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1. On the Support of Anomalous Dissipation Measures

   https://doi.org/10.1007/s00021-024-00894-z  

   De_Rosa, Luigi; Drivas, Theodore D; Inversi, Marco (November 2024, Journal of Mathematical Fluid Mechanics)

   Abstract By means of a unifying measure-theoretic approach, we establish lower bounds on the Hausdorff dimension of the space-time set which can support anomalous dissipation for weak solutions of fluid equations, both in the presence or absence of a physical boundary. Boundary dissipation, which can occur at both the time and the spatial boundary, is analyzed by suitably modifying the Duchon & Robert interior distributional approach. One implication of our results is that any bounded Euler solution (compressible or incompressible) arising as a zero viscosity limit of Navier–Stokes solutions cannot have anomalous dissipation supported on a set of dimension smaller than that of the space. This result is sharp, as demonstrated by entropy-producing shock solutions of compressible Euler (Drivas and Eyink in Commun Math Phys 359(2):733–763, 2018.https://doi.org/10.1007/s00220-017-3078-4; Majda in Am Math Soc 43(281):93, 1983.https://doi.org/10.1090/memo/0281) and by recent constructions of dissipative incompressible Euler solutions (Brue and De Lellis in Commun Math Phys 400(3):1507–1533, 2023.https://doi.org/10.1007/s00220-022-04626-0 624; Brue et al. in Commun Pure App Anal, 2023), as well as passive scalars (Colombo et al. in Ann PDE 9(2):21–48, 2023.https://doi.org/10.1007/s40818-023-00162-9; Drivas et al. in Arch Ration Mech Anal 243(3):1151–1180, 2022.https://doi.org/10.1007/s00205-021-01736-2). For$$L^q_tL^r_x$$ $L_t^q{L}_x^r$suitable Leray–Hopf solutions of the$$d-$$ $d-$dimensional Navier–Stokes equation we prove a bound of the dissipation in terms of the Parabolic Hausdorff measure$$\mathcal {P}^{s}$$ $P^s$, which gives$$s=d-2$$ $s=d-2$as soon as the solution lies in the Prodi–Serrin class. In the three-dimensional case, this matches with the Caffarelli–Kohn–Nirenberg partial regularity. 

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2. 5-dimensional space-periodic solutions of the static vacuum Einstein equations

   https://doi.org/10.1007/JHEP12(2020)002  

   Khuri, Marcus; Weinstein, Gilbert; Yamada, Sumio (December 2020, Journal of High Energy Physics)

   null (Ed.)

   A bstract An affirmative answer is given to a conjecture of Myers concerning the existence of 5-dimensional regular static vacuum solutions that balance an infinite number of black holes, which have Kasner asymptotics. A variety of examples are constructed, having different combinations of ring S 1 × S 2 and sphere S 3 cross-sectional horizon topologies. Furthermore, we show the existence of 5-dimensional vacuum solitons with Kasner asymptotics. These are regular static space-periodic vacuum spacetimes devoid of black holes. Consequently, we also obtain new examples of complete Riemannian manifolds of nonnegative Ricci curvature in dimension 4, and zero Ricci curvature in dimension 5, having arbitrarily large as well as infinite second Betti number. 

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3. Vector-Valued Maximal Inequalities and Multiparameter Oscillation Inequalities for the Polynomial Ergodic Averages Along Multi-dimensional Subsets of Primes

   https://doi.org/10.1007/s00041-024-10119-6  

   Mehlhop, Nathan (December 2024, Journal of Fourier Analysis and Applications)

   Abstract We prove uniform$$\ell ^2$$ ${\ell }^2$-valued maximal inequalities for polynomial ergodic averages and truncated singular operators of Cotlar type modeled over multidimensional subsets of primes. In the averages case, we combine this with earlier one-parameter oscillation estimates (Mehlhop and Słomian in Math Ann, 2023,https://doi.org/10.1007/s00208-023-02597-8) to prove corresponding multiparameter oscillation estimates. This provides a fuller quantitative description of the pointwise convergence of the mentioned averages and is a generalization of the polynomial Dunford–Zygmund ergodic theorem attributed to Bourgain (Mirek et al. in Rev Mat Iberoam 38:2249–2284, 2022). 

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4. Insight into Voting Problem Complexity Using Randomized Classes

   https://doi.org/10.24963/ijcai.2022/42  

   Fitzsimmons, Zack; Hemaspaandra, Edith (July 2022, International Joint Conference on Artificial Intelligence (IJCAI))

   The first step in classifying the complexity of an NP problem is typically showing the problem in P or NP-complete. This has been a successful first step for many problems, including voting problems. However, in this paper we show that this may not always be the best first step. We consider the problem of constructive control by replacing voters (CCRV) introduced by Loreggia et al. [2015, https://dl.acm.org/doi/10.5555/2772879.2773411] for the scoring rule First-Last, which is defined by (1, 0, ..., 0, -1). We show that this problem is equivalent to Exact Perfect Bipartite Matching, and so CCRV for First-Last can be determined in random polynomial time. So on the one hand, if CCRV for First-Last is NP-complete then RP = NP, which is extremely unlikely. On the other hand, showing that CCRV for First-Last is in P would also show that Exact Perfect Bipartite Matching is in P, which would solve a well-studied 40-year-old open problem.Considering RP as an option for classifying problems can also help classify problems that until now had escaped classification. For example, the sole open problem in the comprehensive table from Erdélyi et al. [2021, https://doi.org/10.1007/s10458-021-09523-9] is CCRV for 2-Approval. We show that this problem is in RP, and thus easy since it is widely assumed that P = RP. 

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5. The Dirichlet isospectral problem for trapezoids

   https://doi.org/10.1063/5.0036384  

   Hezari, Hamid; Lu, Z.; Rowlett, J. (May 2021, Journal of Mathematical Physics)

   We show that non-obtuse trapezoids are uniquely determined by their Dirichlet Laplace spectrum. This extends our previous result [Hezari et al., Ann. Henri Poincare 18(12), 3759–3792 (2017)], which was only concerned with the Neumann Laplace spectrum. 

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