Using binary operations to construct a transitive set of block transformations

We study the set of transformations {ΣF : F∈ 𝓑∗(Ω)} implemented by a network Σ with a single binary operation F, where 𝓑∗(Ω) is the set of all binary operations on Ω that are invertible as function of the second variable. We state a criterion of bijectivity of all transformations from the family {ΣF : F∈ 𝓑∗(Ω)} in terms of the structure of the network Σ, identify necessary and sufficient conditions of transitivity of the set of transformations {ΣF : F∈ 𝓑∗(Ω)}, and propose an efficient way of verifying these conditions. We also describe an algorithm for construction of networks Σ with transitive sets of transformations {ΣF : F∈ 𝓑∗(Ω)}.

Grothendieck Universes

Summary The foundation of the Mizar Mathematical Library [2], is first-order Tarski-Grothendieck set theory. However, the foundation explicitly refers only to Tarski’s Axiom A, which states that for every set X there is a Tarski universe U such that X ∈ U. In this article, we prove, using the Mizar [3] formalism, that the Grothendieck name is justified. We show the relationship between Tarski and Grothendieck universe. First we prove in Theorem (17) that every Grothendieck universe satisfies Tarski’s Axiom A. Then in Theorem (18) we prove that every Grothendieck universe that contains a given set X, even the least (with respect to inclusion) denoted by GrothendieckUniverseX, has as a subset the least (with respect to inclusion) Tarski universe that contains X, denoted by the Tarski-ClassX. Since Tarski universes, as opposed to Grothendieck universes [5], might not be transitive (called epsilon-transitive in the Mizar Mathematical Library [1]) we focused our attention to demonstrate that Tarski-Class X ⊊ GrothendieckUniverse X for some X. Then we show in Theorem (19) that Tarski-ClassX where X is the singleton of any infinite set is a proper subset of GrothendieckUniverseX. Finally we show that Tarski-Class X = GrothendieckUniverse X holds under the assumption that X is a transitive set. The formalisation is an extension of the formalisation used in [4].

A novel route to cyclic dominance in voluntary social dilemmas

Cooperation is the backbone of modern human societies, making it a priority to understand how successful cooperation-sustaining mechanisms operate. Cyclic dominance, a non-transitive set-up comprising at least three strategies wherein the first strategy overrules the second, which overrules the third, which, in turn, overrules the first strategy, is known to maintain biodiversity, drive competition between bacterial strains, and preserve cooperation in social dilemmas. Here, we present a novel route to cyclic dominance in voluntary social dilemmas by adding to the traditional mix of cooperators, defectors and loners, a fourth player type, risk-averse hedgers, who enact tit-for-tat upon paying a hedging cost to avoid being exploited. When this cost is sufficiently small, cooperators, defectors and hedgers enter a loop of cyclic dominance that preserves cooperation even under the most adverse conditions. By contrast, when the hedging cost is large, hedgers disappear, consequently reverting to the traditional interplay of cooperators, defectors, and loners. In the interim region of hedging costs, complex evolutionary dynamics ensues, prompting transitions between states with two, three or four competing strategies. Our results thus reveal that voluntary participation is but one pathway to sustained cooperation via cyclic dominance.

A model of second-order arithmetic satisfying AC but not DC

We show that there is a [Formula: see text]-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a [Formula: see text]-assertion, confirming a conjecture of Stephen Simpson. We obtain as a corollary that the Reflection Principle, stating that every formula reflects to a transitive set, can fail in models of [Formula: see text]. This work is a rediscovery by the first two authors of a result obtained by the third author in [V. G. Kanovei, On descriptive forms of the countable axiom of choice, Investigations on nonclassical logics and set theory, Work Collect., Moscow, 3-136 (1979)].

Robust chain transitive vector fields

Let M be a closed n(≥2)-dimensional smooth Riemannian manifold and let X be a vector field on M. In this paper, we show that the robust chain transitive set is hyperbolic if and only if there are a C1-neighborhood [Formula: see text] of X and a compact neighborhood U of the chain transitive set such that for any [Formula: see text], the index of the continuation on ΛY(U) = ⋂t∈ℝYt(U) of every critical point does not change.

Non-Perfect Colorings of Symmetrical Tilings with Color Groups of Index 4

In this work we present a method that will allow for the construction and enumeration of non-perfect colorings of symmetrical tilings. If G is the symmetry group of an uncolored symmetrical tiling, then a coloring of the symmetrical tiling is non-perfect if its associated color group is a proper subgroup of G. The process will facilitate a systematic construction of non-perfect colorings of a wider class of symmetrical tilings where the stabilizer of a tile in the symmetry group G of the uncolored symmetrical tiling is non-trivial and the set of tiles may not form a transitive set under the action of G. This poster discusses results on how to identify and characterize non-perfect colorings arising from the method with associated color groups of index 4. The approach obtained here provides an avenue to model and characterize various chemical structures with atoms of different proportions, and their symmetries. This is relevant particularly for understanding new and emerging structures, such as structural analogues of carbon nanotubes, where a lot of its physical and electronic properties depend on their symmetry.

Fundamental domain of invariant sets and applications

Let $X$ be a compact metric space, $f:X\to X$ a homeomorphism and $\phi \in C(X,\mathbb {R})$. We construct a fundamental domain for the set of points with finite peaks with respect to the induced cocycle $\{\phi _n\}$. As applications, we give sufficient conditions for the transitive set of a non-conservative partially hyperbolic diffeomorphism to have positive Lebesgue measure, i.e., for an accessible partially hyperbolic diffeomorphism, if the set of points with finite peaks for the Jacobian cocycle is not of full volume, then the set of transitive points is of positive volume.

SINGULAR CYCLES AND Ck-ROBUST TRANSITIVE SET ON MANIFOLD WITH BOUNDARY

Let M be a 3-manifold with boundary ∂M. Let X be a C∞, vector field on M, tangent to ∂M, exhibiting a singular cycle associated to a hyperbolic equilibrium σ∈∂M with real eigenvalues λss < λs < 0 < λu satisfying λs - λss - 2λu > 0. We prove under generic conditions and k large enough the existence of a Ck robust transitive set of X, that is, any Ck vector field Ck close to X exhibits a transitive set containing the cycle. In particular, C∞ vector fields exhibiting Ck robust transitive sets, for k large enough, which are not singular-hyperbolic do exist on any compact 3-manifold with boundary.

Partially hyperbolic and transitive dynamics generated by heteroclinic cycles

We study \mathcal{C}^k-diffeomorphisms, k\ge 1, f: M\to M, exhibiting heterodimensional cycles (i.e. cycles containing periodic points of different stable indices). We prove that if f can not be \mathcal{C}^k-approximated by diffeomorphisms with homoclinic tangencies, then f is in the closure of an open set \mathcal{U}\subset \operatorname{Diff}^k(M) consisting of diffeomorphisms g with a non-hyperbolic transitive set \Lambda_g which is locally maximal and strongly partially hyperbolic (the partially hyperbolic splitting at \Lambda_g has three non-trivial directions). As a consequence, in the case of 3-manifolds, we give new examples of open sets of \mathcal{C}^1-diffeomorphisms for which residually infinitely many sinks or sources coexist (\mathcal{C}^1-Newhouse's phenomenon). We also prove that the occurrence of non-hyperbolic dynamics has persistent character in the unfolding of heterodimensional cycles.