Hermite‐Hadamard‐type inequalities for interval‐valued coordinated convex functions involving generalized fractional integrals

In this paper, we define interval‐valued left‐sided and right‐sided generalized fractional double integrals. We establish inequalities of Hermite‐Hadamard like for coordinated interval‐valued convex functions by applying our newly defined integrals.     

Functional zinc(II) phthalocyanines bearing Schiff base complexes as oxidation catalysts for bleaching systems

Oxidation catalysis is used to increase the performance of hydrogen peroxide in laundry bleach applications. Bleach catalysts provide cost‐effective, energy‐saving and environmentally friendly bleach systems yielding perfect stain removal at lower temperatures. This comparative study is based on the synthesis of bis[bis(salicylhydrazonephenoxy)manganese(III)] phthalocyaninatozinc(II) ( 2 ), bis[bis(salicylhydrazonephenoxy)cobalt(III)] phthalocyaninatozinc(II) ( 3 ) and bis[bis(salicylhydrazonephenoxy)iron(III)] phthalocyaninatozinc(II) ( 4 ) as tri‐nuclear complexes consisting of two Schiff base complexes substituting a zinc phthalocyanine. Complexion on the periphery to obtain complexes 2 , 3 , 4 was performed through the reaction of a Schiff base‐substituted phthalocyanine using MnCl₂?4H₂O, CoCl₂?6H₂O or FeCl₃?6H₂O salts in basic condition in dimethylformamide. Fourier transform infrared, ¹H NMR, ¹³C NMR, UV–visible, inductively coupled plasma optical emission and mass spectra were applied to characterize the prepared compounds. The bleach performances of the three phthalocyanine compounds 2 , 3 , 4 were examined by the degradation of morin as hydrophilic dye. The degradation progress in the presence of catalysts 2 , 3 , 4 /H₂O₂ combination in aqueous solution was investigated using an online spectrophotometric method. It was found that the catalysts 2 , 3 , 4 exhibited better bleaching performance at 25 °C than tetraactylethylethylenediamine as bleach activator used in powder detergent formulations for stain removal. Copyright © 2015 John Wiley & Sons, Ltd.     

Corrigendum to: on the inheritance of the strongly π-extending property

In this corrigendum, we provide correct versions of Theorem 2.8 and Corollary 3.2, retract two consequences of the previous form of the Theorem 2.8. To this end, the main results of the article remain after the aforementioned changes.     

On the inheritance of the strongly π-extending property

In this article, we focus on modules with the property that every projection invariant submodule is essential in a fully invariant direct summand. In contrast to π-extending condition, it is shown that the former property is inherited by direct summands and Morita invariant. An application of our results yields that the endomorphism ring of a free module enjoys the property. Moreover, we characterize generalized triangular matrix rings with the aforementioned property and apply to somewhat special cases.     

Novel and Stereospecific Synthesis of (2S)‐3‐(2,4,5‐Trifluorophenyl)propane‐1,2‐diol from D‐Mannitol

A stereospecific synthesis of (2S)‐3‐(2,4,5‐trifluorophenyl)propane‐1,2‐diol from D ‐mannitol has been developed. The reaction of 2,3‐O‐isopropylidene‐D ‐glyceraldehyde with 2,4,5‐trifluorophenylmagnesium bromide gave [(4R)‐2,2‐dimethyl‐1,3‐dioxolan‐4‐yl](2,4,5‐trifluorophenyl)methanol in 65% yield as a mixture of diastereoisomers (1 : 1). The Ph₃P catalyzed reaction of the latter with C₂Cl₆ followed by reduction with Pd/C‐catalyzed hydrogenation gave (2S)‐3‐(2,4,5‐trifluorophenyl)propane‐1,2‐diol with >99% ee and 65% yield.     

Crystal structure and magnetic properties of dinuclear iron(III) complex with ONNO-donor ligand

A new dinuclear Fe(III) complex, [Fe(5-MeOL1)(OH)_0.86(CH₃O)_0.14]₂?2(CH₃OH), [H₂-5-MeOL1 = N,N′-bis(5-methoxy-2-hydroxybenzylidene)-2,2-dimethylpropane-1,3-diamine], 1 has been synthesized and characterized by single crystal structure analysis. The structure of 1 consists of two Fe(III) centers with one tetradentate schiff base ligand (N₂O₂) which are bridged by dihydroxo/dimethoxo groups to yield a Fe₂O₂ core. Complex 1 exhibits weak antiferromagnetic exchange interaction between Fe(III) ions with J = ?0.21 cm^–1.     

Noether-Type Symmetries and Conservation Laws Via Partial Lagrangians

We show how one can construct conservation laws of Euler-Lagrange-type equations via Noether-type symmetry operators associated with what we term partial Lagrangians. This is even in the case when a system does not directly have a usual Lagrangian, e.g. scalar evolution equations. These Noether-type symmetry operators do not form a Lie algebra in general. We specify the conditions under which they do form an algebra. Furthermore, the conditions under which they are symmetries of the Euler-Lagrange-type equations are derived. Examples are given including those that admit a standard Lagrangian such as the Maxwellian tail equation, and equations that do not such as the heat and nonlinear heat equations. We also obtain new conservation laws from Noether-type symmetry operators for a class of nonlinear heat equations in more than two independent variables.     

Exact solutions of some general nonlinear wave equations in elasticity

A similarity analysis of a nonlinear wave equation in elasticity is studied; in particular, one with anharmonic corrections. The symmetry transformation give rise to exact solutions via the method of invariants. In some cases, graphical figure of the solutions are presented. Furthermore, we consider some cases wherein the velocities of the longitudinal and transversal plane waves are variable. Finally, a brief discussion on how a symmetry analysis on a perturbation of the elasticity equation can be pursued.