Spiral algorithm

Given the efficient spiral algorithm for generation of isomer lists, many problems in structure and properties of higher fullerenes have been attacked. It is particularly important to have complete lists because it has turned out that possession of a closed... 

Symmetries are listed for isomers in the order in which they are generated by the spiral algorithm. Even cages are marked by a star.)... 

P. W. Fowler (Exeter University, U.K.). I believe you require 60 heptagons and 60 extra pentagons because they have a plane of symmetry, if you are going icosohedral, and you can do it with a slightly modified spiral algorithm. 

Vol. 51, P. Hansen, P. W. Fowler, and M. Zheng, Eds., American Mathematical Society, Providence, Rhode Island 2000, pp. 175-188. A Generalized Ring Spiral Algorithm for Coding Fullerenes and Other Cubic Polyhedra. 

Using the theorem that the sufficiency condition for mathematical correctness in 3D-reconstruction is fulfilled if all planes intersecting the object have to intersect the source-trajectory at least in one point [8], it is possible to generalise Feldkamp s method. Using projection data measured after changing the sotuce-trajectory from circular to spiral focus orbit it is possible to reconstruct the sample volume in a better way with the Wang algorithm [9]. 

Shortcomings of Wang s method like limited pitch of the spiral and blurring in the vertical direction can be improved by the CFBP-algorithm [10], where gaps in the spiral sampling pattern are filled using X-rays measured from the opposite side. 

Another efficient and practical method for exact 3D-reconstruction is the Grangeat algorithm [11]. First the derivative of the three-dimensional Radon transfomi is computed from the Cone-Beam projections. Afterwards the 3D-Object is reconstructed from the derivative of the Radon transform. At present time this method is not available for spiral orbits, instead two perpendicular circular trajectories are suitable to meet the above sufficiency condition. 

A. Katsevich, Exact Filtered Back Projection (FPB) Algorithm for Spiral Computer Tomography, US Patent 6,574,277 (2003). 

Let us first describe the tracking in the focal plane of the microscope objective. First, the chosen particle has to be selected by, for example, moving a cross on the screen. Its position is given by the barycenter of the white or black zone. To do so. after having found a first white pixel (three-level situation), a recursive algorithm must be applied to find the coordinates of all the other white pixels of the central zone. One thus obtains the position of the panicle at a time t. To determine its new position at time t + elt. we have to turn around on a spiral from the last position known until we find a new white pixel, and the analysis of the new white zone found will give the new position of the particle. Depending... 

One-channel feedback has been first applied to control meandering spiral waves in experiments with the light-sensitive BZ medinm [21]. Later a theory of this control method has been elaborated for rigidly rotating [40, 43] and for meandering [30] spiral waves. In accordance with this control algorithm, the wave activity (e.g. the value of the variable v in Eq. (9.1)) is measured at a particular detection point as a function of time. This value oscillates with time and exceeds the value Ve every time instant ti, when a wave front touches the detector point. A short perturbation is applied to the system immediately at ti or after some time delay r, i.e. at instances = ti + r. The frequency of the periodic sequence of generated... 

Fig. 9.4(b) shows the spiral tip trajectory obtained experimentally under this feedback control. After a short transient the spiral core center drifts in parallel to the line detector. The asymptotic drift trajectory reminds the resonance attractor observed under one-channel control, because a small variation of the initial location of the spiral wave does not change the final distance between the detector and the drift line. To construct the drift velocity field for this control algorithm an Archimedean spiral approximation is used again. Assume the detector line is given as a = 0 and an Archimedean spiral described by Eq. (9.5) is located at a site x,y) with a > 0. A pure geometrical consideration shows that the spiral front touches the detector each time ti satisfying the following equation ... 

Yet, no reasons exist why orderly generation could not potentially be applied, provided that a canonical code exists to uniquely identify fullerenes. Next we describe the spiral canonical code for fullerenes,and we then propose a sketch of a Read-Faradzev orderly generation taken from the algorithms of Fowler and Manolopoulos and Brinkmann. ... 

Benasla L, Behnadani A, Rahli M, Spiral Optimization Algorithm for solving Combined Economic and Emission Dispatch, Electrical Power and Energy Systems 62 (2014) 163-174. 

S. Y. Kung and S. C. Lo. A spiral systolic architecture/algorithm for transitive closure problems. Proc. IEEE Int Conf. on Computer Design, Port Chester NY, pages 622-626, Oct 1985. 

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