Numerically determined transport laws for fingering ("thermohaline") convection in astrophysics

We present the first three-dimensional simulations of fingering convection performed in a parameter regime close to the one relevant for astrophysics, and reveal the existence of simple asymptotic scaling laws for turbulent heat and compositional transport. These laws can straightforwardly be extrapolated to the true astrophysical regime. Our investigation also indicates that thermocompositional "staircases," a key consequence of fingering convection in the ocean, cannot form spontaneously in the fingering regime in stellar interiors. Our proposed empirically-determined transport laws thus provide simple prescriptions for mixing by fingering convection in a variety of astrophysical situations, and should, from here on, be used preferentially over older and less accurate parameterizations. They also establish that fingering convection does not provide sufficient extra mixing to explain observed chemical abundances in RGB stars.Comment: Submitted to ApJ Letters on October 29th. 15 pages, 4 figures. See Garaud 2010 for companion pape

Combinatorial simplex algorithms can solve mean payoff games

A combinatorial simplex algorithm is an instance of the simplex method in which the pivoting depends on combinatorial data only. We show that any algorithm of this kind admits a tropical analogue which can be used to solve mean payoff games. Moreover, any combinatorial simplex algorithm with a strongly polynomial complexity (the existence of such an algorithm is open) would provide in this way a strongly polynomial algorithm solving mean payoff games. Mean payoff games are known to be in NP and co-NP; whether they can be solved in polynomial time is an open problem. Our algorithm relies on a tropical implementation of the simplex method over a real closed field of Hahn series. One of the key ingredients is a new scheme for symbolic perturbation which allows us to lift an arbitrary mean payoff game instance into a non-degenerate linear program over Hahn series.Comment: v1: 15 pages, 3 figures; v2: improved presentation, introduction expanded, 18 pages, 3 figure

The horofunction boundary of the Hilbert geometry

We investigate the horofunction boundary of the Hilbert geometry defined on an arbitrary finite-dimensional bounded convex domain D. We determine its set of Busemann points, which are those points that are the limits of `almost-geodesics'. In addition, we show that any sequence of points converging to a point in the horofunction boundary also converges in the usual sense to a point in the Euclidean boundary of D. We prove that all horofunctions are Busemann points if and only if the set of extreme sets of the polar of D is closed in the Painleve-Kuratowski topology.Comment: 24 pages, 2 figures; minor changes, examples adde

Tropical Fourier-Motzkin elimination, with an application to real-time verification

We introduce a generalization of tropical polyhedra able to express both strict and non-strict inequalities. Such inequalities are handled by means of a semiring of germs (encoding infinitesimal perturbations). We develop a tropical analogue of Fourier-Motzkin elimination from which we derive geometrical properties of these polyhedra. In particular, we show that they coincide with the tropically convex union of (non-necessarily closed) cells that are convex both classically and tropically. We also prove that the redundant inequalities produced when performing successive elimination steps can be dynamically deleted by reduction to mean payoff game problems. As a complement, we provide a coarser (polynomial time) deletion procedure which is enough to arrive at a simply exponential bound for the total execution time. These algorithms are illustrated by an application to real-time systems (reachability analysis of timed automata).Comment: 29 pages, 8 figure

Unsung heroes: who supports social work students on placement?

Since the introduction of the three year degree programme in 2003, social work education has undergone a number of significant changes. The time students spend on placement has been increased to two hundred days, and the range of placement opportunities and the way in which these placements have been configured has significantly diversified. A consistent feature over the years, however, has been the presence of a Practice Educator (PE) who has guided, assessed and taught the student whilst on placement. Unsurprisingly, the role of the PE and the pivotal relationship they have with the student has been explored in the past and features in social work literature. This paper, however, concentrates on a range of other relationships which are of significance in providing support to students on placement. In particular it draws on research to discuss the role of the university contact tutor, the place of the wider team in which the student is sited, and the support offered by family, friends and others. Placements and the work undertaken by PE’s will continue to be integral to the delivery of social work education. It is, however, essential to recognise and value the often over looked role of others in providing support to students on placement

The Firefighter Problem: A Structural Analysis

We consider the complexity of the firefighter problem where b>=1 firefighters are available at each time step. This problem is proved NP-complete even on trees of degree at most three and budget one (Finbow et al.,2007) and on trees of bounded degree b+3 for any fixed budget b>=2 (Bazgan et al.,2012). In this paper, we provide further insight into the complexity landscape of the problem by showing that the pathwidth and the maximum degree of the input graph govern its complexity. More precisely, we first prove that the problem is NP-complete even on trees of pathwidth at most three for any fixed budget b>=1. We then show that the problem turns out to be fixed parameter-tractable with respect to the combined parameter "pathwidth" and "maximum degree" of the input graph

Violator Spaces: Structure and Algorithms

Sharir and Welzl introduced an abstract framework for optimization problems, called LP-type problems or also generalized linear programming problems, which proved useful in algorithm design. We define a new, and as we believe, simpler and more natural framework: violator spaces, which constitute a proper generalization of LP-type problems. We show that Clarkson's randomized algorithms for low-dimensional linear programming work in the context of violator spaces. For example, in this way we obtain the fastest known algorithm for the P-matrix generalized linear complementarity problem with a constant number of blocks. We also give two new characterizations of LP-type problems: they are equivalent to acyclic violator spaces, as well as to concrete LP-type problems (informally, the constraints in a concrete LP-type problem are subsets of a linearly ordered ground set, and the value of a set of constraints is the minimum of its intersection).Comment: 28 pages, 5 figures, extended abstract was presented at ESA 2006; author spelling fixe

Tropically convex constraint satisfaction

A semilinear relation S is max-closed if it is preserved by taking the componentwise maximum. The constraint satisfaction problem for max-closed semilinear constraints is at least as hard as determining the winner in Mean Payoff Games, a notorious problem of open computational complexity. Mean Payoff Games are known to be in the intersection of NP and co-NP, which is not known for max-closed semilinear constraints. Semilinear relations that are max-closed and additionally closed under translations have been called tropically convex in the literature. One of our main results is a new duality for open tropically convex relations, which puts the CSP for tropically convex semilinaer constraints in general into NP intersected co-NP. This extends the corresponding complexity result for scheduling under and-or precedence constraints, or equivalently the max-atoms problem. To this end, we present a characterization of max-closed semilinear relations in terms of syntactically restricted first-order logic, and another characterization in terms of a finite set of relations L that allow primitive positive definitions of all other relations in the class. We also present a subclass of max-closed constraints where the CSP is in P; this class generalizes the class of max-closed constraints over finite domains, and the feasibility problem for max-closed linear inequalities. Finally, we show that the class of max-closed semilinear constraints is maximal in the sense that as soon as a single relation that is not max-closed is added to L, the CSP becomes NP-hard.Comment: 29 pages, 2 figure

Bounds on the Complexity of Halfspace Intersections when the Bounded Faces have Small Dimension

We study the combinatorial complexity of D-dimensional polyhedra defined as the intersection of n halfspaces, with the property that the highest dimension of any bounded face is much smaller than D. We show that, if d is the maximum dimension of a bounded face, then the number of vertices of the polyhedron is O(n^d) and the total number of bounded faces of the polyhedron is O(n^d^2). For inputs in general position the number of bounded faces is O(n^d). For any fixed d, we show how to compute the set of all vertices, how to determine the maximum dimension of a bounded face of the polyhedron, and how to compute the set of bounded faces in polynomial time, by solving a polynomial number of linear programs