Search results for: eigenvalue problems

Authors: Z. Ulukan, E. Demircioğlu,

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Facility location is a complex real-world problem which needs a strategic management decision. This paper provides a general review on studies, efforts and developments in Facility Location Problems which are classical optimization problems having a wide-spread applications in various areas such as transportation, distribution, production, supply chain decisions and telecommunication. Our goal is not to review all variants of different studies in FLPs or to describe very detailed computational techniques and solution approaches, but rather to provide a broad overview of major location problems that have been studied, indicating how they are formulated and what are proposed by researchers to tackle the problem. A brief, elucidative table based on a grouping according to “General Problem Type” and “Methods Proposed” used in the studies is also presented at the end of the work.

Keywords: discrete location problems, exact methods, heuristic algorithms, single source capacitated facility location problems

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Authors: Irena Smetackova

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The important part of school mathematics are word problems which represent the connection between school knowledge and life reality. To find the reasons why students consider word problems to be difficult, it is necessary to take into consideration the motivational settings, besides mathematical knowledge and reading skills. Our goal is to identify whether the familiar or unfamiliar context of math word problem influences solving success rate and if so, whether the reasons are motivational or cognitive. For this purpose, we conducted three steps study in group of fifty pupils 9-10 years old. In the first step, we asked pupils to create ‘the best’ word problems for entered numerical formula. The set of 19 word problems with different contexts were selected. In the second step, pupils were asked to evaluate (without solving) how they like each item and how easy it is for them. The 6 word problems with low preference and low estimated success rate were selected and combined with other 6 problems with high preference and success rate. In the third step, the same pupils were asked to solve the word problems. The analysis showed that pupils attitudes and solving toward word problems varied by the context. The strong gender patterns both in preferred contexts and in estimated success rates were identified however the real success rate did not differ so strongly. The success gap between word problems with and without preferred contexts were stronger than the gap between problems with and without real experience with the context. The hypothesis that motivational factors are more important than cognitive factors was confirmed.

Keywords: mathematics, context of reality, motivation, cognition, word problems

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Authors: Fred N. H. Parker, Nicola T. Fear, S. A. M. Stevelink, L. Rafferty

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Purpose Auditory problems, such as hearing loss and tinnitus, have been associated with mental health problems and alcohol misuse in the UK general population and in the US Armed Forces; however, few studies have examined these associations within the UK Armed Forces. The present study examined the association between auditory problems and probable common mental disorders, post-traumatic stress disorder and alcohol misuse. Methods 5474 serving and ex-service personnel from the UK Armed Forces were examined, selected from those who responded to phase two (data collection 2007–09) and phase three (2014–16) of a military cohort study. Multivariable logistic regression was used to examine the association between auditory problems at phase two and mental health problems at phase three. Results 9.7% of participants reported ever experiencing hearing problems alone, 7.9% reported tinnitus within the last month alone, and 7.8% reported hearing problems with tinnitus. After adjustment, hearing problems with tinnitus at phase two was associated with increased odds of probable common mental disorders (AOR = 1.50, 95% CI 1.09–2.08), post-traumatic stress disorder (AOR = 2.30, 95% CI 1.41–3.76), and alcohol misuse (AOR = 1.94, 95% CI 1.28–2.96) at phase three. Tinnitus alone was associated with probable post-traumatic stress disorder (AOR = 1.80, 95% CI 1.03–3.15); however, hearing problems alone were not associated with any outcomes of interest. Conclusions The association between auditory problems and mental health problems emphasizes the importance of the prevention of auditory problems in the Armed Forces: through enhanced audiometric screening, improved hearing protection equipment, and greater levels of utilization of such equipment.

Keywords: armed forces, hearing problems, tinnitus, mental health, alcohol misuse

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Authors: S. H. Nasseri, A. Ebrahimnejad

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Fuzzy set theory has been applied to many fields, such as operations research, control theory, and management sciences. In this paper, we consider two classes of fuzzy linear programming (FLP) problems: Fuzzy number linear programming and linear programming with trapezoidal fuzzy variables problems. We state our recently established results and develop fuzzy primal simplex algorithms for solving these problems. Finally, we give illustrative examples.

