Earticle

The purpose of the study was to investigate the common errors that could be identified in solving problems involving linear equations for 7th graders, and suggested some ideas to improve problem solving abilities and to suggest efficient ideas to the students and mathematics teachers. The ability to deal with linear equations is important in teaching and learning mathematics since this idea continue to grow as the grade levels elevate such ideas as quadratic equation and cubic equations. Four different patterns were revealed and several suggestions were made to rectify the error in learning and teaching mathematics.

중학교 학생들 중에는 일차방정식, 특히 계수가 분수 또는 소수인 일차 방정식을 푸는 과정에서 학습 부진을 보이는 학생이 많이 있다. 학습 부진아에게 수학적 개념을 이해시키고 성취도를 높이기 위해서는 교수-학습 방법의 개선이 필요하며 그 대안 중의 하나가 CAS 계산기이다. 본 연구에서는 일차방정식에서 학습 부진을 보이는 중학생을 대상으로 CAS 계산기를 활용한 일차방정식 수업을 통해 학생들의 학습 과정의 특징과 성취도 효과를 분석함으로써 향후 일차방정식의 수업 현장에서 학습부진아에 대한 적절하고 효과적인 교수-학습 프로그램을 제시하고자 하였다.

The purpose of this study is to analyze effectiveness of CAS calculator lesson for slow learners in linear equation. For this study, subjects of 4 students(2 boys and 2 girls) at H middle school who are going through difficulties in learning of linear equation were selected. Seven hours of CAS calculator lesson were carried out to investigate features of the learning process and mathematical achievement. The research findings are as follows. First, students understood clearly the meaning of the linear equation. Second, students searched various problem solving methods. Third, students writing ability of mathematical symbol and terminology were improved. Fourth, students had confidence on solving the linear equation. Fifth, students' mathematical achievement were improved.

The present paper, we doing the statistical analysis of insertion sort program and compute its execution times with various parameters. The statistical analysis endorses the specific quadratic pattern of the execution time on the number of items to be sorted. Next, a cursor along the future scope is indicated.

일차방정식의 해결 과정에서 어떤 문항은 등호(‘=‘)관계를 올바르게 표현하고, 다른 문항은 등호(‘=‘)관계를 올바르게 표현하지 못한 것으로 나타났다. 학생들이 등호(‘=‘)관계를 올바르게 표현할 수 있는지, 표현할 수 없는지는 문항에 따라 그 반응이 다르게 나타나므로 여러 문항에 대한 테스트를 한 후에 비교 분석하여 학습지도 방향을 설정하고, 교수ㆍ학습지도가 이루어져야 할 것이다. 일차방정식의 풀이 방법이 제시되지 않은 문항에서 방정식의 해를 구한 대부분의 학생들은 이항을 사용하여 해결 하였다. 등식의 성질을 사용하여 해결하라는 문항에서도 등식의 성질을 사용하여 해결하기 보다는 이항을 사용하여 해결한 학생들이 대부분 이었다. 등식의 성질과 이항을 모두 사용하여 방정식을 해결할 수 있도록 교수ㆍ학습이 이루어져야 할 것이다.

In the process of solving a linear equation, some questions had equal sign(‘=’) relation properly, while other questions did not have equal sign(‘=’) relation properly. Since whether students could express equal sign(‘=’) relation properly or not is determined by questions, the direction for teaching should be instituted, and instruction and teaching should be conducted by comparing and analyzing after conducing tests on may items. Most of students who got the answer for items without the method of solving a linear equation solved the items using binomial. For questions asking to solve using the characteristic of equality, most of students solved the questions using binomial instead of using the characteristic of equality. Therefore, instruction and learning to solve equations using both the characteristic of equality and binomial have to be achieved.

학생들은 변수가 등호의 좌변에 있는 일차방정식보다 우변에 있는 일차방정식 문제를 해결하는데 어려움을 겪고 있다. 이러한 어려움을 학생들이 극복할 수 있도록, 기본적인 여러 유형의 일차방정식 문제를 경험할 수 있는 기회를 제공하여야 할 것이다. 그리고 일차방정식의 교수ㆍ학습에서 여러 유형의 평가 문항을 구성하여 테스트 한 후에 학생들의 풀이 과정을 면밀히 검토하거나, 개별 면담을 통하여 학생들의 학습 상황을 파악하고 이를 토대로 피드백을 통한 오류 교정이 이루어져야 할 필요성이 있다.

