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

Non-backtracking Walk Centrality for Directed Networks

The theory of zeta functions provides an expression for the generating function of nonbacktracking walk counts on a directed network. We show how this expression can be used to produce a centrality measure that eliminates backtracking walks at no cost. We also show that the radius of convergence of the generating function is related to the spectrum of a three-by-three block matrix involving the original adjacency matrix. This gives a means to choose appropriate values of the attenuation parameter. We find that three important additional benefits arise when we use this technique to eliminate traversals around the network that are unlikely to be of relevance. First, we obtain a larger range of choices for the attenuation parameter. Second, a natural approach for determining a suitable parameter range is invariant under the removal of certain types of nodes, we can gain computational efficiencies through reducing the dimension of the resulting eigenvalue problem. Third, the dimension of the linear system defining the centrality measures may be reduced in the same manner. We show that the new centrality measure may be interpreted as standard Katz on a modified network, where self loops are added, and where nonreciprocal edges are augmented with negative weights. We also give a multilayer interpretation, where negatively weighted walks between layers compensate for backtracking walks on the only non-empty layer. Studying the limit as the attenuation parameter approaches its upper bound also allows us to propose a generalization of eigenvector-based nonbacktracking centrality measure to this directed network setting. In this context, we find that the two-by-two block matrix arising in previous studies focused on undirected networks must be extended to a new three-by-three block structure to allow for directed edges. We illustrate the centrality measure on a synthetic network, where it is shown to eliminate a localization effect present in standard Katz centrality. Finally, we give results for real networks

Algebraic arctic curves in the domain-wall six-vertex model

The arctic curve, i.e. the spatial curve separating ordered (or `frozen') and disordered (or `temperate) regions, of the six-vertex model with domain wall boundary conditions is discussed for the root-of-unity vertex weights. In these cases the curve is described by algebraic equations which can be worked out explicitly from the parametric solution for this curve. Some interesting examples are discussed in detail. The upper bound on the maximal degree of the equation in a generic root-of-unity case is obtained.Comment: 15 pages, no figures; v2: metadata correcte

A class of quasi-sparse companion pencils

In this paper, we introduce a general class of quasi-sparse potential companion pencils for arbitrary square matrix polynomials over an arbitrary field, which extends the class introduced in [B. Eastman, I.-J. Kim, B. L. Shader, K.N. Vander Meulen, Companion matrix patterns. Linear Algebra Appl. 436 (2014) 255-272] for monic scalar polynomials. We provide a canonical form, up to permutation, for companion pencils in this class. We also relate these companion pencils with other relevant families of companion linearizations known so far. Finally, we determine the number of different sparse companion pencils in the class, up to permutation.This work has been partially supported by theMinisterio de Economía y Competitividad of Spain through grants MTM2015-68805-REDT and MTM2015-65798-P

On the role of ethylene, auxin and a GOLVEN-like peptide hormone in the regulation of peach ripening

Background: In melting flesh peaches, auxin is necessary for system-2 ethylene synthesis and a cross-talk between ethylene and auxin occurs during the ripening process. To elucidate this interaction at the transition from maturation to ripening and the accompanying switch from system-1 to system-2 ethylene biosynthesis, fruits of melting flesh and stony hard genotypes, the latter unable to produce system-2 ethylene because of insufficient amount of auxin at ripening, were treated with auxin, ethylene and with 1-methylcyclopropene (1-MCP), known to block ethylene receptors. The effects of the treatments on the different genotypes were monitored by hormone quantifications and transcription profiling. Results: In melting flesh fruit, 1-MCP responses differed according to the ripening stage. Unexpectedly, 1-MCP induced genes also up-regulated by ripening, ethylene and auxin, as CTG134, similar to GOLVEN (GLV) peptides, and repressed genes also down-regulated by ripening, ethylene and auxin, as CTG85, a calcineurin B-like protein. The nature and transcriptional response of CTG134 led to discover a rise in free auxin in 1-MCP treated fruit. This increase was supported by the induced transcription of CTG475, an IAA-amino acid hydrolase. A melting flesh and a stony hard genotype, differing for their ability to synthetize auxin and ethylene amounts at ripening, were used to study the fine temporal regulation and auxin responsiveness of genes involved in the process. Transcriptional waves showed a tight interdependence between auxin and ethylene actions with the former possibly enhanced by the GLV CTG134. The expression of genes involved in the regulation of ripening, among which are several transcription factors, was similar in the two genotypes or could be rescued by auxin application in the stony hard. Only GLV CTG134 expression could not be rescued by exogenous auxin. Conclusions: 1-MCP treatment of peach fruit is ineffective in delaying ripening because it stimulates an increase in free auxin. As a consequence, a burst in ethylene production speeding up ripening occurs. Based on a network of gene transcriptional regulations, a model in which appropriate level of CTG134 peptide hormone might be necessary to allow the correct balance between auxin and ethylene for peach ripening to occur is proposed

