A Criterion for an Optimal Control System to have Finite Switchings

A simple criterion is suggested which determines whether ornot an optimal control system has a finite number of switches.The results are developed using as an example the simple system with performance criterion , but are shown to hold for the more general system where f(0, 0) = 0 (i.e. the uncontrolled system has the originas an equilibrium point) subject to a performance criterion where g(0, 0) 0. The cases where g(0, 0)and are zero for all are also discussed. In particular one obtains a very simpleproof of the result that all two dimensional time optimal controlsystems with a single linear control, whose objective set consistsof the (assumed) unique critical point of the uncontrolled system,have a finite number of switches.     

Regular approximation of singular self-adjoint differential operators

Given a singular self-adjoint differential operator of order 2n with real coefficients we constructtwo sequences of regular self-adjoint differential expressions^r which converge to ina generalized sense of resolvent convergence. The first constructionis suitable when no information about the real resolvent setof is available. The second is suitablewhen we know a real point of the resolvent set of .The main application of this construction is in numerical solutionof singular differential equations.     

Convergence Properties of Pad? Approximants to exp (z) and Their Derivatives

Bounds are found for the error involved in representing exp(z) in a by the (p,q) Pad? approximant,, and its derivatives . The behaviour of these bounds is studied using information aboutthe distribution of the zeros of d_p,q(z). It is shown that undersuitable conditions g_p,q(z) and tend to exp (z) along all diagonal, row, and column paths in the Pad?table.     

Moir? Separations

The analysis of the moir? phenomenon by means of elementarygeometry which was begun by Firby (1984) and Firby & Stone(1984) is extended from first order to all integral and fractionalorders, and from consideration of the plane only to considerationof a general finite-dimensional space . The idea of the -curve, which forms the boundaryof the set in in which first-order interference might be observed, is extended to the idea of aboundary in for any given integral or fractional order of interference. Separating setsin , which form the boundary of regions in which two given integral or fractional ordersof interference might dominate relative to one another, areintroduced. The wavelength and contrast analysis previouslycarried out only for first-order interference in is extended to deal with all integral and fractionalorders of interference, and this is used to explain why highintegral orders, and all fractional orders, of interferenceare observed only in exceptional circumstances.     

Heat Equation for an Arbitrary Multiply-Connected Region in R2 with Impedance Boundary Conditions

The basic problem in this paper is that of determining the geometryof an arbitrary multiply-connected region in , together with the impedance boundary conditions,from a complete knowledge of the eigenvalues for the Laplace operator, using the asymptotic expansionof the spectral function as t0.     

Bivariational Methods for Hammerstein Integral Equations

Bivariational methods are presented for nonlinear integral equationsof Ham-merstein type = Kf(). With appropriate conditions onK and f, various upper and lower bounding functionals are derivedfor inner products and associated with the solution . Inthe latter case, suitable choices for g lead to point wise boundson both and its derivative. The methods are tested on a pendulumequation, and encouraging accuracy is obtained using simpletrial vectors.     

Optimal Approximation of the Electrostatic Capacity Matrix of Two Conducting Spheres by a Short-Distance Asymptotic Expansion

An integral representation for the electrostatic capacity matrixC=[c_ij]_i,j=1,2 of two conducting spheres of radii R₁, and R₂is obtained. A short-distance asymptotic expansion is then derivedand its approximation properties for fixed (surface) distancer between the spheres are investigated. An error function is defined for c_ij(r) and its nthorder asymptotic approximant it has the property following from the divergence of the expansion, and thereby shows thatthe optimal approximation of c_ij(r) is achieved by an approximantof finite order n = n_ij(r) depending possibly on r and the indicesi,j. The value gives the quality of approximation of c_ij by the asymptotic expansion for a givendistance r between the spheres. The point sets and are introduced in order to describe the distance ranges where c_ij can be approximatedwithin a given error >0 by an asymptotic approximant of given order n, or at least by theoptimal approximant, respectively. The optimal order n_ij(r)and the -approximation sets and D() are investigated numerically.     

