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https://machinelearningmedium.com/2018/04/10/support-vector-machines/ | Blog Logo
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Index
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### Optimization Objective
The support vector machine objective can seen as a modification to the cost of logistic regression. Consider the sigmoid function, given as,
where $z = \theta^T x$
The cost function of logistic regression as in the post Logistic Regression Model, is given by,
Each training instance contributes to the cost function the following term,
So when $y = 1$, the contributed term is $-log(\frac {1} {1 + e^{-z}})$, which can be seen in the plot below. The cost function of SVM, denoted as $cost_1(z)$, is a modification the former and a close approximation.
import numpy as np
def svm_cost_1(x):
return np.array([0 if _ >= 1 else -0.26*(_ - 1) for _ in x])
x = np.linspace(-3, 3)
plt.plot(x, -np.log10(1 / (1+np.exp(np.negative(x)))))
plt.plot(x, svm_cost_1(x))
plt.legend(['logistic regression cost function', 'modified SVM cost function'])
plt.show()
Similarly, when $y = 0$, the contributed term is $-log(1 - \frac {1} {1 + e^{-z}})$, which can be seen in the plot below. The cost function of SVM, denoted as $cost_0(z)$, is a modification the former and a close approximation.
def svm_cost_0(x):
return np.array([0 if _ <= -1 else 0.26*(_ + 1) for _ in x])
plt.plot(x, -np.log10(1-(1 / (1+np.exp(np.negative(x))))))
plt.plot(x, svm_cost_0(x))
plt.legend(['logistic regression cost function', 'modified SVM cost function'])
plt.show()
While the slope the straight line is not of as much importance, it is the linear approximation that gives SVMs computational advantages that helps in formulating an easier optimization problem.
Regularized version of \eqref{2} can from the post Regularized Logistic Regression can rewritten as,
In order to come up with the cost function for the SVM, \eqref{3} is modified by replacing the corresponding cost terms, which gives,
Following the conventions of SVM the following modifications are made to the cost in \eqref{4}, which effectively is a change in notation but not the underlying logic,
• removing ${1 \over m}$ does not affect the minimization logic at all as the minima of a function is not changed by the linear scaling.
• change the form of parameterization from $A + \lambda B$ to $CA + B$ where it can be intuitively thought that $C = {1 \over \lambda}$.
After applying the above changes, \eqref{4} gives,
The SVM hypothesis does not predict probability, instead gives hard class labels,
### Large Margin Intuition
According to \eqref{5} and the plots of the cost function as shown in the image above, the following are two desirable states for SVM,
• if $y=1$, then $\theta^Tx \geq 1$ (not just $\geq 0$)
• if $y=0$, then $\theta^Tx \leq -1$ (not just $\lt 0$)
Let C in \eqref{5} be a large value. Consequently, in order to minimize the cost, the corresponding term $\sum_{i=1}^m \left[ y^{(i)}\,cost_1(\theta^T x^{(i)}) + (1-y^{(i)})\,cost_0(\theta^T x^{(i)}) \right]$ must be close to 0.
Hence, in order to minimize the cost function, when $y=1$, $cost_1(\theta^T x)$ should be 0, and similarly, when $y=0$, $cost_0(\theta^T x)$ should be 0. And thus, from the plots in Fig.3, it is clear that it can only fulfilled by the two states listed above.
Following the above intuition, the cost function can we written as,
subject to contraints,
What this basically leads to is the selection of a decision boundary that tries to maximize the margin from the support vectors as shown in the plot below. This maximization of the margin as seen for decision boundary A increases the robustness over decision boundaries with lesser margins like B. And it is this property of the SVMs that attributes the name large margin classifier to it.
### Effect of Parameter C
As discussed in the section above, the effect of C can be considered as reciprocal of regularization parameter, $\lambda$. This is more clear from Fig-5. A single outlier, can make the model choose the decision boundary with smaller margin if the value of C is large. A small value of C ensures that the outliers are overlooked and best approximation of large margin boundary is determined.
### Mathematical Background
Vector Inner Product: Consider two vectors, $v$ and $w$, given by,
Then, the inner product or the dot product is defined as $v^Tw = w^Tv$.
Norm of a vector, $v$, denoted as $\lVert v\rVert$ is the euclidean length of the vector given by the pythagoras theorem as,
The inner product can also be defined as,
where $p=\lVert w\rVert \cdot cos \theta$ can be described as the projection of vector $w$ onto vector $v$ which can be either positive or negative signed based on the angle $\theta$ between the vectors as shown in the image below.
SVM Decision Boundary: From \eqref{7}, the optimization statement can be written as,
subject to contraints,
Let $\theta_0 = 0$ and $n=2$, i.e. number of features is 2 for simplicity, then \eqref{10} can be written as,
Using \eqref{9}, $\theta^Tx^{(i)}$ in \eqref{11} can be written as,
The plot of \eqref{13} can be seen below,
Hence, using \eqref{12} and \eqref{13}, the optimization objective in \eqref{10} and the constraints in \eqref{11} are written as,
subject to contraints,
where $p^{(i)}$ is the projection of $x^{(i)}$ onto vector $\theta$.
Consider two decision boundaries, A and B, and their respective perpendicular parameters, $\theta_A$ and $\theta_B$ as shown in the plot below. As a consequence of choosing $\theta_0 = 0$ for simplification, all the corresponding decision boundaries pass through the origin.
Based on the two training examples of either class chosen, close to the boundaries, it can be seen that the magnitude of projection is more in case of $\theta_B$ than $\theta_A$. This basically tells that it would be possible to choose smaller values of $\theta$ and satisfy \eqref{14} and \eqref{15} if the value of projection $p$ is bigger and hence, the decision boundary, B is more favourable to the optimization objective.
Why is decision boundary perpendicular to the $\theta$?
Consider two points $x_1$ and $x_2$ on the decision boundary given by,
Since the two points are on the line, they must satisfy \eqref{16}. Substitution leads to the following,
Subtracting \eqref{18} from \eqref{17},
Since $x_1$ and $x_2$ lie on the line, the vector $(x_1 - x_2)$ is on the line too. Following the property of orthogonal vectors, \eqref{19} is possible only if $\theta$ is orthogonal or perpendicular to $(x_1 - x_2)$, and hence perpendicular to the decision boundary.
### Kernels
When dealing with non-linear decision boundaries, a learning method like logistic regression relies on high order polynomial features to find a complex decision boundary and fit the dataset, i.e. predict $y=1$ if,
where $f_0 = x_0,\, f_1=x_1,\, f_2=x_2,\, f_3=x_1x_2,\, f_4=x_1^2,\, \cdots$.
A natural question that arises is if there are choices of better/different features than in \eqref{20}? A SVM does this by picking points in the space called landmarks and defining functions called similarity corresponding to the landmarks.
Say, there are three landmarks defined, $l^{(1)}$, $l^{(2)}$ and $l^{(3)}$ as shown in the plot above, the for any given x, $f_1$, $f_2$ and $f_3$ are defined as follows,
Here, the similarity function is mathematically termed a kernel. The specific kernel used in \eqref{21} is called the $Gaussian Kernel$. Kernels are sometimes also denoted as $k(x, l^{(i)})$.
Consider $f_1$ from \eqref{21}. If there exists $x$ close to landmark $l^{(1)}$, then $\lVert x - l^{(1)} \rVert \approx 0$ and hence, $f_1 \approx 1$. Similarly for a $x$ far from the landmark, $\lVert x - l^{(1)} \rVert$ will be a larger value and hence exponential fall will cause $f_1 \approx 0$. So effectively the choice of landmarks has helped in increasing the number of features $x$ had from 2 to 3. which can be helpful in discrimination.
For a gaussian kernel, the value of $\sigma$ defines the spread of the normal distribution. If $\sigma$ is small, the spread will be narrower and when its large the spread will be wider.
Also, the intuition is clear about how landmarks help in generating the new features. Along with the values of parameter, $\theta$ and $\sigma$, various different decision boundaries can be achieved.
### How to choose optimal landmarks?
In a complex machine learning problem it would be advantageous to choose a lot more landmarks. This is generally acheived by choosing landmarks at the point of the training examples, i.e. landmarks equal to the number of training examples are chosen, ending up in $l^{(1)}, l^{(2)}, \cdots l^{(m)}$ if there are $m$ training examples. This translates to the fact that each feature is a measure of how close is an instance to the existing points of the class, leading to generation of new feature vectors.
For SVM training, given training examples, $x$, features $f$ are computed, and $y=1$, if $\theta^Tf \geq 0$
The training objective from \eqref{5} is modified as follows,
In this case, $n=m$ in \eqref{5} by the virtue of procedure used to choose $f$.
The regularization term in \eqref{22} can be written as $\theta^T\theta$. But in practice most SVM libraries, instead $\theta^TM\theta$, which can be considered a scaled version is used as it gives certain optimization benefits and scaling to bigger training sets, which will be taken up at a later point in maybe another post.
While the kernels idea can be applied to other algorithms like logistic regression, the computational tricks that apply to SVMs do not generalize as well to other algorithms.
Hence, SVMs and Kernels tend to go particularly well together.
### Bias/Variance
Since $C (= {1 \over \lambda})$,
• Large C: Low bias, High Variance
• Small C: High bias, Low Variance
Regarding $\sigma$,
• Large $\sigma^2$: High Bias, Low Variance (Features vary more smoothly)
• Small $\sigma^2$: Low Bias, High Variance (Features vary less smoothly)
### Choice of Kernels
• Linear Kernel: is equivalent to a no kernel setting giving a standard linear classifier given by,
Linear kernels are used when the number of training data is less but the number of features in the training data is huge.
• Gaussian Kernel: Make a choice of $\sigma^2$ to adjust the bias/variance trade-off.
Gaussian kernels are generally used when the number of training data is huge and the number of features are small.
Feature scaling is important when using SVM, especially Gaussian Kernels, because if the ranges vary a lot then the similarity feature would be dominated by features with higher range of values.
All the kernels used for SVM, must satisfy Mercer’s Theorem, to make sure that SVM optimizations do not diverge.
Some other kernels known to be used with SVMs are:
• Polynomial kernels, $k(x, l) = (x^T l + constant)^degree$
• Esoteric kernels, like string kernel, chi-square kernel, histogram intersection kernel, ..
### Multi-Class Classification
• Most SVM libraries have multi-class classification.
• Alternatively, one may use one-vs-all technique to train $k$ different SVMs and pick class with largest $\theta^Tx$
### Logistic Regression vs SVM
• If $n$ is large relative to $m$, use logistic regression or SVM with linear kernel, like if $n=10000, m=10-1000$
• If $n$ is small and $m$ is intermediate, use SVM with gaussian kernel, like if $n=1-1000, m=10-10000$
• If $n$ is small and $m$ is large, create/add more features, then use logistic regression or SVM with no kernel, as with huge datasets SVMs struggle with gaussian kernels, like if $n=1-1000, m=50000+$
Logistic Regression and SVM without a kernel (with linear kernel) generally give very similar. A neural network would work well on these training data too, but would be slower to train.
Also, the optimization problem of SVM is a convex problem, so the issue of getting stuck in local minima is non-existent for SVMs.
· · · | 2020-07-14 02:39:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 6, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8229305148124695, "perplexity": 676.0483632259184}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593657147917.99/warc/CC-MAIN-20200714020904-20200714050904-00532.warc.gz"} |
https://quant.stackexchange.com/questions/14646/how-to-find-cvar-avar-for-triangular-fuzzy-no | how to find CVaR/AVaR for triangular fuzzy no
While going through different methods of risk measure i came across AVaR/CVaR, while i was calculating AVaR/CVaR in credibilistic environment using VaR, i got stuck in the calculations
eg. For triangular fuzzy numbers $A =(a1,a2,a3)$ $$VaR(\alpha)= 2*(a1-a2)*\alpha -a1;$$ $$\alpha \leq 0.5$$ $$= 2*(a2-a3)*\alpha +a3-2*a2;$$$$\alpha >0.5$$
Now, $$AVaR(\alpha) = (1/\alpha) \int_0^\alpha Var(\beta)d\beta$$
which somehow results in $$AVaR(\alpha)= (a1-a2)*\alpha -a1;$$$$\alpha \leq 0.5$$ $$= (a2-a3)*\alpha +a3-2*a2-1/4*\alpha(a1-2*a2+a3)$$$$\alpha >0.5$$
Plz help me wd d term $$-1/4*\alpha(a1-2*a2+a3).$$ How to get it? | 2019-07-16 04:53:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8884140849113464, "perplexity": 2386.4411002225693}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195524502.23/warc/CC-MAIN-20190716035206-20190716061206-00209.warc.gz"} |
https://www.physicsforums.com/threads/sum-of-squares-of-2-non-commutating-operators.876317/ | # Sum of squares of 2 non-commutating operators
Prof Adams does something rather strange, starting from 14:35 minutes in this lecture -- http://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2013/lecture-videos/lecture-9/
He reminds us that for complex scalars, ##c^2+d^2=(c-id)(c+id)## and then proceeds to do the same with operators,
factorizing ##\frac{\hat{X}^2}{X_0^2}+\frac{\hat{P}^2}{P_0^2}##
in this way :
##=(\frac{\hat{X}}{X_0}-i\frac{\hat{P}}{P_0})(\frac{\hat{X}}{X_0}+i\frac{\hat{P}}{P_0})##
which he re-expands into a sum of squares plus a NON-ZERO commutator.
Is it not true that the identity he started with, i.e. ##c^2+d^2=(c-id)(c+id)## for complex scalars - is valid precisely when (and because) ##icd=idc##? So how does this apply to the operators where ##XP\ne{PX}## ?
Is it not true that the identity he started with, i.e. ##c^2+d^2=(c-id)(c+id)## for complex scalars - is valid precisely when (and because) ##icd=idc##?
Of course, and this is exactly the reason why you get an additional commutator when you try to generalize this formula to non-commuting operators.
Thanks, but I'm still confused.
