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prof. dr hab. inż. Tomasz Szarek
Employment
- Professor at Instytut Matematyki Stosowanej
Keywords Help
- gaussian unitary ensemble
- homeomorphisms, distortion, rotation
- iterated function systems · invariant measures · stability · strong law of large numbers · functional equations
- marchenko-pastur distribution
- markov operators, semigroups of interval homeomorphisms,absolute continuity, singularity, minimal actions.
- markov operatorsequicontinuitymeasuredual bounded lipschitz normweak topologyschur property
- positive partial transpose
- random transformations, law of the iterated logarithm
- random walks, markov operators, central limit theorem, law of the iterated logarithm
- separable states
Publications
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total: 7
Catalog Publications
Year 2023
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Distortion in the group of circle homeomorphisms
PublicationLet G be the group PAff+(R/Z) of piecewise affine circle homeomorphisms or the group Diff∞(R/Z) of smooth circle diffeomorphisms. A constructive proof that all irrational rotations are distorted in G is given.
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Invariant Measures for Uncountable Random Interval Homeomorphisms
PublicationA necessary and sufficient condition for the iterated function system { f (·, ω) | ω ∈ } with probability P to have exactly one invariant measure μ∗ with μ∗((0, 1)) = 1 is given. The main novelty lies in the fact that we only require the transformations f (·, ω) to be increasing homeomorphims, without any smoothness condition, nei- ther we impose conditions on the cardinality of . In particular, positive Lyapunov exponents conditions...
Year 2022
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Limits Theorems for Random Walks on Homeo(S1)
PublicationThe central limit theorem and law of the iterated logarithm for Markov chains corresponding to random walks on the space Homeo(S1) of circle homeomorphisms for centered Lipschitz functions and every starting point are proved.
Year 2021
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Equivalence of equicontinuity concepts for Markov operators derived from a Schur-like property for spaces of measures
PublicationVarious equicontinuity properties for families of Markov operators have been – and still are – used in the study of existence and uniqueness of invariant probability for these operators, and of asymptotic stability. We prove a general result on equivalence of equicontinuity concepts. It allows comparing results in the literature and switching from one view on equicontinuity to another, which is technically convenient in proofs....
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The law of the Iterated Logarithm for random interval homeomorphisms
PublicationA proof of the law of the iterated logarithm for random homeomorphisms of the interval is given.
Year 2020
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Generic invariant measures for iterated systems of interval homeomorphisms
PublicationIt is well known that iterated function systems generated by orientation preserving homeomorphisms of the unit interval with positive Lyapunov exponents at its ends admit a unique invariant measure on (0, 1) provided their action is minimal. With the additional requirement of continuous differentiability of maps on a fixed neighbourhood of {0,1} { 0 , 1 } , we present a metric in the space of such systems which renders it complete....
Year 2017
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Convex set of quantum states with positive partial transpose analysed by hit and run algorithm
PublicationThe convex set of quantum states of a composite K×K system with positive partial transpose is analysed. A version of the hit and run algorithm is used to generate a sequence of random points covering this set uniformly and an estimation for the convergence speed of the algorithm is derived. For K >3 or K=3 this algorithm works faster than sampling over the entire set of states and verifying whether the partial transpose is positive....
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