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In this short paper, we characterise graphs of order $pq$ with $p, q$ prime which are self-complementary and vertex-transitive.

Let $\{U_n\}$ be given by $U_0=1$ and $U_n=-2\sum _{k=1}^{[n/2]} \binom n{2k}U_{n-2k}\ (n\ge 1)$, where $[\cdot ]$ is the greatest integer function. Then $\{U_n\}$ is analogous to the Euler numbers and $U_{2n}=3^{2n}E_{2n}(\frac 13)$, where $E_m(x)$ is the Euler polynomial. In a previous paper we gave many properties of $\{U_n\}$. In this paper we present a summation formula and several congruences involving $\{U_n\}$.

The main aim of this paper is to establish conditions that are necessary and sufficient for the edge-connectivity of the Cartesian product of two graphs to equal the sum of the edge-connectivities of the factors. The paper also clarifies an issue that has arisen in the literature on Cartesian products of graphs.

We give an explicit Krasnoselski–Mann type method for finding common solutions of the following system of equilibrium and hierarchical fixed points: where C is a closed convex subset of a Hilbert space H, G:C×C→ℝ is an equilibrium function, T:C→C is a nonexpansive mapping with Fix(T) its set of fixed points and f:C→C is a ρ-contraction. Our algorithm is constructed and proved using the idea of the paper of [Y. Yao and Y.-C. Liou, ‘Weak and strong convergence of Krasnosel’skiĭ–Mann iteration for hierarchical fixed point problems’, Inverse Problems24 (2008), 501–508], in which only the variational inequality problem of finding hierarchically a fixed point of a nonexpansive mapping T with respect to a ρ-contraction f was considered. The paper follows the lines of research of corresponding results of Moudafi and Théra.

In a recent paper (which is to appear in J. Algebra Appl.) we proved that every element of a right self-injective ring R is a sum of two units if and only if R has no factor ring isomorphic to ℤ2 and hence the unit sum number of a nonzero right self-injective ring is 2, ω or ∞. In this paper we characterise right self-injective rings with unit sum numbers ω and ∞. We prove that the unit sum number of a right self-injective ring R is ω if and only if R has a factor ring isomorphic to ℤ2 but no factor ring isomorphic to ℤ2 × ℤ2, and also in this case every element of R is a sum of either two or three units. It follows that the unit sum number of a right self-injective ring R is ∞ precisely when R has a factor ring isomorphic to ℤ2 × ℤ2. We also answer a question of Henriksen (which appeared in J. Algebra, Question E, page 192), by giving a large class of regular right self-injective rings having the unit sum number ω in which not all non-invertible elements are the sum of two units.

In this paper we investigate the relationship between the existence of parallel semi-Riemannian metrics of a connection and the reducibility of the associated holonomy group. The question as to whether the holonomy group necessarily reduces in the presence of a specified number of independent parallel semi-Riemannian metrics is completely determined by the the signature of the metrics and the dimension d of the manifold, when d ≠ 4. In particular, the existence of two independent, parallel semi-Riemannian metrics, one of which having signature (p,q) with p ≠ q, implies the holonomy group is reducible. The (p,p) cases, however, may allow for more than one parallel metric and yet an irreducible holonomy group: for n = 2m, m ≥ 3, there exist connections on Rn with irreducible infinitesimal holonomy and which have two independent, parallel metrics of signature (m,m). The case of four-dimensional manifolds, however, depends on the topology of the manifold in question: the presence of three parallel metrics always implies reducibility but reducibility in the case of two metrics of signature (2,2) is guaranteed only for simply connected manifolds. The main theorem in the paper is the construction of a topologically non-trivial four-dimensional manifold with a connection that admits two independent metrics of signature (2,2) and yet has irreducible holonomy. We provide a complete solution to the general problem.

We provide a criterion for midpoint locally uniformly rotund renormability of normed spaces involving the class of -slicely continuous maps, recently introduced by Molto, Orihuela, Troyanski and Valdiva in 2003. As a consequence of this result, we obtain a theorem of G. Alexandrov concerning the three space problem for midpoint locally uniformly rotund renormings of Banach spaces. A normed space (X,k · k) (or its norm) is said to be midpoint locally uniformly rotund if for every x 2 X and every sequence (xn)n in X such that kxn + xk!kxk and kxn xk!kxk we have kxnk!0. Recall also that X is locally uniformly rotund if for every x 2 X and every sequence (xn)n X such that lim n kxnk = kxk and lim n kxn +xk = 2kxk we have lim n kxn xk = 0, and that X is strictly convex or rotund (R for short) if x = y whenever x and y are elements of X such that kxk = kyk = (x + y)/2 . It is clear that locally uniformly rotund ) midpoint locally uniformly rotund and that midpoint locally uniformly rotund ) R. In the paper [5], devoted to the renorming of spaces of continuous functions on trees, R. Haydon provides the first example (the only known to date) of midpoint locally uniformly rotund space with no equivalent locally uniformly rotund renorming. There, he also shows that for every tree , the existence of an equivalent strictly convex norm on C() implies midpoint locally uniformly rotund renormability on this space. This coincidence is not true in general: an example of strictly convexifiable space without midpoint locally uniformly rotund renorming is ‘1 (see [2, 6]). Our aim in this paper is to provide a criterion for midpoint locally uniformly rotund renorming of spaces that have images in midpoint locally uniformly rotund spaces through special non linear maps. These are the -slicely continuous maps recently introduced in [11], where a non linear transfer technique for locally uniformly rotund renormability has been developed.

