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1. 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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2. Combining Finite Combination Properties: Finite Models and Busy Beavers

   https://doi.org/10.1007/978-3-031-43369-6_9  

   Toledo, Guilherme V.; Zohar, Yoni; Barrett, Clark (September 2023, Lecture Notes in Computer Science)

   Sattler, Uli; Suda, Martin (Ed.)

   This work is a part of an ongoing effort to understand the relationships between properties used in theory combination. We here focus on including two properties that are related to shiny theories: the finite model property and stable finiteness. For any combination of properties, we consider the question of whether there exists a theory that exhibits it. When there is, we provide an example with the simplest possible signature. One particular class of interest includes theories with the finite model property that are not finitely witnessable. To construct such theories, we utilize the Busy Beaver function. 

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3. Globally Coupled Finite Geometry and Finite Field LDPC Coding Schemes

   https://doi.org/10.1109/TVT.2021.3102178  

   Nasseri, Mona; Xiao, Xin; Vasic, Bane; Lin, Shu (September 2021, IEEE Transactions on Vehicular Technology)
4. Finite permutation resolutions

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   Balmer, Paul; Gallauer, Martin (February 2023, Duke Mathematical Journal)
5. Finite-Word Hyperlanguages

   Bonakdarpour, Borzoo; Sheinvald, Sarai (March 2021, 15th International Conference on Language and Automata Theory and Applications (LATA))

   null (Ed.)