Search results for: mathematical framework

Authors: Jindřiška Šedová

Abstract:

In the past 20 years the economy of the Czech Republic has experienced substantial changes. In the 1990s the development was affected by the transformation which sought to establish the right conditions for privatization and creation of elementary market relations. In the last decade the characteristic elements such as private ownership and corresponding institutional framework have been strengthened. This development was marked by the accession of the Czech Republic to the EU. The Czech Republic is striving to reduce the difference between its level of economic development and the quality of institutional framework in comparison with other developed countries. The process of finding the adequate solutions has been hampered by the negative impact of the world financial crisis on the Czech Republic and the standard of living of its inhabitants. This contribution seeks to address the question of whether and to which extent the economic development of the transitive Czech economy is affected by the change in behaviour of households and their tendency to consumption, i.e. in the sense of reduction or increase in demand for goods and services. It aims to verify whether the increasing trend of household indebtedness and decreasing trend of saving pose a significant risk in the Czech Republic. At a general level the analysis aims to contribute to finding an answer to the question of whether the debt increase of Czech households is connected to the risk of "eating through" the borrowed money and whether Czech households risk falling into a debt trap. In addition to household indebtedness risks in the Czech Republic the analysis will focus on identification of specifics of the transformation phase of the Czech economy in comparison with the EU countries, or selected OECD countries.

Keywords: household indebtedness, household consumption, credits, financial literacy

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Authors: Yasin Tadayonrad, Alassane Ballé Ndiaye

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Most of the supply chain networks have many nodes starting from the suppliers’ side up to the customers’ side that each node sends/receives the raw materials/products from/to the other nodes. One of the major concerns in this kind of supply chain network is finding the best schedule for loading/unloading the shipments through the whole network by which all the constraints in the source and destination nodes are met and all the shipments are delivered on time. One of the main constraints in this problem is the loading/unloading capacity in each source/destination node at each time slot (e.g., per week/day/hour). Because of the different characteristics of different products/groups of products, the capacity of each node might differ based on each group of products. In most supply chain networks (especially in the Fast-moving consumer goods (FMCG) industry), there are different planners/planning teams working separately in different nodes to determine the loading/unloading timeslots in source/destination nodes to send/receive the shipments. In this paper, a mathematical problem has been proposed to find the best timeslots for loading/unloading the shipments minimizing the overall delays subject to respecting the capacity of loading/unloading of each node, the required delivery date of each shipment (considering the lead-times), and working-days of each node. This model was implemented on Python and solved using Python-MIP on a sample data set. Finally, the idea of a heuristic algorithm has been proposed as a way of improving the solution method that helps to implement the model on larger data sets in real business cases, including more nodes and shipments.

Keywords: Supply chain management, transportation, multiple-multiple network, timeslots management, mathematical modeling, mixed integer programming.

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Authors: Amira Abbes Kheriji, Faouzi Bouani, Mekki Ksouri, Mohamed Ben Ahmed

Abstract:

Model Predictive Control (MPC) is increasingly being proposed for real time applications and embedded systems. However comparing to PID controller, the implementation of the MPC in miniaturized devices like Field Programmable Gate Arrays (FPGA) and microcontrollers has historically been very small scale due to its complexity in implementation and its computation time requirement. At the same time, such embedded technologies have become an enabler for future manufacturing enterprises as well as a transformer of organizations and markets. Recently, advances in microelectronics and software allow such technique to be implemented in embedded systems. In this work, we take advantage of these recent advances in this area in the deployment of one of the most studied and applied control technique in the industrial engineering. In fact in this paper, we propose an efficient framework for implementation of Generalized Predictive Control (GPC) in the performed STM32 microcontroller. The STM32 keil starter kit based on a JTAG interface and the STM32 board was used to implement the proposed GPC firmware. Besides the GPC, the PID anti windup algorithm was also implemented using Keil development tools designed for ARM processor-based microcontroller devices and working with C/Cµ langage. A performances comparison study was done between both firmwares. This performances study show good execution speed and low computational burden. These results encourage to develop simple predictive algorithms to be programmed in industrial standard hardware. The main features of the proposed framework are illustrated through two examples and compared with the anti windup PID controller.

Keywords: Embedded systems, Model Predictive Control, microcontroller, Keil tool.

