Abstract:
A Deformable Mirror (DM) is an important component of an Adaptive Optics system. It is known that an on-axis spherical/parabolic optical component, placed at an angle to the incident beam introduces defocus as well as astigmatism in the image plane. Although the former can be compensated by changing the focal plane position, the latter cannot be removed by mere optical re-alignment. Since the DM is to be used to compensate a turbulence-induced curvature term in addition to other aberrations, it is necessary to determine the aberrations induced by such (curved DM surface) an optical element when placed at an angle (other than 0 degree) of incidence in the optical path. To this effect, we estimate to a first order, the aberrations introduced by a DM as a function of the incidence angle and deformation of the DM surface. We record images using a simple setup in which the incident beam is reflected by a 37 channel Micro-machined Membrane Deformable Mirror for various angles of incidence. It is observed that astigmatism is a dominant aberration which was determined by measuring the difference between the tangential and sagital focal planes. We justify our results on the basis of theoretical simulations and discuss the feasibility of using such a system for adaptive optics considering a trade-off between wavefront correction and astigmatism due to deformation.

Abstract:
Intrinsic Delaunay triangulation (IDT) is a fundamental data structure in computational geometry and computer graphics. However, except for some theoretical results, such as existence and uniqueness, little progress has been made towards computing IDT on simplicial surfaces. To date the only way for constructing IDTs is the edge-flipping algorithm, which iteratively flips the non-Delaunay edge to be locally Delaunay. Although the algorithm is conceptually simple and guarantees to stop in finite steps, it has no known time complexity. Moreover, the edge-flipping algorithm may produce non-regular triangulations, which contain self-loops and/or faces with only two edges. In this paper, we propose a new method for constructing IDT on manifold triangle meshes. Based on the duality of geodesic Voronoi diagrams, our method can guarantee the resultant IDTs are regular. Our method has a theoretical worst-case time complexity $O(n^2\log n)$ for a mesh with $n$ vertices. We observe that most real-world models are far from their Delaunay triangulations, thus, the edge-flipping algorithm takes many iterations to fix the non-Delaunay edges. In contrast, our method is non-iterative and insensitive to the number of non-Delaunay edges. Empirically, it runs in linear time $O(n)$ on real-world models.

Abstract:
The standard Grad-Shafranov equation for axisymmetric toroidal plasma equilibrium is customary expressed in cylindrical coordinates with toroidal contours, and through which benchmark equilibria are solved. An alternative approach to cast the Grad-Shafranov equation in spherical coordinates is presented. This equation, in spherical coordinates, is examined for toroidal solutions to describe low $\beta$ Solovev and high $\beta$ plasma equilibria in terms of elementary functions.

Abstract:
The vertices of regular four-dimensional polytopes are used to generate sets of uniformly distributed three-dimensional rotations, which are provided as tables of Euler angles. The spherical moments of these orientational sampling schemes are treated using group theory. The orientational sampling sets may be used in the numerical computation of solid-state nuclear magnetic resonance spectra, and in spherical tensor analysis procedures.

Abstract:
We present the existence of the Kosterlitz-Thouless (KT) transition for $n$-atic tangent-plane order on a deformable spherical surface and investigate the development of quasi-long range $n$-atic order and the continuous shape changes below the KT transition in the low temperature limit. The $n$-atic order parameter $\psi= \exp[in\Theta]$ describes, respectively, vector, nematic, and hexatic order for $n=1,2,$ and 6. We derive a Coulomb gas Hamiltonian to describe it. We then convert it into the sine-Gordon Hamiltonian and find a linear coupling between a scalar field and the Gaussian curvature. After integrating over the shape fluctuations, we obtain the massive sine-Gordon Hamiltonian, where the interaction between vortices is screened. We find, for $n^{2}K_{n}/\kappa \ll 1/4$, there is an effective KT transition. In the ordered phase, tangent-plane $n$-atic order expels the Gaussian curvature. In addition, the total vorticity of orientational order on a surface of genus zero is two. Thus, the ordered phase of an $n$-atic on such a surface will have $2n$ vortices of strength $1/n$, $2n$ zeros in its order parameter, and a nonspherical equilibrium shape. Our calculations are based on a phenomenological model with a gauge-like coupling between $\psi$ and curvature and close to the Abrikosov treatment of a type II superconductor.

Abstract:
Critical values of Wilson lines and general background fields for toroidal compactifications of heterotic string theories are constructed systematically using Dynkin diagrams.

Abstract:
A two dimensional system of discs upon which a triangle of spins are mounted is shown to undergo a sequence of interesting phase transitions as the temperature is lowered. We are mainly concerned with the `solid' phase in which bond orientational order but not positional order is long ranged. As the temperature is lowered in the `solid' phase, the first phase transition involving the orientation or toroidal charge of the discs is into a `gauge toroid' phase in which the product of a magnetic toroidal parameter and an orientation variable (for the discs) orders but due to a local gauge symmetry these variables themselves do not individually order. Finally, in the lowest temperature phase the gauge symmetry is broken and toroidal order and orientational order both develop. In the `gauge toroidal' phase time reversal invariance is broken and in the lowest temperature phase inversion symmetry is also broken. In none of these phases is there long range order in any Fourier component of the average spin. A definition of the toroidal magnetic moment $T_i$ of the $i$th plaquette is proposed such that the magnetostatic interaction between plaquettes $i$ and $j$ is proportional to $T_iT_j$. Symmetry considerations are used to construct the magnetoelectric free energy and thereby to deduce which coefficients of the linear magnetoelectric tensor are allowed to be nonzero. In none of the phases does symmetry permit a spontaneous polarization.

Abstract:
In this paper we present a Deformable Mirror (DM) based on the continuous voltage distribution over a resistive layer. This DM can correct the low order aberrations (defocus, astigmatism, coma and spherical aberration) using three electrodes with nine contacts leading to an ideal device for sensorless applications. We present a mathematical description of the mirror, a comparison between the simulations and the experimental results. In order to demonstrate the effectiveness of the device we compared its performance with the one of a multiactuator DM of similar properties in the correction of an aberration statistics. At the end of the paper an example of sensorless correction is shown.

Abstract:
Intrinsic computation refers to how dynamical systems store, structure, and transform historical and spatial information. By graphing a measure of structural complexity against a measure of randomness, complexity-entropy diagrams display the range and different kinds of intrinsic computation across an entire class of system. Here, we use complexity-entropy diagrams to analyze intrinsic computation in a broad array of deterministic nonlinear and linear stochastic processes, including maps of the interval, cellular automata and Ising spin systems in one and two dimensions, Markov chains, and probabilistic minimal finite-state machines. Since complexity-entropy diagrams are a function only of observed configurations, they can be used to compare systems without reference to system coordinates or parameters. It has been known for some time that in special cases complexity-entropy diagrams reveal that high degrees of information processing are associated with phase transitions in the underlying process space, the so-called ``edge of chaos''. Generally, though, complexity-entropy diagrams differ substantially in character, demonstrating a genuine diversity of distinct kinds of intrinsic computation.

Abstract:
Toroidal templates such as vesicles with hexatic bond orientational order are discussed. The total energy including disclination charges is explicitly computed for hexatic order embedded in a toroidal geometry. Related results apply for tilt or nematic order on the torus in the one Frank constant approximation. Although there is no topological necessity for defects in the ground state, we find that excess disclination defects are nevertheless energetically favored for fat torii or moderate vesicle sizes. Some experimental consequences are discussed.