Intersection of Isomorphic Linear Codes

Given an (n, k) linear code over GF(q), the intersection of with a codeπ( ), whereπS_n, is an (n, k₁) code, where max{0, 2k−n}k₁k. The intersection problem is to determine which integers in this range are attainable for a given code . We show that, depending on the structure of the generator matrix of the code, some of the values in this range are attainable. As a consequence we give a complete solution to the intersection problem for most of the interesting linear codes, e.g. cyclic codes, Reed–Muller codes, and most MDS codes.     

N.m.r. spectra of cyclic amines. II—Factors influencing the chemical shifts of α-protons in aziridines

Proton magnetic resonance spectra of trans and cis-2,3-diphenylaziridine (1 and 2) and their N-ethyl derivatives 3 and 4 were measured in carbon tetrachloride, chloroform, and benzene-d₆ at low temperatures (1 and 3) and in dry conditions (1 and 2). On the basis of these results it was concluded that an N-ethyl group exerts a shielding influence on a cis ring proton and a deshielding influence on a trans ring proton. From results obtained by measuring the ¹H n.m.r. spectra of 1–4 in deuterochlorofom-trifluoroacetic acid it was derived that the lone pair of the aziridine nitrogen exerts a shielding influence on cis related ring hydrogens. In most N-alkylaziridines the effect of the N-alkyl group predominates.     

Application of the generalized internal reference method for the characterization of parameters causing drift in inductively-coupled plasma emission spectrometry

Drift in signals limits the performance of inductively-coupled plasma (i.c.p.) emission spectrometry. The generalized internal reference method was applied to deconvolute the drift of an i.c.p. system. The plasma parameters were correlated to the drift components and the error boundary for the method was found. In the system studied, it was found that fluctuations in the nebulizer uptake rate were responsible for short-term signal changes. Long-term signal drift was caused mainly by flow variations of both cooling gas and aerosol carrier gas. The incident power to the plasma was found to be stable compared to other sources of drift. The method was found to be adequate for evaluating the origins of i.c.p. drift.     

The consistency with CH of some consequences of Martin’s axiom plus2^{\aleph _0 } > \aleph _1

We present here a (weak) axiom which implies some of the consequences of MA, but is consistent with GCH. We use the method of Jensen in his proof of consis (ZFC+GCH+SH). The third author wishes to thank the United States—Israel Binational Foundation for partially supporting this research by Grant 1110.     

A simple classical prediction of quantal resonances in collinear reactive scattering

Quantal collinear reactive scattering computations have shown that in the vicinity of thresholds of reactant or product vibrational states, one finds resonances in the state to state reaction probability. We find that these resonances can be explained classically in terms of energy transfer between adiabatic reactant and product channels. This transfer is attributable to resonant periodic orbits, resonating between reactants and products. The classical condition for a quantal resonance is that the action of the orbit over one period be an integer (in units of h) and that the energy at which this occurs be lower than the adiabatic barrier heights of the resonating states. These conditions suffice for a prediction of the location of the quantal resonance within a 1% accuracy!     

Anti-plane shear fields with discontinuous deformation gradients near the tip of a crack in finite elastostatics

Summary This paper reconsiders the problem of determining the elastostatic field near the tip of a crack in an all-round infinite body deformed by a Mode III loading at infinity to a state of anti-plane shear. The problem is treated for a class of incompressible, homogeneous, isotropic elastic materials whose constitutive laws permit a loss of ellipticity in the governing displacement equation of equilibrium at sufficiently severe shearing strains. The analysis represents a generalization of that reported in an earlier study and, as before, is carried out for the small-scale nonlinear crack problem, in which a crack of finite length is replaced by a semi-infinite one, and the nonlinear field far from the crack-tip is matched to the near field predicted by the linearized theory. The methods employed in the present paper are necessarily largely qualitative, since they apply to all materials in the class considered. The principal feature of the resulting elastic field is the presence of two symmetrically located curves issuing from the crack-tip and bearing discontinuities in displacement gradient and stress.The results communicated in this paper were obtained in the course of an investigation supported in part by Contract N00014-75-C-0196 with the Office of Naval Research in Washington, D.C.