Search results for: approximate arithmetic

Authors: Paola Cattabriga

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According to Gödel’s first incompleteness argument it is possible to construct a formally undecidable proposition in Principia mathematica, a statement that, although true, turns out to be neither provable nor refutable for the system, making therefore incomplete any formal system suitable for the arithmetic of integers. Its features and limitation effects are today widespread basics throughout whole scientific thought. This article brings Gödel’s achievement into question by the definition of the refutability predicate as a number-theoretical statement. We develop proof of invalidity of Theorem VI in Gödel’s 1931, the so-called Gödel’s first incompleteness theorem, in two steps: defining refutability within the same recursive status as provability and showing that as a consequence propositions (15) and (16), derived from definition 8.1 in Gödel’s 1931, are false and unacceptable for the system. The achievement of their falsity blocks the derivation of Theorem VI, which turns out to be therefore invalid, together with all the depending theorems. This article opens up thus new perspectives for mathematical research and for the overall scientific reasoning.

Keywords: Gödel numbering, incompleteness, provability predicate, refutability predicate

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Authors: Salisu ibrahim, Abdalla Rababah

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In this research, we use Weighted G2-Multi-degree reduction of Bezier curve of degree n to a Bezier curve of degree m, m < n. The degree reduction of Bezier curves is used to represent a given Bezier curve of n by a Bezier curve of degree m, m < n. Exact degree reduction is not possible, and degree reduction is approximate process in nature. We derive a weighted degree reducing method that is geometrically continuous at the end points. Different norms will be considered, several error minimizations will be given. The proposed methods produce error function that are less than the errors of existing methods.

Keywords: Bezier curves, multiple degree reduction, geometric continuity, error function

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Authors: Maryam Mosleh, Mahmood Otadi

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The hybrid differential equations have a wide range of applications in science and engineering. In this paper, the homotopy analysis method (HAM) is applied to obtain the series solution of the hybrid differential equations. Using the homotopy analysis method, it is possible to find the exact solution or an approximate solution of the problem. Comparisons are made between improved predictor-corrector method, homotopy analysis method and the exact solution. Finally, we illustrate our approach by some numerical example.

Keywords: Fuzzy number, parametric form of a fuzzy number, fuzzy integrodifferential equation, homotopy analysis method

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Authors: Seree Chadcham, Niphon Sukvilai

Abstract:

This study is the research and development which is intended to: 1) design of the Intelligent Scaffolding Learning Management System (ISLMS) for vocational education, 2) assess the suitability of the Design of Intelligent Scaffolding Learning Management System for Vocational Education. Its methods are divided into 2 phases. Phase 1 is the design of the ISLMS for Vocational Education and phase 2 is the assessment of the suitability of the design. The samples used in this study are work done by 15 professionals in the field of Intelligent Scaffolding, Learning Management System, Vocational Education, and Information and Communication Technology in education selected using the purposive sampling method. Data analyzed by arithmetic mean and standard deviation. The results showed that the ISLMS for vocational education consists of 2 main components which are: 1) the Intelligent Learning Management System for Vocational Education, 2) the Intelligent Scaffolding Management System. The result of the system suitability assessment from the professionals is in the highest range.

Keywords: intelligent, scaffolding, learning management system, vocational education

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Authors: Babatunde J. Falaye

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In this study, we obtain the approximate analytical solutions of the radial Schrödinger equation for the Deng-Fan diatomic molecular potential by using exact quantization rule approach. The wave functions have been expressed by hypergeometric functions via the functional analysis approach. An extension to rotational-vibrational energy eigenvalues of some diatomic molecules are also presented. It is shown that the calculated energy levels are in good agreement with the ones obtained previously E_nl-D (shifted Deng-Fan).

Keywords: Schrödinger equation, exact quantization rule, functional analysis, Deng-Fan potential

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Authors: Karan Sharma, Ajay Kumar

