Results for "Brian Tyrrell"

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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
A New Universal Definition of $\mathbb{F}_q [t]$ in $\mathbb{F}_q (t)$May 14 2019This paper gives a universal definition of $\mathbb{F}_q [t]$ in $\mathbb{F}_q (t)$ using 89 quantifiers, more direct than those that exist in the current literature. The language $\mathcal{L}_{\mbox{rings}, t}$ we consider here is the language of rings ... More
Degree bounds for type-A weight rings and Gelfand--Tsetlin semigroupsDec 03 2008Dec 06 2008A weight ring in type A is the coordinate ring of the GIT quotient of the variety of flags in $\C^n$ modulo a twisted action of the maximal torus in $\SL(n,\C)$. We show that any weight ring in type A is generated by elements of degree strictly less than ... More
"Going back to our roots": second generation biocomputingDec 16 2005Researchers in the field of biocomputing have, for many years, successfully "harvested and exploited" the natural world for inspiration in developing systems that are robust, adaptable and capable of generating novel and even "creative" solutions to human-defined ... More
Degrees of stretched Kostka coefficientsMar 07 2006May 24 2007Given a partition l and a composition b, the stretched Kostka coefficient K_{l, b}(n) is the map sending each positive integer n to the Kostka coefficient indexed by nl and nb. Kirillov and Reshetikhin (1986) have shown that stretched Kostka coefficients ... More
Coefficient functions of the Ehrhart quasi-polynomials of rational polygonsJun 03 2009In 1976, P. R. Scott characterized the Ehrhart polynomials of convex integral polygons. We study the same question for Ehrhart polynomials and quasi-polynomials of \emph{non}-integral convex polygons. Define a \emph{pseudo-integral polygon}, or \emph{PIP}, ... More
Ellipsoidal cones in normed vector spacesJan 29 2015We give two characterizations of cones over ellipsoids in real normed vector spaces. Let $C$ be a closed convex cone with nonempty interior such that $C$ has a bounded section of codimension $1$. We show that $C$ is a cone over an ellipsoid if and only ... More
Quasi-period collapse and GL_n(Z)-scissors congruence in rational polytopesSep 26 2007Quasi-period collapse occurs when the Ehrhart quasi-polynomial of a rational polytope has a quasi-period less than the denominator of that polytope. This phenomenon is poorly understood, and all known cases in which it occurs have been proven with ad ... More
Ehrhart quasi-period collapse in rational polygonsSep 11 2015In 1976, P. R. Scott characterized the Ehrhart polynomials of convex integral polygons. We study the same question for Ehrhart polynomials and quasi-polynomials of *non*-integral convex polygons. Turning to the case in which the Ehrhart quasi-polynomial ... More
Two characterizations of ellipsoidal conesApr 18 2012Mar 06 2013We give two characterizations of cones over ellipsoids. Let $C$ be a closed pointed convex linear cone in a finite-dimensional real vector space. We show that $C$ is a cone over an ellipsoid if and only if the affine span of $\partial C \cap \partial(a ... More
On the Computation of Clebsch-Gordan Coefficients and the Dilation EffectJan 25 2005We investigate the problem of computing tensor product multiplicities for complex semisimple Lie algebras. Even though computing these numbers is #P-hard in general, we show that if the rank of the Lie algebra is assumed fixed, then there is a polynomial ... More
A Bochner Formula for Harmonic Maps into Non-Positively Curved Metric SpacesMay 26 2016We study harmonic maps from Riemannian manifolds into arbitrary non-positively curved metric spaces. First we discuss the domain variation formula with special emphasis on the error terms. Expanding higher order terms of this and other formulas in terms ... More
Shift-preserving maps on $ω^*$May 04 2016The shift map $\sigma$ on $\omega^*$ is the continuous self-map of $\omega^*$ induced by the function $n \mapsto n+1$ on $\omega$. Given a compact Hausdorff space $X$ and a continuous function $f: X \rightarrow X$, we say that $(X,f)$ is a quotient of ... More
Structure of entanglement in regulated Lorentz invariant field theoriesApr 23 2013Regulated Lorentz invariant quantum field theories satisfy an area law for the entanglement entropy $S$ of a spatial subregion in the ground state in $d>1$ spatial dimensions; nevertheless, the full density matrix contains many more than $e^{S}$ non-zero ... More
