Let G be a finite nilpotent group, χ and ψ be irreducible complex characters of G with prime degree. Assume that χ(1) = p. Then, either the product χψ is a multiple of an irreducible character or χψ is the linear combination of at least distinct irreducible characters.

In an earlier paper, for ‘large’ (but otherwise unspecified) subsets , , , of q, Sárközy showed the solvability of the equations a + b = cd with a ∈ , b ∈ , c ∈ , d ∈ . This equation has been studied recently by many other authors. In this paper, we study the solvability of systems of equations of this type using additive character sums.

Regular maps, that is, graph embeddings with the ‘highest level’ of orientation-preserving symmetry, can be identified with two-generator presentations of groups G of the form 〈x, y; xm = y2 = (xy)n = … = 1〉; the parameters m and n are the valence and the covalence of the map, respectively. The element j ∈ Zm* is an exponent of such a map if the assignments x ↦ xj and y ↦ y extend to an automorphism of G. The element −1 is an exponent if and only if the map is reflexible, that is, isomorphic to its mirror image. Non-trivial exponents, j ≠ ±1, induce automorphisms of the underlying graph, which are not map automorphisms but can nevertheless be considered as ‘external symmetries’ of the map. In this paper we show with the help of residual finiteness of triangle groups that for any given m, n≥3 such that 1/m + 1/n ≤ 1/2, there exist infinitely many finite, reflexible and regular maps of valence m and covalence n with no non-trivial exponent.

This paper is inspired by a counter example of J. Kurzweil published in [5], whose intention was to demonstrate that a certain property of linear operators on finite-dimensional spaces need not be preserved in infinite dimension. We obtain a stronger result, which says that no infinite-dimensional Banach space can have the given property. Along the way, we will also derive an interesting proposition related to Dvoretzky's theorem.

In this paper, we establish an extended Loomis–Whitney inequality for positive double John bases, which generalises Ball's result [1]. Moreover, a different extension of the Loomis–Whitney inequality is deduced.

We introduce the principal matrix solution Z(t, s) of the linear Volterra-type vector integro-dynamic equation and prove that it is the unique matrix solution of We also show that the solution of is unique and given by the variation of parameters formula

In this paper, we deal with a tree-shaped network of strings with a fixed root node. By imposing velocity feedback controllers on all vertices except the root node, we show that the spectrum of the system operator consists of all isolated eigenvalues of finite multiplicity and is distributed in a strip parallel to the imaginary axis under certain conditions. Moreover, we prove that there exists a sequence of eigenvectors and generalised eigenvectors that forms a Riesz basis with parentheses, and that the imaginary axis is not an asymptote of the spectrum. Thereby, we deduce that the system is exponentially stable.

We obtain sharp upper and lower bounds on a certain four-dimensional Frobenius number determined by a prime pair (p, q), 2 < p < q, including exact formulae for two infinite subclasses of such pairs. Our work is motivated by the study of compact Riemann surfaces which can be realised as semi-regular pq-fold coverings of surfaces of lower genus. In this context, the Frobenius number is (up to an additive translation) the largest genus in which no surface is such a covering. In many cases it is also the largest genus in which no surface admits an automorphism of order pq. The general t-dimensional Frobenius problem (t ≥ 3) is NP-hard, and it may be that our restricted problem retains this property.

Let L be a convex body in n and z an interior point of L. We associate with L and z a new, convex and centrally symmetric, body CI(L, z). This generalizes the classical intersection bodyI(L, z) (whose radial function at u ∈ Sn−1 is the volume of the hyperplane section of L through z, orthogonal to u). CI(L, z) coincides with I(L, z) if and only if L is centrally symmetric about z. We study the properties of CI(L, z).

In this paper, by using the fixed point index method first we obtain some existence and multiplicity results for sign-changing solutions of an (e1, B)-limit increasing operator equation. The main results can be applied to many non-linear boundary value problems to obtain the existence and multiplicity results for sign-changing solutions. We also give a clear description of locations of these sign-changing solutions through strict lower and upper solutions. As an example, in the last section we obtain some existence and multiplicity results for sign-changing solutions of some Sturm–Liouville differential boundary value problems.

In this paper we prove the existence of closed geodesics in the leaf space of some classes of singular Riemannian foliations (s.r.f.), namely s.r.fs. that admit sections or have no horizontal conjugate points. We also investigate the shortening process with respect to Riemannian foliations.

This paper deals with some non-linear initial-boundary value problems under homogeneous Neumann boundary conditions, in which the solutions may blow up in finite time. Using a first-order differential inequality technique, lower bounds for blow-up time are determined.

Let H ≤ K be subgroups of a group G. We say that H is strongly closed in K with respect to G if whenever ag ∈ K, where a ∈ H, g ∈ G, then ag ∈ H. In this paper, we investigate the structure of a group G under the assumption that every subgroup of order 2m (and 4 if m = 1) of a 2-Sylow subgroup S of G is strongly closed in S with respect to G. Some results related to 2-nilpotence and supersolvability of a group G are obtained. This is a complement to Guo and Wei (J. Group Theory13(2) (2010), 267–276).

