Automated Generation and Reactions of 3‐Hydroxymethylindoles in Continuous‐Flow Microreactors

An automated sequential approach for the generation and reactions of 3‐hydroxymethylindoles in continuous‐flow microreactors is described. Consecutive halogen–magnesium exchanges of four 3‐iodoindoles followed by addition to three aldehydes provided twelve 3‐hydroxymethylindoles in a multi‐microreactor setup. The synthetic flow strategy could be coupled with an in line continuous liquid–liquid extraction workup protocol for each reaction. Further elaboration of each of these indoles within the fluidic setup was achieved by acid‐catalysed nucleophilic substitutions with allyltrimethylsilane and methanol used as nucleophiles. Overall, a set of four 3‐iodoindoles was converted into thirty‐six indole derivatives by a range of transformations including iodo–magnesium exchange/electrophile trapping and acid‐catalysed nucleophilic substitution in a fully automated sequential fashion.     

Needle and seed segmentation in intra-operative 3D ultrasound-guided prostate brachytherapy

In order to guide the needle to the correct location in 3D US-guided brachytherapy, the needle is continuously tracked as it is being inserted. A pre-scan before the needle insertion and a post-scan after the needle insertion are subtracted to obtain a difference image containing the needle. The image is projected along two orthogonal directions approximately perpendicular to the needle, and the 3D needle is reconstructed from the segmented needles in the two projected images. The seeds implanted with the needle are located in the cropped volume along the needle. Thus, the seeds are segmented using a tri-bar model and 3D line segment patterns. Finally, the positions of the seeds are determined using a peak detection technique. Experiments with agar and turkey/chicken phantoms as well as patient data demonstrated that our needle segmentation technique could segment the needle in near real-time with an accuracy of 0.6 mm in position and 1.0 degrees in orientation. The true-positive rate for seed segmentation is 100% for the agar phantom and 93% for the chicken phantom. The average distance to manual seed segmentation was 1.0mm for the agar phantom and 1.7 mm for the chicken phantom.     

Fixed points of compact Kakutani maps with antipodal boundary conditions

We prove a fixed-point result for compact upper semicontinuous compact-convex-valued multifunctions satisfying antipodal boundary conditions on bounded symmetric subsets of a normed space. Two types or antipodal conditions are considered.

     

Type I operators and the approximation of singular two-point boundary value problems

A class of self adjoint operators associated with second order singular ordinary differential expressions, arises naturally when the problem is weakly formulated and integration by parts is performed. We call this class Type I operators. It turns out that this class can be successfully used to tackle numerical approximations of singular two-point boundary value problems. They can also be approximated by regular differential operators in a straightforward manner without having to bring the delicate structure of singular differential operators to the forefront of the investigation.     

Constant‐sign solutions for singular systems of Fredholm integral equations

We consider the system of Fredholm integral equations where T>0 is fixed and the nonlinearities H_i(t, u₁, u₂, …, u_n) can be singular at t=0 and u_j=0 where j∈{1, 2, …, n}. Criteria are offered for the existence of constant‐sign solutions, i.e. θ_iu_i(t)≥0 for t∈[0, 1] and 1≤i≤n, where θ_i∈{1,?1} is fixed. We also include an example to illustrate the usefulness of the results obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

One dimensional φ-Laplacian functional equations

In this paper we study functional φ-Laplacian equations with functional boundary conditions. The non-linearity belongs to a class of completely continuous operators, which includes Carathéodory ones, while the functional boundary conditions are given by compactly-fixed operators. We extend, in particular, a solvability result of C. Bereanu and J. Mawhin.     

Infinite Interval Problems Modeling the Flow of a Gas Through a Semi-Infinite Porous Medium

Various existence results are presented for boundary-value problems on the infinite interval. In particular, our theory includes a discussion of a problem arising in the unsteady flow of a gas through a semi–infinite porous medium.