The asymptotic stability of two types of invariant solutions under a curvature flow in the whole plane is studied. First, by extending the work of others, we prove that the stationary line with nonzero slope will attract the graphical curves which surround it. Then a similar property is obtained for the grim reaper.

A new sharp general Ostrowski type inequality in L∞ norm is established. Some special cases are discussed.

The main result shows that a small perturbation of a univalent function is again a univalent function, hence a univalent function has a neighbourhood consisting entirely of univalent functions. For the particular choice of a linear function in the hypothesis of the main theorem, we obtain a corollary which is equivalent to the classical Noshiro–Warschawski–Wolff univalence criterion. We also present an application of the main result in terms of Taylor series, and we show that the hypothesis of our main result is sharp.

We extend some results known for FC-groups to the class FC* of generalized FC-groups introduced in de Giovanni et al. [‘Groups with restricted conjugacy classes’, Serdica Math. J.28(3) (2002), 241–254]. The main theorems pertain to the join of pronormal subgroups. The relevant role that the Wielandt subgroup plays in an FC*-group is pointed out.

We define and study λ-strict ideals in Banach spaces, which for λ=1 means strict ideals. Strict u-ideals in their biduals are known to have the unique ideal property; we prove that so also do λ-strict u-ideals in their biduals, at least for λ>1/2. An open question, posed by Godefroy et al. [‘Unconditional ideals in Banach spaces’, Studia Math.104 (1993), 13–59] is whether the Banach space X is a u-ideal in Ba(X), the Baire-one functions in X**, exactly when κu(X)=1; we prove that if κu(X)=1 then X is a strict u-ideal in Ba (X) , and we establish the converse in the separable case.

Let Σ be a set and σ be a positive function on Σ. We introduce and study a locally convex topology β1(Σ,σ) on the space ℓ1(Σ,σ) such that the strong dual of (ℓ1(Σ,σ),β1(Σ,σ)) can be identified with the Banach space . We also show that, except for the case where Σ is finite, there are infinitely many such locally convex topologies on ℓ1(Σ,σ). Finally, we investigate some other properties of the locally convex space (ℓ1(Σ,σ),β1(Σ,σ)) , and as an application, we answer partially a question raised by A. I. Singh [‘L∞0(G)* as the second dual of the group algebra L1 (G) with a locally convex topology’, Michigan Math. J.46 (1999), 143–150].

Let R be the ring of linear transformations of a right vector space over a division ring D. Three results are proved: (1) if |D|>4, then for any a∈R there exists a unit u of R such that a+u,a−u and a−u−1 are units of R; (2) if |D|>3 , then for any a∈R there exists a unit u of R such that both a+u and a−u−1 are units of R; (3) if |D|>2 , then for any a∈R there exists a unit u of R such that both a−u and a−u−1 are units of R. The second result extends the main result in H. Chen, [‘Decompositions of countable linear transformations’, Glasg. Math. J. (2010), doi:10.1017/S0017089510000121] and the third gives an affirmative answer to the question raised in the same paper.

In this paper we investigate the analogue of the classical badly approximable setup in which the distance to the nearest integer ‖⋅‖ is replaced by the sup norm |⋅|. In the case of one linear form we prove that the hybrid badly approximable set is of full Hausdorff dimension.

In this paper, we classify the finite groups belonging to the class of cyclic-transitive groups. A group G is said to be cyclic-transitive if the following condition holds: if x, y, z are nonidentity elements of G such that 〈x,y〉 and 〈y,z〉 are both cyclic, then 〈x,z〉 is also cyclic.

The partition monoid is a salient natural example of a *-regular semigroup. We find a Galois connection between elements of the partition monoid and binary relations, and use it to show that the partition monoid contains copies of the semigroup of transformations and the symmetric and dual-symmetric inverse semigroups on the underlying set. We characterize the divisibility preorders and the natural order on the (straight) partition monoid, using certain graphical structures associated with each element. This gives a simpler characterization of Green’s relations. We also derive a new interpretation of the natural order on the transformation semigroup. The results are also used to describe the ideal lattices of the straight and twisted partition monoids and the Brauer monoid.

Let Y be a fixed nonempty subset of a set X and let T(X,Y ) denote the semigroup of all total transformations from X into Y. In 1975, Symons described the automorphisms of T(X,Y ). Three decades later, Nenthein, Youngkhong and Kemprasit determined its regular elements, and more recently Sanwong, Singha and Sullivan characterized all maximal and minimal congruences on T(X,Y ). In 2008, Sanwong and Sommanee determined the largest regular subsemigroup of T(X,Y ) when |Y |≠1 and Y ≠ X; and using this, they described the Green’s relations on T(X,Y ) . Here, we use their work to describe the ideal structure of T(X,Y ) . We also correct the proof of the corresponding result for a linear analogue of T(X,Y ) .

We prove that the modified Benjamin–Ono–Burgers equation is globally well-posed in Hs for s>0. Moreover, we show that the solution of the modified Benjamin–Ono–Burgers equation converges to that of the modified Benjamin–Ono equation in the natural space C([0,T];Hs), s≥1/2, as the dissipative coefficient ϵ goes to zero, provided that the L2 norm of the initial data is sufficiently small.

Let X be a topological space. A family ℬ of nonempty open sets in X is called a π-base of X if for each open set U in X there exists B∈ℬ such that B⊂U. The order of a π-base ℬ at a point x is the cardinality of the family ℬx ={B∈ℬ:x∈B} and the order of the π-base ℬ is the supremum of the orders of ℬ at each point x∈X. A classical theorem of Shapirovskiĭ [‘Special types of embeddings in Tychonoff cubes’, in: Subspaces of Σ-Products and Cardinal Invariants, Topology, Coll. Math. Soc. J. Bolyai, 23 (North-Holland, Amsterdam, 1980), pp. 1055–1086; ‘Cardinal invariants in compact Hausdorff spaces’, Amer. Math. Soc. Transl.134 (1987), 93–118] establishes that the minimum order of a π-base is bounded by the tightness of the space when the space is compact. Since then, there have been many attempts at improving the result. Finally, in [‘The projective π-character bounds the order of a π-base’, Proc. Amer. Math. Soc.136 (2008), 2979–2984], Juhász and Szentmiklóssy proved that the minimum order of a π-base is bounded by the ‘projective π-character’ of the space for any topological space (not only for compact spaces), improving Shapirovskiĭ’s theorem. The projective π-character is in some sense an ‘external’ cardinal function. Our purpose in this paper is, on the one hand, to give bounds of the projective π-character using ‘internal’ topological properties of the subspaces on compact spaces. On the other hand, we give a bound on the minimum order of a π-base using other cardinal functions in the frame of general topological spaces. Open questions are posed.

The main purpose of this paper is to give a new, elementary proof of Flanigan’s theorem, which says that a given ring A has a maximal essential extension ME(A) if and only if the two-sided annihilator of A is zero. Moreover, we discuss the problem of description of ME(A) for a given right ideal A of a ring with an identity.

Real ideals of the ring ℜL of real-valued continuous functions on a completely regular frame L are characterized in terms of cozero elements, in the manner of the classical case of the rings C(X). As an application, we show that L is realcompact if and only if every free maximal ideal of ℜL is hyper-real—which is the precise translation of how Hewitt defined realcompact spaces, albeit under a different appellation. We also obtain a frame version of Mrówka’s theorem that characterizes realcompact spaces.