Almost arcwise connected continua without dense arc components

A continuum M is almost arcwise connected if each pair of nonempty open subsets of M can be joined by an arc in M. An almost arcwise connected plane continuum without a dense arc component can be defined by identifying pairs of endpoints of three copies of the Knaster indecomposable continuum that has two endpoints. In [7] K.R. Kellum gave this example and asked if every almost arcwise connected continuum without a dense arc component has uncountably many arc components. We answer Kellum's question by defining an almost arcwise connected plane continuum with only three arc components none of which are dense. A continuum M is almost Peano if for each finite collection $\text{C}$ of nonempty open subsets of M there is a Peano continuum in M that intersects each element of $\text{C}$. We define a hereditarily unicoherent almost Peano plane continuum that does not have a dense arc component. We prove that every almost arcwise connected planar λ-dendroid has exactly one dense arc component. It follows that every hereditarily unicoherent almost arcwise connected plane continuum without a dense arc component has uncountably many arc components. Using an example of J. Krasinkiewicz and P Minc [8], we define an almost Peano λ-dendroid that do not have a dense arc component. Using a theorem of J.B. Fugate and L. Mohler [3], we prove that every almost arcwise connected λ-dendroid without a dense arc component has uncountably many arc components. In Euclidean 3-space we define an almost Peano continuum with only countably many arc components no one of which is dense. It is not known if the plane contains a continuum with these properties.     

A disjoint disks property for 3-manifolds

We show that the map separation property (MSP), a concept due to H.W. Lambert and R.B. Sher, is an appropriate analogue of J.W. Cannon’s disjoint disks property (DDP) for the class $\text{C}$ of compact generalized 3-manifolds with zero-dimensional singular set, modulo the Poincaré conjecture. Our main result is that the Poincaré conjecture (in dimension three) is equivalent to the conjecture that every X?$\text{C}$ with the MSP is a topological 3-manifold.     

Examples of cell-like decompositions of the infinite-dimensional manifolds σ and Σ

Denote by σ the subspace of Hilbert space {(x_i)?l₂:x_i=0 for all but finitely many i}. Examples of cell-like decompositions of σ are constructed that have decomposition spaces that are not homeomorphic to σ. At one extreme is a cell-like decomposition G of σ produced using ghastly finite dimensional examples such that the decomposition space σ?G contains no embedded 2-cell but (σ?G)×$\text{R}$ is homeomorphic to σ. At the other extreme is a cell-like decomposition G of σ satisfying: (a) the nondegeneracy set N_G={g?G:g≠point} consists of countably many arcs (necessarily tame); (b) the nondegeneracy set N_G is a closed subset of the decomposition space σ?G; (c) each map f:B²→σ?G of a 2-cell into σ?G can be approximated arbitrarily closely by an embedding; (d) σ?G is not homeomorphic to σ but (σ?G)×$\text{R}$ is homeomorphic to σ. The fact that both conditions (a) and (b) can be satisfied (and have (d) hold) is directly attributable to σ’s incompleteness as a topological space.     

Preserving Z-sets by Dranishnikov's resolution

We prove that Dranishnikov's k-dimensional resolution is a UV^n − 1-divider of Chigogidze's k-dimensional resolution ck. This fact implies that preserves Z-sets. A further development of the concept of UV^n − 1-dividers permits us to find sufficient conditions for to be homeomorphic to the Nöbeling space νk or the universal pseudoboundary σk. We also obtain some other applications.     

Homogeneous curves that contain arcs

The author has classified atriodic, homogeneous, one-dimensional continua that contain arcs— they are precisely the solenoids. This paper begins the study of homogeneous, one-dimensional continua that contain an arc and a triod.     

Quadrature rule-based bounds for functions of adjacency matrices

Bounds for entries of matrix functions based on Gauss-type quadrature rules are applied to adjacency matrices associated with graphs. This technique allows to develop inexpensive and accurate upper and lower bounds for certain quantities (Estrada index, subgraph centrality, communicability) that describe properties of networks.     

