Results for "Brian Roark"

total 6155took 0.16s
Compact non-left-recursive grammars using the selective left-corner transform and factoringAug 22 2000The left-corner transform removes left-recursion from (probabilistic) context-free grammars and unification grammars, permitting simple top-down parsing techniques to be used. Unfortunately the grammars produced by the standard left-corner transform are ... More
Efficient probabilistic top-down and left-corner parsingAug 21 2000This paper examines efficient predictive broad-coverage parsing without dynamic programming. In contrast to bottom-up methods, depth-first top-down parsing produces partial parses that are fully connected trees spanning the entire left context, from which ... More
Approximating probabilistic models as weighted finite automataMay 21 2019Weighted finite automata (WFA) are often used to represent probabilistic models, such as $n$-gram language models, since they are efficient for recognition tasks in time and space. The probabilistic source to be represented as a WFA, however, may come ... More
Are All Languages Equally Hard to Language-Model?Jun 10 2018For general modeling methods applied to diverse languages, a natural question is: how well should we expect our models to work on languages with differing typological profiles? In this work, we develop an evaluation framework for fair cross-linguistic ... More
Meaning to Form: Measuring Systematicity as InformationJun 13 2019Jul 26 2019A longstanding debate in semiotics centers on the relationship between linguistic signs and their corresponding semantics: is there an arbitrary relationship between a word form and its meaning, or does some systematic phenomenon pervade? For instance, ... More
What Kind of Language Is Hard to Language-Model?Jun 11 2019How language-agnostic are current state-of-the-art NLP tools? Are there some types of language that are easier to model with current methods? In prior work (Cotterell et al., 2018) we attempted to address this question for language modeling, and observed ... More
Meaning to Form: Measuring Systematicity as InformationJun 13 2019A longstanding debate in semiotics centers on the relationship between linguistic signs and their corresponding semantics: is there an arbitrary relationship between a word form and its meaning, or does some systematic phenomenon pervade? For instance, ... More
The Deconfinement Phase Transition in Proto-Neutron-Star MatterMar 06 2018Nov 12 2018In this work, we study in detail the deconfinement phase transition that takes place in hot/dense nuclear matter in the context of neutron stars and proto-neutron stars (in which lepton fraction is fixed). The possibility of different mixtures of phases ... More
The Age of the UniverseMay 16 1996Globular clusters are the oldest objects in the Galaxy whose age may be accurately determined. As such globular cluster ages provide the best estimate for the age of the universe. The age of a globular cluster is determined by a comparison between theoretical ... More
Halo Star EvolutionSep 08 1995In this review, I will discuss a few problems which point to the need for improved stellar evolution models of halo stars. Current stellar evolution models do not match the observed $^7$Li abundance patterns, suggesting that the input physics and/or the ... More
Deviations from the Circular LawDec 01 2003Consider Ginibre's ensemble of $N \times N$ non-Hermitian random matrices in which all entries are independent complex Gaussians of mean zero and variance $\frac{1}{N}$. As $N \uparrow \infty$ the normalized counting measure of the eigenvalues converges ... More
Multi-Epoch Observations of the Redwing Excess in the Spectrum of 3C279Nov 12 2012It has been previously determined that there is a highly significant correlation between the spectral index from 10 GHz to 1350 $\AA$ and the amount of excess luminosity in the red wing of quasar CIV $\lambda$1549 broad emission lines (BELs). Ostensibly, ... More
Models of the Compact Jet in GRS 1915+105Aug 23 2011In this article, models are constructed of the compact jet in GRS 1915+105 during an epoch of optimal data capture. On April 02, 2003, the object was observed in the hard X-ray/soft gamma ray band (INTEGRAL), hard X-ray band (RXTE), near IR (ESO/New Technology ... More
High Jet Efficiency and Simulations of Black Hole MagnetospheresDec 09 2010This article reports on a growing body of observational evidence that many powerful lobe dominated (FR II) radio sources likely have jets with high efficiency. This study extends the maximum efficiency line (jet power $\approx$ 25 times the thermal luminosity) ... More
X-ray Absorption in Type II Quasars: Implications for the Equatorial Paradigm of Broad Absorption Line QuasarsMay 01 2006In this article, the hydrogen column densities derived from X-ray observations of type II (hidden) quasars and broad absorption line quasars (BALQSOs) are compared. These column densities represent the amount of absorbing material between the X-ray source ... More
