Using a dual variational method we shall show the existence of solutions to the Dirichlet problem without assuming Palais-Smale condition.

Let u be a solution to a second order elliptic equation with singular potentials belonging to the Kato-Fefferman-Phong's class in Lipschitz domains. We prove the boundary unique continuation theorems and the doubling properties for u2 near the boundary under the zero Neumann boundary condition.

Let A ∈ B(H) be a bounded non-compact operator that is not semi-Fredholm. The similarity invariant semigroup generated by A is shown to consist of all operators that are not semi-Fredholm and satisfy obvious inequalities for the nullity and co-nullity.

In the paper we prove a result on the existence of positive solutions for a class of nonvariational elliptic system with nonlocal source by Galerkin methods and a fixed point theorem in finite dimensions. We establish another existence result by the super and subsolution method and a monotone iteration.

In this paper, we characterise the analytic functions ϕ mapping the open unit disk ▵ into itself whose induced composition operator Cϕ: f ↦ f ∘ ϕ is an isometry on the Bloch space. We show that such functions are either rotations of the identity function or have a factorisation ϕ = gB where g is a non-vanishing analytic function from Δ into the closure of ▵, and B is an infinite Blaschke product whose zeros form a sequence{zn} containing 0 and a subsequence satisfying the conditions , and

Ky Fan type inequalities for means of two or more variables are obtained. Refinements and improvements of known inequalities are derived. Applications to symmetric elliptic integrals of the first and second kind are also included.

In previous papers, we used abstract potential theory, as developed by Fuglede and Ohtsuka, to a systematic treatment of rendezvous numbers. We considered Chebyshev constants and energies as two variable set functions, and introduced a modified notion of rendezvous intervals which proved to be rather nicely behaved even for only lower semicontinuous kernels or for not necessarily compact metric spaces.

Here we study the rendezvous and average numbers of possibly infinite dimensional normed spaces. It turns out that very general existence and uniqueness results hold for the modified rendezvous numbers in all Banach spaces. We also observe the connections of these magical numbers to Chebyshev constants, Chebyshev radius and entropy. Applying the developed notions with the available methods we calculate the rendezvous numbers or rendezvous intervals of certain concrete Banach spaces. In particular, a satisfactory description of the case of Lp spaces is obtained for all p > 0.

We show that some results of the Hutchinson-Barnsley theory for finite iterated function systems can be carried over to the infinite case. Namely, if {Fi : i ∈ ℕ} is a family of Matkowski's contractions on a complete metric space (X, d) such that (Fix0)i∈N is bounded for some x0 ∈ X, then there exists a non-empty bounded and separable set K which is invariant with respect to this family, that is, . Moreover, given σ ∈ ℕℕ and x ∈ X, the limit exists and does not depend on x. We also study separately the case in which (X, d) is Menger convex or compact. Finally, we answer a question posed by Máté concerning a finite iterated function system {F1,…, FN} with the property that each of Fi has a contractive fixed point.

Let (L, ⊥, ⊕,0,1) be an effect algebra and X a locally convex space with dual X′. A function μ: L → X is called a measure if μ(a ⊕ b) = μ(a) + μ(b) whenever a⊥b in L and it is bounded if is bounded for each orthogonal sequence {an} in L. We establish five useful conditions that are equivalent to boundedness for vector measures on effect algebras.

We classify the Hopf algebras U (L)#kQ of homological type FP2 where L is a Lie algebra and Q an Abelian group such that L has an Abelian ideal A invariant under the Q-action via conjugation and U (L/A)#kQ is commutative. This generalises the classification of finitely presented metabelian Lie algebras given by J. Groves and R. Bryant.

If A is a set of positive integers, let R1 (n) be the number of solutions of a + a′ = n, a. a′ ∈ A, and let R2(n) and R3(n) denote the number of solutions with the additional restrictions a < a′, and a ≤ a′ respectively. The monotonicity properties of the three functions R1(n), R2(n), and R3(n) are studied and compared.

In this paper we consider the Wośko problem of evaluating, in an infinite-dimensional Banach space X, the infimum of all k ≤ 1 for which there exists a k-ball contractive retraction of the unit ball onto its boundary. We prove that in some classical Banach spaces the best possible value 1 is attained. Moreover we give estimates of the lower H-measure of noncompactness of the retractions we construct.

