of transport solutions and isotropic boundary sources. We study this problem in the stationary regime first, then in the time-harmonic regime. In particular we show, using techniques from functional analysis and stationary phase, that this inverse problem is severely ill-posed in the former setting, whereas it is mildly ill-posed in the latter. In this case, we deduce that making the measurements depend on modulation frequency allows to improve the stability of reconstructions. In the second part, we investigate the inverse problem of reconstructing a tensor-valued conductivity (or diffusion) coefficient in a second-order elliptic partial differential equation, from knowledge of internal measurements of power density type. This problem finds applications in the medical imaging modalities of Electrical Impedance Tomography and Optical Tomography, and the fact that one considers power densities is justified in practice by assuming a coupling of this physical model with ultrasound waves, a coupling assumption that is characteristic of so-called hybrid medical imaging methods. Starting from the famous Calderon's problem (i.e. the same parameter reconstruction problem from knowledge of boundary fluxes of solutions), and recalling its lack of injectivity and severe instability, we show how going from Dirichlet-to-Neumann data to considering the power density operator leads to reconstruction of the full conductivity tensor via explicit inversion formulas. Moreover, such reconstruction algorithms only require the loss of either zero or one derivative from the power density functionals to the unknown, depending on what part of the tensor one wants to reconstruct. The inversion formulas are worked out with the help of linear algebra and differential geometry, in particular calculus with the Euclidean connection. The practical pay-off of such theoretical improvements in injectivity and stability is twofold: (i) the lack of injectivity of CalderÃ³n's problem, no longer existing when using power density measurements, implies that future medical imaging modalities such as hybrid methods may make anisotropic properties of human tissues more accessible; (ii) the improvements in stability for both problems in transport and conductivity may yield practical improvements in the resolution of images of the reconstructed coefficients.Mathematics, Medical imaging and radiologyfm2234Applied Physics and Applied MathematicsDissertationsBordered Heegaard Floer Homology, Satellites, and Decategorification
http://academiccommons.columbia.edu/catalog/ac:146701
Petkova, Tsvetelina Vanevahttp://hdl.handle.net/10022/AC:P:13158Mon, 07 May 2012 00:00:00 +0000We use the methods of bordered Floer homology to provide a formula for both τ and HFK of certain satellite knots. In many cases, this formula determines the 4-ball genus of the satellite knot. In parallel, we explore the structural aspects of the bordered theory, developing the notion of an Euler characteristic for the modules associated to a bordered manifold. The Euler characteristic is an invariant of the underlying space, and shares many properties with the analogous invariants for closed 3-manifolds. We study the TQFT properties of this invariant corresponding to gluing, as well as its connections to sutured Floer homology. As one application, we show that the pairing theorem for bordered Floer homology categorifies the classical Alexander polynomial formula for satellites.Mathematicstvp2103MathematicsDissertationsOn Fourier-Mukai type functors
http://academiccommons.columbia.edu/catalog/ac:146747
Rizzardo, Alicehttp://hdl.handle.net/10022/AC:P:13173Mon, 07 May 2012 00:00:00 +0000In this thesis we study functors between bounded derived categories of sheaves and how they can be expressed in a geometric way, namely whether they are isomorphic to a Fourier-Mukai transform. Specifically, we describe the behavior of a functor between derived categories of smooth projective varieties when restricted to the derived category of the generic point of the second variety, when this last variety is a curve, a point or a rational surface. We also compute in general some sheaves that play the role of the cohomology sheaves of the kernel of a Fourier-Mukai transform and are then able to exhibit a class of functors that are neither faithful nor full, that are isomorphic to a Fourier-Mukai transform.Mathematicsar2532MathematicsDissertationsEigenvarieties and twisted eigenvarieties
http://academiccommons.columbia.edu/catalog/ac:146594
