SOLVEBLOK: A Package for Solving Almost Block Diagonal Linear Systems
In spline approximation and in the numerical solution of (systems of) ordinary differential equations, one is eventually faced with linear systems of a rather characteristic, almost block diagonal, form which reflects the underlying geometry of the problems. While these systems could be treated as banded systems, it is more efficient, both in the generation of the systems and in their numerical solution, to work explicitly with their block structure. The report describes a package of FORTRAN routines for the solution of an almost block diagonal system by Gauss elimination with scaled row pivoting. Two examples are given. The first involves the solution of an m-th order ordinary differential equation by collocation. The second example offers a package of routines for the solution of a system of first order ODEs by collection at Lobatto points which is then used on a somewhat tricky reentry problem.

title: An analysis of the box and trapezoidal schemes for linear singularly perturbed boundary value problems
abstract: Stability and convergence results are derived for the box and trapezoidal schemes applied to boundary value problems for linear singularly perturbed first order systems of o.d.e.'s without turning points.

title: The application of implicit Runge-Kutta and collection methods to boundary-value problems
abstract: The solution of a nonlinear system of first order differential equations with nonlinear boundary conditions by implicit Runge-Kutta methods based on interpolatory quadrature formulae is examined. An equivalence between implicit Runge-Kutta and collocation schemes is established. It is shown that the difference equations obtained have a unique solution in a neighbourhood of an isolated solution of the continuous problem, that this solution can be computed by Newton iteration and that it converges to the isolated solution. The order of convergence is equal to the degree of precision of the related quadra- ture formula plus one. The efficient implementation of the methods is discussed and numerical examples are given. 1.  We investigate the application of certain implicit Runge- Kutta methods (cf. Butcher (2)) to the numerical solution of nonlinear boundary- value problems of the form

title: The convergence of shooting methods
abstract: Shooting methods for nonlinear boundary value problems are examined. It is shown that the methods converge whenever the problem is well posed in the sense that the solution to be computed is isolated.

title: Symmetric graphs with projective subconstituents
abstract: Let P be an undirected graph and G a subgroup of aut(r) acting transitively on the vertex set of P. Let x be an arbitrary vertex of P. We denote by P(x) the set of vertices adjacent to x and by G(x)F(x) the permutation group induced by the stabilizer G (x) of x in G on P(x); G (x)r(X) is called the subconstituent of the pair (P, G). For each i E N let Gi(x) = {a E G(x)Ia E G(y) if a (x, y) 1 then s 3. In , it was shown that s 3. We prove the following partial result:

title: On the Approximation of Fixed Points of Nonlinear Compact Operators
abstract: Certain methods for the approximate solution of $x - Kx = 0$, where K is a compact operator, lead to a family of operator equations $x - K_n x = 0$, $n \geqq 1$, where $K_n $,$n \geqq 1$, are collectively compact and converge pointwise to K. In this paper a new and simplified proof of the existence and uniqueness of a solution of $x - K_n x = 0$ in a neighborhood of an isolated solution of $x - Kx = 0$ is given.

title: A uniqueness lemma for groups generated by 3-transpositions

title: Groups with a $(B,\,N)$-pair and locally transitive graphs
abstract: Let Γ be an undirected graph and G a subgroup of aut (Γ). We denote by d(x, y) the distance between two vertices x and y, by E(Γ) the edge set of Γ, by V(Γ) the vertex set of Γ, by Γ(x) the set of neighbors of the vertex x and by G(x) the permutation group induced by the stabilizer G(x) on Γ(x). For each i e N, let G^x) = {a \ a e G(y) for every y with d(x, y) < ϊ). An s-path is an ordered sequence (x0, ••-,#«) of s + 1 vertices xt with xt e jP(Xi_i) for 1 < i < s and ^ ^ xt_2 for 2 < i < s. For each vertex x, let WS(JC) be the set of s-paths (x0, , xs) with x = x0. We say that the graph Γ is locally (G, s)-transitive if for every vertex x, G(x) acts transitively on Ws(x) but not on Ws+1(x) (compare , ). If, in addition, G acts transitively on V(Γ), then Γ is called (G, s)-transitive; otherwise Γ is bipartite with vertex blocks Vo and Vλ and G acts transitively on both Vo and VΊ, assuming that Γ is connected and s > 1. Now let G be a finite group with a (B, iV)-pair whose Weyl group is a dihedral group Z)2w of order 2n (n > 2) and .Γ be the incidence graph of the associated coset geometry as defined in [3, p. 129] (or [2, (15. 5. 1)]). The graph Γ has the following properties: (A) V(Γ) = Vo U V1 with Vo Π V1 0 and Γ(x) c F ^ for every vertex x e ^ (ί = 0 and 1). For j = 0 and 1 there exists a dt e N such that \Γ(x)\ = dt + 1 for every vertex xe Vi% The diameter of Γ is πand the girth 2n. (B) Γ is locally (G, τι + l)-transitive. A generalized n-gon of order (d0, dλ) is, by definition, an incidence structure whose incidence graph has the properties listed in (A). W. Feit and G. Higman have shown in  that finite generalized ra-gons of order (d0, dλ) with dod1 > 1 exist only for n = 2, 3, 4, 6, 8 and 12, that

title: Permutation groups with projective unitary subconstituents
abstract: AssTRAcT. Let r be a finite directed graph with vertex set V(I) and edge set EQ7) and let G be a subgroup of aut(I) which we assume to act transitively on both V(r) and E(r). Suppose that for some prime power q, the stabilizer G(x) of a vertex x induces on both {yI(x,y) E E(r)) and {wl(w, x) E E)} a group lying between PSU(3, q2) and Pr1U(3, q2). It is shown that if G acts primitively on V(I), then for each edge (x, y), the subgroup of G(x) fixing every vertex in {wI(x, w) or (y, w) E E(r)) is trivial.

title: Presentations for (G,s)-transitive graphs of small valency
abstract: Given a group G , a subgroup H and an element a ∈ G , we define Γ( G, H, a ) to be the graph on the set of left-cosets of H in G , where two left-cosets g 1 H and g 2 H are adjacent whenever . We will consider this construction only in the case that The first of these conditions assures that adjacency is a symmetric relation (i.e. that Γ( G, H, a ) is an undirected graph) and the second assures that Γ( G, H, a ) is connected.

title: Über symmetrische Graphen und die projektiven Gruppen