A modification of the usual extended $N=2$ super symmetry algebra implementing the two dimensional permutation group is performed. It is shown that one can found a multiplet that forms an off-shell realization of this alternative extension of standard super symmetry.

An approach by J. Wu describes homotopy groups $\pi_{\mathrm{n}}(\mathrm{S^2})$ of the standard 2-sphere as isotopy classes of spherical $n+1$-strand Brunnian braids. The case $n=3$ is investigated for applications.

For certain classes of lattice models of nanosystems the eigenvalues of the row-to-row transfer matrix and the components of the corner transfer matrix truncations are algebraic functions of the fugacity and of Boltzmann weights. Such functions can be expanded in Puiseux series using techniques from algebraic geometry. Each successive term in the expansions in powers a Boltzmann weight is obtained exactly without modifying previous terms. We are able to obtain useful analytical expressions for any thermodynamic function for these systems from the series in circumstances in which no exact solutions can be found.

In the Dunkl setting, we establish three continuous uncertainty principles of concentration type, where the sets of concentration are not intervals. The first and the second uncertainty principles are $L^p$ versions and depend on the sets of concentration $T$ and $W$, and on the time function $f$. The time-limiting operators and the Dunkl integral operators play an important role to prove the main results presented in this paper. However, the third uncertainty principle is also $L^p$ version depends on the sets of concentration and he is independent on the band limited function $f$. These uncertainty principles generalize the results obtained for the Fourier transform and the Dunkl transform in the case $p=2$.