Determinants of Kraus matrices associated with power functions

The depth of penetration in laser beam welding of steels has been determined by two different techniques. A 23 factorial experiment was used to calculate the depth of penetration as a function of laser power, welding speed, and focal distance. Using Yates’ algorithm, the main effects, the two-factor, and three-factor interactions were determined. A generalized multivariable least squares analysis of the data resulted in an equation which expressed the depth of penetration as a function of power, speed, focal distance, and all combinations of the those variables. A dimensional analysis of laser beam welding resulted in an functional relationship that expressed the depth of welding as a function of laser power, welding speed, beam diameter, and thermal properties of the steel. Both of these techniques were then used to compare the calculations with sets of experimental data.The depth of penetration in laser beam welding of steels has been determined by two different techniques. A 23 factorial experiment was used to calculate the depth of penetration as a function of laser power, welding speed, and focal distance. Using Yates’ algorithm, the main effects, the two-factor, and three-factor interactions were determined. A generalized multivariable least squares analysis of the data resulted in an equation which expressed the depth of penetration as a function of power, speed, focal distance, and all combinations of the those variables. A dimensional analysis of laser beam welding resulted in an functional relationship that expressed the depth of welding as a function of laser power, welding speed, beam diameter, and thermal properties of the steel. Both of these techniques were then used to compare the calculations with sets of experimental data.

CHAPTER 1. AN INTRODUCTION TO DATA AND FUNCTIONS. 1.1 Describing Single-Variable Data. 1.2 Describing Relationships between Two Variables. 1.3 An Introduction to Functions. 1.4 The Language of Functions. 1.5 Visualizing Functions. CHAPTER 2. RATES OF CHANGE AND LINEAR FUNCTIONS. 2.1 Average Rates of Change. 2.2 Change in the Average Rate of Change. 2.3 The Average Rate of Change Is a Slope. 2.4 Putting a Slant on Data. 2.5 Linear Functions: When Rates of Change Are Constant. 2.6 Visualizing Linear Functions. 2.7 Constructing Graphs and Equations of Linear Functions. 2.8 Special Cases. 2.9 Breaking the Line: Piecewise Linear Functions. 2.10 Constructing Linear Models of Data. 2.11 Looking for Links between Education and Earnings: A Case Study on Using Regression Lines. CHAPTER 3. WHEN LINES MEET: LINEAR SYSTEMS. 3.1 Interpreting Intersection Points: Linear and Nonlinear Systems. 3.2 Visualizing and Solving Linear Systems. 3.3 Reading between the Lines: Linear Inequalities. 3.4 Systems with Piecewise Linear Functions: Tax Plans. CHAPTER 4. THE LAWS OF EXPONENTS AND LOGARITHMS: MEASURING THE UNIVERSE. 4.1 The Numbers of Science: Measuring Time and Space. 4.2 Positive Integer Exponents. 4.3 Zero, Negative, and Fractional Exponents. 4.4 Converting Units. 4.5 Orders of Magnitude. 4.6 Logarithms as Numbers. CHAPTER 5. GROWTH AND DECAY: AN INTRODUCTION TO EXPONENTIAL FUNCTIONS. 5.1 Exponential Growth. 5.2 Exponential Decay. 5.3 Comparing Linear and Exponential Functions. 5.4 Visualizing Exponential Functions. 5.5 Exponential Functions: A Constant Percent Change. 5.6 More Examples of Exponential Growth and Decay. 5.7 Compound Interest and the Number e. 5.8 Semi-Log Plots of Exponential Functions. CHAPTER 6. LOGARITHMIC LINKS: LOGARITHMIC AND EXPONENTIAL FUNCTIONS. 6.1 Using Logarithms to Solve Exponential Equations. 6.2 Using Natural Logarithms to Solve Exponential Equations Base e. 6.3 Visualizing and Applying Logarithmic Functions. 6.4 Using Semi-Log Plots to Construct Exponential Models for Data. C H A P T E R 7. POWER FUNCTIONS. 7.1 The Tension between Surface Area and Volume. 7.2 Direct Proportionality: Power Functions with Positive Powers. 7.3 Visualizing Positive Integer Power Functions. 7.4 Comparing Power and Exponential Functions. 7.5 Inverse Proportionality: Power Functions with Negative Powers. 7.6 Visualizing Negative Integer Power Functions. 7.7 Using Logarithmic Scales to Find the Best Functional Model. CHAPTER 8. QUADRATICS AND THE MATHEMATICS OF MOTION. 8.1 An Introduction to Quadratic Functions: The Standard Form. 8.2 Visualizing Quadratics: The Vertex Form. 8.3 The Standard Form vs. the Vertex Form. 8.4 Finding the Horizontal Intercepts: The Factored Form. 8.5 The Average Rate of Change of a Quadratic Function. 8.6 The Mathematics of Motion. CHAPTER 9. NEW FUNCTIONS FROM OLD. 9.1 Transformations. 9.2 The Algebra of Functions. 9.3 Polynomials: The Sum of Power Functions. 9.4 Rational Functions: The Quotient of Polynomials. 9.5 Composition and Inverse Functions. 9.6 Exploring, Extending & Expanding. APPENDIX Student Data Tables for Exploration 2.1. Data Dictionary for FAM1000 Data. SOLUTIONS For all Algebra Aerobics and Check Your Understanding problems for odd-numbered problems in the Exercises and Chapter Reviews. All solutions are grouped by chapter ANS-1. INDEX.

