Journal of the London Mathematical Society-Second Series最新文献

In this survey article, we explore a central theme in Diophantine approximation inspired by a celebrated result of Besicovitch on the Hausdorff dimension of well approximable real numbers. We outline some of the key developments stemming from Besicovitch's result, with a focus on the mass transference principle, ubiquity and Diophantine approximation on manifolds and fractals. We highlight the subtle yet profound connections between number theory and fractal geometry, and discuss several open problems at their intersection.

In a short paper that appeared in the Journal of the London Mathematical Society in 1934, H. S. M. Coxeter completed the classification of finite Coxeter groups. In this survey, we describe what Coxeter did in this paper and examine an assortment of topics that illustrate the broad and enduring influence of Coxeter's paper on developments in algebra, group theory, and geometry.

In 1935, Philip Hall published what is often referred to as ‘Hall's marriage theorem’ in a short paper (P. Hall, J. Lond. Math. Soc. (1) 10 (1935), no. 1, 26–30.) This paper has been very influential. I state the theorem and outline Hall's proof, together with some equivalent (or stronger) earlier results, and proceed to discuss some the many directions in combinatorics and beyond which this theorem has influenced.

I survey C. T. C. Wall's influential papers, ‘Diffeomorphisms of 4-manifolds’ and ‘On simply-connected 4-manifolds’, published in 1964 on pp. 131–149 of volume 39 of the Journal of the London Mathematical Society.

In this expository article, we review some of the ideas behind the work of Heath–Brown (D. R. Heath-Brown, J. London Math. Soc., (2), 24, (1981), no. 1, 65–78) on upper and lower bounds for moments of the Riemann zeta-function, as well as the impact this work had on subsequent developments in the field. We survey recent results on the topic, which essentially recover the expected rate of growth for all moments — unconditionally for small moments and conditionally on the Riemann hypothesis for all larger moments.

Mary L. Cartwright and John E. Littlewood published a short “preliminary survey” in 1945 describing results of their investigation of the forced van der Pol equation

The GJMS operators, introduced by Graham, Jenne, Mason and Sparling, are a family of conformally invariant linear differential operators with leading term a power of the Laplacian. These operators and their method of construction have had a major impact in geometry, analysis and physics. We describe the GJMS operators and their construction, and briefly survey their importance and impact.

The Davenport–Heilbronn method is a version of the circle method that was developed for studying Diophantine inequalities in the paper (Davenport and Heilbronn, J. Lond. Math. Soc. (1) 21 (1946), 185–193). We discuss the main ideas in the paper, together with an account of the development of the subject in the intervening 80 years.

We consider a regularised Fermi projection of the Hamiltonian of the massless Dirac equation at Fermi energy zero. The matrix-valued symbol of the resulting operator is discontinuous at the origin. For this operator, we prove Szegő-type asymptotics with the spatial cut-off domains given by $�d$-dimensional cubes. For analytic test functions, we obtain a $�d$-term asymptotic expansion and provide an upper bound of logarithmic order for the remaining terms. This bound does not depend on the regularisation. In the special case that the test function is given by a polynomial of degree less or equal than three, we prove a $��(�d�+�1�)�$-term asymptotic expansion with an error term of constant order. The additional term is of logarithmic order and its coefficient is independent of the regularisation.

We correct a claim in the paper cited in the title and delineate our current knowledge regarding related claims.