text:
## Abstract

In this work, data-driven soft sensors are developed for the debutanizer column for online monitoring of butane content in the debutanizer column bottom product. The data set consists of data for seven process inputs and one process output. The total process data were equally divided into a training set and a validation set using the Kennard–Stone maximal intra distance criterion. The training set was used to develop multiple linear regression, principal component regression and back propagation neural network models for the debutanizer column. Performances of the developed models were assessed by simulation with the validation data set. Results show that the neural network model designed using Levenberg–Marquardt algorithm is capable of estimating the product quality with nearly 95% accuracy. The performance of the neural network model reported in this article is found to be better than the performances of least square support vector regression and standard support vector regression models reported in the literature earlier.

## Keywords

Back propagation neural network; Debutanizer column; Principal component analysis; Soft sensor

## Nomenclature

BPNN- back propagation neural network

MAE- mean absolute error

MLR- multiple linear regression

PC- principal component

PCA- principal component analysis

PCR- principal component regression

R- correlation coefficient

RMSE- root mean squared error

x- input variable

X- input data matrix

y- output variable

yi- actual output value for ith observation

${\textstyle {\overset {\mbox{ˆ}}{y}}_{i}}$- model predicted output for ith observation

${\textstyle {\overline {\overset {\mbox{ˆ}}{y}}}}$- average value of model predicted outputs

Y- output data vector

β- regression coefficients of linear regression model

## 1. Introduction

In most of the processes, online monitoring of product quality is difficult or impossible due to lack of hardware sensors or their low reliability. This problem leads to occasional production of low quality products resulting in rejection of the final product and subsequent revenue loss to the industry. Soft sensors are process models which are used for continuous online monitoring of quality variables. In the last decade there has been growing use of soft sensors for quality monitoring in different process industries such as polymer [1] and [2], fermentation, bioprocesses [3], [4] and [5], size reduction [6], [7] and [8], and rotary kiln [9] and [10] to name a few.

In the context of petroleum refinery and petrochemical industries, soft sensing techniques have been proposed for prediction of different quality variables. A survey of the different soft sensors reported in petroleum refinery and petrochemical industries is reported in Table 1.

Table 1. Survey of soft sensor applications reported in petroleum refinery and petrochemical industries.
Author(s) Year Quality variable predicted Technique used
Kresta et al. [11] 1994 Heavy key components in distillate PLS
Chen and Wang [12] 1998 Condensation temperature of light diesel oil BPNN
Park and Han [13] 2000 Toluene composition Multivariate locally weighted regression
Bhartiya and Whiteley [14] 2001 ASTM 95% end point of kerosene BPNN
Fortuna et al. [15] 2003 Hydrogen sulfide and sulfur dioxide in the tail stream of the sulfur recovery unit BPNN and RBFNN
Yan et al. [16] 2004 Freezing point of light diesel oil Standard SVR and LSSVR
Dam and Saraf [17] 2006 Specific gravity, flash point and ASTM temperature of crude fractionator products BPNN
Yan [18] 2008 Naphtha 95% cut point Ridge regression
Kaneko et al. [19] 2009 Distillation unit bottom product composition PLS
Wang et al. [20] 2010 ASTM 90% distillation temperature of the distillate Dynamic PLS
Ge and Song [21] 2010 Hydrogen sulfide and sulfur dioxide in the tail stream of the sulfur recovery unit Relevance vector machine
Soft sensors reported for the debutanizer column
Fortuna et al. [22] 2005 Butane (C4) content in the bottom flow of a debutanizer column BPNN
Ge and Song [21] 2010 Butane (C4) content in the bottom flow of a debutanizer column PLS, standard SVR, LSSVR
Ge [23] 2014 Butane (C4) content in the bottom flow of a debutanizer column PCR
Ge et al. [24] 2014 Butane (C4) content in the bottom flow of a debutanizer column Non-linear semi supervised PCR
Ramli et al. [25] 2014 Top and bottom product composition ANN

A debutanizer column is used to separate the light gases and LPG consisting of mainly butane (C4), from the overhead distillate coming from the distillation (and/or cracking) unit. The control of product quality in a debutanizer column is a difficult problem because of lack of real-time monitoring system for the product quality, process non-linearity and multivariate nature of the process [25]. Fortuna et al. [22] proposed back propagation neural network (BPNN) model of a debutanizer column for predicting the bottom product composition. Ge and Song [21] have reported partial least square (PLS), support vector regression (SVR) and least square support vector regression (LSSVR) soft sensor models and Ge et al. [24] proposed non-linear semi supervised principal component regression (PCR) model of the debutanizer column for prediction of the same process variable.

An important issue in the development of data-driven soft sensor is the design of training set for model development. It has been reported by Pani and Mohanta [26] how a proper design of training set can result in significant improvement in model’s prediction performance. However, till date most of the data driven soft sensors reported in the field of petroleum refinery or other industries are based on randomly constructing the training set from the total data.

In this work, we attempt to address the issue of difficulty in real-time monitoring of the product quality by developing an inferential sensing system for the debutanizer column. Data driven soft sensors are developed for prediction of butane content of the debutanizer column bottom product. The input–output data set for the debutanizer process was obtained from the website which has been shared by Fortuna et al. [27]. This benchmark data set has been used before by Ge and Song [21] and Ge et al., [24] for development of least square support vector regression and principal component regression models for the debutanizer column. Here we report the development of statistical regression and back propagation neural network models of the debutanizer column. The performances of the developed models were assessed by simulation with the validation dataset. From the simulation results, statistical model evaluation parameters, mean absolute error (MAE), root mean squared error (RMSE) and correlation coefficient (R) values were computed. Results indicate that the back propagation neural network model trained by Levenberg–Marquardt algorithm reported in this article performs better than the support vector regression models reported earlier in the literature.

The article is organized as follows. Section 2 gives a brief description of the debutanizer column along with the associated input–output process variables. The procedure for model development is presented in Section 3 followed by analysis and discussion of the simulation results in Section 4. Finally, concluding remarks are presented in Section 5.

## 2. Process description

A debutanizer column is a part of several processing units in a refinery. Wherever there is production of LPG and gasoline e.g. in the atmospheric and vacuum crude distillation unit, cracking and coking units, the debutanizer column is used to remove the lighter fractions from gasoline. The feed to the debutanizer column is the unstabilized naphtha and the products coming out from the column are LPG as the top product and gasoline/stabilized naphtha as the bottom product. The schematic process diagram is shown in Fig. 1.

 Figure 1. Schematic diagram of the debutanizer column.

For improved process performance, the butane (C4) content in the bottom product should be minimized. This requires continuous monitoring of C4 content in the bottom product. A gas chromatograph is used in the process for this purpose. However, the hardware sensor (gas chromatograph) is not installed in the bottom flow line coming from the debutanizer column and instead is located in the overhead of the deisopentanizer column which is located some distance away from the debutanizer column. This introduces a time delay in measurement which is of the order of 30–75 min [22]. Therefore, a soft sensor can be used in the bottom flow of the debutanizer column to overcome the time delay problem of the hardware sensor. The output to be predicted by the soft sensor model is the C4 content present in the debutanizer column bottom product. This output depends on seven process inputs as has been reported in the literature [21] and [22]. The seven process inputs and the output quality variable to be estimated by the soft sensor are mentioned in Table 2. The location of sensors for the seven process inputs, the gas chromatograph used for C4 content measurement and the proposed soft sensor are shown in Fig. 1.

Table 2. Input–Output Process Variables for the debutanizer column.
Variables Description
Inputs x1 Top temperature
x2 Top pressure
x3 Reflux flow
x4 Flow to next process
x5 6th tray temperature
x6 Bottom temperature
x7 Bottom temperature
Output y Butane (C4) content in the debutanizer column bottom

## 3. Model development

A total of 2394 input–output process data values were available for the debutanizer column. This data set, taken from a petroleum refinery is shared by Fortuna et al. [27]. Interested researchers can access the data from the web resource. The available dataset was equally divided into a training set (used for model development) and a validation set (for model evaluation). Each data subset has 1197 input–output data values. From the total data, the training set was obtained by applying the Kennard–Stone algorithm. The required MATLAB code for implementation of the algorithm was adopted from the freely available TOMCAT toolbox [28]. This training set was subsequently used for development of statistical (multiple linear and principal component) and neural network models.

In multiple linear regression (MLR) model, the output is expressed as a linear combination of the inputs. The MLR model for the debutanizer column has the following form:

 ${\displaystyle y={\beta }_{0}+{\beta }_{1}x_{1}+{\beta }_{2}x_{2}+{\beta }_{3}x_{3}+}$${\displaystyle {\beta }_{4}x_{4}+{\beta }_{5}x_{5}+{\beta }_{6}x_{6}+{\beta }_{7}x_{7}}$
(1)

Here, ${\textstyle {\beta }_{0}\ldots {\beta }_{7}}$ are regression coefficients, ${\textstyle x_{1}\ldots x_{7}}$ are process inputs as mentioned in Table 2 and y is the process output i.e. C4 content in the debutanizer column bottom product. The regression coefficients of the above model are determined using the least of squared error criterion as per the equation given below:

 ${\displaystyle \beta ={\left(X^{T}X\right)}^{-1}X^{T}Y}$
(2)

Here, X is the 1197 × 7 input data matrix, Y is the 1197 × 1 output column vector and β is 7 × 1 column vector consisting of the regression coefficients.

For development of principal component regression (PCR) model, initially principal component analysis (PCA) was conducted on the total input data. The principal components are found by calculating the eigenvectors and eigen values of the data covariance matrix. Subsequently, using cumulative variance criterion principal components or latent variables were selected which are the linear combinations of the actual variables. Least square regression model was developed as mentioned earlier using the latent variables as inputs and the output. The sequence of steps for developing a PCR model from the input–output data set is presented in Fig. 2.

 Figure 2. Procedure for development of PCR model for any process.