Let

where ψ denotes the logarithmic derivative of Euler's gamma function. We prove that the functional inequality

holds if and only if 0 < r ≤ 1. And, we show that the converse is valid if and only if r < 0 or r ≥ n + 1.

The Jacobson group of a ring R (denoted by  = (R)) is the normal subgroup of the group of units of R (denoted by G(R)) obtained by adding 1 to the Jacobson radical of R (J(R)). Coleman and Easdown in 2000 showed that the Jacobson group is complemented in the group of units of any finite commutative ring and also in the group of units a n × n matrix ring over integers modulo ps, when n = 2 and p = 2, 3, but it is not complemented when p ≥ 5. In 2004 Wilcox showed that the answer is positive also for n = 3 and p = 2, and negative in all the remaining cases. In this paper we offer a different proof for Wilcox's results and also generalise the results to a matrix ring over an arbitrary finite commutative ring. We show this by studying the generators and relations that define a matrix ring over a field. We then proceed to examine the complementation of the Jacobson group in the matrix rings over graded rings and prove that complementation depends only on the 0-th grade.

The diagonal right (respectively, left) act of a semigroup S is the set S × S on which S acts via (x, y) s = (xs, ys) (respectively, s (x, y) = (sx, sy)); the same set with both actions is the diagonal bi-act. The diagonal right (respectively, left, bi-) act is said to be finitely generated if there is a finite set A ⊆ S × S such that S × S = AS1 (respectively, S × S = S1A, S × S = SlASl).

In this paper we consider the question of finite generation for diagonal acts of certain infinite semigroups of transformations and relations. We show that the semi-groups of full transformations, partial transformations and binary relations on an infinite set each have cyclic diagonal right and left acts. The semigroup of full finite-to-one transformations on an infinite set has a cyclic diagonal right act but its diagonal left act is not finitely generated. The semigroup of partial injections on an infinite set has neither finitely generated diagonal right nor left act, but has a cyclic diagonal bi-act. The semigroup of bijections (symmetric group) on an infinite set does not have any finitely generated diagonal acts.

The recovery of flow curves for non-Newtonian fluids from Couette rheometry measurements involves the solution of a quite simple first kind Volterra integral equation with a discontinuous kernel for which the solution, as a summation of an infinite series, has been known since 1953. Various methods, including an Euler-Maclaurin sum formula, have been proposed for the estimation of the value of the summation. They all involve the numerical differentiation of the observational data. In this paper, the properties of Bernoulli polynomials, in conjunctions with the special structure of the integral equation, are exploited to derive a parametric family of representations for its solution. They yield formulas similar to, but more general than, the previously published Euler-Maclaurin sum formula representations. The parameterisation is then utilised to derive two new classes of approximations. The first yields a family of finite difference approximations, which avoids the direct numerical differentiation of the observational data, while the second generates a framework for the construction of improved power law approximations.

A new representation is obtained for the Riemann ξ function, in the form of a series of integrals, multiplied by an exponential factor capturing the correct decay rate for large imaginary argument. Each term in this series then has a simple stationary-phase asymptote, the total agreeing with the Riemann-Siegel sum.

Recently Calegari and Emerton made a conjecture about the 3-class groups of certain pure cubic fields and their normal closures. This paper proves their conjecture and provides additional insight into the structure of the 3-class groups of pure cubic fields and their normal closures.

A theorem independently due to A.I. Shirshov and E. Witt asserts that every subalgebra of a free Lie algebra (over a field) is free. The main step in Shirshov's proof is a little known but rather remarkable result: if a set of homogeneous elements in a free Lie algebra has the property that no element of it is contained in the subalgebra generated by the other elements, then this subset is a free generating set for the subalgebra it generates. Witt also proved that every subalgebra of a free restricted Lie algebra is free. Later G.P. Kukin gave a proof of this theorem in which he adapted Shirshov's argument. The main step is similar, but it has come to light that its proof contains substantial gaps. Here we give a corrected proof of this main step in order to justify its applications elsewhere.