Computations of instanton invariants using Donaldson-Floer Theory

Summary We compute the Donaldson SU(2)-invariants of the double cover of ² branched over a smooth algebraic curve of degree eight. From this we deduce a formula for the relative invariants of the blow-up of the Gompf nucleusN ₂, and we show how this gives a blow-up formula for a class of 4-manifolds which includes essentially all the simply connected 4-manifolds known to have big diffeomorphism group. We apply the result on the nucleus also to prove a formula for the invariants of minimal simply connected elliptic surfaces which reduces the computation to the case of geometric genus one. In particular, we compute all the Donaldson invariants of minimal simply connected elliptic surfaces without multiple fibers. Our main tool is Donaldson-Floer theory.Oblatum IX-1993 & 26-IV-1994     

An exact algorithm for the capacitated shortest spanning arborescence

We are given a complete and loop-free digraphG=(V, A), whereV={1,...,n} is the vertex set,A={(i, j) :i, j V} the arc set, andr V is a distinguishedroot vertex. For each arc (i, j) A, letc _ij be the associatedcost, and for each vertexi, letq _i 0 be the associateddemand (withq _r =0). Moreover, a nonnegativebranch capacity, Q, is defined.A Capacitated Shortest Spanning Arborescence rooted at r (CSSA _r ) is a minimum cost partial digraph such that: (i) each vertexj r has exactly one entering arc; (ii) for each vertexj r, a path fromr toj exists; (iii) for each branch leaving vertexr, the total demand of the vertices does not exceed the branch capacity,Q. A variant of theCSSA _r problem (calledD-CSSA _r ) arises when the out-degree of the root vertex is constrained to be equal to a given valueD. These problems are strongly NP-hard, and find practical applications in routing and network design. We describe a new Lagrangian lower bound forCSSA _r andD-CSSA _r problems, strengthened in a cutting plane fashion by iteratively adding violated constraints to the Lagrangian problem. We also present a new lower bound based on projection leading to the solution of min-cost flow problems. The two lower bounds are then combined so as to obtain an overall additive lower bounding procedure. The additive procedure is then imbedded in a branch-and-bound algorithm whose performance is enhanced by means of reduction procedures, dominance criteria, feasibility checks and upper bounding. Computational tests on asymmetric and symmetric instances from the literature, involving up to 200 vertices, are given, showing the effectiveness of the proposed approach.     

Surface ablation by excimer laser irradiation of Ti and Ti6Al4V alloy

This paper presents the surface microstructure of Ti and Ti6Al4V alloy irradiated with a high output energy XeCl ( = 308 nm) excimer laser. The treatments are carried out on both materials at two beam fluences and the effects of single- and multiple-pulse irradiation are compared. The results of the scanning electron microscopy and of the X-ray diffraction techniques suggest the possible influence of both time-behaviour and energy fluence of the laser pulse on the relative weight of the ablation rate and of the reaction product deposition rate at the sample surface.     

Sobolev spaces associated to a polyhedron and Fourier integral operations inR n

The theory of the Sobolev spacesH _m ^p (R ⁿ ) (mR,p polyhedron in R ²ⁿ )of [BG]is revisited here in the frame of new classes of pseudodifferential operators related to the same polyhedron p.These operators generalize to corresponding classes of Fourier integral operators, for which we present the main lines of a symbolic calculus and results of continuity on the H _m ^p (R ⁿ ) spaces.