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1. Combining Combination Properties: An Analysis of Stable Infiniteness, Convexity, and Politeness

   https://doi.org/10.1007/978-3-031-38499-8_30  

   Toledo, Guilherme V.; Zohar, Yoni; Barrett, Clark (September 2023, Lecture notes in computer science)

   Pientka, B.; Tinelli, C. (Ed.)

   We make two contributions to the study of theory combination in satisfiability modulo theories. The first is a table of examples for the combinations of the most common model-theoretic properties in theory combination, namely stable infiniteness, smoothness, convexity, finite witnessability, and strong finite witnessability (and therefore politeness and strong politeness as well). All of our examples are sharp, in the sense that we also offer proofs that no theories are available within simpler signatures. This table significantly progresses the current understanding of the various properties and their interactions. The most remarkable example in this table is of a theory over a single sort that is polite but not strongly polite (the existence of such a theory was only known until now for two-sorted signatures). The second contribution is a new combination theorem showing that in order to apply polite theory combination, it is sufficient for one theory to be stably infinite and strongly finitely witnessable, thus showing that smoothness is not a critical property in this combination method. This result has the potential to greatly simplify the process of showing which theories can be used in polite combination, as showing stable infiniteness is considerably simpler than showing smoothness. 

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2. The Nonexistence of Unicorns and Many-Sorted Löwenheim–Skolem Theorems

   https://doi.org/10.1007/978-3-031-71162-6_34  

   Przybocki, Benjamin; Toledo, Guilherme; Zohar, Yoni; Barrett, Clark (September 2024, Springer Nature Switzerland)

   Platzer, André; Rozier, Kristin Yvonne; Pradella, Matteo; Rossi, Matteo (Ed.)

   Abstract Stable infiniteness, strong finite witnessability, and smoothness are model-theoretic properties relevant to theory combination in satisfiability modulo theories. Theories that are strongly finitely witnessable and smooth are calledstrongly politeand can be effectively combined with other theories. Toledo, Zohar, and Barrett conjectured that stably infinite and strongly finitely witnessable theories are smooth and therefore strongly polite. They called counterexamples to this conjectureunicorn theories, as their existence seemed unlikely. We prove that, indeed, unicorns do not exist. We also prove versions of the Löwenheim–Skolem theorem and the Łoś–Vaught test for many-sorted logic. 

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3. 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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4. Residually finite rationally p groups

   https://doi.org/10.1142/S0219199719500160  

   Koberda, Thomas; Suciu, Alexander I. (May 2020, Communications in Contemporary Mathematics)

   In this paper, we develop the theory of residually finite rationally [Formula: see text] (RFR[Formula: see text]) groups, where [Formula: see text] is a prime. We first prove a series of results about the structure of finitely generated RFR[Formula: see text] groups (either for a single prime [Formula: see text], or for infinitely many primes), including torsion-freeness, a Tits alternative, and a restriction on the BNS invariant. Furthermore, we show that many groups which occur naturally in group theory, algebraic geometry, and in 3-manifold topology enjoy this residual property. We then prove a combination theorem for RFR[Formula: see text] groups, which we use to study the boundary manifolds of algebraic curves [Formula: see text] and in [Formula: see text]. We show that boundary manifolds of a large class of curves in [Formula: see text] (which includes all line arrangements) have RFR[Formula: see text] fundamental groups, whereas boundary manifolds of curves in [Formula: see text] may fail to do so. 

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5. Direct serendipity finite elements on cuboidal hexahedra

   https://doi.org/10.1016/j.cma.2024.117500  

   Arbogast, Todd; Wang, Chuning (January 2025, Computer Methods in Applied Mechanics and Engineering)

   We construct direct serendipity finite elements on general cuboidal hexahedra, which are 𝐻1-conforming and optimally approximate to any order. The new finite elements are direct in that the shape functions are directly defined on the physical element. Moreover, they are serendipity by possessing a minimal number of degrees of freedom satisfying the conformity requirement. Their shape function spaces consist of polynomials plus (generally nonpolynomial) supplemental functions, where the polynomials are included for the approximation property and supplements are added to achieve 𝐻1 -conformity. The finite elements are fully constructive. The shape function spaces of higher order 𝑟 ≥ 3 are developed first, and then the lower order spaces are constructed as subspaces of the third order space. Under a shape regularity assumption, and a mild restriction on the choice of supplemental functions, we develop the convergence properties of the new direct serendipity finite elements. Numerical results with different choices of supplements are compared on two mesh sequences, one regularly distorted and the other one randomly distorted. They all possess a convergence rate that aligns with the theory, while a slight difference lies in their performance. 

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