Students have difficulties in solving linear equation problems with a variable on the right side rather than linear equation problems a variable on the left side of the sign of equality. In order for students to overcome such difficulties, opportunities to experience many types of basic linear equation problems would have to be provided. Also, it is necessary to examine the process of students’ problem solving process by constructing various types of evaluation item and test them in instruction and learning of linear equations, or grasp students’ studying statues through individual interview and based on theses, error correction through feedbacks have to be achieved.

Linear and non-linear regressions were used to derive the calibration function for the measurement of roxithromycin plasma concentration. Their results were compared with weighted least squares regression by usual weight factors. In this paper the performance of a non-linear calibration equation with the capacity to account empirically for the curvature, y = ax$^{b}$ + c (b $\neq$ 1) is compared with the commonly used linear equation, y = ax + b, as well as the quadratic equation, y = ax$^{2}$+ bx + c. In the calibration curve (range of 0.01 to 10 ${\mu}g/mL$) of roxithromycin, both heteroscedasticity and nonlinearity were present therefore linear least squares regression methods could result in large errors in the determination of roxithromycin concentration. By the non-linear and weighted least squares regression, the accuracy of the analytical method was improved at the lower end of the calibration curve. This study suggests that the non-linear calibration equation should be considered when a curve is required to be fitted to low dose calibration data which exhibit slight curvature.

본 연구에서는 조작적 수학활동의 관점에서 일차방정식 해결과정에 관한 이론적 분석을 시도하였다. 연구방법으로 TEP(Teaching Experiment Methodology)를 사용하였다. 이 분석은 학생들의 자연수학습과 분수학습에 관해 기존에 연구 되어왔던 이차적 모델들과 연구자 자신들의 수학적 지식을 근거로 한 일차적 분석이 조합되어 이루어 졌으며, 학생들의 Explicitly Nested number Sequence (ENS)로부터 시작하여 일차방정식을 풀기 위해 필요한 그 이상의 수학적 지식에 이르기까지 고찰되었다. 연구 결과, 같은 형태의 일차방정식이라 하더라도, 일차방정식을 구성하고 있는 계수와 상수의 종류에 따라 필요한 수학적 지식이 달라질 수 있음을 보여주고 있으며, 따라서 교수학적 시사점으로서 교사들은 요구되는 수학적 지식에 따라 다양한 일차방정식 문제들을 구분할 수 있어야 하고, 또한 구분된 문제들을 상응하는 수준의 학생들에게 적절히 제시하여야 할 것이다.

In this study, an operational analysis in the context of linear equations is presented. For the analysis, several second-order models concerning students' whole number knowledge and fraction knowledge based on teaching experiment methodology were employed, in addition to our first-order analysis. This ontogenetic analysis begins with students' Explicitly Nested number Sequence (ENS) and proceeds on through various forms of linear equations. This study shows that even in the same representational forms of linear equations, the mathematical knowledge necessary for solving those equations might be different based on the type of coefficients and constants the equation consists of. Therefore, the pedagogical implications are that teachers should be able to differentiate between different types of linear equation problems and propose them appropriately to students by matching the required mathematical knowledge to the students' potential constructs.

In this paper, we first introduce the notions of (2, ${\beta}$)-Banach spaces and we will reformulate the fixed point theorem [10, Theorem 1] in this space. We also show that this theorem is a very efficient and convenient tool for proving the new hyperstability results of the general linear equation in (2, ${\beta}$)-Banach spaces. Our main results state that, under some weak natural assumptions, functions satisfying the equation approximately (in some sense) must be actually solutions to it. Our results are improvements and generalizations of the main results of Piszczek [34], Brzdęk [6, 7] and Bahyrycz et al. [2] in (2, ${\beta}$)-Banach spaces.

In the reversed-phase chromatography, the retention time of sample was investigated based on the molecular structure of compound. Several descriptors that were related to retention factors were selected, and then the values of descriptors were calculated with several softwares. The effect of retention factor was measured with calculated values, and the results were obtained that each descriptors of molecular structure of compound have different effect on the retention factor. Therefore, the empirical equation for seven types of descriptors considered was obtained, and it has high values of correlation coefficient. Furthermore, the experimental data and calculated values have good agreement.

