Search

We study the monic orthogonal polynomials with respect to a singularly perturbed Airy weight. By using Chen and Ismail's ladder operator approach, we derive a discrete system satisfied by the recurrence coefficients for the orthogonal polynomials. We find that the orthogonal polynomials satisfy a second-order linear ordinary differential equation, whose coefficients are all expressed in terms of the recurrence coefficients. By considering the time evolution, we obtain a system of differential-difference equations satisfied by the recurrence coefficients. Finally, we study the asymptotics of the recurrence coefficients when the degrees of the orthogonal polynomials tend to infinity., Comment: 16 pages

We consider the smallest eigenvalue distributions of some Freud unitary ensembles, that is, the probabilities that all the eigenvalues of the Hermitian matrices from the ensembles lie in the interval $(t,\infty)$. This problem is related to the Hankel determinants generated by the Freud weights with a jump discontinuity. By using Chen and Ismail's ladder operator approach, we obtain the discrete systems for the recurrence coefficients of the corresponding orthogonal polynomials. This enables us to derive the large $n$ asymptotics of the recurrence coefficients via Dyson's Coulomb fluid approach. We finally obtain the large $n$ asymptotics of the Hankel determinants and that of the probabilities from their relations to the recurrence coefficients and with the aid of some recent results in the literature., Comment: 35 pages

Can graded meshes yield more accurate numerical solution than uniform meshes? A time-dependent nonlocal diffusion problem with a weakly singular kernel is considered using collocation method. For its steady-state counterpart, under the sufficiently smooth solution, we first clarify that the standard graded meshes are worse than uniform meshes and may even lead to divergence; instead, an optimal convergence rate arises in so-called anomalous graded meshes. Furthermore, under low regularity solutions, it may suffer from a severe order reduction in (Chen, Qi, Shi and Wu, IMA J. Numer. Anal., 41 (2021) 3145--3174). In this case, conversely, a sharp error estimates appears in standard graded meshes, but offering far less than first-order accuracy. For the time-dependent case, however, second-order convergence can be achieved on graded meshes. The related analysis are easily extended for certain multidimensional problems. Numerical results are provided that confirm the sharpness of the error estimates., Comment: 20pages

In this paper, we first establish the separation theorem between a point and a locally geodesic convex set and then prove the existence of a supporting quasi-hyperplane at any point on the boundary of the closed locally geodesic convex set on a Riemannian manifold. As applications, some optimality conditions are obtained for optimization problems with constraints on Riemannian manifolds.

Effective prediction of shale gas production is crucial for strategic reservoir development. However, in new shale gas blocks, two main challenges are encountered: (1) the occurrence of negative transfer due to insufficient data, and (2) the limited interpretability of deep learning (DL) models. To tackle these problems, we propose a novel transfer learning methodology that utilizes domain adaptation and physical constraints. This methodology effectively employs historical data from the source domain to reduce negative transfer from the data distribution perspective, while also using physical constraints to build a robust and reliable prediction model that integrates various types of data. The methodology starts by dividing the production data from the source domain into multiple subdomains, thereby enhancing data diversity. It then uses Maximum Mean Discrepancy (MMD) and global average distance measures to decide on the feasibility of transfer. Through domain adaptation, we integrate all transferable knowledge, resulting in a more comprehensive target model. Lastly, by incorporating drilling, completion, and geological data as physical constraints, we develop a hybrid model. This model, a combination of a multi-layer perceptron (MLP) and a Transformer (Transformer-MLP), is designed to maximize interpretability. Experimental validation in China's southwestern region confirms the method's effectiveness.

To address the issues of stability and fidelity in interpretable learning, a novel interpretable methodology, ensemble interpretation, is presented in this paper which integrates multi-perspective explanation of various interpretation methods. On one hand, we define a unified paradigm to describe the common mechanism of different interpretation methods, and then integrate the multiple interpretation results to achieve more stable explanation. On the other hand, a supervised evaluation method based on prior knowledge is proposed to evaluate the explaining performance of an interpretation method. The experiment results show that the ensemble interpretation is more stable and more consistent with human experience and cognition. As an application, we use the ensemble interpretation for feature selection, and then the generalization performance of the corresponding learning model is significantly improved.

