Estimating Mahler measures using periodic points for the doubling map

The Mahler measure m(P)$m\left(P\right)$ of a polynomial P$P$ is a numerical value which is useful in number theory, dynamical systems and geometry. In this article we show how this can be written in terms of periodic points for the doubling map on the unit interval. This leads to an interesting algorithm for approximating m(P)$m\left(P\right)$ which we illustrate with several examples.     

Periodic solutions in an array of coupled FitzHugh–Nagumo cells

We analyse the dynamics of an array of N²$N^2$ identical cells coupled in the shape of a torus. Each cell is a 2-dimensional ordinary differential equation of FitzHugh–Nagumo type and the total system is _ZN×_ZN${\mathbb{Z}}_N\times {\mathbb{Z}}_N$-symmetric. The possible patterns of oscillation, compatible with the symmetry, are described. The types of patterns that effectively arise through Hopf bifurcation are shown to depend on the signs of the coupling constants, under conditions ensuring that the equations have only one equilibrium state.     

Periodic traveling wave train and point-to-periodic traveling wave for a diffusive predator–prey system with Ivlev-type functional response

A diffusive predator–prey system with Ivlev-type scheme is investigated in this article. The existences of a small amplitude periodic traveling wave train _Γp${\Gamma }_p$ and the traveling wave solution connecting the boundary equilibrium _Eu(1,0)$E_u(1,0)$ to the periodic traveling wave _Γp${\Gamma }_p$ are obtained. The existence of this point-to-periodic solution reveals that the predator invasion leads to the periodic population densities in the coexistence domain, and thus plays a mild role in the evolution of predator–prey communities. The techniques used here are the Hopf bifurcation theorem, the improved shooting method combining with the geometric singular perturbation method.     

Automorphism group actions in complex analysis

In this paper we describe the subject of automorphism groups of domains in complex space. This has been an active area of research for fifty years or more, and continues to be dynamic and developing today. We discuss noncompact automorphism groups, the Bun Wong/Rosay theorem, the Greene/Krantz conjecture, semicontinuity of automorphism groups, the method of scaling, and other current topics. Contributions from geometers, Lie theorists, analysts, and function theorists are described.     

Limit cycle bifurcations of planar piecewise differential systems with three zones

This paper considers the limit cycle bifurcation problem of planar piecewise differential systems with three zones. Some computation formulas studied the problem of limit cycle bifurcations are provided by introducing multiple parameters. As an application to the obtained method, the number of limit cycles of a piecewise linear system with three zones studied in Lima et al. (2017) is discussed and some more limit cycles are found.     

Optical beamforming networks employing phase modulation and direct detection

We propose a novel dispersion-based optical beamforming network scheme employing phase modulation and direct detection. Optical phase modulators have the advantages of simple-structure, low loss and absence of bias. Dispersion-induced phase-to-intensity conversion is utilized to facilitate direct detection. A structure of wideband dispersive device (WDD) cascaded with periodic dispersive device (PDD) is introduced to enhance the system flexibility, so that the delay adjustability and RF response can be properly designed respectively by choosing appropriate dispersions of the WDD and PDD. A concept-proof system with a wideband chirped fiber grating (CFG) as the WDD and two multiband CFGs (MCFG1 and MCFG2) as the PDD separately is built to demonstrate the basic idea. The delay tuning range is 0-1.8 ns with increment of 164.2 ps. The passband center is 30 GHz for MCFG1 and 20 GHz for MCFG2, and the fractional bandwidth is 51.8%. The shot-noise-limited spurious-free dynamic range is also analyzed and measured to be 105.7 dB ⋅ Hz^2/3 when the average photocurrent is 2.7 mA.     

Novel and efficient multifunctional periodic mesoporous organosilica supported benzotriazolium ionic liquids for reusable synthesis of 2,4,5-trisubstituted imidazoles

A series of periodic mesoporous organosilica supported benzotriazolium ionic liquids were synthesized and tested as effective and practical heterogeneous catalysts in the condensation reaction of diphenylethanedione, aromatic aldehydes and ammonium acetate. The catalyst PMO@ILBF4(1.0) showed brilliant catalytic activity for the synthesis of 2,4,5-trisubstituted imidazoles with good to high yields. We also found that the catalytic activity could be significantly influenced by the loading levels and functional anions of the benzotriazolium-cation ionic liquid, probably due to an intensification of intramolecular synergistic effect. Furthermore, the multifunctional catalyst PMO@ILBF4(1.0) could be easily recovered by filtration and recycled for six times with no significant loss in activity, indicating its excellent stability and reusability. This method provides an efficient and environmentally-friendly procedure for the production of 2,4,5-trisubstituted imidazoles.     

Hitting-time limits for some exceptional rare events of ergodic maps

We discuss limit distributions for hitting-time functions of certain “exceptional” families of asymptotically rare events for ergodic probability preserving transformations. The abstract core is an inducing argument. The latter applies, for example, to shrinking intervals around periodic points (both uniformly expanding and neutral) of certain finite measure preserving interval maps. In particular, we give a complete answer to a question raised in Freitas et al. (2016).     

Complex adjoint orbits in Lie theory and geometry

This expository article is an introduction to the adjoint orbits of complex semisimple groups, primarily in the algebro-geometric and Lie-theoretic contexts, and with a pronounced emphasis on the properties of semisimple and nilpotent orbits. It is intended to build a foundation for more specialized settings in which adjoint orbits feature prominently (ex. hyperkähler geometry, Landau–Ginzburg models, and the theory of symplectic singularities). Also included are a few arguments and observations that, to the author’s knowledge, have not yet appeared in the research literature.