In this paper, we study the structure of the limit space of a sequence of almost Einstein manifolds, which are generalizations of Einstein manifolds. Roughly speaking, such manifolds are the initial manifolds of some normalized Ricci flows whose scalar curvatures are almost constants over space-time in the L 1 -sense, Ricci curvatures are bounded from below at the initial time. Under the non-collapsed condition, we show that the limit space of a sequence of almost Einstein manifolds has most properties which is known for the limit space of Einstein manifolds. As applications, we can apply our structure results to study the properties of Kähler manifolds.

Twitter is a widely used online social media. One important characteristic of Twitter is its real-time nature. In this paper, we investigate whether the daily number of tweets that mention Standard & Poor 500 (S&P 500) stocks is correlated with S&P 500 stock indicators (stock price and traded volume) at three different levels, from the stock market to industry sector and individual company stocks. We further apply a linear regression with exogenous input model to predict stock market indicators, using Twitter data as exogenous input. Our preliminary results demonstrate that daily number of tweets is correlated with certain stock market indicators at each level. Furthermore, it appears that Twitter is helpful to predict stock market. Specifically, at the stock market level, we find that whether S&P 500 closing price will go up or down can be predicted more accurately when including Twitter data in the model.

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ReB 3 has been synthesized and was reported to have symmetry of P6 3 /mmc [Acta Chem. Scand. 1960, 14, 733]. However, we find that this structure is not stable due to its positive formation energy. In 2009, IrB 1.35 and IrB 1.1 were synthesized and were considered to be superhard [Chem. Mater. 2007, 21, 1407; ACS Appl.Mater. Interfaces 2010, 2, 581]. Inspired by these results, we explored the possible crystal structures of ReB 3 and IrB 3 by using the developed particle swarm optimization algorithm. We predict that P 6m2-ReB 3 and Amm2-IrB 3 are the ground-state phases of ReB 3 and IrB 3 , respectively. The stability, elastic properties, and electronic structures of the predicted structures were studied by first-principles calculations. The negative calculated formation enthalpies for P 6m2-ReB 3 and P6 3 /mmc-ReB 3 indicate that they are stable and can be synthesized under ambient pressure. Their dynamical stability is confirmed by calculated phonon dispersion curves. The predicted P6 3 /mmc-ReB 3 has the highest hardness among these predicted structures. The calculated density of state shows that these predicted structures are metallic. The chemical bonding features of the predicted ReB 3 and IrB 3 were investigated by analyzing their electronic localization function.