G. Leitmann’s integral transformation for global optimal control

Back in 1967, George Leitmann has published “A Note on Absolute Extrema of Certain Integrals” [1] which later generated much interest in the control systems theory and, in effect, started a new method for global optimal control. In this short paper, some observations, extensions, further developments and perspectives of G. Leitmann’s equivalence transformation are discussed to mark his exceptional contribution to control theory and to facilitate further research in this area following the bright idea of George Leitmann’s basic Lemma on translational symmetry of certain integrals.     

The solution of infinitely ill-conditioned weakly-singular problems

It is demonstrated on examples that a weak singularity (i.e., with converging improper integral) may produce in computations (depending on the algorithm employed) an infinitely ill-conditioned situation when arbitrarily small imprécisions introduced by the algorithm or by a software create divergent approximations for mathematically convergent integrals. The possibility of hidden singularities is shown, and the double error phenomenon is identified and demonstrated in a simple example. Construction of test problems is proposed to check the applicability of existing software prior to its use for the solution of real life problems with weakly-singular equations. It is shown that the application of the integration by parts formula to weakly-singular integrals may create strong singularities (i.e., unbounded terms or divergent improper integrals). Methods of removal of singularities with and without compensation are studied for the numerical solution of infinitely ill-conditioned weakly-singular problems.     

The conditions of existence of first integrals and hamiltonian structures of Lotka-Volterra and Volterra systems

A study of a certain subset of Volterra equations has revealed that some statements about time-independent constants of motion, Hamiltonian functions, and Poisson structure matrices appearing in the Lotka-Volterra equations, either regarded as proven or of the sort that could be proven, are not valid, in fact. Particular cases are given as examples to explain the reasons for the occurring phenomena.     

Synthesis and characterization of new radiopaque microspheres by the dispersion polymerization of an iodinated acrylate monomer for X‐ray imaging applications

Radiopaque microspheres of sizes ranging from 0.2 to 1.4 μm were formed by the dispersion polymerization of the monomer 2‐methacryloyloxyethyl(2,3,5‐triiodobenzoate) in 2‐methoxyethanol. The effects of various polymerization parameters, including the monomer concentration, initiator type and concentration, and stabilizer molecular weight and concentration, on the molecular weight, size, and size distribution of the particles were elucidated. The characterization of these iodinated microspheres was accomplished with routine methods such as Fourier transform infrared, nuclear magnetic resonance, thermogravimetric analysis, differential scanning calorimetry, gel permeation chromatography, scanning electron microscopy, Brunauer–Emmett–Teller measurements, and elemental analysis. Because of the presence of iodine atoms in these microspheres, they were expected to possess a radiopaque nature. The radiopacity of these particles dispersed in water and in the dry state was demonstrated with an imaging technique based on X‐ray absorption usually used in hospitals. These novel radiopaque microspheres may be used for different X‐ray imaging needs, such as blood pooling, body organs, embolization, dental compositions, implants, prostheses, and nanocomposites. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 3859–3868, 2006     

Sensitivity of Pareto Solutions in Multiobjective Optimization

The paper presents a sensitivity analysis of Pareto solutions on the basis of the Karush-Kuhn-Tucker (KKT) necessary conditions applied to nonlinear multiobjective programs (MOP) continuously depending on a parameter. Since the KKT conditions are of the first order, the sensitivity properties are considered in the first approximation. An analogue of the shadow prices, well known for scalar linear programs, is obtained for nonlinear MOPs. Two types of sensitivity are investigated: sensitivity in the state space (on the Pareto set) and sensitivity in the cost function space (on the balance set) for a vector cost function. The results obtained can be used in applications for sensitivity computation under small variations of parameters. Illustrative examples are presented.Research of this author was partially supported by Grant BEC2003-09067-C04-03.Research of this author was partially supported by NSERC Grant RGPIN-3492-00.Research of this author was partially supported by Grant BEC2003-09067-C04-02.     

A separable integer programming problem equivalent to its continual version

The separable integer programming problem with so called nested constraints is shown to be equivalent to its continual version obtained by piecewise linear continuation of the cost functions. A new approach to solution of the latter based on its successive reduction in size is suggested. When applied to the problem with piecewise linear convex functions it leads to two algorithms for its solution applicable also to the similar integer problem. These algorithms turn out more efficient than those obtained by dynamic programming approach.     

Non-periodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons

This paper proves the existence of non-periodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons. For anyn≧3 there exists a corresponding convexn-agon (forn=3 this will be a right triangle with a small acute angle), while in three-dimensional space it will be a prism, then-agon serving as the base. The results are applied for investigating a mechanical system of two absolutely elastic balls on a segment, and also for proving the existence of an infinite number of periodic trajectories in the given polygons. The exchange transformation of two intervals is used for proving the theorems. An arbitrary exchange transformation of any number of intervals can also be modeled by a billiard trajectory in some convex polygon with many sides.     

Some peculiarities of coke formation and burning on catalysts for reforming

Presence of two maxima on the DTA curves of coked catalysts is attributed to the metal-catalyzed oxidation of less polymerized coke in a higher temperature region.

---

.