Xiang, Zhengyuhttp://hdl.handle.net/10022/AC:P:13124Thu, 03 May 2012 00:00:00 +0000For an arbitrary reductive group G, we construct the full eigenvariety E, which parameterizes all p-adic overconvergent cohomological eigenforms of G in the sense of Ash-Stevens and Urban. Further, given an algebraic automorphism a of G, we construct the twisted eigenvariety E^a, a rigid subspace of E, which parameterizes all eigenforms that are invariant under a. In particular, in the case G = GLn, we prove that every self-dual automorphic representation can be deformed into a family of self-dual cuspidal forms containing a Zariski dense subset of classical points. This is the inverse of Ash-Pollack-Stevens conjecture. We also give some hint to this conjecture.Mathematicszx2108MathematicsDissertationsForbidden Substructures in Graphs and Trigraphs, and Related Coloring Problems
http://academiccommons.columbia.edu/catalog/ac:146465
Penev, Irenahttp://hdl.handle.net/10022/AC:P:13082Tue, 01 May 2012 00:00:00 +0000Given a graph G, χ(G) denotes the chromatic number of G, and ω(G) denotes the clique number of G (i.e. the maximum number of pairwise adjacent vertices in G). A graph G is perfect provided that for every induced subgraph H of G, χ(H) = ω(H). This thesis addresses several problems from the theory of perfect graphs and generalizations of perfect graphs. The bull is a five-vertex graph consisting of a triangle and two vertex-disjoint pendant edges; a graph is said to be bull-free provided that no induced subgraph of it is a bull. The first result of this thesis is a structure theorem for bull-free perfect graphs. This is joint work with Chudnovsky, and it first appeared in [12]. The second result of this thesis is a decomposition theorem for bull-free perfect graphs, which we then use to give a polynomial time combinatorial coloring algorithm for bull-free perfect graphs. We remark that de Figueiredo and Maffray [33] previously solved this same problem, however, the algorithm presented in this thesis is faster than the algorithm from [33]. We note that a decomposition theorem that is very similar (but slightly weaker) than the one from this thesis was originally proven in [52], however, the proof in this thesis is significantly different from the one in [52]. The algorithm from this thesis is very similar to the one from [52]. A class G of graphs is said to be χ-bounded provided that there exists a function f such that for all G in G, and all induced subgraphs H of G, we have that χ(H) ≤ f(ω(H)). χ-bounded classes were introduced by Gyarfas [41] as a generalization of the class of perfect graphs (clearly, the class of perfect graphs is χ-bounded by the identity function). Given a graph H, we denote by Forb*(H) the class of all graphs that do not contain any subdivision of H as an induced subgraph. In [57], Scott proved that Forb*(T) is χ-bounded for every tree T, and he conjectured that Forb*(H) is χ-bounded for every graph H. Recently, a group of authors constructed a counterexample to Scott's conjecture [51]. This raises the following question: for which graphs H is Scott's conjecture true? In this thesis, we present the proof of Scott's conjecture for the cases when H is the paw (i.e. a four-vertex graph consisting of a triangle and a pendant edge), the bull, and a necklace (i.e. a graph obtained from a path by choosing a matching such that no edge of the matching is incident with an endpoint of the path, and for each edge of the matching, adding a vertex adjacent to the ends of this edge). This is joint work with Chudnovsky, Scott, and Trotignon, and it originally appeared in [13]. Finally, we consider several operations (namely, "substitution," "gluing along a clique," and "gluing along a bounded number of vertices"), and we show that the closure of a χ-bounded class under any one of them, as well as under certain combinations of these three operations (in particular, the combination of substitution and gluing along a clique, as well as the combination of gluing along a clique and gluing along a bounded number of vertices) is again χ-bounded. This is joint work with Chudnovsky, Scott, and Trotignon, and it originally appeared in [14].Mathematicsip2158Industrial Engineering and Operations Research, MathematicsDissertationsThe Asymptotic Cone of Teichmuller Space: Thickness and Divergence
http://academiccommons.columbia.edu/catalog/ac:146378
Sultan, Harold Markhttp://hdl.handle.net/10022/AC:P:13058Mon, 30 Apr 2012 00:00:00 +0000Using the geometric model of the pants complex, we study the Asymptotic Cone of Teichmüller space equipped with the Weil Petersson metric. In particular, we provide a characterization of the canonical finest pieces in the tree-graded structure of the asymptotic cone of Teichmüller space along the same lines as similar characterizations for right angled Artin groups by Behrstock-Charney and for mapping class groups by Behrstock-Kleiner-Minsky-Mosher. As a corollary of the characterization, we complete the thickness classification of Teichmüller spaces for all surfaces of finite type. In particular, we prove that Teichmüller space of the genus two surface with one boundary component (or puncture) can be uniquely characterized in the following two