I'm not sure how you can equate/replace ##\frac{\hat{X}^2}{X_0^2}+\frac{\hat{P}^2}{P_0^2}##
with
##(\frac{\hat{X}}{X_0}-i\frac{\hat{P}}{P_0})(\frac{\hat{X}}{X_0}+i\frac{\hat{P}}{P_0})##
in the first place, when you know that the cross terms in the latter are not going to cancel?
Staff Emeritus
Thanks, but I'm still confused.
I'm not sure how you can equate/replace ##\frac{\hat{X}^2}{X_0^2}+\frac{\hat{P}^2}{P_0^2}##
with
##(\frac{\hat{X}}{X_0}-i\frac{\hat{P}}{P_0})(\frac{\hat{X}}{X_0}+i\frac{\hat{P}}{P_0})##
in the first place, when you know that the cross terms in the latter are not going to cancel?
They don't cancel, but the cross-terms are a constant. So:
$\frac{\hat{X}^2}{X_0^2}+\frac{\hat{P}^2}{P_0^2} = (\frac{\hat{X}}{X_0}-i\frac{\hat{P}}{P_0})(\frac{\hat{X}}{X_0}+i\frac{\hat{P}}{P_0}) + K$
You can work out what the constant $K$ is.
Thank you. | 2022-08-16 20:16:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8010679483413696, "perplexity": 940.9034792583625}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572515.15/warc/CC-MAIN-20220816181215-20220816211215-00207.warc.gz"} |
https://electronics.stackexchange.com/questions/109533/why-a-lower-voltage-is-better-for-modern-fast-cpu-and-other-similar-chips | # Why a lower voltage is better for modern fast CPU and other similar chips?
I started to use computers in the 1980's. As far as I remember 8bit CPUs of those times like Z80 were often powered by 5 V and used the same voltage for I/O signals. Later CPUs run at higher speeds and started to consume more power. I would expect that to deliver more power to a chip we must use either higher voltage or higher current. And since high currents normally need thick cables I would expect that CPUs would go for higher voltages to keep currents low. But the opposite is true. For example a standard Intel Pentium 4 or Core 2 Quad CPUs I use at home have 95 W TDP which means that they consume more than 100 W at power spikes. Since they run on a very low voltage around 1 V, they actually need to deliver the power using approximately 100 A. So here comes my question: Why is this preffered, why is it efficient?
• The CPU voltages are NOT chosen to optimise power supply factors - they are chosen to suit the characteristics of the ICs used. Modern CPUs use very small line dimensions and correspondingly small insulation layers. The voltages used are appropriate to the requirements of the transistors used in the ICs. – Russell McMahon May 7 '14 at 18:34
In traditional CMOS circuits the power consumption, to first order, followed this expression:
$P \propto f_{CLOCK} \times C_{LOAD} \times V^2_{SUPPLY}$
where the load capacitance was the effective capacitance of the internal wiring and transistor gate oxide. Notice that power consumption is proportional to the square of the supply voltage, so lowering the supply voltage is a powerful way to decrease power consumption. Unfortunately, just lowering the supply voltage tends to make the circuits run slower so other changes, such as scaling the transistors, are necessary to keep achieving higher clock frequencies.
As transistor scaling approached deep submicron feature sizes, say below about 250 nm, transistors stopped behaving like "traditional" CMOS and started to be more leaky. That added a term to the power equation that is proportional only to supply voltage (not voltage squared) and limited the benefit of lowering the supply voltage in order to decrease power consumption. | 2020-02-22 17:23:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5867986679077148, "perplexity": 1146.3093205031425}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145708.59/warc/CC-MAIN-20200222150029-20200222180029-00246.warc.gz"} |
http://thawom.com/subsec-slope.html | ### 11.3.1 Straight Lines
This subsection is not completed yet, but in the meantime here are some practice questions.
Remember:
Slope, which is usually given the letter $m$, is a measure of the steepness of a line. It is the number of units up or down for every unit across, and can be calculated by dividing the rise by the run. A line connecting two points $(x_1,y_1)$ and $(x_2,y_2)$ will have a rise of $\Delta y = y_2 - y_1$ and a run of $\Delta x = x_2 - x_1$, with a slope of: $m = \frac{\Delta y}{\Delta x}$
Remember:
The steeper the line, the bigger the slope, $m$. A slope of $1$ is on a $45^\circ$ angle. A slope of $0$ will be horizontal. If the line is sloping downwards then either $\Delta y$ or $\Delta x$ will be negative, and so will the slope, $m$.
Remember:
To find the slope of a line given its equation, rewrite it in the form $y=mx + c$. For example, \begin{align*} y &= 2 + x\\ &= 1x + 2\\ m &= 1 \end{align*} \begin{align*} y &= 5 - \frac{x}{2}\\ &= -\frac{1}{2} x + 5\\ m &= -\frac{1}{2} \end{align*} \begin{align*} 2x + y + 3 &= 0\\ y &= -2x - 3\\ m &= -2 \end{align*}
Remember:
If a line with slope $m_1$ is perpendicular to another line with slope $m_2$ then $m_1 m_2 = -1$
Remember:
To find the y-intercept of a straight line (which is the $c$ in $y = mx + c$), knowing its slope, $m$, and a point on the line, $(x_1,y_1)$, there are a couple methods. You can say: $y-y_1 = m(x-x_1)$ and then solve for $y$ to get it into the form $y = mx + c$. Or you could sub $(x_1,y_1)$ into the equation $y=mx+c$ to get: $y_1 = mx_1 + c$ and then solve for $c$. In either case, $c = y_1 - mx_1$ will be the conclusion. | 2019-05-25 04:11:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9999606609344482, "perplexity": 165.40440419017054}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232257847.56/warc/CC-MAIN-20190525024710-20190525050710-00551.warc.gz"} |
https://jeopardylabs.com/play/tax-and-more-tax | Assets Income & Gains Deductions Principles Hodgepodge
### 100
What is The Code does not define what a capital asset is, only what it is not.
How does the Code define a capital asset?
### 100
What is $80,000 Amount realized of$135,000 ($90,000 cash +$20,000 art + $25,000 loan assumption) minus$50,000 adj. basis and $5,000 selling expense =$80,000.
### 100
What is when the payment is made
Under the cash method of tax accounting, tax deductions are generally taken when:
### 100
What is $480 Exclusion ratio = 40,000 / 100,000 = 40%. Therefore, 60% of each annuity payment is included in gross income.$800 x .60 = $480. Jane purchased an annuity contract that pays her$800 per month. The annuity cost her $40,000 and it has an expected return of$100,000. How much of each monthly annuity payment is includible in Jane's gross income?
### 200
What is When property that is not like-kind (i.e., cash or other property, known as boot'') is received, then any realized gain is recognized to the extent of the boot received. If there is a realized loss in a boot received situation, such loss will not be recognized.
What is "boot" and how is it treated in like-kind exchanges?
# Tax and more Tax
Press F11 for full-screen mode | 2017-01-23 22:59:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2472459375858307, "perplexity": 6688.276395081936}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560283301.73/warc/CC-MAIN-20170116095123-00448-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://blog.andrewaadland.me/2018-10-11-compiling-and-testing-a-fluentd-plugin/ | Compiling and Testing a FluentD Plugin
This post will be in the context of running FluentD on a VM using the td-agent and filebeat packages.
Background
I’ve been looking into how to optimize FluentD. I want Aggregators that can handle a lot of throughput and utilize each CPU core. FluentD is written in Ruby, and is thus subject to the constraints of the Global Interpretor Lock like Python.
This way seems to work well. As long as you break out each <process> block in your /etc/td-agent/td-agent.conf to handle one input plugin (although not necessary), it’ll work for you. You can break it out into many sub-configurations. Its just a bit harder to maintain and scale.
I tried using the more native workers configuration property, but plugins must explicitly support it.
Beats is a common way to ship logs. I was surprised to find that the FluentD Beats Plugin doesn’t support Multiple Workers. In order to test some changes to the plugin, I needed to be able to compile from source. And off we go with my forked repo…
The Process
1. Prep the file system and clone the plugin repo:
mkdir -p /tmp/fluent-plugin-beats && \
cd /tmp/fluent-plugin-beats && \
git clone --single-branch -b multi-workers https://github.com/chicken231/fluent-plugin-beats
1. Use td-agent’s gem wrapper to build the gem:
td-agent-gem build fluent-plugin-beats.gemspec
1. The build command generates a gem and adds a version number into the file name. Now you can install the gem to make it available to FluentD:
td-agent-gem install fluent-plugin-beats-0.1.4.gem
Configure, Install, Test
Assumptions:
• td-agent is installed and configured. Here’s an excerpt from my /etc/td-agent/td-agent.conf:
# general system config. Note the log format for later.
<system>
workers 4
suppress_config_dump
<log>
format json
</log>
</system>
# beats input
<source>
@type beats
port 5044
bind 0.0.0.0
</source>
# use this when testing to print to stdout and thus the log file
<match **>
@type stdout
</match>
• Filebeat is installed, configured, and enabled to point to Logstash output on localhost:5044 and is capturing files matching /var/log/*.log (default).
1. Start td-agent and filebeat:
systemctl start td-agent filebeat
1. Anything that comes in as an input to FluentD will write into /var/log/td-agent/td-agent.log. So tail it:
tail -f /var/log/td-agent/td-agent.log
1. echo some stuff and append to a file in a directory that filebeat is monitoring:
echo WOOOO \$(date) >> /var/log/temp.log
1. Observe a log printed to td-agent’s log. Note the logs are in JSON. An unflattened example of a message from filebeat:
{
"@timestamp": "2018-10-11T02:26:50.313Z",
"beat": "filebeat",
"type": "doc",
"version": "6.4.2"
},
"input": {
"type": "log"
},
"beat": {
"name": "centos-beats",
"hostname": "centos-beats",
"version": "6.4.2"
},
"host": {
"name": "centos-beats"
},
"source": "/var/log/temp.log",
"offset": 754,
"message": "WOOOO Thu Oct 11 02:26:45 UTC 2018",
"prospector": {
"type": "log"
}
}
And there we are. We downloaded a fork of a plugin that had changes to enable multiple workers. Built it, compiled it, tested it, and watched the messages from Filebeat stream through the FluentD logs. | 2021-05-10 07:57:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1793210357427597, "perplexity": 10416.638221495903}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989115.2/warc/CC-MAIN-20210510064318-20210510094318-00131.warc.gz"} |
https://mathoverflow.net/questions/380734/counting-monomials-in-product-polynomials-part-i | # Counting monomials in product polynomials: Part I
This question is motivated by recent work of R P Stanley, Theorems and conjectures on some rational generating functions. Consider the polynomials $$P_n(x)=\prod_{i=1}^{n-1}(1+x^{3^{i-1}}+x^{3^i}).$$ Define the sequence $$a_n$$ to count the number of monomials of $$P_n(x)$$. For example, \begin{align*} P_2(x)&=x^3 + x + 1 \qquad \qquad \qquad \qquad \qquad \qquad\qquad \,\,\implies \qquad a_2=3, \\ P_3(x)&=x^{12} + x^{10} + x^9 + x^6 + x^4 + 2x^3 + x + 1 \qquad \implies \qquad a_3=8. \end{align*} Recall the Fibonacci numbers $$F_1=F_2=1$$ and $$F_{n+2}=F_{n+1}+F_n$$.
QUESTION. Is it true that $$a_n=F_{2n}$$? How does "ternary expansion" relate to Fibonacci?
Yes, it is true. In other words, you ask whether $$|X_n|=F_{2n}$$ where $$X_n:=\sum_{i=1}^{n-1}\{0,3^{i-1},3^i\}.$$ We have $$X_n=X_{n-1}\cup Y_{n-1}\cup Z_{n-1},\quad (1)$$ where $$Y_{n-1}=X_{n-1}+3^{n-1}$$, $$Z_{n-1}=X_{n-1}+3^n$$. We have $$(X_{n-1}\cup Y_{n-1})\cap Z_{n-1}=\emptyset$$, since $$\min Z_{n-1}=3^n>\max (X_{n-1}\cup Y_{n-1})=2\cdot 3^{n-1}+3^{n-2}+\ldots+3$$. So, we have \begin{align*} |X_n|&=|Z_{n-1}|+|X_{n-1}\cup Y_{n-1}| \\ &=|Z_{n-1}|+|X_{n-1}|+|Y_{n-1}|-|X_{n-1}\cap Y_{n-1}| \\ &=3|X_{n-1}|-|X_{n-1}\cap Y_{n-1}|\\ &=3|X_{n-1}|-|X_{n-2}| \end{align*} (that follows from the decomposition (1) with $$n-1$$ instead $$n$$: $$X_{n-1}\cap Y_{n-1}=Z_{n-2}$$).
That's the recursion for $$F_{2n}$$'s.
Here is another argument. We have $$P_{n+1}(x)=(1+x+x^3)P_n(x^3).$$ Now $$P_n(x^3), xP_n(x^3)$$, and $$x^3P_n(x^3)$$ all have $$a_n$$ monomials. If a monomial $$x^i$$ appears in more than one of them, then it must appear in $$P_n(x^3)$$ and $$x^3P_n(x^3)$$, but not $$xP_n(x^3)$$ (by considering exponents mod 3). Thus we need to subtract off the number of monomials $$x^i$$ that appear in $$P_n(x)$$ such that $$x^{i+1}$$ also appears. By the uniqueness of the ternary expansion, the monomials with such $$x^i$$ or $$x^{i+1}$$ are those appearing in $$(1+x)P_{n-1}(x^3)$$. There are $$2a_{n-1}$$ such monomials, occurring in pairs $$x^i$$ and $$x^{i+1}$$. Hence $$a_{n+1}=3a_n-a_{n-1}$$, the recurrence satisfied by $$F_{2n}$$ (and with the correct initial conditions). | 2021-04-16 06:57:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 38, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9995274543762207, "perplexity": 212.9427545519974}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038088731.42/warc/CC-MAIN-20210416065116-20210416095116-00031.warc.gz"} |
https://math.stackexchange.com/questions/3087773/free-subgroups-of-psl2-mathbbz-of-index-6 | Free subgroups of $PSL(2,\mathbb{Z})$ of index 6
There are two "natural" subgroups of $$PSL(2,\mathbb{Z})\cong C_2\ast C_3$$ of index 6. One is the congruence subgroup $$\Gamma_0(2)$$ which is the kernel of the map $$PSL(2,\mathbb{Z})\to PSL(2,\mathbb{Z}/2\mathbb{Z})$$. The other subgroup $$H$$ is the kernel of the map $$C_2\ast C_3\to C_2\times C_3$$. Here are two similarities between these two subgroups:
• Both $$\Gamma_0(2)$$ and $$H$$ are subgroups of $$PSL(2,\mathbb{Z})$$ of index 6.
