total 105163took 0.21s

Waves in Honeycomb StructuresDec 30 2012We review recent work of the authors on the non-relativistic Schr\"odinger equation with a honeycomb lattice potential, $V$. In particular, we summarize results on (i) the existence of Dirac points, conical singularities in dispersion surfaces of $H_V=-\Delta+V$ ... More

The Brenner-Hochster-Kollár and Whitney Problems for Vector-valued Functions and JetsSep 11 2012Sep 15 2012In this paper, we give analytic methods for finding m (and m+\omega) times continuously differentiable solutions of a finite system of linear equations. Along the way, we also solve a generalized Whitney problem for vector-valued functions and jets.

Honeycomb Lattice Potentials and Dirac PointsFeb 17 2012Jun 08 2012We prove that the two-dimensional Schroedinger operator with a potential having the symmetry of a honeycomb structure has dispersion surfaces with conical singularities (Dirac points) at the vertices of its Brillouin zone. No assumptions are made on the ... More

Non-conservation of dimension in divergence-free solutions of passive and active scalar systemsMay 14 2019For any $h\in(1,2]$, we give an explicit construction of a compactly supported, uniformly continuous, and (weakly) divergence-free velocity field in $\mathbb{R}^2$ that weakly advects a measure whose support is initially the origin but for positive times ... More

The Structure of Sobolev Extension OperatorsJun 09 2012Nov 12 2012Let $L^{m,p}(\R^n)$ denote the Sobolev space of functions whose $m$-th derivatives lie in $L^p(\R^n)$, and assume that $p>n$. For $E \subset \R^n$, denote by $L^{m,p}(E)$ the space of restrictions to $E$ of functions $F \in L^{m,p}(\R^n)$. It is known ... More

Edge states in honeycomb structuresJun 19 2015Sep 06 2016An edge state is a time-harmonic solution of a conservative wave system, e.g. Schroedinger, Maxwell, which is propagating (plane-wave-like) parallel to, and localized transverse to, a line-defect or "edge". Topologically protected edge states are edge ... More

Topologically Protected States in One-Dimensional SystemsMay 19 2014Apr 08 2015We study a class of periodic Schr\"odinger operators, which in distinguished cases can be proved to have linear band-crossings or "Dirac points". We then show that the introduction of an "edge", via adiabatic modulation of these periodic potentials by ... More

Sobolev Extension By Linear OperatorsMay 11 2012May 21 2012Let $L^{m,p}(\R^n)$ be the Sobolev space of functions with $m^{th}$ derivatives lying in $L^p(\R^n)$. Assume that $n< p < \infty$. For $E \subset \R^n$, let $L^{m,p}(E)$ denote the space of restrictions to $E$ of functions in $L^{m,p}(\R^n)$. We show ... More

Defect modes for dislocated periodic mediaOct 13 2018We study defect modes in a one-dimensional periodic medium with a dislocation. The model is a periodic Schrodinger operator on $\mathbb{R}$, perturbed by an adiabatic dislocation of amplitude $\delta\ll 1$. If the periodic background admits a Dirac point ... More

Simultaneous approximation in Lebesgue and Sobolev norms via eigenspacesApr 06 2019We show how to approximate functions defined on smooth bounded domains by elements of eigenspaces of linear operators (e.g. the Laplacian or the Stokes operator) in such a way that the approximations are bounded and converge in both Sobolev and Lebesgue ... More

Fitting a Sobolev function to dataNov 06 2014We exhibit an algorithm to solve the following extension problem: Given a finite set $E \subset \mathbb{R}^n$ and a function $f: E \rightarrow \mathbb{R}$, compute an extension $F$ in the Sobolev space $L^{m,p}(\mathbb{R}^n)$, $p>n$, with norm having ... More

Scalars convected by a 2D incompressible flowJan 30 2001We provide a test for numerical simulations, for several two dimensional incompressible flows, that appear to develop sharp fronts. We show that in order to have a front the velocity has to have uncontrolled velocity growth.

