The total distance (or Wiener index) of a connected graph $G$ is the sum of all distances between unordered pairs of vertices of $G$. DeLaViña and Waller [‘Spanning trees with many leaves and average distance’, Electron. J. Combin.15(1) (2008), R33, 14 pp.] conjectured in 2008 that if $G$ has diameter $D>2$ and order $2D+1$, then the total distance of $G$ is at most the total distance of the cycle of the same order. In this note, we prove that this conjecture is true for 2-connected graphs.

A class of locally convex spaces, a B-subfamily of finite order, is defined and the omega lemma for spaces belonging to this family is proved.

Let G be a finite d-dimensional classical group and p a prime divisor of ∣G∣ distinct from the characteristic of the natural representation. We consider a subfamily of p-singular elements in G (elements with order divisible by p) that leave invariant a subspace X of the natural G-module of dimension greater than d/2 and either act irreducibly on X or preserve a particular decomposition of X into two equal-dimensional irreducible subspaces. We proved in a recent paper that the proportion in G of these so-called p-abundant elements is at least an absolute constant multiple of the best currently known lower bound for the proportion of all p-singular elements. From a computational point of view, the p-abundant elements generalise another class of p-singular elements which underpin recognition algorithms for finite classical groups, and it is our hope that p-abundant elements might lead to improved versions of these algorithms. As a step towards this, here we present efficient algorithms to test whether a given element is p-abundant, both for a known prime p and for the case where p is not known a priori.

It is well known that in general the Jacobi–Perron algorithm (a multi-dimensional analogue of the continued fraction algorithm) may or might not acknowledge the dependence over ℚ of its arguments 1, α1,…, αn by truncating itself down to fewer arguments from some step onwards (if so, the algorithm is said to display an ‘interruption’). We show here that if n = 2 then 1, α1, α2 are linearly dependent over ℚ of and only if the Jacobi–Perron Algorithm displays an interruption. We give examples showing this is not so for any n ≥ 3.

Let $K$ be any field with $\text{char}\,K\neq 2,3$. We classify all cubic homogeneous polynomial maps $H$ over $K$ whose Jacobian matrix, ${\mathcal{J}}H$, has $\text{rk}\,{\mathcal{J}}H\leq 2$. In particular, we show that, for such an $H$, if $F=x+H$ is a Keller map, then $F$ is invertible and furthermore $F$ is tame if the dimension $n\neq 4$.

Let E be an elliptic curve with positive rank over a number field K and let p be an odd prime number. Let $K_{\operatorname {cyc}}$ be the cyclotomic $\mathbb {Z}_p$ -extension of K and $K_n$ its nth layer. The Mordell–Weil rank of E is said to be constant in the cyclotomic tower of K if for all n, the rank of $E(K_n)$ is equal to the rank of $E(K)$ . We apply techniques in Iwasawa theory to obtain explicit conditions for the rank of an elliptic curve to be constant in this sense. We then indicate the potential applications to Hilbert’s tenth problem for number rings.

We use a variation on a construction due to Corner 1965 to construct (Abelian) groups A that are torsion as modules over the ring End (A) of group endomorphisms of A. Some applications include the failure of the Baer-Kaplansky Theorem for Z[X]. There is a countable reduced torsion-free group A such that IA = A for each maximal ideal I in the countable commutative Noetherian integral domain, End (A). Also, there is a countable integral domain R and a countable. R-module A such that (1) R = End(A), (2) T0 ⊗RA ≠ 0 for each nonzero finitely generated (respectively finitely presented) R-module T0, but (3) T ⊗RA = 0 for some nonzero (respectively nonzero finitely generated). R-module T.

It is shown that the dynamical game theoretic mating behaviour of males and females can be modelled by a planar system of autonomous ordinary differential equations. This system occurs in modelling “the battle of the sexes” in evolutionary biology. The existence of a heteroclinic cycle and a continuous family of periodic orbits of the system is established; then the dynamical characteristics of a time-periodic perturbation of the system are investigated. By using the well-known Melnikov's method, a sufficient condition is obtained for the perturbed system to have a transverse heteroclinic cycle and hence to possess chaotic behaviour in the sense of Smale. Finally, subharmonic Melnikov theory is used to obtain a criterion for the existence of subharmonic periodic orbits of the perturbed system.

