Fast algorithms for polynomial evaluation and differentiation at special knots

We present fast evaluating and differentiating algorithms for the Hermite interpolating polynomials with the knots of multiplicity 2, which are generated dynamically in a field K = (K,+,⋅) by the recurrent formula of the formX^i = αx^i-1+β (i=1,2,..,n-1; x^0=γ).As in the case of Lagrange-Newton interpolating algorithms, the running time of these algorithms is C(n) + O(n) base operations from the field K, where C(n) = O(nlog n) denotes the time needed to compute the wrapped convolution in Kn

Fast multidimensional Bernstein-Lagrange algorithms

In this paper we present two fast algorithms for the Bézier curves and surfaces of an arbitrary dimension. The first algorithm evaluates the Bernstein-Bézier curves and surfaces at a set of specific points by using the fast Bernstein-Lagrange transformation. The second algorithm is an inversion of the first one. Both algorithms reduce the initial problem to computation of some discrete Fourier transformations in the case of geometrical subdivisions of the d-dimensional cube. Their orders of computational complexity are proportional to those of corresponding d-dimensional FFT-algorithm, i.e. to O (N logN) + O (dN), where N denotes the order of the Bernstein-Bézier curves

Algorithms for multi-secret hierarchical sharing schemes of shamir type

In this paper there are presented algorithms for multilevel hierarchical threshold secret sharing schemes based on the interpolation of Hermite type, which use either traditional Shamir's keys or polynomial and orthogonal polynomial keys. These algorithms enable to compute the probability of authenticity of shares during the process of recovering the keys. In addition, two models of secret sharing are considered, which enlarge their security against attacks and decrease their rate of computations by applying FFT-algorithm

Fast interpolating algorithms in cryptography

We present two fast polynomial interpolating algorithms with knots generated in a field K by the recurrent formula of the form xf = axj_l + ft (z = 1,2,..,« -1; x0 = . The running time of them is C(n) + 0(n) base operations from K, where C(n) = 0(n\og2 n) denotes the time needed to compute the wrapped convolution in K. Moreover, we give an application of these algorithms to threshold secret sharing schemes in cryptography

The effect of aluminum exposure on reproductive ability in the Bank Vole (Myodes glareolus)

Human impact on the environment is steadily increasing the amounts of aluminum in the ecosystems. This element accumulates in plants and water, potentially exposing herbivores to its harmful effect. In heavily polluted sites, a decrease in the density of small rodent populations has been observed. This decline may be caused by many factors, including decreased fertility. The aim of the presented research was to determine how aluminum, administered at concentrations similar to those recorded in industrial districts (Al I = 3 mg/l, Al II = 200 mg/l), affects the reproductive abilities of small rodents. As the indicators of reproductive abilities, body weight, weight of the testes and accessory sex glands of males, and uterus weight of females were estimated. In females, the number of matured follicles (types 6, 7, and 8) was analyzed, while in males, the quantity and quality (matured, viable, swollen, motile, head abnormalities) of epididymal sperm cells were assessed. Moreover, the development of testes, measured by spermatogenic index, was determined. The model species was the bank vole. Our results have proven that aluminum impairs adult individuals’ reproductive abilities by decreasing the quality and quantity of sperm cells and by causing morphologically abnormal development of the gonads. However, no difference in male organometric parameters was found, and only in females treated with 3 mg/l Al, the uterus weight was higher than control. No differences were found in the total number of matured follicles. These results suggest that the decline in rodent numbers in industrial districts is due, at least in part, to poorer males’ reproductive abilities, resulting from exposure to aluminum contamination

Effect of different polishing methods on anodic titanium dioxide formation

Among various methods of synthesis of nanostructured TiO2, a self-organized anodization is the most commonly used and discussed in the literature. However, different methods of pretreatment of Ti before anodic titanium dioxide (ATO) formation are not often addressed. Therefore, various polishing procedures based on mechanical, chemical, electrochemical, and combined electrochemical with chemical pretreatments were examined to establish whether they represent effective methods for smooth Ti surface preparation before anodization. The ATO layers were prepared via two-step anodization carried out in an ethylene glycol solution containing fluoride ions at 20°C and under the anodizing potential of 60 V. The influence of applied polishing method on the cell size, pore diameter, pore circularity, pore density, and porosity of the top ATO layer was studied. In addition, the effect of polishing procedure on cell arrangement in ATO films was also investigated. The quantitative analyses of the regularity of cell arrangement, based on the regularity ratio derived from bottom-view SEM images, showed that the type of polishing procedure does not affect the cell order

The influence of fie and met1 mutations and in vitro culture conditions on autonomous endosperm development in unfertilized ovules of Arabidopsis thaliana

In flowering plants, seeds are produced both sexually (double fertilization is required) and asexually via apomixis (meiotic reduction and egg fertilization are omitted). An apomictic-like pattern of endosperm development in planta is followed by fis mutants of sexual Arabidopsis thaliana. In our experiments in planta, autonomous endosperm (AE) developed in met1 mutants. Furthermore we obtained autonomous endosperm formation in vitro not only in unfertilized ovules of fie mutants but also in wild genotypes (Col-0, MET1/MET1, FIE/FIE) and met1 mutants. AE induction and development occurred in all genotypes on the each of the media used and in every trial. The frequency of AE was relatively high (51.2% ovaries) and genotype-dependent. AE induced in vitro represents a more advanced stage of development than AE induced in fie mutants in planta. This was manifested by a high number of nuclei surrounded by cytoplasm and organized in nuclear cytoplasmic domains (NCDs), nodule formation, division into characteristic regions, and cellularization. The high frequency of AE observed in homozygous met1 (met1/met1) mutants probably is due to accumulation of hypomethylation as an effect of the met1 mutation and the in vitro conditions. AE development was most advanced in FIE/fie mutants. We suggest that changes in the methylation of one or several genes in the DNA of Arabidopsis genotypes caused by in vitro conditions resulted in AE induction and/or further AE development