相关论文: Martin Gardner's Mistake

A formula is given for the propagation of errors during matrix inversion. An explicit calculation for a 2 by 2 matrix using both the formula and a Monte Carlo calculation are compared. A prescription is given to determine when a matrix with…

The term Gibbons conjecture is widely used in connection with symmetry results for the Allen-Cahn equation. However, its origin is less transparent than its frequent citation suggests. In this note, we revisit its emergence, tracing it to a…

One of the most discussed peculiarities of Einstein's theory of relativity is the twin paradox, the fact that the time between two events in space-time appears to depend on the path between these events. We show that this time discrepancy…

Jing (1995) and Liu et al. (2008) studied the two-sample empirical likelihood and showed it is Bartlett correctable for the univariate and multivariate cases, respectively. We expand its domain to the full parameter space and obtain a…

Bertrand's paradox is a famous problem of probability theory, pointing to a possible inconsistency in Laplace's principle of insufficient reason. In this article we show that Bertrand's paradox contains two different problems: an "easy"…

In this article, we give a particular recreational application of the sequence A000533 and A261544 in "The On-line Encyclopedia of Integer Sequences" (OEIS). The recreational application provides a direct extension to "The Repetitious…

Baird counterexample was proposed by Leemon Baird in 1995, first used to show that the Temporal Difference (TD(0)) algorithm diverges on this example. Since then, it is often used to test and compare off-policy learning algorithms. Gradient…

We prove the two-scale transformation method which allows rigorous homogenisation of problems defined on locally periodic domains by transformation on periodic domains. The idea to consider periodic substitute problems was originally…

We present here a probabilistic approach to the generation of new polynomials in two discrete variables. This extends our earlier work on the 'classical' orthogonal polynomials in a previously unexplored direction, resulting in the…

The purpose of this paper is to investigate the connection between context-free grammars and normal ordering problem, and then to explore various extensions of the Stirling grammar. We present grammatical characterizations of several well…

In 1907, Henry Ernest Dudeney posed a puzzle: ``cut any equilateral triangle \dots\ into as few pieces as possible that will fit together and form a perfect square'' (without overlap, via translation and rotation). Four weeks later, Dudeney…

An argument, perhaps originating with Feyerabend a half century ago, and repeated many times since, purporting to establish that an "ignorance interpretation" of a bipartite pure entangled state leads to logical inconsistency, is incorrect:…

It has long been accepted that the foundations of Grothendieck duality are complicated. This has changed recently. By "Grothendieck duality" we mean what, in the old literature, used to go by the name "coherent duality". This isn't to be…

In this article, we give a polynomial algorithm to decide whether a given permutation $\sigma$ is sortable with two stacks in series. This is indeed a longstanding open problem which was first introduced by Knuth. He introduced the stack…

In Grammatical Error Correction, systems are evaluated by the number of errors they correct. However, no one has assessed whether all error types are equally important. We provide and apply a method to quantify the importance of different…

This is an English translation of the paper in which N. I. Akhiezer discovered his famous orthogonal polynomials on two intervals in a connection with a generalization of the Korkin-Zolotarev (Korkine-Zolotaref) problem (see the small…

The recently introduced backward Monte-Carlo method [Johan Carlsson, arXiv:math.NA/0010118] is validated, benchmarked, and compared to the conventional, forward Monte-Carlo method by analyzing the error in the Monte-Carlo solutions to a…

Machine learning has achieved tremendous success in a variety of domains in recent years. However, a lot of these success stories have been in places where the training and the testing distributions are extremely similar to each other. In…

There is an error in the proof (but not the truth) of Theorem 3.2 in the author's 1985 paper "The Double-Wedge Algebra for Quantum Fields on Schwarzschild and Minkowski Spacetimes" in "Communications in Mathematical Physics". The author…

A generalized Chinese remainder theorem (CRT) for multiple integers from residue sets has been studied recently, where the correspondence between the remainders and the integers in each residue set modulo several moduli is not known. A…