The annihilator of a finite generated β-critical module is called a β-coprimative ideal. A prime ideal P is called β-prime if the Krull dimension of R/P is β. This paper is concerned with the relationship between the set of β-prime ideals and the set of minimal β-coprimitive ideals over a strongly right FBN ring. it is shown that there exists a one-to-one correspondence between the set of β-prime ideals and the set on minimal β-coprimitive ideals over a strongly right FBN ring R for −1 < β ≤ α, where α is the Krull dimension of R.

We show that the automorphism groups of right-angled Artin groups whose defining graphs have at least three vertices are not relatively hyperbolic. We then show that the outer automorphism groups are also not relatively hyperbolic, except for a few exceptional cases. In these cases, the outer automorphism groups are virtually isomorphic to either a finite group, an infinite cyclic group or $\mathrm {GL}_2(\mathbb {Z})$.

We study the relationship between generalisations of P-spaces and Volterra (weakly Volterra) spaces, that is, spaces where every two dense Gδ have dense (nonempty) intersection. In particular, we prove that every dense and every open, but not every closed subspace of an almost P-space is Volterra and that there are Tychonoff nonweakly Volterra weak P-spaces. These results should be compared with the fact that every P-space is hereditarily Volterra. As a byproduct we obtain an example of a hereditarily Volterra space and a hereditarily Baire space whose product is not weakly Volterra. We also show an example of a Hausdorff space which contains a nonweakly Volterra subspace and is both a weak P-space and an almost P-space.

Let $G$ be a finite group. We denote by ${\it\nu}(G)$ the probability that two randomly chosen elements of $G$ generate a nilpotent subgroup and by $\text{Nil}_{G}(x)$ the set of elements $y\in G$ such that $\langle x,y\rangle$ is a nilpotent subgroup. A group $G$ is called an ${\mathcal{N}}$-group if $\text{Nil}_{G}(x)$ is a subgroup of $G$ for all $x\in G$. We prove that if $G$ is an ${\mathcal{N}}$-group with ${\it\nu}(G)>\frac{1}{12}$, then $G$ is soluble. Also, we classify semisimple ${\mathcal{N}}$-groups with ${\it\nu}(G)=\frac{1}{12}$.

We give a sufficient and necessary condition for a Praeger–Xu graph to be a Cayley graph.

In this note we give details of a method by which we can produce an index-0 graph from any unstable graph and use it to show that given any finite group there exists an index-0 graph whose automorphism group is isomorphic, as an abstract group, to the given group. We proceed to construct two infinite families of connected index-0 graphs with connected complements whose automorphism group contains a transposition. This enables us to produce, for any finite group G, an index-0 graph whose automorphism group, isomorphic as an abstract group to C2 × G, contains a transposition.

In this paper, we introduce the notion of weakly weighted sharing of zeros of meromorphic functions ignoring multiplicities, which extends the notion of weakly weighted sharing counting multiplicities, and we also introduce the notion of multiplicity. By using these notions, we prove some results on the uniqueness of meromorphic functions concerning differential polynomials sharing nonzero finite values. The results in this paper extend the results of Yang and Hua, Fang, and Dyavanal. In this paper, we correct defective points in the paper of Wu et al. [‘Uniqueness of meromorphic functions sharing one value’, Bull. Aust. Math. Soc. 85(2012), 280–294].

Let K be a planar, compact, convex set with circumradius R, diameter d, width w and inradius r, and containing no points of the integer lattice. We generalise inequalities concerning the ‘dual’ quantities (2R − d) and (w − 2r) to rectangular lattices. We then use these results to obtain corresponding inequalities for a planar convex set with two interior lattice points. Finally, we conjecture corresponding results for sets containing one interior lattice point.

The proliferation of classifying spaces in recent years owes much to the theorem of Edgar H. Brown, Jr on the representability of homotopy functors. Since the theorem only gives a representation for functors defined on the category of spaces having the homotopy type of a CW complex, there is some interest in finding conditions under which the domain category may be enlarged. It appears that a version of the theorem holds for any small full subcategory of Htp, the category of topological spaces and homotopy classes of continuous maps, but that the resulting classifying space is generally intractable.

Let X be a compactification of the ray with the arc as remainder. The following characterisation of the open images of X is obtained: Let h: X → Y be an open onto map. If Y is not homeomorphic to [0, 1] or the one-point space, then h is a homeomorphism. In 1977 open images of the usual sin (1/x) continuum were characterised by Professor Sam B. Nadler.

We improve the intuition and some ideas of G. D. Faulkner and M. C. Vipera presented in the article “Remainders of compactifications and their relation to a quotient lattice of the topology” [Proc. Amer. Math. Soc. 122 (1994)] in connection with the question about internal conditions for locally compact spaces X and Y under which βX/X ≅ βY/Y or, more generally, under which the remainders of compactifications of X belong to the collection of the remainders of compactifications of Y. We point out the reason why the quotient lattice of the topology considered by Faulkner and Vipera cannot lead to a satisfactory answer to the above question. We replace their lattice by the qoutient lattice of a new equivalence relation on a Wallman base in order to describe a method of constructing a Wallman-type compactification which allows us to deduce more complete solutions to the problems investigated by Faulkner and Vipera.

We provide a geometric interpretation of the well-known A3w condition for regularity in optimal transport.

We study an operation, that we call lifting, creating nonisomorphic monomial curves from a single monomial curve. Our main result says that all but finitely many liftings of a monomial curve have Cohen–Macaulay tangent cones even if the tangent cone of the original curve is not Cohen–Macaulay. This implies that the Betti sequence of the tangent cone is eventually constant under this operation. Moreover, all liftings have Cohen–Macaulay tangent cones when the original monomial curve has a Cohen–Macaulay tangent cone. In this case, all the Betti sequences are just the Betti sequence of the original curve.

Let K1 and K2 be extension fields over a field K with char K = p > 0. Assume L = K1(x1) = K2(x2) ⊃ K where xi is transcendental over Ki, for i = 1, 2. In this paper we prove that if K1 is a perfect field, then K1 = K2.

In this paper, existence criteria for multiple solutions of periodic boundary value problems for the first-order difference equation are established by using the Leggett–Williams multiple fixed point theorem and fixed point theorem of cone expansion and compression. Two examples are also given to illustrate the main results.

For $n\in \mathbb{Z}$ and $A\subseteq \mathbb{Z}$, define $r_{A}(n)$ and ${\it\delta}_{A}(n)$ by $r_{A}(n)=\#\{(a_{1},a_{2})\in A^{2}:n=a_{1}+a_{2},a_{1}\leq a_{2}\}$ and ${\it\delta}_{A}(n)=\#\{(a_{1},a_{2})\in A^{2}:n=a_{1}-a_{2}\}$. We call $A$ a unique representation bi-basis if $r_{A}(n)=1$ for all $n\in \mathbb{Z}$ and ${\it\delta}_{A}(n)=1$ for all $n\in \mathbb{Z}\setminus \{0\}$. In this paper, we prove that there exists a unique representation bi-basis $A$ such that $\limsup _{x\rightarrow \infty }A(-x,x)/\sqrt{x}\geq 1/\sqrt{2}$.

A graph is called arc-regular if its full automorphism group acts regularly on its arc set. In this paper, we completely determine all the arc-regular Frobenius metacirculants of prime valency.