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Achievable ranks of intersections of finitely generated free groupsJan 20 2004We answer a question due to A. Myasnikov by proving that all expected ranks occur as the ranks of intersections of finitely generated subgroups of free groups.

MP users guideApr 19 2010Apr 20 2010MP is a package of ANSI Standard Fortran (ANS X3.9-1966) subroutines for performing multiple-precision floating-point arithmetic and evaluating elementary and special functions. The subroutines are machine independent and the precision is arbitrary, subject ... More

Valid Orderings of Real Hyperplane ArrangementsJun 07 2013Given a real finite hyperplane arrangement A and a point p not on any of the hyperplanes, we define an arrangement vo(A,p), called the *valid order arrangement*, whose regions correspond to the different orders in which a line through p can cross the ... More

Some comments on C. S. Wallace's random number generatorsMay 13 2010We outline some of Chris Wallace's contributions to pseudo-random number generation. In particular, we consider his idea for generating normally distributed variates without relying on a source of uniform random numbers, and compare it with more conventional ... More

Spanning trees and a conjecture of KontsevichJun 10 1998Nov 09 1998Kontsevich conjectured that the number f(G,q) of zeros over the finite field with q elements of a certain polynomial connected with the spanning trees of a graph G is polynomial function of q. We have been unable to settle Kontsevich's conjecture. However, ... More

Increasing and Decreasing Subsequences of Permutations and Their VariantsDec 01 2005We survey the theory of increasing and decreasing subsequences of permutations. Enumeration problems in this area are closely related to the RSK algorithm. The asymptotic behavior of the expected value of the length is(w) of the longest increasing subsequence ... More

The Borwein brothers, Pi and the AGMFeb 21 2018Aug 08 2018We consider some of Jonathan and Peter Borweins' contributions to the high-precision computation of $\pi$ and the elementary functions, with particular reference to their book "Pi and the AGM" (Wiley, 1987). Here "AGM" is the arithmetic-geometric mean ... More

Some integer factorization algorithms using elliptic curvesApr 20 2010Lenstra's integer factorization algorithm is asymptotically one of the fastest known algorithms, and is ideally suited for parallel computation. We suggest a way in which the algorithm can be speeded up by the addition of a second phase. Under some plausible ... More

Equivariant K-theory and refined Vafa-Witten invariantsSep 28 2018Nov 02 2018In [MT2] the Vafa-Witten theory of complex projective surfaces is lifted to oriented $\mathbb C^*$-equivariant cohomology theories. Here we study the K-theoretic refinement. It gives rational functions in $t^{1/2}$ invariant under $t^{1/2}\leftrightarrow ... More

On asymptotic approximations to the log-Gamma and Riemann-Siegel theta functionsSep 13 2016Oct 06 2016We give bounds on the error in the asymptotic approximation of the log-Gamma function $\ln\Gamma(z)$ for complex $z$ in the right half-plane. These improve on earlier bounds by Behnke and Sommer (1962), Spira (1971), and Hare (1997). We show that $|R_{k+1}(z)/T_k(z)| ... More

Some remarks on sign-balanced and maj-balanced posetsNov 06 2002Jan 16 2004Let P be a poset with elements 1,2,...,n. We say that P is sign-balanced if exactly half the linear extensions of P (regarded as permutations of 1,2,...,n) are even permutations, i.e., have an even number of inversions. This concept first arose in the ... More

Irreducible Symmetric Group Characters of Rectangular ShapeSep 14 2001Dec 17 2002We give a new formula for the values of an irreducible character of the symmetric group S_n indexed by a partition of rectangular shape. Some observations and a conjecture are given concerning a generalization to arbitrary shapes.

On the Enumeration of Skew Young TableauxJun 13 2001Dec 18 2001We extend work of McKay, Morse, and Wilf by giving exact formulas and asymptotic formulas for the number of skew Young tableaux T in two situations: (1) the "inside shape" and total number of cells of T are fixed, and (2) the inside shape of T is fixed. ... More

Phase behaviour of a simple model of globular proteinsApr 29 1999Sep 22 1999A simple model of globular proteins which incorporates anisotropic attractions is proposed. It is closely related to models used to model simple hydrogen-bonding molecules such as water. Theories for both the fluid and solid phases are presented, and ... More

