In this paper, we prove a gap theorem for translating solitons to the almost calibrated Lagrangian mean curvature flow More precisely, we prove that there exists a constant λ > 0 such that if | H | 2 ⩽ λ | T | 2 , then the translating soliton must be a plane We also obtain a similar result for symplectic mean curvature flow

A gap theorem for translating solitons to Lagrangian mean curvature flow

Computing the Lyapunov operator φ-functions, with an application to matrix-valued exponential integrators

Stiff delay differential equations are frequently utilized in practice, but their numerical simulations are difficult due to the complicated interaction between the stiff and delay terms. At the moment, only a few low-order algorithms offer acceptable convergent and stable features. Exponential integrators are a type of efficient numerical approach for stiff problems that can eliminate the influence of stiffness on the scheme by precisely dealing with the stiff term. This study is concerned with two exponential multistep methods of Adams type for stiff delay differential equations. For semilinear delay differential equations, applying the linear multistep method directly to the integral form of the equation yields the exponential multistep method. It is shown that the proposed k-step method is stiffly convergent of order k. On the other hand, we can follow the strategy of the Rosenbrock method to linearize the equation along the numerical solution in each step. The so-called exponential Rosenbrock multistep method is constructed by applying the exponential multistep method to the transformed form of the semilinear delay differential equation. This method can be easily extended to nonlinear delay differential equations. The main contribution of this study is that we show that the k-step exponential Rosenbrock multistep method is stiffly convergent of order k+1 within the framework of a strongly continuous semigroup on Banach space. As a result, the methods developed in this study may be utilized to solve abstract stiff delay differential equations and can be served as time matching methods for delay partial differential equations. Numerical experiments are presented to demonstrate the theoretical results.

/pdf/exponential-multistep-methods-for-stiff-delay-differential-17ctm1l2.pdf

Exponential Multistep Methods for Stiff Delay Differential Equations

In this paper, we develop e ﬃ cient and accurate evaluation for the Lyapunov operator function ϕ l ( L A )[ Q ] , where ϕ l ( · ) is the function related to the exponential, L A is a Lyapunov operator and Q is a symmetric and full-rank matrix. An important application of the algorithm is to the matrix-valued exponential integrators for matrix di ﬀ erential equations such as di ﬀ erential Lyapunov equations and di ﬀ erential Riccati equations. The method is exploited by using the modiﬁed scaling and squaring procedure combined with the truncated Taylor series. A quasi-backward error analysis is presented to determine the value of the scaling parameter and the degree of the Taylor approximation. Numerical experiments show that the algorithm performs well in both accuracy and e ﬃ ciency.

Computing the Lyapunov operator \varphi-functions, with an application to matrix-valued exponential integrators

In this work we develop a low-rank algorithm for the computation of low-rank approximations to the large-scale Lyapunov operator ϕ -functions. Such computations constitute the important ingredient to the implementation of matrix-valued exponential integrators for large-scale sti ﬀ matrix di ﬀ erential equations where the (approximate) solutions are of low rank. We evaluate the approximate solutions of LDL T -type based on a scaling and recursive procedure. The key parameters of the method are determined using a quasi-backward error bound combined with the computational cost. Numerical results demonstrate that our method can be used as a basis of matrix-valued exponential integrators for solving large-scale di ﬀ erential Lyapunov equations and di ﬀ erential Riccati equations.

A low-rank algorithm for solving Lyapunov operator $\varphi$-functions within the matrix-valued exponential integrators

In this paper, we develop efficient and accurate algorithms for evaluating both 𝜑 l ( A ) and 𝜑 l ( A ) b , where 𝜑 l ( ⋅ ) is the function related to the exponential defined by 𝜑 l ( z ) ≡ ∑ ∞ k = 0 z k ( l + k )! , A is an N × N matrix and b is an N dimensional vector. Such matrix functions play a key role in a class of numerical methods well-known as exponential integrators. The algorithms use the modified scaling and squaring procedure combined with truncated Taylor series. A quasi-backward error analysis is presented to find the optimal value of the scaling and the degree of the Taylor approximation. Some useful techniques are employed for reducing the computational cost. Numerical comparisons with state-of-the-art algorithms show that the algorithms perform well in both accuracy and efficiency.

Efficient and accurate computation for the ' ‑functions arising from exponential integrators

In this paper we mainly study the relation between |A|2, |H|2 and cosα (α is the Kähler angle) of the blow up flow around the type II singularities of a symplectic mean curvature flow. We also study similar property of an almost calibrated Lagrangian mean curvature flow. We show the nonexistence of type II blow-up flows for a symplectic mean curvature flow satisfying |A|2 ≤ λ|H|2 and cosα ≥ δ > 1− 1 2λ ( 2 ≤ λ ≤ 2), or for an almost calibrated Lagrangian mean curvature flow satisfying |A|2≤λ|H|2 and cosθ≥δ>max{0,1− 1 λ} ( 3 4 ≤λ≤2), where θ is the Lagrangian angle. AMS Subject Classifications: 53C44, 53C21 Chinese Library Classifications: O186.12, O175.26