Let S be a semigroup contained in a locally compact Abelian group G. Let Ĝ denote the Bohr compactification of G. We say that a sequence contained in S is Hartman uniform distributed on G iffor any character χ in Ĝ. Suppose that (Tg)g∈s is a semigroup of measurable measure preserving transformations of a probability space (X, β, μ) and B is an element of the σ-algebra β of positive μ measure. For a map T: X → X and a set A ⊆ X let T−1A denote {x ∈ X: Tx ∈ A}. In an earlier paper, the author showed that if k is Hartman uniform distributed thenIn this paper we show that ≥ cannot be replaced by =. A more detailed discussion of this situation ensues.

A number r > 0 is called a rendezvous number for a metric space (M, d) if for any n ∈ ℕ and any x1,…xn ∈ M, there exists x ∈ M such that . A rendezvous number for a normed space X is a rendezvous number for its unit sphere. A surprising theorem due to O. Gross states that every finite dimensional normed space has one and only one average number, denoted by r (X). In a recent paper, A. Hinrichs solves a conjecture raised by R. Wolf. He proves that for any n-dimensional real normed space. In this paper, we prove the analogous inequality in the complex case for n ≥ 3.

Let G be a compact Lie group. A G-cell of dimension n is a space of the form G/H x D, where H is a closed subgroup of G and D is an n-cell. A G-CW complex X (or an equivariant CW complex in the terminology of [9]) is constructed by iterated attaching of G-cells. It is the union of G-spaces X^> such that X^ is a disjoint union of G-cells of dimension 0, that is, orbits G/H, and X' + 1 ) is obtained from X n ) by attaching G-cells of dimension n + 1 along equivariant attaching maps G/H x dD -* X^K The space X^"\ which is called the n-skeleton of X, is thus the union of all G-cells of dimension at most n (the topological dimension of X^ is in general greater than n). For basic facts about G-complexes see the original papers [5] and [3] or the exposition in [9]. For discrete groups G it is well known that every G-CW complex is also a CW complex with a cellular action of G (this follows for example from [9, Proposition 1.16, p. 102]). For non-discrete groups, Illman [4] gave an example showing that a G-CW complex X does not always admit a CW decomposition, compatible with the given GCW decomposition, and proved that there always exists a homotopy equivalent CW complex Y which is finite if X is a finite G-complex. In this paper we consider the following problem. Given a G-CW complex X, does there exist a G-space Y, G-homotopy equivalent to X, with a CW decomposition such that the action p: G xY -> Y is a. cellular map with respect to some decomposition of G. The existence of such a Y is interesting from the point of view of equivariant homology and cohomology. For example, Greenlees and May showed that for some groups G the generalised Tate cohomology defined in [1] can be calculated from the CW decomposition

G is reduced torsion-free A belian group such that for every direct sum ⊕G of copies of G, Ext(⊕G, ⊕G) = 0 if and only if G is a free module over a rank 1 ring. For every direct product ΠG of copies of G, Ext(ΠG,ΠG) = 0 if and only if G is cotorsion.This paper began as a Research Report of the Department of Mathematics of the University of Western Australia in 1988, and circulated among members of the Abelian group community. However, it was never submitted for publication. The results have been cited, widely, and since copies of the original research report are no longer available, the paper is presented here in its original form in Sections 1 to 5. In Section 6, I survey the progress that has been made in the topic since 1988.