Entanglement Entropy and the Fermi SurfaceAug 12 2009Mar 11 2010Free fermions with a finite Fermi surface are known to exhibit an anomalously large entanglement entropy. The leading contribution to the entanglement entropy of a region of linear size $L$ in $d$ spatial dimensions is $S\sim L^{d-1} \log{L}$, a result ... More
Entanglement does not generally decrease under renormalizationJul 30 2013Aug 01 2013Renormalization is often described as the removal or "integrating out" of high energy degrees of freedom. In the context of quantum matter, one might suspect that quantum entanglement provides a sharp way to characterize such a loss of degrees of freedom. ... More
Experimental Summary on Hadronic Decays: A TAU98 ReviewNov 16 1998Jan 21 1999Selected results on hadronic decays of the tau lepton from the TAU98 Workshop are reviewed. A comprehensive picture emerges for strange particle branching fractions, and exploration of resonant substructure of both strange and non-strange decays is seen ... More
Update on the lattice calculation of $B \to K^* γ$Nov 30 1994We summarise our current results for calculations of the form factors for $B \to K^* \gamma$, and their extrapolation to the physical b-quark mass.
Hyperbolicity of the Cyclic Splitting ComplexDec 12 2012Dec 14 2012We define a new complex on which $Out(F_n)$ acts by simplicial automorphisms, the cyclic splitting complex of $F_n$, and show that it is hyperbolic using a method developed by Kapovich and Rafi.
Evidence on the Origin of Ergospheric Disk Field Line Topology in Simulations of Black Hole AccretionSep 19 2011This Letter investigates the origin of the asymmetric magnetic field line geometry in the ergospheric disk (and the corresponding asymmetric powerful jet) in 3-D perfect magnetohydrodynamic (MHD) numerical simulations of a rapidly rotating black hole ... More
Dynamic Boundaries of Event Horizon MagnetospheresJul 20 2007This Letter analyzes 3-dimensional simulations of Kerr black hole magnetospheres that obey the general relativistic equations of perfect magnetohydrodynamics (MHD). Particular emphasis is on the event horizon magnetosphere (EHM) which is defined as the ... More
3-D Simulations of Ergospheric Disk Driven Poynting JetsApr 05 2007This Letter reports on 3-dimensional simulations of Kerr black hole magnetospheres that obey the general relativistic equations of perfect magnetohydrodynamics (MHD). In particular, we study powerful Poynting flux dominated jets that are driven from dense ... More
Evidence of the Dynamics of Relativistic Jet Launching in QuasarsApr 01 2015Hubble Space Telescope (HST) spectra of the extreme ultraviolet (EUV), the optically thick emission from the innermost accretion flow onto the central supermassive black hole, indicate that RLQs tend to be EUV weak compared to the radio quiet quasars ... More
A Multi-Component Analysis Indicates a Positronic Major Flare in GRS 1915+105Nov 21 2011Dec 21 2011A modeling strategy that is adapted to the study of synchrotron-self absorbed plasmoids that was developed for the quasar, Mrk 231, in Reynolds et al (2009) is applied to the microquasar GRS 1915+105. The major flare from December 1993 shows spectral ... More
The Redshifted Excess in Quasar C IV Broad Emission LinesFeb 25 2010In this Letter, the Evans and Koratkar Atlas of Hubble Space Telescope Faint Object Spectrograph Spectra of Active Galactic Nuclei and Quasars is used to study the redward asymmetry in CIV broad emission lines (BELs). It is concluded that there is a highly ... More
Kinetically Dominated FRII Radio SourcesOct 02 2006The existence of FR II objects that are kinetically dominated, the jet kinetic luminosity, $Q$, is larger than the total thermal luminosity (IR to X-ray) of the accretion flow, $L_{bol}$, is of profound theoretical interest. Such objects are not expected ... More
String Theory on Calabi-Yau ManifoldsFeb 23 1997These lectures are devoted to introducing some of the basic features of quantum geometry that have been emerging from compactified string theory over the last couple of years. The developments discussed include new geometric features of string theory ... More
Asymptotic behavior of the dimension of the Chow varietySep 04 2013Jan 13 2016We analyze the asymptotics of the dimension of components of the Chow variety as degree increases. By analogy with the divisor case, the main goal is to relate the asymptotic behavior with the positivity of the corresponding cycle classes. We also compute ... More