Diffusiophoresis in Cells: a General Non-Equilibrium, Non-Motor Mechanism for the Metabolism-Dependent Transport of Particles in CellsJan 03 2019The more we learn about the cytoplasm of cells, the more we realise that the cytoplasm is not uniform but instead is highly inhomogeneous. In any inhomogeneous solution, there are concentration gradients, and particles move either up or down these gradients ... More

Recent developments in algebraic combinatoricsNov 06 2002Feb 17 2004A survey of three recent developments in algebraic combinatorics: (1) the Laurent phenomenon, (2) Gromov-Witten invariants and toric Schur functions, and (3) toric h-vectors and intersection cohomology. This paper is a continuation of "Recent progress ... More

On the Orbital Evolution of Low Mass Protoplanets in Turbulent, Magnetised DisksAug 23 2005(Abridged).We present the results of MHD simulations of low mass protoplanets interacting with turbulent disks. We calculate the orbital evolution of `planetesimals' and protoplanets with masses in the range 0 < m_p < 30 M_Earth. Planetesimals and protoplanets ... More

Promotion and EvacuationJun 28 2008Jul 21 2008Promotion and evacuation are bijections on the set of linear extensions of a finite poset first defined by Schutzenberger. This paper surveys the basic properties of these two operations and discusses some generalizations.

Multiple-precision zero-finding methods and the complexity of elementary function evaluationApr 20 2010May 30 2010We consider methods for finding high-precision approximations to simple zeros of smooth functions. As an application, we give fast methods for evaluating the elementary functions log(x), exp(x), sin(x) etc. to high precision. For example, if x is a positive ... More

A Simple Approach to Error Reconciliation in Quantum Key DistributionMay 07 2010We discuss the error reconciliation phase in quantum key distribution (QKD) and analyse a simple scheme in which blocks with bad parity (that is, blocks containing an odd number of errors) are discarded. We predict the performance of this scheme and show, ... More

Note on Computing Ratings from EigenvectorsMay 05 2010We consider the problem of computing ratings using the results of games played between a set of n players, and show how this problem can be reduced to computing the positive eigenvectors corresponding to the dominant eigenvalues of certain n by n matrices. ... More

Generalising Tuenter's binomial sumsJul 14 2014Jan 19 2015Tuenter [Fibonacci Quarterly 40 (2002), 175-180] and other authors have considered centred binomial sums of the form \[S_r(n) = \sum_k \binom{2n}{k}|n-k|^r,\] where $r$ and $n$ are non-negative integers. We consider sums of the form \[U_r(n) = \sum_k ... More

Uses of randomness in computationApr 19 2010Apr 20 2010Random number generators are widely used in practical algorithms. Examples include simulation, number theory (primality testing and integer factorization), fault tolerance, routing, cryptography, optimization by simulated annealing, and perfect hashing. ... More

On the periods of generalized Fibonacci recurrencesApr 30 2010We give a simple condition for a linear recurrence (mod 2^w) of degree r to have the maximal possible period 2^(w-1).(2^r-1). It follows that the period is maximal in the cases of interest for pseudo-random number generation, i.e. for 3-term linear recurrences ... More

Nucleation of the crystalline phase of proteins in the presence of semidilute non-adsorbing polymerFeb 14 2001Starting from a protein solution which is metastable with respect to the crystalline phase, the effect of adding semidilute non-adsorbing polymer is considered. It is found to increase the chemical potential of the protein by a few tenths of kT, which ... More

Metastability and nucleation in the dilute fluid phase of a simple model of globular proteinsDec 11 1999The dilute fluid phase of model globular proteins is studied. The model possesses a fluid-fluid transition buried within the fluid-crystal coexistence region, as do some globular proteins. If this fluid-fluid transition is not buried deep inside the fluid-crystal ... More

Generalisation of Levine's prediction for the distribution of freezing temperatures of droplets: A general singular model for ice nucleationJul 29 2013Models without an explicit time dependence, called singular models, are widely used for fitting the distribution of temperatures at which water droplets freeze. In 1950 Levine developed the original singular model. His key assumption was that each droplet ... More

Some long-period random number generators using shifts and xorsApr 19 2010Marsaglia recently introduced a class of xorshift random number generators (RNGs) with periods 2n-1 for n = 32, 64, etc. Here we give a generalisation of Marsaglia's xorshift generators in order to obtain fast and high-quality RNGs with extremely long ... More

