Good covers are algorithmically unrecognizable

A good cover in R^d is a collection of open contractible sets in R^d such that the intersection of any subcollection is either contractible or empty. Motivated by an analogy with convex sets, intersection patterns of good covers were studied intensively. Our main result is that intersection patterns of good covers are algorithmically unrecognizable. More precisely, the intersection pattern of a good cover can be stored in a simplicial complex called nerve which records which subfamilies of the good cover intersect. A simplicial complex is topologically d-representable if it is isomorphic to the nerve of a good cover in R^d. We prove that it is algorithmically undecidable whether a given simplicial complex is topologically d-representable for any fixed d \geq 5. The result remains also valid if we replace good covers with acyclic covers or with covers by open d-balls. As an auxiliary result we prove that if a simplicial complex is PL embeddable into R^d, then it is topologically d-representable. We also supply this result with showing that if a "sufficiently fine" subdivision of a k-dimensional complex is d-representable and k \leq (2d-3)/3, then the complex is PL embeddable into R^d.Comment: 22 pages, 5 figures; result extended also to acyclic covers in version

Effectively Transparent Front Contacts for Optoelectronic Devices

Effectively transparent front contacts for optoelectronic devices achieve a measured transparency of up to 99.9% and a measured sheet resistance of 4.8 Ω sq^(−1). The 3D microscale triangular cross-section grid fingers redirect incoming photons efficiently to the active semiconductor area and can replace standard grid fingers as well as transparent conductive oxide layers in optoelectronic devices

A coupled terrestrial and aquatic biogeophysical model of the Upper Merrimack River watershed, New Hampshire, to inform ecosystem services evaluation and management under climate and land-cover change

Accurate quantification of ecosystem services (ES) at regional scales is increasingly important for making informed decisions in the face of environmental change. We linked terrestrial and aquatic ecosystem process models to simulate the spatial and temporal distribution of hydrological and water quality characteristics related to ecosystem services. The linked model integrates two existing models (a forest ecosystem model and a river network model) to establish consistent responses to changing drivers across climate, terrestrial, and aquatic domains. The linked model is spatially distributed, accounts for terrestrial–aquatic and upstream–downstream linkages, and operates on a daily time-step, all characteristics needed to understand regional responses. The model was applied to the diverse landscapes of the Upper Merrimack River watershed, New Hampshire, USA. Potential changes in future environmental functions were evaluated using statistically downscaled global climate model simulations (both a high and low emission scenario) coupled with scenarios of changing land cover (centralized vs. dispersed land development) for the time period of 1980–2099. Projections of climate, land cover, and water quality were translated into a suite of environmental indicators that represent conditions relevant to important ecosystem services and were designed to be readily understood by the public. Model projections show that climate will have a greater influence on future aquatic ecosystem services (flooding, drinking water, fish habitat, and nitrogen export) than plausible changes in land cover. Minimal changes in aquatic environmental indicators are predicted through 2050, after which the high emissions scenarios show intensifying impacts. The spatially distributed modeling approach indicates that heavily populated portions of the watershed will show the strongest responses. Management of land cover could attenuate some of the changes associated with climate change and should be considered in future planning for the region

How large are the level sets of the Takagi function?

Let T be Takagi's continuous but nowhere-differentiable function. This paper considers the size of the level sets of T both from a probabilistic point of view and from the perspective of Baire category. We first give more elementary proofs of three recently published results. The first, due to Z. Buczolich, states that almost all level sets (with respect to Lebesgue measure on the range of T) are finite. The second, due to J. Lagarias and Z. Maddock, states that the average number of points in a level set is infinite. The third result, also due to Lagarias and Maddock, states that the average number of local level sets contained in a level set is 3/2. In the second part of the paper it is shown that, in contrast to the above results, the set of ordinates y with uncountably infinite level sets is residual, and a fairly explicit description of this set is given. The paper also gives a negative answer to a question of Lagarias and Maddock by showing that most level sets (in the sense of Baire category) contain infinitely many local level sets, and that a continuum of level sets even contain uncountably many local level sets. Finally, several of the main results are extended to a version of T with arbitrary signs in the summands.Comment: Added a new Section 5 with generalization of the main results; some new and corrected proofs of the old material; 29 pages, 3 figure

Critical points for nondifferentiable functions in presence of splitting

AbstractA classical critical point theorem in presence of splitting established by Brézis–Nirenberg is extended to functionals which are the sum of a locally Lipschitz continuous term and of a convex, proper, lower semicontinuous function. The obtained result is then exploited to prove a multiplicity theorem for a family of elliptic variational–hemivariational eigenvalue problems

Singular solutions of fully nonlinear elliptic equations and applications

We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in each cone of $\mathbb{R}^n$, and the solutions are unique in an appropriate sense. We introduce a new method for analyzing the behavior of solutions near certain Lipschitz boundary points, which permits us to classify isolated boundary singularities of solutions which are bounded from either above or below. We also obtain a sharp Phragm\'en-Lindel\"of result as well as a principle of positive singularities in certain Lipschitz domains.Comment: 41 pages, 2 figure

Laser Microdissection of Narrow Sheath Mutant Maize Uncovers Novel Gene Expression in the Shoot Apical Meristem

Microarrays enable comparative analyses of gene expression on a genomic scale, however these experiments frequently identify an abundance of differentially expressed genes such that it may be difficult to identify discrete functional networks that are hidden within large microarray datasets. Microarray analyses in which mutant organisms are compared to nonmutant siblings can be especially problematic when the gene of interest is expressed in relatively few cells. Here, we describe the use of laser microdissection microarray to perform transcriptional profiling of the maize shoot apical meristem (SAM), a ~100-μm pillar of organogenic cells that is required for leaf initiation. Microarray analyses compared differential gene expression within the SAM and incipient leaf primordium of nonmutant and narrow sheath mutant plants, which harbored mutations in the duplicate genes narrow sheath1 (ns1) and narrow sheath2 (ns2). Expressed in eight to ten cells within the SAM, ns1 and ns2 encode paralogous WUSCHEL1-like homeobox (WOX) transcription factors required for recruitment of leaf initials that give rise to a large lateral domain within maize leaves. The data illustrate the utility of laser microdissection-microarray analyses to identify a relatively small number of genes that are differentially expressed within the SAM. Moreover, these analyses reveal potentially conserved WOX gene functions and implicate specific hormonal and signaling pathways during early events in maize leaf development