Matrices provide essential tools in many branches of mathematics and matrix semigroups have applications in various areas. In this paper we give a complete description of all infinite matrix semigroups satisfying a certain combinatorial property defined in terms of annihilator graphs.

In this paper, we consider generalised mixed co-quasi-variational inequalities with noncompact valued mappings and propose an iterative algorithm for computing their approximate solutions. We prove that the approximate solutions obtained by the proposed algorithm converge to the exact solution of our co-quasi-variational inequality. Some special cases are also discussed.

ABS mthods are direct iteration methods for solving linear systems where the i-th iterate satisfies the first i equations, and therefore a system on m equations is solved in at most m ABS steps. In this paper, using a new rank two update of the Abaffian matrix, we introduce a class of ABS-type methods for solving full row rank linear equations, where the i-th iterate solves the first 2i equations. So, termination is achieved in at most ⌊(m + 1)/2⌋ steps. We also show how to decrease the dimension of the Abaffian matrix by choosing appropriate parameters.

In this paper, we consider n (n ≥ 3)-dimensional compact oriented connected hypersurfaces with constant scalar curvature n(n − 1)r in the unit sphere Sn+1(1). We prove that, if r ≥ (n − 2)/(n − 1) and S ≤ (n − 1)(n(r − 1) + 2)/(n − 2) + (n − 2)/(n(r − 1) + 2), then either M is diffeomorphic to a spherical space form if n = 3; or M is homeomorphic to a sphere if n ≥ 4; or M is isometric to the Riemannian product , where c2 = (n − 2)/(nr) and S is the squared norm of the second fundamental form of M.

We show that the Hypercyclicity Criterion coincides with other existing hypercyclicity criteria and prove that a wide class of hypercyclic operators satisfy the Criterion. The results obtained extend or improve earlier work of several authors. We also unify the different versions of the Supercyclicity Criterion and show that operators with dense generalised kernel and dense range are supercyclic.

We introduce a Riemannian invariant and establish general optimal inequalities involving the invariants and the squared mean curvature for Einstein manifolds isometrically immersed in real space forms. We show that these inequalities do not hold for arbitrary submanifolds in real space forms in general. We also provide some immediate applications of the inequalities.

In this paper, we prove that the twisted Alexander polynomial for the Lawrence-Krammer representation of the braid group B4 is trivial. This gives an answer to the problem of whether the twisted Alexander polynomial for given faithful representations is always non-trivial.

We study the asymptotics of fundamental solutions of p-adic pseudo-differential equations of type where f(∂,β) is a pseudo-differential operator with symbol , f is a form of arbitrary degree with coefficients in a p-adic field, λ ≥ 0, and g is a Schwartz-Bruhat function.

For a H-type group G, we first give explicit equations for its shortest sub-Riemannian geodesics. We use properties of sub-Riemannian geodesics in G to characterise the isometry group ISO(G) with respect to the Carnot-Carathéodory metric. It turns out that ISO(G) coincides with the isometry group with respect to the standard Riemannian metric of G.

Let C*(E) = C*(se, pv) be the graph C*-algebra of a directed graph E = (E0, E1) with the vertices E0 and the edges E1. We prove that if E is a finite graph (possibly with sinks) and φE: C*(E) → C*(E) is the canonical completely positive map defined by then Voiculescu's topological entropy ht(φE) of φE is log r(AE), where r(AE) is the spectral radius of the edge matrix AE of E. This extends the same result known for finite graphs with no sinks. We also consider the map φE when E is a locally finite irreducible infinite graph and prove that , where the supremum is taken over the set of all finite subgraphs of E.

We prove that the dual of a Banach space E is isomorphic to an ℓ1(Γ) space if and only if, for a fixed integer m, every m-homogeneous 1-dominated polynomial on E is nuclear. This extends a result for linear operators due to Lewis and Stegall. The same techniques used for this result allow us to prove that, if every m-homogeneous integral polynomial between two Banach spaces is nuclear, then every integral (linear) operator between the same spaces is nuclear.

Let E be an elliptic curve with complex multiplication by R, where R is an order of discriminant D < −4 of an imaginary quadratic field K. If a prime number p is decomposed completely in the ring class field associated with R, then E has good reduction at a prime ideal p of K dividing p and there exist positive integers u and υ such that 4p = u2 – Du;2. It is well known that the absolute value of the trace ap of the Frobenius endomorphism of the reduction of E modulo p is equal to u. We determine whether ap = u or ap = −u in the case where the class number of R is 2 or 3 and D is divisible by 3, 4 or 5.

We consider graphs E which have been obtained by adding one or more sinks to a fixed directed graph G. We classify the C*-algebra of E up to a very strong equivalence relation, which insists, loosely speaking, that C*(G) is kept fixed. The main invariants are vectors WE: G0 → ℕ which describe how the sinks are attached to G; more precisely, the invariants are the classes of the WE in the cokernel of the map A – I, where A is the adjacency matrix of the graph G.

For a hereditary torsion theory τ, a module A is called τ-completedly decomposable if it is a direct sum of modules that are the τ-injective hull of each of their non-zero submodules. We give a positive answer in several cases to the following generalised Matlis' problem: Is every direct summand of a τ-completely decomposable module still τ-completely decomposable? Secondly, for a commutative Noetherian ring R that is not a domain, we determine those torsion theories with the property that every τ-injective module is an essential extension of a (τ-injective) τ-completely decomposable module.