On volumes of hyperideal tetrahedra with constrained edge lengths

Hyperideal tetrahedra are the fundamental building blocks of hyperbolic 3-manifolds with geodesic boundary. The study of their geometric properties (in particular, of their volume) has applications also in other areas of low-dimensional topology, like the computation of quantum invariants of 3-manifolds and the use of variational methods in the study of circle packings on surfaces. The Schl\"afli formula neatly describes the behaviour of the volume of hyperideal tetrahedra with respect to dihedral angles, while the dependence of volume on edge lengths is worse understood. In this paper we prove that, for every $\ell<\ell_0$, where $\ell_0$ is an explicit constant, regular hyperideal tetrahedra of edge length $\ell$ maximize the volume among hyperideal tetrahedra whose edge lengths are all not smaller than $\ell$. This result provides a fundamental step in the computation of the ideal simplicial volume of an infinite family of hyperbolic 3-manifolds with geodesic boundary.Comment: 20 pages, 2 figures, Some minor changes, To appear in Periodica Mathematica Hungaric

Faster Geometric Algorithms via Dynamic Determinant Computation

The computation of determinants or their signs is the core procedure in many important geometric algorithms, such as convex hull, volume and point location. As the dimension of the computation space grows, a higher percentage of the total computation time is consumed by these computations. In this paper we study the sequences of determinants that appear in geometric algorithms. The computation of a single determinant is accelerated by using the information from the previous computations in that sequence. We propose two dynamic determinant algorithms with quadratic arithmetic complexity when employed in convex hull and volume computations, and with linear arithmetic complexity when used in point location problems. We implement the proposed algorithms and perform an extensive experimental analysis. On one hand, our analysis serves as a performance study of state-of-the-art determinant algorithms and implementations. On the other hand, we demonstrate the supremacy of our methods over state-of-the-art implementations of determinant and geometric algorithms. Our experimental results include a 20 and 78 times speed-up in volume and point location computations in dimension 6 and 11 respectively.Comment: 29 pages, 8 figures, 3 table

The simplicial volume of 3-manifolds with boundary

We provide sharp lower bounds for the simplicial volume of compact $3$-manifolds in terms of the simplicial volume of their boundaries. As an application, we compute the simplicial volume of several classes of $3$-manifolds, including handlebodies and products of surfaces with the interval. Our results provide the first exact computation of the simplicial volume of a compact manifold whose boundary has positive simplicial volume. We also compute the minimal number of tetrahedra in a (loose) triangulation of the product of a surface with the interval.Comment: 24 pages, 5 figures. Section 6 has been removed, and will appear in a separate paper by the same authors. This version has been accepted for publication by the Journal of Topolog

D-brane gauge theories from toric singularities of the form C3/ΓC^3/\Gamma and C4/ΓC^4/\Gamma

We discuss examples of D-branes probing toric singularities, and the computation of their world-volume gauge theories from the geometric data of the singularities. We consider several such examples of D-branes on partial resolutions of the orbifolds ${\bf C^3/Z_2\times Z_2}$,${\bf C^3/Z_2\times Z_3}$ and ${\bf C^4/Z_2\times Z_2 \times Z_2}$.Comment: 34+1 pages, References added, Version to appear in Nuclear Physics

Dispersion of Mass and the Complexity of Randomized Geometric Algorithms

How much can randomness help computation? Motivated by this general question and by volume computation, one of the few instances where randomness provably helps, we analyze a notion of dispersion and connect it to asymptotic convex geometry. We obtain a nearly quadratic lower bound on the complexity of randomized volume algorithms for convex bodies in R^n (the current best algorithm has complexity roughly n^4, conjectured to be n^3). Our main tools, dispersion of random determinants and dispersion of the length of a random point from a convex body, are of independent interest and applicable more generally; in particular, the latter is closely related to the variance hypothesis from convex geometry. This geometric dispersion also leads to lower bounds for matrix problems and property testing.Comment: Full version of L. Rademacher, S. Vempala: Dispersion of Mass and the Complexity of Randomized Geometric Algorithms. Proc. 47th IEEE Annual Symp. on Found. of Comp. Sci. (2006). A version of it to appear in Advances in Mathematic

Two-point functions of quenched lattice QCD in Numerical Stochastic Perturbation Theory

We summarize the higher-loop perturbative computation of the ghost and gluon propagators in SU(3) Lattice Gauge Theory. Our final aim is to compare with results from lattice simulations in order to expose the genuinely non-perturbative content of the latter. By means of Numerical Stochastic Perturbation Theory we compute the ghost and gluon propagators in Landau gauge up to three and four loops. We present results in the infinite volume and $a \to 0$ limits, based on a general fitting strategy.Comment: 3 pages, 5 figures, talk at conference QCHS-IX, Madrid 201

An Analytical Solution for Probabilistic Guarantees of Reservation Based Soft Real-Time Systems

We show a methodology for the computation of the probability of deadline miss for a periodic real-time task scheduled by a resource reservation algorithm. We propose a modelling technique for the system that reduces the computation of such a probability to that of the steady state probability of an infinite state Discrete Time Markov Chain with a periodic structure. This structure is exploited to develop an efficient numeric solution where different accuracy/computation time trade-offs can be obtained by operating on the granularity of the model. More importantly we offer a closed form conservative bound for the probability of a deadline miss. Our experiments reveal that the bound remains reasonably close to the experimental probability in one real-time application of practical interest. When this bound is used for the optimisation of the overall Quality of Service for a set of tasks sharing the CPU, it produces a good sub-optimal solution in a small amount of time.Comment: IEEE Transactions on Parallel and Distributed Systems, Volume:27, Issue: 3, March 201