Rota–Baxter Algebras and New Combinatorial Identities

The word problem for an arbitrary associative Rota–Baxter algebra is solved. This leads to a noncommutative generalization of the classical Spitzer identities. Links to other combinatorial aspects are indicated.      

The Sixth Painlevé Equation as Similarity Reduction of $${\widehat{\mathfrak{gl}}_3}$$ Generalized Drinfel’d–Sokolov Hierarchy

Scaling symmetry of -type Drinfel’d–Sokolov hierarchy is investigated. Applying similarity reduction to the hierarchy, one can obtain the Schlesinger equation with (n + 1) regular singularities. Especially in the case of n = 3, the hierarchy contains the three-wave resonant system and the similarity reduction gives the generic case of the Painlevé VI equation. We also discuss Weyl group symmetry of the hierarchy.      

生物质碳点的合成及初步用于盐酸土霉素的测定(英文)

以活性炭为碳源,过氧化氢和醋酸为混合氧化剂,通过化学氧化法合成了碳点。经过乙醚提取纯化,并用聚乙二醇2000进一步钝化,碳点性能得到了改善。最终制得的碳点在316 nm处激发时,最大发射波长出现在435 nm左右,其荧光量子产率为19.6%。盐酸土霉素对碳点有猝灭作用,据此实现了对土霉素的测定。该方法线性关系良好(r=0.998 6),回收率在98.0%~102.0%(RSD=1.7%),检测限达1.4×10-3μg·mL-1。     

Rigidity of proper holomorphic mappings between nonequidimensional bounded symmetric domains

We prove that any proper holomorphic mapping from to is necessarily a totally geodesic isometric embedding with respect to their Bergman metrics and therefore is the standard linear embedding up to their automorphisms. The proof of the main result in this paper will be achieved by a differential-geometric study of a special class of complex geodesic curves on the bounded symmetric domains with respect to their Bergman metrics. Received: 31 October 2000; in final form: 2 July 2001/ Published online: 4 April 2002     

Lattice Points in Three-Dimensional Convex Bodies with Points of Gaussian Curvature Zero at the Boundary

 In this article we investigate the number of lattice points in a three-dimensional convex body which contains non-isolated points with Gaussian curvature zero but a finite number of flat points at the boundary. Especially, in case of rational tangential planes in these points we investigate not only the influence of the flat points but also of the other points with Gaussian curvature zero on the estimation of the lattice rest. Received 19 June 2001; in revised form 17 January 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

Multiplicity of the second Schrödinger eigenvalue on closed surfaces

We improve – roughly by a factor 2 – the known bound on the multiplicity of the second eigenvalue of Schr?dinger operators (i.e. Laplace plus potential) on closed surfaces. This gives four new topological types of surfaces for which Colin de Verdière's conjecture relating the maximal multiplicity to the chromatic number of the surface is verified. The proof goes by defining a space of "nodal splittings” of the surface, equipped with a double covering to which a Borsuk-Ulam type theorem is applied. Received: 19 June 2001; revised version: 18 March 2002 /Published online: 17 June 2002     

Small Universal Graphs for Bounded-Degree Planar Graphs

For all positive integers N and k, let denote the family of planar graphs on N or fewer vertices, and with maximum degree k. For all positive integers N and k, we construct a -universal graph of size . This construction answers with an explicit construction the previously open question of the existence of such a graph. Received July 8, 1998 RID="*" ID="*" Supported by NSF grant CCR98210-58 and ARO grant DAAH04-96-1-0013.     

Large Cliques in -Free Graphs

Dedicated to the memory of Paul Erdős A graph is called -free if it contains no cycle of length four as an induced subgraph. We prove that if a -free graph has n vertices and at least edges then it has a complete subgraph of vertices, where depends only on . We also give estimates on and show that a similar result does not hold for H-free graphs––unless H is an induced subgraph of . The best value of is determined for chordal graphs. Received October 25, 1999 RID="*" ID="*" Supported by OTKA grant T029074. RID="**" ID="**" Supported by TKI grant stochastics@TUB and by OTKA grant T026203.     

Improved Bounds for Acyclic Job Shop Scheduling

In acyclic job shop scheduling problems there are n jobs and m machines. Each job is composed of a sequence of operations to be performed on different machines. A legal schedule is one in which within each job, operations are carried out in order, and each machine performs at most one operation in any unit of time. If D denotes the length of the longest job, and C denotes the number of time units requested by all jobs on the most loaded machine, then clearly lb = max[C,D] is a lower bound on the length of the shortest legal schedule. A celebrated result of Leighton, Maggs, and Rao shows that if all operations are of unit length, then there always is a legal schedule of length O(lb), independent of n and m. For the case that operations may have different lengths, Shmoys, Stein and Wein showed that there always is a legal schedule of length , where the notation is used to suppress terms. We improve the upper bound to . We also show that our new upper bound is essentially best possible, by proving the existence of instances of acyclic job shop scheduling for which the shortest legal schedule is of length . This resolves (negatively) a known open problem of whether the linear upper bound of Leighton, Maggs, and Rao applies to arbitrary job shop scheduling instances (without the restriction to acyclicity and unit length operations). Received June 30, 1998 RID="*" ID="*" Incumbent of the Joseph and Celia Reskin Career Development Chair RID="†" ID="†" Research was done while staying at the Weizmann Institute, supported by a scholarship from the Minerva foundation.     

Transversals in Uniform Hypergraphs with Property (p,2)

Dedicated to the memory of Paul Erdős Let f(r,p,t) (p > t >= 1, r >= 2) be the maximum of the cardinality of a minimum transversal over all r-uniform hypergraphs possessing the property that every subhypergraph of with p edges has a transversal of size t. The values of f(r,p,2) for p = 3, 4, 5, 6 were found in [1] and bounds on f(r,7,2) are given in [3]. Here we prove that for large p and huge r. Received September 23, 1999 RID="*" ID="*" This work was partially supported by the grant 99-01-00581 of the Russian Foundation for Fundamental Research and the Dutch–Russian Grant NWO-047-008-006.