In addition to the MLR and PCR models, back propagation neural network (BPNN) model of the debutanizer column was developed. In a BPNN model, the number of input and output nodes is decided based on process conditions. For the debutanizer column model, the number of input nodes is seven and output node is one. The crucial design step is to optimally determine the number of hidden layer neurons. The activation functions used for the hidden layer and the output layer are, hyperbolic tangent and linear respectively. For deciding optimum number of neurons in hidden layer, the network was initially trained from 3 neurons in hidden layer to 40 neurons in hidden layer using gradient descent training algorithm. The optimum number of neurons was decided as the one which produced lowest error value for the validation data. Subsequently, feed-forward neural networks were created with this optimum number of neurons and trained using three training algorithms. The training algorithms used are as follows: gradient descent, conjugate gradient and Levenberg–Marquardt techniques. The optimum model was the one that produced the lowest error for the validation data i.e. the model with the best generalization capability.

## 4. Results and discussion

The linear regression model developed is as follows:

 ${\displaystyle y=0.64+0.422x_{1}-0.402x_{2}-0.134x_{3}+0.238x_{4}-0.481x_{5}-}$${\displaystyle 0.391x_{6}+0.521x_{7}}$
(3)

The number of principal components was determined by analyzing the variance accounted for by the individual principal components. The results of the principal component analysis for the debutanizer column data are reported in Table 3 and the SCREE plot showing the eigenvalue versus principal components is presented in Fig. 3.

Table 3. PCA results of the input data.
Principal components Eigen value Percentage of eigenvalue Cumulative percentage
PC1 2.8723 41.0329 41.0329
PC2 1.4633 20.9039 61.9368
PC3 1.1421 16.3151 78.2519
PC4 0.7441 10.6300 88.8820
PC5 0.5565 7.9496 96.8316
PC6 0.2130 3.0428 99.8744
PC7 0.0088 0.1256 100.0000
Total 7 100

 Figure 3. SCREE plot for the PCA conducted on the debutanizer column data.

In Table 3, each of the seven principal components is a linear combination of the seven actual input process variables. The usual practice is to retain the number of PCs which have a cumulative variance of more than 70% of the actual total data variance. Therefore, based on the results reported in Table 3 and Fig. 3, five principal components accounting for more than 95% of the total data variance were used for development of PCR model. The principal component regression model obtained has the following form:

 ${\displaystyle y=1.32\times {10}^{-5}-0.00059{\mbox{PC}}_{1}-0.03076{\mbox{PC}}_{2}-}$${\displaystyle 0.02848{\mbox{PC}}_{3}-0.02404{\mbox{PC}}_{4}-0.06074{\mbox{PC}}_{5}}$
(4)

As stated earlier, the optimum number of hidden layer neurons in BPNN model was decided by creating models with different number of neurons, training with gradient descent algorithm and simulating the trained networks with the validation data. The results are reported in terms of mean absolute error (MAE) for the training data and validation data in Fig. 4.

 Figure 4. MAE for different no. of neurons in hidden layer.

From Fig. 4, it is evident that both testing and training error increases if the number of neurons is increased beyond 30. Therefore optimum number of neurons was decided as 30 with hyperbolic tangent activation function. Keeping this optimum number of neurons fixed two more neural networks were trained using conjugate gradient and Levenberg–Marquardt algorithms. The neural network model was designed using mean absolute error (MAE) as the performance criterion. MAE in this work was determined using the formula given below:

 ${\displaystyle {\mbox{MAE}}={\frac {{\sum }_{i=1}^{N}\left|y_{i}-{\overset {\mbox{ˆ}}{y}}_{i}\right|}{N}}}$
(5)

Here yi   and ${\textstyle {\overset {\mbox{ˆ}}{y}}_{i}}$ are actual and model predicted values for the ith observation and N is the number of observations.

A model showing better value of one model evaluation parameter may produce worse value of another model evaluation parameter. Therefore, to choose the best model, in addition to MAE, the parameters root mean squared error (RMSE) and correlation coefficient (R) values were computed for all the developed models. The expressions for RMSE and R are presented below:

 ${\displaystyle {\mbox{RMSE}}={\sqrt {\frac {{\sum }_{i=1}^{N}{\left(y-{\overset {\mbox{ˆ}}{y}}\right)}^{2}}{N}}}}$
(6)
 ${\displaystyle R=\left({\frac {\sum (y-{\overline {y}})({\overset {\mbox{ˆ}}{y}}-{\overline {\overset {\mbox{ˆ}}{y}}})}{\sqrt {\sum {\left(y-{\overline {y}}\right)}^{2}\sum {\left({\overset {\mbox{ˆ}}{y}}-{\overline {\overset {\mbox{ˆ}}{y}}}\right)}^{2}}}}\right)}$
(7)

The performances of the multiple linear regression (MLR) model, principal component regression (PCR) model and the three neural network models (trained by gradient descent, conjugate gradient and Levenberg–Marquardt algorithms) for training and validation data are reported in Tables 4 and 5 respectively.

Table 4. Debutanizer column model performance for training data.
Model type Statistical model evaluation parameter
Mean absolute error (MAE) Root mean squared error (RMSE) Correlation coefficient (R)
Multiple linear regression (MLR) 0.994 1.007 0.313
Principal component regression (PCR) 0.171 0.24 0.015
Back propagation neural network (BPNN) trained by Gradient-descent 0.094 0.144 0.537
Conjugate-gradient 0.066 0.112 0.757
Levenberg–Marquardt 0.046 0.064 0.925

Table 5. Debutanizer column model performance for validation data.
Model type Statistical model evaluation parameter
Mean absolute error (MAE) Root mean squared error (RMSE) Correlation coefficient (R)
Models reported in this work Multiple linear regression (MLR) 0.989 0.999 0.395
Principal component regression (PCR) 0.105 0.1511 0.148
Back propagation neural network (BPNN) trained by Gradient-descent 0.081 0.125 0.553
Conjugate-gradient 0.069 0.111 0.664
Levenberg–Marquardt 0.055 0.076 0.856
Models reported by Ge and Song [21] LSSVR Not reported 0.1418 0.9132
SVR Not reported 0.145 0.6897
PLS Not reported 0.165 0.4035
Model reported by Ge et al. [24] Non-linear semi supervised PCR Not reported 0.1499 Not reported

MAE values are reported because it has been mentioned in the literature that MAE is a better model evaluation parameter as compared to other statistical parameters [29]. Comparison of the MAE values of different models shows that the BPNN model trained using Levenberg–Marquardt algorithm, clearly outperforms all other models. The accuracy of the BPNN model is also quite satisfactory. The average error produced by the model is 4.6% for the training data and 5.5% for the validation data.

The purpose of reporting the other two statistical parameters is for the sake of comparison with values reported in the literature. Usually, comparing the performances of different data driven models is difficult, because the data used are different for different models reported in the literature. However, in this research, use of the same benchmark data set for the debutanizer column soft sensor development by various researchers offers scope for comparison of the different model performances. In the last two rows of Table 5 the different statistical values reported by Ge and Song [21] and Ge et al. [24] are reproduced. The RMSE values mentioned for the model reported in [21] are approximate values since the values were available only in the form of a chart. It may be noted that the R value of the LSSVR model is slightly better than the R value of the best model reported in this work. However the present BPNN model is superior to the LSSVR and PCR models reported earlier in terms of the RMSE value.

Finally in Figure 5 and Figure 6 the prediction results of the present BPNN model are reported for the training and validation data set respectively.

 Figure 5. Prediction of C4 content in the debutanizer bottom product (training data).

 Figure 6. Prediction of C4 content in the debutanizer bottom product (validation data).

The exact neural network model developed in this work is represented in Fig. 7. The model receives information of seven process variables as inputs and produces the estimated value of C4 content in the debutanizer column bottom product.

 Figure 7. Feed forward neural network model of the debutanizer column.

For the sake of verification of the reported results and further research, the architecture of the present optimized neural network model, weight values for the hidden and outer layers and the input–output data used for model development and validation are supplied as supplementary materials.

The architecture of the neural network model is as follows: seven input nodes, thirty hidden layer nodes and 1 output node. Activation functions used are hyperbolic sigmoidal in hidden layer neurons and linear in output neuron. The weights associated with the neural network structure are given below:

• Input to hidden layer: 7 × 30 weight matrix.
• Bias values to hidden layer neurons: 30 × 1 weight vector.
• Hidden layer to output layer: 1 × 30 weight vector.
• Bias value to output layer neuron: 1 × 1 (single scaler value).

All weight values of the trained neural network model are provided as supplementary materials. Interested readers can download the process data supplied by Fortuna et al. [27] from the website, create the feed forward neural network model with the structure and weight values supplied in this article and verify the results. They are also encouraged to bring further improvements in the modeling results.

## 5. Conclusion

For optimum process performance the content of butane in the debutanizer column bottom product should be limited to a minimum. This control of product quality requires online monitoring of the product composition. However, the hardware sensor (gas chromatograph) used for composition monitoring is located some distance far from the column and hence introduces significant time delay in the monitored value. Therefore, a soft sensor installed in the product outlet line can be effectively used for continuous monitoring and control of product quality. A back propagation neural network based soft sensor trained with Levenberg–Marquardt algorithm is reported in this article. Simulation study shows that the reported soft sensor performs better than the multiple linear regression and principal component regression models reported in this work and the earlier reported models of least square support vector regression and non-linear semi supervised principal component regression models reported earlier in the literature.

## Appendix A. Supplementary material

Supplementary data 1.

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# Air Quality - Lesson 5: How Can We Tell What the Quality of the Air Is Today?

Background Information

Air pollution can be reported in several ways: concentration (amount of material in a volume of air in ppm or ppb), mass (weight of material in air in micrograms per cubic meter), or an Air Quality Index (AQI). The AQI is an index from 0 to 500 for reporting daily air quality based on human health considerations.

The U.S. EPA calculates the AQI for five major air pollutants regulated by the Clean Air Act: ground level ozone, particle pollution (also known as particulate matter), carbon monoxide, sulfur dioxide, and nitrogen dioxide. For each of these pollutants, EPA has established National Ambient Air Quality Standards (NAAQS) to protect public health (see teacher resource on National Ambient Air Quality Standards.) The AQI is not a “standard,” but it is related to air quality standards. Note that the AQI is calculated hourly and is derived from data that have not been quality assured.

An AQI value of 100 generally corresponds to the national air quality standard for the pollutant, which is the level EPA has set to protect public health. AQI values below 100 are generally thought of as satisfactory. When AQI values are above 100, air quality is considered to be unhealthy—at first for certain sensitive groups of people, then for everyone as AQI values get higher.