Almost arcwise connectivity in unicoherent continua

K.R. Kellum has proved that a continuum is an almost continuous image of the interval [0, 1] if and only if it is an almost Peano continuum. Hence, a continuum is an almost continuous image of [0, 1] if it has a dense arc component.Our principal result is that any almost arcwise connected, semi-hereditarily unicoherent, metric continuum with only countably many arc components has a dense arc component. An example is given to show that this is not true for unicoherent continua in general. It is also shown that any semi-hereditarily unicoherent continuum with only countably many arc components has at most one dense arc component, and if it has a dense arc component, then every other arc component is nowhere dense. This generalizes results of Fugate and Mohler for λ-dendroids.     

On the fundamental groups of one-dimensional spaces

We study here a number of questions raised by examining the fundamental groups of complicated one-dimensional spaces. The first half of the paper considers one-dimensional spaces as such. The second half proves related results for general spaces that are needed in the first half but have independent interest. Among the results we prove are the theorem that the fundamental group of a separable, connected, locally path connected, one-dimensional metric space is free if and only if it is countable if and only if the space has a universal cover and the theorem that the fundamental group of a compact, one-dimensional, connected metric space embeds in an inverse limit of finitely generated free groups and is shape injective.     

The hyperspace of hereditarily decomposable subcontinua of a cube is the Hurewicz set

The hyperspaces of hereditarily decomposable continua and of decomposable subcontinua without pseudoarcs in the cube of dimension greater than 2 are homeomorphic to the Hurewicz set of all nonempty countable closed subsets of the unit interval [0,1]. Moreover, in such a cube, all indecomposable subcontinua form a homotopy dense subset of the hyperspace of (nonempty) subcontinua.     

Maximal elements in topological and generalized metric spaces

In this note, some problems concerning existence of maximal elements in a topological as well as in a generalized metric space, equipped with an ordering, are studied. The results presented here may be considered as a partial refinement of those established in [2] for uniform structures.     

The Mysterious 2-Crown

We show that the 2-crown is not coproductive, which is to say that the class of those bounded distributive lattices whose Priestley spaces lack any copy of the 2-crown is not productive. We do this by first exhibiting a general construction to handle questions of this sort. We then use a particular instance of this constrution, along with some of the combinatorial features of projective planes, to show that the 2-crown is not coproductive. This paper is dedicated to Walter Taylor. Received November 24, 2004; accepted in final form July 16, 2005. The first author would like to express his thanks for support by project LN 00A056 of the Ministry of Education of the Czech Republic. The second author would like to express his thanks for support by project LN 00A056 of the Ministry of Education of the Czech Republic, by the NSERC of Canada and by the Gudder Trust of the University of Denver. The third author would like to express his thanks for support by the NSERC of Canada and partial support by project LN 00A056 of the Ministry of Education of the Czech Republic.     

Relaxed and stabilized incomplete factorizations for non-self-adjoint linear systems

Two classes of incomplete factorization preconditioners are considered for nonsymmetric linear systems arising from second order finite difference discretizations of non-self-adjoint elliptic partial differential equations. Analytic and experimental results show that relaxed incomplete factorization methods exhibit numerical instabilities of the type observed with other incomplete factorizations, and the effects of instability are characterized in terms of the relaxation parameter. Several stabilized incomplete factorizations are introduced that are designed to avoid numerically unstable computations. In experiments with two-dimensional problems with variable coefficients and on nonuniform meshes, the stabilized methods are shown to be much more robust than standard incomplete factorizations.The work presented in this paper was supported by the National Science Foundation under grants DMS-8607478, CCR-8818340, and ASC-8958544, and by the U.S. Army Research Office under grant DAAL-0389-K-0016.     

