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This study addresses the critical challenges of assessing foundational academic skills by leveraging advancements in natural language processing (NLP). Traditional assessment methods often struggle to provide timely and comprehensive feedback on key cognitive and linguistic aspects, such as coherence, syntax, and analytical reasoning. Our approach integrates multiple state-of-the-art NLP models, including BERT, RoBERTa, BART, DeBERTa, and T5, within an ensemble learning framework. These models are combined through stacking techniques using LightGBM and Ridge regression to enhance predictive accuracy. The methodology involves detailed data preprocessing, feature extraction, and pseudo-label learning to optimize model performance. By incorporating sophisticated NLP techniques and ensemble learning, this study significantly improves the accuracy and efficiency of assessments, offering a robust solution that surpasses traditional methods and opens new avenues for educational technology research focused on enhancing core academic competencies., Comment: 5 pages, 2 figures

In this paper, we present a method for digitally representing the "volume element" and calculating the integral of a function on compact hypersurfaces with or without boundary, and low-dimensional submanifolds in $\mathbb{R}^n$. We also extend such calculation to hypersurfaces in compact Riemannnian manifolds.

Studying the existence of rational functions with given branching datum is a classical problem in the field of complex analysis and algebraic geometry. This problem dates back to Hurwitz and remains open to this day. In this paper, we utilize complex analysis to establish a property of rational functions with 3 branching points on the Riemann sphere. Given two compact Riemann surfaces $M$ and $N$, a pair $(d,\mathcal{D})$ of an integer $d\geq2$ and a collection $\mathcal{D}$ of nontrivial partitions of $d$ is called a candidate branching datum if it satisfies the Riemann-Hurwitz formula. And a candidate branching datum is exceptional if there does not exist a rational function realization it. As applications, we present some new types of exceptional branching datum. These results cover some previous results mentioned in \cite{EKS84,PP06,Zhu19}. We also deduce the realizability of a certain type of candidate branching datum on the Riemann sphere., Comment: 9 pages. We have revised certain sentences, corrected several errors, and added additional content to enhance the clarity and depth of our paper. Comments are welcome

A conformal metric ${\rm d}s^{2}$ with finitely many conical singularities of constant Gaussian curvature $K=1$ on a compact Riemann surface is referred to as a spherical conical metric. When the associated monodromy group of ${\rm d}s^{2}$ is diagonalizable, we refer to ${\rm d}s^{2}$ as a reducible spherical conical metric. The simplest case of a reducible spherical conical metric is a `football', which denotes a 2-sphere with a spherical conical metric that has precisely two singularities separated by a distance of $\pi$. This study delves into the intrinsic geometric structure and existence of reducible spherical conical metrics on compact Riemann surfaces. We demonstrate that any such spherical surface can be divided into a finite number of pieces by cutting along a set of suitable geodesics, which connect the conical singularities and some smooth points of the metric. Especially, each piece is isometric to a portion obtained by cutting a football along a geodesic that joins the two conical singularities. As an application, an angle condition for the existence of such a metric is presented. Most notably, our study demonstrates the existence of a reducible spherical conical metric where all the saddle points of a Morse function are located on the same geodesic., Comment: 47 pages, 17 figures

In \cite{WeiWu22-3} (J.Geom.Anal.32, 199(2022)), we classified HCMU surfaces in 3-dimensional Euclidean space as Weingarten surfaces by holomorphic functions. In this paper, using a totally different method from \cite{WeiWu22-3}, we will classify HCMU surfaces in 3-dimensional space forms as Weingarten surfaces. Moreover we will give a criteria of a Weingarten surface which is also an HCMU surface., Comment: 12pages

On any compact Riemann surface there always exists a singular non-CSC (constant scalar curvature) extremal K$\ddot{a}$hler metric which is called a non-CSC HCMU (the Hessian of the Curvature of the Metric is Umbilical) metric. In this paper, by moving frames, we show that any non-CSC HCMU metric can not be isometrically minimal immersed into 3-dimensional real space forms even locally. In general, any non-CSC HCMU metric can not be isometrically immersed into 3-dimensional real space forms with constant mean curvature (CMC).

A conformal metric $g$ with constant curvature one and finite conical singularities on a compact Riemann surface $\Sigma$ can be thought of as the pullback of the standard metric on the 2-sphere by a multi-valued locally univalent meromorphic function $f$ on $\Sigma\backslash \{{\rm singularities}\}$, called the {\it developing map} of the metric $g$. When the developing map $f$ of such a metric $g$ on the compact Riemann surface $\Sigma$ has reducible monodromy, we show that, up to some M{\" o}bius transformation on $f$, the logarithmic differential $d\,(\log\, f)$ of $f$ turns out to be an abelian differential of 3rd kind on $\Sigma$, which satisfies some properties and is called a {\it character 1-form of} $g$. Conversely, given such an abelian differential $\omega$ of 3rd kind satisfying the above properties, we prove that there exists a unique conformal metric $g$ on $\Sigma$ with constant curvature one and conical singularities such that one of its character 1-forms coincides with $\omega$. This provides new examples of conformal metrics on compact Riemann surfaces of constant curvature one and with singularities. Moreover, we prove that the developing map is a rational function for a conformal metric $g$ with constant curvature one and finite conical singularities with angles in $2\pi\,{\Bbb Z}_{>1}$ on the two-sphere., Comment: Substantially revised. In particular, the definition of abelian metric is replaced by that of reducible metric. Comments welcomed. Submitted

We study the basic structure of a HCMU metric in a K-Surface with prescribed singularities. When the underlying smooth surface is $S^2$, we prove the necessary condition given in [1] for the existence of HCMU metric is also sufficient.