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Poisson approximations for epidemics with two levels of mixing
No full textThis paper is concerned with a stochastic model for the spread of an epidemic among a population of n individuals, labeled , in which a typical infected individual, i say, makes global contacts, with individuals chosen independently and uniformly from the whole population, and local contacts, with individuals chosen independently according to the contact distribution , at the points of independent Poisson processes with rates and , respectively, throughout an infectious period that follows an arbitrary but specified distribution. The population initially comprises infectives and susceptibles. A sufficient condition is derived for the number of individuals who survive the epidemic to converge weakly to a Poisson distribution as . The result is specialized to the households model, in which the population is partitioned into households and local contacts are chosen uniformly within an infective's household; the overlapping groups model, in which the population is partitioned in several ways and local mixing is uniform within the elements of the partitions; and the great circle model, in which
Periodic solutions of discrete Volterra equations
Get PDFIn this paper, we investigate periodic solutions of linear and nonlinear discrete Volterra equations of convolution
or non-convolution type with unbounded memory.
For linear discrete Volterra equations of convolution type, we establish Fredholm’s alternative theorem and for
equations of non-convolution type, and we prove that a unique periodic solution exists for a particular bounded initial
function under appropriate conditions. Further, this unique periodic solution attracts all other solutions with bounded
initial function. All solutions of linear discrete Volterra equations with bounded initial functions are asymptotically
periodic under certain conditions. A condition for periodic solutions in the nonlinear case is established.
© 2003 IMACS. Published by Elsevier B.V. All rights reserved
Reductions of integrable equations: dihedral group
Get PDFWe discuss the algebraic and analytic structure of rational Lax operators. With algebraic reductions of Lax equations we associate a reduction group—a group of automorphisms of the corresponding infinite-dimensional Lie algebra. We present a complete study of dihedral reductions for sl(2,{\bb C}) Lax operators with simple poles and corresponding integrable equations. In the last section we give three examples of dihedral reductions for sl(N,{\bb C}) Lax operators
Properties of American option prices
No full textWe investigate some properties of American option prices in the setting of time- and level-dependent volatility. The properties under consideration are convexity in the underlying stock price, monotonicity and continuity in the volatility and time decay. Some properties are direct consequences of the corresponding properties of European option prices that are already known, and some follow by writing solutions of different stochastic differential equations as time changes of the same Brownian motion
The numerical stability of barycentric Lagrange interpolation
No full textThe Lagrange representation of the interpolating polynomial can be rewritten in two
more computationally attractive forms: a modified Lagrange form and a barycentric
form. We give an error analysis of the evaluation of the interpolating polynomial using
these two forms. The modified Lagrange formula is shown to be backward stable. The
barycentric formula has a less favourable error analysis, but is forward stable for any set of
interpolating points with a small Lebesgue constant. Therefore the barycentric formula can
be significantly less accurate than the modified Lagrange formula only for a poor choice of
interpolating points. This analysis provides further weight to the argument of Berrut and
Trefethen that barycentric Lagrange interpolation should be the polynomial interpolation
method of choice
Shock wave interaction with viscous wake in supersonic flow
Get PDFA theoretical investigation of the breakdown of the viscous wake downstream of a flat plate in supersonic flow is performed in this paper based on the large Reynolds number () asymptotic analysis of the Navier–Stokes equations. The breakdown is provoked by an oblique shock wave impinging on the wake a small distance downstream of the plate trailing edge. Two flow regimes are considered. In the first is assumed to be an quantity in which case the shock impinges on the wake within the region of viscous–inviscid interaction that is known to occupy a vicinity of the trailing edge with longitudinal extent of . Under these conditions the interaction process may be described by the equations of the triple-deck theory. To obtain a numerical solution of these equations we used a rapid matrix Thomas technique in conjunction with Newton iterations. The results of the calculations not only predict the wake breakdown near the shock location but also reveal a hysteresis behaviour of the flow as the shock is moved downstream, giving rise to three solution branches. The second part of the paper is concerned with the flow regime when the shock interacts with the wake further downstream of the trailing edge triple-deck region: . In this case the fluid motion proves to be inviscid to leading order not only in the upper deck of the interaction region but also everywhere inside the wake. Due to this simplification the interaction problem can be reduced to a single integro-differential equation governing the pressure distribution along the interaction region. With known pressure the Bernoulli equation may be used to find the velocity field. The Bernoulli equation also allows us to formulate a simple criterion which may be used to predict the onset of wake breakdown. We found that viscosity becomes important again in a smaller vicinity of the breakdown point where the flow reversal takes place. It is remarkable that the viscous–inviscid interaction problem governing the flow in this vicinity admits an analytical solution
Stochastic bounds for Lévy processes
Get PDFUsing the Wiener–Hopf factorization, it is shown that it is possible to bound the path of an arbitrary Lévy process above and below by the paths of two random walks. These walks have the same step distribution, but different random starting points. In principle, this allows one to deduce Lévy process versions of many known results about the large-time behavior of random walks. This is illustrated by establishing a comprehensive theorem about Lévy processes which converge to ∞ in probability
Deriving information from inconsistent knowledge bases: A completeness proof for "
No full textThe logical consequence relations {eta}{triangleright}{eta} provide a very attractive way of inferring new facts from inconsistent knowledge bases without compromising standards of credibility. In this short note we provide proof theories and completeness theorems for these consequence relations which may have some applicability in small examples
Asymptotic matching constraints for a boundary-layer flow of a power-law fluid
No full textWe reconsider the three-dimensional boundary-layer flow of a power-law (Ostwald�
de Waele) rheology fluid, driven by the rotation of an infinite rotating plane in an
otherwise stationary system. Here we address the problem for both shear-thinning and
shear-thickening fluids and show that there are some fundamental issues regarding
the application of power-law models in a boundary-layer context that have not
been mentioned in previous discussions. For shear-thickening fluids, the leading-order
boundary-layer equations are shown to have no suitable decaying behaviour in the
far field, and the only solutions that exist are necessarily non-differentiable at a
critical location and of �finite thickness�. Higher-order effects are shown to regularize
the singularity at the critical location. In the shear-thinning case, the boundary-layer
solutions are shown to possess algebraic decay to a free-stream flow. This case is
known from the existing literature; however here we shall emphasize the complexity
of applying such solutions to a global flow, describing why they are in general
inappropriate in a traditional boundary-layer context. Furthermore, previously noted
difficulties for fluids that are highly shear thinning are also shown to be associated
with the imposition of incorrect assumptions regarding the nature of the far-field flow.
Based on Newtonian results, we anticipate the presence of non-uniqueness and through
accurate numerical solution of the leading-order boundary-layer equations we locate
several such solutions
Iterated Function System Models of Digital Channels
No full textThis paper introduces a new class of models of digital communications channels. Physically, these models take account of the digital nature of the input. Mathematically, they are iterated function systems. As a consequence of making explicit assumptions about the role of discreteness in the models, it is possible to make general statements about the behaviour of these channels without needing to assume that they are linear. We provide the mathematical background necessary to understand the behaviour of these models and prove a number of results about their observability. We also provide a number of examples intended to demonstrate their connection with linear state-space models, and to suggest how the nonlinear theory might be developed towards applications