Matrices provide essential tools in many branches of mathematics, and matrix semi-groups have applications in various areas. In this paper we give a complete descriptionof all infinite matrix semigroups satisfying a certain combinatorial property definedin terms of power graphs.Research on combinatorial properties of words in groups originates from the fol-lowing well-known theorem due to Bernhard Neumann [12], which was obtained as ananswer to a question of Paul Erdos: a group is centre-by-finite if and only if every infinitesequence contains a pair of elements that commute. Combinatorial properties of groupsand semigroups with all infinite subsets containing certain special elements have beenconsidered by Bell, Blyth, Curzio, de Luca, Gillam, Hall, Higgins, Justin, Longobardi,Maj, Okninski, Piochi, Pirillo, Restivo, Reutenauer, Rhemtulla, Robinson, Sapir, Shumy-atsky, Simon, Varricchio and other authors, and a survey of this direction was given bythe first author in [7] (see also [2, 3, 6, 11]).The following combinatorial property was introduced in [9] using power graphs. Thepower graph Pow(S) of a semigroup S has all elements of S as vertices, and it has edges(u, v) for all u,v € S such that u ^ v and v is a power of u. Let D be a directed graph.We say that an infinite semigroup 5 is power D-saturated if and only if, for every infinitesubset T of 5, the power graph of S has a subgraph isomorphic to D with all verticesin T. In this paper we describe all pairs (£>, S), where D is a directed graph and S is amatrix semigroup, such that S is power D-saturated.The reader is referred to [1, 4, 14] for standard graph, semigroup and group theoreticterminology, respectively. By the word 'graph' we mean a directed graph without loopsor multiple edges. A graph is said to be acyclic if it has no directed cycles.We refer to [10, 13] for preliminaries on fields and matrix semigroups, respectively.For a skew field K, the set of all n x n matrices with entries in K i

A sign pattern matrix A is require said t propertyo P if every matrix in Q(A) hasproperty P, an allowd property P if there exists a matrix in Q{A) which has prop-erty P. There have been a number of papers [3, 4, 5, 6, 7] on sign pattern matrices.In mathematical economics, Quirk and Ruppert [6] studied the stability of sign pat-tern matrices and characterised sign pattern matrices that require negative real parteigenvalues. Maybee and Quirk [5] introduced graph-theoretic methods to solve quali-tative stability of linear systems. Eschenbach and Johnson [2] raised several questionsabout sign pattern matrices that require or allow certain distributions of eigenvalues.They characterised sign pattern matrices that require all real, all nonreal, and all pureimaginary eigenvalues [3], and also characterised sign patterns that require at least ifczero eigenvalues [4]. In [7], Yeh discussed sign pattern matrices that allow a nilpotentmatrix.In this paper, we first decompose the graph of a tree sign pattern matrix intostar tree components, and then we characterise tree sign pattern matrices that requireat least k zero eigenvalues and exactly k zero eigenvalues in terms of these star treecomponents.

In a recent paper, Fitzpatrick and Phelps introduced a new class of operators on a Banach space, the locally maximal monotone operators, and showed that this kind of operator can be approximated by a sequence of nicer maximal monotone operators. We give here an affirmative answer to a question posed in this paper: is the subdifferential of a proper convex lower semicontinuous function necessarily locally maximal monotone? Since a locally maximal operator is maximal monotone, our result represents a strengthening of Rockafellar's maximal monotonicity theorem.

In this paper a formula is obtained for the entries of the diagonal factor in the U D L factorisation of an invertible operator matrix in the case when its inverse has a chordal graph. As a consequence, in the finite dimensional case a determinant formula is obtained in terms of some key principal minors. After a cancellation process this formula leads to a determinant formula from an earlier paper by W.W. Barrett and C.R. Johnson, deriving in this way a different and shorter proof of their result. Finally, an algorithmic method of constructing minimal vertex separators of chordal graphs is presented.

The main purpose of the present paper is to introduce a class of "j-nonexpansive mappings" and to prove fixed point theorems for such mappings. They naturally arise in the existence theory of functional differential equations. It is well known that Edelstein [4] has been successful in replacing Banach's condition d(Tx, Ty) X). In the case when {X, \\-\\x) i a normed space possessing a normal structure, Browder [2], Kirk [7], Gohde [6] have proved the existence of a fixed point under nonexpansive condition ||Ts; — Tj/Hjt ^ \\x — y\\x . There are many papers dealing with these problems (see, [8, 9]). Other authors have generalised some results to the case of locally convex and uniform spaces [5, 13, 12]. Unfortunately, there are no applications of the mentioned fixed point theorems (see also [3]). That is why we shall consider a class of j-nonexpansive mappings in the spaces without specific geometric properties as, for instnace, a uniform convexity. Such assumptions restrict the class of functions in which we can find a solution, and hence the spaces L and L°° do not have normal structure. With a view to applications it is more useful to introduce supplementary conditions on T instead of requiring that the space X possesses certain geometric properties. Let (X, A) be a separated uniform space whose uniformity is generated by a saturated family of pseudometrics A = {da(x, y) : a 6 A}, A being an index set [14]. Let j : A —• A be a mapping of an index set into itself and j(a) = j{j~{^)) stands for the /cth iterate of j and j°(a) = a, a £ A. Let {$Q(

