IFS attractors and Cantor sets

We build a metric space which is homeomorphic to a Cantor set but cannot be realized as the attractor of an iterated function system. We give also an example of a Cantor set K in R³ such that every homeomorphism f of R³ which preserves K coincides with the identity on K.     

The representation dimension of domestic weakly symmetric algebras

Auslander’s representation dimension measures how far a finite dimensional algebra is away from being of finite representation type. In [1], M. Auslander proved that a finite dimensional algebra A is of finite representation type if and only if the representation dimension of A is at most 2. Recently, R. Rouquier proved that there are finite dimensional algebras of an arbitrarily large finite representation dimension. One of the exciting open problems is to show that all finite dimensional algebras of tame representation type have representation dimension at most 3. We prove that this is true for all domestic weakly symmetric algebras over algebraically closed fields having simply connected Galois coverings.     

The measure of a translated ball in uniformly convex spaces

Let (E, ¦·¦) be a uniformly convex Banach space with the modulus of uniform convexity of power type. Let be the convolution of the distribution of a random series inE with independent one-dimensional components and an arbitrary probability measure onE. Under some assumptions about the components and the smoothness of the norm we show that there exists a constant such that |{·<t}–{·+r<t}|r ^q , whereq depends on the properties of the norm. We specify it in the case ofL spaces, >1.     

The complexity of many cells in arrangements of planes and related problems

We consider several problems involving points and planes in three dimensions. Our main results are: (i) The maximum number of faces boundingm distinct cells in an arrangement ofn planes isO(m ^2/3 n logn +n ²); we can calculatem such cells specified by a point in each, in worst-case timeO(m ^2/3 n log³ n+n ² logn). (ii) The maximum number of incidences betweenn planes andm vertices of their arrangement isO(m ^2/3 n logn+n ²), but this number is onlyO(m ^3/5– n ^4/5+2 +m+n logm), for any>0, for any collection of points no three of which are collinear. (iii) For an arbitrary collection ofm points, we can calculate the number of incidences between them andn planes by a randomized algorithm whose expected time complexity isO((m ^3/4– n ^3/4+3 +m) log² n+n logn logm) for any>0. (iv) Givenm points andn planes, we can find the plane lying immediately below each point in randomized expected timeO([m ^3/4– n ^3/4+3 +m] log² n+n logn logm) for any>0. (v) The maximum number of facets (i.e., (d–1)-dimensional faces) boundingm distinct cells in an arrangement ofn hyperplanes ind dimensions,d>3, isO(m ^2/3 n ^d/3 logn+n ^d–1). This is also an upper bound for the number of incidences betweenn hyperplanes ind dimensions andm vertices of their arrangement. The combinatorial bounds in (i) and (v) and the general bound in (ii) are almost tight.Work on this paper by the first author has been supported by Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862 and by NSF Grant CCR-8714565. Work by the third author has been supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grant DCR-82-20085, by grants from the Digital Equipment Corporation, and the IBM Corporation, and by a research grant from the NCRD—the Israeli National Council for Research and Development. An abstract of this paper has appeared in theProceedings of the 13th International Mathematical Programming Symposium, Tokyo, 1988, p. 147.     

Implicitly representing arrangements of lines or segments

Anarrangement ofn lines (or line segments) in the plane is the partition of the plane defined by these objects. Such an arrangement consists ofO(n ²) regions, calledfaces. In this paper we study the problem of calculating and storing arrangementsimplicitly, using subquadratic space and preprocessing, so that, given any query pointp, we can calculate efficiently the face containingp. First, we consider the case of lines and show that with (n) space¹ and (n ^3/2) preprocessing time, we can answer face queries in (n)+O(K) time, whereK is the output size. (The query time is achieved with high probability.) In the process, we solve three interesting subproblems: (1) given a set ofn points, find a straight-edge spanning tree of these points such that any line intersects only a few edges of the tree, (2) given a simple polygonal path , form a data structure from which we can find the convex hull of any subpath of quickly, and (3) given a set of points, organize them so that the convex hull of their subset lying above a query line can be found quickly. Second, using random sampling, we give a tradeoff between increasing space and decreasing query time. Third, we extend our structure to report faces in an arrangement of line segments in (n ^1/3)+O(K) time, given(n ^4/3) space and (n ^5/3) preprocessing time. Lastly, we note that our techniques allow us to computem faces in an arrangement ofn lines in time (m ^2/3 n ^2/3+n), which is nearly optimal.The first author is pleased to acknowledge the support of Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862 and National Science Foundation Grant CCR-8714565. Work on this paper by the fifth author has been supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grant NSF-DCR-83-20085, by grants from the Digital Equipment Corporation, and the IBM Corporation, and by a research grant from the NCRD—the Israeli National Council for Research and Development. The sixth author was supported in part by a National Science Foundation Graduate Fellowship. This work was begun while the non-DEC authors were visiting at the DEC Systems Research Center.     

