A Classification of Minimal Sets of Torus Homeomorphisms

We provide a classification of minimal sets of homeomorphisms of the two-torus, in terms of the structure of their complement. We show that this structure is exactly one of the following types: (1) a disjoint union of topological disks, or (2) a disjoint union of essential annuli and topological disks, or (3) a disjoint union of one doubly essential component and bounded topological disks. Periodic bounded disks can only occur in type 3. This result provides a framework for more detailed investigations, and additional information on the torus homeomorphism allows to draw further conclusions. In the non-wandering case, the classification can be significantly strengthened and we obtain that a minimal set other than the whole torus is either a periodic orbit, or the orbit of a periodic circloid, or the extension of a Cantor set. Further special cases are given by torus homeomorphisms homotopic to an Anosov, in which types 1 and 2 cannot occur, and the same holds for homeomorphisms homotopic to the identity with a rotation set which has non-empty interior. If a non-wandering torus homeomorphism has a unique and totally irrational rotation vector, then any minimal set other than the whole torus has to be the extension of a Cantor set.Comment: Published in Mathematische Zeitschrift, June 2013, Volume 274, Issue 1-2, pp 405-42

Strictly Toral Dynamics

This article deals with nonwandering (e.g. area-preserving) homeomorphisms of the torus $\mathbb{T}^2$ which are homotopic to the identity and strictly toral, in the sense that they exhibit dynamical properties that are not present in homeomorphisms of the annulus or the plane. This includes all homeomorphisms which have a rotation set with nonempty interior. We define two types of points: inessential and essential. The set of inessential points $ine(f)$ is shown to be a disjoint union of periodic topological disks ("elliptic islands"), while the set of essential points $ess(f)$ is an essential continuum, with typically rich dynamics (the "chaotic region"). This generalizes and improves a similar description by J\"ager. The key result is boundedness of these "elliptic islands", which allows, among other things, to obtain sharp (uniform) bounds of the diffusion rates. We also show that the dynamics in $ess(f)$ is as rich as in $\mathbb{T}^2$ from the rotational viewpoint, and we obtain results relating the existence of large invariant topological disks to the abundance of fixed points.Comment: Incorporates suggestions and corrections by the referees. To appear in Inv. Mat

Isoperimetric Inequalities for Minimal Submanifolds in Riemannian Manifolds: A Counterexample in Higher Codimension

For compact Riemannian manifolds with convex boundary, B.White proved the following alternative: Either there is an isoperimetric inequality for minimal hypersurfaces or there exists a closed minimal hypersurface, possibly with a small singular set. There is the natural question if a similar result is true for submanifolds of higher codimension. Specifically, B.White asked if the non-existence of an isoperimetric inequality for k-varifolds implies the existence of a nonzero, stationary, integral k-varifold. We present examples showing that this is not true in codimension greater than two. The key step is the construction of a Riemannian metric on the closed four-dimensional ball B with the following properties: (1) B has strictly convex boundary. (2) There exists a complete nonconstant geodesic. (3) There does not exist a closed geodesic in B.Comment: 11 pages, We changed the title and added a section that exhibits the relation between our example and the question posed by Brian White concerning isoperimetric inequalities for minimal submanifold

Variation in the flowering time orthologs BrFLC and BrSOC1 in a natural population of Brassica rapa.

Understanding the genetic basis of natural phenotypic variation is of great importance, particularly since selection can act on this variation to cause evolution. We examined expression and allelic variation in candidate flowering time loci in Brassica rapa plants derived from a natural population and showing a broad range in the timing of first flowering. The loci of interest were orthologs of the Arabidopsis genes FLC and SOC1 (BrFLC and BrSOC1, respectively), which in Arabidopsis play a central role in the flowering time regulatory network, with FLC repressing and SOC1 promoting flowering. In B. rapa, there are four copies of FLC and three of SOC1. Plants were grown in controlled conditions in the lab. Comparisons were made between plants that flowered the earliest and latest, with the difference in average flowering time between these groups ∼30 days. As expected, we found that total expression of BrSOC1 paralogs was significantly greater in early than in late flowering plants. Paralog-specific primers showed that expression was greater in early flowering plants in the BrSOC1 paralogs Br004928, Br00393 and Br009324, although the difference was not significant in Br009324. Thus expression of at least 2 of the 3 BrSOC1 orthologs is consistent with their predicted role in flowering time in this natural population. Sequences of the promoter regions of the BrSOC1 orthologs were variable, but there was no association between allelic variation at these loci and flowering time variation. For the BrFLC orthologs, expression varied over time, but did not differ between the early and late flowering plants. The coding regions, promoter regions and introns of these genes were generally invariant. Thus the BrFLC orthologs do not appear to influence flowering time in this population. Overall, the results suggest that even for a trait like flowering time that is controlled by a very well described genetic regulatory network, understanding the underlying genetic basis of natural variation in such a quantitative trait is challenging

Thermodynamic behaviour and structural properties of an aqueous sodium chloride solution upon supercooling

