In this article, we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying
The director field adopted by a confined liquid crystal is controlled by a balance between the externally imposed interactions and the liquid’s internal orientational elasticity. While the latter is usually considered to resist all deformations, liquid crystals actually have an intrinsic propensity to adopt saddle-splay arrangements, characterised by the elastic constant
- Award ID(s):
- 1654283
- NSF-PAR ID:
- 10153761
- Publisher / Repository:
- Nature Publishing Group
- Date Published:
- Journal Name:
- Nature Communications
- Volume:
- 10
- Issue:
- 1
- ISSN:
- 2041-1723
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract , for any even integer$$K^2 = 4p_g-8$$ . These surfaces also have unbounded irregularity$$p_g\ge 4$$ q . We carry out our study by investigating the deformations of the canonical morphism , where$$\varphi :X\rightarrow {\mathbb {P}}^N$$ is a quadruple Galois cover of a smooth surface of minimal degree. These canonical covers are classified in Gallego and Purnaprajna (Trans Am Math Soc 360(10):5489-5507, 2008) into four distinct families, one of which is the easy case of a product of curves. The main objective of this article is to study the deformations of the other three, non trivial, unbounded families. We show that any deformation of$$\varphi $$ factors through a double cover of a ruled surface and, hence, is never birational. More interestingly, we prove that, with two exceptions, a general deformation of$$\varphi $$ is two-to-one onto its image, whose normalization is a ruled surface of appropriate irregularity. We also show that, with the exception of one family, the deformations of$$\varphi $$ X are unobstructed even though does not vanish. Consequently,$$H^2(T_X)$$ X belongs to a unique irreducible component of the Gieseker moduli space. These irreducible components are uniruled. As a result of all this, we show the existence of infinitely many moduli spaces, satisfying the strict Beauville inequality , with an irreducible component that has a proper quadruple sublocus where the degree of the canonical morphism jumps up. These components are above the Castelnuovo line, but nonetheless parametrize surfaces with non birational canonical morphisms. The existence of jumping subloci is a contrast with the moduli of surfaces with$$p_g > 2q-4$$ , studied by Horikawa. Irreducible moduli components with a jumping sublocus also present a similarity and a difference to the moduli of curves of genus$$K^2 = 2p_g- 4$$ , for, like in the case of curves, the degree of the canonical morphism goes down outside a closed sublocus but, unlike in the case of curves, it is never birational. Finally, our study shows that there are infinitely many moduli spaces with an irreducible component whose general elements have non birational canonical morphism and another irreducible component whose general elements have birational canonical map.$$g\ge 3$$ -
Abstract Hemiwicking is the phenomena where a liquid wets a textured surface beyond its intrinsic wetting length due to capillary action and imbibition. In this work, we derive a simple analytical model for hemiwicking in micropillar arrays. The model is based on the combined effects of capillary action dictated by interfacial and intermolecular pressures gradients within the curved liquid meniscus and fluid drag from the pillars at ultra-low Reynolds numbers
. Fluid drag is conceptualized via a critical Reynolds number:$${\boldsymbol{(}}{{\bf{10}}}^{{\boldsymbol{-}}{\bf{7}}}{\boldsymbol{\lesssim }}{\bf{Re}}{\boldsymbol{\lesssim }}{{\bf{10}}}^{{\boldsymbol{-}}{\bf{3}}}{\boldsymbol{)}}$$ , where$${\bf{Re}}{\boldsymbol{=}}\frac{{{\bf{v}}}_{{\bf{0}}}{{\bf{x}}}_{{\bf{0}}}}{{\boldsymbol{\nu }}}$$ v 0corresponds to the maximum wetting speed on a flat, dry surface andx 0is the extension length of the liquid meniscus that drives the bulk fluid toward the adsorbed thin-film region. The model is validated with wicking experiments on different hemiwicking surfaces in conjunction withv 0andx 0measurements using Water , viscous FC-70$${\boldsymbol{(}}{{\bf{v}}}_{{\bf{0}}}{\boldsymbol{\approx }}{\bf{2}}\,{\bf{m}}{\boldsymbol{/}}{\bf{s}}{\boldsymbol{,}}\,{\bf{25}}\,{\boldsymbol{\mu }}{\bf{m}}{\boldsymbol{\lesssim }}{{\bf{x}}}_{{\bf{0}}}{\boldsymbol{\lesssim }}{\bf{28}}\,{\boldsymbol{\mu }}{\bf{m}}{\boldsymbol{)}}$$ and lower viscosity Ethanol$${\boldsymbol{(}}{{\boldsymbol{v}}}_{{\bf{0}}}{\boldsymbol{\approx }}{\bf{0.3}}\,{\bf{m}}{\boldsymbol{/}}{\bf{s}}{\boldsymbol{,}}\,{\bf{18.6}}\,{\boldsymbol{\mu }}{\bf{m}}{\boldsymbol{\lesssim }}{{\boldsymbol{x}}}_{{\bf{0}}}{\boldsymbol{\lesssim }}{\bf{38.6}}\,{\boldsymbol{\mu }}{\bf{m}}{\boldsymbol{)}}$$ .$${\boldsymbol{(}}{{\boldsymbol{v}}}_{{\bf{0}}}{\boldsymbol{\approx }}{\bf{1.2}}\,{\bf{m}}{\boldsymbol{/}}{\bf{s}}{\boldsymbol{,}}\,{\bf{11.8}}\,{\boldsymbol{\mu }}{\bf{m}}{\boldsymbol{\lesssim }}{{\bf{x}}}_{{\bf{0}}}{\boldsymbol{\lesssim }}{\bf{33.3}}\,{\boldsymbol{\mu }}{\bf{m}}{\boldsymbol{)}}$$ -
Abstract Positive
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Abstract We perform path-integral molecular dynamics (PIMD), ring-polymer MD (RPMD), and classical MD simulations of H
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