Different Types Of Classes Research Articles

Решение физических задач как инструмент формирования профессиональных компетенций врача и провизора

Solving physical problems can be considered as a goal and method of teaching. Insufficient fundamental knowledge of medical university students will not allow them to meet modern requirements of medical science. The choice of physics tasks, the content of which is focused on the future profession, contributes to the formation of professional and general cultural competencies of a doctor and a pharmacist. It is reasonable to use such tasks in different types of classes: seminars, laboratory workshops, colloquiums. Motivation is increased by selecting tasks taking into account the future medical specialty and identifying interdisciplinary connections. The article provides examples of tasks in accordance with the goals set during the lesson.

Towards the optimal beam dump experiment to search for feebly interacting particles

Future searches for new physics beyond the Standard Model are without doubt in need of a diverse approach and experiments with complementary sensitivities to different types of classes of models. One of the directions that should be explored is feebly interacting particles (FIPs) with masses below the electroweak scale. The interest in FIPs has significantly increased in the last ten years. Searches for FIPs at colliders have intrinsic limitations in the region they may probe, significantly restricting exploration of the mass range mFIP<5-10\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_{\ ext {FIP}} < 5{-}10$$\\end{document} GeV/c2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^2$$\\end{document}. Beam dump-like experiments, characterized by the possibility of extremely high luminosity at relatively high energies and the effective coverage of the production and decay acceptance, are the perfect option to generically explore the “coupling frontier” of the light FIPs. Several proposals for beam-dump detectors are currently being considered by CERN for implementation at the SPS ECN3 beam facility. In this paper, we analyse in depth how the characteristic geometric parameters of a beam dump experiment influence the signal yield. We apply an inclusive approach by considering the phenomenology of different types of FIPs. From the various production modes and kinematics, we demonstrate that the optimal layout that maximises the production and decay acceptance consists of a detector located on the beam-axis, at the shortest possible distance from the target defined by the systems required to suppress the beam-induced backgrounds.

E-Learning and Flipped Classroom in Inclusive Education: The Case of Students with the Psychopathology of Language and Cognition.

The study analyzed modern e-learning and flipped classroom methods in inclusive education, focusing on their impact on motivation, academic performance, and the effectiveness of the educational process. The experiment involved 648 first- and second-year students from Abai Kazakh National Pedagogical University, the Elabuga Institute (Branch) of Kazan Federal University, and I.M. Sechenov First Moscow State Medical University. According to the survey conducted in the first phase, 66% of students reported using e-learning and flipped classroom methods, while 34% indicated that these methods were not employed. Furthermore, 50% of respondents rated the development of these methods in their universities as low, suggesting a lack of attention, familiarity among teachers, or reluctance to adopt them. To study the psychopathology of language and cognitive functions among children with special needs, the authors used the ASEBA questionnaire algorithm. It allowed for a survey among children from specialized classes, mixed classes (E-learning and flipped classroom), and control classes. Each group consisted of 50 people, there were three classes in each category. The ASEBA questionnaire revealed information about various aspects of the psychological functioning of children, including their behavior, emotional state, social adaptation, as well as problems with language and communication. The use of the questionnaire in different types of classes helped to compare the results between groups and identify features and differences in the psychopathology of language and cognitive functions in children with special needs. The findings can contribute to a deeper understanding of the psychopathology of language and cognitive functions in this category of children.

Improved Cyber Attack Detection Using MLB-FQS: A Novel Modified Lagrange Butterfly-based Fuzzy Quasi-Linear SVM Algorithm

With the rapid enhancement of communication and computer networks, internet-based technologies provide numerous convenient and accessible services to the public all over the world. Meanwhile, the types and number of cyber attacks have been increasing day by day and create a serious threat to secure most confidential data. Hence, securing data is considered a crucial task and nowadays intrusion detection system is one of the most effective and promising techniques to provide secure data by detecting malicious data. However, it becomes extremely hard in preventing the network system from attack due to imbalanced data that results in low detection and poor accuracy rates. Therefore, a novel intrusion detection mechanism is proposed to accurately detect cyber intrusion with high detection accuracy rate. The Intrusion detection mechanism comprises three significant phases namely the data pre-processing phase, the feature selection phase as well as attack detection phase. Various pre-processing techniques namely data normalization, data conversion as well as one-hot encoding process are utilized to provide an appropriate format that is suitable for analysis. In the feature selection phase, a Modified Lagrange Butterfly (MLB) optimization is employed to select optimal features for effective detection. Finally, a Fuzzy Quasi-Linear Support Vector Machine (FQL-SVM) classifier is employed to determine the network as two different types of classes namely the abnormal or malicious classes. After examining on malware dataset, the experimental analysis demonstrates a high classification accuracy rate of 98.41% which is better than other comparative techniques.

