The Random Average Process and Random Walk in a Space-Time Random Environment in One Dimension

We study space-time fluctuations around a characteristic line for a one-dimensional interacting system known as the random average process. The state of this system is a real-valued function on the integers. New values of the function are created by averaging previous values with random weights. The fluctuations analyzed occur on the scale n ^1/4, where n is the ratio of macroscopic and microscopic scales in the system. The limits of the fluctuations are described by a family of Gaussian processes. In cases of known product-form invariant distributions, this limit is a two-parameter process whose time marginals are fractional Brownian motions with Hurst parameter 1/4. Along the way we study the limits of quenched mean processes for a random walk in a space-time random environment. These limits also happen at scale n ^1/4 and are described by certain Gaussian processes that we identify. In particular, when we look at a backward quenched mean process, the limit process is the solution of a stochastic heat equation.     

Aluminum(III) complexes containing O,O chelating ligand

The stoichiometric reactions of trimethylaluminum with 2,6‐(MeOCH₂)₂C₆H₃OH (LH) revealed compounds L₃Al ( 1 ) and L₂AlMe ( 2 ). On the other hand reaction of 1 equiv. of LH with trimethylaluminum did not lead to the formation of complex LAlMe₂ ( 3 ), rather 2 together with Me₃Al were observed as a result of a disproportionation of 3 . Compounds 1 and 2 were characterized by elemental analysis, ¹H and ¹³C NMR spectroscopy and in the case of 1 by X‐ray diffraction. Derivative 2 underwent transmetalation with Ph₃SnOH, giving LSnPh₃ ( 4 ) as the result of a migration of ligand L from the aluminum to the tin atom. The identity of 4 was established by elemental analysis, ¹H, ¹³C and ¹¹⁹Sn NMR spectroscopy and ¹H, ¹¹⁹Sn HMBC experiments. The system 2 and B(C₆F₅)₃ in a 1:1 molar ratio was shown to be active in the polymerization of propylene oxide and ε‐caprolactone. Copyright © 2007 John Wiley & Sons, Ltd.     

Coloring mixed hypertrees

A mixed hypergraph is a triple (V,C,D) where V is its vertex set and C and D are families of subsets of V, called C-edges and D-edges, respectively. For a proper coloring, we require that each C-edge contains two vertices with the same color and each D-edge contains two vertices with different colors. The feasible set of a mixed hypergraph is the set of all k's for which there exists a proper coloring using exactly k colors. A hypergraph is a hypertree if there exists a tree such that the edges of the hypergraph induce connected subgraphs of the tree.We prove that feasible sets of mixed hypertrees are gap-free, i.e., intervals of integers, and we show that this is not true for precolored mixed hypertrees. The problem to decide whether a mixed hypertree can be colored by k colors is NP-complete in general; we investigate complexity of various restrictions of this problem and we characterize their complexity in most of the cases.     

Inclusive production of vector mesons in π+ p interactions at 250 GeV/c

We report on a study of ρ⁰, ρ⁺, ω, \(\bar K^{*0} (892)\) andK ^*0 (892) inclusive production in π⁺ p interactions at 250 GeV/c, for ρ⁺, \(\bar K^{*0} (892)\) for the first time in a π⁺ p experiment. The data are compared withK ⁺ p data in the same experiment, with results of other experiments and with quark-parton models. Interesting differences are found between ρ^+,0 and ω production.     

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.     

Crystal structure of m-nitrobenzoic anhydride

The title compound (C₁₄H₈N₂O₇,M _r =306.2) crystallizes in the orthorhombic space group Pbca witha=6.962(1).b=24.688(1), andc=15.890(1)Å,V=2731.0 ų,D _x =1.489 g·cm^–3 forZ=8,=0.98 mm^–1,F(000)=1296,T=293 K. FinalR=0.053 for 1873 observed reflections. The structure was solved by direct methods. Approximately planar molecules lie perpendicular to the [100] direction and show partial stacking. The structure is the first example of a symmetric anhydride which does not retain the symmetry in the crystal state. The two independent nitro groups twist out of the ring planes by 10.5 and 14.8°, respectively.     

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.