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Abstract We investigate a puzzle involving the winning probabilities in a duel of two players. The problem of calculating limiting probabilities leads to the summation of a divergent infinite series. The solution admits a generalization that applies to a wide class of duels.
Abstract The displacement of a parking function measures the total difference between where cars want to park and where they ultimately park. In this article, we prove that the set of parking functions of length n with displacement one is in bijection with the set of ideal states in the famous...
Abstract In general, the Cauchy product of an absolutely convergent series and a conditionally convergent one might converge absolutely. In our note, we provide an easy and quite general method for construction of such pairs of series, a method that is not related to the classic Pringsheim’s...
Abstract What is the expected number of roots of a polynomial whose coefficients are random? More generally, what is the expected number of zeros of a random one-variable function? The Kac-Rice formula is meant to answer such questions. This paper is an introduction to this less familiar formula...
Abstract Stickelberger proved that the discriminant of a number field is congruent to 0 or 1 modulo 4. We generalize this to an arbitrary (not necessarily commutative) ring of finite rank over using techniques from linear algebra. Our proof relies on elementary matrix identities.
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