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A detectability criterion and data assimilation for non-linear differential equationsNov 14 2017In this paper we propose a new sequential data assimilation method for non-linear ordinary differential equations with compact state space. The method is designed so that the Lyapunov exponents of the corresponding estimation error dynamics are negative, ... More

Auslander's Theorem for permutation actions on noncommutative algebrasApr 28 2017Sep 11 2018When $A = \mathbb{k}[x_1, \ldots, x_n]$ and $G$ is a small subgroup of $\operatorname{GL}_n(\mathbb{k})$, Auslander's Theorem says that the skew group algebra $A \# G$ is isomorphic to $\operatorname{End}_{A^G}(A)$ as graded algebras. We prove a generalization ... More

Arithmetic complexity via effective names for random sequencesAug 28 2010Aug 12 2014We investigate enumerability properties for classes of sets which permit recursive, lexicographically increasing approximations, or left-r.e. sets. In addition to pinpointing the complexity of left-r.e. Martin-L\"{o}f, computably, Schnorr, and Kurtz random ... More

How to verify computation with a rational networkJun 19 2016The present paper introduces a practical protocol for provably secure, outsourced computation. Our protocol minimizes overhead for verification by requiring solutions to withstand an interactive game between a prover and challenger. For optimization problems, ... More

The Fermi Paradox and the Aurora Effect: Exo-civilization Settlement, Expansion and Steady StatesFeb 12 2019We model the settlement of the galaxy by space-faring civilizations in order to address issues related to the Fermi Paradox. We explore the problem in a way that avoids assumptions about the intent and motivation of any exo-civilization seeking to settle ... More

The Creation of AGB Fallback ShellsMay 12 2015Jan 22 2016The possibility that mass ejected during Asymptotic Giant Branch (AGB) stellar evolution phases falls back towards the star has been suggested in applications ranging from the formation of accretion disks to the powering of late-thermal pulses. In this ... More

Things that can be made into themselvesAug 03 2012Feb 12 2014One says that a property $P$ of sets of natural numbers can be made into itself iff there is a numbering $\alpha_0,\alpha_1,\ldots$ of all left-r.e. sets such that the index set $\{e: \alpha_e$ satisfies $P\}$ has the property $P$ as well. For example, ... More

Energy Budget and Core-Envelope Motion in Common Envelope EvolutionDec 28 2018Mar 25 2019We analyze a 3D hydrodynamic simulation of common envelope evolution to understand how energy is transferred between various forms and whether theory and simulation are mutually consistent given the setup. Virtually all of the envelope unbinding in the ... More

Extremal Graphs Without 4-CyclesJan 24 2012We prove an upper bound for the number of edges a C4-free graph on q^2 + q vertices can contain for q even. This upper bound is achieved whenever there is an orthogonal polarity graph of a plane of even order q.

Projected Shadowing-based Data AssimilationJul 28 2017In this article we develop algorithms for data assimilation based upon a computational time dependent stable/unstable splitting. Our particular method is based upon shadowing refinement and synchronization techniques and is motivated by work on Assimilation ... More

An Intriguing Failing of Convolutional Neural Networks and the CoordConv SolutionJul 09 2018Dec 03 2018Few ideas have enjoyed as large an impact on deep learning as convolution. For any problem involving pixels or spatial representations, common intuition holds that convolutional neural networks may be appropriate. In this paper we show a striking counterexample ... More

Mass Transfer and Disc Formation in AGB Binary SystemsFeb 20 2017Apr 13 2017We investigate mass transfer and the formation of disc in binary systems using a combination of numerical simulations and theory. We consider six models distinguished by binary separation, secondary mass and outflow mechanisms. Each system consists of ... More

Energy Budget and Core-Envelope Motion in Common Envelope EvolutionDec 28 2018Mar 01 2019We analyze a 3D hydrodynamic simulation of common envelope evolution that unbinds $13$-$14\%$ of the envelope, to understand how energy is transferred between various forms and whether theory and simulation are mutually consistent given the setup. Virtually ... More