• Both $$\Gamma_0(2)$$ and $$H$$ are free groups of rank 2.
However, $$PSL(2,\mathbb{Z}/2\mathbb{Z})\cong S_3$$ and $$C_2\times C_3\cong C_6$$ so these are different subgroups of $$PSL(2,\mathbb{Z})$$. Moreover, $$\Gamma_0(2)$$ is freely generated by the matrices $$\begin{bmatrix}1&2\\0&1\end{bmatrix}$$ and $$\begin{bmatrix}1&0\\2&1\end{bmatrix}$$ whereas $$H$$ is freely generated by the matrices $$\begin{bmatrix}2&1\\1&1\end{bmatrix}$$ and $$\begin{bmatrix}1&1\\1&2\end{bmatrix}$$. This last statement can be seen by noting that if $$a=\begin{bmatrix}0&-1\\0&1\end{bmatrix}$$ generates $$C_2$$ and $$b=\begin{bmatrix}-1&-1\\1&0\end{bmatrix}$$ generates $$C_3$$ then $$H$$ is freely generated by $$abab^2$$ and $$ab^2ab$$.
What is going on here? More precisely,
• Are these two subgroups the largest free subgroups of $$PSL(2,\mathbb{Z})$$?
• Are there any other free subgroups of $$PSL(2,\mathbb{Z})$$ of index 6?
• Is there any reason to expect that $$PSL(2,\mathbb{Z})$$ contains two free subgroups of rank 2 and index 6 with different quotients?
Let $$G = \langle x,y \mid x^2,y^3 \rangle \cong {\rm PSL}(2,\mathbb{Z})$$.
Question 1. Yes. Let $$H < G$$, and consider the permutation action of $$G$$ on the (left or right) cosets of $$H$$ in $$G$$. If $$|G:H| < 6$$, then it is not possible for the images of both $$x$$ and $$y$$ to act fixed point freely, and so $$H$$ contains a conjugate of $$x$$ or $$y$$ and hence cannot be free.
Question 2. No, but the two subgroups that you have found are the only two normal subgroups of index $$6$$ in $$G$$. You can see this by observing that there is essentially only one surjective group homomorphism from $$G$$ to each of $$C_6$$ and $$S_3$$ (i.e. up to equivalence under an automorphism of $$C_6$$ or $$S_3$$), so there are only two possible kernels $$H$$.
By a computer calculation, I found that there is also one conjugacy class of non-normal subgroups $$H$$ with $$|G:H| = 6$$ and with $$H$$ free of rank $$2$$, and a representative of this class is $$H=\langle yx, y^{-1}(xy)^3 \rangle$$. The quotient of $$G$$ by the core of $$H$$ is isomorphic to $$S_4$$, and there are three conjugates of $$H$$ in $$G$$.
Question 3. I can only say here that the reason is that we have a proof that this is the case!
Note that the Kurosh Subgroup Theorem says that any subgroup of $$G$$ is a free product of conjugates of $$\langle x \rangle$$, $$\langle y \rangle$$ and a free subgroup of $$G$$. So, for $$H \lhd G$$, if $$H$$ does not contain $$x$$ or $$y$$, then it must be free. I believe that the rank of free subgroups of free products can be calculated using Euler Characteristics, but I don't know the details.
• Very nice. With regards to your comment on Euler characteristics, the answers to mathoverflow.net/questions/43726/… suggest (using Euler Characteristics or tools from Serre's book on Trees) that a free subgroup of $PSL(2,\mathbb{Z})$ of index 6 necessarily has rank 2. – Thomas Browning Jan 27 at 14:28
• . . . nice! . . – janmarqz Jan 27 at 20:57 | 2019-09-15 13:56:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 66, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.854656994342804, "perplexity": 57.29577892132705}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514571506.61/warc/CC-MAIN-20190915134729-20190915160729-00071.warc.gz"} |
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https://www.scielo.br/j/bjce/a/pjMR6jzkkdr8m4Td46rXDkk/?lang=en | # EXPERIMENTAL APPROACH TO ASSESS EVAPORATIVE COOLING UNDER FORCED AIR FLOW
About the authors
# Abstract
Air blast is one of the most employed industrial chilling methods. It can be enhanced, i.e., increasing heat transfer and reducing cooling time, by superficial evaporative phenomena. This work reports a methodology, including experimental setup and mathematical modelling, to quantify the air chilling enhancement by wetting the surface of the object to be chilled. A spherical metal model was covered by a cotton tissue (wet or dry) and placed into a cold chamber. The effective heat transfer coefficient was determined at different temperature, air velocity, and relative humidity from time-temperature profiles into the sphere. Under the same air conditions, the effective coefficient between sphere and air was increased three-fold by moistening the cotton tissue on the sphere surface. Furthermore, comparing a dry and wet surface showed that evaporative cooling resulted in much shorter chilling times. The proposed approach was able to assess evaporative heat transfer by measuring only the time-temperature profile, and is suitable for industrial applications.
Keywords:
Air blast chilling; Cold chamber; Convection; Heat and mass transfer; Effective heat transfer coefficient
# INTRODUCTION
Chilling processes are very important in many industries, such as in the food industry, in order to inhibit microbial growth and preserve the quality of foodstuffs from the producer to the consumer. Air blast chilling of foods can involve both heat transfer between the product and the chilling medium and mass transfer due to water evaporation (weight loss) at the wet surface (Mekprayoon and Tangduangdee, 2012Mekprayoon, R. and Tangduangdee, C., Influence of Nusselt Number on weight loss during chilling process. Proc. Eng., 32, 90-96 (2012).). Processing time and weight loss are important parameters regarding the design of chilling lines for the quality of many food products. Conventional methods such as slow air, air blast, water immersion, and water spraying have been widely used in industrial chilling of food products (Dincer, 1997aDincer, I., New effective Nusselt-Reynolds correlations for food-cooling applications. J. Food Eng., 31, 59-67 (1997a).; Sun and Wang, 2004Sun, D.-W. and Wang, L., Experimental investigation of performance of vacuum cooling for commercial large cooked meat joints. J. Food Eng., 61, 527-532 (2004).; Laurindo et al., 2010aLaurindo, J. B., Carciofi, B. A. M., Silva, R. R. and Hense, H., On-line monitoring of heat transfer coefficients in a stirred tank from the signatures of the resultant force on a submerged body. Int. J. Refrig., 33, 600-606 (2010a).; Laurindo et al., 2010bLaurindo, J. B., Carciofi, B. A. M., Silva, R. R., Dannenhauer, C. E. and Hense, H., Evaluation of the effects of water agitation by air injection and water recirculation on the heat transfer coefficients in immersion cooling. J. Food Eng., 96, 59-65 (2010b).), each one with its own advantages and disadvantages. Alternatively, rapid chilling methods increase food safety, improve quality, prevent microbial growth, and might reduce evaporative weight losses (Erdogdu et al., 2005Erdogdu, F., Sarkar, A. and Singh, R. P., Mathematical modeling of air-impingement cooling of finite slab shaped objects and effect of spatial variation of heat transfer coefficient. J. Food Eng., 71, 287-294 (2005).).
It is well known that water chilling is a very efficient procedure for food cooling due to the high heat transfer coefficients between the product and chilling water (Laurindo et al., 2010bLaurindo, J. B., Carciofi, B. A. M., Silva, R. R., Dannenhauer, C. E. and Hense, H., Evaluation of the effects of water agitation by air injection and water recirculation on the heat transfer coefficients in immersion cooling. J. Food Eng., 96, 59-65 (2010b).). Some products such as peas, radishes, cantaloupes, peaches, cherries, oranges, shrimps, crabs, sardines, tuna, and poultry carcasses are usually cooled by water immersion (Dincer, 1997bDincer, I., Heat Transfer in Food Cooling Applications. Taylor & Francis, Washington, DC (1997b).; Lucas and Raoult-Wack, 1998Lucas, T. and Raoult-Wack, A. L., Immersion chilling and freezing in aqueous refrigerating media: Review and future trends. Int. J. Refrig., 21, 419-429 (1998).; Teruel et al., 2003Teruel, B., Cortez, L. and Neves Filho, L., Estudo comparativo do resfriamento de laranja valência com ar forçado e com água. Ciênc. Tec. Ali., 23, 174-178 (2003). (In Portuguese).; Amendola and Teruel, 2005Amendola, M. and Teruel, B., Uso de um esquema implícito e de splines para a simulação numérica do processo de resfriamento de frutas esféricas. Rev. Bras. Eng. Agríc. Amb., 9, 78-82 (2005). (In Portuguese).). In the poultry industry, carcasses are traditionally cooled from approximately 40 ºC to 4 ºC by immersion in cold water (or a mixture of ice and water), which is crucial to ensure safe products (James et al., 2006James, C., Vincent, C., Lima, T. I. and James, S. J., The primary chilling of poultry carcasses - a review. Int. J. Refrig., 29, 847-862 (2006).; Carciofi and Laurindo, 2007Carciofi, B. A. M. and Laurindo, J. B., Water uptake by poultry carcasses during cooling by water immersion. Chem. Eng. Proc., 46, 444-450 (2007).; Carciofi and Laurindo, 2010Carciofi, B. A. M. and Laurindo, J. B., Experimental results and modeling of poultry carcass cooling by water immersion. Ciênc. Tec. Ali., 30, 447-453 (2010).). On the other hand, air chilling is gaining in popularity because of the limited availability of water, wastewater discharge restrictions, and governmental regulations on carcass moisture retention (Huezo et al., 2007Huezo, R., Smith, D. P., Northcutt, J. K. and Fletcher, D. L., Effect of immersion or dry air chilling on broiler carcass moisture retention and breast fillet functionality. J. Appl. Poult. Res., 16, 438-447 (2007).). Evaporative air chilling, a mixed type of air chilling and water chilling, combine the advantages of both methods and is an alternative to immersion chilling of poultry carcasses (Mielnik et al., 1999Mielnik, M. B., Dainty, R. H., Lundby, F. and Mielnik, J., The Effect of evaporative air chilling and storage temperature on quality and shelf life of fresh chicken carcasses. Poult Sci., 78, 1065-1073 (1999)., Jeong et al., 2011Jeong, J. Y., Janardhanan, K. K., Booren, A. M., Harte, J. B. and Kang, I., Breast meat quality and consumer sensory properties of broiler carcasses chilled by water, air, or evaporative air. Poult. Sci., 90, 694-700 (2011).). In a typical cold-air meat chilling process, the water vapor pressure at the meat surface is much higher than in the cooling air, resulting in water evaporation (weight and quality losses). Consequently, water from inside diffuses towards the surface as a result of moisture gradients, and the balance between evaporation and diffusion governs the water activity near the surface. The intensity of these coupled phenomena determines the chilling time, weight loss, temperature, and surface water activity (Trujillo and Pham, 2006Trujillo, F. C. and Pham, Q. T., A Computational fluid dynamic model of the heat and moisture transfer during beef chilling. Int. J. Refrig., 29, 998-1009 (2006).). Chuntranuluck et al. (1998)Chuntranuluck, S., Wells, C. M. and Cleland, A. C., Prediction of chilling times of foods in situations where evaporative cooling is significant-Part 1. Model development. J. Food Eng., 37, 111-125 (1998). developed a model for predicting chilling times of foods when evaporative chilling is significant. They used finite difference analysis to simulate transient chilling of food products of simple shapes such as spheres, infinite slabs, and cylinders. Those authors showed how to account for evaporative chilling considering the latent heat of evaporation and the evaporative weight loss rate.
The most significant factors for the design of a refrigerated food chain are the food's thermophysical properties (e.g., specific heat, thermal conductivity, thermal diffusivity) and the chilling heat transfer parameters (heat transfer coefficients) (Dincer, 1997bDincer, I., Heat Transfer in Food Cooling Applications. Taylor & Francis, Washington, DC (1997b).). The convective heat transfer coefficient is a key parameter to calculate chilling time, and, hence, in the design of food-cooling facilities (hydro-chilling, air chilling) (Cuesta et al., 2012Cuesta, F. J., Lamúa, M. and Alique, R., A new exact numerical series for the determination of the biot number: Application for the inverse estimation of the surface heat transfer coefficient in food processing. Int. J. Heat Mass Transf., 55, 4053-4062 (2012).). This coefficient is a local or global parameter, which depends mainly on the velocity of the surrounding fluid, product geometry, flow orientation, surface roughness, and on the packaging (Incropera and Dewitt, 2002Incropera, F. P. and Dewitt, D. P., Fundamentos de transferência de calor e de massa. LTC, Rio de Janeiro, Brazil (2002). (In Portuguese).; Becker and Fricke, 2004Becker, B. R. and Fricke, B. A., Heat transfer coefficients for forced-air cooling and freezing of selected foods. Int. J. Refrig., 27, 540-551 (2004).). At high heat transfer rates, food shape will play a minor role. The effect of these various factors on convection heat transfer can be mathematically expressed as a relationship of dimensionless numbers: Nusselt (Nu), Reynolds (Re), Prandtl (Pr), and Grashof (Gr) (Incropera and Dewitt, 2002; Verboven et al., 2003Verboven, P., Scheerlinck, N. and Nicolai, B. M., Surface heat transfer coefficients to stationary spherical particles in an experimental unit for hydrofluidisation freezing of individual foods. Int. J. Refrig., 26, 328-336 (2003).).
