Expanding time intervals

Feldberg and Goldstein [14] extended BDF to exponentially expanding time intervals, by using a general method for computing the coefficients for any time sequence. This also yielded good results. 

The Peaceman-Rachford ADI method is second-order with respect to time, and performs similarly to Crank-Nicolson. Indeed, Lapidus and Finder write [224, p. 246] ... is a variation of the Crank-Nicolson approximation . It is known to be unconditionally stable [225]. As with CN, ADI may show some error oscillations, as also evidenced by the fact that some habitually use expanding time intervals when employing ADI [226-231], although some of these same workers on occasion also use equal time intervals [232,233]. 

Suppose a small characteristic time interval exists such that y i changes without strongly affecting - i(y 2b 0 so that the latter may be expanded in a Taylor series as... 

This expression demonstrates use of the Einstein summation convention 6. The significance of r is made clear by examining a particle momentarily at rest in a Lorentz system. The components of the vector, transformed dx = (0, 0, 0, icdt ) and dr2 = —(1 /c dx dx = [dt )2. Thus dr is the time interval on a clock travelling with the particle and is therefore referred to as the interval of the particle s proper time or world time. The relationship between dr and an interval of time as measured in a given Lorentz system can be derived directly by expanding the equation... 

The slightly changed notation reflects the choice of states sn is the state of normal operation, and state, Vj is the state of being out of order due to short-circuiting of j number of cells. Thus, /L0o represents normal operation, and A0j represents the breakdown of j number of cells, etc. At is sufficiently small for only one cell to become short-circuited, or repaired within this time interval. For they = 0 case, Eq. (13) expands to... 

A121a cells were exposed to paclitaxel at concentrations of 0.2, 0.8, 2, and 5 ng/mL. Paclitaxel uptake by cells was able to be quantified at each time point under four treatment conditions. The lowest measured intracellular accumulation was 6.5 pg/106 cells, observed 10 min after the addition of 0.2 ng/mL paclitaxel to cells. The detected concentration in this sample corresponded to approximately 9.6 pg/mL drug in the cell lysate, which is nearly twice the LOQ. Foremost concentrations of paclitaxel, it appeared that the intracellular drug concentrations increased rapidly with exposure time and reached a plateau within 1-3 h. However, for cells exposed to the lowest concentration (0.2 ng/mL), the maximum intracellular drug concentration apparently was not achieved within 6 h, which was longest time interval investigated. In future studies, these data will be expanded to include additional concentrations and exposure times, and analyzed according to cellular pharmacokinetic models such as those published previously [12],... 

There are some simpler strategies that might do, and are easier to program. If an experiment such as double pulse or square wave voltammetry is simulated, the sharp changes occur at predictable times, and simple sequences of time intervals, such as exponentially expanding intervals, can be satisfactory, repeating the sequence at the onset of each pulse. 

In the Chronset system (Fig. 11), the driving force for the drug release was an osmotically active layer in the semipermeable vessel, which pushed the cap out off the impermeable vessel after a predetermined time interval.f The complete release of the drug, often problematic in capsular-shaped dosage forms, was ensured by an expanding layer at the bottom of the capsule body. 

The technical terms employed in the figures are as follows Cream Time—the time between the start of the mixing and the point at which the clear mixture turns creamy or cloudy and starts to expand Gel Time—the time interval between the start of mixing and the start of gelation Rise Time—the time interval between the start of mixing and the completion of expansion of the foaming mass and Tack-Free Time— the time interval between the start of mixing and the time to reach a non-sticky state. 

Suppose a small time interval Tc exists such that, during Tc, t/n-i changes without strongly affecting (j/n-al Vn J + Tc) in Eq. (E.IO). In the limit Te —+ 0 we may then expand the left. side of Eq. (E.IO) in a Taylor series according to... 

Expanding the proportional terms using P(t) = ke(t) and the integral term using the Euler integration algorithm valid for small time intervals. At = t - t -i ... 

Consider a short clock-time interval dt. In this time interval a small amount of feed, of volume fb dt, enters the reactor, expands by the factor 6V to volume Svfo dt, and forces an equal volume out of the reactor. In addition, during the same time interval the volume expansion factor pertaining to the contents of the reactor increases by some amount d5w and this causes the whole body of fluid in the reactor to expand from total volume V to V(5v+ d5v)/5v The extra volume Vd6 (5 thus generated is therefore also forced out of the reactor. Combining these two effects, we see that the volume exiting the reactor in time dt is dV = 6vfo dt + Vd(6v/6v). The outlet flow rate, f = dV/dt, is then ... 

Pharmacoinvasive recanalization with fibrinolytic therapy to induce initial reperfusion, conjunctive pharmacological therapy with anticoagulants to enhance the rapidity and extent of lysis and subsequent timely (within 12-24 hours) PCI to prevent reocclusion and reinfarction to eliminate underlying anatomical obstruction and thrombus are particularly promising for most patients with STEMI (30,73-77). This pharmacoinvasive approach for treatment of asymptomatic patients following thrombolysis should expand the interval during which PCI can be effective well beyond the 90-minute interval within which optimal benefits are seen with primary PCI. If so, it would be attractive for patients presenting to... 

Nuclear magnetic resonance (NMR) spectroscopy, arguably the most powerful technique in chemistry, is a relatively insensitive method, but in the short time interval between the appearance of the first and second editions of the Encyclopedia of Analytical Sciences, considerable improvements have been made. Consequently, the range of applications that now become possible is greatly expanded as is illustrated here for the two isotopes of hydrogen, tritium ( H) and deuterium ( H) further developments are also anticipated. 

The statement that a gas consists of a large number of particles is consistent with what we know about the size of individual molecules. And the postulate that the molecules are in constant random motion is consistent with the observation that gases readily expand to fill their containers. The postulates are also consistent with what we said at the outset of this chapter about the origin of gas pressure. The molecules in a gas exert force when they collide with the wall of the container, and this accounts for the relationship between pressure and volume. As the volume of the container increases, the walls move farther apart and so each molecule will strike the wall less often. Thus, each molecule will exert less force on the wall in a given time interval, reducing the pressure. So the kinetic theory is consistent with the inverse relationship between P and Vthat we ve seen in the gas law equation. 

---