Solving polynomial eigenvalue problems by means of the Ehrlich-Aberth method

Given the $n\times n$ matrix polynomial $P(x)=\sum_{i=0}^kP_i x^i$, we consider the associated polynomial eigenvalue problem. This problem, viewed in terms of computing the roots of the scalar polynomial $\det P(x)$, is treated in polynomial form rather than in matrix form by means of the Ehrlich-Aberth iteration. The main computational issues are discussed, namely, the choice of the starting approximations needed to start the Ehrlich-Aberth iteration, the computation of the Newton correction, the halting criterion, and the treatment of eigenvalues at infinity. We arrive at an effective implementation which provides more accurate approximations to the eigenvalues with respect to the methods based on the QZ algorithm. The case of polynomials having special structures, like palindromic, Hamiltonian, symplectic, etc., where the eigenvalues have special symmetries in the complex plane, is considered. A general way to adapt the Ehrlich-Aberth iteration to structured matrix polynomial is introduced. Numerical experiments which confirm the effectiveness of this approach are reported.Comment: Submitted to Linear Algebra App

Locating the Eigenvalues of Matrix Polynomials

Some known results for locating the roots of polynomials are extended to the case of matrix polynomials. In particular, a theorem by Pellet [Bull. Sci. Math. (2), 5 (1881), pp. 393--395], some results from Bini [Numer. Algorithms, 13 (1996), pp. 179--200] based on the Newton polygon technique, and recent results from Gaubert and Sharify (see, in particular, [Tropical scaling of polynomial matrices, Lecture Notes in Control and Inform. Sci. 389, Springer, Berlin, 2009, pp. 291--303] and [Sharify, Scaling Algorithms and Tropical Methods in Numerical Matrix Analysis: Application to the Optimal Assignment Problem and to the Accurate Computation of Eigenvalues, Ph.D. thesis, École Polytechnique, Paris, 2011]). These extensions are applied to determine effective initial approximations for the numerical computation of the eigenvalues of matrix polynomials by means of simultaneous iterations, like the Ehrlich--Aberth method. Numerical experiments that show the computational advantage of these results are presented

Gravitomagnetism in the Kerr-Newman-Taub-NUT spacetime

We study the motion of test particles and electromagnetic waves in the Kerr-Newman-Taub-NUT spacetime in order to elucidate some of the effects associated with the gravitomagnetic monopole moment of the source. In particular, we determine in the linear approximation the contribution of this monopole to the gravitational time delay and the rotation of the plane of the polarization of electromagnetic waves. Moreover, we consider "spherical" orbits of uncharged test particles in the Kerr-Taub-NUT spacetime and discuss the modification of the Wilkins orbits due to the presence of the gravitomagnetic monopole.Comment: 12 pages LaTeX iopart style, uses PicTex for 1 Figur

Test particle motion in a gravitational plane wave collision background

Test particle geodesic motion is analysed in detail for the background spacetimes of the degenerate Ferrari-Ibanez colliding gravitational wave solutions. Killing vectors have been used to reduce the equations of motion to a first order system of differential equations which have been integrated numerically. The associated constants of the motion have also been used to match the geodesics as they cross over the boundary between the single plane wave and interaction zones.Comment: 11 pages, 6 Postscript figure

Neutrino current in a gravitational plane wave collision background

The behaviour of a massless Dirac field on a general spacetime background representing two colliding gravitational plane waves is discussed in the Newman-Penrose formalism. The geometrical properties of the neutrino current are analysed and explicit results are given for the special Ferrari-Ibanez solution.Comment: 17 pages, 6 Postscript figures, accepted by International Journal of Modern Physics

L'autonomia comune: territori e progetti di sviluppo rurale decentrato nel plateau mossi (Burkina Faso)

The research intends to analyse the relationship between territories and decentralised rural development projects in the central region of Burkina Faso (plateau Mossi). The analysis of territorial dynamics has evidenced two reference parameters: the degree of local autonomy and the level of coordination between different territorial projects (exogenous and endogenous). The outcomes of such analysis have allowed to critically read the territorial impact of decentralised rural development projects, to confront them with the theoretical paradigm of local development and to define some elements of interest which need further researches

La lunga decolonizzazione delle città africane

The political project of decolonization was based on a "total, complete, and absolute substitution" (Fanon) of a system with another, but in the urban geography of Sub-Saharan Africa this process has been long, complex and not without ambiguity. On one hand, we observe a radical discontinuity in place names, in the administrative structure and in the monumental landscape of African cities; on the other, postcolonial urban planning and architecture showed strong continuity with the colonial functionalist model. In recent African urban projects as well, we observe the action of a colonial rationality, based on the removal of existing territories and on the production of standardized and fragmented urban spaces

