Geometric information in eight dimensions vs. quantum information

Complementary idempotent paravectors and their ordered compositions, are used to represent multivector basis elements of geometric Clifford algebra for 3D Euclidean space as the states of a geometric byte in a given frame of reference. Two layers of information, available in real numbers, are distinguished. The first layer is a continuous one. It is used to identify spatial orientations of similar geometric objects in the same computational basis. The second layer is a binary one. It is used to manipulate with 8D structure elements inside the computational basis itself. An oriented unit cube representation, rather than a matrix one, is used to visualize an inner structure of basis multivectors. Both layers of information are used to describe unitary operations -- reflections and rotations -- in Euclidian and Hilbert spaces. The results are compared with ones for quantum gates. Some consequences for quantum and classical information technologies are discussed.Comment: 14 pages, presented at International Symposium "Quantum Informatics 2007", October 3rd - 5th, 2007, Moscow Zvenigorod, Russi

Templates for Convex Cone Problems with Applications to Sparse Signal Recovery

This paper develops a general framework for solving a variety of convex cone problems that frequently arise in signal processing, machine learning, statistics, and other fields. The approach works as follows: first, determine a conic formulation of the problem; second, determine its dual; third, apply smoothing; and fourth, solve using an optimal first-order method. A merit of this approach is its flexibility: for example, all compressed sensing problems can be solved via this approach. These include models with objective functionals such as the total-variation norm, ||Wx||_1 where W is arbitrary, or a combination thereof. In addition, the paper also introduces a number of technical contributions such as a novel continuation scheme, a novel approach for controlling the step size, and some new results showing that the smooth and unsmoothed problems are sometimes formally equivalent. Combined with our framework, these lead to novel, stable and computationally efficient algorithms. For instance, our general implementation is competitive with state-of-the-art methods for solving intensively studied problems such as the LASSO. Further, numerical experiments show that one can solve the Dantzig selector problem, for which no efficient large-scale solvers exist, in a few hundred iterations. Finally, the paper is accompanied with a software release. This software is not a single, monolithic solver; rather, it is a suite of programs and routines designed to serve as building blocks for constructing complete algorithms.Comment: The TFOCS software is available at http://tfocs.stanford.edu This version has updated reference

Minimax Current Density Coil Design

'Coil design' is an inverse problem in which arrangements of wire are designed to generate a prescribed magnetic field when energized with electric current. The design of gradient and shim coils for magnetic resonance imaging (MRI) are important examples of coil design. The magnetic fields that these coils generate are usually required to be both strong and accurate. Other electromagnetic properties of the coils, such as inductance, may be considered in the design process, which becomes an optimization problem. The maximum current density is additionally optimized in this work and the resultant coils are investigated for performance and practicality. Coils with minimax current density were found to exhibit maximally spread wires and may help disperse localized regions of Joule heating. They also produce the highest possible magnetic field strength per unit current for any given surface and wire size. Three different flavours of boundary element method that employ different basis functions (triangular elements with uniform current, cylindrical elements with sinusoidal current and conic section elements with sinusoidal-uniform current) were used with this approach to illustrate its generality.Comment: 24 pages, 6 figures, 2 tables. To appear in Journal of Physics D: Applied Physic

Expression Analysis of Genes Involved in Transport Processes in Mice with MPTP-Induced Model of Parkinson’s Disease

Processes of intracellular and extracellular transport play one of the most important roles in the functioning of cells. Changes to transport mechanisms in a neuron can lead to the disruption of many cellular processes and even to cell death. It was shown that disruption of the processes of vesicular, axonal, and synaptic transport can lead to a number of diseases of the central nervous system, including Parkinson’s disease (PD). Here, we studied changes in the expression of genes whose protein products are involved in the transport processes (Snca, Drd2, Rab5a, Anxa2, and Nsf) in the brain tissues and peripheral blood of mice with MPTP (1-methyl-4-phenyl-1,2,3,6-tetrahydropyridine)-induced models of PD. We detected changes in the expressions of Drd2, Anxa2, and Nsf at the earliest modeling stages. Additionally, we have identified conspicuous changes in the expression level of Anxa2 in the striatum and substantia nigra of mice with MPTP-induced models of PD in its early stages. These data clearly suggest the involvement of protein products in these genes in the earliest stages of the pathogenesis of PD