New Eco-gas mixtures for the Extreme Energy Events MRPCs: results and plans

The Extreme Energy Events observatory is an extended muon telescope array, covering more than 10 degrees both in latitude and longitude. Its 59 muon telescopes are equipped with tracking detectors based on Multigap Resistive Plate Chamber technology with time resolution of the order of a few hundred picoseconds. The recent restrictions on greenhouse gases demand studies for new gas mixtures in compliance with the relative requirements. Tetrafluoropropene is one of the candidates for tetrafluoroethane substitution, since it is characterized by a Global Warming Power around 300 times lower than the gas mixtures used up to now. Several mixtures have been tested, measuring efficiency curves, charge distributions, streamer fractions and time resolutions. Results are presented for the whole set of mixtures and operating conditions, %. A set of tests on a real EEE telescope, with cosmic muons, are being performed at the CERN-01 EEE telescope. The tests are focusing on identifying a mixture with good performance at the low rates typical of an EEE telescope.Comment: 8 pages, 6 figures, proceedings for the "XIV Workshop on Resistive Plate Chambers and Related Detectors" (19-23 February 2018), Puerto Vallarta, Jalisco State, Mexic

INFN What Next: Ultra-relativistic Heavy-Ion Collisions

This document was prepared by the community that is active in Italy, within INFN (Istituto Nazionale di Fisica Nucleare), in the field of ultra-relativistic heavy-ion collisions. The experimental study of the phase diagram of strongly-interacting matter and of the Quark-Gluon Plasma (QGP) deconfined state will proceed, in the next 10-15 years, along two directions: the high-energy regime at RHIC and at the LHC, and the low-energy regime at FAIR, NICA, SPS and RHIC. The Italian community is strongly involved in the present and future programme of the ALICE experiment, the upgrade of which will open, in the 2020s, a new phase of high-precision characterisation of the QGP properties at the LHC. As a complement of this main activity, there is a growing interest in a possible future experiment at the SPS, which would target the search for the onset of deconfinement using dimuon measurements. On a longer timescale, the community looks with interest at the ongoing studies and discussions on a possible fixed-target programme using the LHC ion beams and on the Future Circular Collider.Comment: 99 pages, 56 figure

The Extreme Energy Events HECR array: status and perspectives

The Extreme Energy Events Project is a synchronous sparse array of 52 tracking detectors for studying High Energy Cosmic Rays (HECR) and Cosmic Rays-related phenomena. The observatory is also meant to address Long Distance Correlation (LDC) phenomena: the network is deployed over a broad area covering 10 degrees in latitude and 11 in longitude. An overview of a set of preliminary results is given, extending from the study of local muon flux dependance on solar activity to the investigation of the upward-going component of muon flux traversing the EEE stations; from the search for anisotropies at the sub-TeV scale to the hints for observations of km-scale Extensive Air Shower (EAS).Comment: XXV ECRS 2016 Proceedings - eConf C16-09-04.

Multiplicity dependence of jet-like two-particle correlations in p-Pb collisions at sNN\sqrt{s_{NN}} = 5.02 TeV

Two-particle angular correlations between unidentified charged trigger and associated particles are measured by the ALICE detector in p-Pb collisions at a nucleon-nucleon centre-of-mass energy of 5.02 TeV. The transverse-momentum range 0.7 $< p_{\rm{T}, assoc} < p_{\rm{T}, trig} <$ 5.0 GeV/$c$ is examined, to include correlations induced by jets originating from low momen\-tum-transfer scatterings (minijets). The correlations expressed as associated yield per trigger particle are obtained in the pseudorapidity range $|\eta|<0.9$. The near-side long-range pseudorapidity correlations observed in high-multiplicity p-Pb collisions are subtracted from both near-side short-range and away-side correlations in order to remove the non-jet-like components. The yields in the jet-like peaks are found to be invariant with event multiplicity with the exception of events with low multiplicity. This invariance is consistent with the particles being produced via the incoherent fragmentation of multiple parton--parton scatterings, while the yield related to the previously observed ridge structures is not jet-related. The number of uncorrelated sources of particle production is found to increase linearly with multiplicity, suggesting no saturation of the number of multi-parton interactions even in the highest multiplicity p-Pb collisions. Further, the number scales in the intermediate multiplicity region with the number of binary nucleon-nucleon collisions estimated with a Glauber Monte-Carlo simulation.Comment: 23 pages, 6 captioned figures, 1 table, authors from page 17, published version, figures at http://aliceinfo.cern.ch/ArtSubmission/node/161