Shape-preserving interpolation by G1 and G2 PH quintic splines

The interpolation of a planar sequence of points p₀, ..., p_Nby shape-preserving G¹ or G² PH quintic splines with specifiedend conditions is considered. The shape-preservation propertyis secured by adjusting ‘tension’ parameters thatarise upon relaxing parametric continuity to geometric continuity.In the G² case, the PH spline construction is based on applyingNewton–Raphson iterations to a global system of equations,commencing with a suitable initialization strategy—thisgeneralizes the construction described previously in NumericalAlgorithms 27, 35–60 (2001). As a simpler and cheaperalternative, a shape-preserving G¹ PH quintic spline schemeis also introduced. Although the order of continuity is lower,this has the advantage of allowing construction through purelylocal equations.     

Pythagorean-hodograph space curves

We investigate the properties of polynomial space curvesr(t)={x(t), y(t), z(t)} whose hodographs (derivatives) satisfy the Pythagorean conditionx′²(t)+y′²(t)+z′²(t)≡σ²(t) for some real polynomial σ(t). The algebraic structure of thecomplete set of regular Pythagorean-hodograph curves in ℝ³ is inherently more complicated than that of the corresponding set in ℝ². We derive a characterization for allcubic Pythagoreanhodograph space curves, in terms of constraints on the Bézier control polygon, and show that such curves correspond geometrically to a family of non-circular helices. Pythagorean-hodograph space curves of higher degree exhibit greater shape flexibility (the quintics, for example, satisfy the general first-order Hermite interpolation problem in ℝ³), but they have no “simple” all-encompassing characterization. We focus on asubset of these higher-order curves that admits a straightforward constructive representation. As distinct from polynomial space curves in general, Pythagorean-hodograph space curves have the following attractive attributes: (i) the arc length of any segment can be determined exactly without numerical quadrature; and (ii) thecanal surfaces based on such curves as spines have precise rational parameterizations.     

Informationally optimal correlation

This papers studies an optimization problem under entropy constraints arising from repeated games with signals. We provide general properties of solutions and a full characterization of optimal solutions for 2 × 2 sets of actions. As an application we compute the minmax values of some repeated games with signals.     

Efficient Mittag-Leffler Collocation Method for Solving Linear and Nonlinear Fractional Differential Equations

In this paper, a new approximation method for fractional differential equations based on Mittag-Leffler function is developed. Finite Mittag-Leffler function and its fractional-order derivatives are investigated. An efficient technique for solving linear and nonlinear fractional order differential equations is developed. The proposed method combines Mittag-Leffler collocation method and optimization technique. Error estimation of the approximation is stated and proved. We present numerical results and comparisons of previous treatments to demonstrate the efficiency and applicability of the proposed method. Making use of small number of unknowns, the resulting solution converges to the exact one in the linear case and it has a very small error in the nonlinear case.     

Boundary evaluation algorithms for Minkowski combinations of complex sets using topological analysis of implicit curves

Minkowski geometric algebra is concerned with sets in the complex plane that are generated by algebraic combinations of complex values varying independently over given sets in ℂ. This algebra provides an extension of real interval arithmetic to sets of complex numbers, and has applications in computer graphics and image analysis, geometrical optics, and dynamical stability analysis. Algorithms to compute the boundaries of Minkowski sets usually invoke redundant segmentations of the operand-set boundaries, guided by a “matching” criterion. This generates a superset of the true Minkowski set boundary, which must be extracted by the laborious process of identifying and culling interior edges, and properly organizing the remaining edges. We propose a new approach, whereby the matching condition is regarded as an implicit curve in the space ℝⁿ whose coordinates are boundary parameters for the n given sets. Analysis of the topological configuration of this curve facilitates the identification of sets of segments on the operand boundaries that generate boundary segments of the Minkowski set, and rejection of certain sets that satisfy the matching criterion but yield only interior edges. Geometrical relations between the operand set boundaries and the implicit curve in ℝⁿ are derived, and the use of the method in the context of Minkowski sums, products, planar swept volumes, and Horner terms is described.     

