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Irrational numbers associated to sequences without geometric progressionsJul 30 2013Let s and k be integers with s \geq 2 and k \geq 2. Let g_k^{(s)}(n) denote the cardinality of the largest subset of the set {1,2,..., n} that contains no geometric progression of length k whose common ratio is a power of s. Let r_k(\ell) denote the cardinality ... More

Decomposition and Identification of Linear Structural Equation ModelsAug 07 2015In this paper, we address the problem of identifying linear structural equation models. We first extend the edge set half-trek criterion to cover a broader class of models. We then show that any semi-Markovian linear model can be recursively decomposed ... More

Goto Numbers of a Numerical Semigroup Ring and the Gorensteiness of Associated Graded RingsSep 02 2008Oct 26 2009The Goto number of a parameter ideal Q in a Noetherian local ring (R,m) is the largest integer q such that Q : m^q is integral over Q. The Goto numbers of the monomial parameter ideals of R = k[[x^{a_1}, x^{a_2},..., x_{a_{\nu}}]] are characterized using ... More

Strong spin-lattice coupling and negative thermal expansion in a magnetically frustrated spinelDec 14 2018We report on high magnetic field dilatometry measurements of the frustrated spinel CdCr$_2$O$_4$, and discover a distinct negative thermal expansion in the high-field fractional magnetization state. By precisely mapping the phase diagram, and comparing ... More

Bochner-Kahler metricsMar 16 2000Jun 27 2000A Kahler metric is said to be Bochner-Kahler if its Bochner curvature vanishes. This is a nontrivial condition when the complex dimension of the underlying manifold is at least 2. In this article it will be shown that, in a certain well-defined sense, ... More

On the Sum of the Heights of Sturmian FactorsNov 13 2006A binary word is a map W : N --> {0,1}, and the set of factors of W with length n is F_n(W):={(W(i),W(i+1),...,W(i+n-1)) : i >= 0}. A word is Sturmian if |F_n(W)|=n+1 for every n>0. We show that the sum of the heights (also known as hamming weights) of ... More

Nonembedding and nonextension results in special holonomyMay 31 2012Constructions of metrics with special holonomy by methods of exterior differential systems are reviewed and the interpretations of these construction as `flows' on hypersurface geometries are considered. It is shown that these hypersurface 'flows' are ... More

Notes on Projective, Contact, and Null CurvesMay 15 2019These are notes on some algebraic geometry of complex projective curves, together with an application to studying the contact curves in CP^3 and the null curves in the complex quadric Q^3 in CP^4, related by the well-known Klein correspondence. Most of ... More

An $O(n \log n)$ time Algorithm for computing the Path-length Distance between TreesNov 01 2018Tree comparison metrics have proven to be an invaluable aide in the reconstruction and analysis of phylogenetic (evolutionary) trees. The path-length distance between trees is a particularly attractive measure as it reflects differences in tree shape ... More

Sets of integers that do not contain long arithmetic progressionsNov 19 2008Jun 21 2010In 1946, Behrend gave a construction of dense finite sets of integers that do not contain 3-term arithmetic progressions. In 1961, Rankin generalized Behrend's construction to sets avoiding k-term arithmetic progressions, and in 2008 Elkin refined Behrend's ... More

Energy Dissipation Bounds in Autonomous Thermodynamic SystemsMar 15 2019How much free energy is irreversibly lost when a system is moved through thermodynamic space? For systems with deterministic control, lower bounds on energy dissipation are established. Recent literature has also bounded the cost of moving a single degree ... More

A problem of Rankin on sets without geometric progressionsAug 12 2014A geometric progression of length $k$ and integer ratio is a set of numbers of the form $\{a,ar,\dots,ar^{k-1}\}$ for some positive real number $a$ and integer $r\geq 2$. For each integer $k \geq 3$, a greedy algorithm is used to construct a strictly ... More

On sequences without geometric progressionsJun 03 2013Jun 04 2013An improved upper bound is obtained for the density of sequences of positive integers that contain no k-term geometric progression.

