119709952
Product Integration for the Generalized Abel Equation
The solution of the generalized Abel integral equation g(t) f {k(t, s)/(t s)a}f (s) ds, 0 < a < 1, where k(t, s) is continuous, by the product integration analogue of the trapezoidal method is examined. It is shown that this method has order two convergence for a E [al, 1) with al 0.2117. This interval contains the important case a = 2. Convergence of order two for a E (0, al) is discussed and illustrated numerically. The possibility of constructing higher order methods is illustrated with an example.
6860028
SOLVEBLOK: A Package for Solving Almost Block Diagonal Linear Systems
Abstract : 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.
119855015
The application of implicit Runge-Kutta and collection methods to boundary-value problems
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. Introduction. 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
121432014
The convergence of shooting methods
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.
120998826
Symmetric graphs with projective subconstituents
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 [8], it was shown that s 3. We prove the following partial result:
122289259
On the Approximation of Fixed Points of Nonlinear Compact Operators
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.
122390175
An application of p -factorization methods to symmetric graphs
125321998
A characterization of the group Co 3 as a transitive extension of HS
123397959
A uniqueness lemma for groups generated by 3-transpositions
118512387
Groups with a $(B,\,N)$-pair and locally transitive graphs
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 [1], [11]). 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 [3] that finite generalized ra-gons of order (d0, dλ) with dod1 > 1 exist only for n = 2, 3, 4, 6, 8 and 12, that
120974879
Presentations for (G,s)-transitive graphs of small valency
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.