Derivation of Delay Equation Climate Models Using the Mori-Zwanzig FormalismFeb 08 2019Models incorporating delay have been frequently used to understand climate variability phenomena, but often the delay is introduced through an ad-hoc physical reasoning, such as the propagation time of waves. In this paper, the Mori-Zwanzig formalism ... More

Planetary Nebulae Shaped By Common Envelope EvolutionJul 16 2018The morphologies of planetary nebula have long been believed to be due to wind shaping processes in which a fast wind from the central star impacts a previously ejected envelope. Asymmetries assumed to exist in the slow wind envelope lead to inertial ... More

Energy Budget and Core-Envelope Motion in Common Envelope EvolutionDec 28 2018Feb 23 2019We analyze a 3D hydrodynamic simulation of common envelope evolution that unbinds $13$-$14\%$ of the envelope, to understand how energy is transferred between various forms and whether theory and simulation are mutually consistent given the setup. Virtually ... More

Wind-accelerated orbital evolution in binary systems with giant starsMay 04 2017Aug 31 2017Using 3D radiation-hydrodynamic simulations and analytic theory, we study the orbital evolution of asymptotic-giant-branch (AGB) binary systems for various initial orbital separations and mass ratios, and thus different initial accretion modes. The time ... More

Three-dimensional hydrodynamic simulations of L2 PuppisFeb 19 2016Oct 04 2016Recent observations of the L2 Puppis system suggest that this Mira-like variable may be in the early stages of forming a bipolar planetary nebula (PN). As one of nearest and brightest AGB stars, thought be a binary, L2 Puppis serves as a benchmark object ... More

Accretion in Common Envelope EvolutionOct 10 2018Common envelope evolution (CEE) occurs in some binary systems involving asymptotic giant branch (AGB) or red giant branch (RGB) stars, and understanding this process is crucial for understanding the origins of various transient phenomena. CEE has been ... More

Accretion in common envelope evolutionMay 09 2018Common envelope evolution (CEE) is presently a poorly understood, yet critical, process in binary stellar evolution. Characterizing the full 3D dynamics of CEE is difficult in part because simulating CEE is so computationally demanding. Numerical studies ... More

A new cohomological formula for helicity in $\R^{2k+1}$ reveals the effect of a diffeomorphism on helicityMar 08 2009Oct 26 2012The helicity of a vector field is a measure of the average linking of pairs of integral curves of the field. Computed by a six-dimensional integral, it is widely useful in the physics of fluids. For a divergence-free field tangent to the boundary of a ... More

Direct control of the small-scale energy balance in 2D fluid dynamicsDec 19 2014We explore the direct modification of the pseudo-spectral truncation of 2D, incompressible fluid dynamics to maintain a prescribed kinetic energy spectrum. The method provides a means of simulating fluid states with defined spectral properties, for the ... More

On the discriminant of twisted tensor productsJun 09 2016Dec 21 2016We provide formulas for computing the discriminant of noncommutative algebras over central subalgebras in the case of Ore extensions and skew group extensions. The formulas follow from a more general result regarding the discriminants of certain twisted ... More

A Next Generation Low Band Observatory: A Community Study Exploring Low Frequency Options for ngVLAJul 31 2017Aug 10 2017We present a community study exploring the low frequency (5 - 800 MHz) options and opportunities for the ngVLA project and its infrastructure. We describe a Next Generation LOw Band Observatory (ngLOBO) that will provide access to the low frequency sky ... More

Optical Monitoring of the Broad-Line Radio Galaxy 3C390.3Aug 03 2012We have undertaken a new ground-based monitoring campaign on the BLRG 3C390.3 to improve the measurement of the size of the BLR and to estimate the black hole mass. Optical spectra and g-band images were observed in 2005 using the 2.4m telescope at MDM ... More