Landfeld and Houska (2006)Landfeld, A. and Houska, M., Prediction of heat and mass transfer during passage of the chicken through the chilling tunnel. J. Food Eng., 72,108-112 (2006). used a metallic chicken as a simplified physical model to estimate the convective heat transfer coefficient in an air blast chilling tunnel. They reported coefficient values between the metallic model and the chilling media close to 50 W m-2 K-1. Verboven et al. (2003)Verboven, P., Scheerlinck, N. and Nicolai, B. M., Surface heat transfer coefficients to stationary spherical particles in an experimental unit for hydrofluidisation freezing of individual foods. Int. J. Refrig., 26, 328-336 (2003). reported results of convective heat transfer coefficients between stationary spherical particles and an aqueous solution inside a hydrofluidized freezing unit ranging from 154 to 1548 W m-2K-1. These values depended on the body diameter, chilling fluid temperature, and fluid agitation level. The authors attributed the variability of the measured coefficients to non-constant flow and turbulence fields in the aqueous medium.
The lack of actual data about the effect of cooling conditions on the chilling rates, weight loss, and quality is a major barrier to the advancement of technology related to this unit operation (James et al., 2006James, C., Vincent, C., Lima, T. I. and James, S. J., The primary chilling of poultry carcasses - a review. Int. J. Refrig., 29, 847-862 (2006).). In this study, the objective was to assess the heat transfer between the cooling air and a metallic sphere, chosen as a model, covered by a wet or dry cotton tissue and to propose a methodology to obtain an effective heat transfer coefficient taking into account mass transfer phenomena, but without measuring it. The enhancement of this effective coefficient and the cooling rates provided by wetting the surface were evaluated from the experimental temperature analysis.
# MATERIALS AND METHODS
In order to evaluate the contribution of evaporative cooling on the heat transfer between a metallic sphere and air, effective heat transfer coefficients were determined from three different arrangements: i) uncovered aluminum sphere, ii) aluminum sphere covered with a dry cotton tissue, and iii) aluminum sphere covered with a wet cotton tissue. The purpose-built experimental setup, the instruments used to determine the experimental conditions, and the mathematical and statistical approaches are described below.
## Experimental Setup
The experimental setup used in this study consists of a cold chamber (0.95 m x 1.02 m x 0.98 m) equipped with a temperature and relative humidity (RH) control system. For controlling the air temperature inside the cold chamber, a PID controller (NOVUS, model N1100, Porto Alegre, Brazil) acting on an electrical heating element placed inside the cooling chamber was used. Pre-defined temperatures and RH inside the chamber were achieved by controlling the electric heating element, while the compressor capacity was controlled using a frequency inverter (Danfoss, model VLT Micro Drive FC 51, Beijing, China) (Danfoss, model 136 LCZ, Osasco, Brazil). The Field Chart software (Novus, version 1.8, Porto Alegre, Brazil) registered temperature and RH data inside the chamber. Airflow patterns inside the chamber were provided by a fan (WEG, 1000 rpm, Jaraguá do Sul, Brazil) at two average air velocities: 0.92 m s-1 (V1) and 1.42 m s-1 (V2). These average air velocities over the sphere were measured by a digital anemometer (Testo 425, Lenzkirch, Germany) positioned 0.50 m from the fan, as shown in Figure 1.
Figure 1
(a) Schematics of the forced-air chilling process of a sphere. (b) Plan view of the forced-air chilling process of a sphere. (c) Schematics of the sphere covered with cotton, sensor positions, and thermal resistance network: convective (Rconv), conductive in the cotton (Rcot), and conductive in the sphere (Rsph).
An aluminum sphere (diameter, DAl = 0.100 m, and mass, mAl = 1.41 kg) was used as a model to assess the influence of the surface condition (dry or wet) on heat transfer. For that, convective heat transfer coefficient values between the sphere and the cooling air were determined under different conditions: both velocities (V1 and V2) at three RH conditions (30%, 60%, and 90%). Aluminum's physical properties were density, ρAl = 2702 kg m-3; specific heat, cAl = 903 J kg-1 K-1; and thermal conductivity, kAl = 237 W m-1 K-1 (Incropera and Dewitt, 2002Incropera, F. P. and Dewitt, D. P., Fundamentos de transferência de calor e de massa. LTC, Rio de Janeiro, Brazil (2002). (In Portuguese).). A rigid polyethylene rod was used in order to support the sphere during the cooling experiments; this polymer was chosen to prevent the effects of conduction heat transfer through itself.
The sphere's core temperature (TAl) was measured using a T-type thermocouple (IOPE, model 24 AWG, São Paulo, Brazil) inserted into the sphere's geometric center through a channel made with a precision mechanical machine. The channel was filled with thermal paste (IPT300, Implastec, São Paulo, Brazil) and sealed with a commercial epoxy resin (Professional Araldite, Brascola, São Paulo, Brazil). Time-temperature data of cooling air (T) were measured, as well as time-temperature data of internal (Ti, in contact with the aluminum sphere's surface) and external (Te, in contact with air) cotton tissue surfaces. The dry cotton tissue mass (md) was 0.009 kg, while its specific heat was 1300 J kg-1 K-1 (Incropera and Dewitt, 2002Incropera, F. P. and Dewitt, D. P., Fundamentos de transferência de calor e de massa. LTC, Rio de Janeiro, Brazil (2002). (In Portuguese).). The diameter of the sphere covered with cotton tissue was De = 0.102 m. Figure 1c is a schematic that represents the sphere covered with cotton, the sensor positioning, and the three resistances to heat transfer present in the forced air chilling process. Contact resistances between cotton and aluminum were neglected. Rconv, Rcot, and Rsph denote convective thermal resistances (between air and surface), conductive thermal resistances in the cotton, and conductive thermal resistances in the aluminum.
All temperatures were monitored with T-type thermocouples (IOPE, model A-TX-TF-TF-R 30 AWG, São Paulo, Brazil) and recorded at 5 s intervals by a data acquisition system (Agilent, model 34972A, Santa Clara, USA).
## Determination of Heat Transfer Coefficients Between the Sphere and Forced Air
### Aluminum Sphere
Initially, the aluminum sphere was heated in a thermoregulated bath (Tecnal, model TE - 184, Piracicaba, Brazil) up to 38-39 ºC and kept in water to reach a spatially homogeneous temperature. Next, the sphere was quickly introduced into the cold chamber. The cooling process was monitored until the sphere's core temperature had reached 4.0 ºC. This procedure was done in quintuplicate for cooling air at each predetermined air velocity and at 1.0 ºC.
The convective heat transfer coefficient (h) between the sphere and air was determined from experimental time-temperature evolutions at both the sphere's core and air chilling using the well-known lumped capacitance method. This method can be used when the thermal resistance to heat transfer by conduction within one solid is much smaller than the thermal resistance to heat transfer between the solid's surface and fluid flow. In other words, the lumped capacitance method can be applied only when the temperature inside the solid can be considered spatially uniform at any given time during the process, i.e., it is practical only if the Biot number (Eq. (1)) is less than 0.1 (Incropera and Dewitt, 2002Incropera, F. P. and Dewitt, D. P., Fundamentos de transferência de calor e de massa. LTC, Rio de Janeiro, Brazil (2002). (In Portuguese).).
(1) $Bi = Lh k Al$
in which L is the solid's characteristic dimension, defined as the ratio between its volume and interfacial area. For a sphere, L = DAl/6. This way, if Bi < 0.1, the global energy balance in the whole solid is given by Eq. (2).
(2) $− hA Al T Al − T ∞ = m AL C Al dT Al dt$
in which AAl is the interfacial area of the aluminum sphere and t is the cooling time.
Integrating Eq. (2) leads to Eq. (3), from which the h value can be determined using experimental time-temperature data.
(3) $ln T o − T ∞ T Al − T ∞ = hA Al m Al c Al t − t o$
in which To is the solid's temperature at initial time to (initial condition).
### Aluminum Sphere Covered with Cotton Tissue
An effective convective heat transfer coefficient (hef), which has both thermal and evaporative contribution, was determined from time-temperature data and Eq. (4), which represents the overall energy balance in the aluminum sphere covered with the cotton tissue,
(4) $− h ef A e T e − T ∞ = m c c c d T ─ c dt + m Al c Al dT Al dt$
in which Ae is the surface area of the aluminum sphere covered by the wet cotton tissue; and mc and cc are the mass and the specific heat of the wet cotton, respectively.
Assuming the temperature inside the aluminum sphere as spatially uniform at any given time during the process and a pseudo-stationary approach, Eq. (5) allows calculating an average temperature (c) in the cotton spherical shell (rirre, from the aluminum sphere's surface, radius ri, and cotton surface, radius re),
(5) $T ─ c = ∫ r i r e T c r 4 π r 2 dr ∫ r i r e 4 π r 2 dr$
in which Tc(r) represents the temperature in the spherical shell, as given by Eq. (6) for a steady state (Incropera and Dewitt, 2002Incropera, F. P. and Dewitt, D. P., Fundamentos de transferência de calor e de massa. LTC, Rio de Janeiro, Brazil (2002). (In Portuguese).).
(6) $T c r = T i − T i − T e 1 − r i r 1 − r i r e$
Thus, after integration, c is given by Eq. (7).
(7) $T c ─ = T i − T e − T i r e 4 − 3 2 r e 3 r i + 1 2 r e r i 3 r e 3 − r i 3 r e − r i$
Mass and specific heat were assumed to be constant for the aluminum sphere and variable for the wet cotton as a consequence of moisture content in the tissue. Time-dependence was calculated from the cotton tissue's moisture content, as given by Eq. (8),
(8) $m c c c = m d c d + m w c w$
in which mdcd and mwcware the product between mass and specific heat of dry cotton around the aluminum sphere and water soaked into the tissue, respectively. Specific heat of cotton and water were assumed as 1300 J kg-1 K-1 and 4180 J kg-1 K-1, respectively (Incropera and Dewitt, 2002Incropera, F. P. and Dewitt, D. P., Fundamentos de transferência de calor e de massa. LTC, Rio de Janeiro, Brazil (2002). (In Portuguese).).
The dry cotton tissue's mass (md) was constant at 0.009 kg, while water mass decreased during the evaporative cooling. In order to evaluate mw as a function of time, the wet cotton tissue was weighed outside the chamber on a semi-analytical balance (Gehaka, model BG 400, São Paulo, Brazil). At predetermined intervals, the sphere was quickly removed from the chamber and placed on the balance. The result was expressed as a percentage of the water present in the cotton tissue. The initial mass was determined from ten measurements (average value = 0.0338 kg and standard deviation = 0.0003 kg). Time-water mass data were fitted by the empirical Eq. (9),
(9) $m = a exp − bt + c$
in which a, b and c are empirical constants.
One instantaneous coefficient value was calculated from each recorded time interval (∆t) using Eq. (4) re-written as Eq. (10). Thus, one average coefficient was determined from each cooling assay. Finally, the average value of hef is the average from five cooling assays at each experimental setup investigated.
(10) $h ef = 1 A e T e − T ∞ m c c c Δ T c ─ Δ t + m Al c Al Δ T Al Δ t$
in which Te, T, mc, cc, ∆c, and ∆TAl were the experimental results recorded at each ∆t = 5 s.
The hef values when the sphere was covered by the dry cotton tissue were determined as a particular case in which mw was constant and equal to zero.
### Correlation for Air-Sphere Convective Heat Transfer Coefficient
Some well-known empirical correlations can be used to estimate the convective heat transfer coefficient between a solid sphere and a fluid without simultaneous mass transfer. McAdams proposed an empirical correlation to estimate the convection heat transfer coefficient between a sphere and a gas (Eq. (11)) in which all properties are evaluated at the film's temperature. This correlation is limited to 17 < Re < 70,000 (McAdams, 1954McAdams, W. H., Heat Transmission. McGraw-Hill, New York (1954).; Holman, 1992Holman, J. P., Heat Transfer. McGraw-Hill, London, UK (1992).),
(11) $N u = hD k ∞ = 0 , 37 Re 0 , 6 = 0 , 37 ρ ∞ D ν$
in which Nu is the Nusselt Number; D is the sphere's diameter (DAl or De); and k, ρ, and ν are the gas's (air) thermal conductivity, density, and kinematic viscosity, respectively. In the present study, Re values were close to 7,000 and 10,500 at V1 and V2, respectively.
### Statistical Analysis
Estimated parameters were evaluated by one-way ANOVA at 95% probability level. In cases with significant effects (p < 0.05), the average values were compared using Tukey's test.
# RESULTS AND DISCUSSION
## Convective Heat Transfer Coefficients
Table 1 summarizes h and hef values determined from experimental time-temperature data and those estimated from McAdams correlation. It is remarkable that, in all experimental conditions, Bi values were equal to or less than 10-2, indicating that there were no significant temperature gradients inside the metallic sphere and that the lumped method was properly chosen. Estimated values of h and hef had low coefficients of variation (CV < 10%), indicating reproducibility. The coefficient of determination (R2) of the linear regression (Eq. (3)) to experimental data ranged from 0.987 to 0.999.
Table 1
h and hef values determined by experimental time-temperature data and by McAdams correlation for the two air velocities (with and without a cotton tissue) and three RH levels (only when the wet tissue covered the sphere). Results contain: average value ± standard deviation, and coefficient of variation in parentheses.