To make it easier to understand, the AQI is divided into six categories and a specific color is assigned to each AQI category. For example, the color orange means that conditions are “unhealthy for sensitive groups,” while red means that conditions may be “unhealthy for everyone,” and so on. There are different health messages for each of the AQI pollutants.

The six levels of health concern and what they mean are:

• "Good" (Green): AGI between 0 and 50. Air quality is considered satisfactory, and air pollution poses little or no risk.

• "Moderate" (Yellow): AQI between 51 and 100. Air quality is acceptable: however, for some pollutants there may be a moderate health concern for a very small number of people (extremely sensitive people).

• "Unhealthy for Sensitive Groups" (Orange): AQI between 101 and 150. These groups include people with asthma, children and teens, elderly, and people who work or exercise outside strenuously. People with lung disease are at greater risk from exposure to ozone, while people with either lung disease or heart disease are at greater risk from exposure to particle pollution. The general public is not likely to be affected when the AQI is in this range.

• "Unhealthy" (Red): AQI between 151 and 200. The air is unhealthy for everyone and members of sensitive groups may experience more serious health effects.

• "Very Unhealthy" (Purple): AQI between 201 and 300. Everyone should reduce and sensitive groups should avoid prolonged or heavy outdoor exertion.

• "Hazardous" (Maroon): AQI over 300. Everyone should avoid all outdoor exertion.

In Michigan, ozone and fine particles are usually the pollutants that cause the AQI to be elevated. Newspapers, radio, television, and web sites report the daily AQI levels. The daily AQI for Michigan is posted on the MDEQ web site at http://www. deqmiair.org. That site has specific information about the actual values of pollutants at air monitoring stations and details of the air quality forecast. Maps from the U.S. EPA’s AIRNow site are particularly useful for identifying ozone patterns. These maps and AQIs for the whole country can be found at http://www.airnow.gov.

MDEQ meteorologists predict when conditions are favorable for excessive fine particle or ozone formation. Weather conditions conducive to ozone formation are high temperatures, high pressure, clear skies, and no wind. Low-pressure systems tend to dissipate the ozone. Inversions (atmospheric temperature increase with height) can trap pollutants, compounding the problem. High pollutant concentrations upwind can be transported into a different area and can cause substantial increases in air pollution that would not have otherwise occurred. An Action Day is called when weather conditions are likely to combine with pollution to create elevated amounts of ground-level ozone.

The Southeast Michigan Council of Governments and the Clean Air Coalition of Southeast Michigan (http://www.semcog.org) as well as the West Michigan Clean Air Coalition (http://www.wmcac. org/) operate local action programs to disseminate the air quality forecast message to communities. They encourage individuals and companies to make clean air choice on Action Days. Ozone Action and Clean Air Action are voluntary emissions reduction initiatives. These organizations provide teacher resources and tip cards for appropriate actions when DEQ meteorologists announce Action Days.

The UV Index is a next-day forecast of the amount of skin-damaging UV radiation expected to reach the Earth’s surface at the time when the sun is highest in the sky (solar noon). The amount of UV radiation reaching the surface is primarily related to the elevation of the sun in the sky, the amount of ozone in the stratosphere, and the amount of clouds present. The UV index is a good way to link stratospheric ozone depletion with human health effects. The Index predicts UV intensity levels on a scale of 1 to 11+, where low indicates a minimal risk of overexposure and 11+ means an extreme risk. Calculated on a next-day basis for every zip code across the United States, the UV Index takes into account clouds and other local conditions that affect the amount of UV radiation reaching the ground in different parts of the country.

The UV index can be found as a link from http://www.epa.gov/sunwise/uvindex.html

sources: Michigan Department of Environmental Quality. Air Quality Division. Michigan's Air. Retrieved April 1, 2005, from http://www.michigan.gove/deqair/

U.S. Environmental Protection Agency. AIRNow. Retrieved April 1, 2005 from http://www.epq.gove/airnow

U.S. Environmental Protection Agency. What Is the UV Index? Retrieved April 1, 2005, from http://www.epa.gove/sunwise/uvwhat.html

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text:
Owen wrote:

> Puzzle 124: I don't think there's just one functor from \$$\mathcal{D}\$$ to \$$\mathcal{C}\$$. There is just one graph homomorphism, though.

You're right!

I'll fix the puzzle by indicating the functor I'm actually interested in: the one that maps the morphisms \$$\textrm{FriendOf}\$$ and \$$\textrm{FriendOf}^\prime\$$ in \$$\mathcal{D}\$$ to the morphism \$$\textrm{FriendOf}\$$ in \$$\mathcal{C}\$$.

> (Feel free to delete this paragraph after reading it.)

I'll leave it in! I don't want to convey the false impression that I'm infallible. I do, however, want to fix up my "lectures" so that students don't get confused by mistakes.

Anindya wrote:

> Could we not send one or both FriendOf in \$$\mathcal{D}\$$ to the identity map in \$$\mathcal{C}\$$?

Yes, that's one of infinitely many other functors from \$$\mathcal{D}\$$ to \$$\mathcal{C}\$$ I hadn't noticed when first stating Puzzle 124.


text:
## A Benders decomposition based framework for solving cable trench problems.(English)Zbl 1391.90505

Summary: In this work, we present an algorithmic framework based on Benders decomposition for the capacitated $$p$$-cable trench problem with covering. We show that our approach can be applied to most variants of the cable trench problem (CTP) that have been considered in the literature. The proposed algorithm is augmented with a stabilization procedure to accelerate the convergence of the cut loop and with a primal heuristic to derive high-quality primal solutions. Three different variants of the CTP are considered in a computational study which compares the Benders approach with two compact integer linear programming formulations that are solved with CPLEX. The obtained results show that the proposed algorithm significantly outperforms the two compact models and that it can be used to tackle significantly larger instances than previously considered algorithms based on Lagrangean relaxation.

### MSC:

 90C27 Combinatorial optimization 90C35 Programming involving graphs or networks 90C11 Mixed integer programming 90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut 90B10 Deterministic network models in operations research 90B80 Discrete location and assignment

### Software:

OR-Library; OGDF; CPLEX; Algorithm 97
Full Text:

### References:

 [1] Discrete location problems: benchmark library. http://www.math.nsc.ru/AP/benchmarks/english.html; Accessed: 2015-07-14. [2] Beasley, J., OR-library: distributing test problems by electronic mail, J Oper Res Soc, 41, 11, 1069-1072, (1990) [3] Ben-Ameur, W.; Neto, J., Acceleration of cutting-plane and column generation algorithms: applications to network design, Networks, 49, 1, 3-17, (2007) · Zbl 1131.90047 [4] Benders, J., Partitioning procedures for solving mixed-variables programming problems, Numerische Mathematik, 4, 1, 238-252, (1962) · Zbl 0109.38302 [5] Cherkassky, B.; Goldberg, A., On implementing push-relabel method for the maximum flow problem, (Balas, E.; Clausen, J., Proceedings of IPCO IV, LNCS, 920, (1995), Springer), 157-171 [6] Chimani, M.; Gutwenger, C.; Jünger, M.; Klau, G.; Klein, K.; Mutzel, P., The open graph drawing framework (OGDF), Handbook of graph drawing and visualization, 543-569, (2011) [7] Dijkstra, E., A note on two problems in connexion with graphs, Numerische Mathematik, 1, 1, 269-271, (1959) · Zbl 0092.16002 [8] Fischetti, M.; Ljubić, I.; Sinnl, M., Redesigning benders decomposition for large scale facility location, Manage Sci, (2016) [9] Floyd, R., Algorithm 97: shortest path, Commun ACM, 5, 6, (1962) [10] Gouveia, L., A comparison of directed formulations for the capacitated minimal spanning tree, Telecommun Syst, 1, 51-76, (1993) [11] Hakimi, S., Optimum distribution of switching centers in a communication network and some related graph theoretic problems, Oper Res, 13, 3, 462-475, (1965) · Zbl 0135.20501 [12] Jiang, Y.; Zhuang, Z.; Sinusas, A.; Papademetris, X., Vascular tree reconstruction by minimizing a physiological functional cost, 2010 IEEE Computer society conference on computer vision and pattern recognition-workshops, 178-185, (2010), IEEE [13] Jiang, Y.; Zhuang, Z.; Sinusas, A.; Staib, L.; Papademetris, X., Vessel connectivity using murray’s hypothesis, International conference on medical image computing and computer-assisted intervention, 528-536, (2011), Springer [14] Lalla-Ruiz, E.; Schwarze, S.; Voß, S., A matheuristic approach for the p-cable trench problem, Learning and intelligent optimization: 10th international conference, LION 10, 247-252, (2016) [15] Lorena, L.; Senne, E., A column generation approach to capacitated p-Median problems, Comput Oper Res, 31, 6, 863-876, (2004) · Zbl 1048.90132 [16] Marianov, V.; Gutiérrez-Jarpa, G.; Obreque, C.; Cornejo, O., Lagrangean relaxation heuristics for the p-cable-trench problem, Comput Oper Res, 39, 3, 620-628, (2012) · Zbl 1251.90248 [17] Marianov, V.; Gutiérrez-Jarpa, G.; Obreque, C., p-cable trench problem with covering, XXII EURO working group on locational analysis meeting, 75-76, (2015) [18] Minieka, E., The m-center problem, SIAM Rev, 12, 1, 138-139, (1970) · Zbl 0193.24204 [19] Schwarze, S., The multi-commodity cable trench problem, ECIS 2015 Completed research papers, (2015) [20] Vasko, F.; Barbieri, R.; Rieksts, B.; Reitmeyer, K.; Stott, K., The cable trench problem: combining the shortest path and minimum spanning tree problems, Comput Oper Res, 29, 5, 441-458, (2002) · Zbl 0994.90124 [21] Vasko, F.; Landquist, E.; Kresge, G.; Tal, A.; Jiang, Y.; Papademetris, X., A simple and efficient strategy for solving very large-scale generalized cable-trench problems, Networks, 67, 3, 199-208, (2016)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
# IQ and GDP

russ_watters said:
the key to American economic success: optomism. The American economy is strong because optomism is part of American culture.
Lynn and Vanhanen presented the alternative case that that variances between nations in economic success can be positively explained by variances between nations in:

1. National IQ
2. Economic liberality
and,
3. Political liberality
...in that order:

• We also measured the significance of economic and political systems using data on economic freedom ratings (1997) and an index of democratization (1998). The results of multiple correlation analyses show that these two additional variables explain 10-12 percentage points of the variation in the measures of per capita income independently from national IQs. Thus, when the three explanatory variables are taken together, the explained part of variation in the measures of per capita income increases to 52-65 percent and less than half of the variation remains unexplained. Our conclusion is that differences in national intelligence provide the most powerful and fundamental explanation for the gap between rich and poor economies.
(Richard Lynn and Tatu Vanhanen. IQ and the Wealth of Nations. p158.)