Improved Online Square-into-Square PackingJan 22 2014In this paper, we show an improved bound and new algorithm for the online square-into-square packing problem. This two-dimensional packing problem involves packing an online sequence of squares into a unit square container without any two squares overlapping. ... More
Singular differential equationsSep 22 2015Jan 18 2016The purpose of this paper is to study a class of ill-posed differential equations. In some settings, these differential equations exhibit uniqueness but not existence, while in others they exhibit existence but not uniqueness. An example of such a differential ... More
Regularity of minimal submanifolds and mean curvature flows with a common free boundarySep 26 2016Let $N$ be a smooth $(n+l)$-dimensional Riemannian manifold. We show that if $V$ is an area-stationary union of three or more $C^{1,\mu}$ $n$-dimensional submanifolds-with-boundary $M_k \subset N$ with a common boundary $\Gamma$, then $\Gamma$ is smooth ... More
First betti numbers of Kähler manifolds with weakly pseudoconvex boundaryOct 20 2011We study K\"ahler manifolds-with-boundary, not necessarily compact, with weakly pseudoconvex boundary, each component of which is compact. If such a manifold $K$ has $l\ge2$ boundary components (possibly $l=\infty$), it has first betti number at least ... More
ODE Maximum Principle at Infinity and Non-Compact Solutions of IMCF in Hyperbolic SpaceOct 04 2016In this work we extend the ODE Maximum principle of Hamilton to non-compact hypersurfaces using the Omari-Yau maximum principle at infinity. As an application of this result, we investigate Inverse Mean Curvature Flow (IMCF) of non-compact hypersurfaces ... More
Association schemes, classical RCFT's, and centres of monoidal functor categoriesFeb 08 2007Aug 12 2011Here we describe three straightforward examples of what was called a graphic Fourier transformation in [4]. At least two of these examples may be viewed simply as monoidal comonads on suitable monoidal closed functor categories, but the third example, ... More
Generators for Symbolic Powers of Ideals Defining General Points of $P^2$Sep 01 1995Given distinct points $p_1,\cdots,p_r$ of the projective plane $P^2$ and a positive integer $m$, the homogeneous ideal defining the fat point subscheme $Z=m(p_1+\cdots+p_r)$ is the symbolic power $I^{(m)}$ of the homogeneous ideal $I$ defining the smooth ... More
Maximal Function Inequalities and a Theorem of BirchNov 12 2017Dec 05 2017In this paper we prove an analogue of the discrete spherical maximal theorem of Magyar, Stein, and Wainger, an analogue which concerns maximal functions associated to homogenous algebraic surfaces. Let $\mathfrak{p}$ be a homogenous polynomial in $n$ ... More
Sharp Regularity for the Integrability of Elliptic StructuresOct 23 2018Nov 18 2018As part of his celebrated Complex Frobenius Theorem, Nirenberg showed that given a smooth elliptic structure (on a smooth manifold), the manifold is locally diffeomorphic to an open subset of $\mathbb{R}^r\times \mathbb{C}^n$ (for some $r$ and $n$) in ... More
Differential Equations with a Difference QuotientSep 22 2015Jan 02 2017The purpose of this paper is to study a class of ill-posed differential equations. In some settings, these differential equations exhibit uniqueness but not existence, while in others they exhibit existence but not uniqueness. An example of such a differential ... More
On Nagata's ConjectureSep 16 1999Modifying an approach of J. Roe, this paper gives an improved lower bound on the degrees d such that for general points p1,...,pn in P2 and m > 0 there is a plane curve of degree d vanishing at each point pi with multiplicity at least m. In certain cases, ... More
Linked systems of symmetric designsDec 08 2017A linked system of symmetric designs (LSSD) is a $w$-partite graph ($w\geq 2$) where the incidence between any two parts corresponds to a symmetric design and the designs arising from three parts are related. The original construction for LSSDs by Goethals ... More
Integral-Input-Output to State StabilityJun 15 2001A notion of detectability for nonlinear systems is discussed. Within the framework of ``input to state stability'' (ISS), a dual notion of ``output to state stability'' (OSS), and a more complete detectability notion, ``input-output to state stability'' ... More
Decompositions of the tensor products of irreducible sl(2)-modules in characteristic 3Sep 22 2008We completely describe the decompositions (into indecomposable submodules) of the tensor products of irreducible sl(2)-modules in characteristic 3. The answer resembles analogous decompositions for the Lie superalgebra sl(1|1).
