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1. Toward extracting the scattering phase shift from integrated correlation functions. II. A relativistic lattice field theory model

   https://doi.org/10.1103/PhysRevD.110.014504  

   Guo, Peng (July 2024, Physical Review D)

   In the present work, a relativistic relation that connects the difference of interacting and noninteracting integrated two-particle correlation functions in finite volume to infinite volume scattering phase shift through an integral is derived. We show that the difference of integrated finite volume correlation functions converges rapidly to its infinite volume limit as the size of the periodic box is increased. The fast convergence of our proposed formalism is illustrated by analytic solutions of a contact interaction model, the perturbation theory calculation, and also the Monte Carlo simulation of a complex ${\varphi }^4$lattice field theory model. Published by the American Physical Society2024 

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2. Box-ball systems and RSK tableaux

   Drucker, Ben; Garcia, Eli; Gunawan, Emily; Silver, Rose (January 2021, Séminaire lotharingien de combinatoire)

   A box-ball system is a collection of discrete time states. At each state, we have a collection of countably many boxes with each integer from 1 to n assigned to a unique box; the remaining boxes are considered empty. A permutation on n objects gives a box-ball system state by assigning the permutation in one-line notation to the first n boxes. After a finite number of steps, the system will reach a so-called soliton decomposition which has an integer partition shape. We prove the following: if the soliton decomposition of a permutation is a standard Young tableau or if its shape coincides with its Robinson–Schensted (RS) partition, then its soliton decomposition and its RS insertion tableau are equal. We study the time required for a box-ball system to reach a steady state. We also generalize Fukuda’s single-carrier algorithm to algorithms with more than one carrier. 

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3. Thin spectra and singular continuous spectral measures for limit‐periodic Jacobi matrices

   https://doi.org/10.1002/mana.202100561  

   Damanik, David; Fillman, Jake; Wang, Chunyi (June 2023, Mathematische Nachrichten)

   Abstract This paper investigates the spectral properties of Jacobi matrices with limit‐periodic coefficients. We show that generically the spectrum is a Cantor set of zero Lebesgue measure, and the spectral measures are purely singular continuous. For a dense set of limit‐periodic Jacobi matrices, we show that the spectrum is a Cantor set of zero lower box counting dimension while still retaining the singular continuity of the spectral type. We also show how results of this nature can be established by fixing the off‐diagonal coefficients and varying only the diagonal coefficients, and, in a more restricted version, by fixing the diagonal coefficients to be zero and varying only the off‐diagonal coefficients. We apply these results to produce examples of weighted Laplacians on the multidimensional integer lattice having purely singular continuous spectral type and zero‐dimensional spectrum. 

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4. The Novikov conjecture, the group of volume preserving diffeomorphisms and Hilbert-Hadamard spaces

   Gong, Sherry; Wu, Jianchao; Yu, Guoliang (August 2024, Geometric and functional analysis)

   We prove that the Novikov conjecture holds for any discrete group admitting an isometric and metrically proper action on an admissible Hilbert-Hadamard space. Admissible Hilbert-Hadamard spaces are a class of (possibly infinite-dimensional) non-positively curved metric spaces that contain dense sequences of closed convex subsets isometric to Riemannian manifolds. Examples of admissible Hilbert-Hadamard spaces include Hilbert spaces, certain simply connected and non-positively curved Riemannian-Hilbertian manifolds and infinite-dimensional symmetric spaces. Thus our main theorem can be considered as an infinite-dimensional analogue of Kasparov’s theorem on the Novikov conjecture for groups acting properly and isometrically on complete, simply connected and non-positively curved manifolds. As a consequence, we show that the Novikov conjecture holds for geometrically discrete subgroups of the group of volume preserving diffeomorphisms of a closed smooth manifold. This result is inspired by Connes’ theorem that the Novikov conjecture holds for higher signatures associated to the Gelfand-Fuchs classes of groups of diffeormorphisms. 

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5. A General Technique for Searching in Implicit Sets via Function Inversion

   https://doi.org/10.1137/1.9781611977936.20  

   Aronov, Boris; Cardinal, Jean; Dallant, Justin; Iacono, John (January 2024, SIAM)

   Given a function f from the set [N] to a d-dimensional integer grid, we consider data structures that allow efficient orthogonal range searching queries in the image of f, without explicitly storing it. We show that, if f is of the form [N] → [2w]d for some w = polylog(N) and is computable in constant time, then, for any 0 < α < 1, we can obtain a data structure using Õ(N1-α/3) words of space such that, for a given d-dimensional axis-aligned box B, we can search for some x ∈ [N] such that f (x) ∈ B in time Õ(Nα). This result is obtained simply by combining integer range searching with the Fiat-Naor function inversion scheme, which was already used in data-structure problems previously. We further obtain • data structures for range counting and reporting, predecessor, selection, ranking queries, and combinations thereof, on the set f ([N]), • data structures for preimage size and preimage selection queries for a given value of f, and • data structures for selection and ranking queries on geometric quantities computed from tuples of points in d-space. These results unify and generalize previously known results on 3SUM-indexing and string searching, and are widely applicable as a black box to a variety of problems. In particular, we give a data structure for a generalized version of gapped string indexing, and show how to preprocess a set of points on an integer grid in order to efficiently compute (in sublinear time), for points contained in a given axis-aligned box, their Theil-Sen estimator, the kth largest area triangle, or the induced hyperplane that is the kth furthest from the origin. 

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