senses: it is thick of order two, and it has superquadratic yet at most cubic divergence. In addition, we characterize strongly contracting quasi-geodesics in Teichmüller space, generalizing results of Brock-Masur-Minsky. As a tool in the thesis, we develop a natural relative of the curve complex called the complex of separating multicurves which may be of independent interest. The final chapter includes various related and independent results including, under mild hypotheses, a proof of the equivalence of wideness and unconstrictedness in the CAT(0) setting, as well as adapted versions of three preprints. Specifically, in the three preprints we characterize hyperbolic type quasi-geodesics in CAT(0) spaces, we prove that the separating curve complex of the genus two surface satisfies a quasidistance formula and is Gromov-hyperbolic, and we study the net of separating pants decompositions in the pants complex.Mathematicshms2121MathematicsDissertationsDevelopments in the Extended Finite Element Method and Algebraic Multigrid for Solid Mechanics Problems Involving Discontinuities
http://academiccommons.columbia.edu/catalog/ac:146402
Hiriyur, Badri Krishna Jainathhttp://hdl.handle.net/10022/AC:P:13065Mon, 30 Apr 2012 00:00:00 +0000In this dissertation, some contribututions related to computational modeling and solution of solid mechanics problems involving discontinuities are discussed. The main tool employed for discrete modeling of discontinuities is the extended finite element method and the primary solution method discussed is the algebraic multigrid. The extended finite element method has been shown to be effective for both weak and strong discontinuities. With respect to weak discontinuities, a new approach that couples the extended finite element method with Monte Carlo simulations with the goal of quantifying uncertainty in homogenization of material properties of random microstructures is presented. For accelearated solution of linear systems arising from problems involving cracks, several new methods involving the algebraic multigrid are presented. In the first approach, the Schur complement of the linear system arising from XFEM is used to develop a Hybrid-AMG method such that crack-conforming aggregates are formed. Another alternative approach involves transforming the original linear system into a modified system that is amenable for a direct application of algebraic multigrid. It is shown that if only Heaviside-enrichments are present, a simple transformation based on the phantom-node approach is available, which decouples the linear sysem along the discontinuities such that crack conforming aggregates are automatically generated via smoother aggregation algebraic multigrid. Various numerical examples are presented to verify the accuracy of the resuting solutions and the convergence properties of the proposed algorithms. The parallel scalability performance of the implementation are also discussed.Engineering, Mathematicsbkh2112Civil Engineering and Engineering MechanicsDissertationsLimiting Properties of Certain Geometric Flows in Complex Geometry
http://academiccommons.columbia.edu/catalog/ac:146314
Jacob, Adam Joshuahttp://hdl.handle.net/10022/AC:P:13047Thu, 19 Apr 2012 00:00:00 +0000In this thesis, we study convergence results of certain non-linear geometric flows on vector bundles over complex manifolds. First we consider the case of a semi-stable vector bundle E over a compact Kahler manifold X of arbitrary dimension. We show that in this case Donaldson's functional is bounded from below. This allows us to construct an approximate Hermitian-Einstein structure on E along the Donaldson heat flow, generalizing a classic result of Kobayashi for projective manifolds to the Kahler case. Next we turn to general unstable bundles. We show that along a solution of the Yang-Mills flow, the trace of the curvature approaches in L2 an endomorphism with constant eigenvalues given by the slopes of the quotients from the Harder-Narasimhan filtration of E. This proves a sharp lower bound for the Hermitian-Yang-Mills functional and thus the Yang-Mills functional, generalizing to arbitrary dimension a formula of Atiyah and Bott first proven on Riemann surfaces. Furthermore, we show any reflexive extension to all of X of the limiting bundle is isomorphic to the double dual of the graded quotients from the Harder-Narasimhan-Seshadri filtration, verifying a conjecture of Bando and Siu. Our work on semi-stable bundles plays an important part of this result. For the final section of this thesis, we show that, in the case where X is an arbitrary Hermitian manifold equipped with a Gauduchon metric, given a stable Higgs bundle the Donaldson heat flow converges along a subsequence of times to a Hermitian-Einstein connection. This allows us to extend to the non-Kahler case the correspondence between stable Higgs bundles and (possibly) non-unitary Hermitian-Einstein connections first proven by Simpson on Kahler manifolds.Mathematicsajj2107MathematicsDissertationsArithmetic inner product formula for unitary groups
