Path-Fault-Tolerant Approximate Shortest-Path Trees

Let $G=(V,E)$ be an $n$-nodes non-negatively real-weighted undirected graph. In this paper we show how to enrich a {\em single-source shortest-path tree} (SPT) of $G$ with a \emph{sparse} set of \emph{auxiliary} edges selected from $E$, in order to create a structure which tolerates effectively a \emph{path failure} in the SPT. This consists of a simultaneous fault of a set $F$ of at most $f$ adjacent edges along a shortest path emanating from the source, and it is recognized as one of the most frequent disruption in an SPT. We show that, for any integer parameter $k \geq 1$, it is possible to provide a very sparse (i.e., of size $O(kn\cdot f^{1+1/k})$) auxiliary structure that carefully approximates (i.e., within a stretch factor of $(2k-1)(2|F|+1)$) the true shortest paths from the source during the lifetime of the failure. Moreover, we show that our construction can be further refined to get a stretch factor of $3$ and a size of $O(n \log n)$ for the special case $f=2$, and that it can be converted into a very efficient \emph{approximate-distance sensitivity oracle}, that allows to quickly (even in optimal time, if $k=1$) reconstruct the shortest paths (w.r.t. our structure) from the source after a path failure, thus permitting to perform promptly the needed rerouting operations. Our structure compares favorably with previous known solutions, as we discuss in the paper, and moreover it is also very effective in practice, as we assess through a large set of experiments.Comment: 21 pages, 3 figures, SIROCCO 201

On Pythagoras' theorem for products of spectral triples

We discuss a version of Pythagoras theorem in noncommutative geometry. Usual Pythagoras theorem can be formulated in terms of Connes' distance, between pure states, in the product of commutative spectral triples. We investigate the generalization to both non pure states and arbitrary spectral triples. We show that Pythagoras theorem is replaced by some Pythagoras inequalities, that we prove for the product of arbitrary (i.e. non-necessarily commutative) spectral triples, assuming only some unitality condition. We show that these inequalities are optimal, and provide non-unital counter-examples inspired by K-homology.Comment: Paper slightly shortened to match the published version; Lett. Math. Phys. 201

A 3D environment to rebuild virtually the so-called Augusteum in Herculaneum

Computer graphics and three-dimensional modelling techniques have extended the possibilities of archaeologists in the creation of virtual reconstruction of ancient sites and monuments. Modern computational systems allow the implementation of computer-generated scenarios tailored on human cognitive capacities. Although Virtual Archaeology is not a novelty in the panorama of archaeological methods, there is no agreement among scholars on the minimal parameters necessary to virtually rebuild an ancient context, nor is there any requirement needed to guarantee the accuracy and the effectiveness of the final reconstruction; the strength of a model is based mainly on the capacity of the archaeologist to check the final result in terms of comparison between interpretations and hypotheses. The paper aims at exploring how the archaeologists could perform their work in a computational laboratory thanks to shared 3D models. The case study selected is the recent virtual reconstruction of the so-called Basilica in Herculaneum, a monument - 250 years after its discovery - still largely unexplained. The building is completely buried by volcanic lava save for part of its entrance porch. It was extensively explored using tunnels and looted by its early excavators. Different scholars have rebuilt the monument mainly on the basis of two plans, drawn in the 18th century, and few notes taken by the archaeologists during the exploration. The 3D model, carried out by integrating cad modelling with close-range photogrammetry, is intended to highlight some controversial parts of the reconstructions. Metadata associated to the digital replica describe the physical object and register all phases from data-acquisition to data-visualization in order to allow the validation of the model and the use or re-use of the digital resource

The Isospectral Dirac Operator on the 4-dimensional Orthogonal Quantum Sphere

Equivariance under the action of Uq(so(5)) is used to compute the left regular and (chiral) spinorial representations of the algebra of the orthogonal quantum 4-sphere S^4_q. These representations are the constituents of a spectral triple on this sphere with a Dirac operator which is isospectral to the canonical one on the round undeformed four-sphere and which gives metric dimension four for the noncommutative geometry. Non-triviality of the geometry is proved by pairing the associated Fredholm module with an `instanton' projection. We also introduce a real structure which satisfies all required properties modulo smoothing operators.Comment: 40 pages, no figures, Latex. v2: Title changed. Sect. 9 on real structure completely rewritten and results strengthened. Additional minor changes throughout the pape

Fundamental collapse of the exciton-exciton effective scattering

The exciton-exciton effective scattering which rules the time evolution of two excitons is studied as a function of initial momentum difference, scattering angle and electron-to-hole mass ratio. We show that this effective scattering can collapse for energy-conserving configurations provided that the difference between the two initial exciton momenta is larger than a threshold value. Sizeable scatterings then exist in the forward direction only. We even find that, for an electron-to-hole mass ratio close to 1/2, the exciton-exciton effective scattering stays close to zero in all directions when the difference between the initial exciton momenta has a very specific value. This unexpected but quite remarkable collapse comes from tricky compensation between direct and exchange Coulomb processes which originates from the fundamental undistinguishability of the exciton fermionic components.Comment: Revised text version. Accepted for publication in Physical Review

Spectral geometry of κ\kappa-Minkowski space

After recalling Snyder's idea of using vector fields over a smooth manifold as `coordinates on a noncommutative space', we discuss a two dimensional toy-model whose `dual' noncommutative coordinates form a Lie algebra: this is the well known $\kappa$-Minkowski space. We show how to improve Snyder's idea using the tools of quantum groups and noncommutative geometry. We find a natural representation of the coordinate algebra of $\kappa$-Minkowski as linear operators on an Hilbert space study its `spectral properties' and discuss how to obtain a Dirac operator for this space. We describe two Dirac operators. The first is associated with a spectral triple. We prove that the cyclic integral of M. Dimitrijevic et al. can be obtained as Dixmier trace associated to this triple. The second Dirac operator is equivariant for the action of the quantum Euclidean group, but it has unbounded commutators with the algebra.Comment: 23 pages, expanded versio

Dirac Operators on Quantum Projective Spaces

We construct a family of self-adjoint operators D_N which have compact resolvent and bounded commutators with the coordinate algebra of the quantum projective space CP_q(l), for any l>1 and 0<q<1. They provide 0^+ dimensional equivariant even spectral triples. If l is odd and N=(l+1)/2, the spectral triple is real with KO-dimension 2l mod 8.Comment: 54 pages, no figures, dcpic, pdflate

Local Index Formula on the Equatorial Podles Sphere

We discuss spectral properties of the equatorial Podles sphere. As a preparation we also study the `degenerate' (i.e. $q=0$) case (related to the quantum disk). We consider two different spectral triples: one related to the Fock representation of the Toeplitz algebra and the isopectral one. After the identification of the smooth pre-$C^*$-algebra we compute the dimension spectrum and residues. We check the nontriviality of the (noncommutative) Chern character of the associated Fredholm modules by computing the pairing with the fundamental projector of the $C^*$-algebra (the nontrivial generator of the $K_0$-group) as well as the pairing with the $q$-analogue of the Bott projector. Finally, we show that the local index formula is trivially satisfied.Comment: 18 pages, no figures; minor correction

Quantum Isometries of the finite noncommutative geometry of the Standard Model

We compute the quantum isometry group of the finite noncommutative geometry F describing the internal degrees of freedom in the Standard Model of particle physics. We show that this provides genuine quantum symmetries of the spectral triple corresponding to M x F where M is a compact spin manifold. We also prove that the bosonic and fermionic part of the spectral action are preserved by these symmetries.Comment: 29 pages, no figures v3: minor change