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A geometric definition of news tensor on null hypersurfaces in four space-time dimensions is presented. When the conformal Einstein field equations hold, this news tensor yields the correct expression for the radiative components of the rescaled Weyl tensor at infinity with vanishing cosmological constant in arbitrary conformal gauge. Also, a generalised transport equation for the Geroch tensor is derived. Important differences between null hypersurfaces in the bulk of the space-time and null infinity with vanishing cosmological constant are reviewed, and their impact on the role of the news is discussed., Comment: 25 pages

The criterion for existence of gravitational radiation at conformal infinity in the presence of a positive cosmological constant is applied to a general family of exact solutions representing generic (pairs of) black holes of algebraic type D. Our analysis shows that only accelerating black holes generate gravitational radiation measurable at infinity. This very satisfactory result confirms the goodness of the criterion. To that end, a new metric form of the family of exact type D black holes is constructed -- including any cosmological constant and a (double-aligned) non-null electromagnetic field -- whose expression is suitable for investigation of the asymptotic structure of this large family of spacetimes. The family depends on seven physical parameters, namely $m$, $a$, $l$, $\alpha$, $e$, $g$, and $\Lambda$ that characterize mass, specific angular momentum parameter, NUT parameter, acceleration, electric and magnetic charges, and the cosmological constant, respectively., Comment: 35 pages, 5 figures

Let $T:E\rightarrow F$ be a non-necessarily continuous triple homomorphism from a (complex) JB$^*$-triple (respectively, a (real) J$^*$B-triple) to a normed Jordan triple. The following statements hold: (1) $T$ has closed range whenever $T$ is continuous (2) $T$ has closed range whenever $T$ is continuous This result generalises classical theorems of I. Kaplansky and S.B. Cleveland in the setting of C$^*$-algebras and of A. Bensebah and J.P\'erez, L. Rico and A. Rodr'\iguez Palacios in the setting of JB$^*$-algebras.

Hall and Robinson (2009) proposed and analyzed the use of bagged cross-validation to choose the bandwidth of a kernel density estimator. They established that bagging greatly reduces the noise inherent in ordinary cross-validation, and hence leads to a more efficient bandwidth selector. The asymptotic theory of Hall and Robinson (2009) assumes that $N$, the number of bagged subsamples, is $\infty$. We expand upon their theoretical results by allowing $N$ to be finite, as it is in practice. Our results indicate an important difference in the rate of convergence of the bagged cross-validation bandwidth for the cases $N=\infty$ and $N<\infty$. Simulations quantify the improvement in statistical efficiency and computational speed that can result from using bagged cross-validation as opposed to a binned implementation of ordinary cross-validation. The performance of thebagged bandwidth is also illustrated on a real, very large, data set. Finally, a byproduct of our study is the correction of errors appearing in the Hall and Robinson (2009) expression for the asymptotic mean squared error of the bagging selector., Comment: 37 pages, 9 figures

In the search for improved permanent magnets, fueled by the geostrategic and environmental issues associated with rare-earth-based magnets, magnetically hard (high anisotropy)-soft (high magnetization) composite magnets hold promise as alternative magnets that could replace modern permanent magnets, such as rare-earth-based and ceramic magnets, in certain applications. However, so far, the magnetic properties reported for hard-soft composites have been underwhelming. Here, an attempt to further understand the correlation between magnetic and microstructural properties in strontium ferrite-based composites, hard SrFe$_{12}$O$_{19}$ (SFO) ceramics with different contents of Fe particles as soft phase, both in powder and in dense injection molded magnets, is presented. In addition, the influence of soft phase particle dimension, in the nano- and micron-sized regimes, on these properties is studied. While Fe and SFO are not exchange-coupled in our magnets, a remanence that is higher than expected is measured. In fact, in composite injection molded anisotropic (magnetically oriented) magnets, remanence is improved by 2.4% with respect to a pure ferrite identical magnet. The analysis of the experimental results in combination with micromagnetic simulations allows us to establish that the type of interaction between hard and soft phases is of a dipolar nature, and is responsible for the alignment of a fraction of the soft spins with the magnetization of the hard. The mechanism unraveled in this work has implications for the development of novel hard-soft permanent magnets.