Least-biased correction of extended dynamical systems using observational dataNov 21 2014We consider dynamical systems evolving near an equilibrium statistical state where the interest is in modelling long term behavior that is consistent with thermodynamic constraints. We adjust the distribution using an entropy-optimizing formulation that ... More

On the discriminant of twisted tensor productsJun 09 2016We provide formulas for computing the discriminant of noncommutative algebras over central subalgebras in the case of Ore extensions and skew group extensions. The formulas follow from a more general result regarding the discriminants of certain twisted ... More

Measuring and Correcting Wind-Induced Pointing Errors of the Green Bank Telescope Using an Optical Quadrant DetectorApr 08 2011Wind-induced pointing errors are a serious concern for large-aperture high-frequency radio telescopes. In this paper, we describe the implementation of an optical quadrant detector instrument that can detect and provide a correction signal for wind-induced ... More

How powerful are integer-valued martingales?Apr 06 2010Apr 10 2010In the theory of algorithmic randomness, one of the central notions is that of computable randomness. An infinite binary sequence X is computably random if no recursive martingale (strategy) can win an infinite amount of money by betting on the values ... More

The remapped particle-mesh advection schemeJul 26 2006We describe the remapped particle-mesh method, a new mass-conserving method for solving the density equation which is suitable for combining with semi-Lagrangian methods for compressible flow applied to numerical weather prediction. In addition to the ... More

High-Performance Distributed Multi-Model / Multi-Kernel Simulations: A Case-Study in Jungle ComputingMar 01 2012High-performance scientific applications require more and more compute power. The concurrent use of multiple distributed compute resources is vital for making scientific progress. The resulting distributed system, a so-called Jungle Computing System, ... More

An Update on Monitoring Stellar Orbits in the Galactic CenterNov 28 2016Using 25 years of data from uninterrupted monitoring of stellar orbits in the Galactic Center, we present an update of the main results from this unique data set: A measurement of mass of and distance to SgrA*. Our progress is not only due to the eight ... More

Dialog-based Language LearningApr 20 2016Aug 23 2016A long-term goal of machine learning research is to build an intelligent dialog agent. Most research in natural language understanding has focused on learning from fixed training sets of labeled data, with supervision either at the word level (tagging, ... More

The geometry of cyclic hyperbolic polygonsJan 25 2011Jun 30 2015A hyperbolic polygon is defined to be cyclic, horocyclic, or equidistant if its vertices lie on a metric circle, horocycle, or a component of the equidistant locus to a hyperbolic geodesic, respectively. Convex such $n$-gons are parametrized by the subspaces ... More

A sharp analysis of the mixing time for random walk on rooted treesAug 08 2009We define an analog of Plancherel measure for the set of rooted unlabeled trees on n vertices, and a Markov chain which has this measure as its stationary distribution. Using the combinatorics of commutation relations, we show that order n^2 steps are ... More

Commutation relations and Markov chainsDec 09 2007Jan 20 2008It is shown that the combinatorics of commutation relations is well suited for analyzing the convergence rate of certain Markov chains. Examples studied include random walk on irreducible representations, a local random walk on partitions whose stationary ... More

An Inductive Proof of the Berry-Esseen Theorem for Character RatiosMar 11 2005Aug 08 2006Bolthausen used a variation of Stein's method to give an inductive proof of the Berry-Esseen theorem for sums of independent, identically distributed random variables. We modify this technique to prove a Berry-Esseen theorem for character ratios of a ... More

Stein's Method and Minimum Parsimony Distance after ShufflesOct 29 2004Motivated by Bourque and Pevzner's simulation study of the parsimony method for studying genome rearrangement, Berestycki and Durrett used techniques from random graph theory to prove that the minimum parsimony distance after iterating the random transposition ... More

