Chern-Simons functions on toric Calabi-Yau threefolds and Donaldson-Thomas theory

In this paper, we give a construction of the global Chern-Simons functions for toric Calabi-Yau stacks of dimension three using strong exceptional collections. The moduli spaces of sheaves on such stacks can be identified with critical loci of these functions. We give two applications of these functions. First, we prove Joyce's integrality conjecture of generalized DT invariants on local surfaces. Second, we prove a dimension reduction formula for virtual motives, which leads to two recursion formulas for motivic Donaldson-Thomas invariants.Comment: A rewritten versoin. Some references adde

A Plane Wave Virtual Element Method for the Helmholtz Problem

We introduce and analyze a virtual element method (VEM) for the Helmholtz problem with approximating spaces made of products of low order VEM functions and plane waves. We restrict ourselves to the 2D Helmholtz equation with impedance boundary conditions on the whole domain boundary. The main ingredients of the plane wave VEM scheme are: i) a low frequency space made of VEM functions, whose basis functions are not explicitly computed in the element interiors; ii) a proper local projection operator onto the high-frequency space, made of plane waves; iii) an approximate stabilization term. A convergence result for the h-version of the method is proved, and numerical results testing its performance on general polygonal meshes are presented

An intracardiac electrogram model to bridge virtual hearts and implantable cardiac devices

Virtual heart models have been proposed to enhance the safety of implantable cardiac devices through closed loop validation. To communicate with a virtual heart, devices have been driven by cardiac signals at specific sites. As a result, only the action potentials of these sites are sensed. However, the real device implanted in the heart will sense a complex combination of near and far-field extracellular potential signals. Therefore many device functions, such as blanking periods and refractory periods, are designed to handle these unexpected signals. To represent these signals, we develop an intracardiac electrogram (IEGM) model as an interface between the virtual heart and the device. The model can capture not only the local excitation but also far-field signals and pacing afterpotentials. Moreover, the sensing controller can specify unipolar or bipolar electrogram (EGM) sensing configurations and introduce various oversensing and undersensing modes. The simulation results show that the model is able to reproduce clinically observed sensing problems, which significantly extends the capabilities of the virtual heart model in the context of device validation

Hirzebruch-Milnor classes and Steenbrink spectra of certain projective hypersurfaces

We show that the Hirzebruch-Milnor class of a projective hypersurface, which gives the difference between the Hirzebruch class and the virtual one, can be calculated by using the Steenbrink spectra of local defining functions of the hypersurface if certain good conditions are satisfied, e.g. in the case of projective hyperplane arrangements, where we can give a more explicit formula. This is a natural continuation of our previous paper on the Hirzebruch-Milnor classes of complete intersections.Comment: 15 pages, Introduction is modifie