The Weierstrass mean

Theorem.Let the sequences {e _i ⁽ⁿ⁾ },i=1, 2, 3,n=0, 1, 2, ...be defined by where the e ⁽⁰⁾ s satisfy and where all square roots are taken positive. Then where the convergence is quadratic and monotone and where The discussions of convergence are entirely elementary. However, although the determination of the limits can be made in an elementary way, an acquaintance with elliptic objects is desirable for real understanding.     

Local nondeterminism and local times for stable processes

Summary Our main theorem gives sufficient conditions for symmetric stable processes and fields to have a jointly continuous local time. The approach is through the L ^p representation for such processes. We develop a measure of dependence for vectors in a normed linear space and use that to analyze the probabilistic independence of the increments of a stable process. Local nondeterminism is defined for stable processes and shown to be equivalent to locally approximately independent increments. Sufficient conditions for several classes of stable processes to be local nondeterministic are given. These ideas are extended to multidimensional stable random fields and we prove existence of jointly continuous local times. The results extend most Gaussian results to their stable analogs.     

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.     

Partial regularity for minimisers of certain functionals having nonquadratic growth

We prove partial regularity results for local minimisers of

Anticancer agent development: X-ray crystal structure and keto-enol tautomerism of dimethyl 1-hydroxy-6,7-methylenedioxy-4-(3′,4′,5′-trimethoxyphenyl-trans-3,4-dihydronaphthalene-2,3-dicarboxylate

Structural and conformational information obtained from the crystal structure and solution¹H nmr investigations of the title compound are compared. The 4-aryltetralone, C₂₄H₂₄O₁₀, crystallizes as a chloroform solvate in the monoclinic space group, P2₁/n, witha=12.519(4),b=17.938(6),c=12.534(9)Å,=111.90(5)°, and D_calc=1.51 g cm^–3 forZ=4. The data for this compound were collected at –150°C. Least-squares refinement of 2796 observed [F _o5(F _o)] reflections led to the final agreement index ofR=0.062. A threefold static disorder was observed for one of the carboxyl groups. The second carboxyl group participates in an intramolecular hydrogen bond and is thus ordered. The¹H nmr spectrum revealed the title compound to exist as a keto-enol tautomeric mixture in solution. Vicinal hydrogen coupling constant analysis proved reliable in ascertaining B-ring stereochemistry of 2,3-disubstituted-4-aryltetralones.     

Optimal Discrete Approximations for Continuous Outcomes with Applications in Decision and Risk Analysis

In decision and risk analysis, it is common to use discrete probability distributions to approximate uncertain events with continuous outcomes. This paper discusses how these approximations may be selected. A class of approximations based on a modification to Taguchi's work on tolerance analysis is shown to be optimal under assumptions of independent uncertainties with normally distributed outcomes. The approximation procedure is shown to be robust in many other situations and is extremely easy to use in practice. We also show how the approximation may be integrated into the process of subjective probability estimation by a ‘subject-matter expert’.     

An intermediate phase with slow decay of correlations in one dimensional 1/|x−y|2 percolation,Ising and Potts models

We rigorously establish the existence of an intermediate ordered phase in one-dimensional 1/|x–y|² percolation, Ising and Potts models. The Ising model truncated two-point function has a power law decay exponent which ranges from its low (and high) temperature value of two down to zero as the inverse temperature and nearest neighbor coupling vary. Similar results are obtained for percolation and Potts models.Alfred P. Sloan Research Fellow. Research supported in part by NSF Grants No. PHY-8706420 and PHY-8645122Research supported in part by NSF Grant No. DMS-8514834 and AFOSR Contract F49620-86-C0130 at the Arizona Center for Math. Sciences     

Self-assembly of a hydrophobic groove

Heptakis(2,6-di-O-methyl)-β-cyclodextrin interacts with 5,15-diphenylporphine to produce a 2:1 complex in dimethyl sulfoxide. This complex possesses a hydrophobic groove that circumscribes the metal binding site of the porphyrin moiety.     

HYDROXYSTILBAZOLES AND HYDROXYAZAPHENANTHRENES: PHOTOCYCLIZATION AND FLUORESCENCE STUDIES

Abstract— As models for novel fluorescent probes, we have synthesized three isomeric hydroxystilbazole systems (4′-hydroxy-substituted 2-, 3-, and 4-stiIbazoles), examined their photocyclization-oxi-dation to four hydroxyazaphenanthrene systems, and made a preliminary study of their absorption and fluorescence spectra. All three stilbazoles can be prepared easily by addition of the isomeric picoline anions to 4-methoxybenzaldehyde, followed by dehydration and deprotection. Photocyclization proceeds efficiently, furnishing a single product isomer from each of the 2- and 4-stilbazole systems, and two isomeric azaphenanthrenes from the 3-stilbazole. The stilbazoles all have intense UV absorbance bands whose maxima depend upon solvent and pH; all three isomers have relatively similar spectra under neutral conditions and all three show a large red shift in base; in acid, however, the 2-and 4-stilbazole isomers show a greater red shift than the 3-stilbazole. The fluorescence of the stilbazoles is also solvent dependent, shifting to the red in more polar medium; red shifts are also observed in acid and base, but in acid, the 3-stilbazole shows a larger shift. The azaphenanthrene photocyclization products show absorbance spectra typical for quinolines and isoquinolines; their absorptivities are less than the stilbazoles, but their fluorescence is more intense. In general, the benzoquinolines have longer wavelength but weaker fluorescence than the benzoisoquinolines. Also, those isomers in which the resonance effects of the hydroxy and nitrogen groups can reinforce one another show longer wavelength emissions of greater intensity. All seven systems show dual fluorescence in water under neutral conditions, suggesting the emission from both non-ionized and ionized species in the excited state. In one case, the benzoisoquinoline system derived from 4′-hydroxy-4-stilbazole, an emission at 640 nm, observed in water over a wide pH range, is ascribed to a zwitterionic phototautomer. These stilbazoles, benzoquinolines and benzoisoquinolines may prove to be useful spectroscopic probes.