A NEW CONDITION FOR CERTAIN TRIGONOMETRIC SERIES

The regularly present paper proposes a new natural weak condition to replace the quasimonotone and Ovarying quasimonotone conditions, and shows the conclusion of the main result in still holds. Moreover, we prove that any O-regularly varying quasimonotone condition implies this condition, thus the vague implication of quasimonotonieity can be avoided in practice

Readjustment of the Paper [J.Kaur and S.S.Bhatia,Integrability and L^1-Convergence of Double Cosine Trigonometric Series,Anal.Theory Appl.,27(1)(2011),pp.32–39.]

In this paper, we show that new modified double cosine trigonometric sums introduced in [1] are inappropriate, the class of double sequences Jintroduced there is unusable for such sums and consequently the results obtained in it are completely incorrect. We here introduce appropriate modified double cosine trigonometric sums making the class Jusable considering a particular double cosine trigonometric series.

A trigonometric series expansion method for the Orr-Sommerfeld equation

A trigonometric series expansion method and two similar modified methods for the Orr-Sommerfeld equation are presented. These methods use the trigonometric series expansion with an auxiliary function added to the highest order derivative of the unknown function and generate the lower order derivatives through successive integra- tions. The proposed methods are easy to implement because of the simplicity of the chosen basis functions. By solving the plane Poiseuille flow (PPF), plane Couette flow (PCF), and Blasius boundary layer flow with several homogeneous boundary conditions, it is shown that these methods yield results with the same accuracy as that given by the conventional Chebyshev collocation method but with better robustness, and that ob- tained by the finite difference method but with fewer modal number.

L^p integrability of trigonometric series with special varying coefficients

Very recently, Yu, Le and Zhou introduced the so called △B1^＊ and △B2^＊ conditions, which are generalizations of the monotone condition. By applying these two new conditions, the author essentially generalizes the classical results of Chen on the necessary and sufficient conditions of the Lp integrability of trigonometric series. In fact, the present paper gives the first result on the necessary and sufficient conditions of the Lp integrability of trigonometric series, where coefficients may have different signs.

SUMMATION OF TRIGONOMETRIC SERIES BY FOURIER TRANSFORMS

This paper gives the theorems concerning the summation of trigonometric series with the help of Fourier transforms. By means of the known results of Fourier transforms, many difficult and complex problems of summation of trigonometric series can be solved. This method is a comparatively unusual way to find the summation of trigonometric series, and has been used to establish the comprehensive table of summation of trigonometric series. In this table 10 thousand scries arc given, and most of them are new.

COMPARISONS OF ABSOLUTELY CONVERGENT TRIGONOMETRIC SERIES

In this paper, we first discuss the methods of comparing two special absolutely convergentsine series, sinnx and sinnx. We state the theorem in.one dimensional case as follows; Theorem. Let be convergent series with nonnegative terms. SupposeThen for all x∈[0,π]If, in addition, then

Analysis of the causes of primary revision after unicompartmental knee arthroplasty: A case series

BACKGROUND Unicompartmental knee arthroplasty(UKA)has great advantages in the treatment of unicompartmental knee osteoarthritis,but its revision rate is higher than that of total knee arthroplasty.AIM To summarize and analyse the causes of revision after UKA.METHODS This is a retrospective case series study in which the reasons for the first revision after UKA are summarized.We analysed the clinical symptoms,medical histories,laboratory test results,imaging examination results and treatment processes of the patients who underwent revision and summarized the reasons for primary revision after UKA.RESULTS A total of 13 patients,including 3 males and 10 females,underwent revision surgery after UKA.The average age of the included patients was 67.62 years.The prosthesis was used for 3 d to 72 months.The main reasons for revision after UKA were improper suturing of the surgical opening(1 patient),osteophytes(2 patients),intra-articular loose bodies(2 patients),tibial prosthesis loosening(2 patients),rheumatoid arthritis(1 patient),gasket dislocation(3 patients),anterior cruciate ligament injury(1 patient),and medial collateral ligament injury with residual bone cement(1 patient).CONCLUSION The causes of primary revision after UKA were gasket dislocation,osteophytes,intra-articular loose bodies and tibial prosthesis loosening.Avoidance of these factors may greatly reduce the rate of revision after UKA,improve patient satisfaction and reduce medical burden.