Ambient metric construction of Q-curvature in conformal and CR geometriesMar 16 2003Aug 21 2003We give a geometric derivation of Branson's Q-curvature in terms of the ambient metric associated with conformal structures; it naturally follows from the ambient metric construction of conformally invariant operators and can be applied to a large class ... More

Sharp finiteness principles for Lipschitz selections: long versionAug 02 2017Oct 21 2017Let $({\mathcal M},\rho)$ be a metric space and let $Y$ be a Banach space. Given a positive integer $m$, let $F$ be a set-valued mapping from ${\mathcal M}$ into the family of all compact convex subsets of $Y$ of dimension at most $m$. In this paper we ... More

Growth of solutions for QG and 2D Euler equationsJan 30 2001We study the rate of growth of sharp fronts of the Quasi-geostrophic equation and 2D incompressible Euler equations.. The development of sharp fronts are due to a mechanism that piles up level sets very fast. Under a semi-uniform collapse, we obtain a ... More

Continuous linear combinations of polynomialsMar 04 2011We give necessary and sufficient existence criteria, and methods for finding, continuous solutions of linear equations whose coefficients are polynomials.

Sharp finiteness principles for Lipschitz selectionsDec 31 2017Aug 03 2018Let $(M,\rho)$ be a metric space and let $Y$ be a Banach space. Given a positive integer $m$, let $F$ be a set-valued mapping from $M$ into the family of all compact convex subsets of $Y$ of dimension at most $m$. In this paper we prove a finiteness principle ... More

Generators for the $C^m$-closures of IdealsFeb 11 2019Let $\mathscr{R}$ denote the ring of real polynomials on $\mathbb{R}^{n}$. Fix $m\geq 0$, and let $A_{1},\cdots ,A_{M}\in \mathscr{R}$. The $ C^{m}$-closure of $\left( A_{1},\cdots ,A_{M}\right) $, denoted here by $ \left[ A_{1},\cdots ,A_{M};C^{m}\right] ... More

Juhl's Formulae for GJMS Operators and Q-CurvaturesMar 02 2012Direct proofs are given of Juhl's formulae for GJMS operators and Q-curvatures starting from the original construction of GJMS.

The ambient metricOct 04 2007Oct 22 2008This paper provides details of the construction, properties and some applications of the ambient metric associated to a conformal class of metrics on a smooth manifold. Existence and uniqueness of formal expansions defining such metrics are considered. ... More

Solutions to A System of Equations for $C^m$ FunctionsFeb 11 2019Fix $m\geq 0$, and let $A=\left( A_{ij}\left( x\right) \right) _{1\leq i\leq N,1\leq j\leq M}$ be a matrix of semialgebraic functions on $\mathbb{R}^{n}$ or on a compact subset $E \subset \mathbb{R}^n$. Given $f=\left( f_{1},\cdots ,f_{N}\right) \in C^{\infty ... More

Q-Curvature and Poincare MetricsOct 24 2001This article presents a new definition of Branson's Q-curvature in even-dimensional conformal geometry. We derive the Q-curvature as a coefficient in the asymptotic expansion of the formal solution of a boundary problem at infinity for the Laplacian in ... More

On the Whitney Extension-Interpolation-Alignment problem for almost isometries with small distortion in $\Bbb R^D$Nov 10 2014Nov 26 2017Let $D\geq 2$, $S\subset \mathbb R^D$ be finite and let $\phi:S\to \mathbb R^D$ with $\phi$ a small distortion on $S$. We solve the Whitney extension-interpolation-alignment problem of how to understand when $\phi$ can be extended to a function $\Phi:\mathbb ... More

On Smooth Whitney Extensions of almost isometries with small distortion, Interpolation and Alignment in $\Bbb R^D$-Part 1Nov 10 2014Jan 13 2018In this paper, we study the following problem: Let $D\geq 2$ and let $E\subset \mathbb R^D$ be finite satisfying certain conditions. Suppose that we are given a map $\phi:E\to \mathbb R^D$ with $\phi$ a small distortion on $E$. How can one decide whether ... More

