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Academic Commons Search Resultsen-usApproximate converse theorem
https://academiccommons.columbia.edu/catalog/ac:130096
Lee, Min10.7916/D82V2MXKWed, 29 Mar 2017 19:55:56 +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.Mathematicsml2660MathematicsThesesNonlocal models with a finite range of nonlocal interactions
https://academiccommons.columbia.edu/catalog/ac:206858
Tian, Xiaochuanhttp://dx.doi.org/10.7916/D8ZG6XWNMon, 06 Feb 2017 12:14:10 +0000Nonlocal phenomena are ubiquitous in nature. The nonlocal models investigated in this thesis use integration in replace of differentiation and provide alternatives to the classical partial differential equations. The nonlocal interaction kernels in the models are assumed to be as general as possible and usually involve finite range of nonlocal interactions. Such settings on one hand allow us to connect nonlocal models with the existing classical models through various asymptotic limits of the modeling parameter, and on the other hand enjoy practical significance especially for multiscale modeling and simulations.
To make connections with classical models at the discrete level, the central theme of the numerical analysis for nonlocal models in this thesis concerns with numerical schemes that are robust under the changes of modeling parameters, with mathematical analysis provided as theoretical foundations. Together with extensive discussions of linear nonlocal diffusion and nonlocal mechanics models, we also touch upon other topics such as high order nonlocal models, nonlinear nonlocal fracture models and coupling of models characterized by different scales.Mathematics, Calculus, Integral, Mathematical models, Multiscale modeling, Mathematicsxt2156Applied Physics and Applied MathematicsDissertationsA GL(3) Kuznetsov Trace Formula and the Distribution of Fourier Coefficients of Maass Forms
https://academiccommons.columbia.edu/catalog/ac:202236
Guerreiro, João Leitãohttp://dx.doi.org/10.7916/D8GM87JPThu, 15 Sep 2016 18:03:01 +0000We study the problem of the distribution of certain GL(3) Maass forms, namely, we obtain a Weyl’s law type result that characterizes the distribution of their eigenvalues, and an orthogonality relation for the Fourier coefficients of these Maass forms. The approach relies on a Kuznetsov trace formula on GL(3) and on the inversion formula for the Lebedev-Whittaker transform. The family of Maass forms being studied has zero density in the set of all GL(3) Maass forms and contains all self-dual forms. The self-dual forms on GL(3) can also be realised as symmetric square lifts of GL(2) Maass forms by the work of Gelbart-Jacquet. Furthermore, we also establish an explicit inversion formula for the Lebedev-Whittaker transform, in the nonarchimedean case, with a view to applications.Mathematics, Weyl's problem, Eigenvalues, Mathematics, Trace formulasjlg2211MathematicsDissertationsA local relative trace formula for spherical varieties
https://academiccommons.columbia.edu/catalog/ac:202143
Filip, Ioanhttp://dx.doi.org/10.7916/D8HX1CWKFri, 09 Sep 2016 12:16:41 +0000Let F be a local non-Archimedean field of characteristic zero. We prove a Plancherel formula for the symmetric space GL(2,F)\GL(2,E), where E/F is an unramified quadratic extension. Our method relies on intrinsic geometric and combinatorial properties of spherical varieties and constitutes the local counterpart of the global computation of the Flicker-Rallis period as a residue of periods against Eisenstein series. We also give a novel derivation of the Plancherel formula for the strongly tempered variety T\PGL(2) over F (with maximal split torus T) using a canonical smooth asymptotics morphism and a contour shifting method. In this rank one local setting, our proof is similar to Langlands' proof over global fields describing the spectrum of a reductive group in terms of residues of Eisenstein series. Finally, using both L2-decompositions, we develop a local relative trace formula and outline a comparison result in the setting of the unitary rank one Gan-Gross-Prasad conjecture.Mathematics, Trace formulas, Combinatorial geometry, Geometry, Algebraic, Mathematicsif2179MathematicsDissertationsDynamic Algorithms for Shortest Paths and Matching
https://academiccommons.columbia.edu/catalog/ac:202021
Bernstein, Aaronhttp://dx.doi.org/10.7916/D8QF8T2WFri, 19 Aug 2016 19:22:23 +0000There is a long history of research in theoretical computer science devoted to designing efficient algorithms for graph problems. In many modern applications the graph in question is changing over time, and we would like to avoid rerunning our algorithm on the entire graph every time a small change occurs. The evolving nature of graphs motivates the dynamic graph model, in which the goal is to minimize the amount of work needed to reoptimize the solution when the graph changes. There is a large body of literature on dynamic algorithms for basic problems that arise in graphs. This thesis presents several improved dynamic algorithms for two fundamental graph problems: shortest paths, and matching.Computer science, Mathematics, Computer algorithms, Graphic methods, Computer science, Mathematicsab3417Computer Science, Industrial Engineering and Operations ResearchDissertationsEnlargement of Filtration and the Strict Local Martingale Property in Stochastic Differential Equations
https://academiccommons.columbia.edu/catalog/ac:201869
Dandapani, Aditihttp://dx.doi.org/10.7916/D8XW4JZ2Tue, 02 Aug 2016 12:25:23 +0000In this thesis, we study the strict local martingale property of solutions of various types of stochastic differential equations and the effect of an initial expansion of the filtration on this property. For the models we consider, we either use existing criteria or, in the case where the stochastic differential equation has jumps, develop new criteria that can can detect the presence of the strict local martingale property. We develop deterministic sufficient conditions on the drift and diffusion coefficient of the stochastic process such that an enlargement by initial expansion of the filtration can produce a strict local martingale from a true martingale. We also develop a way of characterizing the martingale property in stochastic volatility models where the local martingale has a general diffusion coefficient.Mathematics, Applied mathematics, Martingales (Mathematics), Stochastic differential equationsad2259Applied Physics and Applied Mathematics, StatisticsDissertationsInference On Two-Component Mixtures Under Tail Restrictions
https://academiccommons.columbia.edu/catalog/ac:199769
Jochmans, Koen; Henry, Marc; Salanie, Bernardhttp://dx.doi.org/10.7916/D8B27VCRTue, 07 Jun 2016 17:02:50 +0000Many econometric models can be analyzed as finite mixtures. We focus on two-component mixtures and we show that they are nonparametrically point identified by a combination of an exclusion restriction and tail restrictions. Our identification analysis suggests simple closed-form estimators of the component distributions and mixing proportions, as well as a specification test. We derive their asymptotic properties using results on tail empirical processes and we present a simulation study that documents their finite-sample performance.Economics, Mathematics, Econometrics--Mathematical models, Estimation theory--Asymptotic theorybs2237EconomicsWorking papersHigher-order Properties of Approximate Estimators
https://academiccommons.columbia.edu/catalog/ac:199766
Kristensen, Dennis; Salanie, Bernardhttp://dx.doi.org/10.7916/D8KK9BVXTue, 07 Jun 2016 16:44:08 +0000Many modern estimation methods in econometrics approximate an objective function, for instance, through simulation or discretization. These approximations typically affect both bias and variance of the resulting estimator. We first provide a higher-order expansion of such “approximate” estimators that takes into account the errors due to the use of approximations. We show how a Newton-Raphson adjustment can reduce the impact of approximations. Then we use our expansions to develop inferential tools that take into account approximation errors: we propose adjustments of the approximate estimator that remove its first-order bias and adjust its standard errors. These corrections apply to a class of approximate estimators that includes all known simulation-based procedures. A Monte Carlo simulation on the mixed logic model shows that our proposed adjustments can yield significant improvements at a low computational cost.Statistics, Economics, Mathematics, Computer science, Estimation theory, Econometricsbs2237EconomicsWorking papersHierarchical Bayes models for daily rainfall time series at multiple locations from heterogenous data sources
https://academiccommons.columbia.edu/catalog/ac:199595
Shirley, Kenneth; Vasilaky, Kathryn N.; Greatrex, Helen L.; Osgood, Daniel E.http://dx.doi.org/10.7916/D8QF8SZ4Fri, 03 Jun 2016 14:41:34 +0000We estimate a Hierarchical Bayesian models for daily rainfall that incorporates two novelties for estimating spatial and temporal correlations. We estimate the within site time series correlations for a particular rainfall site using multiple data sources at a given location, and we estimate the across site covariance in rainfall based on location distance. Previous rainfall models have captured cross site correlations as a functions of site specific distances, but not within site correlations across multiple data sources, and not both aspects simultaneously. Further, we incorporate information on the technology used (satellite versus rain gauge) in our estimations, which is also a novel addition. This methodology has far reaching applications in providing more accurate and complex weather insurance contracts based combining information from multiple data sources from a single site, a crucial improvement in the face of climate change. Secondly, the modeling extends to many other data contexts where multiple datasources exist for a given event or variable where both within and between series covariances can be estimated over time.Statistics, Mathematics, Meteorology, Rain and rainfall--Mathematical models, Rain and rainfall--Forecasting, Computer simulationknv4, hlg2124, do2126Earth Institute, International Research Institute for Climate and SocietyReportsWhen Are Nonconvex Optimization Problems Not Scary?
https://academiccommons.columbia.edu/catalog/ac:199718
Sun, Juhttp://dx.doi.org/10.7916/D8251J7HFri, 27 May 2016 11:38:27 +0000Nonconvex optimization is NP-hard, even the goal is to compute a local minimizer. In applied disciplines, however, nonconvex problems abound, and simple algorithms, such as gradient descent and alternating direction, are often surprisingly effective. The ability of simple algorithms to find high-quality solutions for practical nonconvex problems remains largely mysterious.
This thesis focuses on a class of nonconvex optimization problems which CAN be solved to global optimality with polynomial-time algorithms. This class covers natural nonconvex formulations of central problems in signal processing, machine learning, and statistical estimation, such as sparse dictionary learning (DL), generalized phase retrieval (GPR), and orthogonal tensor decomposition. For each of the listed problems, the nonconvex formulation and optimization lead to novel and often improved computational guarantees.
This class of nonconvex problems has two distinctive features: (i) All local minimizer are also global. Thus obtaining any local minimizer solves the optimization problem; (ii) Around each saddle point or local maximizer, the function has a negative directional curvature. In other words, around these points, the Hessian matrices have negative eigenvalues. We call smooth functions with these two properties (qualitative) X functions, and derive concrete quantities and strategy to help verify the properties, particularly for functions with random inputs or parameters. As practical examples, we establish that certain natural nonconvex formulations for complete DL and GPR are X functions with concrete parameters.
Optimizing X functions amounts to finding any local minimizer. With generic initializations, typical iterative methods at best only guarantee to converge to a critical point that might be a saddle point or local maximizer. Interestingly, the X structure allows a number of iterative methods to escape from saddle points and local maximizers and efficiently find a local minimizer, without special initializations. We choose to describe and analyze the second-order trust-region method (TRM) that seems to yield the strongest computational guarantees. Intuitively, second-order methods can exploit Hessian to extract negative curvature directions around saddle points and local maximizers, and hence are able to successfully escape from the saddles and local maximizers of X functions. We state the TRM in a Riemannian optimization framework to cater to practical manifold-constrained problems. For DL and GPR, we show that under technical conditions, the TRM algorithm finds a global minimizer in a polynomial number of steps, from arbitrary initializations.Electrical engineering, Computer science, Mathematics, Nonconvex programming, Mathematical optimizationjs4038Electrical EngineeringDissertationsPeriodic symplectic cohomologies and obstructions to exact Lagrangian immersions
https://academiccommons.columbia.edu/catalog/ac:199078
Zhao, Jingyuhttp://dx.doi.org/10.7916/D8V69JMZMon, 09 May 2016 12:43:14 +0000Given a Liouville manifold, symplectic cohomology is defined as the Hamiltonian Floer homology for the symplectic action functional on the free loop space. In this thesis, we propose two versions of periodic S^1-equivariant homology or S^1-equivariant Tate homology for the natural S^1-action on the free loop space. The first version is called periodic symplectic cohomology. We prove that there is a localization theorem or a fix point property for periodic symplectic cohomology. The second version is called the completed periodic symplectic cohomology which satisfies Goodwillie's excision isomorphism.