Lynn and Vanhanen presented the alternative case that that variances between nations in economic success can be positively explained by variances between nations in:

1. National IQ
2. Economic liberality
and,
3. Political liberality
...in that order:

• We also measured the significance of economic and political systems using data on economic freedom ratings (1997) and an index of democratization (1998). The results of multiple correlation analyses show that these two additional variables explain 10-12 percentage points of the variation in the measures of per capita income independently from national IQs. Thus, when the three explanatory variables are taken together, the explained part of variation in the measures of per capita income increases to 52-65 percent and less than half of the variation remains unexplained. Our conclusion is that differences in national intelligence provide the most powerful and fundamental explanation for the gap between rich and poor economies.
(Richard Lynn and Tatu Vanhanen. IQ and the Wealth of Nations. p158.)
That has been heavily criticized.
Recently Richard Lynn and Tatu Vanhanen have presented evidence that differences in national IQ account for the substantial variation in national per capita income and growth. This paper challenges these findings and claims that, firstly, they simply reflect inappropriate use and interpretations of statistical instruments. Secondly, it is argued that the models presented by Lynn and Vanhanen are under-complex and inadequately specified. More precisely the authors confuse IQ with human capital. The paper concludes that once control variables are introduced and the models are adequately specified, neither an impact of IQ on income nor on growth can be substantiated
http://www.suz.unizh.ch/volken/ThomasVolken/pdfs/IQWealthNation.pdf.

But there is at least one other paper claiming a link between IQ and wealth, assuming the national IQ data from the Lynn and Vanhanen study but with different statistics. But they do not claim that it accounts for more than 50% of the variation.
www.siue.edu/~garjone/JonesSchneApr.pdf[/URL]

The claims are in all cases dependent on that the IQ national data presented by Lynn and Vanhanen are correct. And also their methods for doing that has also been criticzed, for example for using small, old and unrepresentative samples from different times using different tests.

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russ_watters
Mentor
Tom McCurdy said:
Whose ecomony currently is the strongest in the world?
It depends on how you measure it (absolute or per capita, large vs fast growing), but the US is largest in absolute terms and Luxemburg is largest per capita. http://www.worldfactsandfigures.com/gdp_country_desc.php [Broken]

Growth rate is a good indicator of momentum and thus strength (though size is more often used), but can be clouded by small countries having massive short term increases. HERE is some data on that. The most noteable country on the list is China, which has maintained ~8% for some time and is expected to maintain it for some time to come. (note: that data is for one year only, and there can be large year-to-year fluctuations for any country).

Aquamarine - when you see someone puke in the supermarket, report it to someone who works there and a janitor will clean it up. Don't try to do anything about it yourself.

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Volken's phunny phactor rotations, revisited

Aquamarine said:
Lynn and Vanhanen presented the alternative case that that variances between nations in economic success can be positively explained by variances between nations in:
1. National IQ
2. Economic liberality
and,
3. Political liberality
...in that order
That has been heavily criticized.
http://www.suz.unizh.ch/volken/ThomasVolken/pdfs/IQWealthNation.pdf [Broken]

Volken says, "...the authors [Lynn and Vanhanen] confuse IQ with human capital." As I pointed out in the Volken thread, that is a criticism of Arthur Jensen's The g Factor (1998), not of Lynn and Vanhanen. If Volken has an issue with IQ as a major factor in human capital, his issue is with Arthur Jensen's work, and therefore that is where he should be directing his investigations.

The claims are in all cases dependent
That is because science and dogmatic religion are different things. Science is contingent upon empirical data and dogmatic religion is not. Please see Raymond Cattell's A New Morality from Science for a full discussion of that:
http://www.efn.org/~callen/ToC.htm

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Nereid
Staff Emeritus
Gold Member
Two interesting variations: if you print US$(instead of your own currency, in Tom's new country), then could you use these fake notes to buy goods to bring back to your country? Have you then created new wealth out of nothing? (Hint: try imagining that there's a country setatS detinU, as big as the US, and it does this forgery on a massive scale; oh, and that the forged notes are so good no one can tell them from the real). What if Tom's new country were completely autarkic (absolutely no trade with the outside world)? Oh, and a comment on the important of perception and trust ... the value of your new currency Tom, in$US, has a great deal to do with your national accounts - trade balance, domestic interest rates, capital account balance, ... - all of which are more or less open to determination by really smart people with economics degrees. Then there are currency speculators who read the reports of those really smart people, and are prepared to bet billions (even trillions) on your currency, if they think they can make a profit ... the more out of whack your currency is with the economic fundamentals, the greater the chances are that these speculators will start taking positions ... the more liquid the market for your currency is, the more certain this will happen quickly.

You might be interested to know that one suggestion for Japan to pull itself out of its decade-long doldrums was 'print lots of money' ... the thinking being that the resulting inflation would somehow result in the economic engine revving up.

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Nereid
Staff Emeritus
Gold Member
Oh dear, back to Lynn and Vanhannen eh?

For those new to this, we had a very extensive discussion of this infamous book some time ago, here on PF (I'll dig up the URL if anyone's interested).

That there is a correlation between the IQ data L+V present and three economic indicators is undeniable; whether national IQ is a (leading) cause of economic success (as claimed by L+V) is a very different kettle of fish.

First, as Aquamarine said, L+V's data is questionable, to say the least.

Next, L+V's case for a causal connection (IQ -> success) is deeply flawed, even assuming the data they use are correct.

Finally, it's quite easy to find plenty of indicators which correlate more strongly with the three 'success metrics' used by L+V than 'national IQ' - an urban/rural measure of where people live, for example (I'm not sure anyone is suggesting that city living causes economic success!).

loseyourname
Staff Emeritus
Gold Member
Seems to me that a wealthy nation can better educate its populace, thus increasing the IQ. I wouldn't doubt that smarter people are theoretically capable of creating more wealth, but how often does this actually happen? Many of the most intelligent people out there are researchers or professors making next to nothing. Conversely, quite a few wealthy people either were extremely driven or just got lucky, not necessarily a whole lot more intelligent than the average person. Why don't we do an accounting of the personal wealth of posters on this site? Despite our obviously above-average level of intelligence, I'm willing to bet we aren't particularly wealthy.

Volken's individuals-groups confusion revisited

loseyourname said:
I wouldn't doubt that smarter people are theoretically capable of creating more wealth, but how often does this actually happen? Many of the most intelligent people out there are researchers or professors making next to nothing.
Individuals are not groups. One of Volken's mistakes, when he was criticizing Lynn & Vanhanen, was to confuse individual success with group success. And, again, this is old ground that was covered in the Volken thread:

• In addition, nations whose populations have high IQs will have intelligent and efficient personnel in services and public sector employment that contributes indirectly to the strength of the economy such as teachers, doctors, scientists and a variety of public servants responsible for the running of telephones, railroads, electricity supplies and other public utilities....

http://www.rlynn.co.uk/pages/article_intelligence/6.htm [Broken]

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Nereid
Staff Emeritus
Gold Member
The really interesting thing that's not included - well, one of them - is 'mobility of labour'. If trained nurses and doctors can move to the EU, and get paid far, far more than they would in their home countries, how does this affect the 'national IQ' argument? If a significant part of the 'public sector' can be effectively run by a few, highly educated folk, how does this impact Lynn's naive views (think the telecoms sector ... how many telecoms engineers and managers do you need to run an efficient, modern, wireless-based network? Answer: less than 100, even for a country as big as South Africa - don't forget that most of the workforce in telecoms and electricity are engaged in pretty mundane, routine activities).

One of the 'tragedies in waiting' for the US is the dramatic slowdown of intake of science and engineering students from other countries - the US's (former) ability to attract huge numbers of top class scientists contributed in no small way to its present economic strength. Their contribution to the mythical 'national IQ' was minimal; their contribution to the US' per capita GDP and growth enormous.

loseyourname
Staff Emeritus
Gold Member
Individuals are not groups. One of Volken's mistakes, when he was criticizing Lynn & Vanhanen, was to confuse individual success with group success. And, again, this is old ground that was covered in the Volken thread:

Most wealth is held and created by individuals, though, not by groups. Furthermore, there are plenty of individuals that do not even consider personal wealth to be a good measure of their success, and many of these individuals are quite intelligent.

In addition, nations whose populations have high IQs will have intelligent and efficient personnel in services and public sector employment that contributes indirectly to the strength of the economy such as teachers, doctors, scientists and a variety of public servants responsible for the running of telephones, railroads, electricity supplies and other public utilities....

Since when is there a correlation between IQ and efficiency? My IQ is way higher than my fathers, but I could never do his job as efficiently as he does it, simply because I am not as physically capable. This is the case with many public service jobs that require physical labor. In the case of non-physical jobs, I would think that organizational capability, as well as the ability to deal well with stress and multitasking, contribute as much or even more to job efficiency than does IQ.

Just take the most prominent and important job in this country: that of the president. Most scholars would probably argue that Lincoln or perhaps Washington was the most effective president we've had, but Wilson was almost certainly the most intelligent. Furthermore, if we look at some of the people in history with the highest IQ's (Goethe, Mozart, Marilyn vos Savant), none of them were particularly wealthy, nor did they seem to have any desire to be.

loseyourname
Staff Emeritus
Gold Member
Nereid said:
The really interesting thing that's not included - well, one of them - is 'mobility of labour'. If trained nurses and doctors can move to the EU, and get paid far, far more than they would in their home countries, how does this affect the 'national IQ' argument? If a significant part of the 'public sector' can be effectively run by a few, highly educated folk, how does this impact Lynn's naive views (think the telecoms sector ... how many telecoms engineers and managers do you need to run an efficient, modern, wireless-based network? Answer: less than 100, even for a country as big as South Africa - don't forget that most of the workforce in telecoms and electricity are engaged in pretty mundane, routine activities).

