The "paradox" of computability and a recursive relative version of the Busy Beaver function

In this article, we will show that uncomputability is a relative property not only of oracle Turing machines, but also of subrecursive classes. We will define the concept of a Turing submachine, and a recursive relative version for the Busy Beaver function which we will call Busy Beaver Plus function. Therefore, we will prove that the computable Busy Beaver Plus function defined on any Turing submachine is not computable by any program running on this submachine. We will thereby demonstrate the existence of a "paradox" of computability a la Skolem: a function is computable when "seen from the outside" the subsystem, but uncomputable when "seen from within" the same subsystem. Finally, we will raise the possibility of defining universal submachines, and a hierarchy of negative Turing degrees.Comment: 10 pages. 0 figures. Supported by the National Council for Scientific and Technological Development (CNPq), Brazil. Book chapter published in Information and Complexity, Mark Burgin and Cristian S. Calude (Editors), World Scientific Publishing, 2016, ISBN 978-981-3109-02-5, available at http://www.worldscientific.com/worldscibooks/10.1142/10017. arXiv admin note: substantial text overlap with arXiv:1612.0522

Factorization invariants in numerical monoids

Nonunique factorization in commutative monoids is often studied using factorization invariants, which assign to each monoid element a quantity determined by the factorization structure. For numerical monoids (co-finite, additive submonoids of the natural numbers), several factorization invariants have received much attention in the recent literature. In this survey article, we give an overview of the length set, elasticity, delta set, $\omega$-primality, and catenary degree invariants in the setting of numerical monoids. For each invariant, we present current major results in the literature and identify the primary open questions that remain

Quality assessment of butter produced using traditional and mechanized churning methods

Traditional butter/ghee-making, predominantly done by women, is labor-intensive. To reduce this labor and/or increase incomes among these women, a hand-operated churner was previously developed with the capacity to reduce labor eight-fold. The present study was carried out to compare the quality of butter/ghee made using traditional churning in locally harvested plant containers (gourds and calabashes) and mechanized churning in the new device. As opposed to shaking the whole vessel, churning in the new device is achieved through a hand-operated crank connected to mixing baffles. Butter samples were aseptically collected from four locations (Kiboga1, Kiboga2, Kotido, and Ngoma) along the cattle corridor of Uganda. A “control” butter sample was made under laboratory conditions following standard procedure. The five samples were analyzed with respect to microbial safety, type and concentration of free fatty acids, and sensory attributes. Total viable count (TVC), Total coliforms (TC), Staphylococcus aureus, Salmonella, yeasts and molds counts were determined using International Organization for Standardization (ISO) standards. Fatty acid profile was determined by gas chromatography. Sensory evaluation of aroma, smell, taste, mouth feel, and overall acceptability of the products were also conducted. In the sensory evaluation, two commercially marketed ghee products (Sameer, and Lubega brands) were added. Total viable counts in all the samples were in the range of 102-107 cfu/g. Total coliforms were detected in Kiboga samples in the 101-103cfu/g range while none were detected from other regions’ samples. Yeasts and molds were detected in the 102-105 cfu/g range. Staphylococcus aureus was detected only in butter samples from Kiboga region (102 cfu/g) while Salmonella was not detected in any of the samples. The fatty acid profile consisted of saturated fatty acids, monounsaturated fatty acids, polyunsaturated fatty acids, trans-fatty acids, omega 3 fatty acids, omega 6 fatty acids, and omega 9 fatty acids. Saturated fatty acids were most dominant in the butter and ghee samples ranging from 70-82% whereas trans-fatty acids were present in the least concentration. From the overall acceptability dimension, the butter/ghee made using traditional churning and the new device scored the highest. However, Student’s t-test analysis showed no significant difference in the organoleptic parameters analyzed in all the samples (p>0.05). Therefore, the butter/ghee produced using mechanized churning is as acceptable and as microbiologically safe as butter/ghee produced using traditional churning and two representative marketed products.Key words: Butter, ghee, microbial safety, churning, traditional processing, human-centered desig