Inspired by the work of Seidel and Solomon on the existence of dilations on symplectic cohomology, we formulate the notion of Liouville manifolds admitting higher dilations using Goodwillie's excision isomorphism on the completed periodic symplectic cohomology. As an application, we derive obstructions to existence of certain exact Lagrangian immersions in Liouville manifolds admitting higher dilations.Mathematics, Homology theory, Symplectic geometryjz2432MathematicsDissertationsTopology of Reticulate Evolution
https://academiccommons.columbia.edu/catalog/ac:198973
Emmett, Kevin Josephhttp://dx.doi.org/10.7916/D8028RKGFri, 06 May 2016 18:23:54 +0000The standard representation of evolutionary relationships is a bifurcating tree. However, many types of genetic exchange, collectively referred to as reticulate evolution, involve processes that cannot be modeled as trees. Increasing genomic data has pointed to the prevalence of reticulate processes, particularly in microorganisms, and underscored the need for new approaches to capture and represent the scale and frequency of these events.
This thesis contains results from applying new techniques from applied and computational topology, under the heading topological data analysis, to the problem of characterizing reticulate evolution in molecular sequence data. First, we develop approaches for analyzing sequence data using topology. We propose new topological constructions specific to molecular sequence data that generalize standard constructions such as Vietoris-Rips. We draw on previous work in phylogenetic networks and use homology to provide a quantitative measure of reticulate events. We develop methods for performing statistical inference using topological summary statistics.
Next, we apply our approach to several types of molecular sequence data. First, we examine the mosaic genome structure in phages. We recover inconsistencies in existing morphology-based taxonomies, use a network approach to construct a genome-based representation of phage relationships, and identify conserved gene families within phage populations. Second, we study influenza, a common human pathogen. We capture widespread patterns of reassortment, including nonrandom cosegregation of segments and barriers to subtype mixing. In contrast to traditional influenza studies, which focus on the phylogenetic branching patterns of only the two surface-marker proteins, we use whole-genome data to represent influenza molecular relationships. Using this representation, we identify unexpected relationships between divergent influenza subtypes. Finally, we examine a set of pathogenic bacteria. We use two sources of data to measure rates of reticulation in both the core genome and the mobile genome across a range of species. Network approaches are used to represent the population of S. aureus and analyze the spread of antibiotic resistance genes. The presence of antibiotic resistance genes in the human microbiome is investigated.Evolution and development, Mathematics, Topology, Microbial genetics, Evolutionary genetics--Mathematical modelskje2109Physics, Biomedical InformaticsDissertationsNearly Overconvergent Forms and p-adic L-Functions for Symplectic Groups
https://academiccommons.columbia.edu/catalog/ac:198695
Liu, Zhenghttp://dx.doi.org/10.7916/D8ZC82VKThu, 05 May 2016 21:20:06 +0000We reformulate Shimura's theory of nearly holomorphic forms for Siegel modular forms using automorphic sheaves over Siegel varieties. This sheaf-theoretic reformulation allows us to define and study basic properties of nearly overconvergent Siegel modular forms as well as their p-adic families. Besides, it finds applications in the construction, via the doubling method, of p-adic partial standard L-functions associated to Siegel cuspidal Hecke eigensystems. We illustrate how the sheaf-theoretic definition of nearly holomorphic forms and Maass--Shimura differential operators helps with the choice of the archimedean sections for the Siegel Eisenstein series on the doubling group Sp(4n) and the study of the p-adic properties of their restrictions to Sp(2n)*Sp(2n). The selection of archimedean sections, together with p-adic interpolation considerations, then naturally gives the sections at the place p. We compute p-adic zeta integrals corresponding to those sections. Finally, we construct the p-adic standard L-functions associated to ordinary families of Siegel Hecke eigensystems and obtain their interpolation properties.Mathematicszl2283MathematicsDissertationsQuasi-local energy and isometric embedding
https://academiccommons.columbia.edu/catalog/ac:198605
Gimre, Karsten Trevorhttp://dx.doi.org/10.7916/D8765FB1Wed, 04 May 2016 21:30:05 +0000In this thesis, we consider the recent definition of gravitational energy at the quasi-local level provided by Mu-Tao Wang and Shing-Tung Yau. Their definition poses a variational question predicated on isometric embedding of Riemannian surfaces into the Minkowski space; as such, there is a naturally associated Euler-Lagrange equation, which is a fourth-order system of partial differential equations for the embedding functions. We prove a perturbation result for solutions of this Euler-Lagrange equation.Mathematics, Generalized spaces, Isometrics (Mathematics)ktg2117MathematicsDissertationsAn alternative proof of genericity for unitary group of three variables
https://academiccommons.columbia.edu/catalog/ac:198528
Wang, Chonglihttp://dx.doi.org/10.7916/D8C24WF7Wed, 04 May 2016 15:32:46 +0000In this thesis, we prove that local genericity implies globally genericity for the quasi-split unitary group U3 for a quadratic extension of number fields E/F. We follow [Fli1992] and [GJR2001] closely, using the relative trace formula approach. Our main result is the existence of smooth transfer for the relative trace formulae in [GJR2001], which is circumvented there. The basic idea is to compute the Mellin transform of Shalika germ functions and show that they are equal in the unitary case and the general linear case.Mathematics, Trace formulas, Unitary groups, Group theory, Mellin transform, Mathematicscw2639MathematicsDissertationsRelative Trace Formula for SO₂ × SO₃ and the Waldspurger Formula
https://academiccommons.columbia.edu/catalog/ac:198125
Krishna, Rahul Marathehttp://dx.doi.org/10.7916/D8FB52XBFri, 29 Apr 2016 21:15:19 +0000We provide a new relative trace formula approach to the theorem of Waldspurger on toric periods for GL₂, with possible applications to the global Gross-Prasad conjecture for orthogonal groups.Mathematics, Torus (Geometry), Trace formulas, Mathematics, Linear algebraic groupsrmk2138MathematicsDissertationsDualization and deformations of the Bar-Natan—Russell skein module
https://academiccommons.columbia.edu/catalog/ac:197984
Heyman, Andrea L.http://dx.doi.org/10.7916/D85D8RV2Tue, 26 Apr 2016 15:34:28 +0000This thesis studies the Bar-Natan skein module of the solid torus with a particular boundary curve system, and in particular a diagrammatic presentation of it due to Russell. This module has deep connections to topology and categorification: it is isomorphic to both the total homology of the (n,n)-Springer variety and the 0th Hochschild homology of the Khovanov arc ring H^n.
We can also view the Bar-Natan--Russell skein module from a representation-theoretic viewpoint as an extension of the Frenkel--Khovanov graphical description of the Lusztig dual canonical basis of the nth tensor power of the fundamental U_q(sl_2)-representation. One of our primary results is to extend a dualization construction of Khovanov using Jones--Wenzl projectors from the Lusztig basis to the Russell basis.
We also construct and explore several deformations of the Russell skein module. One deformation is a quantum deformation that arises from embedding the Russell skein module in a space that obeys Kauffman--Lins diagrammatic relations. Our quantum version recovers the original Russell space when q is specialized to -1 and carries a natural braid group action that recovers the symmetric group action of Russell and Tymoczko. We also present an equivariant deformation that arises from replacing the TQFT algebra A used in the construction of the rings H^n by the equivariant homology of the two-sphere with the standard action of U(2) and taking the 0th Hochschild homology of the resulting deformed arc rings. We show that the equivariant deformation has the expected rank.
Finally, we consider the Khovanov two-functor F from the category of tangles. We show that it induces a surjection from the space of cobordisms of planar (2m, 2n)-tangles to the space of (H^m, H^n)-bimodule homomorphisms and give an explicit description of the kernel. We use our result to introduce a new quotient of the Russell skein module.Mathematics, Torus (Geometry), Mathematics, Duality theory (Mathematics), Quantum theory--Mathematicsalh2172MathematicsDissertationsConformally invariant random planar objects
https://academiccommons.columbia.edu/catalog/ac:197710
Benoist, Stephanehttp://dx.doi.org/10.7916/D80G3K4TTue, 19 Apr 2016 12:22:43 +0000This thesis explores different aspects of a surprising field of research: the conformally invariant scaling limits of planar statistical mechanics models.
The aspects developed here include the proof of convergence of certain interfaces in the critical Ising magnetization model (joint work with Hugo Duminil-Copin and Clement Hongler), a study of the near-critical behavior of the uniform spanning tree in the scaling limit (joint work with Laure Dumaz and Wendelin Werner), the construction of an interesting measure on continuous loops satisfying a certain stability property under deformation (joint work with Julien Dubedat) as well as some related algebraic considerations, and finally, notes on a paper of Sheffield, that studies a certain coupling of the scaling limits of discrete interfaces - SLE curves - together with random surfaces obtained from the Gaussian free field.Mathematics, Theoretical mathematics, Statistical mechanics, Ising model, Statistical mechanics--Mathematical models, Conformal invariantssb3193MathematicsDissertationsQuantum difference equations for quiver varieties
https://academiccommons.columbia.edu/catalog/ac:197653
Smirnov, Andreyhttp://dx.doi.org/10.7916/D8RN37T6Thu, 14 Apr 2016 12:21:44 +0000For an arbitrary Nakajima quiver variety X, we construct an analog of the quantum dynamical Weyl group acting in its equivariant K-theory. The correct generalization of the Weyl group here is the fundamental groupoid of a certain periodic locally finite hyperplane arrangement in Pic(X)⊗C. We identify the lattice part of this groupoid with the operators of quantum difference equation for X. The cases of quivers of finite and affine type are illustrated by explicit examples.Mathematics, Weyl groups, Quantum theory--Mathematics, K-theory, Difference equationsas4128MathematicsDissertationsDynamics, Graph Theory, and Barsotti-Tate Groups: Variations on a Theme of Mochizuki
https://academiccommons.columbia.edu/catalog/ac:197650
Krishnamoorthy, Rajuhttp://dx.doi.org/10.7916/D88K792NThu, 14 Apr 2016 12:21:26 +0000In this dissertation, we study etale correspondence of hyperbolic curves with unbounded dynamics. Mochizuki proved that over a field of characteristic 0, such curves are always Shimura curves. We explore variants of this question in positive characteristic, using graph theory, l-adic local systems, and Barsotti-Tate groups. Given a correspondence with unbounded dynamics, we construct an infinite graph with a large group of ”algebraic” automorphisms and roughly measures the ”generic dynamics” of the correspondence. We construct a specialization map to a graph representing the actual dynamics. Along the way, we formulate conjectures that etale correspondences with unbounded dynamics behave similarly to Hecke correspondences of Shimura curves. Using graph theory, we show that type (3,3) etale correspondences verify various parts of this philosophy. Key in the second half of this dissertation is a recent p-adic Langlands correspondence, due to Abe, which answers affirmatively the petites camarades conjecture of Deligne in the case of curves. This allows us the build a correspondence between rank 2 l-adic local systems with trivial determinant and Frobenius traces in Q and certain height 2, dimension 1 Barsotti-Tate groups. We formulate a conjecture on the fields of definitions of certain compatible systems of l-adic representations. Relatedly, we conjecture that the Barsotti-Tate groups over complete curves in positive characteristic may be ”algebraized” to abelian schemes.Mathematics, Exponential functions, Geometry, Hyperbolic, Shimura varieties, Mathematicsstk2117MathematicsDissertationsBifurcation perspective on topologically protected and non-protected states in continuous systems
https://academiccommons.columbia.edu/catalog/ac:197647
Lee-Thorp, James Patrickhttp://dx.doi.org/10.7916/D8J38SJRThu, 14 Apr 2016 12:21:18 +0000We study Schrödinger operators perturbed by non-compact (spatially extended) defects. We consider two models: a one-dimensional (1D) dimer structure with a global phase shift, and a two-dimensional (2D) honeycomb structure with a line-defect or "edge''. In both the 1D and 2D settings, the non-compact defects are modeled by adiabatic, domain wall modulations of the respective dimer and honeycomb structures. Our main results relate to the rigorous construction of states via bifurcations from continuous spectra. These bifurcations are controlled by asymptotic effective (homogenized) equations that underlie the protected or non-protected character of the states.