The Decomposition of Lie PowersMay 16 2005Let G be a group, F a field of prime characteristic p and V a finite-dimensional FG-module. Let L(V) denote the free Lie algebra on V regarded as an FG-submodule of the free associative algebra (or tensor algebra) T(V). For each positive integer r, let ... More

On Hamilton Decompositions of Line Graphs of Non-Hamiltonian Graphs and Graphs without Separating TransitionsOct 17 2017In contrast with Kotzig's result that the line graph of a $3$-regular graph $X$ is Hamilton decomposable if and only if $X$ is Hamiltonian, we show that for each integer $k\geq 4$ there exists a simple non-Hamiltonian $k$-regular graph whose line graph ... More

Hamilton path decompositions of complete multipartite graphsOct 13 2017Jul 23 2018We prove that a complete multipartite graph $K$ with $n>1$ vertices and $m$ edges can be decomposed into edge-disjoint Hamilton paths if and only if $\frac m{n-1}$ is an integer and the maximum degree of $K$ is at most $\frac {2m}{n-1}$.

Decompositions of complete graphs into cycles of arbitrary lengthsApr 17 2012Oct 29 2013We show that the complete graph on $n$ vertices can be decomposed into $t$ cycles of specified lengths $m_1,\ldots,m_t$ if and only if $n$ is odd, $3\leq m_i\leq n$ for $i=1,\ldots,t$, and $m_1+\cdots+m_t=\binom n2$. We also show that the complete graph ... More

Determining equilibrium osmolarity in Poly(ethylene glycol) / Chondrotin sulfate gels mimicking articular cartilageAug 20 2013We present an experimentally guided, multi-phase, multi-species polyelectrolyte gel model to make quantitative predictions on the electro-chemical properties of articular cartilage. The mixture theory consists of two different types of polymers: Poly(ethylene ... More

Explicit bounds for primes in arithmetic progressionsJan 31 2018Nov 27 2018We derive explicit upper bounds for various functions counting primes in arithmetic progressions. By way of example, if $q$ and $a$ are integers with $\gcd(a,q)=1$ and $3 \leq q \leq 10^5$, and $\theta(x;q,a)$ denotes the sum of the logarithms of the ... More

Compressed sensing with combinatorial designs: theory and simulationsMar 24 2015May 20 2015In 'An asymptotic result on compressed sensing matrices', a new construction for compressed sensing matrices using combinatorial design theory was introduced. In this paper, we use deterministic and probabilistic methods to analyse the performance of ... More

Asymptotic behaviour of Lie powers and Lie modulesSep 06 2010Dec 03 2010Let $V$ be a finite-dimensional $FG$-module, where $F$ is a field of prime characteristic $p$ and $G$ is a group. We show that, when $r$ is not a power of $p$, the Lie power $L^r(V)$ has a direct summand $B^r(V)$ which is a direct summand of the tensor ... More

Reciprocals of Binary Power SeriesJun 24 2005If A is a set of nonnegative integers containing 0, then there is a unique nonempty set B of nonnegative integers such that every positive integer can be written in the form a+b, where a\in A and b\in B, in an even number of ways. We compute the natural ... More

On the Quirks of Maximum Parsimony and Likelihood on Phylogenetic NetworksMay 26 2015Oct 24 2016Maximum parsimony is one of the most frequently-discussed tree reconstruction methods in phylogenetic estimation. However, in recent years it has become more and more apparent that phylogenetic trees are often not sufficient to describe evolution accurately. ... More

Hadamard Phylogenetic Methods and the n-taxon processJun 09 2008The Hadamard transform of \cite{Hendy89a, Hendy89} provides a way to work with stochastic models for sequence evolution without having to deal with the complications of tree space and the graphical structure of trees. Here we demonstrate that the transform ... More

Generalization of the coupled dipole method to periodic structuresMay 13 2003We present a generalization of the coupled dipole method to the scattering of light by arbitrary periodic structures. This new formulation of the coupled dipole method relies on the same direct-space discretization scheme that is widely used to study ... More

Decompositions of complete multigraphs into cycles of varying lengthsAug 04 2015We establish necessary and sufficient conditions for the existence of a decomposition of a complete multigraph into edge-disjoint cycles of specified lengths, or into edge-disjoint cycles of specified lengths and a perfect matching.

