Our aim in this note is to make some conjectures about extremal densities of daisy-free families, where a ‘daisy’ is a certain hypergraph. These questions turn out to be related to some Turán problems in the hypercube, but they are also natural in their own right. We start by giving the daisy conjectures, and some related problems, and shall then go on to describe the connection with vertex-Turán problems in the hypercube.

Philippe Flajolet, mathematician and computer scientist extraordinaire, the father of analytic combinatorics, suddenly passed away on 22 March 2011, at the prime of his career. He is celebrated for opening new lines of research in the analysis of algorithms, developing powerful new methods, and solving difficult open problems. His research contributions will have an impact for generations, and his approach to research, based on curiosity, discriminating taste, broad knowledge and interests, intellectual integrity, and a genuine sense of camaraderie, will serve as an inspiration to those who knew him, for years to come.

Let (r−1)n ≥ rk and let . Suppose that F1 ∩ ⋅⋅⋅ ∩ Fr ≠ ∅ holds for all Fi ∈ i, 1 ≤ i ≤ r. Then we show that .

Let p be a prime number. For a positive integer n and a p-adic number ξ, let λn(ξ) denote the supremum of the real numbers λ such that there are arbitrarily large positive integers q such that ‖qξ‖p,‖qξ2‖p,…,‖qξn‖p are all less than q−λ−1. Here, ‖x‖p denotes the infimum of |x−n|p as n runs through the integers. We study the set of values taken by the function λn.

Suppose that G is a finite group and H is a subgroup of G. We call H a weakly s-supplementally embedded subgroup of G if there exist a subgroup T of G and an s-quasinormally embedded subgroup Hse of G contained in H such that G = HT and H ∩ T ≤ Hse. We investigate the influence of the weakly s-supplementally embedded property of some minimal subgroups on the structure of finite groups. As an application of our results, some earlier results are generalized.

We show that if the given cotorsion pair in the category of modules is complete and hereditary, then both of the induced cotorsion pairs in the category of complexes are complete. We also give a cofibrantly generated model structure that can be regarded as a generalization of the projective model structure.

Recent research of the author has studied edge-labelled directed trees under a natural multiplication operation. The class of all such trees (with a fixed labelling alphabet) has an algebraic interpretation, as a free object in the class of adequate semigroups. We consider here a natural subclass of these trees, defined by placing a restriction on edge orientations, and show that the resulting algebraic structure is a free object in the class of left adequate semigroups. Through this correspondence we establish some structural and algorithmic properties of free left adequate semigroups and monoids, and consequently of the category of all left adequate semigroups.

Following the derivation of amplitude equations through a new two-time-scale method [O'Malley, R. E., Jr. & Kirkinis, E (2010) A combined renormalization group-multiple scale method for singularly perturbed problems. Stud. Appl. Math. 124, 383–410], we show that a multi-scale method may often be preferable for solving singularly perturbed problems than the method of matched asymptotic expansions. We illustrate this approach with 10 singularly perturbed ordinary and partial differential equations.

We consider a class of operators that contains the strictly singular operators and it is contained in the perturbation class of the upper semi-Fredholm operators PΦ+. We show that this class is strictly contained in PΦ+, solving a question of Friedman. We obtain similar results for the strictly cosingular operators and the perturbation class of the lower semi-Fredholm operators PΦ−. We also characterize in terms of PΦ+ and in terms of PΦ−. As a consequence, we show that and are the biggest operator ideals contained in PΦ+ and PΦ−, respectively.

We define left and right kernels of representations of Hopf algebras. In the case of group algebras, left and right kernels coincide and they are the usual kernels of modules. In the general case, we show that these kernels coincide with the categorical left and right Hopf kernels of morphisms of Hopf algebras defined in Andruskiewitsch and Devoto [Extensions of Hopf algebras, Algebra i Analiz7 (1995), 22–69]. Brauer's theorem for kernels over group algebras is generalised to Hopf algebras.

To any walk in a quiver, we associate a Laurent polynomial. When the walk is the string of a string module over a 2-Calabi–Yau tilted algebra, we prove that this Laurent polynomial coincides with the corresponding cluster character of the string module up to an explicit normalising monomial factor.

We continue our investigation of the general notion of universal enveloping algebra introduced in [A. Ardizzoni, A Milnor–Moore type theorem for primitively generated braided Bialgebras, J. Algebra 327(1) (2011), 337–365]. Namely, we study a universal enveloping algebra when it is of Poincaré–Birkhoff–Witt (PBW) type, meaning that a suitable PBW-type theorem holds. We discuss the problem of finding a basis for a universal enveloping algebra of PBW type: as an application, we recover the PBW basis both of an ordinary universal enveloping algebra and of a restricted enveloping algebra. We prove that a universal enveloping algebra is of PBW type if and only if it is cosymmetric. We characterise braided bialgebra liftings of Nichols algebras as universal enveloping algebras of PBW type.