In 1D, the states we construct are localized solutions. In 2D, they are "edge states'' - time-harmonic solutions which are propagating (plane-wave-like) parallel to a line-defect or "edge'' and are localized transverse to it. The states are described as protected if they persist in the presence of spatially localized (even strong) deformations of the global phase defect (in 1D) or edge (in 2D). The protected states bifurcate from "Dirac points'' (linear/conical spectral band-crossings) in the continuous spectra and are seeded by an effective Dirac equation. The (more conventional) non-protected states bifurcate from spectral band edges are seeded by an effective Schrödinger equation.
Our 2D model captures many aspects of the phenomenon of topologically protected edge states observed in honeycomb structures such as graphene and "artificial graphene''. The protected states we construct in our 1D dimer model can be realized as highly robust TM- electromagnetic modes for a class of photonic waveguides with a phase-defect. We present a detailed computational study of an experimentally realizable photonic waveguide array structure.Applied mathematics, Mathematics, Physics, Schrödinger operator, Bifurcation theory, Honeycomb structures, Schrödinger equationjpl2154Applied Physics and Applied MathematicsDissertationsSimultaneous twists of elliptic curves and the Hasse principle for certain K3 surfaces
https://academiccommons.columbia.edu/catalog/ac:197554
Pal, Vivekhttp://dx.doi.org/10.7916/D81C1WVGMon, 11 Apr 2016 18:25:39 +0000In this thesis we unconditionally show that certain K3 surfaces satisfy the Hasse principle. Our method involves the 2-Selmer groups of simultaneous quadratic twists of two elliptic curves, only with places of good or additive reduction. More generally we prove that, given finitely many such elliptic curves defined over a number field (with rational 2-torsion and satisfying some mild conditions) there exists an explicit quadratic extension such that the quadratic twist of each elliptic curve has essential 2-Selmer rank one. Furthermore, given a 2-covering in each of the 2-Selmer groups, the quadratic extension above can be chosen so that the 2-Selmer group of the quadratic twist of each elliptic curve is generated by the given 2-covering and the image of the 2-torsion.
Our approach to the Hasse Principle is outlined below and was introduced by Skorobogatov and Swinnerton-Dyer. We also generalize the result proved in their paper. If each elliptic curve has a distinct multiplicative place of bad reduction, then we find a quadratic extension such that the quadratic twist of each elliptic curve has essential 2-Selmer rank one. Furthermore, given a 2-covering in each of the 2-Selmer groups, the quadratic extension above can be chosen so that the 2-Selmer group of the quadratic twist of each elliptic curve is generated by the given 2-covering and the image of the 2-torsion. If we further assume the finiteness of the Shafarevich-Tate groups (of the twisted elliptic curves) then each elliptic curve has Mordell-Weil rank one. If K = Q, then under the above assumptions the analytic rank of each elliptic curves is one. Furthermore, with the assumption on the Shafarevich-Tate group (and K = Q), we describe a single quadratic twist such that each elliptic curve has analytic rank zero and Mordell-Weil rank zero, again under some mild assumptions.Mathematics, Curves, Elliptic, Mathematics, Equations, Geometry, Differentialvp2262MathematicsDissertationsKuranishi atlases and genus zero Gromov-Witten invariants
https://academiccommons.columbia.edu/catalog/ac:197454
Castellano, Roberthttp://dx.doi.org/10.7916/D89W0FF0Mon, 11 Apr 2016 18:24:16 +0000Kuranishi atlases were introduced by McDuff and Wehrheim as a means to build a virtual fundamental cycle on moduli spaces of J-holomorphic curves and resolve some of the challenges in this field. This thesis considers genus zero Gromov-Witten invariants on a general closed symplectic manifold. We complete the construction of these invariants using Kuranishi atlases. To do so, we show that Gromov-Witten moduli spaces admit a smooth enough Kuranishi atlas to define a virtual fundamental class in any virtual dimension. In the process, we prove a stronger gluing theorem. Once we have defined genus zero Gromov-Witten invariants, we show that they satisfy the Gromov-Witten axioms of Kontsevich and Manin, a series of main properties that these invariants are expected to satisfy. A key component of this is the introduction of the notion of a transverse subatlas, a useful tool for working with Kuranishi atlases.Mathematics, Invariants, Moduli theory, Pseudoholomorphic curves, Gromov-Witten invariantsrtc2119Mathematics, Mathematics (Barnard College)DissertationsViscosity Characterizations of Explosions and Arbitrage
https://academiccommons.columbia.edu/catalog/ac:197340
Wang, Yinghuihttp://dx.doi.org/10.7916/D8125SMHWed, 06 Apr 2016 18:13:37 +0000This thesis analyzes the viscosity characterizations of the explosion time distribution for diffusions and of the arbitrage function in an equity market model with uncertainty. In the first part, we show that the tail distribution U of the explosion time for a multidimensional diffusion -- and more generally, a suitable function 𝒰 of the Feynman-Kac type involving the explosion time -- is a viscosity solution of an associated parabolic partial differential equation (PDE), provided that the dispersion and drift coefficients of the diffusion are continuous. This generalizes a result of Karatzas and Ruf (2015), who characterize U as a classical solution of a Cauchy problem for the PDE in the one-dimensional case, under the stronger condition of local Hölder continuity on the coefficients. We also extend their result to 𝒰 in the one-dimensional case by establishing the joint continuity of 𝒰. Furthermore, we show that 𝒰 is dominated by any nonnegative classical supersolution of this Cauchy problem. Finally, we consider another notion of weak solvability, that of the distributional (sub/super)solution, and show that 𝒰 is no greater than any nonnegative distributional supersolution of the relevant PDE. In the second part, a more elaborate mathematical finance setting is taken. We show that, in an equity market model with Knightian uncertainty regarding the relative risk and covariance structure of its assets, the arbitrage function -- defined as the reciprocal of the highest return on investment that can be achieved relative to the market using nonanticipative strategies, and under any admissible market model configuration -- is a viscosity solution of an associated Hamilton-Jacobi-Bellman (HJB) equation under appropriate boundedness, continuity and Markovian assumptions on the uncertainty structure. This result generalizes that of Fernholz and Karatzas (2011), who characterized this arbitrage function as a classical solution of a Cauchy problem for this HJB equation under much stronger conditions than those needed here. Our approach and results also extend to the Markovian Market Weight model introduced in Fernholz and Karatzas (2010b).Mathematics, Finance, Arbitrage, Business mathematics, Differential equations, Partial, Viscosity solutions, Hamilton-Jacobi equationsyw2450MathematicsDissertationsDerived Categories of Moduli Spaces of Semistable Pairs over Curves
https://academiccommons.columbia.edu/catalog/ac:197334
Potashnik, Natashahttp://dx.doi.org/10.7916/D8H99542Mon, 04 Apr 2016 18:46:08 +0000The context of this thesis is derived categories in algebraic geometry and geo- metric quotients. Specifically, we prove the embedding of the derived category of a smooth curve of genus greater than one into the derived category of the moduli space of semistable pairs over the curve. We also describe closed cover conditions under which the composition of a pullback and a pushforward induces a fully faithful functor. To prove our main result, we give an exposition of how to think of general Geometric Invariant Theory quotients as quotients by the multiplicative group.Mathematics, Geometry, Algebraic, Moduli theory, Mathematics, Derived categories (Mathematics)np2411MathematicsDissertationsMathematical Extracurricular Activities in Russia
https://academiccommons.columbia.edu/catalog/ac:197313
Marushina, Albinahttp://dx.doi.org/10.7916/D8P55NFJMon, 04 Apr 2016 12:18:19 +0000The dissertation is devoted to the history and practice of extracurricular activities in mathematics in Russia. It investigates both the views expressed by mathematics educators concerning the aims and objectives of extracurricular activities, and the daily organization of such activities, including the pedagogical formats and the mathematical assignments and questions to which extracurricular activities have given rise. Thus the dissertation provides an overview of the history of extracurricular activities over the course of a century, as part of the general development of education (including mathematics education) in Russia.
The study called for a multifaceted investigation of surviving sources, which include practically all available textbooks and teaching manuals, scholarly articles on conducting extracurricular activities, magazine and newspaper articles on conducting extracurricular activities, surviving memoirs of participants and organizers of extracurricular activities, and much else, including methodological materials preserved in archives, which have been located by the author.
Summing up the results of the study, it may be said that two major goals have always been important in extracurricular activities in Russia: the first goal is motivating students; the second goal is preparing the mathematically strongest students and providing them with an opportunity to deepen and enrich their mathematical education. Of course, extracurricular activities have not been aimed exclusively at these two goals, and at different stages of development additional goals (such as ideological preparation) were also formulated. Broadly speaking, it may be said that the history of the Russian system of mathematical extracurricular activities in general has been strongly aligned with the history of the development of the system of Russian school education. The study analyzes the specific character of extracurricular activities at each of the historical stages of Russia's development, in particular, it lists and described the basic forms of extracurricular activities, paying special attention to the indissoluble connection between the so-called mass-scale forms, in which millions of schoolchildren participate, and forms and activities that are engaged in only by a very few. Also provided is a survey of the changes that have occurred in the mathematical problems that are offered to students.
