The local limit theorem describes how the density of a sum of random variables follows the normal curve. However the local limit theorem is often seen as a curiosity of no particular importance when compared with the central limit theorem. Nevertheless the local limit theorem came first and is in fact associated with the foundation of probability theory by Blaise Pascal and Pierre de Fer...

This paper describes our collection of benchmarks for inductive theorem provers. The recent spur of interest in automated inductive theorem proving has increased the demands for evaluation and comparison between systems. We expect the benchmark suite to continually grow as more problems are submitted by the community. New challenge problems will promote further development of provers which will...

In this note, we estimate the upper bound of volume closed positively or nonnegatively curved Alexandrov space $X$ with strictly convex boundary. We also discuss equality case. particular, Boundary Conjecture holds when is achieved. Our theorem can be applied to Riemannian manifolds non-smooth boundary, which generalizes Heintze and Karcher's classical comparison theorem. main tool gradient flo...

In this paper, we discuss a new type of mean-field anticipated backward stochastic differential equation with time-delayed generator (MF-DABSDEs) which extends the results to case limits, and in considers not only present future times but also past time. By using fixed point theorem, shall demonstrate existence uniqueness solutions these equations. Finally, establish comparison theorem for solu...

Social Comparison is important as it can be either encouraging or discouraging for the person who makes it. Although studies concerning social comparison in the classroom context abound, few deal with the relationship between the role of social comparison orientation and tendencies in academic achievement in a competition based educational context. Adopting a non- experimental ex-post facto des...

Freek Wiedijk proposed the well-known theorem about the irrationality of √ 2 as a case study and used this theorem for a comparison of fifteen (interactive) theorem proving systems, which were asked to present their solution (see [48]). This represents an important shift of emphasis in the field of automated deduction away from the somehow artificial problems of the past as represented, for exa...

In this paper, we first establish a new fixed point theorem for a Meir-Keeler type condition. As an application, we derive a simultaneous generalization of Banach contraction principle, Kannan's fixed point theorem, Chatterjea's fixed point theorem and other fixed point theorems. Some new fixed point theorems are also obtained.