Hereditary optimal control problems: Numerical method based upon a padé approximation

In this paper, we consider a particular approximation scheme which can be used to solve hereditary optimal control problems. These problems are characterized by variables with a time-delayed argumentx(t – ). In our approximation scheme, we first replace the variable with an augmented statey(t) x(t - ). The two-sided Laplace transform ofy(t) is a product of the Laplace transform ofx(t) and an exponential factor. This factor is approximated by a first-order Padé approximation, and a differential relation fory(t) can be found. The transformed problem, without any time-delayed argument, can then be solved using a gradient algorithm in the usual way. Four problems are solved to illustrate the validity and usefulness of this technique.This research was supported in part by the National Aeronautics and Space Administration under NASA Grant NCC-2-106.     

Properties of noninteger moments in a first passage time problem

We calculate the moments t ^q , whereq is not necessarily an integer, of the first passage time to trapping for a simple diffusion problem in one dimension. If a characteristic length of the system isL and t _q ~L (q) asL, then we show that there is a phase transition atq=q _c such that whenq<q _c ,(g)=0, and forq>q _c , (q) is a linear function ofq. These analytical results can be used to explain results for large moments for diffusion on a hierarchic structure. We also show how to calculate noninteger moments in terms of characteristic functions.     

Gradient extremals and valley floor bifurcations on potential energy surfaces

Gradient extremals are curves in configuration space denned by the condition that the gradient of the potential energy is an eigenvector of the Hessian matrix. Solutions of a corresponding equation go along a valley floor or along a crest of a ridge, if the norm of the gradient is a minimum, and along a cirque or a cliff or a flank of one of the two if the gradient norm is a maximum. Properties of gradient extremals are discussed for simple 2D model surfaces including the problem of valley bifurcations.     

Optimum gas chromatographic conditions in wall-coated capillary columns. Extended and simplified forms of the Golay-equation

Band broadening in capillary columns is satisfactorily described by the Golay-equation extended to situations of appreciable pressure drop by Giddings. In practice, however, several simplifications are often made. The effect of these simplifications on the calculated values of the minimum plate height and optimum carrier gas velocity are treated systematically.     

二组分混合物的最小流化特性 Ⅰ.混合/分离状态与最小流化空隙率

分析了二组分混合物的３种混合／分离状态及在完全混合时发生连续相转变的极限组成。提出一个判别非等密度体系混合／分离状态的法则与确定混合物空隙率简易可行的随意松散堆积方法。     

Dynamic optimization problems with bounded terminal conditions

Bounded terminal conditions of nonlinear optimization problems are converted to equality terminal conditions via the Valentine's device. In so doing, additional unknown parameters are introduced into the problem. The transformed problems can still be easily solved using the sequential gradient-restoration algorithm (SGRA) via a simple augmentation of the unknown parameter vector . Three example problems with bounded terminal conditions are solved to verify this technique.This research was supported in part by the National Aeronautics and Space Administration under NASA Grant No. NCC 2-106.     

On the statistical mechanics of the traveling salesman problem

We consider the statistical mechanics of the traveling salesman problem (TSP) and develop some representations to study it. In one representation the mean field theory has a simple form and brings out some of the essential features of the problem. It shows that the system has spontaneous symmetry breaking at any nonzero temperature. In general the phase progressively changes as one decreases the temperature. At low temperatures the mean field theory solution is very sensitive to any small perturbations, due to the divergence of some local susceptibilities. This critical region extends down to zero temperature. We perform the quenched average for a nonmetric TSP in the second representation and the resulting problem is more complicated than the infinite-range spin-glass problem, suggesting that the free energy landscape may be more complex. The role played by frustration in this problem appears explicitly through the localization property of a random matrix, which resembles the tight binding matrix of an electron in a random lattice.     

New developments in the synthesis of pyrrolizidinone-based dipeptide isosteres

1,3-Dipolar cycloaddition of acrylamide with the cyclic nitrone derived from proline tert-butyl ester has been employed in the synthesis of bicyclic Gly-(s-cis)Pro isosteres suitably protected for the Fmoc-based solid phase peptide synthesis. (R)-1-Phenylethylamine was introduced as chiral auxiliary to resolve racemic intermediates and obtain enantiopure compounds. Using methacrylamide as dipolarophile, the analogous Ala-Pro mimetics have been prepared in racemic form, whereas the same strategy applied to methyl itaconate failed to give the corresponding Asp-Pro mimetic.     

Model dipeptides incorporating the trans cyclohexane analogues of phenylalanine: further evidence of the relationship between side-chain orientation and β-turn type

In order to study the influence of the side-chain orientation on the peptide backbone conformation we have synthesised the model dipeptides t-BuCO-l-Pro-(1S,2R)-c₆Phe-NHMe and t-BuCO-l-Pro-(1R,2S)-c₆Phe-NHMe, incorporating each enantiomer of the trans cyclohexane analogue of phenylalanine (trans-1-amino-2-phenylcyclohexanecarboxylic acid). The orientation of the aromatic side-chain determines the β-turn type accommodated by these peptides to the point that the (1S,2R)-c₆Phe derivative retains the type I β-turn in the crystalline state, in contrast to the behaviour exhibited by the natural counterpart t-BuCO-l-Pro-l-Phe-NHMe.     

Properties of a genetic algorithm extended by a random self-learning operator and asymmetric mutations: A convergence study for a task of powder-pattern indexing

Genetic algorithms represent a powerful global-optimisation tool applicable in solving tasks of high complexity in science, technology, medicine, communication, etc. The usual genetic-algorithm calculation scheme is extended here by introduction of a quadratic self-learning operator, which performs a partial local search for randomly selected representatives of the population. This operator is aimed as a minor deterministic contribution to the (stochastic) genetic search. The population representing the trial solutions is split into two equal subpopulations allowed to exhibit different mutation rates (so called asymmetric mutation). The convergence is studied in detail exploiting a crystallographic-test example of indexing of powder diffraction data of orthorhombic lithium copper oxide, varying such parameters as mutation rates and the learning rate. It is shown through the averaged (over the subpopulation) fitness behaviour, how the genetic diversity in the population depends on the mutation rate of the given subpopulation. Conditions and algorithm parameter values favourable for convergence in the framework of proposed approach are discussed using the results for the mentioned example. Further data are studied with a somewhat modified algorithm using periodically varying mutation rates and a problem-specific operator. The chance of finding the global optimum and the convergence speed are observed to be strongly influenced by the effective mutation level and on the self-learning level. The optimal values of these two parameters are about 6 and 5%, respectively. The periodic changes of mutation rate are found to improve the explorative abilities of the algorithm. The results of the study confirm that the applied methodology leads to improvement of the classical genetic algorithm and, therefore, it is expected to be helpful in constructing of algorithms permitting to solve similar tasks of higher complexity.