The author believes that familiarity with the longstanding tradition of extracurricular activities in mathematics in Russia may be useful also to the international sphere of mathematics educators, since the issue of motivating students is becoming increasingly important. The study concludes with a discussion of the possibilities and the expediency of putting such experience to use.Mathematics education, Mathematics, Mathematics--Study and teaching, Student activitiesam3582Mathematics EducationDissertationsUser association for energy harvesting relay stations in cellular networks
https://academiccommons.columbia.edu/catalog/ac:196929
Wang, Zhe; Wang, Xiaodong; Aldiab, Motasem; Jaber, Tareqhttp://dx.doi.org/10.7916/D89886X6Wed, 30 Mar 2016 16:36:49 +0000We consider a cellular wireless network enhanced by relay stations that are powered by renewable energy sources. Such a network consists of the macro base stations (BS), relay stations (RSs), and many mobile stations (MSs). In addition to the traditional data/voice transmission between the BS and the MSs, a higher service tier may be provided by using the energy harvesting RSs for some MSs. We propose a network scenario utilizing the energy harvesting relay stations to improve the service quality without taking the additional licensed frequency band and transmission power, and design a user association algorithm for the energy harvesting RSs in such a network. The goal is to assign each MS an RS for relaying its signal to minimize the probability of the relay service outage, i.e, the probability that an MS’s relay service request is rejected. First, we propose a network scenario and develop a mathematical model to estimate the rejection probability for a given user association. We then propose a low-complexity local search algorithm, which balances the computational complexity and the performance, to obtain a locally optimal user association. Simulation results are provided to demonstrate the superior performance of the proposed techniques over the traditional methods.Electrical engineering, Mathematics, Radio relay systems, Wireless communication systems, Energy harvesting, Mobile radio stationszw2231, xw2008Electrical EngineeringArticlesReconstruction of novel transcription factor regulons through inference of their binding sites
https://academiccommons.columbia.edu/catalog/ac:196938
Elmas, Abdulkadir; Wang, Xiaodong; Samoilov, Michael S.http://dx.doi.org/10.7916/D8K07469Wed, 30 Mar 2016 11:00:53 +0000Background
In most sequenced organisms the number of known regulatory genes (e.g., transcription factors (TFs)) vastly exceeds the number of experimentally-verified regulons that could be associated with them. At present, identification of TF regulons is mostly done through comparative genomics approaches. Such methods could miss organism-specific regulatory interactions and often require expensive and time-consuming experimental techniques to generate the underlying data.
Results
In this work, we present an efficient algorithm that aims to identify a given transcription factor’s regulon through inference of its unknown binding sites, based on the discovery of its binding motif. The proposed approach relies on computational methods that utilize gene expression data sets and knockout fitness data sets which are available or may be straightforwardly obtained for many organisms. We computationally constructed the profiles of putative regulons for the TFs LexA, PurR and Fur in E. coli K12 and identified their binding motifs. Comparisons with an experimentally-verified database showed high recovery rates of the known regulon members, and indicated good predictions for the newly found genes with high biological significance. The proposed approach is also applicable to novel organisms for predicting unknown regulons of the transcriptional regulators. Results for the hypothetical protein D d e0289 in D. alaskensis include the discovery of a Fis-type TF binding motif.
Conclusions
The proposed motif-based regulon inference approach can discover the organism-specific regulatory interactions on a single genome, which may be missed by current comparative genomics techniques due to their limitations.Electrical engineering, Genetics, Molecular biology, Mathematics, Transcription factors, Comparative genomics, Bioinformatics, Genomics--Data processing, Computational biologyae2321, xw2008Electrical EngineeringArticlesDynamics of Large Rank-Based Systems of Interacting Diffusions
https://academiccommons.columbia.edu/catalog/ac:195668
Bruggeman, Cameronhttp://dx.doi.org/10.7916/D80G3K1GThu, 10 Mar 2016 12:18:05 +0000We study systems of n dimensional diffusions whose drift and dispersion coefficients depend only on the relative ranking of the processes. We consider the question of how long it takes for a particle to go from one rank to another. It is argued that as n gets large, the distribution of particles satisfies a Porous Medium Equation. Using this, we derive a deterministic limit for the system of particles. This limit allows for direct calculation of the properties of the rank traversal time. The results are extended to the case of asymmetrically colliding particles.
These models are of interest in the study of financial markets and economic inequality. In particular, we derive limits for the performance of some Functionally Generated Portfolios originating from Stochastic Portfolio Theory.Mathematics, Statistics, Diffusion processes, Diffusion--Mathematical models, Dispersion--Mathematical models, Porous materials--Mathematical models, Portfolio management--Mathematical modelscpb2133MathematicsDissertationsSemi-convergence of an Iterative Algorithm
https://academiccommons.columbia.edu/catalog/ac:194857
Vasilaky, Kathryn N.http://dx.doi.org/10.7916/D8SJ1KFXFri, 26 Feb 2016 15:15:49 +0000An iterative method is introduced for solving noisy, ill-conditioned inverse problems. Analysis of the semi-convergence behavior identifies three error components - iteration error, noise error, and initial guess error. A derived expression explains how the three errors are related to each other relative to the number of iterations. The Standard Tikhonov regularization method is just the first iteration of the iterative method and the derived noise damping filter is a generalization of the Standard Tikhonov filter. The derived filter is a function two parameters, a regularization parameter and the iteration number parameter. The new method is tested on image reconstruction from projections simulated data set.Statistics, Mathematics, Inverse problems (Differential equations), Iterative methods (Mathematics), Filters (Mathematics)knv4Earth InstituteReportsDirect covariance measurement of CO2 gas transfer velocity during the 2008 Southern Ocean Gas Exchange Experiment: Wind speed dependency
https://academiccommons.columbia.edu/catalog/ac:194466
Edson, J. B.; Fairall, C. W.; Bariteau, L.; Zappa, Christopher J.; Cifuentes-Lorenzen, A.; McGillis, Wade R.; Pezoa, S.; Hare, J. E.; Helmig, D.http://dx.doi.org/10.7916/D8K937CCTue, 23 Feb 2016 12:23:38 +0000Direct measurements of air-sea heat, momentum, and mass (including CO2, DMS, and water vapor) fluxes using the direct covariance method were made over the open ocean from the NOAA R/V Ronald H. Brown during the Southern Ocean Gas Exchange (SO GasEx) program. Observations of fluxes and the physical processes associated with driving air-sea exchange are key components of SO GasEx. This paper focuses on the exchange of CO2 and the wind speed dependency of the transfer velocity, k, used to model the CO2 flux between the atmosphere and ocean. A quadratic dependence of k on wind speed based on dual tracer experiments is most frequently encountered in the literature. However, in recent years, bubble-mediated enhancement of k, which exhibits a cubic relationship with wind speed, has emerged as a key issue for flux parameterization in high-wind regions. Therefore, a major question addressed in SO GasEx is whether the transfer velocities obey a quadratic or cubic relationship with wind speed. After significant correction to the flux estimates (primarily due to moisture contamination), the direct covariance CO2 fluxes confirm a significant enhancement of the transfer velocity at high winds compared with previous quadratic formulations. Regression analysis suggests that a cubic relationship provides a more accurate parameterization over a wind speed range of 0 to 18 m s−1. The Southern Ocean results are in good agreement with the 1998 GasEx experiment in the North Atlantic and a recent separate field program in the North Sea.Physical oceanography, Mathematics, Atmospheric chemistry, Analysis of covariance, Ocean-atmosphere interaction--Measurement, Atmospheric carbon dioxide--Measurement, Winds--Speedscjz9, wrm2102Lamont-Doherty Earth ObservatoryArticlesPolarized light field under dynamic ocean surfaces: Numerical modeling compared with measurements
https://academiccommons.columbia.edu/catalog/ac:194463
You, Yu; Kattawar, George W.; Voss, Kenneth J.; Bhandari, Purushottam; Wei, Jianwei; Lewis, Marlon; Zappa, Christopher J.; Schultz, Howardhttp://dx.doi.org/10.7916/D8TT4QS8Tue, 23 Feb 2016 12:09:48 +0000As part of the Radiance in a Dynamic Ocean (RaDyO) program, we have developed a numerical model for efficiently simulating the polarized light field under highly dynamic ocean surfaces. Combining the advantages of the three-dimensional Monte Carlo and matrix operator methods, this hybrid model has proven to be computationally effective for simulations involving a dynamic air-sea interface. Given water optical properties and ocean surface wave slopes obtained from RaDyO field measurements, model-simulated radiance and polarization fields under a dynamic surface are found to be qualitatively comparable to their counterparts from field measurements and should be quantitatively comparable if the light field measurement and the wave slope/water optical property measurements are appropriately collocated and synchronized. This model serves as a bridge to connect field measurements of water optical properties, wave slopes and polarized light fields. It can also be used as a powerful yet convenient tool to predict the temporal underwater polarized radiance in a real-world situation. When appropriate surface measurements are available, model simulation is shown to reveal more dynamic features in the underwater light field than direct measurements.Physical oceanography, Mathematics, Optics, Brightness temperature, Polarization (Light)--Measurement, Ocean surface topography, Underwater light--Measurementcjz9Lamont-Doherty Earth ObservatoryArticlesSea surface pCO2 and O2 in the Southern Ocean during the austral fall, 2008
https://academiccommons.columbia.edu/catalog/ac:194451
Moore, T. S.; DeGrandpre, M. D.; Sabine, C. L.; Hamme, R. C.; Zappa, Christopher J.; McGillis, Wade R.; Feely, R. A.; Drennan, W. M.http://dx.doi.org/10.7916/D8CR5T6JMon, 22 Feb 2016 17:45:02 +0000The physical and biological processes controlling surface mixed layer pCO2 and O2 were evaluated using in situ sensors mounted on a Lagrangian drifter deployed in the Atlantic sector of the Southern Ocean (∼50°S, ∼37°W) during the austral fall of 2008. The drifter was deployed three times during different phases of the study. The surface ocean pCO2 was always less than atmospheric pCO2 (−50.4 to −76.1 μatm), and the ocean was a net sink for CO2 with fluxes averaging between 16.2 and 17.8 mmol C m−2 d−1. Vertical entrainment was the dominant process controlling mixed layer CO2, with fluxes that were 1.8 to 2.2 times greater than the gas exchange fluxes during the first two drifter deployments, and was 1.7 times greater during the third deployment. In contrast, during the first two deployments the surface mixed layer was always a source of O2 to the atmosphere, and air-sea gas exchange was the dominant process occurring, with fluxes that were 2.0 to 4.1 times greater than the vertical entrainment flux. During the third deployment O2 was near saturation the entire deployment and was a small source of O2 to the atmosphere. Net community production (NCP) was low during this study, with mean fluxes of 3.2 to 6.4 mmol C m−2 d−1 during the first deployment and nondetectable (within uncertainty) in the third. During the second deployment the NCP was not separable from lateral advection. Overall, this study indicates that in the early fall the area is a significant sink for atmospheric CO2.Physical oceanography, Mathematics, Hydrologic sciences, Chemical oceanography--Data processing, Atmospheric carbon dioxide--Measurementcjz9, wrm2102Lamont-Doherty Earth ObservatoryArticlesIntroduction to special section on Recent Advances in the Study of Optical Variability in the Near-Surface and Upper Ocean
https://academiccommons.columbia.edu/catalog/ac:194448