Focus on 10th grade students with learning problems: What influences their achievements?

AbstractThere is an increased interest in exploring the interaction between knowledge of foreign languages, Reading and Mathematics and correlation of students' achievement in Hungary (e. g. Szirmai, 2003; Garami, 2009; Hegedűs &amp; Sebestyén, 2019). Special learner groups are rarely put in the focus of research, therefore in our paper, we examine achievements in Reading and Mathematics of students with learning disorder or integration, learning and behavioural difficulties (ILBD). We examine data in different types of classes, for example, in special classes with a high number of foreign language lessons. We analyse the data of National Competency Measurement (NCM) of 2019 in the 10th grade (N = 83,751) with SPSS. In this database, there are 3,029 students with learning disorder and 4,284 students with ILBD. 5.9% of the first group and 9% of the latter group learn in classes with a high number of foreign language lessons per week. According to our results, these students have better competence results from Reading and Mathematics than students who learn in classes with “normal” curriculum, and lower achievements than students in classes with a special curriculum.

The project “Colour Made Inclusive” as an example of creative education for youths

The purpose of this study is to present the programme of working with socially maladjusted youngsters, which is tailored to special needs of youths in one of the residential child care institutions. The theoretical background of the programme under discussion is based on cognitive–behavioral theory. The example in question consisted of several different types of classes including art classes and psychological support. Using the qualitative methods, the semi-structured interviews were used to collect the data as this study analyzed the participants’ changes in their behaviour. The most significant results of this study are the following conclusions: 1. the new programmes of working with youths based on cognitive behavioral theory (CBT) concept are necessary for the effective social rehabilitation process and restore these young people to the society, 2. social rehabilitation institutions for young people should provide them with the possibilities of changing their behaviour in the society as in the example presented in this paper.

Лингводидактический потенциал ресурсов интернета при электронном обучении языковым курсам (на примере технического вуза)

The article discusses the educational potential of various Internet resources that can be used in teaching students about language disciplines (English as a foreign language and Russian as a native language) at a technical university. The purpose of the article is to systematize Internet resources that can be productively used in teaching English, speech culture and business communications in the framework of technical education in higher educational institutions. The article describes the possibilities of various sites designed specifically for language learning. The most successful resources for different types of classes, exercises and speech activity have been identified. The general parameters in the use of such resources in language disciplines were determined in the context of the disputes of modern pedagogy related to the “overdose” of the digital format in modern education. The authors of the article drew conclusions about the need for active, but thoughtful use of Internet resources in the process of teaching languages at a technical university.

Real‐Time Medical Image Classification with ML Framework and Dedicated CNN–LSTM Architecture

In the domain of modern deep learning and classification techniques, the convolutional neural network (CNN) stands out as a highly successful and preferred method for image classification in artificial intelligence. Especially in the medical field, CNN has proven to be an ideal approach for analyzing medical data and accurately identifying diseases. Over the recent years, CNN has demonstrated significant potential and success in various computer vision tasks, with medical image classification being one of the prominent applications. In our study, we introduce a novel custom CNN model called MedvCNN, designed for classifying different types of classes. We conduct experiments with various image sizes to explore their versatility. In addition, long short‐term memory (LSTM), a type of recurrent neural network (RNN), is incorporated into our approach. LSTM is specifically tailored to handle sequential data, making it ideal for time series analysis. However, its capabilities extend beyond time series data and are effectively applied to various sequential data types, including sequential vectors derived from image data. One of the key advantages of utilizing LSTM for image classification is its ability to effectively memorize and capture important features in the image data. This feature is particularly advantageous in medical image processing, where precise and accurate identification of key attributes is crucial for successful diagnosis and analysis. Furthermore, our experiments reveal that the hybrid custom LSTM model, MedvLSTM, a RNN algorithm, surpasses other methods in the domain of medical image classification. Our study places significant emphasis on attaining robust classification performance for medical image data through a sophisticated, parameter free approach, complemented by an ablation study, and comprehensive statistical analysis. This comprehensive analysis and evaluation allow us to gain a deeper understanding of the model’s effectiveness and its potential impact in the field of medical image analysis. We compare these two approaches to a baseline CNN architecture, aiming to streamline the classification process, reduce time consumption, and improve cost efficiency. Additionally, we present a real‐time web‐based AutoML framework along with a practical demonstration. Ultimately, our research provides a thorough investigation of the current state‐of‐the‐art in medical image analysis accuracy, focusing on the utilization of neural networks and LSTM.