In the present study, both predicted (McAdams correlation) and experimental coefficients are of the same order of magnitude. Predicted h values between the uncovered sphere and air flowing around it were 23.0% and 25.2% (at V1 and V2, respectively) lower than experimental values. Appling the correlation to the aluminum sphere covered by dry tissue underestimates the coefficients in 33.0% and 28.1% (at V1 and V2, respectively). Although the chosen correlation is appropriate for air flowing around a sphere, errors up to 30% are expected from empirical correlations estimating convective heat transfer coefficients (McAdams, 1954McAdams, W. H., Heat Transmission. McGraw-Hill, New York (1954).; Holman, 1992Holman, J. P., Heat Transfer. McGraw-Hill, London, UK (1992).). McAdams correlation was obtained by a simplified linear regression to a big set of data, including spheres with different surface roughness and air flow under a large range of Reynolds numbers. It leads to differences even between these original data used to estimate the correlation and the predicted values.
Evaporative phenomena increased the heat transfer (resulting in an effective coefficient, hef) more than two-fold in all experimental setups with wet tissue compared with the same setup with dry tissue. Moreover, h and hef results in Table 1 show that increasing air velocity promotes significant coefficient increases (p < 0.05), as expected, while the RH did not affect hef significantly (p > 0.05) at either air velocity.
The difference in water vapor pressure between the wet tissue and the air (quantified by the RH values) is the driving force for mass transfer, promoting evaporation and increasing heat transfer and, consequently, resulting in hef>h with the same experimental setup. Similar results were reported by Rainieri et al. (2009)Rainieri, S., Bozzoli, F. and Pagliarini, G., Effect of a hydrophobic coating on the local heat transfer coefficient in forced convection under wet conditions. Exp. Heat Transf., 22, 163-177 (2009)., who determined local heat transfer coefficients on an aluminum plate coated with a non-wetting material, on which dropwise condensation of water vapor was carried by a humid air stream. The results showed that, for a given air velocity, the heat transfer coefficient between air and the wet surface was up to six times higher than the values determined for the dry surface.
## Chilling Times
Figure 2 presents time-temperature evolutions as an example of the six experimental replicates for the uncovered aluminum sphere (TAl), covered by dry tissue, and covered by wet tissue at both air velocities. The dry tissue works as a thermal insulator, reducing heat transfer and increasing chilling time for both air velocities. On the other hand, a great reduction in cooling time was observed (e.g., more than three fold at RH = 60%), as a consequence of water evaporation on the sphere surface covered by the wet tissue. Evaporative cooling phenomena increased the heat transfer rates despite the thermal insulation associated with the wet tissue. This behavior was observed at all RH conditions and for both air velocities (data not shown).
Figure 2
Time-temperature for the aluminum sphere (TAl), cooling air (T), and internal (Ti) and external (Te) cotton tissue surfaces and time-RH evolution at two air velocities (V1 = 0.92 m s-1 and V2 = 1.42 m s-1): (a) V1 and (b) V2- uncovered sphere; (c) V1 and (d) V2- sphere covered by wet tissue (RH = 60%); (e) V1 and (f) V2- sphere covered by dry tissue.
Figure 3 illustrates the cooling time taking into account the core time-temperature evolution in the sphere covered with wet tissue at different RH and air velocities. As expected, the chilling time of the sphere covered with wet tissue decreased with lower RH and with the increase of air velocity inside of chamber. It shows a time increase close to 50% to reach 15 ºC and a time increase over 70% to reach 4 ºC (V1-90% compared to V2-30%).
Figure 3
Cooling time of core time-temperature evolution in the sphere covered with wet cotton tissue for different RH and air velocities.
Both experimental and adjusted time-water mass evolution of cotton tissue during evaporative cooling at different RH levels (30%, 60%, and 90%) and air velocities (V1 and V2) are presented in Figure 4. An initial water mass equal to 0.0338 kg (average values from experimental measurements) was assumed for all simulations. As illustrated in the figure, the empirical Equation (Eq. (9)) was appropriate to represent the time-water mass evolution in cotton tissue during different experimental cooling conditions.
Figure 4
Experimental and adjusted time-water mass evolution of cotton tissue during cooling of the aluminum sphere covered by wet tissue in different conditions: (a) V1-30%, (b) V2-30%, (c) V1-60%, (d) V2-60%, (e) V1-90%, (f) V2-30%.
Table 2 summarizes experimental results comparing the influence of air velocity and RH on water loss (relative to the initial amount of water and determined by wet tissue weight reduction) and chilling time (time to reach TAl = 4 ºC). As expected, RH impacted water loss during cooling. Nevertheless, the largest difference was lower than 6% (37.6% and 42.1% of initial water content at V2-90% and V1-30%, respectively). The longest chilling time was at V1 and RH = 90% (the worst condition, because of the highest RH and lowest air velocity), requiring 65% more time to reach 4 ºC in the sphere's core than the process performed at V2 and RH = 30% (the best condition, because of the highest air velocity and lowest RH). Comparing experiments at the same air velocity, an RH increase from 30% to 90% extended the cooling time by 38% and 31%, at V1 and V2, respectively. These differences are meaningful and show the power of evaporative phenomena to enhance heat transfer in this kind of chilling process.
Table 2
Water loss by the wet tissue caused by the evaporative cooling, chilling time required to reach 4 ºC, and correlation coefficients (R2) obtained from regression of Eq. (9) to the water loss data. Data contain average value ± standard deviation, and coefficient of variation (in parentheses).
# CONCLUSIONS
The experimental approach proposed in this study allows assessing the effective heat transfer coefficient between a metal sphere covered with a moist cotton tissue and forced air. Under the investigated experimental conditions, an effective heat transfer coefficient approximately three-fold higher was found for the sphere covered with the moist cotton tissue. From a practical point of view, it is possible to increase heat transfer by spraying a solid surface (e.g., solid foods) with clean water, if it is tolerable. Moreover, the experimental results showed that it is possible to control product water loss by changing the velocity and RH of the cooling air. Many industrial processes, such as food cooling by air blast, can take advantage of these results. Moreover, this experimental approach, requiring only temperature measurements, can be tested in industrial plants, in which, in general, airflow and properties are expensive and impractical to be measured.
# ACKNOWLEDGMENT
The authors are thankful for the financial support from CNPq/Brazil (Projects: 482581/2011-5 and 506556/2013-1) and CAPES/Brazil.
NOMENCLATURE
Greek Letters
# REFERENCES
• Amendola, M. and Teruel, B., Uso de um esquema implícito e de splines para a simulação numérica do processo de resfriamento de frutas esféricas. Rev. Bras. Eng. Agríc. Amb., 9, 78-82 (2005). (In Portuguese).
• Becker, B. R. and Fricke, B. A., Heat transfer coefficients for forced-air cooling and freezing of selected foods. Int. J. Refrig., 27, 540-551 (2004).
• Carciofi, B. A. M. and Laurindo, J. B., Water uptake by poultry carcasses during cooling by water immersion. Chem. Eng. Proc., 46, 444-450 (2007).
• Carciofi, B. A. M. and Laurindo, J. B., Experimental results and modeling of poultry carcass cooling by water immersion. Ciênc. Tec. Ali., 30, 447-453 (2010).
• Chuntranuluck, S., Wells, C. M. and Cleland, A. C., Prediction of chilling times of foods in situations where evaporative cooling is significant-Part 1. Model development. J. Food Eng., 37, 111-125 (1998).
• Cuesta, F. J., Lamúa, M. and Alique, R., A new exact numerical series for the determination of the biot number: Application for the inverse estimation of the surface heat transfer coefficient in food processing. Int. J. Heat Mass Transf., 55, 4053-4062 (2012).
• Dincer, I., New effective Nusselt-Reynolds correlations for food-cooling applications. J. Food Eng., 31, 59-67 (1997a).
• Dincer, I., Heat Transfer in Food Cooling Applications. Taylor & Francis, Washington, DC (1997b).
• Erdogdu, F., Sarkar, A. and Singh, R. P., Mathematical modeling of air-impingement cooling of finite slab shaped objects and effect of spatial variation of heat transfer coefficient. J. Food Eng., 71, 287-294 (2005).
• Holman, J. P., Heat Transfer. McGraw-Hill, London, UK (1992).
• Huezo, R., Smith, D. P., Northcutt, J. K. and Fletcher, D. L., Effect of immersion or dry air chilling on broiler carcass moisture retention and breast fillet functionality. J. Appl. Poult. Res., 16, 438-447 (2007).
• Incropera, F. P. and Dewitt, D. P., Fundamentos de transferência de calor e de massa. LTC, Rio de Janeiro, Brazil (2002). (In Portuguese).
• James, C., Vincent, C., Lima, T. I. and James, S. J., The primary chilling of poultry carcasses - a review. Int. J. Refrig., 29, 847-862 (2006).
• Jeong, J. Y., Janardhanan, K. K., Booren, A. M., Harte, J. B. and Kang, I., Breast meat quality and consumer sensory properties of broiler carcasses chilled by water, air, or evaporative air. Poult. Sci., 90, 694-700 (2011).
• Landfeld, A. and Houska, M., Prediction of heat and mass transfer during passage of the chicken through the chilling tunnel. J. Food Eng., 72,108-112 (2006).
• Laurindo, J. B., Carciofi, B. A. M., Silva, R. R. and Hense, H., On-line monitoring of heat transfer coefficients in a stirred tank from the signatures of the resultant force on a submerged body. Int. J. Refrig., 33, 600-606 (2010a).
• Laurindo, J. B., Carciofi, B. A. M., Silva, R. R., Dannenhauer, C. E. and Hense, H., Evaluation of the effects of water agitation by air injection and water recirculation on the heat transfer coefficients in immersion cooling. J. Food Eng., 96, 59-65 (2010b).
• Lucas, T. and Raoult-Wack, A. L., Immersion chilling and freezing in aqueous refrigerating media: Review and future trends. Int. J. Refrig., 21, 419-429 (1998).
• McAdams, W. H., Heat Transmission. McGraw-Hill, New York (1954).
• Mekprayoon, R. and Tangduangdee, C., Influence of Nusselt Number on weight loss during chilling process. Proc. Eng., 32, 90-96 (2012).
• Mielnik, M. B., Dainty, R. H., Lundby, F. and Mielnik, J., The Effect of evaporative air chilling and storage temperature on quality and shelf life of fresh chicken carcasses. Poult Sci., 78, 1065-1073 (1999).
• Rainieri, S., Bozzoli, F. and Pagliarini, G., Effect of a hydrophobic coating on the local heat transfer coefficient in forced convection under wet conditions. Exp. Heat Transf., 22, 163-177 (2009).
• Sun, D.-W. and Wang, L., Experimental investigation of performance of vacuum cooling for commercial large cooked meat joints. J. Food Eng., 61, 527-532 (2004).
• Teruel, B., Cortez, L. and Neves Filho, L., Estudo comparativo do resfriamento de laranja valência com ar forçado e com água. Ciênc. Tec. Ali., 23, 174-178 (2003). (In Portuguese).
• Trujillo, F. C. and Pham, Q. T., A Computational fluid dynamic model of the heat and moisture transfer during beef chilling. Int. J. Refrig., 29, 998-1009 (2006).
• Verboven, P., Scheerlinck, N. and Nicolai, B. M., Surface heat transfer coefficients to stationary spherical particles in an experimental unit for hydrofluidisation freezing of individual foods. Int. J. Refrig., 26, 328-336 (2003).
# Publication Dates
• Publication in this collection
Jan-Mar 2017
# History
• Received
07 July 2015
• Reviewed
06 Nov 2015
• Accepted
22 Dec 2015
Brazilian Society of Chemical Engineering Rua Líbero Badaró, 152 , 11. and., 01008-903 São Paulo SP Brazil, Tel.: +55 11 3107-8747, Fax.: +55 11 3104-4649, Fax: +55 11 3104-4649 - São Paulo - SP - Brazil
E-mail: rgiudici@usp.br | 2022-08-07 17:00:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 22, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.43962085247039795, "perplexity": 5828.388192187429}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882570651.49/warc/CC-MAIN-20220807150925-20220807180925-00781.warc.gz"} |
https://www.physicsforums.com/threads/binomial-expansion-for-rational-index-please-help.117908/ | 1. Apr 18, 2006
### sparsh
Hi
I wanted to know what is the expansion of (1+x)^n when n is a rational number and |x|<1 ...
Please let me know as soon as possible..
Sincerely
Sparsh
2. Apr 18, 2006
### Hootenanny
Staff Emeritus
$$(1+x)^{n} = 1 + nx + \frac{n(n-1)}{1\cdot 2}x^{2} + ... + \frac{n(n-1)...(n-r+1)}{1\cdot 2 ... r}x^{r}$$
Where $|x|<1$ and n is any real number. This can be derived from the general binomial expansion of $(a+b)^n$.
Regards,
~Hoot
3. Apr 18, 2006
### HallsofIvy
Staff Emeritus
I assume you know that, for n a positive integer
$$(1+ x)^n= 1+ nx+ ... + _nC_i x^i+ ...$$
where
$$_nC_i= \frac{n!}{i!(n-i)!}= \frac{n(n-1)...(n-i+1)}{i!}$$
For n a rational number, basically the same formula is true. Only now $_nC_i$ is never 0 so we get an infinite sum.
For example, if n= 1/2 then $_{\frac{1}{2}}C_1= \frac{1}{2}$, $_{\frac{1}{2}}C_2= \frac{\frac{1}{2}(\frac{1}{2}-1)}{2}= -\frac{1}{8}$, $_{\frac{1}{2}}C_3= \frac{\frac{1}{2}(\frac{1}{2}-1)(\frac{1}{2}-2)}{6}= \frac{1}{16}$, etc. so that
$$(1+ x)^{\frac{1}{2}}= 1+ \frac{1}{2}x-\frac{1}{8}x^2+ \frac{1}{16}x^4-...$$
exactly as you would get from the Taylor's series.
4. Apr 19, 2006
### sparsh
Thanks to both .