Dickey, T.; Banner, M. L.; Bhandari, P.; Boyd, T.; Carvalho, L.; Chang, G.; Chao, Y.; Czerski, H.; Darecki, M.; Dong, C.; Farmer, D.; Freeman, S.; Gemmrich, J.; Gernez, P.; Hall-Patch, N.; Holt, B.; Jiang, S.; Jones, C.; Kattawar, G.; LeBel, D.; Lenain, L.; Lewis, M.; Liu, Y.; Logan, L.; Manov, D.; Melville, W. K.; Moline, M. A.; Morison, R.; Nencioli, F.; Pegau, W. S.; Reineman, B.; Robbins, I.; Röttgers, R.; Schultz, H.; Shen, L.; Shinki, M.; Slivkoff, M.; Sokólski, M.; Spada, F.; Statom, N.; Stramski, D.; Sutherland, P.; Twardowski, M.; Vagle, S.; Van Dommelen, R.; Voss, K.; Washburn, L.; Wei, J.; Wijesekera, H.; Wurl, O.; Yang, D.; Yildiz, S.; You, Y.; Yue, D. K. P.; Zaneveld, R.; Zappa, Christopher J.http://dx.doi.org/10.7916/D8WS8T4SMon, 22 Feb 2016 17:11:59 +0000Optical variability occurs in the near-surface and upper ocean on very short time and space scales (e.g., milliseconds and millimeters and less) as well as greater scales. This variability is caused by solar, meteorological, and other physical forcing as well as biological and chemical processes that affect optical properties and their distributions, which in turn control the propagation of light across the air-sea interface and within the upper ocean. Recent developments in several technologies and modeling capabilities have enabled the investigation of a variety of fundamental and applied problems related to upper ocean physics, chemistry, and light propagation and utilization in the dynamic near-surface ocean. The purpose here is to provide background for and an introduction to a collection of papers devoted to new technologies and observational results as well as model simulations, which are facilitating new insights into optical variability and light propagation in the ocean as they are affected by changing atmospheric and oceanic conditions.Physical oceanography, Mathematics, Hydrologic sciences, Surface waves (Oceanography), Optical measurements, Analysis of variancecjz9Lamont-Doherty Earth ObservatoryArticlesStatistics of surface divergence and their relation to air-water gas transfer velocity
https://academiccommons.columbia.edu/catalog/ac:194442
Asher, William E.; Liang, Hanzhuang; Zappa, Christopher J.; Loewen, Mark R.; Mukto, Moniz A.; Litchendorf, Trina M.; Jessup, Andrew T.http://dx.doi.org/10.7916/D8571BVQMon, 22 Feb 2016 16:56:19 +0000Air-sea gas fluxes are generally defined in terms of the air/water concentration difference of the gas and the gas transfer velocity,kL. Because it is difficult to measure kLin the ocean, it is often parameterized using more easily measured physical properties. Surface divergence theory suggests that infrared (IR) images of the water surface, which contain information concerning the movement of water very near the air-water interface, might be used to estimatekL. Therefore, a series of experiments testing whether IR imagery could provide a convenient means for estimating the surface divergence applicable to air-sea exchange were conducted in a synthetic jet array tank embedded in a wind tunnel. Gas transfer velocities were measured as a function of wind stress and mechanically generated turbulence; laser-induced fluorescence was used to measure the concentration of carbon dioxide in the top 300 μm of the water surface; IR imagery was used to measure the spatial and temporal distribution of the aqueous skin temperature; and particle image velocimetry was used to measure turbulence at a depth of 1 cm below the air-water interface. It is shown that an estimate of the surface divergence for both wind-shear driven turbulence and mechanically generated turbulence can be derived from the surface skin temperature. The estimates derived from the IR images are compared to velocity field divergences measured by the PIV and to independent estimates of the divergence made using the laser-induced fluorescence data. Divergence is shown to scale withkLvalues measured using gaseous tracers as predicted by conceptual models for both wind-driven and mechanically generated turbulence.Physical oceanography, Mathematics, Statistics, Surface waves (Oceanography), Ocean-atmosphere interaction, Divergence theorem, Gas flow--Mathematical modelscjz9Lamont-Doherty Earth ObservatoryArticlesAnalyzing the footprints of near-surface aqueous turbulence: An image processing-based approach
https://academiccommons.columbia.edu/catalog/ac:194439
Schnieders, J.; Garbe, C. S.; Peirson, W. L.; Smith, G. B.; Zappa, Christopher J.http://dx.doi.org/10.7916/D8P84BRHMon, 22 Feb 2016 16:41:22 +0000In this contribution, a detailed investigation of surface thermal patterns on the water surface is presented, with wind speeds ranging from 1 to 7 m s − 1 and various surface conditions. Distinct structures can be observed on the surface—small-scale short-lived structures termed fish scales and larger-scale cold streaks that are consistent with the footprints of Langmuir circulations. The structure of the surface heat pattern depends strongly on wind-induced stress. Consistent behavior regarding the spacing of cold streaks can be observed in a range of laboratory facilities when expressed as a function of water-sided friction velocity, u * . This behavior systematically decreased until a point of saturation at u * = 0.7 cm/s. We present a new image processing-based approach to the analysis of the spacing of cold streaks based on a machine learning approach to classify the thermal footprints of near-surface turbulence. Comparison is made with studies of Langmuir circulation and the following key points are found. Results suggest a saturation in the tangential stress, anticipating that similar behavior will be observed in the open ocean. A relation to Langmuir numbers shows that thermal footprints in infrared images are consistent with Langmuir circulations and depend strongly on wind wave conditions.Physical oceanography, Mathematics, Hydrologic sciences, Surface waves (Oceanography), Turbulence--Mathematical models, Image processing--Data processing, Ocean circulation--Data processingcjz9Lamont-Doherty Earth ObservatoryArticlesWave breaking in developing and mature seas
https://academiccommons.columbia.edu/catalog/ac:194433
Gemmrich, Johannes; Zappa, Christopher J.; Banner, Michael L.; Morison, Russel P.http://dx.doi.org/10.7916/D8668D1DMon, 22 Feb 2016 16:28:06 +0000In response to the growing need for robust validation data for Phillips (1985) breaking wave spectral framework, we contribute new field results observed from R/P FLIP for the breaking crest length distributions, Λ, during two different wind-wave conditions, and breaking strength during one wind-wave condition. The first experiment in Santa Barbara Channel had developing seas and the second experiment in the central Pacific Ocean south of Hawaii had mature seas. These are among the first experiments to use dissipation rate measurements probing up into the breaking crest together with simultaneous measurements of breaking crest length distributions. We directly measured the effective breaking strength parameter to be inline image in mature seas with wave age, inline image, of 40–47. We also found that the velocity scale of the breaking dissipation rate peak decreases with increasing wave age. Further, the breaking crest length spectrum falls off slower than the inline image behavior predicted by Phillips (1985). The integrated dissipation rate was consistently higher for mature seas compared to developing seas due to higher energy and momentum fluxes from the wind.Physical oceanography, Mathematics, Hydrologic sciences, Ocean-atmosphere interaction, Surface waves (Oceanography), Wind waves--Mathematical modelscjz9Lamont-Doherty Earth ObservatoryArticlesOptical measurements of small deeply penetrating bubble populations generated by breaking waves in the Southern Ocean
https://academiccommons.columbia.edu/catalog/ac:194418
Randolph, Kaylan; Dierssen, Heidi M.; Twardowski, Michael ; Cifuentes-Lorenzen, Alejandro ; Zappa, Christopher J.http://dx.doi.org/10.7916/D80P0ZVQMon, 22 Feb 2016 14:06:23 +0000Bubble size distributions ranging from 0.5 to 125 μm radius were measured optically during high winds of 13 m s−1 and large-scale wave breaking as part of the Southern Ocean Gas Exchange Experiment. Very small bubbles with radii less than 60 µm were measured at 6–9 m depth using optical measurements of the near-forward volume scattering function and critical scattering angle for bubbles (∼80°). The bubble size distributions generally followed a power law distribution with mean slope values ranging from 3.6 to 4.6. The steeper slopes measured here were consistent with what would be expected near the base of the bubble plume. Bubbles, likely stabilized with organic coatings, were present for time periods on the order of 10–100 s at depths of 6–9 m. Here, relatively young seas, with an inverse wave age of approximately 0.88 and shorter characteristic wave scales, produced lower bubble concentrations, shallower bubble penetration depths, and steep bubble size distribution slopes. Conversely, older seas, with an inverse wave age of 0.70 and longer characteristic wave scales, produced relatively higher bubble concentrations penetrating to 15 m depth, larger bubble sizes, and shallower bubble size distribution slopes. When extrapolated to 4 m depth using a previously published bubble size distribution, our estimates suggest that the deeply penetrating small bubbles measured in the Southern Ocean supplied ∼36% of the total void fraction and likely contributed to the transfer and supersaturation of low-solubility gases.Physical oceanography, Mathematics, Hydrologic sciences, Bubbles, Ocean-atmosphere interaction, Optical measurementscjz9Lamont-Doherty Earth ObservatoryArticlesWave-induced light field fluctuations in measured irradiance depth profiles: A wavelet analysis
https://academiccommons.columbia.edu/catalog/ac:194415
Wei, Jianwei; Lewis, Marlon R.; Dommelen, Ronnie Van; Zappa, Christopher J.; Twardowski, Michael S.http://dx.doi.org/10.7916/D8862G95Mon, 22 Feb 2016 13:50:38 +0000Rapid variations in the intensities of light are commonly observed in profiles of downwelling plane irradiance in the ocean. These fluctuations are often treated as noise and filtered out. Here an effort is made to extract the pertinent statistics to quantify the light field fluctuations from vertical profiles of irradiance measured under clear skies. The irradiance data are collected in oceanic and coastal waters using a traditional free-fall downwelling plane irradiance sensor. The irradiance profiles are transformed into time-frequency domain with a wavelet technique. Two signatures including the dominant frequency (<3.5 Hz) and the coefficient of variation of irradiance fluctuations along the water column are identified from the variance spectrum. Both the dominant frequency and the amplitude decrease as the inverse square root of depth, consistent with simple models of wave focusing and data from other studies. Mechanisms contributing to the observed variations and the observational uncertainties are discussed.Physical oceanography, Mathematics, Optics, Spectral irradiance, Wavelets (Mathematics), Analysis of variancecjz9Lamont-Doherty Earth ObservatoryArticlesVariations in Ocean Surface Temperature due to Near-Surface Flow: Straining the Cool Skin Layer
https://academiccommons.columbia.edu/catalog/ac:194406
Wells, Andrew J.; Cenedese, Claudia; Farrar, J. Thomas; Zappa, Christopher J.http://dx.doi.org/10.7916/D8HQ3ZRHMon, 22 Feb 2016 13:34:38 +0000The aqueous thermal boundary layer near to the ocean surface, or skin layer, has thickness O(1 mm) and plays an important role in controlling the exchange of heat between the atmosphere and the ocean. Theoretical arguments and experimental measurements are used to investigate the dynamics of the skin layer under the influence of an upwelling flow, which is imposed in addition to free convection below a cooled water surface. Previous theories of straining flow in the skin layer are considered and a simple extension of a surface straining model is posed to describe the combination of turbulence and an upwelling flow. An additional theory is also proposed, conceptually based on the buoyancy-driven instability of a laminar straining flow cooled from above. In all three theories considered two distinct regimes are observed for different values of the Péclet number, which characterizes the ratio of advection to diffusion within the skin layer. For large Péclet numbers, the upwelling flow dominates and increases the free surface temperature, or skin temperature, to follow the scaling expected for a laminar straining flow. For small Péclet numbers, it is shown that any flow that is steady or varies over long time scales produces only a small change in skin temperature by direct straining of the skin layer. Experimental measurements demonstrate that a strong upwelling flow increases the skin temperature and suggest that the mean change in skin temperature with Péclet number is consistent with the theoretical trends for large Péclet number flow. However, all of the models considered consistently underpredict the measured skin temperature, both with and without an upwelling flow, possibly a result of surfactant effects not included in the models.Physical oceanography, Mathematics, Hydrologic sciences, Ocean temperature--Research, Thermal boundary layer, Heat budget (Geophysics)cjz9Lamont-Doherty Earth ObservatoryArticlesA Note on the Phillips Spectral Framework for Ocean Whitecaps
https://academiccommons.columbia.edu/catalog/ac:194403