Development of a tool for quantifying need-supportive coaching in technology-mediated exercise classes

Technology-mediated interventions to promote physical activity are growing in popularity and appear to be effective for supporting continued adherence for some people. Some of this efficacy may be related to the cultivation of motivation that is self-determined (i.e., autonomous), which is posited to arise from the satisfaction of three basic psychological needs: competence, relatedness, and autonomy. We developed an observational coding tool for quantifying the frequency of needs-supportive and needs-indifferent coaching during technology-mediated exercise classes. The Peloton Instructor Needs-Supportive Coaching (PINC) tool shows evidence of reliability (average kappa = .91). We also demonstrated the utility of the PINC for characterizing needs-supportive coaching profiles across 4 different types of classes (Beginner, Power Zone, Groove, and High-Intensity Interval Training) and the construct validity of the PINC with respect to examining the relationship of needs-supportive coaching to intrinsic motivation. The PINC offers a useful tool with which future studies could evaluate whether and how instructor coaching impacts self-determined motivation to exercise within a technology-mediated context.

The effect of orthodontic treatment on facial attractiveness: a systematic review and meta-analysis.

Facial and smile attractiveness are significant motivating factor for patients to seek orthodontic treatment. Although there is a general belief that orthodontic treatment improves facial appearance, this has yet not been systematically evaluated. The objective of this study was to assess the current evidence on the effect of orthodontic treatment on facial attractiveness. Systematic and unrestricted search of nine databases were performed up to January 2022. Studies evaluating facial attractiveness before and after orthodontic treatment. Extracted data included study design and setting, sample size and demographics, malocclusion type, treatment modality, and method for outcome assessment. Risk of bias was assessed with the ROBINS-I tool for non-randomized studies and with RoB-2 for randomized controlled trials (RCTs). Random-effects meta-analyses of mean differences and their 95% confidence intervals (CIs) were performed. Twenty studies were included in data synthesis; three randomized controlled clinical trials and 17 non-randomized clinical studies of retrospective or prospective design. One of the RCTs was found to have low risk of bias, one presented some concerns and the third showed a high risk of bias. All non-randomized studies showed either unclear or high risk of bias. Data syntheses showed that orthodontic treatment improved facial attractiveness ratings by 9% when compared with untreated controls (MD: 9.05/95% CI: 4.71; 13.39). A combination of orthodontics and orthognathic surgery also showed a positive effect of 5.5% (MD: 5.51/95% CI: 1.55; 9.47) when compared with orthodontic treatment alone. There was no difference in effect between extraction and non-extraction treatments (MD: -0.89/ 95% CI: -8.72; 6.94) or between different types of Class II correctors (MD: 2.21/95% CI: -16.51; 20.93). With the exception of two RCTs, included studies were of unclear or low quality. Orthodontic treatment has a clinically weak effect on facial attractiveness when compared to no treatment. The same is true when a combined orthodontic/surgical treatment is compared to orthodontics alone. PROSPERO #: CRD42020169904.

