Symmetry2014, 6(1), 90-102; doi:10.3390/sym6010090 - published online 25 February 2014 Show/Hide Abstract
Abstract: We show how transformation group ideas can be naturally used to generate efficient algorithms for scientific computations. The general approach is illustrated on the example of determining, from the experimental data, the dissociation constants related to multiple binding sites. We also explain how the general transformation group approach is related to the standard (backpropagation) neural networks; this relation justifies the potential universal applicability of the group-related approach.
Symmetry2014, 6(1), 67-88; doi:10.3390/sym6010067 - published online 3 February 2014 Show/Hide Abstract
Abstract: Every finite-dimensional unitary representation of the N-extended world line supersymmetry without central charges may be obtained by a sequence of differential transformations from a direct sum of minimal Adinkras, simple supermultiplets that are identifiable with representations of the Clifford algebra. The data specifying this procedure is a sequence of subspaces of the direct sum of Adinkras, which then opens an avenue for the classification of the continuum of the so-constructed off-shell supermultiplets.
Symmetry2014, 6(1), 23-66; doi:10.3390/sym6010023 - published online 3 January 2014 Show/Hide Abstract
Abstract: The timing patterns of animal gaits are produced by a network of spinal neurons called a Central Pattern Generator (CPG). Pinto and Golubitsky studied a four-node CPG for biped dynamics in which each leg is associated with one flexor node and one extensor node, with Ζ2 x Ζ2 symmetry. They used symmetric bifurcation theory to predict the existence of four primary gaits and seven secondary gaits. We use methods from symmetric bifurcation theory to investigate local bifurcation, both steady-state and Hopf, for their network architecture in a rate model. Rate models incorporate parameters corresponding to the strengths of connections in the CPG: positive for excitatory connections and negative for inhibitory ones. The three-dimensional space of connection strengths is partitioned into regions that correspond to the first local bifurcation from a fully symmetric equilibrium. The partition is polyhedral, and its symmetry group is that of a tetrahedron. It comprises two concentric tetrahedra, subdivided by various symmetry planes. The tetrahedral symmetry arises from the structure of the eigenvalues of the connection matrix, which is involved in, but not equal to, the Jacobian of the rate model at bifurcation points. Some of the results apply to rate equations on more general networks.
Symmetry2014, 6(1), 1-22; doi:10.3390/sym6010001 - published online 2 January 2014 Show/Hide Abstract
Abstract: Integrating shape contours in the visual periphery is vital to our ability to locate objects and thus make targeted saccadic eye movements to efficiently explore our surroundings. We tested whether global shape symmetry facilitates peripheral contour integration and saccade targeting in three experiments, in which observers responded to a successful peripheral contour detection by making a saccade towards the target shape. The target contours were horizontally (Experiment 1) or vertically (Experiments 2 and 3) mirror symmetric. Observers responded by making a horizontal (Experiments 1 and 2) or vertical (Experiment 3) eye movement. Based on an analysis of the saccadic latency and accuracy, we conclude that the figure-ground cue of global mirror symmetry in the periphery has little effect on contour integration or on the speed and precision with which saccades are targeted towards objects. The role of mirror symmetry may be more apparent under natural viewing conditions with multiple objects competing for attention, where symmetric regions in the visual field can pre-attentively signal the presence of objects, and thus attract eye movements.
Symmetry2013, 5(4), 344-354; doi:10.3390/sym5040344 - published online 9 December 2013 Show/Hide Abstract
Abstract: Some characteristic features of band structures, like the band degeneracy at high symmetry points or the existence of energy gaps, usually reflect the symmetry of the crystal or, more precisely, the symmetry of the wave vector group at the relevant points of the Brillouin zone. In this paper, we will illustrate this property by considering two-dimensional (2D)-hexagonal lattices characterized by a possible two-fold degenerate band at the K points with a linear dispersion (Dirac points). By combining scanning tunneling spectroscopy and angle-resolved photoemission, we study the electronic properties of a similar system: the Ag/Cu(111) interface reconstruction characterized by a hexagonal superlattice, and we show that the gap opening at the K points of the Brillouin zone of the reconstructed cell is due to the symmetry breaking of the wave vector group.