Variable Notation Research Articles

Influence of magnetohydrodynamics and chemical reactions on oscillatory free convective flow through a vertical channel in a rotating system with variable permeability

This study investigates the impact of variable permeability as well as chemical reactions on the oscillatory free convective flow that passes parallel porous flat plates with fluctuating temperature and concentration in the presence of a magnetic field. A vertical channel is assumed to be rotating at an angular velocity [Formula: see text]. Periodic free stream velocity causes oscillations in one plate, while periodic suction velocity causes oscillations in the other plate. Complex variable notations are used to solve the governing equations. The perturbation technique is used to derive analytical expressions for the temperature, concentration, and velocity fields. In this study, various parameters were investigated in relation to mean velocity, mean temperature, mean concentration, amplitude, and phase difference. The study also examines the impact on secondary velocity, primary velocity, temperature, concentration, and heat transfer rate during transients. The outcomes are presented graphically for the physical parameters of the problem. The findings contribute to optimizing systems and improving efficiency in heat transfer, fluid dynamics, and environmental remediation.

О существовании точек Штейнгауза. Элементарное рассмотрение

The paper discusses issues related to the well-known "square" Steinghaus hypothesis, which has already hit the list of Richard Guy's unsolved problems three times. Consideration is carried out exclusively by elementary methods. That is, only those that could be used in working with schoolchildren on mathematical projects of the International Computer Schools for the Young (ICSS - 4–7, 13–15). The particular results obtained are very extensive: both in the winter and summer sessions of the ICJJ, the task was to obtain the most general (using elementary methods) results with minimal assumptions: we study a system of four Pythagorean equations with seven unknowns. The numbering of formulas and lemmas is by paragraphs; the numbering of picturies and theorems is through. The designations of the variables in the main text are consistent with those used in Appendix 1 and the drawings cited therein. Variable notations and drawings in Appendices 2 and 3 follow Steinhaus's book Tasks and Thoughts.

A learning trajectory in Kindergarten and first grade students’ thinking of variable and use of variable notation to represent indeterminate quantities

Given the importance of understanding and using indeterminate quantities in algebraic thinking, the development of learning trajectories about how Kindergarten and first grade students understand variable and use variable notation in the context of algebraic expressions is critical. Based on an empirically developed learning trajectory, we analyzed children’s responses at three different points in a classroom teaching experiment. Our purpose was to describe levels of thinking among 16 students (eight in Kindergarten and eight in first grade). Our results revealed qualitative changes in the thinking about indeterminate quantities of most student participants. As students progressed through the experiment, we found that they advanced from what we characterized as a “Pre-variable” Level to a “Letters as representing indeterminate quantities as varying unknowns; explicit operations on indeterminate quantities” Level. Learning trajectories such as that developed here hold promise for informing the design of interventions that support young children’s early algebraic thinking.

Growth in children’s understanding of generalizing and representing mathematical structure and relationships

We share here results from a quasi-experimental study that examines growth in students’ algebraic thinking practices of generalizing and representing generalizations, particularly with variable notation, as a result of an early algebra instructional sequence implemented across grades 3–5. Analyses showed that, while there were no significant differences between experimental and control students on a grade 3 pre-assessment measuring students’ capacity for generalizing and representing generalizations, experimental students significantly outperformed control students on post-assessments at each of grades 3–5. Moreover, experimental students were able to more flexibly interpret variable in different roles and were better able to use variable notation in meaningful ways to represent arithmetic properties, expressions and equations, and functional relationships. This study provides important evidence that young children can learn to think algebraically in powerful ways and suggests that the earlier introduction of algebraic concepts and practices is beneficial to students.

A practice of information processing by rearranging matrix of Jacques Bertin – application of diagram in environmental color workshop

