We determine all finite sets of equiangular lines spanning finite-dimensional complex unitary spaces for which the action on the lines of the set-stabiliser in the unitary group is 2-transitive with a regular normal subgroup.

Let K be a closed convex set in the Euclidean plane, with area A(K), which contains in its interior only one point 0 of the integer lattice. If K has other than one or three chords through 0 of one of the following types, it is shown that A(K) ≤ 4, while if K has three of one type, A(K) ≤ 4.5. The types of chords considered are chords which partition K into two regions of equal area, chords which lie midway between parallel supporting lines of K, and chords such that K is invariant under reflection in them. The results are generalised to any lattice in the plane.

Let $p_{\{3, 3\}}(n)$ denote the number of $3$ -regular partitions in three colours. Da Silva and Sellers [‘Arithmetic properties of 3-regular partitions in three colours’, Bull. Aust. Math. Soc. 104(3) (2021), 415–423] conjectured four Ramanujan-like congruences modulo $5$ satisfied by $p_{\{3, 3\}}(n)$ . We confirm these conjectural congruences using the theory of modular forms.

In the present paper the asymptotic and globally asymptotic stability of the zero solution of systems with impulses are investigated. For this purpose piecewise continuous auxiliary functions are used which are analogous to Lyapunov's functions. The theorem of Marachkov on the asymptotic stability of systems without impulses is generalised. The results obtained are formulated in four theorems.

We prove Reilly-type upper bounds for the first nonzero eigenvalue of the Steklov problem associated with the p-Laplace operator on submanifolds of manifolds with sectional curvature bounded from above by a nonnegative constant.

A generating function is derived for the number of transitions to the first passage through a particular vertex of a random walk on a doubly regular tournament. As an application of this result, we obtain a generating function for the number of solutions (q1, …, qk) of qi ≡ r (mod p), where p is a prime of the form 4l + 3 and the qi are quadratic residues modulo p.

A subgroup H of a group G is said to be contranormal in G if the normal closure of H in G is equal to G. In this paper, we consider groups whose nonmodular subgroups (of infinite rank) are contranormal.

We obtain an asymptotically sharp lower bound for the sum of the first k eigenvalues of the biharmonic operator.

We obtain sufficient conditions for the convergence of martingale triangular arrays to infinitely divisible laws with finite variances, without making the usual assumptions of uniform asymptotic negligibility. Our results generalise known results for both the martingale case under a negligibility assumption and the classical (independence) case without such assumptions.

Jeśmanowicz conjectured that $(x,y,z)=(2,2,2)$ is the only positive integer solution of the equation $(*)\; ((\kern1.5pt f^2-g^2)n)^x+(2fgn)^y=((\kern1.5pt f^2+g^2)n)^x$ , where n is a positive integer and f, g are positive integers such that $f>g$ , $\gcd (\kern1.5pt f,g)=1$ and $f \not \equiv g\pmod 2$ . Using Baker’s method, we prove that: (i) if $n>1$ , $f \ge 98$ and $g=1$ , then $(*)$ has no positive integer solutions $(x,y,z)$ with $x>z>y$ ; and (ii) if $n>1$ , $f=2^rs^2$ and $g=1$ , where r, s are positive integers satisfying $(**)\; 2 \nmid s$ and $s<2^{r/2}$ , then $(*)$ has no positive integer solutions $(x,y,z)$ with $y>z>x$ . Thus, Jeśmanowicz’ conjecture is true if $f=2^rs^2$ and $g=1$ , where r, s are positive integers satisfying $(**)$ .

New fixed point theorems are given which have applications in the theory of abstract economies.

We study images of Hankel operators on the Bergman or Hardy space of the poly-disk. We give characterisations of pairs of antiholomorphic symbols inducing Hankel operators whose images are mutually orthogonal.

This paper explores a five-lemma situation in the context of a free product of a family of groups with amalgamated subgroups (that is, a colimit of an appropriate diagram in the category of groups). In particular, for two families {Aα}, {Bα} of groups with amalgamated subgroups {Aαβ}, {Bαβ} and free products A, B we assume the existence of homomorphisms Aα → Bα whose restrictions Aαβ → Bαβ are isomorphisms and which induce an isomorphism A → B between the products. We show that the usual five-lemma conclusion is false, in that the morphisms Aα → Bα are in general neither monic nor epic. However, if all Bα → B are monic, Aα → Bα is always epic; and if Aα → A is monic, for all α, then Aα → Bα is an isomorphism.

The objectives of this paper are the following:

(1) to show that a theorem of Ficker is incorrect;

(2) to show that a stronger version of Ficker's Theorem is valid for a certain class of measures;

(3) characterize all σ-algebras on which every measure is a countable sum of finite measures.

In this paper we prove partial regularity for a weakly stable p-harmonic map from Ω into Sk when k > 2p – 1.

We define the notion of Darboux integrability for linear second order partial differential operators,

.

We then build on certain geometric results of E. Vessiot related to the theory of Monge characteristics to show that the Darboux integrable operators L can be used to obtain a solution of the A2 Toda field theory. This solution is parametrised by four arbitrary functions. The approach presented in this paper thereby represents an alternative means of linearising the A2 Toda equations and may be contrasted with the known linearisation via the Lax pair.

The semigroup of all onto mappings of a set to itself and the semigroup of all one-to-one mappings of a set to itself are shown to have the property that every automorphism is inner.

Mr V.E. Hoggatt, Jr, has pointed out an error in the examples of my paper [2], If Fm denotes the mth Fibonacci number, these examples asserted that

is transcendental. This is in fact false, for by a theorem of Good [1],

for it happens that

is a rational and not a transcendental function of z, so that Theorem 1 of my paper cannot be applied. The value of s follows from (1) on putting .

Hence the following changes have to be made in [2].

On p. 390, lines 7 and 10, the case k = 1 must each time be excluded, and in Theorem 2 the two numbers r and s may not be both be 0.

Let Δ be an affine building of type Ã2 and let Γ be a discrete group of type-rotating automorphisms acting simply transitively on the vertices of δ. We prove that the reduced group C*-algebra is simple. To prove this result we use the sufficient condition for the simplicity of given in a recent paper by M. Bekka, M. Cowling and P. de la Harpe.