Diophantine analysis and torsion on elliptic curves

In a recent paper of Bennett and the author, it was shown thatthe elliptic curve defined by y² = x³ + Ax + B, where A andB are integers, has no rational points of finite order if Ais sufficiently large relative to B (at least if one assumesthe abc Conjecture of Masser and Oesterlé). In the presentarticle we show, perhaps surprisingly, that the rational torsionon the above curve is also quite restricted if B is sufficientlylarge relative to A. In particular, we demonstrate that forany > 0 there is a constant c such that if A and B are integerssatisfying |B| > c |A|⁶⁺, then the elliptic curve definedabove has no rational torsion points, other than a possiblepoint of order 2 (again making use of the abc Conjecture insome cases). We then extend this by proving similar resultsfor elliptic curves admitting non-trivial -isogenies, ellipticcurves written in other forms, and elliptic curves over certainnumber fields. Curiously, the results on isogenies lead to twounexpected irrationality measures for certain algebraic numbers.     

An interpretation of glass chemistry in terms of the optical basicity concept

The basicity of an oxide glass can be measured experimentally from the frequency shifts in the ultra-violet (UV) (s-p) spectra of probe ions such as Pb²⁺ and can be expressed on the numerical scale of optical basicity Λ (ideally Λ lies between zero and unity). It is possible to relate Λ with (i) the constitution, and (ii) the electronegativity of the cations (e.g. Na⁺, Si⁴⁺, etc.) of the glass, and the relationship allows microscopic optical basicities λ to be assigned to individual oxides and oxy-groups in the glass. These microscopic optical basicities are used for interpreting various aspects of the physics and chemistry of glass including refractivity, network coordination number changes, chemical durability, the glass electrode, UV transparency and the host behaviour of glass towards metal ions generally. Changes in glass basicity in going from one alkali metal oxide to another are also discussed. Finally, the concept of optical basicity, both as an experimentally obtained quantity and as a number calculated from glass constitution and electronegativity, is discussed in relation to the traditional approach to acid-base behaviour in glass.     

Closed-form solution of a maximization problem

According to the characterization of eigenvalues of a real symmetric matrix A, the largest eigenvalue is given by the maximum of the quadratic form 〈xA, x〉 over the unit sphere; the second largest eigenvalue of A is given by the maximum of this same quadratic form over the subset of the unit sphere consisting of vectors orthogonal to an eigenvector associated with the largest eigenvalue, etc. In this study, we weaken the conditions of orthogonality by permitting the vectors to have a common inner product r where 0 ≤ r < 1. This leads to the formulation of what appears—from the mathematical programming standpoint—to be a challenging problem: the maximization of a convex objective function subject to nonlinear equality constraints. A key feature of this paper is that we obtain a closed-form solution of the problem, which may prove useful in testing global optimization software. Computational experiments were carried out with a number of solvers. We dedicate this paper to the memory of our great friend and colleague, Gene H. Golub.     

A theory of the mixed-alkali effect in β-alumina

A new theory of the mixed-alkali effect in β-alumina involves replacement of mobile interstitialcy pairs by immobilized mixed-alkali pairs. Applying this concept to data for (Na,K) β-alumina shows only 2.2% of the cations in Na β-alumina are mobile at 150°C, in close agreement with an earlier estimate based on the Haven ration, H_R.     

Mixed alkali effects in Na/Ag β-alumina

Mixed Na/Ag β-aluminas show the characteristic mixed alkali effect. Pronounced minima in conductivity isotherms are accompanied by a corresponding maximum in the activation energy. An entirely new effect is a change in ac response at high frequencies. In the region of the so-called Jonscher dispersion, where σ_ac = Aωⁿ, n falls from about 0.6 in pure Ag and Na β-aluminas to about 0.2 near the conductivity minimum. It is suggested that this reflects the disappearance of collective many-body interactions among the mobile cations.     

Isolation of cholera toxin receptors from a mouse fibroblast and lymphoid cell line by immune precipitation

Cholera toxin receptors have been isolated from both a mouse fibroblast (Balbc/3T3) and mouse lymphoid cell line labeled by the galactose oxidase borotritiide technique. Tritiated receptor-toxin complexes solubilized in NP40 were isolated by addition of toxin antibody followed by a protein A-containing strain of Staphylococcus aureus. In both cell types by far the major species of toxin receptor isolated was ganglioside in nature, although galactoproteins were also present in the immune complexes. Whether the galactoproteins form part of a toxin-receptor complex or are artifacts of the isolation procedure is presently unclear. The relative specificity of cholera toxin for a carbohydrate sequence in a glycolipid suggests that the toxin might prove a useful tool in establishing the function and organization of glycolipids in membranes. For example, interaction of cholera toxin with the mouse lymphoid cell line was shown to result in patching and capping of bound toxin, raising the possibility that the glycolipid receptor interacts indirectly with cytoskeletal elements. Cholera toxin might also be used to select for mutant fibroblasts lacking the toxin receptor and therefore having an altered glycolipid profile. Such mutants might prove useful in establishing the relationship (if any) between modified glycolipid pattern and other aspects of the transformed phenotype. Attempts to isolate mutants, based on the expectation that growth of cells containing the toxin receptor would be inhibited by the increase in cAMP levels normally induced by cholera toxin, proved unsuccessful. Cholera toxin failed to inhibit significantly the growth of either Balbc or Swiss 3T3 mouse fibroblasts although it markedly elevated cAMP levels.     

Noch-nie-getrocknete Baumwollfasern. VI. Glanz

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