Rank 2 proximal Cantor systems are residually scrambled

We offer a criterion for showing that the automorphism group of an ultrahomogeneous structure is topologically 2-generated and even has a cyclically dense conjugacy class. We then show how finite topological rank of the automorphism group of an ω \omega -categorical structure can go down to reducts. Together, those results prove that a large number of ω \omega -categorical structures that appear in the literature have an automorphism group of finite topological rank. In fact, we are not aware of any ω \omega -categorical structure to which they do not apply (assuming the automorphism group has no compact quotients). We end with a few questions and conjectures.

We have added support for associative unification to Maude 2.7.1. Associative unification is infinitary, i.e., there are unification problems \(u =^? v\) such that there is an infinite minimal set of unifiers, whereas associative-commutative unification is finitary. A unique feature of the associative unification algorithm implemented in Maude is that it is guaranteed to terminate with a finite and complete set of associative unifiers for a fairly large class of unification problems occurring in practice. For any problems outside this class, the algorithm returns a finite set of unifiers together with a warning that such set may be incomplete. This paper describes this associative unification algorithm implemented in Maude and also how other symbolic reasoning Maude features such as (i) variant generation; (ii) variant unification; and (iii) narrowing based symbolic reachability analysis have been extended to deal with associativity.

Zero-dimensional almost 1–1 extensions of odometers from graph coverings

A foliation that admits a Cartan geometry as its transversal structure is called a Cartan foliation. We prove that on a manifold M with a complete Cartan foliation ℱ, there exists one more foliation (M, $$\mathcal{O}$$ ), which is generally singular and is called an aureole foliation; moreover, the foliations ℱ and $$\mathcal{O}$$ have common minimal sets. By using an aureole foliation, we prove that for complete Cartan foliations of the type ℊ/ℎ with a compactly embedded Lie subalgebra ℊ in ℎ, the closure of each leaf forms a minimal set such that the restriction of the foliation onto this set is a transversally locally homogeneous Riemannian foliation. We describe the structure of complete transversally similar foliations (M, ℱ). We prove that for such foliations, there exists a unique minimal set ℱ, and ℱ is contained in the closure of any leaf. If the foliation (M, ℱ) is proper, then ℳ is a unique closed leaf of this foliation.

We study ergodic finite and infinite measures defined on the path space $X_{B}$ of a Bratteli diagram $B$ which are invariant with respect to the tail equivalence relation on $X_{B}$. Our interest is focused on measures supported by vertex and edge subdiagrams of $B$. We give several criteria when a finite invariant measure defined on the path space of a subdiagram of $B$ extends to a finite invariant measure on $B$. Given a finite ergodic measure on a Bratteli diagram $B$ and a subdiagram $B^{\prime }$ of $B$, we find the necessary and sufficient conditions under which the measure of the path space $X_{B^{\prime }}$ of $B^{\prime }$ is positive. For a class of Bratteli diagrams of finite rank, we determine when they have maximal possible number of ergodic invariant measures. The case of diagrams of rank two is completely studied. We also include an example which explicitly illustrates the proven results.

As was known to H. Poincare, an orientation preserving circle homeomorphism without periodic points is either minimal or has no dense orbits, and every orbit accumulates on the unique minimal set. In the first case the minimal set is the circle, in the latter case a Cantor set. In this paper we study a two-dimensional analogue of this classical result: we classify the minimal sets of non-resonant torus homeomorphisms; that is, torus homeomorphisms isotopic to the identity for which the rotation set is a point with rationally independent irrational coordinates.

The result of this paper contributes to the classification of triangular maps of the square with zero topological entropy stated by A. N. Sharkovsky in the 1980s. The problem was if a triangular map of the square such that its any ω-limit set contains unique minimal set can be distributionally chaotic. So far such result was disproved only for the class of triangular maps non-decreasing on fibres [L. Paganoni, J. Smítal, Strange distributionally chaotic triangular maps, Chaos Solitons Fractals 26(2) (2005), pp. 581–589]. In this paper, we solve the problem in negative for all triangular maps of the square, correcting the original result from Balibrea and Smítal [Strong distributional chaos and minimal sets, Topology appl. 156 (2009), pp. 1673–1678].

Many problems in multiagent decision making can be addressed using tournament solutions, i.e., functions that associate with each complete and asymmetric relation on a set of alternatives a non-empty subset of the alternatives. For any given tournament solution S, there is another tournament solution S, which returns the union of all inclusion-minimal sets that satisfy S-retentiveness, a natural stability criterion with respect to S. Schwartz's tournament equilibrium set (TEQ) is then defined as TEQ = TEQ. Due to this unwieldy recursive definition, preciously little is known about TEQ. Contingent on a well-known conjecture about TEQ, we show that S inherits a number of important and desirable properties from S. We thus obtain an infinite hierarchy of attractive and efficiently computable tournament solutions that approximate TEQ, which itself is intractable. This hierarchy contains well-known tournament solutions such as the top cycle (TC) and the minimal covering set (MC). We further prove a weaker version of the conjecture mentioned above, which establishes TC as an attractive new tournament solution.

