We give a classification of Lie bialgebra structures on generalized loop Schrödinger-Virasoro algebras $\mathfrak{s}\mathfrak{v}$. Then we find out that not all Lie bialgebra structures on generalized loop Schrödinger-Virasoro algebras sv are triangular coboundary.

We introduce a new concept of μ-pseudo almost automorphic processes in p-th mean sense by employing the measure theory, and present some results on the functional space of such processes like completeness and composition theorems. Under some conditions, we establish the existence, uniqueness, and the global exponentially stability of μ-pseudo almost automorphic mild solutions for a class of nonlinear stochastic evolution equations driven by Brownian motion in a separable Hilbert space.

The Jordan canonical form of matrix is an important concept in linear algebra. And the concept has been extended to three-way arrays case. In this paper, we study the Jordan canonical form of three-way tensor with multilinear rank (4,4,3). For a 4×4×4 tensor ${\mathfrak{G}}_{j}$ with multilinear rank (4,4,3), we show that ${\mathfrak{G}}_{j}$ must be turned into the canonical form if the upper triangular entries of the last three slices of ${\mathfrak{G}}_{j}$ are nonzero. If some of the upper triangular entries of the last three slices of ${\mathfrak{G}}_{j}$ are zeros, we give some conditions to guarantee that ${\mathfrak{G}}_{j}$ can be turned into the canonical form.

We prove a regularity criterion for the 3D Navier-Stokes-Allen-Cahn system in a bounded smooth domain which improves the result obtained by Y. Li, S. Ding, and M. Huang [Discrete Contin. Dyn. Syst. Ser. B, 2016, 21(5): 1507–1523]. We also present a similar result to the 3D Navier-Stokes-Cahn-Hilliard system.

We study the geometry of conformal minimal two spheres immersed in G(2; 7;$\mathbb{R}$ ): Then we classify the linearly full irreducible conformal minimal immersions with constant curvature from S^{2} to G(2; 7; $\mathbb{R}$); or equivalently, a complex hyperquadric Q5 under some conditions. We also completely determine the Gaussian curvature of all linearly full totally unramified irreducible and all linearly full reducible conformal minimal immersions from S^{2} to G(2; 7; $\mathbb{R}$) with constant curvature. For reducible case, we give some examples, up to SO(7) equivalence, in which none of the spheres are congruent, with the same Gaussian curvature.

We show that a closed piecewise flat 2-dimensional Alexandrov space Σ can be bi-Lipschitz embedded into a Euclidean space such that the embedded image of Σ has a tubular neighborhood in a generalized sense. As an application, we show that for any metric space sufficiently close to Σ in the Gromov-Hausdorff topology, there is a Lipschitz Gromov-Hausdorff approxima-tion.

It is known that the Schrödinger-Virasoro algebras, including the original Schrödinger-Virasoro algebra and the twisted Schrödinger-Virasoro algebra, are playing important roles in mathematics and statistical physics. In this paper, we study the tensor products of weight modules over the Schrödinger-Virasoro algebras. The irreducibility criterion for the tensor products of highest weight modules with intermediate series modules over the Schrödinger-Virasoro algebra is obtained.

This paper is devoted to study Frobenius Poisson algebras. We introduce pseudo-unimodular Poisson algebras by generalizing unimodular Poisson algebras, and investigate Batalin-Vilkovisky structures on their cohomology algebras. For any Frobenius Poisson algebra, all Batalin-Vilkovisky operators on its Poisson cochain complex are described explicitly. It is proved that there exists a Batalin-Vilkovisky operator on its cohomology algebra which is induced from a Batalin-Vilkovisky operator on the Poisson cochain complex, if and only if the Poisson structure is pseudo-unimodular. The relation between modular derivations of polynomial Poisson algebras and those of their truncated Poisson algebras is also described in some cases.

We give a recursive algorithm to compute the multivariable Zassenhaus formula ${\mathrm{e}}^{{{\displaystyle X}}_{\mathrm{1}}+{{\displaystyle X}}_{\mathrm{2}}+\mathrm{...}+{{\displaystyle X}}_{n}}={\mathrm{e}}^{{{\displaystyle X}}_{\mathrm{1}}}{\mathrm{e}}^{{{\displaystyle X}}_{\mathrm{2}}}\mathrm{...}{\mathrm{e}}^{{{\displaystyle X}}_{n}}$${{\displaystyle \mathrm{\Pi}}}_{k=\mathrm{2}}^{\infty}{\hspace{0.17em}\mathrm{e}}^{{{\displaystyle W}}_{k}}$ and derive ane effective recursion formula of ${{\displaystyle W}}_{k}$.

This paper is concerned with obtaining an approximate solution for a linear multidimensional Volterra integral equation with a regular kernel. We choose the Gauss points associated with the multidimensional Jacobi weight function $\omega (x)={\Pi i=\mathrm{1}d}_{}^{}{(\mathrm{1}-{xi}_{})}^{\alpha}{(\mathrm{1}+{xi}_{})}^{\beta},-\mathrm{1}<\alpha ,\beta <{\displaystyle \frac{\mathrm{1}}{d}}-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}$ (d denotes the space dimensions) as the collocation points. We demonstrate that the errors of approximate solution decay exponentially. Numerical results are presented to demonstrate the eectiveness of the Jacobi spectral collocation method.

We classify all the indecomposable modules of dimension≤5 over the quantum exterior algebra $k(x,y)/\langle {x}^{\mathrm{2}},{y}^{\mathrm{2}},xy+qyx\rangle $ in two variables, and all the indecomposable modules of dimension≤3 over the quantum complete intersection $k(x,y)/\langle {x}^{m},{y}^{n},xy+qyx\rangle $ in two variables, where m or n≥3, by giving explicitly their diagram presentations.

We study the derivative operator of the generalized spherical mean ${{\displaystyle S}}_{t}^{\gamma}$. By considering a more general multiplier ${{\displaystyle m}}_{\gamma ,b}^{\mathrm{\Omega}}={{\displaystyle V}}_{\frac{n-\mathrm{2}}{\mathrm{2}}+\gamma}(\left|\xi \right|){\left|\xi \right|}^{b}\mathrm{\Omega}(\xi \text{'})$ and finding the smallest $\gamma $ such that ${{\displaystyle m}}_{\gamma ,b}^{\Omega}$ is an H^{p} multiplier, we obtain the optimal range of exponents $(\gamma ,\beta ,p)$ to ensure the ${H}^{p}({\mathbb{R}}^{n})$ boundedness of ${\partial}^{\beta}{{\displaystyle S}}_{\mathrm{1}}^{\gamma}f(x)$. As an application, we obtain the derivative estimates for the solution for the Cauchy problem of the wave equation on ${H}^{p}({\mathbb{R}}^{n})$ spaces.