Distribution of Missing Sums in SumsetsSep 22 2011Dec 21 2012For any finite set of integers X, define its sumset X+X to be {x+y: x, y in X}. In a recent paper, Martin and O'Bryant investigated the distribution of |A+A| given the uniform distribution on subsets A of {0, 1, ..., n-1}. They also conjectured the existence ... More

Spin wave imaging in atomically designed nanomagnetsMar 24 2014The spin dynamics of all ferromagnetic materials are governed by two types of collective excitations: spin waves and domain walls. The fundamental processes underlying these collective modes, such as exchange interactions and magnetic anisotropy, all ... More

Local control of single atom magneto-crystalline anisotropyMay 07 2013Jul 25 2013Individual Fe atoms on a Cu2N/Cu(100) surface exhibit strong magnetic anisotropy due to the crystal field. Using atom manipulation in a low-temperature STM we demonstrate that the anisotropy of one Fe atom is significantly influenced by local strain due ... More

Exploring the phase diagram of the two-impurity Kondo problemNov 17 2014Sep 16 2015A system of two exchange-coupled Kondo impurities in a magnetic field gives rise to a rich phase space hosting a multitude of correlated phenomena. Magnetic atoms on surfaces probed through scanning tunnelling microscopy provide an excellent platform ... More

Binary linear forms over finite sets of integersJan 02 2007Let A be a finite set of integers. For a polynomial f(x_1,...,x_n) with integer coefficients, let f(A) = {f(a_1,...,a_n) : a_1,...,a_n \in A}. In this paper it is proved that for every pair of normalized binary linear forms f(x,y)=u_1x+v_1y and g(x,y)=u_2x+v_2y ... More

Optical/Infrared Observations of the Anomalous X-ray Pulsar 1E 1048.1-5937 During Its 2007 X-Ray FlareNov 14 2007Feb 14 2008We report on optical and infrared observations of the anomalous X-ray pulsar (AXP) 1E 1048.1-5937, made during its ongoing X-ray flare which started in 2007 March. We detected the source in the optical I and near-infrared Ks bands in two ground-based ... More

An ultra-compact low temperature scanning probe microscope for magnetic fields above 30 TMay 03 2018Jun 29 2018We present the design of a highly compact High Field Scanning Probe Microscope (HF-SPM) for operation at cryogenic temperatures in an extremely high magnetic field, provided by a water-cooled Bitter magnet able to reach 38 T. The HF-SPM is 14 mm in diameter: ... More

Controlled complete suppression of single-atom inelastic spin and orbital cotunnellingApr 22 2015Apr 23 2015The inelastic portion of the tunnel current through an individual magnetic atom grants unique access to read out and change the atom's spin state, but it also provides a path for spontaneous relaxation and decoherence. Controlled closure of the inelastic ... More

A new search for distant radio galaxies in the Southern hemisphere -- II. 2.2 micron imagingFeb 03 2009We have compiled a sample of 234 ultra-steep-spectrum (USS) selected radio sources in order to find high-redshift radio galaxies. The sample covers the declination range -40deg < DEC < -30deg in the overlap region between the 1400-MHz NRAO VLA Sky Survey, ... More

Quantitative $μ$PIV Measurements of Velocity ProfilesJul 18 2014Jul 29 2014In Microscopic Particle Image Velocimetry ($\mu$PIV), velocity fields in microchannels are sampled over finite volumes within which the velocity fields themselves may vary significantly. In the past, this has limited measurements often to be only qualitative ... More

