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We study certain six dimensional theories arising on (p, q) brane webs living on R × S. These brane webs are dual to toric elliptically fibered Calabi-Yau threefolds. The compactification of the space on which the brane web lives leads to a deformation of the partition functions equivalent to the elliptic deformation of the Ding-Iohara algebra. We compute the elliptic version Dotsenko-Fateev integrals and show that they reproduce the instanton counting of the six dimensional theory.

Suppose that <i></i>₀ is an unknotted simple closed curve contained in the 3-sphere which happens to be invariant under a subgroup <i>G</i> of the Mbius group of <i>S³</i> = the group (generated by inversions in 2-spheres). It is shown that there is an equivariant isotopy <i></i>_<i>t</i>, 0 <i>t</i> 1, from <i></i>₀ to a round circle <i></i>₁.

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In this paper, we prove that into isometries and disjointness preserving linear maps from<i>C</i>₀(<i>X</i>) into<i>C</i>₀(<i>Y</i>) are essentially weighted composition operators<i>Tf</i>=<i>h</i><i>f</i> for some continuous map and some continuous scalar-valued function<i>h</i>.