Mean crossing

Pimblott and Mozunider consider each ionization subsequent to the first as a random walk of the progenitor electron with probability q = (mean cross sections of ionization and excitation. F(i) is then given by the Bernoulli distribution... 

In a cube of side X, the volume of free space is eX3 so that the mean cross-sectional area for flow is the free volume divided by the height, or eX2. The volume flowrate through this cube is ucX2, so that the average linear velocity through the pores, u, is given by ... 

Here, V is the volume of the hydrocarbon chain(s) of the surfactant, the mean cross-sectional (effective) headgroup surface area, and 4 is the length of the hydrocarbon tail in the all-trans configuration. Surfactants with Pcone-shaped and form spherical micelles. For l/3truncated-cone-shaped, resulting in wormlike micelles (the term wormlike is preferred over rodlike to highlight the highly dynamic nature of these micelles). 

This calculation is for spherical micelles, but a similar calculation could be used to obtain estimates of salt concentrations for ionic wormlike micelles. Such salt concentrations for wormlike micelles are expected to be increased in comparison to spherical micelles. In fact, the addition of counterions or a sufficient increase in surfactant concentration often leads to a transition from spherical micelles to wormlike micelles. As the free counterion concentration in solution increases, so does the counterion binding. As a result, electrostatic repulsion between the charged head-groups is increasingly shielded and the mean cross-sectional (effective) headgroup... 

The more gently sloping lines correspond to a state of expansion in which the mean cross-section is governed by the characteristic active group, and the curves naturally differ according to the nature of this group. 

In this reach, on the day of the spill, the Maipo River carried a discharge of 60 m /s, at a mean depth of 2 m, with a mean cross-sectional area of 60 m2 and a stream slope of 0.002. The longitudinal dispersion coefficient is known to be 20 m2/s. 

The volume distribution function (r) represents the volumetric uptake in a unit interval of pore radii, irrespective of the variation in the number or the length of the pores. When (r) is divided by nr, the mean cross-... 

Radiation cross-linking of polyethylene requires considerably less overall energy and less space, and is faster, more efficient, and environmentally more acceptable. Chemically cross-linked PE contains chemicals, which are by-products of the curing system. These often have adverse effects on the dielectric properties and, in some cases, are simply not acceptable. The disadvantage of electron beam cross-linking is a more or less nonuniform dose distribution. This can happen particularly in thicker objects due to intrinsic dose-depth profiles of electron beams. Another problem can be a nonuniformity of rotation of cylindrical objects as they traverse a scanned electron beam. However, the mechanical properties often depend on the mean cross-link density. ... 

Further calculations with respect to the loading capacity on the surface can be performed knowing the surface area of the support (SA= 999 m2/g) and the mean cross-sectional area of the Al(acac)3 complex (0.60 nm ) [8] A full monolayer would therefore correspond to an Al-loading of 2.7 mmol/g. However, the actual maximal Al-loading obtained here is 0.672 mmol/g, which means only 25% of the monolayer capacity. When comparing this value to the silanol number of 1.89 OH mmol/g, it can be concluded that 36% of the available OH-groups on the surface have reacted with the Al-complex. 

A theoretical model predicting the shape-structure relationship between the monomeric units and their aggregates was developed by Israelachvili and was based on statistical mechanics of phospholipids.23 This model predicts the type of the aggregate formed on the basis of the packing parameter (P), which relates the volume of the molecule (V) to its length (1) and to the mean cross-sectional (effective) head group surface area (a) ... 

However, using rj as the only parameter in the optimization can often be misleading, since the maximum degree of conversion is not only determined by the number of reacting hydroxyl groups, but also by the mean cross-sectional area (A.J of the reacted group (sterical hindrance effects). The maximum number of bonded groups on the surface can be estimated as... 

Incorporating the correction factor for void volume, the am would amount about 2.2/nm2. For spherical molecules, a fast approximation of the mean cross-sectional area is given by the formula of Emmett and Brunauer 56... 

To determine the pressure and velocity distributions, we must relate mean cross-sectional flow velocity (v) to the pressure gradient. This is done by means of Eq. 4.11 for capillary tubes and Eq. 4.22 for packed columns. To work with these equations, we must replace the overall pressure gradient Ap/L by the local pressure gradient - dpldx, where the negative sign arises because pressure decreases as one moves along the positive flow coordinate x. With this substitution, Eq. 4.11 can be rearranged to... 

Assuming a grain diameter of O.l/i, a cloud corresponding to % v 1000 in the Watson-Salpeter model, and a mean cross-section of 10 18 cm2 the mean abundance of complex molecules has been calculated as an equilibrium between production by photochemical surface reactions and destruction (see IV. E) by absorption of UV photons (Breuer, 1971). For a hydrogen density of nH =... 

We see that head-on collisions are most effective in bringing about transition in region 1, and the probability depends on Rx rather weakly, only via Aee (equation 52). The mean cross section al corresponding to equation (56)... 

The plot of F(y) is given in Figure 5.6. It is seen from equation (59) that most effective are the collisions with nonzero (but still small) impact parameters, and the probability strongly depends on R2, decreasing with increasing R2. The mean cross section corresponding to equation (59) is... 

Consider a tree with a leaf area index of 6 and a crown diameter of 6 m. The trunk is 3 m tall, has a mean cross-sectional area of 0.10 m2 of which 5% is xylem tissue, and varies from an average water potential along its length of -0.1 MPa at dawn to -0.5 MPa in the steady state during the daytime. 

The value of A is determined as follows. Under standard atmospheric conditions the mean free path of molecules by the order of magnitude is /free l/(lVaOg), where NA = 2.68 x 1019cm-3 is the number of molecules per cubic centimeter and erg 10 Mcm2 is the mean cross section of molecules collision. For pressure in the range 1-3 bar this gives /free 10 6cm. 

In the Cauchy limit a 1, we can find an approximate result for the mean crossing time as a function of noise strength D. To this end, we start with the... 

To obtain an approximate expression for the mean crossing time, we follow the standard steps [92] and for large values of 1/D make the constant flux approximation assuming that the flux across the barrier is a constant, jo, corresponding to the existence of a stationary solution Pst(x). By integration of the continuity equation, it then follows that equation (112) is satisfied, and Tc = 1 /jo- Due to the low noise strength, we also assume that for all relevant times the normalization = 1 obtains. 

We observe from numerical simulations an exponential decrease of the survival probability Sf(t) in the potential well, at the bottom of which we initialize the process. Moreover, we find that the mean crossing time assumes the scaled form (114) with scaling exponent p being approximately constant in the range 1 < a // 1.6, followed by an increase before the apparent divergence at a = 2, that leads back to the exponential form of the Brownian case, Eq. (113). An analytic calculation in the Cauchy limit a = 1 reproduces, consistently with the constant flux approximation commonly applied in the Brownian case, the scaling Tc 1/D, and, within a few percent error, the numerical value of the mean crossing time Tc. 

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