Theory of imaging a photonic crystal with transmission near-field optical microscopyJan 15 1998While near-field scanning optical microscopy (NSOM) can provide optical images with resolution much better than the diffraction limit, analysis and interpretation of these images is often difficult. We present a theory of imaging with transmission NSOM ... More

A new search for distant radio galaxies in the Southern hemisphere -- III. Optical spectroscopy and analysis of the MRCR--SUMSS sampleFeb 08 2009We have compiled a sample of 234 ultra-steep-spectrum(USS)-selected radio sources in order to find high-redshift radio galaxies (HzRGs). The sample is in the southern sky at -40 deg < DEC < -30 deg which is the overlap region of the 408-MHz Revised Molonglo ... More

On extremals with prescribed Lagrangian densitiesJun 23 1994Consider two manifolds~$M^m$ and $N^n$ and a first-order Lagrangian $L(u)$ for mappings $u:M\to N$, i.e., $L$ is an expression involving $u$ and its first derivatives whose value is an $m$-form (or more generally, an $m$-density) on~$M$. One is usually ... More

Parsimony via concensusApr 04 2007Oct 01 2013The parsimony score of a character on a tree equals the number of state changes required to fit that character onto the tree. We show that for unordered, reversible characters this score equals the number of tree rearrangements required to fit the tree ... More

Conformal geometry and 3-plane fields on 6-manifoldsNov 04 2005Dec 29 2005The purpose of this note is to provide yet another example of the link between certain conformal geometries and ordinary differential equations, along the lines of the examples discussed by Nurowski in math.DG/0406400. In this particular case, I consider ... More

A Complete Annotated Bibliography of Work Related to Sidon SequencesJul 08 2004A Sidon sequence is a sequence of integers a_1 < a_2 < a_3 < ... with the property that the sums a_i+a_j (i\le j) are distinct. This work contains a survey of Sidon sequences and their generalizations, and an extensive annotated and hyperlinked bibliography ... More

On Z.-W. Sun's Disjoint Congruence Classes ConjectureApr 15 2006Apr 19 2006Zhi-Wei Sun conjectures if k congruence classes are disjoint, then two of the moduli have greatest common divisor at least as large as k. We prove this conjecture for k strictly less than 21.

Gaps in the Spectrum of Heights of Projective PointsMar 21 2007May 01 2007Let r mod m be the least positive residue of r modulo m, and set the height of a pair (r,s) of integers, both relatively prime to m, to be the minimum over k, with 0<k<m, of (k r mod m) + (k s mod m). Denote this quantity by h(m,r,s). We give a formula ... More

Remarks on the geometry of almost complex 6-manifoldsAug 23 2005Sep 15 2005This article is mostly a writeup of two talks, the first given in the Besse Seminar at the Ecole Polytechnique in 1998 and the second given at the 2000 International Congress on Differential Geometry in memory of Alfred Gray in Bilbao, Spain. It begins ... More

Harmonic morphisms with fibers of dimension oneJan 03 1997Jan 13 1997I prove three classification results about harmonic morphisms whose fibers have dimension one. All are valid when the domain is at least of dimension 4. (The character of this overdetermined problem is very different when the dimension of the domain is ... More

Complex analysis and a class of Weingarten surfacesMay 27 2011An idea of Hopf's for applying complex analysis to the study of constant mean curvature spheres is generalized to cover a wider class of spheres, namely, those satisfying a Weingarten relation of a certain type, namely H = f(H^2-K) for some smooth function ... More

The link between segregation and phylogenetic diversityAug 27 2010Aug 30 2010We derive an invertible transform linking two widely used measures of species diversity: phylogenetic diversity and the expected proportions of segregating (non-constant) sites. We assume a bi-allelic, symmetric, finite site model of substitution. Like ... More

Computing the Distribution of a Tree MetricOct 06 2008The Robinson-Foulds (RF) distance is by far the most widely used measure of dissimilarity between trees. Although the distribution of these distances has been investigated for twenty years, an algorithm that is explicitly polynomial time has yet to be ... More

Homologically arc-homogeneous ENRsMar 17 2009We prove that an arc-homogeneous Euclidean neighborhood retract is a homology manifold.

