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β-carotene is a powerful antioxidant and dietary precursor of vitamin A whose role in maintaining mental health and cognitive performance, either alone or in combination with other dietary compounds, has been a topic of recent research. However, its effectiveness is still unclear. This systematic review, conducted according to the PRISMA guideline and assisted by the MySLR platform, addressed this issue. A total of 16 eligible original research articles were identified. Dietary intake or β-carotene serum levels were associated with improved measures of cognitive function in 7 out of 10 epidemiological studies included. In intervention studies, β-carotene consumption alone did not promote better cognitive function in the short term, but only in a long-term intervention with a mean duration of 18 years. However, all but one intervention study suggested the beneficial effects of β-carotene supplementation at doses ranging from 6 mg to 50 mg per day in combination with a multicomplex such as vitamin E, vitamin C, zinc, or selenium for a period of 16 weeks to 20 years. Despite the current limitations, the available evidence suggests a potential association between β-carotene dietary/supplementary intake and the maintenance of cognitive function. The β-carotene most probably does not act alone but in synergy with other micronutrients.

We focus on the random generation of SAT instances that have properties similar to real-world instances. It is known that many industrial instances, even with a great number of variables, can be solved by a clever solver in a reasonable amount of time. This is not possible, in general, with classical randomly generated instances. We provide a different generation model of SAT instances, called scale-free random SAT instances. This is based on the use of a non-uniform probability distribution P(i)∼i−β to select variable i, where β is a parameter of the model. This results in formulas where the number of occurrences k of variables follows a power-law distribution P(k)∼k−δ, where δ=1+1/β. This property has been observed in most real-world SAT instances. For β=0, our model extends classical random SAT instances. We prove the existence of a SAT–UNSAT phase transition phenomenon for scale-free random 2-SAT instances with β<1/2 when the clause/variable ratio is m/n=1−2β(1−β)2. We also prove that scale-free random k-SAT instances are unsatisfiable with a high probability when the number of clauses exceeds ω(n(1−β)k). The proof of this result suggests that, when β>1−1/k, the unsatisfiability of most formulas may be due to small cores of clauses. Finally, we show how this model will allow us to generate random instances similar to industrial instances, of interest for testing purposes.