The splitting game and applications

First we define the splitting operator, which is related to the Shapley operator of the splitting game introduced by Sorin (2002). It depends on two compact convex sets C and D and associates to a function defined on C×D a saddle function, extending the usual convexification or concavification operators. We first prove general properties on its domain and its range. Then we give conditions on C and D allowing to preserve continuity or Lipschitz properties, extending the results in Laraki (2001a) obtained for the convexification operator. These results are finally used, through the analysis of the asymptotic behavior of the splitting game, to prove the existence of a continuous solution for the Mertens-Zamir system of functional equations (Mertens and Zamir (1971–72) and (1977)) in a quite general framework. Revised November 2001     

The polymerization of ethylene with a soluble titanium-aluminium complex

The reaction between biscyclopentadienyl titanium dichloride and aluminium alkyls and alkyl chlorides has been examined by ESR spectroscopy, and by the abilities of the various resulting complexes to initiate the polymerization of ethylene at 1 atm and ambient temperature. In general, it is concluded that the active initiating species is a Ti(IV) complex, and not the paramagnetic Ti(III) complexes. Accordingly, the development of an ESR spectrum is accompanied by a fall off in initiating efficiency, decrease in polymerization rate and increased molecular weight of the polymers produced. The reduction of Ti(IV) to Ti(III) during the polymerization accounts for the fall in rate with conversion; addition of an oxidizing agent (1:2-dichloroethane) to the polymerization converts Ti(III) to Ti(IV), as observed by the disappearance of the ESR signal, and increases the efficiency of the polymerization. Block copolymers of ethylene with propylene and butadiene have been prepared with this initiator; the efficiency in producing blocks has been used to study the decomposition of the active sites by a first order reduction process.     

Gas-phase fragmentation study of novel synthetic 1,5-benzodiazepine derivatives using electrospray ionization tandem mass spectrometry

The fragmentation patterns of a series of three novel synthesized 3-hydroxy-4-phenyl-tetrahydro-1,5-benzodiazepin-2-ones (1-3), possessing the same backbone structure, were investigated using electrospray ionization mass spectrometry (ESI-MS) and tandem mass spectrometry (MS/MS) techniques. A simple methodology, based on the use of ESI (positive ion mode) and by increasing the declustering potential in the atmospheric pressure/vacuum interface, collision-induced dissociation (CID), was used to enhance the formation of the fragment ions. In general, the novel synthetic 1,5-benzodiazepine derivatives afforded, in the gas phase, both protonated and sodiated molecules. This led to the confirmation of the molecular masses and chemical structures of the studied compounds. Exact accurate masses were measured using a high-resolution ESI-quadrupole orthogonal time-of-flight (QqToF)-MS/MS hybrid mass spectrometer instrument.The breakdown routes of the protonated molecules were rationalized by conducting low-energy collision CID-MS/MS analyses (product ion- and precursor ion scans) using a conventional quadrupole-hexapole-quadrupole (QhQ) tandem mass spectrometer. All the observed major fragmentations for the 1,5-benzodiazepines occurred in the saturated seven-membered ring containing the nitrogen atoms. These formed a multitude of product ions by different breakdown routes. All the major fragmentations involved cleavages of the N-1-C-2 and C-3-C-4 bonds. These occurred with concomitant eliminations of glyoxal, benzene and ethyl formate, forming the product ion at m/z 119, which was observed in all the studied compounds. In addition, an unique simultaneous CID-MS/MS fragmentation was noticed for the 1,5-benzodiazepines 1 and 3, which occurred by a pathway dictated by the substituent located on the N-1-position. It was evident that the aromatic ring portion of the 1,5-benzodiazepines was resistant to CID-MS/MS fragmentation. Re-confirmation of the various geneses of the product ions was achieved by conducting a series of precursor ion scans. ESI-MS and CID-MS/MS analyses have thus proven to be a specific and very sensitive method for the structural identification of these novel 1,5-benzodiazepine derivatives.     

Homotopy Analysis Method for Solving Biological Population Model

In this paper, the homotopy analysis method (HAM) is applied to solve generalized biological population models. The fractional derivatives are described by Caputo's sense. The method introduces a significant improvement in this field over existing techniques. Results obtained using the scheme presented here agree well with the analytical solutions and the numerical results presented in Ref. [6]. However, the fundamental solutions of these equations still exhibit
useful scaling properties that make them attractive for applications.