http://academiccommons.columbia.edu/catalog/ac:146317
Liu, Yifenghttp://hdl.handle.net/10022/AC:P:13049Thu, 19 Apr 2012 00:00:00 +0000We study central derivatives of L-functions of cuspidal automorphic representations for unitary groups of even variables defined over a totally real number field, and their relation with the canonical height of special cycles on Shimura varieties attached to unitary groups of the same size. We formulate a precise conjecture about an arithmetic analogue of the classical Rallis' inner product formula, which we call arithmetic inner product formula, and confirm it for unitary groups of two variables. In particular, we calculate the Néron-Tate height of special points on Shimura curves attached to certain unitary groups of two variables. For an irreducible cuspidal automorphic representation of a quasi-split unitary group, we can associate it an ε-factor, which is either 1 or -1, via the dichotomy phenomenon of local theta liftings. If such factor is -1, the central L-value of the representation always vanishes and the Rallis' inner product formula is not interesting. Therefore, we are motivated to consider its central derivative, and propose the arithmetic inner product formula. In the course of such formulation, we prove a modularity theorem of the generating series on the level of Chow groups. We also show the cohomological triviality of the arithmetic theta lifting, which is a necessary step to consider the canonical height. As evidence, we also prove an arithmetic local Siegel-Weil formula at archimedean places for unitary groups of arbitrary sizes, which contributes as a part of the local comparison of the conjectural arithmetic inner product formula.MathematicsMathematicsDissertationsMethods for Computing Genus Distribution Using Double-Rooted Graphs
http://academiccommons.columbia.edu/catalog/ac:145943
Khan, Imran Faridhttp://hdl.handle.net/10022/AC:P:12962Thu, 05 Apr 2012 00:00:00 +0000This thesis develops general methods for computing the genus distribution of various types of graph families, using the concept of double-rooted graphs, which are defined to be graphs with two vertices designated as roots (the methods developed in this dissertation are limited to the cases where one of the two roots is restricted to be of valence two). I define partials and productions, and I use these as follows: (i) to compute the genus distribution of a graph obtained through the vertex amalgamation of a double-rooted graph with a single-rooted graph, and to show how these can be used to obtain recurrences for the genus distribution of iteratively growing infinite graph families. (ii) to compute the genus distribution of a graph obtained (a) through the operation of self-vertex-amalgamation on a double-rooted graph, and (b) through the operation of edge-addition on a double-rooted graph, and finally (iii) to develop a method to compute the recurrences for the genus distribution of the graph family generated by the Cartesian product of P3 and Pn.Computer science, Mathematicsifk2103Computer ScienceDissertationsAn Algebraic Circle Method
http://academiccommons.columbia.edu/catalog/ac:134194
Pugin, Thibauthttp://hdl.handle.net/10022/AC:P:10557Mon, 20 Jun 2011 00:00:00 +0000In this thesis we present an adaptation of the Hardy-Littlewood Circle Method to give estimates for the number of curves in a variety over a finite field. The key step in the classical Circle Method is to prove that some cancellation occurs in some exponential sums. Using a theorem of Katz, we reduce this to bounding the dimension of some singular loci. The method is fully carried out to estimate the number of rational curves in a Fermat hypersurface of low degree and some suggestions are given as to how to handle other cases. We draw geometrical consequences from the main estimates, for instance the irreducibility of the space of rational curves on a Fermat hypersurface in a given degree range, and a bound on the dimension of the singular locus of the moduli space.Mathematicstfp2102MathematicsDissertationsKnot Floer Homology and Categorification
http://academiccommons.columbia.edu/catalog/ac:132323