We prove a theorem which shows the existence of at least three ${\pi}$-periodic solutions of the wave equation with asymptotical linearity. We obtain this result by the finite dimensional reduction method which reduces the critical point results of the infinite dimensional space to those of the finite dimensional subspace. We also use the critical point theory and the variational method.

연직 2차원 평면을 대상으로 하는 연속방정식을 수심방향으로 이중적분 하여 수평1차원 파랑방정식을 구하였다. 새 방정식은 복소수 포텐셜 함수로 구성되어 있으며, 파랑의 진폭과 위상경사함수를 도입하여 한 세트의 실수방정식으로도 변형되었다. 파랑진폭과 위상경사함수를 포함한 한 세트의 식은 각각 1차, 2차 상미분방정식이며, 한쪽 경계에서 경계조건을 적절히 지정하여 전 영역에서의 해를 한 방향으로 진행하면서 구할 수 있다. 이때 경계조건으로는 파랑진폭 값, 파랑진폭의 경사, 위상 경사 값이다. 단순한 중앙차분식을 이용하여 식을 차분화 하였다. 새 방정식을 Booij의 경사판, Massel의 부드러운 저면, Bragg의 싸인 함수의 저면에 대하여 적용하여 보았다. 본 방정식은 Massel의 수정완경사방정식, Berkhoff의 완경사방정식, 완전 선형방정식과 비교하여 유사한 결과를 나타내었으며, 유용함을 보였다.

A set of equations for description of transformation of harmonic waves is proposed here. Velocity potential function and separation of variables are introduced for the derivation. The continuity equation is in a vertical plane is integrated through the water so that a horizontal one-dimensional wave equation is produced. The new equation composed of the complex velocity potential function, further be modified into. A set up of equations composed of the wave amplitude and wave phase gradient. The horizontally one-dimensional equations on the wave amplitude and wave phase gradient are the first and second-order ordinary differential equations. They are solved in a one-way marching manner starting from a side where boundary values are supplied, i.e. the wave amplitude, the wave amplitude gradient, and the wave phase gradient. Simple spatially-centered finite difference schemes are adopted for the present set of equations. The equations set is applied to three test cases, Booij's inclined plane slope profile, Massel's smooth bed profile, and Bragg's wavy bed profile. The present equations set is satisfactorily verified against existing theories including Massel's modified mild-slope equation, Berkhoff's mild-slope equation, and the full linear equation.

Near Infrared Reflectance Spectroscopy(NIRS)는 짧은 시간 안에 식물의 다양한 영양소를 동시에 정확하고 빠르게 측정할 수 있다. 본 연구는 creeping bentgrass 'CY2' 엽의 여러 가지 기본 요소의 값을 예측하기 위해서 NIRS(근적의선 분광분석기)를 사용하여 측정하였다. 그 결과, 질소와 수분 그리고 탄수화물의 $r^2$은 각각 0.892, 0.925, 0.971이었다. 검량식에 대한 검증에서 $r^2$이 높은 상관관계를 나타냈으므로, 잔디에서 더 많은 연구를 위한 실용화 가능성을 확인 할 수 있었다.

Near Infrared Reflectance Spectroscopy(NIRS) is a quick, accurate, and non-destructive method to measure multiple nutrient components in plant leaves. This study was to acquire a liner regression equation by evaluating the nutrient contents of 'CY2' creeping bentgrass rapidly and accurately using NIRS. In particular, nitrogen fertility is a primary element to keep maintaining good quality of turfgrass. Nitrogen, moisture, carbohydrate, and starch were assessed and analyzed from 'CY2' creeping bentgrass clippings. A linear regression equation was obtained from accessing NIRS values from NIR spectrophotometer(NIR system, Model XDS, XM-1100 series, FOSS, Sweden) programmed with WinISI III project manager v1.50e and ISIscan(R) (Infrasoft International) and calibrated with laboratory values via chemical analysis from an authorized institute. The equation was formulated as MPLS(modified partial least squares) analyzing laboratory values and mathematically pre-treated spectra. The accuracy of the acquired equation was confirmed with SEP(standard error of prediction), which indicated as correlation coefficient($r^2$) and prediction error of sample unacquainted, followed by the verification of model equation of real values and these monitoring results. As results of monitoring, $r^2$ of nitrogen, moisture, and carbohydrate in 'CY2' creeping bentgrass was 0.840, 0.904, and 0.944, respectively. SEP was 0.066, 1.868, and 0.601, respectively. After outlier treatment, $r^2$ was 0.892, 0.925, and 0.971, while SEP was 0.052, 1.577, and 0.394, respectively, which totally showed a high correlation. However, $r^2$ of starch was 0.464, which appeared a low correlation. Thereof, the verified equation appearing higher $r^2$ of nitrogen, moisture, and carbohydrate showed its higher accuracy of prediction model, which finally could be put into practical use for turf management system.

