Hamiltonian, Energy and Entropy in General Relativity with Non-Orthogonal Boundaries

A general recipe to define, via Noether theorem, the Hamiltonian in any natural field theory is suggested. It is based on a Regge-Teitelboim-like approach applied to the variation of Noether conserved quantities. The Hamiltonian for General Relativity in presence of non-orthogonal boundaries is analysed and the energy is defined as the on-shell value of the Hamiltonian. The role played by boundary conditions in the formalism is outlined and the quasilocal internal energy is defined by imposing metric Dirichlet boundary conditions. A (conditioned) agreement with previous definitions is proved. A correspondence with Brown-York original formulation of the first principle of black hole thermodynamics is finally established.Comment: 29 pages with 1 figur

The effect of particle size on the core losses of soft magnetic composites

In the field of electrical machines, the actual research activities mainly focus on improving the energetic aspects; for this reason, new magnetic materials are currently investigated and proposed, supporting the design and production of magnetic cores. The innovative aspects are related to both hard and soft magnetic materials. In the case of permanent magnets, the use of NdFeB bonded magnets represents a good solution in place of ferrites. For what concerns the soft magnetic materials, the adoption of Soft Magnetic Composites (SMCs) cores permits significant advantages compared to the laminated sheets, such as complex geometries and reduced eddy currents losses. SMC materials are ferromagnetic grains covered with an insulating layer that can be of an organic or inorganic type. The proposed study focuses on the impact of the particle size and distribution on the final material properties. The original powder was cut into three different fractions, and different combinations have been prepared, varying the fractions percentages. The magnetic and energetic properties have been evaluated in different frequency ranges, thus ranking the best combinations. The best specimens were then tested to evaluate the mechanical performances. The preliminary results are promising, but deeper analysis and tests are required to refine the selection and evaluate the improvements against the original composition taken as a reference.In the field of electrical machines, the actual research activities mainly focus on improving the energetic aspects; for this reason, new magnetic materials are currently investigated and proposed, supporting the design and production of magnetic cores. The innovative aspects are related to both hard and soft magnetic materials. In the case of permanent magnets, the use of NdFeB bonded magnets represents a good solution in place of ferrites. For what concerns the soft magnetic materials, the adoption of Soft Magnetic Composites (SMCs) cores permits significant advantages compared to the laminated sheets, such as complex geometries and reduced eddy currents losses. SMC materials are ferromagnetic grains covered with an insulating layer that can be of an organic or inorganic type. The proposed study focuses on the impact of the particle size and distribution on the final material properties. The original powder was cut into three different fractions, and different combinations have been prepared, varying th..

Conserved Quantities from the Equations of Motion (with applications to natural and gauge natural theories of gravitation)

We present an alternative field theoretical approach to the definition of conserved quantities, based directly on the field equations content of a Lagrangian theory (in the standard framework of the Calculus of Variations in jet bundles). The contraction of the Euler-Lagrange equations with Lie derivatives of the dynamical fields allows one to derive a variational Lagrangian for any given set of Lagrangian equations. A two steps algorithmical procedure can be thence applied to the variational Lagrangian in order to produce a general expression for the variation of all quantities which are (covariantly) conserved along the given dynamics. As a concrete example we test this new formalism on Einstein's equations: well known and widely accepted formulae for the variation of the Hamiltonian and the variation of Energy for General Relativity are recovered. We also consider the Einstein-Cartan (Sciama-Kibble) theory in tetrad formalism and as a by-product we gain some new insight on the Kosmann lift in gauge natural theories, which arises when trying to restore naturality in a gauge natural variational Lagrangian.Comment: Latex file, 31 page

The present universe in the Einstein frame, metric-affine R+1/R gravity

We study the present, flat isotropic universe in 1/R-modified gravity. We use the Palatini (metric-affine) variational principle and the Einstein (metric-compatible connected) conformal frame. We show that the energy density scaling deviates from the usual scaling for nonrelativistic matter, and the largest deviation occurs in the present epoch. We find that the current deceleration parameter derived from the apparent matter density parameter is consistent with observations. There is also a small overlap between the predicted and observed values for the redshift derivative of the deceleration parameter. The predicted redshift of the deceleration-to-acceleration transition agrees with that in the \Lambda-CDM model but it is larger than the value estimated from SNIa observations.Comment: 11 pages; published versio

