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We construct a cubic cut-and-join operator description for the partition function of the Chekhov–Eynard–Orantin topological recursion for a local spectral curve with simple ramification points. In particular, this class contains partition functions of all semi-simple cohomological field theories. The cut-and-join description leads to an algebraic version of topological recursion. For the same partition functions we also derive N families of the Virasoro constraints and prove that these constraints, supplemented by a deformed dimension constraint, imply the cut-and-join description. [ABSTRACT FROM AUTHOR]

The Lowest Unique Positive Integer game, a.k.a. Limbo, is among the simplest games that can be played by any number of players and has a nontrivial strategic component. Players independently pick positive integers, and the winner is the player that picks the smallest number nobody else picks. The Nash equilibrium for this game is a mixed strategy, (p (1) , p (2) , ...) , where p(k) is the probability you pick k. A recursion for the Nash equilibrium has been previously worked out in the case where the number of players is Poisson distributed, an assumption that can be justified when there is a large pool of potential players. Here, we summarize previous results and prove that as the (expected) number of players, n, goes to infinity, a properly scaled version of the Nash equilibrium random variable converges in distribution to a Unif(0, 1) random variable. The result implies that for large n, players should choose a number uniformly between 1 and ϕ n ∼ O (n / ln (n)) . Convergence to the uniform is rather slow, so we also investigate a continuous analog of the Nash equilibrium using a differential equation derived from the recursion. The resulting approximation is unexpectedly accurate and is interesting in its own right. Studying the differential equation yields some useful analytical results, including a precise expression for ϕ n , and efficient ways to sample from the continuous approximation. [ABSTRACT FROM AUTHOR]

Weak enough relevant logics are often closed under depth substitutions. To determine the breadth of logics with this feature, we show there is a largest sublogic of R closed under depth substitutions and that this logic can be recursively axiomatized. [ABSTRACT FROM AUTHOR]

Motivated by queueing applications, we study various reflected autoregressive processes with dependencies. Among others, we study cases where the interarrival and service times are proportionally dependent on additive and/or subtracting delay, as well as cases where interarrival times depend on whether the service duration of the previous arrival exceeds or not a random threshold. Moreover, we study cases where the autoregressive parameter is constant as well as a discrete or a continuous random variable. More general dependence structures are also discussed. Our primary aim is to investigate a broad class of recursions of autoregressive type for which several independence assumptions are lifted and for which a detailed exact analysis is provided. We provide expressions for the Laplace transform of the waiting time distribution of a customer in the system in terms of an infinite sum of products of known Laplace transforms. An integer-valued reflected autoregressive process that can be used to model a novel retrial queueing system with impatient customers and a general dependence structure is also considered. For such a model, we provide expressions for the probability generating function of the stationary orbit queue length distribution in terms of an infinite sum of products of known generating functions. A first attempt towards a multidimensional setting is also considered. [ABSTRACT FROM AUTHOR]

"The Story of Proof: Logic and the History of Mathematics" by John Stillwell is a book that explores the development of mathematical proofs throughout history. The author examines the emergence of proofs in antiquity and the long-standing dominance of Euclid's foundations of geometry. Stillwell presents a historical narrative of mathematical achievements and subfields, highlighting the ways in which different areas of mathematics are interconnected. The book also discusses the influence of linguistic and conceptual developments on the evolution of mathematics, including the role of infinity and the impact of notation. While the text assumes some prior knowledge of mathematical proofs, it provides numerous examples and explanations to help readers understand the subject. The book concludes with a discussion of modern foundations of mathematics and the potential for computer-assisted proof and automated proof verification. [Extracted from the article]

It is well known that the Laplace cascade method is an effective tool for constructing solutions to linear equations of hyperbolic type, as well as nonlinear equations of the Liouville type. The connection between the Laplace method and soliton equations of hyperbolic type remains less studied. The article shows that the Laplace cascade also has important applications in the theory of hyperbolic equations of the soliton type. Laplace's method provides a simple way to construct such fundamental objects related to integrability theory as the recursion operator, the Lax pair and Dubrovin-type equations, allowing one to find algebro-geometric solutions. As an application of this approach, previously unknown recursion operators and Lax pairs are found for two nonlinear integrable equations of the sine-Gordon type. [ABSTRACT FROM AUTHOR]

We present an asymptotic method to compute efficiently second-order telegraph equations with high-frequency extrinsic oscillations. This method is based on asymptotic expansions in inverse powers of the oscillatory parameter. Each term of this asymptotic expansion is the sum of at most two coefficients. Each coefficient is derived either by recursion or by solving a non-oscillatory problem. This leads to method which exhibit improved performance with growing frequency of oscillation. [ABSTRACT FROM AUTHOR]