The post by HallsofIvy was particularly useful . | 2016-12-10 13:20:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9113240838050842, "perplexity": 1488.3222468187062}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698543170.25/warc/CC-MAIN-20161202170903-00481-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://princetonuniversity.github.io/Athena-Cversion/AthenaDocsParIntro?action=diff&version=4 | Documentation/UserGuide/Introduction to Particles
The particle code integrates a number of Lagrangian particles in the gas that are subject to the aerodynamic drag. The drag force to the particles is given by
where $m$ is the particle mass, $v_g$ and $v_p$ are the velocity of gas and particles, $t_{stop}$ is the stopping time, characterizing the coupling between gas and particles. Particles can either respond to the gas motion passively, or also provide back reaction to the gas as momentum and energy feedback. Two particle integrators are designed to deal with particles with any $t_{stop}$ from zero to infinity. A full description of the algorithms and test problems can be found in ‘Particle-gas Dynamics with Athena: Method and Convergence’ (Bai & Stone, 2010). The particle integrators and the overall hybrid particle-gas scheme are both second order accurate in space and time.
Main applications of the particle-gas hybrid code include:
1. Aerodynamics of dust grains / solids in protoplanetary disks, with relevance on dust transport, heating, planetesimal/chondrule formation. Momentum feedback can be turned on or off depending on the application.
2. Tracking the motion of fluid parcels in any MHD simulations. In this case, simply use passive particles with $t_{stop}=0$.
3. Proper modifications to the particle integrator can make it suitable to integrate particles that are subject to other forces, such as the Lorentz force (see Lehe, Parrish & Quataert, 2009).
The current implementation of the particles works with MPI, but it does NOT work with static mesh refinement. Moreover, it only works with the CTU integrator. | 2018-01-20 12:53:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6034602522850037, "perplexity": 941.519375302483}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084889617.56/warc/CC-MAIN-20180120122736-20180120142736-00307.warc.gz"} |
https://www.physicsforums.com/threads/average-velocity-of-an-object.154024/ | # Average velocity of an object
1. Jan 31, 2007
### future_vet
Can we say that when the velocity is constant, the average velocity of an object is equal to the instantaneous velocity?
Thanks!
2. Jan 31, 2007
### cristo
Staff Emeritus
If the velocity is constant, then at every point on the particle's path, the velocity will be the same. Thus the average velocity will be equal to the velocity at any point on the particle's path. If you are defining the instantaneous velocity to be the velocity at any point, then the answer to your question is yes.
3. Jan 31, 2007
### future_vet
Thank you so much, you are very very helpful!
4. Jan 31, 2007
### cristo
Staff Emeritus
You're welcome! | 2017-05-30 09:47:41 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8368090987205505, "perplexity": 486.0975882549855}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463614620.98/warc/CC-MAIN-20170530085905-20170530105905-00248.warc.gz"} |
http://mathoverflow.net/revisions/78129/list | I have casually almost (i.e. up to details that shoud work) proved the following discrete version of Brouwer's fixed point theorem. I should have obtained this result as a corollary of quite complicated things and I do not understand if the result is trivial and can be easily proved directly or it deserves to be stressed. I would like to hear your opinion about that.
Let $n\geq1$ be a fixed integer and denote by $X=[-n,n]^2$. X=[-n,n]^2\subseteq\mathbb Z^2$. Given$(x,y)\in X$I denote by$A(x,y)$the set formed by the following at most five points:$(x-1,y),(x,y),(x+1,y),(x,y-1),(x,y+1)$. At most means that if one of those points does not belong to$X$, I will not consider it. The result would be: let$f:X\rightarrow X$such that for all$(x,y)\in X$one has$f(A(x,y))\subseteq A(f(x,y))$. Then$f$has a fixed point. Is that trivial? Thank you in advance, Valerio 2 misprint fixed: y+1 replaced to (x,y+1) I have casually almost (i.e. up to details that shoud work) proved the following discrete version of Brouwer's fixed point theorem. I should have obtained this result as a corollary of quite complicated things and I do not understand if the result is trivial and can be easily proved directly or it deserves to be stressed. I would like to hear your opinion about that. Let$n\geq1$be a fixed integer and denote by$X=[-n,n]^2$. Given$(x,y)\in X$I denote by$A(x,y)$the set formed by the following at most five points:$(x-1,y),(x,y),(x+1,y),(x,y-1),(y+1)$. (x-1,y),(x,y),(x+1,y),(x,y-1),(x,y+1)$. At most means that if one of those points does not belong to $X$, I will not consider it.
The result would be: let $f:X\rightarrow X$ such that $f(A(x,y))\subseteq A(f(x,y))$. Then $f$ has a fixed point.
Is that trivial?
Valerio
1
# A Brouwer fixed point theorem on finite sets
I have casually almost (i.e. up to details that shoud work) proved the following discrete version of Brouwer's fixed point theorem. I should have obtained this result as a corollary of quite complicated things and I do not understand if the result is trivial and can be easily proved directly or it deserves to be stressed. I would like to hear your opinion about that.
Let $n\geq1$ be a fixed integer and denote by $X=[-n,n]^2$. Given $(x,y)\in X$ I denote by $A(x,y)$ the set formed by the following at most five points: $(x-1,y),(x,y),(x+1,y),(x,y-1),(y+1)$. At most means that if one of those points does not belong to $X$, I will not consider it.
The result would be: let $f:X\rightarrow X$ such that $f(A(x,y))\subseteq A(f(x,y))$. Then $f$ has a fixed point.
Is that trivial? | 2013-05-20 08:23:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9459362030029297, "perplexity": 148.14889572461365}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368698646863/warc/CC-MAIN-20130516100406-00050-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://entrelink.hk/event/photoshop-2021-version-22-4-1-with-license-key-free-download-mac-win-march-2022/ | In this book, I walk you through the basics in Photoshop and use it for basic image editing of both raster and vector images. When necessary, you are given helpful tutorials that teach you how to use Photoshop for specific tasks. These tutorials are an ideal way to get a professional feel for Photoshop even though you’re a beginner.
Although I recommend that you master the basics of Photoshop before tackling this book, if you’re already familiar with Photoshop, you can get started with this book right away.
# Chapter 1: Getting Acquainted with Photoshop
After you read this book, you’ll be a Photoshop pro. To help you get acquainted with Photoshop and get started editing images, I give you an overview of Photoshop in this chapter.
# What is Photoshop?
Simply put, Photoshop is a powerful image-editing tool used to create and edit raster images. Photoshop allows you to work on a number of different layers and add and manipulate objects. In this book, I’m using Photoshop CS5, although the basics in this book are applicable to earlier versions of Photoshop as well.
You may have heard the term _vector artwork_ in reference to Adobe Illustrator. Adobe Illustrator is a vector design tool that enables you to work with large graphics. The basic purpose of Illustrator is to provide a smart path for you to create great-looking graphics. The layers that make up the entire design are easily interchangeable, and shapes and colors can be easily deleted or pasted from one graphic to another.
It’s important to note that the term _vector artwork_ may mean two different things:
* **Raster artwork:** Raster artwork is created with a computer program that takes a photograph, scans a drawing, or captures an image to create its raster format.
* **Vector artwork:** Vector artwork is created in a design program that lets you draw a path for any image you want to create. Vector artwork opens easily in most design programs and in Photoshop.
If you’re interested in creating raster artwork, this book is for you. However, if you’re interested in creating vector artwork and don’t need to create raster artwork, you may want to skip the next section and go directly to the next chapter, which covers Adobe Illustrator.
# The Building Blocks of Photoshop
Photoshop can be broken down into three areas: the toolbox, the workspace, and the canvas.
The toolbox is composed of tools
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A blend mode is used to give various effects to an image. In most cases, Photoshop allows you to merge two separate images into one. There are also different blend modes, such as screen, overlay, multiply, lighten, color balance, etc.
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CONCLUSION {#sec5}
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In conclusion, the sEPSP in the subiculum region is likely to be a neural substrate of the antidepressive effects of BES. Persistent sE
## What’s New in the Photoshop 2021 (Version 22.4.1)?
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I am trying to find a series representation for the function
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on \$D(\lambda) = \{z\in \mathbb{C} : |z| Treatment strategy for patients with hemophilia A or B in an era of coagulation factor inhibitors: a statement of the Brazilian Association for Hemophilia and Associated Disorders – ABRASCO.
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Hard Drive: 18 GB available space | 2022-08-14 15:57:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17538653314113617, "perplexity": 2181.949992768372}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572043.2/warc/CC-MAIN-20220814143522-20220814173522-00304.warc.gz"} |
http://mathoverflow.net/questions/12425/canonical-bases-for-modules-over-the-ring-of-symmetric-polynomials?sort=votes | # Canonical bases for modules over the ring of symmetric polynomials
The ring $S=\mathbb{C}[x_1,x_2,\dots,x_n]^{S_n}$ of symmetric polynomials has a number of commonly used bases, but the undisputed world champion of these is the basis consisting of Schur polynomials $s_\lambda$, where $\lambda$ ranges over non-increasing sequences $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n \geq 0$ of non-negative integers. For a partition $\mu$ of $n$, let $V_\mu$ be the corresponding irreducible $S_n$-module, and let $M(\mu)=(\mathbb{C}[x_1,x_2,\dots,x_n] \otimes V_\mu)^{S_n}$ be the ($S$-module of) $S_n$-invariant polynomial functions on $\mathbb{C}^n$ with values in $V_\mu$. Is there a $\mathbb{C}$-basis of $M(\mu)$ that deserves top billing?
(A bit of background: the dimensions of the homogeneous components of $M(\mu)$ can be computed from the exponents of $V_\mu$, that is, the degrees in which it appears in the coinvariant algebra. There are combinatorial expressions known for these numbers---see e.g. Stembridge's paper "On the eigenvalues of representations of reflection groups and wreath products", Pacific J. Math. 140 (1989), 353--396 and the references therein, but they are not obtained by writing down a particularly nice basis.)
-
In my paper Cyclage, catabolism, and the affine Hecke algebra http://arxiv.org/abs/1001.1569 I exhibit a canonical basis for $\mathbb{C}[x_1,x_2,\dots,x_n]$ and more generally a canonical basis for $\mathbb{C}[x_1,x_2,\dots,x_n] \otimes V_\mu$ coming from the extended affine Hecke algebra of type A. The subset of this canonical basis corresponding to cells of shape $(n)$ is a basis for the $S_n$-invariants in this module. For the special case $M(n)$, these canonical basis elements do correspond to Schur functions -- they are Schur functions in the Bernstein generators times $e^+$, where $e^+$ spans the trivial representation for the finite Hecke algebra (see Theorem 6.1).
I have not thought too much about the combinatorics of the canonical basis for $M(\mu)$, but it may be possible to work this out explicitly, including an explicit description of the $S_n$ invariants (see Example 9.21 for a little bit about this). This paper is long and may take some time to get through. Feel free to contact me at the email address at the very bottom of the paper if you have any questions or want to discuss this in detail.
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Thanks! I'll have a look. – GS Apr 7 '10 at 10:54
One profitable thing to look at might be geometric Satake:
Roughly, one can categorify the symmetric polynomials acting on all polynomials as perverse sheaves on $GL(n,\mathbb{C}[[t]])\setminus GL(n,\mathbb{C}((t)))/GL(n,\mathbb{C}[[t]])$ acting on perverse sheaves on $GL(n,\mathbb{C}[[t]])\setminus GL(n,\mathbb{C}((t)))/I$ where $I$ is the Iwahori (matrices in $GL(n,\mathbb{C}[[t]])$ which are upper-triangular mod $t$).
The maps to polynomials are take a sheaf and send it to the sum over sequences $\mathbf{a}$ of $n$ integers of the Euler characteristic of its stalk at the diagonal matrix $t^{\mathbf{a}}$ times the monomial $x^{\mathbf{a}}$.
One nice thing that happens in this picture is the filtration of polynomials by the invariants of Young subgroups appears as a filtration of categories. Thus, one can take quotient categories and get a nice basis, with lots of good positivity, for the isotypic components.
For the multiplicity space, one might be able to do some trick using cells. It's not immediately clear to me how.
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Thanks; I like geometric Satake and hadn't thought of using it! I'm upvoting, but I hope you don't mind if I don't accept the answer in the (likely vain) hope that someone will come along with a totally different answer. – GS Jan 21 '10 at 12:09
It's not even a complete answer, so I certanly wouldn't expect you to accept it. – Ben Webster Jan 21 '10 at 14:15 | 2014-04-17 04:52:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8701393604278564, "perplexity": 163.1717441551998}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00656-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://www.cogitolingua.net/blog/ | ## Statistical Measures
In the stats book that I used at college, A First Course in Probability (sixth ed) by Sheldon Ross, I found two problems that seem paradoxical when juxtaposed. Can you explain the opposite results?
Ch 2 Axioms of Probability, Self-Test Exercise #15.
Show that if $P(A_i) = 1$ for all $i\geq1$, then $P\left(\bigcap\limits_{i=1}^{\infty} A_i\right) = 1$.
Ch 5 Continuous Random Variables, Theoretical Exercise #6.
Define a collection of events $E_a, 0 < a < 1$, having the property that $P(E_a) = 1$ for all $a$, but $P\left(\bigcap\limits_{a} E_a\right) = 0$.
Hint: Let random variable $X$ be uniform over $(0,1)$ and define $E_a$ in terms of $X$.
## Acer Swift 1
I decided to upgrade my laptop, and chose to get the Acer Swift 1 SF113-31-P6XP (the rose gold color). The Acer website indicated that this model would have a keyboard backlight, but it does not. It has 3 stuck pixels and a weird bright spot in the display that looks like a reflection, which I only notice when showing bright colors. Since I changed all my settings over to dark mode, I don’t notice these issues at all.
The laptop itself is slightly underpowered, so occasionally an application will behave as if it got paused for a second. Because my main use case for this device is just surfing the web in bed, I don’t mind that behavior at all, and have come to expect it even on workhorse machines. I blame javascript, plugins, and browser architecture generally. Bonus points: the device has no fan and runs completely silent. I find the trade-off worth it. The N4200 supports bursting up to 2.5GHz and Linux makes good use of that ability. I had no problem streaming videos and playing them full-screen.
My biggest complaint comes from the trackpad, which kept freezing. So, I took some steps to remedy that problem (in Ubuntu).
