Can ideals without ccc be interesting?

AbstractWe study those ideals I of sets in a perfect Polish space X which admit a Borel measurable f: X→X with f-1[\s{x\s}]∉I for each xϵX. A stronger version of that property (when, additionally, X is a group and I an invariant ideal) states that there exist a Borel set B∉I and a perfect P⊆X, such that \s{B + x:x ϵP\s} forms a disjoint family

Ideal convergent subseries in Banach spaces

Assume that $\mathcal{I}$ is an ideal on $\mathbb{N}$, and $\sum_n x_n$ is a divergent series in a Banach space $X$. We study the Baire category, and the measure of the set $A(\mathcal{I}):=\left\{t \in \{0,1\}^{\mathbb{N}} \colon \sum_n t(n)x_n \textrm{ is } \mathcal{I}\textrm{-convergent}\right\}$. In the category case, we assume that $\mathcal{I}$ has the Baire property and $\sum_n x_n$ is not unconditionally convergent, and we deduce that $A(\mathcal{I})$ is meager. We also study the smallness of $A(\mathcal{I})$ in the measure case when the Haar probability measure $\lambda$ on $\{0,1\}^{\mathbb{N}}$ is considered. If $\mathcal{I}$ is analytic or coanalytic, and $\sum_n x_n$ is $\mathcal{I}$-divergent, then $\lambda(A(\mathcal{I}))=0$ which extends the theorem of Dindo\v{s}, \v{S}al\'at and Toma. Generalizing one of their examples, we show that, for every ideal $\mathcal{I}$ on $\mathbb{N}$, with the property of long intervals, there is a divergent series of reals such that $\lambda(A(Fin))=0$ and $\lambda(A(\mathcal{I}))=1$

Implementation of the New Control Methods in Simplification of a Multidimensional Control and Optimization of a Control System Parameters.

The main purpose of this text is to present application of the Largest Lyapunov Exponent (LLE) as a criterion for optimization of the new type of simple controller parameters. Investigated controller is the part of numerically simulated control system. The calculation of LLE was done with a new method [2]. Introduction contains reference to previous publications on inverted pendulum control and Lyapunov stability. Application of the new simple formula for LLE estimation in control systems is discussed. In the next part simulated dynamical system is described and new type of simple controller allowing to control multidimensional system is introduced. In the last part results of the simulation are shown along with conclusions to whole dynamics analysis. Comparison of the proposed regulator with the linearquadratic regulator (LQR) was verified and its better effectiveness with respect to LQR was proved