Probability and expectation inequalities

Summary This paper introduces a mathematical framework within which a wide variety of known and new inequalities can be viewed from a common perspective. Probability and expectation inequalities of the following types are considered: (a)P(ZA) P(ZA) for some class of setsA, (b)k(Z)k(Z) for some class of functionsk, and (c)l(Z)k(Z) for some class of pairs of functionsl andk. It is shown, sometimes using explicit constructions ofZ andZ, that, in several cases, (a) (b) (c); included here are cases of normal and elliptically contoured distributions. A case where (a) (b) (c) is studied and is expressed in terms ofn monotone functions for (some of) which integral representations are obtained. Also, necessary and sufficient conditions for (c) are given.Research supported by the Air Force Office of Scientific Research under Grants AFOSR-75-2796 and AFOSR-80-0080Research supported by the National Science Foundation under Grants MCS78-01240 and MCS81-00748     

Synthesis of 2-substituted benzofurans and indoles using functionalized titanium benzylidene reagents on solid phase

Titanium(IV) benzylidenes bearing a masked oxygen or nitrogen nucleophile in the ortho position were generated from thioacetals, using low-valent titanocene complex, Cp2Ti[P(OEt)3]2. Methylene acetal, alkyl ether, silyl ether, fluoro, tertiary amino, and N-alkyl, N-benzyl, N-prenyl, and N-silyl tert-butyl carbamate groups were tolerated in the titanium alkylidene reagents (Schrock carbenes). Aryl-chlorine bonds were stable to the titanium benzylidene functionality, but there was poor chemoselectivity for the reduction of the thioacetal in the presence of an aryl chloride. The titanium benzylidenes converted Merrifield and Wang resin-bound esters into enol ethers. The oxygen nucleophile was masked as a TMS ether, and when the resin-bound enol ethers bearing this ortho substituent were treated with 1% TFA in dichloromethane, benzofurans were released from resin in high yields. The chameleon catch strategy ensured excellent purity. In a similar way, N-alkylated and N-silylated tert-butyl carbamates were used for the synthesis of N-alkyl and N-Boc indoles, respectively. These traceless solid-phase syntheses of heterocycles are believed to involve postcleavage modification rather than cyclative termination.     

ELECTRON TRANSFER REACTIONS OCCURRING UPON LASER FLASH PHOTOLYSIS OF PHOSPHOLIPID BILAYER SYSTEMS CONTAINING CHLOROPHYLL,BENZOQUINONE AND CYTOCHROME c-II. ELECTRICALLY NEUTRAL AND POSITIVELY-CHARGED VESICLES*

Abstract— The primary and secondary electron transfer reactions which occurred upon laser flash photolysis of electrically neutral and positively-charged lipid bilayer vesicles containing chlorophyll, benzoquinone and cytochrome c were determined by time-resolved difference spectral and kinetic measurements, and compared with previous results obtained with negatively-charged vesicles (Y. Fang and G. Tollin, Photochem. Photobiol. 1988). The extent to which oxidized cytochrome c could function as an electron acceptor from triplet state chlorophyll, and reduced cytochrome c could act as an electron donor to chlorophyll cation radical, decreased from negatively-charged to electrically neutral to positively-charged vesicles, in agreement with expectations based on changes in the ability of cytochrome c to bind to the bilayer. In all three types of vesicles, cytochrome c reduction by benzoquinone anion radical occurred in the aqueous phase.     

On the electronic energy spectrum of disordered systems

The present paper deals with the investigation of the electronic energy spectrum of a one-dimensional model of monatomic disordered systems with rectangular-well type atomic potentials. First, an equation describing the connection between the wavefunctions at three consecutive atomic sites is derived, the method followed for this purpose being based on the use of well-known “transfer matrix” together with an approach reported earlier by the author. This connection equation is further analysed with the help of continued fraction methods as well as a perturbational approach (first used by Phariseau), and explicit equations, giving allowed electronic energies for various one-dimensional models with small deviations from periodicity, are obtained thereby. It is found in general that, for the types of disorder considered here, a band structure of electronic energies exists; in addition, there appear extra localised energy levels. The treatments involving continued fractions show that such extra energies lie unambiguously within the forbidden regions of the band structure, while the extra energies obtained with the help of perturbational approach, are seen to have the possibility of lying both in the allowed and forbidden regions. With regard to localisation of electronic states, we have found that, for all tratments, the electronic states other than those giving the band structure are localised. As for the band states Phariseau's approach predicts them to be localised, while the continued fraction methods fail to predict whether the band states are localised or not.     