Generalized Kahler Taub-NUTs and Two Exceptional InstantonsFeb 19 2016Jun 17 2019We study the one-parameter family of twisted Kahler Taub-NUT metrics (discovered by Donaldson), along with two exceptional Taub-NUT-like instantons, and understand them to the extend that should be sufficient for blow-up and gluing arguments. In particular ... More
On Lattice Barycentric TetrahedraJan 05 2004We say a lattice tetrahedron whose centroid is its only non-vertex lattice point is lattice barycentric. The notation T(a,b,c) describes the lattice tetrahedron with vertices {0, e_1, e_2, a e_1 + b e_2 + c e_3}. Our result is that all such T(a,b,c) are ... More
A Density Result for Real Hyperelliptic CurvesMar 31 2017May 15 2019Let $\{\infty^+, \infty^-\}$ be the two points above $\infty$ on the real hyperelliptic curve $H: y^2 = (x^2 - 1) \prod_{i=1}^{2g} (x - a_i)$. We show that the divisor $([\infty^+] - [\infty^-])$ is torsion in $\operatorname{Jac} J$ for a dense set of ... More
Inverse Mean Curvature Flow and the Stability of the Positive Mass TheoremJul 23 2018We study the stability of the Positive Mass Theorem (PMT) in the case where a sequence of regions of manifolds with positive scalar curvature $U_T^i\subset M_i^3$ are foliated by a smooth solution to Inverse Mean Curvature Flow (IMCF) which may not be ... More
Hyponormal Toeplitz Operators with Non-Harmonic Algebraic SymbolJul 01 2018We generalize recent results of Fleeman and Liaw on the topic of hyponormal Toeplitz operators acting on the Bergman space of the unit disk.
Spline-Based Probability CalibrationSep 20 2018In many classification problems it is desirable to output well-calibrated probabilities on the different classes. We propose a robust, non-parametric method of calibrating probabilities called SplineCalib that utilizes smoothing splines to determine a ... More
Applying Distributional Compositional Categorical Models of Meaning to Language TranslationNov 08 2018The aim of this paper is twofold: first we will use vector space distributional compositional categorical models of meaning to compare the meaning of sentences in Irish and in English (and thus ascertain when a sentence is the translation of another sentence) ... More
Sobolev stability of the PMT and RPI using IMCFAug 23 2018Apr 05 2019We study the Sobolev stability of the Positive Mass Theorem (PMT) and the Riemannian Penrose Inequality (RPI) in the case where a region of a sequence of manifolds $M^3_i$ can be foliated by a smooth solution of Inverse Mean Curvature Flow (IMCF) which ... More
arXiv and the Symbiosis of Physics Preprints and Journal Review ArticlesApr 01 2019May 20 2019New thinking needs to emerge about how to reform publishing along lines that best meet two perennial needs of scientific communication. This paper discusses a model that addresses these two needs with respect to physics. Given the considerable barriers ... More
Compact Embedded Minimal Surfaces of Positive Genus Without Area BoundsAug 22 2003Let M be a 3-manifold (possibly with boundary). We show that, for any positive integer g, there exists an open nonempty set of metrics on M for each of which there are stable compact embedded minimal surfaces of genus g with arbitrarily large area. This ... More
The $\square_b$ Heat Equation and Multipliers via the Wave EquationMay 09 2008Sep 10 2008Recently, Nagel and Stein studied the $\square_b$-heat equation, where $\square_b$ is the Kohn Laplacian on the boundary of a weakly-pseudoconvex domain of finite type in $\C^2$. They showed that the Schwartz kernel of $e^{-t\square_b}$ satisfies good ... More
Inverse Mean Curvature Flow over Non-Star-Shaped SurfacesSep 03 2019We derive an upper bound on the waiting time for a non star-shaped hypersurface in $\mathbb{R}^{n+1}$ moving by Inverse Mean Curvature Flow to become star-shaped. Combining this result with an embeddedness principle for the flow, we provide an upper bound ... More
Complete intersection P-partition ringsFeb 22 2018We present an alternate proof of a result of F\'eray and Reiner characterizing posets whose $P$-partition rings are complete intersections. This shortened proof relates the complete intersection property to a simple structural property of a graph associated ... More
Subdivision rules and the eight model geometriesJul 23 2012Cannon and Swenson have shown that each hyperbolic 3-manifold group has a natural subdivision rule on the space at infinity, and that this subdivision rule captures the action of the group on the sphere. Explicit subdivision rules have also been found ... More
Blow-up in the Parabolic Scalar Curvature EquationMay 25 2007Jun 05 2012The \textit{parabolic scalar curvature equation} is a reaction-diffusion type equation on an $(n-1)$-manifold $\Sigma$, the time variable of which shall be denoted by $r$. Given a function $R$ on $[r_0,r_1)\times\Sigma$ and a family of metrics $\gamma(r)$ ... More
Isometries of optimal pseudo-Riemannian metricsApr 26 2011Apr 29 2011We give a concise proof that large classes of optimal (constant curvature or Einstein) pseudo-Riemannian metrics are maximally symmetric within their conformal class.