Forward-backward multiplicity correlations in σ+ p,K + p andpp collisions at 250 GeV/c

Forward-backward multiplicity correlations in σ⁺,K ⁺ p andpp collisions at 250 GeV/c ( \(\sqrt s \) =22 GeV) are given for all charges and for the different charge combinations. The correlations are found to be caused predominantly by centrally produced particles. It is demonstrated that this result is an agreement with observations at the ISR and the CERNp \(\bar p\) -Collider. The results are compared to expectations from LUND, DPM and FRITIOF Monte Carlo models and a geometrical picture relating correlations in hadron-hadron collisions toe ⁺ e ^? data in terms of impact parameters is tested.     

Flow-injection chemiluminescence determination of epinephrine in pharmaceutical preparations using raw apple juice as enzyme source

Raw apple juice exhibits the activity of polyphenol oxidase which can be employed for the determination of mono-, di- and polyhydric phenols. The chemiluminescence detection in the two-channel manifold was used to develop the new procedure for determination of epinephrine. Epinephrine can be determined by this method in pharmaceutical preparations in concentration ranges 1.0–10.0 and 10.0–25.0 mg l⁻¹, with a detection limit of 0.2 mg l⁻¹. Good selectivity against typical antioxidants and other coexisting substances was achieved. Relatively slow loss of the enzyme activity was observed during 1 week. The proposed method is very simple. Fresh juice solution from one apple can be prepared in 10 min and used for at least 8 h with excellent precision. Consumption of luminol solution was 0.15 ml min⁻¹. The throughput was 20 samples per hour.     

Facts and artifacts about aromatic stability estimation

The stability of a set of 105 five-membered π-electron systems (involving aromatic, non-aromatic and anti-aromatic species) was evaluated using six isodesmic reactions of which two belong to the subclass of homodesmotic reactions, which are based on cyclic and acyclic reference systems. We demonstrate that the ‘Resonance Energies’ derived from isodesmotic schemes have obvious flaws and do not correct or cancel other contributions to the energy, such as the changes of hybridization, homoconjugation of heterosubstituted cyclopentadienes, conjugative interactions of CC or CX (X=N or P) with a π or pseudo π orbital at Y (Y=O, S, NH, PH), strain, etc. as effectively as possible. Likewise, ‘aromatic stabilization energies (ASE)’ derived from homodesmotic schemes based on the acyclic reference compounds do not give satisfactory results. We strongly recommend that only cyclic reference compounds should be used for ASE and other aromaticity evaluations. The analysis is based on ab initio optimized geometries at B3LYP/6-311+G^∗∗.     

Crystal structures and magnetic properties of two low-dimensional materials constructed from [Mn(III)(salen)H2O]+ and [M(CN)8](3-/4-) (M = Mo or W) precursors

The syntheses, X-ray structures, and magnetic behaviors of two new cyano-bridged assemblies, the molecular [Mn(III)(salen)H2O]3[W(V)(CN)8].H2O (1) and one-dimensional [Mn(salen)(H2O)2]2[[Mn(salen)(H2O)][Mn(salen)]2[Mo(CN)(8)]].0.5ClO4.0.5OH.4.5H2O (2), are presented. Compound 1 crystallizes in the monoclinic system, has space group P2(1)/c, and has unit cell constants a = 13.7210(2) A, b = 20.6840(4) A, c = 20.6370(2) A, and Z = 4. Compound 2 crystallizes in the triclinic system, has space group P, and has unit cell dimensions a = 18.428(4) A, b = 18.521(3) A, c = 18.567(4) A, and Z = 2. The structure of 1 consists of the asymmetric V-shaped Mn-NC-W-NC-Mn-O(phenolate)-Mn molecules, where W(V) coordinates with [Mn(salen)H2O] and singly phenolate-bridged [Mn(salen)H2O]2 moieties through the neighboring cyano bridges. The [W(V)(CN)8]3- ion displays distorted square-antiprism geometry. The structure of 2 consists of the cyano-bridged [Mn3(III)Mo(IV)]n- repeating units linked by double phenolate bridges into one-dimensional zigzag chains. The Mn(III) centers are bound to Mo(IV) of square-antiprism geometry through the neighboring cyano bridges. The magnetic studies of 1 reveal the antiferromagnetic intramolecular interactions through the CN and phenolate bridges and the relatively weak intermolecular interactions. Compound 1 becomes antiferromagnetically ordered below TN = 4.6 K. The presence of the magnetic anisotropy is documented with the MH measurements carried out for both polycrystalline and single-crystal samples. At T = 1.9 K, the spin-flop transition is observed in the field of 18 kOe applied parallel to the bc plane, which is the easy plane of magnetization. Field dependence of magnetization of 1 shows field-induced metamagnetic behavior from the antiferromagnetic ground state of ST = 3/2 to the state of ST = 5/2. The magnetic properties of 2 indicate a weak antiferromagnetic interaction between Mn(III) centers in double-phenolate-bridged [Mn(III)(salen)]2 dinuclear subunits and a very weak ferromagnetic interaction between them through the diamagnetic [Mo(IV)(CN)8]4- spacer.