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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. Classification of Finite Depth Objects in Bicommutant Categories via Anchored Planar Algebras

   https://doi.org/10.1007/s00220-025-05548-3  

   Henriques, André; Penneys, David; Tener, James (April 2026, Communications in Mathematical Physics)

   Abstract In our article [arxiv:1511.05226], we studied the commutant$$\mathcal {C}'\subset \operatorname {Bim}(R)$$ $C^{\prime }\subset Bim\left(R\right)$ of a unitary fusion category$$\mathcal {C}$$ $C$ , whereRis a hyperfinite factor of type$$\mathrm II_1$$ $I{I}_1$ ,$$\mathrm II_\infty $$ $I{I}_{\infty }$ , or$$\mathrm III_1$$ $II{I}_1$ , and showed that it is a bicommutant category. In other recent work [arxiv:1607.06041,arxiv:2301.11114] we introduced the notion of a (unitary) anchored planar algebra in a (unitary) braided pivotal category$$\mathcal {D}$$ $D$ , and showed that they classify (unitary) module tensor categories for$$\mathcal {D}$$ $D$ equipped with a distinguished object. Here, we connect these two notions and show that finite depth objects of$$\mathcal {C}'$$ $C^{\prime }$ are classified by connected finite depth unitary anchored planar algebras in$$\mathcal {Z}(\mathcal {C})$$ $Z\left(C\right)$ . This extends the classification of finite depth objects of$$\operatorname {Bim}(R)$$ $Bim\left(R\right)$ by connected finite depth unitary planar algebras. 

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3. The SXS collaboration’s third catalog of binary black hole simulations

   https://doi.org/10.1088/1361-6382/adfd34  

   Scheel, Mark A; Boyle, Michael; Mitman, Keefe; Deppe, Nils; Stein, Leo C; Armaza, Cristóbal; Bonilla, Marceline S; Buchman, Luisa T; Ceja, Andrea; Chaudhary, Himanshu; et al (October 2025, Classical and Quantum Gravity)

   Abstract We present a major update to the Simulating eXtreme Spacetimes (SXSs) Collaboration’s catalog of binary black hole (BBH) simulations. Using highly efficient spectral methods implemented in the Spectral Einstein Code (SpEC), we have nearly doubled the total number of binary configurations from 2018 to 3756. The catalog now more densely covers the parameter space with precessing simulations up to mass ratioq = 8 and dimensionless spins up to $|\overrightarrow{\chi }|\text{⩽}0.8$ with near-zero eccentricity. The catalog also includes some simulations at higher mass ratios with moderate spin and more than 250 eccentric simulations. We have also deprecated and rerun some simulations from our previous catalog (e.g. simulations run with a much older version ofSpECor that had anomalously high errors in the waveform). The median waveform difference (which is similar to the mismatch) between resolutions over the simulations in the catalog is $4\times {10}^{-4}$ . The simulations have a median of 22 orbits, while the longest simulation has 148 orbits. We have corrected each waveform in the catalog to be in the binary’s center-of-mass frame and exhibit gravitational-wave memory. We estimate the total CPU cost of all simulations in the catalog to be 480 000 000 core-hours. We find that using spectral methods for BBH simulations is over 1000 times more efficient than previously published finite-difference simulations. The full catalog is publicly available through thesxsPython package and athttps://data.black-holes.org . 

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4. An almost linear time algorithm testing whether the Markoff graph modulo p is connected

   https://doi.org/10.1007/s40993-024-00592-9  

   Brown, Colby Austin (March 2025, Research in Number Theory)

   Abstract The Markoff graphs modulopwere proven by Chen (Ann Math 199(1), 2024) to be connected for all but finitely many primes, and Baragar (The Markoff equation and equations of Hurwitz. Brown University, 1991) conjectured that they are connected for all primes, equivalently that every solution to the Markoff equation moduloplifts to a solution over$$\mathbb {Z}$$ $Z$ . In this paper, we provide an algorithmic realization of the process introduced by Bourgain et al. [arXiv:1607.01530] to test whether the Markoff graph modulopis connected for arbitrary primes. Our algorithm runs in$$o(p^{1 + \epsilon })$$ $o\left(p^{1+ϵ}\right)$ time for every$$\epsilon > 0$$ $ϵ>0$ . We demonstrate this algorithm by confirming that the Markoff graph modulopis connected for all primes less than one million. 

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5. NeuralSAT: A High-Performance Verification Tool for Deep Neural Networks

   https://doi.org/10.1007/978-3-031-98679-6_19  

   Duong, Hai; Nguyen, ThanhVu; Dwyer, Matthew B (July 2025, Springer Nature Switzerland)

   Abstract Deep Neural Networks (DNNs) are increasingly deployed in critical applications, where ensuring their safety and robustness is paramount. We present$$_\text {CAV25}$$ ${}_{\mathrm{CAV}25}$ , a high-performance DNN verification tool that uses the DPLL(T) framework and supports a wide-range of network architectures and activation functions. Since its debut in VNN-COMP’23, in which it achieved the New Participant Award and ranked 4th overall,$$_\text {CAV25}$$ ${}_{\mathrm{CAV}25}$ has advanced significantly, achieving second place in VNN-COMP’24. This paper presents and evaluates the latest development of$$_\text {CAV25}$$ ${}_{\mathrm{CAV}25}$ , focusing on the versatility, ease of use, and competitive performance of the tool.$$_\text {CAV25}$$ ${}_{\mathrm{CAV}25}$ is available at:https://github.com/dynaroars/neuralsat. 

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