Triangulating a simple polygon in linear time

We give a deterministic algorithm for triangulating a simple polygon in linear time. The basic strategy is to build a coarse approximation of a triangulation in a bottom-up phase and then use the information computed along the way to refine the triangulation in a top-down phase. The main tools used are the polygon-cutting theorem, which provides us with a balancing scheme, and the planar separator theorem, whose role is essential in the discovery of new diagonals. Only elementary data structures are required by the algorithm. In particular, no dynamic search trees, of our algorithm. The author wishes to acknowledge the National Science Foundation for supporting this research in part under Grant CCR-8700917.     

Synthesis of well-defined soluble polystyrene supports: Impact of polymer chain on grafted organotin reagent activity

The synthesis of well-defined soluble polystyrene supports bearing tin hydride functionalities has been achieved in two steps. First, a precursor was prepared by copolymerization of styrene and acetoxystyrene using atom transfer radical polymerization. Prior any modification, this precursor was fully characterized to check its structure. Then in a second part, tin hydride functions were introduced by a four step process. The chemical modification was monitored by IR and ¹H NMR spectroscopies. The soluble support was also characterized by triple detection size exclusion chromatography at each step. Two families of supports were synthesized by varying the molecular weight and the degree of functionalization. The effectiveness of these tin hydride supports was tested through the reduction of 6-bromohexene and compared with the small counterpart Bu₃SnH. Measurements of rate constant for hydrogen transfer were also reported.     

On the rheology of cold drawing. I. Elastic materials

In this, the first paper in a series on neck formation and steady-state drawing of polymeric fibers and films under uniaxial tension, the emphasis is laid on those aspects of the mechanics of cold drawing that are not sensitive to viscoelastic effects and, therefore, can be treated by use of constitutive assumptions appropriate to elastic materials. It is here shown that a unidimensional theory which has been employed to model the mechanics of slender bars in tension₆ can be derived as an approximation for three-dimensional bars and, in a sense which can be made precise, is valid to within an error of the order of the fourth power of the thickness. A particular constitutive equation for incompressible, three-dimensional, elastic materials is explored in detail and is found to yield, for such slender bars as thin fibers and wide (but thin) strips of film, equations of equilibrium whose solutions are in good qualitative accordance with the necks and drawing configurations observed in practice.     

Vibrational properties of hydrogen in compound semiconductors

Hydrogen introduced in compound semiconductors generally forms complexes with atoms of the crystal. They result from the interaction of hydrogen with some dopants, residual impurities and defects. It is shown here how vibrational spectroscopy can inform on the chemical nature of the complexes and on their structure, which can explain some electrical effects. From the experimental data, a kind of classification of the complexes, depending on the binding and location of the H atom, is established. Spectroscopy under uniaxial stress allows to determine the symmetry of the complexes and the relation with theoretical models is discussed. From the stress-induced dichroism one can derive reorientation energies related to the lattice distortion induced by the complex. An overview of the complexes in proton-implanted materials is presented and we finally try to summarize some of the questions not yet answered on the interaction between hydrogen and bulk compound semiconductors.     

Structural characterization of the smectic polymorphism of chiral liquid crystalline polysiloxanes containing biphenyl mesogens

A structural characterization of chiral side chain siloxanes with different average degrees of polymerization, DP_n, was performed by X-ray diffraction experiments on powder and oriented fibre specimens. Polymers (DP_n = 35) and oligomers (DP_n = 4) contained the 4,4'-biphenylene unit with either an (S)-2-methylbutoxy (An, Bn) or an (S)-2-chloro-3-methylbutanoyloxy substituent (Cll). The spacer segment connected to the siloxane backbone had a variable number, n, of methylene groups (n = 5, 8, or 11). Independent of the spacer length and the chiral tail nature, the polysiloxanes underwent the same sequence of phases: C-S_F1 (or S_I1)-S_C1-S_A1-I, whereas in the oligosiloxanes the sequence C-S_B1-S_A1-I (B11) or C-S_F1-S_C1-I (B5) occurred. The influence of the structure of the polysiloxanes on the formation of the smectic (tilted or orthogonal) mesophases was elucidated. The rather large number of reflections (three or four) detected in the X-ray patterns at low angles, allowed a drawing of the projection of the electron density profiles along the layer normal, p(z), and deduction of the most physically acceptable electron density profiles from among the numerous possibilities for each smectic phase. The electron density profiles were in agreement with monolayer smectic phases presenting a microphase separation between the siloxane backbones and the side chains, so constraining the polymer backbones within a thin layer.     

Existence of minimizers for non-quasiconvex integrals

Communicated by H. Brezis     

A generalized inverse method for asymptotic linear programming

Consider a linear program in which the entries of the coefficient matrix vary linearly with time. To study the behavior of optimal solutions as time goes to infinity, it is convenient to express the inverse of the basis matrix as a series expansion of powers of the time parameter. We show that an algorithm of Wilkinson (1982) for solving singular differential equations can be used to obtain such an expansion efficiently. The resolvent expansions of dynamic programming are a special case of this method.