Gilmore, Allison Leighhttp://hdl.handle.net/10022/AC:P:10411Wed, 18 May 2011 00:00:00 +0000With the goal of better understanding the connections between knot homology theories arising from categorification and from Heegaard Floer homology, we present a self-contained construction of knot Floer homology in the language of HOMFLY-PT homology. Using the cube of resolutions for knot Floer homology defined by Ozsváth and Szabó, we first give a purely algebraic proof of invariance that does not depend on Heegaard diagrams, holomorphic disks, or grid diagrams. Then, taking Khovanov's HOMFLY-PT homology as our model, we define a category of twisted Soergel bimodules and construct a braid group action on the homotopy category of complexes of twisted Soergel bimodules. We prove that the category of twisted Soergel bimodules categorifies the Hecke algebra with an extra indeterminate and its inverse adjoined. The braid group action, which is defined via twisted Rouquier complexes, is simultaneously a natural extension of the knot Floer cube of resolutions and a mild modification of the action by Rouquier complexes used by Khovanov in defining HOMFLY-PT homology. Finally, we introduce an operation Qu to play the role that Hochschild homology plays in HOMFLY-PT homology. We conjecture that applying Qu to the twisted Rouquier complex associated to a braid produces the knot Floer cube of resolutions chain complex associated to its braid closure. We prove a partial result in this direction.MathematicsMathematicsDissertationsMonopole Floer homology, link surgery, and odd Khovanov homology
http://academiccommons.columbia.edu/catalog/ac:132266
Bloom, Jonathan Michaelhttp://hdl.handle.net/10022/AC:P:10392Tue, 17 May 2011 00:00:00 +0000We construct a link surgery spectral sequence for all versions of monopole Floer homology with mod 2 coefficients, generalizing the exact triangle. The spectral sequence begins with the monopole Floer homology of a hypercube of surgeries on a 3-manifold Y, and converges to the monopole Floer homology of Y itself. This allows one to realize the latter group as the homology of a complex over a combinatorial set of generators. Our construction relates the topology of link surgeries to the combinatorics of graph associahedra, leading to new inductive realizations of the latter. As an application, given a link L in the 3-sphere, we prove that the monopole Floer homology of the branched double-cover arises via a filtered perturbation of the differential on the reduced Khovanov complex of a diagram of L. The associated spectral sequence carries a filtration grading, as well as a mod 2 grading which interpolates between the delta grading on Khovanov homology and the mod 2 grading on Floer homology. Furthermore, the bigraded isomorphism class of the higher pages depends only on the Conway-mutation equivalence class of L. We constrain the existence of an integer bigrading by considering versions of the spectral sequence with non-trivial U action, and determine all monopole Floer groups of branched double-covers of links with thin Khovanov homology. Motivated by this perspective, we show that odd Khovanov homology with integer coefficients is mutation invariant. The proof uses only elementary algebraic topology and leads to a new formula for link signature that is well-adapted to Khovanov homology.Mathematicsjmb2177MathematicsDissertationsSoergel Diagrammatics for Dihedral Groups
http://academiccommons.columbia.edu/catalog/ac:132257
Elias, Benhttp://hdl.handle.net/10022/AC:P:10389Tue, 17 May 2011 00:00:00 +0000We give a diagrammatic presentation for the category of Soergel bimodules for the dihedral group W, finite or infinite. The (two-colored) Temperley-Lieb category is embedded inside this category as the degree 0 morphisms between color-alternating objects. The indecomposable Soergel bimodules are the images of Jones-Wenzl projectors. When W is finite, the Temperley-Lieb category must be taken at an appropriate root of unity, and the negligible Jones-Wenzl projector yields the Soergel bimodule for the longest element of W.Mathematicsbse2103MathematicsDissertationsModuli Spaces of Dynamical Systems on Pn
http://academiccommons.columbia.edu/catalog/ac:132314
Levy, Alonhttp://hdl.handle.net/10022/AC:P:10408Tue, 17 May 2011 00:00:00 +0000This thesis studies the space of morphisms on Pn defined by polynomials of degree d and its quotient by the conjugation action of PGL(n+1), which should be thought of as coordinate change. First, we construct the quotient using geometric invariant theory, proving that it is a geometric quotient and that the stabilizer group in PGL(n+1) of each morphism is finite and bounded in terms of n and d. We then show that when n = 1, the quotient space is rational over a field of any characteristic. We then study semistable reduction in this space. For every complete curve C in the semistable completion of the quotient space, we can find curves upstairs mapping down to it; this leads to an abstract complete curve D with a projective vector bundle parametrizing maps on the curve. The bundle is trivial iff there exists a complete curve D in the semistable space upstairs mapping down to C; we show that for every n and d we can find a C for which no such D exists. Finally, in the case where D does exist, we show that, whenever it lies in the stable space, the map from D to C is ramified only over points with unusually large stabilizer, which for a fixed rational C will bound the degree of the map from D to C.Mathematicsal2495MathematicsDissertationsBordered Sutured Floer Homology
