Cayley’s decomposition and pólya’sW-property of ordinary linear differential equations

A proof is given for the equivalence of Pólya’sW- property of a linear differential equationL _n (D) y=0 to the possibility of decomposingL _n (D) ≡ Π _n ¹ [D+λ_i(x)] in a given interval. In this case a set ofn independent solutions form a Chebyshev system in the interval. An application determines intervals of non-oscillation for solutions of linear equations of the second order. Research supported by the National Science Foundation Grant No. GP-3897 with the University of Maryland.     

Toluene solubilization induces different modes of mixed micelle growth

The effect of toluene solubilization on the size and mobility of Triton X100 (TX100) micelles and TX100/sodium dodecyl sulfate (SDS) mixed micelles was studied by turbidimetry, dynamic light scattering, and capillary electrophoresis. Micelle growth due to toluene solubilization was observed for both surfactant systems; however, two different modes of growth were seen. Mixed micelles in 0.1 M NaCl are spherical (apparent diameter d(app) = 8 nm) and remain so while taking up 3 mM toluene, with a volume increase per micelle of deltaV(m) = 50 nm3. In 0.5 M NaCl, the large d(app) of both nonionic and mixed micelles (14 and 24 nm, respectively) indicate ellipsoidal or rodlike shapes, and their large increases in d(app) upon addition of 3 mM toluene thus correspond to elongational growth, with the same deltaV(m) = 50 nm3. Further addition of toluene to TX100/SDS in 0.5 M NaCl results in a dramatic increase in micelle size followed by an unexpected bimodal size distribution. The addition of excess toluene leads to the formation of ca. 140 nm toluene droplets, stabilized mainly by monomers of the high critical micelle concentration surfactant, SDS. These microemulsions coexist with the smaller (20 nm) swollen mixed micelles.     

Effect of pore size on adsorption of a polyelectrolyte to porous glass

The adsorption of quaternized poly(vinylpyridine) (QPVP) on controlled pore glass (CPG) size, over the ionic strength range 0.001-0.5 M was found to display nonmonotonic behavior as a function of pore size. Both adsorption kinetics and ionic strength effects deviated dramatically from behavior typical of adsorption on flat surfaces when the ratio of the pore radius Rp to the polymer hydrodynamic radius Rh became smaller than ca. 2. Ionic strength enhancement of adsorption for small pore sizes was observed at much higher salt concentrations than is typical for polycation adsorption on flat surfaces. The amount of polymer adsorbed per unit surface area of glass GammaA, in 0.5 M NaCl, exhibited a shallow maximum at Rp/Rh approximately 5. Since the value of GammaA for small pore size CPG is strongly depressed by the large surface area, an alternative and more interesting observation is that the amount of polymer adsorbed per gram of CPG, Gammaw, displays a strong maximum when Rp is equal to or slightly smaller than Rh. The efficiency with which QPVP binds anionic micelles to (negatively charged) CPG (grams of surfactant/grams of QPVP) increases strongly with diminishing pore size, indicating that the configuration of polycation bound to small pores favors micelle binding. Since the micelles are larger than small pores, the results indicate that when Rp < Rh, adsorbed polycation molecules reside only partially within the pore. The results of this study are supported by simulations of polyelectrolytes within cylindrical cavities.     

The small core of the German corporate board network

We consider the bipartite graph of German corporate boards and identify a small core of directors who are highly central in the entire network while being densely connected among themselves. To identify the core, we compare the actual number of board memberships to a random benchmark, focusing on deviations from the benchmark that span several orders of magnitude. The board appointment decisions of largely capitalized companies appear to be the driving force behind the existence of a core in Germany’s corporate network.     

Effects of slight Fe (III)-doping on structural and optical properties of TiO2 nanoparticles

Journal of Sol-Gel Science and Technology - In this research work, undoped and slight Fe-doped-TiO2 nanoparticles (Fe–TiO2 NPs) were prepared using simple chemical sol–gel method and...     

Conjugacy in Permutation Representations of the Symmetric Group

Although the conjugacy classes of the general linear group are known, it is not obvious (from the canonic form of matrices) that two permutation matrices are similar if and only if they are conjugate as permutations in the symmetric group, i.e., that conjugacy classes of S _n do not unite under the natural representation. We prove this fact, and give its application to the enumeration of fixed points under a natural action of S _n  × S _n . We also consider the permutation representations of S _n which arise from the action of S _n on ordered tuples and on unordered subsets, and classify which of them unite conjugacy classes and which do not.