Geometrical coefficients and the structure of the fixed-point set of asymptotically regular mappings in Banach spaces

It is shown that if E is a separable and uniformly convex Banach space with Opial’s property and C is a nonempty bounded closed convex subset of E, then for some asymptotically regular self-mappings of C the set of fixed points is not only connected but even a retract of C. Our results qualitatively complement, in the case of a uniformly convex Banach space, a corresponding result presented in [T. Domínguez, M.A. Japón, G. López, Metric fixed point results concerning measures of noncompactness mappings, in: W.A. Kirk, B. Sims (Eds.), Handbook of Metric Fixed Point Theory, Kluwer Acad. Publishers, Dordrecht, 2001, pp. 239-268].     

A Class of Solvable Lie Algebras with Triangular Decompositions

So far there has been elementary proof for Frobenius's theorem only in special cases: if the complement is solvable, see e.g. [3], if the complement is of even order, see e.g. [6]. In the first section we consider the case, when the order of the complement is odd. We define a graph the vertices of which are the set K^# of elements of our Frobenius group with 0 fixed points. Two vertices are connected with an edge if and only if the corresponding elements commute. We prove with elementary methods that K is a normal subgroup in G if and only if there exists an element x in K^# such that all elements of K^# belonging to the connected component C of K^# containing x are at most distance 2 from c and N^G(C) is not a -group, where is the set of prime divisors of the Frobenius complement of G. In the second section we generalize the case when the order of the complement is even, proving that the Frobenius kernel is a normal subgroup, if a fixed element a of the complement, the order of which is a minimal prime divisor of the order of the complement, generates a solvable subgroup together with any ofits conjugates. In the third section we prove a generalization of the Glauberman-Thompson normal p-complement theorem, and using this wegive another sufficient condition for the Frobenius kernel to be a normal subgroup for |G| odd, namely we prove this under the conditionthat all the Sylow normalizers in G intersect some of the complements     

The flow complex: A data structure for geometric modeling

We study a special case of the critical point (Morse) theory of distance functions namely, the gradient flow associated with the distance function to a finite point set in . The fixed points of this flow are exactly the critical points of the distance function. Our main result is a mathematical characterization and algorithms to compute the stable manifolds, i.e., the inflow regions, of the fixed points. It turns out that the stable manifolds form a polyhedral complex that shares many properties with the Delaunay triangulation of the same point set. We call the latter complex the flow complex of the point set. The flow complex is suited for geometric modeling tasks like surface reconstruction.     

Partial Actions of Groups on Cell Complexes

 In this paper, we study partial group actions on 2-complexes. Our results include a characterization, in terms of generating sets, of when a partial group action on a connected 2-complex has a connected globalization. Using this result, we give a short combinatorial proof that a group acting without fixed points on a connected 2-complex, with finite quotient, is finitely generated. This result is then generalized to characterize finitely generated groups as precisely those groups having a partial action, without fixed points, on a finite tree, with a connected globalization. Finally, using Bass-Serre theory, we determine when a partial group action on a graph has a globalization which is a tree. The author was supported in part by NSF-NATO postdoctoral fellowship DGE-9972697, by Praxis XXI scholarship BPD 16306 98 and by FCT through Centro de Matemática da Universidade do Porto. Received September 20, 2001; in revised form June 25, 2002     

Connectedness in topological linear spaces

The following properties, well known for normed linear spaces of dimension ≧2, are established for an arbitrary topological linear space of dimension ≧2: (a) every neighborhood of 0 contains one whose complement is connected; (b) the complement of a bounded set has exactly one unbounded component. Research supported by the National Science Foundation, U.S.A. (NSF-GP-378).     

Point determination in graphs

A point determining graph is defined to be a graph in which distinct nonadjacent points have distinct neighborhoods. Those graphs which are critical with respect to this property are studied. We show that a graph is complete if and only if it is connected, point determining, but fails to remain point determining upon the removal of any edge. We also show that every connected, point determining graph contains at least two points, the removal of either of which will result again in a point determining graph. Graphs which are point determining and contain exactly two such points are shown to have the property that every point is adjacent to exactly one of these two points.     

Cut points in some connected topological spaces

We prove that a connected topological space with endpoints has exactly two non-cut points and every cut point is a strong cut point; it follows that such a space is a COTS and the only two non-cut points turn out to be endpoints (in each of the two orders) of the COTS. A non-indiscrete connected topological space with exactly two non-cut points and having only finitely many closed points is proved homeomorphic to a finite subspace of the Khalimsky line. Further, it is shown, without assuming any separation axiom, that in a connected and locally connected topological space X, for a, b in X, S[a,b] is compact whenever it is closed. Using this result we show that an H(i) connected and locally connected topological space with exactly two non-cut points is a compact COTS with end points.     

