Limiting tutorial-group size

Group Theory: A Tutorial

Symmetry means sameness, or indistinguishability, under some transformation. For example, a human face is indistinguishable from its reflected version, and therefore we say that it is reflectionally symmetric. A circle is indistinguishable from any of its rotated versions (about its center), and therefore we say that it is rotationally symmetric. When one puts together all the transformations t...

Social Group Size

Medical nanobiosensors: A tutorial review

A sensitive monitoring of biological analytes, such as biomolecules (protein, lipid, DNA and RNA), and biological cells (blood cell, virus and bacteria), is essential to assess and avoid risks for human health. Nanobiosensors, analytical devices that combine a biologically sensitive element with a nanostructured transducer, are being widely used for molecular detection of biomarkers associated ...

Size-dependent optical limiting behavior of multi-walled carbon nanotubes

The size-fractionation of multi-walled carbon nanotube (MWNT) in the dimethylformamide (DMF) solution of poly(vinylidene fluoride) (PVDF) has been achieved by controlled sonication–centrifugation cycles. The Z-scan experiments and fluence-dependent transmission measurements with nanosecond laser pulses of 532 nm wavelength show that the optical limiting behavior of MWNTs in PVDF/DMF solution is...

Limiting size index distributions for Zipf-type word frequencies

We consider a random linguistic model where words are placed randomly and sequentially in a given text or corpus. The word frequency follows the Zipf-type distribution; that is, the probability with which the ith most popular word occurs is asymptotically proportional to 1/i, α > 0. In this model, we derive the limiting distributions of size indices, where the size index of degree k at time t r...