A solution of the DGLAP equation for gluon at low x

We obtain a solution of the DGLAP equation for the gluon at low x first by expanding the gluon in a Taylor series and then using the method of characteristics. We test its validity by comparing it with that of Glück, Reya and Vogt. The convergence criteria of the approximation used are also discussed. We also calculate εF ₂(x,Q)²/ε In Q ² using its approximate relations with the gluon distribution at low x. The predictions are then compared with the HERA data.     

Enhancing the enzymatic hydrolysis of cellulosic materials using simultaneous ball milling

One of the limiting factors restricting the effective and efficient bioconversion of softwood-derived lignocellulosic residues is the recalcitrance of the substrate following pretreatment. Consequently, the ensuing enzymatic process requires relatively high enzyme loadings to produce monomeric carbohydrates that are readily fermentable by ethanologenic microorganisms. In an attempt to circumvent the need for larger enzyme loadings, a simultaneous physical and enzymatic hydrolysis treatment was evaluated. A ball-mill reactor was used as the digestion vessel, and the extent and rate of hydrolysis were monitored. Concurrently, enzyme adsorption profiles and the rate of conversion during the course of hydrolysis were monitored. α-Cellulose, employed as a model substrate, and SO₂-impregnated steam-exploded Douglas-fir wood chips were assessed as the cellulosic substrates. The softwood-derived substrate was further posttreated with water and hot alkaline hydrogen peroxide to remove >90% of the original lignin. Experiments at different reaction conditions were evaluated, including substrate concentration, enzyme loading, reaction volumes, and number of ball beads employed during mechanical milling. It was apparent that the best conditions for the enzymatic hydrolysis of α-cellulose were attained using a higher number of beads, while the presence of air-liquid interface did not seem to affect the rate of saccharification. Similarly, when employing the lignocellulosic substrate, up to 100% hydrolysis could be achieved with a minimum enzyme loading (10 filter paper units/g of cellulose), at lower substrate concentrations and with a greater number of reaction beads during milling. It was apparent that the combined strategy of simultaneous ball milling and enzymatic hydrolysis could improve the rate of saccharification and/or reduce the enzyme loading required to attain total hydrolysis of the carbohydrate moieties.     

Hydrodynamically limited precision of gradient techniques in flow injection analysis

Based on the law of error propagation, a general expression is derived to study theoretically the hydrodynamically limited precision associated with each single element of fluid in a concentration profile in flow injection analysis. Convolution of the injection and residence time distribution functions is used to obtain response functions for a homogeneously stirred mixing chamber and for a straight capillary tube. The effects of stochastic variations on overall precision by sample introduction, pumping and timing are elucidated and compared with experimental findings. Resulting practical implications for the use of these two dispersing elements are outlined.     

Symmetry reductions and exact solutions of shallow water wave equations

In this paper we study symmetry reductions and exact solutions of the shallow water wave (SWW) equation $$u_{xxxt} + \alpha u_x u_{xt} + \beta u_t u_{xx} - u_{xt} - u_{xx} = 0,$$ whereα andβ are arbitrary, nonzero, constants, which is derivable using the so-called Boussinesq approximation. Two special cases of this equation, or the equivalent nonlocal equation obtained by settingu _x =U, have been discussed in the literature. The caseα=2β was discussed by Ablowitz, Kaup, Newell and Segur (Stud. Appl. Math.,53 (1974), 249), who showed that this case was solvable by inverse scattering through a second-order linear problem. This case and the caseα=β were studied by Hirota and Satsuma (J. Phys. Soc. Japan,40 (1976), 611) using Hirota's bi-linear technique. Further, the caseα=β is solvable by inverse scattering through a third-order linear problem. In this paper, a catalogue of symmetry reductions is obtained using the classical Lie method and the nonclassical method due to Bluman and Cole (J. Math. Mech,18 (1969), 1025). The classical Lie method yields symmetry reductions of (1) expressible in terms of the first, third and fifth Painlevé transcendents and Weierstrass elliptic functions. The nonclassical method yields a plethora of exact solutions of (1) withα=β which possess a rich variety of qualitative behaviours. These solutions all like a two-soliton solution fort < 0 but differ radically fort > 0 and may be viewed as a nonlinear superposition of two solitons, one travelling to the left with arbitrary speed and the other to the right with equal and opposite speed. These families of solutions have important implications with regard to the numerical analysis of SWW and suggests that solving (1) numerically could pose some fundamental difficulties. In particular, one would not be able to distinguish the solutions in an initial-value problem since an exponentially small change in the initial conditions can result in completely different qualitative behaviours. We compare the two-soliton solutions obtained using the nonclassical method to those obtained using the singular manifold method and Hirota's bi-linear method. Further, we show that there is an analogous nonlinear superposition of solutions for two (2+1)dimensional generalisations of the SWW Equation (1) withα=β. This yields solutions expressible as the sum of two solutions of the Korteweg-de Vries equation.     

A vibrationally compensated far infrared laser interferometer for plasma density measurements

A modulated far-infrared laser interferometer which is presently operating on the PDX experiment at Princeton is described. The interferometer geometry permits the characterization of inside D, outside D as well as circular discharges. To achieve this versatility, a titanium corner cube reflector, mounted inside the PDX vacuum vessel is used in conjunction with a second visible wavelength interferometer for vibration corrections. In addition, the use of room temperature quasi-optical Schottky diodes in the far-infrared interferometer is reported. The minimum detectable line average density of the system is 5×10¹¹ cm^–3.