arXiv subjects
Publications and source records attributed to Costas Poulios.
We introduce the concept of $ε$-uncontrollability for random linear systems, i.e. linear system in which the usual matrices have been replaced by random matrices. We also estimate the $ε$-uncontrollability in the case where the matrices come from the Gaussian orthogonal ensemble. Our proof utilizes tools from systems theory, probability theory and convex geometry.
The $3x+1$-problem (or Collatz problem) is a notorious conjecture in arithmetic. It can be viewed as iterating a map and, therefore, it is a dynamical system on the discrete space $\mathbb{N}$ of natural numbers. The emerging dynamical system is studied in the present work with methods from the theory of Koopman operators and $C^*$-algebras. This approach enables us to "lift" the $3x+1$-dynamical system from the state space (i.e the set $\mathbb{N}$) to spaces of functions defined on the state space, i.e. to sequence spaces. The advantage of this lifting is that the Collatz problem can be described via bounded linear operators, which consist an extensively studied area of Analysis. We study the properties of these operators and their relationship to the $3x+1$-problem. Furthermore, we use Fourier transform techniques to investigate the frequency content of the sequences of signs emerging from the trajectories of the Collatz map. This enables us to define an isometry on a Hilbert space. Finally, we utilize the $C^*$-algebra generated by this isometry in order to study how the sequences of signs correlate with each other.
Suppose that $(x_s)_{s\in S}$ is a normalized family in a Banach space indexed by the dyadic tree $S$. Using Stern's combinatorial theorem we extend important results from sequences in Banach spaces to tree-families. More precisely, assuming that for any infinite chain $β$ of $S$ the sequence $(x_s)_{s\inβ}$ is weakly null, we prove that there exists a subtree $T$ of $S$ such that for any infinite chain $β$ of $T$ the sequence $(x_s)_{s\inβ}$ is nearly (resp., convexly) unconditional. In the case where $(f_s)_{s\in S}$ is a family of continuous functions, under some additional assumptions, we prove the existence of a subtree $T$ of $S$ such that for any infinite chain $β$ of $T$, the sequence $(f_s)_{s\inβ}$ is unconditional. Finally, in the more general setting where for any chain $β$, $(x_s)_{s\inβ}$ is a Schauder basic sequence, we obtain a dichotomy result concerning the semi-boundedly completeness of the sequences $(x_s)_{s\inβ}$.
We consider a Banach space, which comes naturally from c0 and it appears in the literature, and we prove that this space has the fixed point property for non-expansive mappings.
We introduce and investigate a class of non-separable tree-like Banach spaces. As a consequence, we prove that we can not achieve a satisfactory extension of Rosenthal's $\ell_1$-theorem to spaces of the type $\ell_1(κ)$, for $κ$ an uncountable cardinal.