On The Prime Numbers In Intervals

Bertrand's postulate establishes that for all positive integers $n>1$ there exists a prime number between $n$ and $2n$. We consider a generalization of this theorem as: for integers $n\geq k\geq 2$ is there a prime number between $kn$ and $(k+1)n$? We use elementary methods of binomial coefficients and the Chebyshev functions to establish the cases for $2\leq k\leq 8$. We then move to an analytic number theory approach to show that there is a prime number in the interval $(kn, (k+1)n)$ for at least $n\geq k$ and $2\leq k\leq 519$. We then consider Legendre's conjecture on the existence of a prime number between $n^2$ and $(n+1)^2$ for all integers $n\geq 1$. To this end, we show that there is always a prime number between $n^2$ and $(n+1)^{2.000001}$ for all $n\geq 1$. Furthermore, we note that there exists a prime number in the interval $[n^2,(n+1)^{2+\varepsilon}]$ for any $\varepsilon>0$ and $n$ sufficiently large. We also consider the question of how many prime numbers there are between $n$ and $kn$ for positive integers $k$ and $n$ for each of our results and in the general case. Furthermore, we show that the number of prime numbers in the interval $(n,kn)$ is increasing and that there are at least $k-1$ prime numbers in $(n,kn)$ for $n\geq k\geq 2$. Finally, we compare our results to the prime number theorem and obtain explicit lower bounds for the number of prime numbers in each of our results.Comment: 71 pages, Master's Thesis (2015

Adaptations to infer fitness interdependence promote the evolution of cooperation

The evolution of cooperation is a major question in the biological and behavioral sciences. While most theoretical studies model cooperation in the context of an isolated interaction (e.g., a Prisoner’s Dilemma), humans live in heterogeneous social environments, characterized by large variations in fitness interdependence—the extent to which one’s fitness is affected by others. Theoretical and experimental work indicates that humans can infer, and respond to, variations in interdependence. In a heterogeneous ancestral environment, these psychological mechanisms to infer fitness interdependence could have provided a selective advantage, allowing individuals to maximize their fitness by deciding when and with whom to cooperate. Yet, to date, the link between cognitive inference, variation in fitness interdependence, and cooperation remains unclear. Here we introduce a theoretical framework to study the evolution of inference and cooperation in heterogeneous social environments, where individuals experience interactions with varying levels of corresponding interests. Using a combination of evolutionary game theory and agent-based modeling, we model the evolution of adaptive agents, who incur a cost to infer interdependence, in populations of fixed-behavior agents who always cooperate or defect. Our results indicate that natural selection could promote the evolution of psychological mechanisms to infer fitness interdependence, provided that there is enough variation in fitness interdependence to offset the cost of inference. Under certain conditions, the fixation of adaptive agents results in higher levels of cooperation. This depends crucially on the type of inference performed and the features of the interdependence landscape

Farm Production Analysis Training for Small Farmers

A partnership between Penn State Cooperative Extension and the Farm Service Agency has developed a successful production training program for more than 367 farmers. Farmers received training in producing planning and budgets, partial budgeting, and livestock and agronomic basics. Significant gains in knowledge occurred in multi-year planning, enterprise budgeting, and use of the Penn State Agronomy Guide. Designing workshop material for low-producing farmers proved difficult because most participants, while finding the topics helpful, also found the material too advanced. Future focus will encourage greater involvement with private-sector institutions and coordination between production and management Extension specialists

Relating trait self-control and forgiveness within prosocials and proselfs: Compensatory versus synergistic models

Ministry of Education, Singapore under its Academic Research Funding Tier

Social Value Orientation, Expectations, and Cooperation in Social Dilemmas:A Meta-analysis

Interdependent situations are pervasive in human life. In these situations, it is essential to form expectations about the others' behaviour to adapt one's own behaviour to increase mutual outcomes and avoid exploitation. Social value orientation, which describes the dispositional weights individuals attach to their own and to another person's outcome, predicts these expectations of cooperation in social dilemmas—an interdependent situation involving a conflict of interests. Yet, scientific evidence is inconclusive about the exact differences in expectations between prosocials, individualists, and competitors. The present meta-analytic results show that, relative to proselfs (individualists and competitors), prosocials expect more cooperation from others in social dilemmas, whereas individualists and competitors do not significantly differ in their expectations. The importance of these expectations in the decision process is further highlighted by the finding that they partially mediate the well-established relation between social value orientation and cooperative behaviour in social dilemmas. In fact, even proselfs are more likely to cooperate when they expect their partner to cooperate

Social Value Orientation, Expectations, and Cooperation in Social Dilemmas:A Meta-analysis

Interdependent situations are pervasive in human life. In these situations, it is essential to form expectations about the others’ behaviour to adapt one’s own behaviour to increase mutual outcomes and avoid exploitation. Social value orientation, which describes the dispositional weights individuals attach to their own and to another person’s outcome, predicts these expectations of cooperation in social dilemmas—an interdependent situation involving a conflict of interests. Yet, scientific evidence is inconclusive about the exact differences in expectations between prosocials, individualists, and competitors. The present meta-analytic results show that, relative to proselfs (individualists and competitors), prosocials expect more cooperation from others in social dilemmas, whereas individualists and competitors do not significantly differ in their expectations. The importance of these expectations in the decision process is further highlighted by the finding that they partially mediate the well-established relation between social value orientation and cooperative behaviour in social dilemmas. In fact, even proselfs are more likely to cooperate when they expect their partner to cooperate

Getting a Grip on the Grapevine: Extension and Factor Structure of the Motives to Gossip Questionnaire

Gossip is condemned but also ubiquitous and thought to be essential for groups. This triggers the question of which motives explain gossip behavior. Hitherto, negative influence, social enjoyment, group protection, and information gathering and validation are established as motives to gossip. However, venting emotions—discussed as a potentially important motive—has been overlooked empirically. Furthermore, a lack of consensus about a definition of gossip may have affected previous conclusions about gossip motives. This study (N = 460) expands the Motives to Gossip Questionnaire (MGQ; Beersma and Van Kleef, 2012) by including a subscale measuring emotion venting, the desire to share emotionally evocative experiences. To validate the five motives to gossip across definitions, we asked participants to report the most recent gossip event they experienced, randomly assigning them to one of three instructions containing different gossip definitions commonly used in the literature: (1) broad instructions (sharing information about third parties who have no knowledge of the exchanged information), (2) narrower instructions (adding that the shared information must be evaluative), and (3) instructions using the word gossip. After participants recalled and described a gossip event, they completed the 25-item measure of five motives to gossip: social enjoyment, information gathering and validation, negative influence, group protection, and emotion venting. Confirmatory factor analysis confirmed the five-factor structure. Multi-group confirmatory factor analysis supported full invariance across the three definition conditions. This indicates the Motives to Gossip Questionnaire successfully measures the five dimensions argued to motivate gossip and can be applied in research conceptualizing gossip both narrowly and broadly

Understanding cooperation through fitness interdependence

Some acts of human cooperation are not easily explained by traditional models of kinship or reciprocity. Fitness interdependence may provide a unifying conceptual framework, in which cooperation arises from the mutual dependence for survival or reproduction, as occurs among mates, risk-pooling partnerships and brothers-in-arms