3,571 research outputs found
Counting homomorphisms from surface groups to finite groups
We prove a result that relates the number of homomorphisms from the
fundamental group of a compact nonorientable surface to a finite group ,
where conjugacy classes of the boundary components of the surface must map to
prescribed conjugacy classes in , to a sum over values of irreducible
characters of weighted by Frobenius-Schur multipliers. The proof is
structured so that the corresponding results for closed and possibly orientable
surfaces, as well as some generalizations, are derived using the same methods.
We then apply these results to the specific case of the symmetric group.Comment: Comments welcome
A relative version of Rochlin's theorem
Rochlin proved \cite{VR} that a closed 4-dimensional connected smooth
oriented manifold with vanishing second Stiefel-Whitney class has
signature divisible by 16. This was generalized by Kervaire and
Milnor \cite{kervaire_milnor_spheres} to the statement that if is an integral lift of an element in that is dual to , and if can be
represented by an embedded sphere in , then the self-intersection number
is divisible by 16. This was subsequently generalized further by
Rochlin (see Theorem \ref{matsumoto} below) and various alternative proofs of
this result where given by Freedman and Kirby \cite{freedman1978geometric},
Matsumoto \cite{matsumoto}, and Kirby \cite{kirbybook}.
We give further generalizations of this result concerning 4-manifolds with
boundary. Given a smooth compact orientable four manifold with integral
homology sphere boundary and a connected orientable characteristic surface with
connected boundary properly embedded in , we prove a theorem relating
the Arf invariant of , and the Arf invariant of , and the
Rochlin invariant of . We then proceed to generalize this result to
the case where is a topological compact orientable 4-manifold (which brings
in the Kirby-Siebenmann invariant), is not connected (which brings
in the condition of being proper as a link), is not orientable (which
brings in Brown invariants), and finally where is an arbitrary
3-manifold (which brings in pin structures). The final result gives a
``combinatorial'' description of the Kirby-Siebenmann invariant of a compact
orientable 4-manifold with nonempty boundary.Comment: Added several generalizations of the main result. Comments welcome
Morphological Phase Diagram for Lipid Membrane Domains with Entropic Tension
Circular domains in phase-separated lipid vesicles with symmetric leaflet composition commonly exhibit three stable morphologies: flat, dimpled, and budded. However, stable dimples (i.e., partially budded domains) present a puzzle since simple elastic theories of domain shape predict that only flat and spherical budded domains are mechanically stable in the absence of spontaneous curvature. We argue that this inconsistency arises from the failure of the constant surface tension ensemble to properly account for the effect of entropic bending fluctuations. Formulating membrane elasticity within an entropic tension ensemble, wherein tension represents the free energy cost of extracting membrane area from thermal bending of the membrane, we calculate a morphological phase diagram that contains regions of mechanical stability for each of the flat, dimpled, and budded domain morphologies
Acoustic suppression of the coffee-ring effect
We study the influence of acoustic fields on the evaporative self-assembly of solute particles suspended inside sessile droplets of complex fluids. The self-assembly process often results in an undesirable ring-like heterogeneous residue, a phenomenon known as the coffee-ring effect. Here we show that this ring-like self-assembly can be controlled acoustically to form homogeneous disc-like or concentrated spot-like residues. The principle of our method lies in the formation of dynamic patterns of particles in acoustically excited droplets, which inhibits the evaporation-driven convective transport of particles towards the contact line. We elucidate the mechanisms of this pattern formation and also obtain conditions for the suppression of the coffee-ring effect. Our results provide a more general solution to suppress the coffee-ring effect without any physiochemical modification of the fluids, the particles or the surface, thus potentially useful in a broad range of industrial and analytical applications that require homogenous solute depositions
Sulfur reduction in sediments of marine and evaporite environments
Transformations of sulfur in sediments of ponds ranging in salinities from that of normal seawater to those of brines saturated with sodium chloride were examined. The chemistry of the sediment and pore waters were focused on with emphasis on the fate of sulfate reduction. The effects of increasing salinity on both forms of sulfur and microbial activity were determined. A unique set of chemical profiles and sulfate-reducing activity was found for the sediments of each of the sites examined. The quantity of organic matter in the salt pond sediments was significantly greater than that occurring in the adjacent intertidal site. The total quantitative and qualitative distribution of volatile fatty acids was also greater in the salt ponds. Volatile fatty acids increased with salinity
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