Necessary and sufficient conditions for existence of exponential dichotomy of impulsive equations

Necessary and sufficient conditions for the existence of an exponential dichotomy of impulsive differential equations in a Hilbert space are found.     

希土元素与三甲胺氧化物配合物的制备及光谱性质

尽管文献报道了三甲胺氧化物与Mn、Co、Ni、Zn、Cd、Cu等多种离子在有机溶剂中形成较稳定的配合物。但对其金属配合物的研究远不如芳香族杂环化合物那样引起人们广泛的注意。本文研究了它与希土元素形成配合物的合成和光谱性质。     

An Unexpected Formation of a 14‐Membered Cyclodepsipeptide

The treatment of diluted solutions of the hydroxy diamides 6a and 6b in toluene with HCl gas at 100° gave the dimeric, 14‐membered cyclodepsipeptide 10 in up to 72% yield (Scheme 3). The same product was formed from the linear dimer of 6b , the depsipeptide 11 , under the same conditions (cf. Scheme 4). All attempts to prepare the cyclic seven‐membered monomer 9 , starting with different precursors and using different lactonization methods failed, and 10 was the only product which was isolated (cf. Scheme 6). For example, the reaction of the ester 20 with NaH in toluene at 80° led exclusively to the cyclodimer 10 . On the other hand, the base‐catalyzed cyclization of the hydroxy diester 22 , which is the ‘O‐analogue' of 20 , yielded neither the seven‐membered dilactone, nor the 14‐membered tetralactone, but only the known trimer 23 and tetramer 24 of 2,2‐dimethylpropano‐3‐lactone (cf. Scheme 7).     

cis‐(6RS,13RS)‐3,3,10,10‐Tetra­methyl‐6,13‐diphenyl‐1,8‐dioxa‐4,11‐diaza­cyclo­tetra­decane‐2,5,9,12‐tetra­one and two of its precursors

The title macrocycle, C₂₆H₃₀N₂O₆, (VI), was obtained by `direct amide cyclization' from the linear precursor 3‐hydr­oxy‐N‐[1‐methyl‐1‐(N‐methyl‐N‐phenyl­carbamoyl)ethyl]‐2‐phenylpropanamide, the N‐methyl­anilide of rac‐2‐methyl‐2‐[(3‐hydroxy‐2‐phenyl­propanoyl)­amino]­propanoic acid, C₁₃H₁₇NO₄, (IV). The reaction proceeds via the inter­mediate rac‐2‐(2‐hydroxy‐1‐phenyl­ethyl)‐4,4‐dimethyl‐1,3‐oxazol‐5(4H)‐one, C₁₃H₁₅NO₃, (V), which was synthesized independently and whose structure was also established. Unlike all previously described analogues, the title macrocycle has the cis‐diphenyl configuration. The 14‐membered ring has a distorted rect­angular diamond‐based [3434] configuration and inter­molecular N—H⋯O hydrogen bonds link the mol­ecules into a three‐dimensional framework. The propanoic acid precursor forms a complex series of inter­molecular hydrogen bonds, each of which involves pairwise association of mol­ecules and which together result in the formation of extended two‐dimensional sheets. The oxazole inter­mediate forms centrosymmetric hydrogen‐bonded dimers in the solid state.     

Reaction of γ‐Hydroxy‐N‐[1‐(dimethylcarbamoyl)ethyl]butanamides under the ‘Direct Amide Cyclization’ Conditions

The preparation of the title compounds was achieved via the ‘azirine/oxazolone method’ starting from the corresponding γ‐hydroxy acids. Upon subjecting the γ‐hydroxy‐N‐[1‐(dimethylcarbamoyl)ethyl]butanamides 4 to the so‐called ‘direct amide cyclization’ (DAC) conditions, chlorinated acids 11 or imino lactones 12 were obtained as the sole products instead of the expected cyclodepsipeptides A or their cyclodimers (Scheme 4). Variation of the substituents in 4 did not affect the outcome of the reaction and a mechanism for the formation of both products from the intermediate oxazolone 13 has been proposed. Under the acidic conditions of the DAC, the imino lactones are formed as their HCl salts 12 , which, in polar solvents or on silica gel, reacted further to give the chlorinated acids 11 . Stabilization of the imino lactones was achieved by increasing the substitution in the five‐membered ring, and their structure, in the form of the hydrochlorides, was established independently by X‐ray crystallography (Fig. 4). A derivative 15 of the imino lactone 12a was prepared by the reaction with the 2H‐azirin‐3‐amine 10a ; its structure was also established by an X‐ray crystal‐structure determination (Fig. 3). Furthermore, the structures of the ω‐chloro acids 11a and 11b were determined by X‐ray crystallography (Fig. 2).     