We study the dynamics of a reaction–diffusion–advection equation $u_{t}=u_{xx}-au_{x}+f(u)$ on the right half-line with Robin boundary condition $u_{x}=au$ at $x=0$, where $f(u)$ is a combustion nonlinearity. We show that, when $0<a<c$ (where $c$ is the travelling wave speed of $u_{t}=u_{xx}+f(u)$), $u$ converges in the $L_{loc}^{\infty }([0,\infty ))$ topology either to $0$ or to a positive steady state; when $a\geq c$, a solution $u$ starting from a small initial datum tends to $0$ in the $L^{\infty }([0,\infty ))$ topology, but this is not true for a solution starting from a large initial datum; when $a>c$, such a solution converges to $0$ in $L_{loc}^{\infty }([0,\infty ))$ but not in $L^{\infty }([0,\infty ))$ topology.

Let $\text{pod}_{-4}(n)$ denote the number of partition quadruples of $n$ where the odd parts in each partition are distinct. We find many arithmetic properties of $\text{pod}_{-4}(n)$ including the following infinite family of congruences: for any integers ${\it\alpha}\geq 1$ and $n\geq 0$,

$$\begin{eqnarray}\text{pod}_{-4}\biggl(3^{{\it\alpha}+1}n+\frac{5\cdot 3^{{\it\alpha}}+1}{2}\biggr)\equiv 0~(\text{mod}~9).\end{eqnarray}$$

$\text{pod}_{-4}(n)$

A basis ${\mathcal{B}}=\{u_{i}\}_{i\in I}$ of a commutative or anticommutative algebra $\mathfrak{C},$ over an arbitrary base field $\mathbb{F}$, is called multiplicative if for any $i,j\in I$ we have that $u_{i}u_{j}\in \mathbb{F}u_{k}$ for some $k\in I$. We show that if a commutative or anticommutative algebra $\mathfrak{C}$ admits a multiplicative basis then it decomposes as the direct sum $\mathfrak{C}=\bigoplus _{j}\mathfrak{i}_{j}$ of well-described ideals each one of which admits a multiplicative basis. Also the minimality of $\mathfrak{C}$ is characterised in terms of the multiplicative basis and it is shown that, under a mild condition, the above direct sum is indexed by the family of its minimal ideals admitting a multiplicative basis.

An Oseén type linearization of the Navier-Stokes equations is made with respect to a uniform shear flow at the trailing edge of a flat plate. Asymptotic expansions are obtained to describe a symmetrical merging flow for distances from the trailing edge that are, in a certain sense, large. Expansions for three regions are found:

(i) a wake region,

(ii) an inviscid region, and

(iii) an upstream lower order boundary layer.

The results are compared with those of Hakkinen and O'Neil (Douglas Aircraft Co. Report, 1967) and Stewartson (Proc. Roy. Soc. Ser. A 306 (1968)). They are further related to the results of Stewartson (Mathematika 16 (1969)) and Messiter (SIAM J. Appl. Math. 18 (1970)).

In this paper we consider a nonlinear differential-integral equation which arises in the problems of ecology and study the questions of convergent solutions and asymptotic equivalence of the solutions.

Let R be a ring in which for each x, y in R there exists a positive integer n = n(x, y) such that (xy)n − (yx)n is in the center of R. Then R has a nil commutator ideal.

A method of construction of designs with parameters v1 = r2 = p2, r1 = v2 = p + 1, b = p(p+1), k = p which may be used for the two-way elimination of heterogeneity is discussed. These designs were first studied in connection with estimating tobacco mosaic virus. Our designs have the advantage that every treatment occurs at most once in a row or column. We give the designs explicitly for p = 3, 4, 5.

We obtain an inequality concerning the width and diameter of a planar convex set with interior containing no point of the rectangular lattice. We then use the result to obtain a corresponding inequality for a planar convex set with interior containing exactly two points of the integral lattice.

There is an approach due to Bergman describing ordinary polynomial identities for Mn (K) by means of commutative algebra. In the paper we apply this approach to the matrix algebra of even order M2n (K, *) with symplectic involution *.

Let A be a noncommutative C*-algebra other than M2(I). We show that there exists a completely positive map φ of norm one on A and an element a ɛ A such that φ(a) = a, φ(a*a) = a*a, but φ(aa*) ≠ aa*.

We prove that every topologically amenable locally compact quantum group is amenable. This answers an open problem by Bédos and Tuset [‘Amenability and co-amenability for locally compact quantum groups’, Internat. J. Math.14 (2003), 865–884].