For SO2 the first coefficient, G 0), is 0.75% per 10 μg/m3 for the single pollutant fit and 0.73% per 10 μg/m3 for the fit with two pollutants; the last coefficient, G(1096) in this case, is 1.28 days/ μg/m) for the single pollutant fit and 0.3 days/ μg/m) for the fit with two pollutants.
For PM10 the first coefficient, G 0), is 0.21% per 10 μg/m3 for the single pollutant fit and 0.19% per 10 μg/m3 for the fit with two pollutants; the last coefficient, G(1096) in this case, multiplied by Tday = 1 day, is 3.14 days/ μg/m) for the single pollutant fit and 1.74 days/ μg/m) for the fit with two pollutants.
Approximations to two terms are found in each region, and approximate value for all Fourier coefficients also found, and compared very favourably with numerical curve fits for the first coefficients given in reference [1].
In particular, if we let r → 1- in the growth estimate, it gives the bound |a2| ≤ B1/2 for the second coefficient of functions in K s.
Remark 1 By setting μ = 0 in Theorem 1, we get the sharp estimate for the third coefficient of functions in K s : ∣ a 3 ∣ ≤ 1 ∕ 3 + ( B 1 ∕ 3 ) max ( 1, ∣ B 2 ∣ ∕ B 1 ), while the limiting case μ → ∞ gives the sharp estimate |a2| ≤ B1/2.
Recently, the expansion of the first Melnikov function appearing by perturbing an integrable and reversible system with a homoclinic loop passing through a nilpotent singular point was investigated in [8], and the authors got the formulas for computing the first coefficients of the expansion.
For the present application, especially the second coefficient c20 is important (supplementary material Fig. S3A,B).
Experimental scaling laws for the second virial coefficient (A2), the third virial coefficient (A3), the radius of gyration (⟨R2G⟩1/2) and the intrinsic viscosity were determined.
We shall see that while for the first few coefficients (L_m) with m odd the vertex terms cancel, they do not for m even.
For the first experiment, coefficients of variation were calculated for the mean signal intensity for the 6 slides (i.e, two dyes × three biological replicates).
