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In this paper, we give an example of a statement concerning two-player zero-sum games which is undecidable, meaning that it can neither be proven or disproven by the standard axioms of mathematics. Earlier work has shown that there exist "paradoxical" two-player zero-sum games with unbounded payoffs, in which a standard calculation of the two players' expected utilities of a mixed strategy profile yield a positive sum. We show that whether or not a modified version of this paradoxical situation, with bounded payoffs and a weaker measurability requirement, exists is an unanswerable question. Our proof relies on a mixture of techniques from set theory and ergodic theory. [ABSTRACT FROM AUTHOR]

In this paper, we present a general principle for the Lebesgue measure theory of limsup sets defined by rectangles under the hypothesis of ubiquity for rectangles. Applications are given for simultaneous Diophantine approximation, Khintchine-Groshev theorem for rectangles and higher dimensional shrinking target problems. [ABSTRACT FROM AUTHOR]

Given a family F ⊂ 2 [ n ] , its diversity is the number of sets not containing an element with the highest degree. The concept of diversity has proven to be very useful in the context of k -uniform intersecting families. In this paper, we study (different notions of) diversity in the context of other extremal set theory problems. One of the main results of the paper is a sharp stability result for cross-intersecting families in terms of diversity and, slightly more generally, sharp stability for the Kruskal–Katona theorem. We also answer a question of Huang on the diversity of non-uniform intersecting families. [ABSTRACT FROM AUTHOR]

2-nondegenerate real hypersurfaces in complex manifolds play an important role in CR-geometry and the theory of Hermitian Symmetric Domains. In this paper, we obtain a complete convergent normal form for everywhere 2-nondegenerate real-analytic hypersurfaces in complex 3-space. We do so by entirely reproducing the Chern-Moser theory in the 2-nondegenerate setting. This seems to be the first such construction for hypersurfaces of infinite Catlin multitype. We in particular discover chains in an everywhere 2-nondegenerate hypersurface, the tangent lines to which at a point form the so-called canonical cone. Our approach is based on using a rational (nonpolynomial) model for everywhere 2-nondegenerate hypersurfaces, which is the local realization due to Fels-Kaup of the well known tube over the light cone. For the convergence of the normal form, we use an argument due to Zaitsev, based on building a canonical direction field in an appropriate bundle over a hypersurface. As an application, we obtain, in the spirit of Chern-Moser theory, a criterion for the local sphericity (i.e. local equivalence to the model) for a 2-nondegenerate hypersurface in terms of its normal form. As another application, we obtain an explicit description of the moduli space of everywhere 2-nondegenerate hypersurfaces. [ABSTRACT FROM AUTHOR]

To mitigate the conflict of economic development and ecological protection, China government initiates its national park plan in recent years. In this stage, addressing the benefit contradictories among stakeholders is of importance. Around this issue, this paper focuses on the one of most keen-edged topics in China, which is how to deal well with the coordination issue among local government, tourism development enterprise and local residents. To disentangle the complexly dynamic relationships among them, a set of mathematical models are built and derived, based on evolutionary game theory. The corresponding results indicate that the intensity of regulation by local governments, the degree of compensation to ecotourism development projects by tourism development enterprises, and the participation degree of local residents to the projects are closely related to ultimate game stabilization; some propositions are gained accordingly, which suggest the bounded conditions to achieve the optimal stabilization. To validate and broaden the results, the relevant numerical simulation and discussion are provided. • The stakeholders relationship about China's National Park Ecotourism is discussed. • The corresponding evolutionary game models are built and analyzed. • The optimal conditions of advancing the ecotourism of national parks are put forth. [ABSTRACT FROM AUTHOR]

Grinberg defined Nyldon words as those words which cannot be factorized into a sequence of lexicographically nondecreasing smaller Nyldon words. He was inspired by Lyndon words, defined the same way except with "nondecreasing" replaced by "nonincreasing." Charlier, Philibert, and Stipulanti proved that, like Lyndon words, any word has a unique nondecreasing factorization into Nyldon words. They also show that the Nyldon words form a right Lazard set, and equivalently, a right Hall set. In this paper, we provide a new proof of unique factorization into Nyldon words related to Hall set theory and resolve several questions of Charlier, Philibert, and Stipulanti. In particular, we prove that Nyldon words of a fixed length form a circular code, we prove a result on factorizing powers of words into Nyldon words, and we investigate the Lazard procedure for generating Nyldon words. We show that these results generalize to a new class of Hall sets, of which Nyldon words are an example, that we name "Nyldon-like sets", and show how to generate these sets easily. [ABSTRACT FROM AUTHOR]