The power function is treated as the law relating response time to practice trials. However, the evidence for a power law is flawed, because it is based on averaged data. We report a survey that assessed the form of the practice function for individual learners and learning conditions in paradigms that have shaped theories of skill acquisition. We fit power and exponential functions to 40 sets of data representing 7,910 learning series from 475 subjects in 24 experiments. The exponential function fit better than the power function in all the unaveraged data sets. Averaging produced a bias in favor of the power function. A new practice function based on the exponential, the APEX function, fit better than a power function with an extra, preexperimental practice parameter. Clearly, the best candidate for the law of practice is the exponential or APEX function, not the generally accepted power function. The theoretical implications are discussed.

Changes in blood and plasma volumes during growth

Studies of allometry, or relative growth, have used the power function, y = axb, where a and b are constants, almost exclusively to describe measurements made on some organ y in terms of measurements made on a reference organ x. For this type of application the exclusive use of the power function is questioned on both empirical and logical grounds, and use of the exponential function y = abx is suggested as an alternative. Data representing measurements on army-ant larvae are presented in which a better fit is obtained through use of the exponential function. Fictitious data are also presented to illustrate a situation where failure to consider using the exponential function could lead to a possible misapplication of the power function. Logically, it is demonstrated that the power and exponential functions can be differentiated on the basis of how x and y vary separately as a function of time. If both organs (x and y) are exponential functions of time, the allometric relationship is of the power form. If the organ y is exponential and the reference organ x is linear (as functions of time), the allometric relationship is exponential. Finally, it is suggested that the choice of a function to describe an allometric relationship should also be based, in part, on statistical criteria: What function provides an adequate and simple fit to the data? In particular, the power function should not be regarded as an inherent law of relative growth.

The changes in soil properties and root traits caused by plant growth might have great effects on the process of soil detachment by overland flow. On this basis, two typical herbaceous plants, Bothriochloa ischcemum (Linn.). Keng (BI; fibrous root system) and Artemisia vestita Wall. ex Bess (AG; tap root system), from the Loess Plateau were studied for one year under six planted densities of 5 plants m−2, 10 plants m−2, 15 plants m−2, 20 plants m−2, 25 plants m−2, and 30 plants m−2 to determine how the soil detachment rate responds to soil properties and plant root traits. In total, 24 steel tanks were planted, and two plots were used as bare soil controls. Their soil detachment rates were tested under a constant overland flow (1.5 l s−1) on a 26.2 % slope. The results showed that the soil detachment rate under the six planted densities ranged from 0.034 kg m2 s−1 to 0.112 kg m2 s−1 for BI and was ranged from 0.053 m2 s−1 to 0.132 m2 s−1 for AG, which all greatly reduced soil detachment rate and were 68.17 % to 92.33 % and 69.20 % to 87.27 % less than that of the control. In general, BI was more effective in reducing the soil detachment rate than AG, achieving a mean soil detachment rate that was 23.75 % lower. With increasing plant density, the soil detachment rate decreased as a power function (R2 = 0.23, p < 0.01). The overland flow hydraulic characteristics, soil properties and root traits influenced by plant density were positively or negatively correlated with the soil detachment rate. Specifically, the soil detachment rate decreased with velocity, bulk density, root length density, and increased with shear stress and the Darcy–Weisbach friction factor as power or exponential functions (R2 ranged from 0.16 to 0.54, p < 0.01). On this basis, the soil detachment rate (Dr) can be satisfactorily estimated by the overland flow velocity (v), soil bulk density (BD) and root length density (RLD) as a power function (Dr = 5.636v0.118 × BD−19.917 × RLD−0.170; R2 = 0.58; NSE = 0.78; p < 0.01).