http://academiccommons.columbia.edu/catalog/ac:132302
Zarev, Rumenhttp://hdl.handle.net/10022/AC:P:10404Tue, 17 May 2011 00:00:00 +0000We investigate the relationship between two versions of Heegaard Floer homology for 3-manifolds with boundary--the sutured Floer homology of Juhasz, and the bordered Heegaard Floer homology of Lipshitz, Ozsvath, and Thurston. We define a new invariant called Bordered sutured Floer homology which encompasses these two invariants as special cases. Using the properties of this new invariant we prove a correspondence between the original bordered and sutured homologies. In one direction we prove that for a 3-manifold Y with connected boundary F = dY , and sutures Gamma in dY , we can compute the sutured Floer homology SFH(Y ) from the bordered invariant CFA(Y )A(F ) . The chain complex SFC(Y, Gamma) defining SFH is quasi-isomorphic to the derived tensor product CFA(Y )xCFD(Gamma) where A(F )CFD(Gamma) is a module associated to Gamma. In the other direction we give a description of the bordered invariants in terms of sutured Floer homology. If F is a closed connected surface, then the bordered algebra A(F) is a direct sum of certain sutured Floer complexes. These correspond to the 3-manifold (F \ D2;)Ã—[0,1], where the sutures vary in a finite collection. Similarly, if Y is a connected 3-manifold with boundary dY = F , the module CFA(Y)A(F) is a direct sum of sutured Floer complexes for Y where the sutures on dY vary over a finite collection. The multiplication structure on A(F) and the action of A(F) on CFA(Y) correspond to a natural gluing map on sutured Floer homology. (Further work of the author shows that this map coincides with the one defined by Honda, Kazez, and Matic, using contact topology and open book decompositions).Mathematicsriz2102MathematicsDissertationsCorrector Theory in Random Homogenization of Partial Differential Equations
http://academiccommons.columbia.edu/catalog/ac:132034
Jing, Wenjiahttp://hdl.handle.net/10022/AC:P:10336Wed, 11 May 2011 00:00:00 +0000We derive systematically a theory for the correctors in random homogenization of partial differential equations with highly oscillatory coefficients, which arise naturally in many areas of natural sciences and engineering. This corrector theory is of great practical importance in many applications when estimating the random fluctuations in the solution is as important as finding its homogenization limit. This thesis consists of three parts. In the first part, we study some properties of random fields that are useful to control corrector in homogenization of PDE. These random fields mostly have parameters in multi-dimensional Euclidean spaces. In the second part, we derive a corrector theory systematically that works in general for linear partial differential equations, with random coefficients appearing in their zero-order, i.e., non-differential, terms. The derivation is a combination of the studies of random fields and applications of PDE theory. In the third part of this thesis, we derive a framework of analyzing multiscale numerical algorithms that are widely used to approximate homogenization, to test if they succeed in capturing the limiting corrector predicted by the theory.Mathematics, Applied mathematicswj2136Applied Physics and Applied MathematicsDissertationsOn Using Graphical Calculi: Centers, Zeroth Hochschild Homology and Possible Compositions of Induction and Restriction Functors in Various Diagrammatical Algebras
http://academiccommons.columbia.edu/catalog/ac:132037
Brichard, Joellehttp://hdl.handle.net/10022/AC:P:10337Wed, 11 May 2011 00:00:00 +0000This thesis is divided into three chapters, each using certain graphical calculus in a slightly different way. In the first chapter, we compute the dimension of the center of the 0-Hecke algebra Hn and of the Nilcoxeter algebra NCn using a calculus of diagrams on the Moebius band. In the case of the Nilcoxeter algebra, this calculus is shown to produce a basis for Z(NCn) and the table of multiplication in this basis is shown to be trivial. We conjecture that a basis for Z(Hn) can also be obtained in a specic way from this topological calculus. In the second chapter, we also use a calculus of diagrams on the annulus and the Moebius band to determine the zeroth Hochschild Homology of Kuperberg's webs for rank two Lie algebras. We use results from Sikora and Westbury to prove the linear independence of these webs on these surfaces. In the third chapter, we use other diagrams to attempt to find explicitely the possible compositions of the induction and restriction functors in the cyclotomic quotients of the NilHecke algebra. We use a computer program to obtain partial results.Mathematicsjb2543MathematicsDissertationsBounds for the Spectral Mean Value of Central Values of L-functions
