Some new examples for nonuniqueness of the evolution problem of harmonic maps

We find some new examples to show nonuniquence for the heat flow of harmonic maps where weak solutions satisfy the same monotonicity property.

Theory of modulational instability in Bragg gratings with quadratic nonlinearity

Modulational instability in optical Bragg gratings with a quadratic nonlinearity is studied. The electric field in such structures consists of forward and backward propagating components at the fundamental frequency and its second harmonic. Analytic continuous wave (CW) solutions are obtained, and the intricate complexity of their stability, due to the large number of equations and number of free parameters, is revealed. The stability boundaries are rich in structures and often cannot be described by a simple relationship. In most cases, the CW solutions are unstable. However, stable regions are found in the nonlinear Schrodinger equation limit, and also when the grating strength for the second harmonic is stronger than that of the first harmonic. Stable CW solutions usually require a low intensity. The analysis is confirmed by directly simulating the governing equations. The stable regions found have possible applications in second-harmonic generation and dark solitons, while the unstable regions maybe useful in the generation of ultrafast pulse trains at relatively low intensities. [S1063-651X(99)03005-6].

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.

Weak-force detection with superposed coherent states

We investigate the utility of nonclassical states of simple harmonic oscillators, particularly a superposition of coherent states, for sensitive force detection. We find that like squeezed states, a superposition of coherent states allows displacement measurements at the Heisenberg limit. Entangling many superpositions of coherent states offers a significant advantage over a single-mode superposition state with the same mean photon number.

Large amplitude folding in finely layered viscoelastic rock structures

We analyze folding phenomena in finely layered viscoelastic rock. Fine is meant in the sense that the thickness of each layer is considerably smaller than characteristic structural dimensions. For this purpose we derive constitutive relations and apply a computational simulation scheme (a finite-element based particle advection scheme; see MORESI et al., 2001) suitable for problems involving very large deformations of layered viscous and viscoelastic rocks. An algorithm for the time integration of the governing equations as well as details of the finite-element implementation is also given. We then consider buckling instabilities in a finite, rectangular domain. Embedded within this domain, parallel to the longer dimension we consider a stiff, layered plate. The domain is compressed along the layer axis by prescribing velocities along the sides. First, for the viscous limit we consider the response to a series of harmonic perturbations of the director orientation. The Fourier spectra of the initial folding velocity are compared for different viscosity ratios. Turning to the nonlinear regime we analyze viscoelastic folding histories up to 40% shortening. The effect of layering manifests itself in that appreciable buckling instabilities are obtained at much lower viscosity ratios (1:10) as is required for the buckling of isotropic plates (1:500). The wavelength induced by the initial harmonic perturbation of the director orientation seems to be persistent. In the section of the parameter space considered here elasticity seems to delay or inhibit the occurrence of a second, larger wavelength. Finally, in a linear instability analysis we undertake a brief excursion into the potential role of couple stresses on the folding process. The linear instability analysis also provides insight into the expected modes of deformation at the onset of instability, and the different regimes of behavior one might expect to observe.

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.

Temperature safety profile of laparoscopic devices: Harmonic ACE (ACE), Ligasure V (LV), and plasma trisector (PT)

Background Reports of iatrogenic thermal injuries during laparoscopic surgery using new generation vessel-sealing devices, as well as anecdotal reports of hand burn injuries during hand-assisted surgeries, have evoked questions about the temperature safety profile and the cooling properties of these instruments. Methods This study involved video recording of temperatures generated by different instruments (Harmonic ACE [ACE], Ligasure V [LV], and plasma trisector [PT]) applied according the manufacturers` pre-set settings (ACE setting 3; LV 3 bars, and the PT TR2 50W). The video camera used was the infrared Flex Cam Pro directed to three different types of swine tissue: (1) peritoneum (P), (2) mesenteric vessels (MV), and (3) liver (L). Activation and cooling temperature and time were measured for each instrument. Results The ACE device produced the highest temperatures (195.9 degrees +/- 14.5 degrees C) when applied against the peritoneum, and they were significantly higher than the other instruments (LV = 96.4 degrees +/- 4.1 degrees C, and PT = 87 degrees +/- 2.2 degrees C). The LV and PT consistently yielded temperatures that were < 100 degrees C independent of type of tissue or ""on""/ ""off"" mode. Conversely, the ACE reached temperatures higher than 200 degrees C, with a surprising surge after the instrument was deactivated. Moreover, temperatures were lower when the ACE was applied against thicker tissue (liver). The LV and PT cooling times were virtually equivalent, but the ACE required almost twice as long to cool. Conclusions The ACE increased the peak temperature after deactivation when applied against thick tissue (liver), and the other instruments inconsistently increased peak temperatures after they were turned off, requiring few seconds to cool down. Moreover, the ACE generated very high temperatures (234.5 degrees C) that could harm adjacent tissue or the surgeon`s hand on contact immediately after deactivation. With judicious use, burn injury from these instruments can be prevented during laparoscopic procedures. Because of the high temperatures generated by the ACE device, particular care should be taken when it is used during laparoscopy.