The changes in soil properties and root traits caused by plant growth might have great effects on the process of soil detachment by overland flow. On this basis, two typical herbaceous plants, Bothriochloa ischcemum (Linn.). Keng (BI; fibrous root system) and Artemisia vestita Wall. ex Bess (AG; tap root system), from the Loess Plateau were studied for one year under six planted densities of 5 plants m−2, 10 plants m−2, 15 plants m−2, 20 plants m−2, 25 plants m−2, and 30 plants m−2 to determine how the soil detachment rate responds to soil properties and plant root traits. In total, 24 steel tanks were planted, and two plots were used as bare soil controls. Their soil detachment rates were tested under a constant overland flow (1.5 l s−1) on a 26.2 % slope. The results showed that the soil detachment rate under the six planted densities ranged from 0.034 kg m2 s−1 to 0.112 kg m2 s−1 for BI and was ranged from 0.053 m2 s−1 to 0.132 m2 s−1 for AG, which all greatly reduced soil detachment rate and were 68.17 % to 92.33 % and 69.20 % to 87.27 % less than that of the control. In general, BI was more effective in reducing the soil detachment rate than AG, achieving a mean soil detachment rate that was 23.75 % lower. With increasing plant density, the soil detachment rate decreased as a power function (R2 = 0.23, p < 0.01). The overland flow hydraulic characteristics, soil properties and root traits influenced by plant density were positively or negatively correlated with the soil detachment rate. Specifically, the soil detachment rate decreased with velocity, bulk density, root length density, and increased with shear stress and the Darcy–Weisbach friction factor as power or exponential functions (R2 ranged from 0.16 to 0.54, p < 0.01). On this basis, the soil detachment rate (Dr) can be satisfactorily estimated by the overland flow velocity (v), soil bulk density (BD) and root length density (RLD) as a power function (Dr = 5.636v0.118 × BD−19.917 × RLD−0.170; R2 = 0.58; NSE = 0.78; p < 0.01).

The changes in soil properties and root traits caused by plant growth might have great effects on the process of soil detachment by overland flow. On this basis, two typical herbaceous plants, Bothriochloa ischcemum (Linn.). Keng (BI; fibrous root system) and Artemisia vestita Wall. ex Bess (AG; tap root system), from the Loess Plateau were studied for one year under six planted densities of 5 plants m−2, 10 plants m−2, 15 plants m−2, 20 plants m−2, 25 plants m−2, and 30 plants m−2 to determine how the soil detachment rate responds to soil properties and plant root traits. In total, 24 steel tanks were planted, and two plots were used as bare soil controls. Their soil detachment rates were tested under a constant overland flow (1.5 l s−1) on a 26.2 % slope. The results showed that the soil detachment rate under the six planted densities ranged from 0.034 kg m2 s−1 to 0.112 kg m2 s−1 for BI and was ranged from 0.053 m2 s−1 to 0.132 m2 s−1 for AG, which all greatly reduced soil detachment rate and were 68.17 % to 92.33 % and 69.20 % to 87.27 % less than that of the control. In general, BI was more effective in reducing the soil detachment rate than AG, achieving a mean soil detachment rate that was 23.75 % lower. With increasing plant density, the soil detachment rate decreased as a power function (R2 = 0.23, p < 0.01). The overland flow hydraulic characteristics, soil properties and root traits influenced by plant density were positively or negatively correlated with the soil detachment rate. Specifically, the soil detachment rate decreased with velocity, bulk density, root length density, and increased with shear stress and the Darcy–Weisbach friction factor as power or exponential functions (R2 ranged from 0.16 to 0.54, p < 0.01). On this basis, the soil detachment rate (Dr) can be satisfactorily estimated by the overland flow velocity (v), soil bulk density (BD) and root length density (RLD) as a power function (Dr = 5.636v0.118 × BD−19.917 × RLD−0.170; R2 = 0.58; NSE = 0.78; p < 0.01).

