On path lengths modulo three

We show that between any two nodes of a cubic, planar, three-connected graph there are three paths whose lengths are 0, 1, and 2 modulo 3, respectively. The proof is by a rather extensive case analysis. Counterexamples show that all three hypotheses (i.e., planarity, degree-three, and three-connectivity) are necessary.     

Third‐order sensitivity analysis for robust aerodynamic design using continuous adjoint

Robust design problems in aerodynamics are associated with the design variables, which control the shape of an aerodynamic body, and also with the so‐called environmental variables, which account for uncertainties. In this kind of problems, the set of design variables, which leads to optimal performance, taking into account possible variations in the environmental variables, is sought. One of the possible ways to solve this problem is by means of the second‐order second‐moment approach, which requires first‐order and second‐order derivatives of the objective function with respect to the environmental variables. Should the minimization problem be solved using a gradient‐based method, algorithms for the computation of up to third‐order sensitivity derivatives (twice with respect to the environmental variables and once with respect to the shape controlling design variables) must be devised. In this paper, a combination of the continuous adjoint variable method and direct differentiation to compute the third‐order sensitivities is proposed. This is shown to be the most efficient among all alternative methods provided that the environmental variables are much less than the design ones. Apart from presenting the method formulation, this paper focuses on the assessment of the so‐computed up‐to third‐order mixed derivatives through comparison with costly finite‐difference schemes. To this end, the robust design of a two‐dimensional duct is performed. Then, using the validated method, the robust design of a two‐dimensional cascade airfoil is demonstrated. Although both cases are handled as inverse design problems, the method can be extended to other objective functions or three‐dimensional problems in a straightforward manner. Copyright © 2012 John Wiley & Sons, Ltd.     

Hidden Markov models revealing the stress field underlying the earthquake generation

The application of the hidden Markov models (HMMs) is attempted for revealing key features for the earthquake generation which are not accessible to direct observation. Considering that the states of the HMM correspond to levels of the stress field, our objective is to identify these states. The observations are considered after grouping earthquake magnitudes and the cases of different number of states are examined. The problems of HMMs theory are solved and the ensuing HMMs are compared on the basis of Akaike and Bayesian information criteria. A new insight on the evaluation of future seismic hazard is given by calculating the mean number of steps for the first visit to a particular state, along with the respective variance. We further calculate an estimator of the mean number of steps for the first visit to a particular state and we construct its confidence interval. Additionally, a second approach to the problem is followed by assuming a different determination of observations. The HMMs applied to both approaches, contribute significantly to seismic hazard assessment via revealing the number of the stress levels as well as the way in which these levels are associated with certain earthquake occurrence.     

Continuous and discrete adjoint approaches for aerodynamic shape optimization with low Mach number preconditioning

Discrete and continuous adjoint approaches for use in aerodynamic shape optimization problems at all flow speeds are developed and assessed. They are based on the Navier–Stokes equations with low Mach number preconditioning. By alleviating the large disparity between acoustic waves and fluid speeds, the preconditioned flow and adjoint equations are numerically solved with affordable CPU cost, even at the so‐called incompressible flow conditions. Either by employing the adjoint to the preconditioned flow equations or by preconditioning the adjoint to the ‘standard’ flow equations (under certain conditions the two formulations become equivalent, as proved in this paper), efficient optimization methods with reasonable cost per optimization cycle, even at very low Mach numbers, are derived. During the mathematical development, a couple of assumptions are made which are proved to be harmless to the accuracy in the computed gradients and the effectiveness of the optimization method. The proposed approaches are validated in inviscid and viscous flows in external aerodynamics and turbomachinery flows at various Mach numbers. Copyright © 2007 John Wiley & Sons, Ltd.     

A rotating disc electrode study of oxygen reduction at platinised nickel and cobalt coatings

Platinized nickel and cobalt coatings, Pt(Ni) and Pt(Co), have been prepared on glassy carbon, GC, rotating disc electrode substrates by a two-step room temperature procedure that involved the electrodeposition of nickel and cobalt layers and their spontaneous partial replacement by platinum (“transmetalation”) when immersed into a chloroplatinic acid solution. By tuning the quantity of initially deposited nickel and cobalt, Pt(Ni) and Pt(Co) bimetallic coatings having a 26% atom Ni and 30% atom Co composition have been prepared. For both materials typical Pt surface electrochemistry was recorded during fast voltammetry in deaerated acid, pointing to the existence of a continuous Pt skin over a Pt–Ni and Pt–Co core. Oxygen reduction at the Pt(Ni)/GC and Pt(Co)/GC electrodes was studied by means of steady-state voltammetry at a rotating disc electrode and the construction of Tafel plots from corresponding voltammetric data. It was found that, when the initial potential of the voltammetric sweep allowed the formation of a complete Pt oxide monolayer, then oxygen reduction was hindered for low overpotentials at Pt(Ni) and Pt(Co), compared to pure bulk Pt. On the other hand, when the initial potential was less positive (thus leading to the formation of a fraction of surface oxide monolayer) the presence of Ni and Co enhanced the kinetics of oxygen reduction. The former behaviour is attributed to a decrease in oxide reduction ability of Pt in the presence of Ni and Co, while the latter to an increase in dissociative oxygen chemisorption due to Ni and Co.     

Effect of multivalent cation substitution on the capacity fading of lithium manganese spinel cathodes

The introduction of Co, Al, and Ti into LiMn₂O₄ cathode powders has been studied in this work. The substitution of 25% Mn with either of these metals is expected to improve the cycle ability of the cathode, while the Jahn–Teller effect is suppressed. The solid-state reaction method is used in order to produce the cathode materials, which are prepared at 750 or 800 °C. The materials are characterized by X-ray data and, electrochemically, by galvanostatic cycling at the 4-V range. Their capacity fading behavior is evaluated, and a further investigation into cycling at a higher voltage range is also presented.