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   Miruna Oprescu, Jacob Dorn (January 2023, International Conference on Machine Learning)
2. Upper bounds on the spectral gaps of quasi-periodic Schrödinger operators with Liouville frequencies

   https://doi.org/10.4171/jst/275  

   Liu, Wencai; Shi, Yunfeng (January 2019, Journal of Spectral Theory)
3. Computable error bounds for quasi-Monte Carlo using points with non-negative local discrepancy

   https://doi.org/10.1093/imaiai/iaae021  

   Gnewuch, Michael; Kritzer, Peter; Owen, Art_B; Pan, Zexin (August 2024, Information and Inference: A Journal of the IMA)

   Abstract Let $$f:[0,1]^{d}\to{\mathbb{R}}$$ be a completely monotone integrand as defined by Aistleitner and Dick (2015, Acta Arithmetica, 167, 143–171) and let points $$\boldsymbol{x}_{0},\dots ,\boldsymbol{x}_{n-1}\in [0,1]^{d}$$ have a non-negative local discrepancy (NNLD) everywhere in $$[0,1]^{d}$$. We show how to use these properties to get a non-asymptotic and computable upper bound for the integral of $$f$$ over $$[0,1]^{d}$$. An analogous non-positive local discrepancy property provides a computable lower bound. It has been known since Gabai (1967, Illinois J. Math., 11, 1–12) that the two-dimensional Hammersley points in any base $$b\geqslant 2$$ have NNLD. Using the probabilistic notion of associated random variables, we generalize Gabai’s finding to digital nets in any base $$b\geqslant 2$$ and any dimension $$d\geqslant 1$$ when the generator matrices are permutation matrices. We show that permutation matrices cannot attain the best values of the digital net quality parameter when $$d\geqslant 3$$. As a consequence the computable absolutely sure bounds we provide come with less accurate estimates than the usual digital net estimates do in high dimensions. We are also able to construct high-dimensional rank one lattice rules that are NNLD. We show that those lattices do not have good discrepancy properties: any lattice rule with the NNLD property in dimension $$d\geqslant 2$$ either fails to be projection regular or has all its points on the main diagonal. Complete monotonicity is a very strict requirement that for some integrands can be mitigated via a control variate. 

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4. Boundary behavior of compact manifolds with scalar curvature lower bounds and static quasi-local mass of tori

   https://doi.org/10.1090/proc/17223  

   Alaee, Aghil; Hung, Pei-Ken; Khuri, Marcus (March 2025, Proceedings of the American Mathematical Society)

   A classic result of Shi and Tam states that a 2-sphere of positive Gauss and mean curvature bounding a compact 3-manifold with nonnegative scalar curvature must have total mean curvature not greater than that of the isometric embedding into Euclidean 3-space, with equality only for domains in this reference manifold. We generalize this result to 2-tori of Gauss curvature greater than $-<\#comment/>1$, which bound a compact 3-manifold having scalar curvature not less than $-<\#comment/>6$and at least one other boundary component satisfying a ‘trapping condition’. The conclusion is that the total weighted mean curvature is not greater than that of an isometric embedding into the Kottler manifold, with equality only for domains in this space. Examples are given to show that the assumption of a secondary boundary component cannot be removed. The result gives a positive mass theorem for the static Brown-York mass of tori, in analogy to the Shi-Tam positivity of the standard Brown-York mass, and represents the first such quasi-local mass positivity result for nonspherical surfaces. Furthermore, we prove a Penrose-type inequality in this setting. 

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5. Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States

   https://doi.org/10.1007/s00023-021-01086-5  

   Nachtergaele, Bruno; Sims, Robert; Young, Amanda (August 2021, Annales Henri Poincaré)

   Abstract We study the stability with respect to a broad class of perturbations of gapped ground-state phases of quantum spin systems defined by frustration-free Hamiltonians. The core result of this work is a proof using the Bravyi–Hastings–Michalakis (BHM) strategy that under a condition of local topological quantum order (LTQO), the bulk gap is stable under perturbations that decay at long distances faster than a stretched exponential. Compared to previous work, we expand the class of frustration-free quantum spin models that can be handled to include models with more general boundary conditions, and models with discrete symmetry breaking. Detailed estimates allow us to formulate sufficient conditions for the validity of positive lower bounds for the gap that are uniform in the system size and that are explicit to some degree. We provide a survey of the BHM strategy following the approach of Michalakis and Zwolak, with alterations introduced to accommodate more general than just periodic boundary conditions and more general lattices. We express the fundamental condition known as LTQO by means of an indistinguishability radius, which we introduce. Using the uniform finite-volume results, we then proceed to study the thermodynamic limit. We first study the case of a unique limiting ground state and then also consider models with spontaneous breaking of a discrete symmetry. In the latter case, LTQO cannot hold for all local observables. However, for perturbations that preserve the symmetry, we show stability of the gap and the structure of the broken symmetry phases. We prove that the GNS Hamiltonian associated with each pure state has a non-zero spectral gap above the ground state. 

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