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Authors: L.A du Plessis

Abstract:

South Africa is facing a crisis with not being able to produce enough graduates in the scarce skills areas to sustain economic growth. The crisis is fuelled by a school system that does not produce enough potential students with Mathematics, Accounting and Science. Since the introduction of the new school curriculum in 2008, there is no longer an option to take pure maths on a standard grade level. Instead, only two mathematical subjects are offered: pure maths (which is on par with higher grade maths) and mathematical literacy. It is compulsory to take one or the other. As a result, lees student finishes Grade 12 with pure mathematics every year. This national problem needs urgent attention if South Africa is to make any headway in critical skills development as mathematics is a gateway to scarce skills professions. Higher education institutions initiated several initiatives in an attempt to address the above, including preparatory courses, bridging programmes and extended curricula with foundation provisions. In view of the above, and government policy directives to broaden access in the scarce skills areas to increase student throughput, foundation provision was introduced for Commerce and Information Technology programmes at the Vaal Triangle Campus (VTC) of North-West University (NWU) in 2010. Students enrolling for extended programmes do not comply with the minimum prerequisites for the normal programmes. The question then arises as to whether these programmes have the intended impact? This paper reports the results of a two year longitudinal study, tracking the first year academic achievement of the two cohorts of enrolments since 2010. The results provide valuable insight into the structuring of an extended programme and its potential impact.

Keywords: Access, extended programmes, foundation provision, mathematics.

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Authors: Vasilescu M., Rusu L., Nestianu V., Romanescu F., Dragomir M

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The research was designed to examine the relationship between the development of muscle fatigue and the effect it has on sport performance, specifically during maximal voluntary contraction. This kind of this investigation using simultaneous electrophysiological and mechanical recordings, based on advanced mathematical processing, allows us to get parameters, and indexes in a short time, and finally, the mapping to use for the thorough investigation of the muscle contraction force, respectively the phenomenon of local muscle fatigue, both for athletes and other subjects.

Keywords: Electromyography, mechanomyography, musclefatigue

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Authors: S. Sundaravalli, M. C. Majumder, G. K. Vijayaraghavan

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The study focuses to investigate the thermal response of delaminations and develop mathematical models using numerical results to obtain the optimum heat requirement and time to identify delaminations in GLARE type of Fibre Metal Laminates (FML) in both reflection mode and through-transmission (TT) mode of step pulsed active thermography (SPAT) method in the type of nondestructive testing and evaluation (NDTE) technique. The influence of applied heat flux and time on various sizes and depth of delaminations in FML is analyzed to investigate the thermal response through numerical simulations. A finite element method (FEM) is applied to simulate SPAT through ANSYS software based on 3D transient heat transfer principle with the assumption of reflection mode and TT mode of observation individually.

The results conclude that the numerical approach based on SPAT in reflection mode is more suitable for analysing smaller size of near-surface delaminations located at the thermal stimulator side and TT mode is more suitable for analysing smaller size of deeper delaminations located far from thermal stimulator side or near thermal detector/Infrared camera side. The mathematical models provide the optimum q and T at the required MRTD to identify unidentified delamination 7 with 25015.0022W/m2 at 2.531sec and delamination 8 with 16663.3356 W/m2 at 1.37857sec in reflection mode. In TT mode, the delamination 1 with 34954W/m2 at 13.0399sec, delamination 2 with 20002.67W/m2 at 1.998sec and delamination 7 with 20010.87 W/m2 at 0.6171sec could be identified.

Keywords: Step pulsed active thermography (SPAT), NDTE, FML, Delaminations, Finite element method.

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Authors: A. A. Penin, A. S. Sidorenko

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A convenient and physically sound mathematical model of the external or I - V characteristic of solar cells generators is presented in this paper. This model is compared with the traditional model of p-n junction. The direct analytical calculation of load regime leads to a quadratic equation, which is importantly to simplify the calculations in the real time.

Keywords: A solar cell generator, I−V characteristic, activetwo-pole.

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Authors: Martin Steinberger, Martin Horn

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In this article, a high vacuum system for the evaporation of organic semiconductors is introduced and a mathematical model is given. Based on the exact input output linearization a deposition rate controller is designed and tested with different evaporation materials.

Keywords: Effusion cell, organic semiconductors, deposition rate, exact linearization.