Li and Yorke chaos with respect to the cardinality of the scrambled sets

In the present paper we study Li and Yorke chaos on several spaces in connection with the cardinality of its scrambled sets. We prove that there is a map on a Cantor set and a map on a two-dimensional arcwise connected continuum (with empty interior) such that each scrambled set contains exactly two points.     

Implicit iterative algorithms for asymptotically nonexpansive mappings in the intermediate sense and Lipschitz-continuous monotone mappings

In this paper, we introduce some implicit iterative algorithms for finding a common element of the set of fixed points of an asymptotically nonexpansive mapping in the intermediate sense and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. These implicit iterative algorithms are based on two well-known methods: extragradient and approximate proximal methods. We obtain some weak convergence theorems for these implicit iterative algorithms. Based on these theorems, we also construct some implicit iterative processes for finding a common fixed point of two mappings, such that one of these two mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the other mapping is asymptotically nonexpansive.     

扭转映射的一个不动点定理

本文利用集连通理论，给出了非保面积的扭转映射至少有两个不动点的一个新定理．     

Harmonische Involutionen und Kreisspiegelungen in Möbiusebenen

An involutory automorphism of an inversive plane whose set of fixed points consists of exactly two points resp. of a circle is called a harmonic involution resp. an inversion. In this paper we study inversive planes with sufficiently many such involutions. Herrn Walter Benz zum 75. Geburtstag gewidmet     

Laguerre-Ebenen mit Symmetrien an Punktepaaren und Zykeln

Let ∞ be a point of a Laguerre plane, such that

1. For any cycle containing ∞ there exists an automorphism of order 2 whose set of fixed points is exactly z.
2. For any point X, not parallel to ∞, there exists an automorphism of order 2 whose set of fixed points is exactly {∞,X}.

Then the give Laguerre plane is a Miquelian one of characteristik ≠ 2.     

Matroids with a unique non-common circuit containing a fixed element

In this paper, we define a matroid operation that generalizes the circuit-hyperplane relaxation. This operation is used to characterize when a pair of connected matroids over the same ground set have exactly one non-common circuit containing a fixed element.     

The Number of Cyclic Configurations of Type (v3) and the Isomorphism Problem

A configuration of points and lines is cyclic if it has an automorphism that permutes its points in a full cycle. A closed formula is derived for the number of nonisomorphic connected cyclic configurations of type (v₃), i.e. which have v points and lines, and each point/line is incident with exactly three lines/points. In addition, a Bays–Lambossy type theorem is proved for cyclic configurations if the number of points is a product of two primes or a prime power.     

Graphs which are locally a cube

We prove that there are exactly two connected graphs which are locally a cube: a graph on 15 vertices which is the complement of the (3×5)-grid and a graph on 24 vertices which is the 1-skeleton of a certain 4-dimensional regular polytope called the 24-cell.     

ON STABLE LINE SEGMENTS IN ALL TRIANGULATIONS OF A PLANAR POINT SET

Abstract. Let S be a set of finite plauar points. A llne segment L(p, q) with p, q E Sis called a stable line segment of S, if there is no Line segment with two endpoints in S intersecting L(p, q). In this paper, some geometric properties of the set of all stable line segments     

On certain remarkable curves of genus five

The aim of this note is twofold. First to show the existence of genus five curves having exactly twenty four Weierstrass points, which constitute the set of fixed points of three distinct elliptic involutions on them. Second to characterize these curves, in fact we prove that all such curves are bielliptic double cover of Fermat's quartic.     

Fixed-point free automorphisms of abelian varieties

An automorphismf of an abelian varietyX is called fixed point free if it admits no fixed points other than the origin and this is of multiplicity one. It is well known that the elliptic curve withj-invariant 0 is the only elliptic curve admitting a fixed point free automorphism. In this note, this result is extended to abelian varieties of higher dimensions and some connected commutative algebraic groups.Supported by DFG-contract La 318/4 and EC-contract SC1-0398-C(A).     

Ned sets on a hyperplane

Under study are the sets in ℝ ⁿ (NED sets) each of which does not affect the conformal capacity of any condenser with connected plates disjoint from this set. These sets are removable singularities of quasiconformal mappings, which explains our interest in them. For compact sets on a hyperplane we obtain a geometric criterion of the NED property; we point out a simple sufficient condition for an NED set in terms of the connected attainability of its points from its complement in the hyperplane. For compact sets on a hypersphere we obtain a criterion for an NED set in terms of the reduced module at a pair of points in its complement. We establish that a compact set on a hypersphere S, removable for the capacity in at least one spherical ring concentric with S and containing S, is an NED set.