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A resolution of Stewartson's quarter-infinite plate problem
No full textWe revisit a problem originally considered by Stewartson in 1961: the incompressible,
high-Reynolds-number flow past a quarter-infinite plate, with a leading edge that is perpendicular to, and a side edge that is parallel to, an undisturbed oncoming freestream. Particular emphasis is placed on the key region close to the side edge, where the flow is (superficially) three-dimensional, although the use of similarity variables reduces the dimensionality of the problem down to two. As noted by Stewartson, this problem has several intriguing features; it includes singularities and is also of a mixed parabolic type, with edge conditions influencing the solution in both directions across the flow domain. These features serve to greatly complicate the (numerical) solution process (the problem is of course also highly non-linear), and computation was clearly infeasible in 1961. In the present paper, a detailed computational study is presented, answering many of the questions that arose from the 1961 study. We present detailed numerical results together with asymptotic analyses of the key locations in the flow
Continuous-Time Revenue Management in Carparks
Get PDFIn this paper, we study optimal revenue management applied to carparks, with primary objective to maximize
revenues under a continuous-time framework. We develop a stochastic discrete-time model and propose a
rejection algorithm that makes optimal decisions (accept/reject) according to the future expected revenues
generated and on the opportunity cost that arises before each sale. For this aspect of the problem, a Monte
Carlo approach is used to derive optimal rejection policies. We then extend this approach to show that there
exists an equivalent continuous-time methodology that yields to a partial differential equation (PDE). The
nature of the PDE, as opposed to theMonte Carlo approach, generates the rejection policies quicker and causes
the optimal surfaces to be significantly smoother. However, because the solution to the PDE is considered not
to solve the `full' problem, we propose an approach to generate optimal revenues using the discrete-time
model by exploiting the information coming from the PDE. We give a worked example of how to generate
near-optimal revenues with an order of magnitude decrease in computation speed
Stable and Efficient Spectral Divide and Conquer Algorithms for the Symmetric Eigenvalue Decomposition and the SVD
Get PDFSpectral divide and conquer algorithms solve the eigenvalue problem by recursively computing an invariant subspace for a subset of the spectrum and using it to decouple the problem into two smaller subproblems. A number of such algorithms have been developed over the last forty years, often motivated by parallel computing and, most recently, with the aim of achieving minimal communication costs. However, none of the existing algorithms has been proved to be backward stable, and they all have a significantly higher arithmetic cost than the standard algorithms currently used. We present new spectral divide and conquer algorithms for the symmetric eigenvalue problem and the singular value decomposition that are backward stable, achieve lower bounds on communication costs recently derived by Ballard, Demmel, Holtz, and Schwartz, and have operation counts within a small constant factor of those for the standard algorithms. The new algorithms are built on the polar decomposition and exploit the recently developed QR-based dynamically weighted Halley algorithm of Nakatsukasa, Bai, and Gygi, which computes the polar decomposition using a cubically convergent iteration based on the building blocks of QR factorization and matrix multiplication. The algorithms have great potential for efficient, numerically stable computations on computing architectures where the cost of communication dominates the cost of arithmetic
Characterisation of multiple substrate-specific (d)ITP/(d)XTPase and modelling of deaminated purine nucleotide metabolism
Get PDFAccumulation of modified nucleotides is defective to various cellular processes, especially those involving DNA and RNA. To be viable, organisms possess a number of (deoxy)nucleotide phosphohydrolases, which hydrolyze these nucleotides removing them from the active NTP and dNTP pools. Deamination of purine bases can result in accumulation of such nucleotides as ITP, dITP, XTP and dXTP. E. coli RdgB has been characterised as a deoxyribonucleoside triphosphate pyrophosphohydrolase that can act on these nucleotides. S. cerevisiae homologue encoded by YJR069C was purified and its (d)NTPase activity was assayed using fifteen nucleotide substrates. ITP, dITP, and XTP were identified as major sub- strates and kinetic parameters measured. Inhibition by ATP, dATP and GTP were established. On the basis of experimental and published data, modelling and simulation of ITP, dITP, XTP and dXTP metabolism was performed. (d)ITP/(d)XTPase is a new example of enzyme with multiple substrate-specificity demonstrating that multispecificity is not a rare phenomeno
The border collision normal form with stochastic switching surface
Get PDFThe deterministic border collision normal form describes the bifurcations of a discrete time dynamical system as a fixed point moves across the switching surface with changing parameter. If the position of the switching surface varies randomly, but within some bounded region, we give conditions which imply that the attractor close to the bifurcation point is the attractor of an Iterated Function System. The proof uses an equivalent metric to the Euclidean metric because the functions involved are never contractions in the Euclidean metric. If the conditions do not hold then a range of possibilities may be realized, including local instability, and some examples are investigated numerically
Finite and Infinite Elementary Divisors of Matrix Polynomials: A Global Approach
Get PDFThere is general agreement on the definition of the finite elementary divisors of a matrix polynomial Q(\l)\in\F[\l]^{m\times n}, where \F an arbitrary field.
Regarding the elementary divisors at infinity, or infinite elementary divisors, such an agreement has not been so unanimous.
We define the infinite elementary divisors of Q(\l) to be the elementary divisors of \l^\ell Q(\l^{-1}) at 0, where is the degree of Q(\l).
We show that this is the most natural definition if one applies the usual geometric technique of using homogeneous coordinates to deal with the point at infinity.
We call our approach global because the homogeneous invariant factors of Q(\l) are defined for all points of the projective line and to distinguish it from
another possible approach that, using local rings, leads to the same conclusions
Finite groups admitting a Frobenius group of automorphisms with fixed-point-free kernel
Get PDFSuppose that a finite group admits a Frobenius group of
automorphisms with kernel and complement such that
. There are good reasons to expect many properties and
parameters of to be close to the same properties and
parameters of (possibly, also depending on ). We discuss
several recent results in this direction. The properties and parameters in question
include the order, rank, Fitting height, nilpotency class, and the exponent
Finite groups and Lie rings admitting a Frobenius group of automorphisms with fixed-point-free kernel
Get PDFSuppose that a finite group admits a Frobenius group of
automorphisms with kernel and complement such that
. There are good reasons to expect many properties and
parameters of to be close to the same properties and
parameters of (possibly, also depending on ). We discuss
several recent results in this direction. The properties and parameters in question
include the order, rank, Fitting height, nilpotency class, and the exponent
Geometric structure and the local Langlands conjecture
Get PDFWe prove that a strengthened form of the local Langlands conjecture is valid throughout the principal series of any connected split reductive -adic group. The method of proof is to establish the presence of a very simple geometric structure, in both the smooth dual and the Langlands parameters. We prove that this geometric structure is present, in the same way, for the general linear group, including all of its inner forms. With these results as evidence, we give a detailed formulation of a general geometric structure conjecture
Sheaves as essentially algebraic objects
Get PDFWe develop the notion of essentially algebraic theories from [1]. We associate with each Grothendieck site a corresponding essentially algebraic theory whose models are the sheaves on that site. This is used to classify locally finitely presented toposes, and to show that the category of modules over a ring object in a locally finitely presented topos is also locally finitely presentable