The changes in soil properties and root traits caused by plant growth might have great effects on the process of soil detachment by overland flow. On this basis, two typical herbaceous plants, Bothriochloa ischcemum (Linn.). Keng (BI; fibrous root system) and Artemisia vestita Wall. ex Bess (AG; tap root system), from the Loess Plateau were studied for one year under six planted densities of 5 plants m−2, 10 plants m−2, 15 plants m−2, 20 plants m−2, 25 plants m−2, and 30 plants m−2 to determine how the soil detachment rate responds to soil properties and plant root traits. In total, 24 steel tanks were planted, and two plots were used as bare soil controls. Their soil detachment rates were tested under a constant overland flow (1.5 l s−1) on a 26.2 % slope. The results showed that the soil detachment rate under the six planted densities ranged from 0.034 kg m2 s−1 to 0.112 kg m2 s−1 for BI and was ranged from 0.053 m2 s−1 to 0.132 m2 s−1 for AG, which all greatly reduced soil detachment rate and were 68.17 % to 92.33 % and 69.20 % to 87.27 % less than that of the control. In general, BI was more effective in reducing the soil detachment rate than AG, achieving a mean soil detachment rate that was 23.75 % lower. With increasing plant density, the soil detachment rate decreased as a power function (R2 = 0.23, p < 0.01). The overland flow hydraulic characteristics, soil properties and root traits influenced by plant density were positively or negatively correlated with the soil detachment rate. Specifically, the soil detachment rate decreased with velocity, bulk density, root length density, and increased with shear stress and the Darcy–Weisbach friction factor as power or exponential functions (R2 ranged from 0.16 to 0.54, p < 0.01). On this basis, the soil detachment rate (Dr) can be satisfactorily estimated by the overland flow velocity (v), soil bulk density (BD) and root length density (RLD) as a power function (Dr = 5.636v0.118 × BD−19.917 × RLD−0.170; R2 = 0.58; NSE = 0.78; p < 0.01).

The changes in soil properties and root traits caused by plant growth might have great effects on the process of soil detachment by overland flow. On this basis, two typical herbaceous plants, Bothriochloa ischcemum (Linn.). Keng (BI; fibrous root system) and Artemisia vestita Wall. ex Bess (AG; tap root system), from the Loess Plateau were studied for one year under six planted densities of 5 plants m−2, 10 plants m−2, 15 plants m−2, 20 plants m−2, 25 plants m−2, and 30 plants m−2 to determine how the soil detachment rate responds to soil properties and plant root traits. In total, 24 steel tanks were planted, and two plots were used as bare soil controls. Their soil detachment rates were tested under a constant overland flow (1.5 l s−1) on a 26.2 % slope. The results showed that the soil detachment rate under the six planted densities ranged from 0.034 kg m2 s−1 to 0.112 kg m2 s−1 for BI and was ranged from 0.053 m2 s−1 to 0.132 m2 s−1 for AG, which all greatly reduced soil detachment rate and were 68.17 % to 92.33 % and 69.20 % to 87.27 % less than that of the control. In general, BI was more effective in reducing the soil detachment rate than AG, achieving a mean soil detachment rate that was 23.75 % lower. With increasing plant density, the soil detachment rate decreased as a power function (R2 = 0.23, p < 0.01). The overland flow hydraulic characteristics, soil properties and root traits influenced by plant density were positively or negatively correlated with the soil detachment rate. Specifically, the soil detachment rate decreased with velocity, bulk density, root length density, and increased with shear stress and the Darcy–Weisbach friction factor as power or exponential functions (R2 ranged from 0.16 to 0.54, p < 0.01). On this basis, the soil detachment rate (Dr) can be satisfactorily estimated by the overland flow velocity (v), soil bulk density (BD) and root length density (RLD) as a power function (Dr = 5.636v0.118 × BD−19.917 × RLD−0.170; R2 = 0.58; NSE = 0.78; p < 0.01).