Keywords: fuzzy linear programming, fuzzy numbers, duality, sensitivity analysis

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Authors: Zhidong Zhang, Zhi-Chao Zhang

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In this study, three Chinese versions of the Achenbach systems of empirically based assessment (ASEBA) scales were used to examine adolescent psychological and behavioral problems. These three scales are CBCL, TRF, and YSR. In order to further understand the robustness of these scales, their reliability and construct validity have been examined. Each scale consists of about 113 items plus relevant background variables. These 113 items were further classified into 8 psychological and behavioral problems: emotionally reactive, anxious/depressed, somatic complaints, withdrawn, attention problems, aggressive behavior, social problems, thought problems, and association problems. The study explored the item and construct correlation relations and the correlations between the corresponding constructs among three scales. The results indicated that the associations between item and constructs varied. The construct validities were very robust.

Keywords: ASEBA, construct validity, psychological and behavioral problems, reliability

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Authors: Ayşe Yurdakul, Eckehard Schnieder

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Nowadays, there are multilingual and multidisciplinary communication problems because of the increasing technical progress. Each technical field has its own specific terminology and in each particular language, there are differences in relation to definitions of terms. Besides, there could be several translations in the certain target language for one term of the source language. First of all, these problems of semantic relations between terms include the synonymy, antonymy, hypernymy/hyponymy, ambiguity, risk of confusion and translation problems. Therefore, the iglos terminology management system of the Institute for Traffic Safety and Automation Engineering of the Technische Universität Braunschweig has the goal to avoid these problems by a methodological standardisation of term definitions on the basis of the iglos sign model and iglos relation types. The focus of this paper should be on standardisation of navigation terminology as an example.

Keywords: iglos, localisation, methodological approaches, navigation, positioning, definition problems, terminology

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Authors: Theddeus Tochukwu Akano, Omotayo Abayomi Fakinlede

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Proper understanding of the behavior of compliant mechanisms use by athletes is important in order to avoid catastrophic failure. Such compliant mechanisms like the flex-run require the knowledge of their dynamic response and deformation behavior under quickly varying loads. The modeling of finite deformations of the compliant athletic system is described by Neo-Hookean model under contact-impact conditions. The dynamic impact-contact governing equations for both the target and impactor are derived based on the updated Lagrangian approach. A method where contactor and target are considered as a united body is applied in the formulation of the principle of virtual work for the bodies. In this paper, methods of continuum mechanics and nonlinear finite element method were deployed to develop a model that could capture the behavior of the compliant athletic system under quickly varying loads. A hybrid system of symbolic algebra (AceGEN) and a compiled back end (AceFEM) were employed, leveraging both ease of use and computational efficiency. The simulated results reveal the effect of the various contact-impact conditions on the deformation behavior of the impacting compliant mechanism.

Keywords: eigenvalue problems, finite element method, robin boundary condition, sturm-liouville problem

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Authors: Abdelhamid Zerroug, Mlle Ismahene Sehili

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Spectral methods are usually applied to solve uni-dimensional boundary value problems. With the advantage of the creation of multidimensional basis, we propose a new spectral method for bi-dimensional problems. In this article, we start by creating bi-spectral basis by different ways, we developed also a new relations to determine the expressions of spectral coefficients in different partial derivatives expansions. Finally, we propose the principle of a new bi-spectral method for the bi-dimensional problems.

Keywords: boundary value problems, bi-spectral methods, bi-dimensional Legendre basis, spectral method

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Authors: J. Ritonja, B. Grcar

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For the synchronous generator simulation and analysis and for the power system stabilizer design and synthesis a mathematical model of synchronous generator is needed. The model has to accurately describe dynamics of oscillations, while at the same time has to be transparent enough for an analysis and sufficiently simplified for design of control system. To study the oscillations of the synchronous generator against to the rest of the power system, the model of the synchronous machine connected to an infinite bus through a transmission line having resistance and inductance is needed. In this paper, the linearized reduced order dynamic model of the synchronous generator connected to the infinite bus is presented and analysed in details. This model accurately describes dynamics of the synchronous generator only in a small vicinity of an equilibrium state. With the digression from the selected equilibrium point the accuracy of this model is decreasing considerably. In this paper, the equations’ descriptions and the parameters’ determinations for the linearized reduced order mathematical model of the synchronous generator are explained and summarized and represent the useful origin for works in the areas of synchronous generators’ dynamic behaviour analysis and synchronous generator’s control systems design and synthesis. The main contribution of this paper represents the detailed analysis of the accuracy of the linearized reduced order dynamic model in the entire synchronous generator’s operating range. Borders of the areas where the linearized reduced order mathematical model represents accurate description of the synchronous generator’s dynamics are determined with the systemic numerical analysis. The thorough eigenvalue analysis of the linearized models in the entire operating range is performed. In the paper, the parameters of the linearized reduced order dynamic model of the laboratory salient poles synchronous generator were determined and used for the analysis. The theoretical conclusions were confirmed with the agreement of experimental and simulation results.

