New perturbation analyses for the Cholesky factorization

We present new perturbation analyses for the Cholesky factorizationA = R^T R of a symmetric positive definite matrix A. The analysesmore accurately reflect the sensitivity of the problem thanprevious normwise results. The condition numbers here are alteredby any symmetric pivoting used in PAP^T = R^TR, and both numericalresults and an analysis show that the standard method of pivotingis optimal in that it usually leads to a condition number veryclose to its lower limit for any given A. It follows that thecomputed R will probably have greatest accuracy when we usethe standard symmetric pivoting strategy. Initially we give a thorogh analysis to obtain both first-orderand strict normwise perturbation bounds which are as tight aspossible, leading to a definition of an optimal condition numberfor the problem. Then we use this approach to obtain reasonablyclear first-order and strict componentwise perturbation bounds. We complete the work by giving a much simpler normwise analysiswhich provides a somewhat weaker bound, but which allows usto estimate the condition of the problem quite well with anefficient computation. This simpler analysis also shows whythe factorization is often less sensitive than we previouslythought, and adds further insight into why pivoting usuallygives such good results. We derive a useful upper bound on thecondition of the problem when we use pivoting. This research was supported by the Natural Sciences and EngineeringResearch Ciuncil of Canada Grant OGP0009236. This research was supported in part by the US National ScienceFoundation under grant CCR 95503126.     

Forced convection over a semi-infinite flat plate

The problem of forced convection heat transfer over a semi-infinite flat plate is treated by the method of series truncation, so as to yield results valid from leading edge to far downstream (0 < -R ₀< ∞). Results are presented for Prandtl number Pr = 0.1, 0.7, and 10. It is found that the effect of leading edge on heat transfer is smaller than on skin friction.     

Amplitude Bounds on Periodic and Aperiodic Oscillations: Second-order Systems

The global behaviour of the control systems described by thepair of differential equations x{dot} = –f(x)±g(y)+p(t), y{dot} = –f(x)±g(y)+p(t) has been investigated. Here f(x), g(y), h(x) and k(y) are polynomialsof odd degree with leading coefficients positive and p(t) andq(t) are bounded functions of time. Sufficient conditions havebeen found under which the trajectories of the above systemmay eventually be confined in a subset of (x, y, t)-space, thusgiving bounds on the amplitude of periodic as well as aperiodicoscillations. Further bounds on the amplitude of oscillationshave been investigated by finding regions in (x,y,t)-space fromwhich all trajectories eventually leave and into which no trajectoriesenter. Thus sufficient conditions have been derived for theexistence of an annulus in which oscillatory behaviour may beconfined.     

Interaction of Weak Shock Waves with Magnetohydrodynamic Boundary Layers

The interaction of weak shock waves with magnetohydrodynamicboundary layers is investigated. An approximate model is usedin which the main stream is supersonic, and the laminar boundarylayer is replaced by a stationary or shear layer in the xy-plane.The inviscid semi-infinite fluid is continuous and perfectly-electricallyconducting. The fluid velocity U(y) varies monotonically fromzero at the non-conducting wall, y = 0, to a supersonic valueat the outer edge of the shear layer where y = , beyond whichthe supersonic velocity remains uniform. In practice, we shallassume that the velocity profile in the region y < is linear;also that the fluid density in the basic flow, and the soundspeed c, are independent of y. A uniform magnetic field, alignedparallel to the free stream at infinity, is assumed to be appliedthroughout the flow field. Perturbations to the flow are assumedto be small so that the governing equations can be linearized.Disturbances to the flow are found to propagate upstream farmore than in the corresponding non-magnetic problem, and anincident compression wave is reflected locally as a compressionwave upstream and an expansion wave downstream of the incidentdisturbance.     

Implicit finite difference schemes for a linear model of well-reservoir coupling

Standard reservoir models usually consider wells as Dirac measuresover an interval length. Moreover, the well-reservoir couplingis taken into account under quite simplified assumptions. Mostrecently, however, attention has been drawn to the fact thatin some situations, such as those related to non-vertical wells,these simplifications do not allow us to model some relevantmechanisms of the coupled flow. Therefore, more complex alternativemodels have been proposed recently in the oil reservoir simulationliterature. A linearized version for the well-reservoir couplingcan be written, in an appropriate functional setting, in theform U'(t) + AU(t) = F(t). In this work we discuss implicitin time discretizations of this equation, of the form { Uⁿ⁺¹_h - Uⁿ_h÷+A_hUⁿ⁺¹_h = Fⁿ⁼¹_h, U⁰_h=U_0.h We propose two different approximations, corresponding to first-and second-order spatial truncation errors, and we establishthe convergence of both approximations.     

Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes

This paper is devoted to the study of an error estimate of thefinite volume, approximation to the solution u L(R_N x R) ofthe equation u_t + div(Vf(u)) = 0, where v is a vector functiondepending on time and space. A 'h' error estimate for an initialvalue in BV(R^N) is shown for a large variety of finite volumemonotonous flux schemes, with an explicit or implicit time discretization.For this purpose, the error estimate is given for the generalsetting of approximate entropy solutions, where the error isexpressed in terms of measures in R^N and R^N x R. The study ofthe implicit schemes involves the study of the existence anduniqueness of the approximate solution. The cases where an 'h'error estimate can be achieved are also discussed.     

A qualocation method for parabolic partial differential equations

In this paper a qualocation method is analysed for parabolicpartial differential equations in one space dimension. Thismethod may be described as a discrete H¹-Galerkin method inwhich the discretization is achieved by approximating the integralsby a composite Gauss quadrature rule. An O (h^4-i) rate of convergencein the W^i.p norm for i = 0, 1 and 1 p is derived for a semidiscretescheme without any quasi-uniformity assumption on the finiteelement mesh. Further, an optimal error estimate in the H² normis also proved. Finally, the linearized backward Euler methodand extrapolated Crank-Nicolson scheme are examined and analysed.     

Nonoscillation of Elliptic Integrals Related to Cubic Polynomials with Symmetry of Order Three

We study zeros of elliptic integrals I(h)=_HhR(x,y)dxdy, whereH(x,y) is a real cubic polynomial with a symmetry of order three,and R(x,y) is a real polynomial of degree at most n. It turnsout that the vector space A_n formed by such integrals is a Chebishevsystem: the number of zeros of each elliptic integral I(h)A_nis less than the dimension of the vector space A_n. 1991 MathematicsSubject Classification 34C10.     

Funk-Hecke Formula for Orthogonal Polynomials on Spheres and on Balls

Analogues of the Funk–Hecke formula for spherical harmonicsare proved for Dunkl's h-harmonics associated to the reflectiongroups, and for orthogonal polynomials related to h-harmonicson the unit ball. In particular, an analogue and its applicationare discussed for the weight function (1–|x|²)^µ–1/2on the unit ball in R^d. 2000 Mathematics Subject Classification33C50, 33C55, 42C10.     

Sums of Connectivity Functions on Rn

A function f: Rⁿ R is a connectivity function if the graphof its restriction f|C to any connected C Rⁿ is connected inRⁿ x R. The main goal of this paper is to prove that every functionf: Rⁿ R is a sum of n + 1 connectivity functions (Corollary2.2). We will also show that if n > 1, then every functiong: Rⁿ R which is a sum of n connectivity functions is continuouson some perfect set (see Theorem 2.5) which implies that thenumber n + 1 in our theorem is best possible (Corollary 2.6). Toprove the above results, we establish and then apply the followingtheorems which are of interest on their own. For every dense G-subset G of Rⁿ there are homeomorphisms h₁,..., h_n of Rⁿ such that Rⁿ = G h₁(G) ... h_n(G) (Proposition2.4). For every n > 1 and any connectivity function f: Rⁿ R, ifx Rⁿ and > 0 then there exists an open set U Rⁿ such thatx U Bⁿ(x, ), f|bd(U) is continuous, and |(x) – f(y)|< for every y bd(U) (Proposition 2.7). 1991 MathematicsSubject Classification: 26B40, 54C30, 54F45.     

On the Non-linear Evolution of Gortler Vortices in Non-parallel Boundary Layers

The non-linear development of finite amplitude Görtlervortices in a non-parallel boundary layer on a curved wall isinvestigated using perturbation methods based on the smallnessof e, the non-dimensional wavelength of the vortices. The crucialstage in the growth or decay of the vortices takes place inan interior viscous layer of thickness O(²) and length O().In this region the downstream velocity component of the perturbationcontains a mean flow correction of the same order of magnitudeas the fundamental which is driving it. Moreover, these functionssatisfy a pair of coupled non-linear partial differential equationswhich must be solved subject to some initial conditions imposedat a given downstream location. It is found that, dependingon whether the boundary layer is more or less unstable downstreamof this location, the initial disturbance either grows intoa finite amplitude Görtler vortex or decays to zero. Forthe Blasius boundary layer on a concave wall it is found thatGörtler vortices can only develop if the rate of increaseof curvature of the wall is sufficiently large. In this casethe finite amplitude solution which develops initially in an-neighbourhood of the position where the disturbance is introducedchanges its structure further downstream. This structure isinvestigated at a distance O() (with 0< <1) downstreamof the above -neighbourhood. In this régime the downstreamfundamental velocity component has an elliptical profile overmost of the flow field. However, in two thin boundary layerslocated symmetrically either side of the centre of the viscouslayer the fundamental velocity component decays exponentiallyto zero. The locations of these layers are determined by aneigenvalue problem associated with the one-dimensional diffusionequation. The mean flow correction persists both sides of theboundary layer and ultimately decays exponentially to zero. This large amplitude motion is not sensitive to the imposedinitial conditions and appears to be the ultimate state of anyinitial disturbance. However, in the initial stages of the growthof the vortex, some surprising flows are possible. For example,it is possible to set up a vortex flow similar to that observedby Wortmann (1969) which consists of a sequence of cells inclinedat an angle to the vertical.     

