Properties of bounded stochastic processes employed in biophysics

Abstract

Realistic stochastic modeling is increasingly requiring the use of bounded noises. In this work, properties and relationships of commonly employed bounded stochastic processes are investigated within a solid mathematical ground. Four families are object of investigation: the Sine-Wiener (SW), the Doering–Cai–Lin (DCL), the Tsallis–Stariolo–Borland (TSB), and the Kessler–Sørensen (KS) families. We address mathematical questions on existence and uniqueness of the processes defined through Stochastic Differential Equations, which often conceal non-obvious behavior, and we explore the behavior of the solutions near the boundaries of the state space. The expression of the time-dependent probability density of the Sine-Wiener noise is provided in closed form, and a close connection with the Doering–Cai–Lin noise is shown. Further relationships among the different families are explored, pathwise and in distribution. Finally, we illustrate an analogy between the Kessler–Sørensen family and Bessel processes, which allows to relate the respective local times at the boundaries.     

Biosorption of Chromium(III) by Biomass of Seaweed Sargassum sp. in a Fixed-Bed Column

This work aimed at modeling chromium biosorption using the biomass of seaweed Sargassum sp. in a fixed-bed column. The mathematical model used was obtained from the mass balance of the component in the liquid phase and in the biosorbent material. The effects of both axial dispersion in the column and the resistance to mass transfer in the solid were considered for the solution of the partial differential equations of the model, using the Galerkin method on finite elements. To represent the equilibrium data of the batch system the Langmuir isotherm were used. The chromium ion adsorption capacity of the seaweed Sargassum sp., at a temperature of 30°C and pH 3.5, was 2.61 mmol/g. The model performance was evaluated from experimental data obtained at 30°C for flow rates of 2, 6 and 8 mL/min. The parameters of the model, mass transfer and axial dispersion coefficients, were adjusted from these experimental data. The model proved adequate to describe chromium biosorption dynamics in fixed-bed columns.     

Ion implantation induced by Cu ablation at high laser fluence

High energy laser plasma-produced Cu ions have been implanted in silicon substrates placed at different distances and angles with respect to the normal to the surface of the ablated target. The implanted samples have been produced using the iodine high power Prague Asterix Laser System (PALS) using 438 nm wavelength irradiating in vacuum a Cu target. The high laser pulse energy (up to 230 J) and the short pulse duration (400 ps) produced a non-equilibrium plasma expanding mainly along the normal to the Cu target surface. Time-of-flight (TOF) technique was employed, through an electrostatic ion energy analyzer (IEA) placed along the target normal, in order to measure the ion energy, the ion charge state, the energy distribution and the charge state distribution. Ions had a Boltzmann energy distributions with an energy increasing with the charge state. At a laser fluence of the order of 6 × 10⁶ J/cm², the maximum ion energy was about 600 keV and the maximum charge state was about 27+.In order to investigate the implantation processes, Cu depth profiles have been performed with Rutherford backscattering spectrometry (RBS) of 1.5 MeV helium ions, Auger electron spectroscopy (AES) with 3 keV electron beam and 1 keV Ar sputtering ions in combination with scanning electron microscopy (SEM). Surface analysis results indicate that Cu ions are implanted within the first surface layers and that the ion penetration ranges are in agreement with the ion energy measured with IEA analysis.     

Positive and Nontrivial Solutions for the Urysohn Integral Equation

We establish new criteria for the existence of either positive or nonzero solutions of the Urysohn integral equation. We also discuss the existence of an interval of positive eigenvalues and sufficient conditions for the existence of at least a positive eigenvalue with a nonzero or positive eigenfunction for the Urysohn integral operator. Among others, we employ techniques based on fixed point index theory for compact maps, which are new for this type of equation.     

Unique satisfiability of Horn sets can be solved in nearly linear time

If a Horn set I has a single satisfying truth assignment or model then that model is said to be unique for I. The question of determining whether a unique model exists for a given Horn set I is shown to be solved in O((L)*L) time, where L is the sum of the lengths of the clauses in I and is the inverse Ackermann function. It is also shown that if LA*log (A) where A is the number of distinct proposition letters then unique satisfiability can be determined in O(L) time.