Fast and backward stable computation of roots of polynomials

A stable algorithm to compute the roots of polynomials is presented. The roots are found by computing the eigenvalues of the associated companion matrix by Francis's implicitly shifted QR algorithm. A companion matrix is an upper Hessenberg matrix that is unitary-plus-rankone, that is, it is the sum of a unitary matrix and a rank-one matrix. These properties are preserved by iterations of Francis's algorithm, and it is these properties that are exploited here. The matrix is represented as a product of 3n - 1 Givens rotators plus the rank-one part, so only O(n) storage space is required. In fact, the information about the rank-one part is also encoded in the rotators, so it is not necessary to store the rank-one part explicitly. Francis's algorithm implemented on this representation requires only O(n) flops per iteration and thus O(n2) flops overall. The algorithm is described, normwise backward stability is proved, and an extensive set of numerical experiments is presented. The algorithm is shown to be about as accurate as the (slow) Francis QR algorithm applied to the companion matrix without exploiting the structure. It is faster than other fast methods that have been proposed, and its accuracy is comparable or better

Yet another algorithm for the symmetric eigenvalue problem

In this paper we present a new algorithm for solving the symmetric matrix eigenvalue problem that works by first using a Cayley transformation to convert the symmetric matrix into a unitary one and then uses Gragg’s implicitly shifted unitary QR algorithm to solve the resulting unitary eigenvalue problem. We prove that under reasonable assumptions on the symmetric matrix this algorithm is backward stable and also demonstrate that this algorithm is comparable with other well known implementations in terms of both speed and accuracy

Fast and stable unitary QR algorithm

A fast Fortran implementation of a variant of Gragg's unitary Hessenberg QR algorithm is presented. It is proved, moreover, that all QR- And QZ-like algorithms for the unitary eigenvalue problems are equivalent. The algorithm is backward stable. Numerical experiments are presented that confirm the backward stability and compare the speed and accuracy of this algorithm with other methods

Roots of Polynomials: on twisted QR methods for companion matrices and pencils

Two generalizations of the companion QR algorithm by J.L. Aurentz, T. Mach, R. Vandebril, and D.S. Watkins, SIAM Journal on Matrix Analysis and Applications, 36(3): 942--973, 2015, to compute the roots of a polynomial are presented. First, we will show how the fast and backward stable QR algorithm for companion matrices can be generalized to a QZ algorithm for companion pencils. Companion pencils admit a greater flexibility in scaling the polynomial and distributing the matrix coefficients over both matrices in the pencil. This allows for an enhanced stability for polynomials with largely varying coefficients. Second, we will generalize the pencil approach further to a twisted QZ algorithm. Whereas in the classical QZ case Krylov spaces govern the convergence, the convergence of the twisted case is determined by a rational Krylov space. A backward error analysis to map the error back to the original pencil and to the polynomial coefficients shows that in both cases the error scales quadratically with the input. An extensive set of numerical experiments supports the theoretical backward error, confirms the numerical stability and shows that the computing time depends quadratically on the problem size

Annual Report Town of Bowdoinham Maine 2013

Perceptual processes play an important role in motor learning. While it is evident that visual information greatly contributes to learning new movements, much less is known about provision of prescriptive proprioceptive information. Here, we investigated whether passive (proprioceptively-based) movement training was comparable to active training for learning a new bimanual task. Three groups practiced a bimanual coordination pattern with a 1∶2 frequency ratio and a 90° phase offset between both wrists with Lissajous feedback over the course of four days: 1) passive training; 2) active training; 3) no training (control). Retention findings revealed that passive as compared to active training resulted in equally successful acquisition of the frequency ratio but active training was more effective for acquisition of the new relative phasing between the limbs in the presence of augmented visual feedback. However, when this feedback was removed, performance of the new relative phase deteriorated in both groups whereas the frequency ratio was better preserved. The superiority of active over passive training in the presence of augmented feedback is hypothesized to result from active involvement in processes of error detection/correction and planning.status: publishe

Constructible motivic functions and motivic integration

We introduce a direct image formalism for constructible motivic functions. One deduces a very general version of motivic integration for which a change of variables theorem is proved. These constructions are generalized to the relative framework, in which we develop a relative version of motivic integration. These results have been announced in math.AG/0403349 and math.AG/0403350. Main results and statements unchanged. Many minor slips corrected and some details added.Comment: Final versio

Integrating evolutionary theory and social-ecological systems research to address the sustainability challenges of the Anthropocene

