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1. A weaker notion of the finite factorization property

   Jiang, Henry; Kanungo, Shihan; Kim, Harry (April 2024, Communications of the Korean Mathematical Society)

   An (additive) commutative monoid is called atomic if every given non-invertible element can be written as a sum of atoms (i.e., irreducible elements), in which case, such a sum is called a factorization of the given element. The number of atoms (counting repetitions) in the corresponding sum is called the length of the factorization. Following Geroldinger and Zhong, we say that an atomic monoid M is a length-finite factorization monoid if each b ∈ M has only finitely many factorizations of any prescribed length. An additive submonoid of ℝ≥0 is called a positive monoid. Factorizations in positive monoids have been actively studied in recent years. The main purpose of this paper is to give a better understanding of the non-unique factorization phenomenon in positive monoids through the lens of the length-finite factorization property. To do so, we identify a large class of positive monoids which satisfy the length-finite factorization property. Then we compare the length-finite factorization property to the bounded and the finite factorization properties, which are two properties that have been systematically investigated for more than thirty years. 

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2. The finite dual coalgebra as a quantization of the maximal spectrum

   https://doi.org/10.1016/j.jalgebra.2023.12.035  

   Reyes, Manuel L. (January 2024, Journal of Algebra)

   In pursuit of a noncommutative spectrum functor, we argue that the Heyneman-Sweedler finite dual coalgebra can be viewed as a quantization of the maximal spectrum of a commutative affine algebra, integrating prior perspectives of Takeuchi, Batchelor, Kontsevich-Soibelman, and Le Bruyn. We introduce fully residually finite-dimensional algebras A as those with enough finite-dimensional representations to let A^o act as an appropriate depiction of the noncommutative maximal spectrum of A; importantly, this class includes affine noetherian PI algebras. In the case of prime affine algebras that are module-finite over their center, we describe how the Azumaya locus is represented in the finite dual. This is used to describe the finite dual of quantum planes at roots of unity as an endeavor to visualize the noncommutative space on which these algebras act as functions. Finally, we discuss how a similar analysis can be carried out for other maximal orders over surfaces. 

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3. Determinant of the Finite Volume Laplacian

   https://doi.org/10.1007/s00454-022-00429-1  

   Doehrman, Thomas; Glickenstein, David (August 2022, Discrete & Computational Geometry)

   The finite volume Laplacian can be defined in all dimensions and is a natural way to approximate the operator on a simplicial mesh. In the most general setting, its definition with orthogonal duals may require that not all volumes are positive; an example is the case corresponding to two-dimensional finite elements on a non-Delaunay triangulation. Nonetheless, in many cases two- and three-dimensional Laplacians can be shown to be negative semidefinite with a kernel consisting of constants. This work generalizes work in two dimensions that gives a geometric description of the Laplacian determinant; in particular, it relates the Laplacian determinant on a simplex in any dimension to certain volume quantities derived from the simplex geometry. 

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4. Connections between finite difference and finite element approximations

   https://doi.org/https://doi.org/10.1080/00036811.2021.2005785  

   Constantin Bacuta; Cristina Bacuta (October 2021, Applicable analysis)

   Taylor And Francis Online (Ed.)

   We present useful connections between the finite difference and the finite element methods for a model boundary value problem. We start from the observation that, in the finite element context, the interpolant of the solution in one dimension coincides with the finite element approximation of the solution. This result can be viewed as an extension of the Green function formula for the solution at the continuous level. We write the finite difference and the finite element systems such that the two corresponding linear systems have the same stiffness matrices and compare the right hand side load vectors for the two methods. Using evaluation of the Green function, a formula for the inverse of the stiffness matrix is extended to the case of non-uniformly distributed mesh points. We provide an error analysis based on the connection between the two methods and estimate the energy norm of the difference of the two solutions. Interesting extensions to the 2D case are provided. 

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5. The spread of a finite group

   https://doi.org/doi.org/10.4007/annals.2021.193.2.5  

   Burness, Tim; Guralnick, Robert; Harper, Scott (January 2021, Annals of mathematics)

   null (Ed.)

   A group G is said to be 3/2-generated if every nontrivial element belongs to a generating pair. It is easy to see that if G has this property, then every proper quotient of G is cyclic. In this paper we prove that the converse is true for finite groups, which settles a conjecture of Breuer, Guralnick and Kantor from 2008. In fact, we prove a much stronger result, which solves a problem posed by Brenner and Wiegold in 1975. Namely, if G is a finite group and every proper quotient of G is cyclic, then for any pair of nontrivial elements x1, x2 ϵ G, there exists y ϵ G such that G = ⟨x1, y⟩ = ⟨x2, y⟩. In other words, s(G) ⩾ 2, where s(G) is the spread of G. Moreover, if u(G) denotes the more restrictive uniform spread of G, then we can completely characterise the finite groups G with u(G) = 0 and u(G) = 1. To prove these results, we first establish a reduction to almost simple groups. For simple groups, the result was proved by Guralnick and Kantor in 2000 using probabilistic methods, and since then the almost simple groups have been the subject of several papers. By combining our reduction theorem and this earlier work, it remains to handle the groups with socle an exceptional group of Lie type, and this is the case we treat in this paper. 

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