The MVE estimator is an important estimator in robust statistics and characterized by an ellipsoid which contains inside 100β percent of given points in IR n and attains the minimal volume, where β ∈ [0.5, 1.0) is of usual interest. The associated minimal ellipsoid can be considered as a generalization of the minimum volume covering ellipsoid which covers all the given points since the latter e...

BACKGROUND Kidney volume (KV) becomes clinically relevant in autosomal dominant polycystic kidney disease (ADPKD) management. KV can be conveniently estimated (ceKV) using ellipsoid volume equations with three axes measurements; however, the accuracy and reliability are unknown. METHODS KVs of 347 kidneys in 177 consecutive ADPKD patients were determined with a volumetric method (standard-KV)...

We study the relationship between unit-distance representations and the Lovász theta number of graphs, originally established by Lovász. We derive and prove min-max theorems. This framework allows us to derive a weighted version of the hypersphere number of a graph and a related min-max theorem. Then, we connect to sandwich theorems via graph homomorphisms. We present and study a generalization...

In this Letter, we report on the densest-known packings of congruent ellipsoids. The family of new packings consists of crystal arrangements of spheroids with a wide range of aspect ratios, and with density phi always surpassing that of the densest Bravais lattice packing phi approximately equal to 0.7405. A remarkable maximum density of phi approximately equal to 0.7707 is achieved for maximal...

Lennard-Jones-Devonshire equation of state is an old but theoretical based EOS. The concept of the nearest neighboring molecules or coordination number is proposed to be a function of temperature and volume, whereas it is a constant in the original. The dilute gas and hard sphere limits of molecules are employed to determine this function. Improvement of this modification is demonstrated by pro...

Associated with each body K in Euclidean n-space Rn is an ellipsoid 02K called the Legendre ellipsoid of K . It can be defined as the unique ellipsoid centered at the body’s center of mass such that the ellipsoid’s moment of inertia about any axis passing through the center of mass is the same as that of the body. In an earlier paper the authors showed that corresponding to each convex body K ⊂...

Linear programming is a very powerful tool for attacking hard combinatorial optimization problems. Methods such as the ellipsoid algorithm have shown that linear programming is solvable in polynomial time. Linear programming also plays a central role in the design of approximation algorithms. In fact, it is known that linear programming is P-complete, and this implies that if NP = P then for ev...

We describe a new proof of the complete integrability of the two related dynamical systems: the billiard inside the ellipsoid and 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 nonparameterized geodesics coincide with those of the standard metric. This new metric is induced by the hyperb...