We introduce a type of zero-dimensional dynamical system (a pair consisting of a totally disconnected compact metrizable space along with a homeomorphism of that space), which we call “fiberwise essentially minimal”, that is a class that includes essentially minimal systems and systems in which every orbit is minimal. We prove that the associated crossed product [Formula: see text]-algebra of such a system is an A[Formula: see text]-algebra. Under the additional assumption that the system has no periodic points, we prove that the associated crossed product [Formula: see text]-algebra has real rank zero, which tells us that such [Formula: see text]-algebras are classifiable by [Formula: see text]-theory. The associated crossed product [Formula: see text]-algebras to these nontrivial examples are of particular interest because they are non-simple (unlike in the minimal case).

Let IP{A, c} denote the family of integer programs of the form Min cx: Ax = b, x ∈ Nn obtained by varying the right-hand side vector b but keeping A and c fixed. A test set for IPA, c is a set of vectors in Zn such that for each nonoptimal solution α to a program in this family, there is at least one element g in this set such that α − g has an improved cost value as compared to α. We describe a unique minimal test set for this family called the reduced Gröbner basis of IP{A, c}. An algorithm for its construction is presented which we call a geometric Buchberger algorithm for integer programming and we show how an integer program may be solved using this test set. The reduced Gröbner basis is then compared with some other known test sets from the literature. We also indicate an easy procedure to construct test sets with respect to all cost functions for a matrix A ∈ Z(n−2)×n of full row rank.

A vertex set D in a graph G is called a geodetic set if each vertex of G is lying on some shortest u−v path of G, where u, v∈D. The geodetic number of a graph G is the minimum cardinality among all geodetic sets. A subset S of a geodetic set D is called a forcing subset of D if D is the unique geodetic set containing S. The forcing geodetic number of D is the minimum cardinality of a forcing subset of D, and the lower and the upper forcing geodetic numbers of a graph G are the minimum and the maximum forcing geodetic numbers, respectively, among all minimum geodetic sets of G. In this article, we find out the geodetic numbers, the lower and the upper forcing geodetic numbers of complete n-partite graphs, n-dimensional meshes and tori.

In this paper the feasibility of identification of adiscontinuous source term in a parabolic equation isinvestigated. It is shown that a minimal set of data accordingto uniqueness of the inverse problem is given by flux measurements in time at two distinct points on the boundary.Besides the uniqueness result iterative regularization schemesare developed. The methods are based on the domain derivativeand a general existence theory and a representation of thedomain derivative for parabolic equations is derived.

Nonwandering points of monotone local dendrite maps revisited

Networks have been widely used to model the structure of various biological systems. Currently, a series of approaches have been developed to construct reliable biological networks. However, the ultimate understanding of a biological system is to steer its states to the desired ones by imposing signals. The control process is dominated by the intrinsic structure and the dynamic propagation. To understand the underlying mechanisms behind the life process, the control theory can be applied to biological networks with specific target requirements. In this article, we first introduce the structural controllability of complex networks and discuss its advantages and disadvantages. Then, we review the effective control to meet the specific requirements for complex biological networks. Moreover, we summarize the existing methods for finding the unique minimum set of driver nodes via the optimal control for complex networks. Finally, we discuss the relationships between biological networks and structural controllability, effective control and optimal control. Moreover, potential applications of general control principles are pointed out.

Intensive livestock farming has raised issues about environmental impacts and food security during the past 20 years. As a consequence, there is a strong social demand for sustainable livestock systems. Sustainable livestock systems should indeed be environmentally friendly, economically viable for farmers, and socially acceptable, notably for animal welfare. For that goal, many sustainability indicators and methods have been developed at the farm level. The main challenge is using a transparent selection process to avoid assessment subjectivity. Here, we review typologies of sustainability indicators. We set guidelines for selecting indicators in a data-driven context, by reviewing selection criteria and discussing methodological issues. A case study is presented. The selected set of indicators mainly includes (1) environmental indicators focusing on farmer practices; (2) quantitative economic indicators; and (3) quantitative social indicators with a low degree of aggregation. The selection of indicators should consider (1) contextualization to determine purpose, scales, and stakeholders involved in the assessment; (2) the comparison of indicators based on various criteria, mainly data availability; and (3) the selection of a minimal, consistent, and sufficient set of indicators. Finally, we discuss the following issues: topics for which no indicators are measurable from available data should explicitly be mentioned in the results. A combination of means-based indicators could be used to assess a theme, but redundancy must be avoided. The unit used to express indicators influences the results and has therefore to be taken into account during interpretation. To compare farms from indicators, the influence of the structure on indicator values has to be carefully studied.