Linked Grassmannians and crude limit linear seriesMay 15 2006In math.AG/0407496, a new construction of limit linear series is presented which functorializes and compactifies the original construction of Eisenbud and Harris, using a new space called the linked Grassmannian. The boundary of the compactification consists ... More
Deformations of covers, Brill-Noether theory, and wild ramificationDec 01 2004In this paper, we give a simple description of the deformations of a map between two smooth curves with partially prescribed branching, in the cases that both curves are fixed, and that the source is allowed to vary. Both descriptions work equally well ... More
Extending Landau's Theorem on Dirichlet Series with Non-Negative CoefficientsSep 01 2010A classical theorem of Landau states that, if an ordinary Dirichlet series has non-negative coefficients, then it has a singularity on the real line at its abscissae of absolute convergence. In this article, we relax the condition on the coefficients ... More
Geometric Structures on Spaces of Weighted SubmanifoldsNov 02 2009In this paper we use a diffeo-geometric framework based on manifolds that are locally modeled on "convenient" vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold $(M,\omega)$, we construct ... More
Limit linear series moduli stacks in higher rankMay 12 2014In order to prove new existence results in Brill-Noether theory for rank-2 vector bundles with fixed special determinant, we develop foundational definitions and results for limit linear series of higher-rank vector bundles. These include two entirely ... More
Topological Change in Mean Convex Mean Curvature FlowJul 23 2011Oct 28 2013Consider the mean curvature flow of an (n+1)-dimensional, compact, mean convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We prove that elements of the m-th homotopy group of the complementary region can die only if there is a shrinking ... More
The (unexpected) importance of knowing $α$Feb 19 2005In order to determine the Hilbert function of the ideal of a fat point subscheme of projective space, we show that it is enough to determine, both for the subscheme itself and the subschemes obtained from it by successively adjoining to it additional ... More
*-Autonomous categories in quantum theoryMay 01 2006Nov 26 2011*-Autonomous categories were initially defined by M. Barr to describe a type of duality carried by many monoidal closed categories. Later they were generalised by the current author to include *-autonomous promonoidal categories. Together, these structures ... More
Scoring Strategies for the Underdog: A general, quantitative method for determining optimal sports strategiesNov 02 2011When facing a heavily-favored opponent, an underdog must be willing to assume greater-than-average risk. In statistical language, one would say that an underdog must be willing to adopt a strategy whose outcome has a larger-than-average variance. The ... More
All finite subdivision rules are combinatorially equivalent to three-dimensional subdivision rulesDec 01 2015Finite subdivision rules in high dimensions can be difficult to visualize and require complex topological structures to be constructed explicitly. In many applications, only the history graph is needed. We characterize the history graph of a subdivision ... More
Ideals and idempotents in the uniform ultrafiltersMay 08 2015If $S$ is a discrete semigroup, then $\beta S$ has a natural, left-topological semigroup structure extending $S$. Under some very mild conditions, $U(S)$, the set of uniform ultrafilters on $S$, is a two-sided ideal of $\beta S$, and therefore contains ... More
Highly entangled quantum systems in 3+1 dimensionsMar 11 2010Many systems exhibit boundary law scaling for entanglement entropy in more than one spatial dimension. Here I describe three systems in 3+1 dimensions that violate the boundary law for entanglement entropy. The first is free Weyl fermions in a magnetic ... More
Conformal Field Theory on the Fermi SurfaceFeb 25 2010The Fermi surface may be usefully viewed as a collection of 1+1 dimensional chiral conformal field theories. This approach permits straightforward calculation of many anomalous ground state properties of the Fermi gas including entanglement entropy and ... More