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Epileptic seizure is a type of disease due to which electrical charge in the brain flows abruptly resulting in abnormal activity by the subject. One percent of total world population gets epileptic seizure attacks.Due to abrupt flow of charge, EEG (Electroencephalogram) waveforms change. On the display appear a lot of spikes and sharp waves in the EEG signals. Detection of epileptic seizure by using conventional methods is time-consuming. Many methods have been evolved that detect it automatically. The initial part of this paper provides the review of techniques used to detect epileptic seizure automatically. The automatic detection is based on the feature extraction and classification patterns. For better accuracy decomposition of the signal is required before feature extraction. A number of parameters are calculated by the researchers using different techniques e.g. approximate entropy, sample entropy, Fuzzy approximate entropy, intrinsic mode function, cross-correlation etc. to discriminate between a normal signal & an epileptic seizure signal.The main objective of this review paper is to present the variations in the EEG signals at both stages (i) Interictal (recording between the epileptic seizure attacks). (ii) Ictal (recording during the epileptic seizure), using most appropriate methods of analysis to provide better healthcare diagnosis. This research paper then investigates the effects of a noninvasive healing therapy on the subjects by studying the EEG signals using latest signal processing techniques. The study has been conducted with Reiki as a healing technique, beneficial for restoring balance in cases of body mind alterations associated with an epileptic seizure. Reiki is practiced around the world and is recommended for different health services as a treatment approach. Reiki is an energy medicine, specifically a biofield therapy developed in Japan in the early 20th century. It is a system involving the laying on of hands, to stimulate the body’s natural energetic system. Earlier studies have shown an apparent connection between Reiki and the autonomous nervous system. The Reiki sessions are applied by an experienced therapist. EEG signals are measured at baseline, during session and post intervention to bring about effective epileptic seizure control or its elimination altogether.

Keywords: EEG signal, Reiki, time consuming, epileptic seizure

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Authors: Walaa M.Abd-Elhafiez, Wajeb Gharibi

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In this paper a novel color image compression technique for efficient storage and delivery of data is proposed. The proposed compression technique started by RGB to YCbCr color transformation process. Secondly, the canny edge detection method is used to classify the blocks into edge and non-edge blocks. Each color component Y, Cb, and Cr compressed by discrete cosine transform (DCT) process, quantizing and coding step by step using adaptive arithmetic coding. Our technique is concerned with the compression ratio, bits per pixel and peak signal to noise ratio, and produce better results than JPEG and more recent published schemes (like, CBDCT-CABS and MHC). The provided experimental results illustrate the proposed technique which is efficient and feasible in terms of compression ratio, bits per pixel and peak signal to noise ratio.

Keywords: image compression, color image, q-coder, quantization, edge-detection

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Authors: A. M. Sagir

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This paper derived four newly schemes which are combined in order to form an accurate and efficient block method for parallel or sequential solution of third order ordinary differential equations of the form y^'''= f(x,y,y^',y^'' ), y(α)=y_0,〖y〗^' (α)=β,y^('' ) (α)=μ with associated initial or boundary conditions. The implementation strategies of the derived method have shown that the block method is found to be consistent, zero stable and hence convergent. The derived schemes were tested on stiff and non-stiff ordinary differential equations, and the numerical results obtained compared favorably with the exact solution.

Keywords: block method, hybrid, linear multistep, self-starting, third order ordinary differential equations

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Authors: Melusi Khumalo, Anastacia Dlamini

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In this paper, we consider a numerical solution for nonlinear Fredholm integral equations of the second kind. We work with uniform mesh and use the Lagrange polynomials together with the Galerkin finite element method, where the weight function is chosen in such a way that it takes the form of the approximate solution but with arbitrary coefficients. We implement the finite element method to the nonlinear Fredholm integral equations of the second kind. We consider the error analysis of the method. Furthermore, we look at a specific example to illustrate the implementation of the finite element method.

Keywords: finite element method, Galerkin approach, Fredholm integral equations, nonlinear integral equations

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Authors: Syukri Himran, Erwin Eka Putra, Nanang Roni

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This study explains the natural convection of viscous fluid flowing on semi-infinite vertical plate. A set of the governing equations describing the continuity, momentum and energy, have been reduced to dimensionless forms by introducing the references variables. To solve the problems, the equations are formulated by explicit finite-difference in time dependent form and computations are performed by Fortran program. The results describe velocity, temperature profiles both in transient and steady state conditions. An approximate value of heat transfer coefficient and the effects of Pr on convection flow are also presented.

Keywords: natural convection, vertical plate, velocity and temperature profiles, steady and unsteady

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Authors: Assad Maalouf, Lunjin Lu, James Lynott

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We present a taint analysis that can automatically detect when string operations result in a string that is free of taints, where all the tainted patterns have been removed. This is an improvement on the conservative behavior of previous taint analyzers, where a string operation on a tainted string always leads to a tainted string unless the operation is manually marked as a sanitizer. The taint analysis is built on top of a string analysis that uses finite state automata to approximate the sets of values that string variables can take during the execution of a program. The proposed approach has been implemented as an extension of FlowDroid and experimental results show that the resulting taint analyzer is much more precise than the original FlowDroid.

