ECG Classification Based on Wasserstein Scalar Curvature.

Electrocardiograms (ECG) analysis is one of the most important ways to diagnose heart disease. This paper proposes an efficient ECG classification method based on Wasserstein scalar curvature to comprehend the connection between heart disease and the mathematical characteristics of ECG. The newly proposed method converts an ECG into a point cloud on the family of Gaussian distribution, where the pathological characteristics of ECG will be extracted by the Wasserstein geometric structure of the statistical manifold. Technically, this paper defines the histogram dispersion of Wasserstein scalar curvature, which can accurately describe the divergence between different heart diseases. By combining medical experience with mathematical ideas from geometry and data science, this paper provides a feasible algorithm for the new method, and the theoretical analysis of the algorithm is carried out. Digital experiments on the classical database with large samples show the new algorithm's accuracy and efficiency when dealing with the classification of heart disease.

Application of Statistical K-Means Algorithm for University Academic Evaluation.

With the globalization of higher education, academic evaluation is increasingly valued by the scientific and educational circles. Although the number of published papers of academic evaluation methods is increasing, previous research mainly focused on the method of assigning different weights for various indicators, which can be subjective and limited. This paper investigates the evaluation of academic performance by using the statistical K-means (SKM) algorithm to produce clusters. The core idea is mapping the evaluation data from Euclidean space to Riemannian space in which the geometric structure can be used to obtain accurate clustering results. The method can adapt to different indicators and make full use of big data. By using the K-means algorithm based on statistical manifolds, the academic evaluation results of universities can be obtained. Furthermore, through simulation experiments on the top 20 universities of China with the traditional K-means, GMM and SKM algorithms, respectively, we analyze the advantages and disadvantages of different methods. We also test the three algorithms on a UCI ML dataset. The simulation results show the advantages of the SKM algorithm.

University Academic Performance Development Prediction Based on TDA.

With the rapid development of higher education, the evaluation of the academic growth potential of universities has received extensive attention from scholars and educational administrators. Although the number of papers on university academic evaluation is increasing, few scholars have conducted research on the changing trend of university academic performance. Because traditional statistical methods and deep learning techniques have proven to be incapable of handling short time series data well, this paper proposes to adopt topological data analysis (TDA) to extract specified features from short time series data and then construct the model for the prediction of trend of university academic performance. The performance of the proposed method is evaluated by experiments on a real-world university academic performance dataset. By comparing the prediction results given by the Markov chain as well as SVM on the original data and TDA statistics, respectively, we demonstrate that the data generated by TDA methods can help construct very discriminative models and have a great advantage over the traditional models. In addition, this paper gives the prediction results as a reference, which provides a new perspective for the development evaluation of the academic performance of colleges and universities.

Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data Processing.

The Wasserstein distance, especially among symmetric positive-definite matrices, has broad and deep influences on the development of artificial intelligence (AI) and other branches of computer science. In this paper, by involving the Wasserstein metric on SPD(n), we obtain computationally feasible expressions for some geometric quantities, including geodesics, exponential maps, the Riemannian connection, Jacobi fields and curvatures, particularly the scalar curvature. Furthermore, we discuss the behavior of geodesics and prove that the manifold is globally geodesic convex. Finally, we design algorithms for point cloud denoising and edge detecting of a polluted image based on the Wasserstein curvature on SPD(n). The experimental results show the efficiency and robustness of our curvature-based methods.

A Geometric Approach to Average Problems on Multinomial and Negative Multinomial Models.

This paper is concerned with the formulation and computation of average problems on the multinomial and negative multinomial models. It can be deduced that the multinomial and negative multinomial models admit complementary geometric structures. Firstly, we investigate these geometric structures by providing various useful pre-derived expressions of some fundamental geometric quantities, such as Fisher-Riemannian metrics, α -connections and α -curvatures. Then, we proceed to consider some average methods based on these geometric structures. Specifically, we study the formulation and computation of the midpoint of two points and the Karcher mean of multiple points. In conclusion, we find some parallel results for the average problems on these two complementary models.

The Bayesian Inference of Pareto Models Based on Information Geometry.

