On Norm-Based Estimations for Domains of Attraction in Nonlinear Time-Delay Systems

For nonlinear time-delay systems, domains of attraction are rarely studied despite their importance for technological applications. The present paper provides methodological hints for the determination of an upper bound on the radius of attraction by numerical means. Thereby, the respective Banach space for initial functions has to be selected and primary initial functions have to be chosen. The latter are used in time-forward simulations to determine a first upper bound on the radius of attraction. Thereafter, this upper bound is refined by secondary initial functions, which result a posteriori from the preceding simulations. Additionally, a bifurcation analysis should be undertaken. This analysis results in a possible improvement of the previous estimation. An example of a time-delayed swing equation demonstrates the various aspects.Comment: 33 pages, 8 figures, "This is a pre-print of an article published in 'Nonlinear Dynamics'. The final authenticated version is available online at https://doi.org/10.1007/s11071-020-05620-8

Dynamics of immersed molecules in superfluids

The dynamics of a molecule immersed in a superfluid medium are considered. Results are derived using a classical hydrodynamic approach followed by canonical quantization. The classical model, a rigid body immersed in incompressible fluid, permits a thorough analysis; its effective Hamiltonian generalizes the usual rigid-rotor Hamiltonian. In contrast to the free rigid rotor, the immersed body is shown to have chaotic dynamics. Quantization of the classical model leads to new and experimentally verifiable features. It is shown, for instance, that chiral molecules can behave as "quantum propellers": the rotational-translational coupling induced by the superfluid leads to a nonzero linear momentum in the ground state. Hydrogen peroxide is a strong candidate for experimental detection of this effect. The signature is a characteristic splitting of rotational absorption lines. The 1_{01} --> 1_{10} line in hydrogen peroxide, for example, is predicted to split into three lines separated by as much as 0.01 cm^{-1}, which is about the experimental linewidth.Comment: 10 pages, 3 figure

Random projections and the optimization of an algorithm for phase retrieval

Iterative phase retrieval algorithms typically employ projections onto constraint subspaces to recover the unknown phases in the Fourier transform of an image, or, in the case of x-ray crystallography, the electron density of a molecule. For a general class of algorithms, where the basic iteration is specified by the difference map, solutions are associated with fixed points of the map, the attractive character of which determines the effectiveness of the algorithm. The behavior of the difference map near fixed points is controlled by the relative orientation of the tangent spaces of the two constraint subspaces employed by the map. Since the dimensionalities involved are always large in practical applications, it is appropriate to use random matrix theory ideas to analyze the average-case convergence at fixed points. Optimal values of the gamma parameters of the difference map are found which differ somewhat from the values previously obtained on the assumption of orthogonal tangent spaces.Comment: 15 page

On norm-based estimations for domains of attraction in nonlinear time-delay systems

For nonlinear time-delay systems, domains of attraction are rarely studied despite their importance for technological applications. The present paper provides methodological hints for the determination of an upper bound on the radius of attraction by numerical means. Thereby, the respective Banach space for initial functions has to be selected and primary initial functions have to be chosen. The latter are used in time-forward simulations to determine a first upper bound on the radius of attraction. Thereafter, this upper bound is refined by secondary initial functions, which result a posteriori from the preceding simulations. Additionally, a bifurcation analysis should be undertaken. This analysis results in a possible improvement of the previous estimation. An example of a time-delayed swing equation demonstrates the various aspects

A method for dense packing discovery

The problem of packing a system of particles as densely as possible is foundational in the field of discrete geometry and is a powerful model in the material and biological sciences. As packing problems retreat from the reach of solution by analytic constructions, the importance of an efficient numerical method for conducting \textit{de novo} (from-scratch) searches for dense packings becomes crucial. In this paper, we use the \textit{divide and concur} framework to develop a general search method for the solution of periodic constraint problems, and we apply it to the discovery of dense periodic packings. An important feature of the method is the integration of the unit cell parameters with the other packing variables in the definition of the configuration space. The method we present led to improvements in the densest-known tetrahedron packing which are reported in [arXiv:0910.5226]. Here, we use the method to reproduce the densest known lattice sphere packings and the best known lattice kissing arrangements in up to 14 and 11 dimensions respectively (the first such numerical evidence for their optimality in some of these dimensions). For non-spherical particles, we report a new dense packing of regular four-dimensional simplices with density $\phi=128/219\approx0.5845$ and with a similar structure to the densest known tetrahedron packing.Comment: 15 pages, 5 figure

The Arabidopsis AtRaptor genes are essential for post-embryonic plant growth

BACKGROUND: Flowering plant development is wholly reliant on growth from meristems, which contain totipotent cells that give rise to all post-embryonic organs in the plant. Plants are uniquely able to alter their development throughout their lifespan through the generation of new organs in response to external signals. To identify genes that regulate meristem-based growth, we considered homologues of Raptor proteins, which regulate cell growth in response to nutrients in yeast and metazoans as part of a signaling complex with the target of rapamycin (TOR) kinase. RESULTS: We identified AtRaptor1A and AtRaptor1B, two loci predicted to encode Raptor proteins in Arabidopsis. Disruption of AtRaptor1B yields plants with a wide range of developmental defects: roots are thick and grow slowly, leaf initiation and bolting are delayed and the shoot inflorescence shows reduced apical dominance. AtRaptor1A AtRaptor1B double mutants show normal embryonic development but are unable to maintain post-embryonic meristem-driven growth. AtRaptor transcripts accumulate in dividing and expanding cells and tissues. CONCLUSION: The data implicate the TOR signaling pathway, a major regulator of cell growth in yeast and metazoans, in the maintenance of growth from the shoot apical meristem in plants. These results provide insights into the ways in which TOR/Raptor signaling has been adapted to regulate plant growth and development, and indicate that in plants, as in other eukaryotes, there is some Raptor-independent TOR activity