A K-theoretic Fulton classSep 28 2018Fulton defined classes in the Chow group of a quasi-projective scheme $M$ which reduce to its Chern classes when $M$ is smooth. When $M$ has a perfect obstruction theory, Siebert gave a formula for its virtual cycle in terms of its total Fulton class. ... More

Some Combinatorial Properties of Hook Lengths, Contents, and Parts of PartitionsJul 02 2008Apr 10 2009This paper proves a generalization of a conjecture of Guoniu Han, inspired originally by an identity of Nekrasov and Okounkov. The main result states that certain sums over partitions p of n, involving symmetric functions of the squares of the hook lengths ... More

George Forsythe's last paperMay 06 2010May 07 2010We describe von Neumann's elegant idea for sampling from the exponential distribution, Forsythe's generalization for sampling from a probability distribution whose density has the form exp(-G(x)), where G(x) is easy to compute (e.g. a polynomial), and ... More

Smith Normal Form in CombinatoricsJan 30 2016Apr 02 2016This paper surveys some combinatorial aspects of Smith normal form, and more generally, diagonal form. The discussion includes general algebraic properties and interpretations of Smith normal form, critical groups of graphs, and Smith normal form of random ... More

Recent Progress in Algebraic CombinatoricsOct 23 2000Dec 19 2001A survey of recent progress in three areas of algebraic combinatorics: (1) the Saturation Conjecture for Littlewood-Richardson coefficients, (2) the n! and (n+1)^{n-1} conjectures, and (3) longest increasing subsequences of permutations.

Asymptotic approximation of central binomial coefficients with rigorous error boundsAug 17 2016Sep 15 2016We show that a well-known asymptotic series for the logarithm of the central binomial coefficient is strictly enveloping, so the error incurred in truncating the series is of the same sign as the next term, and is bounded in magnitude by that term. We ... More

The Rank and Minimal Border Strip Decompositions of a Skew PartitionSep 14 2001Nazarov and Tarasov recently generalized the notion of the rank of a partition to skew partitions. We give several characterizations of the rank of a skew partition and one possible characterization that remains open. One of the characterizations involves ... More

Stability of fast algorithms for structured linear systemsMay 05 2010We survey the numerical stability of some fast algorithms for solving systems of linear equations and linear least squares problems with a low displacement-rank structure. For example, the matrices involved may be Toeplitz or Hankel. We consider algorithms ... More

Unrestricted algorithms for elementary and special functionsApr 21 2010We describe some "unrestricted" algorithms which are useful for the computation of elementary and special functions when the precision required is not known in advance. Several general classes of algorithms are identified and illustrated by examples. ... More

Old and new algorithms for piMar 12 2013This is a letter to the editor concerning Semjon Adlaj's article "An eloquent formula for the perimeter of an ellipse", AMS Notices 59, 8 (2012), 1094-1099.

Some Linear Recurrences Motivated by Stern's Diatomic ArrayJan 15 2019We define a triangular array closely related to Stern's diatomic array and show that for a fixed integer $r\geq 1$, the sum $u_r(n)$ of the $r$th powers of the entries in row $n$ satisfy a linear recurrence with constant coefficients. The proof technique ... More

Skinning mapsDec 19 2006Jul 01 2009Let M be a hyperbolic 3-manifold with nonempty totally geodesic boundary. We prove that there are upper and lower bounds on the diameter of the skinning map of M that depend only on the volume of the hyperbolic structure with totally geodesic boundary, ... More

The complexity of multiple-precision arithmeticApr 21 2010In studying the complexity of iterative processes it is usually assumed that the arithmetic operations of addition, multiplication, and division can be performed in certain constant times. This assumption is invalid if the precision required increases ... More

On asymptotic approximations to the log-Gamma and Riemann-Siegel theta functionsSep 13 2016We give new bounds on the error in the asymptotic approximation of the log-Gamma function $\ln\Gamma(z)$ (also $\ln\Gamma(z+\frac{1}{2}))$ for complex $z$ in the right half-plane. These improve on bounds by Hare (1997) and Spira (1971). We show that $|R_{k+1}(z)| ... More

Liquids that form due to dynamics of the molecules that depend on the local densityMar 24 2015Mar 26 2015RNA molecules in living cells form what look like liquid droplets formed by liquid/liquid phase separation. But unlike the molecules in conventional phase separating mixtures, RNA molecules are transported by molecular motors that consume energy and so ... More