A New Bound for Kloosterman SumsMay 22 2001Jun 21 2001We give generating functions for Gauss sums for finite general linear and unitary groups. For the general linear case only our method of proof is new, but we deduce a bound on Kloosterman sums which is sometimes sharper than Deligne's bound from algebraic ... More

Applications of the Brauer complex: card shuffling, permutation statistics, and dynamical systemsFeb 14 2001May 09 2001By algebraic group theory, there is a map from the semisimple conjugacy classes of a finite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on conjugacy classes ... More

A Probabilistic Proof of the Rogers Ramanujan IdentitiesJan 13 2000Nov 21 2000The asymptotic probability theory of conjugacy classes of the finite general linear and unitary groups leads to a probability measure on the set of all partitions of natural numbers. A simple method of understanding these measures in terms of Markov chains ... More

Stein's method, heat kernel, and traces of powers of elements of compact Lie groupsMay 07 2010Combining Stein's method with heat kernel techniques, we show that the trace of the jth power of an element of U(n,C), USp(n,C) or SO(n,R) has a normal limit with error term of order j/n. In contrast to previous works, here j may be growing with n. The ... More

Bounding the area of a centered dual two-cell below, given lower bounds on its side lengthsAug 31 2015Suppose $C$ is a compact, $n$-edged two-cell of the centered dual decomposition of a locally finite set in the hyperbolic plane, a coarsening of the Delaunay tessellation which was introduced in the author's prior work. We describe an effectively computable ... More

Covariant bandlimitation from Generalized Uncertainty PrinciplesMar 28 2019It is widely believed that combining the uncertainty principle with gravity will lead to an effective minimum length scale. A particular challenge is to specify this scale in a coordinate-independent manner so that covariance is not broken. Here we examine ... More

Coupling the Dirac and Einstein equations through geometryMar 28 2019We show that the Clifford bundle over a curved spacetime can be used as framework in which both the Dirac and the Einstein equations can be obtained. These equations, and their coupling, follow from the variational principle applied to a Lagrangian constructed ... More

Speculative Parallel Evaluation Of Classification Trees On GPGPU Compute EnginesNov 06 2011We examine the problem of optimizing classification tree evaluation for on-line and real-time applications by using GPUs. Looking at trees with continuous attributes often used in image segmentation, we first put the existing algorithms for serial and ... More

On the close interaction between algorithmic randomness and constructive/computable measure theoryDec 08 2018Mar 16 2019This is a survey of constructive and computable measure theory with an emphasis on the close connections with algorithmic randomness. We give a brief history of constructive measure theory from Brouwer to the present, emphasizing how Schnorr randomness ... More

New Electrochemical Characterization Methods for Nanocomposite Supercapacitor ElectrodesJun 02 2014Apr 19 2016Supercapacitor electrodes fabricated from a nanocomposite consisting of multiwall carbon nanotubes and titanium oxide nanoparticles were characterized electrochemically. Conventional electrochemical characterizations cyclic voltammetry and galvanostatic ... More

Global Strichartz Estimates for Solutions to the Wave Equation Exterior to a Convex ObstacleOct 15 2002Jul 20 2004In this paper, we show that certain local Strichartz estimates for solutions of the wave equation exterior to a convex obstacle can be extended to estimates that are global in both space and time. This extends the work that was done previously by H. Smith ... More

Global existence for semilinear wave equations exterior to nontrapping obstaclesOct 16 2002Feb 21 2003In this paper, we show global existence, in spatial dimensions greater than or equal to four, for semilinear wave equations with quadratic nonlinearities exterior to a nontrapping obstacle. This extends the previous work of Shibata-Tsutsumi and Hayashi. ... More

Efficient Normal-Form Parsing for Combinatory Categorial GrammarJun 02 1996Under categorial grammars that have powerful rules like composition, a simple n-word sentence can have exponentially many parses. Generating all parses is inefficient and obscures whatever true semantic ambiguities are in the input. This paper addresses ... More

