Spectra of Large Random Trees

We analyze the eigenvalues of the adjacency matrices of a wide variety of random trees. Using general, broadly applicable arguments based on the interlacing inequalities for the eigenvalues of a principal submatrix of a Hermitian matrix and a suitable notion of local weak convergence for an ensemble of random trees that we call probability fringe convergence, we show that the empirical spectral distributions for many random tree models converge to a deterministic (model-dependent) limit as the number of vertices goes to infinity. Moreover, the masses assigned by the empirical spectral distributions to individual points also converge in distribution to constants. We conclude for ensembles such as the linear preferential attachment models, random recursive trees, and the uniform random trees that the limiting spectral distribution has a set of atoms that is dense in the real line. We obtain lower bounds on the mass assigned to zero by the empirical spectral measures via the connection between the number of zero eigenvalues of the adjacency matrix of a tree and the cardinality of a maximal matching on the tree. In particular, we employ a simplified version of an algorithm due to Karp and Sipser to construct maximal matchings and understand their properties. Moreover, we show that the total weight of a weighted matching is asymptotically equivalent to a constant multiple of the number of vertices when the edge weights are independent, identically distributed, nonnegative random variables with finite expected value, thereby significantly extending a result obtained by Aldous and Steele in the special case of uniform random trees. We greatly generalize a celebrated result obtained by Schwenk for the uniform random trees by showing that if any ensemble converges in the probability fringe sense and a very mild further condition holds, then, with probability converging to one, the spectrum of a realization is shared by at least one other (nonisomorphic) tree. For the linear preferential attachment model with parameter a>?1, we show that for any fixed k, the k largest eigenvalues jointly converge in distribution to a nontrivial limit when rescaled by $n^{1/2\gamma_{a}}$ , where ?? _a =a+2 is the Malthusian rate of growth parameter for an associated continuous-time branching process.     

Testing for Constant Nonparametric Effects in General Semiparametric Regression Models with Interactions

We consider the problem of testing for a constant nonparametric effect in a general semiparametric regression model when there is a potential for interaction between the parametrically and nonparametrically modeled variables. The work was originally motivated by a unique testing problem in genetic epidemiology (Chatterjee et al., 2006) that involved a typical generalized linear model but with an additional term reminiscent of the Tukey 1-degree-of-freedom formulation, and their interest was in testing for main effects of the genetic variables, while gaining statistical power by allowing for a possible interaction between genes and the environment. Later work (Maity et al., 2009) involved the possibility of modeling the environmental variable nonparametrically, but they focused on whether there was a parametric main effect for the genetic variables. In this paper, we consider the complementary problem, where the interest is in testing for the main effect of the nonparametrically modeled environmental variable. We derive a generalized likelihood ratio test for this hypothesis, show how to implement it, and provide evidence that our method can improve statistical power when compared to standard partially linear models with main effects only. We use the method for the primary purpose of analyzing data from a case-control study of colorectal adenoma.     

On the relationship between distinction and irreducibility of parabolic induction

Let $U_{2n}$ denote the quasi-split unitary group over 2n variables with respect to a quadratic extension $E/F$ of p-adic fields. In this short note, we relate ${\mathrm{GL}}_n(F)$-distinction of ladder representations of ${\mathrm{GL}}_n(E)$ with irreducibility of its Siegel parabolic induction in $U_{2n}$.     

Investigation of the effects of silicate modification on polymer‐layered silicate nanocomposite morphology

The morphological behavior of a series of polymer‐layered silicate nanocomposites (PLSNs) has been investigated. The goal was to probe the effect of “textured” silicate surfaces on PLSN morphology. The nanocomposites were fabricated by mixing montmorillonite clay that was carefully modified with tailor‐made polystyrene (PS) surfactants into a PS homopolymer matrix, where the chemical similarity of the matrix polymer and surfactants assures complete miscibility of surfactant and homopolymer. To examine the effect of silicate surface “texture,” clay was modified with combinations of long and short surfactants. The samples were then direct melt annealed to allow the equilibrium morphology to develop, and characterized by small‐angle X‐ray scattering. Based on the implications of the Balazs model and other work on the wetting behavior of polymer melts with longer surfactants and textured surfaces we expected that the intercalation of the homopolymer matrix material into the modified clay would be promoted. Extensive characterization of both the modified clays as well as the resultant nanocomposites clearly show that the modified clays exhibit a high degree of order, but also that only phase‐separated morphologies are formed in the corresponding nanocomposites. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 4075–4083, 2004     

Fabrication techniques for high-quality optical fibers

Various vapor deposition processes are being used to manufacture optical fibers in different parts of the world. This paper provides an introduction to the common vapor deposition technology, reviews the different processes, and compares them in terms of performance and cost potential.     