Abstract. The goal of this research is to apply diagrams to environmental color studies. The experience of environmental color is not only visual experience, but also the relationship of the color and the form of the object, the material, the place, the human perception experience. Diagrams records these factors along with the human perception experiences. These diagrams make it possible to study and analyze environmental color more deeply and more easily. In this paper, we first reviewed the "case studies of environmental color workshops" and "theory of information processing by rearranging matrix of Jacques Bertin" which are related to this research. Next, we conducted an environmental color workshop and evaluated the application of diagrams at workshop by students and experts (design teachers and designers). As a result, we confirmed that the diagrams proposed by us are effective for the environmental color workshop. In particular, it was confirmed that these diagrams are effective for group discussion (analysis).The workshop was held at the South China University of Technology (Guangzhou, China) on October 14 – October 17, 2016 (Figure.1). The participants were 5 students. First, students walked along a path near the natural lake in the campus and recorded the observed color objects and their perception experiences. Next, the students (A, B, C, D, E) entered the observation results into the matrix type diagram (a) (Figure.2). In diagram (a), the colors observed by each student were displayed on the left side of the observation route, and the photographs of the color objects, the perception experiences on the color objects, and the landscape photographs of the observation place were arranged on the right side of the observation route. Next, according to Jacques Bertin's theory (visual variables and rearrangement of the matrix), diagram (a) was transformed into diagram (b) (Figure.3) and group discussion and information analysis were carried out. In diagram (b), color objects are represented as "rows", and various attributes of color objects as "columns". The attributes of color objects are the keywords extracted from the "perception experiences for color objects" in diagram (a). In the notation of visual variables, ordinary impression was displayed as gray dot, good impression was displayed as orange dot, bad impression was indicated as blue dot, and the number of observers was indicated by the size of the dot. Looking at Figure.3-left, the most common thing is finding color objects by "brightness", and the number of observers who recorded the "brightness" of bicycle is the largest. By rearranging the matrix in diagram (b), we found that the color objects can be divided into "gray", "white" and "yellow" (Figure.3-middle), there were many good impressions for "yellow and white (flowers)", and there were many bad impressions for "gray and white (street lights)" (Figure.3-right).

Linear Programming Models: Identifying Common Errors in Engineering Students’ Work with Complex Word Problems

In linear programming, many students find it difficult to translate a verbal description of a problem into a valid mathematical model. To better understand this, we examine the existing characteristics of college engineering students’ errors across linear programming (LP) problems. We examined textbooks to identify the types of problems typically found in introductory linear programming courses. We then developed a comprehensive set of tasks and analyzed students’ work to create a taxonomy of the errors and issues that students exhibited. From our findings, we define four categories of identified error types: (1) decision variable errors, (2) variable relationship errors, (3) notation errors, and (4) form errors. This study contributes to the research by investigating students’ work in an area of undergraduate mathematics that has not been heavily explored before. Findings suggest specific areas of focus for future work in helping students develop their understanding of linear programming models and mathematical modeling in word problems in general.

Low-Dissipation Simulation Methods and Models for Turbulent Subsonic Flow

The simulation of turbulent flows by means of computational fluid dynamics is highly challenging. The costs of an accurate direct numerical simulation (DNS) are usually too high, and engineers typically resort to cheaper coarse-grained models of the flow, such as large-eddy simulation (LES). To be suitable for the computation of turbulence, methods should not numerically dissipate the turbulent flow structures. Therefore, energy-conserving discretizations are investigated, which do not dissipate energy and are inherently stable because the discrete convective terms cannot spuriously generate kinetic energy. They have been known for incompressible flow, but the development of such methods for compressible flow is more recent. This paper will focus on the latter: LES and DNS for turbulent subsonic flow. A new theoretical framework for the analysis of energy conservation in compressible flow is proposed, in a mathematical notation of square-root variables, inner products, and differential operator symmetries. As a result, the discrete equations exactly conserve not only the primary variables (mass, momentum and energy), but also the convective terms preserve (secondary) discrete kinetic and internal energy. Numerical experiments confirm that simulations are stable without the addition of artificial dissipation. Next, minimum-dissipation eddy-viscosity models are reviewed, which try to minimize the dissipation needed for preventing sub-grid scales from polluting the numerical solution. A new model suitable for anisotropic grids is proposed: the anisotropic minimum-dissipation model. This model appropriately switches off for laminar and transitional flow, and is consistent with the exact sub-filter tensor on anisotropic grids. The methods and models are first assessed on several academic test cases: channel flow, homogeneous decaying turbulence and the temporal mixing layer. As a practical application, accurate simulations of the transitional flow over a delta wing have been performed.

A progression in first-grade children’s thinking about variable and variable notation in functional relationships

Recent research suggests that children in elementary grades have some facility with variable and variable notation in ways that warrant closer attention. We report here on an empirically developed progression in first-grade children’s thinking about these concepts in functional relationships. Using learning trajectories research as a framework for the study, we developed and implemented an instructional sequence designed to foster children’s understanding of functional relationships. Findings suggest that young children can learn to think in sophisticated ways about variable quantities and variable notation. This challenges assumptions that young children are not “ready” for a study of such concepts and raises the question of whether difficulties adolescents exhibit might be ameliorated by an earlier introduction to these ideas.