The effect of different types of class F fly ashes on the mechanical properties of geopolymers cured at ambient environment

This research investigates the effect of the physicochemical characteristics of five cognate fly ashes on the mechanical properties of Class F fly ash-based geopolymers synthesized with the same mix proportion and cured at ambient conditions. X-ray diffraction, scanning electron microscopy, energy dispersive spectroscopy, and low-field nuclear magnetic resonance were conducted to create the quantitative connection for geopolymers between micro-characteristics and macro-mechanical properties. With detailed experimental study, the research reveals that the difference in the inherent characteristics of fly ash such as specific gravity, median particle size, and amorphous phase content causes the formation of geopolymers with varied micro-characteristics and hence the discrepancy in the mechanical properties. The compressive strength increases with increasing geopolymers' amorphous phase content and decreasing porosity and Si/Al molar ratio of geopolymer binder, expressed with the reliable numerical relationships. Besides, a single and comprehensive reactivity index is proposed to assess the suitability of fly ashes for geopolymers. The results show that reactivity index of more than 20 (corresponding amorphous phase content, specific gravity, and median particle size of fly ash is >84%, >2.35, and <10 μm, respectively) leads to the high compressive strength of 60–80 MPa (28 d). Such a conclusion proves that high-temperature curing is not the prerequisite of activating the fly ash for geopolymer preparation. The approach of producing high-quality geopolymers under ambient conditions will contribute to the development of fly ash utilization and Class F fly ash-based geopolymers’ industrialization.

РАБОТА НАД СИТУАТИВНЫМИ ТЕКСТАМИ НА ЗАНЯТИЯХ ПО РУССКОМУ ЯЗЫКУ КАК ИНОСТРАННОМУ

The whole system of education is aimed at activating the speech-cogitative activity of students. To this end, the teacher is looking for different forms of teaching. Within the framework of the personal-activity and communicative approaches in teaching a foreign language, innovative learning models are developed and different types of classes are used to achieve all these learning goals. In this article, we consider the features of one of the principles of learning - situational-thematic. It involves knowledge of lexical and grammatical material, which is grouped around certain topics and situations of communication. The issue raised in the article concerns the situational topic of communication "At the doctor's". It is known that the acquisition of various kinds of knowledge, communicative and speech skills, as well as the formation of personality are the principles that play a significant role in the learning process. The authors of the article tried to show that properly selected textual material, which is of interest to students, makes it possible to fulfill the learning goals set for teachers and students.

Research on Characteristics of Teachers’ Feedback Discourse in High-Quality English Classes in Junior High Schools: Evidence from Thirty-Two Teachers’ Classroom Observation

Feedback Discourse is teachers’ response to students’ learning performance, which plays an extremely important part in teachers’ discourse, especially in language teaching classes. Taking the teaching videos of thirty-two English teachers in middle school exemplary lessons competition as the subjects, this study is intended to explore the characteristics of feedback discourse of junior high school English teachers from the proportion of teachers’ feedback discourse, the types of teachers’ feedback discourse, as well as the discourse structure of teachers’ feedback discourse.The results are as follows: (1) Teachers of high-quality classes observed pay attention to the importance of classroom feedback, and the proportion of teachers’ feedback discourse in high-quality classes accounted for 21% teachers’ classroom discourse. What’s more, the proportion of teachers’ feedback discourse in different types of classes is relatively consistent. (2) As for the feedback discourse used by teachers in high-quality classes observed, the positive feedback discourse is more frequently used than the negative feedback discourse. The positive feedback most commonly used is “Request for Clarification” and the negative feedback is“Explicit Correction”. (3)The teachers in high-quality classes used more separate discourse structure than continuous discourse structure which is suggested to use more. It is hoped that this study could provide some reference for the future study on junior high school English teachers’ feedback discourse.

Оцінка функціональних показників у пацієнтів з дистальним прикусом при нормальній і порушеній функції зовнішнього дихання

