In 1982 Yamnitsky and Levin gave a variant of the ellipsoid method which uses simplices instead of ellipsoids. Unlike the ellipsoid method this simplices method can be implemented in rational arithmetic. We show, however, that this results in a non-polynomial method since the storage requirement may grow exponentially with the size of the input. Nevertheless , by introducing a rounding procedur...

We show that strictly abnormal geodesics arise in graded nilpotent Lie groups. We construct such a group, for which some Carnot geodesics are strictly abnormal and, in fact, not normal in any subgroup. In the 2-step case we also prove that these geodesics are always smooth. Our main technique is based on the equations for the normal and abnormal curves, that we derive (for any Lie group) explic...

The deterministic problem for finding an aircraft’s optimal risk trajectory in a threat environment has been formulated. The threat is associated with the risk of aircraft detection by radars or similar sensors. The model considers an aircraft’s trajectory in three-dimensional (3-D) space and represents the aircraft by a symmetrical ellipsoid with the axis of symmetry directing the trajectory. ...

A complete error calibration model with 12 independent parameters is established by analyzing the three-axis magnetometer error mechanism. The said model conforms to an ellipsoid restriction, the parameters of the ellipsoid equation are estimated, and the ellipsoid coefficient matrix is derived. However, the calibration matrix cannot be determined completely, as there are fewer ellipsoid parame...

Using Langevin dynamics simulations, we investigate the dynamics of a flexible polymer translocation into a confined area under a driving force through a nanopore. We choose an ellipsoidal shape for the confinement and consider the dependence of the asymmetry of the ellipsoid measured by the aspect ratio on the translocation time. Compared with an isotropic confinement (sphere), an anisotropic ...

A classification is obtained of Borel measurable, GL(n) covariant, symmetric matrix valued valuations on the space of n-dimensional convex polytopes. The only ones turn out to be the moment matrix corresponding to the classical Legendre ellipsoid and the matrix corresponding to the ellipsoid recently discovered by Lutwak, Yang, and Zhang. A classical concept from mechanics is the Legendre ellip...

We give a deterministic O(log n) algorithm for the Shortest Vector Problem (SVP) of a lattice under any norm, improving on the previous best deterministic bound of n for general norms and nearly matching the bound of 2 for the standard Euclidean norm established by Micciancio and Voulgaris (STOC 2010). Our algorithm can be viewed as a derandomization of the AKS randomized sieve algorithm, which...

It has often been claimed in recent papers that one can find a degree d Sum-of-Squares proof if one exists via the Ellipsoid algorithm. In [16], Ryan O’Donnell notes this widely quoted claim is not necessarily true. He presents an example of a polynomial system with bounded coefficients that admits low-degree proofs of non-negativity, but these proofs necessarily involve numbers with an exponen...

We consider the N-vortex problem on a ellipsoid of revolution. Applying standard techniques of classical perturbation theory we construct a sequence of conformal transformations from the ellipsoid into the complex plane. Using these transformations the equations of motion for the N-vortex problem on the ellipsoid are written as a formal series on the eccentricity of the ellipsoid’s generating e...

Finite strain and vorticity analyses were carried out in the deformed conglomerate and micro-conglomerate of Goushti shear zone in the Sanandaj-Sirjan metamorphic belt (Zagros Mountains Iran). These deformed rocks are bounded between two NE and SW major thrust faults. Finite strain measurements on three principal planes of strain ellipsoid show the higher amounts of strain ratios near the major...