Constructing holographic spacetimes using entanglement renormalizationSep 14 2012We elaborate on our earlier proposal connecting entanglement renormalization and holographic duality in which we argued that a tensor network can be reinterpreted as a kind of skeleton for an emergent holographic space. Here we address the question of ... More
Towards a General-Purpose Belief Maintenance SystemMar 27 2013There currently exists a gap between the theories proposed by the probability and uncertainty and the needs of Artificial Intelligence research. These theories primarily address the needs of expert systems, using knowledge structures which must be pre-compiled ... More
A Novel Approach to Quantum Heuristics for Structured Database SearchJun 27 2001An algorithm for structured database searching is presented and used to solve the set partition problem. O(n) oracle calls are required in order to obtain a solution, but the probability that this solution is optimal decreases exponentially with problem ... More
3C 216: A Powerful FRII Seyfert 1 GalaxySep 07 2006Sep 12 20063C 216 has a weak accretion flow luminosity, well below the Seyfert1/QSO dividing line, weak broad emission lines (BELs) and powerful radio lobes. As a consequence of the extreme properties of 3C 216, it is the most convincing example known of an FR II ... More
An Independent Derivation of the Oxford Jet Kinetic Luminosity FormulaMar 11 2005This letter presents a theoretical derivation of an estimate for a radio source jet kinetic luminosity. The expression yields jet powers that are quantitatively similar to a more sophisticated empirical relation published by the Willott, Blundell and ... More
Generalized Kahler Taub-NUTs and Two Exceptional InstantonsFeb 19 2016We study the one-parameter family of twisted Kahler Taub-NUT metrics (discovered by Donaldson), along with two exceptional Taub-NUT-like instantons, and understand them to the extend that is sufficient for blow-up and gluing-type arguments. In particular ... More
On the universality of the countable random graph: a density theoremSep 02 2016It is proved that for almost every graph on $\mathbb{N}$, every $A \subseteq \mathbb{N}$ of positive upper density contains every countable graph as an induced subgraph. This can be viewed as a stronger, Ramsey-theoretic version of the well-known embedding ... More
Path Results for Symmetric Jump ProcessesFeb 28 2012We consider a class of jump processes in euclidean space which are associated to a certain non-local symmetric Dirichlet form. We prove a lower bound on the occupation times of sets, and that a support theorem holds for these processes.
Dark Discrete Gauge SymmetriesJun 30 2010Jul 23 2010We investigate scenarios in which dark matter is stabilized by an abelian Z_N discrete gauge symmetry. Models are surveyed according to symmetries and matter content. Multi-component dark matter arises when N is not prime and Z_N contains one or more ... More
Fast Radio Bursts: Constraints on the Dispersing MediumMar 10 2014Fast radio bursts appear to exhibit large dispersion measures, typically exceeding any expected galactic interstellar contribution, especially along the moderate to high-galactic-latitude directions in which such events have been most often observed. ... More
Co-Variant Derivatives And The Renormalisation GroupMar 11 1994The renormalisation group equation for $N$-point correlation functions can be interpreted in a geometrical manner as an equation for Lie transport of amplitudes in the space of couplings. The vector field generating the diffeomorphism has components given ... More
A dual first-postulate basis for special relativityMar 12 2018An overlooked straightforward application of velocity reciprocity to a triplet of inertial frames in collinear motion identifies the ratio of their cyclic relative velocities' sum to the negative product as a cosmic invariant, whose inverse square root ... More
Evolution of curves and surfaces by mean curvatureDec 01 2002This article describes the mean curvature flow, some of the discoveries that have been made about it, and some unresolved questions.