Solutions to a System of Equations for $C^m$ FunctionsFeb 11 2019Feb 12 2019Fix $m\geq 0$, and let $A=\left( A_{ij}\left( x\right) \right) _{1\leq i\leq N,1\leq j\leq M}$ be a matrix of semialgebraic functions on $\mathbb{R}^{n}$ or on a compact subset $E \subset \mathbb{R}^n$. Given $f=\left( f_{1},\cdots ,f_{N}\right) \in C^{\infty ... More

Efficient Algorithms for Approximate Smooth SelectionMay 08 2019In this paper we provide efficient algorithms for approximate $\mathcal{C}^m(\mathbb{R}^n, \mathbb{R}^D)-$selection. In particular, given a set $E$, constants $M_0 > 0$ and $0 <\tau \leq \tau_{\max}$, and convex sets $K(x) \subset \mathbb{R}^D$ for $x ... More

Local existence for the non-resistive MHD equations in nearly optimal Sobolev spacesFeb 08 2016This paper establishes the local-in-time existence and uniqueness of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations in $\mathbb{R}^d$, $d=2,3$, with initial data $B_0\in H^s(\mathbb{R}^d)$ and $u_0\in H^{s-1+\varepsilon}(\mathbb{R}^d)$ ... More

Testing the Manifold HypothesisOct 01 2013Dec 19 2013The hypothesis that high dimensional data tend to lie in the vicinity of a low dimensional manifold is the basis of manifold learning. The goal of this paper is to develop an algorithm (with accompanying complexity guarantees) for fitting a manifold to ... More

Finiteness Principles for Smooth SelectionNov 16 2015Nov 27 2015In this paper we prove finiteness principles for $C^{m}\left( \mathbb{R}^{n}, \mathbb{R}^{D}\right) $-selection, and for $C^{m-1,1}\left( \mathbb{R}^{n}, \mathbb{R}^{D}\right) $-selection, in particular providing a proof for a conjecture of Brudyni-Shvartsman ... More

On the absence of "splash" singularities in the case of two-fluid interfacesDec 10 2013Dec 16 2013We show that "splash" singularities cannot develop in the case of locally smooth solutions of the two-fluid interface in two dimensions. More precisely, we show that the scenario of formation of singularities discovered by Castro-C\'{o}rdoba-Fefferman-Gancedo-G\'{o}mez-Serrano ... More

Interpolation of data by smooth non-negative functionsMar 07 2016We prove a finiteness principle for interpolation of data by nonnegative Cm functions. Our result raises the hope that one can start to understand constrained interpolation problems in which e.g. the interpolating function F is required to be nonnegative. ... More

Finitness Principles for Smooth SelectionMar 07 2016In this paper we prove finiteness principles for $C^{m}\left( \mathbb{R}^{n}, \mathbb{R}^{D}\right) $-selection, and for $C^{m-1,1}\left( \mathbb{R}^{n}, \mathbb{R}^{D}\right) $-selection, in particular providing a proof for a conjecture of Brudyni-Shvartsman ... More

Photonic realization of topologically protected bound states in domain-wall waveguide arraysFeb 12 2016Mar 11 2016We present an analytical theory of topologically protected photonic states for the two-dimensional Maxwell equations for a class of continuous periodic dielectric structures, modulated by a domain wall. We further numerically confirm the applicability ... More

Breakdown of smoothness for the Muskat problemJan 12 2012Jan 29 2012In this paper we show that there exist analytic initial data in the stable regime for the Muskat problem such that the solution turns to the unstable regime and later breaks down i.e. no longer belongs to $C^4$.