Volume-type functions for numerical cycle classesJan 13 2016A numerical equivalence class of k-cycles is said to be big if it lies in the interior of the closed cone generated by effective classes. We construct analogues for arbitrary cycle classes of the volume function for divisors which distinguishes big classes ... More
General Properties of Quiescent NovaeJun 26 2002The observed properties of novae before and after eruption are discussed. The distribution of orbital periods of novae shows a concentration near 3.2 h, which resembles that of magnetic cataclysmic variables, and there is some evidence that many of the ... More
Rapid Oscillations in Cataclysmic VariablesDec 06 2003I give an overview of the rich phenomenology of dwarf nova oscillations (DNOs) and Quasi-periodic Oscillations (QPOs) observed in cataclysmic variable stars (CVs). The favoured interpretation of these rapid brightness modulations (3 - >1000 s time scales) ... More
Globular Cluster Distance DeterminationsAug 19 1998The present status of the distance scale to Galactic globular clusters is reviewed. Six distance determination techniques which are deemed to be most reliable are discussed in depth. These different techniques are used to calibrate the absolute magnitude ... More
The Primordial Abundance of $^6$Li and $^9$beMay 31 1994Light element ($^6$Li, $^7$Li and $^9$Be) depletion isochrones for halo stars have been calculated with standard stellar evolution models. These models include the latest available opacities and are computed through the sub-giant branch. If $^6$Li is ... More
The Age of the UniverseAug 19 1998A minimum age of the universe can be estimated directly by determining the age of the oldest objects in the our Galaxy. These objects are the metal-poor stars in the halo of the Milky Way. Recent work on nucleochronology finds that the oldest stars are ... More
Formal Verification of Monad TransformersJul 13 2012We present techniques for reasoning about constructor classes that (like the monad class) fix polymorphic operations and assert polymorphic axioms. We do not require a logic with first-class type constructors, first-class polymorphism, or type quantification; ... More
Measurement of the Top Quark Mass with a Matrix Element Method in the Lepton Plus Jets Channel at CDFMay 17 2006We present a measurement of the mass of the top quark from ppbar collisions at 1.96 TeV observed with the Collider Detector at Fermilab (CDF) at the Fermilab Tevatron Run II. The events have the decay signature of ppbar to ttbar in the lepton plus jets ... More
The Cheeger Constant, Isoperimetric Problems, and Hyperbolic SurfacesSep 30 2015Jan 06 2016We give a brief literature review of the isoperimetric problem and discuss its relationship with the Cheeger constant of Riemannian $n$-manifolds. For some non-compact, finite area 2-manifolds, we prove the existence and regularity of subsets whose isoperimetric ... More
Sub-Hermitian Geometry and the Quantitative Newlander-Nirenberg TheoremOct 26 2018Jan 04 2019Given a finite collection of $C^1$ complex vector fields on a $C^2$ manifold $M$ such that they and their complex conjugates span the complexified tangent space at every point, the classical Newlander-Nirenberg theorem gives conditions on the vector fields ... More
Deformation Formulas for Parameterizable HypersurfacesNov 29 2017We investigate one-parameter deformations of functions on affine space which define parametrizable hyper surfaces. With the assumption of isolated polar activity at the origin, we are able to completely express the L\^e numbers of the special fiber in ... More
On the Evoution of the Light Elements I. D, He-3, and He-4Dec 08 1995The light elements D, \he3, \he4, and \li7 are produced in big bang nucleosynthesis and undergo changes in their abundances due to galactic processing. Since one may observe most of these elements only in contemporary environments, knowledge of the intervening ... More
A Note on Bounded Biclique Coverings of Complete GraphsNov 09 2015An undirected biclique $K_{a,b}$ is a graph with vertices partitioned into two sets: a set $A$ containing $a$ vertices and a set $B$ containing $b$ vertices such that every vertex in set $A$ is connected to every vertex in set $B$, and such that no two ... More
Development of a VO Registry Subject Ontology using Automated MethodsFeb 20 2015We report on our initial work to automate the generation of a domain ontology using subject fields of resources held in the Virtual Observatory registry. Preliminary results are comparable to more generalized ontology learning software currently in use. ... More