You know, I didn't even consider it, but you just made me realize that many remote service jobs are now held by workers overseas. Take, for instance, financial managers living in India. They do a great deal to increase the wealth of the US, but they certainly do nothing to contribute to the national IQ.

Nereid
Staff Emeritus
Gold Member
loseyourname said:
You know, I didn't even consider it, but you just made me realize that many remote service jobs are now held by workers overseas. Take, for instance, financial managers living in India. They do a great deal to increase the wealth of the US, but they certainly do nothing to contribute to the national IQ.
You've introduced another dimension which Lynn is apparently wholly blind to ... the economic sector composition of a country. For example, the correlation between sector mix and 'national IQ' is truly amazing (countries with low 'national IQs' tend to have most of their workforce engaged in agriculture etc - the 'primary' sector; those with high, mostly in services - the tertiary sector). Lynn et al would have you believe that the causation is IQ -> economic sector. To see how nonsensical this is, consider (as you have hinted) India and Ireland, the Flynn effect, and the economic sector distributions of countries 50 and 100 years ago. If you go back even further, you find that most folk lived in rural areas, and were engaged in agriculture (and other primary sector activities) - yet a mere instant (in terms of genetic composition of populations) later the descendants of almost all those folk live in cities and are engaged in services!

Goethe and the wealth of nations

loseyourname said:
services and public sector employment that contributes indirectly to the strength of the economy such as teachers, doctors, scientists
Most wealth is held and created by individuals, though....

...if we look at some of the people in history with the highest IQ's (Goethe, Mozart, Marilyn vos Savant), none of them were particularly wealthy, nor did they seem to have any desire to be.
And the point was that the creators of wealth in a society do not necessarily enjoy proportional ownership of it. The high-IQ members of a society such as the teacher who creates human capital, and the scientist who creates technological capital, normally do not personally profit as much from their productiveness as do the investors in business enterprises that employ that created human and technological capital. The wealth of a nation may be largely owned by those business investors, but that does not necessarily mean that the high-IQ citizens of that nation – such as teachers, scientists, Goethe, Mozart, and Marilyn vos Savant – did not contribute more than their per capita share of the productiveness leading to its creation.

Since when is there a correlation between IQ and efficiency?
According to Arthur Jensen, the g factor is the most important factor predicting occupational productivity:

• In the world of work, g is the main cognitive correlate and best single predictor of success in job training and job performance. Its validity is not nullified or replaced by formal education (independent of g), nor is it decreased by increasing experience on the job.
(Arthur Jensen. The g Factor. Chapter 9: The Practical Validity of g. p270.)

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Nereid
Staff Emeritus
Gold Member
Just to put the Jensen quote (thanks hitssquad) in context: he is referring primarily to the US, and (if you study the literature) indirectly to what we'd call 'developed economies'. It would be interesting to see how much this 'best single predictor of success' varies throughout the world, and (if we had a time machine) since the Industrial Revolution. There are some obvious, glaring exceptions: in a great many occupations and places, the best single predictor would likely be general health and/or physical strength.

Well, I was reading somewhere [forget where] that UK and France were "printing" more money then they can cover during the first World War, this was at the beginning of the modern era of money printing. Both countires were almost going to lose the war, and they were having this choice to reduce the lack of money the goverments were facing [I know, they were before the war and after the war rich empires, but the economy was at a big risk]

I was reading that the Islamic Caliphate country during the centuries before its distruction in the world war I [As Ottoman's State at that time] was using a "smart" tactic to keep the trust in the currency and "notes" they were spreading. They cover it by both silver and Gold at the same time, so if they lack one sometimes, the other may help is rescuing the situation.

loseyourname
Staff Emeritus
Gold Member
Nereid said:
Just to put the Jensen quote (thanks hitssquad) in context: he is referring primarily to the US, and (if you study the literature) indirectly to what we'd call 'developed economies'. It would be interesting to see how much this 'best single predictor of success' varies throughout the world, and (if we had a time machine) since the Industrial Revolution. There are some obvious, glaring exceptions: in a great many occupations and places, the best single predictor would likely be general health and/or physical strength.

Note that Jensen says g is the single most important "cognitive" correlate of success. He says nothing of non-cognitive correlates, such as health, strength, or even motivation.

Nereid
Staff Emeritus
Gold Member
loseyourname said:
Note that Jensen says g is the single most important "cognitive" correlate of success. He says nothing of non-cognitive correlates, such as health, strength, or even motivation.
Hmm, it guess that's one read of the sentence; however, I read it differently ("In the world of work, g is the [...] best single predictor of success in job training and job performance") - 'main cognitive correlate' refers (solely?) to correlation.

You also point to another interesting facet - I believe you're correct that Jensen does not regard 'motivation' as 'cognitive'; I wonder if 'psychometricians' study motivation? Another example of the great divide in this field of psychology?

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Lynn and Vanhanen presented the alternative case that that variances between nations in economic success can be positively explained by variances between nations in:

1. National IQ
2. Economic liberality
and,
3. Political liberality
...in that order:
I just spent the past 3 days in a conference with Richard Lynn and had a private conversation with him only yesterday. We were discussing the organization of the Mongoloid brain and did not discuss the book he wrote with Vanhanen. There was, however, a paper presented by Michael McDaniel (yesterday) which demonstrated that Lynn's association of national GDP with IQ was not sensitive to the estimates of IQ for nations below 90. If all of the lower ones (down to and below 60) are assumed to have means of 90, the curve fit is virtually identical. He also showed that it is quite likely that Lynn's estimate of this relationship understated the relationship between IQ and GDP. His conclusion: "IQ is a very robust predictor of the wealth of nations." Bouchard pointed out that Lynn's conclusions are especially believable in light of what we otherwise know about IQ.

Lynn presented a paper, but it was on another subject.

Staff Emeritus
Gold Member
Dearly Missed
Nereid said:
You also point to another interesting facet - I believe you're correct that Jensen does not regard 'motivation' as 'cognitive'; I wonder if 'psychometricians' study motivation? Another example of the great divide in this field of psychology?

Going through Jensen's chapter on The Practical Validity of g, we find that the various tests, like the ASVAB*, that he uses did not include motivation. However there was an interview as part of the testing for pilot training:

Ratings based on structured interviews (which systematically assess subject attributes including educational background, self-confidence and leadership, flying motivation) were also found to have signifigant predictive validity for success in pilot training. However, when the interview ratings were included in a multiple correlation along with the ASVAB to predict training success, the interview ratings proved to have no incremental validity over the ASVAB score. This finding indicates that whatever predictive validity the interview had was due to its overlapping variance with the predictive component of the ability factors tapped by the ASVAB, which is largely g.
(p 283).

*ASVAB subtests: Arithmetic reasoning, numerical operations, paragraph comprehension, word knowledge, coding speed, general science, mathematics knowledge, electronics information, mechanical comprehension, automotive-shop information.

Nereid said:
Lynn et al would have you believe that the causation is IQ -> economic sector.
Lynn has noted that there is a robust relationship between national GDP and IQ. Do you dispute that? If so, would you supply data to demonstrate it?

To see how nonsensical this is, consider (as you have hinted) India and Ireland, the Flynn effect,
The Flynn Effect??? I have previously posted information here which supports the finding that the Flynn Effect is not a Jensen Effect. If it is not, it has nothing to do with intelligence, but is instead related to specific abilities that do not correlate with g. I discussed this two days ago with Rushton, who was one of the first to demonstrate that the Flynn Effect is not a Jensen effect.

and the economic sector distributions of countries 50 and 100 years ago. If you go back even further, you find that most folk lived in rural areas, and were engaged in agriculture (and other primary sector activities) - yet a mere instant (in terms of genetic composition of populations) later the descendants of almost all those folk live in cities and are engaged in services!
Do you think the above observation disproves the relationship between GDP and IQ? That relationship has existed for centuries. Have you read the conclusions reported by Murray in his book about Human Accomplishment? If so, perhaps you can see a connection? If you can't see the connection, I suggest that you need to give it some serious thought.

Nereid said:
That there is a correlation between the IQ data L+V present and three economic indicators is undeniable; whether national IQ is a (leading) cause of economic success (as claimed by L+V) is a very different kettle of fish.
Only when the observer wishes to take a mind set and ignore the full body of knowledge that pertains to the issue.
First, as Aquamarine said, L+V's data is questionable, to say the least.
As I have pointed out, the conclusion is not sensitive to the data and is probably understated. I challenge you to demonstrate otherwise.

Next, L+V's case for a causal connection (IQ -> success) is deeply flawed, even assuming the data they use are correct.
This is incorrect. Lynn's conclusions are consistent with both the findings of intelligence research and with economic principles. Economist Edward Miller commented that McDaniel's analysis was consistent with the causation effects that have been discussed in detail by Gordon and Gottfredson. These are basically that daily life choices are slightly g loaded and that the variances add enough to accumulate large effects.

loseyourname said:
Seems to me that a wealthy nation can better educate its populace, thus increasing the IQ.
There are no data showing that g can be increased, even a tiny bit, by education. Knowledge and intelligence are different commodities.

I wouldn't doubt that smarter people are theoretically capable of creating more wealth, but how often does this actually happen?
On a statistical basis it can be seen within our country, and within other countries, and between countries. This point has been demonstrated over and over. Yesterday, Dave Lubinski presented very impressive results showing the grossly disproportionate financial success of the top 1/4 of the top 1% of talent search students as compared to the bottom 1/4 of that same top 1%. The numbers are amazingly large, even over that tiny separation.

Many of the most intelligent people out there are researchers or professors making next to nothing.
The people who achieve success in academics were also discussed by Lubinski. He measured their success by tenure and age at tenure. Again, the differences were amazingly large.

Conversely, quite a few wealthy people either were extremely driven or just got lucky,
Some probably got lucky. This subject relates to statistics, not to exceptions and impressions. I suggest that you consider examining real numbers.

not necessarily a whole lot more intelligent than the average person. Why don't we do an accounting of the personal wealth of posters on this site?
You could do that, if you think the results would be meaningful. They would not.