A Survey of Exoplanetary Detection TechniquesMay 07 2018Exoplanets, or planets outside our own solar system, have long been of interest to astronomers; however, only in the past two decades have scientists had the technology to characterize and study planets so far away from us. With advanced telescopes and ... More

Polynomial Bridgeland Stable Objects and Reflexive SheavesDec 19 2011Aug 01 2012On a smooth projective threefold, we show that there are only two isomorphism types for the moduli of stable objects with respect to Bayer's standard polynomial Bridgeland stability - the moduli of Gieseker-stable sheaves and the moduli of PT-stable objects ... More

Isomorphisms of some quantum spacesOct 31 2012May 16 2013We consider a series of questions that grew out of determining when two quantum planes are isomorphic. In particular, we consider a similar question for quantum matrix algebras and certain ambiskew polynomial rings. Additionally, we modify a result by ... More

Rationality and the Jordan-Gatti-Viniberghi decompositionJan 19 2012Sep 14 2012We verify the conjecture of [10] and use it to prove that the semisimple parts of the rational Jordan-Kac-Vinberg decompositions of a rational vector all lie in a single rational orbit.

Fluctuations for the Ginzburg-Landau $\nabla φ$ Interface Model on a Bounded DomainFeb 02 2010Jun 08 2010We study the massless field on $D_n = D \cap \tfrac{1}{n} \Z^2$, where $D \subseteq \R^2$ is a bounded domain with smooth boundary, with Hamiltonian $\CH(h) = \sum_{x \sim y} \CV(h(x) - h(y))$. The interaction $\CV$ is assumed to be symmetric and uniformly ... More

Dynamics of the evolving Bolthausen-Sznitman coalescentDec 12 2011Consider a population of fixed size that evolves over time. At each time, the genealogical structure of the population can be described by a coalescent tree whose branches are traced back to the most recent common ancestor of the population. As time goes ... More

Rigorous results for a population model with selection I: evolution of the fitness distributionJul 01 2015We consider a model of a population of fixed size $N$ undergoing selection. Each individual acquires beneficial mutations at rate $\mu_N$, and each beneficial mutation increases the individual's fitness by $s_N$. Each individual dies at rate one, and ... More

The distribution of r-free numbers in arithmetic progressionsFeb 25 2013A positive integer n is called r-free if n is not divisible by the r-th power of a prime. Generalizing earlier work of Orr, we provide an upper bound of Bombieri-Vinogradov type for the r-free numbers in arithmetic progressions.

Denominators and Differences of Boundary Slopes for (1,1)-KnotsJan 26 2013Jul 23 2014We show that every nonzero integer occurs in the denominator of a boundary slope for infinitely many (1,1)-knots and that infinitely many (1,1)-knots have boundary slopes of arbitrarily small difference. Specifically, we prove that for any integers m, ... More

The exponential of the spin representation of the Lorentz algebraJan 29 2012As discussed in a previous article, any (real) Lorentz algebra element possess a unique orthogonal decomposition as a sum of two mutually annihilating decomposable Lorentz algebra elements. In this article, this concept is extended to the spin representation ... More

Rotations in three, four, and five dimensionsMar 08 2011The geometry of rotations in dimensions 3, 4, and 5 is discussed using the matrix exponential map. Explicit closed formulas for the exponential of an antisymmetric matrix, as well as the logarithm of a rotation, are given for these dimensions.