Synthesis of symmetrical trisubstituted olefins by cross metathesis

[reaction: see text] Trisubstituted alkenes have been prepared via intermolecular olefin cross-metathesis (CM) between alpha-olefins and symmetrically 1,1-disubstituted olefins using an imidazolylidene ruthenium benzylidene complex. Of particular interest is the synthesis of isoprenoid/prenyl groups by a simple solvent-free CM reaction with isobutylene. In addition, prenyl groups can also be installed by a cross-metathesis of 2-methyl-2-butene with a variety of alpha-olefins at room temperature with low catalyst loadings.     

Drying of alkoxide gels—Observation of an alternate phenomenology

The process of drying of a porous material as per the current phenomenological theory can be divided into two stages. At first the body shrinks by an amount equal to the volume of liquid that evaporates, and the liquid-vapor interface remains at the exterior surface of the body. The second stage begins when the body becomes too stiff to shrink and the liquid recedes into the interior, leaving air filled pores near the surface. We shall refer to this phenomenology as the drying front model. In our investigation of drying of alkoxide silica gels of less than 50 Angstroms pore radius, we have observed a different drying pattern, in which even after the gel body stops shrinking, drying continues to occur by evaporation on the exterior surface of the gel body, causing spontaneous nucleation of partially or fully dried opaque clusters, randomly distributed in the interior parts of the gel. These clusters than increase in number and size till they coalesce to form an opaque body. Upon further drying, the gel returns to its transparent form. We postulate that this is possible only if the rate of fluid flow in the pores by diffusion is faster than that by Darcy's flow, as well as the evaporation rate at the surface of the gel body. We shall refer to this as the cluster drying model. We shall present results of pin-hole drying experiments on cylindrical alkoxide gels showing that for identical gels the evaporation rate can be increased to change the phenomenology from cluster drying to one that exhibits both phenomenology simultaneously and finally to that of the drying front phenomenology. We shall also show the effect of gel pore size distribution on the phenomenology of drying under identical drying conditions. Finally, we will present evidence that for successful drying of large cylindrical alkoxide gels, drying conditions favoring cluster drying phenomenology is desirable.     

Exploration of an arabinogalactan isolated from Odina wodier Roxb.: Physicochemical,compositional characterisations and functional attributes

Exudate gum polysaccharides have a diverse range of functionalities in food, cosmetics, textiles, biomedical, pharmaceutical and other industries for centuries. The potentiality of gum odina as tablet binder, coacervates (chitosan‐gum odina complex) for colon‐targeted drug delivery system and also as prebiotic with immunomodulating properties was reported earlier. Since no detail study of the physicochemical, functional properties of the gum has been reported, the present investigation deals with physicochemical, compositional and functional characterisations of purified gum odina (PGO) for adopting in food and pharmaceutical industry. PGO, an arabinogalactan, was obtained by ethanol precipitation from exudates (gum odina) of tropical deciduous plant Odina wodier Roxb. Colour profiling of PGO including L* (87.74 ± 0.42), a* (1.73 ± 0.65) and b* (7.79 ± 0.58) was determined. Physicochemical parameters revealed good flow ability and compressibility desired for an excipient. Concentration‐dependent surface tension was measured by du Noüy ring method. Rheological study showed pseudoplastic behaviour of PGO dispersion. Sugar analysis by gas liquid chromatography indicated presence of arabinogalactan in PGO. Size exclusion chromatography of PGO revealed two high‐molecular‐weight components PGO‐I (95%, Arabinose:Galactose :: 1:1.6) and PGO‐II (5%, Arabinose:Galactose :: 1:4). Further characterisations of PGO by means of CHNS, FTIR, differential scanning calorimetry, X‐ray diffraction, transmission electron microscopy, scanning electron microscopy and energy‐dispersive X‐ray diffraction, conductivity, pH, zeta potential analysis and antioxidant activity indicated typical polysaccharide characteristics. Collectively, this work established the fundamental properties of PGO and the results presented here will facilitate the applications of PGO as sustainable food additive, pharmaceutical excipient for commercial adoption.     

Rh(III)‐Catalyzed Denitrogenative [4+2] Annulation of Benzamides and 3‐Diazoindolin‐2‐imines: Expedient Access to Indolo[2,3‐c] isoquinolin‐5‐ones

An effective and pragmatic strategy for the synthesis of structurally diverse indolo[2,3‐c]isoquinolin‐5‐ones has been developed via a Rh(III)‐catalyzed C?H activation and [4+2] annulation reaction of N‐methoxybenzamides and 3‐diazoindolin‐2‐imines. The reaction involves the efficient formation of two new (one C?C and one C?N) bonds under operationally simple conditions and has the benefits of a broad substrate scope.