This paper discusses the problem that making the air flowing into the room maximized by rotating the windows. Two models are built for rotary windows with 2 and 3 sashes respectively. Based on these models and the assumption of wind blowing from different directions with equal chance, the optimal way of opening windows is calculated by using Pascal programs on computer. Finally, the optimal design for window ventilation is found via comparing maximum value of wind quantity in two situations.

The stochastic gradient (SG) method can quickly solve a problem with a large number of components in the objective, or a stochastic optimization problem, to a moderate accuracy. The block coordinate descent/update (BCD) method, on the other hand, can quickly solve problems with multiple (blocks of) variables. This paper introduces a method that combines the great features of SG and BCD for problems with many components in the objective and with multiple (blocks of) variables. This paper proposes a block SG (BSG) method for both convex and nonconvex programs. BSG generalizes SG by updating all the blocks of variables in the Gauss–Seidel type (updating the current block depends on the previously updated block), in either a fixed or randomly shuffled order. Although BSG has slightly more work at each iteration, it typically outperforms SG because of BSG’s Gauss–Seidel updates and larger step sizes, the latter of which are determined by the smaller per-block Lipschitz constants. The convergence of BSG is established for both convex and nonconvex cases. In the convex case, BSG has the same order of convergence rate as SG. In the nonconvex case, its convergence is established in terms of the expected violation of a first-order optimality condition. In both cases our analysis is nontrivial since the typical unbiasedness assumption no longer holds. BSG is numerically evaluated on the following problems: stochastic least squares and logistic regression, which are convex, and low-rank tensor recovery and bilinear logistic regression, which are nonconvex. On the convex problems, BSG performed significantly better than SG. On the nonconvex problems, BSG significantly outperformed the deterministic BCD method because the latter tends to stagnate early near local minimizers. Overall, BSG inherits the benefits of both SG approximation and block coordinate updates and is especially useful for solving large-scale nonconvex problems.

We develop foundational theory for the Laplacian flow for closed G2 structures which will be essential for future study. (1). We prove Shi-type derivative estimates for the Riemann curvature tensor Rm and torsion tensor T along the flow, i.e. that a bound on Λ(x,t)=(|∇T(x,t)|^2g(t)+|Rm(x,t)|^2g(t))^{1/2} will imply bounds on all covariant derivatives of Rm and T. (2). We show that Λ(x,t) will blow up at a finite-time singularity, so the flow will exist as long as Λ(x,t) remains bounded. (3). We give a new proof of forward uniqueness and prove backward uniqueness of the flow, and give some applications. (4). We prove a compactness theorem for the flow and use it to strengthen our long time existence result from (2) to show that the flow will exist as long as the velocity of the flow remains bounded. (5). Finally, we study soliton solutions of the Laplacian flow.

Japan’s lunar probe satellite “Kaguya”, China’s lunar probe satellite “Chang’ e-1 lunar probe” left the earth on September 14th, 2007 and October 24th, 2007, began the exploration to the moon. The launching of the China’s lunar satellite is the proud to the global Chinese. However, at the same time as we are proud of the event, we must admit that the pictures published which were sent back by Japanese are clearer and more beautiful than Chinese. While we are saying that the Japan has a better technology, maybe we also need to say that the Japanese can hold better chance than Chinese when taking the pictures. The following two pictures were sent back by the satellite. Picture one is sent back by Japan’s satellite, Picture two is sent back by China’s. It can be seen that there is a long way for China if we want to become stronger at science technology. We also hope that we can have some contribution in the prosperity of China now and in the future.

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