Criteria of strong nearest-cross points and strong best approximation pairs

[EN] The concept of strong nearest-cross point (strong n.c. point) is introduced, which is the generalization of strong uniqueness of best approximation from a single point. The relation connecting to localization is discussed. Some criteria of strong n.c. points are given. The strong best approximation pairs are also studied.The corresponding author Jingshi Xu was supported by the NNSF (No. 60474070) of ChinaPan, W.; Xu, J. (2006). Criteria of strong nearest-cross points and strong best approximation pairs. Applied General Topology. 7(1):41-50. https://doi.org/10.4995/agt.2006.1931SWORD41507

Tris(ethyl­enediamine-κ2 N,N′)cobalt(III) aqua­tris­(oxalato-κ2 O 1,O 2)indate(III)

In the cation of the title compound, [Co(C2H8N2)3][In(C2O4)3(H2O)], the CoIII atom is coordinated by six N atoms from three ethyl­enediamine mol­ecules. The CoIII—N bond lengths lie in the range 1.956 (4)–1.986 (4) Å. In the anion, the InIII atom is seven-coordinated by six O atoms from three oxalate ligands and by a water mol­ecule. The cations and anions are linked by extensive O—H⋯O and N—H⋯O hydrogen bonds, forming a supermolecular network

Permanence and almost periodic solution of a multispecies Lotka-Volterra mutualism system with time varying delays on time scales

In this paper, we consider the almost periodic dynamics of a multispecies Lotka-Volterra mutualism system with time varying delays on time scales. By establishing some dynamic inequalities on time scales, a permanence result for the model is obtained. Furthermore, by means of the almost periodic functional hull theory on time scales and Lyapunov functional, some criteria are obtained for the existence, uniqueness and global attractivity of almost periodic solutions of the model. Our results complement and extend some scientific work in recent years. Finally, an example is given to illustrate the main results.Comment: 31page

Universal presence of time-crystalline phases and period-doubling oscillations in one-dimensional Floquet topological insulators

In this work, we reported a ubiquitous presence of topological Floquet time crystal (TFTC) in one-dimensional periodically-driven systems. The rigidity and realization of spontaneous discrete time-translation symmetry (DTS) breaking in our model require necessarily coexistence of anomalous topological invariants (0 modes and $\pi$ modes), instead of the presence of disorders or many-body localization. We found that in a particular frequency range of the underlying drive, the anomalous Floquet phase coexistence between zero and pi modes can produce the period-doubling (2T, two cycles of the drive) that breaks the spontaneously, leading to the subharmonic response ($\omega/2$, half the drive frequency). The rigid period-oscillation is topologically-protected against perturbations due to both non-trivially opening of 0 and $\pi$-gaps in the quasienergy spectrum, thus, as a result, can be viewed as a specific "Rabi oscillation" between two Floquet eigenstates with certain quasienergy splitting $\pi/T$. Our modeling of the time-crystalline 'ground state' can be easily realized in experimental platforms such as topological photonics and ultracold fields. Also, our work can bring significant interests to explore topological phase transition in Floquet systems and to bridge the gap between Floquet topological insulators and photonics, and period-doubled time crystals.Comment: 31 pages, 6 figures, 2 supplementary figure