The rapid, human-induced changes in the Earth system during the Anthropocene present humanity with critical sustainability challenges. Social-ecological systems (SES) research provides multiple approaches for understanding the complex interactions between humans, social systems, and environments and how we might direct them towards healthier and more resilient futures. However, general theories of SES change have yet to be fully developed. Formal evolutionary theory has been applied as a dynamic theory of change of complex phenomena in biology and the social sciences, but rarely in SES research. In this paper, we explore the connections between both fields, hoping to foster collaboration. After sketching out the distinct intellectual traditions of SES research and evolutionary theory, we map some of their terminological and theoretical connections. We then provide examples of how evolutionary theory might be incorporated into SES research through the use of systems mapping to identify evolutionary processes in SES, the application of concepts from evolutionary developmental biology to understand the connections between systems changes and evolutionary changes, and how evolutionary thinking may help design interventions for beneficial change. Integrating evolutionary theory and SES research can lead to a better understanding of SES changes and positive interventions for a more sustainable Anthropocene. This article is part of the theme issue 'Evolution and sustainability: gathering the strands for an Anthropocene synthesis'

Exploring hypotheses of the actions of TGF-beta 1 in epidermal wound healing using a 3D computational multiscale model of the human epidermis

In vivo and in vitro studies give a paradoxical picture of the actions of the key regulatory factor TGF-beta 1 in epidermal wound healing with it stimulating migration of keratinocytes but also inhibiting their proliferation. To try to reconcile these into an easily visualized 3D model of wound healing amenable for experimentation by cell biologists, a multiscale model of the formation of a 3D skin epithelium was established with TGF-beta 1 literature-derived rule sets and equations embedded within it. At the cellular level, an agent-based bottom-up model that focuses on individual interacting units ( keratinocytes) was used. This was based on literature-derived rules governing keratinocyte behavior and keratinocyte/ECM interactions. The selection of these rule sets is described in detail in this paper. The agent-based model was then linked with a subcellular model of TGF-beta 1 production and its action on keratinocytes simulated with a complex pathway simulator. This multiscale model can be run at a cellular level only or at a combined cellular/subcellular level. It was then initially challenged ( by wounding) to investigate the behavior of keratinocytes in wound healing at the cellular level. To investigate the possible actions of TGF-beta 1, several hypotheses were then explored by deliberately manipulating some of these rule sets at subcellular levels. This exercise readily eliminated some hypotheses and identified a sequence of spatial-temporal actions of TGF-beta 1 for normal successful wound healing in an easy-to-follow 3D model. We suggest this multiscale model offers a valuable, easy-to-visualize aid to our understanding of the actions of this key regulator in wound healing, and provides a model that can now be used to explore pathologies of wound healing

Primordial black holes in braneworld cosmologies: Formation, cosmological evolution and evaporation

We consider the population evolution and evaporation of primordial black holes in the simplest braneworld cosmology, Randall-Sundrum type II. We demonstrate that black holes forming during the high-energy phase of this theory (where the expansion rate is proportional to the density) have a modified evaporation law, resulting in a longer lifetime and lower temperature at evaporation, while those forming in the standard regime behave essentially as in the standard cosmology. For sufficiently large values of the AdS radius, the high-energy regime can be the one relevant for primordial black holes evaporating at key epochs such as nucleosynthesis and the present. We examine the formation epochs of such black holes, and delimit the parameter regimes where the standard scenario is significantly modified.Comment: 9 pages RevTeX4 file with four figures incorporated, minor changes to match published versio

Exploring the Foundation of Genomics: A Northern Blot Reference set for the Comparative Analysis of Transcript Profiling Technologies

In this paper we aim to create a reference data collection of Northern blot results and demonstrate how such a collection can enable a quantitative comparison of modern expression profiling techniques, a central component of functional genomics studies. Historically, Northern blots were the de facto standard for determining RNA transcript levels. However, driven by the demand for analysis of large sets of genes in parallel, high-throughput methods, such as microarrays, dominate modern profiling efforts. To facilitate assessment of these methods, in comparison to Northern blots, we created a database of published Northern results obtained with a standardized commercial multiple tissue blot (dbMTN). In order to demonstrate the utility of the dbMTN collection for technology comparison, we also generated expression profiles for genes across a set of human tissues, using multiple profiling techniques. No method produced profiles that were strongly correlated with the Northern blot data. The highest correlations to the Northern blot data were determined with microarrays for the subset of genes observed to be specifically expressed in a single tissue in the Northern analyses. The database and expression profiling data are available via the project website (http://www.cisreg.ca). We believe that emphasis on multitechnique validation of expression profiles is justified, as the correlation results between platforms are not encouraging on the whole. Supplementary material for this article can be found at: http://www.interscience.wiley.com/jpages/1531-6912/suppma