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Modular representation theory of blocks with trivial intersection defect groups
Get PDFWe show that Uno's refinement of the projective conjecture of Dade holds every block whose defect groups intersect trivially modulo the maximal normal p-subgroup. This corresponds to the block having p-local rank one as defined by Jianbei An and Eaton. An immediate consequence is that Dade's projective conjecture, Alperin-McKay conjecture and Puig's nilpotent block conjecture hold for all trivial intersection blocks
The Small World Network Structure of Boards of Directors
No full textIn this paper we present a random graph model to explain the network structure of boards of directors. We investigate the conditions under which corporate boards can be said to be a "small-world". Our empirical results show that the random graph model is remarkably good at explaining board structure and connectedness in the United States, the United Kingdom and Germany. Although there are small-world traits such as "clustering" and "short-paths" in the corporate world, they are no more pronounced than would be expected by chance in a statistically similar, but randomly assembled corporate universe. Finally, our results show the existence of positive degree correlation: directors who sit on many boards do so
in the company of other directors who sit on many boards. This result helps explain the distribution of board interlocks
Bundles of acceleration on Banach manifolds
No full textWe consider an infinite dimensional manifold M modelled on a
Banach space E and we construct smooth fiber bundle
structures on the tangent bundle of order two T^2M, which
consists of all smooth curves of M that agree up to their
acceleration, as well as on the corresponding second order frame
bundle L^2M. These bundles prove to be associated with
respect to the identity representation of the general linear group GL(E}) that serves as the structure group of both of them. Moreover, a
bijective correspondence between linear connections on T^2M and
connection forms of L^2M is revealed
Infinite dimensional second order differential equations via
No full textThe vector bundle structure obtained on the second order
(acceleration) tangent bundle T^2M of a smooth manifold M by
means of a linear connection on the base provides an alternative
way for the study of second order differential equations on
manifolds of finite and infinite dimension. Second order vector
fields and their integral curves provide a new way of solving a
wide class of second order differential equations on Frechet
manifolds and may be used also to describe geodesic curves on a
Riemannian manifold. The new technique proposed is illustrated by
concrete examples within the framework of Banach and Frechet
spaces as well as on Lie groups
The Ehrlich--Aberth Method for the Nonsymmetric Tridiagonal Eigenvalue Problem
Get PDFAn algorithm based on the Ehrlich--Aberth iteration is presented for the computation of the zeros of , where is a real irreducible nonsymmetric tridiagonal matrix. The algorithm requires the evaluation of , which is done by exploiting the QR factorization of and the semiseparable structure of . The choice of initial approximations relies on a divide-and-conquer strategy, and some results motivating this strategy are given. Guaranteed a posteriori error bounds based on a running error analysis are proved. A Fortran 95 module implementing the algorithm is provided and numerical experiments that confirm the effectiveness and the robustness of the approach are presented. In particular, comparisons with the LAPACK subroutine \texttt{dhseqr} show that our algorithm is faster for large dimensions
Ideals in mod-R and the ω-radical
Get PDFLet be an artin algebra, and let mod- denote the category of finitely presented right -modules. The radical {\rm rad}={\rm rad}({\rm mod}\mbox{-}R) of this category and its finite powers play a major role in the representation theory of . The intersection of these finite powers is denoted , and the nilpotence of this ideal has been investigated, in , for instance. In , arbitrary transfinite powers, , of rad were defined and linked to the extent to which morphisms in {\rm mod}\mbox{-}R may be factorised. In particular, it has been shown that if is an artin algebra, then the transfinite radical, , the intersection of all ordinal powers of rad, is non-zero if and only if there is a ‘factorisable system’ of morphisms in rad and, in that case, the Krull–Gabriel dimension of {\rm mod}\mbox{-}R equals (that is, is undefined). More precise results on the index of nilpotence of rad for artin algebras were proved in , ,
Transient growth in developing plane and Hagen Poiseuille flow
