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1. Dynamical degrees of Hurwitz correspondences

   https://doi.org/10.1017/etds.2018.125  

   RAMADAS, ROHINI (July 2020, Ergodic Theory and Dynamical Systems)

   null (Ed.)

   Let $$\unicode[STIX]{x1D719}$$ be a post-critically finite branched covering of a two-sphere. By work of Koch, the Thurston pullback map induced by $$\unicode[STIX]{x1D719}$$ on Teichmüller space descends to a multivalued self-map—a Hurwitz correspondence $${\mathcal{H}}_{\unicode[STIX]{x1D719}}$$ —of the moduli space $${\mathcal{M}}_{0,\mathbf{P}}$$ . We study the dynamics of Hurwitz correspondences via numerical invariants called dynamical degrees . We show that the sequence of dynamical degrees of $${\mathcal{H}}_{\unicode[STIX]{x1D719}}$$ is always non-increasing and that the behavior of this sequence is constrained by the behavior of $$\unicode[STIX]{x1D719}$$ at and near points of its post-critical set. 

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2. Resource Optimization for Quantum Dynamics with Tensor Networks: Quantum and Classical Algorithms

   https://doi.org/10.1021/acs.jpca.4c03407  

   Dwivedi, Anurag; Lopez-Ruiz, Miguel Angel; Iyengar, Srinivasan S (August 2024, The Journal of Physical Chemistry A)

   The exponential scaling of the quantum degrees of freedom with the size of the system is one of the biggest challenges in computational chemistry and particularly in quantum dynamics. We present a tensor network approach for the time-evolution of the nuclear degrees of freedom of multiconfigurational chemical systems at a reduced storage and computational complexity. We also present quantum algorithms for the resultant dynamics. To preserve the compression advantage achieved via tensor network decompositions, we present an adaptive algorithm for the regularization of nonphysical bond dimensions, preventing the potentially exponential growth of these with time. While applicable to any quantum dynamical problem, our method is particularly valuable for dynamical simulations of nuclear chemical systems. Our algorithm is demonstrated using ab initio potentials obtained for a symmetric hydrogen-bonded system, namely, the protonated 2,2′-bipyridine, and compared to exact diagonalization numerical results. 

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3. NOPE-SAC: Neural One-Plane RANSAC for Sparse-View Planar 3D Reconstruction

   https://doi.org/10.1109/TPAMI.2023.3314745  

   Tan, Bin; Xue, Nan; Wu, Tianfu; Xia, Gui-Song (January 2023, IEEE Transactions on Pattern Analysis and Machine Intelligence)

   This paper studies the challenging two-view 3D reconstruction problem in a rigorous sparse-view configuration, which is suffering from insufficient correspondences in the input image pairs for camera pose estimation. We present a novel Neural One-PlanE RANSAC framework (termed NOPE-SAC in short) that exerts excellent capability of neural networks to learn one-plane pose hypotheses from 3D plane correspondences. Building on the top of a Siamese network for plane detection, our NOPE-SAC first generates putative plane correspondences with a coarse initial pose. It then feeds the learned 3D plane correspondences into shared MLPs to estimate the one-plane camera pose hypotheses, which are subsequently reweighed in a RANSAC manner to obtain the final camera pose. Because the neural one-plane pose minimizes the number of plane correspondences for adaptive pose hypotheses generation, it enables stable pose voting and reliable pose refinement with a few of plane correspondences for the sparse-view inputs. In the experiments, we demonstrate that our NOPE-SAC significantly improves the camera pose estimation for the two-view inputs with severe viewpoint changes, setting several new state-of-the-art performances on two challenging benchmarks, i.e., MatterPort3D and ScanNet, for sparse-view 3D reconstruction. The source code is released at https://github.com/IceTTTb/NopeSAC for reproducible research. 

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4. Supersymmetry on the lattice: Geometry, Topology, and Spin Liquids

   https://doi.org/https://doi.org/10.48550/arXiv.2207.09475  

   Roychowdhury, Krishanu; Attig, Jan; Trebst, Simon; Lawler, Michael J. (July 2022, ArXivorg)

   In quantum mechanics, supersymmetry (SUSY) posits an equivalence between two elementary degrees of freedom, bosons and fermions. Here we show how this fundamental concept can be applied to connect bosonic and fermionic lattice models in the realm of condensed matter physics, e.g., to identify a variety of (bosonic) phonon and magnon lattice models which admit topologically nontrivial free fermion models as superpartners. At the single-particle level, the bosonic and the fermionic models that are generated by the SUSY are isospectral except for zero modes, such as flat bands, whose existence is undergirded by the Witten index of the SUSY theory. We develop a unifying framework to formulate these SUSY connections in terms of general lattice graph correspondences and discuss further ramifications such as the definition of supersymmetric topological invariants for generic bosonic systems. Notably, a Hermitian form of the supercharge operator, the generator of the SUSY, can itself be interpreted as a hopping Hamiltonian on a bipartite lattice. This allows us to identify a wide class of interconnected lattices whose tight-binding Hamiltonians are superpartners of one another or can be derived via squaring or square-rooting their energy spectra all the while preserving band topology features. We introduce a five-fold way symmetry classification scheme of these SUSY lattice correspondences, including cases with a non-zero Witten index, based on a topological classification of the underlying Hermitian supercharge operator. These concepts are illustrated for various explicit examples including frustrated magnets, Kitaev spin liquids, and topological superconductors. 

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5. Fast Two-View Motion Segmentation Using Christoffel Polynomials.

   Ozbay, B. (November 2022, 2022 European Computer Vision Conference)

   Avidan, S. (Ed.)

   We address the problem of segmenting moving rigid objects based on two-view image correspondences under a perspective camera model. While this is a well understood problem, existing methods scale poorly with the number of correspondences. In this paper we propose a fast segmentation algorithm that scales linearly with the number of correspondences and show that on benchmark datasets it offers the best trade-off between error and computational time: it is at least one order of magnitude faster than the best method (with comparable or better accuracy), with the ratio growing up to three orders of magnitude for larger number of correspondences. We approach the problem from an algebraic perspective by exploiting the fact that all points belonging to a given object lie in the same quadratic surface. The proposed method is based on a characterization of each surface in terms of the Christoffel polynomial associated with the probability that a given point belongs to the surface. This allows for efficiently segmenting points “one surface at a time” in O(number of points) 

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