Primitive radon partitions for cyclic polytopes

It is proved that the setA∪B is a primitive Radon partition in the cyclic polytopeC(n,d) if and only if thed+2 points ofA andB alternate along the moment curve.     

Determining starshaped sets and unions of starshaped sets by their sections

Let S be a compact set in R^d. Let p be a fixed point of S and let k be a fixed integer, 1 k <d. Then S is starshaped with p ker S if and only if for every k-dimensional flat F through p, F S is starshaped. Moreover, an analogue of this result holds for unions of starshaped sets as well.     

j-partitions for visible shorelines

Summary LetC be a compact set inR ². A setS R ² C is said to have aj-partition relative toC if and only if there existj or fewer pointsc ₁,, c _j inC such that each point ofS sees somec _i via the complement ofC. Letm, j be fixed integers, 3 m, 2 j, and writem (uniquely) asm = qj + r, where 1 r j. Assume thatC is a convexm-gon in R², withS R ² C. Forq = 0 orq = 1, the setS has aj-partition relative toC. Forq 2,S has aj-partition relative toC if and only if every (qj + 1)-member subset ofS has aj-partition relative toC, and the Helly numberqj + 1 is best possible.IfC is a disk, no such Helly number exists.     

Unions of orthogonally convex or orthogonally starshaped polygons

LetT be a simply connected orthogonal polygon having the property that for every three points ofT, at least two of these points see each other via staircases inT. ThenT is a union of three orthogonally convex polygons. The number three is best possible.ForT a simply connected orthogonal polygon,T is a union of two orthogonally convex polygons if and only if for every sequencev ₁,...,v _n,v _n+1 =v ₁ inT, n odd, at least one consecutive pairv _i ,v _i+1 sees each other via staircase paths inT, 1 i n. An analogous result says thatT is a union of two orthogonal polygons which are starshaped via staircase paths if and only if for every odd sequence inT, at least one consecutive pair sees a common point via staircases inT.Supported in part by NSF grants DMS-8908717 and DMS-9207019.     

Some tilings of the plane whose singular points are uncountable and unbounded

Let I be a tiling of the plane such that for every tile T of I there correspond a tile T of I (not necessarily unique) and an integer k(T, T) (depending on T and T), k(T, T)>2, such that T meets T in k(T, T) connected components. Tiles T and T satisfying this condition are called associated tiles in I. Various properties concerning I and its singular points are obtained. First, it is not possible that every tile in I have a unique associated tile. In fact, there exist infinite families of tiles {F} {F _n:n1} such that F is the unique associated tile for every F _n. Next, if x is a singular point of I, then every neighborhood of x contains uncountably many singular points of I. Finally, the set of singular points of I is unbounded.     

Points of local nonconvexity,clear visibility,and starshaped sets in Rd

Let S be a compact, connected, locally starshaped set in R^d, S not convex. For every point of local nonconvexity q of S, define A_q to be the subset of S from which q is clearly visible via S. Then ker S = {conv A_q: q lnc S}. Furthermore, if every d+1 points of local nonconvexity of S are clearly visible from a common d-dimensional subset of S, then dim ker S = d.     

A characterization theorem for tilings having countably many singular points

LetT be a tiling of the plane. At most countably many points of U{bdry T T inT} fail to lie in a nondegenerate edge ofT if and only ifT has at most countably many singular points. The result fails without the requirement that the edges be nondegenerate. Moreover, countably many cannot be replaced by finitely many in the theorem.     

An improved Krasnosel'skii theorem for nonclosed sets in the plane

Let S be a nonempty bounded set in R² whose complement consists of a finite number of locally compact components. Assume that every 3 or fewer points in S see a common point via S. Then for some point p in cl S, the set A ≡ {s ∶ s in S and (p,s]? S} is nowhere dense in S. The number 3 is best possible.     

High-performance,solution-processed organic thin film transistors from a novel pentacene precursor

The Lewis acid-catalyzed Diels-Alder reaction of the organic semiconductor pentacene with N-sulfinylacetamide yields a soluble adduct. Spin-coated thin films of this adduct undergo solid-phase conversion to form thin films of pentacene at moderate temperatures. Organic thin film transistors fabricated by spin-coating this adduct, followed by thermal conversion to pentacene, exhibit the highest mobility reported to date for a solution-processed organic semiconductor.     

Atom probe tomography of size‐controlled phosphorus doped silicon nanocrystals

Doping of silicon nanocrystals is essential to control their electronic and optical properties. The incorporation of an impurity into a silicon nanovolume is a nontrivial task due to the self‐purification effect. Here, a systematic atom probe tomography study of the phosphorus distribution and incorporation in size‐controlled silicon nanocrystals embedded in silicon dioxide is presented. Qualitatively, it turns out that the phosphorus distribution in the system follows a universal, nanocrystal‐size independent trend: phosphorus‐enrichment at the interface with a substantial phosphorus‐incorporation in the silicon nanocrystal as small as 2 nm in diameter. This clearly contradicts strict self‐purification. These observations are explained by the bulk‐solubility and ‐segregation behaviour, kinetic effects related to the diffusion lengths, and nanoscale interface strain. The quantitative determination of the amount of phosphorus atoms per quantum dot enables a systematic understanding of phosphorus‐induced effects on optical and electronic properties of silicon nanovolumes.