Despite our obviously above-average level of intelligence, I'm willing to bet we aren't particularly wealthy.
Self-selecting groups are often very poor representations. Mensa, for example, has collected data showing that its members have higher incomes than average, but even so, it is likely that Mensans are self-selected in the down direction for success. Successful people are more likely to be working than playing. Lubinski showed that the people in the TS group worked very long hours, in addition to being super intelligent.

Nereid said:
Just to put the Jensen quote (thanks hitssquad) in context: he is referring primarily to the US, and (if you study the literature) indirectly to what we'd call 'developed economies'. It would be interesting to see how much this 'best single predictor of success' varies throughout the world, and (if we had a time machine) since the Industrial Revolution. There are some obvious, glaring exceptions: in a great many occupations and places, the best single predictor would likely be general health and/or physical strength.
It happens that there was a paper presented by Arden and Hill on Friday that showed intelligence to be significant even in preliterate society. The study was one of the most impressive I have seen. Intelligence was determined by the Raven's Color set. The people were Ache (living in Paraguay) and were living at the most primitive level imaginable. Through very well designed methods the researchers found that tribal status was determined by intelligence and not any other factor (of many tested), including hunting ability.

loseyourname said:
Note that Jensen says g is the single most important "cognitive" correlate of success. He says nothing of non-cognitive correlates, such as health, strength, or even motivation.
Really? How did you come to that conclusion? It is simply not true. Have you read The _g_ Factor? Have you read a good sampling of Jensen's papers? Jensen has discussed factors other than intelligence and commented on their importance.

FWIW, I had the wonderful experience of several private conversations with Jensen yesterday and the day before. One lasted for over 1/2 hour. He is brilliant and held in the highest esteem by his colleagues (there are no peers).

loseyourname
Staff Emeritus
Gold Member
Mandrake said:
There are no data showing that g can be increased, even a tiny bit, by education. Knowledge and intelligence are different commodities.

That may be true, but hitssquad was claiming a correlation between national IQ, not g. The results a person gets on an IQ test depends on more than their intelligence. It depends on understanding of the language, experience taking tests, and attention span as well. A very intelligent person that cannot concentrate will underperform.

The people who achieve success in academics were also discussed by Lubinski. He measured their success by tenure and age at tenure. Again, the differences were amazingly large.

I'm glad to hear that, but again, that is outside of what seemed to be going on in this thread thus far. Although I will note, even then, that tenure is no guarantee of self-satisfaction, which would seem to me the best indicator of success. Any attempt to objectively quantify success seems ill-fated. You can quantify whether or not certain goals have been attained, but you cannot quantify the feeling that a person is really after.

Some probably got lucky. This subject relates to statistics, not to exceptions and impressions. I suggest that you consider examining real numbers.

Well, to be honest, I'd imagine (granted, my assessment isn't all that scientific) that the single greatest predictor of personal wealth is the wealth of one's parents. In order to remove this factor from any study I was personally conducting, I would include only self-made subjects. Do you have any idea if this was considered?

You could do that, if you think the results would be meaningful. They would not.

It would confirm what you said about people that spend a lot of their time playing (or on an internet forum) not being as likely to make big bucks regardless of their intelligence. In fact, I'd like to see a study correlating the percentage of time spend attempting to acquire wealth with personal wealth. I'd be willing to bet (though it's just a hunch) that you'd see a greater correlation here than you would with intelligence. But granted, I'm no expert on the subject, and I'm not attempting to make any authoritative statements here.


text:
# Research

My research interest includes combinatorics and model theory. My Erdős number is at most 3. Below are some selected papers and preprints.

(A complete list of papers and preprints is available here.)

Minimal and nearly minimal measure expansions in connected unimodular groups (with Yifan Jing), arXiv:2006.01824, preprint, 2020.

Answering a question asked by Kemperman in 1964, we classify connected and compact groups $$G$$  and nonempty compact $$A, B \subseteq G$$ with $$\mu_G(AB) =\mu_G(A)+\mu_G(B)$$.  We also get a near equality version of the above result with sharp bound (up to a constant factor) confirming conjectures asked by Griesmer and by Tao. As an application, we obtain a measure expansion gap result for connected compact simple Lie groups.

A nonabelian Brunn-Minkowski inequality (with Yifan Jing and Ruixiang Zhang), arXiv:2101.07782, submitted, 2021

In 1953, Henstock and Macbeath asked whether there is a Brunn–Minkowski inequality in all locally compact groups. In this paper, we obtain such an inequality and prove it is sharp for helix-free groups (including linear algebraic groups, semisimple groups with a finite center, solvable groups, Nash groups, etc). This also answers questions asked by Hrushovski and by Tao.

Étale-open topology and the stable field conjecture (with Will Johnson, Erik Walsberg, and Jinhe Ye), arXiv:2101.07782, submitted, 2021

We introduce the étale-open system of topologies for a given field, show that this recover the canonical system of topologies in many situations, and obtain characterization of field-theoretic properties in term of the étale-open system of topologies. As an application, we confirm the special case of the longstanding stable fields conjecture for the class of large fields which contains all known model-theoretically tame fields.

Zarankiewicz’s problem for semilinear hypergraphs (with Abdul Basit, Artem Chernikov, Sergei Starchenko, and Terence Tao), arxiv:2009.02922, submitted, 2020

For a $$K_{k,k}$$-free bipartite graph $$(V_1, V_2, E)$$ is  with $$V_1, V_2 \subseteq \mathbb{R}^n$$ and $$E$$ the subset of $$V_1 \times V_2$$ satisfying a fixed system of linear inequalities of complexity $$s$$, we show that  $$|E| \leq \alpha|V|( \log|V| )^ \beta$$ with $$\alpha =\alpha(k,s)$$ and $$\beta=\beta(s)$$. We also obtain a version of this result for hypergraph, show that the appearance of log is necessary in general, and connect this to o-minimal trichotomy in model theory.

Interpolative Fusions (with Alex Kruckman and Erik Walsberg), Journal of Mathematical Logic. 2020 Nov 28:2150010

We define the interpolative fusion of multiple theories over a common reduct, a notion that aims to provide a general framework to study model-theoretic properties of structures with randomness. In the special case where the theories involved are model complete, their interpolative fusion is precisely the model companion of their union. Several theories of model-theoretic interest are shown to be canonically bi-interpretable with interpolative fusions of simpler theories. We initiate a systematic study of interpolative fusions by also giving general conditions for their existence and relating their properties to those of the individual theories from which they are built.


text:
# Convert a beta distribution to Poisson distribution

I need a Poisson distribution in an application, but in that application the only available functions for generating distributions are Beta, Uniform, and Normal functions. Now, I am thinking is there a way I could draw a beta distribution, but change the parameters in a way so as to approximate a Poisson distribution? Thanks.

• Since a Beta(1,1) is a Uniform(0,1), you can code any generic Poisson random generator using that Beta generator. Or you can use the connection between Poisson and Exponential, plus the fact that a $\chi^2_2$ is an Exponential $\cal{E}(1/2)$. – Xi'an Apr 4 '18 at 7:37

Or you can use the connection between Poisson and Exponential, plus the fact that a $\chi^2_2$ is an Exponential $\cal{E}(1/2)$.


text:
## anonymous one year ago Please help me with the question : show that Q[x]/(x^2-2) is not isomorphic with Q[x]/(x^2-3)

• This Question is Open
1. misty1212

hmm i think i remember this gimmick but it was with $$\mathbb{Q}(\sqrt2)$$ and $$\mathbb{Q}(\sqrt3)$$

2. misty1212

since they are the same thing, it must be the same the hint is to show that if you had such an isomorphims, since $$\phi(1)=1$$ it must be the case that $$\phi(\sqrt2)=\sqrt2$$ but $$\sqrt 2\notin \mathbb{Q}(\sqrt3)$$

3. anonymous

you are right it is similar , but this one is little bit more trickier!


text:
# Codeforces 1606F - Tree Queries (virtual tree + tree dp)

Obviously, the points we choose to delete together with \ (u \) will form a connected block. Otherwise, if we choose not to delete the points that are not in the same connected block with \ (u \), the answer must be better.

Note that if we choose to delete a son \ (V \) of \ (u \), the increment of the answer is \ (chd_v-1-k \), where \ (chd_v \) is the number of sons of node \ (V \). The initial answer is \ (chd_ \) is a fixed value, so our task can be equivalent to assigning a point weight \ (chd_-1-k \) to each point, and then finding a connected block on the tree with \ (u \) as the root, so that the sum of the weights of other points in the connected block except point \ (u \) is as large as possible.

Therefore, we have a method of complexity \ (\ Theta(n^2) \): run the tree \ (DP \) for each \ (K \), that is, let \ (dp_ \) represent the maximum value of the sum of weights of connected blocks rooted in \ (U \), then there is obviously a transfer \ (dp_ = chd_-1-k + \ sum \ limits {V \ in son_} \ max (dp_v, 0) \). Note that the point weights of the points on the boundary of the connected block on the tree (that is, all the points satisfying \ (x \) are in the connected block on the tree, but all the sons of \ (x \) are not in the connected block on the tree) must be positive. Otherwise, we deduct those points whose point weights are not positive and on the boundary of the connected block, and the answer must become better. Therefore, for each \ (K \), take out all points with positive point weight to build a virtual tree - because \ (\ sum \ limits {I = 1} ^ nchd_i = n-1 \), for all \ (K \), the total number of points with positive point weight is \ (\ Theta(n) \), and then run a tree \ (DP \) for them, and then query the maximum value of the sum of point weights of connected blocks with root \ (U \) each time, if point \ (U \) It is on the virtual tree itself. Just return the DP value of \ (U \) directly. Otherwise, we can find the node \ (V \) below the chain where \ (U \) is located on the virtual tree - this can take a set to store the DFS order of all nodes on the virtual tree, and then directly lower it in the set_ bound. Then the maximum value of the point weight sum of the connected block with \ (U \) as the root is the point weight sum of the path sum of \ (dp_v \) plus \ (fa_v\to u \).

