Blowups with log canonical singularities

We show that the minimum weight of a weighted blowup of Ad with [epsilon]- log canonical singularities is bounded by a constant depending only on [epsilon] and d . This was conjectured by Birkar. Using the recent classification of 4_ dimensional empty simplices by Iglesias-Valiño and Santos, we work out an explicit bound for blowups of A4 with terminal singular-ities: the smallest weight is always at most 32, and at most 6 in all but finitely many cases.Some background on birational geometry was supplied to Sankaran by Anne-Sophie Kaloghiros. The explanations here relating these results to their wider context are largely hers, but errors and omissions in such explanations are definitely ours. Parts of this work were carried out while Sankaran was visiting Fukuoka University and KIAS, Seoul: he thanks both for hospitality and a helpful environment. Work of Santos was supported by grant MTM2017-83750-P of the Spanish Ministry of Economy and Competitiveness and by the Einstein Foundation Berlin under grant EVF-2015-230. We also thank the organisers of MEGA 2019 (Madrid), where the two authors first met and discussed these questions

Regular cylindrical algebraic decomposition

We show that a strong well-based cylindrical algebraic decomposition P of a bounded semi-algebraic set S is a regular cell decomposition, in any dimension and independently of the method by which P is constructed. Being well-based is a global condition on P that holds for the output of many widely used algorithms. We also show the same for S of dimension at most 3 and P a strong cylindrical algebraic decomposition that is locally boundary simply connected: this is a purely local extra condition.</p

Lazard-style CAD and Equational Constraints

McCallum-style Cylindrical Algebra Decomposition (CAD) is a major improvement on the original Collins version, and has had many subsequent advances, notably for total or partial equational constraints. But it suffers from a problem with nullification. The recently-justified Lazard-style CAD does not have this problem. However, transporting the equational constraints work to Lazard-style does reintroduce nullification issues. This paper explains the problem, and the solutions to it, based on the second author's Ph.D. thesis and the Brown--McCallum improvement to Lazard. With a single equational constraint, we can gain the same improvements in Lazard-style as in McCallum-style CAD . Moreover, our approach does not fail where McCallum would due to nullification. Unsurprisingly, it does not achieve the same level of improvement as it does in the non-nullified cases. We also consider the case of multiple equational constraints.Comment: 9 page

Regular cylindrical algebraic decomposition

We show that a strong well-based cylindrical algebraic decomposition P of a bounded semi-algebraic set S is a regular cell decomposition, in any dimension and independently of the method by which P is constructed. Being well-based is a global condition on P that holds for the output of many widely used algorithms. We also show the same for S of dimension at most 3 and P a strong cylindrical algebraic decomposition that is locally boundary simply connected: this is a purely local extra condition.</p

Lattice Boltzmann simulations of low-Reynolds-number flows past fluidized spheres : effect of inhomogeneities on the drag force

This work is supported by a grant from the ExxonMobil Research & Engineering Co., and by a fellowship awarded to G.J.R. by the National Science Foundation (DGE-1148900).Peer reviewedPostprin

On Benefits of Equality Constraints in Lex-Least Invariant CAD (Extended Abstract

There are two relevant methods for CAD: McCallum [1984] which used order invariant CAD's and Lazard [Lazard1994, McCallumetal2019] which used lex-least invariant CADs, and doesn't have the nullification problem of McCallum [1984]. McCallum [1999] was the first to prove a CAD operator based on McCallum [1984], that took advantage of an equational constraint.In this paper, we do the same for Lazard's method. This takes in a lex-least invariant CAD of \RR^{n-1} as input and outputs a sign invariant CAD of \RR^n: consequently, it cannot be used recursively, but only for $x_n$, the first variable to be projected. In the further steps of the projection phase, we use Lazard's original projection operator. Nonetheless, reducing the output in the first step has a domino effect throughout the remaining steps, which significantly reduces the complexity. The long-term goal is to find a general projection operator that takes advantage of the equality constraint and can be used recursively, and this operator gives an important first step in that direction.<br/