Hyperbolic Coordinate System Research Articles

A VLSI Implementation of Independent Component Analysis for Biomedical Signal Separation Using CORDIC Engine.

This study aims to design and implement a very large scale integration (VLSI) chip of the extend InfoMax independent component analysis (ICA) algorithm which can separate the super-Gaussian source signals. In order to substantially reduce the circuit area, the proposed circuit utilizes the time sharing matrix multiplication array (MMA) to realize a series of matrix multiplication operations and employs the coordinate rotation digital computer (CORDIC) algorithm to calculate the hyperbolic functions sinh(θ) and cosh(θ) with the rotation of the hyperbolic coordinate system. Also, the rotation of the linear coordinate system of the CORDIC is adopted for the design of a divider used for obtaining the required function value of tanh(θ) simply by evaluating sinh(θ)/cosh(θ). Implemented in a TSMC 90-nm CMOS technology, the proposed ICA has an operation frequency of 100MHz with 90.8K gate counts. Furthermore, the measurement results show the ICA core can be successfully applied to separating mixed medical signals into independent sources.

A PARALLEL AND PIPELINED ARCHITECTURE FOR CORDIC ALGORITHM

The COordinate Rotation DIgital Computer (CORDIC) algorithm is an efficient algorithm to calculate the iteratively phase and magnitude or the vector rotations in linear, hyperbolic and circular coordinate system. The existing CORDIC method takes less clock frequency with high delay. To overcome this problem, a new version of updated parallel and pipelined architecture is designed without degrading the performance. It provides highest maximum frequency with less delay by splitting the critical path into several smaller delay paths with enhanced circuit processing time. The designed architecture in this study can be used in navigation application. This method is implemented in the Xilinx ISE tool.

A mass-independent conformal quantum cloak

An obstacle to constructing a quantum cloak with the transformation design method for steering a matter wave is the device involving the anisotropic modulation of the particle mass. It is the purpose of this paper to show that a narrow-acceptance-angle quantum cloak generated by a kind of conformal mapping could be mass-independent. This greatly simplifies the construction of the device since it can now be finished only by applying a potential field to a region of the shell. As demonstrations of the conclusion, an elliptical and a parabolic conformal quantum cloak are designed from the elliptical and parabolic coordinate systems. Their performance is inspected by probability currents. It is shown that the conformal devices can serve as novel splitters for a 2D matter wave. Finally, we discuss the difficulty of constructing a steering device from the hyperbolic coordinate system.

Hyperbolic-symmetry vector fields.

We present and construct a new kind of orthogonal coordinate system, hyperbolic coordinate system. We present and design a new kind of local linearly polarized vector fields, which is defined as the hyperbolic-symmetry vector fields because the points with the same polarization form a series of hyperbolae. We experimentally demonstrate the generation of such a kind of hyperbolic-symmetry vector optical fields. In particular, we also study the modified hyperbolic-symmetry vector optical fields with the twofold and fourfold symmetric states of polarization when introducing the mirror symmetry. The tight focusing behaviors of these vector fields are also investigated. In addition, we also fabricate micro-structures on the K9 glass surfaces by several tightly focused (modified) hyperbolic-symmetry vector fields patterns, which demonstrate that the simulated tightly focused fields are in good agreement with the fabricated micro-structures.

Theoretical study of diffusion processes around a non-rotating neutron star

The general relativistic diffusion process on curved space-time manifold around a non-rotating neutron star has been analyzed. The general relativistic diffusion equation of diffusive particles around non-rotating neutron star is derived by constructing phase space in the parametrization of observer time in the hyperbolic coordinate system. This diffusion equation describes the stochastic dynamic of particles around non-rotating neutron stars. In this work we also have studied the diffusion processes around a non-rotating neutron star for asymptotic case.

Fluctuations in Tat copy number when it counts the most: a possible mechanism to battle the HIV latency

The HIV-1 virus can enter a dormant state and become inactive, which reduces accessibility by antiviral drugs. We approach this latency problem from an unconventional point of view, with the focus on understanding how intrinsic chemical noise (copy number fluctuations of the Tat protein) can be used to assist the activation process of the latent virus. Several phase diagrams have been constructed in order to visualize in which regions of the parameter space noise can drive the activation process. Essential to the study is the use of a hyperbolic coordinate system, which greatly facilitates quantification of how the various reaction rate combinations shape the noise behavior of the Tat protein feedback system. We have designed a mathematical manual of how to approach the problem of activation quantitatively, and introduce the notion of an “operating point” of the virus. For both noise-free and noise-based strategies we show how operating point off-sets induce changes in the number of Tat molecules. The major result of the analysis is that for every noise-free strategy there is a noise-based strategy that requires lower dosage, but achieves the same anti-latency effect. It appears that the noise-based activation is advantageous for every operating point.