Keywords: eigenvalue analysis, mathematical model, power system stability, synchronous generator

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Authors: Zhidong Zhang, Zhi-Chao Zhang

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This study examines adolescent psychological and behavioral problems. The Achenbach systems of empirically based assessment (ASEBA) were used as the instrument. The problem framework consists of internal, external and social behavioral problems which are theoretically developed based on about 113 items plus relevant background variables. In this study, the sample consist of 1,975 sixth and seventh grade students in Northeast China. Stratified random sampling method was used to collect the data, meaning that samples were from different school districts, schools, and classes. The researchers looked at both macro and micro effect. Therefore, multilevel analysis techniques were used in the data analysis. The parts of the research results indicated that the background variables such as extracurricular activities were directly related to students’ internal problems.

Keywords: behavioral problems, anxious/depressed problems, internalizing problems, mental health, under-represented groups, empirically-based assessment, hierarchical modeling, ASEBA, multilevel analysis

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Authors: Ayşe Dilek Maden

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For a given a simple connected graph, we present some new bounds via a new approach for a special topological index given by the sum of the real number power of the non-zero normalized Laplacian eigenvalues. To use this approach presents an advantage not only to derive old and new bounds on this topic but also gives an idea how some previous results in similar area can be developed.

Keywords: degree Kirchhoff index, normalized Laplacian eigenvalue, spanning tree, simple connected graph

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Authors: Jana Kožárová

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Even the behavior problems in pre-school children might be considered as a transitional problem which may disappear by their transition into elementary school; it is an issue that needs a lot of attention because of the fact that the behavioral patterns are adopted in the children especially in this age. Common issue in the process of elimination of the behavior problems in the group of pre-school children is a difference in the perception of the importance and gravity of the symptoms. The underestimation of the children's problems by parents often result into conflicts with kindergarten teachers. Thus, the child does not get the support that his/her problems require and this might result into a school failure and can negatively influence his/her future school performance and success. The research sample consisted of 4 children with behavior problems, their teachers and parents. To determine the most problematic area in the child's behavior, Child Behavior Checklist (CBCL) filled by parents and Caregiver/Teacher Form (CTF-R) filled by teachers were used. Scores from the CBCL and the CTR-F were compared with Pearson correlation coefficient in order to find the differences in the perception of behavior problems in pre-school children.

Keywords: behavior problems, Child Behavior Checklist, Caregiver/Teacher Form, Pearson correlation coefficient, pre-school age

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Authors: Zhidong Zhang, Zhi-Chao Zhang, Georgianna Duarte

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This study investigated the attention problems with the instrument of Achenbach System of Empirically Based Assessment (ASEBA). Two thousand eight hundred and ninety-four adolescents were surveyed by using a stratified sampling method. We examined the relationships between relevant background variables and attention problems. Multiple regression models were applied to analyze the data. Relevant variables such as sports activities, hobbies, age, grade and the number of close friends were included in this study as predictive variables. The analysis results indicated that educational environments and extracurricular activities are important factors which influence students’ attention problems.

Keywords: adolescents, ASEBA, attention problems, educational environments, stratified sampling

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Authors: Dona Grace Jeyaseeli

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Water quality conditions in a pond are controlled by both natural processes and human influences. Natural factors such as the source of the pond water and the types of rock and soil in the pond watershed will influence some water quality characteristics. These factors are difficult to control but usually cause few problems. Instead, most serious water quality problems originate from land uses or other activities near or in the pond. The effects of these activities can often be minimized through proper management and early detection of problems through testing. In the present study a survey of three ponds in Coimbatore city, Tamilnadu, India were analyzed and found that water quality problems in their ponds, ranging from muddy water to fish kills. Unfortunately, most pond owners have never tested their ponds, and water quality problems are usually only detected after they cause a problem. Hence the present study discusses some common water quality parameters that may cause problems in ponds and how to detect through testing for better management of pond.

