The quantum-rotors model can be regarded as an effective model for the low-temperature behavior of the quantum Heisenberg antiferromagnets. Here, we consider a d-dimensional model in the spherical approximation confined to a general geometry of the form L dϪdЈ ϫϱ dЈ ϫL z (L-linear space size and L -temporal size͒ and subjected to periodic boundary conditions. Due to the remarkable opportunity it offers for rigorous study of finite-size effects at arbitrary dimensionality this model may play the same role in quantum critical phenomena as the popular Berlin-Kac spherical model in classical critical phenomena. Close to the zerotemperature quantum critical point, the ideas of finite-size scaling are utilized to the fullest extent for studying the critical behavior of the model. For different dimensions 1ϽdϽ3 and 0рdЈрd a detailed analysis, in terms of the special functions of classical mathematics, for the susceptibility and the equation of state is given. Particular attention is paid to the two-dimensional case.

Abstract. Twenty-two samples from different beers have been investigated in two colour systems -XYZ and SIELab -and have been characterised according to their colour parameters. The goals of the current study were to conduct correlation and discriminant analysis and to find the inner relation between the studied indices. K-means cluster has been used to compare and group the tested types of beer based on their similarity. To apply the K-Cluster analysis it is required that the number of clusters be determined in advance. The variant K = 4 was worked out. The first cluster unified all bright beers, the second one contained samples with fruits, the third one contained samples with addition of lemon, the fourth unified the samples of dark beers. By applying the discriminant analysis it is possible to help selections in the establishment of the type of beer. The proposed model correctly describes the types of beer on the Bulgarian market and it can be used for determining the affiliation of the beer which is not used in obtained model. One sample has been chosen from each cluster and the digital image has been obtained. It confirms the color parameters in the color system XYZ and SIELab. These facts can be used for elaboration for express estimation of beer by color.

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The current research focused on algae from the waters of the Black Sea—Chaetomorpha linum, Ulva intestinalis, Ericaria crinita, and bioreactors—Chlorella spp. and Arthrospira platensis/cyanobacterium/. Pigment content, total phenolic content, and antioxidant capacity were investigated for their use as pharmaceutical, food, and cosmetic ingredients. E. crinita exhibited the highest antioxidant activity by ORAC and HORAC (463.3 µmol TE/g and 463.3 µmol GAE/g) and the highest total content of polyphenols and rutin of the investigated algae. Lower protein content was found in saltwater algae than in freshwater algae. For the first time, biologically active substances from the Bulgarian A. platensis and Chlorella spp., produced in a bioreactor, have been quantitatively identified. A. platensis contained rutin (141.25 mg/100 g), naringenin (42.17 mg/100 g), quercetin (26.74 mg/100 g), kaempferol, and quercetin-3-β-glycoside. Phenolic acids were isolated: neochlorogenic (172.27 mg/100 g) for lyophilized and 5783 mg/100 g for convection-dried A. platensis. It has been found that the protein content in A. platensis (convection dried) and Chlorella spp. (lyophilized) was nearly 54% higher than that of green algae U. intestinalis from Varna. The lyophilization process reduced the protein content of A. platensis samples by almost 20%. The high protein content of convection-dried A. platensis (43.4%) and lyophilized Chlorella spp. (43.7%) identified them as suitable emulsifiers in colloidal and emulsion systems.