Background. Class II malocclusion is one of the most common problems in orthodontic practice and is accompanied by certain morphological, functional, and aesthetic changes: a violation of the dynamic balance of muscles of a peri-oral area and tongue, which affects many functions of the child: there are disorders of external breathing, speech, chewing and swallowing functions. Among vital functions, breathing can be called one of the most important, because the exchange of air between a living being, and the external atmosphere is necessary to meet the metabolic needs of the body. During the respiratory process, the tissues of the oral cavity are the first structures that come into contact with the air moving to the upper respiratory tract. Thus, a malocclusion, especially in Class II with retrognathia mandible, can lead to narrowing of the upper respiratory tract. And these changes, in turn, can adversely affect lung function. Therefore, the correction of retrognathia mandible in Class II Malocclusion by functional appliances can have a beneficial effect on the functions of the lower respiratory tract. Objective. Study of functional (cephalometric and spirometric indicators) in patients aged 7–13 years with skeletal class II malocclusion with normal and impaired function of external breathing. Research methods. Cephalometric analysis of the assessment of the position of the jaws and upper respiratory tract (nasopharynx, oropharynx and hypopharynx) by the McNamara method. Spirometric analysis to assess lung function. Results. The conducted study did not reveal a correlation between the types of jaw growth profile and the volume of the upper respiratory tract in all experimental groups compared to patients in the control group. We also found a low correlation between the nasopharyngeal respiratory function and the position of the lower jaw in different types of class II malocclusion, while changes in the position of the lower jaw significantly affect the volume of the oropharynx and hypopharynx. We found a worsening of the results of FVC, FEV1, IT%(FEV1/FVC) PEF and FEF25–75% in children with narrowing of the upper airways (I-1 and II-1 subgroups) compared to all I-2 and II-2 subgroups and control group (p &lt; 0.05), In addition, a non-significant (p &gt; 0.05) decrease in spirometric measurements was noted in children without narrowing of the upper respiratory tract (I-2, II-2 subgroups) compared to control group. Conclusions. In order to determine the effectiveness of orthodontic treatment of class II malocclusion, it is necessary to conduct a cephalometric and spirometric study before and after the treatment to determine its effectiveness. Key words: class II malocclusion, distal occlusion, impaired external breathing, cephalometric, spirometer.

Optimization of Deep Learning Model for Plant Disease Detection Using Particle Swarm Optimizer

Plant diseases are a major impendence to food security, and due to a lack of key infrastructure in many regions of the world, quick identification is still challenging. Harvest losses owing to illnesses are a severe problem for both large farming structures and rural communities, motivating our mission. Because of the large range of diseases, identifying and classifying diseases with human eyes is not only time-consuming and labor intensive, but also prone to being mistaken with a high error rate. Deep learning-enabled breakthroughs in computer vision have cleared the road for smartphone-assisted plant disease and diagnosis. The proposed work describes a deep learning approach for detection plant disease. Therefore, we proposed a deep learning model strategy for detecting plant disease and classification of plant leaf diseases. In our research, we focused on detecting plant diseases in five crops divided into 25 different types of classes (wheat, cotton, grape, corn, and cucumbers). In this task, we used a public image database of healthy and diseased plant leaves acquired under realistic conditions. For our work, a deep convolutional neural model AlexNet and Particle Swarm optimization was trained for this task we found that the metrics (accuracy, specificity, Sensitivity, precision, and F-score) of the tested deep learning networks achieves an accuracy of 98.83%, specificity of 98.56%, Sensitivity of 98.78%, precision of 98.67%, and F-score of 98.47%, demonstrating the feasibility of this approach.

БІЛІНГВАЛЬНЕ НАВЧАННЯ ПРИ ПІДГОТОВЦІ МАЙБУТНІХ УЧИТЕЛІВ ПРИРОДНИЧИХ ДИСЦИПЛІН

The article is dedicated to the issue of enhancement of the professional training of the would-be Science teachers via the introduction of bilingual studying as a result of internationalization and globalization. The analysis of the national and international scientific research has shown that bilingual studying enables the formation of the future specialist’s professional competence which later facilitates their integration into the world science community. With the help of the research into the scientific and pedagogical as well as methodological literature the retrospective analysis of the notion «bilingual studying» and its interpreting concepts have been conducted, bilinguism levels and basic models of bilingual studying have been outlined. Specific features of bilingual studying and would-be teachers’ bilingual competence formation are outlined. The problems of implementing bilingual studying in Ukraine’s higher educational establishments have been identified. It is highlighted that the issue of bilingual competence formation of would-be Science teachers requires further research. Specific methodological features of effective implementation of bilingual training for would-be science teachers have been identified (enhancement of students’ motivation to bilingual studying, gradual implementation of bilingual studying in the curriculum, availability of studying and methodology sets, which would include bilingual studying materials; implementing active studying methods; applying innovative information technologies; two-way interaction between the subjects of education process). It is emphasized, that forming bilingual competence is an integral process and is to be carried out not separately, but integrating the use of English as an inseparable means of educational activity. The importance of the development of the subjects’ studying and methodology sets as well as methodological guidelines for conducting different types of classes with the aim of training would-be teachers on bilingual basis has been highlighted.