Variables in Elementary Mathematics Education

AbstractIn this article, I analyze episodes from two third-grade classrooms drawn from a larger classroom teaching experiment to explore how these students began to incorporate nonnumerical symbols in their mathematical expressions when asked to represent indeterminate quantities. The article addresses two research questions: What understandings did these third-grade students construct when they used nonnumerical symbols to represent indeterminate quantities, and how did these understandings vary during the course of working on a single task? What were some of the challenges these third-grade students faced when they first used nonnumerical symbols to represent indeterminate quantities, and how did these challenges vary while working on the Candy Boxes task? Using the constructs of semantic space and form/function relationships, I argue that teaching and learning environments that encourage children’s use of nonnumerical symbols, such as variable notation, to represent indeterminate quantities can support...

Injection/suction effect on MHD convective flow through porous medium filled in a vertical porous channel with heat radiation and chemical reaction

The paper attempt to investigate the effect of injection/suction on the flow of electrically conducting fluid through a porous channel filled with porous material. A magnetic field of uniform strength is applied perpendicular to the plane of the porous channel. The effect of heat radiation, heat source and chemical reaction has been taken into consideration. The governing equations are solved analytically by using complex variable notation. The expressions for velocity, temperature, concentration, skin friction coefficient, rate of heat transfer and rate of mass transfer have been obtained. The effect of various parameters entering the governing equation have been evaluated numerically and expressed graphically and in tabular form. A detailed discussion is given in the result and discussion section of the research paper.

Oscillatory MHD Convective Flow of Second Order Fluid Through Porous Medium in a Vertical Rotating Channel in Slip-Flow Regime with Heat Radiation

An analysis of an oscillatory magnetohydrodynamic (MHD) convective flow of a second order (viscoelastic), incompressible, and electrically conducting fluid through a porous medium bounded by two infinite vertical parallel porous plates is presented. The two porous plates with slip-flow condition and the no-slip condition are subjected respectively to a constant injection and suction velocity. The pressure gradient in the channel varies periodically with time. A magnetic field of uniform strength is applied in the direction perpendicular to the planes of the plates. The induced magnetic field is neglected due to the assumption of a small magnetic Reynolds number. The temperature of the plate with no-slip condition is non-uniform and oscillates periodically with time and the temperature difference of the two plates is assumed high enough to induce heat radiation. The entire system rotates in unison about the axis perpendicular to the planes of the plates. Adopting complex variable notations, a closed form solution of the problem is obtained. The analytical results are evaluated numerically and then presented graphically to discuss in detail the effects of different parameters of the problem. The velocity, temperature and the skin-friction in terms of its amplitude and phase angle have been shown graphically to observe the effects of the viscoelastic parameter γ, rotation parameter Ω, suction parameter λ , Grashof number Gr, Hartmann number M, the pressure A, Prandtl number Pr, radiation parameter N and the frequency of oscillation ω .

A first grade student’s exploration of variable and variable notation / Una alumna de primer grado explora las variables y su notación

This paper presents a case study of a first grade student to illustrate the diversity of her understandings related to variables and variable notation. While prior research has documented secondary school students’ difficulties with variables and variable notation, we identify many productive understandings in this much younger student, leading us to question the prevailing argument that students might have difficulties with variables due mostly to their own limitations. We draw our data from a teaching experiment that explored functional relationships. Individual interviews were carried out with a subset of the students in the experiment prior to, as well as mid-way through and at the end of the experiment. This paper focuses on a set of three interviews with one of the first grade students. We illustrate the shifts that occurred in the student’s understandings about variables and variable notation across as well as within each of the three interviews.

Children’s Use of Variables and Variable Notation to Represent Their Algebraic Ideas

In this article, we analyze a first grade classroom episode and individual interviews with students who participated in that classroom event to provide evidence of the variety of understandings about variable and variable notation held by first grade children approximately six years of age. Our findings illustrate that given the opportunity, children as young as six years of age can use variable notation in meaningful ways to express relationships between co-varying quantities. In this article, we argue that the early introduction of variable notation in children’s mathematical experiences can offer them opportunities to develop familiarity and fluency with this convention as groundwork for ultimately powerful means of representing general mathematical relationships.