The changes in soil properties and root traits caused by plant growth might have great effects on the process of soil detachment by overland flow. On this basis, two typical herbaceous plants, Bothriochloa ischcemum (Linn.). Keng (BI; fibrous root system) and Artemisia vestita Wall. ex Bess (AG; tap root system), from the Loess Plateau were studied for one year under six planted densities of 5 plants m−2, 10 plants m−2, 15 plants m−2, 20 plants m−2, 25 plants m−2, and 30 plants m−2 to determine how the soil detachment rate responds to soil properties and plant root traits. In total, 24 steel tanks were planted, and two plots were used as bare soil controls. Their soil detachment rates were tested under a constant overland flow (1.5 l s−1) on a 26.2 % slope. The results showed that the soil detachment rate under the six planted densities ranged from 0.034 kg m2 s−1 to 0.112 kg m2 s−1 for BI and was ranged from 0.053 m2 s−1 to 0.132 m2 s−1 for AG, which all greatly reduced soil detachment rate and were 68.17 % to 92.33 % and 69.20 % to 87.27 % less than that of the control. In general, BI was more effective in reducing the soil detachment rate than AG, achieving a mean soil detachment rate that was 23.75 % lower. With increasing plant density, the soil detachment rate decreased as a power function (R2 = 0.23, p < 0.01). The overland flow hydraulic characteristics, soil properties and root traits influenced by plant density were positively or negatively correlated with the soil detachment rate. Specifically, the soil detachment rate decreased with velocity, bulk density, root length density, and increased with shear stress and the Darcy–Weisbach friction factor as power or exponential functions (R2 ranged from 0.16 to 0.54, p < 0.01). On this basis, the soil detachment rate (Dr) can be satisfactorily estimated by the overland flow velocity (v), soil bulk density (BD) and root length density (RLD) as a power function (Dr = 5.636v0.118 × BD−19.917 × RLD−0.170; R2 = 0.58; NSE = 0.78; p < 0.01).

The changes in soil properties and root traits caused by plant growth might have great effects on the process of soil detachment by overland flow. On this basis, two typical herbaceous plants, Bothriochloa ischcemum (Linn.). Keng (BI; fibrous root system) and Artemisia vestita Wall. ex Bess (AG; tap root system), from the Loess Plateau were studied for one year under six planted densities of 5 plants m−2, 10 plants m−2, 15 plants m−2, 20 plants m−2, 25 plants m−2, and 30 plants m−2 to determine how the soil detachment rate responds to soil properties and plant root traits. In total, 24 steel tanks were planted, and two plots were used as bare soil controls. Their soil detachment rates were tested under a constant overland flow (1.5 l s−1) on a 26.2 % slope. The results showed that the soil detachment rate under the six planted densities ranged from 0.034 kg m2 s−1 to 0.112 kg m2 s−1 for BI and was ranged from 0.053 m2 s−1 to 0.132 m2 s−1 for AG, which all greatly reduced soil detachment rate and were 68.17 % to 92.33 % and 69.20 % to 87.27 % less than that of the control. In general, BI was more effective in reducing the soil detachment rate than AG, achieving a mean soil detachment rate that was 23.75 % lower. With increasing plant density, the soil detachment rate decreased as a power function (R2 = 0.23, p < 0.01). The overland flow hydraulic characteristics, soil properties and root traits influenced by plant density were positively or negatively correlated with the soil detachment rate. Specifically, the soil detachment rate decreased with velocity, bulk density, root length density, and increased with shear stress and the Darcy–Weisbach friction factor as power or exponential functions (R2 ranged from 0.16 to 0.54, p < 0.01). On this basis, the soil detachment rate (Dr) can be satisfactorily estimated by the overland flow velocity (v), soil bulk density (BD) and root length density (RLD) as a power function (Dr = 5.636v0.118 × BD−19.917 × RLD−0.170; R2 = 0.58; NSE = 0.78; p < 0.01).

