A Continuous Analogue of the Upper Bound Theorem

For an absolutely continuous probability measure μ on R ^d and a nonnegative integer k , let \tilde s _k (μ ,\origin ) denote the probability that the convex hull of k+d+1 random points which are i.i.d. according to μ contains the origin \bf 0 . For d and k given, we determine a tight upper bound on \tilde s _k (μ ,\origin ) , and we characterize the measures in R ^d which attain this bound. As we will see, this result can be considered a continuous analogue of the Upper Bound Theorem for the maximal number of faces of convex polytopes with a given number of vertices. For our proof we introduce so-called h -functions, continuous counterparts of h -vectors of simplicial convex polytopes. Received April 14, 2000, and in revised form October 6, 2000. Online publication June 20, 2001.     

Euclidean minimum spanning trees and bichromatic closest pairs

We present an algorithm to compute a Euclidean minimum spanning tree of a given setS ofN points inE ^d in timeO(F _d (N,N) log ^d N), whereF _d (n,m) is the time required to compute a bichromatic closest pair amongn red andm green points inE ^d . IfF _d (N,N)=Ω(N ^1+ε), for some fixed ɛ>0, then the running time improves toO(F _d (N,N)). Furthermore, we describe a randomized algorithm to compute a bichromatic closest pair in expected timeO((nm logn logm)^2/3+m log² n+n log² m) inE ₃, which yields anO(N ^4/3 log^4/3 N) expected time, algorithm for computing a Euclidean minimum spanning tree ofN points inE ³. Ind≥4 dimensions we obtain expected timeO((nm)^{1−1/([d/2]+1)+ε}+m logn+n logm) for the bichromatic closest pair problem andO(N ^{2−2/([d/2]+1)ε}) for the Euclidean minimum spanning tree problem, for any positive ɛ. The first, second, and fourth authors acknowledge support from the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), a National Science Foundation Science and Technology Center under NSF Grant STC 88-09648. The second author's work was supported by the National Science Foundation under Grant CCR-8714565. The third author's work was supported by the Deutsche Forschungsgemeinschaft under Grant A1 253/1-3, Schwerpunktprogramm “Datenstrukturen und effiziente Algorithmen”. The last two authors' work was also partially supported by the ESPRIT II Basic Research Action of the EC under Contract No. 3075 (project ALCOM).     

Oxo-biodegradable full carbon backbone polymers - biodegradation behaviour of thermally oxidized polyethylene in an aqueous medium

Several demonstrations of the effective biodegradation in soil of pro-oxidant activated polyethylene (PE) have been reported recently. Nevertheless a comprehensive understanding of the ultimate fate in the environment of the oxidized fragments of oxo-biodegradable polyethylene materials needs the extension of the studies to other natural environments and in particular to aqueous media (river, lake, brackish and marine waters) where accidental plastic littering and the resulting degraded fragments eventually may end up.In this respect, as part of our continuing activity in the area of oxo-biodegradable polymeric materials, in the present paper we wish to report on the results attained in an ongoing investigation on the biodegradation in a water medium of thermally pre-oxidized low density polyethylene (LDPE) film samples containing pro-oxidant additives.Thermally oxidized LDPE-film samples and corresponding acetone extractable fractions were submitted to the effect of microorganism flora present in river water. The effective biodegradation was assessed by monitoring the amount of CO₂ developed over time in a respirometer apparatus. Levels of biodegradation up to 12 and 48% for the degraded fragments and corresponding fractions extracted with boiling acetone were detected on a 100-day time frame.     

X-ray diffraction study of the smectic mesophase of some azobenzene-containing polyacrylates

An X-ray study is performed on powder specimens and on stretched oriented fibres of two liquid crystalline polyacrylates containing the azobenzene mesogenic unit with either a 4'-n-pentyloxy (sample 1-4) or 4'-n-hexyloxy (sample 1-5) substituent. The X-ray diffraction patterns of both samples showed the presence of a bilayer smectic C mesophase with the mesogneic groups tilted by an angle β ∼ 45° (1-4) or β ∼ 38° (1-5) with respect to the layer normal. The electron density profile p(z) along the direction normal to the smectic layers was calculated by Fourier inversion and possible structural models of the smectic mesophase are discussed. A partial interdigitation of the terminal alkyloxy substituents appears to occur.     

Symmetric graphs and interpretations

The mechanism of interpretation (subgraph homomorphism, homomorphic embedding, general coloring) gives rise to a “geography” of graphs. The position of symmetric graphs in this geography is investigated.     

Polyurethane Based Materials for the Production of Biomedical Materials

Polyurethanes (PUs) composed by hard and soft segments have been extensively used in the manufacturing of biocompatible prosthesis and medical devices. A broad variety of PUs can be obtained by modifying the balance between both segments. In the present work, different basically-flexible PUs have been prepared by employing different combinations of aliphatic hexamethylene diisocyanate, poly(ethylene glycol) (Mw 400 Da), poly(ϵ-caprolactone) diol (Mw 530 Da), and 1,4-butanediol. Thermal analysis of the synthesized PUs demonstrated high thermal stability and the assumption of glassy state well below room temperature, in agreement with their marked flexibility. Morphological characterization of PUs films indicated that films prepared by spin coating were smoother and more homogeneous than those obtained by casting. Biological assays performed by using 3T3/BALB-C mouse embryo fibroblast cell line confirmed the absence of toxicity and hence the biocompatibility of PU-films.     

The rank of sparse random matrices over finite fields

Let M be a random (n×n)-matrix over GF[q] such that for each entry M_ij in M and for each nonzero field element α the probability Pr[M_ij=α] is p/(q−1), where p=(log n−c)/n and c is an arbitrary but fixed positive constant. The probability for a matrix entry to be zero is 1−p. It is shown that the expected rank of M is n−𝒪(1). Furthermore, there is a constant A such that the probability that the rank is less than n−k is less than A/q^k. It is also shown that if c grows depending on n and is unbounded as n goes to infinity, then the expected difference between the rank of M and n is unbounded. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 10 , 407–419, 1997     

Biodegradable hybrid polymer films based on poly(vinyl alcohol) and collagen hydrolyzate

Hybrid blends of poly(vinyl alcohol) (PVA) and collagen hydrolyzate (CH), an added value waste from leather indutry, have been converted by blown molding extrusion, to environmentally degradable films. Blown extruded films comprising 5-15% of CH, were tested as sel fertilizing mulching films and analyzed for their propensity to enviromental degradation. PVA/CH films rapidly disintegrate when buried in soil, and resulted promising for application such as transplanting films, with additional fertilizing action of CH.