No full textThe stability of developing entry flow in both two-dimensional channels and circular pipes is investigated for large Reynolds numbers. The basic flow is generated by uniform flow entering a channel/pipe, which then provokes the growth of boundary layers on the walls, until (far downstream) fully developed flow is attained; the length for this development is well known to be O(Reynolds number)×the channel/pipe width/diameter. This enables the use of high-Reynolds-number theory, leading to boundary-layer-type equations which govern the flow; as such, there is no need to impose heuristic parallel-flow approximations. The resulting base flow is shown to be susceptible to significant, three-dimensional, transient (initially algebraic) growth in the streamwise direction, and, consequently, large amplifications to flow disturbances are possible (followed by ultimate decay far downstream). It is suggested that this initial amplification of disturbances is a possible and alternative mechanism for flow transition
Berezinians, exterior powers and recurrent sequences
No full textAbstract We study power expansions of the characteristic function of a linear operator A in a p|q-dimensional superspace V. We show that traces of exterior powers of A satisfy universal recurrence relations of period q. ‘Underlying’ recurrence relations hold in the Grothendieck ring of representations of GL(V). They are expressed by vanishing of certain Hankel determinants of order q+1 in this ring, which generalizes the vanishing of sufficiently high exterior powers of an ordinary vector space. In particular, this allows to express the Berezinian of an operator as a ratio of two polynomial invariants. We analyze the Cayley–Hamilton identity in a superspace. Using the geometric meaning of the Berezinian we also give a simple formulation of the analog of Cramer’s rul
Superreplication of options on several underlying assets
No full textWe investigate the conditions on a hedger, who overestimates the (time- and level-dependent) volatility, to superreplicate a convex claim on several underlying assets. It is shown that the classic Black-Scholes model is the only model, within a large class, for which overestimation of the volatility yields the desired superreplication property. This is in contrast to the one-dimensional case, in which it is known that overestimation of the volatility with any time- and level-dependent model guarantees superreplication of convex claims
Topics in Matrix Computations: Stability and Efficiency of Algorithms
Get PDFNumerical algorithms are considered for three distinct areas of
numerical linear algebra: hyperbolic matrix computations, condition numbers
of structured matrices, and trigonometric matrix functions.
We first consider hyperbolic rotations and show how to construct them
accurately. A new accurate representation is devised which also avoids
overflow. We show how to apply hyperbolic rotations directly, in
mixed form, and by the OD procedure, with a rounding error analysis
that shows the latter two methods are stable. A rounding
error analysis for combining a sequence of nonoverlapping hyperbolic
rotations applied in mixed form or by the OD procedure is then given.
Applying a hyperbolic rotation directly is generally thought to be
unstable but no proof has previously been given. However, using
numerical experiments we prove that it is unstable. We describe several
methods of applying fast
hyperbolic rotations and unified rotations, giving a rounding error
analysis and numerical experiments to show which are stable and which
are not. Hyperbolic Householder transformations are briefly
discussed.
We then consider the hyperbolic QR factorization for which
we present new results for the existence of the closely related HR
factorization, and then use these to prove new theorems for the
existence of the hyperbolic QR factorization. We describe how
nonoverlapping hyperbolic rotations can be used to compute the
hyperbolic QR factorization, with a rounding error analysis to show
that this method is stable. Two applications of the hyperbolic QR
factorization are also discussed.
For an tridiagonal matrix we exploit the structure of its
QR factorization to devise two new algorithms for computing the 1-norm
condition number in operations. The algorithms avoid
underflow and overflow, and are simpler than existing algorithms since
tests are not required for degenerate cases. An error analysis of the
first algorithm is given, while the second algorithm is shown to be
competitive in speed with existing algorithms. We then turn our
attention to an diagonal-plus-semiseparable matrix, ,
for which several algorithms have recently been developed to solve
in operations. We again exploit the QR factorization of
the matrix to present an algorithm that computes the 1-norm condition
number in operations.
We also consider algorithms for computing the matrix cosine. The
algorithms scale a matrix by a power of two to make the norm of the
scaled matrix small, use a Pad\'e approximation to compute the cosine
of the scaled matrix, and recover the cosine of the original matrix
using the double angle formula . We make
several improvements to an algorithm of Higham and Smith to derive
new algorithms, which are shown by theory and numerical experiments to
bring increased efficiency and accuracy. We also consider an
algorithm for simultaneously computing and that
extends the ideas for the cosine and intertwines the cosine and sine
double angle recurrences