In this paper, we determine the semigroup End D of endomorphisms of the infinite dihedral group D, and give a multiplication table for it. We determine the additive structure of the near-rings E(D) generated by the endomorphisms of D, A(D) generated by the automorphisms of D and I(D) generated by the inner automorphisms of D, and determine their radicals and all their maximal right ideals.

Using a variational approach we obtain the existence of at least three periodic solutions for discontinuous perturbations of the vector p-Laplacian operator .

Results by Simader, Brézis and Cycon of the genre ‘locally essentially self-adjoint implies globally essentially self-adjoint’ are generalised to Schrödinger operators that are not necessarily bounded from below.

We prove optimal lower bounds for arbitrary eigenvalue ratios (μm/μn) of the Sturm–Liouville operator with Dirichlet and Neumann boundary conditions. These imply optimal bounds for the eigenvalue gaps (μm – μn) of the corresponding problem. The method can be generalised to consider general separated endpoint boundary conditions.

We consider the weak Neumann problem for logarithmic potentials in plane domains. We prove that this problem can be treated by the Fredholm–Radon method if and only if the boundary of the corresponding domain is formed by finitely many curves fulfilling specified regularity conditions.

Linear second order differential equations of the form d2w/dz2 − {u2f(u, z) + g(z)}w = 0 are studied, where |u| → ∞ and z lies in a complex bounded or unbounded domain D. If f(u, z) and g(z) are meromorphic in D, and f(u, z) has no zeros, the classical Liouville-Green/WKBJ approximation provides asymptotic expansions involving the exponential function. The coefficients in these expansions either multiply the exponential or in an alternative form appear in the exponent. The latter case has applications to the simplification of turning point expansions as well as certain quantum mechanics problems, and new computable error bounds are derived. It is shown how these bounds can be sharpened to provide realistic error estimates, and this is illustrated by an application to modified Bessel functions of complex argument and large positive order. Explicit computable error bounds are also derived for asymptotic expansions for particular solutions of the nonhomogeneous equations of the form d2w/dz2 − {u2f(z) + g(z)}w = p(z).

The Fibonacci groups F(r, n) have been studied by various authors, chiefly in order to determine which ones are finite. This article contains a summary of the known results about this problem, followed by some further results obtained by the author. In particular, the orders of the groups F(r, 3) for r ≡ 2 (mod 3) and F(r, 4) for r ≡ 2 (mod 4) are determined, and various other Fibonacci groups are proved infinite by methods similar to those of Chalk and Johnson.

We introduce the totally multiplicatively prime algebras as those normed algebras for which there exists a positive number K such that K‖F‖‖a‖ ≤ ‖WF,a‖ for all F in M(A) (the multiplication algebra of A) and a in A, where WF,a denotes the operator from M(A) into A defined by WF,a(T) = FT(a) for all T in M(A). These algebras are totally prime and their multiplication algebra is ultraprime. We get the stability of the class of totally multiplicatively prime algebras by taking central closure. We prove that prime H*-algebras are totally multiplicatively prime and that the ℓ1-norm is the only classical norm on the free non-associative algebras for which these are totally multiplicatively prime.

In this communication an attempt is made to correlate the different members of the Lower Carboniferous Limestone Series of the East of Scotland with what has been chosen as the type section at Hurlet, near Paisley. For close on fifty years Scottish geologists have been endeavouring to correlate the Lower Carboniferous Limestones of Fife and the Lothians, and in general the Carboniferous Rocks of Scotland, with the Hurlet Section, but up till the present time the various attempts have only met with a partial success, the reason for this being, that our knowledge of the Hurlet section has by no means been either sufficiently detailed or accurate enough to admit of comparison with other widely separated Scottish sections. The important question also arises as to what extent the Hurlet Section can be considered as typical of the group of deposits as a whole.