Nonexistence of Blow-Up Flows for Symplectic and Lagrangian Mean Curvature Flows

References: [1] Drinfel’d, V. G., Quantum groups, Proceeding of the International Congress of Mathematicians, Vol. 1,2, Berkeley, Calif. 1986, 1 and 2, 798-820 (1987), Providence, RI: American Mathematical Society, Providence, RI [2] Etingof, P.; Schiffmann, O., Lectures on Quantum Groups (2002), International press · Zbl 1106.17015 [3] Drinfel’d, V. G., Constant quasiclassical solutions of the Yang-Baxter quantum equation, Soviet Mathematics-Doklady, 273, 667-671 (1983) · Zbl 0553.58038 [4] Drinfel’d, V. G., On some unsolved problems in quantum group theory, Quantum Groups (1992), Berlin, Heidelberg: Springer, Berlin, Heidelberg · Zbl 0765.17014 · doi:10.1007/BFb0101175 [5] Etingof, P.; Kazhdan, D., Quantization of Lie bialgebras I, Selecta Mathematica, 2, 1, 1-41 (1996) · Zbl 0863.17008 · doi:10.1007/BF01587938 [6] Chen, H.; Dai, X.; Yang, H., Lie bialgebra structures on generalized loop Schrödinger-Virasoro algebras, Frontiers of Mathematics in China, 14, 2, 239-260 (2019) · Zbl 1471.17042 · doi:10.1007/s11464-019-0761-0 [7] Han, J.; Li, J.; Su, Y., Lie bialgebra structures on the Schrödinger-Virasoro Lie algebra, Journal of Mathematical Physics, 50, 8, 083504-083820 (2009) · Zbl 1234.17016 · doi:10.1063/1.3187784 [8] Yang, H.; Su, Y., Lie super-bialgebra structures on generalized super-Virasoro algebras, Acta Mathematica Scientia, 30, 1, 225-239 (2010) · Zbl 1224.17030 · doi:10.1016/S0252-9602(10)60040-9 [9] Wu, H.; Wang, S.; Yue, X., Lie bialgebras of Generalized loop Virasoro algebras, Chinese. Ann. Math., 36 (2013) [10] Dai, X.; Xin, B., Lie super-bialgebras on generalized loop super-virasoro algebras, Bulletin of the Korean Mathematical Society, 53, 6, 1685-1695 (2016) · Zbl 1402.17033 · doi:10.4134/BKMS.b150884 [11] Yuan, L.; He, C., Lie super-bialgebra and quantization of the super Virasoro algebra, Journal of Mathematical Physics, 57, 5, 053508-053515 (2016) · Zbl 1348.81273 · doi:10.1063/1.4952701 [12] Hu, N.; Wang, X., Quantizations of generalized-Witt algebra and of Jacobson-Witt algebra in the modular case, Journal of Algebra, 312, 2, 902-929 (2007) · Zbl 1170.17002 · doi:10.1016/j.jalgebra.2007.02.019 [13] Bo, J.; Yu, L.; Su, Y., Quantizations of the $W$ Algebra $W\left( 2 , 2\right)$, Acta Mathematica Sinica, English Series, 27, 647-656 (2011) [14] Cheng, Y.; Shi, Y., Quantization of the q-analog Virasoro-like algebras, Journal of Generalized Lie Theory and Applications, 4, 1, 98-109 (2010)· doi:10.4172/1736-4337.1000172 [15] Grunspan, C., Quantizations of the Witt algebra and of simple Lie algebras in characteristic $p$, Journal of Algebra, 280, 145-161 (2004) · Zbl 1137.17303 [16] Song, G.; Su, Y.; Wu, Y., Quantization of generalized Virasoro-like algebras, Linear Algebra and its Applications, 428, 11-12, 2888-2899 (2008) · Zbl 1137.17019 · doi:10.1016/j.laa.2008.01.020 [17] Su, Y.; Yue, X.; Zhu, X., Quantization of the Lie superalgebra of Witt type, Journal of Geometry and Physics, 160, article 103987 (2021) · Zbl 07299626 · doi:10.1016/j.geomphys.2020.103987 [18] Yue, X.; Jiang, Q.; Xin, B., Quantization of Lie algebras of generalized Weyl type, Algebra Colloquium, 16, 3, 437-448 (2009) · Zbl 1235.17014 · doi:10.1142/S100538670900042X [19] Barnich, G.; Donnay, L.; Matulich, J.; Troncoso, R., Super-BM $S_3$ invariant boundary theory from three-dimensional flat supergravity, Journal of High Energy Physics, 1, 1-15 (2017) · Zbl 1373.83080 [20] Chen, H.; Dai, X.; Liu, Y.; Su, Y., Non-weight modules over the super-BMS_3 algebra (2019), http://arxiv.org/abs/1911.09651 [21] Cheng, X.; Sun, J.; Yang, H., Lie super-bialgebra structures on the super-BMS_3algebra, Algebra Colloquium, 26, 3, 437-458 (2019) · Zbl 1430.17065 · doi:10.1142/S1005386719000324 [22] Drinfel’d, V. G., Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of classical Yang-Baxter equations, Soviet Mathematics-Doklady, 27, 222-225 (1983) · Zbl 0526.58017 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically

Quantization of the super- BMS 3 algebra. (English)

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Efficient and accurate computation for the ' ‑functions arising from exponential integrators

Nonexistence of Blow-Up Flows for Symplectic and Lagrangian Mean Curvature Flows

Quantization of the super- BMS 3 algebra. (English)