The smallest ideal of βS is not closed

Let S be an infinite discrete semigroup which can be embedded algebraically into a compact topological group and let βS be the Stone–Čech compactification of S. We show that the smallest ideal of βS is not closed.     

Metrizable refinements of group topologies and the pseudo-intersection number

The pseudo-intersection number, denoted p, is the minimum cardinality of a family A⊆P(ω) having the strong finite intersection property but no infinite pseudo-intersection. For every countable topologizable group G, let pG denote the minimum character of a nondiscrete Hausdorff group topology on G which cannot be refined to a nondiscrete metrizable group topology. We show that pG=p.     

Representation of a Monotone Curve by a Contour with Regular Change in Curvature

The problem of modelling a smooth contour with a regular change in curvature representing a monotone curve with specified accuracy is solved in this article. The contour was formed within the area of the possible location of a convex curve, which can interpolate a point series. The assumption that if a sequence of points can be interpolated by a monotone curve, then the reference curve on which these points have been assigned is monotone, provides the opportunity to implement the proposed approach to estimate the interpolation error of a point series of arbitrary configuration. The proposed methods for forming a convex regular contour by arcs of ellipses and B-spline ensure the interpolation of any point series in parts that can be interpolated by a monotone curve. At the same time, the deflection of the contour from the boundaries of the area of the possible location of the monotone curve can be controlled. The possibilities of the developed methods are tested while solving problems of the interpolation of a point series belonging to monotone curves. The problems are solved in the CAD system of SolidWorks with the use of software application created based on the methods developed in the research work.     

Large rectangular semigroups in Stone-Cech compactifications

We show that large rectangular semigroups can be found in certain Stone-Cech compactifications. In particular, there are copies of the rectangular semigroup in the smallest ideal of , and so, a semigroup consisting of idempotents can be embedded in the smallest ideal of if and only if it is a subsemigroup of the rectangular semigroup. In fact, we show that for any ordinal with cardinality at most , contains a semigroup of idempotents whose rectangular components are all copies of the rectangular semigroup and form a decreasing chain indexed by , with the minimum component contained in the smallest ideal of .

As a fortuitous corollary we obtain the fact that there are -chains of idempotents of length in . We show also that there are copies of the direct product of the rectangular semigroup with the free group on generators contained in the smallest ideal of .

     

Topologies on groups determined by discrete subsets

Let G be an infinite group. Given a filter F on G, let T[F] denote the largest left invariant topology on G in which F converges to the identity. In this paper, we study the topology T[F] in case when F contains the Fréchet filter and there is such that all the subsets xM(x), where x∈G, are pairwise disjoint. We show that T[F] possesses interesting extremal properties. We consider also the question whether T[F] can be a group topology.     

Optical spectroscopy of polysilylene films

Luminescence of poly[cyclohexyl(methyl)silylene] films was studied by time-resolved emission spectroscopy at temperatures from 15 K to 300 K. The main short-wavelength peak, shifted with rising temperature to lower energies, showed time features typical of an energy donor. Three weak emission centres were found in the visible spectral region. They decayed more slowly having the time features of an energy acceptor. The rise in temperature made both the energy transfer and extension of the effective -conjugation length more efficient. Model calculations of the chain conformation energy showed the close-to-all-trans and gauche helix minima for iso- and syndiotactic sequences and barriers between them.     

Periodic Mesoporous Organosilica Nanoparticles with Controlled Morphologies and High Drug/Dye Loadings for Multicargo Delivery in Cancer Cells

Despite the worldwide interest generated by periodic mesoporous organosilica (PMO) bulk materials, the design of PMO nanomaterials with controlled morphology remains largely unexplored and their properties unknown. In this work, we describe the first study of PMO nanoparticles (NPs) based on meta‐phenylene bridges, and we conducted a comparative structure–property relationship investigation with para‐phenylene‐bridged PMO NPs. Our findings indicate that the change of the isomer drastically affects the structure, morphology, size, porosity and thermal stability of PMO materials. We observed a much higher porosity and thermal stability of the para‐based PMO which was likely due to a higher molecular periodicity. Additionally, the para isomer could generate multipodal NPs at very low stirring speed and upon this discovery we designed a phenylene–ethylene bridged PMO with a controlled Janus morphology. Unprecedentedly high payloads could be obtained from 40 to 110 wt % regardless of the organic bridge of PMOs. Finally, we demonstrate for the first time the co‐delivery of two cargos by PMO NPs. Importantly, the cargo stability in PMOs did not require the capping of the pores, unlike pure silica, and the delivery could be autonomously triggered in cancer cells by acidic pH with nearly 70 % cell killing.     

The Rb7Bi3−3xSb3xCl16 Family: A Fully Inorganic Solid Solution with Room‐Temperature Luminescent Members

Low‐dimensional ns²‐metal halide compounds have received immense attention for applications in solid‐state lighting, optical thermometry and thermography, and scintillation. However, these are based primarily on the combination of organic cations with toxic Pb²⁺ or unstable Sn²⁺, and a stable inorganic luminescent material has yet to be found. Here, the zero‐dimensional Rb₇Sb₃Cl₁₆ phase, comprised of isolated [SbCl₆]^3? octahedra and edge‐sharing [Sb₂Cl₁₀]^4? dimers, shows room‐temperature photoluminescence (RT PL) centered at 560 nm with a quantum yield of 3.8±0.2 % at 296 K (99.4 % at 77 K). The temperature‐dependent PL lifetime rivals that of previous low‐dimensional materials with a specific temperature sensitivity above 0.06 K^?1 at RT, making it an excellent thermometric material. Utilizing both DFT and chemical substitution with Bi³⁺ in the Rb₇Bi_3?3xSb_3xCl₁₆ (x≤1) family, we present the edge‐shared [Sb₂Cl₁₀]^4? dimer as a design principle for Sb‐based luminescent materials.     

Fixed point compactifications

A fixed point compactification of a locally compact noncompact group G is a faithful semigroup compactification S such that \(ap=pa=p\) for all \(p\in S\setminus G\) and \(a\in G\). Since the right translations are continuous, the remainder of a fixed point compactification is a right zero semigroup. Among all fixed point compactifications of G there is a largest one, denoted \(\theta G\). We show that if G is \(\sigma \)-compact, then \(\theta G\setminus G\) contains a copy of \(\beta \omega \setminus \omega \). In contrast, if G is not \(\sigma \)-compact, then \(\theta G\) is the one-point compactification.