Intersections and joins of free groupsJan 31 2008Aug 20 2008Let H and K be subgroups of a free group of ranks h and k \geq h. We prove the following strong form of Burns' inequality: rank(H \cap K) - 1 \leq 2(h-1)(k-1) - (h-1)(rank(H \vee K) -1). A corollary of this, also obtained by L. Louder and D. B. McReynolds, ... More

Totally geodesic boundaries of knot complementsMar 25 2004Jul 21 2004Given a compact orientable 3-manifold M whose boundary is a hyperbolic surface and a simple closed curve C in its boundary, every knot in M is homotopic to one whose complement admits a complete hyperbolic structure with totally geodesic boundary in which ... More

A fast vectorised implementation of Wallace's normal random number generatorApr 19 2010Wallace has proposed a new class of pseudo-random generators for normal variates. These generators do not require a stream of uniform pseudo-random numbers, except for initialisation. The inner loops are essentially matrix-vector multiplications and are ... More

Towards a fully predictive model of flight paths in pigeons navigating in the familiar area: prediction across differing individualsApr 20 2016This paper will detail the basis of our previously developed predictive model for pigeon flight paths based on observations of the specific individual being predicted. We will then describe how this model can be adapted to predict the flight of a new, ... More

Model specification via sequential coherence and backward inductionFeb 21 2015This paper describes how to specify probability models for data analysis via a backward induction procedure. The new approach yields coherent, prior-free uncertainty assessment. After presenting some intuition-building examples, the new approach is applied ... More

On computing factors of cyclotomic polynomialsApr 30 2010For odd square-free n > 1 the n-th cyclotomic polynomial satisfies an identity of Gauss. There are similar identity of Aurifeuille, Le Lasseur and Lucas. These identities all involve certain polynomials with integer coefficients. We show how these coefficients ... More

The myth of equidistribution for high-dimensional simulationMay 08 2010A pseudo-random number generator (RNG) might be used to generate w-bit random samples in d dimensions if the number of state bits is at least dw. Some RNGs perform better than others and the concept of equidistribution has been introduced in the literature ... More

Fast normal random number generators on vector processorsApr 19 2010Apr 20 2010We consider pseudo-random number generators suitable for vector processors. In particular, we describe vectorised implementations of the Box-Muller and Polar methods, and show that they give good performance on the Fujitsu VP2200. We also consider some ... More

Ordering Events in Minkowski SpaceJan 17 2005May 27 2005We are given k points (events) in (n+1)-dimensional Minkowski space. Using the theory of hyperplane arrangments and chromatic polynomials, we obtain information the number of different orders in which the events can occur in different reference frames ... More

Longest alternating subsequences of permutationsNov 16 2005The length is(w) of the longest increasing subsequence of a permutation w in the symmetric group S_n has been the object of much investigation. We develop comparable results for the length as(w) of the longest alternating subsequence of w, where a sequence ... More

On the precision attainable with various floating-point number systemsApr 20 2010For scientific computations on a digital computer the set of real number is usually approximated by a finite set F of "floating-point" numbers. We compare the numerical accuracy possible with difference choices of F having approximately the same range ... More

Diffusiophoresis in Cells: a General Non-Equilibrium, Non-Motor Mechanism for the Metabolism-Dependent Transport of Particles in CellsJan 03 2019Mar 04 2019The more we learn about the cytoplasm of cells, the more we realise that the cytoplasm is not uniform but instead is highly inhomogeneous. In any inhomogeneous solution, there are concentration gradients, and particles move either up or down these gradients ... More

Alternating permutations and symmetric functionsMar 21 2006Jul 05 2006We use the theory of symmetric functions to enumerate various classes of alternating permutations w of {1,2,...,n}. These classes include the following: (1) both w and w^{-1} are alternating, (2) w has certain special shapes, such as (m-1,m-2,...,1), ... More

Two Enumerative Results on Cycles of PermutationsJan 14 2009Answering a question of Bona, it is shown that for n>1 the probability that 1 and 2 are in the same cycle of a product of two n-cycles on the set {1,2,...,n} is 1/2 if n is odd and 1/2 - 2/(n-1){n+2) if n is even. Another result concerns the generating ... More

Generalized Riffle Shuffles and Quasisymmetric FunctionsDec 03 1999Mar 30 2001This paper concerns a probability distribution on the symmetric group generalizing the riffle shuffle of Bayer, Diaconis, and others. There are close connections with the theory of quasisymmetric and symmetric functions.

