Anthropology in the Real World: Year VI

2012 Petersheim Academic Expositio

Transcendental Numbers

The numbers e and π are transcendental numbers, meaning each of them are not the root of any polynomial with rational coefficients. We prove that e and π are transcendental numbers. The original proofs use the Fundamental Theorem of Symmetric Polynomials and Stirling’s Formula, which we develop and prove. Since π is not algebraic, neither is √π, which answers the ancient question of whether one can square a circle. The proof that π is transcendental is a beautiful example of how higher level mathematics can be used to answer ancient questions

IAF 15 Draft Paper

With the International Space Station Program transition from assembly to utilization, focus has been placed on the optimization of essential resources. This includes resources both resupplied from the ground and also resources produced by the ISS. In an effort to improve the use of two of these, the ISS Engineering teams, led by the ISS Program Systems Engineering and Integration Office, undertook an effort to modify the techniques use to perform several key on-orbit events. The primary purposes of this endeavor was to make the ISS more efficient in the use of the Russian-supplied fuel for the propulsive attitude control system and also to minimize the impacts to available ISS power due to the positioning of the ISS solar arrays. Because the ISS solar arrays are sensitive to several factors that are present when propulsive attitude control is used, they must be operated in a manner to protect them from damage. This results in periods of time where the arrays must be positioned, rather than autonomously tracking the sun, resulting in negative impacts to power generated by the solar arrays and consumed by both the ISS core systems and payload customers. A reduction in the number and extent of the events each year that require the ISS to use propulsive attitude control simultaneously accomplishes both these goals. Each instance where the ISS solar arrays normal sun tracking mode must be interrupted represent a need for some level of powerdown of equipment. As the magnitude of payload power requirements increases, and the efficiency of the ISS solar arrays decreases, these powerdowns caused by array positioning, will likely become more significant and could begin to negatively impact the payload operations. Through efforts such as this, the total number of events each year that require positioning of the arrays to unfavorable positions for power generation, in order to protect them against other constraints, are reduced. Optimization of propulsive events and transitioning some of them to non-propulsive CMG control significantly reduces propellant usage on the ISS leading to the reduction of the propellant delivery requirement. This results in move available upmass that can be used for delivering critical dry cargo, additional water, air, crew supplies and science experiments

Consecutive primes which are widely digitally delicate and Brier numbers

Making use of covering systems and a theorem of D. Shiu, the first and second authors showed that for every positive integer $k$, there exist $k$ consecutive widely digitally delicate primes. They also noted that for every positive integer $k$, there exist $k$ consecutive primes which are Brier numbers. We show that for every positive integer $k$, there exist $k$ consecutive primes that are both widely digitally delicate and Brier numbers

Totémisme duel, totémisme pluriel. Un exemple de Nouvelle-Guinée

L'auteur analyse deux formes de totémisme parmi les Yafar, une petite société forestière des basses terres de Papouasie Nouvelle-Guinée : un totémisme duel fondé sur l'organisation de la société en moitiés rituelles, et un totémisme clanique dans lequel chaque patriclan ancien exhibe, à certaines occasions rituelles, un emblème peint sur masque. Le premier totémisme réfère à la cosmogonie qui fait commencer l'univers à partir d'un couple parental ancestral, ainsi qu'à deux espèces de palmiers qui sont donnés respectivement comme féminin-maternel et masculin-paternel. Le second renvoie à des espèces naturelles diverses, comestibles ou non, ne faisant pas système entre elles, contrairement aux totems de moitiés. Les deux systèmes se combinent au cours de la cérémonie Yangis dont la fonction symbolique est : d'une part, le renouvellement des espèces totémiques duelles et la recréation de la Société ; d'autre part, la perpétuation sociale des patriclans.Among the Yafar, a small society in the lowland forests of Papua New Guinea, there are two forms of totemism: a totemism based on the organization of the society in two ritual moieties and a clan-based totemism whereby, on certain ceremonial occasions, each patriclan of long standing exhibits a painted emblem on a mask. The first form of totemism refers to the cosmogony, according to which the universe started out from an ancestral couple and from two species of palm trees presented as being respectively feminine-maternal and masculine-paternal. The second form refers to diverse natural species, whether edible or not, that do not form a system together, unlike the totems related to the moieties. The two systems combine during the Yangis ceremony which has the symbolic function, on the one hand, of renewing the dual totemic species and recreating Society; and, on the other, of perpetuating the patriclans

Widely Digitally Stable Numbers and Irreducibility Criteria For Polynomials With Prime Values

This dissertation considers three different topics. In the first part of the dissertation, we show for an integer b \u3e 2 that if a polynomial f(x) with non-negative integer coefficients is such that f(b) is prime, then there are explicit bounds M1(b), M2(b), and M3(b) such that if the coefficients of f(x) are each ≤ M1(b), then f(x) is irreducible; if the coefficients of f(x) are each ≤ M2(b) and f(x) is reducible, then it is divisible by the shifted cyclotomic polynomial Φ3(x−b) for 3 ≤ b ≤ 5, and divisible by Φ4(x − b) for b \u3e 5; and if the coefficients of f(x) are each ≤ M3(b) and f(x) is reducible, then it is divisible by at least one of Φ3(x−b) and Φ4(x−b). Furthermore, if b \u3e 69 and the coefficients of f(x) are each ≤ M4(b), then f(x) is either irreducible or divisible by at least one of Φ3(x − b), Φ4(x − b), and Φ6(x − b). In the second part of the dissertation, we show that there are only finitely many values of t such that the truncated binomial polynomial of degree 6, q6,t(x) = X 6 j=0 t j ! x j . has Galois group P GL2(5), a transitive subgroup of S6 isomorphic to S5. When the Galois group of the truncated binomial of degree 6 is not P GL2(5), it has been shown to be S6. Additionally, we show that the truncated binomial of degree 6 is irreducible for all values of t. In the third part of the dissertation, we show that there are infinitely many composite numbers, N, with the property that inserting a digit between any two digits in base 10 of N, including between any two of the infinitely many leading zeros and to the right of N, always results in a composite number. We show that the same result holds for bases b ∈ {2, 3, · · · , 8, 9, 11, 31}