Levi-flat Minimal Hypersurfaces in Two-dimensional Complex Space FormsSep 27 1999The purpose of this article is to classify the real hypersurfaces in complex space forms of dimension 2 that are both Levi-flat and minimal. The main results are as follows: When the curvature of the complex space form is nonzero, there is a 1-parameter ... More

S.-S. Chern's study of almost-complex structures on the six-sphereMay 14 2014In 2003, S.-s. Chern began a study of almost-complex structures on the 6-sphere, with the idea of exploiting the special properties of its well-known almost-complex structure invariant under the exceptional group $G_2$. While he did not solve the (currently ... More

On the convex Pfaff-Darboux Theorem of Ekeland and NirenbergDec 22 2015The classical Pfaff-Darboux Theorem, which provides local `normal forms' for 1-forms on manifolds, has applications in the theory of certain economic models. However, the normal forms needed in these models come with an additional requirement of convexity, ... More

Recent Advances in the Theory of HolonomyOct 11 1999Oct 12 1999This article is a report on the status of the problem of classifying the irriducibly acting subgroups of GL(n,R) that can appear as the holonomy of a torsion-free affine connection. In particular, it contains an account of the completion of the classification ... More

Rigidity and quasi-rigidity of extremal cycles in Hermitian symmetric spacesJun 24 2000Mar 05 2001I use local differential geometric techniques to prove that the algebraic cycles in certain extremal homology classes in Hermitian symmetric spaces are either rigid (i.e., deformable only by ambient motions) or quasi-rigid (roughly speaking, foliated ... More

Pseudo-Riemannian metrics with parallel spinor fields and vanishing Ricci tensorApr 11 2000I discuss geometry and normal forms for pseudo-Riemannian metrics with parallel spinor fields in some interesting dimensions. I also discuss the interaction of these conditions for parallel spinor fields with the condition that the Ricci tensor vanish ... More

Sets of natural numbers with proscribed subsetsOct 18 2014Jun 13 2015Fix $A$, a family of subsets of natural numbers, and let $G_A(n)$ be the maximum cardinality of a subset of $\{1,2,..., n\}$ that does not have any subset in $A$. We consider the general problem of giving upper bounds on $G_A(n)$ and give some new upper ... More

Vertex-transitive graphs that have no Hamilton decompositionAug 22 2014Nov 12 2014It is shown that there are infinitely many connected vertex-transitive graphs that have no Hamilton decomposition, including infinitely many Cayley graphs of valency 6, and including Cayley graphs of arbitrarily large valency.

Real hypersurfaces in unimodular complex surfacesJul 27 2004A unimodular complex surface is a complex 2-manifold X endowed with a holomorphic volume form. A strictly pseudoconvex real hypersurface M in X inherits not only a CR-structure but a canonical coframing as well. In this article, this canonical coframing ... More

'Bureaucratic' set systems, and their role in phylogeneticsJun 09 2011We say that a collection $\Cc$ of subsets of $X$ is {\em bureaucratic} if every maximal hierarchy on $X$ contained in $\Cc$ is also maximum. We characterise bureaucratic set systems and show how they arise in phylogenetics. This framework has several ... More

On surfaces with prescribed shape operatorJul 11 2001Jul 20 2001The problem of immersing a simply connected surface with a prescribed shape operator is discussed. From classical and more recent work, it is known that, aside from some special degenerate cases, such as when the shape operator can be realized by a surface ... More

Notes on exterior differential systemsMay 13 2014These are notes for a very rapid introduction to the basics of exterior differential systems and their connection with what is now known as Lie theory, together with some typical and not-so-typical applications to illustrate their use.

