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

Psychological adaptations for fitness interdependence underlie cooperation across human ecologies

Humans evolved to solve adaptive problems with kin and nonkin across fitness-relevant domains, including childcare and resource sharing, among others. Therefore, there is a great diversity in the types of interdependences humans experience across activities, relationships, and ecologies. To identify human psychological adaptations for cooperation, we argue that researchers must accurately characterize human fitness interdependence (FI). We propose a theoretical framework for assessing variation in FI that applies to the social interactions humans would have experienced across situations, relationships, and ecologies in the ancestral past and continue to experience today. According to this model, FI is characterized along four dimensions: (a) corresponding versus conflicting interests (b) mutual dependence versus independence, (c) asymmetrical versus symmetrical dependence (i.e., power), and (d) coordination. Because humans evolved to be highly mutually dependent on others to solve myriad adaptive problems, even compared to our closest living relatives, there is immense variability in the types of interdependences humans experience in daily life. Here, we describe the kinds of variation in interdependence humans experience, paying particular attention to social life in small-scale societies. In demonstrating the diversity of conflicts and coordination problems humans manage, we contend that humans evolved psychological adaptations to infer from signals, cues, and properties of the environment the four dimensions of FI under degrees of uncertainty to reduce the costs of cooperation. We conclude by discussing the theoretical implications of FI theory and emphasize that when individuals understand that others depend on them, it gives way to a new means of leverage to influence how others behave toward them.Security and Global Affair

National parochialism is ubiquitous across 42 nations around the world

Cooperation within and across borders is of paramount importance for the provision of public goods. Parochialism – the tendency to cooperate more with ingroup than outgroup members – limits contributions to global public goods. National parochialism (i.e., greater cooperation among members of the same nation) could vary across nations and has been hypothesized to be associated with rule of law, exposure to world religions, relational mobility and pathogen stress. We conduct an experiment in participants from 42 nations (N = 18,411), and observe cooperation in a prisoner’s dilemma with ingroup, outgroup, and unidentified partners. We observe that national parochialism is a ubiquitous phenomenon: it is present to a similar degree across the nations studied here, is independent of cultural distance, and occurs both when decisions are private or public. These findings inform existing theories of parochialism and suggest it may be an obstacle to the provision of global public goods

Prosocial and punishment behaviors in everyday life

Funding: This research was funded by an ERC Consolidator Grant (864519) awarded to Daniel Balliet.Theory and experiments suggest people have different strategies (1) to condition their prosocial behavior in ways that maximize individual benefits and (2) to punish others who have exploited their own and others’ prosocial behaviors. To date, most research testing existing theories has relied on experiments. However, documenting prosocial and punishment behaviors outside of the laboratory via experience sampling and diary methods can yield additional, rich insights. Recent work demonstrates these methods can describe social behaviors in daily life and be used to test theory about how behaviors change across situations and relationships. These methods have exposed discrepancies between what people experience in daily life and the problems researchers want to solve to understand the nature of human prosociality.Peer reviewe

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

Assessing Graphical Robot Aids for Interactive Co-working

The shift towards more collaborative working between humans and robots increases the need for improved interfaces. Alongside robust measures to ensure safety and task performance, humans need to gain the confidence in robot co-operators to enable true collaboration. This research investigates how graphical signage can support human–robot co-working, with the intention of increased productivity. Participants are required to co-work with a KUKA iiwa lightweight manipulator on a manufacturing task. The three conditions in the experiment differ in the signage presented to the participants – signage relevant to the task, irrelevant to the task, or no signage. A change between three conditions is expected in anxiety and negative attitudes towards robots; error rate; response time; and participants’ complacency, suggested by facial expressions. In addition to understanding how graphical languages can support human–robot co-working, this study provides a basis for further collaborative research to explore human–robot co-working in more detail

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