social anarchism sometimes referred to as socialist anarchism is generally considered tobe the branch of anarchism which sees individual freedom as being dependent upon mutual aid. socialanarchist consideration generally emphasizes community and social equality. society in the modernperiod is in a critical situation of not only economy and social system, but also in the remarkableaspects of the...

Social anarchism sometimes referred to as socialist anarchism is generally considered tobe the branch of anarchism which sees individual freedom as being dependent upon mutual aid. Socialanarchist consideration generally emphasizes community and social equality. Society in the modernperiod is in a critical situation of not only economy and social system, but also in the remarkableaspects of the...

Meshalkin’s theorem states that a class of ordered p-partitions of an n-set has at most max ( n a1,...,ap ) members if for each k the k parts form an antichain. We give a new proof of this and the corresponding LYM inequality due to Hochberg and Hirsch, which is simpler and more general than previous proofs. It extends to a common generalization of Meshalkin’s theorem and Erdős’s theorem about ...

We saw the following in the previous lecture.

Write R(x, y) = Γ(x + y) Γ(x). Inequalities for this ratio have interesting applications, and have been considered by a number of writers over a long period. In a Monthly article [7], Wendel showed that x(x + y) y−1 ≤ R(x, y) ≤ x y for 0 ≤ y ≤ 1. (1) Wendel's method was an ingenious application of Hölder's inequality to the integral definition of the gamma function. Note that both inequalities ...

Before we venture into Chernoff bound, let us recall Chebyshev's inequality which gives a simple bound on the probability that a random variable deviates from its expected value by a certain amount.

The second last inequality follows from the observation that the event E i (t) was defined as μ̂i(t) > xi, At time τk+1 for k ≥ 1, μ̂i(τk+1) = Si(τk+1) k+1 ≤ Si(τk+1) k , where latter is simply the average of the outcomes observed from k i.i.d. plays of arm i, each of which is a Bernoulli trial with mean μi. Using Chernoff-Hoeffding bounds (Fact 1), we obtain that Pr(μ̂i(τk + 1) > xi) ≤ Pr(ik k > ...

Communities can intuitively be defined as subsets of nodes of a graph with a dense structure in the corresponding subgraph. However, for mining such communities usually only structural aspects are taken into account. Typically, no concise nor easily interpretable community description is provided. For tackling this issue, this paper focuses on description-oriented community detection using subg...