Stability switches in a first-order complex neutral delay equation

Stability of the first-order neutral delay equation x’ (t) + ax’ (t – τ) = bx(t) + cx(t – τ) with complex coefficients is studied, by analyzing the existence of stability switches.

On some solutions of a functional equation related to the partial sums of the Riemann zeta function

In this paper, we prove that infinite-dimensional vector spaces of α-dense curves are generated by means of the functional equations f(x)+f(2x)+⋯+f(nx)=0, with n≥2, which are related to the partial sums of the Riemann zeta function. These curves α-densify a large class of compact sets of the plane for arbitrary small α, extending the known result that this holds for the cases n=2,3. Finally, we prove the existence of a family of solutions of such functional equation which has the property of quadrature in the compact that densifies, that is, the product of the length of the curve by the nth power of the density approaches the Jordan content of the compact set which the curve densifies.

Nitrate reduction at Pt(1 0 0) single crystals and preferentially oriented nanoparticles in neutral media

The electroreduction of nitrate on Pt(1 0 0) electrodes in phosphate buffer neutral solution, pH 7.2, is reported. The sensitivity of the reaction to the crystallographic order of the surface is studied through the controlled introduction of defects by using stepped surfaces with (1 0 0) terraces of different length separated by monoatomic steps, either with (1 1 1) or (1 1 0) symmetry. The results of this study show that nitrate reduction occurs mainly on the well defined (1 0 0) terraces in the potential region where H adsorption starts to decrease, allowing the nitrate anion to access the surface. Adsorbed NO has been detected as a stable intermediate in this media. An oxidation process observed at 0.8 V has been identified as leading to the formation of adsorbed NO and being responsible for a secondary reduction process observed in the subsequent negative scan. Using in situ FTIRS, ammonium was found to be the main product of nitrate reduction. This species can be oxidized at high potentials resulting in adsorbed NO and nitrate (probably with nitrite as intermediate).

Numerical resolution of Emden's equation using Adomian polynomials

Purpose: In this paper the authors aim to show the advantages of using the decomposition method introduced by Adomian to solve Emden's equation, a classical non‐linear equation that appears in the study of the thermal behaviour of a spherical cloud and of the gravitational potential of a polytropic fluid at hydrostatic equilibrium. Design/methodology/approach: In their work, the authors first review Emden's equation and its possible solutions using the Frobenius and power series methods; then, Adomian polynomials are introduced. Afterwards, Emden's equation is solved using Adomian's decomposition method and, finally, they conclude with a comparison of the solution given by Adomian's method with the solution obtained by the other methods, for certain cases where the exact solution is known. Findings: Solving Emden's equation for n in the interval [0, 5] is very interesting for several scientific applications, such as astronomy. However, the exact solution is known only for n=0, n=1 and n=5. The experiments show that Adomian's method achieves an approximate solution which overlaps with the exact solution when n=0, and that coincides with the Taylor expansion of the exact solutions for n=1 and n=5. As a result, the authors obtained quite satisfactory results from their proposal. Originality/value: The main classical methods for obtaining approximate solutions of Emden's equation have serious computational drawbacks. The authors make a new, efficient numerical implementation for solving this equation, constructing iteratively the Adomian polynomials, which leads to a solution of Emden's equation that extends the range of variation of parameter n compared to the solutions given by both the Frobenius and the power series methods.

Exploring the interfacial neutral pH region of Pt(111) electrodes

The interfacial properties of Pt(111) single crystal electrodes have been investigated in the pH range 3 < pH < 5 in order to obtain information about the acidity of electrosorbed water. Proper experimental conditions are defined to avoid local pH changes while maintaining the absence of specifically adsorbed anions and preserving the cleanliness of the solution. For this purpose, buffer solutions resulting from mixtures of NaF and HClO4 are used. Total charge curves are obtained at different pHs from the integration of the voltammetric currents in combination with CO charge displacement experiments. Analysis of the composition of the interphase as a function of the pH provides information for the understanding of the notion of interfacial pH.