We present the results of a molecular dynamics simulation study of thermodynamic and structural properties upon supercooling of a low concentration sodium chloride solution in TIP4P water and the comparison with the corresponding bulk quantities. We study the isotherms and the isochores for both the aqueous solution and bulk water. The comparison of the phase diagrams shows that thermodynamic properties of the solution are not merely shifted with respect to the bulk. Moreover, from the analysis of the thermodynamic curves, both the spinodal line and the temperatures of maximum density curve can be calculated. The spinodal line appears not to be influenced by the presence of ions at the chosen concentration, while the temperatures of maximum density curve displays both a mild shift in temperature and a shape modification with respect to bulk. Signatures of the presence of a liquid-liquid critical point are found in the aqueous solution. By analysing the water-ion radial distribution functions of the aqueous solution we observe that upon changing density, structural modifications appear close to the spinodal. For low temperatures additional modifications appear also for densities close to that corresponding to a low density configurational energy minimum.Comment: 10 pages, 13 figures, 2 tables. To be published in J. Chem. Phy

Global surfaces of section in the planar restricted 3-body problem

The restricted planar three-body problem has a rich history, yet many unanswered questions still remain. In the present paper we prove the existence of a global surface of section near the smaller body in a new range of energies and mass ratios for which the Hill's region still has three connected components. The approach relies on recent global methods in symplectic geometry and contrasts sharply with the perturbative methods used until now.Comment: 11 pages, 1 figur

Molecular architecture of softwood revealed by solid-state NMR

Economically important softwood from conifers is mainly composed of the polysaccharides cellulose, galactoglucomannan and xylan, and the phenolic polymer, lignin. The interactions between these polymers lead to wood mechanical strength and must be overcome in biorefining. Here, we use 13C multidimensional solid-state NMR to analyse the polymer interactions in never-dried cell walls of the softwood, spruce. In contrast to some earlier softwood cell wall models, most of the xylan binds to cellulose in the two-fold screw conformation. Moreover, galactoglucomannan alters its conformation by intimately binding to the surface of cellulose microfibrils in a semi-crystalline fashion. Some galactoglucomannan and xylan bind to the same cellulose microfibrils, and lignin is associated with both of these cellulose-bound polysaccharides. We propose a model of softwood molecular architecture which explains the origin of the different cellulose environments observed in the NMR experiments. Our model will assist strategies for improving wood usage in a sustainable bioeconomy

Molecular modeling of a tandem two pore domain potassium channel reveals a putative binding Site for general anesthetics

[Image: see text] Anesthetics are thought to mediate a portion of their activity via binding to and modulation of potassium channels. In particular, tandem pore potassium channels (K2P) are transmembrane ion channels whose current is modulated by the presence of general anesthetics and whose genetic absence has been shown to confer a level of anesthetic resistance. While the exact molecular structure of all K2P forms remains unknown, significant progress has been made toward understanding their structure and interactions with anesthetics via the methods of molecular modeling, coupled with the recently released higher resolution structures of homologous potassium channels to act as templates. Such models reveal the convergence of amino acid regions that are known to modulate anesthetic activity onto a common three- dimensional cavity that forms a putative anesthetic binding site. The model successfully predicts additional important residues that are also involved in the putative binding site as validated by the results of suggested experimental mutations. Such a model can now be used to further predict other amino acid residues that may be intimately involved in the target-based structure–activity relationships that are necessary for anesthetic binding

Spatiotemporal Controls on Observed Daytime Ozone Deposition Velocity over Northeastern U.S. Forests During Summer

Spatiotemporal variability in ozone dry deposition is often overlooked despite its implications for interpreting and modeling tropospheric ozone concentrations accurately. Understanding the influences of stomatal versus nonstomatal deposition processes on ozone deposition velocity is important for attributing observed changes in the ozone depositional sink and associated damage to ecosystems. Here, we aim to identify the stomatal versus nonstomatal deposition processes driving observed variability in ozone deposition velocity over the northeastern United States during June–September. We use ozone eddy covariance measurements from Harvard Forest in Massachusetts, which span a decade, and from Kane Experimental Forest in Pennsylvania and Sand Flats State Forest in New York, which span one growing season each, along with observation‐driven modeling. Using a cumulative precipitation indicator of soil wetness, we infer that high soil uptake during dry years and low soil uptake during wet years may contribute to the twofold interannual variability in ozone deposition velocity at Harvard Forest. We link stomatal deposition and humidity to variability in ozone deposition velocity on daily timescales. The humidity dependence may reflect higher uptake by leaf cuticles under humid conditions, noted in previous work. Previous work also suggests that uptake by leaf cuticles may be enhanced after rain, but we find that increases in ozone deposition velocity on rainy days are instead mostly associated with increases in stomatal conductance. Our analysis highlights a need for constraints on subseasonal variability in ozone dry deposition to soil and fast in‐canopy chemistry during ecosystem stress

Periodic orbits of period 3 in the disc

Let f be an orientation preserving homeomorphism of the disc D2 which possesses a periodic point of period 3. Then either f is isotopic, relative the periodic orbit, to a homeomorphism g which is conjugate to a rotation by 2 pi /3 or 4 pi /3, or f has a periodic point of least period n for each n in N*.Comment: 7 page