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1. CR embeddability of quotients of the Rossi sphere via spectral theory

   https://doi.org/10.1142/S0129167X22500148  

   Bosch, Henry; Gonzales, Tyler; Spinelli, Kamryn; Udell, Gabe; Zeytuncu, Yunus E. (February 2022, International Journal of Mathematics)

   We look at the action of finite subgroups of [Formula: see text] on [Formula: see text], viewed as a CR manifold, both with the standard CR structure as the unit sphere in [Formula: see text] and with a perturbed CR structure known as the Rossi sphere. We show that quotient manifolds from these actions are indeed CR manifolds, and relate the order of the subgroup of [Formula: see text] to the asymptotic distribution of the Kohn Laplacian’s eigenvalues on the quotient. We show that the order of the subgroup determines whether the quotient of the Rossi sphere by the action of that subgroup is CR embeddable. Finally, in the unperturbed case, we prove that we can determine the size of the subgroup by using the point spectrum. 

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2. Computing cohomology groups that classify bundles of strongly self-absorbing C∗-algebras

   https://doi.org/10.1142/S1793525324500110  

   Dadarlat, Marius; McClure, James E; Pennig, Ulrich (March 2024, Journal of Topology and Analysis)

   Locally trivial bundles of [Formula: see text]-algebras with fiber [Formula: see text] for a strongly self-absorbing [Formula: see text]-algebra [Formula: see text] over a finite CW-complex [Formula: see text] form a group [Formula: see text] that is the first group of a cohomology theory [Formula: see text]. In this paper, we compute these groups by expressing them in terms of ordinary cohomology and connective [Formula: see text]-theory. To compare the [Formula: see text]-algebraic version of [Formula: see text] with its classical counterpart we also develop a uniqueness result for the unit spectrum of complex periodic topological [Formula: see text]-theory. 

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3. Marchenko–Pastur law with relaxed independence conditions

   https://doi.org/10.1142/s2010326321500404  

   Bryson, Jennifer; Vershynin, Roman; Zhao, Hongkai (January 2021, Random Matrices: Theory and Applications)

   null (Ed.)

   We prove the Marchenko–Pastur law for the eigenvalues of [Formula: see text] sample covariance matrices in two new situations where the data does not have independent coordinates. In the first scenario — the block-independent model — the [Formula: see text] coordinates of the data are partitioned into blocks in such a way that the entries in different blocks are independent, but the entries from the same block may be dependent. In the second scenario — the random tensor model — the data is the homogeneous random tensor of order [Formula: see text], i.e. the coordinates of the data are all [Formula: see text] different products of [Formula: see text] variables chosen from a set of [Formula: see text] independent random variables. We show that Marchenko–Pastur law holds for the block-independent model as long as the size of the largest block is [Formula: see text], and for the random tensor model as long as [Formula: see text]. Our main technical tools are new concentration inequalities for quadratic forms in random variables with block-independent coordinates, and for random tensors. 

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4. Linear instability of Z-pinch in plasma: Inviscid case

   https://doi.org/10.1142/S0218202521500093  

   Bian, Dongfen; Guo, Yan; Tice, Ian (February 2021, Mathematical Models and Methods in Applied Sciences)

   null (Ed.)

   The [Formula: see text]-pinch is a classical steady state for the MHD model, where a confined plasma fluid is separated by vacuum, in the presence of a magnetic field which is generated by a prescribed current along the [Formula: see text] direction. We develop a variational framework to study its stability in the absence of viscosity effect, and demonstrate for the first time that such a [Formula: see text]-pinch is always unstable. Moreover, we discover a sufficient condition such that the eigenvalues can be unbounded, which leads to ill-posedness of the linearized MHD system. 

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5. Games with filters I

   https://doi.org/10.1142/S021906132450003X  

   Foreman, Matthew; Magidor, Menachem; Zeman, Martin (August 2023, Journal of Mathematical Logic)

   Chong, Chita; Feng, Qi; Slaman, Theodore_A; Woodin, W_Hugh (Ed.)

   This paper has two parts. The first is concerned with a variant of a family of games introduced by Holy and Schlicht, that we call Welch games. Player II having a winning strategy in the Welch game of length [Formula: see text] on [Formula: see text] is equivalent to weak compactness. Winning the game of length [Formula: see text] is equivalent to [Formula: see text] being measurable. We show that for games of intermediate length [Formula: see text], II winning implies the existence of precipitous ideals with [Formula: see text]-closed, [Formula: see text]-dense trees. The second part shows the first is not vacuous. For each [Formula: see text] between [Formula: see text] and [Formula: see text], it gives a model where II wins the games of length [Formula: see text], but not [Formula: see text]. The technique also gives models where for all [Formula: see text] there are [Formula: see text]-complete, normal, [Formula: see text]-distributive ideals having dense sets that are [Formula: see text]-closed, but not [Formula: see text]-closed. 

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