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

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

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

Surgical treatment of gastrointestinal stromal tumors of the duodenum. A literature review

Background: Gastrointestinal stromal tumors (GIST) are the most frequent mesenchymal tumours in the digestive tract. The duodenal GIST (dGIST) is the rarest subtype, representing only 4–5% of all GIST, but up to 21% of the resected ones. The diagnostic and therapeutic management of dGIST may be difficult due to the rarity of this tumor, its anatomical location, and the clinical behavior that often mimic a variety of conditions; moreover, there is lack of consent for their treatment. This study has evaluated the scientific literature to provide consensus on the diagnosis of dGIST and to outline possible options for surgical treatment. Methods: An extensive research has been carried out on the electronic databases MEDLINE, Scopus, EMBASE and Cochrane to identify all clinical trials that report an event or case series of dGIST. Results: Eighty-six studies that met the inclusion criteria were identified with five hundred forty-nine patients with dGIST: twenty-seven patients were treated with pancreatoduodenectomy and ninety-six with only local resection (segmental/wedge resections); in four hundred twenty-six patients it is not possible identify the type of treatment performed (pancreatoduodenectomy or segmental/wedge resections). Conclusions: dGISTs are a very rare subset of GISTs. They may be asymptomatic or may involve symptoms of upper GI bleeding and abdominal pain at presentation. Because of the misleading clinical presentation the differential diagnosis may be difficult. Tumours smaller than 2 cm have a low biological aggressiveness and can be followed annually by endoscopic ultrasound. The biggest ones should undergo radical surgical resection (R0). In dGIST there is no uniformly adopted surgical strategy because of the low incidence, lack of experience, and the complex anatomy of the duodenum. Therefore, individually tailored surgical approach is recommended. R0 resection with 1–2 cm clear margin is required. Lymph node dissection is not recommended due to the low incidence of lymphatic metastases. Tumor rupture should be avoided

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

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

The spectral distance on the Moyal plane

We study the noncommutative geometry of the Moyal plane from a metric point of view. Starting from a non compact spectral triple based on the Moyal deformation A of the algebra of Schwartz functions on R^2, we explicitly compute Connes' spectral distance between the pure states of A corresponding to eigenfunctions of the quantum harmonic oscillator. For other pure states, we provide a lower bound to the spectral distance, and show that the latest is not always finite. As a consequence, we show that the spectral triple [20] is not a spectral metric space in the sense of [5]. This motivates the study of truncations of the spectral triple, based on M_n(C) with arbitrary integer n, which turn out to be compact quantum metric spaces in the sense of Rieffel. Finally the distance is explicitly computed for n=2.Comment: Published version. Misprints corrected and references updated; Journal of Geometry and Physics (2011

Connes distance and optimal transport

We give a brief overview on the relation between Connes spectral distance in noncommutative geometry and the Wasserstein distance of order 1 in optimal transport. We first recall how these two distances coincide on the space of probability measures on a Riemannian manifold. Then we work out a simple example on a discrete space, showing that the spectral distance between arbitrary states does not coincide with the Wasserstein distance with cost the spectral distance between pure states

The Objective Buttocks Assessment Scale (OBAS): a new and complete method to assess the gluteal region

Introduction: New treatment methods to improve and enhance buttocks appearance require globally accepted scales for aesthetic research and patient evaluation. The purpose of our study was to develop a set of grading scales for objective assessment of the gluteal region and assess their reliability and validity. Materials and methods: Twelve photonumeric grading scales were created. Eleven aesthetic experts rated photographs of 650 women in 2 validation sessions. Responses were analyzed to assess inter-rater and intra-rater reliability. The Rasch model was used as part of the validation process. Results: All the scales exceeded criteria for acceptability, reliability and validity. Overall inter-rater reliability and intra-rater reliability were both “almost perfect” (p=0.15 and p=0.16 respectively). Conclusion: Consistent outcomes between raters and by individual raters at 2 time points confirm the reliability of the Objective Buttocks Assessment Scale in female patients and suggest it will be a valuable tool for use in research and clinical practice