Regularity of coupled two-dimensional nonlinear Fokker-Planck and Navier-Stokes systemsMay 10 2006We consider systems of particles coupled with fluids. The particles are described by the evolution of their density, and the fluid is described by the Navier-Stokes equations. The particles add stress to the fluid and the fluid carries and deforms the ... More

Splash singularities for the one-phase Muskat problem in stable regimesNov 29 2013Feb 07 2015This paper shows finite time singularity formation for the Muskat problem in a stable regime. The framework we found is with a dry region, where the density and the viscosity are set equal to $0$ (the gradient of the pressure is equal to $(0,0)$) in the ... More

Reconstruction and interpolation of manifolds I: The geometric Whitney problemAug 04 2015Nov 17 2018We study the geometric Whitney problem on how a Riemannian manifold $(M,g)$ can be constructed to approximate a metric space $(X,d_X)$. This problem is closely related to manifold reconstruction where a smooth $n$-dimensional submanifold $S\subset {\mathbb ... More

Reconstruction of a Riemannian manifold from noisy intrinsic distancesMay 17 2019We consider reconstruction of a manifold, or, invariant manifold learning, where a smooth Riemannian manifold $M$ is determined from intrinsic distances (that is, geodesic distances) of points in a discrete subset of $M$. In the studied problem the Riemannian ... More

Reconstruction and interpolation of manifolds I: The geometric Whitney problemAug 04 2015We study the geometric Whitney problem on how a Riemannian manifold $(M,g)$ can be constructed to approximate a metric space $(X,d_X)$. This problem is closely related to manifold interpolation (or manifold learning) where a smooth $n$-dimensional surface ... More

Honeycomb Schroedinger operators in the strong binding regimeOct 16 2016In this article, we study the Schr\"odinger operator for a large class of periodic potentials with the symmetry of a hexagonal tiling of the plane. The potentials we consider are superpositions of localized potential wells, centered on the vertices of ... More

Splash singularities for the free boundary Navier-Stokes equationsApr 10 2015In this paper, we prove the existence of smooth initial data for the 2D free boundary incompressible Navier-Stokes equations, for which the smoothness of the interface breaks down in finite time into a splash singularity.

Bifurcations of edge states -- topologically protected and non-protected -- in continuous 2D honeycomb structuresSep 29 2015This paper summarizes and extends the authors' work on the bifurcation of topologically protected edge states in continuous two-dimensional honeycomb structures. We consider a family of Schr\"odinger Hamiltonians consisting of a bulk honeycomb potential ... More

Splash singularities for the free boundary Navier-Stokes equationsApr 10 2015May 12 2019In this paper, we prove the existence of smooth initial data for the 2D free boundary incompressible Navier-Stokes equations, for which the smoothness of the interface breaks down in finite time into a splash singularity.

Finite time singularities for the free boundary incompressible Euler equationsDec 09 2011Sep 29 2012In this paper, we prove the existence of smooth initial data for the 2D free boundary incompressible Euler equations (also known for some particular scenarios as the water wave problem), for which the smoothness of the interface breaks down in finite ... More

Turning waves and breakdown for incompressible flowsNov 27 2010We consider the evolution of an interface generated between two immiscible incompressible and irrotational fluids. Specifically we study the Muskat and water wave problems. We show that starting with a family of initial data given by $(\al,f_0(\al))$, ... More

Rayleigh-Taylor breakdown for the Muskat problem with applications to water wavesFeb 09 2011Jun 10 2011The Muskat problem models the evolution of the interface given by two different fluids in porous media. The Rayleigh-Taylor condition is natural to reach the linear stability of the Muskat problem. We show that the Rayleigh-Taylor condition may hold initially ... More

Structural stability for the splash singularities of the water waves problemJan 24 2014In this paper we show a structural stability result for water waves. The main motivation for this result is that we would like to exhibit a water wave whose interface starts as a graph and ends in a splash. Numerical simulations lead to an approximate ... More

Splash singularity for water wavesJun 10 2011Oct 03 2011We exhibit smooth initial data for the 2D water wave equation for which we prove that smoothness of the interface breaks down in finite time. Moreover, we show a stability result together with numerical evidence that there exist solutions of the 2D water ... More

Finite time singularities for water waves with surface tensionApr 30 2012Oct 19 2012Here we consider the 2D free boundary incompressible Euler equation with surface tension. We prove that the surface tension does not prevent a finite time splash or splat singularity, i.e. that the curve touches itself either in a point or along an arc. ... More