Forces on a Clifford bundleMar 29 2019In a companion article, the Clifford bundle over spacetime was used as a geometric framework for obtaining coupled Dirac and Einstein equations. Other forces may be incorporated using minimal coupling. Here the fundamental forces that are allowed within ... More

Learning Parameters for Weighted Matrix Completion via Empirical EstimationDec 31 2014Apr 02 2015Recently theoretical guarantees have been obtained for matrix completion in the non-uniform sampling regime. In particular, if the sampling distribution aligns with the underlying matrix's leverage scores, then with high probability nuclear norm minimization ... More

Iterative Hard Thresholding for Weighted Sparse ApproximationDec 12 2013Jan 07 2015Recent work by Rauhut and Ward developed a notion of weighted sparsity and a corresponding notion of Restricted Isometry Property for the space of weighted sparse signals. Using these notions, we pose a best weighted sparse approximation problem, i.e. ... More

Weak Convergence of the Scaled Median of Independent Brownian MotionsJul 26 2005Aug 02 2006We consider the median of n independent Brownian motions, and show that this process, when properly scaled, converges weakly to a centered Gaussian process. The chief difficulty is establishing tightness, which is proved through direct estimates on the ... More

Every AF-algebra is Morita equivalent to a graph algebraOct 27 2003We show how to modify any Bratteli diagram $E$ for an AF-algebra $A$ to obtain a Bratteli diagram $KE$ for $A$ whose graph algebra $C^*(KE)$ contains both $A$ and $C^*(E)$ as full corners.

Contextuality from missing and versioned dataAug 10 2017Traditionally categorical data analysis (e.g. generalized linear models) works with simple, flat datasets akin to a single table in a database with no notion of missing data or conflicting versions. In contrast, modern data analysis must deal with distributed ... More

The loop-erased random walk and the uniform spanning tree on the four-dimensional discrete torusFeb 23 2006Jul 29 2007Let x and y be points chosen uniformly at random from $\Z_n^4$, the four-dimensional discrete torus with side length n. We show that the length of the loop-erased random walk from x to y is of order $n^2 (\log n)^{1/6}$, resolving a conjecture of Benjamini ... More

The Rogers-Ramanujan Identities, the Finite General Linear Groups, and the Hall-Littlewood PolynomialsDec 09 1997The Rogers-Ramanujan identities have been studied from the viewpoints of combinatorics, number theory, affine Lie algebras, statistical mechanics, and quantum field theory. This note connects the Rogers-Ramanujan identities with the finite general linear ... More

Fourier-Mukai transforms of slope stable torsion-free sheaves and stable 1-dimensional sheaves on Weierstrass elliptic threefoldsOct 10 2017We focus on a class of Weierstrass elliptic threefolds that allows the base of the fibration to be a Fano surface or a numerically $K$-trivial surface. In the first half of this article, we define the notion of limit tilt stability, which is closely related ... More

T-structures on elliptic fibrationsSep 10 2015We consider t-structures that naturally arise on elliptic fibrations. By filtering the category of coherent sheaves on an elliptic fibration using the torsion pairs corresponding to these t-structures, we prove results describing equivalences of t-structures ... More

Moduli of PT-semistable objects IINov 29 2010May 04 2011We generalise the techniques of semistable reduction for flat families of sheaves to the setting of the derived category $D^b(X)$ of coherent sheaves on a smooth projective three-fold $X$. Then we construct the moduli of PT-semistable objects in $D^b(X)$ ... More

The number of small blocks in exchangeable random partitionsNov 09 2009Jul 12 2010Suppose $\Pi$ is an exchangeable random partition of the positive integers and $\Pi_n$ is its restriction to $\{1, ..., n\}$. Let $K_n$ denote the number of blocks of $\Pi_n$, and let $K_{n,r}$ denote the number of blocks of $\Pi_n$ containing $r$ integers. ... More

Rigorous results for a population model with selection II: genealogy of the populationJul 01 2015We consider a model of a population of fixed size $N$ undergoing selection. Each individual acquires beneficial mutations at rate $\mu_N$, and each beneficial mutation increases the individual's fitness by $s_N$. Each individual dies at rate one, and ... More

