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Some results and problems on complex germs with definable Mittag-Leffler stars
Get PDFWorking in an o-minimal expansion of the real field, we investigate
when a germ (around 0 say) of a complex analytic function has a
definable analytic continuation to its Mittag-Leffler star.
As an application we show that any algebro-logarithmic function
that is complex analytic in a neighbourhood of the origin in C has an
analytic continuation to all but finitely many points in C
On Coprimality Graphs for Symmetric Groups
Get PDFFor a group , a subset of and a set of positive integers we define a graph whose vertex set is with joined by an edge provided and the order of is in . Here we investigate when is a finite symmetric group and is a -conjugacy class of elements of order , a prime
Cocycle twists and extensions of braided doubles
Get PDFIt is well known that central extensions of a group G correspond to 2-cocycles on G. Cocycles can be used to construct extensions of G-graded algebras via a version of the Drinfeld twist introduced by Majid. We show how to define the second cohomology group of an abstract monoidal category C, generalising the Schur multiplier of a finite group and the lazy cohomology of a Hopf algebra, recently studied by Schauenburg, Bichon, Carnovale and others. A braiding on C leads to analogues of Nichols algebras in C, and we explain how the recent work on twists of Nichols algebras by Andruskiewitsch, Fantino, Garcia and Vendramin fits in our context.
In the second part of the paper we propose an approach to twisting the multiplication in braided doubles, which are a class of algebras with triangular decomposition over G. Braided doubles are not G-graded, but may be embedded in a double of a Nichols algebra, where a twist is carried out. This is a source of new algebras with triangular decomposition. As an example, we show how to twist the rational Cherednik algebra of the symmetric group by the cocycle arising from the Schur covering group, obtaining the spin Cherednik algebra introduced by Wang
Computing the Frechet Derivative of the Matrix Logarithm and Estimating the Condition Number
Get PDFThe most popular method for computing the matrix logarithm is the
inverse scaling and squaring method, which is the basis of the recent
algorithm of [A. H. Al-Mohy and N. J. Higham, \emph{Improved inverse
scaling and squaring algorithms for the matrix logarithm}, SIAM J.
Sci.\ Comput., 34 (2012), pp.~C152--C169].
For real matrices
we develop a version of the latter algorithm
that works entirely in real arithmetic
and is twice as fast as and more accurate than the original algorithm.
We show that by differentiating the algorithms we obtain backward stable
algorithms for computing the Fr\'echet derivative.
We demonstrate experimentally that our two algorithms are more
accurate and efficient than existing algorithms for computing the
Fr\'echet derivative
and we also show how the algorithms can be used to produce reliable
estimates of the condition number of the matrix logarithm
The hyperbolic Schur decomposition
No full textWe propose a hyperbolic counterpart of the Schur decomposition, with the emphasis on the preservation of structures related to some given hyperbolic scalar product. We give results regarding the existence of such a decomposition and research the properties of its block triangular factor for various structured matrices
Topics in Dynamical Systems
No full textIn this thesis I explore three new topics in Dynamical Systems. In Chapters 2 and 3 I investigate the dynamics of a family of asynchronous linear systems. These sys- tems are of interest as models for asynchronous processes in economics and computer science and as novel ways to solve linear equations. I find a tight sandwich of bounds relating the Lyapunov exponents of these asynchronous systems to the eigenvalue of their synchronous counterparts. Using ideas from the theory of IFSs I show how the random behavior of these systems can be quickly sampled and go some way to characterizing the associated probability measures.
In Chapter 4 I consider another family of random linear dynamical system but this time over the Max-plus semi-ring. These models provide a linear way to model essentially non-linear queueing systems. I show how the topology of the queue net- work impacts on the dynamics, in particular I relate an eigenvalue of the adjacency matrix to the throughput of the queue.
