Narrow Results

Recently, George Beck conjectured and George Andrews proved a handful of novel congruences concerning the rank statistic of Dyson and the crank statistic of Andrews and Garvan. In this paper, we shall prove two identities concerning the weighted rank and crank moments, from which more congruences of the Andrews–Beck type may be deduced.

Let L_S denote the language of (right) S-acts over a monoid S and let Σ_S be a set of sentences in L_S which axiomatises S-acts. A general result of model theory says that Σ_S has a model companion, denoted by T_S, precisely when the class of existentially closed S-acts is axiomatisable and in this case, T_S axiomatises . It is known that T_S exists if and only if S is right coherent. Moreover, by a result of Ivanov, T_S has the model-theoretic property of being stable.

In the study of stable first order theories, superstable and totally transcendental theories are of particular interest. These concepts depend upon the notion of type: we describe types over T_S algebraically, thus reducing our examination of T_S to consideration of the lattice of right congruences of S. We indicate how to use our result to confirm that T_S is stable and to prove another result of Ivanov, namely that T_S is superstable if and only if S satisfies the maximal condition for right ideals. The situation for total transcendence is more complicated but again we can use our description of types to ascertain for which right coherent monoids S we have that T_S is totally transcendental and is such that the U-rank of any type coincides with its Morley rank.

Linear preserver problems are questions about characterising linear maps on spaces of matrices or spaces of operators (or more generally on rings or algebras) that preserve certain properties. We present an exposition of three such problems on preserving invertibility or commutativity or rank one.

This is an introduction to a probabilistic model for the arithmetic of elliptic curves, a model developed in a series of articles of the author with Bhargava, Kane, Lenstra, Park, Rains, Voight, and Wood. We discuss the theoretical evidence for the model, and we make predictions about elliptic curves based on corresponding theorems proved about the model. In particular, the model suggests that all but finitely many elliptic curves over ℚ have rank ≤ 21, which would imply that the rank is uniformly bounded.

We discuss applications of Clifford's theorem in the situation where a finite group G admits a Frobenius group of automorphisms F H with complement H and fixed-point-free kernel F (that is, such that C_G(F) = 1). The strong hypothesis makes it easier to apply Clifford's theorem, but the desired results must also have strong conclusions. The aim is proving that various properties of G are close to the corresponding properties of C_G(H). It is shown that the fixed points of H in any F H-invariant section are covered by the fixed points of H in G. This implies a bound for |G| in terms of |H| and |C_G(H)|, and a bound for the rank of G in terms of |H| and the rank of C_G(H). It is shown that G is nilpotent if C_G(H) is nilpotent. These results do not need any coprimeness assumptions on the orders of G and FH. Under the additional condition that (|G|, |H|) = 1 it is proved that the Fitting height of G is equal to the Fitting height of the fixed-point subgroup C_G(H) and the Fitting series of C_G(H) coincides with its intersections with the Fitting series of G. Further recent results on bounding the exponent and nilpotency class of G are mentioned, which use different Lie ring methods.

In this survey we discuss recent progress in the study of groups and Lie rings admitting a Frobenius group of automorphisms F H with kernel F and complement H. Our main objective is the case, where a finite group G is acted on by F H in such a way that the fixed-point subgroup of the Frobenius kernel is trivial: C_G(F) = 1. Recent results of Khukhro, Makarenko and Shumyatsky show that in this situation various properties of the group G such as the order, rank, nilpotency, nilpotent length, exponent, nilpotency class are close to the corresponding properties of the fixed-point subgroup of the Frobenius complement, C_G(H). Special emphasis is placed on Lie-theoretic methods used for bounding the nilpotency class and exponent of G.

In this paper,we give a necessary and sufficient condition for the solvability to a system of linear matrix equations A₁X₁ = C₁, X₁B₁ = C₂, A₂X₂ = C₃. X₂B₂ = C₄, A₃X₃ = C₅, X₃B₃ = C₆, A₄X₁B₄+A₅X₂B₅+A₆X₃B₆ = C₇ over an arbitrary division ring. The findings of this paper extend some known results in the literature.