A novel population-based optimization method is proposed in this paper, the Calico Salmon Migration Algorithm (CSMA), which is inspired by the natural behavior of calico salmon during their migration for mating. The CSMA optimization process comprises four stages: selecting the search space by swimming into the river, expanding the search space from the river into the ocean, performing precise search during the migrating process, and breeding new subspecies by the remaining calico salmon population. To evaluate the effectiveness of the new optimizer, we conducted a series of experiments using different optimization problems and compared the results with various optimization algorithms in the literature. The numerical experimental results for benchmark functions demonstrate that the proposed CSMA outperforms other competing optimization algorithms in terms of convergence speed, accuracy, and stability. Furthermore, the Friedman ranking test shows that the CSMA is ranked first among similar algorithms.

We study the asymptotic properties of monic orthogonal polynomials (OPs) with respect to some Freud weights when the degree of the polynomial tends to infinity, including the asymptotics of the recurrence coefficients, the nontrivial leading coefficients of the monic OPs, the associated Hankel determinants and the squares of $L^2$-norm of the monic OPs. These results are derived from the combination of the ladder operator approach, Dyson's Coulomb fluid approach and some recent results in the literature.

Artificial Intelligence (AI), a cornerstone of 21st-century technology, has seen remarkable growth in China. In this paper, we examine China's AI development process, demonstrating that it is characterized by rapid learning and differentiation, surpassing the export-oriented growth propelled by Foreign Direct Investment seen in earlier Asian industrializers. Our data indicates that China currently leads the USA in the volume of AI-related research papers. However, when we delve into the quality of these papers based on specific metrics, the USA retains a slight edge. Nevertheless, the pace and scale of China's AI development remain noteworthy. We attribute China's accelerated AI progress to several factors, including global trends favoring open access to algorithms and research papers, contributions from China's broad diaspora and returnees, and relatively lax data protection policies. In the vein of our research, we have developed a novel measure for gauging China's imitation of US research. Our analysis shows that by 2018, the time lag between China and the USA in addressing AI research topics had evaporated. This finding suggests that China has effectively bridged a significant knowledge gap and could potentially be setting out on an independent research trajectory. While this study compares China and the USA exclusively, it's important to note that research collaborations between these two nations have resulted in more highly cited work than those produced by either country independently. This underscores the power of international cooperation in driving scientific progress in AI.

It is very difficult to forecast the production rate of oil wells as the output of a single well is sensitive to various uncertain factors, which implicitly or explicitly show the influence of the static, temporal and spatial properties on the oil well production. In this study, a novel machine learning model is constructed to fuse the static geological information, dynamic well production history, and spatial information of the adjacent water injection wells. There are 3 basic modules in this stacking model, which are regarded as the encoders to extract the features from different types of data. One is Multi-Layer Perceptron, which is to analyze the static geological properties of the reservoir that might influence the well production rate. The other two are both LSTMs, which have the input in the form of two matrices rather than vectors, standing for the temporal and the spatial information of the target well. The difference of the two modules is that in the spatial information processing module we take into consideration the time delay of water flooding response, from the injection well to the target well. In addition, we use Symbolic Transfer Entropy to prove the superiorities of the stacking model from the perspective of Causality Discovery. It is proved theoretically and practically that the presented model can make full use of the model structure to integrate the characteristics of the data and the experts' knowledge into the process of machine learning, greatly improving the accuracy and generalization ability of prediction.

Machine learning approaches are widely studied in the production prediction of CBM wells after hydraulic fracturing, but merely used in practice due to the low generalization ability and the lack of interpretability. A novel methodology is proposed in this article to discover the latent causality from observed data, which is aimed at finding an indirect way to interpret the machine learning results. Based on the theory of causal discovery, a causal graph is derived with explicit input, output, treatment and confounding variables. Then, SHAP is employed to analyze the influence of the factors on the production capability, which indirectly interprets the machine learning models. The proposed method can capture the underlying nonlinear relationship between the factors and the output, which remedies the limitation of the traditional machine learning routines based on the correlation analysis of factors. The experiment on the data of CBM shows that the detected relationship between the production and the geological/engineering factors by the presented method, is coincident with the actual physical mechanism. Meanwhile, compared with traditional methods, the interpretable machine learning models have better performance in forecasting production capability, averaging 20% improvement in accuracy.

