Perfect matchings in random s-uniform hypergraphs

Let E = {X₁, X₂…, X_m} where the X_i ? V for 1 ≤ i ≤ m are distinct. The hypergraph G = (V, E) is said to be s-uniform if |X₁| = s for 1 ≤ i ≤ m. A set of edges M = {X_i : i ? I } is a perfect matching if (i) i ≠ j ? I implies X_i ∩ X_i = 0, and (ii) ∪_i?I X_i = V. In this article we consider the question of whether a random s-uniform hypergraph contains a perfect matching. Let s ≥ 3 be fixed and m/n^4/3 → ∞. We show that an s-uniform hypergraph with m edges chosen uniformly from [74] contains a perfect matching with high probability. This improves an earlier result of Schmidt and Shamir who showed that m/n^3/2 → ∞ suffices. © 1995 John Wiley & Sons, Inc.     

The asymptotic distributions of incomplete U-statistics

Summary An incomplete U-statistic is obtained by sampling the terms of an U-statistic. This paper derives the asymptotic distribution (if the variance is finite). Depending on the number of sampled terms, the resulting distribution is either the same as for the U-statistic, a normal distribution, or something intermediate. Also the case of a non-random sampling of the terms is treated. As an example, a non-parametric test of the independence of two circular random variables is studied. The results are generalized to generalized U-statistics.     

The Cut Metric,Random Graphs,and Branching Processes

In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the sequence of matrices of edge probabilities converges to an appropriate limit object (a kernel), but only in a very weak sense, namely in the cut metric. Our results thus generalize previous results on the phase transition in the already very general inhomogeneous random graph model introduced by the present authors in Random Struct. Algorithms 31:3–122 (2007), as well as related results of Bollobás, Borgs, Chayes and Riordan (Ann. Probab. 38:150–183, 2010), all of which involve considerably stronger assumptions. We also prove corresponding results for random hypergraphs; these generalize our results on the phase transition in inhomogeneous random graphs with clustering (Random Struct. Algorithms, 2010, to appear).     

Sesqui-type branching processes

We consider branching processes consisting of particles (individuals) of two types (type $L$ and type $S$) in which only particles of type $L$ have offspring, proving estimates for the survival probability and the (tail of) the distribution of the total number of particles. Such processes are in some sense closer to single- than to multi-type branching processes. Nonetheless, the second, barren, type complicates the analysis significantly. The results proved here (about point and survival probabilities) are a key ingredient in the analysis of bounded-size Achlioptas processes in a recent paper by the last two authors.     

Compact Zernike phase contrast x-ray microscopy using a single-element optic

We demonstrate Zernike phase contrast in a compact soft x-ray microscope using a single-element optic. The optic is a combined imaging zone plate and a Zernike phase plate and does not require any additional alignment or components. Contrast is increased and inversed in an image of a test object using the Zernike zone plate. This type of optic may be implemented into any existing x-ray microscope where phase contrast is of interest.     

Adsorption of sodium dodecyl sulfate and sodium dodecyl phosphate on aluminum, studied by QCM-D, XPS, and AAS

The adsorption of two anionic surfactants, sodium dodecyl sulfate (SDS) and sodium dodecyl phosphate (SDP), at surfaces of aluminum and aluminum oxide has been studied by means of atomic absorption spectrometry (AAS), X-ray photoelectron spectroscopy (XPS), and quartz crystal microbalance with dissipation monitoring (QCM-D). It was shown that more SDP than SDS binds to the surface and that SDP prevents dissolution of aluminum in water whereas SDS does not. This was not obvious, since the adsorption isotherms of the two surfactants to aluminum pigment powder are quite similar, as shown in an earlier work. The decreased aluminum dissolution with SDP compared to SDS was explained by the formation of a more compact protective layer with less permeability on the aluminum surface with SDP than with SDS. This is explained by differences in complexing ability between the surfactants and the aluminum pigment surface. While SDP is expected to form an inner-sphere complex with aluminum, leading to a lower accessibility of aluminum sites to water, SDS is likely to form a weaker outer-sphere complex.     

Competition between lipases and monoglycerides at interfaces

Tensiometry (the pendant drop technique), interfacial shear rheology, and ellipsometry have been used to study the effect of polar lipids that are generated during fat digestion on the behavior of lipases at the oil-water interface. Both Sn-1,3 regiospecific and nonregiospecific lipases have been used, and a noncatalytically active protein, beta-lacloglobulin, has been used as reference in the interfacial shear rheology experiments. The results from the pendant drop measurements and the interfacial rheology studies were in agreement with each other and demonstrated that the Sn-2 monoglyceride, which is one of the lipolysis products generated when a Sn-1,3 regiospecific lipase catalyzes triglyceride hydrolysis, is very interfacially active and efficiently expels the enzyme from the interface. Ellipsometry conducted at the liquid-liquid interface showed that the lipase forms a sublayer in the aqueous phase, just beneath the monoglyceride-covered interface. Sn-1/3 monoglycerides do not behave this way because they are rapidly degraded to fatty acid and glycerol and the fatty acid (or the fatty acid salt) does not have enough interfacial activity to expel the lipase from the interface. Since the lipases present in the gastrointestinal tract are highly Sn-1,3 regiospecific, we believe that the results obtained can be transferred to the in vivo situation. The formation of stable and amphiphilic Sn-2 monoglycerides can be seen as a self-regulatory process for fat digestion.     

A production-transportation problem with stochastic demand and concave production costs

Well known extensions of the classical transportation problem are obtained by including fixed costs for the production of goods at the supply points (facility location) and/or by introducing stochastic demand, modeled by convex nonlinear costs, at the demand points (the stochastic transportation problem, [STP]). However, the simultaneous use of concave and convex costs is not very well treated in the literature. Economies of scale often yield concave cost functions other than fixed charges, so in this paper we consider a problem with general concave costs at the supply points, as well as convex costs at the demand points. The objective function can then be represented as the difference of two convex functions, and is therefore called a d.c. function. We propose a solution method which reduces the problem to a d.c. optimization problem in a much smaller space, then solves the latter by a branch and bound procedure in which bounding is based on solving subproblems of the form of [STP]. We prove convergence of the method and report computational tests that indicate that quite large problems can be solved efficiently. Problems up to the size of 100 supply points and 500 demand points are solved. Received October 11, 1993 / Revised version received July 31, 1995 Published online November 24, 1998     

Precise Logarithmic Asymptotics for the Right Tails of Some Limit Random Variables for Random Trees

For certain random variables that arise as limits of functionals of random finite trees, we obtain precise asymptotics for the logarithm of the right-hand tail. Our results are based on the facts (i) that the random variables we study can be represented as functionals of a Brownian excursion and (ii) that a large deviation principle with good rate function is known explicitly for Brownian excursion. Examples include limit distributions of the total path length and of the Wiener index in conditioned Galton-Watson trees (also known as simply generated trees). In the case of Wiener index (where we recover results proved by Svante Janson and Philippe Chassaing by a different method) and for some other examples, a key constant is expressed as the solution to a certain optimization problem, but the constant’s precise value remains unknown. Research supported by NSF grants DMS-0104167 and DMS-0406104 and by The Johns Hopkins University’s Acheson J. Duncan Fund for the Advancement of Research in Statistics.