Planetary Migration in Protoplanetary DisksApr 27 2018The known exoplanet population displays a great diversity of orbital architectures, and explaining the origin of this is a major challenge for planet formation theories. The gravitational interaction between young planets and their protoplanetary disks ... More

Some Schubert shenanigansApr 04 2017Apr 05 2017We give a conjectured evaluation of the determinant of a certain matrix $\tilde{D}(n,k)$. The entries of $\tilde{D}(n,k)$ are either 0 or specializations $\mathfrak{S}_w(1,\dots,1)$ of Schubert polynomials. The conjecture implies that the weak order of ... More

Finding D-optimal designs by randomised decomposition and switchingDec 20 2011Aug 11 2012The Hadamard maximal determinant (maxdet) problem is to find the maximum determinant D(n) of a square {+1, -1} matrix of given order n. Such a matrix with maximum determinant is called a saturated D-optimal design. We consider some cases where n > 2 is ... More

Return Optimization Securities and Other Remarkable Structured Investment Products: Indicators of Future Outcomes for U.S. Treasuries?Apr 03 2018We analyze four structured products that have caused severe losses to investors in recent years. These products are: return optimization securities, yield magnet notes, reverse exchangeable securities, and principal-protected notes. We describe the basic ... More

Parallel algorithms in linear algebraApr 30 2010This report provides an introduction to algorithms for fundamental linear algebra problems on various parallel computer architectures, with the emphasis on distributed-memory MIMD machines. To illustrate the basic concepts and key issues, we consider ... More

The Smith Normal Form of a Specialized Jacobi-Trudi MatrixAug 19 2015Aug 26 2015Let $\mathrm{JT}_\lambda$ be the Jacobi-Trudi matrix corresponding to the partition $\lambda$, so $\det\mathrm{JT}_\lambda$ is the Schur function $s_\lambda$ in the variables $x_1,x_2,\dots$. Set $x_1=\cdots=x_n=1$ and all other $x_i=0$. Then the entries ... More

Distance Correlation: A New Tool for Detecting Association and Measuring Correlation Between Data SetsAug 14 2017The difficulties of detecting association, measuring correlation, and establishing cause and effect have fascinated mankind since time immemorial. Democritus, the Greek philosopher, underscored well the importance and the difficulty of proving causality ... More

Algebraic methods toward higher-order probability inequalities, IIOct 06 2004Let (L,\preccurlyeq) be a finite distributive lattice, and suppose that the functions f_1,f_2:L\to R are monotone increasing with respect to the partial order \preccurlyeq. Given \mu a probability measure on L, denote by E(f_i) the average of f_i over ... More

Further analysis of the binary Euclidean algorithmMar 12 2013The binary Euclidean algorithm is a variant of the classical Euclidean algorithm. It avoids multiplications and divisions, except by powers of two, so is potentially faster than the classical algorithm on a binary machine. We describe the binary algorithm ... More

A Survey of Alternating PermutationsDec 21 2009This survey of alternating permutations and Euler numbers includes refinements of Euler numbers, other occurrences of Euler numbers, longest alternating subsequences, umbral enumeration of classes of alternating permutations, and the cd-index of the symmetric ... More

The Descent Set and Connectivity Set of a PermutationJul 11 2005The descent set D(w) of a permutation w of 1,2,...,n is a standard and well-studied statistic. We introduce a new statistic, the connectivity set C(w), and show that it is a kind of dual object to D(w). The duality is stated in terms of the inverse of ... More

An equivalence relation on the symmetric group and multiplicity-free flag h-vectorsAug 17 2012We consider the equivalence relation ~ on the symmetric group S_n generated by the interchange of two adjacent elements a_i and a_{i+1} of w=a_1 ... a_n in S_n such that |a_i - a_{i+1}|=1. We count the number of equivalence classes and the sizes of equivalence ... More

An asymptotic expansion inspired by RamanujanApr 30 2010Corollary 2, Entry 9, Chapter 4 of Ramanujan's first notebook claims that a certain sum is asymptotic to ln(x) + gamma, where x is a real variable in the sum and gamma is Euler's constant. Ramanujan's claim is known to be correct for the case n = 1, but ... More

A conjectured combinatorial interpretation of the normalized irreducible character values of the symmetric groupJun 19 2006Jul 05 2006In math.CO/0109093 the author obtained a formula for the value of an irreducible symmetric group character indexed by a partition of rectangular shape. In the present paper this formula is (conjecturally) generalized to arbitrary shapes.