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Authors: S. Rozhkova, O. Rozhkova, A. Harlova, V. Lasukov

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For optimal unbiased filter as mean-square and in the case of functioning anomalous noises in the observation memory channel, we have proved insensitivity of filter to inaccurate knowledge of the anomalous noise intensity matrix and its equivalence to truncated filter plotted only by non anomalous components of an observation vector.

Keywords: Mathematical expectation, filtration, anomalous noise, memory.

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Authors: Xiaocong Liu, Huazhen Wang, Ting He, Xiaozheng Li, Weihan Zhang, Jian Chen

Abstract:

The utilization of electronic medical record (EMR) data to establish the disease diagnosis model has become an important research content of biomedical informatics. Deep learning can automatically extract features from the massive data, which brings about breakthroughs in the study of EMR data. The challenge is that deep learning lacks semantic knowledge, which leads to impracticability in medical science. This research proposes a method of incorporating lexical-semantic knowledge from abundant entities into a convolutional neural network (CNN) framework for pediatric disease diagnosis. Firstly, medical terms are vectorized into Lexical Semantic Vectors (LSV), which are concatenated with the embedded word vectors of word2vec to enrich the feature representation. Secondly, the semantic distribution of medical terms serves as Semantic Decision Guide (SDG) for the optimization of deep learning models. The study evaluates the performance of LSV-SDG-CNN model on four kinds of Chinese EMR datasets. Additionally, CNN, LSV-CNN, and SDG-CNN are designed as baseline models for comparison. The experimental results show that LSV-SDG-CNN model outperforms baseline models on four kinds of Chinese EMR datasets. The best configuration of the model yielded an F1 score of 86.20%. The results clearly demonstrate that CNN has been effectively guided and optimized by lexical-semantic knowledge, and LSV-SDG-CNN model improves the disease classification accuracy with a clear margin.

Keywords: lexical semantics, feature representation, semantic decision, convolutional neural network, electronic medical record

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Authors: Sajith Sajeev

Abstract:

The application of Computational Fluid Dynamics (CFD) is widespread in engineering and industry, including aerospace, automotive, and energy. CFD simulations necessitate the use of intricate mathematical models and a substantial amount of computational power to accurately describe the behavior of fluids. The implementation of CFD projects can be difficult, and a well-structured approach to project management is required to assure the timely and cost-effective delivery of high-quality results. This paper's objective is to provide an overview of project management in CFD, including its problems, methodologies, and best practices. The study opens with a discussion of the difficulties connected with CFD project management, such as the complexity of the mathematical models, the need for extensive computational resources, and the difficulties associated with validating and verifying the results. In addition, the study examines the project management methodologies typically employed in CFD, such as the Traditional/Waterfall model, Agile and Scrum. Comparisons are made between the advantages and disadvantages of each technique, and suggestions are made for their effective implementation in CFD projects. The study concludes with a discussion of the best practices for project management in CFD, including the utilization of a well-defined project scope, a clear project plan, and effective teamwork. In addition, it highlights the significance of continuous process improvement and the utilization of metrics to monitor progress and discover improvement opportunities. This article is a resource for project managers, researchers, and practitioners in the field of CFD. It can aid in enhancing project outcomes, reducing risks, and enhancing the productivity of CFD projects. This paper provides a complete overview of project management in CFD and is a great resource for individuals who wish to implement efficient project management methods in CFD projects.

Keywords: Project management, Computational Fluid Dynamics, Traditional/Waterfall methodology, agile methodology, scrum methodology.

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Authors: Yesubai Rubavathi Charles, Ravi Ramraj

Abstract:

In order to retrieve images efficiently from a large database, a unique method integrating color and texture features using genetic programming has been proposed. Opponent color histogram which gives shadow, shade, and light intensity invariant property is employed in the proposed framework for extracting color features. For texture feature extraction, fast discrete curvelet transform which captures more orientation information at different scales is incorporated to represent curved like edges. The recent scenario in the issues of image retrieval is to reduce the semantic gap between user’s preference and low level features. To address this concern, genetic algorithm combined with relevance feedback is embedded to reduce semantic gap and retrieve user’s preference images. Extensive and comparative experiments have been conducted to evaluate proposed framework for content based image retrieval on two databases, i.e., COIL-100 and Corel-1000. Experimental results clearly show that the proposed system surpassed other existing systems in terms of precision and recall. The proposed work achieves highest performance with average precision of 88.2% on COIL-100 and 76.3% on Corel, the average recall of 69.9% on COIL and 76.3% on Corel. Thus, the experimental results confirm that the proposed content based image retrieval system architecture attains better solution for image retrieval.