Banner, Michael L.; Zappa, Christopher J.; Gemmrich, Johannes R.http://dx.doi.org/10.7916/D8S46RTPMon, 22 Feb 2016 13:17:12 +0000There has been a recent upsurge in interest in quantifying kinematic, dynamic, and energetic properties of wave breaking in the open ocean, especially in severe sea states. The underpinning observational and modeling framework is provided by the seminal paper of O. M. Phillips. In this note, a fundamental issue contributing to the scatter in results between investigators is highlighted. This issue relates to the choice of the independent variable used in the expression for the spectral density of the mean breaking crest length per unit area. This note investigates the consequences of the different choices of independent variable presently used by various investigators for validating Phillips model predictions for the spectral density of the breaking crest length per unit area and the associated spectral breaking strength coefficient. These spectral measures have a central role in inferring the associated turbulent kinetic energy dissipation rate and the momentum flux to the upper ocean from breaking wave observations.Physical oceanography, Mathematics, Meteorology, Wind waves--Mathematical models, Turbulence, Oceanography--Remote sensing, Ocean circulationmlb2121, cjz9Lamont-Doherty Earth ObservatoryArticlesA parameter model of gas exchange for the seasonal sea ice zone
https://academiccommons.columbia.edu/catalog/ac:194400
Loose, B.; McGillis, Wade R.; Perovich, D.; Zappa, Christopher J.; Schlosser, Peterhttp://dx.doi.org/10.7916/D81N810BMon, 22 Feb 2016 12:53:37 +0000Carbon budgets for the polar oceans require better constraint on air–sea gas exchange in the sea ice zone (SIZ). Here, we utilize advances in the theory of turbulence, mixing and air–sea flux in the ice–ocean boundary layer (IOBL) to formulate a simple model for gas exchange when the surface ocean is partially covered by sea ice. The gas transfer velocity (k) is related to shear-driven and convection-driven turbulence in the aqueous mass boundary layer, and to the mean-squared wave slope at the air–sea interface. We use the model to estimate k along the drift track of ice-tethered profilers (ITPs) in the Arctic. Individual estimates of daily-averaged k from ITP drifts ranged between 1.1 and 22 m d−1, and the fraction of open water (f) ranged from 0 to 0.83. Converted to area-weighted effective transfer velocities (keff), the minimum value of keff was 10−55 m d−1 near f = 0 with values exceeding keff = 5 m d−1 at f = 0.4. The model indicates that effects from shear and convection in the sea ice zone contribute an additional 40% to the magnitude of keff, beyond what would be predicted from an estimate of keff based solely upon a wind speed parameterization. Although the ultimate scaling relationship for gas exchange in the sea ice zone will require validation in laboratory and field studies, the basic parameter model described here demonstrates that it is feasible to formulate estimates of k based upon properties of the IOBL using data sources that presently exist.Physical oceanography, Mathematics, Meteorology, Sea ice, Ocean-atmosphere interaction, Atmospheric turbulence--Mathematical modelswrm2102, cjz9, ps10Lamont-Doherty Earth Observatory, Earth and Environmental EngineeringArticlesSuperspace and Subspace Identification of Bilinear Models by Discrete-Level Inputs
https://academiccommons.columbia.edu/catalog/ac:192778
Phan, Minh Q.; Vicario, Francesco; Longman, Richard W.; Betti, Raimondohttp://dx.doi.org/10.7916/D8XS5V4CTue, 05 Jan 2016 18:01:48 +0000When excited by an input consisting of a number of discrete levels, a bilinear system becomes a linear time-varying system whose dynamics switches from one linear subsystem to another depending on the input level. This paper describes an identification method that uses the concept of a superstate of a linear switching system as a superstate of the bilinear system. In a superspace method, these superstates are used directly to identify a bilinear system model. In a subspace method, two or more superstate representations are intersected to find a reduced dimension subspace prior to identification of a bilinear model.Mechanical engineering, Mathematics, Aerospace engineering, Space flight--Mathematical models, State-space methods, Bilinear formsfv2157, rwl4, rb68Mechanical Engineering, Civil Engineering and Engineering MechanicsConferencesOKID as a Unified Approach to System Identification
https://academiccommons.columbia.edu/catalog/ac:192629
Vicario, Francesco; Phan, Minh Q.; Betti, Raimondo; Longman, Richard W.http://dx.doi.org/10.7916/D869739XTue, 05 Jan 2016 17:38:15 +0000This paper presents a unified approach for the identification of linear state-space models from input-output measurements in the presence of noise. It is based on the established Observer/Kalman filter IDentification (OKID) method of which it proposes a new formulation capable of transforming a stochastic identification problem into a (simpler) deterministic problem, where the Kalman filter corresponding to the unknown system and the unknown noise covariances is identified. The system matrices are then recovered from the identified Kalman filter. The Kalman filter can be identified with any deterministic identification method for linear state-space models, giving rise to numerous new algorithms and establishing the Kalman filter as the unifying bridge from stochastic to deterministic problems in system identification.Mechanical engineering, Mathematics, Aerospace engineering, System identification, Space flight--Mathematical models, Kalman filtering, State-space methodsfv2157, rb68, rwl4Mechanical Engineering, Civil Engineering and Engineering MechanicsConferencesBilinear Observer/Kalman Filter Identification
https://academiccommons.columbia.edu/catalog/ac:192626
Vicario, Francesco; Phan, Minh Q.; Betti, Raimondo; Longman, Richard W.http://dx.doi.org/10.7916/D8FQ9WCDTue, 05 Jan 2016 17:15:05 +0000Bilinear systems are important per se since several phenomena in engineering and other fields are inherently bilinear. Even more interestingly, bilinear systems can approximate more general nonlinear systems, providing a promising approach to handle various nonlinear identification and control problems, such as satellite attitude control. This paper develops and demonstrates via numerical examples a method for discrete-time state-space model identification for bilinear systems in the presence of noise in the process and in the measurements. The formulation relies on a bilinear observer which is proven to have properties similar to the linear Kalman filter under the sole additional assumption of stationary white excitation input, and on a novel approach to system identification based on the estimation of the observer residuals. The latter are used to construct a new, noise-free identification problem, in which the observer is identified and the matrices of the system state-space model are recovered. The resulting method represents the bilinear counterpart of the Observer/Kalman filter Identification (OKID) approach for linear systems, originally developed for the identification of lightly-damped structures and distributed by NASA.Mechanical engineering, Mathematics, Aerospace engineering, Bilinear forms, Discrete-time systems--Mathematical models, Kalman filtering, Space flight--Mathematical modelsfv2157, rb68, rwl4Mechanical Engineering, Civil Engineering and Engineering MechanicsConferencesA Linear-Time-Varying Approach for Exact Identification of Bilinear Discrete-Time Systems by Interaction Matrices
https://academiccommons.columbia.edu/catalog/ac:192623
Vicario, Francesco; Phan, Minh Q.; Longman, Richard W.; Betti, Raimondohttp://dx.doi.org/10.7916/D8Q81CT0Tue, 05 Jan 2016 16:39:02 +0000Bilinear systems offer a promising approach for nonlinear control because a broad class of nonlinear problems can be reformulated in bilinear form. In this paper system identification is shown to be a technique to obtain such a bilinear approximation of a nonlinear system. Recent discrete-time bilinear model identification methods rely on Input-Output-to-State Representations. These IOSRs are exact only for a certain class of bilinear systems, and they are also limited by high dimensionality and explicit bounds on the input magnitude. This paper offers new IOSRs where the bilinear system is treated as a linear time-varying system through the use of specialized input signals. All the mentioned limitations are overcome by the new approach, leading to more accurate and less computationally demanding identification methods for bilinear discrete-time models, which are also shown via examples to be applicable to the identification of bilinear models approximating more general nonlinear systems.Mechanical engineering, Mathematics, Astrophysics, Astrodynamics, Bilinear forms, Discrete-time systems--Mathematical models, Matricesfv2157, rwl4, rb68Mechanical Engineering, Civil Engineering and Engineering MechanicsConferencesBilinear System Identification by Minimal-Order State Observers
https://academiccommons.columbia.edu/catalog/ac:192617
Vicario, Francesco; Phan, Minh Q.; Longman, Richard W.; Betti, Raimondohttp://dx.doi.org/10.7916/D87944DVTue, 05 Jan 2016 15:44:01 +0000Bilinear systems offer a promising approach for nonlinear control because a broad class of nonlinear problems can be reformulated and approximated in bilinear form. System identification is a technique to obtain such a bilinear approximation for a nonlinear system from input-output data. Recent discrete-time bilinear model identification methods rely on Input-Output-to-State Representations (IOSRs) derived via the interaction matrix technique. A new formulation of these methods is given by establishing a correspondence between interaction matrices and the gains of full-order bilinear state observers. The new interpretation of the identification methods highlights the possibility of utilizing minimal-order bilinear state observers to derive new IOSRs. The existence of such observers is discussed and shown to be guaranteed for special classes of bilinear systems. New bilinear system identification algorithms are developed and the corresponding computational advantages are illustrated via numerical examples.Mechanical engineering, Mathematics, Aerospace engineering, Orbital mechanics, Space flight--Mathematical models, Bilinear forms, Observers (Control theory)fv2157, rwl4, rb68Mechanical Engineering, Civil Engineering and Engineering MechanicsConferencesIdentification in Separable Matching with Observed Transfers
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Salanie, Bernardhttp://dx.doi.org/10.7916/D847498HFri, 25 Sep 2015 15:12:03 +0000Imposing a separability assumption on the joint surplus in tranfer- able utility matching models has proved very useful in empirical work. Yet when only “who matches whom” is observed, the distributions of unobserved heterogeneity cannot be identified separately. This note derives the distribution of equiilibrium transfers and shows that if the distribution of transfers within cells is observed, the distribution of heterogeneity can often be recovered, separability can be tested, and complementarities in surplus inferred.Economics, Mathematicsbs2237EconomicsWorking papersHigher-order Properties of Approximate Estimators
https://academiccommons.columbia.edu/catalog/ac:188409
Kristensen, Dennis; Salanie, Bernardhttp://dx.doi.org/10.7916/D89886BKFri, 18 Sep 2015 13:22:20 +0000Many modern estimation methods in econometrics approximate an objective function, for instance, through simulation or discretization. These approximations typically affect both bias and variance of the resulting estimator. We first provide a higher-order expansion of such "approximate" estimators that takes into account the errors due to the use of approximations. We show how a Newton-Raphson adjustment can reduce the impact of approximations. Then we use our expansions to develop inferential tools that take into account approximation errors: we propose adjustments of the approximate estimator that remove its first-order bias and adjust its standard errors. These corrections apply to a class of approximate estimators that includes all known simulation-based procedures. A Monte Carlo simulation on the mixed logit model shows that our proposed adjustments can yield spectacular improvements at a low computational cost.Statistics, Economics, Mathematics, Computer sciencebs2237EconomicsWorking papersBifurcation of On-site and Off-site Solitary Waves of Discrete Nonlinear Schrödinger Type Equations
https://academiccommons.columbia.edu/catalog/ac:189391
Jenkinson, Michael Jameshttp://dx.doi.org/10.7916/D8J102F0Wed, 19 Aug 2015 12:13:41 +0000A feature of immeasurable interest in nonlinear systems is that of spatially localized traveling pulses, or solitary waves - states which persist indefinitely in time, focus energy, and facilitate its transfer. Furthermore, in many lattice systems, discreteness effects are important and play a key role in these dynamics.
In this thesis, we construct the multiple families of solitary standing (time-periodic) waves of the discrete, focusing cubically nonlinear Schrödinger equation (DNLS). These states are related to the so-called Peierls-Nabarro energy barrier, which refers to the energy difference between these distinct states and is thought to be responsible for the absence of indefinitely traveling, non-deforming solitary (spatially localized) waves of arbitrary velocity in many (non-dissipative) discrete systems. Instead, one observes that traveling waves of many discrete equations radiate energy and deform until they eventually cease to propagate and settle to a stationary time-periodic standing wave centered at a vertex.