Constraints on Black Hole Jet Models Used As Diagnostic Tools of Event Horizon Telescope Observations of M87Jun 14 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
Seshadri constants and very ample divisors on algebraic surfacesMar 05 2001Apr 04 2001A broadly applicable geometric approach for constructing nef divisors on blow ups of algebraic surfaces at n general points is given; it works for all surfaces in all characteristics for any n. This construction is used to obtain substantial improvements ... More
Zero Spacings of Paraorthogonal Polynomials on the Unit CircleJul 02 2019We prove some new results about the spacing between neighboring zeros of paraorthogonal polynomials on the unit circle. Our methods also provide new proofs of some existing results. The main tool we will use is a formula for the phase of the appropriate ... More
Self maps of P^1 with prescribed ramification in characteristic pJul 26 2004Using limit linear series and a result controlling degeneration from separable maps to inseparable maps, we give a formula for the number of self-maps of the projective line with ramification to order e_i at general points P_i, in the case that all e_i ... More
Relative dimension of morphisms and dimension for algebraic stacksMay 27 2013Motivated by applications in moduli theory, we introduce a flexible and powerful language for expressing lower bounds on relative dimension of morphisms of schemes, and more generally of algebraic stacks. We show that the theory is robust and applies ... More
Linked Hom spacesAug 31 2010In this note, we describe a theory of linked Hom spaces which complements that of linked Grassmannians. Given two chains of vector bundles linked by maps in both directions, we give conditions for the space of homomorphisms from one chain to the other ... More
Which Ambient Spaces Admit Isoperimetric Inequalities for Submanifolds?Apr 24 2008Dec 16 2008We give simple conditions on an ambient manifold that are necessary and sufficient for isoperimetric inequalities (for submanifolds) to hold.
Algebraic bounds on analytic multiplier idealsSep 21 2011Jun 12 2013Given a pseudo-effective divisor L we construct the diminished ideal of L, a "continuous" extension of the asymptotic multiplier ideal for big divisors to the pseudo-effective boundary. For most pseudo-effective divisors L the multiplier ideal of the ... More
Asymptotics of linear systems, with connections to line arrangementsMay 28 2017The main focus of these notes is recent work on linear systems in which line arrangements play a role, including problems such as semi-effectivity, containment problems of symbolic powers of homogeneous ideals in their powers, bounded negativity, and ... More
Numerical triviality and pullbacksSep 20 2011Jan 12 2012Let f: X \to Z be a surjective morphism of smooth complex projective varieties with connected fibers. Suppose that L is a pseudo-effective divisor on X that is f-numerically trivial. We show that there is a divisor D on Z such that L is numerically equivalent ... More
Non-Rigidity of Cyclic Automorphic Orbits in Free GroupsAug 05 2011We say a subset $\Sigma \subseteq F_N$ of the free group of rank $N$ is \emph{spectrally rigid} if whenever $T_1, T_2 \in \cv_N$ are $\mathbb{R}$-trees in (unprojectivized) outer space for which $|\sigma|_{T_1} = |\sigma|_{T_2}$ for every $\sigma \in ... More
Coordinates Adapted to Vector Fields III: Real AnalyticityAug 14 2018Oct 23 2018Given a finite collection of $C^1$ vector fields on a $C^2$ manifold which span the tangent space at every point, we consider the question of when there is locally a coordinate system in which these vector fields are real analytic. We give necessary and ... More
One-Pass Graphic Approximation of Integer SequencesDec 14 2017Dec 18 2017A variety of network modeling problems begin by generating a degree sequence drawn from a given probability distribution. If the randomly generated sequence is not graphic, we give a new approach for generating a graphic approximation of the sequence. ... More
A Sufficient Condition for Graphic Sequences with Given Largest and Smallest Entries, Length, and SumJul 13 2016Jun 25 2018We give a sufficient condition for a degree sequence to be graphic based on its largest and smallest elements, length, and sum. This bound generalizes a result of Zverovich and Zverovich.
