Dichotomies for Lorentz spaces

Assume that L ^p,q , $L^{p_1 ,q_1 } ,...,L^{p_n ,q_n } $ are Lorentz spaces. This article studies the question: what is the size of the set $E = \{ (f_1 ,...,f_n ) \in L^{p_{1,} q_1 } \times \cdots \times L^{p_n ,q_n } :f_1 \cdots f_n \in L^{p,q} \} $ . We prove the following dichotomy: either $E = L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $ or E is σ-porous in $L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $ , provided 1/p ≠ 1/p ₁ + … + 1/p _n . In general case we obtain that either $E = L^{p_1 ,q_1 } \times \cdots \times L^{p_n ,q_n } $ or E is meager. This is a generalization of the results for classical L ^p spaces.     

Dichotomies for C0(X) and C b (X) spaces

Jachymski showed that the set $$\left\{ {(x,y) \in {{\rm{c}}_0} \times {{\rm{c}}_0}:\left( {\sum\limits_{i = 1}^n {\alpha (i)x(i)y(i)} } \right)_{n = 1}^\infty {\rm{ is bounded}}} \right\}$$ is either a meager subset of c ₀ × c ₀ or is equal to c ₀ × c ₀. In the paper we generalize this result by considering more general spaces than c ₀, namely C ₀(X), the space of all continuous functions which vanish at infinity, and C _b (X), the space of all continuous bounded functions. Moreover, we replace the meagerness by σ-porosity.     

Continuity of Stochastic Convolutions

Let B be a Brownian motion, and let be the space of all continuous periodic functions f with period 1. It is shown that the set of all f such that the stochastic convolution does not have a modification with bounded trajectories, and consequently does not have a continuous modification, is of the second Baire category.     

Cyrtologies of Convergences,II: Sequential Convergences

Relationships between sequential convergences and topologies, pretopologies and pseudotopologies are investigated. A study of appropriate GALOIS connections and of fundamental properties of elementary filters constitutes a base for those investigations. Characterizations of pseudotopological, pretopological and topological inducibility are provided, in particular the theorem of Kantorovich/Vulih/Pinsker, Kisyński and Dolcher is recovered.     

Hockey-stick liquid crystalline 6-oxoverdazyl§

A series of hockey-stick shaped 6-oxoverdazyl radicals 1[n]–3[n], containing CH₃ (a), CF₃ (b) and CN (c) groups in the short ‘arm’, were prepared and their physical properties were investigated. Mesogenic behaviour was found only in the CN derivatives with the COO (1[n]c) and N=N (2[12]c) linking groups, which exhibited a nematic phase. Analysis of binary mixtures of selected compounds with bent-core nematic host 4[12] gave virtual nematic-isotropic transition temperatures, [T_NI], which follow the order 1[12]b < 1[12]c < 1[12]a in one series and 1[12]c < 2[12]c < (3[12]c) in another. The observed effectiveness of the short-arm substituent, CF₃ < CN < CH₃, was related to the magnitude and orientation of the molecular dipole moment calculated with density functional theory methods.     

Long-Range Static and Dynamic Previtreous Effects in Supercooled Squalene—Impact of Strong Electric Field

This article presents evidence for the long-range previtreous changes of two static properties: the dielectric constant (ε) and its strong electric field related counterpart, the nonlinear dielectric effect (NDE). Important evidence is provided for the functional characterizations of ε(T) temperature changes by the ‘Mossotti Catastrophe’ formula, as well as for the NDE vs. T evolution by the relations resembling those developed for critical liquids. The analysis of the dynamic properties, based on the activation energy index, excluded the Vogel–Fulcher–Tammann (VFT) relation as a validated tool for portraying the evolution of the primary relaxation time. This result questions the commonly applied ‘Stickel operator’ routine as the most reliable tool for determining the dynamic crossover temperature. In particular, the strong electric field radically affects the distribution of the relaxation times, the form of the evolution of the primary relaxation time, and the fragility. The results obtained in this paper support the concept of a possible semi-continuous phase transition hidden below T_g. The studies were carried out in supercooled squalene, a material with an extremely low electric conductivity, a strongly elongated molecule, and which is vitally important for biology and medicine related issues.     

