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The Sensitivity of Computational Control Problems
Get PDFIt is well-known that many factors contribute to the accurate and
efficient numerical solution of mathematical problems such as
those arising in computational control system design. In simple
terms these are the arithmetic of the machine on which the
calculations are carried out, sensitivity (or conditioning) of the
mathematical model to small changes of the data and the numerical
stability of the algorithms. It happens quite often that these
concepts are confused. We define these concepts and demonstrate
some of the subtleties that often lead to confusion. In particular
we demonstrate with several examples what may happen when a
problem is modularized, i.e., split into subproblems for which
computational modules are available.
For three classical problems in computational control, pole
placement, linear quadratic control and optimal
control, we then discuss the conditioning of the problems and
point out sources of difficulties. We give some ill-conditioned
examples for which even numerically stable methods fail.
We also stress the need for condition and error estimators that
supplement the numerical algorithm and inform the user about
potential or actual difficulties, and we explain what can be done
to avoid these difficulties.
We finally describe the current status of available condition
estimators in numerical methods for control system design
such as those available in the SLICOT library and we point
out important areas of future research that need
to be adressed
On point orbits of the Co2-minimal parabolic geometry
No full textThe orbits of certain subgroups of Co2, Conway's second largest simple group, on the points of Co2's minimal parabolic geometry are determined. This information is used in [6,7] to analyse a grap associated with the Baby Monster group
Dynamics and processivity of 40S ribosome scanning on mRNA in yeast
No full textThe eukaryotic 40S ribosomal subunit locates the translation initiation codon on an mRNA via the so-called scanning process that follows 40S binding to the capped 5' end. This key step in translation is required for the expression of almost all eukaryotic genes, yet the mechanism and dynamics of scanning are unknown. We have performed quantitative studies in vivo and in vitro of the movement of yeast 40S ribosomes along 5' untranslated regions (UTRs) of different lengths. 40S subunits perform cap-dependent scanning with high processivity for more than 1700 nucleotides in cells of Saccharomyces cerevisiae. Moreover, the observed rates of expression indicate that scanning is performed by an untethered 40S subunit that has been released from the 5' cap complex. Unexpectedly, the capability to maintain scanning competence on a long 5' UTR is more dependent on the Ded1/Dbp1 type of helicase than on eIF4A or eIF4B. In a yeast cell-free extract, scanning shows reduced processivity, with an estimated net 5'3' rate of approximately 10 nucleotides per second at 26°C. We have developed a biased bidirectional walking model of ribosomal scanning that provides a framework for understanding the above observations as well as other known quantitative and qualitative features of this process
The relation between local and global dual pairs
No full textIn this note we clarify the relationship between the local and global definitions of dual pairs in Poisson geometry. It turns out that these are not equivalent. For the passage from local to global one needs a connected fiber hypothesis (this is well known), while the converse requires a dimension condition (which appears not to be known). We also provide examples illustrating the necessity of the extra conditions
Homotopy Decompositions and K-theory of Bott Towers
No full textWe describe Bott towers as sequences of toric manifolds , and identify the omniorientations which correspond
to their original construction as complex varieties. We
show that the suspension of M^k is homotopy equivalent to
a wedge of Thom complexes, and display its complex K-theory as an algebra over the coefficient ring. We extend the results to KO-theory for several families of examples,
and compute the effects of the realification homomorphism; these calculations breathe geometric life into Bahri and Bendersky's analysis of the Adams Spectral Sequence. By
way of application we consider the enumeration of stably complex structures on M^k, obtaining estimates for those which arise from omniorientations and those which are
almost complex. We conclude with observations on the role
of Bott towers in complex cobordism theory
On odd Laplace Operators II
Get PDFWe analyze geometry of the second order differential operators, having in
mind applications to Batalin–Vilkovisky formalism in quantum field theory.
As we show, an exhaustive picture can be obtained by considering pencils of
differential operators acting on densities of all weights simultaneously. The
algebra of densities, which we introduce here, has a natural invariant scalar
product. Using it, we prove that there is a one-to-one correspondence between
second-order operators in this algebra and the corresponding brackets.
A bracket on densities incorporates a bracket on functions, an “upper connection”
in the bundle of volume forms, and a term similar to the “Brans–Dicke
field” of the Kaluza–Klein formalism. These results are valid for even operators
on a usual manifold as well as for odd operators on a supermanifold. For
an odd operator ∆ we show that conditions on the order of the operator ∆^2
give an hierarchy of properties such as flatness of the upper connection and
the Batalin–Vilkovisky master equation. In particular, we obtain a complete
description of generating operators for an arbitrary odd Poisson bracket
Finite presentation and purity in categories σ[M]
Get PDFFor any module M over an associative ring R, let σ[M] denote the smallest Grothendieck subcatgory of Mod-R containing M. If σ[M] is locally finitely presented the notions of purity and pure injectivity are defined in σ[M]. In this paper the relationship between these notions and the corresponding notions defined in Mod-R are investigated, and the connection between the resulting Ziegler spectra is discussed. An example is given of an M such that σ[M] does not contain any nonzero finitely presented objects
An efficient solver for the fully-coupled solution of large-displacement fluid-structure interaction problems
No full textThis paper is concerned with the fully coupled (‘monolithic’) solution of large-displacement fluid–structure interaction problems by Newton’s method. We show that block-triangular approximations of the Jacobian matrix, obtained by neglecting selected fluid–structure interaction blocks, provide good preconditioners for the solution of the linear systems with GMRES. We present an efficient approximate implementation of the preconditioners, based on a Schur complement approximation for the Navier–Stokes block and the use of multigrid approximations for the solution of the computationally most expensive operations. The performance of the the preconditioners is examined in representative steady and unsteady simulations which show that the GMRES iteration counts only display a mild dependence on the Reynolds number and the mesh size. The final part of the paper demonstrates the importance of consistent stabilisation for the accurate simulation of fluid–structure interaction problems
Tridiagonal-diagonal reduction of symmetric indefinite pairs
Get PDFWe consider the reduction of a symmetric indefinite matrix pair (A,B), with B nonsingular, to tridiagonal-diagonal form by congruence transformations. This is an important reduction in solving polynomial eigenvalue problems with symmetric coefficient matrices and in frequency response computations. The pair is first reduced to symmetric-diagonal form. We describe three methods for reducing the symmetric-diagonal pair to tridiagonal-diagonal form. Two of them employ more stable versions of Brebner and Grad's pseudosymmetric Givens and pseudosymmetric Householder reductions, while the third is new and based on a combination of Householder reflectors and hyperbolic rotations. We prove an optimality condition for the transformations used in the third reduction. We present numerical experiments that compare the different approaches and show improvements over Brebner and Grad's reductions
Nucleation of waves in excitable media by noise
No full textWe are interested in reaction-diffusion equations that model excitable media under the influence of an additive noise. In many models of this type, the homogeneous zero state is stable, and interesting dynamics are observed only for certain initial data. In the presence of noise, the excitable media is stimulated, and the noise may be sufficiently large to nucleate wave forms. In computations, we see for small noise that only target waves are nucleated when the time scales for excitation and inhibition are sufficiently separated. We provide a theorem that supports this observation