Mathematical Research Articles

A WKB approach to the quantum multiband electron dynamics in the kinetic formalism

We derive a WKB-like asymptotic expansion of the multiband Wigner function. The model, derived in the envelope function theory, is designed to describe the dynamics in semiconductor devices when the interband conduction-valence transition cannot be neglected. We derive a hierarchy of equations that to lowest order consist of two HamiltonJacobi equations corresponding to the classical dynamics of point particles with positive and negative kinetic energy. Our methodology is based on the Van Vleck approach and a WKB-like asymptotic expansion procedure is used to reduce the numerical complexity of the Wigner multiband evolution system. An approximate closed-form solution is obtained by an iterative procedure that exploits the different time scales on which the intraband and interband dynamical variables evolve. The interband tunneling mechanism appearing to the first order of the expansion is expressed in a very simple mathematical form. By exploiting the highly oscillating behaviour of the multiband Wigner functions we derive a asymptotic expression of the interband transition probability. The resulting formulation reveals particularly close to the classical description of the particles motion and this formal analogy is useful to gain new physical insight and to profit of the numerical method developed for classical systems. The approximates evolution equations are used to simulate the evolution of the Wigner quasi-distribution function in a IRTD diode.

Mathematical Substantiation of the Law on Managing the Company’s Advanced Development

Mathematical Substantiation of the Law on Managing the Company’s Advanced Development

Combined Flexibility and Energy Analysis of Retrofit Actions for Heat Exchanger Networks

Retrofitting of industrial process heat recovery systems can contribute significantly to meeting energy efficiency targets for industrial process plants. One issue to consider when screening retrofit design options is that industrial heat recovery systems must be able to handle external variations, e.g. in ambient temperature, in such a way that operational targets are reached. There exist different approaches to incorporate flexibility considerations in the design process of retrofit proposals for heat exchanger networks (HEN). However, due to mathematical complexity, lack of suitable cost data, and difficulty to handle large-scale systems, the adoption of those methods in industrial retrofit projects has been limited. Therefore, this paper proposes to decouple the design and analysis steps in retrofitting processes. This allows well-proven retrofit design methods to be used in the design step to generate different alternatives. These design alternatives are thereafter evaluated in a separate analysis step in which the initial set of designs is narrowed down to one or several design options that are operable and energy efficient for a priori defined variations of operating conditions. The proposed approach is based upon traditional flexibility analysis combined with energy performance analysis. With such performance data available, a fair evaluation over different operating points can be obtained. The proposed approach is used for analysing the flexibility and energy performance of a HEN case study to illustrate its application.

Student Error Analysis in Learning Algebraic Expression: A Study in Secondary School Putrajaya

Algebraic expressions are an essential topic in the mathematics curriculum as they are interrelated in the application of problem-solving in science and mathematics. Previous studies showed that most students have a low level of understanding of algebraic expressions. This study was conducted to identify the type of student error in learning Algebra Expression topics among Form 4 students. A total of 67 students from Secondary School Putrajaya, comprising 31 males and 36 females, were involved in the study. The instrument used in this study was the Algebraic Expression Diagnostic Test with a reliability value of α = 0.819. This diagnostic test covers two critical concepts in algebraic expressions, namely the development and functioning of algebraic expressions. The types of errors are classified according to Newman’s Error Hierarchical Model, such as the reading error, comprehension, transformation, process skills, encoding or negligence. Data were analyzed using descriptive statistics, namely mean, frequency, and percentage. Inferential statistics, namely the Pearson r correlation. The findings show that students often make the wrong kind of process skills. No students made any reading mistakes. The number of students who form the wrong kind of comprehension, transformation, encoding, and negligence is small. Student’s error in the topic of algebraic expressions is due to students’ weaknesses in finding the most significant common factor, cross-sectional processes, and weaknesses in mastering fractions and negative numbers. It is hoped that the information obtained in this study will help students and teachers to solve learning problems related to the topic of algebraic expressions.

Математическое моделирование динамических процессов в гидравлических и газовых трактах при запуске ЖРД с дожиганием генераторного газа

Математическое моделирование динамических процессов в гидравлических и газовых трактах при запуске ЖРД с дожиганием генераторного газа

An Application of Ergodic Markov Chain to the Process of Teaching Mathematics

The principles of social constructivism for learning have become very popular in the recent decades for teaching mathematics. In the present paper a mathematical representation is created of the process of teaching mathematics (based on those principles) by applying an ergodic Markov chain on its steps. This enables the instructor to evaluate the student difficulties in the classroom and therefore to reorganize his (her) plans for teaching the same subject in future. A classroom application to teaching the Conic Sections to engineering students is also presented illustrating the usefulness of our Markov chain model in practice.

