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$K$-theory
We introduce two new bases of the ring of polynomials and study their relations to known bases. The first basis is the quasi-Lascoux basis, which is simultaneously both a 
$K$-theoretic deformation of the quasi-key basis and also a lift of the 
$K$-analogue of the quasi-Schur basis from quasi-symmetric polynomials to general polynomials. We give positive expansions of this quasi-Lascoux basis into the glide and Lascoux atom bases, as well as a positive expansion of the Lascoux basis into the quasi-Lascoux basis. As a special case, these expansions give the first proof that the 
$K$-analogues of quasi-Schur polynomials expand positively in multifundamental quasi-symmetric polynomials of T. Lam and P. Pylyavskyy.
The second new basis is the kaon basis, a 
$K$-theoretic deformation of the fundamental particle basis. We give positive expansions of the glide and Lascoux atom bases into this kaon basis.
Throughout, we explore how the relationships among these 
$K$-analogues mirror the relationships among their cohomological counterparts. We make several “alternating sum” conjectures that are suggestive of Euler characteristic calculations.
We consider the inertia-dominated rise of a meniscus around a vertical circular cylinder. Previous experiments and scaling analysis suggest that the height of the meniscus, 
$h_{m}$ , grows with the time following the initiation of rise, 
$t$ , like 
$h_{m}\propto t^{1/2}$ . This is in contrast to the rise on a vertical plate, which obeys the classic capillary–inertia scaling 
$h_{m}\propto t^{2/3}$ . We highlight a subtlety in the scaling analysis that yielded 
$h_{m}\propto t^{1/2}$ and investigate the consequences of this subtlety. We develop a potential flow model of the dynamic problem, which we solve using the finite element method. Our numerical results agree well with previous experiments but suggest that the correct early time behaviour is, in fact, 
$h_{m}\propto t^{2/3}$ . Furthermore, we show that at intermediate times the dynamic rise of the meniscus is governed by two parameters: the contact angle and the cylinder radius measured relative to the capillary length scale, 
$t^{2/3}$ . This result allows us to collapse previous experimental results with different cylinder radii (but similar static contact angles) onto a single master curve.