Results on the isolation of the radiative degrees of freedom of the gravitational field with a positive cosmological constant in full General Relativity are put forward. Methods employed in a recent geometric characterisation of gravitational radiation are used and, inspired by Ashtekar's work on asymptotically flat space-times, a space of connections is defined. Ground differences emerge due to the space-like character of the conformal boundary, and one has to put into play a fundamental result by Friedrich concerning the initial value problem for space-times with a positive cosmological constant. Based on this, half of the radiative degrees of freedom are identified; remarkably, they utterly determine the gravitational radiation content for space-times with algebraically special rescaled Weyl tensor at infinity. Directions for defining the phase space in the general case are proposed., Comment: 26 pages, 2 figures, typos corrected

We introduce the Jordan-strict topology on the multipliers algebra of a JB$^*$-algebra, a notion which was missing despite the fourty years passed after the first studies on Jordan multipliers. In case that a C$^*$-algebra $A$ is regarded as a JB$^*$-algebra, the J-strict topology of $M(A)$ is precisely the well-studied C$^*$-strict topology. We prove that every JB$^*$-algebra $\mathfrak{A}$ is J-strict dense in its multipliers algebra $M(\mathfrak{A})$, and that latter algebra is J-strict complete. We show that continuous surjective Jordan homomorphisms, triple homomorphisms, and orthogonality preserving operators between JB$^*$-algebras admit J-strict continuous extensions to the corresponding type of operators between the multipliers algebras. We characterize J-strict continuous functionals on the multipliers algebra of a JB$^*$-algebra $\mathfrak{A}$, and we establish that the dual of $M(\mathfrak{A})$ with respect to the J-strict topology is isometrically isomorphic to $\mathfrak{A}^*$. We also present a first applications of the J-strict topology of the multipliers algebra, by showing that under the extra hypothesis that $\mathfrak{A}$ and $\mathfrak{B}$ are $\sigma$-unital JB$^*$-algebras, every surjective Jordan $^*$-homomorphism (respectively, triple homomorphism or continuous orthogonality preserving operator) from $\mathfrak{A}$ onto $\mathfrak{B}$ admits an extension to a surjective J-strict continuous Jordan $^*$-homomorphism (respectively, triple homomorphism or continuous orthogonality preserving operator) from $M(\mathfrak{A})$ onto $M(\mathfrak{B})$.

A method for deriving the asymptotic behaviour of any physical field is presented. This leads to a geometrically meaningful derivation of the peeling properties for arbitrary values of the cosmological constant. Application to the outstanding case of the physical Weyl tensor provides the explicit form of all terms that determine its asymptotic behaviour along arbitrary lightlike geodesics. The results follow under the assumption of a conformal completion \`a la Penrose. The only freedom available is the choice of a null vector at the conformal boundary of the space-time (which determines the lightlike geodesic arriving there)., Comment: 10 pages, 1 figure

This is the second of two papers that study the asymptotic structure of space-times with a non-negative cosmological constant $\Lambda$. This paper deals with the case $\Lambda >0$. Our approach is founded on the `tidal energies' built with the Weyl curvature and, specifically, we use the asymptotic super-Poynting vector computed from the rescaled Bel-Robinson tensor at infinity to provide a covariant, gauge-invariant, criterion for the existence, or absence, of gravitational radiation at infinity. The fundamental idea we put forward is that the physical asymptotic properties are encoded in $(\scri,h_{ab},D_{ab})$, where the first element of the triplet is a 3-dimensional manifold, the second is a representative of a conformal class of Riemannian metrics on $\scri$, and the third element is a traceless symmetric tensor field on $\scri$. We similarly propose a no-incoming radiation criterion based also on the triplet $(\scri,h_{ab},D_{ab})$ and on radiant supermomenta deduced from the rescaled Bel-Robinson tensor too. We search for news tensors and argue that any news-like object must be associated to, and depends on, 2-dimensional cross-sections of $\scri$. We identify one component of news for every such cross-section and present a general strategy to find the second component. We also introduce the concept of equipped $\scri$, consider the limit $\Lambda\rightarrow 0$ and apply all our results to selected exact solutions of Einstein Field Equations. The full-length abstract is available in the paper., Comment: 150 pages, 10 figures, second of two papers, full-length abstract available in the paper, minor corrections, some references added, new table added, accepted version for publication

This is the first of two papers devoted to the asymptotic structure of space-time in the presence of a non-negative cosmological constant $\Lambda$. This first paper is concerned with the case of $\Lambda =0$. Our approach is fully based on the tidal nature of the gravitational field and therefore on the `tidal energies' built with the Weyl curvature. In particular, we use the (radiant) asymptotic supermomenta computed from the rescaled Weyl tensor at infinity to provide a novel characterisation of radiation escaping from, or entering into, the space-time. Our new criterion is easy to implement and shown to be fully equivalent to the classical one based on the news tensor. One of its virtues is that its formulation can be easily adapted to the case with $\Lambda>0$ covered in the second paper. We derive the general energy-momentum-loss formulae including the matter terms and all factors associated to the choices of arbitrary foliation and of super-translation. We also revisit and present a full reformulation of the traditional peeling behaviour with a neat geometrical construction that leads, in particular, to an asymptotic alignment of the supermomenta in accordance with the radiation criterion., Comment: 45 pages, first of two papers, minor corrections, accepted version for publication