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

Aspirin and the risk of colorectal and other digestive tract cancers: an updated meta-analysis through 2019

Aspirin has been associated with a reduced risk of colorectal cancer, and possibly of a few other digestive tract cancers. The quantification of risk reduction and the optimal dose and duration of aspirin use for the prevention of colorectal and other digestive tract cancers remains unclear

De Sitter Thermodynamics from Diamonds's Temperature

The thermal time hypothesis proposed by Rovelli [1] regards the physical basis for the flow of time as thermodynamical and provides a definition of the temperature for some special cases. We verify this hypothesis in the case of de Sitter spacetime by relating the uniformly accelerated observer in de Sitter spacetime to the diamond in Minkowski spacetime. Then, as an application of it, we investigate the thermal effect for the uniformly accelerated observer with a finite lifetime in dS spacetime, which generalizes the corresponding result for the case of Minkowski spacetime [2]. Furthermore, noticing that a uniformly accelerated dS observer with a finite lifetime corresponds to a Rindler observer with a finite lifetime in the embedding Minkowski spacetime, we show that the global-embedding-Minkowski-spacetime (GEMS) picture of spacetime thermodynamics is valid in this case. This is a rather nontrivial and unexpected generalization of the GEMS picture, as well as a further verification of both the thermal time hypothesis and the GEMS picture.Comment: 10 pages, 3 figures, LaTeX; v2: reorganized with a new section added concerning a generalization of the GEMS picture from our result; v3: version with minor corrections, to appear in JHE

An algebraic Birkhoff decomposition for the continuous renormalization group

This paper aims at presenting the first steps towards a formulation of the Exact Renormalization Group Equation in the Hopf algebra setting of Connes and Kreimer. It mostly deals with some algebraic preliminaries allowing to formulate perturbative renormalization within the theory of differential equations. The relation between renormalization, formulated as a change of boundary condition for a differential equation, and an algebraic Birkhoff decomposition for rooted trees is explicited

mycosis fungoides in childhood description and study of two siblings

Primary cutaneous T-cell lymphomas are exceedingly rare in children and adolescents. However, mycosis fungoides (MF) is the most frequent primary cutaneous lymphoma diagnosed in childhood. Two cases of MF in siblings (a 14-year-old boy and his 10-year-old sister) are reported. On the basis of clinical features (histopathological and immunophenotypical findings) a diagnosis of MF patch lesions was made in both siblings. Since recent data in the literature have underlined a high frequency of the HLA-DQB1*03 allele in patients with familial MF (including child patients), the HLA profile of the patients was analysed, indicating the presence of a haplotype (HLA-DQB1*03,*03 in the girl, HLA-DQB1*02,*03 in the boy) corresponding with that described in recent literature. Two rare and exceptional cases of MF in siblings are reported, highlighting the presence of a peculiar haplotype

Minimal length in quantum space and integrations of the line element in Noncommutative Geometry

We question the emergence of a minimal length in quantum spacetime, comparing two notions that appeared at various points in the literature: on the one side, the quantum length as the spectrum of an operator L in the Doplicher Fredenhagen Roberts (DFR) quantum spacetime, as well as in the canonical noncommutative spacetime; on the other side, Connes' spectral distance in noncommutative geometry. Although on the Euclidean space the two notions merge into the one of geodesic distance, they yield distinct results in the noncommutative framework. In particular on the Moyal plane, the quantum length is bounded above from zero while the spectral distance can take any real positive value, including infinity. We show how to solve this discrepancy by doubling the spectral triple. This leads us to introduce a modified quantum length d'_L, which coincides exactly with the spectral distance d_D on the set of states of optimal localization. On the set of eigenstates of the quantum harmonic oscillator - together with their translations - d'_L and d_D coincide asymptotically, both in the high energy and large translation limits. At small energy, we interpret the discrepancy between d'_L and d_D as two distinct ways of integrating the line element on a quantum space. This leads us to propose an equation for a geodesic on the Moyal plane.Comment: 29 pages, 2 figures. Minor corrections to match the published versio