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1. Entropy Dissipation for Degenerate Stochastic Differential Equations via Sub-Riemannian Density Manifold

   https://doi.org/10.3390/e25050786  

   Feng, Qi; Li, Wuchen (May 2023, Entropy)

   We studied the dynamical behaviors of degenerate stochastic differential equations (SDEs). We selected an auxiliary Fisher information functional as the Lyapunov functional. Using generalized Fisher information, we conducted the Lyapunov exponential convergence analysis of degenerate SDEs. We derived the convergence rate condition by generalized Gamma calculus. Examples of the generalized Bochner’s formula are provided in the Heisenberg group, displacement group, and Martinet sub-Riemannian structure. We show that the generalized Bochner’s formula follows a generalized second-order calculus of Kullback–Leibler divergence in density space embedded with a sub-Riemannian-type optimal transport metric. 

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2. On the computation of Kähler differentials and characterizations of Galois extensions with independent defect

   https://doi.org/10.1002/mana.202300532  

   Cutkosky, Steven_Dale; Kuhlmann, Franz‐Viktor; Rzepka, Anna (March 2025, Mathematische Nachrichten)

   Abstract For important cases of algebraic extensions of valued fields, we develop presentations of the associated Kähler differentials of the extensions of their valuation rings. We compute their annihilators as well as the associated differents. We then apply the results to Galois defect extensions of prime degree. Defects can appear in finite extensions of valued fields of positive residue characteristic and are serious obstructions to several problems in positive characteristic. A classification of defects (dependent vs. independent) has been introduced by the second and the third author. It has been shown that perfectoid fields and deeply ramified fields only admit extensions with independent defect. We give several characterizations of independent defect, using ramification ideals, Kähler differentials, and traces of the maximal ideals of valuation rings. All of our results are for arbitrary valuations; in particular, we have no restrictions on their ranks or value groups. 

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3. Deformed algebraic structure of angular momenta: GUP perspective

   Bhandari, Gaurav; Pathak, S D; Sharma, Manabendra; Wang, Anzhong (December 2024, arXivorg)

   The prediction of a minimal length scale by various quantum gravity candidates (such as string/M theory, Doubly Special Relativity, Loop Quantum Gravity and others) have suggested modification of Heisenberg Uncertainty Principle (HUP), resulting in the Generalized Uncertainty Principle (GUP). In this short review, we investigate the origins of the GUP and examine higher-order models, focusing on the linear plus quadratic form of the GUP. We extend the concept of minimal length to minimal angular resolution, which plays a crucial role in modifying angular momentum and its associated algebra. A comparison is made between the standard angular momentum commutator algebra and that modified by the GUP. Finally, we review its application in the hydrogen atom spectra and and discuss future endeavors. 

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4. Liouville closed HT-fields

   Kaplan, Elliot (April 2023, Journal of algebra)

   Let T be a complete, model complete o-minimal theory extending the theory of real closed ordered fields. An HT-field is a model K of T equipped with a T-derivation ∂ such that the underlying ordered differential field of (K,∂) is an H-field. We study HT-fields and their extensions. Our main result is that if T is power bounded, then every HT-field K has either exactly one or exactly two minimal Liouville closed HT-field extensions up to K-isomorphism. The assumption of power boundedness can be relaxed to allow for certain exponential cases, such as T = Th(Ran,exp). 

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5. Quantum dynamics in Krylov space: Methods and applications

   https://doi.org/10.1016/j.physrep.2025.05.001  

   Nandy, Pratik; Matsoukas-Roubeas, Apollonas S; Martínez-Azcona, Pablo; Dymarsky, Anatoly; del_Campo, Adolfo (June 2025, Physics Reports)

   The dynamics of quantum systems unfolds within a subspace of the state space or operator space, known as the Krylov space. This review presents the use of Krylov subspace methods to provide an efficient description of quantum evolution and quantum chaos, with emphasis on nonequilibrium phenomena of many-body systems with a large Hilbert space. It provides a comprehensive update of recent developments, focused on the quantum evolution of operators in the Heisenberg picture as well as pure and mixed states. It further explores the notion of Krylov complexity and associated metrics as tools for quantifying operator growth, their bounds by generalized quantum speed limits, the universal operator growth hypothesis, and its relation to quantum chaos, scrambling, and generalized coherent states. A comparison of several generalizations of the Krylov construction for open quantum systems is presented. A closing discussion addresses the application of Krylov subspace methods in quantum field theory, holog- raphy, integrability, quantum control, and quantum computing, as well as current open problems. 

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