arXiv subjects
Publications and source records attributed to A. Segal.
We study a few approaches to identify inclusion (up to a shift) between two convex bodies in ${\mathbb R}^n$. To this goal we use mixed volumes and fractional linear maps. We prove that inclusion may be identified by comparing volume or surface area of all projective positions of the sets. We prove similar results for Minkowski sums of the sets.
The purpose of this paper is to show stability of order preserving/reversing transforms on the class of non-negative convex functions in ${\mathbb R}^n$, and its subclass, the class of non-negative convex functions attaining $0$ at the origin (these are called "geometric convex functions"). We show that transforms that satisfy conditions which are weaker than order preserving transforms, are essentially close to the order preserving transforms on the mentioned structures.
We present a direct analytic method towards an estimate for the rate of convergence (to the Euclidean Ball) of Steiner symmetrizations. To this end we present a modified version of a known stability property of the Steiner symmetrization.
Universality of the extraordinary Hall effect scaling was tested in granular three-dimensional Ni-SiO2 films across the metal-insulator transition. Three types of magnetotransport behavior have been identified: metallic, weakly insulating and strongly insulating. Scaling between both the ordinary and extraordinary Hall effects and material resistivity is absent in the weakly insulating range characterized by logarithmic temperature dependence of conductivity. The results provide compelling experimental confirmation to recent models of granular metals predicting transition from logarithmic to exponential conductivity temperature dependence when inter-granular conductance drops below the quantum conductance value and loss of Hall effect scaling when inter-granular conductance is higher than the quantum one. The effect was found at high temperatures and reflects the granular structure of material rather than low temperature quantum corrections.
We report on a proof-of-concept study of split-cell magnetic storage in which multi-bit magnetic memory cells are composed of several multilevel ferromagnetic dots with perpendicular magnetic anisotropy. Extraordinary Hall effect is used for reading the data. Feasibility of the approach is supported by realization of four-, eight- and sixteen- state cells.
Voltages parallel and transverse to electric current in slightly inhomogeneous superconductors can contain components proportional to the field and temperature derivatives of the longitudinal and Hall resistivities. We show that these anomalous contributions can be the origin of the zero field and even-in-field transverse voltage occasionally observed at the superconductor to normal state transition. The same mechanism can also cause an anomaly in the odd-in-field transverse voltage interfering the Hall effect signal.
We study an asymmetric in field magnetoresistance that is frequently observed in magnetic films and, in particular, the odd longitudinal voltage peaks that appear during magnetization reversal in ferromagnetic films, with out-of-plane magnetic anisotropy. We argue that the anomalous signals result from small variation of magnetization and Hall resistivity along the sample. Experimental data can be well described by a simple circuit model, the latter being supported by analytic and numerical calculations of current and electric field distribution in films with a gradual variation of the magnetization and Hall resistance.