UNIFORM CONVERGENCE OF CES■RO MEANS OF NEGATIVE ORDER OF DOUBLE TRIGONOMETRIC FOURIER SERIES

In this paper we prove that if f ∈ C （[-π, π]^2） and the function f is bounded partial p-variation for some p ∈[1, ＋∞）, then the double trigonometric Fourier series of a function f is uniformly （C;-α,-β） summable （α＋β 〈 1/p,α,β 〉 0） in the sense of Pringsheim. If α ＋ β ≥ 1/p, then there exists a continuous function f0 of bounded partial p-variation on [-π,π]^2 such that the Cesàro （C;-α,-β） means σn,m^-α,-β（f0;0,0） of the double trigonometric Fourier series of f0 diverge over cubes.

THE FRACTAL GEOMETRY AND TRIGONOMETRIC SERIES

ＴＨＥＦＲＡＣＴＡＬＧＥＯＭＥＴＲＹＡＮＤＴＲＩＧＯＮＯＭＥＴＲＩＣＳＥＲＩＥＳ￥ＹｕＪｉａｒｏｎｇＤｅｐｔ．ｏｆＭａｔｈ．；ＷｕｈａｎＵｎｉｖ（Ｗｕｈａｎ４３００７０）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐａｐｅｒｗｅｉｎｄｒｏｄｕｃｅｓｏｍｅｒｅｃｅｎ...

Defect Detection Model Using Time Series Data Augmentation and Transformation

Time-series data provide important information in many fields,and their processing and analysis have been the focus of much research.However,detecting anomalies is very difficult due to data imbalance,temporal dependence,and noise.Therefore,methodologies for data augmentation and conversion of time series data into images for analysis have been studied.This paper proposes a fault detection model that uses time series data augmentation and transformation to address the problems of data imbalance,temporal dependence,and robustness to noise.The method of data augmentation is set as the addition of noise.It involves adding Gaussian noise,with the noise level set to 0.002,to maximize the generalization performance of the model.In addition,we use the Markov Transition Field(MTF)method to effectively visualize the dynamic transitions of the data while converting the time series data into images.It enables the identification of patterns in time series data and assists in capturing the sequential dependencies of the data.For anomaly detection,the PatchCore model is applied to show excellent performance,and the detected anomaly areas are represented as heat maps.It allows for the detection of anomalies,and by applying an anomaly map to the original image,it is possible to capture the areas where anomalies occur.The performance evaluation shows that both F1-score and Accuracy are high when time series data is converted to images.Additionally,when processed as images rather than as time series data,there was a significant reduction in both the size of the data and the training time.The proposed method can provide an important springboard for research in the field of anomaly detection using time series data.Besides,it helps solve problems such as analyzing complex patterns in data lightweight.

Cross-Dimension Attentive Feature Fusion Network for Unsupervised Time-Series Anomaly Detection

Time series anomaly detection is crucial in various industrial applications to identify unusual behaviors within the time series data.Due to the challenges associated with annotating anomaly events,time series reconstruction has become a prevalent approach for unsupervised anomaly detection.However,effectively learning representations and achieving accurate detection results remain challenging due to the intricate temporal patterns and dependencies in real-world time series.In this paper,we propose a cross-dimension attentive feature fusion network for time series anomaly detection,referred to as CAFFN.Specifically,a series and feature mixing block is introduced to learn representations in 1D space.Additionally,a fast Fourier transform is employed to convert the time series into 2D space,providing the capability for 2D feature extraction.Finally,a cross-dimension attentive feature fusion mechanism is designed that adaptively integrates features across different dimensions for anomaly detection.Experimental results on real-world time series datasets demonstrate that CAFFN performs better than other competing methods in time series anomaly detection.

Reservoir characteristics and formation model of Upper Carboniferous bauxite series in eastern Ordos Basin,NW China

