On the Jager-Kaul theorem concerning harmonic maps

In 1983, Jager and Kaul proved that the equator map u*(x) = (x/\x\,0) : B-n --> S-n is unstable for 3 less than or equal to n less than or equal to 6 and a minimizer for the energy functional E(u, B-n) = integral B-n \del u\(2) dx in the class H-1,H-2(B-n, S-n) with u = u* on partial derivative B-n when n greater than or equal to 7. In this paper, we give a new and elementary proof of this Jager-Kaul result. We also generalize the Jager-Kaul result to the case of p-harmonic maps.

A time-harmonic inverse methodology for the design of RF coils in MRI

An inverse methodology is described to assist in the design of radio-frequency (RF) coils for magnetic resonance imaging (MRI) applications. The time-harmonic electromagnetic Green's functions are used to calculate current on the coil and shield cylinders that will generate a specified internal magnetic field. Stream function techniques and the method of moments are then used to implement this theoretical current density into an RF coil. A novel asymmetric coil operating for a 4.5 T MRI machine was designed and constructed using this methodology and the results are presented.

On the minimality of the p-harmonic map - x/vertical bar x vertical bar : B-n -> Sn-1

We prove that for any real number p with 1 p less than or equal to n - 1, the map x/\x\ : B-n --> Sn-1 is the unique minimizer of the p-energy functional integral(Bn) \delu\(p) dx among all maps in W-1,W-p (B-n, Sn-1) with boundary value x on phiB(n).

NMR shim coil design utilising a rapid spherical harmonic calculation method

A rapid spherical harmonic calculation method is used for the design of Nuclear Magnetic Resonance shim coils. The aim is to design each shim such that it generates a field described purely by a single spherical harmonic. By applying simulated annealing techniques, coil arrangements are produced through the optimal positioning of current-carrying circular arc conductors of rectangular cross-section. This involves minimizing the undesirable harmonies in relation to a target harmonic. The design method is flexible enough to be applied for the production of coil arrangements that generate fields consisting significantly of either zonal or tesseral harmonics. Results are presented for several coil designs which generate tesseral harmonics of degree one.

On S-asymptotically omega-periodic functions on Banach spaces and applications

This paper is devoted to the study of the class of continuous and bounded functions f : [0, infinity] -> X for which exists omega > 0 such that lim(t ->infinity) (f (t + omega) - f (t)) = 0 (in the sequel called S-asymptotically omega-periodic functions). We discuss qualitative properties and establish some relationships between this type of functions and the class of asymptotically omega-periodic functions. We also study the existence of S-asymptotically omega-periodic mild solutions of the first-order abstract Cauchy problem in Banach spaces. (C) 2008 Elsevier Inc. All rights reserved.

Existence Results for Second Order Differential Equations with Nonlocal Conditions in Banach Spaces

This work is concerned with implicit second order abstract differential equations with nonlocal conditions. Assuming that the involved operators satisfy sonic compactness properties, we establish the existence of local mild solutions, the existence of global mild solutions and the existence of asymptotically almost periodic solutions.

Controllability of Volterra-Fredholm type systems in Banach spaces

We show the results in Chalishajar [Controllability of mixed Volterra-Fredholm-type integro-differential systems in Banach space, J. Franklin Inst. 344(1) (2007) 12-21] and Chang and Chalishajar [Controllability of mixed Volterra-Fredholm type integro-differential systems in Banach space, J. Franklin Inst., doi:10.1016/j. jfranklin.2008.02.002] are only valid for ordinary differential control systems. As a result the examples provided cannot be recovered as applications of the abstract results. (C) 2008 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Endoscopic treatment of Zenker`s diyerticulum by harmonic scalpel

Background: Zenker`s diverticulum (ZD) is a rare condition with a reported prevalence of 0.01% to 0.11% in the general population. Endoscopic treatment consists of the division of the septum between the diverticulum and the esophagus, within which the cricopharyngeal muscle is contained. Diathermic monopolar current, argon plasma coagulation, and laser have been used to incise the muscular septum with satisfactory results. The main limitation of endoscopic treatment is the occurrence of complications. Perforation and hemorrhage are reported in as many as 23% and 10% of patients, respectively. Objective: The aim of this study was to use the technique of endoscopic diverticulotomy by using a harmonic scalpel in patients with ZD and to demonstrate the feasibility of using flexible and rigid devices in ZD treatment. Design: Case series study. Standard protocol was used for patient management, endoscopic procedure, and data collection. Setting: Single endoscopist demonstrating preliminary results. Patients: Five patients (4 men; median standard deviation [SD] age 69.6 +/- 9.06 years, range 59-83 years) with ZD were treated with this technique. All patients reported dysphagia and halitosis. The diagnosis was based on clinical, endoscopic, and radiographic findings. Interventions: All patients received general anesthesia and were placed in the left lateral position. A standard videogastroscope (9.8 mm) and a stiff guidewire were used to insert and achieve an adequate exposure of the ZD septum. The septum was divided using a harmonic scalpel under thin endoscope (5.2 mm) visualization through a soft diverticuloscope. Main Outcome Measurement: Feasibility of an endoscopic technique by using rigid and flexible devices to treat ZD. Results: Four patients (80%) were successfully treated in 1 session. The median SD size of the diverticulum was 3.6 +/- 0.89 cm (range 3-5 cm). Median SD procedure time was 17.33 +/- 2.33 minutes (range 15-20 minutes) in 6 procedures. No hemorrhage or perforation occurred. One patient (20%) required a second session to complete dissection of the ZD septum. All patients demonstrated improvement of dysphagia score after treatment. Limitations: Small case series design. Conclusions: Endoscopic treatment of ZD by harmonic scalpel through a soft diverticuloscope was feasible and effective in this small case series. Larger studies are warranted to further evaluate this technique.