Regularity of optimal sets for some functional involving eigenvalues of an operator in divergence form

In this paper we consider minimizers of the functional$\min \{{\lambda }_1(\mathrm{\Omega })+\cdots +{\lambda }_k(\mathrm{\Omega })+\mathrm{\Lambda }|\mathrm{\Omega }|,:\mathrm{\Omega }\subset D\text{ open}\}$ where $D\subset {\mathbb{R}}^d$ is a bounded open set and where $0<{\lambda }_1(\mathrm{\Omega })\le \cdots \le {\lambda }_k(\mathrm{\Omega })$ are the first k eigenvalues on Ω of an operator in divergence form with Dirichlet boundary condition and with Hölder continuous coefficients. We prove that the optimal sets ${\mathrm{\Omega }}^{⁎}$ have finite perimeter and that their free boundary $\partial {\mathrm{\Omega }}^{⁎}\cap D$ is composed of a regular part, which is locally the graph of a $C^{1,\alpha }$-regular function, and a singular part, which is empty if $d<d^{⁎}$, discrete if $d=d^{⁎}$ and of Hausdorff dimension at most $d-d^{⁎}$ if $d>d^{⁎}$, for some $d^{⁎}\in \{5,6,7\}$.     

全局优化问题的一个新的无参数填充函数

求全局最优化问题的填充函数算法被提出以来，参数的选取和调整一直是制约算法有效性的因素。如何在实际的计算过程中选取合适的参数，直接影响和决定了运算速度和效率。因此，构造不含参数的填充函数就显得极为重要。提出一个新的无参数的填充函数，对其理论性质进行了分析，并给出相应的填充函数算法，数值计算验证了算法的有效性。     

Performance guarantees of forward and reverse greedy algorithms for minimizing nonsupermodular nonsubmodular functions on a matroid

This letter studies the problem of minimizing increasing set functions, equivalently, maximizing decreasing set functions, over the base matroid. This setting has received great interest, since it generalizes several applied problems including actuator and sensor placement problems in control, task allocation problems, video summarization, and many others. We study two greedy heuristics, namely, the forward and reverse greedy. We provide two novel performance guarantees for the approximate solutions obtained by them depending on both submodularity ratio and curvature.     

基于流量测点的热网变动阻力系数优化辨识

随着供热计量系统的普及，系统可以根据负荷的变化进行调整，管网的阻力系数随即发生变化。对变动阻力系数进行优化辨识是了解供热管网实时运行状况的有效手段。提出一种基于流量监测数据的供热管网变动阻力系数优化辨识方法，并利用遗传算法进行求解。对洛阳市某小区供暖季管网实际运行数据进行验证，辨识结果的相对误差在5%以内。结果表明：该方法可以在只有流量观测数据时得到精度较高的供热管网变动阻力系数，为供热系统的运行调节提供指导。     

基于贝叶斯参数优化的无模型自适应硅单晶直径控制

直拉硅单晶的生长过程涉及多场多相耦合与复杂的物理化学变化,其中工艺参数的波动是导致晶体直径不均匀的重要原因,如何实现工艺参数的控制以获得理想的、均匀的晶体直径具有重要的研究意义。本文分析现有控制方法存在不稳定以及控制效果不佳的问题后,提出基于贝叶斯参数优化的无模型自适应控制模型来控制硅单晶生长过程中的晶体直径。首先以坩埚上升速度与加热器的功率作为控制输入参数,晶体直径作为输出,搭建无模型自适应控制模型,并分析算法的稳定性。其次将控制模型进行仿真实验,发现硅单晶直径控制模型中不同的超参数设定会影响控制过程的迭代次数以及控制效果。最后,利用贝叶斯优化超参数的取值范围,并进行最终的仿真实验,结果表明,经贝叶斯参数优化后的控制模型计算快、迭代次数少,输出的晶体直径稳定,同时将生长工艺参数控制在实际生产要求范围内。因此,基于贝叶斯参数优化的无模型自适应控制实现了硅单晶直径均匀稳定的有效控制,具有结合工程背景的实际应用前景。     

The chromatic number of random Borsuk graphs

We study a model of random graph where vertices are n i.i.d. uniform random points on the unit sphere S^d in , and a pair of vertices is connected if the Euclidean distance between them is at least 2??. We are interested in the chromatic number of this graph as n tends to infinity. It is not too hard to see that if ?>0 is small and fixed, then the chromatic number is d+2 with high probability. We show that this holds even if ?→0 slowly enough. We quantify the rate at which ? can tend to zero and still have the same chromatic number. The proof depends on combining topological methods (namely the Lyusternik–Schnirelman–Borsuk theorem) with geometric probability arguments. The rate we obtain is best possible, up to a constant factor—if ?→0 faster than this, we show that the graph is (d+1)‐colorable with high probability.25