Rational Homology Manifolds and Hypersurface NormalizationsApr 25 2018Aug 10 2018We prove a criterion for determining whether the normalization of a complex analytic space on which the constant sheaf is perverse is a rational homology manifold, using a perverse sheaf known as the multiple-point complex. This perverse sheaf is naturally ... More
Topology of Kähler manifolds with weakly pseudoconvex boundaryOct 20 2011Oct 10 2018We study Kahler 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 $l=\infty$), then it has first betti number at least ... More
Regularity and convergence of 4-dimensional extremal Kahler metricsApr 16 2011May 10 2011We establish a regularity result for the metric on any 4-dimensional extremal K\"ahler manifold, and a weak compactness theorem on the space of such metrics. Specifically, the sectional curvature at a point is bounded when the quantity $L^2(|\Riem|)$ ... More
Curvature Estimates for Critical 4-Manifolds with a Lower Ricci Curvature BoundSep 12 2013We draw elliptic regularity results for 4-manifolds with an elliptic system, without Sobolev constant control. Direct use of analysis is circumvented; the results come mainly through geometric and topological arguments. In contrast to our previous paper, ... More
Energy and Asymptotics of Ricci-Flat 4-Manifolds with a Killing FieldAug 22 2013Oct 10 2018Given a complete, Ricci-flat 4-manifold with a Killing field, we give an estimate on the manifold's energy in terms of a certain asymptotic quantity of the Killing field. If the Killing field has no zeros and satisfies a certain asymptotic condition, ... More
Unlabeled Signed Graph ColoringNov 24 2015Feb 22 2018We extend the work of Hanlon on the chromatic polynomial of an unlabeled graph to define the unlabeled chromatic polynomial of an unlabeled signed graph. Explicit formulas are presented for labeled and unlabeled signed chromatic polynomials as summations ... More
Subsequent Singularities in Mean-Convex Mean Curvature FlowMar 08 2011Apr 07 2013We use Ilmanen's elliptic regularization to prove that for an initially smooth mean convex hypersurface in Euclidean n-space moving by mean curvature flow, the surface is very nearly convex in a spacetime neighborhood of every singularity. Previously ... More
The Maximum Principle for Minimal Varieties of Arbitrary CodimensionMay 31 2009Nov 12 2010We prove that an m-dimensional minimal variety in a Riemannian manifold cannot touch the boundary at a point where the sum of the smallest m principal curvatures is greater than 0. We also prove an analogous result for varieties with bounded mean curvature. ... More
Brill-Noether loci with fixed determinant in rank 2May 04 2010Aug 24 2011In the 1990's, Bertram, Feinberg and Mukai examined Brill-Noether loci for vector bundles of rank 2 with fixed canonical determinant, noting that the dimension was always bigger in this case than the naive expectation. We generalize their results to treat ... More
Zeros of non-Baxter paraorthogonal polynomials on the unit circleMay 27 2010Nov 03 2010We provide leading order asymptotics for the size of the gap in the zeros around 1 of paraothogonal polynomials on the unit circle whose Verblunsky coefficients satisfy a slow decay condition and are inside the interval (-1,0). We also include related ... More
Sharp Regularity for the Integrability of Elliptic StructuresOct 23 2018Jul 24 2019As 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
Lattice-point generating functions for free sums of convex setsJun 30 2012Mar 08 2013Let $\J$ and $\K$ be convex sets in $\R^{n}$ whose affine spans intersect at a single rational point in $\J \cap \K$, and let $\J \oplus \K = \conv(\J \cup \K)$. We give formulas for the generating function {equation*} \sigma_{\cone(\J \oplus \K)}(z_1,..., ... 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
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
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
Anticanonical Rational SurfacesSep 01 1995Jan 24 1996A determination of the fixed components, base points and irregularity is made for arbitrary numerically effective divisors on any smooth projective rational surface having an effective anticanonical divisor. All of the results are proven over an algebraically ... 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
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
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
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
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
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).
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
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
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
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
*-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