A graph $\Gamma $ is called $(G, s)$ -arc-transitive if $G \le \text{Aut} (\Gamma )$ is transitive on the set of vertices of $\Gamma $ and the set of s-arcs of $\Gamma $ , where for an integer $s \ge 1$ an s-arc of $\Gamma $ is a sequence of $s+1$ vertices $(v_0,v_1,\ldots ,v_s)$ of $\Gamma $ such that $v_{i-1}$ and $v_i$ are adjacent for $1 \le i \le s$ and $v_{i-1}\ne v_{i+1}$ for $1 \le i \le s-1$ . A graph $\Gamma $ is called 2-transitive if it is $(\text{Aut} (\Gamma ), 2)$ -arc-transitive but not $(\text{Aut} (\Gamma ), 3)$ -arc-transitive. A Cayley graph $\Gamma $ of a group G is called normal if G is normal in $\text{Aut} (\Gamma )$ and nonnormal otherwise. Fang et al. [‘On edge transitive Cayley graphs of valency four’, European J. Combin.25 (2004), 1103–1116] proved that if $\Gamma $ is a tetravalent 2-transitive Cayley graph of a finite simple group G, then either $\Gamma $ is normal or G is one of the groups $\text{PSL}_2(11)$ , ${\rm M} _{11}$ , $\text{M} _{23}$ and $A_{11}$ . However, it was unknown whether $\Gamma $ is normal when G is one of these four groups. We answer this question by proving that among these four groups only $\text{M} _{11}$ produces connected tetravalent 2-transitive nonnormal Cayley graphs. We prove further that there are exactly two such graphs which are nonisomorphic and both are determined in the paper. As a consequence, the automorphism group of any connected tetravalent 2-transitive Cayley graph of any finite simple group is determined.

A strongly concave composition of $n$ is an integer partition with strictly decreasing and then increasing parts. In this paper we give a uniform asymptotic formula for the rank statistic of a strongly concave composition introduced by Andrews et al. [‘Modularity of the concave composition generating function’, Algebra Number Theory7(9) (2013), 2103–2139].

For any finite abelian group$G$with$|G|=m$,$A\subseteq G$and$g\in G$, let$R_{A}(g)$be the number of solutions of the equation$g=a+b$,$a,b\in A$. Recently, Sándor and Yang [‘A lower bound of Ruzsa’s number related to the Erdős–Turán conjecture’, Preprint, 2016,arXiv:1612.08722v1] proved that, if$m\geq 36$and$R_{A}(n)\geq 1$for all$n\in \mathbb{Z}_{m}$, then there exists$n\in \mathbb{Z}_{m}$such that$R_{A}(n)\geq 6$. In this paper, for any finite abelian group$G$with$|G|=m$and$A\subseteq G$, we prove that (a) if the number of$g\in G$with$R_{A}(g)=0$does not exceed$\frac{7}{32}m-\frac{1}{2}\sqrt{10m}-1$, then there exists$g\in G$such that$R_{A}(g)\geq 6$; (b) if$1\leq R_{A}(g)\leq 6$for all$g\in G$, then the number of$g\in G$with$R_{A}(g)=6$is more than$\frac{7}{32}m-\frac{1}{2}\sqrt{10m}-1$.

The logarithmic coefficients$\unicode[STIX]{x1D6FE}_{n}$of an analytic and univalent function$f$in the unit disc$\mathbb{D}=\{z\in \mathbb{C}:|z|with the normalisation$f(0)=0=f^{\prime }(0)-1$are defined by$\log (f(z)/z)=2\sum _{n=1}^{\infty }\unicode[STIX]{x1D6FE}_{n}z^{n}$. In the present paper, we consider close-to-convex functions (with argument 0) with respect to odd starlike functions and determine the sharp upper bound of$|\unicode[STIX]{x1D6FE}_{n}|$,$n=1,2,3$, for such functions $f$.

Diophantine problems involving recurrence sequences have a long history and is an actively studied topic within number theory. In this paper, we connect to the field by considering the equation \begin{align*} B_mB_{m+d}\dots B_{m+(k-1)d}=y^\ell \end{align*} in positive integers $m,d,k,y$ with $\gcd(m,d)=1$ and $k\geq 2$, where $\ell\geq 2$ is a fixed integer and $B=(B_n)_{n=1}^\infty$ is an elliptic divisibility sequence, an important class of non-linear recurrences. We prove that the above equation admits only finitely many solutions. In fact, we present an algorithm to find all possible solutions, provided that the set of $\ell$-th powers in $B$ is given. (Note that this set is known to be finite.) We illustrate our method by an example., Comment: To appear in Bulletin of Australian Math Society

Let G be a 2-transitive permutation group of a set Ω of n points and let P be a Sylow p-subgroup of G where p is a prime dividing |G|. If we restrict the lengths of the orbits of P, can we correspondingly restrict the order of P? In the previous two papers of this series we were concerned with the case in which all P–orbits have length at most p; in the second paper we looked at Sylow p–subgroups of a two point stabiliser. We showed that either P had order p, or G ≥ An, G = PSL(2, 5) with p = 2, or G = M11 of degree 12 with p = 3. In this paper we assume that P has a subgroup Q of index p and all orbits of Q have length at most p. We conclude that either P has order at most p2, or the groups are known; namely PSL(3, p) ≤ G ≤ PGL(3, p), ASL(2, p) ≤ G ≤ AGL(2, p), G = PΓL,(2, 8) with p = 3, G = M12 with p = 3, G = PGL(2, 5) with p = 2, or G ≥ An with 3p ≤ n < 2p2; all in their natural representations.

