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This review discusses the main factors that govern the oxidation processes of lignins into aromatic aldehydes and acids using hydrogen peroxide. Aromatic aldehydes and acids are produced in the oxidative degradation of lignin whereas mono and dicarboxylic acids are the main products. The stability of hydrogen peroxide under the reaction conditions is an important factor that needs to be addressed for selectively improving the yield of aromatic aldehydes. Hydrogen peroxide in the presence of heavy metal ions readily decomposes, leading to minor degradation of lignin. This degradation results in quinones which are highly reactive towards peroxide. Under these reaction conditions, the pH of the reaction medium defines the reaction mechanism and the product distribution. Under acidic conditions, hydrogen peroxide reacts electrophilically with electron rich aromatic and olefinic structures at comparatively higher temperatures. In contrast, under alkaline conditions it reacts nucleophilically with electron deficient carbonyl and conjugated carbonyl structures in lignin. The reaction pattern in the oxidation of lignin usually involves cleavage of the aromatic ring, the aliphatic side chain or other linkages which will be discussed in this review. [ABSTRACT FROM AUTHOR]

The Eisenbud–Mazur conjecture states that given an equicharacteristic zero, regular local ring ( R , m ) and a prime ideal P ⊂ R , we have that P ( 2 ) ⊆ m P . In this paper, we computationally prove that the conjecture holds in the special case of certain prime ideals in formal power series rings. [ABSTRACT FROM AUTHOR]

Abstract: We study the question of obtaining uniform bounds for the growth of symbolic powers of ideals in Noetherian rings. First we use a generalized version of Artinʼs strong approximation theorem to obtain uniform bounds for prime ideals in the integral closure of a finite Galois field extension of the fraction field of an excellent Henselian regular local ring. As a corollary, we obtain a similar result for a complete Noetherian local domains. We also obtain linear uniform bounds for the growth of symbolic powers in certain normal subrings of equicharacteristic regular rings under mild conditions. [Copyright &y& Elsevier]

Abstract: The friction factor, evaluated from the Colebrook and White equation, is traditionally computed iteratively. An analytical solution of the Colebrook and White equation for the friction factor can be obtained, using the Lambert function. Also, the equation relating the outlet pressure to the inlet pressure of an ideal gas flowing through a straight pipe under isothermal, steady state conditions has been hitherto considered in literature to be implicit in these variables, probably due to its inherent non-linear nature. However, it can be shown that an analytical solution to the above equation for the pressure drop (or alternatively, outlet pressure) can also be obtained using the Lambert function. [Copyright &y& Elsevier]