Bilateral semidirect product decompositions of transformation monoids

In this paper we consider the monoid OR(n) of all full transformations on a chain with n elements that preserve or reverse the orientation, as well as its submonoids OD(n) of all order-preserving or order-reversing elements, OP(n) of all orientation-preserving elements and O(n) of all order-preserving elements. By making use of some well known presentations, we show that each of these four monoids is a quotient of a bilateral semidirectproduct of two of its remarkable submonoids.

A Hexanuclear Mixed-Valence Oxovanadium(IV,V) Complex as a Highly Efficient Alkane Oxidation Catalyst

The new hexanuclear mixed-valence vanadium complex [V3O3(OEt)(ashz)(2)(mu-OEt)](2) (1) with an N,O-donor ligand is reported. It acts as a highly efficient catalyst toward alkane oxidations by aqueous H2O2. Remarkably, high turnover numbers up to 25000 with product yields of up to 27% (based on alkane) stand for one of the most active systems for such reactions.

Greener Selective Cycloalkane Oxidations with Hydrogen Peroxide Catalyzed by Copper-5-(4-pyridyl)tetrazolate Metal-Organic Frameworks

Microwave assisted synthesis of the Cu(I) compound [Cu(µ4-4-ptz)]n [1, 4-ptz = 5-(4-pyridyl)tetrazolate] has been performed by employing a relatively easy method and within a shorter period of time compared to its sister compounds. The syntheses of the Cu(II) compounds [Cu3(µ3-4-ptz)4(µ2-N3)2(DMF)2]n∙(DMF)2n (2) and [Cu(µ2-4-ptz)2(H2O)2]n (3) using a similar method were reported previously by us. MOFs 1-3 revealed high catalytic activity toward oxidation of cyclic alkanes (cyclopentane, -hexane and -octane) with aqueous hydrogen peroxide, under very mild conditions (at room temperature), without any added solvent or additive. The most efficient system (2/H2O2) showed, for the oxidation of cyclohexane, a turnover number (TON) of 396 (TOF of 40 h−1), with an overall product yield (cyclohexanol and cyclohexanone) of 40% relative to the substrate. Moreover, the heterogeneous catalytic systems 1–3 allowed an easy catalyst recovery and reuse, at least for four consecutive cycles, maintaining ca. 90% of the initial high activity and concomitant high selectivity.

Spectral and weak polynomial completeness for the product of nonsingular matrices

Let F be a field with at least four elements. In this paper, we identify all the pairs (A, B) of n x n nonsingular matrices over F, satisfying the following property: for every monic polynomial f (x) = x(n) + a(n-1)x(n-1) +... + a(1)x + a(0) over F, with a root in F and a(0) = (-1)(n) det(AB), there are nonsingular matrices X, Y is an element of F-nxn such that XAX(-1)Y BY-1 has characteristic polynomial f (x).