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This review article is devoted to analyze the main properties characterizing the cosmological singularity associated to the homogeneous and inhomogeneous Mixmaster model. After the introduction of the main tools required to treat the cosmological issue, we review in details the main results got along the last forty years on the Mixmaster topic. We firstly assess the classical picture of the homogeneous chaotic cosmologies and, after a presentation of the canonical method for the quantization, we develop the quantum Mixmaster behavior. Finally, we extend both the classical and quantum features to the fully inhomogeneous case. Our survey analyzes the fundamental framework of the Mixmaster picture and completes it by accounting for recent and peculiar outstanding results.are obtained requiring the action for the matter S m introduced in Eq. (2.1.11) to be invariant under diffeomorphisms. For the Einstein equations, the Bianchi identities ∇ i G ij = 0 may be viewed as a consequence of the invariance of the Hilbert action under diffeomorphisms. Let us now list the principal aspects of the three fields under consideration.• The energy-momentum tensor of a perfect fluid is given bywhere u i is a unit time-like vector field representing the four-velocity of the fluid. The scalar functions p and ρ are the energy density and the pressure, respectively, as2.8b) where j k is the current density four-vector of electric charge and [ ] denote the antisymmetric sum, or in the forms language equations (2.2.8) are written as d ⋆ F = 4π ⋆ j (2.2.9a) dF = 0 (2.2.9b)where d is the exterior derivative and ⋆ is the Hodge star operator [505,370].

We explore thick accretion disks around rotating attractors. We detail the configurations analysing the fluid angular momentum and finally providing a characterization of the disk morphology and different possible topologies. Investigating the properties of orbiting disks, a classification of attractors, possibly identifiable in terms of their spin-mass ratio, is introduced; then an attempt to characterize dynamically a series of different disk topologies is discussed, showing that some of the toroidal configuration features are determined by the ratio of the angular momentum of the orbiting matter and the spin mass-ratio of the attractor. Then we focus on "multi-structured" disks, constituted by two o more rings of matter orbiting the same attractor, and we proved that some structures are constrained in the dimension of rings, spacing, location and an upper limit of ring number is provided. Finally, assuming a polytropic equation of state we study some specific cases.K ∈]0, 8/9] there are the solutions ( = 0, y = 3M ) and ( > 0, y = M ), see Figs. (18). The limiting case = 0 is shown here, with increasing a B-configuration emerges.

It has been shown that in hypertensive patients the degree of target organ damage correlates more closely with average blood pressure as recorded by ambulatory monitoring (ABPM) throughout 24 h than with clinic blood pressure. We examined a group of 91 clinically healthy subjects, 23 normotensives and 68 hypertensives according to clinic blood pressure. Cardiac anatomy was investigated by echocardiography. As an index of arterial structural changes forearm minimal vascular resistance was calculated from mean arterial pressure and maximal postischemic blood flow, as assessed by venous occlusion plethysmography. The results were correlated to clinic blood pressure or ABPM values (measured by noninvasive ABPM ICR 5200, Spacelabs, Bellevue, CA). Left ventricular mass was correlated more closely with the average blood pressure recorded during 24 h, or during daytime or nighttime periods, than with clinic blood pressure. Minimal vascular resistance was also significantly correlated to ABPM values, but the correlation was similar to that observed with clinic blood pressure. Minimal vascular resistance was significantly correlated to blood pressure variability, as evaluated by the standard deviation of the mean. Minimal vascular resistance and left ventricular mass were higher in a subgroup of patients in whom blood pressure was not significantly reduced during the night. The results of this study confirm that elevated average ABPM values are associated to higher left ventricular mass; in addition, they suggest that increased blood pressure variability may be associated with vascular structural changes, as evaluated by minimal vascular resistance. It remains to be clarified whether cardiac hypertrophy and/or vascular structural changes are a cause or consequence of increased blood pressure values and variability.

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The Hubble constant (H 0) tension between Type Ia supernovae (SNe Ia) and Planck measurements ranges from 4 to 6σ. To investigate this tension, we estimate H 0 in the ΛCDM and w 0 w a CDM (cold dark matter) models by dividing the Pantheon sample, the largest compilation of SNe Ia, into 3, 4, 20, and 40 bins. We fit the extracted H 0 values with a function mimicking the redshift evolution: g ( z ) = H 0 ( z ) = H ˜ 0 / ( 1 + z ) α , where α indicates an evolutionary parameter and H ˜ 0 = H 0 at z = 0. We set the absolute magnitude of SNe Ia so that H 0 = 73.5 km s − 1 Mpc − 1 , and we fix fiducial values for Ω 0 m Λ CDM = 0.298 and Ω 0 m w 0 w a CDM = 0.308 . We find that H 0 evolves with redshift, showing a slowly decreasing trend, with α coefficients consistent with zero only from 1.2 to 2.0σ. Although the α coefficients are compatible with zero in 3σ, this however may affect cosmological results. We measure locally a variation of H 0 ( z = 0 ) − H 0 ( z = 1 ) = 0.4 km s − 1 Mpc − 1 in three and four bins. Extrapolating H 0 ( z ) to z = 1100, the redshift of the last scattering surface, we obtain values of H 0 compatible in 1σ with Planck measurements independent of the cosmological models and number of bins we investigated. Thus, we have reduced the H 0 tension in the range from 54% to 72% for both cosmological models. If the decreasing trend of H 0 ( z ) is real, it could be due to astrophysical selection effects or to modified gravity.

We analyze the dynamics of the Mixmaster Universe on the base of a standard ArnowittDeser-Misner Hamiltonian approach showing how its asymptotic evolution to the cosmological singularity is isomorphic to a billiard on the Lobachevsky plane. The key result of our study consists in the temporary gauge invariance of the billiard representation, once provided the use of very general Misner-Chitré-like variables.

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