Triple product integrals evaluate the shading at a point by factoring the reflection equation into incident illumination, visibility, and BRDF. By densely sampling the space of incident directions, this approach is capable of highly accurate rendering scenes lit by high-frequency environment lighting, containing complex materials and featuring intricate shadows. Efficient evaluation of triple p...

An exact analytical method for the calculation of volume of overlapping spheres is presented. In the considered procedure volume of overlapping spheres is expressed as surface integrals over closed regions. Using a natural continuous correspondence between the points of the sphere surface and the points of the plane the surface integrals are computed by the next transformation onto double integ...

We consider a class of nonconforming nite element approximations of a simply laminated microstructure which minimizes the nonconvex variational problem for the deformation of martensitic crystals which can undergo either an orthorhombic to monoclinic (double well) or a cubic to tetragonal (triple well) transformation. We rst establish a series of error bounds in terms of elastic energies for th...

The use of Mellin-Barnes integrals became a widespread technique for calculating Feynman diagrams throughout the last years [ 1], in particular to calculate double and triple box-diagrams. In Ref. [ 2], it was possible to expand the scalar two-loop two-point function in all orders in the dimensional regularization parameter ε, using additionally the gluing operation of Feynman diagrams, defined...

We give a proof of Green’s theorem which captures the underlying intuition and which relies only on the mean value theorems for derivatives and integrals and on the change of variables theorem for double integrals.

We introduce and study the triple of a quasitriangular Lie bialgebra as a natural extension of the Drinfeld double. The triple is itself a quasitriangular Lie bialgebra. We prove several results about the algebraic structure of the triple, analogous to known results for the double. Among them, we prove that in the factorisable case the triple is isomorphic to a twisting of g ⊕ g ⊕ g by a certai...

BACKGROUND Tacrolimus (Tac) and mycophenolate mofetil (MMF) are newly approved immunosuppressive agents. However, the safety and efficacy of the combination of MMF and Tac in primary liver transplantation has not been determined. METHODS An Institutional Review Board-approved, open-label prospective randomized protocol was initiated to study the efficacy and toxicity of Tac and steroids (doub...