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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

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

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

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

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

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

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

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

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

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

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

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

Regularity of minimal hypersurfaces with a common free boundarySep 24 2013Let $N$ be a Riemannian manifold and consider a stationary union of three or more $C^{1,\mu}$ hypersurfaces-with-boundary $M_k$ in $N$ with a common boundary $\Gamma$. We show that if $N$ is smooth, then $\Gamma$ is smooth and each $M_k$ is smooth up ... 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

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

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.

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

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

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

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

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

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

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

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

Connectedness of Brill-Noether loci via degenerationsFeb 22 2017We show that limit linear series spaces for chains of curves are reduced. Using new advances in the foundations of limit linear series, we then use degenerations to study the question of connectedness for spaces of linear series with imposed ramification ... 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.

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

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

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

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

Vertices of Gelfand-Tsetlin PolytopesSep 19 2003Sep 23 2003This paper is a study of the polyhedral geometry of Gelfand-Tsetlin patterns arising in the representation theory $\mathfrak{gl}_n \C$ and algebraic combinatorics. We present a combinatorial characterization of the vertices and a method to calculate the ... More

Regularized Stokeslet rings - an efficient method for axisymmetric Stokes flow, with application to the growing pollen tubeFeb 27 2019The method of regularized Stokeslets, based on the divergence-free exact solution to the equations of highly viscous flow due to a spatially-smoothed concentrated force, is widely employed in biological fluid mechanics. Many problems of interest are axisymmetric, ... More

The Minimum Period of the Ehrhart Quasi-polynomial of a Rational PolytopeOct 16 2003If $P\subset \R^d$ is a rational polytope, then $i_P(n):=#(nP\cap \Z^d)$ is a quasi-polynomial in $n$, called the Ehrhart quasi-polynomial of $P$. The period of $i_P(n)$ must divide $\LL(P)= \min \{n \in \Z_{> 0} \colon nP \text{is an integral polytope}\}$. ... 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

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

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

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

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

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

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

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

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).

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

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

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

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

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.

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

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

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

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

*-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

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

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

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

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

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

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

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