Data-parallel distributed training of very large models beyond GPU capacityNov 29 2018GPUs have limited memory and it is difficult to train wide and/or deep models that cause the training process to go out of memory. It is shown in this paper how an open source tool called Large Model Support (LMS) can utilize a high bandwidth NVLink connection ... More

Geodesic behavior for Finsler metrics of constant positive flag curvature on $S^2$Oct 10 2017We study non-reversible Finsler metrics with constant flag curvature 1 on S^2 and show that the geodesic flow of every such metric is conjugate to that of one of Katok's examples, which form a 1-parameter family. In particular, the length of the shortest ... More

Some remarks on G_2-structuresMay 08 2003Feb 01 2005This article consists of some loosely related remarks about the geometry of G_2-structures on 7-manifolds and is partly based on old unpublished joint work with two other people: F. Reese Harvey and Steven Altschuler. Much of this work has since been ... More

Thick subsets that do not contain arithmetic progressionsDec 08 2009Jun 24 2010We adapt the construction of subsets of {1, 2, ..., N} that contain no k-term arithmetic progressions to give a relatively thick subset of an arbitrary set of N integers. Particular examples include a thick subset of {1, 4, 9, ..., N^2} that does not ... More

On the conformal volume of 2-toriJul 06 2015This note provides a proof of a 1985 conjecture of Montiel and Ros about the conformal volume of tori. (This material is not really new; I'm making it available now because of requests related to recent interest in the conjecture.)

Golden-Ratio-Based Rectangular TilingsNov 01 2016A golden-ratio-based rectangular tiling of the first quadrant of the Euclidean plane is constructed by drawing vertical and horizontal grid lines which are located at all even powers of $\phi$ along one axis, and at all odd powers of $\phi$ on the other ... More

Projectively flat Finsler 2-spheres of constant curvatureNov 25 1996I classify the Finsler structures on the 2-sphere that have constant Finsler-Gauss curvature and whose geodesics are the great circles. Modulo diffeomorphism, there is a 2-parameter family of such Finsler structures, only one of which is homogeneous or ... More

Second order families of special Lagrangian 3-foldsJul 21 2000May 01 2001A second order family of special Lagrangian submanifolds of complex m-space is a family characterized by the satisfaction of a set of pointwise conditions on the second fundamental form. For example, the set of ruled special Lagrangian submanifolds of ... More

Canonical dimension of projective PGL_1(A)-homogeneous varietiesMar 30 2009Let A be a central division algebra over a field F with ind A = n. In computing canonical p-dimension of projective PGL_1(A)-homogeneous varieties, for p prime, we can reduce to the case of generalized Severi-Brauer varieties X_e(A) with ind A a power ... More

Probing quantum nanostructures with near-field optical microscopy and (vice versa)Oct 15 1997A theory is presented to show how near-field optical microscopy can be used to probe quantum nanostructures. Calculations are done for a quantum dot. Results for different tip/dot configurations and sizes show that near-field excitation can enhance light-hole ... More

Steiner triple systems without parallel classesJul 22 2014Aug 06 2014We construct Steiner triple systems without parallel classes for an infinite number of orders congruent to $3 \pmod{6}$. The only previously known examples have order $15$ or $21$.

Gradient Kahler Ricci SolitonsJul 27 2004Aug 02 2004Some observations about the local and global generality of gradient Kahler Ricci solitons are made, including the existence of a canonically associated holomorphic volume form and vector field, the local generality of solutions with a prescribed holomorphic ... More

Some examples of special Lagrangian toriFeb 12 1999Mar 30 1999I point out some very elementary examples of special Lagrangian tori in certain Calabi-Yau manifolds that occur as hypersurfaces in complex projective space. All of these are constructed as real slices of smooth hypersurfaces defined over the reals. This ... More

Calibrated embeddings in the special Lagrangian and coassociative casesDec 31 1999Jan 02 2000Every closed, oriented, real analytic Riemannian 3-manifold can be isometrically embedded as a special Lagrangian submanifold of a Calabi-Yau 3-fold, even as the real locus of an antiholomorphic, isometric involution. Every closed, oriented, real analytic ... More

Jointly periodic points in cellular automata: computer explorations and conjecturesJul 07 2006We develop a rather elaborate computer program to investigate the jointly periodic points of one-dimensional cellular automata. The experimental results and mathematical context lead to questions, conjectures and a contextual theorem.

