Distances between classes in $$W^{1,1}(\Omega ;{\mathbb {S}}^{1})$$

We introduce an equivalence relation on the space \(W^{1,1}(\Omega ;{\mathbb {S}}^1)\) which classifies maps according to their “topological singularities”. We establish sharp bounds for the distances (in the usual sense and in the Hausdorff sense) between the equivalence classes. Similar questions are examined for the space \(W^{1,p}(\Omega ;{\mathbb {S}}^1)\) when \(p>1\).     

Emergence of tables as first-graders cope with modelling tasks

In this action research, first-graders were challenged to cope with a sequence of modelling tasks involving an analysis of given situations and choices of mathematical tools. In the course of the sequence, they underwent a change in the nature of their problem-solving processes and developed modelling competencies. Moreover, during the task sequence, the first-graders spontaneously discovered the power of organizing problem data in a table. They did not merely use their existing mathematical knowledge, but also ‘reinvented’ tables as a new mathematical tool. This paper describes the gradual development of this tool as the children moved along the task sequence. Notably, the first-graders exhibited this progress in spite of having relatively little mathematical knowledge.     

Linear cover time is exponentially unlikely

We show that the probability that a simple random walk covers a finite, bounded degree graph in linear time is exponentially small. We conjecture that the same holds for any simple graph.     

Diffusion Limited Aggregation on a Cylinder

We consider the DLA process on a cylinder . It is shown that this process “grows arms”, provided that the base graph G has small enough mixing time. Specifically, if the mixing time of G is at most , the time it takes the cluster to reach the m ^th layer of the cylinder is at most of order . In particular we get examples of infinite Cayley graphs of degree 5, for which the DLA cluster on these graphs has arbitrarily small density. In addition, we provide an upper bound on the rate at which the “arms” grow. This bound is valid for a large class of base graphs G, including discrete tori of dimension at least 3. It is also shown that for any base graph G, the density of the DLA process on a G-cylinder is related to the rate at which the arms of the cluster grow. This implies that for any vertex transitive G, the density of DLA on a G-cylinder is bounded by 2/3.     

Conditioned Diffusions which are Brownian Bridges

Let X _t be a one-dimensional diffusion of the form dX _t=dB _t+(X _t)dt. Let Tbe a fixed positive number and let be the diffusion process which is X _t conditioned so that X ₀=X _T=x. If the drift is constant, i.e., , then the conditioned diffusion process is a Brownian bridge. In this paper, we show the converse is false. There is a two parameter family of nonlinear drifts with this property.     

Quantum chemical modeling of ethene epoxidation with hydrogen peroxide: the effect of microsolvation with water

Quantum chemical calculations were performed to study the mechanism of ethene epoxidation with hydrogen peroxide. The calculations were carried out at the B3LYP/6-311+G(d,p) level of theory. The applicability of this functional to the problem at hand, including basis set effects, was validated by CCSD(T) and CASSCF based multireference MP2 calculations. A mechanism was determined where hydrogen peroxide becomes polarized in the transition state upon binding to the ethene molecule. The distant hydroxide fragment of the attached hydrogen peroxide molecule becomes partly negatively charged, while the other part of the molecule involves a proton and becomes partly positively charged. In the absence of water an activation energy of 139.7 kJ mol(-1) was determined for the isolated H(2)O(2) + C(2)H(4) system. By microsolvating with water, the impact of a hydrogen-bonded network on the activation energy was addressed. A 43.7 kJ mol(-1) lowering of the activation energy, DeltaE(a), was observed when including up to 4 water molecules in the model. This effect results from the stabilization of the proton and hydroxide fragments in the transition state. The findings are discussed in the context of previous theoretical studies on similar systems. Effects of adding or removing a proton to mimic acidic and alkaline conditions are addressed and the limitations of the model in solvating the excess charge are discussed.     

High temporal resolution dynamic contrast MRI in a high risk group for placenta accreta.

Antenatal diagnosis of placenta accreta with MR is not easy even now because T2-weighted images (T2WI) cannot differentiate chorionic villi from decidua basalis. We performed dynamic contrast MRI to study whether trophoblastic villi could be separately demonstrated from the decidua basalis, and whether the contrast resolution between the placenta and myometrium could improve compared to T2WI. Six pregnant women with prior cesarean section were examined at 34-38 gestational weeks. Sagittal T2-weighted images with fast spin echo sequences and dynamic contrast studies with fast field echo sequence every 10-14 s after contrast injection were performed. We analyzed the enhancing pattern of the placenta and compared the contrast between placenta and myometrium. We reviewed medical records to identify complications during the placental delivery and the complications of their newborns. In the early phase after contrast enhancement, multiple foci of the strong lobular enhancement were observed in all cases. Other parts of placenta were slowly but strongly enhanced following them. We speculated that the former corresponded to intervillous space and the latter decidua basalis. The contrast between placenta and myometrium tended to be distinct near the inner cervical os on both T2WI and dynamic contrast study. On the other hand, it was indistinct in the upper part of the uterine body on T2WI despite it was clearly demonstrated on dynamic contrast study. The placentae were delivered without any complication in all cases. Although two neonates showed fetal distress, none of the infant remained any sequelae at the time of the discharge. The other four were well although one of them complicated with meconium staining. As dynamic contrast MRI can differentiate chorionic villi and decidua basalis, and can provide excellent contrast between placenta and myometrium at anywhere within the uterus, it may be a promising technique for antepartum diagnosis of the placenta accreta.     

Combinatorial invariants and covariants as tools for conical intersections

The combinatorial invariant and covariant are introduced as practical tools for analysis of conical intersections in molecules. The combinatorial invariant is a quantity depending on adiabatic electronic states taken at discrete nuclear configuration points. It is invariant to the phase choice (gauge) of these states. In the limit that the points trace a loop in nuclear configuration space, the value of the invariant approaches the corresponding Berry phase factor. The Berry phase indicates the presence of an odd or even number of conical intersections on surfaces bounded by these loops. Based on the combinatorial invariant, we develop a computationally simple and efficient method for locating conical intersections. The method is robust due to its use of gauge invariant nature. It does not rely on the landscape of intersecting potential energy surfaces nor does it require the computation of nonadiabatic couplings. We generalize the concept to open paths and combinatorial covariants for higher dimensions obtaining a technique for the construction of the gauge-covariant adiabatic-diabatic transformation matrix. This too does not make use of nonadiabatic couplings. The importance of using gauge-covariant expressions is underlined throughout. These techniques can be readily implemented by standard quantum chemistry codes.