http://academiccommons.columbia.edu/catalog/ac:132025
Lu, Qinghttp://hdl.handle.net/10022/AC:P:10333Wed, 11 May 2011 00:00:00 +0000We prove two results about the boundedness of spectral mean value of Rankin-Selberg L-functions at s = 1/2, which is an analogue for Eisenstein series of X. Li's result for Hecke-Maass forms.Mathematicsql2132MathematicsDissertationsQuantitative Modeling of Credit Derivatives
http://academiccommons.columbia.edu/catalog/ac:131549
Kan, Yu Hanghttp://hdl.handle.net/10022/AC:P:10272Thu, 05 May 2011 00:00:00 +0000The recent financial crisis has revealed major shortcomings in the existing approaches for modeling credit derivatives. This dissertation studies various issues related to the modeling of credit derivatives: hedging of portfolio credit derivatives, calibration of dynamic credit models, and modeling of credit default swap portfolios. In the first part, we compare the performance of various hedging strategies for index collateralized debt obligation (CDO) tranches during the recent financial crisis. Our empirical analysis shows evidence for market incompleteness: a large proportion of risk in the CDO tranches appears to be unhedgeable. We also show that, unlike what is commonly assumed, dynamic models do not necessarily perform better than static models, nor do high-dimensional bottom-up models perform better than simpler top-down models. On the other hand, model-free regression-based hedging appears to be surprisingly effective when compared to other hedging strategies. The second part is devoted to computational methods for constructing an arbitrage-free CDO pricing model compatible with observed CDO prices. This method makes use of an inversion formula for computing the aggregate default rate in a portfolio from expected tranche notionals, and a quadratic programming method for recovering expected tranche notionals from CDO spreads. Comparing this approach to other calibration methods, we find that model-dependent quantities such as the forward starting tranche spreads and jump-to-default ratios are quite sensitive to the calibration method used, even within the same model class. The last chapter of this dissertation focuses on statistical modeling of credit default swaps (CDSs). We undertake a systematic study of the univariate and multivariate properties of CDS spreads, using time series of the CDX Investment Grade index constituents from 2005 to 2009. We then propose a heavy-tailed multivariate time series model for CDS spreads that captures these properties. Our model can be used as a framework for measuring and managing the risk of CDS portfolios, and is shown to have better performance than the affine jump-diffusion or random walk models for predicting loss quantiles of various CDS portfolios.Finance, Mathematicsyk2246Industrial Engineering and Operations ResearchDissertationsOptimal Trading Strategies Under Arbitrage
http://academiccommons.columbia.edu/catalog/ac:131477
Ruf, Johannes Karl Dominikhttp://hdl.handle.net/10022/AC:P:10250Fri, 29 Apr 2011 00:00:00 +0000This thesis analyzes models of financial markets that incorporate the possibility of arbitrage opportunities. The first part demonstrates how explicit formulas for optimal trading strategies in terms of minimal required initial capital can be derived in order to replicate a given terminal wealth in a continuous-time Markovian context. Towards this end, only the existence of a square-integrable market price of risk (rather than the existence of an equivalent local martingale measure) is assumed. A new measure under which the dynamics of the stock price processes simplify is constructed. It is shown that delta hedging does not depend on the "no free lunch with vanishing risk" assumption. However, in the presence of arbitrage opportunities, finding an optimal strategy is directly linked to the non-uniqueness of the partial differential equation corresponding to the Black-Scholes equation. In order to apply these analytic tools, sufficient conditions are derived for the necessary differentiability of expectations indexed over the initial market configuration. The phenomenon of "bubbles," which has been a popular topic in the recent academic literature, appears as a special case of the setting in the first part of this thesis. Several examples at the end of the first part illustrate the techniques contained therein. In the second part, a more general point of view is taken. The stock price processes, which again allow for the possibility of arbitrage, are no longer assumed to be Markovian, but rather only It^o processes. We then prove the Second Fundamental Theorem of Asset Pricing for these markets: A market is complete, meaning that any bounded contingent claim is replicable, if