We address two specific cases of DNLS: (1) nearest-neighbor coupling on a cubic lattice in dimensions d = 1,2,3, and (2) long-range site coupling in dimension d = 1. These states are obtained via a bifurcation analysis about the continuum nonlinear Schrödinger equation (NLS) limit, with respect to a natural small parameter. Depending on the spatial dimension, these may be vertex-, bond-, cell-, or face-centered. In the first case of nearest-neighbor coupling, we construct an explicit asymptotic expansion. In the second case of one-dimensional long-range coupling when the decay of the site coupling with respect to distance is sufficiently slow, the continuum limiting NLS equation has Laplacian of fractional power. Finally, we show that the energy difference among distinct states of the same frequency is exponentially small with respect to the small parameter beyond all polynomial orders. This provides a rigorous bound for the Peierls-Nabarro barrier.Applied mathematics, Mathematics, Physicsmjj2122Applied Physics and Applied MathematicsDissertationsA Minkowski-Type Inequality for Hypersurfaces in the Reissner-Nordstrom-Anti-deSitter Manifold
https://academiccommons.columbia.edu/catalog/ac:187980
Wang, Zhuhaihttp://dx.doi.org/10.7916/D86H4GGNWed, 13 May 2015 12:21:47 +0000We prove a sharp Minkowski-type inequality for hypersurfaces in the n-dimensional Reissner-Nordström-Anti-deSitter(AdS) manifold for n ≥ 3. This inequality generalizes the one for hypersurfaces in the uncharged AdS-Schwarzschild manifold proved in 5. With the Minkowski inequality, we prove a charged Gibbons-Penrose inequality for a large class of (n - 1)-dimensional spacelike surfaces in the Reissner-Nordström spacetime.Mathematicszw2175MathematicsDissertationsThe Parity of Analytic Ranks among Quadratic Twists of Elliptic Curves over Number Fields
https://academiccommons.columbia.edu/catalog/ac:186977
Balsam, Nava Kaylahttp://dx.doi.org/10.7916/D87P8XF4Thu, 07 May 2015 00:16:25 +0000The parity of the analytic rank of an elliptic curve is given by the root number in the functional equation L(E,s). Fixing an elliptic curve over any number eld and considering the family of its quadratic twists, it is natural to ask what the average analytic rank in this family is. A lower bound on this number is given by the average root number. In this paper, we investigate the root number in such families and derive an asymptotic formula for the proportion of curves in the family that have even rank. Our results are then used to support a conjecture about the average analytic rank in this family of elliptic curves.MathematicsMathematicsDissertationsA Proof of Looijenga's Conjecture via Integral-Affine Geometry
https://academiccommons.columbia.edu/catalog/ac:186926
Engel, Philiphttp://dx.doi.org/10.7916/D8028QGQFri, 24 Apr 2015 18:34:12 +0000A cusp singularity is a surface singularity whose minimal resolution is a reduced cycle of smooth rational curves meeting transversely. Cusp singularities come in naturally dual pairs. In 1981, Looijenga proved that whenever a cusp singularity is smoothable, the minimal resolution of the dual cusp is an anticanonical divisor of some smooth rational surface. He conjectured the converse. This dissertation provides a proof of Looijenga's conjecture based on a combinatorial criterion for smoothability given by Friedman and Miranda in 1983, and explores the geometry of the space of smoothings. The key tool in the proof is the use of integral-affine surfaces, two-dimensional manifolds whose transition functions are valued in the integral-affine transformation group. Motivated by the proof and recent work in mirror symmetry, we make a conjecture regarding the structure of the smoothing components of a cusp singularity.MathematicsMathematicsDissertationsQuantum Algebras and Cyclic Quiver Varieties
https://academiccommons.columbia.edu/catalog/ac:186938
Negut, Andreihttp://dx.doi.org/10.7916/D8J38RGFFri, 24 Apr 2015 18:33:17 +0000The purpose of this thesis is to present certain viewpoints on the geometric representation theory of Nakajima cyclic quiver varieties, in relation to the Maulik-Okounkov stable basis. Our main technical tool is the shuffle algebra, which arises as the K-theoretic Hall algebra of the double cyclic quiver. We prove the isomorphism between the shuffle algebra and the quantum toroidal algebra U_[q,t](sl_n), and identify the quotients of Verma modules for the shuffle algebra with the K-theory groups of Nakajima cyclic quiver varieties, which were studied by Nakajima and Varagnolo-Vasserot.
The shuffle algebra viewpoint allows us to construct the universal R-matrix of the quantum toroidal algebra U_[q,t](sl_n), and to factor it in terms of pieces that arise from subalgebras isomorphic to quantum affine groups U_q(gl_m), for various m. This factorization generalizes constructions of Khoroshkin-Tolstoy to the toroidal case, and matches the factorization that Maulik-Okounkov produce via the stable basis in the K-theory of Nakajima quiver varieties. We connect the two pictures by computing formulas for the root generators of U_[q,t](sl_n) acting on the stable basis, which provide a wide extension of Murnaghan-Nakayama and Pieri type rules from combinatorics.Mathematicsan2534MathematicsDissertationsPartial differential equations and variational approaches to constant scalar curvature metrics in Kähler geometry
https://academiccommons.columbia.edu/catalog/ac:186467
Rubin, Daniel Ilanhttp://dx.doi.org/10.7916/D8HD7TMGThu, 23 Apr 2015 12:24:35 +0000In this thesis we investigate two approaches to the problem of existence of metrics of constant scalar curvature in a fixed Kähler class. In the first part, we
examine the equation for constant scalar curvature under the assumption of toric symmetry, thus reducing the problem to a fourth order nonlinear degenerate elliptic equation for a convex function defined in a polytope in ℝ^n. We obtain partial results on this equation using an associated Monge-Ampère equation to determine the boundary behavior of the solution. In the second part, we consider the asymptotics of certain energy functionals and their relation to stability and the existence of minimizers. We derive explicit formulas for their asymptotic slopes, which allows one to determine whether or not (X,L) is stable, and in some cases rule out the existence of a canonical metric.Mathematicsdr2525MathematicsDissertationsExcluding Induced Paths: Graph Structure and Coloring
https://academiccommons.columbia.edu/catalog/ac:186521
Maceli, Peter Lawsonhttp://dx.doi.org/10.7916/D8WW7GK4Mon, 20 Apr 2015 12:17:03 +0000An induced subgraph of a given graph is any graph which can be obtained by successively deleting vertices, possible none. In this thesis, we present several new structural and algorithmic results on a number of different classes of graphs which are closed under taking induced subgraphs.
The first result of this thesis is related to a conjecture of Hayward and Nastos on the structure of graphs with no induced four-edge path or four-edge antipath. They conjectured that every such graph which is both prime and perfect is either a split graph or contains a certain useful arrangement of simplicial and antisimplicial vertices. We give a counterexample to their conjecture, and prove a slightly weaker version. This is joint work with Maria Chudnovsky, and first appeared in Journal of Graph Theory.
The second result of this thesis is a decomposition theorem for the class of all graphs with no induced four-edge path or four-edge antipath. We show that every such graph can be obtained from pentagons and split graphs by repeated application of complementation, substitution, and split graph unification. Split graph unification is a new graph operation we introduced, which is a generalization of substitution and involves "gluing" two graphs along a common induced split graph. This is a combination of joint work with Maria Chudnovsky and Irena Penev, together with later work of Louis Esperet, Laetitia Lemoine and Frederic Maffray, and first appeared in.
The third result of this thesis is related to the problem of determining the complexity of coloring graphs which do not contain some fixed induced subgraph. We show that three-coloring graphs with no induced six-edge path or triangle can be done in polynomial-time. This is joint work with Maria Chudnovsky and Mingxian Zhong, and first appeared in. Working together with Flavia Bonomo, Oliver Schaudt, and Maya Stein, we have since simplified and extended this result.Operations research, Mathematics, Computer scienceplm2109Operations Research, Industrial EngineeringDissertationsSingular Solutions to the Monge-Ampere Equation
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Mooney, Connor R.http://dx.doi.org/10.7916/D89K4955Fri, 10 Apr 2015 15:20:22 +0000This thesis contains the author's results on singular solutions to the Monge-Ampere equation \det D^2u = 1. We first prove that solutions are smooth away from a small closed singular set of Hausdorff (n-1)-dimensional measure zero. We also construct solutions with a singular set of Hausdorff dimension n-1, showing that this result is optimal. As a consequence we obtain unique continuation for the Monge-Ampere equation. Finally, we prove an interior W^{2,1} estimate for singular solutions, and we construct an example to show that this estimate is optimal.Mathematicscrm2181MathematicsDissertationsOn a Spectral Bound for Congruence Subgroup Families in SL(3,Z)
https://academiccommons.columbia.edu/catalog/ac:184064
Heath, Timothy Christopherhttp://dx.doi.org/10.7916/D8XW4HNMTue, 24 Feb 2015 12:16:08 +0000Spectral bounds on Maass forms of congruence families in algebraic groups are important ingredients to proving almost prime results for these groups. Extending the work of Gamburd [Gamburd, 2002] and Magee [Magee, 2013], we produce a condition under which such a bound exists in congruence subgroup families of SL(3,Z), uniformly and even when these groups are thin, i.e. of infinite index. The condition is analogous to the cusp and collar lemmas in Gamburd's work and is expected to hold for families whose Hausdorff dimension of the limit set is large enough.MathematicsMathematicsDissertationsBifurcation of localized eigenstates of perturbed periodic Schrödinger operators
https://academiccommons.columbia.edu/catalog/ac:182973
Vukicevic, Ivahttp://dx.doi.org/10.7916/D88C9V2ZThu, 05 Feb 2015 18:20:43 +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
https://academiccommons.columbia.edu/catalog/ac:180940
Zong, Zhengyuhttp://dx.doi.org/10.7916/D8513WZCThu, 04 Dec 2014 09:40:48 +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.MathematicsMathematicsDissertationsA General Class of Heuristics for Minimum Weight Perfect Matching and Fast Special Cases with Doubly and Triply Logarithmic Errors
https://academiccommons.columbia.edu/catalog/ac:177681
Imielinska, Celina Z.; Kalantari, B.http://dx.doi.org/10.7916/D8SJ1J5XMon, 29 Sep 2014 11:26:06 +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
https://academiccommons.columbia.edu/catalog/ac:177678
Imielinska, Celina Z.; Kalantari, Bahmanhttp://dx.doi.org/10.7916/D8222SB3Mon, 29 Sep 2014 11:14:06 +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 InformaticsArticlesGene regulatory network inference by point-based Gaussian approximation filters incorporating the prior information
https://academiccommons.columbia.edu/catalog/ac:194860