Let k be a cardinal number. A group G is said to be k-free if fo C.r G any set Swilh cardinality k, the subgroup of G generated by 5 is free (of some rank ^ k). Notethat a Ai-free group is in particular fc'-free for every k' < 1-fre k. A ife grou andp isonly if it is torsion-free.It follows from a result proved by B. Baumslag in [1] that a free product withamalgamation G = F *c F', where the factors F and F' are free groups and theamalgamated subgroup C is a maximal cyclic subgroup of each factor, is 2-free. It is aconsequence (Corollary 4.2) of the main theorem of this paper that such a group G is infact 3-free. Our proof makes strong use of the structure theorem [6, 11] for subgroupsof free products with amalgamation, which was not available when [1] was written.Our results apply to a broader class of groups than the one described above. Indeed,it follows from Corollary 4.1 below that if a group G can be built up from free groups byrepeatedly using the operation of forming a free product with an amalgamated subgroupwhich is a maximal cyclic subgroup of each factor, then G is 3-free. Furthermore, thisremains true if the operations of forming a free product and an ascending union arealso allowed.These results are proved by defining a natural class of 3-free groups which includesfree groups and is closed under the operations mentioned above. (Some evidence thatour class is indeed "natural" will be offered in the appendix to this paper.)In order to describe the appropriate class of groups, we begin by recalling somestandard definitions.If a group H is given by a finite presentation deficiency, the of the presentation isdefined to be the integer m—n, where m is the number of generators in the presentation

There is a familiar construction with two finite, transitive permutation groups as input and a finite, transitive permutation group, called their wreath product, as output. The corresponding ‘imprimitive wreath decomposition’ concept is the first subject of this paper. A formal definition is adopted and an overview obtained for all such decompositions of any given finite, transitive group. The result may be heuristically expressed as follows, exploiting the associative nature of the construction. Each finite transitive permutation group may be written, essentially uniquely, as the wreath product of a sequence of wreath-indecomposable groups, amid the two-factor wreath decompositions of the group are precisely those which one obtains by bracketing this many-factor decomposition.If both input groups are nontrivial, the output above is always imprimitive. A similar construction gives a primitive output, called the wreath product in product action, provided the first input group is primitive and not regular. The second subject of the paper is the ‘product action wreath decomposition’ concept dual to this. An analogue of the result stated above is established for primitive groups with nonabelian socle.Given a primitive subgroup G with non-regular socle in some symmetric group S, how many subgroups W of S which contain G and have the same socle, are wreath products in product action? The third part of the paper outlines an algorithm which reduces this count to questions about permutation groups whose degrees are very much smaller than that of G.

In a previous paper (Acta Arith. 21 (1972), 89–97), I had proved that the sum of the absolute values of the coefficients of the mth transformation polynomial Fm (u, v) of the Weber modular function j(ω) of level 1 is not greater than 2(36n+57)2n when m = 2n is a power of 2. The aim of the present paper is to give an analogous bound for the case of general m. This upper bound is much less good and of the form where c > 0 is an absolute constant which can be determined effectively. It seems probable that also in the general case an upper bound of the form eO(m10gm) should hold, but I have not so far succeeded in proving such a result.

In this paper we investigate the spectrum of a variety of quasigroups satisfying the 2-variable identity x(xy) = yx, called Stein quasigroups. Stein quasigroups are known to be self-orthogonal and have been given a considerable amount of attention because of this property. It is known that there are no Stein quasigroups of order 2, 3, 6, 7, 8, 10, 12, 14. The object of this paper is to show that for all but 36 values of n ≥ 15 there exists a Stein quasigroup of order n. In particular, the spectrum of Stein quasigroups contains all n ≥ 191.

Let G be a doubly transitive permutation group on a finite set Ω, and for α in Ω suppose that Gα has a set Σ of non-trivial blocks of imprimitivity in Ω - {α}. If Gα is 2-transitive but not faithful on Σ, when is it true that the stabiliser in Gα of a block of Σ does not act faithfully on that block (that is, there is a nontrivial element in Gα which fixes every point of the block)? In a previous paper this question was answered when is the alternating or symmetric group, or a Mathieu group in its usual representation. In this paper we answer the question when , permuting the q + 1 points of the projective line, for some prime power q. We show that the only groups which arise satisfy either(i) PSL(3, q) ≤ G ≤ PΓL(3, q) in its natural representation, or(ii) G is a group of collineations of an affine translation plane of order q, and contains the translation group.

