On the minimal energy of unicyclic Hückel molecular graphs possessing Kekulé structures

Let G be any unicyclic Hückel molecular graph with Kekulé structures on n vertices where n≥8 is an even number. In [W. Wang, A. Chang, L. Zhang, D. Lu, Unicyclic Hückel molecular graphs with minimal energy, J. Math. Chem. 39 (1) (2006) 231-241], Wang et al. showed that if G satisfies certain conditions, then the energy of G is always greater than the energy of the radialene graph. In this paper we prove that this inequality actually holds under a much weaker condition.     

On the numerical solution of Plateau's problem

The present paper is dedicated to the numerical computation of minimal surfaces by the boundary element method. Having a parametrization γ of the boundary curve over the unit circle at hand, the problem is reduced to seeking a reparametrization κ of the unit circle. The Dirichlet energy of the harmonic extension of γ○κ has to be minimized among all reparametrizations. The energy functional is calculated as boundary integral that involves the Dirichlet-to-Neumann map. First and second order necessary optimality conditions of the underlying minimization problem are formulated. Existence and convergence of approximate solutions is proven. An efficient algorithm is proposed for the computation of minimal surfaces and numerical results are presented.     

Some Remarks on “Language-Created Entities”

Some entities, such as fictional characters, propositions, properties, events and numbers are prima facie promising candidates for owing their existence to our linguistic and conceptual practices. However, it is notoriously hard to pin down just what sets such allegedly “language-created” entities apart from ordinary entities. The present paper considers some of the features that are supposed to distinguish between entities of the two kinds and argues that, on an independently plausible account of what it takes to individuate objects, the criteria let in more than friends of the strategy might be happy with.

类呋喃香豆素的合成与DNA的结合及细胞毒性研究

呋喃香豆素类化合物无紫外光辅助照射时抗肿瘤活性低,为提高其正常情况下的抗肿瘤活性,依据“最小嵌入”假说对其进行结构改造。把呋喃香豆素结构中的呋喃环拆分出来,使其与香豆素由原来稠环相并的结合方式转变为通过化学键相连,得到合成简化的类呋喃香豆素。利用DNA熔解曲线、吸收光谱、荧光发射光谱和粘度测试考察了这些类呋喃香豆素与DNA的相互作用。综合DNA溶解曲线、光谱法和粘度测试的结果,推测除目标产物5b是一DNA嵌入剂外,其他化合物嵌入DNA能力下降,5a以部分嵌入方式与DNA结合,5c和5d是通过极为少见、鲜有报道的“桥型结构”与DNA相结合。利用“罗丹明B蛋白染色法”考察了目标化合物的体外细胞毒性,测试结果显示,与对照品补骨脂素相比这些化合物对肿瘤细胞体外生长抑制作用明显增强,并且非经典嵌入结合的化合物活性增强更明显。该研究拓展了“最小嵌入”假说的应用范围,同时为呋喃香豆素类化合物结构改造提供了依据。     

An improved bound for disjoint directed cycles

We show that every directed graph with minimum out-degree at least $18k$ contains at least $k$ vertex disjoint cycles. This is an improvement over the result of Alon who showed this result for digraphs of minimum out-degree at least $64k$. The main benefit of the argument is that getting better results for small values of $k$ allows for further improvements to the constant.     

Some theoretical limitations of second-order algorithms for smooth constrained optimization

In second-order algorithms, we investigate the relevance of the constant rank of the full set of active constraints in ensuring global convergence to a second-order stationary point under a constraint qualification. We show that second-order stationarity is not expected in the non-constant rank case if the growth of so-called tangent AKKT2 sequences is not controlled. Since no algorithm controls their growth, we argue that there is a theoretical limitation of algorithms in finding second-order stationary points without constant rank assumptions.     

On a free boundary problem and minimal surfaces

From minimal surfaces such as Simons' cone and catenoids, using refined Lyapunov–Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension 8, using variational arguments, we also obtain solutions which are global minimizers of the corresponding energy functional. This shows that the theorem of Valdinoci et al. [41], [42] is optimal.     

Double circulant matrices

Double circulant matrices are introduced and studied. By a matrix-theoretic method, the rank r of a double circulant matrix is computed, and it is shown that any consecutive r rows of the double circulant matrix are linearly independent. As a generalization, multiple circulant matrices are also introduced. Two questions on square double circulant matrices are posed.