On a graph of ideals

We associate a graph Γ₊(R) to a ring R whose vertices are nonzero proper right ideals of R and two vertices I and J are adjacent if I+J=R. Then we try to translate properties of this graph into algebraic properties of R and vice versa. For example, we characterize rings R for which Γ₊(R) respectively is connected, complete, planar, complemented or a forest. Also we find the dominating number of Γ₊(R).     

Aqueous ionic and complementary zwitterionic soluble surfactants: molecular dynamics simulations and sum frequency generation spectroscopy of the surfaces

An aqueous ionic surfactant, 1-dodecyl-4-(dimethylamino)pyridinium (DMP) bromide, and the corresponding zwitterion 2-[4-(dimethylamino)pyridinio]dodecanoate (DPN) were explored by means of molecular dynamics (MD) simulations and, for the ionic system, by infrared-visible sum frequency generation (IR-vis SFG). The molecular structure of the interfacial layer was investigated for the ionic and zwitterionic systems as a function of surfactant concentration, both in water and in salt (KF or KBr) solutions, by MD simulations in a slab geometry. The buildup of the surface monolayer and a sublayer was monitored, and density and orientational profiles of the surfactants were evaluated. The difference between the ionic and zwitterionic systems and the effect of the added salt were analyzed at the molecular level. The results of MD simulations were compared to those of nonlinear optical spectroscopy measurements. IR-vis SFG was employed to study the DMP ionic surfactant in water and upon addition of simple salts. The influence of added salts on the different molecular moieties at the interface was quantified in detail experimentally.     

Killing vector fields on Riemannian and Lorentzian 3-manifolds

We give a complete local classification of all Riemannian 3-manifolds $(M,g)$ admitting a nonvanishing Killing vector field T. We then extend this classification to timelike Killing vector fields on Lorentzian 3-manifolds, which are automatically nonvanishing. The two key ingredients needed in our classification are the scalar curvature S of g and the function $\text{Ric}(T,T)$, where Ric is the Ricci tensor; in fact their sum appears as the Gaussian curvature of the quotient metric obtained from the action of T. Our classification generalizes that of Sasakian structures, which is the special case when $\text{Ric}(T,T)=2$. We also give necessary, and separately, sufficient conditions, both expressed in terms of $\text{Ric}(T,T)$, for g to be locally conformally flat. We then move from the local to the global setting, and prove two results: in the event that T has unit length and the coordinates derived in our classification are globally defined on ${\mathbb{R}}^3$, we give conditions under which S completely determines when the metric will be geodesically complete. In the event that the 3-manifold M is compact, we give a condition stating when it admits a metric of constant positive sectional curvature.     

Novel swelling/shrinking behaviors of glucose-binding hydrogels and their potential use in a microfluidic insulin delivery system

Hydrogels based on lightly crosslinked poly(methacrylamidophenylboronic acid-co-acylamide) swell monototically at pH 7.4 with exposure to increasing concentrations of glucose. At pH 10 these hydrogels first shrink and then reswell with increasing glucose concentration. A mechanism for these disctinct behaviors is proposed and preliminary results validating this mechanism are presented. A simple microfabricated glucose-sensitive valve based on this hydrogel is described.