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1. The growth rate of surface area measure for noncompact convex sets with prescribed asymptotic cone

   https://doi.org/10.1090/tran/9470  

   Semenov, Vadim; Zhao, Yiming (May 2025, Transactions of the American Mathematical Society)

   The Minkowski problem for a class of unbounded closed convex sets is considered. This is equivalent to a Monge-Ampère equation on a bounded convex open domain with possibly non-integrable given data. A complete solution (necessary and sufficient condition for existence and uniqueness) in dimension 2 is presented. In higher dimensions, partial results are demonstrated. 

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2. On the local converse theorem for split 𝑆𝑂_{2𝑙}

   https://doi.org/10.1090/ert/687  

   Hazeltine, Alexander; Liu, Baiying (March 2025, Representation Theory)

   In this paper, we prove the local converse theorem for split even special orthogonal groups over a non-Archimedean local field of characteristic $p\ne <\#comment/>2$. This is the only case left on local converse theorems of split classical groups and the difficulty is the existence of the outer automorphism. We apply a new idea by considering a certain sum of partial Bessel functions to overcome this difficulty. As a direct application, we obtain a weak rigidity theorem for irreducible generic cuspidal representations of split even special orthogonal groups. 

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3. Inversion formula and range conditions for a linear system related with the multi‐interval finite Hilbert transform in L ²

   https://doi.org/10.1002/mana.201800567  

   Katsevich, Alexander; Bertola, Marco; Tovbis, Alexander (June 2021, Mathematische Nachrichten)

   Abstract Givenndisjoint intervals on together withnfunctions , , and an matrix , the problem is to find anL²solution , , to the linear system , where , is a matrix of finite Hilbert transforms with defined on , and is a matrix of the corresponding characteristic functions on . Since we can interpret , as a generalized multi‐interval finite Hilbert transform, we call the formula for the solution as “the inversion formula” and the necessary and sufficient conditions for the existence of a solution as the “range conditions”. In this paper we derive the explicit inversion formula and the range conditions in two specific cases: a) the matrix Θ is symmetric and positive definite, and; b) all the entries of Θ are equal to one. We also prove the uniqueness of solution, that is, that our transform is injective. In the case a), that is, when the matrix Θ is positive definite, the inversion formula is given in terms of the solution of the associated matrix Riemann–Hilbert Problem. In the case b) we reduce the multi interval problem to a problem onncopies of and then express our answers in terms of the Fourier transform. We also discuss other cases of the matrix Θ. 

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4. Nearly magic rectangles

   https://doi.org/10.1002/jcd.21667  

   Chai, Feng‐Shun; Singh, Rakhi; Stufken, John (July 2019, Journal of Combinatorial Designs)

   Abstract Magic squares have been extremely useful and popular in combinatorics and statistics. One generalization of magic squares ismagic rectangleswhich are useful for designing experiments in statistics. A necessary and sufficient condition for the existence of magic rectangles restricts the number of rows and columns to be either both odd or both even. In this paper, we generalize magic rectangles to even by oddnearly magic rectangles. We also prove necessary and sufficient conditions for the existence of a nearly magic rectangle, and construct one for each parameter set for which they exist. 

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5. On dimensions supporting a rational projective plane

   https://doi.org/10.1142/S1793525319500237  

   Kennard, Lee; Su, Zhixu (July 2019, Journal of Topology and Analysis)

   A rational projective plane ([Formula: see text]) is a simply connected, smooth, closed manifold [Formula: see text] such that [Formula: see text]. An open problem is to classify the dimensions at which such a manifold exists. The Barge–Sullivan rational surgery realization theorem provides necessary and sufficient conditions that include the Hattori–Stong integrality conditions on the Pontryagin numbers. In this paper, we simplify these conditions and combine them with the signature equation to give a single quadratic residue equation that determines whether a given dimension supports a [Formula: see text]. We then confirm the existence of a [Formula: see text] in two new dimensions and prove several non-existence results using factorization of the numerators of the divided Bernoulli numbers. We also resolve the existence question in the Spin case, and we discuss existence results for the more general class of rational projective spaces. 

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