Error estimation for numerical differential equations

Given a capable human being and a computer, it is possible to make an approximation to the solution of a nonlinear differential equation. However, under the (usually correct) assumption that the equation is analytically intractable, the result of the computation is not the exact solution; indeed it may be so far from the exact solution as to be completely useless. We are interested in the relationship between the effort expended by the human and the computer, and the duality of the computed approximation to a partial or ordinary differential equation. To be specific, we would like to think in terms of a cost-benefit analysis. The cost of the computation is a combination of the human effort and computer resources used to obtain the approximation. The benefit includes, of course, the computed approximation, but it also includes an estimate of the quality of the approximation, that is, an error estimate. It is our opinion that in computational science, as with the experimental sciences, results should always be presented with some estimate of their accuracy. In addition, however, there is another facet to error estimation: one cannot even attempt a cost-benefit analysis or efficiency comparison of methods without an error estimate to evaluate the results     

Comparison of magnetic resonance volume flow rates, angiography, and carotid Dopplers. Preliminary results

BACKGROUND AND PURPOSE: We compared the results of conventional angiography, carotid Doppler, and magnetic resonance angiography volume flow rates to determine the clinical utility of volume flow rate assessment of blood flow to the anterior circulation in patients with carotid occlusive disease. METHODS: From 11 symptomatic patients, a total of 22 extracranial carotid arteries were studied with all three techniques. The studies were independently read, and regression analysis was used to compare the measurements. RESULTS: Carotid Doppler measurements of the distal extracranial carotid arteries were proportional to the inverse of the extracranial carotid volume flow rate (r = .53, R2 = 29%, P < .01), volume flow rates were proportional to the inverse of measured percent stenosis on angiography (r = .84, R2 = 71%, P < .01), and Dopplers were proportional to angiography (r = .94, R2 = 90%, P < .01). Symptomatic Doppler systolic velocity was significantly higher (P < .002), symptomatic measured stenosis was significantly higher (P < .002), and symptomatic volume flow rate was significantly lower (P < .01) than their respective asymptomatic-side values. These preliminary observations, however, may well change once a large data set, especially one in which more patients with high-grade carotid stenosis are included, is studied. CONCLUSIONS: Assessment of carotid volume flow rates by magnetic resonance angiography quantifies flow reduction secondary to atherosclerotic occlusive disease. The easily obtained flow data add both documentation of arterial flow characteristics related to internal carotid stenosis and information regarding the adequacy of collateral pathways.     

Angiotensin II receptor type 1 mRNA is upregulated in atria of patients with end-stage heart failure

There is increasing evidence that pathological changes in the myocardium during chronic heart failure (CHF) are partly regulated through the activation of the renin-angiotensin system (RAS), an effect mediated by the angiotensin II type 1 receptor (AT1R). We examined the expression of cardiac AT1R mRNA in normal (atria, n=7; ventricle, n=3) and end-stage CHF human hearts (atria, n=8; ventricle, n=14). Tissue was snap-frozen immediately after explantation during orthotopic cardiac transplantation; control specimens were obtained from healthy donor hearts rejected for technical reasons. Northern blots of purified total mRNA from each tissue were hybridized with a random primed radiolabeled probe for the coding sequence of AT1R. Stringent conditions were used for both hybridization (5X SSC, 65 degrees C) and washing (0.5X SSC, 0.1% SDS, 65 degrees C) of the membrane. Left and right atrial tissue showed low levels of AT1R mRNA expression in the controls, with statistically significant upregulation of expression in tissue from pathological hearts; CHF atria 1.28+/-0.86 optical density (OD) units, control atria 0.56+/-0.31 OD units, P=0.05 (mean+/-s.d.). There were undetectable levels in ventricles from either control (2/2) or dilated hearts (7/7). The results were independent of the etiology of the heart failure and suggest that increased levels of atrial AT1R mRNA may occur in response to elevated atrial pressures in heart failure.     

Catalyst instabilities during the liquid phase partial oxidation of methane

A promising catalytic system for the low temperature oxidation of methane to a methanol derivative has been investigated under both batch and semi-continuous operation in two different reactor types. The system comprises of a bimetallic palladium and copper(II) chloride catalyst contained in a trifluoroacetic acid (TFA) and an aqueous phase. Methane, oxygen and a co-reductant carbon monoxide constitute the gas phase. Typical operating conditions were a temperature of 85 °C and a pressure of 83 bar.

The yields of the methyl trifluoroacetate product observed in this present work were less than those obtained in other batch autoclave works, which employed only 4 ml of liquid phase, compared with 50 ml in this study. Furthermore, an encouraging initial product formation rate of ca. 40 mol/m³ h, quickly decreased after the first hour, and came to an apparent end after only 2 h. This observation had not been reported previously.

Work performed in a semi-continuous porous tube reactor (300 ml of re-circulating liquid phase) also showed the same reaction characteristics as in the batch reactor. Thus, the deteriorating product formation rate cannot be attributed to gaseous reactant depletion (batch operation). The results suggest problems associated with catalyst instabilities, e.g. with the previously elucidated Wacker chemistry.     

Fundamental properties of radiation from a leaky mode excited on a planar transmission line

By using a specific normalization, it is shown that the TM/sub 0/ radiation field from a leaky mode that is excited by a source on any printed-circuit structure can be represented in a unified manner. The general properties of the normalized leaky-mode radiation field are examined in detail for a variety of phase and attenuation constants, as well as distances from the source, and for different types of source excitations. The normalized leaky-mode radiation field is compared to similarly normalized geometrical-optics and far-field radiation expressions to provide further insight into the nature of the radiation fields in the near- and far-field regions. The results provide a general view of leaky-mode radiation properties independent of the type of planar transmission line or background structure.