The simplex algorithm was the first algorithm proposed for linear programming, and although the algorithm is quite fast in practice, no variant of it is known to be polynomial time. The Ellipsoid algorithm is the first polynomial-time algorithm discovered for linear programming. The Ellipsoid algorithm was proposed by the Russian mathematician Shor in 1977 for general convex optimization proble...

-Compared to normal learning algorithms, for example backpropagation, the optimal bounded ellipsoid (OBE) algorithm has some better properties, such as faster convergence, since it has a similar structure as the Kalman filter algorithm. Optimal bounded ellipsoid algorithm has some better properties than the Kalman filter training, one is that the noise is not required to be Guassian. In this pa...

The simplex algorithm was the first algorithm proposed for linear programming, and although the algorithm is quite fast in practice, no variant of it is known to be polynomial time. The Ellipsoid algorithm is the first polynomial-time algorithm discovered for linear programming. The Ellipsoid algorithm was proposed by the Russian mathematician Shor in 1977 for general convex optimization proble...

The Ellipsoid NDF is a new normal distribution function (NDF) that we introduce and use in the accompanying paper. It can be used with micro-facet BRDF to model the reflection and refraction of light from surfaces. This new distribution is a generalization of the widely used isotropic GGX/Trowbridge-Reitz distribution. The Ellipsoid model is based on the surface statistics of an arbitrary 3D el...

The criticisms presented in the preceding letter from Cantù et al. are essentially attributable to a serious misunderstanding of the structural parameters shown by Hirai et al. (1996a). This is evidently derived from some unsuitable expressions in a part of the structural parameters (scattering densities) in the above paper, which should be corrected by Hirai et al. Such corrections of the expr...

Robust control systems synthesis is generally recast as a class of robust feasibility problems which is to find a solution satisfying a set of parameter-dependent convex constraints for all possible parameter values. For this class of the problems, a stochastic ellipsoid method with multiple cuts each of which corresponds to each of the constraint is proposed, where a new update rule is present...

Equal-time comparison. Stochastic light culling has a trade-off between the performance and quality, which is controlled by the user-specified parameter δ. Therefore, our ellipsoid-based approach can render higher-quality images than the sphere-based approach when comparing at the same computation time. Fig. 2 shows the equal-time comparison between the bounding sphere and our bounding ellipsoi...

We give a new proof of the complete integrability of the geodesic flow on the ellipsoid (in Euclidean, spherical or hyperbolic space). The proof is based on the construction of a metric on the ellipsoid whose non-parameterized geodesics coincide with those of the standard metric. This new metric is induced by the hyperbolic metric inside the ellipsoid (Klein’s model). Mathematics Subject Classi...

A system of soft ellipsoid molecules confined between two planar walls is studied using classical density-functional theory. Both the isotropic and nematic phases are considered. The excess free energy is evaluated using two different Ansätze and the intermolecular interaction is incorporated using two different direct correlation functions (DCF's). The first is a numerical DCF obtained from si...

Spectral domain optical coherence tomography (SD-OCT) is a useful non-invasive imaging method which is used in the diagnosis and follow-up of various macular diseases. Photoreceptor disruption detected with OCT might be demonstrated the loss of integrity or absence of external limiting membrane, ellipsoid zone and interdigitation zone. It has been considered that the disruptions of these outer ...