A Presentation of the Austrian Mathematical Society

A Presentation of the Austrian Mathematical Society

Teaching Ethics in Mathematics

Teaching Ethics in Mathematics

TEACHER-STUDENTS RELATIONSHIP AND ITS POTENTIAL IMPACT ON MATHEMATICS LEARNING

This article portrays my lived experience and is an exploration of my pedagogical practices as a learner, as a teacher and as a facilitator focusing on the relationship between teachers and students shifting from traditional to transformative approach in teaching and learning. Based on my lived experiences as a learner from school level to higher study, and as a teacher in different places in different time, the aim of this article is to examine and explore deep settled behavioral practices and seek to change towards transformative/constructivist approach of learning and teaching in terms of teacher-student relationship to maintain quality of education for future generation in Nepal. Subscribing interpretive, critical, and postmodern research paradigms to embrace multi- paradigmatic research design, I used auto-ethnography as a fusion research methodology in this study. Further, the auto-ethnographic inquiry also helped me to examine the pedagogical, cultural and contextual learning from different perspectives as students, teachers, and researcher thereby offering space for interpretation, transformation and envisioning. I predicted that students’ active participation in learning, social and cultural enactment and transformative pedagogy promote our practice to be more meaningful, and learner centered which in turn develops a cordial relationship. My vision to develop the cordial relationship between teacher-students is focused a bit differently in this article.

Hilary Putnam on Logic and Mathematics, edited by Geoffrey Hellman and Roy T. Cook

<i>Hilary Putnam on Logic and Mathematics</i>, edited by Geoffrey Hellman and Roy T. Cook

INFLUENCE OF TEACHER CHARACTERISTICS ON LEARNING READINESS AMONG 5-6 YEARS OLD IN PUBLIC PRE-SCHOOLS IN KIMILILI-BUNGOMA SUB-COUNTY, KENYA

Learner readiness enhances academic achievement and predicts children’s performance in the near future. Children who are ready are less likely to experience behavioural and emotional problems and score higher in reading, mathematics and fine motor skills than those who are not ready. However, teachers are still struggling with learners from preschool who are not well prepared for grade one. This paper provides empirical evidence of the influence of teacher characteristics on learning readiness among 5-6 years old using data obtained from 356 pre-schoolers and 55 preschool teachers in public pre-schools in Kimilili-Bungoma Sub County, Kenya. The multiple regression analysis results indicate that selected teacher characteristics variables (age, years of schooling and number of workshops attended) are statistically significant in explaining variations in learning readiness among pre-schoolers. It is recommended that the Ministry of Education, County and Sub-County Education office and public preschools should ensure that preschool teachers managing learners are qualified and attend more in service training. Article visualizations:

The characteristics didactic gestures of a Study and Research Path involving mathematics and physics at secondary school

This paper analyses the development of a Study and Research Path (SRP) involving mathematics and physics by means of the theoretical construct proposed by the Anthropological Theory of Didactics (ATD) called dialectics . The SRP was implemented in five secondary school math courses and 116 students participated in total. The data analysis was performed using qualitative techniques and constructing nominal variables and their modalities, which also allows the use of descriptive statistical techniques. In this work, absolute frequencies are analysed and some considerations are made - in the light of the theory - to interpret these results. Article visualizations:

Examining Students Mathematical Understanding of Patterns by Pirie-Kieren Model

Bu calismanin amaci ogrencilerin oruntulere iliskin matematiksel anlamalarini arastirmaktir. Bir devlet okulunda ogrenim goren uc yedinci sinif ogrencisi calismaya katilmis ve oruntulerle ilgili sorulari cozmustur. Bu ogrencilerle cozumlerine yonelik yari yapilandirilmis gorusmeler gerceklestirilmistir. Veriler Pirie-Kieren teorisi kullanilarak analiz edilmistir. Elde edilen bulgular, ogrencilerin matematiksel anlamalarinin on bilgiden gozlem yapmaya kadar ilk alti duzey arasinda cesitlilik gosterdigini ve genellikle imaj olusturma ile formullestirme asamalarinda gerceklestigini ortaya koymaktadir. Teoriye gore, ogrencilerin ilk ve ikinci ihtiyac duyulmayan sinirlarin ilerisine gidebildikleri fakat ucuncu siniri gecemedikleri gorulmustur. Sonuclar ogrencilerin oruntulere iliskin bilgilerinin oldugunu ve oruntunun genel formulunu bulmak icin cogunlukla bir kural belirlemeye ve bunun dogrulunu ilk uc terim icin kontrol etmeye calistiklarini gostermektedir.