Keywords: Content based image retrieval, Curvelet transform, Genetic algorithm, Opponent color histogram, Relevance feedback.

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Authors: Anthony Wachs, Guillaume Vinay, Gilles Ferrer, Jacques Kouakou, Calin Dan, Laurence Girolami

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An original Direct Numerical Simulation (DNS) method to tackle the problem of particulate flows at moderate to high concentration and finite Reynolds number is presented. Our method is built on the framework established by Glowinski and his coworkers [1] in the sense that we use their Distributed Lagrange Multiplier/Fictitious Domain (DLM/FD) formulation and their operator-splitting idea but differs in the treatment of particle collisions. The novelty of our contribution relies on replacing the simple artificial repulsive force based collision model usually employed in the literature by an efficient Discrete Element Method (DEM) granular solver. The use of our DEM solver enables us to consider particles of arbitrary shape (at least convex) and to account for actual contacts, in the sense that particles actually touch each other, in contrast with the simple repulsive force based collision model. We recently upgraded our serial code, GRIFF 1 [2], to full MPI capabilities. Our new code, PeliGRIFF 2, is developed under the framework of the full MPI open source platform PELICANS [3]. The new MPI capabilities of PeliGRIFF open new perspectives in the study of particulate flows and significantly increase the number of particles that can be considered in a full DNS approach: O(100000) in 2D and O(10000) in 3D. Results on the 2D/3D sedimentation/fluidization of isometric polygonal/polyedral particles with collisions are presented.

Keywords: Particulate flow, distributed lagrange multiplier/fictitious domain method, discrete element method, polygonal shape, sedimentation, distributed computing, MPI

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Authors: Kejun Zhuang, Hailong Zhu

Abstract:

The dynamics of a delayed mathematical model for Hes1 oscillatory expression are investigated. The linear stability of positive equilibrium and existence of local Hopf bifurcation are studied. Moreover, the global existence of large periodic solutions has been established due to the global bifurcation theorem.

Keywords: Hes1, Hopf bifurcation, time delay, transcriptional repression loop

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Authors: Raymond Xu, Ashley Hua, Andrew Wang, Yuru Lin

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During COVID-19, the depression rate has increased dramatically. Young adults are most vulnerable to the mental health effects of the pandemic. Lower-income families have a higher ratio to be diagnosed with depression than the general population, but less access to clinics. This research aims to achieve early depression detection at low cost, large scale, and high accuracy with an interdisciplinary approach by incorporating clinical practices defined by American Psychiatric Association (APA) as well as multimodal AI framework. The proposed approach detected the nine depression symptoms with Natural Language Processing sentiment analysis and a symptom-based Lexicon uniquely designed for young adults. The experiments were conducted on the multimedia survey results from adolescents and young adults and unbiased Twitter communications. The result was further aggregated with the facial emotional cues analyzed by the Convolutional Neural Network on the multimedia survey videos. Five experiments each conducted on 10k data entries reached consistent results with an average accuracy of 88.31%, higher than the existing natural language analysis models. This approach can reach 300+ million daily active Twitter users and is highly accessible by low-income populations to promote early depression detection to raise awareness in adolescents and young adults and reveal complementary cues to assist clinical depression diagnosis.

Keywords: Artificial intelligence, depression detection, facial emotion recognition, natural language processing, mental disorder.

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Authors: M. Iommi, G. Losco

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Current building sector is focused on reduction of energy requirements, on renewable energy generation and on regeneration of existing urban areas. These targets need to be solved with a systemic approach, considering several aspects simultaneously such as climate conditions, lighting conditions, solar radiation, PV potential, etc. The solar access analysis is an already known method to analyze the solar potentials, but in current years, simulation tools have provided more effective opportunities to perform this type of analysis, in particular in the early design stage. Nowadays, the study of the solar access is related to the easiness of the use of simulation tools, in rapid and easy way, during the design process. This study presents a comparison of three simulation tools, from the point of view of the user, with the aim to highlight differences in the easy-to-use of these tools. Using a real urban context as case study, three tools; Ecotect, Townscope and Heliodon, are tested, performing models and simulations and examining the capabilities and output results of solar access analysis. The evaluation of the ease-to-use of these tools is based on some detected parameters and features, such as the types of simulation, requirements of input data, types of results, etc. As a result, a framework is provided in which features and capabilities of each tool are shown. This framework shows the differences among these tools about functions, features and capabilities. The aim of this study is to support users and to improve the integration of simulation tools for solar access with the design process.

