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1. A driven Kerr oscillator with two-fold degeneracies for qubit protection

   https://doi.org/10.1073/pnas.2311241121  

   Venkatraman, Jayameenakshi; Cortiñas, Rodrigo G; Frattini, Nicholas E; Xiao, Xu; Devoret, Michel H (June 2024, Proceedings of the National Academy of Sciences)

   We present the experimental finding of multiple simultaneous two-fold degeneracies in the spectrum of a Kerr oscillator subjected to a squeezing drive. This squeezing drive resulting from a three-wave mixing process, in combination with the Kerr interaction, creates an effective static two-well potential in the phase space rotating at half the frequency of the sinusoidal drive generating the squeezing. Remarkably, these degeneracies can be turned on-and-off on demand, as well as their number by simply adjusting the frequency of the squeezing drive. We find that when the detuning Δ between the frequency of the oscillator and the second subharmonic of the drive equals an even multiple of the Kerr coefficientK, $\mathrm{\Delta }/K=2m$, the oscillator displays $m+1$exact, parity-protected, spectral degeneracies, insensitive to the drive amplitude. These degeneracies can be explained by the unusual destructive interference of tunnel paths in the classically forbidden region of the double well static effective potential that models our experiment. Exploiting this interference, we measure a peaked enhancement of the incoherent well-switching lifetime, thus creating a protected cat qubit in the ground state manifold of our oscillator. Our results illustrate the relationship between degeneracies and noise protection in a driven quantum system. 

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2. Characterization of Turbulent Fluctuations in the Sub-Alfvénic Solar Wind

   https://doi.org/10.3847/1538-4357/ad34ab  

   Zank, G P; Zhao, L-L; Adhikari, L; Telloni, D; Baruwal, Prashant; Baruwal, Prashrit; Zhu, Xingyu; Nakanotani, M; Pitňa, A; Kasper, J C; et al (April 2024, The Astrophysical Journal)

   Abstract Parker Solar Probe (PSP) observed sub-Alfvénic solar wind intervals during encounters 8–14, and low-frequency magnetohydrodynamic (MHD) turbulence in these regions may differ from that in super-Alfvénic wind. We apply a new mode decomposition analysis to the sub-Alfvénic flow observed by PSP on 2021 April 28, identifying and characterizing entropy, magnetic islands, forward and backward Alfvén waves, including weakly/nonpropagating Alfvén vortices, forward and backward fast and slow magnetosonic (MS) modes. Density fluctuations are primarily and almost equally entropy- and backward-propagating slow MS modes. The mode decomposition provides phase information (frequency and wavenumberk) for each mode. Entropy density fluctuations have a wavenumber anisotropy ofk_∥≫k_⊥, whereas slow-mode density fluctuations havek_⊥>k_∥. Magnetic field fluctuations are primarily magnetic island modes (δBⁱ) with anO(1) smaller contribution from unidirectionally propagating Alfvén waves (δB^A+) giving a variance anisotropy of $〈{\delta B^i}^2〉/〈{\delta B^A}^2〉=4.1$. Incompressible magnetic fluctuations dominate compressible contributions from fast and slow MS modes. The magnetic island spectrum is Kolmogorov-like $k_{\perp }^{-1.6}$in perpendicular wavenumber, and the unidirectional Alfvén wave spectra are $k_{\parallel }^{-1.6}$and $k_{\perp }^{-1.5}$. Fast MS modes propagate at essentially the Alfvén speed with anticorrelated transverse velocity and magnetic field fluctuations and are almost exclusively magnetic due toβ_p≪ 1. Transverse velocity fluctuations are the dominant velocity component in fast MS modes, and longitudinal fluctuations dominate in slow modes. Mode decomposition is an effective tool in identifying the basic building blocks of MHD turbulence and provides detailed phase information about each of the modes. 

