Tree-like continua and 2-to-1 maps

It is not known if there is a 2-to-1 map from a continuum onto a tree-like continuum. In fact, it is not known if there is a 2-to-1 map onto a hereditarily decomposable tree-like continuum. We show that the domain of such a map would have to contain an indecomposable continuum.

     

Exactly -to-1 maps and hereditarily indecomposable tree-like continua

In 1947, W.H. Gottschalk proved that no dendrite is the continuous, exactly -to-1 image of any continuum if . Since that time, no other class of continua has been shown to have this same property. It is shown that no hereditarily indecomposable tree-like continuum is the continuous, exactly -to-1 image of any continuum if .

     

A hereditarily indecomposable tree-like continuum without the fixed point property

A hereditarily indecomposable tree-like continuum without the fixed point property is constructed. The example answers a question of Knaster and Bellamy.

     

Each locally one-to-one map from a continuum onto a tree-like continuum is a homeomorphism

In 1977 T. Mackowiak proved that each local homeomorphism from a continuum onto a tree-like continuum is a homeomorphism. Recently, J. Rogers proved that each locally one-to-one (not necessarily open) map from a hereditarily decomposable continuum onto a tree-like continuum is a homeomorphism, and this paper removes ``hereditarily decomposable" from the hypothesis of Rogers' theorem.

     

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.     

A continuum whose hyperspace of subcontinua is not its continuous image

We construct a metric continuum such that the hyperspace of subcontinua, , of is not a continuous image of . This answers a question by I. Krzeminska and J. R. Prajs.

     

An example of a nonarcwise connected rim-countable locally connected continuum

An example of a rim-countable, locally connected, but nonarcwise connected continuum is constructed, which is important for the classification of locally connected continua. The point of interest in this result is that there are no metrizable locally connected continua with the above properties. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 659–666, May, 1999.     

Reduced 2-to-1 maps and decompositions of graphs with no 2-to-1 cut sets

A graph has an increasing ear decomposition if it can be constructed from a simple closed curve by attaching arcs in stages with the endpoints of each arc attached to different points so that at least one new branch point is formed at each stage. A reduced 2-to-1 map is a 2-to-1 map that does not have a restriction that is 2-to-1. A 2-to-1 cut set of a graph G is a finite subset B such that G⧹B has at least 2|B| components. A graph has an increasing ear decomposition if and only if it does not have a 2-to-1 cut set, and a graph is the image of a reduced 2-to-1 map if and only if it does not have a 2-to-1 cut set.     

On the n-fold symmetric product suspensions of a continuum

In 1979 Sam B. Nadler, Jr. defined the hyperspace suspension of a continuum. We define the n-fold symmetric product suspensions of a continuum using n-fold symmetric products. We study some properties of this hyperspace: unicoherence, local connectedness, arcwise connectedness.     

Open colorings, the continuum and the second uncountable cardinal

The purpose of this article is to analyze the cardinality of the continuum using Ramsey theoretic statements about open colorings or ``open coloring axioms.' In particular it will be shown that the conjunction of two well-known axioms, and , implies that the size of the continuum is .

     

Molecular lines and continuum from W51A (I) —CO and NH3 spectra and 3 mm continuum

As for the 5′ × 4′(∼llpc × 9pc) region centered at W51 lRSl the observations of the 3.4 mm continuum, CO (J = 1-0) line and simultaneous NH₃(1,1), (2,2), (3,3), (4,4) inverse lines were made for studying the massive star formation region located in the main spiral arms of the Galaxy. In the directions of W51 IRS1, IRS2 and el/e2 in 3.4 mm continuum, analyses of the line profiles show that the absorption lines of ammonia, which arise from the gas in front of the HII region, are red-shifted with respect to the emission lines, which arise from the surrounding cloud. Furthermore, a radiation transfer and statistical equilibrium calculation of ammonia molecules show that the densities increase by 3–10 times from the eastern border to the center. These points hint that the collapse is happening in the molecular cloud core obscured in optical wavelengths. The effects of the radiation fields from radio, infrared and UCHII sources is non-negligible on the excitation of various molecules (e.g. NH₃) within the circle of radius 40″ centered at IRS1. The profiles of the COJ = 1–0 line in the circle change from double peaks ( ∼ 60, ∼ 68 km. s^-1) to triple peaks, i.e. the component ∼53 km·s⁻¹, which associates with UCHII, also appears in the spectra. There are indications that the circle of radius 40″ centered at IRSI is a region of massive star forming activity     

