arXiv subjects

Ben Smith

Publications and source records attributed to Ben Smith.

23 records · Page 2Linked to original sources

arXiv2020

Matching fields and lattice points of simplices

We show that the Chow covectors of a linkage matching field define a bijection between certain degree vectors and lattice points, and we demonstrate how one can recover the linkage matching field from this bijection. This resolves two open questions from Sturmfels and Zelevinsky (1993) on linkage matching fields. For this, we give an explicit construction that associates a bipartite incidence graph of an ordered partition of a common set to each lattice point in a dilated simplex. Given a triangulation of a product of two simplices encoded by a set of spanning trees on a bipartite node set, we similarly prove that the bijection from left to right degree vectors of the trees is enough to recover the triangulation. As additional results, we show a cryptomorphic description of linkage matching fields and characterise the flip graph of a linkage matching field in terms of its prodsimplicial flag complex. Finally, we relate our findings to transversal matroids through the tropical Stiefel map.

math.CO↗

arXiv2019

Computing Tropical Varieties in Macaulay2

We introduce a package for doing tropical computations in Macaulay2. The package draws on the functionality of Gfan and Polymake while making the process as simple as possible for the end user. This provides a powerful and user friendly tool for computing tropical varieties requiring little prerequisite knowledge.

math.AG↗

arXiv2018

Commutative Algebra of Generalised Frobenius Numbers

We study commutative algebra arising from generalised Frobenius numbers. The $k$-th (generalised) Frobenius number of natural numbers $(a_1,\dots,a_n)$ is the largest natural number that cannot be written as a non-negative integral combination of $(a_1,\dots,a_n)$ in $k$ distinct ways. Suppose that $L$ is the lattice of integers points of $(a_1,\dots,a_n)^{\perp}$. Taking cue from the concept of lattice modules due to Bayer and Sturmfels, we define generalised lattice modules $M_L^{(k)}$ whose Castelnuovo-Mumford regularity captures the $k$-th Frobenius number of $(a_1,\dots,a_n)$. We study the sequence $\{M_L^{(k)}\}_{k=1}^{\infty}$ of generalised lattice modules providing an explicit characterisation of their minimal generators. We show that there are only finitely many isomorphism classes of generalized lattice modules. As a consequence of our commutative algebraic approach, we show that the sequence of generalised Frobenius numbers forms a generalised arithmetic progression. We also construct an algorithm to compute the $k$-th Frobenius number.

math.AC↗

arXiv2018

EMME: a formal tool for ECMAScript Memory Model Evaluation

Nearly all web-based interfaces are written in JavaScript. Given its prevalence, the support for high performance JavaScript code is crucial. The ECMA Technical Committee 39 (TC39) has recently extended the ECMAScript language (i.e., JavaScript) to support shared memory accesses between different threads. The extension is given in terms of a natural language memory model specification. In this paper we describe a formal approach for validating both the memory model and its implementations in various JavaScript engines. We first introduce a formal version of the memory model and report results on checking the model for consistency and other properties. We then introduce our tool, EMME, built on top of the Alloy analyzer, which leverages the model to generate all possible valid executions of a given JavaScript program. Finally, we report results using EMME together with small test programs to analyze industrial JavaScript engines. We show that EMME can find bugs as well as missed opportunities for optimization.

cs.LO↗

arXiv2011

On the nonlocal viscosity kernel of mixtures

In this report we investigate the multiscale hydrodynamical response of a liquid as a function of mixture composition. This is done via a series of molecular dynamics simulations where the wave vector dependent viscosity kernel is computed for three mixtures each with 7-15 different compositions. We observe that the nonlocal viscosity kernel is dependent on composition for simple atomic mixtures for all the wave vectors studied here, however, for a model polymer melt mixture the kernel is independent of composition for large wave vectors. The deviation from ideal mixing is also studied. Here it is shown that a Lennard-Jones mixture follows the ideal mixing rule surprisingly well for a large range of wave vectors, whereas for both the Kob-Andersen mixture and the polymer melt large deviations are found. Furthermore, for the polymer melt the deviation is wave vector dependent such that there exists a critical length scale at which the ideal mixing goes from under-estimating to over-estimating the viscosity.

cond-mat.soft↗