Voltammetry and electrocatalytic properties of cobalt (1,4,8,12-tetraazacyclopentadecane) in aqueous solution

Summary The spectrochemical, electrochemical and electrocatalytic properties of Co[15]aneN₄ ([15]aneN₄ = 1,4,8,12-tetraazacyclopentadecane) have been investigated. The results show that, in aqueous solution, this compound mainly exists as three species whose axial coordination positions are occupied by water and/or hydroxy ligands; it is marginal whether other substrates such as Cl^– and NO _inf3 ^sup– interact with the central ion in acid-base solutions. The approximate Pourbaix diagram of Co^III/II[15]ane N₄ was determined. There is an electrochemically-induced isomerization between two trans conformational isomers of the Co[15]aneN₄ complexes in acid and netural solutions. The Co[15]aneN₄ complex has electrocatalytic properties for reduction of nitrate and nitrite only in strong alkaline solution.     

A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient

We consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient, such as

$?F(x,u,Du,D^{2}u)=\lambda c(x)u+〈M(x)Du,Du〉+h(x)$

in a bounded domain with a Dirichlet boundary condition; here $\lambda \in \mathbb{R}$, $c,h\in L^p(\mathrm{\Omega })$, $p>n\ge 1$, $c?0$ and the matrix M satisfies $0<{\mu }_{1}I\le M\le {\mu }_{2}I$. Recently this problem was studied in the “coercive” case $\lambda c\le 0$, where uniqueness of solutions can be expected; and it was conjectured that the solution set is more complex for noncoercive equations. This conjecture was verified in 2015 by Arcoya, de Coster, Jeanjean and Tanaka for equations in divergence form, by exploiting the integral formulation of the problem. Here we show that similar phenomena occur for general, even fully nonlinear, equations in nondivergence form. We use different techniques based on the maximum principle.We develop a new method to obtain the crucial uniform a priori bounds, which permit to us to use degree theory. This method is based on basic regularity estimates such as half-Harnack inequalities, and on a Vázquez type strong maximum principle for our kind of equations.     

Performance monitoring of a mammalian cell based bioprocess using Raman spectroscopy

Being able to predict the final product yield at all stages in long-running, industrial, mammalian cell culture processes is vital for both operational efficiency, process consistency, and the implementation of quality by design (QbD) practices. Here we used Raman spectroscopy to monitor (in terms of glycoprotein yield prediction) a fed-batch fermentation from start to finish. Raman data were collected from 12 different time points in a Chinese hamster ovary (CHO) based manufacturing process and across 37 separate production runs. The samples comprised of clarified bioprocess broths extracted from the CHO cell based process with varying amounts of fresh and spent cell culture media. Competitive adaptive reweighted sampling (CoAdReS) and ant colony optimization (ACO) variable selection methods were used to enhance the predictive ability of the chemometric models by removing unnecessary spectral information. Using CoAdReS accurate prediction models (relative error of predictions between 2.1% and 3.3%) were built for the final glycoprotein yield at every stage of the bioprocess from small scale up to the final 5000 L bioreactor. This result reinforces our previous studies which indicate that media quality is one of the most significant factors determining the efficiency of industrial CHO-cell processes. This Raman based approach could thus be used to manage production in terms of selecting which small scale batches are progressed to large-scale manufacture, thus improving process efficiency significantly.     

Stability of the general exponent of nonlinear impulsive differential equations in a Banach space

A theorem is proved which gurantees stability under small perturbations of the general exponent of impulsive nonlinear systems in a Banach space.