In this work we derive by $\Gamma$-convergence techniques a model for brittle fracture linearly elastic plates. Precisely, we start from a brittle linearly elastic thin film with positive thickness $\rho$ and study the limit as $\rho$ tends to $0$. The analysis is performed with no a priori restrictions on the admissible displacements and on the geometry of the fracture set. The limit model is characterized by a Kirchhoff-Love type of structure.

The period function of Liénard systems ẋ = −y + F(x), ẏ = g(x) associated with a centre is considered and sufficient conditions are presented for the period function to be strictly and globally monotone increasing and to have at least one critical point. Examples of applications and remarks on related work are also given.

The Grunsky map is known to be holomorphic on the universal Teichmüller space. In this paper it is proved that the Grunsky map induces a holomorphic map on the asymptotic Teichmüller space. The Caratháodory and Kobayashi metrics on the asymptotic Teichmüller space are studied as applications.

We study Berman's problem

subjects to the conditions f(0) =f″(0) = f′(1) = f(1)− 1 = 0, which arises from the study of laminar flows in channels with porous walls. Positive (negative) Re denotes the case of suction (injection) flows. Preliminary numerical studies indicated the existence of three different types of solutions and a small portion of mathematical evidence was provided. In this paper, we are able to show that there exist connected sets in the Re-K plane on which different types of solutions occur. In particular, our result verifies that the problem possesses all three types of suction solutions for sufficiently large Re. Moreover, the limiting injection solution is also obtained.

The paper presents sufficient conditions on the coefficients of second and fourth order differential equations to ensure that there exists at least one pair of conjugate points on an interval (a, b), −∞≦ a <b ≦ ∞. Oscillation criteria related to the equation (p(x)y″)″ + q(x)y = 0, 0 < x < ∞, are proved with no sign restrictions on q(x).

In this paper we establish Perron and Krein–Rutman-like theorems for an operator mapping a cone into the interior of the cone, by considering the discrete dynamical system for the induced operator on the projective space (= sphere). Existence of a positive eigenvector reduces to showing that the ω-limit set of the induced operator consists of a single equilibrium. A special feature of our approach is that the convexity of the cone is needed only for establishing the non-emptiness of the w-limit set. This allows us in finite dimensions to establish an abstract Perron Theorem for non-convex cones.

In this work we discuss the Cauchy problem for a class of nonlinear dissipative equations as well as the existence of a global attractor and we estimate its dimension in the sense of the Hausdorff (or fractal) dimension.

The first boundary value problem for a linear second order parabolic equation is studied under the assumption that the inhomogeneous term is continuous in space and time and Hölder-continuous only with respect to the space variables.

A comprehensive appraisal of the title problem is presented in terms of a characterizing nondimensional co-ordinate ξ which is based upon the half excess of the momentum of the jet, J. Perturbation features of the problem appear as regular and singular boundary conditions in ξ upstream and downstream respectively. The conservation of momentum excess provides a monitor on the consistency of regular and singular perturbation series solutions. In particular the conservation constraint on the downstream singular perturbation solution confirms the inadequacy of expansions in inverse half powers of ξ and justifies formally the introduction of logarithmic terms.

The formulation provides the basis for a complete numerical integration over the semi-infinite region. Accordingly detailed knowledge of velocity excess along the axis of the jet is obtained and an undetermined coefficient in the asymptotic downstream perturbation solution may be estimated.

Taking the argument used by De Giorgi to obtain the rectifiability of the reduced boundary of a set of finite perimeter in ℝN, we prove that a set E of finite perimeter in an open set Ω of ℝN may be approached, in the sense of the BV(Ω) norm, by sets whose boundary is included in a finite union of 1 hypersurfaces; more precisely, arbitrarily large parts (for the N−1 measure in Ω) of the essential boundaries of E and of the approximating set coincide and are included in a single 1 hypersurface.

Consider the differential expression

where p and w > 0 are real-valued and q is complex-valued on I. A number of criteria are established for certain extensions of the minimal operator generated by τ in the weighted Hilbert space to be maximal dissipative.