Geodesically reversible Finsler 2-spheres of constant curvatureJul 29 2004Aug 02 2004A Finsler space is said to be geodesically reversible if each oriented geodesic can be reparametrized as a geodesic with the reverse orientation. A reversible Finsler space is geodesically reversible, but the converse need not be true. In this note, building ... More

Sturmian Words and the Permutation that Orders Fractional PartsNov 13 2002A Sturmian word is a map W from the natural numbers into {0,1} for which the set of {0,1}-vectors F_n(W):={(W(i),W(i+1),...,W(i+n-1))^T : i \ge 0} has cardinality exactly n+1 for each positive integer n. Our main result is that the volume of the simplex ... More

What Do People See in a Twenty-Second Glimpse of Bivariate Vector Field Visualizations?May 07 2019May 19 2019Little is known about how people learn from a brief glimpse of three-dimensional (3D) bivariate vector field visualizations and about how well visual features can guide behavior. Here we report empirical study results on the use of color, texture, and ... More

Slice Implies Mutant Ribbon for Odd, 5-Stranded Pretzel KnotsNov 22 2015Jan 18 2016A pretzel knot $K$ is called $odd$ if all its twist parameters are odd, and $mutant$ $ribbon$ if it is mutant to a simple ribbon knot. We prove that the family of odd, 5-stranded pretzel knots satisfies a weaker version of the Slice-Ribbon Conjecture: ... More

SO(n)-invariant special Lagrangian submanifolds of C^{n+1} with fixed lociFeb 12 2004Mar 25 2004Let SO(n) act in the standard way on C^n and extend this action in the usual way to C^{n+1}. It is shown that a nonsingular special Lagrangian submanifold L in C^{n+1} that is invariant under this SO(n)-action intersects the fixed line C in a nonsingular ... More

The sequence of fractional parts of rootsOct 10 2014Jun 09 2015We study the function M(t,n) = Floor[ 1 / {t^(1/n)} ], where t is a positive real number, Floor[.] and {.} are the floor and fractional part functions, respectively. In a recent article in the Monthly, Nathanson proved that if log(t) is rational, then ... More

A noisy system with a flattened Hamiltonian and multiple time scalesFeb 02 2003We consider a two-dimensional weakly dissipative dynamical system with time-periodic drift and diffusion coefficients. The average of the drift is governed by a degenerate Hamiltonian whose set of critical points has an interior. The dynamics of the system ... More

Laplacian Flow for Closed $G_2$-Structures: Short Time BehaviorJan 11 2011We prove short time existence and uniqueness of solutions to the Laplacian flow for closed $G_2$ structures on a compact manifold $M^7$. The result was claimed in \cite{BryantG2}, but its proof has never appeared.

Some remarks on Finsler manifolds with constant flag curvatureJul 31 2001This article is an exposition of four loosely related remarks on the geometry of Finsler manifolds with constant positive flag curvature. <p> The first remark is that there is a canonical Kahler structure on the space of geodesics of such a manifold. ... More

Fraenkel's Partition and Brown's DecompositionMay 09 2003Denote the sequence ([ (n-x') / x ])_{n=1}^\infty by B(x, x'), a so-called Beatty sequence. Fraenkel's Partition Theorem gives necessary and sufficient conditions for B(x, x') and B(y, y') to tile the positive integers, i.e., for B(x, x') \cap B(y, y') ... More

A low-temperature scanning tunneling microscope capable of microscopy and spectroscopy in a Bitter magnet at up to 34 TJul 10 2017We present the design and performance of a cryogenic scanning tunneling microscope (STM) which operates inside a water-cooled Bitter magnet, which can attain a magnetic field of up to 38 T. Due to the high vibration environment generated by the magnet ... More

