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1. Numerical semigroups, polyhedra, and posets II: locating certain families of semigroups

   https://doi.org/10.1515/advgeom-2021-0024  

   Autry, Jackson; Ezell, Abigail; Gomes, Tara; O’Neill, Christopher; Preuss, Christopher; Saluja, Tarang; Davila, Eduardo Torres (January 2022, Advances in Geometry)

   Abstract Several recent papers have examined a rational polyhedronP_mwhose integer points are in bijection with the numerical semigroups (cofinite, additively closed subsets of the non-negative integers) containingm. A combinatorial description of the faces ofP_mwas recently introduced, one that can be obtained from the divisibility posets of the numerical semigroups a given face contains. In this paper, we study the faces ofP_mcontaining arithmetical numerical semigroups and those containing certain glued numerical semigroups, as an initial step towards better understanding the full face structure ofP_m. In most cases, such faces only contain semigroups from these families, yielding a tight connection to the geometry ofP_m. 

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2. Complete Logarithmic Sobolev inequality via Ricci curvature bounded below II

   https://doi.org/10.1142/S1793525321500461  

   Brannan, Michael; Gao, Li; Junge, Marius (September 2023, Journal of Topology and Analysis)

   We study the “geometric Ricci curvature lower bound”, introduced previously by Junge, Li and LaRacuente, for a variety of examples including group von Neumann algebras, free orthogonal quantum groups [Formula: see text], [Formula: see text]-deformed Gaussian algebras and quantum tori. In particular, we show that Laplace operator on [Formula: see text] admits a factorization through the Laplace–Beltrami operator on the classical orthogonal group, which establishes the first connection between these two operators. Based on a non-negative curvature condition, we obtain the completely bounded version of the modified log-Sobolev inequalities for the corresponding quantum Markov semigroups on the examples mentioned above. We also prove that the “geometric Ricci curvature lower bound” is stable under tensor products and amalgamated free products. As an application, we obtain a sharp Ricci curvature lower bound for word-length semigroups on free group factors. 

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3. Numerical Semigroups, Polyhedra, and Posets III: Minimal Presentations and Face Dimension

   https://doi.org/10.37236/10380  

   Gomes, Tara; O'Neill, Christopher; Torres_Davila, Eduardo (April 2023, The Electronic Journal of Combinatorics)

   This paper is the third in a series of manuscripts that examine the combinatorics of the Kunz polyhedron $$P_m$$, whose positive integer points are in bijection with numerical semigroups (cofinite subsemigroups of $$\mathbb Z_{\ge 0}$$) whose smallest positive element is $$m$$. The faces of $$P_m$$ are indexed by a family of finite posets (called Kunz posets) obtained from the divisibility posets of the numerical semigroups lying on a given face. In this paper, we characterize to what extent the minimal presentation of a numerical semigroup can be recovered from its Kunz poset. In doing so, we prove that all numerical semigroups lying on the interior of a given face of $$P_m$$ have identical minimal presentation cardinality, and we provide a combinatorial method of obtaining the dimension of a face from its corresponding Kunz poset. 

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4. Relativistic Faddeev 3D equations for three-body bound states without two-body t -matrices

   https://doi.org/10.1093/ptep/ptad153  

   Mohammadzadeh, M.; Radin, M.; Hadizadeh, M. R. (December 2023, Progress of Theoretical and Experimental Physics)

   Abstract This paper explores a novel revision of the Faddeev equation for three-body (3B) bound states, as initially proposed in Ref. [J. Golak, K. Topolnicki, R. Skibiński, W. Glöckle, H. Kamada, A. Nogga, Few Body Syst. 54, 2427 (2013)]. This innovative approach, referred to as t-matrix-free in this paper, directly incorporates two-body (2B) interactions and completely avoids the 2B transition matrices. We extend this formalism to relativistic 3B bound states using a three-dimensional (3D) approach without using partial wave decomposition. To validate the proposed formulation, we perform a numerical study using spin-independent Malfliet–Tjon and Yamaguchi interactions. Our results demonstrate that the relativistic t-matrix-free Faddeev equation, which directly implements boosted interactions, accurately reproduces the 3B mass eigenvalues obtained from the conventional form of the Faddeev equation, referred to as t-matrix-dependent in this paper, with boosted 2B t-matrices. Moreover, the proposed formulation provides a simpler alternative to the standard approach, avoiding the computational complexity of calculating boosted 2B t-matrices and leading to significant computational time savings. 

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5. A Flexible Extended Krylov Subspace Method for Approximating Markov Functions of Matrices

   https://doi.org/10.3390/math11204341  

   Xu, Shengjie; Xue, Fei (October 2023, Mathematics)

   A flexible extended Krylov subspace method (F-EKSM) is considered for numerical approximation of the action of a matrix function f(A) to a vector b, where the function f is of Markov type. F-EKSM has the same framework as the extended Krylov subspace method (EKSM), replacing the zero pole in EKSM with a properly chosen fixed nonzero pole. For symmetric positive definite matrices, the optimal fixed pole is derived for F-EKSM to achieve the lowest possible upper bound on the asymptotic convergence factor, which is lower than that of EKSM. The analysis is based on properties of Faber polynomials of A and (I−A/s)−1. For large and sparse matrices that can be handled efficiently by LU factorizations, numerical experiments show that F-EKSM and a variant of RKSM based on a small number of fixed poles outperform EKSM in both storage and runtime, and usually have advantages over adaptive RKSM in runtime. 

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