Symbolic calculus for Toeplitz operators with half-formsFeb 08 2006This paper is devoted to the use of half-form bundles in the symbolic calculus of Berezin-Toeplitz operators on Kahler manifolds. We state the Bohr-Sommerfeld conditions and relate them to the functional calculus of Toeplitz operators, a trace formula ... More

Higher order commutator estimates and local existence for the non-resistive MHD equations and related modelsJan 20 2014This paper establishes the local-in-time existence and uniqueness of strong solutions in $H^{s}$ for $s > n/2$ to the viscous, non-resistive magnetohydrodynamics (MHD) equations in $\mathbb{R}^{n}$, $n=2, 3$, as well as for a related model where the advection ... More

Toeplitz operators and Hamiltonian torus actionMay 07 2004Feb 02 2006This paper is devoted to semi-classical aspects of symplectic reduction. Consider a compact prequantizable Kahler manifold M with a Hamiltonian torus action. Guillemin and Sternberg introduced an isomorphism between the invariant part of the quantum space ... More

Semi-classical properties of geometric quantization with metaplectic correctionFeb 08 2006The geometric quantization of a symplectic manifold endowed with a prequantum bundle and a metaplectic structure is defined by means of an integrable complex structure. We prove that its semi-classical limit does not depend on the choice of the complex ... More

Wave packets in Honeycomb Structures and Two-Dimensional Dirac EquationsDec 25 2012In a recent article [10], the authors proved that the non-relativistic Schr\"odinger operator with a generic honeycomb lattice potential has conical (Dirac) points in its dispersion surfaces. These conical points occur for quasi-momenta, which are located ... More

Quantum vs Classical Proofs and Subset VerificationOct 22 2015We study the ability of efficient quantum verifiers to decide properties of exponentially large subsets given either a classical or quantum witness. We develop a general framework that can be used to prove that QCMA machines, with only classical witnesses, ... More

Shapes of Molecular Cloud Cores and the Filamentary Mode of Star FormationJun 18 2002Using recent dust continuum data, we generate the intrinsic ellipticity distribution of dense, starless molecular cloud cores. Under the hypothesis that the cores are all either oblate or prolate randomly-oriented spheroids, we show that a satisfactory ... More

Estimating heights using auxiliary functionsAug 18 2014Several recent papers construct auxiliary polynomials to bound the Weil height of certain classes of algebraic numbers from below. Following these techniques, the author gave a general method for introducing auxiliary polynomials to problems involving ... More

Lower bounds on the projective heights of algebraic pointsAug 21 2014If $\alpha_1,\ldots,\alpha_r$ are algebraic numbers such that $$N=\sum_{i=1}^r\alpha_i \ne \sum_{i=1}^r\alpha_i^{-1}$$ for some integer $N$, then a theorem of Beukers and Zagier gives the best possible lower bound on $$\sum_{i=1}^r\log h(\alpha_i)$$ where ... More

The finiteness of computing the ultrametric Mahler measureAug 21 2014Recent work of Fili and the author examines an ultrametric version of the Mahler measure, denoted $M_\infty(\alpha)$ for an algebraic number $\alpha$. We show that the computation of $M_\infty(\alpha)$ can be reduced to a certain search through a finite ... More

Counting Exceptional Points for Rational Numbers Associated to the Fibonacci SequenceJul 07 2016If $\alpha$ is a non-zero algebraic number, we let $m(\alpha)$ denote the Mahler measure of the minimal polynomial of $\alpha$ over $\mathbb Z$. A series of articles by Dubickas and Smyth, and later by the author, develop a modified version of the Mahler ... More

Quantum Merlin Arthur with Exponentially Small GapJan 08 2016We study the complexity of QMA proof systems with inverse exponentially small promise gap. We show that this class can be exactly characterized by PSPACE, the class of problems solvable with a polynomial amount of memory. As applications we show that ... More

Constraints on Field Theoretical Models for Variation of the Fine Structure ConstantAug 26 2003Mar 18 2004Recent theoretical ideas and observational claims suggest that the fine structure constant alpha may be variable. We examine a spectrum of models in which alpha is a function of a scalar field. Specifically, we consider three scenarios: oscillating alpha, ... More

