We provide a Faddeev–Reshetikhin–Takhtajan’s RTT approach to the quantum group $\mathrm{Fun}(\mathrm{GL}_{r,s}(n))$ and the quantum enveloping algebra $U_{r,s}(\mathfrak {gl}_n)$ corresponding to the two-parameter $R$-matrix. We prove that the quantum determinant ${\det }_{r,s}T$ is a quasi-central element in $\mathrm{Fun}(\mathrm{GL}_{r,s}(n))$ generalizing earlier results of Dipper–Donkin and Du–Parshall–Wang. The explicit formulation provides an interpretation of the deforming parameters, and the quantized algebra $U_{r,s}(R)$ is identified to $U_{r,s}(\mathfrak {gl}_n)$ as the dual algebra. We then construct $n-1$ quasi-central elements in $U_{r,s}(R)$ which are analogs of higher Casimir elements in $U_q(\mathfrak {gl}_n)$.

In this note we present various extensions of Obata’s rigidity theorem concerning the Hessian of a function on a Riemannian manifold. They include general rigidity theorems for the generalized Obata equation, and hyperbolic and Euclidean analogs of Obata’s theorem. Besides analyzing the full rigidity case, we also characterize the geometry and topology of the underlying manifold in more general situations.

We give a brief survey on aspects of the local index theory as developed from the mathematical works of V. K. Patodi. It is dedicated to the 70th anniversary of Patodi.

For discrete spectrum of 1D second-order differential/difference operators (with or without potential (killing), with the maximal/minimal domain), a pair of unified dual criteria are presented in terms of two explicit measures and the harmonic function of the operators. Interestingly, these criteria can be read out from the ones for the exponential convergence of four types of stability studied earlier, simply replacing the ‘finite supremum’ by ‘vanishing at infinity’. Except a dual technique, the main tool used here is a transform in terms of the harmonic function, to which two new practical algorithms are introduced in the discrete context and two successive approximation schemes are reviewed in the continuous context. All of them are illustrated by examples. The main body of the paper is devoted to the hard part of the story, the easier part but powerful one is delayed to the end of the paper.

Let $M$ be a complete Kähler surface and $\Sigma $ be a symplectic surface which is smoothly immersed in $M$. Let $\alpha $ be the Kähler angle of $\Sigma $ in $M$. In the previous paper Han and Li (JEMS 12:505–527, 2010) 2010, we study the symplectic critical surfaces, which are critical points of the functional $L=\int _{\Sigma }\frac{1}{\cos \alpha }d\mu $ in the class of symplectic surfaces. In this paper, we calculate the second variation of the functional $L$ and derive some consequences. In particular, we show that, if the scalar curvature of $M$ is positive, $\Sigma $ is a stable symplectic critical surface with $\cos \alpha \ge \delta >0$, whose normal bundle admits a holomorphic section $X\in L^2(\Sigma )$, then $\Sigma $ is holomorphic. We construct symplectic critical surfaces in $\mathbf{C}^2$. We also prove a Liouville theorem for symplectic critical surfaces in $\mathbf{C}^2$.

We study a class of nonlocal-diffusion equations with drifts, and derive a priori $\Phi $-Hölder estimate for the solutions by using a purely probabilistic argument, where $\Phi $ is an intrinsic scaling function for the equation.

A subgroup $E$ of a finite group $G$ is called hypercyclically embedded in $G$ if every chief factor of $G$ below $E$ is cyclic. Let $A$ be a subgroup of a group $G$. Then we call any chief factor $H/A_{G}$ of $G$ a $G$-boundary factor of $A$. For any $G$-boundary factor $H/A_{G}$ of $A$, we call the subgroup $(A\cap H)/A_{G}$ of $G/A_{G}$ a $G$-trace of $A$. On the basis of these notions, we give some new characterizations of hypercyclically embedded subgroups.

A uniform logarithmic Sobolev inequality, a uniform Sobolev inequality and a uniform $\kappa $-noncollapsing estimate along the Ricci flow are established in the situation that a certain smallest eigenvalue for the initial metric is zero.

We address the well-posedness of the 2D (Euler)–Boussinesq equations with zero viscosity and positive diffusivity in the polygonal-like domains with Yudovich’s type data, which gives a positive answer to part of the questions raised in Lai (Arch Ration Mech Anal 199(3):739–760, 2011). Our analysis on the the polygonal-like domains essentially relies on the recent elliptic regularity results for such domains proved in Bardos et al. (J Math Anal Appl 407(1):69–89, 2013) and Di Plinio (SIAM J Math Anal 47(1):159–178, 2015).

Let $\Sigma $ be a $C^3$ compact symmetric convex hypersurface in $\mathbf {R}^{8}$. We prove that when $\Sigma $ carries exactly four geometrically distinct closed characteristics, then all of them must be symmetric. Due to the example of weakly non-resonant ellipsoids, our result is sharp.

The Krätzel function has many applications in applied analysis, so this function is used as a base to create a density function which will be called the Krätzel density. This density is applicable in chemical physics, Hartree–Fock energy, helium isoelectric series, statistical mechanics, nuclear energy generation, etc., and also connected to Bessel functions. The main properties of this new family are studied, showing in particular that it may be generated via mixtures of gamma random variables. Some basic statistical quantities associated with this density function such as moments, Mellin transform, and Laplace transform are obtained. Connection of Krätzel distribution to reaction rate probability integral in physics, inverse Gaussian density in stochastic processes, Tsallis statistics and superstatistics in non-extensive statistical mechanics, Mellin convolutions of products and ratios thereby to fractional integrals, synthetic aperture radar, and other areas are pointed out in this article. Finally, we extend the Krätzel density using the pathway model of Mathai, and some applications are also discussed. The new probability model is fitted to solar radiation data.

We propose a method that combines isogeometric analysis (IGA) with the discontinuous Galerkin (DG) method for solving elliptic equations on 3-dimensional (3D) surfaces consisting of multiple patches. DG ideology is adopted across the patch interfaces to glue the multiple patches, while the traditional IGA, which is very suitable for solving partial differential equations (PDEs) on (3D) surfaces, is employed within each patch. Our method takes advantage of both IGA and the DG method. Firstly, the time-consuming steps in mesh generation process in traditional finite element analysis (FEA) are no longer necessary and refinements, including $h$-refinement and $p$-refinement which both maintain the original geometry, can be easily performed by knot insertion and order-elevation (Farin, in Curves and surfaces for CAGD, 2002). Secondly, our method can easily handle the cases with non-conforming patches and different degrees across the patches. Moreover, due to the geometric flexibility of IGA basis functions, especially the use of multiple patches, we can get more accurate modeling of more complex surfaces. Thus, the geometrical error is significantly reduced and it is, in particular, eliminated for all conic sections. Finally, this method can be easily formulated and implemented. We generally describe the problem and then present our primal formulation. A new ideology, which directly makes use of the approximation property of the NURBS basis functions on the parametric domain rather than that of the IGA functions on the physical domain (the former is easier to get), is adopted when we perform the theoretical analysis including the boundedness and stability of the primal form, and the error analysis under both the DG norm and the ${\mathcal {L}}^{2}$ norm. The result of the error analysis shows that our scheme achieves the optimal convergence rate with respect to both the DG norm and the ${\mathcal {L}}^{2}$ norm. Numerical examples are presented to verify the theoretical result and gauge the good performance of our method.