We are concerned with the monic orthogonal polynomials with respect to a singularly perturbed Laguerre-type weight. By using the ladder operator approach, we derive a complicated system of nonlinear second-order difference equations satisfied by the recurrence coefficients. This allows us to derive the large $n$ asymptotic expansions of the recurrence coefficients. In addition, we also obtain a system of differential-difference equations for the recurrence coefficients.

This paper studies the monic semi-classical Laguerre polynomials based on previous work by Boelen and Van Assche \cite{Boelen}, Filipuk et al. \cite{Filipuk} and Clarkson and Jordaan \cite{Clarkson}. Filipuk, Van Assche and Zhang proved that the diagonal recurrence coefficient $\alpha_n(t)$ satisfies the fourth Painlev\'{e} equation. In this paper we show that the off-diagonal recurrence coefficient $\beta_n(t)$ fulfills the first member of Chazy II system. We also prove that the sub-leading coefficient of the monic semi-classical Laguerre polynomials satisfies both the continuous and discrete Jimbo-Miwa-Okamoto $\sigma$-form of Painlev\'{e} IV. By using Dyson's Coulomb fluid approach together with the discrete system for $\alpha_n(t)$ and $\beta_n(t)$, we obtain the large $n$ asymptotic expansions of the recurrence coefficients and the sub-leading coefficient. The large $n$ asymptotics of the associate Hankel determinant (including the constant term) is derived from its integral representation in terms of the sub-leading coefficient., Comment: 20 pages

Incorporating fresh members in teams is considered a pathway to team creativity. However, whether freshness improves team performance or not remains unclear, as well as the optimal involvement of fresh members for team performance. This study uses a group of authors on the byline of a publication as a proxy for a scientific team. We extend an indicator, i.e., team freshness, to measure the extent to which a scientific team incorporates new members, by calculating the fraction of new collaboration relations established within the team. Based on more than 43 million scientific publications covering more than a half-century of research from Microsoft Academic Graph, this study provides a holistic picture of the current development of team freshness by outlining the temporal evolution of freshness, and its disciplinary distribution. Subsequently, using a multivariable regression approach, we examine the association between team freshness and papers'short-term and long-term citations.The major findings are as follows: (1)team freshness in scientific teams has been increasing in the past half-century; (2)there exists an inverted-U-shaped association between team freshness and papers' citations in all the disciplines and in different periods;(3)the inverted-U-shaped relationship between team freshness and papers' citations is only found in small teams, while, in large teams, team freshness is significantly positively related to papers' citations.

We study the monic polynomials orthogonal with respect to a symmetric perturbed Gaussian weight $$ w(x;t):=\mathrm{e}^{-x^2}\left(1+t\: x^2\right)^\lambda,\qquad x\in \mathbb{R}, $$ where $t> 0,\;\lambda\in \mathbb{R}$. This weight is related to the single-user MIMO systems in information theory. It is shown that the recurrence coefficient $\beta_n(t)$ is related to a particular Painlev\'{e} V transcendent, and the sub-leading coefficient $\mathrm{p}(n,t)$ satisfies the Jimbo-Miwa-Okamoto $\sigma$-form of the Painlev\'{e} V equation. Furthermore, we derive the second-order difference equations satisfied by $\beta_n(t)$ and $\mathrm{p}(n,t)$, respectively. This enables us to obtain the large $n$ full asymptotic expansions for $\beta_n(t)$ and $\mathrm{p}(n,t)$ with the aid of Dyson's Coulomb fluid approach. We also consider the Hankel determinant $D_n(t)$, generated by the perturbed Gaussian weight. It is found that $H_n(t)$, a quantity allied to the logarithmic derivative of $D_n(t)$, can be expressed in terms of $\beta_n(t)$ and $\mathrm{p}(n,t)$. Based on this result, we obtain the large $n$ asymptotic expansion of $H_n(t)$ and then that of the Hankel determinant $D_n(t)$., Comment: 28 pages

