On parametric nonlinear programming

In this tutorial survey we study finite dimensional optimization problems which depend on parameters. It is our aim to work out several basic connections with different mathematical areas. In particular, attention will be paid to unfolding and singularity theory, structural analysis of families of constraint sets, constrained optimization problems and semi-infinite optimization.     

Stability for linearly constrained optimization problems

We deal with finite dimensional differentiable optimization problems under linear constraints. Stability of stationary solutions under restricted perturbations of the constraints will be characterized. The restriction on the constraint perturbations is given by means of a certain rank condition; in particular, righthandside perturbations are allowed.Corresponding author.     

A general model for the absorption of ultrasound by biological tissues and experimental verification

In this paper, a closed-form expression is derived for the absorption of ultrasound by biological tissues. In this expression, the viscothermal and viscoelastic theories of relaxation processes are combined. Three relaxation time distribution functions are introduced, and it is assumed that each of these distributions can be described by an identical and simple hyperbolic function. Several simplifying assumptions had to be made to enable the experimental verification of the derived closed-form expression of the absorption coefficient. The simplified expression leaves two degrees of freedom and it was fitted to the experimental data obtained from homogenized beef liver. The model produced a considerably better fit to the data than other, more pragmatic models for the absorption coefficient as a function of frequency that could be found in the literature. Scattering in beef liver was estimated indirectly from the difference between attenuation in in vitro liver tissue as compared to absorption in a homogenate. The frequency dependence of the scattering coefficient could be described by a power law with a power of the order of 2. A comparable figure was found in direct backscattering measurements, performed at our laboratory with the same liver samples [Van den Aarssen et al., J. Acoust. Soc. Am. (to be published)]. A model for scattering recently proposed by Sehgal and Greenleaf [Ultrason. Imag. 6, 60-80 (1984)] was fitted to the scattering data as well. This latter model enabled the estimation of a maximum scatterer distance, which appeared to be of the order of 25 micron.     

Mesomorphic behaviour of praseodymium(III) alkanoates

In this paper, we describe the synthesis, characterization and thermal behaviour of praseodymium(III) alkanoates. The compounds have the stoichiometry [Pr(C _x H_2x+1COO)₃], where x = 5-19, and were characterized by elemental analysis and infrared spectroscopy. The thermal behaviour was investigated by hot-stage polarizing optical microscopy, DSC and high temperature X-ray diffraction. A highly viscous mesophase M and a smectic A phase were observed for the shorter chain compounds (x = 5-8), whereas only a smectic A phase was observed for the longer chain compounds of this type of metallomesogens. The chain length has a pronounced effect on the transition temperatures.     

Thermal behaviour of lanthanum(III) alkanoates

The mesophase behaviour of the lanthanum(III) alkanoates [La(C_xH_2x+1COO)₃] (x =3-19) has been investigated by hot-stage polarizing optical microscopy, differential scanning calorimetry and high-temperature X-ray diffraction. Lanthanum(III) butyrate monohydrate shows no mesomorphism, whereas for the remaining short chain homologues (x = 4-9) a highly viscous mesophase M and a smectic A phase were observed. The longer chain lanthanum(III) soaps (x = 10-19) exhibit only a smectic A phase. However, the chain length has a pronounced effect on the transition temperatures. The thermal behaviour of lanthanum(III) alkanoates is compared with that of other lanthanide(III) alkanoates.     

Use of seeds to control precipitation of calcium carbonate and determination of seed nature

Understanding and controlling precipitation reactions is a major challenge for industrial crystallization. Calcium carbonate is a widely studied system: more than 3000 papers have been devoted to the subject over the past 10 years. The first step of the precipitation of calcium carbonate, from relatively concentrated solutions (0.01 mol/L), involves the formation of an initial gel phase which later transforms into calcite, vaterite, or a mixture of both phases. Our work aimed at controlling this first step. Nanosized seeds (8 nm), formed in situ, were used in order to control the often chaotic nucleation step which normally leads to poor phase selection and broad particle size distributions. Seeding has often been used to avoid spontaneous nucleation in metastable solutions for growth mechanism investigations of single-crystal calcium carbonate. Here the ability of a seeding method to control the precipitation reaction evolution even in the case of high supersaturation is demonstrated. The seeds and the presence of a polymeric additive (poly(acrylic acid)) allow the control of the precipitated polymorph and the specific surface area, while maintaining a narrow particle size distribution in the submicron range. Direct characterization methods did not succeed in identifying these nanoseeds; indirect methods using solubility calculations are used to demonstrate their existence and quantify size and number density of the nanosized seeds.     

Global Optimization: Local Minima and Transition Points

We consider the minimization of smooth functions of the Euclidean space with a finite number of stationary points having moderate asymptotic behavior at infinity. The crucial role of transition points of first order (i.e., saddle points of index 1) is emphasized. It is shown that (generically) any two local minima can be connected via an alternating sequence of local minima and transition points of first order. In particular, the graph with local minima as its nodes and first order transition points representing the edges turns out to be connected (Theorem A). On the other hand, any connected (finite) graph can be realized in the above sense by means of a smooth function of three variables having a minimal number of stationary points (Theorem B).