and only if the stochastic discount factor is unique. Conditions under which a contingent claim can be perfectly replicated in an incomplete market are established. Then, precise conditions under which relative arbitrage and strong relative arbitrage with respect to a given trading strategy exist are explicated. In addition, it is shown that if the market is quasi-complete, meaning that any bounded contingent claim measurable with respect to the stock price filtration is replicable, relative arbitrage implies strong relative arbitrage. It is further demonstrated that markets are quasi-complete, subject to the condition that the drift and diffusion coefficients are measurable with respect to the stock price filtration.Mathematics, Financejkr2115Statistics, MathematicsDissertationsTwo Approaches to Non-Zero-Sum Stochastic Differential Games of Control and Stopping
http://academiccommons.columbia.edu/catalog/ac:131462
Li, Qinghuahttp://hdl.handle.net/10022/AC:P:10245Fri, 29 Apr 2011 00:00:00 +0000This dissertation takes two approaches - martingale and backward stochastic differential equation (BSDE) - to solve non-zero-sum stochastic differential games in which all players can control and stop the reward streams of the games. Existence of equilibrium stopping rules is proved under some assumptions. The martingale part provides an equivalent martingale characterization of Nash equilibrium strategies of the games. When using equilibrium stopping rules, Isaacs' condition is necessary and sufficient for the existence of an equilibrium control set. The BSDE part shows that solutions to BSDEs provide value processes of the games. A multidimensional BSDE with reflecting barrier is studied in two cases for its solution: existence and uniqueness with Lipschitz growth, and existence in a Markovian system with linear growth rate.Mathematicsql2133Statistics, MathematicsDissertationsLattice Subdivisions and Tropical Oriented Matroids, Featuring Products of Simplices
http://academiccommons.columbia.edu/catalog/ac:131450
Piechnik, Lindsay C.http://hdl.handle.net/10022/AC:P:10240Fri, 29 Apr 2011 00:00:00 +0000Subdivisions of products of simplices, and their applications, appear across mathematics. In this thesis, they are the tie between two branches of my research: polytopal lattice subdivisions and tropical oriented matroid theory. The first chapter describes desirable combinatorial properties of subdivisions of lattice polytopes, and how they can be used to address algebraic questions. Chapter two discusses tropical hyperplane arrangements and the tropical oriented matroid theory they inspire, paying particular attention to the previously uninvestigated distinction between the generic and non-generic cases. The focus of chapter three is products of simplices, and their connections and applications to ideas covered in the first two chapters.Mathematicslp2149Mathematics (Barnard College), MathematicsDissertationsF-virtual Abelian Varieties and Rallis Inner Product Formula
http://academiccommons.columbia.edu/catalog/ac:131441
Wu, Chenyanhttp://hdl.handle.net/10022/AC:P:10237Fri, 29 Apr 2011 00:00:00 +0000This thesis consists of two topics. First we study F-virtual Abelian varieties of GL2-type where F is a number field. We show the relation between these Abelian varieties and those defined over F. We compare their l-adic representations and study the modularity of F-virtual Abelian varieties of GL2-type. Then we construct their moduli spaces and in the case where the moduli space is a surface we give criteria when it is of general type. We also give two examples of surfaces that are rational and one that is neither rational nor of general type. Second we prove a crucial case of Siegel-Weil formula for orthogonal groups and metaplectic groups. With this we can compute the pairing of theta functions and show in this case that it is related to the central value of Langlands L-function. This new case of Rallis inner product formula enables us to relate nonvanishing of L-value to the nonvanishing of theta lifting.Mathematicscw2314MathematicsDissertationsRandomness
http://academiccommons.columbia.edu/catalog/ac:130602
Luccio, Fabrizio; Pagli, Lindahttp://hdl.handle.net/10022/AC:P:10105Thu, 31 Mar 2011 00:00:00 +0000MathematicsItalian AcademyWorking papersApproximate converse theorem
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Lee, Minhttp://hdl.handle.net/10022/AC:P:9949Fri, 11 Mar 2011 00:00:00 +0000The theme of this thesis is an "approximate converse theorem" for globally unramified cuspidal representations of PGL(n, A), n ≥ 1. For a given set of Langlands parameters for some places of Q, we can compute ε > 0 such that there exists a genuine globally unramified cuspidal representation, whose Langlands parameters are within ε of the given ones for finitely many places.Mathematicsml2660MathematicsDissertations