Time complexity \ (\ Theta(n\log n) \).

const int MAXN=2e5;
const int LOG_N=18;
int n,qu,hd[MAXN+5],to[MAXN*2+5],nxt[MAXN*2+5],ec=0;
int chd[MAXN+5],dep[MAXN+5],fa[MAXN+5][LOG_N+2];
int dfn[MAXN+5],tim=0,rid[MAXN+5],edt[MAXN+5];
void dfs0(int x,int f){
fa[x][0]=f;rid[dfn[x]=++tim]=x;
for(int e=hd[x];e;e=nxt[e]){
int y=to[e];if(y==f) continue;
chd[x]++;dep[y]=dep[x]+1;dfs0(y,x);
} edt[x]=tim;
}
int sum_chd[MAXN+5];
void dfs_chd(int x,int f){
for(int e=hd[x];e;e=nxt[e]){
int y=to[e];if(y==f) continue;
sum_chd[y]=sum_chd[x]+chd[y];
dfs_chd(y,x);
}
}
void lca_init(){
dep[1]=1;dfs0(1,0);
for(int i=1;i<=LOG_N;i++) for(int j=1;j<=n;j++)
fa[j][i]=fa[fa[j][i-1]][i-1];
}
int getlca(int x,int y){
if(dep[x]<dep[y]) swap(x,y);
for(int i=LOG_N;~i;i--) if(dep[x]-(1<<i)>=dep[y]) x=fa[x][i];
if(x==y) return x;
for(int i=LOG_N;~i;i--) if(fa[x][i]!=fa[y][i]) x=fa[x][i],y=fa[y][i];
return fa[x][0];
}
vector<int> pos[MAXN+5];
int stk[MAXN+5],tp=0;set<int> st;
vector<pii> qv[MAXN+5];
ll calc_sum(int u,int v,int k){//u is ancestor of v
return (sum_chd[v]-sum_chd[u])-1ll*(k+1)*(dep[v]-dep[u]);
}
int calc_val(int u,int k){return chd[u]-1-k;}
vector<int> g[MAXN+5];
ll dp[MAXN+5],res[MAXN+5];
void insert(int x){
if(!tp) return stk[++tp]=x,void();
int lc=getlca(x,stk[tp]);
//	printf("LCA of %d %d is %d\n",x,stk[tp],lc);
//	printf("stack: ");
//	for(int i=1;i<=tp;i++) printf("%d%c",stk[i]," \n"[i==tp]);
while(tp>1&&dep[stk[tp-1]]>dep[lc]){
g[stk[tp-1]].pb(stk[tp]);--tp;
}
if(tp&&dep[stk[tp]]>dep[lc]){
g[lc].pb(stk[tp--]);
}
if(!tp||stk[tp]!=lc) stk[++tp]=lc;
stk[++tp]=x;
}
void fin(){
while(tp>=2){
g[stk[tp-1]].pb(stk[tp]);--tp;
} tp=0;
}
void dfs_dp(int x,int k){
dp[x]=calc_val(x,k);st.insert(dfn[x]);
for(int y:g[x]){
dfs_dp(y,k);
dp[x]+=max(dp[y]+calc_sum(x,y,k)-calc_val(y,k),0ll);
//		printf("calc_sum %d %d %d = %lld\n",x,y,k,calc_sum(x,y,k));
} //printf("DP %d %lld\n",x,dp[x]);
}
void clr(int x){for(int y:g[x]) clr(y);g[x].clear();}
int main(){
scanf("%d",&n);
lca_init();sum_chd[1]=chd[1];dfs_chd(1,0);scanf("%d",&qu);
//	for(int i=1;i<=n;i++) printf("chd[%d]=%d\n",i,chd[i]);
for(int i=1;i<=n;i++) for(int j=0;j<chd[i];j++) pos[j].pb(i);
for(int i=1,u,k;i<=qu;i++) scanf("%d%d",&u,&k),qv[k].pb(mp(u,i));
for(int i=0;i<=MAXN;i++){
st.clear();
if(!pos[i].empty()){
//			printf("solving %d\n",i);
sort(pos[i].begin(),pos[i].end(),[&](int x,int y){return dfn[x]<dfn[y];});
if(pos[i][0]!=1) insert(1);
for(int x:pos[i]) insert(x);
fin();dfs_dp(1,i);
}
for(pii p:qv[i]){
int u=p.fi,id=p.se;
set<int>::iterator it=st.lower_bound(dfn[u]);
if(it==st.end()||(*it)>edt[u]) res[id]=chd[u];
else{
int pt=rid[*it];
//				printf("%d %d\n",id,pt);
//				printf("%lld\n",calc_sum(fa[u][0],fa[pt][0],i));
res[id]=chd[u]+max(0ll,dp[pt]+calc_sum(fa[u][0],fa[pt][0],i)-calc_val(u,i));
}
}
clr(1);
}
for(int i=1;i<=qu;i++) printf("%lld\n",res[i]);
return 0;
}
/*
11
1 2
2 3
2 4
1 5
5 6
5 7
7 8
7 9
7 10
1 11
4
1 0
1 1
5 2
11 0
*/

Posted by Salsaboy on Mon, 01 Nov 2021 03:51:40 -0700


text:
# Difference of right and left Riemann sum

Why is it that Rn-Ln = $\Delta x$(f(b)-f(a)) on [a,b]? I've tried manipulating summations but I can't seem to figure it out.
The simplest way is to say that except for the first summand of $L_n$ (it is $f(a)\Delta x$) and the last summand of $R_n$ (it is $f(b)\Delta x$), all the other terms appear in both sums. When you subtract, you are left with $R_n - L_n = \Delta x (f(b)-f(a))$.


text:
History

Help on query formulation
A method to find the sums of polynomial functions at positive integer values. (English)
Parabola 51, No. 2, 4 p., electronic only (2015).
From the text: When learning the intuition behind definite integration, calculus students often learn how to find the area under a curve by using a Riemann sum. Often, when attempting to find the area under polynomial curves by this method, students are limited by how many formulas they know for the sums of monomials of a positive integer degree $n$. The most commonly known formula of this variety is for the sum of monomials of degree 1, namely $\sum \limits_{i=0}^n i=\frac{n(n+1)}{2}$. Less common are the formulas for the sum of monomials of degree 2 and 3, namely $\sum \limits_{i=0}^n i^2=\frac{n(n+1)(2n+1)}{6}$ and $\sum \limits_{i=0}^n i^3=\frac{n^2(n+1)^2}{4}$. Most calculus students are taught to memorize these formulas, and are thus not able to find the sums of polynomials of degrees higher than 3. Further research into the area yields Faulhaber’s formula, which involves more complex concepts such as the Bernoulli numbers, with which students are often unfamiliar. In this paper, I show a method for deriving these summations for polynomials of higher degrees without using these complex concepts.
Classification: H20 I20 I30 I50


text:
# How can I make a 3D Plot with NDSolve?

I have a system of coupled ODEs which I am solving using Mathematica. I am solving these using NDSolve:

solved=NDSolve[{Cs'[t] == DL*Stuff,
Cx1'[t] == DL*Stuff,
Cx2'[t] == DL*Stuff, Cs[0] == 10, Cx1[0] == 25,
Cx2[0] == 7}, {Cs, Cx1, Cx2}, {t, 0, 1000}]

Where DL is a constant. Generally following this I will plot them as follows:

Plot[Evaluate[{Cs[t], Cx1[t], Cx2[t]} /. solved], {t, 0, 1000},
PlotRange -> All]

However, what I would like to do is make a 3D plot which shows how the solution to the system changes for different values of the constant DL. How can I do this?

Constraint: The equations mentioned above cannot be solved analytically, or decoupled or whatever.

Full code after attempting changes:

(First, I will Define Constants)

Subscript[Y, sx1] = 1/.14;
U1max = .5;
Subscript[KM, 1] = Subscript[KM, 2] = 10;
Subscript[Y, x1x2] = 2 ;
U2max = .11;

(Next, I will define the prior algebraic expressions)

Clear[Cs]

U1 = (U1max*Cs[t])/(Subscript[KM, 1] + Cs[t]);

U2 = (U2max*Cs[t])/(Subscript[KM, 2] + Cs[t]);

(Now I put in the remaining algebraic expressions)

rgx1 = U1*Cx1[t];

rgx2 = U2*Cx2[t];

(Now we can enter the coupled ODEs)

solved[DL_] =
NDSolve[{Cs'[t] == DL*(250 - Cs[t]) - Subscript[Y, sx1]*rgx1,
Cx1'[t] == -DL*Cx1[t] + rgx1 - Subscript[Y, x1x2]*rgx2,
Cx2'[t] == -DL*Cx2[t] + rgx2, Cs[0] == 10, Cx1[0] == 25,
Cx2[0] == 7}, {Cs, Cx1, Cx2}, {t, 0, 1000}]

Plot3D[Evaluate[Cx2[t] /. solved], {t, 0, 1}, {DL, .01, .1},
PlotRange -> All]

I get two errors:

(1) NDSolve::ndnum: Encountered non-numerical value for a derivative at t == 0.. >>

(2) ReplaceAll::reps: {solved} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing. >>

Here you go:

solved[DL_] :=
NDSolve[{Cs'[t] == DL*(250 - Cs[t]) - Subscript[Y, sx1]*rgx1,
Cx1'[t] == -DL*Cx1[t] + rgx1 - Subscript[Y, x1x2]*rgx2,
Cx2'[t] == -DL*Cx2[t] + rgx2, Cs[0] == 10, Cx1[0] == 25,
Cx2[0] == 7}, {Cs, Cx1, Cx2}, {t, 0, 1000}]

Plot3D[Cx2[tt] /. solved[DL]], {tt, 0, 1000}, {DL, 0.01, 0.1},
PlotRange -> All]

Some important points or it won't work:

(1) the value tt for time has to have a different name to the value t for time inside the NDSolve

(2) the SetDelayed in the definition of solved is important

(3) it's horribly inefficient

But it works nicely.

Comment:

Usually the above approach isn't very useful; e.g. you would find it more useful to plot the final state as a function of DL.

