On the construction of group invariant non expansive operators

In questa tesi vengono illustrati metodi di topologia computazionale negli ambiti di topological data analysis e shape comparison. Nello specifico, dato uno spazio topologico X e un sottogruppo G di Homeo(X), viene studiato un set di dati Φ composto da funzioni definite su X, continue e limitate, a valori reali. Per farlo si utilizzano degli operatori G-invarianti non espansivi (GINO), che si sono dimostrati efficaci per approssimare la pseudo-distanza naturale. In particolare in questo lavoro vengono studiati metodi per la costruzione di tali operatori, sfruttando le proprietà algebriche delle variabili del problema. È importante osservare che il gruppo G viene sempre considerato come variabile, in quanto un cambio dell’osservatore può generalmente coincidere con un cambio dell’invarianza a cui si è interessati

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

Constraints on the χ_(c1) versus χ_(c2) polarizations in proton-proton collisions at √s = 8 TeV

The polarizations of promptly produced χ_(c1) and χ_(c2) mesons are studied using data collected by the CMS experiment at the LHC, in proton-proton collisions at √s=8  TeV. The χ_c states are reconstructed via their radiative decays χ_c → J/ψγ, with the photons being measured through conversions to e⁺e⁻, which allows the two states to be well resolved. The polarizations are measured in the helicity frame, through the analysis of the χ_(c2) to χ_(c1) yield ratio as a function of the polar or azimuthal angle of the positive muon emitted in the J/ψ → μ⁺μ⁻ decay, in three bins of J/ψ transverse momentum. While no differences are seen between the two states in terms of azimuthal decay angle distributions, they are observed to have significantly different polar anisotropies. The measurement favors a scenario where at least one of the two states is strongly polarized along the helicity quantization axis, in agreement with nonrelativistic quantum chromodynamics predictions. This is the first measurement of significantly polarized quarkonia produced at high transverse momentum