Innovative Methodology of Teaching Legal Studies

The article deals with current methods of teaching legal studies. Particular attention is paid to innovative methodology and learning disciplines at law faculties. It is substantiated that the following methods are inherent in the creative technique: analysis of critical situations; «brainstorming»; Socratic method; business (role) game; «taking a position»; commenting, evaluation (or self-evaluation) of participants’ actions; master classes; project method; modelling; «training ground»; work in small groups; method of a business trip, etc. Innovative methods arise from the needs of society in the development of jurisprudence and correspond to its nature and level of activity in a given period; these are new perspective tendencies, processes and approaches in the development of modern higher education. The distinction between innovative and classical methods is made, the possibilities of conducting different types of classes with the use of other methods are discussed. It is established that the transition to the methods inherent in innovative methods involves a combination of practical, educational and research functions in the educational process, which helps to resolve the contradiction between the knowledge provided by the university and the requirements of real life.

Partnership pedagogy

The purpose of the investigation is to study the concept of pedagogy of partnership in the EU and determine the implementation of Western experience in Ukraine on the basis of a sociological survey. The sociological survey was conducted among 400 teachers - practitioners representing different types of classes (NUS members, inclusive practitioners, small-numbered schools, national minorities). It has been established that the key issues for the implementation of partnership pedagogy for NUS classes are the reluctance of parents to participate in school management; low level of “triangular” connection; low guarantee of understanding of the partnership-based educational approach. Respondents-teachers of national minority classes have identified the following problems, namely: prejudice against the minority; insufficient willingness of parents to help the educational process; reluctance of children to communicate. The academic paper indicates the lack of comprehensive research in the relevant field.

Platelet-Derived Growth Factor-Modulated Guided Tissue Regeneration with a Bioresorbable Membrane in Class III Furcation Defects: A Histometric Study in the Monkey.

It was the aim of this study to histometrically evaluate guided tissue regeneration (bioresorbable membrane plus bone mineral) (GTR) with or without platelet-derived growth factor (PDGF) in two different types of class III furcation defects (small keyhole defects and horizonal defects) in monkeys. In six cynomolgus monkeys, two types of class III furcation defects were created and allowed to chronify for 5 months in mandibular first and second molars. After a hygiene program the molars were assigned to GTR group (collagen membrane plus bovine bone mineral), PDGF group (collagen membrane plus bovine bone mineral plus PDGF), or negative control group (flap reposition only). Histologic sections were made after 7 months of healing and descriptive statistics were provided from the histometric parameters. Postoperative healing was uneventful despite marginal membrane exposures in the GTR and PDGF group. Bone regeneration of 23–35% of the original defect area was found in the two treatment groups. In none of the evaluated key parameters (formation of bone, root cementum, connective tissue, or epithelium) differences were detected between GTR and PDGF groups. However, the negative control teeth exhibited better bone regeneration than the treatment groups. The type of class III defect did not influence the regenerative outcome. Within the limits of this study PDGF was not able to enhance the histologic regeneration of class III furcation areas in monkeys compared to bone mineral enhanced GTR treatment regardless of the defect configuration. Membrane exposure during early healing might have influenced these outcomes.

Trees classification based on Fourier coefficients of the sapflow density flux

In this paper we study the possibility to use the artificial neural networks for trees classification based on real and approximated values of the sap flow density flux describing water transport in trees. The data sets were generated by means of a new tree monitoring system TreeTalker. The Fourier series-based model is used for fitting the data sets with periodic patterns. The multivariate regression model defines the functional dependencies between sap flow density and temperature time series. The paper shows that Fourier coefficients can be successfully used as elements of the feature vectors required to solve different classification problems. Here we train multilayer neural networks to classify the trees according to different types of classes. The quality of the developed model for prediction and classification is verified by numerous numerical examples.