Nonunique tangent maps at isolated singularities of harmonic mapsJan 01 1992Shoen and Uhlenbeck showed that ``tangent maps'' can be defined at singular points of energy minimizing maps. Unfortunately these are not unique, even for generic boundary conditions. Examples are discussed which have isolated singularities with a continuum ... More
Density of Complex Zeros of a System of Real Random PolynomialsJun 20 2010We study the density of complex zeros of a system of real random SO($m+1$) polynomials in several variables. We show that the density of complex zeros of this random polynomial system with real coefficients rapidly approaches the density of complex zeros ... More
Black Hole Initial Data with a Horizon of Prescribed GeometryOct 04 2007The purpose of this work is to construct asymptotically flat, time symmetric initial data with an apparent horizon of prescribed intrinsic geometry. To do this, we use the parabolic partial differential equation for prescribing scalar curvature. In this ... More
Elimination of HIV in South Africa through expanded access to antiretroviral therapy: Cautions, caveats and the importance of parsimonyMar 26 2014In a recent article Hontelez and colleagues investigate the prospects for elimination of HIV in South Africa through expanded access to antiretroviral therapy (ART) using STDSIM, a micro-simulation model. One of the first published models to suggest that ... More
The Physical World as a Virtual RealityJan 02 2008Jan 05 2008This paper explores the idea that the universe is a virtual reality created by information processing, and relates this strange idea to the findings of modern physics about the physical world. The virtual reality concept is familiar to us from online ... More
Properties of the donor impurity band in mixed valence insulatorsJul 03 2019In traditional semiconductors with large effective Bohr radius, an electron donor creates a hydrogen-like bound state just below the conduction band edge. The properties of the impurity band arising from such hydrogenic impurities have been studied extensively ... More
Constraints on Black Hole Jet Models Used As Diagnostic Tools of Event Horizon Telescope Observations of M87Jun 14 2019Jun 26 2019Jet models of Event Horizon Telescope (EHT) data should also conform to the observed jet profiles just downstream. This study evaluates conformance of models of black hole jets to images of the innermost jet of M87. This is a basic test that should be ... More
A note on discrete spherical averages over sparse sequencesAug 11 2018Sep 18 2018This note presents an example of an increasing sequence $(\lambda_l)_{l=1}^\infty$ such that the maximal operators associated to normalized discrete spherical convolution averages \[ \sup_{l\geq 1}\frac{1}{r(\lambda_l)}\left|\sum_{|x|^2=\lambda_l}f(y-x)\right|\] ... More
Discrete multilinear spherical averagesAug 05 2018In this note we give a characterization of $\ell^{p}\times ...\times \ell^{p}\to\ell^q$ boundedness of maximal operators associated to multilinear convolution averages over spheres in $\mathbb{Z}^n$.
Comparing numerical dimensionsMar 02 2011Aug 20 2015The numerical dimension is a numerical measure of the positivity of a pseudo-effective divisor $L$. There are several proposed definitions of the numerical dimension due to Nakayama (2004) and Boucksom et al. (2004). We prove the equality of these notions ... More
On Eckl's pseudo-effective reduction mapMar 05 2011Sep 21 2011Suppose that X is a complex projective variety and L is a pseudo-effective divisor. A numerical reduction map is a quotient of X by all subvarieties along which L is numerically trivial. We construct two variants: the L-trivial reduction map and the pseudo-effective ... More
Free Resolutions of Fat Point Ideals on $P^2$Sep 01 1995Jun 18 1998By defining a fat point subscheme of $P^2$ to be a 0-dimensional subscheme defined by a sheaf of integrally closed ideals one extends the notion of fat point subschemes to allow infinitely near points. With this notion of fat points, this preprint determines ... More
The Completion of the Manifold of Riemannian Metrics with Respect to its $L^2$ MetricApr 01 2009This is the author's Ph.D. thesis, submitted to the University of Leipzig. It deals with the $L^2$ Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold. The main body of the thesis is a description ... More
Embedded minimal disks with prescribed curvature blowupAug 05 2004We construct a sequence of compact embedded minimal disks in a ball in Euclidean 3-space, whose boundaries lie in the boundary of the ball, such that the curvature blows up only at a prescribed discrete (and hence, finite) set of points on the x_3-axis. ... More
