Decomposition of Trees and Paths via Correlation

We study the problem of decomposing (clustering) a tree with respect to costs attributed to pairs of nodes, so as to minimize the sum of costs for those pairs of nodes that are in the same component (cluster). For the general case and for the special case of the tree being a star, we show that the problem is NP-hard. For the special case of the tree being a path, this problem is known to be polynomial time solvable. We characterize several classes of facets of the combinatorial polytope associated with a formulation of this clustering problem in terms of lifted multicuts. In particular, our results yield a complete totally dual integral (TDI) description of the lifted multicut polytope for paths, which establishes a connection to the combinatorial properties of alternative formulations such as set partitioning.Comment: v2 is a complete revisio

Combinatorial persistency criteria for multicut and max-cut

In combinatorial optimization, partial variable assignments are called persistent if they agree with some optimal solution. We propose persistency criteria for the multicut and max-cut problem as well as fast combinatorial routines to verify them. The criteria that we derive are based on mappings that improve feasible multicuts, respectively cuts. Our elementary criteria can be checked enumeratively. The more advanced ones rely on fast algorithms for upper and lower bounds for the respective cut problems and max-flow techniques for auxiliary min-cut problems. Our methods can be used as a preprocessing technique for reducing problem sizes or for computing partial optimality guarantees for solutions output by heuristic solvers. We show the efficacy of our methods on instances of both problems from computer vision, biomedical image analysis and statistical physics

Corrigendum to "Topological entropy for impulsive differential equations" [Electron. J. Qual. Theory Differ. Equ. 2020, No. 68, 1-15]

The aim of this corrigendum is two-fold: (i) to indicate the incorrect parts in two propositions of our recent paper with the same title, (ii) to state the correct statements

Semi-periodic solutions of difference and differential equations

International audienceThe spaces of semi-periodic sequences and functions are examined in the relationship to the closely related notions of almost-periodicity, quasi-periodicity and periodicity. Besides the main theorems, several illustrative examples of this type are supplied. As an application, the existence and uniqueness results are formulated for semi-periodic solutions of quasi-linear difference and differential equations

CFD Simulation of a Dry Scroll Vacuum Pump Including Leakage Flows

One challenge for the numerical simulation of a scroll compressor is the discretization of the chamber volume, which changes with time. In addition, the flow characteristics are very complex including the leakage flow caused by radial and axial gaps between the rotors and the housing. For engineers and designers it is imperative to understand the influence of such leakage flows on the efficiency of the scroll compressor. This paper describes the workflow for Computational Fluid Dynamics (CFD) of a dry scroll vacuum pump and shows some preliminary results. The computational grids for the time dependent flow volume are generated by the grid generator TwinMesh. The meshing software generates and optimises all necessary grids for each time step prior to the actual simulation and calculates the mesh quality to assure high quality numerical results. Furthermore, the grid generation accurately considers axial gaps. The transient numerical simulations are performed by the commercial CFD software code ANSYS CFX, which is able to handle complex flow characteristics. The simulations consider compressibility, turbulence, and heat transfer effects using ideal gas properties for the fluid. The pressure ratio between inlet and outlet is varied as well as the gap size. The results are compared with analytical methods. This paper shows time dependent information such as pressure, velocity, and temperature for specific locations in the chamber volume and also integral values such as power, torque, and mass flow. Finally, cross-sectional views are presented for different positions and time steps

Topological entropy and differential equations

summary:On the background of a brief survey panorama of results on the topic in the title, one new theorem is presented concerning a positive topological entropy (i.e. topological chaos) for the impulsive differential equations on the Cartesian product of compact intervals, which is positively invariant under the composition of the associated Poincaré translation operator with a multivalued upper semicontinuous impulsive mapping