Preconditioned Subspace Descent Method for Nonlinear Systems of Equations

Nonlinear least squares iterative solver is considered for real-valued sufficiently smooth functions. The algorithm is based on successive solution of orthogonal projections of the linearized equation on a sequence of appropriately chosen low-dimensional subspaces. The bases of the latter are constructed using only the first-order derivatives of the function. The technique based on the concept of the limiting stepsize along normalized direction (developed earlier by the author) is used to guarantee the monotone decrease of the nonlinear residual norm. Under rather mild conditions, the convergence to zero is proved for the gradient and residual norms. The results of numerical testing are presented, including not only small-sized standard test problems, but also larger and harder examples, such as algebraic problems associated with canonical decomposition of dense and sparse 3D tensors as well as finite-difference discretizations of 2D nonlinear boundary problems for 2nd order partial differential equations.

Estimating Linking Functions for Response Model Parameters

Parameter linking in item response theory is generally necessary to adjust for differences between the true values for the same item and ability parameters due to the use of different identifiability restrictions in different calibrations. The research reported in this article explores a precision-weighted (PW) approach to the problem of estimating the linking functions for the common dichotomous logistic response models. Asymptotic standard errors (ASEs) of linking for the new approach are derived and compared to those of the mean/mean and mean/sigma linking methods to which it has a superficial similarity and to the Haebara and Stocking and Lord response function methods. Empirical examples from a few recent linking studies are presented. It is demonstrated that the new approach has smaller ASE than the mean/mean and mean/sigma methods and comparable ASE to the response function methods. However, when some of the item parameters have large estimation error relative to the others, all current methods appear to violate the rather obvious requirement of monotone decrease in ASE with the number of common items in the linking design while the ASE of the PW method demonstrates monotone decrease with the number of common items. The PW method also has the benefits of simple calculation and an ASE which is additive in the contribution of each item, useful for optimal linking design. We conclude that the proposed approach to estimating linking parameters holds promise and warrants further research.

Influence of Metal-Containing Nanoparticles on the Content of Photosynthetic Pigments of Unicellular Alga Chlorella vulgaris Baijer

The research studies concentration effects and influence of nCeO2, nZnO, nNi and nPt on photosynthetic pigments in Chlorella v. B. in conditions of homeostated lab cultivation. It was shown that dependency of the content of chlorophylls and carotenoids in chlorella cells on concentrations of nCeO2, nZnO and nPt has non-linear dual-phase character. Growing concentrations of nNi in the culture solution caused monotone decrease of all photosynthetic pigments in chlorella cells. Adding 0.1 mg/L nZnO in the suspension caused a statistically significant increase of the chlorophyll a and b content (27.1 % and 64.2 % respectively) in comparison with the control sample. NPs Pt at 1 mg/L induced stimulating effect on chlorophylls a (17.4 % compared to the control) and b (23.6 % compared to the control) in chlorella cells. The number of carotenoids had a statistically significant decrease in chlorella cells by 25-70 % at high concentrations (1-10 mg/L) for all tested NPs.

TRAJECTORY SCALING FUNCTION OF POLYNOMIAL CIRCLE MAPS

The universality of trajectory scaling function (TSF) for the polynomial form of the circle map in the quasiperiodic-to-chaos transition is discussed. It is found that the TSF gets bumpier with increase in the degree of inflection z, for both Golden and Silver Mean cases, and shows multiple scaling with increase in z, reflecting the monotone decrease of α with increasing z. The effect of noise is also studied.