Soit (M, g0) une variété riemannienne compacte de courbure sectionnelle négative. Soit g1 une autre métrique riemannienne sur M de rang 1. On montre que l'égalité des spectres marqués des longueurs de g0 et g1 implique que le flot géodésique de g0 est un facteur de celui de g1.

We describe a type of Lie colour algebra, which we call generic, whose universal enveloping algebra is a domain with finite global dimension. Moreover, it is an iterated Ore extension. We provide an application and show Gröbner basis methods can be used to study universal enveloping algebras of factors of generic Lie colour algebras.

Let A = {a1, a2,…} (a1 < a2 < …) be an infinite sequence of positive integers. Let A(n) be the number of elements of A not exceeding n, and denote by R2(n) the number of solutions of ai + aj = n, i ≤ j. In 1986, Erdős, Sárközy and Sós proved that if (n − A(n))/log n → ∞(n → ∞), then . In this paper, we generalise this theorem and give its quantitative form. For example, one of our conclusions implies that if limsup(n − A(n))/log n = ∞, then for infinitely many positive integers N.

We restrict primes and prime powers to sets . Let . Then the error in θH(x) has, unconditionally, the expected order of magnitude . However, if then ψH (x) = x log 2 + O (log x). Some reasons for and consequences of these sharp results are explored. A proof is given of the “harmonic prime number theorem”, πH (x)/π (x) → log 2.

Lower and upper bounds for CM-dimension, called CM*-dimension and CM*-dimension, will be defined for any finitely generated module M over a local Noetherian ring R. Both CM* and CM*-dimension reflect the Cohen–Macaulay property of rings. Our results will show that these dimensions have the expected basic properties parallel to those of the homological dimensions. In particular, they satisfy an analog of the Auslander-Buchsbaum formula.

Interior point methods are not only the most effective methods for solving optimisation problems in practice but they also have polynomial time complexity. However, there is still a gap between the practical behavior of the interior point method algorithms and their theoretical complexity results. In this paper, by focusing on linear programming problems, we introduce a new family of kernel functions that have some simple and easy to check properties. We present a simplified analysis to obtain the complexity of generic interior point methods based on the proximity functions induced by these kernel functions. Finally, we prove that this family of kernel functions leads to improved iteration bounds of the large-update interior point methods.

If the inner mapping group of a loop is a finite Abelian group, then the loop is centrally nilpotent. We first investigate the structure of those finite Abelian groups which are not isomorphic to inner mapping groups of loops and after this we show that if the inner mapping group of a loop is isomorphic to the direct product of two cyclic groups of the same odd prime power order pn, then our loop is centrally nilpotent of class at most n + 1.

The natural character π of a finite transitive permutation group G has the form 1G + θ where θ is a character which affords a rational representation of G. We call G a QI-group if this representation is irreducible over ℚ. Every 2-transitive group is a QI-group, but the latter class of groups is larger. It is shown that every QI-group is ¾-transitive and primitive, and that it is either almost simple or of affine type. QI-groups of affine type are completely determined relative to the 2-transitive affine groups, and partial information is obtained about the socles of simply transitive almost simple QI-groups. The only known simply transitive almost simple QI-groups are of degree 2k−1(2k − 1) with 2k − 1 prime and socle isomorphic to PSL(2, 2k).

This paper develops a one term Edgeworth expansion under minimal conditions for elementary symmetric polynomials of any degree based on trimmed samples. These statistics are special cases of trimmed U-statistics and natural extensions of the trimmed mean.

We consider the problem of determining all power integral bases for the maximal real subfield Q (ζ + ζ−1) of the p-th cyclotomic field Q (ζ), where p ≥ 5 is prime and ζ is a primitive p-th root of unity. The ring of integers is Z[ζ+ζ−1] so a power integral basis always exists, and there are further non-obvious generators for the ring. Specifically, we prove that if or one of the Galois conjugates of these five algebraic integers. Up to integer translation and multiplication by −1, there are no additional generators for p ≤ 11, and it is plausible that there are no additional generators for p > 13 as well. For p = 13 there is an additional generator, but we show that it does not generalise to an additional generator for 13 < p < 1000.

The group of Poisson automorphisms of the coordinate ring of Poisson symplectic 2n-space is isomorphic to the algebraic torus (κ✶)n+1 and it confirms that the algebra constructed by K.L. Horton (2003) is a quantisation of the coordinate ring of Poisson symplectic 2n-space.

We construct generating functions for the entries of Hankel determinant formula for the Okamoto polynomials which characterise a class of rational solutions to the Painlevé IV equation. Generating functions are characterised as asymptotic expansions of log derivative of Ai and Bi, which are solutions of the Airy equation.