On the interpretation of random forces derived by projection operators

We study the derivation of a Langevin equation from a microscopic basis in order to elucidate the nature of the random force. We arrive at the conclusion that the consistent interpretation of the microscopic Langevin equation in terms of a stochastic differential equation (SDE) is according to I o rules. In addition, the random force is in general not Gaussian, and it is hence not completely characterized by its second moments.     

Determination of pyrene and benzo(a)pyrene residues in water by derivative synchronous solid-phase spectrofluorimetry

Benzo(a)pyrene (BaP) and pyrene (Pyr) are two polycyclic aromatic hydrocarbons (PAHs) showing native fluorescence in solution. Both compounds have been determined in water at trace levels by solid-phase spectrofluorimetry, in which BaP and Pyr are fixed on Sephadex G-25 gel and the relative fluorescence intensity is measured after the system is packed in a 1-mm silica cell. First-derivative synchronous spectra obtained at =38 nm were used to determine BaP and Pyr in the presence of other potentially interferent PAHs. The spectral characteristics of the PAHs-gel system are described, the applicable concentration ranges being 0.4–2.5 ng/ml for BaP and 0.7–4.5 ng/ml for Pyr. The relative standard deviations were 1.1% and 1.4% for BaP and Pyr respectively. The detection limits were 0.04 ng/ml for BaP and 0.1 ng/ml for Pyr. The method was applied to the analysis of both compounds in water at trace levels and a recovery study on tap, natural and sea waters was carried out.     

Potassium analysis by beta counting using a Geiger-Müller system

A technique for quantitative analysis of different soils, ceramics, feldspars and natural halide samples is presented, based on the measurement of -activities using a Geiger-Müller system. The system was calibrated with KCl, KC₈H₅O₄, KNO₃ and K₂SO₄ standards and a potassium content of 1% yields a net -count rate (background subtracted) of about 55 cph. Precision values of less than ±0.4% K can be achieved in counting times of about 4 hours. The results agree with those obtained by means of other more common analytical methods such as flame photometry, atomic absorption and -spectrometry. In comparison with these methods, this approach is direct, precise and non-destructive, because the samples do not require prior treatment.     

On the number of triple points of the tangent developable

We study the singularities of the tangent developable of a generic curve in ³ and prove that if the curve has no torsion zero points, the number of triple points of that surface must be even.     

Inequalities for tails of adapted processes with an application to Wald's lemma

In this paper we introduce a new tail probability version of Wald's lemma for expectations of randomly stopped sums of independent random variables. We also make a connection between the works of Klass^{(18, 19)} and Gundy⁽¹¹⁾ on Wald's lemma. In making the connection, we develop new Lenglart and Good Lambda inequalities comparing the tails of various types of adapted processes. As a consequence of our Good Lambda inequalities we include the following result. Let {d _i }, {e _i } be two sequences of variables adapted to the same increasing sequence of σ-fields ? _n ↗?, (e.g., ? _n =σ({d _i } _i=1 ⁿ , {E _i } _i=1 ⁿ ), and letN?∞ be a stopping time adapted to {? _n }. Then for allp>0, there exists a constant 0<C _p <∞ depending onp only, such that $$\mathop {\overline {\lim } }\limits_\lambda \lambda ^p P\left( {\mathop {\sup }\limits_{1 \leqslant n \leqslant N} \left\| {\sum\limits_{i = 1}^n {d_i } } \right\| > \lambda } \right) \leqslant C_p \mathop {\overline {\lim } }\limits_\lambda \lambda ^p P\left( {\mathop {\sup }\limits_{1 \leqslant n \leqslant N} \left\| {\sum\limits_{i = 1}^n {e_i } } \right\| > \lambda } \right)$$ This result holds when the sequences are real, tangent, and either conditionally symmetric or nonnegative, or alternatively, if {d _i } is a sequence of independent random variables and {e _i } is an independent copy of {d _i }, withN a stopping time adapted to the filtration generated by {d _i } only. Other examples include Hilbert space valued differentially subordinate conditionally symmetric martingale differences. The result is true for more general operators applied to sequences as shown by an example comparing the square function of a conditionally symmetric sequence to the maximum of its absolute partial sums.