“The paper derives bilinear tau forms of the canonically quantized Painlevé equations, relating them to those previously obtained from the $\mathbb{C}^2/\mathbb{Z}_2$ blowup relations.”

“The paper focuses on Chern-Simons theory in 3D, motivated by its applications in condensed matter physics, gravity, and black hole physics, and explores its connection to asymptotic symmetries and integrable systems.”

“The eigenfunctions admit an expansion with universal coefficients so that the dependence on the twist $a$ is hidden only in these ground state eigenfunctions, and we suggest a general formula that allows one to construct these eigenfunctions from non-symmetric Macdonald polynomials.”

“The paper's core argument is that gauge theory and integrability are complementary manifestations of a shared coherence principle, an ongoing journey from gauge symmetry toward mathematical unity.”

“The Zakharov-Shabat equations themselves recover the Lax operators under suitable change of independent variables in the case of the KP hierarchy and the modified KP hierarchy (in the matrix formulation).”

“The paper finds that even in the absence of U(1) symmetry, the Bethe and zero roots still exhibit a highly structured pattern.”

“All models are exactly-solvable, admit algebraic forms for the Hamiltonian and integrals, have polynomial eigenfunctions, hidden algebraic structure, and possess a polynomial algebra of integrals.”

“The method achieved superior performance compared to state-of-the-art methods on BraTS2020, with average Dice scores of 87.55, 79.36, and 62.67 for WT, TC, and ET, respectively, across fifteen missing modality combinations.”

“The paper proves that the family of Bethe subspaces extends regularly to the minimal wonderful model of the toric arrangement.”

“The paper introduces a new class of integrable 3D lattice models, possessing continuous families of commuting layer-to-layer transfer matrices.”

“The paper discusses two correspondences: one based on Hamiltonian reduction and its quantum counterpart, and another involving non-trivial dualities like Fourier and Legendre transforms.”

“The paper proposes integrable space discretizations of the sine-Gordon equation in three distinct cases of non-characteristic coordinates.”

“It exposes the exact moment a network switches from memorization to generalization ("grokking") by monitoring the geometric arrangement of embeddings in real-time.”

“The library's capabilities are validated across four canonical physical regimes: dispersive linear wave propagation, static topological kink preservation in phi-fourth theory, integrable breather dynamics in the sine-Gordon model, and non-integrable kink-antikink collisions.”

“The paper derives moderate deviation estimates for the height function in both the upper and lower tail regimes, with sharp exponents and constants.”

“The paper establishes a correspondence between the auxiliary field model of Russo and Townsend and the Ivanov--Zupnik formalism in four-dimensional electrodynamics.”

“The excitation velocities can be calculated from the knowledge of the matrices of compressibility and the Drude weights, as their squares are the eigenvalues of the product of the two matrices.”

“Non-local couplings facilitate faster operator spreading and more intricate dynamical behavior, enabling these systems to approach maximal chaos more readily than their local counterparts.”

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“The article's context focuses on the application of integrable perturbation theory to the defocusing nonlinear Schrödinger equation.”

“The study focuses on the ergodically constrained nonintegrable transverse-field Ising model.”

“Stäckel problem for non-diagonal Killing tensors.”

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“An integrable hierarchy associated with loop extension of $\mathbb{Z}_2^2$-graded $\mathfrak{osp}(1|2)$”