Distributed optimization for big data applications

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Bestvina–Brady Morse theory on hyperbolic manifolds

Inspired by the work of Jankiewicz, Norin, and Wise, in this thesis we describe that given a hyperbolic right-angled polytope, a colouring and a set of moves, it produces a hyperbolic manifold M with a map f: M→S¹. We apply this to a family of hyperbolic polytopes studied by Potyagailo and Vinberg, and by analyzing the resulting map we obtain a 5-manifold fibering over the circle, a 6-manifold with a perfect circle-valued Morse function, and a 7-manifold and a 8-manifold which fiber algebraically. These results are joint work with Giovanni Italiano and Bruno Martelli

On the shape factor of interaction laws for a non-local approximation of the Sobolev norm and the total variation

We consider the family of non-local and non-convex functionals introduced by H. Brezis and H.-M. Nguyen in a recent paper. These functionals Gamma-converge to a multiple of the Sobolev norm or the total variation, depending on a summability exponent, but the exact values of the constants are unknown in many cases. We describe a new approach to the Gamma-convergence result that leads in some special cases to the exact value of the constants, and to the existence of smooth recovery families.Comment: Compte-rendu that summarizes the strategy developed in ArXiv:1708.01231 and ArXiv:1712.04413. This version extends the previous one keeping into account the changes in the above papers. 9 page

Hyperbolic 5-manifolds that fiber over S1S^1

We exhibit some finite-volume cusped hyperbolic 5-manifolds that fiber over the circle. These include the smallest hyperbolic 5-manifold known, discovered by Ratcliffe and Tschantz. As a consequence, we build a finite type subgroup of a hyperbolic group that is not hyperbolic.Comment: 35 pages, 13 figure

Hyperbolic manifolds that fiber algebraically up to dimension 8

We construct some cusped finite-volume hyperbolic $n$-manifolds $M_n$ that fiber algebraically in all the dimensions $5\leq n \leq 8$. That is, there is a surjective homomorphism $\pi_1(M_n) \to \mathbb Z$ with finitely generated kernel. The kernel is also finitely presented in the dimensions $n=7, 8$, and this leads to the first examples of hyperbolic $n$-manifolds $\widetilde M_n$ whose fundamental group is finitely presented but not of finite type. These $n$-manifolds $\widetilde M_n$ have infinitely many cusps of maximal rank and hence infinite Betti number $b_{n-1}$. They cover the finite-volume manifold $M_n$. We obtain these examples by assigning some appropriate colours and states to a family of right-angled hyperbolic polytopes $P_5, \ldots, P_8$, and then applying some arguments of Jankiewicz, Norin, Wise and Bestvina, Brady. We exploit in an essential way the remarkable properties of the Gosset polytopes dual to $P_n$, and the algebra of integral octonions for the crucial dimensions $n=7,8$.Comment: 40 pages, 21 figure

Hyperbolic 5-manifolds that fiber over S^1

We exhibit some finite-volume cusped hyperbolic 5-manifolds that fiber over the circle. These include the smallest hyperbolic 5-manifold known, discovered by Ratcliffe and Tschantz. As a consequence, we build a finite type subgroup of a hyperbolic group that is not hyperbolic

Hyperbolic manifolds that fibre algebraically up to dimension 8

We construct some cusped finite-volume hyperbolic n-manifolds M-n that fibre algebraically in all the dimensions 5 <= n <= 8. That is, there is a surjective homomorphism pi(1)(M-n) -> Z with finitely generated kernel. The kernel is also finitely presented in the dimensions n = 7,8, and this leads to the first examples of hyperbolic n-manifolds (M) over tilde (n) whose fundamental group is finitely presented but not of finite type. These n-manifolds (M) over tilde (n) have infinitely many cusps of maximal rank and, hence, infinite Betti number b(n-1). They cover the finite-volume manifold M-n. We obtain these examples by assigning some appropriate colours and states to a family of right-angled hyperbolic polytopes P-5, ..., P-8, and then applying some arguments of Jankiewicz, Norin and Wise [18] and Bestvina and Brady [7]. We exploit in an essential way the remarkable properties of the Cosset polytopes dual to P-n, and the algebra of integral octonions for the crucial dimensions n = 7,8

Throughput-optimal Resource Allocation in LTE-Advanced with Distributed Antennas

Distributed antennas are envisaged for LTE-Advanced deployments in order to improve the coverage and increase the cell throughput. The latter in turn depends on how resources are allocated to the User Equipments (UEs) at the MAC layer. In this paper we discuss how to allocate resources to UEs so as to maximize the cell throughput, given that UEs may re-ceive from several antennas simultaneously. We first show that the problem is both NP-hard and APX-hard, i.e. no polynomial-time algorithm exists that approximates the opti-mum within a constant factor. Hence, we pro-pose and evaluate two polynomial-time heuristics whose complexity is feasible for practical purposes. Our simulative analysis shows that, in practical scenarios, the two heuristics are highly accurate