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We establish a new criterion for exponential mixing of random dynamical systems. Our criterion is applicable to a wide range of systems, including in particular dispersive equations. Its verification is in nature related to several topics, i.e., asymptotic compactness in dynamical systems, global stability of evolution equations, and localized control problems. As an initial application, we exploit the exponential mixing of random nonlinear wave equations with degenerate damping, critical nonlinearity, and physically localized noise. The essential challenge lies in the fact that the weak dissipation and randomness interact in the evolution.

In this paper, we introduce notions of (proto-, quasi-)twilled Lie triple systems and give their equivalent descriptions using the controlling algebra and bidegree convention. Then we construct an $L_\infty$-algebra via a twilled Lie triple system. Besides, we establish the twisting theory of Lie triple systems and then characterize the twisting as a Maurer-Cartan element in the constructed $L_\infty$-algebra. Finally, we clarify the relationship between twilled Lie triple systems and matched pairs and clarify the relationship between twilled Lie triple systems and relative Rota-Baxter operators respectively so that we obtain the relationship between matched pairs of Lie triple systems and relative Rota-Baxter operators.

The study of kagome materials has attracted much attention in the past few years due to the presence of many electron-electron interaction-driven phases in a single material.In this work, we report the discovery of intrinsic spin-polarized p-wave superconductivity in the thin-flake kagome material RbV$_3$Sb$_5$. Firstly, when an in-plane magnetic field is swept in opposite directions, we observe a unique form of hysteresis in magnetoresistance which is different from the hysteresis induced by extrinsic mechanisms such as flux-trapping or superheating and supercooling effects. The unconventional hysteresis indicates the emergence of an intrinsic time-reversal symmetry-breaking superconducting phase. Strikingly, at a fixed magnetic field, the finite-resistance state can be quenched to the zero-resistance state by applying a large current. Secondly, at temperatures around 400 mK, the re-entrance of superconductivity occurs during an in-plane field-sweeping process with a fixed sweeping direction. This kind of re-entrance is asymmetric about the zero field axis and observed in all field directions for a fixed current direction, which is different from the re-entrance observed in conventional superconductors. Moreover, the angle-dependent in-plane critical field measurements reveal a two-fold symmetry that deviates from the original, centrosymmetric D$_{6h}$ point group symmetry of the crystal. These findings put very strong constraints on the possible superconducting pairing symmetry of RbV$_3$Sb$_5$. We point out that the pairing symmetry, which is consistent with the crystal symmetry and all the observed novel properties, is a time-reversal symmetry-breaking, p-wave pairing with net spin polarization. Importantly, this p-wave pairing gives rise to a nodal topological superconducting state with Majorana flat bands on the sample edges., Comment: 21 pages, 4 figures

This paper mainly deals with the Sturm-Liouville operator \begin{equation*} \mathbf{H}=\frac{1}{w(x)}\left( -\frac{\mathrm{d}}{\mathrm{d}x}p(x)\frac{ \mathrm{d}}{\mathrm{d}x}+q(x)\right) ,\text{ }x\in \Gamma \end{equation*} acting in $L_{w}^{2}\left( \Gamma \right) ,$ where $\Gamma $ is a metric graph. We establish a relationship between the bottom of the spectrum and the positive solutions of quantum graphs, which is a generalization of the classical Allegretto-Piepenbrink theorem. Moreover, we prove the Persson-type theorem, which characterizes the infimum of the essential spectrum.

In this paper, we propose a general numerical framework to derive structure-preserving reduced order models for thermodynamically consistent PDEs. Our numerical framework has two primary features: (a) a systematic way to extract reduced order models for thermodynamically consistent PDE systems while maintaining their inherent thermodynamic principles and (b) a strategic process to devise accurate, efficient, and structure-preserving numerical algorithms to solve the forehead reduced-order models. The platform's generality extends to various PDE systems governed by embedded thermodynamic laws. The proposed numerical platform is unique from several perspectives. First, it utilizes the generalized Onsager principle to transform the thermodynamically consistent PDE system into an equivalent one, where the transformed system's free energy adopts a quadratic form of the state variables. This transformation is named energy quadratization (EQ). Through EQ, we gain a novel perspective on deriving reduced order models. The reduced order models derived through our method continue to uphold the energy dissipation law. Secondly, our proposed numerical approach automatically provides numerical algorithms to discretize the reduced order models. The proposed algorithms are always linear, easy to implement and solve, and uniquely solvable. Furthermore, these algorithms inherently ensure the thermodynamic laws. In essence, our platform offers a distinctive approach to derive structure-preserving reduced-order models for a wide range of PDE systems abiding by thermodynamic principles.

In this paper, we first construct a differential graded Lie algebra that controls deformations of a Lie-Yamaguti algebra. Furthermore, a relative Rota-Baxter operator on a Lie-Yamaguti algebra is characterized as a Maurer-Cartan element in an appropriate $L_\infty$-algebra that we build through the graded Lie bracket of Lie-Yamaguti algebra's controlling algebra, and gives rise to a twisted $L_\infty$-algebra that controls its deformation. Next we establish the cohomology theory of relative Rota-Baxter operators on Lie-Yamaguti algebras via the Yamaguti cohomology. Then we clarify the relationship between the twisted $L_\infty$-algebra and the cohomology theory. Finally as byproducts, we classify certain deformations on Lie-Yamaguti algebras using the cohomology theory., Comment: This paper surpercedes the preprint arXiv: 2204.04872. We only consider cohomology and deformations of relative Rota-Baxter operators on Lie-Yamaguti algebras in the preprint arXiv: 2204.04872. Not only do we consider cohomology and deformations but also controlling algebras of Lie-Yamaguti algebras and their relative Rota-Baxter operators are constructed in the present paper

In this paper, we introduce the notions of relative Rota-Baxter operators of weight $1$ on Lie-Yamaguti algebras, and post-\LYA s, which is an underlying algebraic structure of relative Rota-Baxter operators of weight $1$. We give the relationship between these two algebraic structures. Besides, we establish the cohomology theory of relative Rota-Baxter operators of weight $1$ via the Yamaguti cohomology. Consequently, we use this cohomology to characterize linear deformations of relative Rota-Baxter operators of weight $1$ on Lie-Yamaguti algebras. We show that if two linear deformations of a relative Rota-Baxter operator of weight $1$ are equivalent, then their infinitesimals are in the same cohomology class in the first cohomology group. Moreover, we show that an order $n$ deformation of a relative Rota-Baxter operator of weight $1$ can be extended to an order $n+1$ deformation if and only if the obstruction class in the second cohomology group is trivial., Comment: arXiv admin note: substantial text overlap with arXiv:2303.15744

In this paper, we introduce the notion of crossed homomorphisms between Lie-Yamaguti algebras and establish the cohomology theory of crossed homomorphisms via the Yamaguti cohomology. Consequently, we use this cohomology to characterize linear deformations of crossed homomorphisms between Lie-Yamaguti algebras. We show that if two linear or formal deformations of a crossed homomorphism are equivalent, then their infinitesimals are in the same cohomology class in the first cohomology group. Moreover, we show that an order $n$ deformation of a crossed homomorphism can be extended to an order $n+1$ deformation if and only if the obstruction class in the second cohomology group is trivial.