Barium alginate capsules for 3D immobilisation of living cells: morphology, membrane properties and permeability

Encapsulation in a barium alginate membrane is a promising strategy to obtain a three dimensional culture of living cells: membrane properties are crucial for a realistic clinical application. A one-step encapsulation technique, recently developed for controlled release of boar semen, was employed to prepare barium alginate and protamine-alginate membranes: permeability to two model molecules (haemoglobin and glucose) was evaluated. Capsules were evaluated for technological properties and scanning electron microscopy was used to examine the external morphology of the capsules and the 3D distribution of the cells within the core. The results indicate that 3D arrangement and cell shape are maintained, capsule dimensions and mechanical properties can be modulated, as well as their permeability to model molecules such as haemoglobin and glucose

The lagoonal harbour of Portus Pisanus (N Tyrrhen- ian Sea, Italy): a long history of human adaptation to changing coastline

During the last millennia human and natural processes have become increasingly intertwined, especially in the Mediterranean coastal and alluvial plains where major urban and trade centres developed since protohistoric times. The construction of ports represents one of the human activities that have mostly contributed to modify coastal environments, inducing a variety of hydrodynamic and hydrochemical changes especially since Roman times (Marriner et al., 2014). Exceptions in this common manner to plan harbours have been recognised along the N Tyrrhenian coast, where no high-impact defense works are explicitly documented by either historical sources or archaeological excavations for three main harbours developed during Etruscan-Roman times (IV-I century BC): Portus Lunae (Bini et al., 2012), Portus Pisanus and Vada Volterrana. Roman literary sources (i.e., Itinerarium Maritimum 501; Rutilio Namaziano) mentioned Portus Pisanus as a flourishing commercial site within a natural protected area (called Sinus Pisanus by Tacito) characterized by Posidonia meadows and located at the foot of Leghorn hills, ca. 18.5 km south of the Pisa city. Accordingly, recent excavations undertaken close to the hills slope, 3 km inland from modern coastline, unearthed a wooden palisade, stone piers and a warehouse dated to the Roman period (Pasquinucci, 2013; Morhange et al., 2015). However, the precise location of the lagoonal harbour basin is still controversial. This study aims to contribute to fill this knowledge gap and shed new light on the main stages of harbour history in the framework of the mid-late Holocene palaeogeographic evolution of the Pisa Plain. The application of a multidisciplinary approach (sedimentological and micropalaeontological core analyses, radiocarbon dating, geomorphological field survey, remote sensing and historical cartography) has revealed that a wide lagoonal basin formed in the study area during the marine transgression peak (ca. 8000 cal yr BP). This basin, recorded by a m-thick subsurface succession of soft grey clays with brackish meiofauna, persisted for several millennia and corresponds to Sinus Pisanus. The available stratigraphic data document that during Roman times the lagoon became progressively less connected to the sea and turned into a coastal lake/pond. Filling processes started two-three millennia later respect to the lagoon occupying the Pisa city area during the Holocene (Rossi et al., 2011). This seaward facies shift forced the westward transferring of the Middle Ages harbour. These results show that natural sheltered conditions along with the distance from coeval Arno River made more advantageous for humans following the shoreline changes, rather than making high-impact interventions. 2012, Bini M., Bruckner H., Chelli A., Da Prato S., Gervasini L., Palaeogeographies of the Magra Valley coastal plain to costrain the location of the Roman harbour of Luna (NW Italy), Palaeogeography, Palaeoclimatology, Palaeoecology, 337-338, 37–51. 2014, Marriner N., Morhange C., Kaniewski D., Carayon N., Ancient harbour infrastructure in the Levant: tracking the birth and rise of new forms of anthropogenic pressure, Nature Scientific Reports, 4, 5554. 2015, Morhange C., Marriner N., Baralis A., Blot M.L., Bony G., Carayon N., Carmona P., Flaux C., Giaime M., Goiran J.-P., Kouka M., Lena A., Oueslati A., Pasquinucci M., Porotov A., Dynamiques géomorphologiques et typologie géoarcheologique des ports antiques en contextes lagunaires, Quaternaire, 26, (2), 117–139. 2013, Pasquinucci M., Guida all’archeologia delle coste livornesi. Nardini Editore. Provincia di Livorno. 271 pp. 2011, Rossi V., Amorosi A., Sarti G., Potenza M., Influence of inherited topography on the Holocene sedimentary evolution of coastal systems: An example from Arno coastal plain (Tuscany, Italy), Geomorphology, 135 (1-2), 117–128