The orderability of nonarchimedean spaces

Let X be a nonarchimedean space and C be the union of all compact open subsets of X. The following conditions are listed in increasing order of generality. (Conditions 2 and 3 are equivalent.) 1. X is perfect; 2. C is an F_σ in X; 3. C? is metrizable; 4. X is orderable. It is also shown that X is orderable if $\text{C}\text{?}\text{?C}$ is scattered or X is a GO space with countably many pseudogaps. An example is given of a non-orderable, totally disconnected, GO space with just one pseudogap.     

Homology Groups of Conormal Varieties

Let G be a real Lie group acting on real analytic manifold with finitely many orbits. We prove that the characteristic cycle map is a surjective homomorphism from the K-group of G-equivariant sheaves on X to the top homology group of the conormal variety of the G-action on X. We also show that the top homology group of the G-action on X is a free -module of rank equal to the number of G-orbits. This work was completed with the support of the Ministry of Science, Education and Sport of Croatia, and Infodesign d.o.o., Varaždin.     

Classifying simple closed curve pairs in the 2-sphere and a generalization of the Schoenflies theorem

A classification theory is developed for pairs of simple closed curves (A,B) in the sphere S², assuming that A∩B has finitely many components. Such a pair of simple closed curves is called an SCC-pair, and two SCC-pairs (A,B) and (A^′,B^′) are equivalent if there is a homeomorphism from S² to itself sending A to A^′ and B to B^′. The simple cases where A and B coincide or A and B are disjoint are easily handled. The component code is defined to provide a classification of all of the other possibilities. The component code is not uniquely determined for a given SCC-pair, but it is straightforward that it is an invariant; i.e., that if (A,B) and (A^′,B^′) are equivalent and C is a component code for (A,B), then C is a component code for (A^′,B^′) as well. It is proved that the component code is a classifying invariant in the sense that if two SCC-pairs have a component code in common, then the SCC-pairs are equivalent. Furthermore code transformations on component codes are defined so that if one component code is known for a particular SCC-pair, then all other component codes for the SCC-pair can be determined via code transformations. This provides a notion of equivalence for component codes; specifically, two component codes are equivalent if there is a code transformation mapping one to the other. The main result of the paper asserts that if C and C^′ are component codes for SCC-pairs (A,B) and (A^′,B^′), respectively, then (A,B) and (A^′,B^′) are equivalent if and only if C and C^′ are equivalent. Finally, a generalization of the Schoenflies theorem to SCC-pairs is presented.     

The combinatorially regular polyhedra of index 2

Summary We investigate polyhedral realizations of regular maps with self-intersections in E³, whose symmetry group is a subgroup of index 2 in their automorphism group. We show that there are exactly 5 such polyhedra. The polyhedral sets have been more or less known for about 100 years; but the fact that they are realizations of regular maps is new in at least one case, a self-dual icosahedron of genus 11. Our polyhedra are closely related to the 5 regular compounds, which can be interpreted as discontinuous polyhedral realizations of regular maps.The author was born on March 5, 1937; so exactly half a century after Otto Haupt.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday.     

Decompositions into codimension two spheres and approximate fibrations

Let M denote an orientable (n+2)-manifold (n?1) and G an upper semicontinuous decomposition of M into compacta having the shape of the n-sphere. In this context it is shown that the decomposition space is a 2-manifold. Moreover, it is established that the decomposition map is an approximate fibration for n>1, while for n=1 the map is an approximate fibration over the complement of a locally finite set.     

Spline approximation methods for multidimensional periodic pseudodifferential equations

We investigate several numerical methods for solving the pseudodifferential equationAu=f on the n-dimensional torusT ⁿ . We examine collocation methods as well as Galerkin-Petrov methods using various periodical spline functions. The considered spline spaces are subordinated to a uniform rectangular or triangular grid. For given approximation method and invertible pseudodifferential operatorA we compute a numerical symbol ^C , resp. ^G , depending onA and on the approximation method. It turns out that the stability of the numerical method is equivalent to the ellipticity of the corresponding numerical symbol. The case of variable symbols is tackled by a local principle. Optimal error estimates are established.The second author has been supported by a grant of Deutsche Forschungsgemeinschaft under grant namber Ko 634/32-1.