Astrophysical Observations: Lensing and Eclipsing Einstein's TheoriesMar 15 2005Albert Einstein postulated the equivalence of energy and mass, developed the theory of special relativity, explained the photoelectric effect, and described Brownian motion in five papers, all published in 1905, 100 years ago. With these papers, Einstein ... More

Critical Comparison of 3-d Imaging Approaches for NGSTAug 22 1999Currently three imaging spectrometer architectures, tunable filter, dispersive, and Fourier transform, are viable for imaging the universe in three dimensions. There are domains of greatest utility for each of these architectures. The optimum choice among ... More

Continued fraction expansions in connection with the metric Mahler measureAug 07 2015Aug 01 2016The metric Mahler measure was first studied by Dubickas and Smyth in 2001 as a means of phrasing Lehmer's conjecture in topological language. More recent work of the author examined a parametrized family of generalized metric Mahler measures that gives ... More

The Weil height in terms of an auxiliary polynomialAug 21 2014Recent theorems of Dubickas and Mossinghoff use auxiliary polynomials to give lower bounds on the Weil height of an algebraic number $\alpha$ under certain assumptions on $\alpha$. We prove a theorem which introduces an auxiliary polynomial for giving ... More

Subelliptic Spin_C Dirac operators, IMay 11 2007We consider modifications of the classical dbar-Neumann conditions that define Fredholm problems for the Spin_C Dirac operator. In part II, we use boundary layer methods to obtain subelliptic estimates for these boundary value problems. Using these results, ... More

Anderson Localization, Non-linearity and Stable Genetic DiversityFeb 28 2006In many models of genotypic evolution, the vector of genotype populations satisfies a system of linear ordinary differential equations. This system of equations models a competition between differential replication rates (fitness) and mutation. Mutation ... More

Effects of Biases in Virial Mass Estimation on Cosmic Synchronization of Quasar AccretionApr 04 2011Recent work using virial mass estimates and the quasar mass-luminosity plane has yielded several new puzzles regarding quasar accretion, including a sub-Eddington boundary on most quasar accretion, near-independence of the accretion rate from properties ... More

Embedded, Self-Gravitating Equilibria in Sheetlike and Filamentary Molecular CloudsMay 13 2000Numerical solutions of the isothermal Lane-Emden equation are presented, corresponding to self-gravitating gaseous cores embedded within a finite density envelope of overall cylindrical symmetry. These structures may be members of a fragmentation hierarchy ... More

Entanglement entropy and Berezin-Toeplitz operatorsMar 08 2018Oct 28 2018We consider Berezin-Toeplitz operators on compact Kahler manifolds whose symbols are characteristic functions. When the support of the characteristic function has a smooth boundary, we prove a two-term Weyl law, the second term being proportional to the ... More

On Quantum ObfuscationFeb 04 2016Encryption of data is fundamental to secure communication in the modern world. Beyond encryption of data lies obfuscation, i.e., encryption of functionality. It is well-known that the most powerful means of obfuscating classical programs, so-called ``black-box ... More

The Power of Quantum Fourier SamplingJul 20 2015A line of work initiated by Terhal and DiVincenzo and Bremner, Jozsa, and Shepherd, shows that quantum computers can efficiently sample from probability distributions that cannot be exactly sampled efficiently on a classical computer, unless the PH collapses. ... More

Pseudorandom generators and the BQP vs. PH problemJul 02 2010Dec 22 2010It is a longstanding open problem to devise an oracle relative to which BQP does not lie in the Polynomial-Time Hierarchy (PH). We advance a natural conjecture about the capacity of the Nisan-Wigderson pseudorandom generator [NW94] to fool AC_0, with ... More

The infimum in the metric Mahler measureAug 18 2014Dubickas and Smyth defined the metric Mahler measure on the multiplicative group of non-zero algebraic numbers. The definition involves taking an infimum over representations of an algebraic number $\alpha$ by other algebraic numbers. We verify their ... More

