Abstract:
We give a topological and geometrical description of focus-focus singularities of integrable Hamiltonian systems. In particular, we explain why the monodromy around these singularities is non-trivial, a result obtained before by J.J. Duistermaat and others for some concrete systems.

Abstract:
In this paper the local singularities of integrable Hamiltonian systems with two degrees of freedom are studied. The topological obstruction to the existence of focus-focus singularity with given complexity was found. It has been showed that only simple focus-focus singularities can appear in a typical mechanical system. The model examples of mechanical systems with complex focus-focus singularity are given.

Abstract:
The power law range for the velocity gradient probability density function in forced Burgers turbulence has been an issue of discussion recently. It is shown in [chao-dyn/9901006] that the negative exponent in the assumed power law range has to be strictly larger than 3. Here we give another direct argument for that result, working with finite viscosity. At the same time we compute viscous correction to the power law range. This should answer the questions raised by Kraichnan in [chao-dyn/9901023] regarding the results of [chao-dyn/9901006].

Abstract:
A new type of combinations of Bernstein operators is given in [1]. Here, we introduce another one, which can be used to approximate the functions with singularities. The direct and inverse results of the weighted approximation of this new type combinations are given.

Abstract:
We give two different simple proofs for the removable singularities of the heat equation in $(\Omega\setminus\{x_0\})\times (0,T)$ with $n\ge 3$. We also give a necessary and sufficient condition for removable singularities of the heat equation in $(\Omega\setminus\{x_0\})\times (0,T)$ for the case $n=2$.

Abstract:
For a weighted quasihomogeneous two dimensional hypersurface singularity, we define a smoothing with unipotent monodromy and an isolated graded normal singularity. We study the natural weighted blow up of both the smoothing and the surface. In particular, we describe our construction for the quasihomogeneous singularities of type I, the 14 unimodal exceptional singularities and we relate it to their stable replacement.

Abstract:
This is a note on toroidalization, formulated as the problem of resolution of singularities of morphisms in the logarithmic category. It is submitted to the proceedings of the Barrett conference held at the University of Tennessee at Knoxville in April 2002.

Abstract:
We prove, assuming that the Bohr-Sommerfeld rules hold, that the joint spectrum near a focus-focus critical value of a quantum integrable system determines the classical Lagrangian foliation around the full focus-focus leaf. The result applies, for instance, to h-pseudodifferential operators, and to Berezin-Toeplitz operators on prequantizable compact symplectic manifolds.

Abstract:
This article gives a classification, up to symplectic equivalence, of singular Lagrangian foliations given by a completely integrable system of a 4-dimensional symplectic manifold, in a full neighbourhood of a singular leaf of focus-focus (=nodal) type.

Abstract:
This paper proposes a new proof of Dilworth's theorem. The proof is based upon the minflow/maxcut property in flow networks. In relation to this proof, a new method to find both a Dilworth decomposition and a maximal antichain is presented. 1. Introduction Several proofs are known for Dilworth's theorem. This theorem says that, in a poset , a maximal antichain and a minimal path cover have equal size. Shortly after Dilworth's seminal paper [1] a “Note” [2] was published containing an algorithmic proof, that is, a proof which also gives a method to find a combination of a maximal antichain and a minimal path cover. The other proofs [1, 3–5] are nonalgorithmic. The key issue in [2] is the relation between a minimal path cover and a maximal antichain in on the one hand and a maximal matching and a minimal vertex cover (in this order) in an associated bipartite graph on the other hand. Dilworth's theorem is proved in [2] using K？nig's theorem stating that, in a bipartite graph, a maximal matching and a minimal vertex cover have equal size. The combination of a maximal matching and a minimal vertex cover in corresponds to a maxflow/mincut combination in a flow network akin to . So the obvious algorithm for a minimal path cover along with a maximal antichain is executing a maxflow/mincut algorithm in associated indirectly to . In the current paper a shortcut is proposed between maxflow/mincut and an optimal path cover jointly with an antichain. To a given poset we associate a flow network which is much simpler than graph constructed via a matching/vertex cover instance. A similar idea for finding a maximal antichain is found in [6]. However, the discussion in that paper was not connected with Dilworth's theorem. Other more complex algorithms in this domain can be found in [7–10]. For an application of the maximal antichain we refer to [11, 12]. 2. Some Preliminaries A poset is a partially ordered set , where denotes a transitive order relation. An ordered pair with is called transitive if there exists an element such that . A poset can be transformed into a directed acyclic graph in a straightforward manner. Each nontransitive pair with generates an arc . An example of a graph derived from a poset is given in Figure 1. Figure 1: A directed acyclic graph (DAG). An antichain in a directed acyclic graph is a set of nodes, no two of which are included in any path of . A path cover in is a collection of paths, not necessarily disjoint, such that every node is included in at least one path. The number of paths is called the size of the path cover. Now we can