Through core observation,thin section identification,X-ray diffraction analysis,scanning electron microscopy,and low-temperature nitrogen adsorption and isothermal adsorption experiments,the lithology and pore characteristics of the Upper Carboniferous bauxite series in eastern Ordos Basin were analyzed to reveal the formation and evolution process of the bauxite reservoirs.A petrological nomenclature and classification scheme for bauxitic rocks based on three units(aluminum hydroxides,iron minerals and clay minerals)is proposed.It is found that bauxitic mudstone is in the form of dense massive and clastic structures,while the(clayey)bauxite is of dense massive,pisolite,oolite,porous soil and clastic structures.Both bauxitic mudstone and bauxite reservoirs develop dissolution pores,intercrystalline pores,and microfractures as the dominant gas storage space,with the porosity less than 10% and mesopores in dominance.The bauxite series in the North China Craton can be divided into five sections,i.e.,ferrilite(Shanxi-style iron ore,section A),bauxitic mudstone(section B),bauxite(section C),bauxite mudstone(debris-containing,section D)and dark mudstone-coal section(section E).The burrow/funnel filling,lenticular,layered/massive bauxite deposits occur separately in the karst platforms,gentle slopes and low-lying areas.The karst platforms and gentle slopes are conducive to surface water leaching,with strong karstification,well-developed pores,large reservoir thickness and good physical properties,but poor strata continuity.The low-lying areas have poor physical properties but relatively continuous and stable reservoirs.The gas enrichment in bauxites is jointly controlled by source rock,reservoir rock and fractures.This recognition provides geological basis for the exploration and development of natural gas in the Upper Carboniferous in the study area and similar bauxite systems.

RIEMANN SUMMABILITY OF MULTI-DIMENSIONAL TRIGONOMETRIC-FOURIER SERIES

The d-dimensional classical Hardy spaces H_p (T^d) are introduced and it is shown that the maximal operator of the Riemann sums of a distribution is bounded from H_p(T^d)to L_p(T^2) (d/(d+1)<p≤∞) and is of weak type (1, 1) provided that the supremum in the maximal operator is taken over a positive cone. The same is proved for the conjugate Riemann sums. As a consequence we obtain that every function f∈L_1(T^d)is a.e. Riemann summable to f, provided again that the limit is taken over a positive cone.

Unsupervised Time Series Segmentation: A Survey on Recent Advances

Time series segmentation has attracted more interests in recent years,which aims to segment time series into different segments,each reflects a state of the monitored objects.Although there have been many surveys on time series segmentation,most of them focus more on change point detection(CPD)methods and overlook the advances in boundary detection(BD)and state detection(SD)methods.In this paper,we categorize time series segmentation methods into CPD,BD,and SD methods,with a specific focus on recent advances in BD and SD methods.Within the scope of BD and SD,we subdivide the methods based on their underlying models/techniques and focus on the milestones that have shaped the development trajectory of each category.As a conclusion,we found that:(1)Existing methods failed to provide sufficient support for online working,with only a few methods supporting online deployment;(2)Most existing methods require the specification of parameters,which hinders their ability to work adaptively;(3)Existing SD methods do not attach importance to accurate detection of boundary points in evaluation,which may lead to limitations in boundary point detection.We highlight the ability to working online and adaptively as important attributes of segmentation methods,the boundary detection accuracy as a neglected metrics for SD methods.

Improved Responses with Multitaper Spectral Analysis for Magnetotelluric Time Series Data Processing:Examples from Field Data

In order to attain good quality transfer function estimates from magnetotelluric field data(i.e.,smooth behavior and small uncertainties across all frequencies),we compare time series data processing with and without a multitaper approach for spectral estimation.There are several common ways to increase the reliability of the Fourier spectral estimation from experimental(noisy)data;for example to subdivide the experimental time series into segments,taper these segments(using single taper),perform the Fourier transform of the individual segments,and average the resulting spectra.

Trigonometric Regularization and Continuation Method Based Time-Optimal Control of Hypersonic Vehicles

Aiming at the time-optimal control problem of hypersonic vehicles(HSV)in ascending stage,a trigonometric regularization method(TRM)is introduced based on the indirect method of optimal control.This method avoids analyzing the switching function and distinguishing between singular control and bang-bang control,where the singular control problem is more complicated.While in bang-bang control,the costate variables are unsmooth due to the control jumping,resulting in difficulty in solving the two-point boundary value problem(TPBVP)induced by the indirect method.Aiming at the easy divergence when solving the TPBVP,the continuation method is introduced.This method uses the solution of the simplified problem as the initial value of the iteration.Then through solving a series of TPBVP,it approximates to the solution of the original complex problem.The calculation results show that through the above two methods,the time-optimal control problem of HSV in ascending stage under the complex model can be solved conveniently.