Keywords: water quality, pond, test, problem

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Authors: Banu Ozturk Kurtaslan, Ruhugul Ozge Ocak

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In this study, educational problems related to landscape design education which is an important study area of landscape architecture discipline. It is important to research about the problems in S.U. Agriculture Faculty Landscape Architecture Department which is a new department, started its B.Sc. education in 2011; and developing some suggestions on this issue in terms of future of the department. In the context of the study a questionnaire has been developed to conduct to the B.Sc. students. The questions has been prepared under the topics of education program, instructor, student, physical infrastructure and other problems.

Keywords: landscape design, landscape design education, problems, Selcuk University Landscape Architecture Department

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Authors: Terence Soule, Tami Al Ghamdi

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To study how to transfer knowledge from multiple source problems to the target problem, we modeled the Transfer Learning (TL) process using Genetic Algorithms as the model solver. TL is the process that aims to transfer learned data from one problem to another problem. The TL process aims to help Machine Learning (ML) algorithms find a solution to the problems. The Genetic Algorithms (GA) give researchers access to information that we have about how the old problem is solved. In this paper, we have five different source problems, and we transfer the knowledge to the target problem. We studied different scenarios of the target problem. The results showed combined knowledge from multiple source problems improves the GA performance. Also, the process of combining knowledge from several problems results in promoting diversity of the transferred population.

Keywords: transfer learning, genetic algorithm, evolutionary computation, source and target

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Authors: Peter L. Varkonyi, Andras A. Sipos

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The quasi-static growth of elastic fibers is studied in the presence of distributed contact with an immobile surface, subject to isotropic dry or viscous friction. Unlike classical problems of elastic stability modelled by autonomous dynamical systems with multiple time scales (slowly varying bifurcation parameter, and fast system dynamics), this problem can only be formulated as a non-autonomous system without time scale separation. It is found that the fibers initially converge to a trivial, straight configuration, which is later replaced by divergence reminiscent of buckling phenomena. In order to capture the loss of stability, a new definition of exponential stability against infinitesimal perturbations for systems defined over finite time intervals is developed. A semi-analytical method for the determination of the critical length based on eigenvalue analysis is proposed. The post-critical behavior of the fibers is studied numerically by using variational methods. The emerging post-critical shapes and the asymptotic behavior as length goes to infinity are identified for simple spatial distributions of growth. Comparison with physical experiments indicates reasonable accuracy of the theoretical model. Some applications from modeling plant root growth to the design of soft manipulators in robotics are briefly discussed.

Keywords: buckling, elastica, friction, growth

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Authors: Mouhamadou Dosso

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This paper presents methods of spectral dichotomy of a matrix which compute spectral projectors on the subspace associated with the eigenvalues external to the parabolas described by a general equation. These methods are modifications of the one proposed in [A. N. Malyshev and M. Sadkane, SIAM J. MATRIX ANAL. APPL. 18 (2), 265-278, 1997] which uses the spectral dichotomy method of a matrix with respect to the imaginary axis. Theoretical and algorithmic aspects of the methods are developed. Numerical results obtained by applying methods presented on matrices are reported.

Keywords: spectral dichotomy method, spectral projector, eigensubspaces, eigenvalue

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Authors: Akinyemi Olufunminiyi Akinbobola

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This paper analytically discusses the disciplinary problems among Adeyemi College of Education Students in the Ondo State of Nigeria. The paper posits that the causes and types of disciplinary problems experienced by the students are determinacy of disciplinary measures to be taken. The study used a questionnaire titled: Disciplinary Problem Questionnaire (DPQ) to collect data. Five hundred (500) students were randomly sampled in the five schools in the college. The results showed that drug addiction, school curriculum, cultism, peer group influence, overcrowded classroom, political, social, and economic among others are disciplinary problems experienced in the study area. The study made recommendations on how to improve the situation.

Keywords: challenges in higher institutions, disciplinary problems, social vices, students’ indiscipline

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Authors: Manana Chumburidze, David Lekveishvili

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We have considered the harmonic oscillations and general dynamic (pseudo oscillations) systems of theory generalized Green-Lindsay of couple-stress thermo-elasticity for isotropic, homogeneous elastic media. Approximate solution of some mixed boundary value problems for finite domain, bounded by the some closed surface are constructed.

