Transient Stochastic Analysis of Biodegradable Contaminant Transport: First-Order Decay

A stochastic methodology was used to analyze the field-scale transport of solutes in heterogeneous aquifers with first-order biodegradation. Spectral methods and perturbation techniques were utilized to develop expressions for the field-scale effective parameters in the mean transport model. Expressions were obtained for the longitudinal and transverse macrodispersivity coefficients, and effective velocity and an effective decay parameter for statistically anisotropic, and isotropic, heterogeneous porous medium, respectively. The behavior of these parameters was described as function of time and log K correlation scale. The expressions for asymptotic values of the dispersivity coefficients and effective velocity and decay parameters were also derived for the isotropic case.     

Advective and dispersive mixing in stratified formations

The mixing process in fluid flow is presented as the bending and stretching of material lines or filaments. A mixing exponent, which quantifies their specific rate of stretching, is defined and analyzed for the case of groundwater flow though stratified formations characterized by a Gaussian autocovariance function. The analysis is performed for purely advective mixing as well as for advective-dispersive mixing. The mixing exponent was found to be proportional to the variance of hydraulic conductivity and inversely proportional to the correlation scale of hydraulic conductivity and to the pore-level dispersion coefficient.     

Primes represented by quadratic polynomials via exceptional characters

Archiv der Mathematik - We estimate the number of primes represented by a general quadratic polynomial with discriminant $$\Delta $$ , assuming that the corresponding real character is exceptional.     

Pairs of integers which are mutually squares

We derived an asymptotic formula for the number of pairs of integers which are mutually squares. Earlier results dealt with pairs of integers subject to the restriction that they are both odd, co-prime and squrefree. Here we remove all these restrictions and prove (similar to the best known one with restrictions) that the number of such pair of integers upto a large real X is asymptotic to \(\frac{{c{X^2}}}{{\log X}}\) with an absolute constant c which we give explicitly. Our error term is also compatible to the best known one.