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A direct proof is given for Akiyama and Tanigawa’s algorithm for computing Bernoulli numbers.. The proof uses a closed formula for Bernoulli numbers expressed in terms of
In fact, as we will see at the end of the article, our method of proof is exible enough and can be adapted to get a smaller error term (of square root type) for finer domains.. In
Theorem 1.2. Proof of Theorem 1.1. We will use the following auxiliary results in the proof.. Let SvwT denote the graph obtained from disjoint graphs S, T by adding an edge joining
In the case of the Potts model, the present theorem reproduces earlier results of the authors, with a simplified proof, in the case of the symmetric Ising model (where the
We consider a discrete time financial market with proportional transaction costs. These markets have already been widely studied. In particular, a proof of the fundamental theorem
The extension of sublinearity on average to pointwise sublinearity is the main novel part of the proof which, unfortunately, still makes non-trivial use of the “heat-kernel
We present a short probabilistic proof of a weak convergence result for the number of cuts needed to isolate the root of a random recursive tree.. The proof is based on a
The following example, adapted from [4, Chapter 7], shows that in general the uniform exponential stability is by no means a necessary condition for the existence of a