Evaluation of the Dawson function and its antiderivative needed for the Gaussian broadening of piecewise polynomial functions

The broadening of a sharp (unbroadened) dielectric function is a fruitful approach to the construction of models of dielectric response of materials. It naturally includes structural disorder or finite state lifetime and allows parameterization of such effects. The unbroadened function is often taken as a piecewise polynomial. Broadening it with the Lorentzian then leads to relatively simple analytical formulae. The Gaussian broadening, however, requires evaluation of several special functions, including the antiderivative of the Dawson function which is not generally available in mathematical libraries. Recently, the authors described the simple recurrent formulae for the construction of a Gaussian-broadened piecewise polynomial model of a complex dielectric function using three special functions, the error function, the Dawson function, and its antiderivative. In this paper, for the Dawson function and its antiderivative an efficient evaluation method is developed enabling the utilization of this model in optical spectra fitting. The effectiveness of this approach is illustrated using elementary and real-world examples of complex dielectric function models.

Glycemic control influences on academic performance in youth with Type 1 diabetes.

This study examined how children's and adolescents' with Type 1 diabetes mellitus glucose levels during and prior to academic assessment contributed to performance on reading, writing, and mathematics tasks. Participants had a mean age of 13.69 (SD = 2.10); 44.6% were female and 58.1% reported their ethnicity to be Hispanic, Latino, or Mexican. They wore a continuous glucose monitor for approximately 6 days and completed a neurobehavioral evaluation that consisted of tasks assessing basic reading skills, reading fluency, reading comprehension, math fact fluency, math calculation, spelling, and writing fluency. Results indicated that individuals whose glucose levels were suboptimal (>140 mg/dL) or hyperglycemic (>180 mg/dL) had significantly lower scores on reading fluency (ηp2 = .16) and writing fluency (ηp2 = .28) subtests than those in the target range (70-140 mg/dL). Moreover, more time spent hypoglycemic (<70 mg/dL) within the 12 hr prior to the evaluation increased the risk for impaired performance on academic tasks. These findings support the need to move beyond considering only overall glycemic control to review temporal influences of glucose levels on academic performance. By tracking how fluctuations impact academic performance, school based practitioners can better determine necessary accommodations to buffer glycemic dysregulation effects. In particular, individuals whose glucose levels are frequently outside of the target range are at greatest risk for performing below their true academic potential. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

Mathematical Programming for Heat Exchanger Network Design with Temperature-Dependent Specific Heat Capacity

Heat Exchanger Network (HEN) optimization by mathematical programming has been developed for half of century. Stage-wise superstructure by Yee and Grossmann (1990) is one of the most popular HEN design models dividing process temperature into stages under assumption of constant specific heat capacity (Cp). To make model more realistic, the model is developed under temperature-dependent Cp which is a cubic equation of temperature. To simplify this circumstance, non-linearity of Cp is linearized by applying partitions of linear equations of Cp at different temperature intervals. HEN from the modified stage-wise superstructure model with variable Cp by GAMS software version 24.2 is validated with crude preheat train case from commercial simulation program (Pro/II software version 9.1). The Cp from Pro/II software as a function of temperature is divided into three linear equations of Cp at three temperature intervals. A case study of crude preheat train without HEN consists of five hot product streams and one cold crude oil stream, consuming hot and cold utilities of 175,353.53 kW and 107,945.89 kW, respectively. The new model synthesizes HEN for the crude preheat train, consuming less hot and cold utilities of 82,100.93 kW and 14,654.38 kW. The synthesized HEN is validated by Pro/II software, giving duties of hot and cold utilities of 81,835.30 kW and 14,417.20 kW. Percent error of hot and cold utilities of HEN from GAMS model are only 0.31 % and 1.65 %, compared to ones from Pro/II software. Compared to Pro/II results, percent error of process HEN area and utility HEN area from new model are 1.22 % and 2.91 %, respectively. New model with variable Cp assumption gives a good agreement with validated HEN by Pro/II. Compared to examples from other publications, the model shows that it generates HEN design with less TAC and less computational time.