Atomic spin chain realization of a model for quantum criticalityApr 19 2016The ability to manipulate single atoms has opened up the door to constructing interesting and useful quantum structures from the ground up. On the one hand, nanoscale arrangements of magnetic atoms are at the heart of future quantum computing and spintronic ... More

Coexistence of orbital and quantum critical magnetoresistance in FeSe$_{1-x}$S$_{x}$Mar 13 2019The recent discovery of a non-magnetic nematic quantum critical point (QCP) in the iron chalcogenide family FeSe$_{1-x}$S$_{x}$ has raised the prospect of investigating, in isolation, the role of nematicity on the electronic properties of correlated metals. ... More

Constant distortion embeddings of Symmetric DiversitiesApr 07 2016Nov 09 2016Diversities are like metric spaces, except that every finite subset, instead of just every pair of points, is assigned a value. Just as there is a theory of minimal distortion embeddings of finite metric spaces into $L_1$, there is a similar, yet undeveloped, ... More

Spin relaxation of a donor electron coupled to interface statesOct 20 2017Nov 28 2018An electron spin qubit in a silicon donor atom is a promising candidate for quantum information processing because of its long coherence time. To be sensed with a single-electron transistor, the donor atom is usually located near an interface, where the ... More

The supremum of autoconvolutions, with applications to additive number theoryJul 31 2008Apr 01 2009We adapt a number-theoretic technique of Yu to prove a purely analytic theorem: if f(x) is in L^1 and L^2, is nonnegative, and is supported on an interval of length I, then the supremum of the convolution f*f is at least 0.631 \| f \|_1^2 / I. This improves ... More

Neutral Gas Distribution and Kinematics of the Nearly Face-on Spiral Galaxy NGC 1232Jul 20 1999We have analyzed high velocity resolution HI synthesis observations of the nearly face-on Sc galaxy NGC 1232. The neutral gas distribution extends well beyond the optical extent of the galaxy. As expected, local peaks in the HI column density are associated ... More

Constant distortion embeddings of Symmetric DiversitiesApr 07 2016Diversities are like metric spaces, except that every finite subset, instead of just every pair of points, is assigned a value. Just as there is a theory of minimal distortion embeddings of finite metric spaces into $L_1$, there is a similar, yet undeveloped, ... More

An asymptotic existence result on compressed sensing matricesMar 12 2014Feb 09 2015For any rational number $h$ and all sufficiently large $n$ we give a deterministic construction for an $n\times \lfloor hn\rfloor$ compressed sensing matrix with $(\ell_1,t)$-recoverability where $t=O(\sqrt{n})$. Our method uses pairwise balanced designs ... More

Factorisation of Lie ResolventsJun 06 2005Let $G$ be a group, $F$ a field of prime characteristic $p$ and $V$ a finite-dimensional $FG$-module. Let $L(V)$ denote the free Lie algebra on $V$, regarded as an $FG$-module, and, for each positive integer $r$, let $L^r(V)$ be the $r$th homogeneous ... More

Constructions of Generalized Sidon SetsAug 05 2004Feb 21 2005We give explicit constructions of sets S with the property that for each integer k, there are at most g solutions to k=s_1+s_2, s_i\in S; such sets are called Sidon sets if g=2 and generalized Sidon sets if g\ge 3. We extend to generalized Sidon sets ... More

Dynamic micro-Hall detection of superparamagnetic beads in a microfluidic channelSep 25 2009We report integration of an InAs quantum well micro-Hall sensor with microfluidics and real-time detection of moving superparamagnetic beads for biological applications. The detected positive and negative signals correspond to beads moving within and ... More

Diversities and the Geometry of HypergraphsDec 19 2013Apr 18 2014The embedding of finite metrics in $\ell_1$ has become a fundamental tool for both combinatorial optimization and large-scale data analysis. One important application is to network flow problems in which there is close relation between max-flow min-cut ... More