The relaxed Newton method derivative: Its dynamics and non-linear properties

The dynamic behaviour of the one-dimensional family of maps f(x)=_c2[(a−1)x+_c1]^{−λ/(α−1)}$f\left(x\right)=c_2{[(a-1)x+c_1]}^{-\lambda /(\alpha -1)}$ is examined, for representative values of the control parameters a,_c1$a,c_1$, _c2$c_2$ and λ$\lambda$. The maps under consideration are of special interest, since they are solutions of the relaxed Newton method derivative being equal to a constant a$a$. The maps f(x)$f\left(x\right)$ are also proved to be solutions of a non-linear differential equation with outstanding applications in the field of power electronics. The recurrent form of these maps, after excessive iterations, shows, in an _xn$x_n$ versus λ$\lambda$ plot, an initial exponential decay followed by a bifurcation. The value of λ$\lambda$ at which this bifurcation takes place depends on the values of the parameters a,_c1$a,c_1$ and _c2$c_2$. This corresponds to a switch to an oscillatory behaviour with amplitudes of f(x)$f\left(x\right)$ undergoing a period doubling. For values of a$a$ higher than 1 and at higher values of λ$\lambda$ a reverse bifurcation occurs. The corresponding branches converge and a bleb is formed for values of the parameter _c1$c_1$ between 1 and 1.20. This behaviour is confirmed by calculating the corresponding Lyapunov exponents.     

On the argument of zeta-functions of certain cusp forms near the critical line

We prove a limit theorem for the argument of zeta-functions of holomorphic normalized Hecke-eigen cusp forms near the critical line.     

Pareto set approximation by the method of adjustable weights and successive lexicographic goal programming

A nonlinear multiobjective optimization problem is considered. Two methods are proposed to generate solutions with an approximately uniform distribution in a Pareto set. The first method is supposed to find the solutions as minimizers of weighted sums of objective functions where the weights are properly selected using a branch and bound type algorithm. The second method is based on lexicographic goal programming. The proposed methods are compared with several metaheuristic methods using two and three-criteria tests and applied problems.     

P-algorithm based on a simplicial statistical model of multimodal functions

A well-recognized one-dimensional global optimization method is generalized to the multidimensional case. The generalization is based on a multidimensional statistical model of multimodal functions constructed by generalizing computationally favorable properties of a popular one-dimensional model—the Wiener process. A simplicial partition of a feasible region is essential for the construction of the model. The basic idea of the proposed method is to search where improvements of the objective function are most probable; a probability of improvement is evaluated with respect to the statistical model. Some results of computational experiments are presented.     

Axiomatic approach to statistical models and their use in multimodal optimization theory

This paper summarizes the results of axiomatic constructing statistical models of complicated multimodal functions. It is shown that an optimization algorithm may be constructed on the basis of a statistical model and some ideas of the rational choice theory. A brief review of related algorithms and reports on investigations of their efficiency is given.     

On justification of use of stochastic functions for multimodal optimization models

A model of a complicated function under uncertainty is constructed axiomatically, formalizing suppositions on rationality of information on a considered function.     

Formation of 2-oxatricyclo[4.3.1.03,8]decanes in intramolecular cyclization of α-halobicyclo[3.3.1]nonanones

Transformations of α-chloro- and α-bromobicyclo[3.3.1]nonanones under conditions of the Favorskii reaction were studied. The interaction of dihalodiketones with MeONa gives 2-oxatricyclo[4.3.1.0^3.8]decane (oxaprotoadamantane) derivatives as a result of intramolecular cyclization, whereas 3-bromobicyclononanone undergoes only nucleophilic substitution of bromine. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 2, pp. 342–344, February, 1997.     

Limit theorems for the Estermann zeta-function. II

A limit theorem in the sense of the weak convergence of probability measures in the space of meromorphic functions for the Estermann zeta-function is obtained. Partially supported by Lithuanian Foundation of Studies and Science     

Value distribution of the Lerch zeta-function with algebraic irrational parameter. III

We obtain a limit theorem in the sense of weak convergence of probability measures in the space of analytic functions for the Lerch zeta-function with algebraic irrational parameter. Partially supported by the Lithuanian Studies and Science Foundation.