An Explicit Formula for the Matrix Logarithm

We present an explicit polynomial formula for evaluating the principal logarithm of all matrices lying on the line segment $\{I(1-t)+At:t\in [0,1]\}$ joining the identity matrix $I$ (at $t=0$) to any real matrix $A$ (at $t=1$) having no eigenvalues on the closed negative real axis. This extends to the matrix logarithm the well known Putzer's method for evaluating the matrix exponential.Comment: 6 page

Insights on the changing dynamics of cemetery use in the neolithic and chalcolithic of southern Portugal. Radiocarbon dating of Lugar do Canto Cave (Santarém)

Lugar do Canto Cave is one of the most relevant Neolithic burial caves in Portugal given not only its extraordinary preservation conditions at the time of discovery but also the quality of the field record obtained during excavation. Its material culture immediately pointed to a Middle Neolithic cemetery but recent radiocarbon determinations also allowed the recognition of an apparent two step phasing of its use within the period (ca. 4000-3400 cal BC): an older one characterized by a single burial and a later reoccupation as a collective necropolis. Comparisons with other well-dated cave cemeteries in Southern Portugal permitted the recognition of changing funerary practices and strategies of cemetery use during the later stages of the Neolithic and the Chalcolithic: 1) ca. 3800 cal BC as the possible turning point from the practice of individual to collective burials; 2) alternating periods of intensive use and deliberate abandonment of cemeteries (evidenced by their intentional closure). Research avenues to investigate the social organization and ideological context underlying these aspects of the Neolithic communities in greater depth are tentatively pointed out in this paper.FEDER funds through the Programa Operacional Factores de Competividade (COMPETE

On generic parametrizations of spinning black-hole geometries

The construction of a generic parametrization of spinning geometries which can be matched continuously to the Kerr metric is an important open problem in General Relativity. Its resolution is more than of academic interest, as it allows to parametrize and quantify possible deviations from the no-hair theorem. Various approaches to the problem have been proposed, all with their own (severe) limitations. Here we discuss the metric recently proposed by Johannsen and Psaltis, showing that (i) the original metric describes only corrections that preserve the horizon area-mass relation of nonspinning geometries; (ii) this unnecessary restriction can be relaxed by introducing a new parameter that in fact dominates in both the post-Newtonian and strong-field regimes; (iii) within this framework, we construct the most generic spinning black-hole geometry which contains twice as many the (infinite) parameters of the original metric; (iv) in the strong-field regime, all parameters are roughly equally important. This fact introduces a severe degeneracy problem in the case of highly-spinning black holes. Finally, we prove that even our generalization fails to describe the few known spinning black-hole metrics in modified gravity.Comment: 8 pages, 2 figures. v2: extended discussion, to appear in Phys. Rev.

Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields

In the present study we consider planar piecewise linear vector fields with two zones separated by the straight line $x=0$. Our goal is to study the existence of simultaneous crossing and sliding limit cycles for such a class of vector fields. First, we provide a canonical form for these systems assuming that each linear system has center, a real one for $y<0$ and a virtual one for $y>0$, and such that the real center is a global center. Then, working with a first order piecewise linear perturbation we obtain piecewise linear differential systems with three crossing limit cycles. Second, we see that a sliding cycle can be detected after a second order piecewise linear perturbation. Finally, imposing the existence of a sliding limit cycle we prove that only one adittional crossing limit cycle can appear. Furthermore, we also characterize the stability of the higher amplitude limit cycle and of the infinity. The main techniques used in our proofs are the Melnikov method, the Extended Chebyshev systems with positive accuracy, and the Bendixson transformation.Comment: 24 pages, 7 figure