Curvatures of embedded minimal disks blow up on subsets of C^1 curvesMar 29 2011Feb 19 2015Any sequence of properly embedded minimal disks in an open subset U of Euclidean 3-space has a subsequence such that the curvatures blow up on a relatively closed subset K of U and such that the disks converge in the complement of K to a minimal lamination ... More
Sobolev spaces associated to singular and fractional Radon transformsMar 02 2015May 13 2016The purpose of this paper is to study the smoothing properties (in $L^p$ Sobolev spaces) of operators of the form $f\mapsto \psi(x) \int f(\gamma_t(x)) K(t)\: dt$, where $\gamma_t(x)$ is a $C^\infty$ function defined on a neighborhood of the origin in ... More
Geodesics, distance, and the CAT(0) property for the manifold of Riemannian metricsNov 05 2010Jul 27 2011Given a fixed closed manifold M, we exhibit an explicit formula for the distance function of the canonical L^2 Riemannian metric on the manifold of all smooth Riemannian metrics on M. Additionally, we examine the (metric) completion of the manifold of ... More
Multi-parameter singular Radon transforms I: the $L^2$ theoryMay 24 2010Apr 13 2011The purpose of this paper is to study the $L^2$ boundedness of operators of the form \[ f\mapsto \psi(x) \int f(\gamma_t(x)) K(t) dt, \] where $\gamma_t(x)$ is a $C^\infty$ function defined on a neighborhood of the origin in $(t,x)\in \R^N\times \R^n$, ... More
An Algebra Containing the Two-Sided Convolution OperatorsFeb 13 2008We present an intrinsically defined algebra of operators containing the right and left invariant Calder\'on-Zygmund operators on a stratified group. The operators in our algebra are pseudolocal and bounded on L^p (1<p<\infty). This algebra provides an ... More
Frobenius-unstable bundles and p-curvatureSep 16 2004We use the theory of p-curvature of connections to analyze stable vector bundles of rank 2 on curves of genus 2 which pull back to unstable bundles under the Frobenius morphism. We take two approaches, first using explicit formulas for p-curvature to ... More
Special determinants in higher-rank Brill-Noether theoryAug 24 2011Continuing our previous study of modified expected dimensions for rank-2 Brill-Noether loci with prescribed special determinants, we introduce a general framework which applies a priori for arbitrary rank, and use it to prove modified expected dimension ... More
Predicting the Integer Decomposition Property via Machine LearningJul 23 2018In this paper we investigate the ability of a neural network to approximate algebraic properties associated to lattice simplices. In particular we attempt to predict the distribution of Hilbert basis elements in the fundamental parallelepiped, from which ... More
Translated points for prequantization spaces over monotone toric manifoldsNov 25 2018Mar 19 2019We prove a version of Sandon's conjecture on the number of translated points of contactomorphisms for the case of a prequantization bundle over a closed monotone toric manifold. Namely we show that any contactomorphism of this prequantization bundle lying ... More
A transport theorem for nonconvecting open sets on an embedded manifoldAug 24 2018Most transport theorems---that is, a formula for the rate of change of an integral in which both the integrand and domain of integration depend on time---involve domains that evolve according to a flow map. Such domains are said to be convecting. Here ... More
Coconuts and Islanders: A Statistics-First Guide to the Boltzmann DistributionApr 07 2019The Boltzmann distribution is one of the key equations of thermal physics and is widely used in machine learning as well. Here I derive a Boltzmann distribution in a simple pedagogical example using only tools from a first-year probability course. The ... More
On the Bumpy Metrics Theorem for Minimal SubmanifoldsMar 05 2015This paper proves several natural generalizations of the theorem that for a generic, $C^k$ Riemannian metric on a smooth manifold, there are no closed, embedded, minimal submanifolds with nontrivial jacobi fields.
A Regression-based Adjusted Plus-Minus Statistic for NHL PlayersJun 22 2010Nov 01 2010The goal of this paper is to develop an adjusted plus-minus statistic for NHL players that is independent of both teammates and opponents. We use data from the shift reports on NHL.com in a weighted least squares regression to estimate an NHL player's ... More
Existence and regularity of multivalued solutions to elliptic equations and systemsSep 24 2013We extend the work of Simon and Wickramasekera, who constructed a large class of $C^{1,\mu}$ multivalued solutions to the minimal surface equation, to produce $C^{1,\mu}$ multivalued solutions to more general classes of elliptic equations and systems, ... More
Lectures on Minimal Surface TheoryAug 15 2013Jan 17 2016An article based on a four-lecture introductory minicourse on minimal surface theory given at the 2013 summer program of the Institute for Advanced Study and the Park City Mathematics Institute.