Almost analytic extensions of ultradifferentiable functions with applications to microlocal analysis

We review and extend the description of ultradifferentiable functions by their almost analytic extensions, i.e., extensions to the complex domain with specific vanishing rate of the $\bar \partial$-derivative near the real domain. We work in a general uniform framework which comprises the main classical ultradifferentiable classes but also allows to treat unions and intersections of such. The second part of the paper is devoted to applications in microlocal analysis. The ultradifferentiable wave front set is defined in this general setting and characterized in terms of almost analytic extensions and of the FBI transform. This allows to extend its definition to ultradifferentiable manifolds. We also discuss ultradifferentiable versions of the elliptic regularity theorem and obtain a general quasianalytic Holmgren uniqueness theorem.Comment: 48 pages; minor changes, accepted for publication in Journal of Mathematical Analysis and Applications; some typos correcte

Ultradifferentiable classes of entire functions

We study classes of ultradifferentiable functions defined in terms of small weight sequences violating standard growth and regularity requirements. First, we show that such classes can be viewed as weighted spaces of entire functions for which the crucial weight is given by the associated weight function of the so-called conjugate weight sequence. Moreover, we generalize results from M. Markin from the so-called small Gevrey-setting to arbitrary convenient families of (small) sequences and show how the corresponding ultradifferentiable function classes can be used to detect boundedness of normal linear operators on Hilbert spaces (associated to an evolution equation problem). Finally, we study the connection between small sequences and the recent notion of dual sequences introduced in the PhD-thesis of J. Jim\'{e}nez-Garrido.Comment: 28 page

Heuristics for a project management problem withincompatibility and assignment costs

Consider a project which consists of a set of jobs to be performed, assuming each job has a duration of at most one time period. We assume that the project manager provides a set of possible durations (in time periods) for the whole project. When a job is assigned to a specific time period, an assignment cost is encountered. In addition, for some pairs of jobs, an incompatibility cost is encountered if they are performed at the same time period. Both types of cost depend on the duration of the whole project, which also has to be determined. The goal is to assign a time period to each job while minimizing the costs. We propose a tabu search heuristic, as well as an adaptive memory algorithm, and compare them with other heuristics on large instances, and with an exact method on small instances. Variations of the problems are also discusse

A solution method for a car fleet management problem with maintenance constraints

The problem retained for the ROADEF'99 international challenge was an inventory management problem for a car rental company. It consists in managing a given fleet of cars in order to satisfy requests from customers asking for some type of cars for a given time period. When requests exceed the stock of available cars, the company can either offer better cars than those requested, subcontract some requests to other providers, or buy new cars to enlarge the available stock. Moreover, the cars have to go through a maintenance process at a regular basis, and there is a limited number of workers that are available to perform these maintenances. The problem of satisfying all customer requests at minimum cost is known to be NP-hard. We propose a solution technique that combines two tabu search procedures with algorithms for the shortest path, the graph coloring and the maximum weighted independent set problems. Tests on benchmark instances used for the ROADEF'99 challenge give evidence that the proposed algorithm outperforms all other existing methods (thirteen competitors took part to this contest

Locally Checkable Problems Parameterized by Clique-Width

We continue the study initiated by Bonomo-Braberman and Gonzalez in 2020 on r-locally checkable problems. We propose a dynamic programming algorithm that takes as input a graph with an associated clique-width expression and solves a 1-locally checkable problem under certain restrictions. We show that it runs in polynomial time in graphs of bounded clique-width, when the number of colors of the locally checkable problem is fixed. Furthermore, we present a first extension of our framework to global properties by taking into account the sizes of the color classes, and consequently enlarge the set of problems solvable in polynomial time with our approach in graphs of bounded clique-width. As examples, we apply this setting to show that, when parameterized by clique-width, the [k]-Roman domination problem is FPT, and the k-community problem, Max PDS and other variants are XP

On optimal solutions of the Borel problem in the Roumieu case

The Borel problem for Denjoy--Carleman and Braun--Meise--Taylor classes has well-known optimal solutions. The unified treatment of these ultradifferentiable classes by means of one-parameter families of weight sequences allows to compare these optimal solutions. We determine the relations among them and give conditions for their equivalence in the Roumieu case.Comment: 18 pages; minor changes, Remark 4.7 adde

The Borel map in the mixed Beurling setting

The Borel map takes a smooth function to its infinite jet of derivatives (at zero). We study the restriction of this map to ultradifferentiable classes of Beurling type in a very general setting which encompasses the classical Denjoy-Carleman and Braun-Meise-Taylor classes. More precisely, we characterize when the Borel image of one class covers the sequence space of another class in terms of the two weights that define the classes. We present two independent solutions to this problem, one by reduction to the Roumieu case and the other by dualization of the involved Fr\'echet spaces, a Phragm\'en-Lindel\"of theorem, and H\"ormander's solution of the $\bar{\partial}$-problem.Comment: 33 pages; some typos correcte

P5-free augmenting graphs and the maximum stable set problem

AbstractThe complexity status of the maximum stable set problem in the class of P5-free graphs is unknown. In this paper, we first propose a characterization of all connected P5-free augmenting graphs. We then use this characterization to detect families of subclasses of P5-free graphs where the maximum stable set problem has a polynomial time solution. These families extend several previously studied classes

Some combinatorial optimization problems in graphs with applications in telecommunications and tomography

The common point between the different chapters of the present work is graph theory. We investigate some well known graph theory problems, and some which arise from more specific applications. In the first chapter, we deal with the maximum stable set problem, and provide some new graph classes, where it can be solved in polynomial time. Those classes are hereditary, i.e. characterized by a list of forbidden induced subgraphs. The algorithms proposed are purely combinatorial. The second chapter is devoted to the study of a problem linked to security purposes in mobile telecommunication networks. The particularity is that there is no central authority guaranteeing security, but it is actually managed by the users themselves. The network is modelled by an oriented graph, whose vertices represent the users, and whose arcs represent public key certificates. The problem is to associate to each vertex a subgraph with some requirements on the size of the subgraphs, the number of times a vertex is taken in a subgraph and the connectivity between any two users as they put their subgraphs together. Constructive heuristics are proposed, bounds on the optimal solution and a tabu search are described and tested. The third chapter is on the problem of reconstructing an image, given its projections in terms of the number of occurrences of each color in each row and each column. The case of two colors is known to be polynomially solvable, it is NP-complete with four or more colors, and the complexity status of the problem with three colors is open. An intermediate case between two and three colors is shown to be solvable in polynomial time. The last two chapters are about graph (vertex-)coloring. In the fourth, we prove a result which brings a large collection of NP-hard subcases, characterized by forbidden induced subgraphs. In the fifth chapter, we approach the problem with the use of linear programming. Links between different formulations are pointed out, and some families of facets are characterized. In the last section, we study a branch and bound algorithm, whose lower bounds are given by the optimal value of the linear relaxation of one of the exposed formulations. A preprocessing procedure is proposed and tested

Nuclear global spaces of ultradifferentiable functions in the matrix weighted setting

We prove that the Hermite functions are an absolute Schauder basis for many global weighted spaces of ultradifferentiable functions in the matrix weighted setting and we determine also the corresponding coefficient spaces, thus extending previous work by Langenbruch. As a consequence we give very general conditions for these spaces to be nuclear. In particular, we obtain the corresponding results for spaces defined by weight functions