Self-commutator inequalities in higher dimension

Three natural multi-dimensional substitutes for the self-commutator of a Hilbert space operator are introduced and generalizations of Putnam's inequality to tuples of operators with semidefinite self-commutators are indicated. In addition, a Riesz transform model is developed and investigated.

     

Asymptotic approximation of functions and their derivatives by Müller’s Gamma operators

We obtain the complete asymptotic expansion of the image functions of Müller’s Gamma operators and of their derivatives. All expansion coefficients are explicitly calculated. Moreover, we study linear combinations of Gamma operators having a better degree of approximation than the operators themselves. Using divided differences we define general classes of linear combinations of which special cases were recently introduced and investigated by other authors.     

A complete asymptotic expansion of power means

The complete asymptotic expansion of power means in terms of Bell polynomials is obtained. Some results recently obtained by M. Bjelica are generalized.     

Some remarks on the behaviour of the solution in dynamic processes for rate-type models

This work examines a dynamic problem in the study of semilinear rate-type models for which the plastic rate of deformation depends also on a parameter . The continuous dependence of the solution with respect to is obtained and the problem of finite time stability is also discussed. In the case when is interpreted as the absolute temperature, the dynamic problem is studied in the context of a Cattaneo-type heat law and also using the classical Fourier law. In the case when is interpreted as an internal state variable an existence and uniqueness result is obtained using a fixed point method and the finite time stability is also investigated.     

Colloidal nano-MOFs nucleate and stabilize ultra-small quantum dots of lead bromide perovskites

The development of synthetic routes to access stable, ultra-small (i.e. <5 nm) lead halide perovskite (LHP) quantum dots (QDs) is of fundamental and technological interest. The considerable challenges include the high solubility of the ionic LHPs in polar solvents and aggregation to form larger particles. Here, we demonstrate a simple and effective host–guest strategy for preparing ultra-small lead bromide perovskite QDs through the use of nano-sized MOFs that function as nucleating and host sites. Cr₃O(OH)(H₂O)₂(terephthalate)₃ (Cr-MIL-101), made of large mesopore-sized pseudo-spherical cages, allows fast and efficient diffusion of perovskite precursors within its pores, and promotes the formation of stable, ∼3 nm-wide lead bromide perovskite QDs. CsPbBr₃, MAPbBr₃ (MA⁺ = methylammonium), and (FA)PbBr₃ (FA⁺ = formamidinium) QDs exhibit significantly blue-shifted emission maxima at 440 nm, 446 nm, and 450 nm, respectively, as expected for strongly confined perovskite QDs. Optical characterization and composite modelling confirm that the APbBr₃ (A = Cs, MA, FA) QDs owe their stability within the MIL-101 nanocrystals to both short- and long-range interfacial interactions with the MOF pore walls.

We demonstrate a simple and effective host–guest strategy for preparing ultra-small lead bromide perovskite QDs through the use of nano-sized MOFs that function as nucleating and host sites.     

Synthesis and characterization of some carbazole-based imine polymers

Five imine polymers, all containing 3,6-disubstituted carbazole ring in the main polymer chain, were synthesized by solution polycondensation of 3,6-diformyl (N-hexyl) carbazole with hydrazine, 1,4-diaminobenzene, 4,4′-diaminobiphenyl, 1,5-diaminonaphthalene and 3,6-(N-hexyl) diaminocarbazole. The polymers were analyzed by spectroscopic methods and compared with two imine models.     

Extended applications of electric cell-substrate impedance sensing for assessment of the structure-function of α2β1 integrin

Electric cell-substrate impedance sensing (ECIS) was applied to assess the structure-function of α2β1 integrin, receptor for collagen and laminin. On collagen-coated gold electrodes, expression of this integrin on human rhabdomyosarcoma (RD) cells (RDX2C2) yielded a five-fold increase in resistance when compared with mock transfected RD (RDpF) cells (34.5±5.2 versus 6.5±0.8 Ω/cell). An intermediate level of 16±2 Ω/cell was measured upon expression of an α2β1 mutant that lacked the α2 cytoplasmic domain (RDX2CO). On laminin, the resistance measured for RDX2C2 cells was also higher but only twice that of RDpF cells at 71±4 and 37±4 Ω/cell, respectively. In comparison, RDX2CO cells (38±4 Ω/cell), exhibiting no enhanced adhesive function, yielded a similar result to that of RDpF cells. On fibronectin, RDX2C2 and RDpF cells, exhibiting comparable levels of adhesion, were similar in resistance measurements at 85±5and 89±7 Ω/cell, respectively. It has been shown that deletion of α2 cytoplasmic domain results in dysregulated recruitment of the α2β1 mutant to focal adhesion complexes that mediate binding of fibronectin. RDX2CO cells on fibronectin, exhibiting reduced adhesive function, was associated with noticeably lower resistance (60±4 Ω/cell). Monitoring electroporation of the RD plasma membrane also indirectly validated cell attachment as reflected by the resistance measured. Results from this study demonstrated the potential of ECIS for study of the structure-function of βl integrin adhesion receptors.     

Aperiodic metal–organic frameworks

Metal–organic frameworks (MOFs) represent one of the most diverse structural classes among solid state materials, yet few of them exhibit aperiodicity, or the existence of long-range order in the absence of translational symmetry. From this apparent conflict, a paradox has emerged: even though aperiodicity frequently arises in materials that contain the same bonding motifs as MOFs, aperiodic structures and MOFs appear to be nearly disjoint classes. In this perspective, we highlight a subset of the known aperiodic coordination polymers, including both incommensurate and quasicrystalline structures. We further comment upon possible reasons for the absence of such structures and propose routes to potentially access aperiodic MOFs.

This perspective discusses progress and future directions in metal–organic frameworks with aperiodic structures. Reported quasicrystalline and incommensurate materials are presented, and pathways towards designing new such materials are provided.