A Polynomial Bound on the Regularity of an Ideal in Terms of Half of the SyzygiesDec 01 2011Let K be a field and let S = K[x_1, ..., x_n] be a polynomial ring. Consider a homogenous ideal I in S. Let t_i denote reg(Tor_i (S/I, K)), the maximal degree of an ith syzygy of S/I. We prove bounds on the numbers t_i for i > n/2 purely in terms of the ... More

The minimum rank problem over finite fieldsJan 18 2008The structure of all graphs having minimum rank at most k over a finite field with q elements is characterized for any possible k and q. A strong connection between this characterization and polarities of projective geometries is explained. Using this ... More

Stillman's Question for Exterior Algebras and Herzog's Conjecture on Betti Numbers of Syzygy ModulesJul 30 2013Jan 24 2018Let K be a field of characteristic 0 and consider exterior algebras of finite dimensional K-vector spaces. In this short paper we exhibit principal quadric ideals in a family whose Castelnuovo-Mumford regularity is unbounded. This negatively answers the ... More

Inter-critical NLS: critical $\dot{H}^s$-bounds imply scatteringSep 20 2012We consider a class of power-type nonlinear Schr\"odinger equations for which the power of the nonlinearity lies between the mass- and energy-critical exponents. Following the concentration-compactness approach, we prove that if a solution $u$ is bounded ... More

A Minimal Subsystem of the Kari-Culik TilingsOct 06 2014Sep 29 2015The Kari-Culik tilings are formed from a set of 13 Wang tiles that tile the plane only aperiodically. They are the smallest known set of Wang tiles to do so and are not as well understood as other examples of aperiodic Wang tiles. We show that the $\mathbb{Z}^2$ ... More

The combinatorics of biased riffle shufflesDec 09 1997This paper studies biased riffle shuffles, first defined by Diaconis, Fill, and Pitman. These shuffles generalize the well-studied Gilbert-Shannon-Reeds shuffle and convolve nicely. An upper bound is given for the time for these shuffles to converge to ... More

On the close interaction between algorithmic randomness and constructive/computable measure theoryDec 08 2018This is a survey of constructive and computable measure theory with an emphasis on the close connections with algorithmic randomness. We give a brief history of constructive measure theory from Brouwer to the present, emphasizing how Schnorr randomness ... More

The Kodaira dimension of spaces of rational curves on low degree hypersurfacesMay 29 2003For a hypersurface in complex projective space $X\subset \PP^n$, we investigate the singularities and Kodaira dimension of the Kontsevich moduli spaces $\Kbm{0,0}{X,e}$ parametrizing rational curves of degree $e$ on $X$. If $d+e \leq n$ and $X$ is a general ... More

Construction of Regular Non-Atomic Strictly-Positive Measures in Second-Countable Locally Compact Non-Atomic Hausdorff SpacesJul 14 2018Feb 21 2019This paper presents a constructive proof of the existence of a regular non-atomic strictly-positive measure on any second-countable locally compact non-atomic Hausdorff space. This construction involves a sequence of finitely-additive set functions defined ... More

Free Diffusions and Property AOJul 07 2009We consider von Neumann algebras generated by the stationary laws of free stochastic differential equations of the form $dX_t = dS_t -1/2 DV(X_t)$ for a suitably convex multivariate noncommutative polynomial $V$. Using techniques of Guionnet and Shlyakhtenko, ... More

Jørgensen Number and ArithmeticityMay 08 2009A J{\o}rgensen group is a non-elementary Kleinian group that can be generated by two elements for which equality holds in J{\o}rgensen's Inequality. This paper shows that the only torsion-free J{\o}rgensen group is the figure-eight knot group, identifies ... More

When does randomness come from randomness?Aug 20 2015Mar 08 2016A result of Shen says that if $F\colon2^{\mathbb{N}}\rightarrow2^{\mathbb{N}}$ is an almost-everywhere computable, measure-preserving transformation, and $y\in2^{\mathbb{N}}$ is Martin-L\"of random, then there is a Martin-L\"of random $x\in2^{\mathbb{N}}$ ... More

Improving on bold play when the gambler is restrictedDec 18 2004Suppose a gambler starts with a fortune in (0,1) and wishes to attain a fortune of 1 by making a sequence of bets. Assume thay whenever the gambler stakes the amount s, the gambler's fortune increases by s with probability w and decreases by s with probability ... More

Totally geodesic surfaces and homologyJan 23 2006Apr 23 2009We construct examples of hyperbolic rational homology spheres and hyperbolic knot complements in rational homology spheres containing closed embedded totally geodesic surfaces.

