COMPLEXATION BEHAVIOR OF POLY(ACRYLIC ACID) AND POLY(ETHYLENE OXIDE) IN WATER AND WATER-METHANOL

To investigate the effects of solvent type and temperature on the interpolymer complexation via hydrogen bonding, a study was made on the complex system of poly(acrylic acid) (PAA) and poly(ethylene oxide) (PEO) in two kinds of solvent systems, pure water and water-MeOH (30 wt%) mixed solvent, at various temperatures using the Ubbelohde viscometer, pH-meter, and UV spectrophotometer. The repeating unit mole ratio at the most optimum complexation as confirmed by the reduced viscosity measurement was shifted from [PEO]/[PAA] ≈ 1.25:1 to 1.5:1 by the addition of methanol to water. From the UV measurement, the deviation from the “isosbestic point” (where the absorbance of the solution remains constant) has presented another evidence for the solvent effect on complexation. In addition, the analysis of the changes in thermodynamic properties upon complexation as well as the fraction of carboxyls associated with PEO oxygens and the complex stability constant as estimated by potentiometric titration at several temperatures reveals that the complex formation in mixed solvent became more unfavorable compared to that in pure solvent at high temperatures above 30°C. This could be explained by considering that in water the hydrophobic interaction as well as the hydrogen bonding may greatly contribute to the stabilization of the polymer complex formed, while in water-methanol the main stabilizing force would be the hydrogen bonding alone.     

The spectral curve theory for (k,l)-symmetric CMC surfaces

Constant mean curvature surfaces in $S^3$ can be studied via their associated family of flat connections. In the case of tori this approach has led to a deep understanding of the moduli space of all CMC tori. For compact CMC surfaces of higher genus the theory is far more involved due to the non abelian nature of their fundamental group. In this paper we extend the spectral curve theory for tori developed in Hitchin (1990), Pinkall and Sterling (1989) and for genus 2 surfaces (Heller, 2014) to CMC surfaces in $S^3$ of genus $g=k\cdot l$ with commuting ${\mathbb{Z}}_{k+1}$ and ${\mathbb{Z}}_{l+1}$ symmetries. We determine their associated family of flat connections via certain flat line bundle connections parametrized by the spectral curve. We generalize the flow on spectral data introduced in Heller (2015) and prove the short time existence of this flow for certain families of initial surfaces. In this way we obtain countably many $1-$parameter families of new CMC surfaces of higher genus with prescribed branch points and prescribed umbilics.     

Hilbert series of invariants, constant terms and Kostka-Foulkes polynomials

A problem that arose in the study of the mass of the neutrino led us to the evaluation of a constant term with a variety of ramifications into several areas from Invariant Theory, Representation Theory, the Theory of Symmetric Functions and Combinatorics. A significant by-product of our evaluation is the construction of a trigraded Cohen Macaulay basis for the Invariants under an action of SL_n(C) on a space of 2n+n² variables.     

Electrophoresis of a toroid along the axis of a cylindrical pore

The electrophoresis of a toroid (doughnut-shaped entity) along the axis of a long cylindrical pore is analyzed under the conditions of low surface potential and weak applied electric field. The system under consideration is capable of modeling the electrophoretic behavior of various types of biocolloid such as bacterial DNA, plasmid DNA, and anabaenopsis, in a confined space. The influences of the key parameters of the problem, including the sizes of a toroid, the radius of a pore, and the thickness of the double layer, on the electrophoretic mobility of a toroid are discussed. We show that the electrophoretic behavior of a toroid under typical conditions can be different from that of an integrated entity. For instance, although the presence of the pore wall has the effect of retarding the movement of a particle, it becomes advantageous if a toroid is sufficiently close to the boundary. Several interesting behaviors are also observed, for example, the mobility of a toroid when the boundary effect is significant can be larger than that when it is insignificant.     

Surfaces with high contact number and their characterization

The notion of contact number of a Euclidean submanifold was introduced in an earlier article (Proc. Edinb. Math. Soc. 47:69–100, 2004) as the highest order of contact of geodesics and normal sections on the submanifold. It was proved in (Proc. Edinb. Math. Soc. 47:69–100, 2004) that the contact number relates closely with the notions of isotropic submanifolds and holomorphic curves. One important problem concerning contact number is to construct Euclidean submanifolds with high contact number. The purpose of this article is thus to construct Euclidean surfaces with high contact number and to provide simple geometric characterization of such surfaces. Mathematics Subject Classification (2000) Primary 53C40, 53A10 Secondary 53B25, 53C42     

On the nondegeneracy of constant mean curvature surfaces

We prove that many complete, noncompact, constant mean curvature (CMC) surfaces are nondegenerate; that is, the Jacobi operator Δ_f + | A_f |² has no L² kernel. In fact, if ∑ has genus zero with k ends, and if f (∑) is embedded (or Alexandrov immersed) in a half-space, then we find an explicit upper bound for the dimension of the L² kernel in terms of the number of non-cylindrical ends. Our main tool is a conjugation operation on Jacobi fields which linearizes the conjugate cousin construction. Consequences include partial regularity for CMC moduli space, a larger class of CMC surfaces to use in gluing constructions, and a surprising characterization of CMC surfaces via spinning spheres. R.K. partially supported by NSF grants DMS-0076085 at GANG/UMass and DMS-9810361 at MSRI, and by a FUNCAP grant in Fortaleza, Brazil. J.R. partially supported by an NSF VIGRE grant at Utah. Received: January 2005; Accepted: June 2005     

On the scalar curvature of constant mean curvature hypersurfaces in space forms

In this paper we study the behavior of the scalar curvature S of a complete hypersurface immersed with constant mean curvature into a Riemannian space form of constant curvature, deriving a sharp estimate for the infimum of S. Our results will be an application of a weak Omori-Yau maximum principle due to Pigola, Rigoli, Setti (2005) [17].     

On spacelike hypersurfaces with constant scalar curvature in locally symmetric Lorentz spaces

The purpose of this paper is to study compact or complete spacelike hypersurfaces with constant normalized scalar curvature in a locally symmetric Lorentz space satisfying some curvature conditions. We give an optimal estimate of the squared norm of the second fundamental form of such hypersurfaces. Furthermore, the totally umbilical hypersurfaces are characterized.     

Natural convection boundary layer flow of fluid with temperature-dependent viscosity from a horizontal elliptical cylinder with constant surface heat flux

This study deals with the temperature-dependent viscosity effects on the natural convection boundary layer on a horizontal elliptical cylinder with constant surface heat flux. The mathematical problem is reduced to a pair of coupled partial differential equations for the temperature and the stream function, and the resulting nonlinear equations are solved numerically by cubic spline collocation method. Results for the heat transfer characteristics are presented as functions of eccentric angle for various values of viscosity variation parameters, Prandtl numbers and aspect ratios. Results show that an increase in the viscosity variation parameter tends to accelerate the fluid flow near the surface and increase the maximum velocity, thus decreasing the velocity boundary layer thickness. As the viscosity variation parameter is increased, the surface temperature tends to decrease, thus increasing the local Nusselt number. Moreover, the local Nusselt number of the elliptical cylinder increases as the Prandtl number of the fluid is increased.