Symmetric M-treeApr 23 2010The M-tree is a paged, dynamically balanced metric access method that responds gracefully to the insertion of new objects. To date, no algorithm has been published for the corresponding Delete operation. We believe this to be non-trivial because of the ... More

Chains in the Bruhat orderFeb 16 2005We study a family of polynomials whose values express degrees of Schubert varieties in the generalized complex flag manifold G/B. The polynomials are given by weighted sums over saturated chains in the Bruhat order. We derive several explicit formulas ... More

The Smith Normal Form Distribution of a Random Integer MatrixMay 30 2015Sep 08 2015We show that the density $\mu$ of the Smith normal form (SNF) of a random integer matrix exists and equals a product of densities $\mu_{p^s}$ of SNF over $\mathbb{Z}/p^s\mathbb{Z}$ with $p$ a prime and $s$ some positive integer. Our approach is to connect ... More

Bottom Schur functionsNov 21 2003Sep 20 2004We give a basis for the space V spanned by the lowest degree part \hat{s}_\lambda of the expansion of the Schur symmetric functions s_\lambda in terms of power sums, where we define the degree of the power sum p_i to be 1. In particular, the dimension ... More

Minorities report: optimal incentives for collective intelligenceNov 11 2016Collective intelligence is the ability of a group to perform more effectively than any individual alone. Diversity among group members is a key condition for the emergence of collective intelligence, but maintaining diversity is challenging in the face ... More

Degeneracy loci, virtual cycles and nested Hilbert schemes IIFeb 11 2019We express nested Hilbert schemes of points and curves on a smooth projective surface as "virtual resolutions" of degeneracy loci of maps of vector bundles on smooth ambient spaces. We show how to modify the resulting obstruction theories to produce the ... More

Degeneracy loci, virtual cycles and nested Hilbert schemes ISep 18 2017Feb 11 2019Given a map of vector bundles on a smooth variety, consider the deepest degeneracy locus where its rank is smallest. We show it carries a natural perfect obstruction theory whose virtual cycle can be calculated by the Thom-Porteous formula. We show nested ... More

Global models of planetary system formation in radiatively-inefficient protoplanetary discsDec 13 2011(Abridged) We present the results of N-body simulations of planetary systems formation in radiatively-inefficient disc models, where positive corotation torques may counter the rapid inward migration of low mass planets driven by Lindblad torques. The ... More

On the ionisation fraction in protoplanetary disks III. The effect of X-ray flares on gas-phase chemistryMay 11 2006Context. Recent observations of the X-ray emission from T Tauri stars in the Orion nebula have shown that they undergo frequent outbursts in their X-ray luminosity. These X-ray flares are characterised by increases in luminosity by two orders of magnitude, ... More

On the Ionisation Fraction in Protoplanetary Disks I: Comparing Different Reaction NetworksSep 19 2005Nov 09 2005We calculate the ionisation fraction in protostellar disk models using a number of different chemical reaction networks, including gas-phase and gas-grain reaction schemes. The disk models we consider are conventional alpha-disks, which include viscous ... More

Three-dimensional simulations of multiple protoplanets embedded in a protostellar discNov 26 2008Protoplanet eccentricities of e >~ H/r can slow or reverse migration, but previous 2D studies have shown that gravitational scattering cannot maintain significant planet eccentricities against disc-induced damping. We simulate the evolution of low-mass ... More

On the evolution of multiple low mass planets embedded in a circumbinary discDec 06 2007Previous work has shown that the tidal interaction between a binary system and a circumbinary disc leads to the formation of a large inner cavity in the disc. Subsequent formation and inward migration of a low mass planet causes it to become trapped at ... More

On the Ionisation Fraction in Protoplanetary Disks II: The Effect of Turbulent Mixing on Gas--phase ChemistrySep 19 2005We calculate the ionisation fraction in protostellar disk models using two different gas-phase chemical networks, and examine the effect of turbulent mixing by modelling the diffusion of chemical species vertically through the disk. The aim is to determine ... More

The great trinomial huntMay 11 2010We describe a search for primitive trinomials of high degree and its interaction with the Great Internet Mersenne prime search (GIMPS). The search is complete for trinomials whose degree is the exponent of a Mersenne prime, for all 47 currently known ... More

