We describe all directed graphs with graph algebras isomorphic to syntactic semigroups of languages.

Fix and elliptic curve y2 = x3 + Ax + B, satisfying A, B ∈ ℤ A ≥ |B| > 0. We prove that the associated quadratic family contains infinitely many elliptic curves with trivial Mordell-Weil group.

We treat a class of functionals which satisfy the Cauchy–Schwarz inequality. This appears to be a natural unifying concept and subsumes inter alia isotonic linear functional and sublinear positive isotonic functionals. Striking superadditivity and monotonicity properties are derived.

Let P be an extraspecial p-group which is neither dihedral of order 8, nor of odd order p3 and exponent p. Let G be a finite group having P as a Sylow p-subgroup. Then the mod-p cohomology ring of G coincides with that of the normaliser NG (P).

In this paper it is shown that a particular iteration scheme converges weakly to a common fixed point of a finite set of nonexpansive mappings. This result is an improvement of two related theorems in the literature.

Let G be a finite group and let N be a normal subgroup of G. If G/N is solvable and H/N is a nilpotent injector of G/N, then there exists a canonical basis of the complex space of the class functions of G which vanish off the G-conjugates of H.

For a finite cyclic P–extension L/K of a rational function field K = κ(x) over an algebraically closed field κ of characteristic P > 0 such that every ramified prime divisor is fully ramified, we find a basis of the κ[G]-module structure of ωL(0) in terms of indecomposable modules.

The minimal faithful permutation degree μ(G) of a finite group G is the least positive integer n such that G is isomorphic to a subgroup of the symmetric group Sn. Let N be a normal subgroup of a finite group G. We prove that μ(G/N) ≤ μ(G) if G/N has no nontrivial Abelian normal subgroup. There is an as yet unproved conjecture that the same conclusion holds if G/N is Abelian. We investigate the structure of a (hypothetical) minimal counterexample to this conjecture.

Let M be a right R-module. It is shown that M is a locally Noetherian module if every finitely generated module in σ[M] is a direct sum of a projective module and a CS-module. Moreover, if every module in σ[M] is a direct sum of a projective module and a CS-module, then every module in σ[M] is a direct sum of modules which are either indecomposable projective or uniform Σ-quasi-injective. In particular, if every module in σ[M] is a direct sum of a projective module and a quasi-continuous module, then every module in σ[M] is a direct sum of a projective module and a quasi-injective module.

This paper introduces a new concept of duality of generalised random fields using the geometric properties of Sobolev spaces of integer order. Under this duality condition, the covariance operators of a generalised random field and its dual can be factorised. The paper also defines a concept of generalised white noise relative to the geometries of the Sobolev spaces, and via the covariance factorisation, obtains a representation of the generalised random field as a stochastic equation driven by a generalised white noise. This representation is unique except for isometric isomorphisms on the parameter space.

The semigroup OT (X) of all order-preserving total transformations of a finite chain X is known to be regular. We extend this result to subchains of Z; and we characterise when OT (X) is regular for an interval X in R. We also consider the corresponding idea for partial transformations of arbitrary chains and posets.

Dedicated to Lamiae and Lucas Zakaria with great affection.

We give a sharp lower bound for the Ricci curvature of bounded complete hypersurfaces of space forms. This leads to several applications.

Let G be a finite group and let p be a prime number. We consider the Submatrix of the character table of G whose rows are indexed by the characters in blocks of maximal defect, and whose columns are indexed by the conjugacy classes of P′-size. We prove that this matrix has maximum rank.

We determine the irreducible subgroups G of GL (r,ℂ), where r is prime and all proper subgroups of G are reducible.