The Optimal Design of Welded Plate Heat Exchanger with Intensified Heat Transfer for Ammonia Synthesis Column

The modification of heat exchanger networks of industrial enterprises targeting energy saving solutions requires proper heat transfer equipment. The estimation of the optimal design parameters for heat exchangers requires reliable mathematical models for the description of the thermo-hydraulic processes inside the channels, and adequate optimisation methods. This work proposes the novel mathematical model and optimisation algorithm for the selection of welded plate heat exchanger (WPHE) operating in ammonia synthesis column. It enables finding the optimal design with the specified shape of the corrugated plates, distribution of flows and number of plates and passes. The developed algorithm is implemented in Mathcad software. The application of the proposed approach is illustrated by example in which the resulted WPHE with the cross flow in one pass and overall symmetric counterflow of streams has shown a reduction of heat transfer area 25 % compared to previously tested in industry WPHE with unsymmetric passes arrangement.

Rate-Based Modelling and Validation of an Absorber and Stripper in an Amine-Based Post-Combustion CO 2 Capture Process

Carbon emissions capture and storage is regarded as an essential approach to mitigate the greenhouse effect led by carbon dioxide emissions from the fossil fuels combustion. Amine-based post-combustion CO2 capture technologies have great potential for retrofitting existing coal-fired power plants. In this paper, a mathematical model for a packed absorber and stripper in a CO2 capture process are developed. Typical aqueous monoethanolamine (MEA) with 30 wt.% concentration is selected as the absorbent. A rate-based method is implemented to represent the mass and heat transfer in carbon dioxide absorption and desorption processes. In addition, the heat of CO2 is derived by Gibbs-Helmholtz equation. In this way, the effect of chemical equilibrium and thermodynamic equilibrium on absorption capacity and absorption heat can be obtained. Finally, these models are validated by literature data in different plant operating conditions. The present mathematical models can be used to analyse the mass and heat transfer behaviour and to improve the overall performance of the CO2 process.

Optimization of Water Network in a Viscose Fiber Plant

In the production process of viscose staple fiber, a large amount of freshwater, steam and desalinated water will be consumed, and a large amount of acidic and alkaline wastewater will be generated, bringing great pressure on water resource and environmental protection. In this paper, the water using system of a viscose staple fiber plant in northwestern China is studied. The key constraints for water using in this system are pH, oxygen demand (COD) and suspended solid (SS). Since pH and COD are non-conserved quantities, their constraints are property-based. So, the optimization of the water network needs to be carried out from both conventional water system integration and property integration. In this paper, the Mathematical Programming method is used to solve the optimization problem. By analyzing the various constraints and taking the corresponding integration methods, the mathematical model is constructed to target the minimum freshwater flowrate, the minimum regenerated water flowrate and relevant parameters. The optimized network can reduce the freshwater consumption 93.1 m³/h and the wastewater discharge 131 m³/h, achieving obvious water-saving and emission reduction effects.

A Simulation-Optimisation Method for Targeting the Optimal Placement of Heat Pumps in Heat Exchanger Networks

Heat pumps can enhance the utilization of low-grade thermal energy. Setting heat pumps correctly in a heat exchanger network can reduce the consumption of cold and hot utilities, achieving energy saving and pollution reduction. Therefore, targeting the optimal placement of heat pumps is of great interest in both the academia and industry. Focusing on the vapour compression heat pump, this work proposes a simulation-optimization method to identify the energy-optimal placement of heat pumps in heat exchanger networks. Based on the powerful fluid property database, the vapour compression heat pump system is first modelled in the Aspen HYSYS V10. Next, a mathematical model is built in the Matlab R2018b platform using the Grand Composite Curve of heat exchanger network as constraints for heat pump placement. The two platforms, Aspen HYSYS V10 and Matlab R2018b, are then coupled to realize the data transfer between the simulation and mathematical models. The genetic algorithm is used to target the energy-optimal placement of heat pump (evaporating temperature, condensing temperature, and working fluid flowrate) in the heat exchanger network. A case study is performed to illustrate the applicability of the proposed method. The proposed simulation-optimization method can be extended to other types of heat pumps.