Keywords: android, static analysis, string analysis, taint analysis

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Authors: Maria Hussin, Malik Zawwar Hussain, Mubashrah Saddiqa

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We present a trigonometric scheme to approximate a circular arc with its two end points and two end tangents/unit tangents. A rational cubic trigonometric Bézier curve is constructed whose end control points are defined by the end points of the circular arc. Weight functions and the remaining control points of the cubic trigonometric Bézier curve are estimated by variational approach to reproduce a circular arc. The radius error is calculated and found less than the existing techniques.

Keywords: control points, rational trigonometric Bézier curves, radius error, shape measure, weight functions

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Authors: Anirudh Bhaikhel

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The exclusive objective of this paper is to develop a device which can pass on the interpreted result of the sensed information to the interfaced communicable devices to avoid or minimise accidents. This device may also be used in case of emergencies like kidnapping, robberies, medical emergencies etc. The present era has seen a rapid metamorphosis in the automobile industry with increasing use of technology and speed. The increase in purchasing power of customers and price war of automobile companies has made an easy access to the automobile users. The use of automobiles has increased tremendously in last 4-5 years thus causing traffic congestions and thus making vehicles more prone to accidents. This device can be an effective measure to counteract cases of abduction. Risks of accidents can be decreased tremendously through the notifications received by these alerts. It will help to detect the upcoming emergencies. This paper includes the simulation of the communication and sensing device required in automobiles using VHDL.

Keywords: automobiles, communication, component, cyclic redundancy check (CRC), modulo-2 arithmetic, parity bits, receiver, sensors, transmitter, turns, VHDL (VHSIC hardware descriptive language)

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Authors: Panita Wannapiroon, Noppadon Phumeechanya, Sitthichai Laisema

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This study is the research and development which is intended to: 1) design the ubiquitous cloud learning management system and: 2) assess the suitability of the design of the ubiquitous cloud learning management system. Its methods are divided into 2 phases. Phase 1 is the design of the ubiquitous cloud learning management system, phase 2 is the assessment of the suitability of the design the samples used in this study are work done by 25 professionals in the field of Ubiquitous cloud learning management systems and information and communication technology in education selected using the purposive sampling method. Data analyzed by arithmetic mean and standard deviation. The results showed that the ubiquitous cloud learning management system consists of 2 main components which are: 1) the ubiquitous cloud learning management system server (u-Cloud LMS Server) including: cloud repository, cloud information resources, social cloud network, cloud context awareness, cloud communication, cloud collaborative tools, and: 2) the mobile client. The result of the system suitability assessment from the professionals is in the highest range.

Keywords: learning management system, cloud computing, ubiquitous learning, ubiquitous learning management system

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Authors: Satyanarayana Engu, Ahmed Mohd, V. Murugan

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We study the large time asymptotics of solutions to the Cauchy problem for a forced Burgers equation (FBE) with the initial data, which is continuous and summable on R. For which, we first derive explicit solutions of FBE assuming a different class of initial data in terms of Hermite polynomials. Later, by violating this assumption we prove the existence of a solution to the considered Cauchy problem. Finally, we give an asymptotic approximate solution and establish that the error will be of order O(t^(-1/2)) with respect to L^p -norm, where 1≤p≤∞, for large time.

Keywords: Burgers equation, Cole-Hopf transformation, Hermite polynomials, large time asymptotics

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Authors: Somveer Singh, Vineet Kumar Singh

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We present a new model based on viscoelasticity for the Non-Newtonian fluids.We use a matrix formulated algorithm to approximate solutions of a class of partial integro-differential equations with the given initial and boundary conditions. Some numerical results are presented to simplify application of operational matrix formulation and reduce the computational cost. Convergence analysis, error estimation and numerical stability of the method are also investigated. Finally, some test examples are given to demonstrate accuracy and efficiency of the proposed method.

Keywords: Legendre Wavelets, operational matrices, partial integro-differential equation, viscoelasticity

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Authors: Alexander S. Andreev, Olga A. Peregudova

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In this paper, we propose a discrete tracking control of nonholonomic mobile robots with two degrees of freedom. The electro-mechanical model of a mobile robot moving on a horizontal surface without slipping, with two rear wheels controlled by two independent DC electric, and one front roal wheel is considered. We present back-stepping design based on the Euler approximate discrete-time model of a continuous-time plant. Theoretical considerations are verified by numerical simulation. The work was supported by RFFI (15-01-08482).

