In this article we prove that there is a positive solution in of the equation −Δu + λu = |u|p−2u in Ω where Ω is the half space with a hole, λ > 0 and 2 < p < 2N/(N−2).

This paper studies the unique range set of meromorphic functions and shows that the set S = {w | w13 + w11 + 1 = 0} is unique range set of meromorphic functions with 13 elements.

We prove that varieties of algebras equivalent to Mendelsohn and Steiner quadruple systems can be defined by single identities.

A ring R is a right SI-ring if every singular right R-module is injective, while R is a right S3I-ring if every singular semisimple right R-module is injective. In this paper, we investigate and characterise several analogues of the two notions to modules, with many illustrative examples included.

We show that a radical has a semisimple essential cover if and only if it is hereditary and has a complement in the lattice of hereditary radicals. In 1971 Snider gave a full description of supernilpotent radicals which have a complement. Recently Beidar, Fong, Ke, and Shum have determined radicals with semisimple essential covers. Using their results, we are able to provide a lower radical representation of complemented subidempotent radicals. This completes Snider's description of hereditary complemented radicals.

A recent result of Huang and Reyn on quadratic systems is reformulated and given a clearer proof.

It is shown that two polycyclic-by-finite groups G and H, satisfying the same sentences with one alternation of quantifiers, are commensurable. In fact we show something stronger: given n > 1 there is a subgroup Hn of H and a subgroup Gn of G such that Hn≃G, Gn≃H and the indices [G: Gn] and [H: Hn] are finite and prime to n.

We prove that any finite subdirectly irreducible algbra in a congruence modular variety with trivial Frattini congruence is critical. We also show that if A and B are critical algebras which generate the same congruence modular variety, then the variety generated by the proper sections of A equals the variety generated by the proper sections of B.

In this paper the structure of finite groups which are the product of two totally permutable subgroups is studied. In fact we can obtain the -residual, where is a formation, -projectors and -normalisers, where is a saturated formation, of the group from the corresponding subgroups of the factor subgroups.

Given a finite set S of cardinality N, the minimum number of j-subsets of S needed to cover all the r-subsets of S is called the covering number C(N, j, r). While Erdös and Hanani's conjecture that was proved by Rödl, no nontrivial upper bound for C(N, j, r) was known for finite N. In this note we obtain a nontrivial upper bound by showing that for finite N,

Neuman and Reid describe a 2-cusped hyperbolic 3-orbifold in which the cusps are geometrically isolated. Based on numerical evidence provided by Jeff Weeks' “SnapPea” program, they conjecture that the cusps are strongly geometrically isolated, a fact which we establish here. We also give a parameterisation of the Dehn Surgery Space of this orbifold which has amusing properties.

Using the methods of KKM theory, almost fixed point and best approximations theorems in H-spaces are proved.

The aim of this paper is to prove the following result: if X is a 2-dimensional symmetric real Banach space, then its self-length is greater than or equal to 2π. Moreover, the minimum value 2π is uniquely attained (up to isometries) by euclidean space.

Using an old result of Von Staudt on sums of consecutive integer powers, we shall show by an elementary method that the Diophantine equation 1k + 2k + … + (x − l)k = axk has no solutions (a, x, k) with k > 1, . For a = 1 this equation reduces to the Erdös-Moser equation and the result to a result of Moser. Our method can also be used to deal with variants of the equation of the title, and two examples will be given. For one of them there are no integer solutions with

We prove an existence theorem for differential inclusions in Banach spaces. Here {A (t): t ∈ [0,T]} is a family of linear operators generating a continuous evolution operator K (t, s). We concentrate on maps F with F (t,·) weakly sequentially hemi-continuous.

Moreover, we show a compactness of the set of all integral solutions of the above problem. These results are also applied to a semilinear optimal control problem. Some corollaries, important in the theory of optimal control, are given too. We extend in several ways theorems existing in the literature.

We characterise reflexive modules over the rings R such that each finitely generated submodule of E(RR) is torsionless (left QF-3″ rings) by means of a suitable linear compactness condition relative to the Lambek torsion theory.

We construct frame starters in Z2n − {0, n}, for n ≡ 0, 1 mod 4, where Z2n denotes the cyclic group of order 2n. We also construct left frame starters in Q2n − {e, αn}, where Q2n is the dicyclic group of order 4n and αn is the unique element of order 2 in Q2n.

ℳ-harmonic functions with ℳ-harmonic square are proved to be either holomorphic or antiholomorphic in the unit ball of complex n-space under certain additional conditions. For example, if u and u2 are ℳ-harmonic in the unit ball of ℂ2 and if u is continuously differentiable up to the boundary then u is either holomorphic or antiholomorphic.

In the case of continuous time systems with bounded operators (coefficients) the following result, of Perron type is well known: “The linear differential system ẋ = Ax + f(t) has, for every function f continuous and bounded on ℝ, a unique bounded solution on ℝ, if and only if the spectrum of the operator A has no points on the imaginary axis”.

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.