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Didactic Transformation
Get PDFMathematics is different:
* In no other living science is the part of presentation, of the transformation of disciplinary knowledge to knowledge as it is to be taught (transformation didactique) so important at a research level.
* In no other discipline, however, is the distance between the taught and the new so large
H(div) Preconditioning for a Mixed Finite Element Formulation of the Stochastic Diffusion Problem
Get PDFWe study H(div) preconditioning for the saddle-point systems that arise in a stochastic Galerkin mixed formulation of the steady-state diffusion problem with random data. The key ingredient is a multigrid V-cycle for a weighted, stochastic H(div) operator, acting on a certain tensor product space of random fields with
finite variance. We build on the Arnold-Falk-Winther multigrid algorithm presented in \cite{arnold1997} by varying the spatial discretization from grid to grid whilst keeping the stochastic discretization fixed. We extend the deterministic analysis to accommodate the modified H(div) operator and establish spectral equivalence bounds with a new multigrid V-cycle operator
that are independent of the spatial and stochastic discretization parameters. We implement multigrid within a block-diagonal preconditioner for the full stochastic saddle-point problem, derive eigenvalue bounds for the preconditioned system matrices and investigate the impact of all the discretization parameters on the convergence rate of preconditioned minres
Computing , and Related Matrix Functions by Contour Integrals
Get PDFNew methods are proposed for the numerical evaluation of
f(\A) or f(\A) b, where f(\A) is a function such as \sqrt
\A or \log (\A) with singularities in and
\A is a matrix with eigenvalues on or near .
The methods are based on combining contour integrals
evaluated by the periodic trapezoid rule with conformal
maps involving Jacobi elliptic functions. The convergence
is geometric, so that the computation of f(\A)b is
typically reduced to one or two dozen linear system solves
A centralized gene-based HIV-1 vaccine elicits broad cross-clade cellular immune responses in rhesus monkeys
No full textOne of the major challenges that must be met in developing an HIV-1 vaccine is devising a strategy to generate cellular immunity with sufficient breadth to deal with the extraordinary genetic diversity of the virus. Amino acids in the envelopes of viruses from the same clade can differ by >15%, and those from different clades can differ by >30%. It has been proposed that creating immunogens using centralized HIV-1 gene sequences might provide a practical solution to this problem. Such centralized genes can be generated by employing a number of different strategies: consensus, ancestral, or center of tree sequences. These computer-generated sequences are a shorter genetic distance from any two contemporary virus sequences than those contemporary sequences are from each other. The present study was initiated to evaluate the breadth of cellular immunity generated through immunization of rhesus monkeys with vaccine constructs expressing either an HIV-1 global consensus envelope sequence (CON-S) or a single patient isolate clade B envelope sequence (clade B). We show that vaccine immunogens expressing the single centralized gene CON-S generated cellular immune responses with significantly increased breadth compared with immunogens expressing a wild-type virus gene. In fact, CON-S immunogens elicited cellular immune responses to 3- to 4-fold more discrete epitopes of the envelope proteins from clades A, C, and G than did clade B immunogens. These findings suggest that immunization with centralized genes is a promising vaccine strategy for developing a global vaccine for HIV-1 as well as vaccines for other genetically diverse viruses
Second order structures for sprays and connections on Fréchet manifolds
Get PDFAmbrose, Palais and Singer introduced the concept of second order structures on finite dimensional manifolds. Kumar and Viswanath extended these results to the category of Banach manifolds. In the present paper all of these results are generalized to a large class of Fréchet manifolds. It is proved that the existence of
Christoffel and Hessian structures, connections, sprays and
dissections are equivalent on those Fréchet manifolds which
can be considered as projective limits of Banach manifolds. These concepts provide also an alternative way for the study of ordinary differential equations on non-Banach infinite dimensional manifolds. Concrete examples of the structures are provided using direct and flat connections
Simple laminar dynamos: from two rolls to one
No full textThe study of laminar flows with simple streamlines in conducting fluids that might support
kinematic dynamo action has a long history. Early successful examples include the Gailitis
(1970) dynamo, which has two meridional rolls. We show here that a single meridional roll,
considered as a limit of a modified Gailitis-like dynamo, can also support kinematic dynamo
action in a deep spherical shell. This flow has a claim to be the simplest such flow possible
Updating the QR factorization and the least squares problem
Get PDFIn this paper we treat the problem of updating the
QR factorization, with applications to the least
squares problem. Algorithms are presented that compute
the factorization A1 = Q1 R1, where A1 is the matrix
A = QR after it has had a number of rows or columns
added or deleted. This is achieved by updating the
factors Q and R, and we show this can be much faster
than computing the factorization of A1 from scratch.
We consider algorithms that exploit the Level 3 BLAS
where possible and place no restriction on the dimensions
of A or the number of rows and columns added or deleted.
For some of our algorithms we present Fortran 77
LAPACK-style code and show the backward error of our
updated factors is comparable to the error bounds of the
QR factorization of A1
Flux balance analysis: A geometric perspective
Get PDFAdvances in the field of bioinformatics have led to reconstruction of genome-scale networks for a number of key organisms. The application of physicochemical constraints to these stoichiometric networks allows researchers, through methods such as flux balance analysis, to highlight key sets of reactions necessary to achieve particular objectives. The key benefits of constraint-based analysis lie in the minimal knowledge required to infer systemic properties. However, network degeneracy leads to a large number of flux distributions that satisfy any objective; moreover, these distributions may be dominated by biologically irrelevant internal cycles. By examining the geometry underlying the problem, we define two methods for finding a unique solution within the space of all possible flux distributions; such a solution contains no internal cycles, and is representative of the space as a whole. The first method draws on typical geometric knowledge, but cannot be applied to large networks because of the high computational complexity of the problem. Thus a second method, an iteration of linear programs which scales easily to the genome scale, is defined. The algorithm is run on four recent genome-scale models, and unique flux solutions are found. The algorithm set out here will allow researchers in flux balance analysis to exchange typical solutions to their models in a reproducible format. Moreover, having found a single solution, statistical analyses such as correlations may be performed
Towards Computability over Effectively Enumerable Topological Spaces
Get PDFIn this paper we study different approaches to computability over effectively enumerable topological spaces. We introduce and investigate the notions of computable function, strongly-computable function and weakly-computable function. Under natural assumptions on effectively enumerable topological spaces the notions of computability and weakly-computability coincide
Learning to divide and conquer: applying the algorithm to automate assume-guarantee reasoning.
Get PDFAssume-guarantee reasoning enables a “divide-and-conquer” approach to the verification of large systems that checks system components separately while using assumptions about each component’s environment. Developing appropriate assumptions used to be a difficult and manual process. Over the past five years, we have developed a framework for performing assume-guarantee verification of systems in an incremental and fully automated fashion. The framework uses an off-the-shelf learning algorithm to compute the assumptions. The assumptions are initially approximate and become more precise by means of counterexamples obtained by model checking components separately. The framework supports different assume-guarantee rules, both symmetric and asymmetric. Moreover, we have recently introduced alphabet refinement, which extends the assumption learning process to also infer assumption alphabets. This refinement technique starts with assumption alphabets that are a subset of the minimal interface between a component and its environment, and adds actions to it as necessary until a given property is shown to hold or to be violated in the system. We have applied the learning framework to a number of case studies that show that compositional verification by learning assumptions can be significantly more scalable than non-compositional verification