In this note, we give a new proof of the existence of an open book decomposition for a closed non-orientable 3–manifold. This open book decomposition is analogous to a planar open book decomposition for a closed orientable 3–manifold. More precisely, in this note, we give an open book decomposition of a given closed non-orientable 3–manifold with the pages punctured Möbius bands. We also give an algorithm to determine the monodromy of this open book.

In this paper, by using the Leggett–Williams fixed point theorem, the existence of three positive periodic solutions for differential equations with piecewise constant argument and impulse on time scales is investigated. Some easily verifiable sufficient criteria are established. Finally, an example is given to illustrate the results.

Let Ω be a bounded domain with smooth boundary in ℝ2, q∈[1,2) and x1, x2,. . .,xm ∈ Ω. In this paper we are concerned with the following type of problem:

\[ -\Delta u-\lambda|\nabla u|^q = \rho^{2}e^{u}, \]

The main result of this paper is the following theorem. Let q be a prime and A be an elementary abelian group of order q3. Suppose that A acts as a coprime group of automorphisms on a profinite group G in such a manner that CG(a)′ is periodic for each a ∈ A#. Then G′ is locally finite.

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.

In this paper, we characterize surjective isometries on certain classes of noncommutative spaces associated with semi-finite von Neumann algebras: the Lorentz spaces $L^{w,1}$ , as well as the spaces $L^1+L^\infty$ and $L^1\cap L^\infty$ . The technique used in all three cases relies on characterizations of the extreme points of the unit balls of these spaces. Of particular interest is that the representations of isometries obtained in this paper are global representations.

In this paper, we define a Cayley–Dickson process for k-coalgebras proving some results that describe the properties of the final coalgebra, knowing the properties of the initial one. Namely, after applying the Cayley–Dickson process for k-coalgebras to a coassociative coalgebra, we obtain a coalternative one. Moreover, the first coalgebra is cocommutative if and only if the final coalgebra is coassociative. Finally we extend these results to a more general approach of D-coalgebras, where D is a k-coalgebra, presenting a class of examples of coalternative (non-coassociative) coalgebras obtained from group D-coalgebras.

There does not exist an algorithm that can determine whether or not a group presented by commutators is a right-angled Artin group.

We consider in this paper the problem (1) where Ω is the unit ball in ℝN centred at the origin, 0 ≤ α < pN,β > 0, N ≥ 3. Suppose qϵ → q as ϵ → 0+ and qϵ, q satisfy, respectively, we investigate the asymptotic estimates of the ground-state solutions (uϵ, vϵ) of (1) as β → + ∞ with p, qϵ fixed. We also show the symmetry-breaking phenomenon with α, β fixed and qϵ → q as ϵ → 0+. In addition, the ground-state solution is non-radial provided that ϵ > 0 is small or β is large enough.

We continue our investigation of the real space H of Hermitian matrices in $${M_n}(\mathbb{C})$$ with respect to norms on $${\mathbb{C}^n}$$. We complete the commutative case by showing that any proper real subspace of the real diagonal matrices on $${\mathbb{C}^n}$$ can appear as H. For the non-commutative case, we give a complete solution when n=3 and we provide various illustrative examples for n ≥ 4. We end with a short list of problems.

We show that over every countable algebraically closed field $\mathbb{K}$ there exists a finitely generated $\mathbb{K}$-algebra that is Jacobson radical, infinite-dimensional, generated by two elements, graded and has quadratic growth. We also propose a way of constructing examples of algebras with quadratic growth that satisfy special types of relations.

Let E be a Hausdorff locally convex space with continuous dual E1 and let M be a subspace of the algebraic dual E* such that M ∩ E1 = {0} and dim M = ℵ0. In the terminology of [4] the Mackey topology τ(E, E1 + M) is called a countable enlargement of τ(E, E1). There has been some interest in the question of when barrelledness is preserved under countable enlargements (see [4], [5], [6], [8], [9]). In this note we are concerned with the preservation of the quasidistinguished property for normed spaces under countable enlargements; this was posed as on open question by B. Tsirulnikov in [7]. According to [7] a Hausdorff locally convex space E is quasidistinguished if every bounded subset of its completion Ê is contained in the completion of a bounded subset of Ê (equivalently, in the closure in Ê of a bounded subset of E). Any normed space is clearly quasidistinguished and remains so under a finite enlargement (dim M < χ0) since the enlarged topology is normable. (See the Main Theorem of [7] for a general result on the preservation of the quasidistinguished property under finite enlargements.) We shall write QDCE for a countable enlargement which preserves the quasidistinguished property.

