The theory of classical waves in periodic high contrast photonic and acoustic media leads to the spectral problem

$$%-\Delta u=\lambda\varepsilon u $$%,

where the dielectric constant $\varepsilon(x)$ is a periodic function which assumes a large value $\varepsilon$ near a periodic graph $\Sigma$ in $\R^2$ and is equal to 1 otherwise. Existence and locations of spectral gaps are of primary interest. The high contrast asymptotics naturally leads to pseudodifferential operators of the Dirichlet-to-Neumann type on graphs and on more general structures. Spectra of these operators are studied numerically and analytically. New spectral effects are discovered, among them the "almost discreteness" of the spectrum for a disconnected graph and the existence of "almost localized" waves in some connected purely periodic structures.

We consider families of analytic area-preserving maps depending on two parameters: the perturbation strength $\varepsilon$ and the characteristic exponent h of the origin. For $\varepsilon=0$, these maps are integrable with a separatrix to the origin, whereas they asymptote to flows with homoclinic connections as $h\rightarrow 0^{+}$. For fixed $\varepsilon\neq 0$ and small h, we show that these connections break up. The area of the lobes of the resultant turnstile is given asymptotically by $\varepsilon \exp(-\pi^{2}/h)@\AreaFunc (h)$, where $\AreaFunc (h)$ is an even Gevrey-1 function such that $\AreaFunc (0)\neq 0$ and the radius of convergence of its Borel transform is $2\pi^{2}$. As $\varepsilon\rightarrow 0$, the function $\AreaFunc $ tends to an entire function $\MelnFunc $. This function $\MelnFunc $ agrees with the one provided by Melnikov theory, which cannot be applied directly, due to the exponentially small size of the lobe area with respect to h.

These results are supported by detailed numerical computations; we use multiple-precision arithmetic and expand the local invariant curves up to very high order.

L'ellipsoïde tridimensionnel est pris comme modèle d'un système engendrant une configuration globale de singularités interconnectées de l'espace $R^{4}$. La caustique, enveloppe des normales à l'ellipsoïde, est analysée et calculée. En particulier, nous montrons qu'elle possède quatre courbes fermées d'ombilics hyperboliques. Nos résultats suggèrent certaines propriétés de la caustique de l'ellipsoïde de l'espace multidimensionnel général.

The 3D ellipsoid is used as a model for a system generating a global configuration of interconnected singularities in $R^{4}$. The caustic obtained as the envelope of the normals to the ellipsoid is analyzed and calculated. In particular, we show that it has four closed curves of hyperbolic umbilics. Our results suggest some properties of the caustic of the ellipsoid in $R^{n}$.

We define a sequence of polynomials $P_d \in \funnyZ[x,y]$, such that $P_d$ is absolutely irreducible, of degree d, has low height, and has at least $d^2+2d+3$ integral solutions to $P_d(x,y)=0$. We know of no other nontrivial family of polynomials of increasing degree with as many integral solutions in terms of their degree.

Let p be a prime congruent to -1 modulo 4, $\kron{n}{p}$ the Legendre symbol and $S(k) = \sum_{n=1}^{p-1}n^k \kron{n}{p}$. The problem of finding a prime p such that $S(3) >0$ was one of the motivating forces behind the development of several of Shanks' ideas for computing in algebraic number fields, although neither he nor D. H. and Emma Lehmer were ever successful in finding such a p. In this paper we exhibit some techniques which were successful in producing, for each k such that $3\le k \le 2000$, a value for p such that $S(k) >0$.

The theory of (embedded) normal surfaces is a powerful technique in 3-manifold topology. There has been much recent interest in extending the theory to immersed surfaces, in particular to attack the word problem for 3-manifolds. Progress in this area has been hindered by the lack of nontrivial examples. This paper and the related work [Matsumoto and Rannard 1997] cover a particular example in depth, using methods which may be generalized. We give detailed information on the existence of immersed surfaces in the figure-eight knot complement using nontrivial computational techniques. After an introduction to the theory, introducing some new concepts, we discuss some strategies for enumerating surfaces of low genus, which have been implemented in software written by the author. The results are tabulated and an unusual example discussed.

Let M be a quadruply-punctured sphere with boundary components A,B,C,D. The $\Slt$-character variety of M consists of equivalence classes of homomorphisms $\rho$ of $\pi_1(M)\longrightarrow\Slt$ and can be identified with a quartic hypersurface in $\funnyC^7$. For fixed $a,b,c,d\in\funnyC$, the subset $\va$ corresponding to representations $\rho$ with \thickmuskip 5mu plus 2mu minus 2mu $\tr(\rho(A)) = a$, $\,\tr(\rho(B)) = b$, $\,\tr(\rho(C)) = c$, $\,\tr(\rho(D)) = d$ is a cubic surface in $\funnyC^3$. We determine the singular points of $\va$ and classify its set $\Var$ of $\funnyR$-points into six topological types, at least when this set is nonsingular. $\Var$ contains a compact component if and only if $-2 < a,b,c,d < 2$. For certain values of (a,b,c,d), this component corresponds to representations in $\Slr$.