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1. Monotone T-convex T-differential fields

   https://doi.org/10.2140/mt.2025.4.55  

   Kaplan, Elliot; Pynn-Coates, Nigel (January 2025, Model Theory)

   Let T be a complete, model complete o-minimal theory extending the theory of real closed ordered fields and assume that T is power bounded. Let K be a model of T equipped with a T-convex valuation ring O and a T-derivation ∂ such that ∂ is monotone, i.e., weakly contractive with respect to the valuation induced by O. We show that the theory of monotone T-convex T-differential fields, i.e., the common theory of such K, has a model completion, which is complete and distal. Among the axioms of this model completion, we isolate an analogue of henselianity that we call T∂-henselianity. We establish an Ax-Kochen/Ershov theorem and further results for monotone T-convex T-differential fields that are T∂-henselian. 

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2. Some bounds arising from a polynomial ideal associated to any $$t$$-design

   https://doi.org/10.13069/jacodesmath.729446  

   MARTIN, William J; STINSON, Douglas R (May 2020, Journal of Algebra Combinatorics Discrete Structures and Applications)

   null (Ed.)

   We consider ordered pairs (X,B) where X is a finite set of size v and B is some collection of k-element subsets of X such that every t-element subset of X is contained in exactly λ "blocks'' b ∈B for some fixed λ. We represent each block b by a zero-one vector c_b of length v and explore the ideal I(B) of polynomials in v variables with complex coefficients which vanish on the set { c_b ∣ b ∈ B}. After setting up the basic theory, we investigate two parameters related to this ideal: γ1(B) is the smallest degree of a non-trivial polynomial in the ideal I(B) and γ2(B) is the smallest integer s such that I(B) is generated by a set of polynomials of degree at most s. We first prove the general bounds t/2 < γ1(B) ≤ γ2(B) ≤ k. Examining important families of examples, we find that, for symmetric 2-designs and Steiner systems, we have γ2(B) ≤ t. But we expect γ2(B) to be closer to k for less structured designs and we indicate this by constructing infinitely many triple systems satisfying γ2(B) = k. 

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3. Revisiting closed asymptotic couples

   https://doi.org/10.1017/S0013091522000219  

   Aschenbrenner, Matthias; van den Dries, Lou; van der Hoeven, Joris (May 2022, Proceedings of the Edinburgh Mathematical Society)

   Abstract Every discrete definable subset of a closed asymptotic couple with ordered scalar field $${\boldsymbol {k}}$$ is shown to be contained in a finite-dimensional $${\boldsymbol {k}}$$ -linear subspace of that couple. It follows that the differential-valued field $$\mathbb {T}$$ of transseries induces more structure on its value group than what is definable in its asymptotic couple equipped with its scalar multiplication by real numbers, where this asymptotic couple is construed as a two-sorted structure with $$\mathbb {R}$$ as the underlying set for the second sort. 

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4. Bounding cohomology classes over semiglobal fields

   https://doi.org/10.1007/s11856-023-2549-x  

   Harbater, David; Hartmann, Julia; Krashen, Daniel (November 2023, Israel Journal of Mathematics)

   We provide a uniform bound for the index of cohomology classes over semiglobal fields (i.e., over one-variable function fields over a complete discretely valued field K). The bound is given in terms of the analogous data for the residue field of K and its finitely generated extensions of transcendence degree at most one. We also obtain analogous bounds for collections of cohomology classes. Our results provide recursive formulas for function fields over higher rank complete discretely valued fields, and explicit bounds in some cases when the information on the residue field is known. In the process, we prove a splitting result for cohomology classes of degree 3 in the context of surfaces over finite fields. ∗ 

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5. Noncollinear polar magnet Fe2(SeO3)3(H2O)3 with inequivalent Fe3+ sites

   https://doi.org/10.1063/5.0241243  

   Oyeka, Ebube_E; Huai, Xudong; Marshall, Madalynn; Winiarski, Michał_J; Błachowski, Artur; Cao, Huibo; Tran, Thao_T (December 2024, APL Materials)

   The emergence of novel magnetic states becomes more likely when the inversion symmetry of the crystal field, relative to the center between two spins, is broken. We propose that placing magnetic spins in inequivalent sites in a polar lattice can promote a realization of nontrivial magnetic states and associated magnetic properties. To test our hypothesis, we study Fe2(SeO3)(H2O)3 as a model system that displays two distinct Fe(1) and Fe(2) magnetic sites in a polar structure (R3c space group). At low fields μ0H≤ 0.06 T, the material undergoes an antiferromagnetic ordering with TN1 = 77 K and a second transition at TN2≈ 4 K. At μ0H≥ 0.06 T and 74 K ≤T≤ 76 K, a positive entropy change of ∼0.12 mJ mol−1 K−1 can be associated with a metamagnetic transition to possibly nontrivial spin states. At zero field, Fe(1) is nearly fully ordered at T≈ 25 K, while Fe(2) features magnetic frustration down to T = 4 K. The magnetic ground state, a result corroborated by single-crystal neutron diffraction and 57Fe Mössbauer spectroscopy, is a noncollinear antiparallel arrangement of ferrimagnetic Fe(1)–Fe(2) dimers along the c-axis. The results demonstrate that placing distinct magnetic sites in a polar crystal lattice can enable a new pathway to modifying spin, orbital, and lattice degrees of freedom for unconventional magnetism. 

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