Bayesian methods have been rapidly developed due to the important role of explicable causality in practical problems. We develope geometric approaches to Bayesian inference of Pareto models, and give an application to the analysis of sea clutter. For Pareto two-parameter model, we show the non-existence of α-parallel prior in general, hence we adopt Jeffreys prior to deal with the Bayesian inference. Considering geodesic distance as the loss function, an estimation in the sense of minimal mean geodesic distance is obtained. Meanwhile, by involving Al-Bayyati's loss function we gain a new class of Bayesian estimations. In the simulation, for sea clutter, we adopt Pareto model to acquire various types of parameter estimations and the posterior prediction results. Simulation results show the advantages of the Bayesian estimations proposed and the posterior prediction.

Rigid Shape Registration Based on Extended Hamiltonian Learning.

Shape registration, finding the correct alignment of two sets of data, plays a significant role in computer vision such as objection recognition and image analysis. The iterative closest point (ICP) algorithm is one of well known and widely used algorithms in this area. The main purpose of this paper is to incorporate ICP with the fast convergent extended Hamiltonian learning (EHL), so called EHL-ICP algorithm, to perform planar and spatial rigid shape registration. By treating the registration error as the potential for the extended Hamiltonian system, the rigid shape registration is modelled as an optimization problem on the special Euclidean group S E ( n ) ( n = 2 , 3 ) . Our method is robust to initial values and parameters. Compared with some state-of-art methods, our approach shows better efficiency and accuracy by simulation experiments.

A Matrix Information-Geometric Method for Change-Point Detection of Rigid Body Motion.

A matrix information-geometric method was developed to detect the change-points of rigid body motions. Note that the set of all rigid body motions is the special Euclidean group S E ( 3 ) , so the Riemannian mean based on the Lie group structures of S E ( 3 ) reflects the characteristics of change-points. Once a change-point occurs, the distance between the current point and the Riemannian mean of its neighbor points should be a local maximum. A gradient descent algorithm is proposed to calculate the Riemannian mean. Using the Baker-Campbell-Hausdorff formula, the first-order approximation of the Riemannian mean is taken as the initial value of the iterative procedure. The performance of our method was evaluated by numerical examples and manipulator experiments.

Riemannian means on special euclidean group and unipotent matrices group.

Among the noncompact matrix Lie groups, the special Euclidean group and the unipotent matrix group play important roles in both theoretic and applied studies. The Riemannian means of a finite set of the given points on the two matrix groups are investigated, respectively. Based on the left invariant metric on the matrix Lie groups, the geodesic between any two points is gotten. And the sum of the geodesic distances is taken as the cost function, whose minimizer is the Riemannian mean. Moreover, a Riemannian gradient algorithm for computing the Riemannian mean on the special Euclidean group and an iterative formula for that on the unipotent matrix group are proposed, respectively. Finally, several numerical simulations in the 3-dimensional case are given to illustrate our results.

A priori and a posteriori error analysis of the first order hyperbolic equation by using DG method.

In this research article, a discontinuous Galerkin method with a weighted parameter θ and a penalty parameter γ is proposed for solving the first order hyperbolic equation. The key aim of this method is to design an error estimation for both a priori and a posteriori error analysis on general finite element meshes. It is also exposed to the reliability and effectiveness of both parameters in the order of convergence of the solutions. For a posteriori error estimation, residual adaptive mesh- refining algorithm is employed. A series of numerical experiments are illustrated that demonstrate the efficiency of the method.

Theoretical design and screening of alkyne bridged triphenyl zinc porphyrins as sensitizer candidates for dye-sensitized solar cells.

Alkyne bridged porphyrins have been proved very promising sensitizers for dye-sensitized solar cells (DSSCs) with the highest photo-to-electric conversion efficiencies of 11.9% solely and 12.3% co-sensitized with other sensitizers achieved. Developing better porphyrin sensitizers with wider electronic absorption spectra to further improve the efficiencies of corresponding solar cells is still of great significance for the application of DSSCs. A series of triphenyl zinc porphyrins (ZnTriPP) differing in the nature of a pendant acceptor group and the conjugated bridge between the porphyrin nucleus and the acceptor unit were modeled and their electronic and spectral properties calculated using density functional theory. As compared with each other and the experimental results of the compounds used in DSSCs previously, the molecules with a relatively longer conjugative linker and a strong electron-withdrawing group such as cyanide adjacent to the carboxyl acid group seem to provide wider electronic absorption spectra and higher photo-to-electric conversion efficiencies. The dye candidates ZnTriPPE, ZnTriPPM, ZnTriPPQ, ZnTriPPR and ZnTriPPS designed in the current work were found promising to provide comparable photo-to-electric conversion efficiencies to the record 11.9% of the alkyne bridged porphyrin sensitizer YD2-o-C8 reported previously.