Tethered Blatter Radical for Molecular Grafting: Synthesis of 6-Hydroxyhexyloxy,Hydroxymethyl, and Bis(hydroxymethyl) Derivatives and Their Functionalization

Synthetic access to 7-CF₃-1,4-dihydrobenzo[e][1,2,4]triazin-4-yl radicals containing 4-(6-hydroxyhexyloxy)phenyl, 4-hydroxymethylphenyl or 3,5-bis(hydroxymethyl)phenyl groups at the C(3) position and their conversion to tosylates and phosphates are described. The tosylates were used to obtain disulfides and an azide with good yields. The Blatter radical containing the azido group underwent a copper(I)-catalyzed azide–alkyne cycloaddition with phenylacetylene under mild conditions, giving the [1,2,3]triazole product in 84% yield. This indicates the suitability of the azido derivative for grafting Blatter radical onto other molecular objects via the CuAAC “click” reaction. The presented derivatives are promising for accessing surfaces and macromolecules spin-labeled with the Blatter radical.     

The Oxime Ethers with Heterocyclic,Alicyclic and Aromatic Moiety as Potential Anti-Cancer Agents

Chemotherapy is one of the most commonly used methods of cancer disease treatment. Due to the acquisition of drug resistance and the possibility of cancer recurrence, there is an urgent need to search for new molecules that would be more effective in destroying cancer cells. In this study, 1-(benzofuran-2-yl)ethan-1-one oxime and 26 oxime ethers containing heterocyclic, alicyclic or aromatic moiety were screened for their cytotoxicity against HeLa cancer cell line. The most promising derivatives with potential antitumor activity were 2-(cyclohexylideneaminoxy)acetic acid (18) and (E)-acetophenone O-2-morpholinoethyl oxime (22), which reduced the viability of HeLa cells below 20% of control at concentrations of 100–250 μg/mL. Some oxime ethers, namely thiazole and benzothiophene derivatives (24–27), also reduced HeLa cell viability at similar concentrations but with lower efficiency. Further cytotoxicity evaluation confirmed the specific toxicity of (E)-acetophenone O-2-morpholinoethyl oxime (22) against A-549, Caco-2, and HeLa cancer cells, with an EC50 around 7 μg/mL (30 μM). The most potent and specific compound was (E)-1-(benzothiophene-2-yl)ethanone O-4-methoxybenzyl oxime (27), which was selective for Caco-2 (with EC50 116 μg/mL) and HeLa (with EC50 28 μg/mL) cells. Considering the bioavailability parameters, the tested derivatives meet the criteria for good absorption and permeation. The presented results allow us to conclude that oxime ethers deserve more scientific attention and further research on their chemotherapeutic activity.     

Does Social Distancing Matter for Infectious Disease Propagation? An SEIR Model and Gompertz Law Based Cellular Automaton

In this paper, we present stochastic synchronous cellular automaton defined on a square lattice. The automaton rules are based on the SEIR (susceptible → exposed → infected → recovered) model with probabilistic parameters gathered from real-world data on human mortality and the characteristics of the SARS-CoV-2 disease. With computer simulations, we show the influence of the radius of the neighborhood on the number of infected and deceased agents in the artificial population. The increase in the radius of the neighborhood favors the spread of the pandemic. However, for a large range of interactions of exposed agents (who neither have symptoms of the disease nor have been diagnosed by appropriate tests), even isolation of infected agents cannot prevent successful disease propagation. This supports aggressive testing against disease as one of the useful strategies to prevent large peaks of infection in the spread of SARS-CoV-2-like diseases.