A general method is described for constructing examples of soluble groups whose normal subgroups form a well-ordered chain under the ordering of inclusion. This method is a variant of one introduced in a recent paper by Heineken and Wilson. Each of the resulting groups is obtained by an embedding procedure from a pair of iterated wreath products A1 wr A2 wr … wr An, B1 wr B2 wr … wr Bn, where the constituent groups Ai., Bi are each either cyclic of prime power order or quasicyclic. Here n may be chosen arbitrarily, and the choice of constituent groups is subject only to a condition on the sequences of prime numbers that may occur as orders of elements in the groupsrespectively. The construction is applied to give certain examples which illustrate the limitations of results on particular classes of soluble groups satisfying the minimal condition for normal subgroups obtained in recent papers by Hartley, McDougall, and the present author.

This paper considers an aspect of the general problem of how the structure of a group influences the structure of its automorphisms group. A recent result of Beisiegel shows that if P is a p-group then the central automorphisms group of P has no normal subgroups of order prime to p. So, roughly speaking, most of the central automorphisms are of p-power order. This generalizes an old result of Hopkins that if Aut P is abelian (so every automorphisms is central), then Aut P is a p-group.This paper uses a different approach to consider the case when P is a π-group. It is shown that the central automorphism group of P has a normal. π′-subgroup only if P has an abelian direct factor whose automorphism group has such a subgroup.

This paper shows that amongst the p-groups of order p5, where p denotes an odd prime, there is only one group whose automorphism group is again a p-group. This automorphism group has order p6 and it is shown that this is the smallest order a p-group may have when it occurs as an automorphism group. The paper also shows that all groups of order p5 have an automorphism of order 2 apart from the group above and three other related groups.

Let p, q, u, and v be any four positive integers, and let δ be a number in the interval 0 < δ ≤ 2. In one of his papers, Kurt Mahler, Bull. Austral. Math. Soc. 10 (1974), 325–335, proved that if q satisfies the inequalitiesthenIn this note, by a slightly different treatment of some inequalities in Mahler's paper, we easily obtain the same result with q only restricted by the first condition.

We determine a lower gap property for the growth of an unbounded $\mathbb{Z}$-valued $k$-regular sequence. In particular, if $f:\mathbb{N}\to \mathbb{Z}$ is an unbounded $k$-regular sequence, we show that there is a constant $c>0$ such that $|f(n)|>c\log n$ infinitely often. We end our paper by answering a question of Borwein, Choi and Coons on the sums of completely multiplicative automatic functions.

Let $\Omega $ be the set of unit vectors and $w$ be a radial weight on the plane. We consider the weighted directional maximal operator defined by $$\begin{eqnarray*}{M}_{\Omega , w} f(x): = \sup _{x\in R\in \mathcal{B} _{\Omega }}\frac{1}{w(R)} \int \nolimits \nolimits_{R} \vert f(y)\vert w(y)\hspace{0.167em} dy,\end{eqnarray*}$$ where ${ \mathcal{B} }_{\Omega } $ denotes the set of all rectangles on the plane whose longest side is parallel to some unit vector in $\Omega $ and $w(R)$ denotes $\int \nolimits \nolimits_{R} w$. In this paper we prove an almost-orthogonality principle for this maximal operator under certain conditions on the weight. The condition allows us to get the weighted norm inequality $$\begin{eqnarray*}\Vert {M}_{\Omega , w} f\mathop{\Vert }\nolimits_{{L}^{2} (w)} \leq C\log N\Vert f\mathop{\Vert }\nolimits_{{L}^{2} (w)} ,\end{eqnarray*}$$ when $w(x)= \vert x\hspace{-1.2pt}\mathop{\vert }\nolimits ^{a} $, $a\gt 0$, and when $\Omega $ is the set of unit vectors on the plane with cardinality $N$ sufficiently large.

For a given integer$n$and a set$ \mathcal{S} \subseteq \mathbb{N} $, denote by${ R}_{h, \mathcal{S} }^{(1)} (n)$the number of solutions of the equation$n= {s}_{{i}_{1} } + \cdots + {s}_{{i}_{h} } $,${s}_{{i}_{j} } \in \mathcal{S} $,$j= 1, \ldots , h$. In this paper we determine all pairs$( \mathcal{A} , \mathcal{B} )$,$ \mathcal{A} , \mathcal{B} \subseteq \mathbb{N} $, for which${ R}_{3, \mathcal{A} }^{(1)} (n)= { R}_{3, \mathcal{B} }^{(1)} (n)$from a certain point on. We discuss some related problems.

