Homogeneous kernel

The fragmentation equation can be scaled into the following time-invariant form for the case of homogeneous kernels ... 

Considerable attention has been devoted to aggregation processes, the physics of which is well described by the Smoluchowski equation with so-called homogeneous kernels (i.e., with Kaitaj=a.xKitj). Here, parameter A is the degree of homogeneity. To describe the size distribution of aggregates in a finite system, functions have been proposed [67] in the form... 

As mentioned previously, the most relevant to polymer science seems to be the product homogeneous kernel Kitj=(ij)a>, (the homogeneity degree A=2co). According to van Dongen and Ernst [69], exponents in the scaling relation given by Eq. (135) are... 

The results just presented show that steady state particle size distributions are possible for a wide variety of coagulation-fragmentation conditions. In the case of homogeneous kernels the shape of these distributions is a slightly distorted Gaussian, and it approaches experimental results. In spite of the fact that the analytical solutions derived in this work are necessarily restricted, it can be expected that solutions with shapes farther from normal distributions are possible once the fragmentation and aggregation kernels are modified. 

We now assume that H h) = /z", a form that is strictly true only for a homogeneous kernel of degree m however, in view of the ad hoc nature of assumption (6.2.5), we must regard this form of if as a power-law approximation for the dynamic variation of the scaling particle size h t). The differential equation (6.2.7) is of course analytically solvable so that the estimation of is more readily facilitated. 

Muralidhar, R. and D. Ramkrishna Inverse Problems in Agglomeration Kinetics-II. Non-homogeneous Kernels, J. Colloid Interf Scl 131, 503-513 (1989). 

Several works over the past decade or so (Ziff, 1991 Ziff and McGrady, 1986 McGrady and Ziff, 1987,1988 and Williams, 1990) have addressed the behavior of systems with specified breakup kernels. Certain specific forms for the breakup kernels lead to analytical solutions for the cluster size distribution. For example, Ziff (1991) obtained explicit forms of the size distribution for homogeneous breakup kernels of the form... 

Homogeneous breakage kernels with positive homogeneity index (A>0) produce self-similar size distributions at long times... 

Scaling solutions, as in the case of fragmentation, are possible if the kernel K(x, y) is homogeneous, which requires that... 

Hence, the decision to use a heated substrate with simultaneous evaporation of the component metals as an aid to homogenization requires consideration of whether or not it might have an adverse effect, i.e., causing preferential nucleation of one component, which interdiffusion may not be able to remedy. It was believed (60) that in preparing Pd-Rh alloys by simultaneous deposition on a substrate at 400°C, rhodium nucleated preferentially and that crystallites grew by the addition of palladium (and rhodium) atoms. The diffusion of palladium atoms into this kernel formed a phase with 88 =t 5% Rh (phase II). The outer shell of the crystallite, phase I, was in effect a solid solution deficient in rhodium compared with the overall film composition, and the Rh content of phase I therefore increased as the Rh flux was increased. 

The solution must satisfy a homogeneous Fredholm integral of the second kind with Hermitian kernel ... 

Almond (200 g) apricot kernel (200 g) peach kernel (200 g) apple seed (100 g) distilled water (1000 mL) ethyl acetate (1000 mL) acetone (500 mL) homogenizer one 100 mL Buchi funnel filter paper. 

The P. dulcis (almond) kernels were soaked in distilled water for 2 h, peeled, air-dried and then powdered in cold (0 °C) ethyl acetate with a homogenizer. The powder was defatted by a further three washes with ethyl acetate and two washes with acetone and then stored at 4 °C. 

Ripened Ume fruit (P. mume) was taken and the fleshy cover was removed to obtain the seeds. The upper layer of the seeds was cracked with a hammer to give the soft kernels inside. Those kernels were collected and crushed in the process homogenizer at 4 °C, with 10 mM potassium phosphate buffer, pH 6.0, to give a milky suspension. The suspension was filtered through four layers of cheesecloth to remove the insoluble part. The HNL activity shown by P. mume extract was 6.9 U mg in the milky suspension. ... 

This picture is usually known as the heterogeneous scenario. The distribution of relaxation times g (In r) can be obtained from < (t) by means of inverse Laplace transformation methods (see, e.g. [158] and references therein) and for P=0.5 it has an exact analytical form. It is noteworthy that if this scenario is not correct, i.e. if the integral kernel, exp(-t/r), is conceptually inappropriate, g(ln r) becomes physically meaningless. The other extreme picture, the homogeneous scenario, considers that all the particles in the system relax identically but by an intrinsically non-exponential process. 

Recall from Section 1.5 that any function in the kernel of the Laplacian (on any space of functions) is called a harmonic function. In other words, a function f is harmonic if V / = 0. The harmonic functions in the example just above are the harmonic homogeneous polynomials of degree two. We call this vector space In Exercise 2,23 we invite the reader to check that the following set is a basis of H/ ... 

In Section 7.1 we will use this characterization of homogeneous harmonic polynomials as a kernel of a linear transformation (along with the Fundamental Theorem of Linear Algebra, Proposition 2.5) to calculate the dimensions of the spaces of the spherical harmonics. 

All of the concepts of this section — kernel, image. Fundamental Theorem, homogeneous harmonic polynomials and isomorphisms — come up repeatedly in the rest of the text. 

In a 12-1. round-bottom flask are placed 1500 g. of corn cobs (ground to about the size of corn kernels), 5 1. of 10 per cent sulfuric acid and 2 kg. of salt. The flask is shaken in order to secure a homogeneous mixture and is then connected with an upright tube water condenser and return tube as shown in Fig. 4 (p. 50). Heat is applied from a ring burner, the flame being adjusted so that the liquid distils at a rapid rate. 

Exemplifying the convenience of the fractional approach, we address the imposition of boundary value problems on the fractional diffusion equation which was demonstrated in Ref. 62. In this force-fiee case for which the kernel, Eq. (27), takes on the homogeneous form K(x,x u) = uw(u) (2( c — x ) — <5(x))/(l - w(u)), one can apply the method of images in order to construct the solution [12]. 

It should be appreciated here that the singularities at r =a and r =b makes the kernel of this equation practically separable of rank two. Fredholm theory shows that a solution to the homogeneous equation (23) requires a singular reciprocal kernel. Thus it holds that... 

This compound was synthesized by Dr. Nowacki by reacting 11+C-IAA-imidazole with myo-inositol (40). Application of these labeled compounds to corn kernels, followed immediately by homogenization of the tissue in acetone permitted us to determine the amounts of each constiuent in the kernel by the isotope dilution method of Rittenberg and Foster (41). An extension of this method, whereby the kernels are incubated for varying periods of time after application of the isotopically labeled compound permits determination of the "turnover" of the pool. Such data are shown in... 

When the linear isotropic dielectric medium used in the standard model is replaced with a linear homogeneous medium with Green kernel Ge, and when the charge distribution is entirely supported inside the cavity, the reaction potential inside the cavity still has a simple integral representation ... 

Only E3 is new and rather unusual. It describes the overpopulation of ion pairs [D+ A-], produced by electron transfer in closely situated products of ion recombination in the bulk, excited before their separation. Since the density of the correlated pairs is enhanced through bulk recombination, the ionization becomes faster than at the entirely homogeneous distribution of the reactants. The terms involving S3 account for this effect. However, unlike other kernels, E3 vanishes in the lowest-order approximation with respect to 7o- Therefore,... 

---