In this paper we study non-complemented spaces of operators and the embeddability of ℓ∞ in the spaces of operators L(X, Y), K(X, Y) and Kw*(X*, Y). Results of Bator and Lewis [2, 3] (Bull. Pol. Acad. Sci. Math.50(4) (2002), 413–416; Bull. Pol. Acad. Sci. Math.549(1) (2006), 63–73), Emmanuele [8–10] (J. Funct. Anal.99 (1991), 125–130; Math. Proc. Camb. Phil. Soc.111 (1992), 331–335; Atti. Sem. Mat. Fis. Univ. Modena42(1) (1994), 123–133), Feder [11] (Canad. Math. Bull.25 (1982), 78–81) and Kalton [16] (Math. Ann.208 (1974), 267–278), are generalised. A vector measure result is used to study the complementation of the spaces W(X, Y) and K(X, Y) in the space L(X, Y), as well as the complementation of K(X, Y) in W(X, Y). A fundamental result of Drewnowski [7] (Math. Proc. Camb. Phil. Soc. 108 (1990), 523–526) is used to establish a result for operator-valued measures, from which we obtain as corollaries the Vitali–Hahn–Saks–Nikodym theorem, the Nikodym Boundedness theorem and a Banach space version of the Phillips Lemma.

For a subfunction u, associated with the stationary Schrödinger operator, which is dominated on the boundary by a certain function on a cone, we generalise the classical Phragmén-Lindelöf theorem by making an a-harmonic majorant of u.

We first study the regularised version of a modified two-component Camassa–Holm shallow water system and obtain the energy estimates of the corresponding approximate solutions. Then, we present a sufficient condition which guarantees that these approximate solutions converge to a low regularity weak solution of the modified two-component Camassa–Holm shallow water system.

For each k ≥ 2, we determine the asymptotic behaviour of the sequence of cardinalities of finitely generated free objects in , the variety consisting of all k-testable semigroups.

We give a simple proof of a statement extending Fu's (J.H.G. Fu, Erratum to ‘some remarks on legendrian rectiable currents’, Manuscripta Math. 113(3) (2004), 397–401) result: ‘If Ω is a set of locally finite perimeter in ℝ2, then there is no function f ∈ C1(ℝ2) such that ∇f(x1, x2) = (x2, 0) at a.e. (x1, x2) ∈ Ω’. We also prove that every measurable set can be approximated arbitrarily closely in L1 by subsets that do not contain enhanced density points. Finally, we provide a new proof of a Poincaré-type lemma for locally finite perimeter sets, which was first stated by Delladio (S. Delladio, Functions of class C1 subject to a Legendre condition in an enhanced density set, to appear in Rev. Mat. Iberoamericana).

For a group H and a subset X of H, we let HX denote the set {hxh−1 | h ∈ H, x ∈ X}, and when X is a free-generating set of H, we say that the set HX is a Whitehead subset of H. For a group F and an element r of F, we say that r is Cohen–Lyndon aspherical in F if F{r} is a Whitehead subset of the subgroup of F that is generated by F{r}. In 1963, Cohen and Lyndon (D. E. Cohen and R. C. Lyndon, Free bases for normal subgroups of free groups, Trans. Amer. Math. Soc. 108 (1963), 526–537) independently showed that in each free group each non-trivial element is Cohen–Lyndon aspherical. Their proof used the celebrated induction method devised by Magnus in 1930 to study one-relator groups. In 1987, Edjvet and Howie (M. Edjvet and J. Howie, A Cohen–Lyndon theorem for free products of locally indicable groups, J. Pure Appl. Algebra45 (1987), 41–44) showed that if A and B are locally indicable groups, then each cyclically reduced element of A*B that does not lie in A ∪ B is Cohen–Lyndon aspherical in A*B. Their proof used the original Cohen–Lyndon theorem. Using Bass–Serre theory, the original Cohen–Lyndon theorem and the Edjvet–Howie theorem, one can deduce the local-indicability Cohen–Lyndon theorem: if F is a locally indicable group and T is an F-tree with trivial edge stabilisers, then each element of F that fixes no vertex of T is Cohen–Lyndon aspherical in F. Conversely, by Bass–Serre theory, the original Cohen–Lyndon theorem and the Edjvet–Howie theorem are immediate consequences of the local-indicability Cohen–Lyndon theorem. In this paper we give a detailed review of a Bass–Serre theoretical form of Howie induction and arrange the arguments of Edjvet and Howie into a Howie-inductive proof of the local-indicability Cohen–Lyndon theorem that uses neither Magnus induction nor the original Cohen–Lyndon theorem. We conclude with a review of some standard applications of Cohen–Lyndon asphericity.

We will state some conditions under which if a quotient of a module M satisfies the radical formula of degree k (s.t.r.f of degree k), so does M. Especially, we will introduce some particular modules M′ such that M′ ⊕ M″ s.t.r.f of degree k, when M″ s.t.r.f of degree k. Furthermore, we will show that, under certain conditions, if the completion of a module M s.t.r.f of degree k, then there is a non-negative integer k′ such that M s.t.r.f. of degree k′. Moreover, we state a corrected version of Leung and Man's theorem (K. H. Leung and S. H. Man, On commutative Noetherian rings which satisfy the radical formula, Glasgow Math. J. 39 (1997), 285–293) on Noetherian rings that satisfies the radical formula.