Bio-inspired propulsion: Downstream travelling wave motion of a surface with finite & infinite extension

The thrust force on a surface that performs a fish-like travelling wave motion downstream to an oncoming flow is discussed. Unsteady potential flow, with vortex shedding from the trailing edge, is known to explain the generation of thrust. Contrarily, fish swimming has been related to the flow over an infinitely extended surface. To interlink both problems, the potential flow over the surface of finite length is considered in the limit of high wave numbers. It turns out that the leading order, space-periodic pressure does not contribute to thrust. Thus, the perturbation pressure is essential for propulsion. Besides, laminar flow is considered in the space-periodic setting. The present results reveal – in contrast to literature – that the surface force is always drag. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical Methods for y' =f(x, y) via Rational Approximations for the Cosine

To investigate stability and phase lag, a numerical method isapplied to the test equation y' = –²y. Frequently, thecharacteristic equation of the resulting recurrence relationhas the form ²– 2R_nm(v²) + 1 = 0, where v = h, with hthe steplength, and R_nm(v²) is a rational approximation forcos v. In this paper, properties of such approximations areused to provide a general framework for the study of stabilityintervals and orders of dispersion of a variety of one- andtwo-step methods. Upper bounds on the intervals of periodicityof explicit methods with maximum order of dispersion are established.It is shown that the order of dispersion of a P-stable method,for given n and m, cannot exceed 2m; a consequence is that,of the Pad? approximants for cos v, only the [0/2m] approximantshave modulus less than unity for all v² >0. A complete characterizationof P-stable methods of fourth order corresponding to the rationalapproximation R₂₂(v²) is followed by several results for methodswhich have finite intervals of periodicity; in particular, weidentify methods which have order of dispersion 6 or 8 withlarge intervals of periodicity. There is also a detailed discussionof P-stable methods of sixth order corresponding to the rationalapproximation R₃₃(v²).     

Amplitude-dependent Stability of Boundary-layer Flow with a Strongly Non-linear Critical Layer

The amplitude-dependent neutral stability properties, mainlyof an accelerating boundary-layer flow, are studied theoreticallyfor large Reynolds numbers when the disturbance size is sufficientlylarge to provoke a strongly non-linear critical layer withinthe flow field. The theory has a rational basis aimed at a detailedunderstanding of the delicate physical balances controllingstability. It shows that when the fundamental disturbance size rises to O(R^-1/3, where R is the Reynolds number based on theboundary-layer thickness, the neutral wavelength shortens andthe wavespeed increases in such a way that they become comparablewith the typical thickness and speed, respectively, of the basicflow. In this Rayleigh-like situation a new (previously negligible)feature emerges, that of a substantial pressure variation acrossthe critical layer, which strongly affects the jump conditionson the Rayleigh solutions holding outside the critical layer.As a result of the strong non-linearity the total velocity jumpis affected non-linearly by the critical layer vorticity, whilein contrast the phase shift remains linearly dependent on thevorticity. Furthermore, it is shown that the phase shift, notthe total velocity jump, dictates the neutral stability criteria. Also, flow reversal occurs near the wall where the disturbanceis greater than the basic flow. The link between the viscouseffects in the wall layers and in the critical layer fixes theamplitude-dependence of the neutral modes throughout. As thedisturbance amplitude increases the critical layer with vorticitytrapped within it moves toward the edge of the boundary layerand is forced to leave the boundary layer when exceeds O(R^-1/3,if neutral stability is to be maintained. This departure israther abrupt, involving a dependence on (scaled amplitude)^–12.A study of the more practical application to temporally growingdisturbances should be interesting.     

The forced convection flow of a uniform stream over a flat surface with a convective surface boundary condition

The forced convection heat transfer resulting from the flow of a uniform stream over a flat surface on which there is a convective boundary condition is considered. In previous papers [5], [6], [7], [8] it was assumed that the convective heat transfer parameter h_f associated with the hot surface depended on x, where x measures distance along the surface, so that problem could be reduced to similarity form. Here it is assumed that this heat transfer parameter h_f is a constant, with the result that the temperature profiles and overall heat transfer characteristics evolve as the solution develops from the leading edge. The heat transfer near the leading edge (small x), which we find to be dominated by the surface heat flux, the solution at large distances along the surface (large x), which dominated by the surface temperature, are discussed. A numerical solution to the full problem is then obtained for a range of values of the Prandtl number to join these two solution regimes.     

