Higher Hopf formulae for homology via Galois Theory

We use Janelidze's Categorical Galois Theory to extend Brown and Ellis's higher Hopf formulae for homology of groups to arbitrary semi-abelian monadic categories. Given such a category A and a chosen Birkhoff subcategory B of A, thus we describe the Barr-Beck derived functors of the reflector of A onto B in terms of centralization of higher extensions. In case A is the category Gp of all groups and B is the category Ab of all abelian groups, this yields a new proof for Brown and Ellis's formulae. We also give explicit formulae in the cases of groups vs. k-nilpotent groups, groups vs. k-solvable groups and precrossed modules vs. crossed modules.     

A note on some chain conditions

We answer a question raised by Othman Echi: Is an E ₁ (resp., a C ₁) ring an E (resp., a C) ring? We construct a C ₁ (thus E ₁) ring which is not an E ₂ (thus not a C ₂) ring. Received: 11 June 2007     

An integro-differential equation model for alignment and orientational aggregation

We study an integro-differential equation modeling angular alignment of interacting bundles of cells or filaments. A bifurcation analysis of the related stationary problem was done by Geigant and Stoll in [E. Geigant, M. Stoll, Bifurcation analysis of an orientational aggregation model, J. Math. Biol. 46 (6) (2003) 537-563]. Here we analyze the time-dependent problem and prove that the type of alignment (one- or multi-directional) depends on the initial distribution, the interaction potential, and the preferred optimal orientation of the bundles of cells or filaments. Our main technical tool is the analysis of the evolution of suitable functionals for the cell density, which allows to also specify the direction(s) where the final alignment takes place.     

Mixed-ligand complexes with trans-1-(2-pyridyl)-2-(4-pyridyl)ethylene terminal and different aromatic polycarboxyl linkers: Synergistic modulation of metallosupramolecular architectures via coordinative and secondary interactions

A series of Co^II, Ni^II, and Cu^II complexes with trans-1-(2-pyridyl)-2-(4-pyridyl)ethylene (bpe) and various polycarboxyl co-ligands have been prepared under general condition and characterized by IR, elemental analysis, and TG-DTA techniques. Single-crystal X-ray diffraction indicates that these complexes display multifarious binuclear, 1-D, and 2-D coordination motifs in virtue of the bridging polycarboxyl building blocks, in which the bpe ligand uniformly adopts the unidentate coordination by using its 4-pyridyl group. Remarkably, higher-dimensional extended networks are further formed with the aid of additional secondary interactions based on bpe (such as H-bonding and π–π stacking). These results demonstrate that bpe is a reliable bifunctional tecton to construct diverse supramolecular architectures via synergistic effect of multiple intermolecular interactions.     

On a theorem of Burnside on fixed-point-free automorphisms

We prove that a finite group having a fixed-point-free automorphism in the Fitting subgroup of its automorphism group must be abelian of rather restricted structure. As a consequence, no finite nonabelian group could have a fixed-point-free automorphism in the Frattini subgroup of its automorphism group. Received: 21 April 2007, Revised: 18 May 2007     

On the low-lying zeros of Hasse-Weil L-functions for elliptic curves

In this paper, we obtain an unconditional density theorem concerning the low-lying zeros of Hasse-Weil L-functions for a family of elliptic curves. From this together with the Riemann hypothesis for these L-functions, we infer the majorant of 27/14 (which is strictly less than 2) for the average rank of the elliptic curves in the family under consideration. This upper bound for the average rank enables us to deduce that, under the same assumption, a positive proportion of elliptic curves have algebraic ranks equaling their analytic ranks and finite Tate-Shafarevich group. Statements of this flavor were known previously [M.P. Young, Low-lying zeros of families of elliptic curves, J. Amer. Math. Soc. 19 (1) (2005) 205-250] under the additional assumptions of GRH for Dirichlet L-functions and symmetric square L-functions which are removed in the present paper.     

A separable manifold failing to have the homotopy type of a CW-complex

We show that (a variation of) the Prüfer surface, which is an example of a separable non-metrizable 2-manifold, does not have the homotopy type of a CW-complex. Received: 6 February 2007 Revised: 21 August 2007     

Above the Glauberman correspondence

Let S be a finite solvable group, and suppose S acts on the finite group N, and they have coprime orders. Then, the celebrated Glauberman correspondence provides a natural bijection from the set IrrS(N) of irreducible characters of N which are invariant under the action of S to the set Irr(CN(S)) of all irreducible characters of the centralizer of S in N. Suppose, further, that the semidirect product SN is a normal subgroup of a finite group G. Let θ∈IrrS(N), and let ψ∈Irr(CN(S)) be its Glauberman correspondent. We prove that there is a bijection with good compatibility properties from the set Irr(G,θ) of the irreducible characters of G above θ to Irr(NG(S),ψ) such that, in the case when S is a p-group for some prime p, it preserves fields of values and Schur indices over Qp, the field of p-adic numbers. Using this result, we also prove a strengthening of the McKay Conjecture for all p-solvable groups.     

On spherical expansions of zonal functions on Euclidean spheres

This paper describes a new approach to the problem of computing spherical expansions of zonal functions on Euclidean spheres. We derive an explicit formula for the coefficients of the expansion expressing them in terms of the Taylor coefficients of the profile function rather than (as done usually) in terms of its integrals against Gegenbauer polynomials. Our proof of this result is based on a polynomial identity equivalent to the canonical decomposition of homogeneous polynomials and uses only basic properties of this decomposition together with simple facts concerning zonal harmonic polynomials. As corollaries, we obtain direct and apparently new derivations of the so-called plane wave expansion and of the expansion of the Poisson kernel for the unit ball. Received: 26 January 2007