Volume 26, Issue 1

Abstract. It is proved that the series of all Brauer monoids generates the pseudovariety of all finite monoids while the series of their aperiodic analogues, the Jones monoids (also called Temperly–Lieb monoids), generates the pseudovariety of all finite aperiodic monoids. The proof is based on the analysis of wreath product decomposition and Krohn–Rhodes theory. The fact that the Jones monoids form a generating series for the pseudovariety of all finite aperiodic monoids can be viewed as solution of an old problem popularized by J.-É. Pin. For the latter, the relationship between the Jones monoids and the monoids of order preserving mappings of a chain of length n is investigated.

Abstract. Let K be an imaginary quadratic field of discriminant . We deal with problems of constructing normal bases between abelian extensions of K by making use of singular values of Siegel functions. First, we find normal bases of ring class fields of orders of bounded conductors depending on d K over K by using a criterion deduced from the Frobenius determinant relation. Next, denoting by the ray class field modulo N of K for an integer we consider the field extension for a prime and a positive integer m relatively prime to p and then find normal bases of all intermediate fields over by utilizing Kawamoto's arguments. We further investigate certain Galois module structure of the field extension with , which would be an extension of Komatsu's work.

Abstract. By definition, the perfect alignment between the vorticity vector and the vortex stretching one means that there is the same perfect alignment between the vorticity vector and the strain matrix eigenvector. We prove the following. For the “restricted Euler equations”, if there is a time instant such that the aforesaid perfect vector alignment happens at some point inside the flow, it continues then permanently and keeps forever in the sense that one happens successively at points which belongs to the given point trajectory generated by the flow. Further, if such a trajectory exists, depending on initial data, one can either “blow up” for a finite time or not. The aforementioned blows up can be very different. What is interesting in doing so is that there is such kind the finite-time blows up that eigenvalues of the strain matrix all go at infinity which can be as positive as negative whereas the vorticity remains bounded at the same time. This is, generally speaking, unexpected if to take into account the well-known BKM criterion of finite-time blows up for solutions to the Euler equations. As for the (original) Euler equations, if their solution is analytic, and: (1) there is a finite while (let very small even) such that the perfect vector alignment happens at both some point inside the flow and points belonging to the given point trajectory fraction which is generated by the flow during the course of the given while, then it continues permanently and keeps forever in the same sense as above; (2) there is no such while and the perfect vector alignment happens at a time instant only, a question respecting of the subsequent perfect vector alignment existence remains open.

Abstract. Suppose that a finite group G admits a Frobenius group of automorphisms with kernel F and complement H such that the fixed-point subgroup of F is trivial: . In this situation various properties of G are shown to be close to the corresponding properties of . By using Clifford's theorem it is proved that the order is bounded in terms of and , the rank of G is bounded in terms of and the rank of , and that G is nilpotent if is nilpotent. Lie ring methods are used for bounding the exponent and the nilpotency class of G in the case of metacyclic . The exponent of G is bounded in terms of and the exponent of by using Lazard's Lie algebra associated with the Jennings–Zassenhaus filtration and its connection with powerful subgroups. The nilpotency class of G is bounded in terms of and the nilpotency class of by considering Lie rings with a finite cyclic grading satisfying a certain `selective nilpotency' condition. The latter technique also yields similar results bounding the nilpotency class of Lie rings and algebras with a metacyclic Frobenius group of automorphisms, with corollaries for connected Lie groups and torsion-free locally nilpotent groups with such groups of automorphisms. Examples show that such nilpotency results are no longer true for non-metacyclic Frobenius groups of automorphisms.

Abstract. Let G be an exceptional Lie group with a maximal torus T . Based on the Schubert presentation of the cohomology ring of the flag manifold G / T obtained by the authors in a previous paper, we present a unified approach to the structure of as a Hopf algebra over the Steenrod algebra .

Abstract. In this paper we will give a unified proof of several results on the sovability of systems of certain equations over finite fields, which were recently obtained by Fourier analytic methods. Roughly speaking, we show that almost all systems of norm, bilinear or quadratic equations over finite fields are solvable in any large subset of vector spaces over finite fields.

Abstract. Deformation K -theory associates to each discrete group G a spectrum built from spaces of finite dimensional unitary representations of G . In all known examples, this spectrum is 2-periodic above the rational cohomological dimension of G (minus 2), in the sense that T. Lawson's Bott map is an isomorphism on homotopy in these dimensions. We establish a periodicity theorem for crystallographic subgroups of the isometries of k -dimensional Euclidean space. For a certain subclass of torsion-free crystallographic groups, we prove a vanishing result for the homotopy groups of the stable moduli space of representations, and we provide examples relating these homotopy groups to the cohomology of G . These results are established as corollaries of the fact that for each , the one-point compactification of the moduli space of irreducible n -dimensional representations of G is a CW complex of dimension at most k . This is proven using real algebraic geometry and projective representation theory.

Abstract. Let f be a Maass cusp form for SL 3 (ℤ) with Fourier coefficients . We consider the sum , where . A bound better than is proved to be valid for certain transcendental numbers . This bound improves Miller's result (2006) for these α. For α close to a rational number with , the smooth sum is further proved to decay rapidly. This extends a result of Booker (2000 and 2005) on a smooth sum of without the exponential function. These bounds manifest a strange vibration nature of Maass cusp forms. The main techniques include rational approximation of real numbers, a Voronoi summation formula for SL 3 (ℤ), and its asymptotic expansion.

Abstract. Let m and be positive integers. The set of links of codimension , , is the set of smooth isotopy classes of smooth embeddings . Haefliger showed that is a finitely generated abelian group with respect to embedded connected summation and computed its rank in the case of knots, i.e. . For and for restrictions on the rank of this group can be computed using results of Haefliger or Nezhinsky. Our main result determines the rank of the group in general. In particular we determine precisely when is finite. We also accomplish these tasks for framed links. Our proofs are based on the Haefliger exact sequence for groups of links and the theory of Lie algebras.

Abstract. In this paper we introduce a common framework for describing the topological part of the Baum–Connes conjecture for a wide class of groups. We compute the Bredon homology for groups with aspherical presentation, one-relator quotients of products of locally indicable groups, extensions of ℤ n by cyclic groups, and fuchsian groups. We take advantage of the torsion structure of these groups to use appropriate models of the universal space for proper actions which allow us, in turn, to extend some technology defined by Mislin in the case of one-relator groups.

Abstract. In this paper first we survey the Turán type inequalities and related problems for the Bessel functions of the first kind. Then we extend the known higher order Turán type inequalities for Bessel functions of the first kind to real parameters and we deduce new closed integral representation formulas for the second kind Neumann type series of Bessel functions of the first kind occurring in the study of Turán determinants of Bessel functions of the first kind. At the end of the paper we prove a Turán type inequality for the Bessel functions of the second kind.