A selection limiter of DSMC for near continuum flows

A selection limiter for the direct simulation monte carlo (DSMC) method is proposed to simulate near continuum flows. The selection limiter is calculated according to a continuum breakdown parameter and is used to limit the number of potential collision pairs. A Couette flow, a supersonic flow into a pitot probe and a nozzle plume flow are studied and compared with the standard DSMC to validate present method. It is found that its computational cost is about 35% of that of the standard DSMC method with satisfactory accuracy in the near continuum regime.     

The Space of subcontinua of a 2-dimensional continuum is infinite dimensional

Let be a metric continuum and let denote the space of subcontinua of with the Hausdorff metric. We settle a longstanding problem showing that if then . The special structure and properties of hereditarily indecomposable continua are applied in the proof.

     

Almost continuous images of Peano continua

We give a necessary and sufficient condition that a 2nd countable space be the almost continuous image of a Peano continuum.     

Multicyclic treelike reflexive graphs

A simple graph is reflexive if its second largest eigenvalue does not exceed 2. A graph is treelike (sometimes also called a cactus) if all its cycles (circuits) are mutually edge-disjoint. In a lot of cases one can establish whether a given graph is reflexive by identifying and removing a single cut-vertex (Theorem 1). In this paper we prove that, if this theorem cannot be applied to a connected treelike reflexive graph G and if all its cycles do not have a common vertex (do not form a bundle), such a graph has at most five cycles (Theorem 2). On the same conditions, in Theorem 3 we find all maximal treelike reflexive graphs with four and five cycles.     

Wave-packet continuum discretization method for solving the three-body scattering problem

We generalize the wave-packet continuum discretization method previously developed for the scattering problem to the three-body system. For each asymptotic channel, we construct a basis of three-body wave packets given by square-integrable functions. We show that the projections of the channel resolvents on the subspace of three-body wave packets are determined by diagonal matrices, whose eigenvalues we find explicitly. We express the amplitudes of 2→2 processes explicitly in terms of “wave-packet” finite-dimensional projections of the full resolvent. To illustrate our formalism, we calculate the differential cross section of elastic deuteron scattering on a heavy nucleus above the three-body breakup threshold and the s-wave quartet (n-d)-scattering amplitude. The results of the calculations agree well with the results obtained by other methods. In terms of complexity, the proposed scheme for solving the three-body scattering problem is comparable to solving a similar problem for bound states. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 3, pp. 473–497, March, 2007.     

Remarks on continuum cardinals on Boolean algebras

We give some results concerning various generalized continuum cardinals. The results answer some natural questions which have arisen in preparing a new edition of 5 . To make the paper self‐contained we define all of the cardinal functions that enter into the theorems here. There are many problems concerning these new functions, and we formulate some of the more important ones.     

A unifying formulation of the discrete and continuum approximations for embedded discontinuities

A methodology for the numerical implementation of embedded discontinuities into the finite element method is developed. This is applicable for the discrete and continuum approximations of discontinuities. The variational formulation of the problem of a solid with discontinuities is established for both approximations, yielding the equations used in this methodology. Three sets of equations are obtained by applying this methodology; all are suitable to be numerically implemented. To show the application potential of this method, the numerical simulation of the formation and propagation of a discontinuity in a concrete specimen is carried out and the results are compared with those from the physical experiment, demonstrating the adequacy of the methodology and its corresponding implementations to model discontinuities. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A universal continuum of weight

We prove that every continuum of weight  is a continuous image of the Cech-Stone-remainder  of the real line. It follows that under  the remainder of the half line is universal among the continua of weight  -- universal in the `mapping onto' sense.

We complement this result by showing that 1) under  every continuum of weight less than  is a continuous image of , 2) in the Cohen model the long segment of length  is not a continuous image of , and 3)  implies that is not a continuous image of , whenever  is a -saturated ultrafilter.

We also show that a universal continuum can be gotten from a -saturated ultrafilter on , and that it is consistent that there is no universal continuum of weight  .