Combinatorial optimization for an improved transition from fluence optimization to fluence delivery in IMRT treatment planning

The intensity modulated radiation therapy (IMRT) treatment planning problem is usually divided into three smaller problems that are solved sequentially: geometry problem, intensity problem and realization problem. There are many models and algorithms that address each one of the problems in a satisfactory way. However, these problems cannot be seen separately, because strong links exist between them. While the linkage between the geometry problem and the intensity problem is straightforward, the linkage between the intensity problem and the realization problem is all but simple and will determine the quality of the treatment planning. In practice, the linkage between these problems is, most of the times, done in a rather simple way, usually by rounding. This can lead to a significant deterioration of the treatment plan quality. We propose a combinatorial optimization approach to enable an improved transition from optimized to delivery fluence maps in IMRT treatment planning. Two clinical examples of head and neck cancer cases are used, both to present numerical evidences of the resulting deterioration of plan quality if a simplistic approach is used, and also to highlight a combinatorial optimization approach as a valuable alternative when linking the intensity problem and the realization problem.     

利用组合群验方法确认运动员违禁物质

组合群验设计属于非适应性的组合群试,非适应性的组合群试算法可以同时进行所有测试,而且不必考虑各次测试之间的影响,它省时快捷,已被广泛应用于低发生率的实验设计中.利用组合群验方法给出了确认运动员体内含有违禁物质的检测与判定方法.     

A discussion on the conservatism of robust linear optimization problems

In 2004, Bertsimas and Sim proposed a robust approach that can control the degree of conservatism by applying a limitation Γ to the maximum number of parameters that are allowed to change. However, the robust approach can become extremely conservative even when Γ is relatively small. In this paper, we provide a theoretical analysis to explain why this extreme conservatism occurs. We further point out that the robust approach does not reach an extremely conservative state when Γ is less than k, where k is the number of nonzero components of the optimal solution of the extremely conservative robust approach. This research also shows that care must be taken when adjusting the value of Γ to control the degree of conservatism because the approach may result in greater conservatism than was intended. We subsequently apply our analysis to additive combinatorial optimization problems. Finally, we illustrate our results on numerical simulations.     

完备化Witt代数中自然出现的一些非有限阶化李代数的表示

讨论了完备化Witt代数的三类非有限阶化子代数的不可约非权模的实现,并利用它们的表示得到一些有趣的组合恒等式.     

Perturbations of Banach Frames and Atomic Decompositions

Banach frames and atomic decompositions are sequences that have basis-like properties but which need not be bases. In particular, they allow elements of a Banach space to be written as linear combinations of the frame or atomic decomposition elements in a stable manner. In this paper we prove several functional — analytic properties of these decompositions, and show how these properties apply to Gabor and wavelet systems. We first prove that frames and atomic decompositions are stable under small perturbations. This is inspired by corresponding classical perturbation results for bases, including the Paley — Wiener basis stability criteria and the perturbation theorem el kato. We introduce new and weaker conditions which ensure the desired stability. We then prove quality properties of atomic decompositions and consider some consequences for Hilbert frames. Finally, we demonstrate how our results apply in the practical case of Gabor systems in weighted L² spaces. Such systems can form atomic decompositions for L²_w(IR), but cannot form Hilbert frames but L²_w(IR) unless the weight is trivial.     

On local optima in multiobjective combinatorial optimization problems

In this article, local optimality in multiobjective combinatorial optimization is used as a baseline for the design and analysis of two iterative improvement algorithms. Both algorithms search in a neighborhood that is defined on a collection of sets of feasible solutions and their acceptance criterion is based on outperformance relations. Proofs of the soundness and completeness of these algorithms are given.     

Riemann Zeta函数的六阶和

利用概率论与组合数学的方法,研究了与Riemann-zeta函数ξ(k)的部分和ξ_n(k)有关的一些级数,计算出了一些重要的和式．特别的,Euler的著名结果5ξ(4)= 2ξ~2(2)能够从四阶和式直接推出．因此,通过计算全部的11个六阶和式,研究它们之间的非平凡关系,就有可能得到ξ(3)的数值．     

The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains

We introduce certain Sobolev-Besov spaces which are particularly well adapted for measuring the smoothness of data and solutions of mixed boundary value problems in Lipschitz domains. In particular, these are used to obtain sharp well-posedness results for the Poisson problem for the Laplacian with mixed boundary conditions on bounded Lipschitz domains which satisfy a suitable geometric condition introduced by R.Brown in (1994). In this context, we obtain results which generalize those by D.Jerison and C.Kenig (1995) as well as E.Fabes, O.Mendez and M.Mitrea (1998). Applications to Hodge theory and the regularity of Green operators are also presented.

     

Amalgams and χ‐Boundedness

A class of graphs is hereditary if it is closed under isomorphism and induced subgraphs. A class of graphs is χ‐bounded if there exists a function such that for all graphs , and all induced subgraphs H of G, we have that . We prove that proper homogeneous sets, clique‐cutsets, and amalgams together preserve χ‐boundedness. More precisely, we show that if and are hereditary classes of graphs such that is χ‐bounded, and such that every graph in either belongs to or admits a proper homogeneous set, a clique‐cutset, or an amalgam, then the class is χ‐bounded. This generalizes a result of [J Combin Theory Ser B 103(5) (2013), 567–586], which states that proper homogeneous sets and clique‐cutsets together preserve χ‐boundedness, as well as a result of [European J Combin 33(4) (2012), 679–683], which states that 1‐joins preserve χ‐boundedness. The house is the complement of the four‐edge path. As an application of our result and of the decomposition theorem for “cap‐free” graphs from [J Graph Theory 30(4) (1999), 289–308], we obtain that if G is a graph that does not contain any subdivision of the house as an induced subgraph, then .