1. apt install xserver-xorg-input-synaptics, for some reason this does not install with xserver-xorg-input-all. It’s presence opens up a bunch of configuration options regarding click behavior, scrolling, palm detection, etc.
2. Create a script that will cycle the touchpad when it freezes and create global keyboard shortcut to run it. If the touchpad freezes, at least you have a button to get it back.
#!/bin/bash declare -i ID ID=xinput list | grep -Eio '(touchpad|glidepoint)\s*id\=[0-9]{1,2}' | grep -Eo '[0-9]{1,2}' xinput disable $ID sleep 0.1 xinput enable$ID
I spent a day using this setup and must have hit the cycle button at least 50 times. Though it was quick, it got really annoying.
3. One time the touchpad didn’t respond after resuming from sleep. So I dug deeper to see if I could virtually unplug and replug it.
If the touchpad doesn’t come back after using the above script, then you can cycle the responsible kernel module.
sudo modprobe -r hid_multitouch sudo modprobe hid_multitouch
4. After some more research, I learned that other Acer models had similar issues, but they could be fixed with a change to the bios settings. During bootup press F2 to access the bios, then switch Main > Touchpad from Advanced to Basic.
For the past five days, I have not had to cycle the touchpad (step 2) since changing the bios flag (step 4).
## Debunking the Intrinsic Value Argument
I have to admit to having updated my mind about the “intrinsic value” argument that many people cite as a justification for treating gold as a money (vs paper currency). I’ve previously attempted to explain away this argument as a side-effect of other properties[Gold is Money] or to dismiss it as an unrelated feature[The Commodity Money Myth]. Now I have some good reasons to believe that the entire argument is unsound.
First, a conversation that I had with a fellow camper at the Jackalope festival.
Person: Gold is money and Bitcoin only a currency.
Me: Ok, what’s the difference?
Person: Well, money can operate as a store of value.
Me: Interesting, how do you store something subjective?
Person: *mumble something about intrinsic value that I find unconvincing and irrelevant*
If you take the Subjective theory of value seriously, then it’s obvious that “intrinsic value” is an illusion. Gold has held its value for a long time, sure, but that’s because people, individuals, continue to have a high subjective value for that material. I don’t see a big problem expecting similar valuations in the future, but that position says much more about human preferences than it does about a shiny yellowish metal.
Next, a dismantling of the argument’s structure.
To say that gold makes a good money because it has some other uses (jewelry for the Ancients, electronics also for modern society) is to cite competing non-monetary uses! Do you really find it convincing to hear someone say “Y is a good X because its useful for non-X” or “Let’s trade with this substance instead of putting it to these other uses”? Consider some of the implications:
• If the other uses become more highly valued than facilitation of trade, your commodity money will disappear from circulation.
• Those other uses have to compete with use as money, making them have a higher price than they otherwise would.
Wouldn’t the world be better off to use that gold industrially or culturally rather than sequester it away in a vault? Cryptocoins can help with that liberation, for they have no competing uses. By explicit design, their highest value use is to facilitate trade.
Furthermore, under the theory of intrinsic value: the more competing uses a substance has, the better a money it becomes. Ridiculous! The very structure of the intrinsic value argument undermines what it attempts to buttress.
## Cognitive Bias in Artificial Intelligence
I believe that artificial intelligence will suffer from cognitive biases, just as humans do. They might be altogether different kinds of bias, I won’t speculate about the details. I came to this conclusion by reading “Thinking Fast and Slow” by psychologist Daniel Kahneman, which proposes the brain has two modes of analysis: a “snap judgement” or “first impression” system and a more methodical or calculating system. Often we engage the quick system out of computational laziness. Why wouldn’t a machine do the same?
Researchers in machine learning already take careful steps to avoid many biases: data collection bias, overfitting, initial connection bias in the neural net, etc. But, I haven’t yet heard of any addressing computation biases in the resulting neural net. I think precursors of biased behavior have been observed already, but was explained away as being present in the input data or as resulting from the reward function during training, or some other statistical inadequacy.
Let me give a simplified example (and admittedly poor example for my argument) of cognitive bias present in humans and reflect on why it would be difficult to filter out such bias in a machine learning algorithm.
In the Muller-Lyer Illusion, which consists of a pair of arrows with fins pointing away or toward the center. Each shaft has the same length, but one appears longer. As a human familiar with this illusion, I will report that the shafts have equal length. Yet, subjectively, I do indeed perceive them as being different. My familiarity with the illusion allows me to report accurate information, lying about my subjective experience.
Now suppose that we train a neural net to gauge linear distances. And we have a way of asking it whether the lines in the Muller-Lyer diagram have the same length. What will it report? Well that depends, being a machine it might have a better mechanism for measuring lines directly in pixels and thus be immune to the extraneous information presented by the fins on the ends of those lines. But, humans ought to have that functionality as well on the cellular sensory level, yet we don’t. But, if the Muller-Lyer Illusion doesn’t fool the neural net, does a different picture confuse it? So far, yes, such things happen: the ML categorizes incorrectly when a human wouldn’t. We tend to interpret this as a one-off “mistake” rather than a “bias”. But the researchers succumb to evidence bias: they have only one example of incorrect categorization and they don’t perform a follow-up investigation into whether that example represents a whole class, demonstrating a cognitive bias in the neural net.
Now suppose the researcher do perform the diligence necessary and discover a cognitive bias. They generate new examples and retrain the net. Now it performs correct categorization for those examples. Have they really removed the bias at a fundamental level? or does the net now have a corrective layer, like I do? I presume the answer here depends on the computation capacity of the net: simple nets will have been retrained, while more complex ones might only have trained a fixer circuit, which identifies the image as being a specific kind of illusion. Thus, the more capable the neural net, the more likely it starts looking like a human: with a first impression followed by a second guess.
How ought research approach this problem? Should the biases get identified one at a time and subsequently be removed with additional training? Due to the large number of biases (c.f. all of Less Wrong, or this list of cognitive biases), I think that approach doesn’t scale well. Especially considering that biases result from cognitive architecture and trained neural nets differ from human brains, I think the biases in ML will be new to us. Those should be exciting discoveries! I propose training with multiple adversarial nets, each trying to confuse the categorizer. This approach contains architectural symmetry, so it probably won’t work for biases that result from differences in wet-ware vs. hard-ware computation. Those should be even more interesting discoveries!
Humans clearly have a large reliance on contextual clues and the whole point of investing in ML is to capture and replicate that level of cognition. But contextual clues can mislead as easily as they help. So ML ought to have cognitive bias, as humans do, but very likely different kinds. Efforts to train out that bias might even be met with repulsion. Humans feel comfort with the familiar, so cognition which has our biases removed should feel viscerally unwelcome. For example, robots which lack biases associated with empathy will be perceived as sociopathic.
## Your vote doesn’t count, but it does matter.
Under their current political system, the American chattel have a “civic duty” to voice their opinion about who they want as a representative. Every 4 years potential presidents spend billions on campaigns to excite the plebeians to “get out and vote!” to “make their voice heard!”. That money would certainly have more impact if spent on the actual causes that Team Red and Team Blue claim to care about. Rather than offer direct assistance, both parties choose instead to promulgate the most basic falsehood of possible: that your vote counts in the national election for president. Nothing excites people more than sports that matter least.
Let’s count the ways that the system ensures your vote does not count.
First, gerrymandered districts ensure predictable voting outcomes. Politicians regularly carve up their constituency in ways designed to support the current power balance, usually to protect the incumbent. From the national perspective, these districts make predictable state outcomes, whether Red or Blue.
Second, either others outnumber your vote when you hold the minority opinion or you vote with the tide. “In either case, your vote does not decide the outcome. In all of American history, a single vote has never determined the outcome of a presidential election”[Reason, 2012].
Third, the Electoral College can ignore the popular vote. “There is no national election for president, only separate state elections. For a candidate to become president, he or she must win enough state elections to garner a majority of electoral votes.”[Walbert, 2004]. Electoral delegates have no obligation to vote the same way as the popular vote of the state they represent, but they usually remain faithful.
Fourth, in the event that a state doesn’t have a clear position, the Supreme Court might decide. In 2000, the state of Florida did not have a clear preference, even after multiple recounts. When hearing the lawsuit over whether the recounts should continue, the Supreme Court accepted the de-facto power to decide the outcome of the election.
Fifth, Congress can decide. According to the rules of the Electoral College, “If no candidate wins a majority of the electoral votes or if the top two candidates are tied, the House of Representatives selects a president from among the five candidates with the most votes.”[Walbert, 2004]. According to this rule, Libertarian Gary Johnson has a chance in 2016 if he can win his home state of New Mexico [Wilson, 2016].
Now I’ve given reasons why your vote doesn’t count, let me address why it does matter.
South Africa endured many years of violence under the Apartheid regime. Many people and countries worldwide boycotted Apartheid, but the US government insisted on supporting the Apartheid regime, saying that while the US abhorred Apartheid, the regime was the legitimate government of South Africa. Then the Apartheid regime held another election. No more than 7% of South Africans voted. Suddenly everything changed. No longer could the US or anyone else say that the Apartheid regime had the consent of the governed. That was when the regime began to make concessions. Suddenly the ANC, formerly considered to be a terrorist group trying to overthrow a legitimate government, became freedom fighters against an illegitimate government. It made all the difference in the world, something that decades more of violence could never have done.
In Cuba, when Fidel Castro’s small, ragged, tired band were in the mountains, the dictator Batista held an election (at the suggestion of the US, by the way). Only 10% of the population voted. Realizing that he had lost the support of 90% of the country, Batista fled. Castro then, knowing that he had the support of 90% of the country, proceeded to bring about a true revolution.
In Haiti, when the US and US-sponsored regimes removed the most popular party from the ballot, in many places only 3% voted. The US had to intervene militarily, kidnap Aristide, and withhold aid after the earthquake to continue to control Haiti, but nobody familiar with the situation thought that the US-backed Haitian government had the consent of the governed or was legitimate.
You’ve Got to Stop Voting by Mark E. Smith
Whether your candidate has a chance or not, your participation in the vote directly demonstrates your “consent to be governed”. The politicians have a system of elaborate and arcane rules, which they deliberately devised to disenfranchise your voice. The political class cares far more about you checking a box than they do about which box you check.
“Boycotting elections alone will not oust the oligarchy, but it is the only proven non-violent way to delegitimize a government.”[Smith, 2012].
## Notes: Market Failure: An argument both for and against government (David Friedman)
I just attended the Young American’s for Liberty state convention yesterday in order to hear the venerable David Friedman speak. Below are my notes highlighting the main points of the talk. You may recognize some examples and positions if you’ve been following his work.
Economists studying market failure make legitimate arguments against laisse-faire, but those arguments make a stronger case against government. Let’s define market failure as those circumstances in which individual rationality doesn’t lead to global rationality. For example, suppose we were part of an army standing on the battle field. I think, there’s only a minuscule chance that it affects the battle if I defect, and I will almost surely live as the others delay the opposition by fighting. All of us execute that logic, we all run, and the opposition kills us. To take another example, I am warmed greatly by burning coal and only make a minuscule contribution to the London fog. But, it’s possible (though not always), to engineer around the failure with a change of the rules. For example, the arab’s deadlocked in the open desert making no progress to the nearby oasis as they insist on winning the “who’s got the slowest camel” competition. Economists largely assume that people act in their own rational self-interest and that’s generally the case.
But what about Public Goods? (aside: The government often produces private goods, such as the post, while many public goods are produced privately, such as education and libraries.) Let’s define a public good as one that’s open to consumption, where the producer cannot capture payment from the consumer. For example, the beauty of the Sears Tower, listening to an unencrytped radio broadcast, or watching a TV program. For some cases, the market arrived at a clever solution: couple the public good (radio program) with a public bad (advertisement) and let the baddies subsidize the good for the enjoyment of all.
There are often externalities, in both directions. Some the costs outweigh the benefit and others where the cost is less than the benefit but the producer can’t collect enough to make it worth the effort. For example, take a resturant, a movie theater, and a store. It might not be worth running any individual enterprise, as they impose foot traffic on neighbors (negative externality). But if they occur together, say in a mall, then the traffic is mutually beneficial to all stores (externality becomes positive) and the rent for a shopwindow captures some of that.
Not all problems are solveable with laisse-faire and the market result is often less than ideal (comparison to the ideal is how the market ‘failed’ even when the outcome was considered ‘good enough’ by the people). With perfect information, you might be able to obtain the ideal, but we are in very short supply of good dictators.
Democracy also has its failure modes. For example, voters should be knowledgeable and informed when they cast their ballots. But proposals are seldom transparent. For example, the Farm Bill is never advertised as a money transfer program. Plus, in a representative system, the voter has a double-indirection problem. First they must know how good/bad each bill is and then they have to know how the candidates voted. Often that’s unknown, because the candidate is new (actually, all upcoming bills are unknown). Given that the chances of changing the election are slim, how much should a voter invest in becoming informed? Not much, they should be rationally ignorant, with two exceptions. One, the think politics is fun and do it out of intrinsic interest and two, they have influence or represent a special interest and have a high stake in the result. In this market, we see concentrated special interests winning benefits over dispersed victims.
What about long term planning? In the market, why should I plant black walnut trees which won’t bear nuts until I’m long dead? Well, ten years from now I can sell them! to a person that doesn’t want to wait as long. In turn, they could sell years later to a still more impatient person. But the transfer needs strong and secure property right. I must expect that the field is still mine to sell after ten years. Politicians, in contrast, have very insecure property rights. They will often be out of office before the benefits of a bill become evident, and it might be the other party is in office at that time and claims all credit! So they have a strong aversion to paying large amounts now when the benefits are far away in the future. You can see that their rhetoric does not reflect their actual behavior, because they often promise without making delivery in that uncertain future.