Plot[Evaluate[{Cs[1000], Cx1[1000], Cx2[1000]} /. solved[DL]], {DL, .01, .1}]

• I have tried to include the full code. My apologies if it is difficult to read. After attempting to take these steps I have received two errors: (1) NDSolve::ndnum: Encountered non-numerical value for a derivative at t == 0.. >> (2) ReplaceAll::reps: {solved} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing. >> – Nick L. Mar 15 '15 at 8:55
• I suggest you edit your code so it can easily be copied and pasted into Mathematica. You need to use SetDelayed and solved[DL] as shown in my edits. – djp Mar 15 '15 at 10:24
• Thank you very much. The code works very nicely. I also appreciate the tips which you gave me. I would up-vote you but apparently my reputation is too low. – Nick L. Mar 15 '15 at 18:52

Rewriting the code:

Ysx1 = 1/.14;
U1max = .5;
KM1 = KM2 = 10;
Yx1x2 = 2;
U2max = .11;
Clear[Cs]
U1 = (U1max*Cs[t])/(KM1 + Cs[t]);
U2 = (U2max*Cs[t])/(KM2 + Cs[t]);
rgx1 = U1*Cx1[t];
rgx2 = U2*Cx2[t];
sol = ParametricNDSolve[{Cs'[t] == DL*(250 - Cs[t]) - Ysx1*rgx1,
Cx1'[t] == -DL*Cx1[t] + rgx1 - Yx1x2*rgx2,
Cx2'[t] == -DL*Cx2[t] + rgx2, Cs[0] == 10, Cx1[0] == 25,
Cx2[0] == 7}, {Cs, Cx1, Cx2}, {t, 0, 1000}, {DL}]

Assuming the desired 3D plot relates to {DL, t, Cx2}:>

f[DL_, t_] := (Cx2[DL] /. sol)[t]
Plot3D[Evaluate[f[x, y]], {y, 0, 1000}, {x, 0.01, 0.1},
PlotRange -> All, AxesLabel -> {"t", "DL", "Cx2"}, Mesh -> False]

• Thank you for the code. I am very appreciative that you took the time to do this. This will be very helpful to me. – Nick L. Mar 15 '15 at 18:55


text:
# Fractional SPDEs driven by spatially correlated noise: existence of the solution and smoothness of its density

Abstract : In this paper we study a class of stochastic partial differential equations in the whole space $\mathbb{R}^{d}$, with arbitrary dimension $d\geq 1$, driven by a Gaussian noise white in time and correlated in space. The differential operator is a fractional derivative operator. We show the existence, uniqueness and H\"{o}lder's regularity of the solution. Then by means of Malliavin calculus, we prove that the law of the solution has a smooth density with respect to the Lebesgue measure.
Type de document :
Article dans une revue
Osaka Journal of Mathematics, Osaka University, 2010, 47 (10), pp.41-65
Domaine :

https://hal.archives-ouvertes.fr/hal-00325068
Contributeur : Mohamed Mellouk <>
Soumis le : vendredi 26 septembre 2008 - 10:25:54
Dernière modification le : mercredi 12 octobre 2016 - 01:17:07

### Citation

Mohamed Mellouk. Fractional SPDEs driven by spatially correlated noise: existence of the solution and smoothness of its density. Osaka Journal of Mathematics, Osaka University, 2010, 47 (10), pp.41-65. <hal-00325068>

### Métriques

Consultations de la notice


text:
# Any deep cause of magnetic field

1. Jun 7, 2009

### Nikaro

when we apply potential difference across any conductor the electric field set up in the conductor which apply force on charged particals and set up moving current.While the current is moving the charge which is at the outer surface of conductor also make elecricfield which is moving along the conductor.
Is this moving electric field is the cause of magnetic field ?
If it is not than please please tell me the reason of magnetic field?
And is there is any deep relation between magnetic and electric field.

2. Jun 7, 2009

### Cyosis

Not sure what you mean by a moving electrical field. Moving charges produce a magnetic field, but also an electrical field that changes with time produces a magnetic field.

I think one of the nicest and deepest understanding of magnetic fields arise from special relativity. Where a magnetic and electric field are part of the same phenomenon. Different observers will see different type of fields.

3. Jun 7, 2009

### Bob S

One reason is that curl H = J, meaning that currents are the cause of magnetic fields. Another reason (which you point out) is that a Lorentz transformation on an electric field creates a magnetic field. A time-dependent magnetic field does create electric fields (curl E = - dB/dt), but a time dependent electric field (=dE/dt) cannot create a magnetic field.

4. Jun 7, 2009

### Cyosis

Could you elaborate on this seeing as

$$\nabla \times \vec{B}=\mu_0 \vec{J}+\mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}$$.

5. Jun 7, 2009

### Bob S

Mea culpa. I should have included the displacement current term, which makes the Curl E and Curl H equations more symmetric, and is very important in electromagnetic radiation, especially when there are no conductors around.

6. Jun 7, 2009

### cabraham

E & H are mutually inclusive under dynamic conditions, i.e. one cannot exist w/o the other. If one is time-changing, the other must be non-zero. Einstein in his 1905 paper, described that neither one is the "seat" ("seat" meaning basis or primary).

Claude

7. Jun 9, 2009

### Bob S

There is another source of a magnetic field. A transverse electric field in one reference frame can be converted into a transverse magnetic field in another reference frame by a Lorentz transformation:

BT' = γBT - (γ/c2) v x ET

α β γ δ ε ζ η θ ι κ λ μ ν ξ ο π ρ ς σ τ υ φ χ ψ ω . . . . . Γ Δ Θ Λ Ξ Π Σ Φ Ψ Ω


text:
# Rocketguts.com's FIRST kit!

### Help Support The Rocketry Forum:

#### rocketsonly

##### Well-Known Member
Greetings.

Well, here are some pictures of our first kit that we will be test launching tomorrow morning at 8! We plan on selling these to fundraise for out '04-'05 TARC project.

Here's some information about the kit:

Diameter: 3.9" (PML phenolic)

Height: 32"

Fin mount type: fin can

Fin material: aircraft 5ply birch plywood (very strong!)

Motor mounts: 2x 24mm, 3x 29mm

Recommended motor configurations:
1x 29mm (1x G40 or 1x G80)
2x 29mm (2x G40 or 2x G80)
3x 24mm (2x D12 and 1x E9)
5x 24mm (4x D12 and 1x E9)

Obviously there are many more possible configurations that you may try on your own.

Motor plugs and adapters will be included so that the rocketeer can launch multiple times on different motor configurations.

We'll be finishing the kit after we test launch it! More pictures to come when it's finished. We'll also try and get some pictures of the test launch.

Please leave any comments. We do realize the fins seem very fragile! We'll see if they break or not upon landing tomorrow.

Thanks,
Matthew

#### rocketsonly

##### Well-Known Member
Detail picture of fin.

#### rocketsonly

##### Well-Known Member
Another detailed picture of fin.
NB: slots in fin.

#### rocketsonly

##### Well-Known Member
Picture of fin can.

#### rocketsonly

##### Well-Known Member
Another picture of fin can.
NB: motors!

#### rocketsonly

##### Well-Known Member
Picture of fin removed from centering ring.
NB: slots on fins AND centering ring

#### rocketsonly

##### Well-Known Member
Picture of fin inserting into centering ring.
NB: perfect fit!

#### rocketsonly

##### Well-Known Member
Another view of the fin can!

#### rocketsonly

##### Well-Known Member
Picture of fin slots.
NB: PML phenolic tube

#### brianc

##### Well-Known Member
OOOooo!! NICE!

Those fins are just calling out for red-orange-yellow and
maybe a touch of blue paint!

How thick is the 5-ply?

Good luck and DON'T break a fin!

#### rocketsonly

##### Well-Known Member
Picture of fin can being inserted to the body tube.
NB: again, perfect fit

#### rocketsonly

##### Well-Known Member
Another view of fully mounted fin can.

#### rocket trike

##### Well-Known Member
I have not personally seen this kit but I do know the work that they have put into this kit. I have had them do allot of fins and centering rings cutting for me and by the way if you need fins or centering rings cut these are the people that you need to see YOU WILL NOT BE DISAPPOINTED. Ok back to the rocket by the work that I have seen them do I know this will be a top quaility kit that you will not want to miss out on.

Tom

#### rocketsonly

##### Well-Known Member
Originally posted by brianc
OOOooo!! NICE!

Those fins are just calling out for red-orange-yellow and
maybe a touch of blue paint!

How thick is the 5-ply?

Good luck and DON'T break a fin!
Fins are 1/8" thick.

#### jetra2

##### Well-Known Member
It looks really cool. Very unique, and you can be pretty confident that it'll be the only one out there that day!

I would probably glass the flames of the fins or do *something* to them to make them less prone to breaking. I guess your only alternative to this would be using a big parachute to keep the descent very very slow.

Any ideas at the cost of this kit?

Jason

#### rocketsonly

##### Well-Known Member
Originally posted by jetra2
It looks really cool. Very unique, and you can be pretty confident that it'll be the only one out there that day!

I would probably glass the flames of the fins or do *something* to them to make them less prone to breaking. I guess your only alternative to this would be using a big parachute to keep the descent very very slow.

Any ideas at the cost of this kit?

Jason
We're still discussing about the prices, but most likely around $70 with out parachute, or$80 with parachute (30" ripstop nylon). Again, these are just estimates, prices may be higher or lower. Kits will be limited to about 20 and preorders will be taken once we successfully test launch the rocket.

#### eugenefl

##### Well-Known Member
TRF Supporter
Not to be a buzzkill, but I just don't see how those fins will stay intact whether in flight or upon landing. If the fins were somehow laminated in a clear epoxy so that the gaps inbetween the flames made them complete fins, then I'd say they would withstand anything. I suppose the kit is still in BETA so no one really knows, but I'd bet something will happen to those fins. Otherwise, I give an A for creativity. Best of luck.

#### Stymye

##### Well-Known Member
hey,can you show a pic of the fin can? > J/K

can you guys cut G-10? that might hold up better

#### Justin Horne

##### Well-Known Member
Will it? I mean, it's kinda flexable.. I suppose lots stronger than ply though in terms of bending....
Quite nice design.

But 70-80 dollars? Too much for me... That's nosecone, phenolic tube, rings, fins ( i see the time cost here) , and motor tubes. plus your profitit... I dunno...

Oh well, While it's too much for me, i'm sure many will buy it. Quite nice design.

#### rocketsonly

##### Well-Known Member
Originally posted by Justin Horne
Will it? I mean, it's kinda flexable.. I suppose lots stronger than ply though in terms of bending....
Quite nice design.

But 70-80 dollars? Too much for me... That's nosecone, phenolic tube, rings, fins ( i see the time cost here) , and motor tubes. plus your profitit... I dunno...

Oh well, While it's too much for me, i'm sure many will buy it. Quite nice design.