Saturation of Berge hypergraphs

Given a graph $F$, a hypergraph is a Berge- $F$ if it can be obtained by expanding each edge in $F$ to a hyperedge containing it. A hypergraph $H$ is Berge-$F$-saturated if $H$ does not contain a subhypergraph that is a Berge-$F$, but for any edge $e\in E\left(\overline{H}\right)$, $H+e$ does. The $k$-uniform saturation number of Berge-$F$ is the minimum number of edges in a $k$-uniform Berge-$F$-saturated hypergraph on $n$ vertices. For $k=2$ this definition coincides with the classical definition of saturation for graphs. In this paper we study the saturation numbers for Berge triangles, paths, cycles, stars and matchings in $k$-uniform hypergraphs.     

Effect of initial conditions on decaying grid turbulence at low R λ

The transport equations for the second-order velocity structure functions 〈(δu)²〉 and 〈(δq)²〉 are used as a scale-by-scale budget to quantify the effect of initial conditions at low Reynolds numbers, typical of grid turbulence. The validity of these equations is first investigated via hot-wire measurements of velocity and transverse vorticity fluctuations. The transport equation for 〈(δq)²〉 is shown to be balanced at all scales, while anisotropy of the large scales leads to a significant imbalance in the equation for 〈(δu)²〉. The effect of using similarity to evaluate the transport equation is rigorously tested. This approach has the desirable benefit of requiring less extensive measurements to calculate the inhomogeneous term of the transport equation. The similarity form of the 〈(δq)²〉 equation produces nearly identical results as those obtained without the similarity assumption. In the case of the 〈(δu)²〉 equation, the similarity method forces a balance at large separation, although the imbalance due to large scale anisotropy remains. The initial conditions of the turbulence at constant R _M ≃ 10,400 (28≤ R _λ≤ 55) are changed by using three grids of different geometries. Initial conditions affect the shape and magnitude of the second- and third-order structure functions, as well as the anisotropy of the large scales. The effect of initial conditions on the scale-by-scale budget is restricted to the inhomogeneous term of the transport equations, while the dissipation term remains unaffected despite the low R _λ. Scales as small as λ are affected by the changes in initial conditions.     

Addressing the role of working memory in mathematical word-problem solving when designing intervention for struggling learners

ZDM – Mathematics Education - The focus of this article is the well documented association between low working memory capacity and difficulty with mathematical word-problem solving. We begin...     

General perspectives on model construction and evaluation for stochastic estimation,with application to a blunt trailing edge wake

The technique of stochastic estimation is examined as a specific application of linear least squares modelling. Factors that are relevant to the objectives of estimation in fluids, such as the number of sensors, the use of multiple time lags, and the strength of linear correlations, are discussed in the context of a general regression formulation. We consolidate the established findings of several research fields in order to outline clearly the potential pitfalls and reasonable performance expectations of these empirical strategies. Experimental measurements of velocity and fluctuating pressure in the wake of a blunt trailing edge body are used for quantitative illustration of key considerations for model construction and performance evaluation. It is emphasized that estimator accuracy is influenced strongly by the physical relationships among the measured variables, in addition to their correlation with the estimated variable. The evaluation of several performance metrics on an independent test set provides valuable information for the selection of a suitably complex model. In particular, “variance inflation” is interpreted as an indicator of the potential amplification of noise by a stochastic estimator.     

Aerodynamic optimization of the flat-plate leading edge for experimental studies of laminar and transitional boundary layers

This work is concerned with the design of a leading edge for a flat-plate model used to study laminar and transitional boundary layers. For this study, the flow over the complete boundary-layer model, including leading edge, flat section, and trailing-edge flap, is modeled. The effect of important geometrical features of the leading edge on the resulting pressure distribution, starting from the well-known symmetric modified super ellipse, is investigated. A minimal pressure gradient on the measurement side of the plate is achieved using an asymmetrical configuration of modified super ellipses, with a thickness ratio of 7/24. An aerodynamic shape optimization is performed to obtain a novel leading edge shape that greatly reduces the length of the non-zero pressure gradient region and the adverse pressure gradient region compared to geometries defined by ellipses. Wind tunnel testing is used to validate the numerical solutions.