Nonassociative Product Research Articles

Quasigroups Connected with Clifford Groups

In Clifford groups, a nonassociative product is defined which leads to the definition of nonassociative groups. These nonassociative groups have matrix representations on the condition that the “row by column” product of two matrices is replaced by the “column by column” product. A nonassociative group of transformations connected with the Lorentz group is determined, together with its irreducible, double-valued matrix representation, whose matrices undergo the “column by column” product.

Algebraic structure of linear and non-linear models of open quantum systems

Abstract We show that all the linear and nonlinear evolution equations proposed so far for the density operator of open quantum systems admit a common algebraic structure in the form of a generalized commutator, which is the nonassociative product of a Lie-admissible algebra.

Measurement of the rate coefficients for the bimolecular and termolecular ion–molecule reactions of He2+ with selected atomic and molecular species

This work reports the measurement of bimolecular and termolecular charge transfer reactions of He2+ into nonassociative product channels. In this study ion destruction frequencies have been experimentally determined from the selectively excited fluorescence of N2+ in high pressure afterglows of mixed gases excited by intense electron beam discharges. Data have been obtained as functions of helium pressure over the range from 400 to 1500 torr and as functions of the partial pressure of reactant from 50 to 200 mtorr. From these data pressure-dependent rate coefficients have been extracted and subsequently resolved into contributions from bimolecular and termolecular components for reactions of He2+ with Kr, H2, O2, NO, HBr, HCl, H2O, N2O, NO2, C3H8, NH3, and CCl2F2, selected because of their widely varying values of polarizability and dipole moment. The bimolecular components have been found to agree with the NOAA flowing afterglow results, where available, and to compare favorably with theoretical values obtained from Langevin, ADO, and locked dipole approximations. The sensitivity of the method has been sufficient to detect termolecular components as small as 2×10−30 cm6 sec−1 and values were found to range widely from 9×10−30 cm6 sec−1 for H2 to 140×10−30 cm6 sec−1 for HBr. A termolecular analog to the Langevin limit has been constructed which explains these values in terms of the rates at which third-body encounters change glancing collisions into inwardly spiraling trajectories. The data reported are in general agreement with this model and indicate that the reaction probabilities are near unity in both bimolecular and termolecular channels. The size of these termolecular rates suggests the general importance of three-body ion–molecule reactions in higher pressure plasmas such as those found in e-beam lasers.

Measurement of the rate coefficients for the bimolecular and termolecular charge transfer reactions of He2+ with Ne, Ar, N2, CO, CO2, and CH4

This work reports the measurement of termolecular charge transfer reactions of He2+ into nonassociative product channels. In this study ion destruction frequencies have been experimentally determined from the selectively excited fluorescence of N2+ in high pressure afterglows of mixed gases excited by an intense electron beam discharge. Data have been obtained as functions of helium pressure over the range from 300 to 1500 torr and as functions of the partial pressure of reactant from 25 to 400 μ Hg. From these data, pressure-dependent rate coefficients have been extracted and subsequently resolved into contributions from bimolecular and termolecular components for reactions of He2+ with Ne, Ar, N2, CO, CO2, and CH4. The bimolecular components have been found to agree with the ESSA flowing afterglow results to within experimental error, and in some cases the values reported here represent an improvement in precision. The sensitivity of the method has been sufficient to detect termolecular components as small as 2×10−30 cm6/sec and values were found to range widely from 2×10−30 for Ne to 67×10−30 cm6/sec for CO2. The size of these termolecular rates suggests the general importance of three-body ion–molecule reactions in higher pressure plasmas such as those found in e-beam lasers.

On Non-Associative Systems

The present paper is concerned with the “logarithmetic”, or arithmetic of shapes of non-associative combinations as defined by Etherington in ref. (1). The shape of a non-associative product is defined as “the manner of association of its factors without regard to their identity”. Thus, for a binary non-communicative operation, the products ((AB)C)D and ((BA)C)D and ((AA)A)A all have the same shape, while D((AB)C) has a different shape. The sum of the two shapes a and b is defined as the shape of the product of two expressions, of shapes a and b respectively, in the original system of non-associative combination. The product of two shapes a and b is defined as the shape of any expression obtained by replacing every factor in an expression of shape b by an expression of shape a. It is readily shown that these definitions are unambiguous.

Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for non-associative products

Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for non-associative products

XV.—On Non-Associative Combinations

Numerous combinatory problems arise in connection with a set of elements subject to a non-associative process of composition—let us say of multiplication—commutative or non-commutative.Non-associative products may be classified according to their shape. By the shape of a product I mean the manner of association of its factors without regard to their identity. Shapes will be called commutative or non-commutative according to the type of multiplication under consideration. Thus if multiplication is non-commutative, the products (AB.C)D and (BA.C)D are distinct but have the same shape, while D(AB.C) has a different shape. The three expressions, however, have the same commutative shape. I confine attention to products (like these) in which the factors are combined only two at a time.