Forward-Backward Splitting in Deformable Image Registration: A Demons Approach

Efficient non-linear image registration implementations are key for many biomedical imaging applications. By using the classical demons approach, the associated optimization problem is solved by an alternate optimization scheme consisting of a gradient descent step followed by Gaussian smoothing. Despite being simple and powerful, the solution of the underlying relaxed formulation is not guaranteed to minimize the original global energy. Implicitly, however, this second step can be recast as the proximal map of the regularizer. This interpretation introduces a parallel to the more general Forward-Backward Splitting (FBS) scheme consisting of a forward gradient descent and proximal step. By shifting entirely to FBS, we can take advantage of the recent advances in FBS methods and solve the original, non-relaxed deformable registration problem for any type of differentiable similarity measure and convex regularization associated with a tractable proximal operator. Additionally, global convergence to a critical point is guaranteed under weak restrictions. For the first time in the context of image registration, we show that Tikhonov regularization breaks down to the simple use of B-Spline filtering in the proximal step. We demonstrate the versatility of FBS by encoding spatial transformation as displacement fields or free-form B-Spline deformations. We use state-of-the-art FBS solvers and compare their performance against the classical demons, the recently proposed inertial demons and the conjugate gradient optimizer. Numerical experiments performed on both synthetic and clinical data show the advantage of FBS in image registration in terms of both convergence and accuracy

Boundary Conditions, Energies and Gravitational Heat in General Relativity (a Classical Analysis)

The variation of the energy for a gravitational system is directly defined from the Hamiltonian field equations of General Relativity. When the variation of the energy is written in a covariant form it splits into two (covariant) contributions: one of them is the Komar energy, while the other is the so-called covariant ADM correction term. When specific boundary conditions are analyzed one sees that the Komar energy is related to the gravitational heat while the ADM correction term plays the role of the Helmholtz free energy. These properties allow to establish, inside a classical geometric framework, a formal analogy between gravitation and the laws governing the evolution of a thermodynamic system. The analogy applies to stationary spacetimes admitting multiple causal horizons as well as to AdS Taub-bolt solutions.Comment: Latex file, 31 pages; one reference and two comments added, misprints correcte

The dynamical equivalence of modified gravity revisited

We revisit the dynamical equivalence between different representations of vacuum modified gravity models in view of Legendre transformations. The equivalence is discussed for both bulk and boundary space, by including in our analysis the relevant Gibbons-Hawking terms. In the f(R) case, the Legendre transformed action coincides with the usual Einstein frame one. We then re-express the R+f(G) action, where G is the Gauss-Bonnet term, as a second order theory with a new set of field variables, four tensor fields and one scalar and study its dynamics. For completeness, we also calculate the conformal transformation of the full Jordan frame R+f(G) action. All the appropriate Gibbons-Hawking terms are calculated explicitly.Comment: 17 pages; v3: Revised version. New comments added in Sections 3 & 5. New results added in Section 6. Version to appear in Class. Quantum Gravit

Extended Loop Quantum Gravity

We discuss constraint structure of extended theories of gravitation (also known as f(R) theories) in the vacuum selfdual formulation introduced in ref. [1].Comment: 7 pages, few typos correcte

Modelling the Frequency of Interarrival Times and Rainfall Depths with the Poisson Hurwitz-Lerch Zeta Distribution

The Poisson-stopped sum of the Hurwitz-Lerch zeta distribution is proposed as a model for interarrival times and rainfall depths. Theoretical properties and characterizations are investigated in comparison with other two models implemented to perform the same task: the Hurwitz-Lerch zeta distribution and the one inflated Hurwitz-Lerch zeta distribution. Within this framework, the capability of these three distributions to fit the main statistical features of rainfall time series was tested on a dataset never previously considered in the literature and chosen in order to represent very different climates from the rainfall characteristics point of view. The results address the Hurwitz-Lerch zeta distribution as a natural framework in rainfall modelling using the additional random convolution induced by the Poisson-stopped model as a further refinement. Indeed the Poisson contribution allows more flexibility and depiction in reproducing statistical features, even in the presence of very different climates

Expansion-induced contribution to the precession of binary orbits

We point out the existence of new effects of global spacetime expansion on local binary systems. In addition to a possible change of orbital size, there is a contribution to the precession of elliptic orbits, to be added to the well-known general relativistic effect in static spacetimes, and the eccentricity can change. Our model calculations are done using geodesics in a McVittie metric, representing a localized system in an asymptotically Robertson-Walker spacetime; we give a few numerical estimates for that case, and indicate ways in which the model should be improved.Comment: revtex, 7 pages, no figures; revised for publication in Classical and Quantum Gravity, with minor changes in response to referees' comment