For $b\in \mathrm{L}\mathrm{i}\mathrm{p}({R}^{n})$, the Calderón commutator with variable kernel is defined by$$[b,{{\displaystyle T}}_{\mathrm{1}}]f(x)=p.v.{{\displaystyle \int}}_{{R}^{n}}{\displaystyle \frac{\mathrm{\Omega}(x,x-y)}{{\left|x-y\right|}^{n+\mathrm{1}}}}(b(x)-b(y))f(y)\mathrm{d}y.$$In this paper, we establish the ${L}^{\mathrm{2}}({R}^{n})$ boundedness for $[b,{{\displaystyle T}}_{\mathrm{1}}]$ with $\Omega (x,{z}^{\text{'}})\in {L}^{\infty}({R}^{n})\times {L}^{q}({S}^{n-\mathrm{1}})(q\rangle \mathrm{2}(n-\mathrm{1})/n)$ satisfying certain cancellation conditions. Moreover, the exponent $q\rangle \mathrm{2}(n-\mathrm{1})/n$ is optimal. Our main result improves a previous result of Calderón.

We construct a sequence of transient random walks in random environments and prove that by proper scaling, it converges to a diffusion process with drifted Brownian potential. To this end, we prove a counterpart of convergence for transient random walk in non-random environment, which is interesting itself.

For the principle eigenvalue of discrete weighted p-Laplacian on the set of nonnegative integers, the convergence of an approximation procedure and the inverse iteration is proved. Meanwhile, in the proof of the convergence, the monotonicity of an approximation sequence is also checked. To illustrate these results, some examples are presented.

Let ${{\displaystyle \{{{\displaystyle S}}_{i}\}}}_{i=\mathrm{1}}^{l}$ be an iterated function system (IFS) on ${?}^{d}$ with an attractor K. Let (Σ, σ) denote the one-sided full shift over the finite alphabet {1, 2, . . . , l}, and let π: Σ → K be the coding map. Given an asymptotically (sub)-additive sequence of continuous functions $F={{\displaystyle \{{{\displaystyle f}}_{n}\}}}_{n\ge \mathrm{1}}$, we define the asymptotically additive projection pressure P_{π}($F$) and show the variational principle for P_{π}($F$) under certain affine IFS. We also obtain variational principle for the asymptotically sub-additive projection pressure if the IFS satisfies asymptotically weak separation condition (AWSC). Furthermore, when the IFS satisfies AWSC, we investigate the zero temperature limits of the asymptotically sub-additive projection pressure P_{π}(β$F$) with positive parameter β.

A hybridization technique is applied to the weak Galerkin finite element method (WGFEM) for solving the linear elasticity problem in mixed form. An auxiliary function, the Lagrange multiplier defined on the boundary of elements, is introduced in this method. Consequently, the computational costs are much lower than the standard WGFEM. Optimal order error estimates are presented for the approximation scheme. Numerical results are provided to verify the theoretical results.

We prove an Artin-Schelter regularity result for the method of twisted tensor products under a certain form. Such twisted tensor products, whose twisting maps are determined by the action on the generators, include Ore extensions and double Ore extensions. It is helpful to construct highdimensional Artin-Schelter regular algebras.

A subgroup H of a finite group G is called a c^{#}-normal subgroup of G if there exists a normal subgroup K of G such that G = HK and H ∩ K is a CAP-subgroup of G. In this paper, we investigate the influence of fewer c^{#}-normal subgroups of Sylow p-subgroups on the p-supersolvability, p-nilpotency, and supersolvability of finite groups. We obtain some new sufficient and necessary conditions for a group to be p-supersolvable, p-nilpotent, and supersolvable. Our results improve and extend many known results.

Let g= W_{1} be the Witt algebra over an algebraically closed field k of characteristic p >3, and let $C(g)$ = {(x, y) ∈g ×g | [x, y] = 0}be the commuting variety of g. In contrast with the case of classical Lie algebras of P. Levy [J. Algebra, 2002, 250: 473–484], we show that the variety $C(g)$ is reducible, and not equidimensional. Irreducible components of $C(g)$ and their dimensions are precisely given. As a consequence, the variety $C(g)$ is not normal.

We introduce two Hopf algebroids associated to a proper and holomorphic Lie group action on a complex manifold. We prove that the cyclic cohomology of each Hopf algebroid is equal to the Dolbeault cohomology of invariant differential forms. When the action is cocompact, we develop a generalized complex Hodge theory for the Dolbeault cohomology of invariant differential forms. We prove that every cyclic cohomology class of these two Hopf algebroids can be represented by a generalized harmonic form. This implies that the space of cyclic cohomology of each Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we generalize the Serre duality and prove a Kodaira type vanishing theorem.

Based on an explicit representation of moments of hitting times for single death processes, the criteria on ergodicity and strong ergodicity are obtained. These results can be applied for an extended class of branching processes. Meanwhile, some sufficient and necessary conditions for recurrence and exponential ergodicity as well as extinction probability for the processes are presented.

We extend the Riemann-Hilbert approach to the TD equation, which is a highly nonlinear differential integrable equation. Zero boundary condition at infinity for the TD equation is not suitable. Inverse scattering transform for this equation involves the singular Riemann-Hilbert problem, which means that the sectionally analytic functions have singularities on the boundary curve. Regularization procedures of the singular Riemann-Hilbert problem for two cases, the general case and the case for reflectionless potentials, are considered. Solitonic solutions to the TD equation are given.