Multivariate Time Series Anomaly Detection Based on Spatial-Temporal Network and Transformer in Industrial Internet of Things

In the Industrial Internet of Things(IIoT),sensors generate time series data to reflect the working state.When the systems are attacked,timely identification of outliers in time series is critical to ensure security.Although many anomaly detection methods have been proposed,the temporal correlation of the time series over the same sensor and the state(spatial)correlation between different sensors are rarely considered simultaneously in these methods.Owing to the superior capability of Transformer in learning time series features.This paper proposes a time series anomaly detection method based on a spatial-temporal network and an improved Transformer.Additionally,the methods based on graph neural networks typically include a graph structure learning module and an anomaly detection module,which are interdependent.However,in the initial phase of training,since neither of the modules has reached an optimal state,their performance may influence each other.This scenario makes the end-to-end training approach hard to effectively direct the learning trajectory of each module.This interdependence between the modules,coupled with the initial instability,may cause the model to find it hard to find the optimal solution during the training process,resulting in unsatisfactory results.We introduce an adaptive graph structure learning method to obtain the optimal model parameters and graph structure.Experiments on two publicly available datasets demonstrate that the proposed method attains higher anomaly detection results than other methods.

An Innovative Deep Architecture for Flight Safety Risk Assessment Based on Time Series Data

With the development of the integration of aviation safety and artificial intelligence,research on the combination of risk assessment and artificial intelligence is particularly important in the field of risk management,but searching for an efficient and accurate risk assessment algorithm has become a challenge for the civil aviation industry.Therefore,an improved risk assessment algorithm(PS-AE-LSTM)based on long short-term memory network(LSTM)with autoencoder(AE)is proposed for the various supervised deep learning algorithms in flight safety that cannot adequately address the problem of the quality on risk level labels.Firstly,based on the normal distribution characteristics of flight data,a probability severity(PS)model is established to enhance the quality of risk assessment labels.Secondly,autoencoder is introduced to reconstruct the flight parameter data to improve the data quality.Finally,utilizing the time-series nature of flight data,a long and short-termmemory network is used to classify the risk level and improve the accuracy of risk assessment.Thus,a risk assessment experimentwas conducted to analyze a fleet landing phase dataset using the PS-AE-LSTMalgorithm to assess the risk level associated with aircraft hard landing events.The results show that the proposed algorithm achieves an accuracy of 86.45%compared with seven baseline models and has excellent risk assessment capability.

Periodic signal extraction of GNSS height time series based on adaptive singular spectrum analysis

Singular spectrum analysis is widely used in geodetic time series analysis.However,when extracting time-varying periodic signals from a large number of Global Navigation Satellite System(GNSS)time series,the selection of appropriate embedding window size and principal components makes this method cumbersome and inefficient.To improve the efficiency and accuracy of singular spectrum analysis,this paper proposes an adaptive singular spectrum analysis method by combining spectrum analysis with a new trace matrix.The running time and correlation analysis indicate that the proposed method can adaptively set the embedding window size to extract the time-varying periodic signals from GNSS time series,and the extraction efficiency of a single time series is six times that of singular spectrum analysis.The method is also accurate and more suitable for time-varying periodic signal analysis of global GNSS sites.

A Time Series Short-Term Prediction Method Based on Multi-Granularity Event Matching and Alignment

Accurate forecasting of time series is crucial across various domains.Many prediction tasks rely on effectively segmenting,matching,and time series data alignment.For instance,regardless of time series with the same granularity,segmenting them into different granularity events can effectively mitigate the impact of varying time scales on prediction accuracy.However,these events of varying granularity frequently intersect with each other,which may possess unequal durations.Even minor differences can result in significant errors when matching time series with future trends.Besides,directly using matched events but unaligned events as state vectors in machine learning-based prediction models can lead to insufficient prediction accuracy.Therefore,this paper proposes a short-term forecasting method for time series based on a multi-granularity event,MGE-SP(multi-granularity event-based short-termprediction).First,amethodological framework for MGE-SP established guides the implementation steps.The framework consists of three key steps,including multi-granularity event matching based on the LTF(latest time first)strategy,multi-granularity event alignment using a piecewise aggregate approximation based on the compression ratio,and a short-term prediction model based on XGBoost.The data from a nationwide online car-hailing service in China ensures the method’s reliability.The average RMSE(root mean square error)and MAE(mean absolute error)of the proposed method are 3.204 and 2.360,lower than the respective values of 4.056 and 3.101 obtained using theARIMA(autoregressive integratedmoving average)method,as well as the values of 4.278 and 2.994 obtained using k-means-SVR(support vector regression)method.The other experiment is conducted on stock data froma public data set.The proposed method achieved an average RMSE and MAE of 0.836 and 0.696,lower than the respective values of 1.019 and 0.844 obtained using the ARIMA method,as well as the values of 1.350 and 1.172 obtained using the k-means-SVR method.