An O(M(n) log n) algorithm for the Jacobi symbolApr 13 2010Jun 02 2010The best known algorithm to compute the Jacobi symbol of two n-bit integers runs in time O(M(n) log n), using Sch\"onhage's fast continued fraction algorithm combined with an identity due to Gauss. We give a different O(M(n) log n) algorithm based on ... More

Unavoidable Multicoloured Families of ConfigurationsSep 15 2014Sep 29 2014Balogh and Bollob\'as [{\em Combinatorica 25, 2005}] prove that for any $k$ there is a constant $f(k)$ such that any set system with at least $f(k)$ sets reduces to a $k$-star, an $k$-costar or an $k$-chain. They proved $f(k)<(2k)^{2^k}$. Here we improve ... More

A Distributive Lattice Connected with Arithmetic Progressions of Length ThreeDec 19 2013Aug 17 2014Let $\mathcal{T}$ be a collection of 3-element subsets $S$ of $\{1, \ldots,n\}$ with the property that if $i<j<k$ and $a<b<c$ are two 3-element subsets in $S$, then there exists an integer sequence $x_1 < x_2 < \cdots < x_n$ such that $x_i, x_j, x_k$ ... More

A Note on the Symmetric Powers of the Standard Representation of S_nFeb 14 2000In this paper, we prove that the dimension of the space spanned by the characters of the symmetric powers of the standard n-dimensional representation of the symmetric group S_n is asymptotic to n^2/2. This is proved by using generating functions to obtain ... More

A note on the asymptotics of the number of O-sequences of given lengthOct 17 2018Apr 01 2019We look at the number $L(n)$ of $O$-sequences of length $n$. Recall that an $O$-sequence can be defined algebraically as the Hilbert function of a standard graded $k$-algebra, or combinatorially as the $f$-vector of a multicomplex. The sequence $L(n)$ ... More

Polynomial Coefficient EnumerationNov 21 2008Let $f(x_1,...,x_k)$ be a polynomial over a field $K$. This paper considers such questions as the enumeration of the number of nonzero coefficients of $f$ or of the number of coefficients equal to $\alpha\in K^*$. For instance, if $K=\ff_q$ then a matrix ... More

On the growth and stability of Trojan planetsNov 27 2008We investigate the stability of those low-mass Trojan planets that form in a protoplanetary disc and subsequently accrete gas to become gas giants. We calculate their evolution before, during, and after gas disc dispersal. A two-dimensional hydrodynamics ... More

Turbulent transport and its effect on the dead zone in protoplanetary discsFeb 29 2008Protostellar accretion discs have cool, dense midplanes where externally originating ionisation sources such as X-rays or cosmic rays are unable to penetrate. This suggests that for a wide range of radii, MHD turbulence can only be sustained in the surface ... More

On the accumulation of solid bodies in global turbulent protoplanetary disc modelsSep 23 2005We study the migration of solid bodies in turbulent protoplanetary accretion discs by means of global MHD simulations. The bodies range in size from 5 centimetres up to 1 metre, and so include objects whose migration is expected to be the most rapid due ... More

Migration and gas accretion scenarios for the Kepler 16, 34 and 35 circumbinary planetsJul 02 2013Several circumbinary planets have been detected by the Kepler mission. Recent work has emphasized the difficulty of forming these planets at their observed locations. It has been suggested that these planets formed further out in their discs and migrated ... More

On the formation and migration of giant planets in circumbinary discsMar 13 2008We present the results of hydrodynamic simulations of the formation and subsequent orbital evolution of giant planets embedded in a circumbinary disc. We assume that a 20 earth masses core has migrated to the edge of the inner cavity formed by the binary ... More

Constraints on resonant-trapping for two planets embedded in a protoplanetary discFeb 14 2008We investigate the evolution of two-planet systems embedded in a protoplanetary disc, which are composed of a Jupiter-mass planet plus another body located further out in the disc. We consider outermost planets with masses ranging from 10 earth masses ... More

On the rank function of a differential posetNov 18 2011Apr 28 2012We study $r$-differential posets, a class of combinatorial objects introduced in 1988 by the first author, which gathers together a number of remarkable combinatorial and algebraic properties, and generalizes important examples of ranked posets, including ... More

Quintic threefolds and Fano elevenfoldsOct 24 2014Nov 17 2015The derived category of coherent sheaves on a general quintic threefold is a central object in mirror symmetry. We show that it can be embedded into the derived category of a certain Fano elevenfold. Our proof also generates related examples in different ... More