Toolset for Supporting the Research of Lattice Based Number Expansions

The world of generalized number systems contains many challenging areas. Computer experiments often support the theoretical research. In this paper we introduce a toolset that helps to analyze some properties of lattice based number expansions. The toolset is able to (1) analyze the expansions, (2) decide the number system property, (3) classify and visualize the periodic points. The toolset is implemented in Python, published alongside with a database that stores plenty of special expansions, and is able to store the custom properties like signature, operator eigenvalues, etc. Researchers can connect to the server and request/upload data, or perform experiments on them. We present an introductory usage of the toolset and detail some results that has been observed by the toolset. The toolset can be downloaded from http://numsys.info domain.

Mapping the surgery exact sequence for topological manifolds to analysis

In this paper we prove the existence of a natural mapping from the surgery exact sequence for topological manifolds to the analytic surgery exact sequence of Higson and Roe. This generalizes the fundamental result of Higson and Roe, but in the treatment given by Piazza and Schick, from smooth manifolds to topological manifolds. Crucial to our treatment is the Lipschitz signature operator of Teleman. We also give a generalization to the equivariant setting of the product defined by Siegel in his Ph.D. thesis. Geometric applications are given to stability results for rho classes. We also obtain a proof of the APS delocalized index theorem on odd dimensional manifolds, both for the spin Dirac operator and the signature operator, thus extending to odd dimensions the results of Piazza and Schick. Consequently, we are able to discuss the mapping of the surgery sequence in all dimensions.