Discretisation for odd quadratic twists

The discretisation problem for even quadratic twists is almost understood, with the main question now being how the arithmetic Delaunay heuristic interacts with the analytic random matrix theory prediction. The situation for odd quadratic twists is much more mysterious, as the height of a point enters the picture, which does not necessarily take integral values (as does the order of the Shafarevich-Tate group). We discuss a couple of models and present data on this question.Comment: To appear in the Proceedings of the INI Workshop on Random Matrix Theory and Elliptic Curve

Maximizing the hyperpolarizability of one-dimensional systems

Previous studies have used numerical methods to optimize the hyperpolarizability of a one-dimensional quantum system. These studies were used to suggest properties of one-dimensional organic molecules, such as the degree of modulation of conjugation, that could potentially be adjusted to improve the nonlinear-optical response. However, there were no conditions set on the optimized potential energy function to ensure that the resulting energies were consistent with what is observed in real molecules. Furthermore, the system was placed into a one-dimensional box with infinite walls, forcing the wavefunctions to vanish at the ends of the molecule. In the present work, the walls are separated by a distance much larger than the molecule's length; and, the variations of the potential energy function are restricted to levels that are more typical of a real molecule. In addition to being a more physically-reasonable model, our present approach better approximates the bound states and approximates the continuum states - which are usually ignored. We find that the same universal properties continue to be important for optimizing the nonlinear-optical response, though the details of the wavefunctions differ from previous result.Comment: 10 pages, 5 figure

Synthesis, characterization and crystal structures of two new phenolic mannich bases

Two new Mannich bases, 5-methyl-2-((4-(pyridin-2-yl)piperazin-1-yl)methyl)phenol (1) and 5-methyl-2-((4-(4-nitrophenyl)piperazin-1-yl)methyl)phenol (2), were prepared and characterized structurally with elemental analysis, IR, UV and NMR spectroscopic techniques as well as single crystal X-ray diffraction. Compound I crystallizes in the monoclinic space group P21/c with unit cell dimensions a = 6.6726(2) Å, b =   17.0542(6)   Å, c = 13.3222(4) Å, β = 100.832(1)°, V = 1489.00 (8) Å3, Z = 4, R1 = 0.0408, wR2 = 0.1143. Compound II crystallizes in the monoclinic space P21 with unit cell dimensions a = 5.9519(2) Å, b = 17.3315(8) Å, c = 15.7237(7) Å, β = 90.348(2)°, V = 1621.95(12) Å3, Z = 4, R1 = 0.0353, wR2 = 0.0965. Both compounds have their structures stabilized by hydrogen bonding and π∙∙∙π interactions.               KEY WORDS: Mannich base, Piperazine, X-ray diffraction, Hydrogen bonds Bull. Chem. Soc. Ethiop. 2019, 33(2), 341-348.DOI: https://dx.doi.org/10.4314/bcse.v33i2.1

A two step algorithm for learning from unspecific reinforcement

We study a simple learning model based on the Hebb rule to cope with "delayed", unspecific reinforcement. In spite of the unspecific nature of the information-feedback, convergence to asymptotically perfect generalization is observed, with a rate depending, however, in a non- universal way on learning parameters. Asymptotic convergence can be as fast as that of Hebbian learning, but may be slower. Moreover, for a certain range of parameter settings, it depends on initial conditions whether the system can reach the regime of asymptotically perfect generalization, or rather approaches a stationary state of poor generalization.Comment: 13 pages LaTeX, 4 figures, note on biologically motivated stochastic variant of the algorithm adde

Macroscopic control parameter for avalanche models for bursty transport

Similarity analysis is used to identify the control parameter RA for the subset of avalanching systems that can exhibit self-organized criticality (SOC). This parameter expresses the ratio of driving to dissipation. The transition to SOC, when the number of excited degrees of freedom is maximal, is found to occur when RA-->0. This is in the opposite sense to (Kolmogorov) turbulence, thus identifying a deep distinction between turbulence and SOC and suggesting an observable property that could distinguish them. A corollary of this similarity analysis is that SOC phenomenology, that is, power law scaling of avalanches, can persist for finite RA with the same RA-->0 exponent if the system supports a sufficiently large range of lengthscales, necessary for SOC to be a candidate for physical (RA finite) systems

Statistical characteristics of total electron content intensifications on global ionospheric maps

Global ionospheric total electron content (TEC) maps exhibit TEC intensifications and depletions of various sizes and shapes. Characterizing key features on TEC maps and understanding their dynamic coupling with external drivers can significantly benefit space weather forecasting. However, comprehensive analysis of ionospheric structuring over decades of TEC maps is currently lacking due to large data volume. We develop feature extraction software based on image processing techniques to extract TEC intensification regions, that is, contiguous regions with sufficiently elevated TEC values than surrounding areas, from global TEC maps. Applying the software to the Jet Propulsion Laboratory Global Ionospheric Map data, we generate a TEC intensification data set for years 2003–2022 and carry out a statistical study on the number and strength of TEC intensifications. We find that the majority of the TEC maps (about 86%) are characterized with one or two intensification(s), while the rest of the TEC maps have three or more intensifications. Both the number and strength of TEC intensifications exhibit semi-annual variation that peaks near equinoxes and dips near solstices, as well as an annual asymmetry with larger values around December solstice compared to June solstice. The number and strength of intensifications increase with enhanced solar extreme-violet irradiance. The strength of intensifications also increases with elevated geomagnetic activity, but the number of intensifications does not. In addition, the number of intensifications is not correlated with the strength of intensifications

Studies on optimizing potential energy functions for maximal intrinsic hyperpolarizability

We use numerical optimization to study the properties of (1) the class of one-dimensional potential energy functions and (2) systems of point charges in two-dimensions that yield the largest hyperpolarizabilities, which we find to be within 30% of the fundamental limit. We investigate the character of the potential energy functions and resulting wavefunctions and find that a broad range of potentials yield the same intrinsic hyperpolarizability ceiling of 0.709.Comment: 9 pages, 9 figure