In this paper, we consider the following Robin problem:$$\begin{eqnarray*}\displaystyle \left\{ \begin{array}{ @{}ll@{}} \displaystyle - \Delta u= \mid x{\mathop{\mid }\nolimits }^{\alpha } {u}^{p} , \quad & \displaystyle x\in \Omega , \\ \displaystyle u\gt 0, \quad & \displaystyle x\in \Omega , \\ \displaystyle \displaystyle \frac{\partial u}{\partial \nu } + \beta u= 0, \quad & \displaystyle x\in \partial \Omega , \end{array} \right.&&\displaystyle\end{eqnarray*}$$where$\Omega $is the unit ball in${ \mathbb{R} }^{N} $centred at the origin, with$N\geq 3$,$p\gt 1$,$\alpha \gt 0$,$\beta \gt 0$, and$\nu $is the unit outward vector normal to$\partial \Omega $. We prove that the above problem has no solution when$\beta $is small enough. We also obtain existence results and we analyse the symmetry breaking of the ground state solutions.

It is known that $\zeta (1+ it)\ll \mathop{(\log t)}\nolimits ^{2/ 3} $ when $t\gg 1$. This paper provides a new explicit estimate $\vert \zeta (1+ it)\vert \leq \frac{3}{4} \log t$, for $t\geq 3$. This gives the best upper bound on $\vert \zeta (1+ it)\vert $ for $t\leq 1{0}^{2\cdot 1{0}^{5} } $.

The objective of this paper is to find some sufficient conditions to ensure the conjugacy of supplements of a normal subgroup of a soluble group.

In 1977 Hartwig and Luh asked whether an element $a$ in a Dedekind-finite ring $R$ satisfying $aR= {a}^{2} R$ also satisfies $Ra= R{a}^{2} $. In this paper, we answer this question in the negative. We also prove that if $a$ is an element of a Dedekind-finite exchange ring $R$ and $aR= {a}^{2} R$, then $Ra= R{a}^{2} $. This gives an easier proof of Dischinger’s theorem that left strongly $\pi $-regular rings are right strongly $\pi $-regular, when it is already known that $R$ is an exchange ring.

In this paper, we prove that, if $N$ is a positive odd number with $r$ distinct prime factors such that $N\mid \sigma (N)$, then $N\lt {2}^{{4}^{r} - {2}^{r} } $ and $N{\mathop{\prod }\nolimits}_{p\mid N} p\lt {2}^{{4}^{r} } $, where $\sigma (N)$ is the sum of all positive divisors of $N$. In particular, these bounds hold if $N$ is an odd perfect number.

In this paper, we determine when $\mathop{({\Gamma }_{I} (L))}\nolimits ^{c} $, the complement of the zero divisor graph ${\Gamma }_{I} (L)$ with respect to a semiprime ideal $I$ of a lattice $L$, is connected and also determine its diameter, radius, centre and girth. Further, a form of Beck’s conjecture is proved for ${\Gamma }_{I} (L)$ when $\omega (\mathop{({\Gamma }_{I} (L))}\nolimits ^{c} )\lt \infty $.

Let $T(S)$ be the Teichmüller space of a hyperbolic Riemann surface $S$. Suppose that $\mu $ is an extremal Beltrami differential at a given point $\tau $ of $T(S)$ and $\{ {\phi }_{n} \} $ is a Hamilton sequence for $\mu $. It is an open problem whether the sequence $\{ {\phi }_{n} \} $ is always a Hamilton sequence for all extremal differentials in $\tau $. S. Wu [‘Hamilton sequences for extremal quasiconformal mappings of the unit disk’, Sci. China Ser. A 42 (1999), 1033–1042] gave a positive answer to this problem in the case where $S$ is the unit disc. In this paper, we show that it is also true when $S$ is a doubly-connected domain.

Recently, Pollack and Shevelev [‘On perfect and near-perfect numbers’, J. Number Theory 132 (2012), 3037–3046] introduced the concept of near-perfect numbers. A positive integer $n$ is called near-perfect if it is the sum of all but one of its proper divisors. In this paper, we determine all near-perfect numbers with two distinct prime factors.

A $\ast $-ring $R$ is called (strongly) $\ast $-clean if every element of $R$ is the sum of a unit and a projection (that commute). Vaš [‘$\ast $-Clean rings; some clean and almost clean Baer $\ast $-rings and von Neumann algebras’, J. Algebra 324(12) (2010), 3388–3400] asked whether there exists a $\ast $-ring that is clean but not $\ast $-clean and whether a unit regular and $\ast $-regular ring is strongly $\ast $-clean. In this paper, we answer these two questions. We also give some characterisations related to $\ast $-regular rings.