The Extracts of Polygonum cuspidatum Root and Rhizome Block the Entry of SARS-CoV-2 Wild-Type and Omicron Pseudotyped Viruses via Inhibition of the S-Protein and 3CL Protease

COVID-19, resulting from infection by the SARS-CoV-2 virus, caused a contagious pandemic. Even with the current vaccines, there is still an urgent need to develop effective pharmacological treatments against this deadly disease. Here, we show that the water and ethanol extracts of the root and rhizome of Polygonum cuspidatum (Polygoni Cuspidati Rhizoma et Radix), a common Chinese herbal medicine, blocked the entry of wild-type and the omicron variant of the SARS-CoV-2 pseudotyped virus into fibroblasts or zebrafish larvae, with IC₅₀ values ranging from 0.015 to 0.04 mg/mL. The extracts were shown to inhibit various aspects of the pseudovirus entry, including the interaction between the spike protein (S-protein) and the angiotensin-converting enzyme II (ACE2) receptor, and the 3CL protease activity. Out of the chemical compounds tested in this report, gallic acid, a phytochemical in P. cuspidatum, was shown to have a significant anti-viral effect. Therefore, this might be responsible, at least in part, for the anti-viral efficacy of the herbal extract. Together, our data suggest that the extracts of P. cuspidatum inhibit the entry of wild-type and the omicron variant of SARS-CoV-2, and so they could be considered as potent treatments against COVID-19.     

Kinetics of crystal growth of strontium tungstate from solutions in sodium tungstate melts by continuous cooling

The crystallisation kinetics of strontium tungstate from unstirred saturated solutions in sodium tungstate melts was studied by continuous cooling from initial crystallisation temperatures T₀ = 1000° to 800°C to room temperature at cooling rates R_T = 0.67° to 3.3°C min⁻¹. The main crystal growth was diffusion rate-controlled; the final crystal growth was rate-controlled by the development rate of excess solute concentration. The estimated diffusion rate constant (k_D) values increased with cooling rates and initial crystallisation temperatures. They are higher than the rate constants for diffusion-controlled growth of calcium tungstate from sodium tungstate melts, but very much smaller than those for strontium tungstate from lithium chloride melts.     

Evolution of Artificial Arginine Analogues—Fluorescent Guanidiniocarbonyl-Indoles as Efficient Oxo-Anion Binders

In this article, we present fluorescent guanidiniocarbonyl-indoles as versatile oxo-anion binders. Herein, the guanidiniocarbonyl-indole (GCI) and methoxy-guanidiniocarbonyl-indole (MGCI) were investigated as ethylamides and compared with the well-known guanidiniocarbonyl-pyrrole (GCP) concerning their photophysical properties as well as their binding behavior towards oxo-anions. Hence, a variety of anionic species, such as carboxylates, phosphonates and sulfonates, have been studied regarding their binding properties with GCP, GCI and MGCI using UV-Vis titrations, in combination with the determination of the complex stoichiometry using the Job method. The emission properties were studied in relation to the pH value using fluorescence spectroscopy as well as the determination of the photoluminescence quantum yields (PLQY). Density functional theory (DFT) calculations were undertaken to obtain a better understanding of the ground-lying electronic properties of the investigated oxo-anion binders. Additionally, X-ray diffraction of GCP and GCI was conducted. We found that GCI and MGCI efficiently bind carboxylates, phosphonates and sulfonates in buffered aqueous solution and in a similar range as GCP (K_ass ≈ 1000–18,000 M⁻¹, in bis-tris buffer, pH = 6); thus, they could be regarded as promising emissive oxo-anion binders. They also exhibit a visible fluorescence with a sufficient PLQY. Additionally, the excitation and emission wavelength of MGCI was successfully shifted closer to the visible region of the electromagnetic spectrum by introducing a methoxy-group into the core structure, which makes them interesting for biological applications.     

Characterizing polymer macrostructures by identifying and locating microstructures along their chains with the kerr effect

In this brief report, we demonstrate that Kerr effect measurements, which determine the excess birefringence contributed by polymer solutes in dilute solutions observed under a strong electric field, are highly sensitive to and capable of determining their microstructures, as well as their locations along the macromolecular backbone. Specifically, using atactic triblock copolymers with the same overall composition of styrene (S) and p-bromostyrene (pBrS) units, but with two different block arrangements, that is, pBrS₉₀-b-S₁₂₀-b-pBrS₉₀ (I) and S₆₀-b-pBrS₁₈₀-b-S₆₀ (II), which are indistinguishable by NMR, we detected a dramatic difference in their molar Kerr constants (_mK), in agreement with those previously estimated. Although similar in magnitude, their Kerr constants differ in sign, with _mK(II) positive and _mK(I) negative. In addition, S/pBrS random and gradient copolymers synthesized by reversible addition-fragmentation chain-transfer (RAFT) polymerization exhibit a heretofore unexpected enhanced enchainment of racemic (r) pBrS-pBrS diads. Comparison of their observed and calculated _mKs suggests that the gradient S/pBrS copolymers possess an unanticipated additional gradient in stereosequence that parallels their comonomer gradient, that is, as the concentration of pBrs units decreases from one end of the copolymer chain to the other, so does the content of r diads. This conclusion could only be reached by comparison of observed and calculated Kerr effects, which access the global properties of macromolecules, and not NMR, which is only sensitive to local polymer structural environments, but not to their locations on the copolymer chains. Molar Kerr constants are characteristic of entire polymer chains and are highly sensitive to their constituent microstructures and their distribution along the chain. They may be used to both identify constituent microstructures and locate them along the polymer chain, thereby enabling, for the first time, characterization of their complete macrostructures. © 2013 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2013