Gorenstein n-X-Injective and n-X-Flat Modules with Respect to a Special Finitely Presented Module

Let R be a ring,X a class of R-modules and n ≥ 1 an integer.We intro-duce the concepts of Gorenstein n-X-injective and n-X-flat modules via special finitely presented modules.Besides,we obtain some equivalent properties of these modules on n-X-coherent rings.Then we investigate the relations among Gorenstein n-X-injective,n-X-flat,injective and fiat modules on X-FC-rings (i.e.,self n-X-injective and n-X-coherent rings).Several known results are generalized to this new context.     

Le phosphate de cobalt et de lithium à valence mixte Li4+xCo2−x(P2O7)2 (x = 0,03): étude structurale et analyse de distribution de charge

The title compound, namely lithium cobalt(II/III) bis(diphosphate), Li_4.03Co_1.97(P₂O₇)₂, is a new mixed‐valent lithium/cobalt(II/III) phosphate. Three metal sites out of seven are occupied simultaneously by Li⁺ and Co^II/III ions. This disorder was established both from an analysis of the atomic displacement ellipsoids and Li/Co—O bond distances, and by means of a charge‐distribution (CHARDI) model, which provides satisfactory agreement on the computed charges (Q) for all the cations.     

Bounded Sequences,Orbits and Invariant Subspaces

Let {x _n } be a sequence of complex numbers and let \({\Delta^nx_j = \sum\nolimits_{k=0}^{n} (-1)^k\break\left(\begin{array}{l}n\\ k\\\end{array} \right)x_{n-k+j}}\) . In this paper, we will show that if \({ |x_n| = O(n^k)}\) , as n → ∞ for some positive integer k, and \({n|\Delta^n x_j|^{\frac{1}{n}} \to 0}\) as n→ ∞, then \({\Delta^{k+1} x_j = 0}\) . More importantly, applications to the orbits of operators and invariant subspace problem are also given; this helps to improve former results obtained by Gelfand–Hille, Mbekhta–Zemánek and others.     

4‐[Bis(1,5‐dimethyl‐1H‐pyrazol‐3‐yl­meth­yl)­amino]phenol monohydrate

The crystal structure of the title compound, C₁₈H₂₃N₅O·H₂O, shows mol­ecules containing a phenol group linked perpendicularly to a roughly planar fragment comprising two pyrazole rings. Mol­ecules are stacked perpendicular to the [101] direction, with their phenol groups disposed alternately. The mol­ecular packing in the crystal is stabilized by hydrogen bonding involving water mol­ecules.     

Nonequilibrium Josephson effect through helical edge states

We study Josephson junctions between superconductors connected through the helical edge states of a two-dimensional topological insulator in the presence of a magnetic barrier. As the equilibrium Andreev bound states of the junction are 4π periodic in the superconducting phase difference, it was speculated that, at finite dc bias voltage, the junction exhibits a fractional Josephson effect with half the Josephson frequency. Using the scattering matrix formalism, we show that his effect is absent in the average current. However, clear signatures can be seen in the finite-frequency current noise. Furthermore, we discuss other manifestations of the Majorana bound states forming at the edges of the superconductors.     

Solving Fredholm integral equations by approximating kernels by spline quasi-interpolants

We study two methods for solving a univariate Fredholm integral equation of the second kind, based on (left and right) partial approximations of the kernel K by a discrete quartic spline quasi-interpolant. The principle of each method is to approximate the kernel with respect to one variable, the other remaining free. This leads to an approximation of K by a degenerate kernel. We give error estimates for smooth functions, and we show that the method based on the left (resp. right) approximation of the kernel has an approximation order O(h ⁵) (resp. O(h ⁶)). We also compare the obtained formulae with projection methods.