Keywords: the couple-stress thermoelasticity, boundary value problems, dynamic problems, approximate solution

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Authors: Ezayra Dubria, Leonora Yambao

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Mother Tongue–Based Multilingual Education (MTB-MLE) is a salient part of the recent reform in the country’s Education system which is the implementation of the K to 12 Basic Education Program. Its importance is highlighted by the passing of Republic Act 10523, otherwise known as the ‘Enhanced Basic Education Act of 2013’. However, teachers, especially new teachers encounter problems in using mother tongue as medium of instruction. Fortunately, teachers are able to create strategies which address these problems. Specifically, this paper gathered the viewpoints of teachers in using mother tongue and analyzed the different problems and strategies used. The problems encountered by teachers are lack of instructional materials written in mother tongue, especially books, lack of vocabulary, lack of teacher training, and influences of social media to learners. The strategies which address these problems are translation of literary pieces and other instructional materials, vocabulary enrichment through the use of word-of-the-day and picture-word association, remedial class, storytelling, differentiated instruction, explicit teaching, individual and group activities, and utilization of multilingual teaching.

Keywords: mother tongue-based instruction, multilingualism, problems, strategies

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Authors: Hussah M. AlShayea

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This study aimed to identify the potential of cloud computing, "Google Drive" in overcoming the problems of collective learning from the viewpoint of Princess Noura University students. The study included (92) students from the College of Education. To achieve the goal of the study, several steps have been taken. First, the most important problems of collective learning were identified from the viewpoint of the students. After that, a survey identifying the potential of cloud computing "Google Drive" in overcoming the problems of collective learning was distributed among the students. The study results showed that the students believe that the use of Google Drive contributed to overcoming these problems. In the light of those results, the researcher presented a set of recommendations and proposals, including: encouraging teachers and learners to employ cloud computing to overcome the problems and constraints of collective learning.

Keywords: cloud computing, collective learning, Google drive, Princess Noura University

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Authors: Nur Aidya Hanum Aizam, Vikneswary Uvaraja

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The agenda of showing the scheduled time for performing certain tasks is known as timetabling. It widely used in many departments such as transportation, education, and production. Some difficulties arise to ensure all tasks happen in the time and place allocated. Therefore, many researchers invented various programming model to solve the scheduling problems from several fields. However, the studies in developing the general integer programming model for many timetabling problems are still questionable. Meanwhile, this thesis describe about creating a general model which solve different types of timetabling problems by considering the basic constraints. Initially, the common basic constraints from five different fields are selected and analyzed. A general basic integer programming model was created and then verified by using the medium set of data obtained randomly which is much similar to realistic data. The mathematical software, AIMMS with CPLEX as a solver has been used to solve the model. The model obtained is significant in solving many timetabling problems easily since it is modifiable to all types of scheduling problems which have same basic constraints.

Keywords: AIMMS mathematical software, integer linear programming, scheduling problems, timetabling

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Authors: T. A. Biala

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This paper deals with the numerical integration of the general second order multipoint boundary value problems. This has been achieved by the development of a continuous linear multistep method (LMM). The continuous LMM is used to construct a main discrete method to be used with some initial and final methods (also obtained from the continuous LMM) so that they form a discrete analogue of the continuous second order boundary value problems. These methods are used as boundary value methods and adapted to cope with the integration of the general second order multipoint boundary value problems. The convergence, the use and the region of absolute stability of the methods are discussed. Several numerical examples are implemented to elucidate our solution process.

Keywords: linear multistep methods, boundary value methods, second order multipoint boundary value problems, convergence

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Authors: Veronica Diaz Quezada

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The society of the 21st century requires training of engineers capable of solving routine and non-routine problems in applications of the limit of real functions, as part of the course Calculus I. For this purpose, research was conducted with a methodological design that combines quantitative and qualitative procedures and that aims, to identify and to characterize the types of problems according to their nature and context, through the application of a mathematics test; to know— through a questionnaire— the opinion of difficulties in their solution, previous and missing knowledge of some students of three engineering careers of a state university in Chile. This research is completed with three case studies. The results favor the performance of students in solving problems of a fantasist and realistic context, but these do not guarantee mathematical skills which are necessary to solve non-routine problems of limit applications. In conclusion, through this research, it became clear that the students of the three engineerings do not have all the necessary skills to solve problems of application of the limit of a function of the real variable.

Keywords: case studies, engineering program, limits, problem solving

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Authors: Liberto Pombal, Christian Dieter Jaekel

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The objective of this work is to establish theoretical and numerical conditions for Solving Extended Linear Complementarity Problems (XLCP), with emphasis on the Horizontal Linear Complementarity Problem (HLCP). Two new strategies for solving complementarity problems are presented, using differentiable and penalized functions, which resulted in a natural formalization for the Linear Horizontal case. The computational results of all suggested strategies are also discussed in depth in this paper. The implication in practice allows solving and optimizing, in an innovative way, the (forestry) problems of the value chain of the industrial wood sector in Angola.