Subelliptic Spin_C Dirac operators, II Basic EstimatesMay 11 2007Nov 02 2007We assume that the manifold with boundary, X, has a Spin_C-structure with spinor bundle S. Along the boundary, this structure agrees with the structure defined by an infinite order integrable almost complex structure and the metric is Kahler. The induced ... More

Metric Mahler measures over number fieldsMay 22 2017For an algebraic number $\alpha$, the metric Mahler measure $m_1(\alpha)$ was first studied by Dubickas and Smyth in 2001 and was later generalized to the $t$-metric Mahler measure $m_t(\alpha)$ by the author in 2010. The definition of $m_t(\alpha)$ involves ... More

Extensions in $\mathbb R^D$Nov 10 2014May 27 2015In this paper we consider the following interesting question. Suppose we are given an almost isometry $\phi:E\to \mathbb R^D$ where $E$ is a finite subset of $\mathbb R^D$. Can $\phi$ be extended to an almost isometry to the whole of $\mathbb R^D$? In ... More

A collection of metric Mahler measuresAug 21 2014Let $M(\alpha)$ denote the Mahler measure of the algebraic number $\alpha$. In a recent paper, Dubickas and Smyth constructed a metric version of the Mahler measure on the multiplicative group of algebraic numbers. Later, Fili and the author used similar ... More

Metric Heights on an Abelian GroupNov 08 2012Aug 18 2014Suppose $m(\alpha)$ denotes the Mahler measure of the non-zero algebraic number $\alpha$. For each positive real number $t$, the author studied a version $m_t(\alpha)$ of the Mahler measure that has the triangle inequality. The construction of $m_t$ is ... More

Discrete ambiguities in the determination of the CP anglesOct 26 2000Oct 27 2000We review briefly the problem of the discrete ambiguities in the determination of the CKM angles, by classifying them into different categories. We then focus on cos(2beta) and the extraction of alpha using QCD factorization.

Cobordism, Relative Indices and Stein FillingsMay 11 2007In this paper we build on the framework developed in "Subelliptic Boundary Value Problems for the Spin_C Dirac Operator, I, II, III" to obtain a more complete understanding of the gluing properties for indices of boundary value problems for the SpinC-Dirac ... More

The parametrized family of metric Mahler measuresAug 21 2014Let $M(\alpha)$ denote the (logarithmic) Mahler measure of the algebraic number $\alpha$. Dubickas and Smyth, and later Fili and the author, examined metric versions of $M$. The author generalized these constructions in order to associate, to each point ... More

Subelliptic SpinC Dirac Operators, IV Proof of the Relative Index ConjectureMar 25 2012We prove the relative index conjecture, which in turn implies that the set of embeddable deformations of a strictly pseudoconvex CR-structure on a compact 3-manifold is closed in the C\infty-topology.

Subelliptic Spin_c Dirac operators, III The Atiyah-Weinstein conjectureJul 26 2005May 11 2007In this paper we show that there is a well defined modified dbar-Neumann problem for a spin_c manifold with a strictly pseudoconvex boundary (in the contact geometry sense). We show that the index of the associated boundary value problem can be computed ... More

On the Collapse of Tubes Carried by 3D Incompressible FlowsJan 30 2001Feb 02 2001We povide a test for numerical simulations for the collapse of regular tubes carried by a 3D incompressible flow. In particular, we obtain necessary conditions for 3D Euler to have a vortex tube collapse in finite time.

Extensions, interpolation and matching in $R^D$Nov 10 2014May 27 2015Suppose we are given an almost isometry $\phi:E\to \mathbb R^D$ where $E$ is a finite subset of $\mathbb R^D$. Is it possible to decide if $\phi$ can be extended to an almost isometry to the whole of $\mathbb R^D$? In this paper, we study this interesting ... More

The Quasar Mass-Luminosity Plane I: A Sub-Eddington Limit for QuasarsNov 06 2009Nov 29 2010We use 62185 quasars from the Sloan Digital Sky Survey Data Release 5 sample to explore the relationship between black hole mass and luminosity. Black hole masses were estimated based on the widths of their H{\beta}, MgII and CIV lines and adjacent continuum ... More

