On a class of matrices generated by certain generalized permutation matrices and applications

ABSTRACT

In this paper, we study a particular class of matrices generated by generalized permutation matrices corresponding to a subgroup of some permutation group. As applications, we first present a technique from which we can get closed formulas for the roots of many families of polynomial equations with degree between 5 and 10, inclusive. Then, we describe a tool that shows how to find solutions to Fermat's last theorem and Beal's conjecture over the square integer matrices of any dimension. Finally, simple generalizations of some of the concepts in number theory to integer square matrices are presented.     

Numerical computation of natural convection in an isosceles triangular cavity with a partially active base and filled with a Cu–water nanofluid

This paper discusses the results of a study related to natural convection cooling of a heat source located on the bottom wall of an inclined isosceles triangular enclosure filled with a Cu water-nanofluid. The right and left walls of the enclosure are both maintained cold at constant equal temperatures, while the remaining parts of the bottom wall are insulated. The study has been carried out for a Rayleigh number in the range 10⁴ ≤ Ra ≤ 10⁶, for a heat source length in the range 0.2 ≤ ε ≤0.8, for a solid volume fraction in the range 0 ≤ ?≤0.06 and for an inclination angle in the range 0° ≤ δ≤45°. Results are presented in the form of streamline contours, isotherms, maximum temperature at the heat source surface and average Nusselt number. It is noticed that the addition of Cu nanoparticles enhances the heat transfer rate and therefore cooling effectiveness for all values of Rayleigh number, especially at low values of Ra. The effect of the inclination angle becomes more noticeable as one increases the value of Ra. For high Rayleigh numbers, a critical value for the inclination angle of δ = 15° is found for which the heat source maximum temperature is highest.     

Weak carbon-hydrogen-nitrogen interactions affect the heterocyclic ligand bonding modes in barium complexes containing eta2-tetrazolato and eta2-pentazolato ligands

Treatment of Ba[N(SiMe3)2]2(THF)2 with 2 equiv of dimethylaminotetrazole or diisopropylaminotetrazole and 1 equiv of 18-crown-6 afforded Ba[CN4(NMe2)]2(18-crown-6) (87%) and Ba[CN4(NiPr2)]2(18-crown-6) (79%) as colorless crystalline solids. Ba[CN4(NMe2)]2(18-crown-6) contains two 1,2-eta2-tetrazolato ligands and one eta6-18-crown-6 ligand. The molecular structure of Ba[CN4(NiPr2)]2(18-crown-6) is similar to that of Ba[CN4(NMe2)]2(18-crown-6), except that the tetrazolato ligands exhibit the isomeric 2,3-eta2-coordination mode and the tetrazolato ligand CN4 cores are bent significantly toward the 18-crown-6 ligands. Molecular orbital calculations were carried out on the model complexes Ba(azolate)2(18-crown-6) (azolate = 1,2-eta2-CHN4, 2,3-eta2-CHN4, and eta2-N5) and demonstrate that the ligand coordination modes are influenced by intramolecular interactions between filled nitrogen orbitals on the azolato ligands and empty C-H sigma* orbitals on the 18-crown-6 ligands.     

Chemistry of 1,4-dioxene II : A new method for the two-carbon homologation of aldehydes and ketones to α-hydroxymethyl ketones

1,4-Dioxen-2-yl lithium $\text{2}$ reacts with ketones or aldehydes to give alcohols $\text{3}$ which lead after an allylic rearrangement under mild conditions, followed by reduction and hydrolysis to α-hydroxymethyl ketones in fair yields.     

Formal synthesis of (+/-)-guanacastepene A: a tandem ring-closing metathesis approach

A concise route to a key intermediate in the total synthesis of guanacastepene A is described. The main features include the simultaneous construction of the seven- and six-membered rings, using a tandem ring-closing metathesis and a stereoselective introduction of the oxygenated function at the C5 position. [reaction: see text]     

Preparation and reactivity of lithium 1,4-dioxenyl cuprates

Reactions of lower and higher order cuprates generated from lithiated 1,4-dioxene and copper (I) species, with epoxides and ,β-unsaturated ketones are desribed.     

Physicochemical Principles and Applications of Supercritical Fluid Chromatography (SFC). New analytical methods (19)

In supercritical fluid chromatography (SFC) compressed gases in the region of their critical temperature are used as mobile phases. SFC has important advantages over gas chromatography (GC) for the separation of low-volatile or thermally unstable substances. Like high pressure liquid chromatography (HPLC) and gel chromatography, it is used for various special applications and preparative separations, e.g. in the petroleum industry and in the separation of oligomers. SFC is of great interest in fundamental research on fluid extraction and for the determination of the physicochemical properties of fluid systems. In this contribution the most important physicochemical, methodological, and instrumental principles of SFC are summarized; characteristic physicochemical applications are the determination of capacity ratios, partition coefficients, partial molar volumes, interaction second virial coefficients, and difusion coefficients.     

Complexes of hydrazide Schiff bases

Summary 1,2-Diphenyl-2-(benzoylhydrazino)ethanol (DBE) and 1,2-diphenyl-2-(p-nitrobenzoylhydrazino)ethanol (DPNBE) form complexes of general formulae [M(H₂L)₂]X₂ and [M₂L₂] (where M = Co or Ni, H₂L and L are the neutral and dibasic forms of the ligands and X = Cl or NO₃). The analytical, spectral, magnetic and physical studies show the complexes to be of two types, namely monomeric cationic species prepared in neutral media and neutral binuclear oxygen-bridged species formed in alkaline solution. The ligands are tridentate, coordinating through carbonyl (or enolic) oxygen, azomethine nitrogen and alcoholic oxygen atoms.     

Rate acceleration of anionic oxy-cope rearrangements induced by an additional unsaturation

[reaction: see text] In sharp contrast to their isopropyl counterparts, a variety of (Z)-isopropenyl tertiary bicyclo[2.2.2]octenols undergo facile anionic oxy-Cope rearrangements allowing the stereoselective incorporation of an isopropenyl group into polycyclic skeletons such as the tricyclic system of vinigrol, bicyclo[5.3.1]undecane, and cis-decalin frameworks. This rate acceleration is probably due to the stabilization of the transition state by the additional unsaturation on the terminal position.