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1. Constructing nonproxy small test modules for the complete intersection property

   https://doi.org/10.1017/nmj.2021.7  

   Briggs, Benjamin; Grifo, Eloísa; Pollitz, Josh (June 2021, Nagoya Mathematical Journal)

   Abstract A local ring R is regular if and only if every finitely generated R -module has finite projective dimension. Moreover, the residue field k is a test module: R is regular if and only if k has finite projective dimension. This characterization can be extended to the bounded derived category $$\mathsf {D}^{\mathsf f}(R)$$ , which contains only small objects if and only if R is regular. Recent results of Pollitz, completing work initiated by Dwyer–Greenlees–Iyengar, yield an analogous characterization for complete intersections: R is a complete intersection if and only if every object in $$\mathsf {D}^{\mathsf f}(R)$$ is proxy small. In this paper, we study a return to the world of R -modules, and search for finitely generated R -modules that are not proxy small whenever R is not a complete intersection. We give an algorithm to construct such modules in certain settings, including over equipresented rings and Stanley–Reisner rings. 

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2. A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case

   Ongie, Greg; Willett, Rebecca; Soudry, Daniel; Srebro, Nathan (October 2019, International Conference on Learning Representations (ICLR 2020))

   We give a tight characterization of the (vectorized Euclidean) norm of weights required to realize a function f : Rd → R as a single hidden-layer ReLU network with an unbounded number of units (infinite width), extending the univariate characterization of Savarese et al. (2019) to the multivariate case. 

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3. Heterotrimetallic {LnOVPt} complexes with antiferromagnetic Ln–V coupling and magnetic memory

   https://doi.org/10.1039/D0CC04334F  

   Beach, Stephanie A.; Guillet, Jesse L.; Lagueux, Sydney P.; Perfetti, Mauro; Livesay, Brooke N.; Shores, Matthew P.; Bacon, Jeffrey W.; Rheingold, Arnold L.; Arnold, Polly L.; Doerrer, Linda H. (September 2020, Chemical Communications)

   null (Ed.)

   The new PtVO(SOCR) 4 lantern complexes, 1 (R = CH 3 ) and 2 (R = Ph) behave as neutral O-donor ligands to Ln(OR) 3 with Ln = Ce, Nd. Four heterotrimetallic complexes with linear {LnOVPt} units were prepared: [Ln(ODtbp) 3 {PtVO(SOCR) 4 }] (Ln = Ce, 3Ce (R = CH 3 ), 4Ce (R = Ph); Nd, 3Nd (R = CH 3 ), 4Nd (R = Ph); ODtbp = 2,6-ditertbutylphenolate). Magnetic characterization confirms slow magnetic relaxation behaviour and suggests antiferromagnetic coupling across {Ln–OV} in all four complexes, with variations tunable as a function of Ln and R. 

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4. A Buchsbaum theory for tight closure

   https://doi.org/10.1090/tran/8762  

   Ma, Linquan; Quy, Pham (November 2022, Transactions of the American Mathematical Society)

   A Noetherian local ring ( R , m ) (R,\frak {m}) is called Buchsbaum if the difference ℓ ( R / q ) − e ( q , R ) \ell (R/\mathfrak {q})-e(\mathfrak {q}, R) , where q \mathfrak {q} is an ideal generated by a system of parameters, is a constant independent of q \mathfrak {q} . In this article, we study the tight closure analog of this condition. We prove that in an unmixed excellent local ring ( R , m ) (R,\frak {m}) of prime characteristic p > 0 p>0 and dimension at least one, the difference e ( q , R ) − ℓ ( R / q ∗ ) e(\mathfrak {q}, R)-\ell (R/\mathfrak {q}^*) is independent of q \mathfrak {q} if and only if the parameter test ideal τ p a r ( R ) \tau _{\mathrm {par}}(R) contains m \frak {m} . We also provide a characterization of this condition via derived category which is analogous to Schenzel’s criterion for Buchsbaum rings. 

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5. Orthogonal rational functions with real poles, root asymptotics, and GMP matrices

   https://doi.org/10.1090/btran/117  

   Eichinger, Benjamin; Lukić, Milivoje; Young, Giorgio (January 2023, Transactions of the American Mathematical Society, Series B)

   There is a vast theory of the asymptotic behavior of orthogonal polynomials with respect to a measure on R \mathbb {R} and its applications to Jacobi matrices. That theory has an obvious affine invariance and a very special role for ∞ \infty . We extend aspects of this theory in the setting of rational functions with poles on R ¯ = R ∪ { ∞ } \overline {\mathbb {R}} = \mathbb {R} \cup \{\infty \} , obtaining a formulation which allows multiple poles and proving an invariance with respect to R ¯ \overline {\mathbb {R}} -preserving Möbius transformations. We obtain a characterization of Stahl–Totik regularity of a GMP matrix in terms of its matrix elements; as an application, we give a proof of a conjecture of Simon – a Cesàro–Nevai property of regular Jacobi matrices on finite gap sets. 

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