Cohomological Hall algebras, semicanonical bases and Donaldson-Thomas invariants for 22-dimensional Calabi-Yau categories (with an appendix by Ben Davison)

We discuss semicanonical bases from the point of view of Cohomological Hall algebras via the "dimensional reduction" from 3-dimensional Calabi-Yau categories to 2-dimensional ones. Also, we discuss the notion of motivic Donaldson-Thomas invariants (as defined by M. Kontsevich and Y. Soibelman) in the framework of 2-dimensional Calabi-Yau categories. In particular we propose a conjecture which allows one to define Kac polynomials for a 2-dimensional Calabi-Yau category (this is a theorem of S. Mozgovoy in the case of preprojective algebras).Comment: The revised version contains the Appendix written by Ben Davison about the relationship of Kontsevich-Soibelman product with the one of Schiffmann-Vassero

Multilevel sequence detection for dynamic mode atomic force microscopy

The atomic force microscope is an instrument that is widely used in fields such as biology, chemistry and medicine for imaging at the atomic level. In this work, we consider a specific mode of AFM usage, known as the dynamic mode where the AFM cantilever probe is forced sinusoidally. In the absence of interaction with the sample being imaged, the cantilever follows a predictable sinusoidal trajectory. The deflection of the cantilever probe changes when it interacts with the sample being imaged and imaging is performed by interpreting these changes. In this work, we present a sequence detection based algorithm that allows for resolving topographic features into one of three possible levels at a fast speed. We demonstrate the effectiveness of our algorithm via simulation results and by comparing it to a lower bound that is obtained by considering a genie aided detector