The complexity of cutting complexes

This paper investigates the combinatorial and computational aspects of certain extremal geometric problems in two and three dimensions. Specifically, we examine the problem of intersecting a convex subdivision with a line in order to maximize the number of intersections. A similar problem is to maximize the number of intersected facets in a cross-section of a three-dimensional convex polytope. Related problems concern maximum chains in certain families of posets defined over the regions of a convex subdivision. In most cases we are able to prove sharp bounds on the asymptotic behavior of the corresponding extremal functions. We also describe polynomial algorithms for all the problems discussed.Bernard Chazelle wishes to acknowledge the National Science Foundation for supporting this research in part under Grant No. MCS83-03925. Herbert Edelsbrunner is pleased to acknowledge the support of Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862.     

Seizure warning algorithm based on optimization and nonlinear dynamics

There is growing evidence that temporal lobe seizures are preceded by a preictal transition, characterized by a gradual dynamical change from asymptomatic interictal state to seizure. We herein report the first prospective analysis of the online automated algorithm for detecting the preictal transition in ongoing EEG signals. Such, the algorithm constitutes a seizure warning system. The algorithm estimates STL_max, a measure of the order or disorder of the signal, of EEG signals recorded from individual electrode sites. The optimization techniques were employed to select critical brain electrode sites that exhibit the preictal transition for the warning of epileptic seizures. Specifically, a quadratically constrained quadratic 0-1 programming problem is formulated to identify critical electrode sites. The automated seizure warning algorithm was tested in continuous, long-term EEG recordings obtained from 5 patients with temporal lobe epilepsy. For individual patient, we use the first half of seizures to train the parameter settings, which is evaluated by ROC (Receiver Operating Characteristic) curve analysis. With the best parameter setting, the algorithm applied to all cases predicted an average of 91.7% of seizures with an average false prediction rate of 0.196 per hour. These results indicate that it may be possible to develop automated seizure warning devices for diagnostic and therapeutic purposes.Mathematics Subject Classification (1991):20E28, 20G40, 20C20     

A correspondence principle connecting IBVPs of wave propagation and heat conduction

A wave-heat correspondence principle is established by utilizing previous operational relations of van der Pol and Bremmer. By the suggested procedure, heat conduction problems can be solved by using an auxiliary wave-propagation problem in conjunction with the correspondence principle. A check on this procedure was provided by solving a mixed BVP of transient heat flow. The present correspondence principle is particularly useful in dealing with IBVPs involving wedge-shaped domains, and characteristic lengths.     

UPML formulation for truncating conductive media in curvilinear coordinates

This work presents a formulation based on UPML for truncating conductive media by using a local and non-orthogonal coordinate system to solve Maxwell’s equations by the FDTD method. The detailed procedure for obtaining the UPML equations for this case is shown and the complete equation set is provided.     

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.     

Investigation of the separation of scandium and rare earth elements from red mud by use of reversed-phase HPLC

A chromatographic method has been developed for separation and determination of scandium (Sc) and rare earth elements (REEs) in samples from a red mud (RM)-utilization process. Reversed-phase high-performance liquid chromatography (RP-HPLC) with post-column derivatization using 4-(2-pyridylazo)-resorcinol (PAR) and UV–visible detection at 520 nm was tested using different gradient elution profiles and pH values to optimize separation and recovery, primarily for Sc but also for yttrium and the individual lanthanides, from iron present in the samples. The separation was performed in less than 20 min by use of a mobile phase gradient. The concentration of -hydroxyisobutyric acid (-HIBA), as eluent, was altered from 0.06 to 0.4 mol L^–1 (pH 3.7) and 0.01 mol L^–1 sodium salt n-octane sulfonic acid (OS) was used as modifier. Very low detection limits in the nanogram range and a good resolution for Sc and REEs except for Y/Dy were achieved. Before application of the method to the red mud samples and to the corresponding bauxites, Sc and REEs were leached from red mud with 0.6 mol L^–1 HNO₃ and mostly separated, as a group, from the main elements by ion exchange/selective elution (6 mol L^–1 HNO₃) in accordance with a pilot-plant process developed in this laboratory. After evaporation of the eluent to dryness the extracted elements were re-dissolved in the mobile phase. By use of this chromatographic method Sc, which is the most expensive of the elements investigated and occurs in economically interesting concentrations in red mud, could be separated not only from co-existing Fe but also from Y/Dy, Yb, Er, Ho, Gd, Eu, Sm, Nd, Pr, Ce and La. All the elements investigated were individually recovered. Their recoveries were found to be nearly quantitative.     

Stereoselective ring closure of 6-[[(methylamino)carbonyl]oxy]-2H-pyran-3(6H)-ones. Formation of the 5H-pyrano[3,2-d]oxazole-2,6-dione ring system

The stereospecific formation of the 5H-pyrano[3,2-d]oxazole-2,6-dione ring system from 2,2-disubstituted-6-[[(methylamino)carbonyl]oxy]-2H-pyran-3(6H)-one via an enolization at C-5 is presented. The outcome of the same reaction in the case of 2-monosubstituted 2H-pyran-3(6H)-ones is also discussed.     

Solid-phase synthesis of an oxalic acid amide library

Monoamides of oxalic acid are of interest as bioisosteric replacements for phosphate groups in the design of new enzyme inhibitors. Here, we have demonstrated the use of oxalic acid as a linker to the Wang resin to synthesize individual or libraries of phosphate biosteres. The highly reactive resin-bound acid chloride reacts with arylamines to yield resin-bound N-aryloxamic acids (oxanilic acids). This methodology is especially useful for the rapid synthesis of 2-(oxalylamino)benzoic acids (OBAs), because it can be utilized for library synthesis and eliminates the intermediate purification step necessary in solution-phase reactions. The products are cleaved off the resin with trifluoroacetic acid in dichloromethane in good yields.