The Decay of Eddy-currents in Thin Sheets and Related Water-wave Problems

The decay of the eddy-currents that are induced in a thin, uniform,imperfectly-conducting sheet by switching off the source ofan external magnetic field is investigated. For the two-dimensionalproblem of an infinite strip the (non-dimensional) decay constants_n and eddy-current distributions i_n(x) are the eigenvalues andeigenfunctions of the integral equation with the constraint. For the circular disc the corresponding equation is where and K and E are complete elliptic integrals. For both problemsthe initial eddy-currents have inverse-square-root singularitiesat the edges but during their decay the eddy currents are finiteat the edges and the normal magnetic fields have logarithmicsingularities there. Numerical results are given for variousinitial-value problems. The eddy current problems are closely related to water-waveproblems in which there is a strip-shaped or circular aperturein a horizontal rigid dock. If _n and _n are the decay constantsand magnetic scalar potentials for the strip and _n and _n theangular frequencies and velocity potentials for the normal modesin the strip-shaped aperture, then _n =_n² and _n and _n are thereal and imaginary parts respectively of a holomorphic function.The velocities in the normal modes are deduced from the solutionof the eddy-current problem and are found to agree with resultsgiven in Miles (1972). For circular geometries the eigenvaluesand eigenfunctions of the axisymmetric eddy-current problemare the same as those of the water-wave problem that has angularvariation eⁱ; where (, , z) are cylindrical polar co-ordinateslocated at the centre of the basin.     

An analogue of Berry's conjecture for the phase in fractal obstacle scattering

In this paper, the authors consider the high-frequency asymptoticsof the phase s() of acoustic waves scattered by an obstacleRⁿ with fractal boundary. Under certain conditions, it is provedthat if is –Minkowski measurable with –Minkowskimeasure µ then there exists a positive constant C_n, dependingonlyon n and such that where     

The Effects of Rounding Error on an Algorithm for Downdating a Cholesky Factorization

Let the positive definite matrix A have a Cholesky factorizationA= R^TR. For a given vector xsuppose that ? =A - xx^T has a Choleskyfactorization ? = ^T.This paper considers an algorithm for computing from R and x and an extension for removing a row from the QR factorizationof a regression problem. It is shown that the algorithm is stablein the presence of rounding errors. However, it is also shownthat the matrix can be a very ill-conditioned function of R and x.     

Periodic solutions and oscillation of discrete non-linear delay population dynamics model with external force

^** Email: emelabbasy{at}mans.edu.eg^*** Email: shsaker{at}mans.edu.eg In this paper, we consider the discrete non-linear delay populationdynamics model [graphic: see PDF] where m is a positive integer, p(n), Q(n) and (n) are positiveperiodic sequences of period . By the method that involves theapplication of the Gaines and Mawhins coincidence degree theory,we prove that there exists a positive -periodic solution (n). We prove that every positive solutionof (*) which does not oscillate about (n)satisfies lim_t[y(n)–(n)]=0.We establish some necessary and sufficient conditions for theoscillation of every positive solution about (n), and finally, we establish the lower and upperbounds of the oscillatory solutions.     

Steady-state solution for electroplating

At steady state, electroplating processes are governed by thedimensionless equations where d_i, e_i, and u_i arerespectively the diffusion coefficient, charge, and concentrationof the ith species. The extra electroneutrality condition will determine the electric potential. This system of nonlinear differential equations is subjectto the nonlinear boundary conditions modelling the actual electrodekinetics. The authors prove the existence of the solution andconstruct a computational algorithm. Numerical experiments areperformed on practical data.     

The Bicanonical Map of Surfaces with pg = 0 and K2 >= 7

A minimal surface of general type with p_g(S) = 0 satisfies 1 K² 9, and it is known that the image of the bicanonical map is a surface for , whilst for , the bicanonical map is always a morphism. In this paper it is shown that is birationalif , and that the degree of is at most 2 if or By presenting two examples of surfaces S with and 8 and bicanonical map of degree 2, it is alsoshown that this result is sharp. The example with is, to our knowledge, a new example of a surfaceof general type with p_g = 0. The degree of is also calculated for two other known surfacesof general type with p_g = 0 and . In both cases, the bicanonical map turns out to be birational.     