An exact solution of unsteady free convective MHD flow past a hot vertical porous plate with variable temperature in slip flow regime

An analysis of an unsteady free convective flow of a viscous, incompressible, electrically conducting fluid past an infinite hot vertical porous plate in slip flow regime has been carried out under the following assumptions: (i)the suction velocity normal to the plate is constant, (ii) the plate temperature is spanwise cosinusoidal fluctuating with time, (iii) the difference between the temperature of the plate and the free stream is moderately large causing free convection currents and (iv) the transversely applied magnetic field and magnetic Reynold number are really small and, hence, the induced magnetic field is negligible. The governing equations have been solved by adopting complex variable notations and the expressions for velocity and temperature fields obtained. The transient velocity, temperature, and amplitude the shear stress and heat transfer at the plate have been discussed.

3D Guidance of Unmanned Aerial Vehicles Using Sliding Mode Approach

3D Guidance of Unmanned Aerial Vehicles Using Sliding Mode Approach

MHD Oscillatory Free Convection Flow Past Parallel Plates with Periodic Temperature and Concentration

This communication investigates the effect of magnetic field on unsteady free convection oscillatory flow through vertical parallel porous flat plates, when free stream velocity, temperature and concentration oscillates in time about a non zero constant mean. The governing equations are solved by adopting complex variable notations. The analytical expression for velocity, temperature and concentration fields have been obtained using perturbation technique. The effect of various parameters on mean flow velocity, transient velocity, transient temperature, transient concentration, mean skin frication, amplitude and phase of skin-friction and heat transfer have been discussed and shown graphically.

MHD Flow though Rotating Porous Medium with Radiating Heat Transfer in the Presence of Fluctuating Thermal Diffusion

This communication investigates the effect of magnetic field and radiating heat transfer on unsteady free convection viscous incompressible electric conducting fluid past a vertical surface in a rotating porous medium. It is assumed that surface is rotating with angular velocity Ω and the porous vertical surface is subjected to constant suction velocity. The variable heat flux is also assumed on the vertical surface varies with time; the governing equations are solved by adopting complex variable notations. The analytical expressions for velocity and temperature fields are obtained. The effects of various parameters on mean velocity, mean temperature, transient velocity and transient temperature have been discussed and shown graphically.

Stata Tip 106: With or without Reference

A convenient way to define a set of indicator variables (often called dummy variables) is to use Stata’s factor variable notation ([U] 11.4.3 Factor variables). In that case, the default is to leave one category out, the so-called reference category. However, the factor variable notation also allows one to include an indicator variable for the reference category. This can provide a useful alternative representation of the same model. The estimation and interpretation of these models is best explained using examples, like the ones below.

A Fast Algorithm for Nonunitary Joint Diagonalization and Its Application to Blind Source Separation

A fast algorithm, named Complex-Valued Fast Frobenius DIAGonalization (CVFFDIAG), is proposed for seeking the nonunitary approximate joint diagonalizer of a given set of complex-valued target matrices. It adopts a multiplicative update to minimize the Frobenius-norm formulation of the approximate joint diagonalization problem. At each of multiplicative iterations, a strictly diagonally dominant updated matrix is obtained. This scheme ensures the invertibility of the diagonalizer. The CVFFDIAG relaxes several constraints on the target matrices and thus has much general applications. Furthermore, the special approximation of the cost function, the ingenious utilization of some structures and the adequate notation of concerned variables lead to the high computational efficiency of the proposed algorithm. Numerical simulations are conducted to illustrate good performances of the CVFFDIAG.

Three Dimensional Fluctuating Flow and Heat Transfer through a Porous Medium with Variable Permeability

This communication investigates the effect of permeability variation on the heat transfer and the flow through a highly porous medium bounded by an infinite flat porous plate with constant suction. The permeability of the porous medium varies in space and tune both. The problem becomes three dimensional due to the periodic variation of permeability in the transverse direction. The governing equations are solved by adopting complex variable notations and the expressions for the velocity and temperature fields are obtained. The wall shear stress and rate of heat transfer are finally discussed. It is found that the amplitudes |L|, |M|, |H|, respectively, of the skin friction components in the main and transverse directions and the rate of heat transfer, all decrease with the increase of the permeability of the porous medium, K 0 , or the frequency of the permeability fluctuations, ω.