Related papers: Wallis's Formula and the Probability Integral
A method of random integral representation, that is, a method of representing a given probability measure as the probability distribution of some random integral, was quite successful in the past few decades. In this note we show that a…
We illustrate the concept of mathematical proof.
The note contains a short elementary proof of Cayley's formula for labeled trees.
The motivation of this paper is to construct the theory of vector calculus of multivariate arithmetical functions. We prove analogues of integral theorems and Poincare's lemma.
Based on the probability distribution observed in complex systems and an assumption that the probability distributions of complex systems satisfy a new generalized multiplication, it is proved that the statistical theory of complex systems…
Bell's theorem is a fundamental result in quantum mechanics: it discriminates between quantum mechanics and all theories where probabilities in measurement results arise from the ignorance of pre-existing local properties. We give an…
We show that the asymptotic formula for $\pi$, the Wallis formula, that was related with quantum mechanics and the hydrogen atom in \cite{HF}, can also be related to the harmonic oscillator using a quantum duality between these two systems.…
The idea of fully accepting statements when the evidence has rendered them probable enough faces a number of difficulties. We leave the interpretation of probability largely open, but attempt to suggest a contextual approach to full belief.…
One of the greatest difficulties encountered by all in their first proof intensive class is subtly assuming an unproven fact in a proof. The purpose of this note is to describe a specific instance where this can occur, namely in results…
In the paper, in light of the generating function of the complete Bell polynomials and other techniques, the author presents concise and elegant proofs of three formulas for the complete Bell polynomials.
Working from definitions and an elementarily obtained integral formula for the Euler-Mascheroni constant, we give an alternative proof of the classical Puiseux representation of the exponential integral.
An observation on Hall-Littlewood polynomials.
Bell tests are of profound statistical nature. Besides physical considerations, the proper understanding of their implications should involve detailed statistical analyses. In this regard, recent works have shown that their consequences and…
We note that an argument by Rogers (1958) gives a proof of Vaaler's theorem (1979) about sections of the cube and allows certain generalizations of the theorem.
Recently, it has been emphasized that the possibility theory framework allows us to distinguish between i) what is possible because it is not ruled out by the available knowledge, and ii) what is possible for sure. This distinction may be…
In this note, we give an explicit expression for the quantile of a mixture of two random variables. We carefully examine all possible cases of discrete and continuous variables with possibly unbounded support. The result is useful for…
In this article we use the Desargues' theorem and its reciprocal to solve two problems.
We give a proof of Cox's Theorem on the product rule and sum rule for conditional plausibility without assuming continuity or differentiablity of plausibility. Instead, we extend the notion of plausibility to apply to unknowns giving them…
In this note, we give a simple proof of the pointwise BMO estimate for Poisson's equation. Then the Calder\'{o}n-Zygmund estimate follows by the interpolation and duality.
We analyze selected iterated conditionals in the framework of conditional random quantities. We point out that it is instructive to examine Lewis's triviality result, which shows the conditions a conditional must satisfy for its probability…