On the Centred Hausdorff Measure

Let v be a measure on a separable metric space. For t, q R,the centred Hausdorff measures µ^h with the gauge functionh(x, r) = r^t(vB(x, r))^q is studied. The dimension defined bythese measures plays an important role in the study of multifractals.It is shown that if v is a doubling measure, then µ^h isequivalent to the usual spherical measure, and thus they definethe same dimension. Moreover, it is shown that this is trueeven without the doubling condition, if q 1 and t 0 or ifq 0. An example in R² is also given to show the surprisingfact that the above assertion is not necessarily true if 0 <q < 1. Another interesting question, which has been askedseveral times about the centred Hausdorff measure, is whetherit is Borel regular. A positive answer is given, using the aboveequivalence for all gauge functions mentioned above.     

Buckling of an axisymmetric vesicle under compression: the effects of resistance to shear

We consider the axisymmetric deformation of an initially spherical,porous vesicle with incompressible membrane having finite resistanceto in-plane shearing, as the vesicle is compressed between parallelplates. We adopt a thin-shell balance-of-forces formulationin which the mechanical properties of the membrane are describedby a single dimensionless parameter, C, which is the ratio ofthe membrane's resistance to shearing to its resistance to bending.This results in a novel free-boundary problem which we solvenumerically to obtain vesicle shapes as a function of plateseparation, h. For small deformations, the vesicle contactseach plate over a small circular area. At a critical value ofplate separation, h_TC, there is a transcritical bifurcationfrom which a new branch of solutions emerges, representing buckledvesicles which contact each plate along a circular curve. Forthe values of C investigated, we find that the transcriticalbifurcation is subcritical and that there is a further saddle-nodebifurcation (fold) along the branch of buckled solutions ath = h_SN (where h_SN > h_TC). The resulting bifurcation structureis commensurate with a hysteresis loop in which a sudden transitionfrom an unbuckled solution to a buckled one occurs as h is decreasedthrough h_TC and a further sudden transition, this time froma buckled solution to an unbuckled one, occurs as h is increasedthrough h_SN. We find that h_SN and h_TC increase with C, thatis, vesicles that resist shear are more prone to buckling.     

Global h Fourier Integral Operators with Complex-Valued Phase Functions

The author considers globally defined h Fourier integral operators(h FIO) with complex-valued phase functions. Symbolic calculusof h FIO is considered and, using a new complex Gauss transform,the composition of h pseudodifferential operators (h PDO) andh FIO is considered. For a self-adjoint h PDO A(h) and h PDOP(h) and Q(h) with compactly supported symbols, the resultsare applied to approximate the kernel of the operator by a single, globally defined h-oscillatoryintegral. 2000 Mathematics Subject Classification 81Q20, 35S30.     

The Harmonious Chromatic Number of Bounded Degree Graphs

A harmonious colouring of a simple graph G is a proper vertexcolouring such that each pair of colours appears together onat most one edge. The harmonious chromatic number h(G) is theleast number of colours in such a colouring. Let d be a fixed positive integer, and >0. We show that thereis a natural number M such that if G is any graph with mM edgesand maximum degree at most d, then the harmonious chromaticnumber h(G) satisfies      

Influence of initial stresses on fracture of composite materials containing interacting cracks

Considered in this study are the axially-symmetric problems of fracture of composite materials with interacting cracks, which are subjected to initial (residual) stresses acting along the cracks planes. An analytical approach within the framework of three-dimensional linearized mechanics of solids is used. Two geometric schemes of cracks location are studied: a circular crack is located parallel to the surface of a semi-infinite composite with initial stresses, and two parallel co-axial penny-shaped cracks are contained in an infinite composite material with initial stresses. The cracks are assumed to be under a normal or a radial shear load. Analysis involves reducing the problems to systems of second-kind Fredholm integral equations, where the solutions are identified with harmonic potential functions. Representations of the stress intensity factors near the cracks edges are obtained. These stress intensity factors are influenced by the initial stresses. The presence of the free boundary and the interaction between cracks has a significant effect on the stress intensity factors as well. The parameters of fracture for two types of composites (a laminar composite made of aluminum/boron/silicate glass with epoxy-maleic resin and a carbon/plastic composite with stochastic reinforcement by short ellipsoidal carbon fibers) are analyzed numerically. The dependence of the stress intensity factors on the initial stresses, physical-mechanical parameters of the composites, and the geometric parameters of the problem are investigated.