Keywords: Solar access analysis, energy building design tools, urban planning, solar potential.

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Authors: D. Moussazadeh, A. Aytug

Abstract:

In this study, aesthetics, which is architecture-dependent, covers the interpretable, debatable, and mathematical features. The purpose of this study is to provide a different perspective on the values of formal aesthetics and to analyze architectural forms to examine the factors that are related to the form of architectural works. In this study, the formal factors of aesthetics have been objectively studied and analyzed.

Keywords: Architecture design, Gestalt, architectural principle, museum.

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Authors: Alexey Kulik

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This article deals with selection standards for national sport teams. The author examines the legal framework for selection criteria and suggests using the most honest criteria.

Keywords: National teams, Standards of forming teams, Selection standards, Sport legislations.

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Authors: Tadeusz Nowicki, Radosław Pytlak, Robert Waszkowski, Jerzy Bertrandt, Anna Kłos

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This paper presents methods for formal modeling of activities in the area of sanitary inspectors outbreak of food-borne diseases. The models allow you to measure the characteristics of the activities of sanitary inspection and as a result allow improving the performance of sanitary services and thus food security.

Keywords: Food-borne disease, epidemic, sanitary inspection, mathematical models.

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Authors: Pasayev, A., C. Askerov, R. Sadiqov, C. Ardil

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In contrast to existing of calculation of temperature field of a profile part a blade with convective cooling which are not taking into account multi connective in a broad sense of this term, we develop mathematical models and highly effective combination (BIEM AND FDM) numerical methods from the point of view of a realization on the PC. The theoretical substantiation of these methods is proved by the appropriate theorems.

Keywords: multi coherent systems, method of the boundary integrated equations, singular operators, gas turbines

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Authors: Fabrice Poirion, Benedicte Puig

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Mathematical justifications are given for a simulation technique of multivariate nonGaussian random processes and fields based on Rosenblatt-s transformation of Gaussian processes. Different types of convergences are given for the approaching sequence. Moreover an original numerical method is proposed in order to solve the functional equation yielding the underlying Gaussian process autocorrelation function.

Keywords: Simulation, nonGaussian, random field, multivariate, stochastic process.

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Authors: Yan-Fang Niu

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In view of current IT integration development of SOA, this paper examines AIS design based on SOA, including information sources collection, accounting business process integration and real-time financial reports. The main objective of this exploratory paper is to facilitate AIS research combing the Web Service, which is often ignored in accounting and computer research. It provides a conceptual framework that clarifies the interdependency between SOA and AIS, and also presents the major SOA functions in different areas of AIS

Keywords: AIS, SOA, Web Service

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Authors: James E. Ponder

Abstract:

The requirement for consistency in physics can sometimes offer a common ground between disciplines such that their fundamental equations share a common parameter set and mathematical method for equation extraction. The parameter set shared by Relativity and Quantum Wave Mechanics enables an analysis which will be seen to be very straightforward, primarily classical in nature using linear algebra concepts, yet deriving a theoretical estimate of the value of the Gravitational Constant along with dependencies never before known.

Keywords: Gravitational Constant, Physical Consistency, Quantum Mechanics, Relativity.

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Authors: Akbar Zamaniyan, Alireza Behroozsarand, Hadi Ebrahimi

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Partial oxidation (POX) of light hydrocarbons (e.g. methane) is occurred in the first part of the autothermal reformer (ATR). The results of the detailed modeling of the reformer based on the thermodynamic model of the POX and 1D heterogeneous catalytic model for the fixed bed section are considered here. According to the results, the overall performance of the ATR can be improved by changing the important feed parameters.

Keywords: Autothermal Reformer, Partial Oxidation, Mathematical Modeling, Process Simulation, Syngas.

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Authors: Merab Khokhobaia

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Within the framework of this research, the regulatory documents, which are in force in relation to this industry, were analyzed. The main attention is turned to their modernization and necessity of their compliance with European standards. It is a current issue to direct the efforts of state policy on support of business by implementing infrastructural projects, as well as by development of human resources, which may be possible by supporting the relevant higher and vocational studying-educational programs.

Keywords: Regional Development, Tourism Industry, Tourism Policy, Transition.