Corner flow in free liquid films

A lubrication-flow model for a free film in a corner is presented. The model, written in the hyperbolic coordinate system ξ = x 2 −y 2, η = 2xy, applies to films that are thin in the η-direction. The lubrication approximation yields two coupled evolution equations for the film thickness and the velocity field which, to lowest order, describes plug flow in the hyperbolic coordinates. A free film in a corner evolving under surface tension and gravity is investigated. The rate of thinning of a free film is compared to that of a film evolving over a solid substrate. Viscous shear and normal stresses are both captured in the model and are computed for the entire flow domain. It is shown that normal stress dominates over shear stress in the far field, while shear stress dominates close to the corner.

Coordinate systems for conformal cerebellar flat maps

Methods Quasi-conformal flat maps of the human cerebellum (1,4) were created from a high-resolution Tl-weighted MRI volume (5). A topologically correct surface was produced from a cerebellar volume defined by a plane parallel to the posterior commisure-obex line and orthogonal to a plane passing through the vermal midline. Our quasi-conformal flattening procedure (2,3) was applied to this surface to produce flat maps; the cortical surface was parcellated according to (6), and activations produced by a target interception task were imposed. Eleven readily identifiable anatomical landmarks, e.g., the apex of the fourth ventricle and the rostral-most tip of the lingula, were defined. For the hyperbolic coordinate system, one landmark was used as the map center and a second to specify the map orientation. We also used a polar coordinate system on the hyperbolic flat map, with equiangular grid lines radiating from the map focus and equidistant circles surrounding the focus. For the spherical coordinate system, one landmark was used as the north pole, one as the south pole, and a third as a distinguished equatorial point. We used the usual latitude and longitude coordinates, similar to (7). Results and Conclusions A major benefit of (quasi-)conformal flat maps is that they can be used to generate canonical surface-based coordinate systems for a cortical surface. Since our fiat maps to the hyperbolic plane and the sphere are l-l and onto, the coordinate systems produced by these flat maps can be used to quantitatively localize structure and function. Furthermore, these flat maps are mathematically unique. Different choices of landmarks and coordinate systems will increase the utility of flat maps for studying functional imaging dam. We believe that the use of cerebellar flat maps and canonical coordinate systems based on reproducible anatomical landmarks will improve our ability to localize functional activity on the cerebellar cortex and to quantify anatomical and functional differences between individual subjects and groups of subjects.

Regularity in chaotic reaction paths. I. Ar6

We scrutinize the saddle crossings of a simple cluster of six atoms to show (a) that it is possible to choose a coordinate system in which the transmission coefficient for the classical reaction path is unity at all energies up to a moderately high energy, above which the transition state is chaotic; (b) that at energies just more than sufficient to allow passage across the saddle, all or almost all the degrees of freedom of the system are essentially regular in the region of the transition state; and (c) that the degree of freedom associated with the reaction coordinate remains essentially regular through the region of the transition state, even to moderately high energies. Microcanonical molecular dynamics simulation of Ar6 bound by pairwise Lennard-Jones potentials reveals the mechanics of passage. We use Lie canonical perturbation theory to construct the nonlinear transformation to a hyperbolic coordinate system which reveals these regularities. This transform “rotates away” the recrossings and nonregular behavior, especially of the motion along the reaction coordinate, leaving a coordinate and a corresponding dividing surface in phase space which minimize recrossings and mode–mode mixing in the transition state region. The action associated with the reactive mode tends to be an approximate invariant of motion through the saddle crossings throughout a relatively wide range of energy. Only at very low energies just above the saddle could any other approximate invariants of motion be found for the other, nonreactive modes. No such local invariants appeared at energies at which the modes are all chaotic and coupled to one another.

The Local Product Structure of Expansive Automorphisms of Solenoids and Their Associated C*-Algebras

AbstractAn explicit description of a hyperbolic canonical coordinate system for an expansive automorphism of a compact connected abelian group is given. These dynamical systems are factors of subshifts of finite type. Some properties of the associated crossed product C*-algebra are discussed. In these examples, the C* -algebras of Ruelle are crossed product algebras.

A simple method for illustrating the difference between the homogeneous and inhomogeneous Lorentz groups

A method is proposed for illustrating the difference between the Poincaré (inhomogeneous Lorentz) and homogeneous Lorentz groups. Representations of the Poincaré group are constructed from solutions of the relativistic harmonic oscillator equation whose physical wave functions have been effective in describing basic high-energy hadronic features in the relativistic quark model. It is shown that the Poincaré group can be represented by solutions of the relativistic oscillator equation in a ’’moving cylindrical’’ coordinate system in which the time-separation variable in the hadronic rest frame is treated separately. Representations which are diagonal in the Casimir operators of the homogeneous Lorentz group are also constructed from solutions of the same oscillator differential equation in a hyperbolic coordinate system. It is pointed out that the difference between the Poincaré and homogeneous Lorentz groups mainfests itself in the coordinate systems in which the oscillator differential equation in separable.