Computing steady-state solutions for a free boundary problem modeling tumor growth by Stokes equation

We consider a free boundary problem modeling tumor growth where the model equations include a diffusion equation for the nutrient concentration and the Stokes equation for the proliferation of tumor cells. For any positive radius R$R$, it is known that there exists a unique radially symmetric stationary solution. The proliferation rate μ$\mu$ and the cell-to-cell adhesiveness γ$\gamma$ are two parameters for characterizing “aggressiveness” of the tumor. We compute symmetry-breaking bifurcation branches of solutions by studying a polynomial discretization of the system. By tracking the discretized system, we numerically verified a sequence of μ/γ$\mu /\gamma$ symmetry breaking bifurcation branches. Furthermore, we study the stability of both radially symmetric and radially asymmetric stationary solutions.     

Numerically Computing Real Points on Algebraic Sets

Given a polynomial system f, a fundamental question is to determine if f has real roots. Many algorithms involving the use of infinitesimal deformations have been proposed to answer this question. In this article, we transform an approach of Rouillier, Roy, and Safey El Din, which is based on a classical optimization approach of Seidenberg, to develop a homotopy based approach for computing at least one point on each connected component of a real algebraic set. Examples are presented demonstrating the effectiveness of this parallelizable homotopy based approach.     

Categorical Skew Lattices

Categorical skew lattices are a variety of skew lattices on which the natural partial order is especially well behaved. While most skew lattices of interest are categorical, not all are. They are characterized by a countable family of forbidden subalgebras. We also consider the subclass of strictly categorical skew lattices.     

Foam droplet separation for nanoparticle synthesis

A novel approach to nanoparticle synthesis was developed whereby foam bubble bursting produced aerosol droplets, an approach patterned after the marine foam aerosol cycle. The droplets were dried to remove solvent, leaving nanometer-sized particles composed of precursor material. Nanoparticles composed of sodium chloride (mean diameter, 100 nm), phosphotungstic acid ( 55 nm), and bovine insulin ( 5–30 nm) were synthesized. Foam droplet separation can be carried out at ambient temperature and pressure. The ‘soft’ nature of the process makes it compatible with a wide range of materials.     

Electrolyte Effects on the Electrocatalytic Performance of Iridium-Based Nanoparticles for Oxygen Evolution in Rotating Disc Electrodes

Proton exchange membrane water electrolysers are very promising renewable energy conversion devices that produce hydrogen from sustainable feedstocks. These devices are mainly limited by the sluggish kinetics of the oxygen evolution reaction (OER). Ir-based nanoparticles are both reasonably active and stable for the OER in acidic media. The electrolyte composition and the pH may play a crucial role in electrocatalysis, yet they have been widely overlooked for the OER. Herein, we present a study on the effects of the composition and concentration of the electrolyte on commercial Ir black nanoparticles using concentrations of 0.05 M, 0.1 M and 0.5 M of both sulphuric and perchloric acid. The results show an important effect of the electrolyte composition on the catalytic performance of the Ir nanoparticles. The concentration of H₂SO₄ interferes on the oxidation of Ir and decreases the catalytic performance of the catalyst. HClO₄ does not show strong interferences in the electrochemistry of Ir. Higher catalytic performances are observed in HClO₄ electrolytes in comparison to H₂SO₄ with little effect of the concentration of HClO₄.     

The nature of the Extended Analog Computer

This paper defines the relationship between Rubel’s Extended Analog Computer (EAC) model and Indiana University’s implementation of it, a fundamental distinction that is necessary to understand the design, implementation, operation, and applications of the EAC. Beginning with the detailed structure of the current EAC, its method of computing is illustrated with diagrams and three applications. Next, the Δ-digraph, a labeled directed graph, is introduced to show how unconventional and conventional computers relate nature, mathematics and computer architecture. The Δ-digraph defines the paradigms of analogy and algorithm, illustrates how applications for the EAC are analogies developed by choosing the semantics for a machine configuration, and suggests how partial differential equations might be compiled to EAC configurations (a difficult problem compared to compiling a digital computer program, and one that is still not solved). The Δ-digraph also distinguishes between the EAC’s explicit functions (whose operation is defined in the structure of its components) and implicit functions (whose operation is inherent in the properties of matter and energy). Brief case studies show how industry and academia are adopting the EAC. The paper closes with topics and open questions for future research.     

Some rules forS(-2k) dipole sums?

Stieltjes conditions and the ratio test provide necessary but not sufficient conditions onS(-2k) dipole sums. If the dipole sums are accurate the associated [n, n –1] Padé approximant provides a better representation of (), the frequency-dependent dipole polarizability, than a truncated series expression and, in addition, should bound () below. It is shown how constraints on the dipole sums effect the form of the [2,1] Padé approximant and an additional constraint is derived that ensures the analyticity of the approximant on 0 < ₁. There then follows a discussion of the reliability of available literature dipole sum values for small molecules containing H, C, N and O.     

Deformation gradients for continuum mechanical analysis of atomistic simulations

We present an expression developed for calculating an atomic-scale deformation gradient within atomistic simulations. This expression is used to analyze the deformation fields for a one-dimensional atomic chain, a biaxially stretched thin film containing a surface ledge, and a FCC metal subject to indentation loading from a nanometer-scale indenter. The analyses presented show that the metric established here is consistent with the continuum mechanical concept of deformation gradient (which is known to have a zero curl for compatible deformations) in most instances. However, our metric does yield non-zero values of curl for atoms near loaded geometric inhomogeneities, such as those that form the ledges themselves and those beneath or adjacent to the indentation contact region. Also, we present expressions for higher order gradients of the deformation field and discuss the requirements for their calculation. These expressions are necessary for linking atomistic simulation results with advanced continuum mechanics theories such as strain gradient plasticity, thereby enabling fundamental, atomic-scale information to contribute to the formulation and parameterization of such theories.     

Application of Pitzer's equations on the system HCl+MnCl2+H2O at various temperatures. V.

Activity coefficients of hydrochloric acid in mixed solutions of manganous chloride at twelve ionic strengths, from 0.1 to 5.0 mole-kg^–1, were obtained from emf measurements of cells without liquid junction at five temperatures from 5 to 45°C. The data were interpreted in terms of the simple and convenient Pitzer treatment. Activity coefficients for manganous chloride in the mixtures were also derived using Pitzer's equations. Hydrochloric acid follows Harned's rule fromI=0.1 to 3.0 mole-kg^–1, as concluded by Downes, whereas quadratic terms are warranted fromI=3.5 to 5.0 mole-kg^–1. Contrary to Downes' conclusion, Harned's rule clearly does not hold true for manganous chloride at most ionic strengths.