Segment Intersection Searching Problems in General Settings

   Abstract. We consider segment intersection searching amidst (possibly intersecting) algebraic arcs in the plane. We show how to preprocess n arcs in time O(n ^2+ɛ ) into a data structure of size O(n ^2+ɛ ) , for any ɛ >0 , such that the k arcs intersecting a query segment can be counted in time O( log n) or reported in time O( log n+k) . This problem was extensively studied in restricted settings (e.g., amidst segments, circles, or circular arcs), but no solution with comparable performance was previously presented for the general case of possibly intersecting algebraic arcs. Our data structure for the general case matches or improves (sometimes by an order of magnitude) the size of the best previously presented solutions for the special cases. As an immediate application of this result, we obtain an efficient data structure for the triangular windowing problem, which is a generalization of triangular range searching. As another application, the first substantially subquadratic algorithm for a red—blue intersection counting problem is derived. We also describe simple data structures for segment intersection searching among disjoint arcs, and for ray shooting among possibly intersecting arcs.     

Phenomena Affecting the Equilibrium of Al(III) and Zn(II) Extraction with Winsor II Microemulsions

The extraction of Al(III) and Zn(II) from an aqueous solution with two water-in-oil microemulsions, one containing di(2-ethylhexyl)phosphoric acid (DEHPA), was investigated to aid the understanding of the role of the extractant and the metal specific characteristics in the mechanism of microemulsion extraction. The extraction of Al with the DEHPA microemulsion increased by a factor of about 10 with respect to that in the conventional DEHPA system, whereas the extraction of Zn was lower than that in the single DEHPA system. Extraction with the DEHPA-free microemulsion was very low, showing that metal ion solubilization was not important in the mechanism of microemulsion extraction. It is proposed that the effect of the mixed microemulsion on the metal distribution coefficient is the result of the balance between a decrease in the complexation reaction yield due to the interaction between butanol and DEHPA, and the adsorption of the metal complex at the macro- and microinterfaces. The former leads to a decrease in Zn(II) extraction and the latter to Al(III) extraction synergism. Copyright 2000 Academic Press.     

On Overlays and Minimization Diagrams

The overlay of 2≤m≤d minimization diagrams of n surfaces in ℝ ^d is isomorphic to a substructure of a suitably constructed minimization diagram of mn surfaces in ℝ ^d+m−1. This elementary observation leads to a new bound on the complexity of the overlay of minimization diagrams of collections of d-variate semi-algebraic surfaces, a tight bound on the complexity of the overlay of minimization diagrams of collections of hyperplanes, and faster algorithms for constructing such overlays. Further algorithmic implications are discussed. Work by V. Koltun was supported by NSF CAREER award CCF-0641402. Work by M. Sharir was supported by NSF Grants CCR-00-98246 and CCF-05-14079, by a grant from the Israeli Academy of Sciences for a Center of Excellence in Geometric Computing at Tel Aviv University, and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University.     

Cutting Triangular Cycles of Lines in Space

We show that n lines in 3-space can be cut into O(n^2-1/69log^16/69n) pieces, such that all depth cycles defined by triples of lines are eliminated. This partially resolves a long-standing open problem in computational geometry, motivated by hidden-surface removal in computer graphics.     

Papers to appear in this volume

Papers to appear in this volume     

Investigation of the reflectivity of highly doped gallium arsenide in a wide spectral range

The optical properties of highly doped single crystals and diffused layers of p-type gallium arsenide were investigated in the spectral range from 0.2 to 25. A technique has been suggested and experimentally tested; it provides for transparent selective coatings and their application, as well as for a modification of the surface microrelief for reducing the reflection coefficient in the desired spectral range.     

Elastic pion-deuteron scattering in the resonance region as a three-body problem

We study elastic pion-deuteron scattering in the Δ(1236) energy region by means of the three-body Faddeev equations. We present a compact angular momentum reduction of the Faddeev integral equation for separable amplitudes, neglecting the nucleon spin, and solve the resulting coupled integral equations. We examine the dependence of the elastic scattering amplitude on the deuteron structure, on the pion-nucleon scattering amplitude, and on the various orders of multiple scattering. The differential cross section is very sensitive to multiple scattering effects at backward angles. We find that a number of conventional approximations do not well reproduce these multiple scattering effects in the resonance region.     

Ring scission of diastereomeric 4-butylspiropentylcarbinyl radicals as a chemical model for identifying enzyme-catalyzed FAD adducts resulting from spiropentylacetyl-CoA

Both diastereomeric 4-butylspiropentylcarbinyl bromides (14a and 14b) were synthesized in seven steps starting from 1-heptyne, and the stereochemical assignments based upon NOE experiments were confirmed by converting their immediate alcohol precursors (13a and 13b) to 1,4-dibutylspiropentanes (17a and 17b) with C(1) and C(2) symmetry. Each bromide was used to generate its corresponding spiropentylcarbinyl radical (18a and 18b) via its AIBN-initiated tri-n-butyltin hydride reduction. The radical-trapped products are identified, the preferred ring scission mode is identified (C1[bond] C2 bond cleavage), and the estimated rates for the ring opening of 4-butylspiropentylcarbinyl radical (18, k(25) degrees C > or = approximately 5 x 10(9) s(-1)) and 2-butyl-1-vinylcyclopropylcarbinyl radical (33, k(25) degrees C approximately 5 x 10(8) s(-1)) are reported. High-level ab initio calculations addressing the ring-opening isomerizations of cyclopropylcarbinyl and spiropentylcarbinyl radicals also are presented. These results in conjunction with a previous study enable us to propose two structures for the enzyme-catalyzed FAD adducts resulting from spiropentylacetic acid-CoA, a synthetic byproduct of fatty acid metabolism.