In this article, three types of joins are introduced for subspaces of a vector space. Decompositions of the Gra{ss}mannian into joins are discussed. This framework admits a generalization of large set recursion methods for block designs to subspace d
esigns. We construct a $2$-$(6,3,78)_5$ design by computer, which corresponds to a halving $operatorname{LS}_5[2](2,3,6)$. The application of the new recursion method to this halving and an already known $operatorname{LS}_3[2](2,3,6)$ yields two infinite two-parameter series of halvings $operatorname{LS}_3[2](2,k,v)$ and $operatorname{LS}_5[2](2,k,v)$ with integers $vgeq 6$, $vequiv 2mod 4$ and $3leq kleq v-3$, $kequiv 3mod 4$. Thus in particular, two new infinite series of nontrivial subspace designs with $t = 2$ are constructed. Furthermore as a corollary, we get the existence of infinitely many nontrivial large sets of subspace designs with $t = 2$.
We present robust radio frequency (rf) pulses that tolerate a factor of six inhomogeneity in the B1 field, significantly enhancing the potential of toroid cavity resonators for NMR spectroscopic applications. Both point-to-point (PP) and unitary rota
tion (UR) pulses were optimized for excitation, inversion, and refocusing using the gradient ascent pulse engineering (GRAPE) algorithm based on optimal control theory. In addition, the optimized parameterization (OP) algorithm applied to the adiabatic BIR-4 UR pulse scheme enabled ultra-short (50 microsec) pulses with acceptable performance compared to standard implementations. OP also discovered a new class of non-adiabatic pulse shapes with improved performance within the BIR-4 framework. However, none of the OP-BIR4 pulses are competitive with the more generally optimized UR pulses. The advantages of the new pulses are demonstrated in simulations and experiments. In particular, the DQF COSY result presented here represents the first implementation of 2D NMR spectroscopy using a toroid probe.
