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The density of algebraic points on certain Pfaffian surfaces
No full textWe prove some instances of Wilkie's conjecture on the density of rational points on sets definable in the real exponential field. In particular, we prove that this conjecture is true for surfaces defined using restricted exponentiation, and that it is true for a general Pfaffian surface provided that the surface admits a certain kind of parameterization
Regulating Industries under Exogenous Uncertainty
Get PDFWe present a quantitative method to find jointly optimal strategies for an industry regulator and a firm, who operate under exogenous uncertainty. The firm controls its operating policy in order to maximize its expected future profits, whilst taking account of regulatory fines. The regulator aims to control the probability that the firm enters a given undesirable state, such as ceasing production, by imposing a fine which is as low as possible, while achieving the required reduction in probability.
The exogenous uncertainty is modeled using a stochastic differential equation, and we show this implies that the firm�s behavior can be solved via the Hamilton-
Jacobi-Bellman equation, and the regulatory fine can be obtained via the Feynman-Kac formula. We discuss both analytic and numerical solution methods. Our results
are illustrated for a security of supply problem for vaccine production where future production costs are uncertain and, using empirical data, for an abandonment problem in a gold mining operation where future commodity prices are uncertain. The method determines the level of fine which establishes a Nash equilibrium in these
nonzero-sum games, under uncertainty
Extended quotients in the principal series of reductive p-adic groups
Get PDFThe geometric conjecture developed by the authors in [1,2,3,4] applies to the smooth dual Irr(G) of any reductive p-adic group G. It predicts a definite geometric structure -- the structure of an extended quotient -- for each component in the Bernstein decomposition of Irr(G).
In this article, we prove the geometric conjecture for the principal series in any split connected reductive p-adic group G. The proof proceeds via Springer parameters and Langlands parameters. As a consequence of this approach, we establish strong links with the local Langlands correspondence.
One important feature of our approach is the emphasis on
two-sided cells in extended affine Weyl groups
An interview with Paul Van Dooren
Get PDFInterviews and dialogues have been a special way to transmit science since Socrates. They are maybe the only way to get the stories behind most discoveries. I was so positively impressed by the interview of Gene Golub by
Nicholas J. Higham [1] that the idea of doing one with Paul Van Dooren obsessed me for a while. After a few attempts to arrange a meeting, I finally got an occasion during the Householder Symposium at Tahoe City. On June 15, 2011, when most participants were on an excursion (Paul did have a kind of cold), we sat in a corner of the Granhall of the Granlibakken Conference Center and we did the interview. This document provides an edited transcript of that dialogue. The bibliography contains some refernces in the
interview
Modelling acidosis and the cell cycle in multicellular tumour spheroids
Get PDFA partial differential equation model is developed to understand the effect that nutrient and acidosis have on the distribution of proliferating and quiescent cells and dead cell material (necrotic and apopotic) within a multicellular tumour spheroid. The rates of cell quiescence and necrosis depend upon the local nutrient and acid concentrations and quiescent cells are assumed to consume less nutrient and produce less acid than proliferating cells. Analysis of the differences in nutrient consumption and acid production by quiescent and proliferating cells shows low nutrient levels do not necessarily lead to increased acid concentration via anaerobic metabolism. Rather, it is the balance between proliferating and quiescent cells within the tumour which is important; decreased nutrient levels lead to more quiescent cells, which produce less acid than proliferating cells. We examine this effect via a sensitivity analysis which also includes a quantification of the effect that nutrient and acid concentrations have on the rates of cell quiescence and necrosis
Standard Triples of Structured Matrix Polynomials
Get PDFThe notion of standard triples plays a central role in the theory of matrix polynomials.
We study such triples for matrix polynomials
with structure , where is the
Hermitian, symmetric, -even, -odd,
-palindromic or -antipalindromic structure (with ).
We introduce the notion of -structured standard triple. With the exception of -(anti)palindromic matrix polynomials of even degree
with both and as eigenvalues,
we show that has structure if and only if
admits an -structured standard triple,
and moreover that every standard triple of a matrix polynomial with
structure is -structured.
We investigate the important special case of -structured
Jordan triples
Investigating the Performance of Asynchronous Jacobi's Method for Solving Systems of Linear Equations
No full textEver-increasing core counts create the need to develop parallel algorithms that avoid closely-coupled execution across cores. In this paper we present two case studies investigating the performance of several parallel asynchronous implementations of Jacobi's method for solving systems of linear equations. Although conditions for the convergence of asynchronous Jacobi are well known, what drives its rate of convergence is less well understood. The first case study investigates the algorithm's performance when executed on large numbers of processors on a Cray XE6, while the second explores the effect of varying the number of synchronous and asynchronous processors. We observe that the performance of parallel asynchronous Jacobi is highly implementation, problem and architecture-dependent
The Jeffery-Hamel similarity solution and its relation to flow in a diverging channel
No full textWe explore the relevance of the idealized Jeffery�Hamel similarity solution to the
practical problem of flow in a diverging channel of finite (but large) streamwise
extent. Numerical results are presented for the two-dimensional flow in a wedge of
separation angle 2α, bounded by circular arcs at the inlet/outlet and for a net radial
outflow of fluid. In particular, we show that in a finite domain there is a sequence
of nested neutral curves in the (Re, α) plane, each corresponding to a midplane
symmetry-breaking (pitchfork) bifurcation, where Re is a Reynolds number based on
the radial mass flux. For small wedge angles we demonstrate that the first pitchfork
bifurcation in the finite domain occurs at a critical Reynolds number that is in
agreement with the only pitchfork bifurcation in the infinite-domain similarity solution,
but that the criticality of the bifurcation differs (in general). We explain this apparent
contradiction by demonstrating that, for α
1, superposition of two (infinite-domain)
eigenmodes can be used to construct a leading-order finite-domain eigenmode. These
constructed modes accurately predict the multiple symmetry-breaking bifurcations of
the finite-domain flow without recourse to computation of the full field equations. Our
computational results also indicate that temporally stable, isolated, steady solutions
may exist. These states are finite-domain analogues of the steady waves recently
presented by Kerswell, Tutty, & Drazin (J. Fluid Mech., vol. 501, 2004, pp. 231�250)
for an infinite domain. Moreover, we demonstrate that there is non-uniqueness of
stable solutions in certain parameter regimes. Our numerical results tie together, in a
consistent framework, the disparate results in the existing literature