Keywords: complementarity, box constrained, optimality conditions, wood and environment

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Authors: Rana Zeina

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Objective: The association between Sensory problems and cognitive abilities has been studied in individuals with Autism Spectrum Disorders (ASDs). In this study, we used a neuropsychological test to evaluate memory and attention in ASDs children with sensory problems compared to the ASDs children without sensory problems. Methods: Four visual memory tests of Cambridge Neuropsychological Test Automated Battery (CANTAB) including Big/Little Circle (BLC), Simple Reaction Time (SRT), Intra/Extra Dimensional Set Shift (IED), Spatial Recognition Memory (SRM), were administered to 14 ASDs children with sensory problems compared to 13 ASDs without sensory problems aged 3 to 12 with IQ of above 70. Results: ASDs Individuals with sensory problems performed worse than the ASDs group without sensory problems on comprehension, learning, reversal and simple reaction time tasks, and no significant difference between the two groups was recorded in terms of the visual memory and visual comprehension tasks. Conclusion: The findings of this study suggest that ASDs children with sensory problems are facing deficits in learning, comprehension, reversal, and speed of response to stimuli.

Keywords: visual memory, attention, autism spectrum disorders, CANTAB eclipse

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Authors: Lian Tong, Yan Ye, Qiong Yan

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Background: Children with attention-deficit/hyperactivity disorder (ADHD) often experience sleep problems, but the comorbidity mechanism has not been sufficiently studied. This study aimed to determine the comorbidity of ADHD and sleep problems as well as the moderating effects of bedtime activities and depression/anxiety symptoms on the relationship between ADHD and sleep problems. Methods: We recruited 934 primary students from third to fifth grade and their parents by stratified random sampling from three primary schools in Shanghai, China. This study used parent-reported versions of the ADHD Rating Scale-IV, Children’s Sleep Habits Questionnaire, and Achenbach Child Behavior Checklist. We used hierarchical linear regression analysis to clarify the moderating effects of bedtime activities and depression/anxiety symptoms. Results: We found that children with more ADHD symptoms had shorter sleep durations and more sleep problems on weekdays. Screen time before bedtime strengthened the relationship between ADHD and sleep-disordered breathing. Children with more screen time were more likely to have sleep onset delay, while those with less screen time had more sleep onset problems with increasing ADHD symptoms. The high bedtime eating group experienced more night waking with increasing ADHD symptoms compared with the low bedtime eating group. Anxiety/depression exacerbated total sleep problems and further interacted with ADHD symptoms to predict sleep length and sleep duration problems. Conclusions: Bedtime activities and emotional problems had important moderating effects on the relationship between ADHD and sleep problems. These findings indicate that appropriate bedtime management and emotional management may reduce sleep problems and improve sleep duration for children with ADHD symptoms.

Keywords: ADHD, sleep problems, anxiety/depression, bedtime activities, children

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Authors: Wang Xingbo

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Computation of determinant in the form |I-X| is primary and fundamental because it can help to compute many other determinants. This article puts forward a time-reducible approach to compute determinant |I-X|. The approach is derived from the Newton’s identity and its time complexity is no more than that to compute the eigenvalues of the square matrix X. Mathematical deductions and numerical example are presented in detail for the approach. By comparison with classical approaches the new approach is proved to be superior to the classical ones and it can naturally reduce the computational time with the improvement of efficiency to compute eigenvalues of the square matrix.

Keywords: algorithm, determinant, computation, eigenvalue, time complexity

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Authors: Sammani Danwawu Abdullahi

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Vertex Enumeration Algorithms explore the methods and procedures of generating the vertices of general polyhedra formed by system of equations or inequalities. These problems of enumerating the extreme points (vertices) of general polyhedra are shown to be NP-Hard. This lead to exploring how to count the vertices of general polyhedra without listing them. This is also shown to be #P-Complete. Some fully polynomial randomized approximation schemes (fpras) of counting the vertices of some special classes of polyhedra associated with Down-Sets, Independent Sets, 2-Knapsack problems and 2 x n transportation problems are presented together with some discovered open problems.

Keywords: counting with uncertainties, mathematical programming, optimization, vertex enumeration

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