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1. Effective Density for Inhomogeneous Quadratic Forms I: Generic Forms and Fixed Shifts

   https://doi.org/10.1093/imrn/rnaa206  

   Ghosh, Anish; Kelmer, Dubi; Yu, Shucheng (August 2020, International Mathematics Research Notices)

   Abstract We establish effective versions of Oppenheim’s conjecture for generic inhomogeneous quadratic forms. We prove such results for fixed shift vectors and generic quadratic forms. When the shift is rational we prove a counting result, which implies the optimal density for values of generic inhomogeneous forms. We also obtain a similar density result for fixed irrational shifts satisfying an explicit Diophantine condition. The main technical tool is a formula for the 2nd moment of Siegel transforms on certain congruence quotients of $$SL_n(\mathbb{R}),$$ which we believe to be of independent interest. In a sequel, we use different techniques to treat the companion problem concerning generic shifts and fixed quadratic forms. 

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2. The Kloosterman circle method and weighted representation numbers of quadratic forms

   https://doi.org/10.1007/s40687-025-00544-4  

   Jones, Edna (August 2025, Research in the Mathematical Sciences)

   Abstract We develop a version of the Kloosterman circle method with a bump function that is used to provide asymptotics for weighted representation numbers of nonsingular integral quadratic forms. Unlike many applications of the Kloosterman circle method, we explicitly state some constants in the error terms that depend on the quadratic form. This version of the Kloosterman circle method uses Gauss sums, Kloosterman sums, Salié sums, and a principle of nonstationary phase. We briefly discuss a potential application of this version of the Kloosterman circle method to a proof of a strong asymptotic local–global principle for certain Kleinian sphere packings. 

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3. Kinship Practices Among Alternative Family Forms in Western Industrialized Societies

   https://doi.org/10.1111/jomf.12712  

   Furstenberg, Frank F.; Harris, Lauren E.; Pesando, Luca Maria; Reed, Megan N. (August 2020, Journal of Marriage and Family)

   Abstract ObjectiveThis article discusses how kinship is construed and enacted in diverse forms of the family that are now part of the culturally pluralistic family system of Western societies. BackgroundThis study is the second in a pair documenting changes over the past century in the meaning and practice of kinship in the family system of Western societies with industrialized economies. While the first paper reviewed the history of kinship studies, this companion piece shifts the focus to research explorations of kinship in alternative family forms, those that depart from the standard nuclear family structure. MethodThe review was conducted running multiple searches on Google Scholar and Web of Science directly targeting nonstandard family forms, using search terms such as “cohabitation and kinship,” “same‐sex family and kinship,” and “Artificial Reproductive Technology and kinship,” among others. About 70% of studies focused on the United States, while the remaining 30% focused on other industrialized Western societies. ResultsWe identified three general processes by which alternative family forms are created and discussed how kinship practices work in each of them. Thefirstcluster of alternative family forms comes about throughvariations of formal marriage or its absence, including sequential marriages, plural marriages, consensual unions, single parenthood, and same‐sex marriages and partnerships. Thesecondcluster is formed as a result ofalterations in the reproduction process, when a child is not the product of sexual intercourse between two people. Thethirdcluster results from theformation of voluntary bondsthat are deemed to be kinship‐like, in which affiliation rests on neither biological nor legal bases. ConclusionFindings from this study point to a broad cultural acceptance of an inclusive approach to incorporating potential kin in “family relationships.” It is largely left to individuals to decide whether they recognize or experience the diffuse sense of emotional connectedness and perceived obligation that characterize the bond of kinship. Also, family scripts and kinship terms often borrow from the vocabulary and parenting practices observed in the standard family form in the West. Concurrently, the cultural importance of biology remains strong. ImplicationsThis study concludes by identifying important gaps in the kinship literature and laying out a research agenda for the future, including building ademography of kinship. 

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4. Shuffle Algebras and Their Integral Forms: Specialization Map Approach in Types B_n and G_2

   https://doi.org/10.1093/imrn/rnae029  

   Hu, Yue; Tsymbaliuk, Alexander (February 2024, International Mathematics Research Notices)

   Abstract We construct a family of PBWD (Poincaré-Birkhoff-Witt-Drinfeld) bases for the positive subalgebras of quantum loop algebras of type $$B_{n}$$ and $$G_{2}$$, as well as their Lusztig and RTT (for type $$B_{n}$$ only) integral forms, in the new Drinfeld realization. We also establish a shuffle algebra realization of these $${\mathbb {Q}}(v)$$-algebras (proved earlier in [26] by completely different tools) and generalize the latter to the above $${{\mathbb {Z}}}[v,v^{-1}]$$-forms. The rational counterparts provide shuffle algebra realizations of positive subalgebras of type $$B_{n}$$ and $$G_{2}$$ Yangians and their Drinfeld-Gavarini duals. All of this generalizes the type $$A_{n}$$ results of [30]. 

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5. On realization of isometries for higher rank quadratic lattices over number fields

   https://doi.org/10.1090/tran/8670  

   Chan, Wai Kiu; Li, Han (July 2022, Transactions of the American Mathematical Society)

   Let F F be a number field, and n ≥ 3 n\geq 3 be an integer. In this paper we give an effective procedure which (1) determines whether two given quadratic lattices on F n F^n are isometric or not, and (2) produces an invertible linear transformation realizing the isometry provided the two given lattices are isometric. A key ingredient in our approach is a search bound for the equivalence of two given quadratic forms over number fields which we prove using methods from algebraic groups, homogeneous dynamics and spectral theory of automorphic forms. 

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