Supercooling and Superheating Effects in Phase Transitions

In the one-dimensional Stefan problem, the standard equilibriumcondition ; = 0 at the free boundary x = s(t) is here replacedby the kinetic law s'(t) = ß((s(t), t)), where ß:R R is continuous and increasing and ß(0) = 0. Thisrepresents supercooling and superheating effects. The standardStefan problem is then obtained in the limit as ß'(0) + A similar condition is considered for a radially symmetric system,taking also account of the surface tension effect. A kineticcondition is introduced also for phase transitions in binaryalloys, represented by means of the system of the Fourier'sand Fick's laws. In the case of several space dimensions, denoting by [0, 1]the concentration of the more energetic phase, the followinglaw is considered this is also extendedto binary systems. For all of the previous models of phase transitions, existenceresults are proved for the variational problems obtained bycoupling the free boundary condition with the energy conservationequation (and with the mass diffusion equation, for alloys).For heterogeneous systems, also a different model based on "non-equilibriumthermodynamics" is considered. This paper reviews the results of Visintin [IMA J. appl. Math.(1985) 34, 225–245] and announces those of Visintin (1985,to appear in Q appl. Math, and in Ann. Mat. pura appl.).     

Application of JMN Approximants to Numerical Initial-value Problems in Linear Differential-algebraic Systems

For f: [0,) R, the J_MN approximant of f(t) is where M and N are non-negative integers and ß_k, _i,K_iare defined constants. Under appropriate conditions on f andprovided Re(_i) > 0 The approximants are the bases of recursions for numerical initial-valueproblems in linear differential-algebraic systems with constantcoefficients. The recursions are stable when N M N–2.Each step of a recursion involves mainly the solution of N/2uncoupled algebraic systems.     

Asymptotic Expansion of a Double Integral with a Curve of Stationary Points

An asymptotic expansion is constructed for the double integral as +, where f(x, y) hasa curve of stationary points in D. The first two coefficientsof the expansion are explicitly calculated.     

A Criticality Criterion for Thermal Explosions in Well-stirred Systems with Reactant Consumption

The autonomous differential equations for the temperature andreactant consumption in a first-order well-stirred exothermicreaction are considered. An examination of the phase-plane solutionsallows the qualitative behaviour of the Semenov number as afunction of maximum temperature rise ^* to be established. Inthe limit of infinite adiabatic temperature rise (B) and zeroactivation energy parameter ( = 0), the relationship between and stationary temperature _s is known to be e_¹ = _s. Criticalityarises at the maximum of (_s) and leads to the critical Semenovvalues (_s)_cr = 1, _cr = e^–1. For sufficiently large B,it is shown that the (^*) curve has a bifurcation at ^* = 1, withthe upper branch monotonically increasing and the lower branchmonotonically decreasing for ^* > 1. In the limit B thesebecome respectively the straight line = e^–1, _s 1 andthe unstable branch of = _se^–1, _s 1 and the unstablebranch of = _s e^–. Criticality for finite B is definedas occurring at the bifurcation, namely ^*_cr = 1, with _cr(B)the value of at this point. Values of these Semenoy numbersare obtainable from the numerical calculations of Boddingtonet al. [Proc. R. Soc. Lond. (1983), 390, 13–30]. The newcriterion is applied to an approximate phase-plane solution.The corresponding critical parameter is found to be _cr = e^–1[1+B^–(2–e^–1)+O(B^–1)].     

Bifurcation of an equation arising in porous-medium combustion

We study the bifurcation of positive solutions of a steady-stateproblem arising in porous-medium combustion: We give explicit criteria for a uniquesolution for all >0 when a = 0 and for an S-shaped bifurcationcurve when a>0. We also prove a conjecture of Norbury &Stuart (1987, IMA J. Appl. Math. 39, 241–57).     

Evaluation of Some Integrals Encountered in Calculations in Many-fermion Systems

The integrals and (l(x)=3j₁(x)/x) and j₁(x) the spherical Bessel function of thefirst order) are evaluated analytically     

The Complex Plank Problem

It is shown that if is a sequence of norm 1 vectors in a complex Hilbert space and is a sequence of non-negative numbers satisfying then there is a unit vector z for which for every j. The result is a strong,complex analogue of the author's real plank theorem.