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3. CscK metrics near the canonical class

   https://doi.org/10.1515/crelle-2024-0096  

   Guo, Bin; Jian, Wangjian; Shi, Yalong; Song, Jian (January 2025, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let 𝑋 be a Kähler manifold with semiample canonical bundle $K_X$K_{X}.It is proved in [W. Jian, Y. Shi and J. Song, A remark on constant scalar curvature Kähler metrics on minimal models,Proc. Amer. Math. Soc.147(2019), 8, 3507–3513] that, for any Kähler class 𝛾, there exists $\delta >0$\delta>0such that, for all $t\in (0,\delta )$t\in(0,\delta), there exists a unique cscK metric $g_t$g_{t}in $K_X+t\gamma$K_{X}+t\gamma.In this paper, we prove that ${\left\{(X,g_t)\right\}}_{t\in (0,\delta )}$\{(X,g_{t})\}_{t\in(0,\delta)}have uniformly bounded Kähler potentials, volume forms and diameters.As a consequence, these metric spaces are pre-compact in the Gromov–Hausdorff sense. 

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4. Better Hardness Results for the Minimum Spanning Tree Congestion Problem

   https://doi.org/10.1007/s00453-024-01278-5  

   Luu, Huong; Chrobak, Marek (January 2025, Algorithmica)

   Abstract In the spanning tree congestion problem, given a connected graphG, the objective is to compute a spanning treeTinGthat minimizes its maximum edge congestion, where the congestion of an edgeeofTis the number of edges inGfor which the unique path inTbetween their endpoints traversese. The problem is known to be$$\mathbb{N}\mathbb{P}$$ $NP$-hard, but its approximability is still poorly understood, and it is not even known whether the optimum solution can be efficiently approximated with ratioo(n). In the decision version of this problem, denoted$${\varvec{K}-\textsf {STC}}$$ $K-\mathrm{STC}$, we need to determine ifGhas a spanning tree with congestion at mostK. It is known that$${\varvec{K}-\textsf {STC}}$$ $K-\mathrm{STC}$is$$\mathbb{N}\mathbb{P}$$ $NP$-complete for$$K\ge 8$$ $K\ge 8$, and this implies a lower bound of 1.125 on the approximation ratio of minimizing congestion. On the other hand,$${\varvec{3}-\textsf {STC}}$$ $3-\mathrm{STC}$can be solved in polynomial time, with the complexity status of this problem for$$K\in { \left\{ 4,5,6,7 \right\} }$$ $K\in \left(4,,,5,,,6,,,7\right)$remaining an open problem. We substantially improve the earlier hardness results by proving that$${\varvec{K}-\textsf {STC}}$$ $K-\mathrm{STC}$is$$\mathbb{N}\mathbb{P}$$ $NP$-complete for$$K\ge 5$$ $K\ge 5$. This leaves only the case$$K=4$$ $K=4$open, and improves the lower bound on the approximation ratio to 1.2. Motivated by evidence that minimizing congestion is hard even for graphs of small constant radius, we also consider$${\varvec{K}-\textsf {STC}}$$ $K-\mathrm{STC}$restricted to graphs of radius 2, and we prove that this variant is$$\mathbb{N}\mathbb{P}$$ $NP$-complete for all$$K\ge 6$$ $K\ge 6$. 

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5. Higher resonance schemes and Koszul modules of simplicial complexes

   https://doi.org/10.1007/s10801-024-01313-2  

   Aprodu, Marian; Farkas, Gavril; Raicu, Claudiu; Sammartano, Alessio; Suciu, Alexander I (March 2024, Journal of Algebraic Combinatorics)

   Abstract Each connected graded, graded-commutative algebraAof finite type over a field$$\Bbbk $$ $k$of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the(higher) Koszul modulesofA. In this note, we investigate the geometry of the support loci of these modules, called theresonance schemesof the algebra. When$$A=\Bbbk \langle \Delta \rangle $$ $A=k⟨\Delta ⟩$is the exterior Stanley–Reisner algebra associated to a finite simplicial complex$$\Delta $$ $\Delta$, we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group. 

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