Pathogenic E. coli associated post-weaning diarrhea (PWD) and edema disease are common diseases in commercially-housed weanling pigs. An enterotoxigenic E. coli (ETEC) oral challenge model has been used to mimic the physiological responses observed in commercial conditions. However, an oral challenge procedure has two major limitations: (1) the ETEC cell density is unknown at the point of oral inoculation, and (2) blending ETEC with traditional TSB (trypticase soy broth) is not palatable and hence decreases acceptability by piglets. Therefore, the purposes of this study were to (1) establish a regression equation that can be used for estimation of ETEC concentration in dilution media using the spectrophotometric measurement of cell density; and (2) examine survival of ETEC after blending either with TSB, sweetener or dextrose. A strain of ETEC (serogroup beta-hemolytic E. coli O149; K91; F4; toxins LT, STa, STb) was grown in TSB for 3.5 hours, centrifuged, the supernatant was discarded, and the ETEC pellet was then blended either with TSB (100 mL), sweetener (60 mL TSB + 40 mL fruit flavored concentrate), or dextrose (50 mL TSB + 50 mL dextrose; 0.5g/mL dextrose). Cell density was measured using the colorimetric method and also plated on a 5% sheep blood agar for counting of ETEC colony forming units at 0, 5, 35, 65 and 125 min after blending. The optical density at 600 nm explained 83% of ETEC colony forming units, indicating that the established linear equation (y= 6E+08x - 4E+07, P<0.004) can be used for robust quantification of ETEC cell density in TSB, sweetener and dextrose media. When ETEC was blended with sweetener and dextrose, survival of ETEC was decreased by 45% and 72% within 5 min post-blending. Therefore, further research is required to find out the suitable medium that has potential to improve palatability without compromising survival of ETEC.

In this paper, the radial oscillation of the solutions of higher order homogeneous linear differential equation $$f^{(k)}+A_{n-2}(z)f^{(k-2)}+{\cdots}+A_1(z)f^{\prime}+A_0(z)f=0$$ with transcendental entire function coefficients is studied. Results are obtained to extend some results in [Z. Wu and D. Sun, Angular distribution of solutions of higher order linear differential equations, J. Korean Math. Soc. 44 (2007), no. 6, 1329-1338].

We give some results on hyperstability for the general linear equation. Namely, we show that a function satisfying the linear equation approximately (in some sense) must be actually the solution of it.

Out of all types of motions the critical motions leading to capsize is roll. The dynamic amplification in case of roll motion may be large for ships as roll natural frequency generally falls within the frequency range of wave energy spectrum typical used for estimation of motion spectrum. Roll motion is highly non-linear in nature. Den are various representations of non-linear damping and restoring available in literature. In this paper an uncoupled non-linear roll equations with three representation of damping and cubic restoring term is solved using a perturbation technique. Damping moment representations are linear plus quadratic velocity damping, angle dependant damping and linear plus cubic velocity dependant damping. Numerical value of linear damping coefficient is almost same for all types but non-linear damping is different. Linear and non-linear damping coefficients are obtained form free roll decay tests. External rolling moment is assumed as deterministic with sinusoidal form. Maximum roll amplitude of non-linear roll equation with various representations of damping is calculated using analytical procedure and compared with experimental results, which are obtained form forced tests in regular waves by varying frequency with three wave heights. Experiments indicate influence of non-linearity at resonance frequency. Both experiment and analytical results indicates increase in maximum roll amplitude with wave slope at resonance. Analytical results are compared with experiment results which indicate maximum roll amplitude analytically obtained with angle dependent and cubic velocity damping are equal and difference from experiments with these damping are less compared to non-linear equation with quadratic velocity damping.

In this paper, the blow-up rate of $L^2$-norm for the semi-linear wave equation with a power nonlinearity is obtained in the bounded domain for any p > 1. We also get the blow-up rate of the derivative under the condition 1 < p < $1+\frac{4}{N-1}$ for $N{\geq}2$ or 1 < p < 5 for N = 1.