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Publications (28)
Statistical Linkage Learning (SLL) is a part of many state-of-the-art optimizers. The purpose of SLL is to discover variable interdependencies. It has been shown that the effectiveness of SLL-using optimizers is highly dependent on the quality of SLL-based problem decomposition. Thus, understanding what kind of problems are hard or easy to decompos...
Benchmarks are essential tools for the optimizer's development. Using them, we can check for what kind of problems a given optimizer is effective or not. Since the objective of the Evolutionary Computation field is to support the tools to solve hard, real-world problems, the benchmarks that resemble their features seem particularly valuable. Theref...
The knowledge about variable interactions is frequently employed in state-of-the-art research concerning Genetic Algorithms (GA). Whether these interactions are known a priori (gray-box optimization) or are discovered by the optimizer (black-box optimization), they are used for many purposes, including proposing more effective mixing operators. Fre...
A special class of doubly stochastic (Markov) operators is constructed. In a sense these operators come from measure preserving transformations and inherit some of their properties, namely ergodicity and positivity of entropy, yet they may have no pointwise factors.
A COMMENT ON ERGODIC THEOREM FOR AMENABLE GROUPS - BARTOSZ FREJ, DAWID HUCZEK
We give a sufficient condition for a symbolic topological dynamical system with action of a countable amenable group to be an extension of the full shift, a problem analogous to those studied by Ashley, Marcus, Johnson and others for actions of $\mathbb{Z}$ and $\mathbb{Z}^d$ .
A special class of doubly stochastic (Markov) operators is constructed. These operators come from measure preserving transformations and inherit some of their properties, namely ergodicity and positivity of entropy, yet they may have no pointwise factors.
We prove a version of ergodic theorem for an action of an amenable group, where a F{\o} lner sequence needs not to be tempered. Instead, it is assumed that a function satisfies certain mixing condition.
We study doubly stochastic operators with zero entropy. We generalize three famous theorems: the Rokhlin's theorem on genericity of zero entropy, the Kushnirenko's theorem on equivalence of discrete spectrum and nullity and the Halmos-von Neumann's theorem on representation of maps with discrete spectrum as group rotations.
We study doubly stochastic operators with zero entropy. We generalize three famous theorems: the Rokhlin's theorem on genericity of zero entropy, the Kushnirenko's theorem on equivalence of discrete spectrum and nullity and the Halmos-von Neumann's theorem on representation of maps with discrete spectrum as group rotations.
We extend the result of Downarowicz (Israel J Math 165:189–210, 2008) to the case of amenable group actions, by showing that every face in the simplex of invariant measures on a zero-dimensional dynamical system with free action of an amenable group G can be modeled as the entire simplex of invariant measures on some other zero-dimensional dynamica...
We prove that every face in the simplex of invariant measures on a zero-dimensional dynamical system with free action of an amenable group $G$ can be modeled as the entire simplex of invariant measures on some other zero-dimensional dynamical system with free action of $G$.
We review subbadditivity properties of Shannon entropy, in particular, from the Shearer's inequality we derive the "infimum rule" for actions of amenable groups. We briefly discuss applicability of the "infimum formula" to actions of other groups. Then we pass to topological entropy of a cover. We prove Shearer's inequality for disjoint covers and...
We review subbadditivity properties of Shannon entropy, in particular, from
the Shearer's inequality we derive the "infimum rule" for actions of amenable
groups. We briefly discuss applicability of the "infimum formula" to actions of
other groups. Then we pass to topological entropy of a cover. We prove
Shearer's inequality for disjoint covers and...
Giordano et al. (2010) showed that every minimal free Zd-action of a Cantor space X is orbit equivalent to some Z-action. Trying to avoid the K-theory used there and modifying Forrest’s (2000) construction of a Bratteli diagram, we show how to define a (one-dimensional) continuous and injective map F on X \n {one point} such that for a residual sub...
We prove that on a metrizable, compact, zero-dimensional space every free
action of an amenable group is measurably isomorphic to a minimal $G$-action
with the same, i.e. affinely homeomorphic, simplex of measures.
In this paper we introduce and study an information function for doubly stochastic operators. This generalization of the notion, which is well known for classical dynamical systems, follows the earlier results on the entropy of operators by Downarowicz, Frej and Frej. As a main result we prove two versions of the Shannon–McMillan theorem for doubly...
A new formula for entropy of doubly stochastic operators is presented. It is also checked that this formula fulfills the axioms of the axiomatic definition of operator entropy, introduced in an earlier paper of Downarowicz and Frej. As an application of the formula the 'product rule' is obtained, i.e. it is shown that the entropy of a product is th...
We prove that on a metrizable, compact, zero-dimensional space every ℤd-action with no periodic points is measurably isomorphie to a minimal ℤd-action with the same, i.e. affinely homeomorphic, simplex of measures.
The paper deals with the notion of entropy for doubly stochastic operators. It is shown that the entropy defined by Maličky and Riečan in [M-R] is equal to the operator entropy proposed in [D-F]. Moreover, some continuity properties of the [M-R] entropy are established.
We study entropy of actions on function spaces with the focus on doubly stochastic operators on probability spaces and Markov operators on compact spaces. Using an axiomatic approach to entropy we prove that there is basically only one reasonable measure-theoretic entropy notion on doubly stochastic operators. By "reasonable" we mean extending the...
On a compact metric space X one defines a transition system to be a lower semicontinuous map X→2 X . It is known that every Markov operator on C(X) induces a transition system on X and that commuting of Markov operators implies commuting of the induced transition systems. We show that even in finite spaces a pair of commuting transition systems may...
