Level curve

High and low stands of sea level are directly recorded as sedimentary coastal onlap sequences and as erosional terraces. These records are complicated in regions of crustal instability, and the rate and nature of crustal deformation determines whether evidence of short-term or long-term sea-level fluctuations are preserved and how easily this evidence is interpreted. Because continental basement warps and fractures through time, and because evidence of sea level is erased by erosion, the interpretation of this evidence to produce sea-level curves for the Phanerozoic has been a subject of considerable debate. 

Fig. 18 Blood-level curves of identical doses of a drug by parenteral route (I) and by oral route as a solution (II) and as a tablet (III).

RL Oberle, GL Amidon. The influence of variable gastric emptying and intestinal transit rates on the plasma level curve of cimetidine An explanation for the double peak phenomenon. J Pharmacok Biopharm 15 529-544, 1987. 

Three correlation levels have been defined and categorized in descending order of the ability of the correlation to reflect the entire plasma drug concentration-time curve that will result from administration of a dosage form. The relationship of the entire in vitro dissolution curve to the entire plasma level curve defines the correlation. 

Level B utilizes the principles of statistical moment analysis. The mean in vitro dissolution time is compared to either the mean residence time or the mean in vivo dissolution time. Like correlation Level A, Level B utilizes all of the in vitro and in vivo data, but unlike Level A it is not a point-to-point correlation because it does not reflect the actual in vivo plasma level curve. It should also be kept in mind that there are a number of different in vivo curves that will produce similar mean residence time values, so a unique correlation is not guaranteed. 

Clinically, penamecillin has exhibited the expected activity against infections normally responding to benzylpenicillin [20-22], with the advantage that doses can be given at 8-hourly (instead of 4—6 hourly) intervals and still provide relatively smooth blood-level curves. Organisms producing penicillinases, or those normally tolerant to penicillin G, do not, of course, respond to penamecillin. 

Equations (19)-(22) offer a method for generating level curves of the rs-catastrophe surface by solving a quadratic equation, rather than the quartic g(0a) = 0. This is done by choosing a fixed rate and then calculating S for several values of a. The t i and a2 coordinates are then calculated from (22). 

When a) l/n3, the field required for ionization is E = 1/9n4, and as a> approaches l/n3 it falls to E=0.04n. These observations can be explained qualitatively in the following way. At low n, so that a> 1/n3, the microwave field induces transitions between the Stark states of the same n and m by means of the second order Stark effect. With only a first order Stark shift a state always has the same dipole moment and wavefunction, as indicated by the constant slope dW/d of the energy level curve. Thus when the field reverses, — — , the Rydberg electron s orbit does not change. With a second order Stark shift as well, the slope dW/d is not the same at E and —E, and as a result the dipole moment and wavefunction are not the same. If the field is reversed suddenly a single Stark state in the field E is projected onto several Stark states of the same n and m when E — - E. Since all the Stark states of the same n make transitions among themselves they ionize once the field is adequate to ionize one of them, the red one, at E = 1/9n4 for m n. 

It is based on the idea of electromagnetic knot, introduced in 1990 [27-29] and developed later [30-32], An electromagnetic knot is defined as a standard electromagnetic field with the property that any pair of its magnetic lines, or any pair of its electric lines, is a link with linking number i (which is a measure of the extent to which the force lines curl themselves around one another, i.e., of the helicity of the field). These lines coincide with the level curves of a pair of complex scalar fields , 0. The physical space and the complex plane are compactified to Si and S2, so that the scalars can be... 

We will admit that the total energy is finite, which implies, of course, that B and E go to zero at infinity. The simplest way for this condition to be achieved is requiring that the limit of c)j when r —> oo does not depend on the direction or, stated otherwise, that 4> takes only one value at infinity. There are certainly other ways we could, for instance, ask that is real or that its real part is a function of its imaginary part at r = oo. In this work the first and simplest possibility is explored and so, after assuming that the magnetic lines are the level curves of the scalar , it will be admitted also that is one-valued at infinity. 

This means that we take the complex number f) as a coordinate in 52.] Note that <() in (6) indicates pullback by the corresponding map and should not be mistaken for the complex conjugate of < ), which we denote as . As we see, there is a 2-form closely associated with the scalar, the level curves of which coincide with the magnetic lines. Since both 4> a and the Faraday 2-form -j Fyn,dxyl A dxv are closed, it seems natural to identify the two, up to a normalization constant factor that, for later convenience, we write as — yfa. More precisely, we assume that... 

As long as no charges are present, we can play the same game with the electric field E and a scalar field 0, the level curves of which coincide with the electric lines. In that case, if the pullback of the area 2-form in S2 by 0 is... 

To summarize this subsection, the description of the dynamics of the force lines as the level curves of two maps. S 3i -rS2, given by two complex functions topological structure, in such a way that the mere existence of a pair of such functions guarantee that the corresponding pullbacks of the area 2-form in S2 automatically obey the Maxwell s equations in empty space. 

A very important property is that the magnetic and electric lines of an electromagnetic knot are the level curves of the scalar fields 4>(r, t) and 0(r, f), respectively. Another is that the magnetic and the electric helicities are topological constants of the motion, equal to the common Hopf index of the corresponding pair of dual maps constant with dimensions of action times velocity. 

The level curves of < >0 must be orthogonal, in each point, to the level curves of 0o, since we know that electromagnetic knots are singular fields (E B = 0). This condition can be written as... 

As the knots are radiation fields, the level curves of the two scalars of a dual pair 4>, 0 must be mutually orthogonal (i.e., form two fibrations of the 3-space, orthogonal to one another). This means the they must obey the differential condition... 

We end this section with a comment referring to the Cauchy data for the scalars. In standard Maxwell theory, the Cauchy data are the eight functions A(i,6o<4M, and there is gauge invariance. In this topological model, they are the four complex functions (r, 0), 0 (r. 0), that is, eight real functions, constrained by the two conditions x V< >k) (V0 x V0 ) =0, k = 1,2, to ensure that the level curves of k will be orthogonal to those of 0. It is not necessary to prescribe the time derivatives 9o4>, 000 since they are determined by the duality conditions (138), as explained above. 

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