Non-integrable Ising Models in Cylindrical Geometry: Grassmann Representation and Infinite Volume Limit

In this paper, meant as a companion to Antinucci et al. (Energy correlations of non-integrable Ising models: the scaling limit in the cylinder, 2020. arXiv: 1701.05356), we consider a class of non-integrable 2D Ising models in cylindrical domains, and we discuss two key aspects of the multiscale construction of their scaling limit. In particular, we provide a detailed derivation of the Grassmann representation of the model, including a self-contained presentation of the exact solution of the nearest neighbor model in the cylinder. Moreover, we prove precise asymptotic estimates of the fermionic Green’s function in the cylinder, required for the multiscale analysis of the model. We also review the multiscale construction of the effective potentials in the infinite volume limit, in a form suitable for the generalization to finite cylinders. Compared to previous works, we introduce a few important simplifications in the localization procedure and in the iterative bounds on the kernels of the effective potentials, which are crucial for the adaptation of the construction to domains with boundaries.

OCD.py - Characterizing immunoglobulin inter-domain orientations

Inter-domain orientations between immunoglobulin domains are important for the modeling and engineering of novel antibody therapeutics. Previous tools to describe these orientations are applicable only to the variable domains of antibodies and T-cell receptors. We present the 'Orientation of Cylindrical Domains (OCD)' tool, which employs a transferable approach to calculate inter-domain orientations for all immunoglobulin domains. Based on a reference structure, the OCD tool automatically builds a suitable reference coordinate system for each domain. Through alignment, the reference coordinate systems are transferred onto the sample to calculate six measures which fully characterize the inter-domain orientation. Availability and implementation: The OCD approach is implemented as a stand-alone Python script, OCD.py, which can handle multiple types of data input for the analysis of single structures and molecular dynamics trajectories alike. OCD.py is available at https://github.com/liedllab/OCD under MIT license.

Optimization of non-cylindrical domains for the exact null controllability of the 1D wave equation

This work is concerned with the null controllability of the one-dimensional wave equation over non-cylindrical distributed domains. The controllability in that case has been obtained by Castro, C\^indea and M\"unch in SIAM J. Control Optim., 52 (2014) for domains satisfying the usual geometric optic condition. We analyze the problem of optimizing the non-cylindrical support $q$ of the control of minimal $L^2(q)$-norm. In this respect, we prove a uniform observability inequality for a class of domains $q$ satisfying the geometric optic condition. The proof based on the d'Alembert formula relies on arguments from graph theory. Numerical experiments are discussed and highlight the influence of the initial condition on the optimal domains.

Pullback attractors for 2D MHD equations on time-varying domains

<p style='text-indent:20px;'>In the present paper, we consider the asymptotic dynamics of 2D MHD equations defined on the time-varying domains with homogeneous Dirichlet boundary conditions. First we introduce some coordinate transformations to construct the invariance of the divergence operators in any <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional spaces and establish some equivalent estimates of the vectors between the time-varying domains and the cylindrical domains. Then, we apply these estimates to overcome the difficulties caused by the variations of the spatial domains, including the processing of the pressure <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula> and the definition of weak solutions. Detailed arguments of converting the equations on the time-varying domains into the corresponding equations on the cylindrical domains are presented. Finally, we show the well-posedness of weak solutions and the existence of a compact pullback attractor for the 2D MHD equations.</p>