In Chapter 5 I consider non-smooth systems which can be used to model a wide variety of physical systems in engineering as well as systems in control and computer science. I introduce the Moving Average Transformation which allows us to systematically �smooth� these systems enabling us to apply standard techniques that rely on some smoothness, for example computing Lyapunov exponents from time series data
Some recent work in Frechet geometry
Get PDFSome recent work in Frechet geometry is briefly reviewed. In particular an earlier result on the structure of second tangent bundles in the finite dimensional case was extended to infinite dimensional Banach manifolds and Frechet manifolds that could be represented as projective limits of Banach manifolds. This led to further results concerning the characterization of second tangent bundles and differential equations in the more general Frechet structure needed for applications. A summary is given of recent results on hypercyclicity of operators on Frechet spaces
Theoretical study of an inviscid transonic flow near a discontinuity in wall curvature (Part 1)
Get PDFThe work provides an extensive theoretical study of an inviscid transonic flow in the vicinity of a wall curvature
discontinuity. Depending on the ratio of the curvatures upstream and downstream of the break, several physically
different regimes can exist, including a special type of supersonic flows which decelerate to subsonic speeds without a shock wave, transonic Prandtl--Meyer flow and supersonic flows with a weak shock. Using a new numerical technique of solving the Karman--Guderley equation in the ODE form, we perform computations and then employ the \emph{hodograph method} along with the \emph{phase portrait technique} to obtain a complete theoretical description of the flow. It appears that if the flow can be extended beyond the \emph{limiting characteristic}, it subsequently develops a shock wave. As a consequence, a fundamental link between the local and the global flow patterns is observed in our problem (a detailed description of this is given in Part 2). The curvature discontinuity leads to singular pressure gradients upstream and downstream of the break point, respectively. Analytical expressions for the amplitude coefficients are derived as functions of the ratio of the curvatures. These results are important for a subsequent study of the boundary layer separation
Yeast 5 � an Expanded Reconstruction of the Saccharomyces Cerevisiae Metabolic Network
Get PDFBackground: Efforts to improve the computational reconstruction of the Saccharomyces cerevisiae biochemical reaction network and to refine the stoichiometrically constrained metabolic models that can be derived from such a reconstruction have continued since the first stoichiometrically constrained yeast genome scale metabolic model was published in 2003. Continuing this ongoing process, we have constructed an update to the Yeast Consensus Reconstruction, Yeast 5. The Yeast Consensus Reconstruction is a product of efforts to forge a community-based reconstruction emphasizing standards compliance and biochemical accuracy via evidence-based selection of reactions. It draws upon models published by a variety of independent research groups as well as information obtained from biochemical databases and primary literature.
Results: Yeast 5 refines the biochemical reactions included in the reconstruction, particularly reactions involved in sphingolipid metabolism; updates gene-reaction annotations; and emphasizes the distinction between reconstruction and stoichiometrically constrained model. Although it was not a primary goal, this update also improves the accuracy of model prediction of viability and auxotrophy phenotypes and increases the number of epistatic interactions. This update maintains an emphasis on standards compliance, unambiguous metabolite naming, and computer-readable annotations available through a structured document format. Additionally, we have developed MATLAB scripts to evaluate the model�s predictive accuracy and to demonstrate basic model applications, such as simulating aerobic and anaerobic growth. These scripts, which provide an independent tool for evaluating the performance of various
stoichiometrically constrained yeast metabolic models using flux balance analysis, are included as additional files.
Conclusions: Yeast 5 expands and refines the computational reconstruction of yeast metabolism and improves the predictive accuracy of a stoichiometrically constrained yeast metabolic model. It differs from previous reconstructions and models by emphasizing the distinction between the yeast metabolic reconstruction and the stoichiometrically constrained model, and makes both available as additional files and at http://yeast.sf.net/ as separate systems biology markup language (SBML) files. Through this separation, we intend to make the modeling process more accessible, explicit, transparent, and reproducible
Providing Dependability and Performance in the Cloud: Case Studies
Get PDFCloud Computing promises a variety of opportunities but also brings up several challenges. The three case studies presented in the following are examples on how challenges in the field of capacity management, dependability, and scalability can be addressed and how opportunities of Cloud Computing can be leveraged to, e.g., maintain performance requirements or to increase dependability