Slow convergence of sequences of linear operators II: Arbitrarily slow convergence

AbstractWe study the rate of convergence of a sequence of linear operators that converges pointwise to a linear operator. Our main interest is in characterizing the slowest type of pointwise convergence possible. This is a continuation of the paper Deutsch and Hundal (2010) [14]. The main result is a “lethargy” theorem (Theorem 3.3) which gives useful conditions that guarantee arbitrarily slow convergence. In the particular case when the sequence of linear operators is generated by the powers of a single linear operator, we obtain a “dichotomy” theorem, which states the surprising result that either there is linear (fast) convergence or arbitrarily slow convergence; no other type of convergence is possible. The dichotomy theorem is applied to generalize and sharpen: (1) the von Neumann–Halperin cyclic projections theorem, (2) the rate of convergence for intermittently (i.e., “almost” randomly) ordered projections, and (3) a theorem of Xu and Zikatanov

Micro- and Nanofluidics for Bionanoparticle Analysis

Bionanoparticles such as microorganisms and exosomes are recoganized as important targets for clinical applications, food safety, and environmental monitoring. Other nanoscale biological particles, includeing liposomes, micelles, and functionalized polymeric particles are widely used in nanomedicines. The recent deveopment of microfluidic and nanofluidic technologies has enabled the separation and anslysis of these species in a lab-on-a-chip platform, while there are still many challenges to address before these analytical tools can be adopted in practice. For example, the complex matrices within which these species reside in create a high background for their detection. Their small dimension and often low concentration demand creative strategies to amplify the sensing signal and enhance the detection speed. This Special Issue aims to recruit recent discoveries and developments of micro- and nanofluidic strategies for the processing and analysis of biological nanoparticles. The collection of papers will hopefully bring out more innovative ideas and fundamental insights to overcome the hurdles faced in the separation and detection of bionanoparticles