We study the real Monge-Ampère equation in two and three dimensions, both from the point of view of the SYZ conjecture , where solutions give rise to semi-flat Calabi-Yau's and in affine differential geometry, where solutions yield parabolic affine sphere hypersurfaces. We find explicit examples, connect the holo-morphic function representation to Hitchin's description of special Lagrangian mod...

By use of approximations based on physical reasoning radar cross-section results for bodies of revolution are found. In the Rayleigh region (wavelength large with respect to object dimensions) approximate solutions are found. Examples given include a finite cone, a lens, an ellipticogive,.a spindle and a finite cylinder. In the physical optics region (wavelength very small with respect to all r...

We identify Melrose's suspended algebra of pseudodiierential operators with a subalgebra of the algebra of parametric pseudodiierential operators with parameter space R. For a general algebra of parametric pseudodiierential operators, where the parameter space may now be a cone ? R p , we construct a unique \symbol valued trace", which extends the L 2 {trace on operators of small order. This al...

In this paper, we present a novel technique which simulates directional light scattering for more realistic interactive visualization of volume data. Our method extends the recent directional occlusion shading model by enabling light source positioning with practically no performance penalty. Light transport is approximated using a tilted cone-shaped function which leaves elliptic footprints in...

in this paper, we introduce the cone normed spaces and cone bounded linear mappings. among other things, we prove the baire category theorem and the banach--steinhaus theorem in cone normed spaces.

Let M be a compact smooth Riemannian man-ifold without boundary and let D = ad 0 0 + bb 1 d 1 ? E on the space of smooth sections of the cotangent bundle where a and b are positive constants and where E is an endomorphism. We use functorial methods and the pseudo-diierential operator calculus to compute the quadratic term a 4 (D) in the asymptotic expansion of the heat equation trace. MOS class...

We compute the Hodge numbers of the polarised (pure) variation of Hodge structure V = grn−1 R f! Z of the Landau-Ginzburg model f : Y → C mirror-dual to a weighted projective space wPn in terms of a variant of Reid’s age function of the anticanonical cone over wPn. This implies, for instance, that wPn has canonical singularities if and only if hn−1,0 V = 1. We state a conjectural formula for th...

We establish existence and qualitative properties of saddle-shaped solutions of the elliptic fractional equation (−∆)u = f(u) in all the space R, where f is of bistable type. These solutions are odd with respect to the Simons cone and even with respect to each coordinate. More precisely, we prove the existence of a saddle-shaped solution in every even dimension 2m, as well as its monotonicity p...