Theoretical screening of novel alkyne bridged zinc porphyrins as sensitizer candidates for dye-sensitized solar cells.

Alkyne bridged porphyrin sensitizers have attracted great attention in the field of dye-sensitized solar cells (DSSCs) because of their excellent photo-to-electric conversion efficiencies, among which YD2 has reached 11% while YD2-o-C8 has reached 11.9% solely and 12.3% co-sensitized with other sensitizers. Design and screening of porphyrin sensitizer candidates with wider electronic absorption spectra to further improve the photo-to-electric conversion efficiencies of corresponding solar cells is still very important. Twenty novel alkyne bridged zinc porphyrin sensitizer candidates composed of the donors diarylamino-, tri-4-methylphenyl-, tri-hydroxyl- and tri-amino-substituted zinc porphyrins as well as the selected acceptors E, M, Q, R and S have been designed and calculated at the density functional B3LYP level. YD2 and YD2-o-C8 are also calculated at the same level for comparison. The result shows that the sensitizer candidates all have smaller HOMO-LUMO gaps as well as wider and red-shifted absorption bands than those of YD2 and YD2-o-C8. Most of the sensitizer candidates have appropriate HOMO and LUMO energy levels relative to the redox potential of the mediator and the TiO2 conduction band, showing that they are promising to provide comparable or even higher photo-to-electric conversion efficiencies than 11% of YD-2 or 11.9% of YD2-o-C8.

Substituent effects on zinc phthalocyanine derivatives: a theoretical calculation and screening of sensitizer candidates for dye-sensitized solar cells.

A series of unsymmetrical phthalocyanine sensitizer candidates with different donor and acceptor substituents, namely ZnPcBPh, ZnPcBOPh, ZnPcBtBu, ZnPcBN(Ph)2, ZnPcBNHPh, ZnPcBNH2, ZnPcBNHCH3 and ZnPcBN(CH3)2, were designed and calculated using density functional theory (DFT) and time-dependent DFT calculations. The molecular orbital energy levels, the molecular orbital spatial distributions and the electronic absorption spectra of the ZnPcB series molecules were compared with those of TT7 and TT8 to reveal the substituent effects of different donor and acceptor groups on the phthalocyanine compounds and select good sesitizer candidates. The results show that some of these compounds have considerably smaller orbital energy gaps, red-shifted absorption bands and better charge-separated states, causing them to absorb photons in the lower energy region. Several new absorption bands emerge in the 400-600 nm region, which makes it possible for them to become panchromatic sensitizers. This characteristic is superior to the phthalocyanine sensitizers reported previously, including the current record holder, PcS6. The sensitizer candidates screened in the current work are very promising for providing good performance and might even challenge the photon-to-electricity conversion efficiency record of 4.6% for phthalocyanine sensitizers.

Theoretical design and screening of panchromatic phthalocyanine sensitizers derived from TT1 for dye-sensitized solar cells.

Computational screening of new dyes is becoming an extremely powerful tool, especially when associated with experimental synthetic efforts that might eventually lead to new and more efficient products. Nine novel unsymmetrical zinc phthalocyanine complexes derived from TT1 were designed as sensitizer candidates for dye-sensitized solar cells with three peripheral -CH3, -OH, -OCH3, -OPh, -NH2, -NHCH3, -N(CH3)2, -NHPh and -N(Ph)2 substituents as the donors and a carboxyl group as the acceptor. The molecular orbital and the electronic absorption spectra properties of these compounds were studied and compared to those of TT1 using the density functional theory and time-dependent density functional theory calculations at B3LYP level with the LANL2DZ basis set. The novel candidates bearing the -NH2, -NHCH3, -N(CH3)2, -NHPh and -N(Ph)2 moieties as the donors were found to be very promising for providing higher efficiencies than that of TT1 or even the current 4.6% efficiency record held by PcS6. They have higher LUMO levels, smaller energy gaps and red-shifted absorption bands compared to those of TT1. The new absorption bands emerging in 450-600 nm regions may promote ZnPcL-NH2, ZnPcL-NHCH3, ZnPcL-N(CH3)2, ZnPcL-NHPh and ZnPcL-N(Ph)2 from near infrared to panchromatic sensitizers. Further experimental synthetic efforts are in progress in our group to validate the predictions in this report.