Dimensional regularization continuation

Starting from the continuous chain model as the dimensionally regularized theory we write the renormalization factors as... 

In field theoretic context the method of dimensional regularization and minimal subtraction has been proposed, in [tHV72]. To the level considered here it is discussed in standard textbooks [ZJ89, Ami84], There the explicit calculation of the -factors can also be found. The method has been applied directly to the polymer system in the continuous chain limit in [Dup86a], where different versions of the approach are compared,... 

Let us now explain the main idea behind this technique. An important point is that the applicability of the integration-by-parts requires that Feynman integrals are regularized dimensionally, so that we work in continuous space-time dimension D = 4 — 2e, where e is the regularization parameter. Both, ultraviolet and infra-red divergences show up as poles in e. If dimensional regularization is adopted, one observes that the following relation ... 

Thus, for 4 > d > 2, the value of 1p(k) given by (10.4.2) is just the analytic continuation with respect to d, of the integral giving 2p k) for d < 2. As was seen above, this analytic continuation can be obtained with the help of subtractions, and the process is quite general (see Appendices G and H) in field theory, it is called dimensional regularization . 

This analytic continuation is generally called dimensional regularization... 

In Refs. [82,83] the elegant method of renormalization at zero mass and non-zero external momenta was developed, which avoids the additional renormalization conditions. Here, the vertex functions are analytically continued in the dimensional parameter d leading to a so called dimensionally regularized theory, where the cutoff Aq in 69 can be removed. 

A dimensional regularization is one of the methods of regularization, based on the idea to use the space dimensionality d as a continuous variable. Application of the method begins with calculating the integrals at d < 2 to yield forms that are subsequently extrapolated to any d (Kholodenko and Freed, 1983). 

The d-dimensional regularization is a method for extracting a non-singular contribution and based is on the idea of using d as a continuous variable. The corresponding integrals are calculated for d < 2, and, then, such forms of their representation are sought for which can be analytically extended to any d. In particular, Kholodenko and Freed (1983) used this way to derive the expression... 

Calculation of dependence of o on the conducting filler concentration is a very complicated multifactor problem, as the result depends primarily on the shape of the filler particles and their distribution in a polymer matrix. According to the nature of distribution of the constituents, the composites can be divided into matrix, statistical and structurized systems [25], In matrix systems, one of the phases is continuous for any filler concentration. In statistical systems, constituents are spread at random and do not form regular structures. In structurized systems, constituents form chainlike, flat or three-dimensional structures. 

There are also several possibilities for the temporal distribution of releases. Although some releases, such as those stemming from accidents, are best described as instantaneous release of a total amount of material (kg per event), most releases are described as rates kg/sec (point source), kg/sec-m (line source), kg/sec-m (area source). (Note here that a little dimensional analysis will often indicate whether a factor or constant in a fate model has been inadvertently omitted.) The patterns of rates over time can be quite diverse (see Figure 3). Many releases are more or less continuous and more or less uniform, such as stack emissions from a base-load power plant. Others are intermittent but fairly regular, or at least predictable, as when a coke oven is opened or a chemical vat... 

The region over which this balance is invoked is the heterogeneous porous catalyst pellet which, for the sake of simplicity, is described as a pscudohomoge-ncous substitute system with regular pore structure. This virtual replacement of the heterogeneous catalyst pellet by a fictitious continuous phase allows a convenient representation of the mass and enthalpy conservation laws in the form of differential equations. Moreover, the three-dimensional shape of the catalyst pellet is replaced by assuming a one-dimensional model... 

The electron density in a crystal, p (xyz), is a continuous function, and it can be evaluated at any point x,y,z in the unit cell by use of the Fourier series in Equations 9.1 and 9.2. It is convenient (because of the amount of computing that would otherwise be required) to confine the calculation of electron density to points on a regularly spaced three-dimensional grid, as shown in Figure 9.3, rather than try to express the entire continuous three-dimensional electron-density function. The electron-density map resulting from such a calculation consists of numbers, one at each of a series of grid points. In order to reproduce the electron density properly, these grid points should sample the unit cell at intervals of approximately one third of the resolution of the diffraction data. They are therefore typically 0.3 A apart in three dimensions for the crystal structures of small molecules where the resolution is 0.8 A. 

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