Abstract. This paper deals with exploiting symmetry for solving linear and integer programming problems. Basic properties of linear representations of finite groups can be used to reduce symmetric linear programming to solving linear programs of lower dimension. Combining this approach with knowledge of the geometry of feasible integer solutions yields an algorithm for solving highly symmetric integer linear programs which only takes time which is linear in the number of constraints and quadratic in the dimension.

This paper deals with smooth stable planes which generalize the notion of differentiable (affine or projective) planes [7). It is intended to be the first one of a series of papers on smooth incidence geometry based on the Habilitationsschrift of the author. It contains the basic definitions and results which are needed to build up a foundation for a systematic study of smooth planes. We define smooth stable planes, and we prove that point rows and line pencils are closed submanifolds of the point set and line set, respectively (Theorem (1.6». Moreover, the flag space is a closed submanifold of the product manifold PxI. (Theorem (1.14», and the smooth structure on the set P of points and on the set I. of lines is uniquely determined by the smooth structure of one single line pencil. In the second section it is shown that for any point pEP the tangent space T. P carries the structure of a locally compact affine translation plane A., see Theorem (2.5). Dually, we prove in Section 3 that for any line LEI. the tangent space TLI. together with the set SL={TLI.plpEL} gives rise to some shear plane. It turned out that the translation planes Ap are one of the most important tools in the investigation of smooth incidence geometries. The linearization theorems (3.9), (3.11), and (4.4) can be viewed as the main results of this paper. In the closing section we investigate some homogeneity properties of smooth projective planes.

Abstract. We consider homomorphisms between abstract, topological, and smooth generalized polygons. It is shown that a continuous homomorphismis either injective or locally constant. A continuous homomorphism between smooth generalized polygons is always a smooth embedding. We apply this result to isoparametric submanifolds.Mathematics Subject Classifications (1991): 51A25, 51 E24, 51H20, 51 H25.The aim of this paper is to investigate homomorphisms of abstract, topological, and smooth generalized polygons. The first two sections deal with generalized polygons and their homomorphisms. We have tried to make this paper self-contained, so we give a short exposition of the coordinatization and the algebraic operations (+, -, o,/) of a generalized polygon. The most important theorems at this stage are the characterization (2.7) of injective homomorphisms, and the proof (2.9) of Pasini's theorem [27] that the fibers of a non-injective homomorphism are infinite.Topological polygons are introduced in Section 3. Here, the main result is that a homomorphism is either injective or locally constant (3.4); in particular, a connected polygon admits only injective homomorphisms. For topological projective planes, this has been proved by Breitsprecher [4]. This result may be compared to Pasini's theorem [27] that finite polygons admit only injective homomorphisms.In the last section, we introduce smooth polygons. The main result of this paper states that a continuous homomorphism between smooth polygons is always a smooth embedding (4.7). In particular, every continuous automorphism of a smooth polygon is smooth, and thus the topological automorphism group of the polygon, endowed with the compact-open topology, is a smooth Lie transformation group (4.9). As an application, we prove a strong inhomogeneity theorem for a class of isoparametric hypersurfaces.

Smooth stable planes have been introduced in [4]. We show that every continuous collineation between two smooth stable planes is in fact a smooth collineation. This implies that the group $ of all continuous collineations of a smooth stable plane is a Lie transformation group on both the set P of points and the set $ of lines. In particular, this shows that the point and line sets of a (topological) stable plane $ admit at most one smooth structure such that $ becomes a smooth stable plane. The investigation of central and axial collineations in the case of (topological) stable planes due to R. Löwen ([25], [26], [27]) is continued for smooth stable planes. Many results of [26] which are only proved for low dimensional planes $ are transferred to smooth stable planes of arbitrary finite dimension. As an application of these transfers we show that the stabilizers $ and $ (see (3.2) Notation) are closed, simply connected, solvable subgroups of $ (Corollary (4.17)). Moreover, we show that $ is even abelian (Theorem (4.18)). In the closing section we investigate the behaviour of reflections.

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