Keywords: actuator dynamics, back stepping, discrete-time controller, Lyapunov function, wheeled mobile robot

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Authors: Sakchai Rachniyom

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Owing to the ASEAN community will formally take place in the few months; therefore, Thais should realize about the importance of English language. Since, it is regarded as a working language in the community. To promote Science students’ English proficiency, teacher should be able to teach in English language appropriately and effectively. The purposes of the quasi – experimental research are (1) to measure the learning achievement, (2) to evaluate students’ satisfaction on the teaching and learning and (3) to study the consequences of co – teaching method in order comprehend the learning achievement and improvement. The participants were 40 general science students teacher. Two types of research instruments were included; (1) an achievement test, and (2) a questionnaire. This research was conducted for 1 semester. The statistics used in this research were arithmetic mean and standard deviation. The findings of the study revealed that students’ achievement score was significantly increased at statistical level .05 and the students satisfied the teaching and learning at the highest level . The students’ involvement and teachers’ support were promoted. It was also reported students’ learning was improved by co – teaching method.

Keywords: co – teaching method, learning science in english, teacher, education

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Authors: Obaid Algahtani

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An interval may be defined as a convex combination as follows: I=[a,b]={x_α=(1-α)a+αb: α∈[0,1]}. Consequently, we may adopt interval operations by applying the scalar operation point-wise to the corresponding interval points: I ∙J={x_α∙y_α ∶ αϵ[0,1],x_α ϵI ,y_α ϵJ}, With the usual restriction 0∉J if ∙ = ÷. These operations are associative: I+( J+K)=(I+J)+ K, I*( J*K)=( I*J )* K. These two properties, which are missing in the usual interval operations, will enable the extension of the usual linear system concepts to the interval setting in a seamless manner. The arithmetic introduced here avoids such vague terms as ”interval extension”, ”inclusion function”, determinants which we encounter in the engineering literature that deal with interval linear systems. On the other hand, these definitions were motivated by our attempt to arrive at a definition of interval random variables and investigate the corresponding statistical properties. We feel that they are the natural ones to handle interval systems. We will enable the extension of many results from usual state space models to interval state space models. The interval state space model we will consider here is one of the form X_((t+1) )=AX_t+ W_t, Y_t=HX_t+ V_t, t≥0, where A∈ 〖IR〗^(k×k), H ∈ 〖IR〗^(p×k) are interval matrices and 〖W 〗_t ∈ 〖IR〗^k,V_t ∈〖IR〗^p are zero – mean Gaussian white-noise interval processes. This feeling is reassured by the numerical results we obtained in a simulation examples.

Keywords: interval analysis, interval matrices, state space model, Kalman Filter

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Authors: Syed Talib Abbas Raza, Tahseen Ahmed Jilani, Saleem Abdullah

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This paper presents interval-valued intuitionistic fuzzy soft sets based TOPSIS method for group decision making. The interval-valued intuitionistic fuzzy soft set is a mutation of an interval-valued intuitionistic fuzzy set and soft set. In group decision making problems IVIFSS makes the process much more algebraically elegant. We have used weighted arithmetic averaging operator for aggregating the information and define a new Hybrid Score Function as metric tool for comparison between interval-valued intuitionistic fuzzy values. In an illustrative example we have applied the developed method to a criminological problem. We have developed a group decision making model for integrating the imprecise and hesitant evaluations of multiple law enforcement agencies working on target killing cases in the country.

Keywords: group decision making, interval-valued intuitionistic fuzzy soft set, TOPSIS, score function, criminology

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Authors: Garry A. Einicke, Langford B. White

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This paper describes an optimum filter and smoother for recovering a Markov process message from noisy measurements. The developments follow from an equivalence between a state space model and a hidden Markov chain. The ensuing filter and smoother employ transition probability matrices and approximate probability distribution vectors. The properties of the optimum solutions are retained, namely, the estimates are unbiased and minimize the variance of the output estimation error, provided that the assumed parameter set are correct. Methods for estimating unknown parameters from noisy measurements are discussed. Signal recovery examples are described in which performance benefits are demonstrated at an increased calculation cost.