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Authors: R. Masarova, M. Juhas, B. Juhasova, Z. Sutova

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In dynamic system theory a mathematical model is often used to describe their properties. In order to find a transfer matrix of a dynamic system we need to calculate an inverse matrix. The paper contains the fusion of the classical theory and the procedures used in the theory of automated control for calculating the inverse matrix. The final part of the paper models the given problem by the Matlab.

Keywords: Dynamic system, transfer matrix, inverse matrix, modeling.

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Authors: Suman Senapati, Goutam Saha

Abstract:

Real world Speaker Identification (SI) application differs from ideal or laboratory conditions causing perturbations that leads to a mismatch between the training and testing environment and degrade the performance drastically. Many strategies have been adopted to cope with acoustical degradation; wavelet based Bayesian marginal model is one of them. But Bayesian marginal models cannot model the inter-scale statistical dependencies of different wavelet scales. Simple nonlinear estimators for wavelet based denoising assume that the wavelet coefficients in different scales are independent in nature. However wavelet coefficients have significant inter-scale dependency. This paper enhances this inter-scale dependency property by a Circularly Symmetric Probability Density Function (CS-PDF) related to the family of Spherically Invariant Random Processes (SIRPs) in Log Gabor Wavelet (LGW) domain and corresponding joint shrinkage estimator is derived by Maximum a Posteriori (MAP) estimator. A framework is proposed based on these to denoise speech signal for automatic speaker identification problems. The robustness of the proposed framework is tested for Text Independent Speaker Identification application on 100 speakers of POLYCOST and 100 speakers of YOHO speech database in three different noise environments. Experimental results show that the proposed estimator yields a higher improvement in identification accuracy compared to other estimators on popular Gaussian Mixture Model (GMM) based speaker model and Mel-Frequency Cepstral Coefficient (MFCC) features.

Keywords: Speaker Identification, Log Gabor Wavelet, Bayesian Bivariate Estimator, Circularly Symmetric Probability Density Function, SIRP.

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Authors: Hamed Fawareh

Abstract:

The recent development in learning technologies leads to emerge many learning management systems (LMS). In this study, we concentrate on the specifications and characteristics of LMSs. Furthermore, this paper emphasizes on the feature of e-learning management systems. The features take on the account main indicators to assist and evaluate the quality of e-learning systems. The proposed indicators based of ten dimensions.

Keywords: E-Learning, System Requirement, Social Requirement, Learning Management System.

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Authors: Dante J. Dorantes-González, Aldrin Balsa-Yepes

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Existing learning techniques such as problem-based learning, project-based learning, or case study learning are learning techniques that focus mainly on technical details, but give no specific guidelines on learner’s experience and emotional learning aspects such as arousal salience and valence, being emotional states important factors affecting engagement and retention. Some approaches involving emotion in educational settings, such as social and emotional learning, lack neuroscientific rigorousness and use of specific neurobiological mechanisms. On the other hand, neurobiology approaches lack educational applicability. And educational approaches mainly focus on cognitive aspects and disregard conditioning learning. First, authors start explaining the reasons why it is hard to learn thoughtfully, then they use the method of neurobiological mapping to track the main limbic system functions, such as the reward circuit, and its relations with perception, memories, motivations, sympathetic and parasympathetic reactions, and sensations, as well as the brain cortex. The authors conclude explaining the major finding: The mechanisms of nonconscious learning and the triggers that guarantee long-term memory potentiation. Afterward, the educational framework for practical application and the instructors’ guidelines are established. An implementation example in engineering education is given, namely, the study of tuned-mass dampers for earthquake oscillations attenuation in skyscrapers. This work represents an original learning technique based on nonconscious learning mechanisms to enhance long-term memories that complement existing cognitive learning methods.

Keywords: Emotion, emotion-enhanced memory, learning technique, STEM.

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Authors: Changjin Xu

Abstract:

In this paper, a delayed prototype model is studied. Regarding the delay as a bifurcation parameter, we prove that a sequence of Hopf bifurcations will occur at the positive equilibrium when the delay increases. Using the normal form method and center manifold theory, some explicit formulae are worked out for determining the stability and the direction of the bifurcated periodic solutions. Finally, Computer simulations are carried out to explain some mathematical conclusions.

Keywords: Prototype model, Stability, Hopf bifurcation, Delay, Periodic solution.

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