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No description available.

No description available.

We consider a one-parameter family of invertible maps of a twodimensional lattice, obtained by applying round-o to planar rotations. All orbits of these maps are conjectured to be periodic. We let the angle of rotation approach =2, and show that the limit of vanishing discretisation is described by an integrable piecewise-a ne Hamiltonian ow, whereby the plane foliates into families of invariant polygons with an increasing number of sides. Considered as perturbations of the ow, the lattice maps assume a di erent character, described in terms of strip maps: a variant of those found in outer billiards of polygons. Furthermore, the flow is nonlinear (unlike the original rotation), and a suitably chosen Poincar e return map satisfi es a twist condition. The round-o perturbation introduces KAM-type phenomena: we identify the unperturbed curves which survive the perturbation, and show that they form a set of positive density in the phase space. We prove this considering symmetric orbits, under a condition that allows us to obtain explicit values for densities. Finally, we show that the motion at in finity is a dichotomy: there is one regime in which the nonlinearity tends to zero, leaving only the perturbation, and a second where the nonlinearity dominates. In the domains where the nonlinearity remains, numerical evidence suggests that the distribution of the periods of orbits is consistent with that of random dynamics, whereas in the absence of nonlinearity, the fluctuations result in intricate discrete resonant structures.

We are interested in rigorous asymptotic results pertaining to three different differential equations which lie in the Heun class (see [1] §31). The Heun class contains those ordinary linear second-order differential equations with four regular singularities. We first investigate the Lamé equation d²w/dz² +(h − v(v + 1)k² sn²(z, k)) w = 0, z ∈ [−K,K], where 0 < k < 1, sn(z, k) is a Jacobi elliptic function, and K = ∫ 1 0 dz/√(1 − z²)(1 − k²z²) is the complete elliptic integral of the first kind. We obtain rigorous uniform asymptotic approximations complete with error bounds for the Lamé functions Ecm/v (z, k²) and Esm/v+¹ (z, k²) for z ∈ [0,K] and m ∈ N0, and rigorous approximations for their respective eigenvalues am/v and bm/v+¹, as v →∞. Then we obtain asymptotic expansions for the Lamé functions complete with error bounds, which hold only in a shrinking neighbourhood of the origin as v →∞. We also find corresponding expansions for the eigenvalues complete with order estimates for the errors. Then finally we give rigorous result for the exponentially small difference between the eigenvalues bm/v+¹ and am/v as v →∞. Second we investigate Mathieu’s equation d²w/dz² + (λ − 2h² cos 2z) w = 0, z ∈ [0, π], and obtain analogous results for the Mathieu functions cem(h, z) and sem+1(h, z) and their corresponding eigenvalues am and bm+1 for m ∈ N0 as h→∞, which are derived from those of Lamé ’s equation by considering a limiting case. Lastly we investigate the spheroidal wave equation d /dz ((1 − z²) dw/dz) + ( λ+ γ²(1 − z²) − μ²/1 − z²) w = 0, z ∈ [−1, 1], and consider separately the cases where γ² > 0 and γ² < 0. In the first case we give similar results to those previously for the prolate spheroidal wave functions Ps(z, γ²) and their corresponding eigenvalues λm/n for m, n ∈ N0 and n ≥ m as γ² →∞, and in the second we discuss the gap in theory which makes it difficult to obtain rigorous results as γ² → -∞, and how one would bridge this gap to obtain these.

The fractional calculus is a generalisation of the calculus of Newton and Leibniz. The substitution of fractional differential operators in ordinary differential equations substantially increases their modelling power. Fractional differential operators set exciting new challenges to the computational mathematician because the computational cost of approximating fractional differential operators is of a much higher order than that necessary for approximating the operators of classical calculus. 1. We present a new formulation of the fractional integral. 2. We use this to develop a new method for reducing the computational cost of approximating the solution of a fractional differential equation. 3. This method can be implemented with two levels of sophistication. We compare their rates of convergence, their algorithmic complexity, and their weight set sizes so that an optimal choice, for a particular application, can be made. 4. We show how linear multiterm fractional differential equations can be approximated as systems of fractional differential equations of order at most 1.

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No description available.

In Chapter I we begin by considering a theorem of J. Dieudonne on the minimum radius of starlikeness of a class of analytic functions. We give a simple new proof of this theorem. By this new proof also we find the minimum radius of univalence of this class and we determine all the cases which give the minimum radius of univalence and the minimum radius of starlikeness. We then use a method similar to that in this new proof to obtain the minimum radius of univalence and the minimum radius of starlikeness of some other classes of analytic functions. For each class we determine all the cases giving the minimum radius of univalence and the minimum radius of starlikeness. Then we give some similar results for the minimum radius of convexity. In Chapter II first we deal with Heawood's Lemma which was established and used by P.J. Heawood to prove the theorem known as the Grace-Heawood Theorem. The same lemma was used by S. Kakeya in the proof of another theorem. We show that Heawood's Lemma is false and we give new proofs of these results. Then for some special cases we improve the value of the radius of univalencegiven by Kakeya's Theorem. In this connection we firstgive L.N. Cakalov's result and then we obtain some improvements of his result. In Appendix I we give some examples related with Chapter I and Chapter II. In Appendix II we give an example which shows that there is an error in a paper by M. Robertson.

The close-coupling approximation method involves the numerical solution of systems of coupled second order ordinary differential equations. The solutions can display instability which is made apparent by dependence of the resonance energy on H (step-size). This instability has been examined and corrected. The comparative efficiency, time-wise and storage-wise, of a number of algorithms for the integration of the system of equations is presented.