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     

Application of the equilibrium theory to ternary moving bed configurations (4+4, 5+4, 8 and 9 zones). II. Langmuir case

In this article, the overall methodology used to determine the working flow-rates of a true moving bed (TMB) processing langmuirian isotherms compounds is explained. Then it is applied to different ternary configurations (4+4, 5+4, 8 or 9 zones TMB) in order to characterize their performances. Finally the results obtained on all the configurations are compared on a given example. This comparison allows the choice of the more suitable configuration to be used for a given set of compounds.     

Computational Fluid Dynamic Design of Jet Stirred Reactors for Measuring Intrinsic Kinetics of Gas‐Phase and Gas‐Solid Reactions

Nonreactive and reactive computational fluid dynamic simulations were applied to optimize the design of a laboratory scale jet stirred reactor for measuring intrinsic kinetics of gas‐phase and gas‐solid reactions, i.e. kinetics determined by chemical steps only and not by heat or mass transfer. In the past these reactors were designed and tested based on empirical design criteria and residence time distribution experiments. This work shows that these do not always capture important local effects that are vital for kinetic studies. First the degree of macro–mixing was evaluated for three different geometries (down case, 45° case and 90° case) by performing in silico residence time distribution experiments at 900 K, showing that with these type of experiments only minor differences are observed. However, the ethane steam cracking simulations revealed major differences, with the 45° case being the most uniform in terms of temperature and the 90° case being by far the worst. The species nonuniformity in all geometries was acceptable and was in some cases even partly masked by important shortcut streams such as those observed in the 90° case. The existing gradients on the substrate surface are sufficiently small to be neglected in modeling efforts. As temperature is the major parameter determining the rate of the surface reactions, the 45° case is suggested as the best geometry for measuring intrinsic kinetics.     

A Positive Functional Bound for Certain Sine Sums

The positivity of the sine sums

Heat transfer due to harmonic variation in the free stream temperatures in a flat plate exposed on both sides

The heat transfer through an infinite flat plate is studied when the temperatures of the two free streams surrounding it are varying harmonically with time and out of phase, with a delay period τ_d. The configuration is a simplified model for the heat transfer through the separating wall in the isochoric counter-current heat exchanger. The results show that apart from the τ_d effect, the perturbation parameters depend mainly on the cavity passing frequency f. At the thick plate solution, the combined passing frequency–delay time influences are significant only when the dimensionless frequency is smaller than 10. Within this range τ_d affects seriously not only the temperature perturbation amplitudes (which determine the thermal stresses) but also the heat fluxes and the accumulated energy ones. When f ≥ 10, the plate behaves as two separate semi-infinite slabs. Heat penetration delays greater than one cavity passing period may be possible.     

On the fragmentation modes in PMO analyses

The molecular species 1,1- and 1,2-disubstituted alkenes have been used as model systems for a comparative discussion of the results obtained with quantitative orbital analyses using different fragmentation modes. It is shown that when indices of the overall energy effects are used, the results of a quantitative orbital analysis are independent of the chosen fragmentation mode. On the other hand, the results of such analysis can depend on the fragmentation mode when indices of partial energy effects are used.     

A new strategy for the stereoselective synthesis of unnatural α-amino acids

A new method for the synthesis of racemic non-proteinogenic α-amino acids has been developed, which involves (i) hetero-Diels-Alder addition of ethyl 2-nitrosoacrylate to electron rich alkenes such as enol ethers, enamines and allylsilanes, (ii) NaCNBH₃ reduction of the CN bond in the oxazines thus generated, the stereochemistry of the products being controlled by epimerisation of the thermodynamically less stable isomer to the more stable one, (iii) protection of the N-H group as N-Boc and (iv) finally, N-O bond cleavage of both free and protected products to give proline or bis-homoserine derivatives, respectively. An example with concomitant reduction of the carboxylate group, resulting in the formation of the respective amino alcohol is reported. Applying this methodology to a homochiral enol ether, the protected parent d-proline was prepared in enantiomerically pure form, whereas the asymmetric synthesis of the respective bis-homoserine was unsuccessful.     

A Sharp Inequality for a Trigonometric Sum

We prove that the inequality $$-\frac{1}{2}\leq {\sum\limits_{k=1}^{n}} \left( \frac{{\rm cos}(2kx)}{2k - 1}+\frac{{\rm sin}((2k - 1)x)}{2k} \right)$$ holds for all natural numbers n and real numbers x with ${x \in [0, \pi]}$ . The sign of equality is valid if and only if n =  1 and x =  π /2.     

Updating Insights into the Catalytic Domain Properties of Plant Cellulose synthase (CesA) and Cellulose synthase-like (Csl) Proteins

The wall is the last frontier of a plant cell involved in modulating growth, development and defense against biotic stresses. Cellulose and additional polysaccharides of plant cell walls are the most abundant biopolymers on earth, having increased in economic value and thereby attracted significant interest in biotechnology. Cellulose biosynthesis constitutes a highly complicated process relying on the formation of cellulose synthase complexes. Cellulose synthase (CesA) and Cellulose synthase-like (Csl) genes encode enzymes that synthesize cellulose and most hemicellulosic polysaccharides. Arabidopsis and rice are invaluable genetic models and reliable representatives of land plants to comprehend cell wall synthesis. During the past two decades, enormous research progress has been made to understand the mechanisms of cellulose synthesis and construction of the plant cell wall. A plethora of cesa and csl mutants have been characterized, providing functional insights into individual protein isoforms. Recent structural studies have uncovered the mode of CesA assembly and the dynamics of cellulose production. Genetics and structural biology have generated new knowledge and have accelerated the pace of discovery in this field, ultimately opening perspectives towards cellulose synthesis manipulation. This review provides an overview of the major breakthroughs gathering previous and recent genetic and structural advancements, focusing on the function of CesA and Csl catalytic domain in plants.