Throughout this paper all rings are assumed commutative with identity. Among integral domains, Dedekind domains are characterized by the property that every ideal is a product of prime ideals. For a history and proof of this result the reader is referred to Cohen [2, pp. 31–32]. More generally, Mori [5] has shown that a ring has the property that every ideal is a product of prime ideals if and only if it is a finite direct product of Dedekind domains and special principal ideal rings (SPIRS). Rings with this property are called general Z.P.I.-rings.

Let K be a field of characteristic different from 2, and a, b quadratically independent elements of K. Put J= K(√a, √b). In [4], Jensen and Yui discuss the question of quaternionic (Q8) extensions of J, and give a survey of known results. In [8], Ware discusses (among other things) some general conditions for, and relations between, the existence of Q8 and D4 (dihedral) extensions of K. A general theorem of Witt [9] says that J will have a quaternionic extension J(√u) if and only if there exists a 3 × 3 matrix P over K such that PPt = diag(a, b, 1/ab), and an appropriate value for u is given in terms of the entries of P. The problem of actually finding P in a particular case is not trivial.

The norm K(G) of a group G is the subgroup of elements of G which normalize every subgroup of G. Under the name kern this subgroup was introduced by Baer [1]. The norm is Dedekindian in the sense that all its subgroups are normal. A theorem of Dedekind [5] describes the structure of such groups completely: if not abelian they are the direct product of a quaternion group of order eight and an abelian group with no element of order four. Baer [2] proves that a 2-group with non-abelian norm is equal to its norm.

Birkhoff and Pierce [2] introduced the class f-rings—those lattice-ordered rings R which satisfy the additional condition that if a, b, and c are positive elements of R and if a ∧ b = 0, then ac ∧ b = 0 = ca ∧ b. They showed that f-rings may be characterized as lattice-ordered rings which are subdirect products of totally-ordered rings.

1. Every family of subnormal operators in a Hilbert space fulfils the Halmos-Bram condition on a suitable dense subset of its domain [2], [3]. In [2] and [4] it is shown that the generalized commutation relation implies the Halmos-Bram condition for one operator. In this paper it is proved that the generalized commutation relation implies the Halmos-Bram condition for infinite families of operators (in a special case Jorgensen proved it in a different way for finite families of operators, see [2]) and as an example of the application of this property it is shown that every family of generalized creation operators in the Bargmann space of an infinite order, indexed by mutually orthogonal vectors from I2 is subnormal. See [1] for the definitions.

Let F be an orientable surface with or without boundary and let M(F) be the mapping class group of F, i.e. the group of isotopy classes of orientation preserving diffeomorphisms of F. To each essential simple closed curve c on F we can associate an element C of M(F) called the Dehn twist about c. We refer the reader to [1] for definitions. It is well known (see [1]) that, at least in the case where F has no more than one boundary component, M(F) is generated by Dehn twists. Further, there are important subgroups of M(F) which are also generated by Dehn twists or simple products of Dehn twists; for example the Torelli group, the kernel of the homology action map M(F)→ Aut(H1(F;Z)) = Sp(H1(F;Z)), where Sp(H1(F;Z)) denotes the symplectic group, is known to be generated by Dehn twists about bounding curves and by “bounding pairs”. See [8] for proofs and definitions. Also Dehn twists crop up as geometric monodromy maps associated to Picard–Lefschetz vanishing cycles for plane curve singularities (see [5]).

In this note we show that any derivation of a simple C*-algebra, whose range is not dense, is closable. We also derive a necessary and sufficient condition for a *-derivation of a C*-algebra, which is defined on the domain of a closed *- derivation, to be closed.

On the setting of the unit ball of the complex n-space, some characterizations of M-harmonic Bloch functions are obtained. As an application, Carleson measures are characterized by means of Berezin type integrals of M-harmonic Bloch functions. As one may expect, these results carry over to M-harmonic little Bloch functions and vanishing Carleson measures.