Generalized equivariant homology on simplicial complexesMar 08 2011A careful account is given of generalized equivariant homology theories on the category of topological pairs acted on by a group. In particular, upon restriction to the category of equivariant simplicial complexes, the equivalence of equivariant simplicial ... More

A relation between higher-rank PT stable objects and quotients of coherent sheavesOct 02 2018On a smooth projective threefold, we construct an essentially surjective functor $\mathcal{F}$ from a category of two-term complexes to a category of quotients of coherent sheaves, and describe the fibers of this functor. Under a coprime assumption on ... More

Stability and Fourier-Mukai transforms on elliptic fibrationsJun 19 2012Jan 17 2014We systematically develop Bridgeland's and Bridgeland-Maciocia's techniques for studying elliptic fibrations, and identify criteria that ensure 2-term complexes are mapped to torsion-free sheaves under a Fourier-Mukai transform. As an application, we ... More

Stein's Method and Non-Reversible Markov ChainsDec 09 1997Aug 17 2004Let W be either the number of descents or inversions of a permutation. Stein's method is applied to show that W satisfies a central limit theorem with error rate n^(-1/2). The construction of an exchangeable pair (W,W') used in Stein's method is non-trivial ... More

On some moduli of complexes on K3 surfacesMar 07 2012We consider moduli stacks of Bridgeland semistable objects that previously had only set-theoretic identifications with Uhlenbeck compactification spaces. On a K3 surface $X$, we give examples where such a moduli stack is isomorphic to a moduli stack of ... More

Fourier-Mukai transforms of slope stable torsion-free sheaves on a product elliptic threefoldOct 09 2017On the product elliptic threefold $X = C \times S$ where $C$ is an elliptic curve and $S$ is a K3 surface of Picard rank 1, we define a notion of limit tilt stability, which satisfies the Harder-Narasimhan property. We show that under the Fourier-Mukai ... More

Relations among conditional probabilitiesAug 08 2008We describe a Groebner basis of relations among conditional probabilities in a discrete probability space, with any set of conditioned-upon events. They may be specialized to the partially-observed random variable case, the purely conditional case, and ... More

Descent algebras, hyperplane arrangements, and shuffling cardsJan 20 1998Jul 15 1999Two notions of riffle shuffling on finite Coxeter groups are given: one using Solomon's descent algebra and another using random walk on chambers of hyperplane arrangements. These coincide for types $A$,$B$,$C$, $H_3$, and rank two groups. Both notions ... More

The Doorways Problem and Sturmian WordsMay 01 2017The doorways problem considers adjacent parallel hallways of unit width each with a single doorway (aligned with integer lattice points) of unit width. It then asks, what are the properties of lines that pass through each doorway? Configurations of doorways ... More

On the maximal graded shifts of ideals and modulesJan 24 2018We generalize a result of Eisenbud-Huneke-Ulrich on the maximal graded shifts of a module with prescribed annihilator and prove a linear regularity bound for ideals in a polynomial ring depending only on the first $p - c$ steps in the resolution, where ... More

The Delaunay tessellation in hyperbolic spaceAug 22 2013Aug 05 2016The Delaunay tessellation of a locally finite subset of hyperbolic space is constructed using convex hulls in Euclidean space of one higher dimension. For finite and lattice-invariant sets it is proven to be a polyhedral decomposition, and versions (necessarily ... More