We study the moment-generating functions (MGF) for linear eigenvalue statistics of Jacobi unitary, symplectic and orthogonal ensembles. By expressing the MGF as Fredholm determinants of kernels of finite rank, we show that the mean and variance of the suitably scaled linear statistics in these Jacobi ensembles are related to the sine kernel in the bulk of the spectrum, whereas they are related to the Bessel kernel at the (hard) edge of the spectrum. The relation between the Jacobi symplectic/orthogonal ensemble (JSE/JOE) and the Jacobi unitary ensemble (JUE) is also established., Comment: 27 pages

We study orthogonal polynomials and Hankel determinants generated by a symmetric semi-classical Jacobi weight. By using the ladder operator technique, we derive the second-order nonlinear difference equations satisfied by the recurrence coefficient $\beta_n(t)$ and the sub-leading coefficient $\mathrm{p}(n,t)$ of the monic orthogonal polynomials. This enables us to obtain the large $n$ asymptotics of $\beta_n(t)$ and $\mathrm{p}(n,t)$ based on the result of Kuijlaars et al. [Adv. Math. \textbf{188} (2004) 337-398]. In addition, we show the second-order differential equation satisfied by the orthogonal polynomials, with all the coefficients expressed in terms of $\beta_n(t)$. From the $t$ evolution of the auxiliary quantities, we prove that $\beta_n(t)$ satisfies a second-order differential equation and $R_n(t)=2n+1+2\alpha-2t(\beta_n(t)+\beta_{n+1}(t))$ satisfies a particular Painlev\'{e} V equation under a simple transformation. Furthermore, we show that the logarithmic derivative of the associated Hankel determinant satisfies both the second-order differential and difference equations. The large $n$ asymptotics of the Hankel determinant is derived from its integral representation in terms of $\beta_n(t)$ and $\mathrm{p}(n,t)$., Comment: 26 pages

The development of inventions is theorized as a process of searching and recombining existing knowledge components. Previous studies under this theory have examined myriad characteristics of recombined knowledge and their performance implications. One feature that has received much attention is technological knowledge age. Yet, little is known about how the age of scientific knowledge influences the impact of inventions, despite the widely known catalyzing role of science in the creation of new technologies. Here we use a large corpus of patents and derive features characterizing how patents temporally search in the scientific space. We find that patents that cite scientific papers have more citations and substantially more likely to become breakthroughs. Conditional on searching in the scientific space, referencing more recent papers increases the impact of patents and the likelihood of being breakthroughs. However, this positive effect can be offset if patents cite papers whose ages exhibit a low variance. These effects are consistent across technological fields.

In this paper, we study the orthogonal polynomials with respect to a singularly perturbed Pollaczek-Jacobi type weight $$ w(x,t):=(1-x^2)^\alpha\mathrm{e}^{-\frac{t}{1-x^{2}}},\qquad x\in[-1,1],\;\;\alpha>0,\;\;t>0. $$ By using the ladder operator approach, we establish the second-order difference equations satisfied by the recurrence coefficient $\beta_n(t)$ and the sub-leading coefficient $\mathrm{p}(n,t)$ of the monic orthogonal polynomials, respectively. We show that the logarithmic derivative of $\beta_n(t)$ can be expressed in terms of a particular Painlev\'{e} V transcendent. The large $n$ asymptotic expansions of $\beta_n(t)$ and $\mathrm{p}(n,t)$ are obtained by using Dyson's Coulomb fluid method together with the related difference equations. Furthermore, we study the associated Hankel determinant $D_n(t)$ and show that a quantity $\sigma_n(t)$, allied to the logarithmic derivative of $D_n(t)$, can be expressed in terms of the $\sigma$-function of a particular Painlev\'{e} V. The second-order differential and difference equations for $\sigma_n(t)$ are also obtained. In the end, we derive the large $n$ asymptotics of $\sigma_n(t)$ and $D_n(t)$ from their relations with $\beta_n(t)$ and $\mathrm{p}(n,t)$., Comment: 29 pages