En la modelación del ahusamiento y del crecimiento en altura dominante con datos de series de tiempo, es muy común la presencia de heterocedasticidad y autocorrelación de los errores. Funciones de varianza (varFunc) y estructuras de correlación (corStruct) para corregir la heterocedasticidad y modelar dependencia de los errores, respectivamente. Estas fueron combinadas y evaluadas en ecuaciones de ahusamiento y crecimiento en altura de Pinus teocote en Durango, México. La base de datos se obtuvo de 51 análisis troncales con 768 observaciones de ahusamiento y 634 de altura. Las varFunc utilizadas fueron: 1) función de potencia (varPower); 2) función exponencial (varExp); 3) función constante y de potencia (varConstPower); y 4) función combinada de potencia y exponencial (varComb). Las corStruct incluyeron: simetría compuesta (corCompSymm), autorregresiva de orden 1 (corAR1), autorregresiva continua (corCAR1), autorregresiva de media móvil (corARMA2-0), corARMA1-1, corARMA2-1, corARMA2-2, corARMA3-1 y corARMA3-2. Las ecuaciones se ajustaron por mínimos cuadrados generalizados no lineales; y se evaluaron con un sistema de calificación con los estadísticos de ajuste: RMSE, R2, AIC, BIC, LogLik, CV y sesgo promedio. Con base en la calificación, las mejores combinaciones para el ahusamiento y crecimiento en altura fueron 1-9, 2-5, 3-8 y 4-6 y 1-6, 2-9, 3-7 y 4-4, respectivamente. En el ahusamiento solo la combinación 2-5 fue homocedástica con residuales independientes al igual que las ecuaciones de altura seleccionadas y las varFunc y corStruct presentaron influencia en la trayectoria de las curvas de índice de sitio construidas.

The changes in soil properties and root traits caused by plant growth might have great effects on the process of soil detachment by overland flow. On this basis, two typical herbaceous plants, Bothriochloa ischcemum (Linn.). Keng (BI; fibrous root system) and Artemisia vestita Wall. ex Bess (AG; tap root system), from the Loess Plateau were studied for one year under six planted densities of 5 plants m−2, 10 plants m−2, 15 plants m−2, 20 plants m−2, 25 plants m−2, and 30 plants m−2 to determine how the soil detachment rate responds to soil properties and plant root traits. In total, 24 steel tanks were planted, and two plots were used as bare soil controls. Their soil detachment rates were tested under a constant overland flow (1.5 l s−1) on a 26.2 % slope. The results showed that the soil detachment rate under the six planted densities ranged from 0.034 kg m2 s−1 to 0.112 kg m2 s−1 for BI and was ranged from 0.053 m2 s−1 to 0.132 m2 s−1 for AG, which all greatly reduced soil detachment rate and were 68.17 % to 92.33 % and 69.20 % to 87.27 % less than that of the control. In general, BI was more effective in reducing the soil detachment rate than AG, achieving a mean soil detachment rate that was 23.75 % lower. With increasing plant density, the soil detachment rate decreased as a power function (R2 = 0.23, p < 0.01). The overland flow hydraulic characteristics, soil properties and root traits influenced by plant density were positively or negatively correlated with the soil detachment rate. Specifically, the soil detachment rate decreased with velocity, bulk density, root length density, and increased with shear stress and the Darcy–Weisbach friction factor as power or exponential functions (R2 ranged from 0.16 to 0.54, p < 0.01). On this basis, the soil detachment rate (Dr) can be satisfactorily estimated by the overland flow velocity (v), soil bulk density (BD) and root length density (RLD) as a power function (Dr = 5.636v0.118 × BD−19.917 × RLD−0.170; R2 = 0.58; NSE = 0.78; p < 0.01).

In the work of city planning, the planning and design personnel should take city roads, city transportations, drainage system and all kinds of infrastructures into account, which are closely linked with spatial position. In the area of the application of urban planning, with its powerful function in representation and analysis, GIS have remarkable advantages compared with normal information management systems. These powerful GIS function can solve the problem such as calculate and estimate of plot ratio and greening rate, the adaptability evaluation of ground and environment. In addition, GIS bring convince for urban planning work with its powerful function in scene simulation, three-dimensional display and analysis. Construction a programmed decision-making system based on GIS with the function of urban planning information management, model construction and decision-making can provide scientific decision and reference for urban planning personnel.