Keywords: optimal filtering, smoothing, Markov chains

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Authors: F. Ahwide, Y. Bouker, K. Hatem

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This paper presents long term wind data analysis in terms of annual and diurnal variations at different areas of Libya. The data of the wind speed and direction are taken each ten minutes for a period, at least two years, are used in the analysis. ‘WindPRO’ software and Excel workbook were used for the wind statistics and energy calculations. As for Derna, average speeds are 10 m, 20 m, and 40 m, and respectively 6.57 m/s, 7.18 m/s, and 8.09 m/s. Highest wind speeds are observed at SSW, followed by S, WNW and NW sectors. Lowest wind speeds are observed between N and E sectors. Most frequent wind directions are NW and NNW. Hence, wind turbines can be installed against these directions. The most powerful sector is NW (29.4 % of total expected wind energy), followed by 19.9 % SSW, 11.9% NNW, 8.6% WNW and 8.2% S. Furthermore in Al-Maqrun: the most powerful sector is W (26.8 % of total expected wind energy), followed by 12.3 % WSW and 9.5% WNW. While in Goterria: the most powerful sector is S (14.8 % of total expected wind energy), followed by SSE, SE, and WSW. And Misalatha: the most powerful sector is S, by far represents 28.5% of the expected power, followed by SSE and SE. As for Tarhuna, it is by far SSE and SE, representing each one two times the expected energy of the third powerful sector (NW). In Al-Asaaba: it is SSE by far represents 50% of the expected power, followed by S. It can to be noted that the high frequency of the south direction winds, that come from the desert could cause a high frequency of dust episodes. This fact then, should be taken into account in order to take appropriate measures to prevent wind turbine deterioration. In Excel workbook, an estimation of annual energy yield at position of Derna, Al-Maqrun, Tarhuna, and Al-Asaaba meteorological mast has been done, considering a generic wind turbine of 1.65 MW. (mtORRES, TWT 82-1.65MW) in position of meteorological mast. Three other turbines have been tested. At 80 m, the estimation of energy yield for Derna, Al-Maqrun, Tarhuna, and Asaaba is 6.78 GWh or 3390 equivalent hours, 5.80 GWh or 2900 equivalent hours, 4.91 GWh or 2454 equivalent hours and 5.08 GWh or 2541 equivalent hours respectively. It seems a fair value in the context of a possible development of a wind energy project in the areas, considering a value of 2400 equivalent hours as an approximate limit to consider a wind warm economically profitable. Furthermore, an estimation of annual energy yield at positions of Misalatha, Azizyah and Goterria meteorological mast has been done, considering a generic wind turbine of 2 MW. We found that, at 80 m, the estimation of energy yield is 3.12 GWh or 1557 equivalent hours, 4.47 GWh or 2235 equivalent hours and 4.07GWh or 2033 respectively . It seems a very poor value in the context of possible development of a wind energy project in the areas, considering a value of 2400 equivalent hours as an approximate limit to consider a wind warm economically profitable. Anyway, more data and a detailed wind farm study would be necessary to draw conclusions.

Keywords: wind turbines, wind data, energy yield, micrositting

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Authors: Abdulhakim Ali Mosleh Alzubidy

Abstract:

The study's aim was to establish a proposed training program for kindergarten teachers according to their modern professional demands so that they could effectively teach children through movement education and play. The sample, which consisted of (46) teachers and administrators selected at random from the Ibb governorate, represented the study population of kindergarten teachers and administrators. The researcher developed three survey forms as a tool for data collection, and the forms were used with the research sample. The researcher used the descriptive method due to its applicability and the nature of the study, and he also used the appropriate statistical treatment of the data, which is to extract the percentage and the percentage of agreement. The study came to the following conclusions: ● The proposed program is of great importance in preparing the kindergarten teacher in an appropriate manner that keeps pace with modern developments in this field. ● The field of movement education is a necessity for the kindergarten teacher, through which she will be able to prepare the child physically and kinetically and teach him effectively the principles of reading, writing, and numerical and arithmetic concepts.

Keywords: training program, professional needs, kindergarten, kindergarten teacher

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Authors: U. Chattaraj, K. Dhusiya, M. Raviteja

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Perception reaction time of drivers is an outcome of human thought process, which is vague and approximate in nature and also varies from driver to driver. So, in this study a fuzzy logic based model for prediction of the same has been presented, which seems suitable. The control factors, like, age, experience, intensity of driving of the driver, speed of the vehicle and distance of stimulus have been considered as premise variables in the model, in which the perception reaction time is the consequence variable. Results show that the model is able to explain the impacts of the control factors on perception reaction time properly.

