Solutions of xqk+⋯+xq+x=ax^{q^k}+\cdots+x^{q}+x=a in GF2nGF{2^n}

Though it is well known that the roots of any affine polynomial over a finite field can be computed by a system of linear equations by using a normal base of the field, such solving approach appears to be difficult to apply when the field is fairly large. Thus, it may be of great interest to find an explicit representation of the solutions independently of the field base. This was previously done only for quadratic equations over a binary finite field. This paper gives an explicit representation of solutions for a much wider class of affine polynomials over a binary prime field

Betalain, Acid Ascorbic, Phenolic Contents and Antioxidant Properties of Purple, Red, Yellow and White Cactus Pears

Commercialization of cactus pears based on their antioxidant properties can generate competitive advantages, and these can turn into business opportunities and the development of new products and a high-value ingredient for the food industry. This work evaluated the antioxidant activities (1,1-diphenyl-2-picrylhydrazyl radical-scavenging, protection against oxidation of a β-carotene-linoleic acid emulsion, and iron (II) chelation), the content of total phenolic compounds, ascorbic acid, betacyanin, betaxanthin and the stability of betacyanin pigments in presence of Cu (II)-dependent hydroxyl radicals (OH•), in 18 cultivars of purple, red, yellow and white cactus pear from six Mexican states. Our results indicated that the antiradical activities from yellow and white cactus pear cultivars were not significantly different (p < 0.05) and were lower than the average antiradical activities in red and purple cultivars. The red cactus pear from the state of Zacatecas showed the highest antioxidant activity. The free radical scavenging activity for red cactus pears was significantly correlated (p < 0.05) to the concentration of total phenolic compounds (R2 = 0.90) and ascorbic acid (R2 = 0.86). All 18 cultivars of cactus pears studied showed significant chelating activity of ferrous ions. The red and purple cactus pears showed a great stability when exposed to OH•

Neuroprotective activity of Hypericum perforatum and its major components

Hypericum perforatum is a perennial plant, with worldwide distribution, commonly known as St. Johns wort. It has been used for centuries in traditional medicine for the treatment of several disorders, such as minor burns, anxiety, and mild to moderate depression. In the past years, its antidepressant properties have been extensively studied. Despite that, other H. perforatum biological activities, as its neuroprotective properties have also been evaluated. The present review aims to provide a comprehensive summary of the main biologically active compounds of H. perforatum, as for its chemistry, pharmacological activities, drug interactions and adverse reactions and gather scattered information about its neuroprotective abilities. As for this, it has been demonstrated that H. perforatum extracts and several of its major molecular components have the ability to protect against toxic insults, either directly, through neuroprotective mechanisms, or indirectly, through is antioxidant properties. H. perforatum has therefore the potential to become an effective neuroprotective therapeutic agent, despite further studies that need to be carried out.This work was supported by Fundacao para a Ciencia e Tecnologia (FCT), projects PTDC/AGR-ALI/105169/2008, PEst-OE/AGR/UI4033/2014. AO was supported by Escola Superior de Tecnologia da Saude do Porto and Instituto Politecnico do Porto (Programa de Formacao Avancada de Docentes).info:eu-repo/semantics/publishedVersio