We study the $\alpha $-harmonic map flow with free boundary for the Sacks–Uhlenbeck functional on a compact Riemann surface with smooth boundary and prove that this flow exists globally for smooth initial data and converges to an $\alpha $-harmonic map with free boundary as $t\rightarrow \infty $.

Inspired by the recent works of Stelzig and Khovanov–Qi on the structure of bounded double complex, we improve further this theory with more emphasis on the bounded double complexes possibly without real structures. We calculate out six types of indecomposable double complexes in the complex decomposition, to characterize various (weak) $\partial \overline{\partial }$-lemmata by de Rham, Dolbeault, Bott–Chern, Appeli and Varouchas’ cohomology groups. In particular, we obtain new characterizations of $\partial \overline{\partial }$-lemma and Frölicher-type inequalities, and also several new injectivity results on complex manifolds. From the double complex perspective, one can easily see the mechanism behind the validity or invalidity of the (weak) $\partial \overline{\partial }$-lemmata and injectivity.

We study the non-local Poisson equation with a parameter and in the whole space $\mathbb {R}^{d_1}$. Regularities of the solution are obtained under mild assumptions on the coefficients. The result is then used to study the asymptotic behavior of the solution of a singularly perturbed non-local partial differential equation. As an application, we obtain the averaging principle for a fast-slow stochastic dynamical system with Hölder continuous coefficients and driven by multiplicative jump noise. Furthermore, the rate of convergence is also obtained.

Evaluations are frequent, significant cognitive activities in human society. People make comprehensive judgments and rank the evaluated objects based on multiple factors, including relevant information and alternative data. Existing integrated evaluation methods assign weights based on expert opinions or data dispersion and provide evaluated rankings of objects through simple mapping. These methods are easy to implement, but the models produce biased results when the differences of all features in the evaluated objects are not significant enough. In order to reduce this error, we borrow the ideas from ranking methods and train the model based on ordering relations. We re-formulate the evaluation problem, construct a ranking model, and then design a new method named Ranking by Siamese Neural Network (RankSNN) adaptive to the model. This method uses ordering relations of sample pairs as labels to train the model, and measure the differences between samples by Siamese structure to get rankings. In this way, a trained scoring function can accurately correspond to the ordering relations among the sample pairs by calculating evaluation values and obtaining the best evaluation ranking results. In order to verify the validity and rationality of RankSNN, we conducted experiments on the public dataset of learning to rank and compared our results with the baseline ranking method. Then, we use RankSNN in integrated evaluation problems related to China’s urban business environment for the purpose of conducting empirical analyses, and the results of our method are superior to existing methods.

In this paper, the author is concerned with the positive radial solutions for a double singular $k_i$-Hessian system with a parameter $\lambda $. We establish a criterion on the existence of a positive radial solution for small $\lambda $, a criterion on the multiplicity of positive radial solutions for certain ranges of $\lambda $, and a criterion on the nonexistence of positive radial solutions in the case $\lambda \gg 1$. The dependence of the positive radial solution on the parameter $\lambda $ is also studied.

We extend the framework of the Karush–Kuhn–Tucker (KKT) limiter from second-order nonlinear scalar equations to complex systems of equations and construct time-implicit high-order accurate positivity-preserving local discontinuous Galerkin methods for the Navier-Stokes(NS) and Navier-Stokes-Korteweg(NSK) equations, which are second-order and third-order nonlinear systems, respectively. For the NS and NSK equations, a meaningful numerical approximation must ensure that the density and pressure are non-negative. Borrowing from the idea of the KKT limiter, we use Lagrange multipliers to couple the positivity constraints of density and pressure with local discontinuous Galerkin discretizations combined with a diagonally implicit Runge–Kutta time integration method and obtain high-order accurate positivity-preserving schemes for the NS and NSK equations. The use of diagonally implicit Runge–Kutta time integration methods greatly enlarges the time step in the numerical simulation of the NS and NSK equations. Different from the original application of the KKT limiter, the nonlinearity of pressure with respect to the conservative variables increases the difficulty in implementation. Numerical examples validate the efficiency of these methods.

In this paper, we study a class of bimodal cubic polynomials for which its critical points have the same $\omega $-limit set which is an invariant Cantor set. These maps have generalized Fibonacci combinatorics in terms of generalized renormalization on the twin principal nest. It is proved that such maps possess ‘decay of geometry’ in the sense that the scaling factor of the twin principal nest decreases at least exponentially fast. As an application, we prove that they have no Cantor attractor.

Recently (Ahmed et al. in Stat Theory Methods 47:324–343, 2018) have discussed the problem of estimating the means of two sensitive variables simultaneously by using two uncorrelated scrambling variables. In this study, we introduce a new idea that use of correlated scrambling variables in randomized response techniques play an important role to increase the efficiency of the estimators of population means of sensitive variables. We found that the simultaneous estimation of the means of the two sensitive variables by using two correlated scrambling variables results in gain in the efficiency of the estimators. This idea is appealing because it costs nothing to the investigator and does not effect the privacy of a respondent as well. Empirical evidences are given based on application to a real secondary data set.