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.     

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.     

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.     

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.     

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     

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.     

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.     

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.     

Some Relations Between Packing Premeasure and Packing Measure

Let K be a compact subset of Rⁿ, 0 s n. Let , P^s denote s-dimensional packing premeasure andmeasure, respectively. We discuss in this paper the relationbetween and P^s. We prove:if , then ; and if , then for any > 0, there exists a compact subset F of K such that and P^s(F) P^s(K) – .1991 Mathematics Subject Classification 28A80, 28A78.     

Near-minimax Interpolation by a Polynomial in z and z-1 on a Circular Annulus

A polynomial of degree n in z^–1 and n–1 in z isdefined by an interpolation projection from the space A(N_p) of functions f analytic in the circular annulusp^–1 < <p and continuous on itsboundaries = p^–1, p. The points ofinterpolation are chosen to be spaced at equal angles aroundthe two boundaries, with arguments on the inner boundary midwaybetween those on the outer boundary. By calculating the Lebesgueconstants numerically, is found to be close to a minimax approximation for all p 1and all degrees n in the range 1 n 15. In the limiting casesp = 1 and, it is proved that is asymptotic to 2^–1 log n. More specifically and , where _nis the Lebesgueconstant of Gronwall for equally spaced interpolation on a circleby a polynomial of degree n. It is also demonstrated that is not in general monotonic in p, and that is not everywhere differentiable in p.     

Global Branch of Solutions for Non-Linear Schrodinger Equations with Deepening Potential Well

We consider the stationary non-linear Schrödinger equation where > 0 and the functionsf and g are such that and for some bounded open set R^N. We use topological methods to establish the existenceof two connected sets D^± of positive/negative solutionsin R x W², ^p R^N where that cover the interval (, ()) in the sense that and furthermore, The number () is characterized as the unique value of in theinterval (, ) for which the asymptotic linearization has a positiveeigenfunction. Our work uses a degree for Fredholm maps of indexzero. 2000 Mathematics Subject Classification 35J60, 35B32,58J55.     

The Model Plasma Problem: Existence of a Solution Branch

We are interested in the model plasma problem –u = u⁺in ,u = –d on , _au⁺ dx=j where is a bounded domain in with boundary ; here, j isa given positive number, the function u and the positive number are the unknowns of the problem, and d is a real parameter.Using a variant of the implicit function theorem, we can provethe existence of a global solution branch parametrized by d.The method has the advantage that it can be used for analysingthe approximation of the above problem by a finite-element method.     

A new methodology for the stability analysis of pairwise triangularizable and related switching systems

We consider the asymptotic stability of the time-varying dynamicsystem : = A(t)x, A(t) R^{n x n}, A(t) A= {A₁, ..., A_m}, where A_i is Hurwitz and where a set of non-singularmatrices T_{i j} exist such that any pair of matrices {T_{i j} A_iT_{i j}^–1, T_{i j} A_j T_{i j}^–1}, i, j {1, ..., m}, areupper triangular. Switching systems of this form are referredto as pairwise triangularizable switching systems. It can beestablished that (a) pairwise triangularizability is not sufficientto guarantee the existence of a common quadratic Lyapunov functionfor the linear time-invariant dynamic systems A_i : = A_i x; (b) additional conditions can be specified which guaranteeasymptotic stability of the switching system . In this paperwe also show that pairwise triangularizability is not even sufficientto guarantee asymptotic stability of the switching system .We also show that the method of proof of stability in (b), whichdoes not assume the existence of a common quadratic Lyapunovfunction, can be used to prove the asymptotic stability of moregeneral switching systems (systems that are not pairwise triangularizable).Finally, we show that our results can be used as the basis forthe design of practical control systems; namely, for the designof an automobile speed switched controller with guaranteed stabilityproperties.     

Eigenvalues of the Laplacian: an Extension to Higher Dimensions

The trace function where are the eigenvalues of the three-dimensional Laplacian, is studied for a variety of domains. The dependenceof (t) on the connectivity of domains and the boundary conditionsis analysed. Particular attention is given for a spherical shell.     

Determinants of Lattices induced by Rational Subspaces

Let L and be orthogonal complementary rational linear subspaces of Eⁿ, and let = L Zⁿ and $$\stackrel{\&macr;}{\Lambda}$$ = Zⁿ be the sublatticesof the usual integer lattice Zⁿ induced by L and . Then the determinants of and are equal. The samerelationship holds between the determinants of the lattices and obtained by orthogonal projection of Zⁿ on to L and .     

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.     

Ergodicity of a Class of Cocycles Over Irrational Rotations

It is proved that if is irrational and L²(S¹) with o(l/n)then for each mZ\{0} the corresponding skew product is ergodic. The rigidity of specialflows over irrational rotations with roof functions whose Fouriercoefficients are in o(l/n) is also shown.     

Asymptotic Solution of Hyperbolic Systems with Constant Coefficients

An integral representation of the exact solution of the initialvalue problem for the hyperbolic equation of the form is derived. Here A^o, A^v, B, and Care constant m x m matrices, u(t, X; ) is an m-component columnvector, and is a positive parameter. Various conditions areimposed on the coefficient matrices that permit the applicationof the method of stationary phase in several variables to theintegral representation of the exact solution. The leading termof the asymptotic expansion as of the exact solution is obtainedfor several types of initial data and source functions whichdepend on the parameter .