Let $R$ be a ring and $Z(R)$ be the set of all zero-divisors of $R$. The total graph of $R$, denoted by $T(\Gamma (R))$ is a graph with all elements of $R$ as vertices, and two distinct vertices $x, y\in R$ are adjacent if and only if $x+ y\in Z(R)$. Let the regular graph of $R$, $\mathrm{Reg} (\Gamma (R))$, be the induced subgraph of $T(\Gamma (R))$ on the regular elements of $R$. In 2008, Anderson and Badawi proved that the girth of the total graph and the regular graph of a commutative ring are contained in the set $\{ 3, 4, \infty \} $. In this paper, we extend this result to an arbitrary ring (not necessarily commutative). We also prove that if $R$ is a reduced left Noetherian ring and $2\not\in Z(R)$, then the chromatic number and the clique number of $\mathrm{Reg} (\Gamma (R))$ are the same and they are ${2}^{r} $, where $r$ is the number of minimal prime ideals of $R$. Among other results, we show that if $R$ is a semiprime left Noetherian ring and $\mathrm{Reg} (R)$ is finite, then $R$ is finite.

A natural number $n$ is called $k$-perfect if $\sigma (n)= kn$. In this paper, we show that for any integers $r\geq 2$ and $k\geq 2$, the number of odd $k$-perfect numbers $n$ with $\omega (n)\leq r$ is bounded by $\left({\lfloor {4}^{r} { \mathop{ \log } \nolimits }_{3} 2\rfloor + r\atop r} \right){ \mathop{ \sum } \nolimits }_{i= 1}^{r} \left({\lfloor kr/ 2\rfloor \atop i} \right)$, which is less than ${4}^{{r}^{2} } $ when $r$ is large enough.

We say that two arithmetic functions f and g form a Mobius pair if f(n) = \sum_{d \mid n} g(d) for all natural numbers n. In that case, g can be expressed in terms of f by the familiar Mobius inversion formula of elementary number theory. In a previous paper, the first-named author showed that if the members f and g of a Mobius pair are both finitely supported, then both functions vanish identically. Here we prove two significantly stronger versions of this uncertainty principle. A corollary is that in a nonzero Mobius pair, either \sum_{n \in supp(f)} 1/n or \sum_{n \in supp(g)} 1/n diverges., Comment: 10 pages

Let ${ \mathcal{T} }_{X} $ be the full transformation semigroup on a set $X$ and $E$ be a nontrivial equivalence relation on $X$. Denote $$\begin{eqnarray*}{T}_{\exists } (X)= \{ f\in { \mathcal{T} }_{X} : \forall x, y\in X, (f(x), f(y))\in E\Rightarrow (x, y)\in E\} ,\end{eqnarray*}$$ so that ${T}_{\exists } (X)$ is a subsemigroup of ${ \mathcal{T} }_{X} $. In this paper, we endow ${T}_{\exists } (X)$ with the natural partial order and investigate when two elements are related, then find elements which are compatible. Also, we characterise the minimal and maximal elements.

The stable rank of Leavitt path algebras of row-finite graphs was computed by Ara and Pardo. In this paper we extend this to an arbitrary directed graph. In part our computation proceeds as for the row-finite case, but we also use knowledge of the row-finite setting by applying the desingularising method due to Drinen and Tomforde. In particular, we characterise purely infinite simple quotients of a Leavitt path algebra.

Let$\alpha $be any radical of associative rings. A radical$\gamma $is called$\alpha $-like if, for every$\alpha $-semisimple ring$A$, the polynomial ring$A[x] $is$\gamma $-semisimple. In this paper we describe properties of$\alpha $-like radicals and show how they can be used to solve some open problems in radical theory.

Given an integer $q\ge 2$, a $q$-normal number is an irrational number $\eta $ such that any preassigned sequence of $\ell $ digits occurs in the $q$-ary expansion of $\eta $ at the expected frequency, namely $1/q^\ell $. In a recent paper we constructed a large family of normal numbers, showing in particular that, if $P(n)$ stands for the largest prime factor of $n$, then the number $0.P(2)P(3)P(4)\ldots ,$ the concatenation of the numbers $P(2), P(3), P(4), \ldots ,$ each represented in base $q$, is a $q$-normal number, thereby answering in the affirmative a question raised by Igor Shparlinski. We also showed that $0.P(2+1)P(3+1)P(5+1) \ldots P(p+1)\ldots ,$ where $p$ runs through the sequence of primes, is a $q$-normal number. Here, we show that, given any fixed integer $k\ge 2$, the numbers $0.P_k(2)P_k(3)P_k(4)\ldots $ and $0. P_k(2+1)P_k(3+1)P_k(5+1) \ldots P_k(p+1)\ldots ,$ where $P_k(n)$ stands for the $k{\rm th}$ largest prime factor of $n$, are $q$-normal numbers. These results are part of more general statements.