On Riesz Transforms Characterization of <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript> Spaces Associated with Some Schrödinger Operators

Let \(\mathcal Lf(x)=-\Delta f (x)+V(x)f(x)\), V?≥?0, \(V\in L^1_{loc}(\mathbb R^d)\), be a non-negative self-adjoint Schrödinger operator on \(\mathbb R^d\). We say that an L ¹-function f is an element of the Hardy space \(H^1_{\mathcal L}\) if the maximal function

$ \mathcal M_{\mathcal L} f(x)=\sup\limits_{t>0}|e^{-t\mathcal L} f(x)| $

belongs to \(L^1(\mathbb R^d)\). We prove that under certain assumptions on V the space \(H^1_{\mathcal L}\) is also characterized by the Riesz transforms \(R_j=\frac{\partial}{\partial x_j}\mathcal L^{-1\slash 2}\), j?=?1,...,d, associated with \(\mathcal L\). As an example of such a potential V one can take any V?≥?0, \(V\in L^1_{loc}\), in one dimension.

     

Properties of β-sitostanol/DPPC monolayers studied with Grazing Incidence X-ray Diffraction (GIXD) and Brewster Angle Microscopy

Although the influence of structurally modified sterols on artificial membranes has been intensively investigated, studies on the properties of stanols, which are saturated analogs of sterols, are very rare. Therefore, we have performed Grazing Incidence X-ray Diffraction (GIXD) experiments aimed at studying in-plane organization of a plant stanol-β-sitostanol monolayer and its mixtures with 1,2-dipalmitoyl-sn-glycero-3-phosphocholine - DPPC at the air/water interface. The collected GIXD data, resulting in-plane parameters and BAM images provide information on molecular organization and in-plane ordering of the investigated films. It was found that the lateral organization of β-sitostanol/DPPC monolayers depends on their composition. The oblique structure of the in-plane lattice of tilted hydrophobic region of molecules, found for DPPC film, is maintained at 10 mol% of stanol in the system. However, at 30 and 90 mol% of stanol in the mixture, the arrangement of molecules is hexagonal and they are oriented perpendicularly to the interface. With the addition of stanol the extend of the in-plane order of the monolayers decreases. Moreover, in mixtures the ordered domains consist of both monolayer's components. Additionally, β-sitostanol film is of similar in-plane organization as the corresponding sterol monolayer (β-sitosterol) and stanol induces condensing effect on DPPC.     

Evaluation of 1‐arylpiperazine derivative of hydroxybenzamides as 5‐HT1A and 5‐HT7 serotonin receptor ligands: An experimental and molecular modeling approach

The synthesis and evaluation as 5‐HT_1A and 5‐HT₇ serotonin receptor ligands of the two sets of O‐substituted hydroxybenzamides, structurally related to 2‐{3‐[4‐(2‐methoxyphenyl)piperazin‐1‐yl]propoxy}benzamide ( 1 ), (K_i 5‐HT_1A = 21 nM, 5‐HT₇ = 234 nM) are reported. To affect the affinity for 5‐HT_1A and 5‐HT₇ receptors, an amide moiety ( 2 , 3 , 4 , 5 , 6 ) and a hydrocarbon chain length ( 7 , 8 , 9 , 10 ) were modified. The serotonergic activity of compounds 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 was generally higher in the case of 5‐HT_1A receptors compared with 5‐HT₇ ones; the most active 5‐HT_1A ligands being meta‐isomer 2 (K_i = 7 nM) and both analogs of 1 with the longest spacer, i.e., penta‐ and hexa‐methylene derivatives 9 and 10 (K_i = 4 and 3 nM, respectively). The observed biological properties of compounds 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 were elucidated using molecular modeling procedures. J. Heterocyclic Chem., (2010).     

Nonlinear vibrations and time delay control of an extensible slowly rotating beam

Nonlinear Dynamics - This paper investigates the time-varying formation tracking (TVFT) problem of multiple heterogeneous Euler–Lagrange agents (MHELAs) with predefined-time convergence...     

The Existing Methods and Novel Approaches in Mycotoxins’ Detection

Mycotoxins represent a wide range of secondary, naturally occurring and practically unavoidable fungal metabolites. They contaminate various agricultural commodities like cereals, maize, peanuts, fruits, and feed at any stage in pre- or post-harvest conditions. Consumption of mycotoxin-contaminated food and feed can cause acute or chronic toxicity in human and animals. The risk that is posed to public health have prompted the need to develop methods of analysis and detection of mycotoxins in food products. Mycotoxins wide range of structural diversity, high chemical stability, and low concentrations in tested samples require robust, effective, and comprehensible detection methods. This review summarizes current methods, such as chromatographic and immunochemical techniques, as well as novel, alternative approaches like biosensors, electronic noses, or molecularly imprinted polymers that have been successfully applied in detection and identification of various mycotoxins in food commodities. In order to highlight the significance of sampling and sample treatment in the analytical process, these steps have been comprehensively described.     

Synthesis and Luminescent Properties of s-Tetrazine Derivatives Conjugated with the 4H-1,2,4-Triazole Ring

New derivatives obtained by the combination of unique 1,2,4,5-tetrazine and 4H-1,2,4-triazole rings have great application potential in many fields. Therefore, two synthetic few-step methodologies, which make use of commercially available 4-cyanobenzoic acid (method A) and ethyl diazoacetate (method B), were applied to produce two groups of the aforementioned heterocyclic conjugates. In both cases, the target compounds were obtained in various combinations, by introducing electron-donating or electron-withdrawing substituents into the terminal rings, together with aromatic or aliphatic substituents on the triazole nitrogen atom. Synthesis of such designed systems made it possible to analyze the influence of individual elements of the structure on the reaction course, as well as the absorption and emission properties. The structure of all products was confirmed by conventional spectroscopic methods, and their luminescent properties were also determined.     

Infrared and Raman Spectroscopy of Methylcyanodiacetylene (CH3C5N)

A spectroscopic study combining IR absorption and Raman scattering is presented for methylcyanodiacetylene (CH₃C₅N). Gas‐phase, cryogenic matrix‐isolated, and pure solid‐phase substance was analyzed. Out of 16 normal vibrational modes, 14 were directly observed. The analysis of the spectra was assisted by quantum chemical calculations of vibrational frequencies, IR absorption intensities, and Raman scattering activities at density functional theory and ab initio levels. Previous assignments of gas‐phase IR absorption bands were revisited and extended.     

Dynamics of Phase Separation in Polymer Blends Revisited: Morphology,Spinodal, Noise,and Nucleation

The properties of the model B of mesoscopic dynamic with the Flory–Huggins free energy for the homopolymer blend are discussed. We focus on the rescaling of the spatial coordinates in the model and demonstrate that the commonly used rescaling of the spatial coordinates by the function vanishing at the spinodal leads to the unphysical pinning of the domains. The proper rescaling function for the spatial variables which avoids the unphysical pinning of the domain growth at the spinodal is proposed. We discuss in detail the evolution of morphological measures during separation in homopolymer blends and the problem of nucleation close to the spinodal.