Jia, Bin; Wang, Xiaodonghttp://dx.doi.org/10.7916/D8833QGNTue, 09 Sep 2014 00:58:39 +0000The extended Kalman filter (EKF) has been applied to inferring gene regulatory networks. However, it is well known that the EKF becomes less accurate when the system exhibits high nonlinearity. In addition, certain prior information about the gene regulatory network exists in practice, and no systematic approach has been developed to incorporate such prior information into the Kalman-type filter for inferring the structure of the gene regulatory network. In this paper, an inference framework based on point-based Gaussian approximation filters that can exploit the prior information is developed to solve the gene regulatory network inference problem. Different point-based Gaussian approximation filters, including the unscented Kalman filter (UKF), the third-degree cubature Kalman filter (CKF3), and the fifth-degree cubature Kalman filter (CKF5) are employed. Several types of network prior information, including the existing network structure information, sparsity assumption, and the range constraint of parameters, are considered, and the corresponding filters incorporating the prior information are developed. Experiments on a synthetic network of eight genes and the yeast protein synthesis network of five genes are carried out to demonstrate the performance of the proposed framework. The results show that the proposed methods provide more accurate inference results than existing methods, such as the EKF and the traditional UKF.Bioinformatics, Biostatistics, Mathematics, Gene regulatory network, Kalman filtering, Gaussian distribution, Iterative methods (Mathematics)xw2008Electrical EngineeringArticlesMultiple Dirichlet Series for Affine Weyl Groups
https://academiccommons.columbia.edu/catalog/ac:176818
Whitehead, Ianhttp://dx.doi.org/10.7916/D8BK19HTMon, 07 Jul 2014 11:54:13 +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.MathematicsMathematicsDissertations'Value Creation' Through Mathematical Modeling: Students' Mathematics Dispositions and Identities Developed in a Learning Community
https://academiccommons.columbia.edu/catalog/ac:176803
Park, Joo younghttp://dx.doi.org/10.7916/D87S7KXXMon, 07 Jul 2014 11:53:38 +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 Education, Mathematics, Science, and TechnologyDissertationsDemazure-Lusztig Operators and Metaplectic Whittaker Functions on Covers of the General Linear Group
https://academiccommons.columbia.edu/catalog/ac:176190
Puskas, Annahttp://dx.doi.org/10.7916/D8J964J6Mon, 07 Jul 2014 11:51:49 +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.MathematicsMathematicsDissertationsSequential Optimization in Changing Environments: Theory and Application to Online Content Recommendation Services
https://academiccommons.columbia.edu/catalog/ac:176086
Gur, Yonatanhttp://dx.doi.org/10.7916/D8639MWFMon, 07 Jul 2014 11:48:10 +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, MathematicsBusinessDissertationsTowards a definition of Shimura curves in positive characteristics
https://academiccommons.columbia.edu/catalog/ac:176077
Xia, Jiehttp://dx.doi.org/10.7916/D8ZP448CMon, 07 Jul 2014 11:47:28 +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.Mathematicsjx2149MathematicsDissertationsOn a triply-graded generalization of Khovanov homology
https://academiccommons.columbia.edu/catalog/ac:175996
Putyra, Krzysztofhttp://dx.doi.org/10.7916/D86971RXMon, 07 Jul 2014 11:44:36 +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
https://academiccommons.columbia.edu/catalog/ac:175993
Pan, Xuanyuhttp://dx.doi.org/10.7916/D8KK98X0Mon, 07 Jul 2014 11:42:57 +0000We prove that the moduli space of rational normal curves on a low degree complete intersection passing several suitable points is a complete intersection.MathematicsMathematicsDissertationsA Spacetime Alexandrov Theorem
https://academiccommons.columbia.edu/catalog/ac:175978
Wang, Ye-Kaihttp://dx.doi.org/10.7916/D8MG7MN2Mon, 07 Jul 2014 11:41:44 +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.MathematicsMathematicsDissertationsProbabilistic Approaches to Partial Differential Equations with Large Random Potentials
https://academiccommons.columbia.edu/catalog/ac:175891
Gu, Yuhttp://dx.doi.org/10.7916/D82R3PTDMon, 07 Jul 2014 11:40:10 +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 MathematicsDissertationsSelf-duality and singularities in the Yang-Mills flow
https://academiccommons.columbia.edu/catalog/ac:175717
Waldron, Alexhttp://dx.doi.org/10.7916/D81V5C3RMon, 07 Jul 2014 11:38:13 +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
https://academiccommons.columbia.edu/catalog/ac:175711
Koziol, Karolhttp://dx.doi.org/10.7916/D89C6VKVMon, 07 Jul 2014 11:38:05 +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.MathematicsMathematicsDissertationsConstant Scalar Curvature of Toric Fibrations
https://academiccommons.columbia.edu/catalog/ac:175513
Nyberg, Thomashttp://dx.doi.org/10.7916/D8TH8JVHMon, 07 Jul 2014 11:34:39 +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.Mathematicstwn2103MathematicsDissertationsCanonical Metrics in Sasakian Geometry
https://academiccommons.columbia.edu/catalog/ac:175504
Collins, Tristanhttp://dx.doi.org/10.7916/D86Q1VCSMon, 07 Jul 2014 11:34:18 +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.Mathematicstcc2119MathematicsDissertationsThe arithmetic and geometry of genus four curves
https://academiccommons.columbia.edu/catalog/ac:175469
Xue, Hanghttp://dx.doi.org/10.7916/D87P8WHMMon, 07 Jul 2014 11:33:44 +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.Mathematicshx2119MathematicsDissertationsBordered Heegaard Floer Homology and Graph Manifolds
https://academiccommons.columbia.edu/catalog/ac:175430
Hanselman, Jonathanhttp://dx.doi.org/10.7916/D8NZ85TFMon, 07 Jul 2014 11:32:47 +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 pantsMathematicsMathematicsDissertationsThree-Manifold Mutations Detected by Heegaard Floer Homology
https://academiccommons.columbia.edu/catalog/ac:175403
Clarkson, Corrinhttp://dx.doi.org/10.7916/D8GF0RNGMon, 07 Jul 2014 11:31:33 +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.MathematicsMathematicsDissertationsConvex Optimization Algorithms and Recovery Theories for Sparse Models in Machine Learning
https://academiccommons.columbia.edu/catalog/ac:175385
Huang, Bohttp://dx.doi.org/10.7916/D8VM49DMMon, 07 Jul 2014 11:31:19 +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 ResearchDissertationsA One-Pass Sequential Monte Carlo Method for Bayesian Analysis of Massive Datasets
https://academiccommons.columbia.edu/catalog/ac:173899
Balakrishnan, Suhrid; Madigan, David B.http://dx.doi.org/10.7916/D8B56GTPThu, 15 May 2014 11:51:51 +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, Statisticsdm2418StatisticsArticlesA Characterization of Markov Equivalence Classes for Acyclic Digraphs
https://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 11:28:36 +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 mathematicsdm2418StatisticsArticlesCorrection: Separation and completeness properties for AMP chain graph Markov models
https://academiccommons.columbia.edu/catalog/ac:173887
Madigan, David B.; Levitz, Michael; Perlman, Michael D.http://dx.doi.org/10.7916/D8QF8R05Wed, 14 May 2014 19:42:28 +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, Statisticsdm2418StatisticsArticles[Bayesian Analysis in Expert Systems]: Comment: What's Next?
https://academiccommons.columbia.edu/catalog/ac:173856
Madigan, David B.http://dx.doi.org/10.7916/D8W37TFJTue, 13 May 2014 17:59:40 +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, Statisticsdm2418StatisticsArticlesSeparation and Completeness Properties for Amp Chain Graph Markov Models
https://academiccommons.columbia.edu/catalog/ac:173847
Levitz, Michael; Perlman, Michael D.; Madigan, David B.http://dx.doi.org/10.7916/D8X34VJGTue, 13 May 2014 16:30:46 +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[Least Angle Regression]: Discussion
https://academiccommons.columbia.edu/catalog/ac:173841
Madigan, David B.; Ridgeway, Greghttp://dx.doi.org/10.7916/D81V5C29Tue, 13 May 2014 16:15:23 +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 Hierarchical Model for Association Rule Mining of Sequential Events: An Approach to Automated Medical Symptom Prediction
https://academiccommons.columbia.edu/catalog/ac:173838
McCormick, Tyler H.; Rudin, Cynthia; Madigan, David B.http://dx.doi.org/10.7916/D89C6VJDTue, 13 May 2014 15:27:01 +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, Medicinedm2418StatisticsArticlesA Note on Equivalence Classes of Directed Acyclic Independence Graphs
https://academiccommons.columbia.edu/catalog/ac:173826
Madigan, David B.http://dx.doi.org/10.7916/D8TB150CTue, 13 May 2014 14:46:04 +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, Statisticsdm2418StatisticsArticlesLocation Estimation in Wireless Networks: A Bayesian Approach
https://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 14:25:34 +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 Flexible Bayesian Generalized Linear Model for Dichotomous Response Data with an Application to Text Categorization
https://academiccommons.columbia.edu/catalog/ac:173817
Eyheramendy, Susana; Madigan, David B.http://dx.doi.org/10.7916/D86M34ZFTue, 13 May 2014 14:04:25 +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 chaptersSurface wave phase velocities of the Western United States from a two-station method
https://academiccommons.columbia.edu/catalog/ac:172091
Foster, Anna; Ekström, Göran; Nettles, Meredith K.http://dx.doi.org/10.7916/D87W6979Thu, 13 Mar 2014 10:38:49 +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, mn2237Earth and Environmental Sciences, Lamont-Doherty Earth ObservatoryArticlesArrival-angle anomalies across the USArray Transportable Array
https://academiccommons.columbia.edu/catalog/ac:172088
Foster, Anna; Ekström, Göran; Hjorleifsdottir, Valahttp://dx.doi.org/10.7916/D8CJ8BJ9Thu, 13 Mar 2014 10:08:27 +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, ge21Earth and Environmental Sciences, Lamont-Doherty Earth ObservatoryArticlesInvisible Mathematics in Italo Calvino's Le città invisibili
https://academiccommons.columbia.edu/catalog/ac:166630
Moreno-Viqueira, Ileanahttp://hdl.handle.net/10022/AC:P:22021Fri, 18 Oct 2013 13:41:44 +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, MathematicsItalianDissertationsOn Lagrange-Hermite Interpolation
https://academiccommons.columbia.edu/catalog/ac:166460
Traub, Joseph F.http://hdl.handle.net/10022/AC:P:21987Thu, 10 Oct 2013 14:40:22 +0000Mathematics, Applied mathematicsjft2Computer ScienceArticlesGeneralized Sequences with Applications to the Discrete Calculus
https://academiccommons.columbia.edu/catalog/ac:166457
Traub, Joseph F.http://hdl.handle.net/10022/AC:P:21986Thu, 10 Oct 2013 14:28:53 +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 ScienceArticlesConstruction of Globally Convergent Iteration Functions for the Solution of Polynomial Equations
https://academiccommons.columbia.edu/catalog/ac:166454
Traub, Joseph F.http://hdl.handle.net/10022/AC:P:21985Thu, 10 Oct 2013 14:21:49 +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 ScienceArticlesAssociated Polynomials and Uniform Methods for the Solution of Linear Problems
https://academiccommons.columbia.edu/catalog/ac:166451
Traub, Joseph F.http://hdl.handle.net/10022/AC:P:21984Thu, 10 Oct 2013 14:16:02 +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 ScienceArticlesA Class of Globally Convergent Iterations for the Solution of Polynomial Equations
https://academiccommons.columbia.edu/catalog/ac:166448
Traub, Joseph F.http://hdl.handle.net/10022/AC:P:21983Thu, 10 Oct 2013 14:08:15 +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
https://academiccommons.columbia.edu/catalog/ac:166445
Jenkins, M.A.; Traub, Joseph F.http://hdl.handle.net/10022/AC:P:21982Thu, 10 Oct 2013 13:59:32 +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 ScienceArticlesComputational Complexity of Iterative Processes
https://academiccommons.columbia.edu/catalog/ac:166442
Traub, Joseph F.http://hdl.handle.net/10022/AC:P:21981Thu, 10 Oct 2013 13:17:03 +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 ScienceArticles