Keywords: driver, fuzzy logic, perception reaction time, premise variable

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Authors: Michal Spilacek, Martin Lisy, Marek Balas, Zdenek Skala

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This article is trying to determine the status of flue gas that is entering the KWH heat exchanger from combustion chamber in order to calculate the heat transfer ratio of the heat exchanger. Combination of measurement, calculation, and computer simulation was used to create a useful way to approximate the heat transfer rate. The measurements were taken by a number of sensors that are mounted on the experimental device and by a thermal imaging camera. The results of the numerical calculation are in a good correspondence with the real power output of the experimental device. Results show that the research has a good direction and can be used to propose changes in the construction of the heat exchanger, but still needs enhancements.

Keywords: heat exchanger, heat transfer rate, numerical calculation, thermal images

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Authors: Hudson Akewe

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In this study, we introduce a modified Jungck (Dual Jungck) iterative scheme and use the scheme to approximate the unique common fixed point of a pair of generalized contractive-like operators in a Banach space. The iterative scheme is also shown to be stable with respect to the maps (S,T). An example is taken to justify the convergence of the scheme. Our result is a generalization and improvement of several results in the literature on single map T.

Keywords: generalized contractive-like operators, modified Jungck iterative scheme, stability results, weakly compatible maps, unique common fixed point

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Authors: Collins Ekpiwre Augustine

Abstract:

Taking into account the impact of teacher training on primary and secondary teachers’ classroom competencies and practices, as revealed by many empirical studies, this study investigated the impact of teacher qualifications on the pedagogical competencies of university teachers in Northwest Nigeria.Four research questions were answered while four hypotheses were tested. Both descriptive statistic of frequencies/arithmetic mean and inferential statistic oft-test were used to analyze the data collected. In order to provide a focus to the study,an observational rating scale titled “University Teachers’ Pedagogical Competency Observation Rating Scale” (UTPCORS) was used to collect data for the study. The population for the study comprised all the university teachers in the three Federal Universities in Northwest Nigeria totaling about 3,401. However, this pilot study was administered on 8 teachers - with 4 participants in each comparison group in Bayero University, Kano.The findings of the study revealed that there was no significant difference in the four hypotheses postulated for the study.

Keywords: impact, university teachers, teachers' qualifications, competencies

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Authors: Soumitr Sanjay Dubey, Shubhajit Roy Chowdhury, Rahul Shrestha

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In this paper, we have presented a novel approach of calculating eigenvalues of any matrix for the first time on Field Programmable Gate Array (FPGA) using Triangular Systolic Arra (TSA) architecture. Conventionally, additional computation unit is required in the architecture which is compliant to the algorithm for determining the eigenvalues and this in return enhances the delay and power consumption. However, recently reported works are only dedicated for symmetric matrices or some specific case of matrix. This works presents an architecture to calculate eigenvalues of any matrix based on QR algorithm which is fully implementable on FPGA. For the implementation of QR algorithm we have used TSA architecture, which is further utilising CORDIC (CO-ordinate Rotation DIgital Computer) algorithm, to calculate various trigonometric and arithmetic functions involved in the procedure. The proposed architecture gives an error in the range of 10−4. Power consumption by the design is 0.598W. It can work at the frequency of 900 MHz.

Keywords: coordinate rotation digital computer, three angle complex rotation, triangular systolic array, QR algorithm

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Authors: Almas Mukhametov, Dina Dautkanova, Moldir Yerbulekova, Gulim Tuyakova, Raziya Zhakudaeva, Makpal Seisenaly, Asemay Kazhymurat

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The composition of samples of mixtures of vegetable oils of Kazakhstan origin, consisting of sunflower, safflower and linseed oils, has been experimentally substantiated. With an approximate optimal ratio of w-6:w-3 fatty acids in 80:15:05 triacylglycerols, providing its therapeutic and prophylactic properties. The resulting mixture can be used in the development of functional products. The result was also identified and evaluated by physical and chemical quality indicators, the content of vitamin E, and the concentration of ions of copper (Cu), iron (Fe), cadmium (Cd), lead (Pb), arsenic (As), nickel (Ni), as well as mercury (Hg).

Keywords: vegetable oil, sunflower, safflower, linseed, mixture, fatty acid composition, heavy metals

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Authors: H. N. Agiza, M. A. Sohaly, M. A. Elfouly

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Parkinson's disease (PD) is a heterogeneous disorder with common age of onset, symptoms, and progression levels. In this paper we will solve analytically the PD model as a non-linear delay differential equation using the steps method. The step method transforms a system of delay differential equations (DDEs) into systems of ordinary differential equations (ODEs). On some numerical examples, the analytical solution will be difficult. So we will approximate the analytical solution using Picard method and Taylor method to ODEs.

Keywords: Parkinson's disease, step method, delay differential equation, two delays

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