Coupled Wire Model of Z4 Orbifold Quantum Hall StatesApr 06 2018We introduce a coupled wire model for a sequence of non-Abelian quantum Hall states that generalize the Z4 parafermion Read Rezayi state. The Z4 orbifold quantum Hall states occur at filling factors \nu = 2/(2m-p) for odd integers $m$ and $p$, and have ... More

The $t$-metric Mahler measures of surds and rational numbersAug 18 2014A. Dubickas and C. Smyth introduced the metric Mahler measure $$ M_1(\alpha) = \inf\left\{\sum_{n=1}^N M(\alpha_n): N \in \mathbb N, \alpha_1 \cdots \alpha_N = \alpha\right\}, $$ where $M(\alpha)$ denotes the usual (logarithmic) Mahler measure of $\alpha ... More

Electromagnetic radiation from relativistic nuclear collisionsJun 11 2003Jun 16 2003We review some of the results obtained in the study of the production of electromagnetic radiation in relativistic nuclear collisions. We concentrate on the emission of real photons and dileptons from the hot and dense strongly interacting phases of the ... More

Properties of the phi meson at finite temperatureJan 06 1994We calculate the $\phi$-meson propagator at finite temperature at the one--loop order. The real and imaginary parts are studied separately in full kinematic ranges. From this activity we investigate how temperature affects such things as decay widths ... More

Determining the metric of the Cosmos: stability, accuracy, and consistencySep 06 2007Mar 28 2008The ultimate application of Einstein's field equations is to empirically determine the geometry of the Universe from its matter content, rather than simply assuming the Universe can be represented by a homogeneous model on all scales. Choosing an LTB ... More

On the non-Archimedean metric Mahler measureAug 21 2014Recently, Dubickas and Smyth constructed and examined the metric Mahler measure and the metric na\"ive height on the multiplicative group of algebraic numbers. We give a non-Archimedean version of the metric Mahler measure, denoted $M_\infty$, and prove ... More

Period derivative of the M15 X-ray Binary AC211/X2127+119Aug 28 1998We have combined Rossi X-ray Timing Explorer observations of X2127+119, the low-mass X-ray binary in the globular cluster M15, with archival X-ray lightcurves to study the stability of the 17.1 hr orbital period. We find that the data cannot be fit by ... More

Restricted Euler dynamics along trajectories of small inertial particles in turbulenceAug 08 2016The fate of small particles in turbulent flows depends strongly on the surrounding fluid's velocity gradient properties such as rotation and strain-rates. For non-inertial (fluid) particles, the Restricted Euler model provides a simple, low-dimensional ... More

Tempering the polylogarithmNov 08 2006We show that the function Li_s(e^x) extends to an entire function of the complex variable s, taking values in tempered distributions in x on the whole real line. That the classical polylogarithm extends similarly, as an entire function taking values in ... More

Debye Sources and the Numerical Solution of the Time Harmonic Maxwell EquationsAug 25 2008Mar 03 2009In this paper, we develop a new representation for outgoing solutions to the time harmonic Maxwell equations in unbounded domains in $\bbR^3.$ This representation leads to a Fredholm integral equation of the second kind for solving the problem of scattering ... More

On Extensions of e DiffeomorphismsMay 26 2015Oct 02 2015Let $n\geq 1$, $U\subset\mathbb R^n$ open and $\phi:U\to \mathbb R^n$ be $C^1$ map. If $E\subset U$ is compact and the restriction of $\phi$ to $E$ is an almost isometry, we study the question of how to decide when $\phi$ extends to a smooth distorted ... More

On the Whitney distortion extension problem for $C^m(\mathbb R^n)$ and $C^{\infty}(\mathbb R^n)$ and its applications to interpolation and alignment of data in $\mathbb R^n$May 26 2015Nov 26 2017Let $n,m\geq 1$, $U\subset\mathbb R^n$ open. In this paper we provide a sharp solution to the following Whitney distortion extension problems: (a) Let $\phi:U\to \mathbb R^n$ be a $C^m$ map. If $E\subset U$ is compact (with some geometry) and the restriction ... More