So politicians tend to 1. Promote policies that sound good on the surface (easy to advertise), 2. enact bills that benefit special, concentrated interests at the expense of dispersed victims, and 3. take short-term actions. Finally, and what’s worse, they make decisions that have very widespread effects (externalities) without knowing the outcome. For example, a panel of judges making a decision about a vaccination program (for polio?) treated an annual recurring cost as a one-time payment, mis-pricing the program by a factor of 40, negatively affecting thousands of people.
We must conclude then, that Market Failure is structurally endemic to Politics and the theory of market failure is a better predictor of government behavior than it is of free market behavior.
The conclusion generalizes to everyday life. For example, for those with a spouse, who does dishes after dinner? There are two options: 1. one cooks, the other cleans, and 2. the same person cooks and cleans. Everyone chooses option 2, because then the cooker controls how much cleaning takes place (makes a meal with fewer dishes). Also, this is why we tell children to clean up their own mess, rather than the messes of others.
There’s also the silent student problem. When the instructor asks “does every one get it?” they never get a response. Mostly, because the cost of asking is to look dumb in front of everyone (but, rationally, in a large classroom, there must be others who didn’t understand), but also because the benefits of asking will be dispersed to everyone present (they might do better on the exam). One solution is to use a button on the floor, which can be inconspicuously pressed, activating a signal at the back of the class visible to the teacher, but not the students.
Or economics and law. Suppose a proposal that makes armed robbery a capital crime, assuming murder is already a capital crime. Many people might be for it, as “being tough” and providing a strong dis-incentive for armed robbery. But the economist will ask: Do you really want all armed robbers to murder their victims? Because if the cost of robbery is the same as robbery + murder, then I, as a robber, will surely kill my victims for the punishment (cost) is no different and there is a benefit that the dead won’t identify me to the police.
Questions.
What should we do? Given that rationally ignorant voters make decisions on freely available information, politicians complicate the issue by advertising bills with plausible deniability. For example, the auto tariff is about protecting jobs and hurting foreigners. The auto-workers union doesn’t ask for a bill that’s a direct transfer payment. How should we respond? One: change the body of free information. Spread more accurate information and ideas. Two: create alternatives. Run a business that competes, showing that the government-provided solution isn’t any good (quality or quantity).
What about replacing the government entirely? Dealt with this in 3rd part of Machinery of Freedom, discussing private security insurance firms that operate under the discipline of repeated interaction. Customers have their choice of firm.
With increasing productivity, what about ensuring work/jobs? First, we should think of jobs and workers as two distinct numbers, and then notice they are always very close to each other. Sometimes apart (depression era) sometimes close, but very strongly correlated (both increased over the last century). This implies they are in an equilibrium situation, so we should not worry too much. Also, look at the fixes: the racism inherent in the minimum wage (historical union). And be careful: if you give the government power to do XYZ, how will they *actually* use that power?
What about education? School voucher program implies that most schools would be comparable in quality. That’s not really different from being centrally managed. Instead, if we value diversity, we should remove single organizational administration and control, we should decentralize.
What about Rothbard and 100% reserves? It’s actually not desirable for a bank to have 100% backing, especially if it has other liquidate-able reserves. It could then, when faced with a run, sell the other assets in exchange for the backing material (e.g. gold or silver) and make good on the original agreement. Personally, would prefer some electronic, anonymous, cash-like system.
What about the incentives to incarcerate faced by private prison operators? Well, those same incentives are faced by state-run prisons. It sounds good to be “tough on crime” and the taxpayers foot the bill. Refer to David Skarbek’s book, The Social Order of the Underworld: How Prison Gangs Govern the American Penal System, to see how even outlaws have created an ordered society. Also, read Poul Anderson The Margin of Profit for an answer on “how to get people to stop doing bad things” (answ: make it unprofitable), and my book Law’s Order: What Economics Has To Do With Law and Why It Matters for an examination on the costs and benefits of a property system.
## Notes: Problems with Libertarianism
David Friedman gave a talk Problems with Libertarianism: Hard Problems (and how to avoid them).
1. What rights do you have against a criminal?
If your only right is to re-claim the stolen property, then at-worst the thief breaks even.
To what extent can we meter out determent punishment.
What’s the factor of retribution, why 2x? not 3x or 1.5x?
What if you only catch 1/10th of the thieves? Then 10x?
But that pushing the guy caught for the crimes of those not caught.
Also, the number of people caught is a function of how much spent to try to catch.
What if a mistake is made? How much trouble to avoid making them?
2. What are you entitled to do to defend your rights?
Capital punishment for petty theft?
3. Human shield problem.
Can you shoot back, and risk killing the innocent shield?
Can the voluntary defense fund aim nuclear weapons at Moscow?
Possibly killing innocent victims (more so than you) of the Soviet Union?
If it’s acceptable to place run roughshod over innocents in attacking the aggressor, then it’s acceptable to draft.
4. Absolute property rights.
Trespassing photons across absentee-held land.
You think they don’t damage, but I, the owner, gets to decide that.
Allow you to breath in, but not out, because I don’t like the CO2.
5. Distribute the risk of injury should I crash my airplane?
I get the benefit of flight, at cost to you without your permission.
6. Property in land, not derived from owning self+labor.
I could walk across before you build up the house and path, so my use is not in conflict with your labor.
7. Very large part of land in the world is stolen.
8. Public good problem.
If something is desirable, then market will provide. Can only say Maybe.
def: good that producer cannot control who gets it, ex: radio broadcast
Combine the good (pos value to customer + pos cost of production) with another public good (neg value to customer + neg cost of production), ex: adverts
What about national defense? (defense against nations) Hard to stop missile in flight by determining if target has paid for defense.
Is answer to aggress the funds, or to surrender?
soln: Assuming problem doesn’t exist.
Soviets have no interest in attacking, only have tanks to prevent us from doing so.
soln: Somewhere there’s a proof that market will provide.
Nobody’s found it.
soln: It’s a lifeboat problem.
Still have to find the answer, we do live on a spaceship after all.
soln: The is-ought dichotomy.
But then in some way you’re defending the ability to do whatever is necessary.
soln: Pooling money.
The good is worth X but whether I spend is only a fraction of the funds and I receive the benefits regardless.
9. Privatizing the government property.
soln: sell it. But if government doesn’t own it, what right have they to sell it?
You can also avoid the problems by changing the subject.
## Bidding to establish Terms of a Contract
As a preliminary, I feel obligated to mention that writing a contract only protects yourself on paper. The real world has such complexity, and with innumerate contingencies, that any contract will fail to enumerate them all. Establishing an explicit contract often also cements distrust between parties, due to its impersonal nature. The contract just gives a written record of what the parties agreed to, so that, should the end up in arbitration, others can more readily see the terms of the agreement. Because of these factors, writing a contract is a costly affair.. So, what should go into it?
Suppose two parties, even after recognizing the costs, still wish two write a contract. They agree on many standard items and write them down (or borrow them from a template), but the parties still have some remaining details about which they disagree. Each has an interest in investing a some time to sort out these details in order to avoid unpredictable future conflict. But how shall they obtain an agreement on what actually gets written down?
I thought of one approach: holding a silent auction. For every issue on which they have a disagreement, they can write down all the mutually-exclusive options and hold a silent Vickrey auction.
## The Problem and Its Solutions
If we were living in a perfect world, the business logic would be separated from the presentation layer. Since Rave sits atop a rich GUI, where event handlers can execute arbitrary code, there exists a strong temptation to put business logic in the presentation layer. The fact that we code both parts using the same language (C++) makes this temptation doubly hard to resist. Indeed, sometimes a clear cut separation doesn’t exist. So we shouldn’t find it at all surprising that our founding coders may not have kept up a wall of separation between GUI and Business Logic.
Let’s walk through an example that I have adapted from Martin Fowler’s post on GUI Architectures.
Suppose we have a system requirement that the GUI must display a red box when a seat is disconnected, a green box when connected, and a yellow box for a slow connection. Suppose further that the application already uses an integer to represent the connection state, and presents it through the function linkRate(). It ranges from 0Mbs (disconnected) to 1000Mbs (full connection), taking various intermediate values depending on a measured traffic rate (not just the OS ethernet link state). The green box represents any measured rate above 700Mbs, while the red box represents any rate below 5Mbs.
Where should the logic for choosing the box color reside?
Where should the listing of boundary values for each category reside?
Where should the listing of colors reside?
If you had to write tests to prove your solution worked, would you change your mind on where to place that code?
Logic Placement Description
GUI The GUI contains all the smarts. It reads the value of linkRate() from the application and then performs its own calculation to determine the color.
Shared The GUI and application share responsibility. The application provides a linkRateState() that presents an enum which the GUI then maps to a color.
Application The application contains all the smarts. It does the heavy lifting and provides a linkRateColor() method that tells a really dumb GUI what color to show.
For some variation of value preferences, all of the above can be reasonable decisions. I have a bias towards testing making the GUI as easy to test as possible. You might think that idealism would incline me to favor the dumbest GUI possible, and you’d be right in most cases, but I want to draw out some reasons to make an exception for this example.
#### The Case for the Dumb GUI
Mostly I want a dumb GUI because testing it is very hard. To test the GUI, I must launch it within a harness that intermediates all the events, introducing programmability to events like clicks, drags, and keyboard presses. The harness requires a full simulation of the application, including connections to external services (database, file system, SCUs, etc). Finally, at least with squish and RAVE, the test scripts execute at glacially slow human speeds, sleeping for entire seconds to allow for menu animations and other GUI renderings.
Having the dumbest possible GUI would mean having a presentation that so incredibly lightweight that it would be very improbable to get wrong. When the application tells it what color to show, the GUI has very little opportunity to do wrong. The mapping logic of linkRateColor() would have a unit test in the application, ensuring conformance to system requirements. With a thin enough GUI, I wouldn’t care that it didn’t have automated tests.
But placing linkRateColor() in the application muddies the purity of the application. For now it must always and forever include a link to whatever library provides QColor. I can no longer build the application without some GUI library. If I want to re-use that component, I drag the dependency along with it. And, finally, no part of the application actually uses linkRateColor(), it only exists to support the GUI.
#### The Case for the Shared Responsibility
I have nothing to say here but “eeewww gross”. Unless there is an application-side consumer for the linkRateState() it’s not worth coupling the GUI and application with such a specific API. Should the specification change the details about the boundary values between colors, then both the GUI and the application will need updating. We shouldn’t use designs that increase our maintenance overhead.
#### The Case for the Smarter GUI
If the application never has a need to know the boundary values for the different link rates, then we can assume that those values represent a specific presentation requirement. Given that circumstance, I favor placing the logic into the presentation layer.
Yes, this solution increases the GUI’s responsibilities, making it more complicated to test. Counter-intuitively, the increased testing difficulty of a more complex GUI has pushed me to advocate for making the GUI a stand-alone piece. The situation just serves to make my next point, about a stricter separation between presentation and application logic stronger.
## The Wall of Separation (between GUI and Application Logic)
In an ideal world, the business logic carried out by the application and the presentation logic carried out by the GUI remain strictly separated. So separated, that we can pull apart the two pieces and test them separately. We can even build a second GUI (for a new customer) without impacting the underlying application. With this separation, the application acts as a data Model while the GUI(s) merely present a View of that data.
For testability purposes, let’s pull apart the two pieces an envision a wall between them. The only communication link through that wall is an API, depicted as a network socket. The application (network server/data model) responds only to specific messages (requests for and updates to data) sent over the socket. It keeps the GUI informed about changes by emitting other messages (events).
The clear separation between application and GUI serves dual purposes:
1. It makes us think harder about which piece (GUI or app) should receive new logic
2. It allows tests for each piece to remain laser-focused on that piece without getting distracted by the other parts of the system.
#### Testing the Application
To test the application, we simply fake the GUI. Because of the separation I’ve made here, it amounts to just implementing a network client that generates a sequence of data updates or requests, and asserts that it receives expected data-update events and delivery of requested data. In a different world (the real one), where the API exits as method calls instead of a network socket, we create a headless driver, that makes the calls and receives the events. Even more granularity can be achieved by single-stepping the event loop (when that makes sense), to assert that certain events do NOT occur.
In our example, we have the fake GUI assert that it receives a linkRateChanged() event, after the test, using an internal update function, modifies the linkRate variable. If the GUI can set the linkRate, then we can also test round-trip in 3 steps:
1. Have the GUI send the update data request
2. Step the application event loop
3. Assert that the fake GUI receives the expected data changed event.
In both circumstances, we assert that the application generates events according to a specified protocol. With a large enough suite of individualized tests, we cover the application’s behavior for all the actions the GUI can take. When we miss an action, we simply record it in a new test as an expected event/response sequence.
#### Testing the GUI
To test the GUI, we simply fake the application. A test harness drives the GUI from one end, clicking and dragging on widgets and buttons, while the application that it links to provides a pre-programmed series of responses. If we generate the GUI events directly, e.g. by calling on the event handlers for specific widgets, we can even drive the GUI in a headless environment (by virtualizing X11). The tests remain focused on accuracy of presentation.
In our example, we have the fake application emit a linkRatechanged() changed event, and assert that the GUI updates the color according to presentation requirements. If the GUI can set the linkRate, then we can also test round-trip, using a similar 3 steps:
1. Drive the GUI to go through the update link rate dialogs/widgets.
2. Assert that the updateLinkRate() event is received by the fake application and respond with a pre-programmed linkRateChanged() event
3. Assert that the real GUI updates the rendered color.
In both circumstances, we assert that the GUI performs renderings according to the events it receives from the fake application. Again, a large enough suite of individualized test covers the presentation layer for all data states the application can take. We still record any missed behaviors into a new test taking the form of an expected event/render sequence.
## What the Wall of Separation Achieves
Separating the GUI from the application, and treating it as a View or presentation layer only (with the application taking the role of a data model) gives use the ability to separately each pieces. The wall itself represents an expected set of behaviors to command/response stimuli. In ordinary implementation we have direct C++ API calls, but that just muddies the idealized separation, and motivated me to start out with a network messaging description. Conceptually, testing the GUI can be approached with the same techniques as testing a client/server implementing a network protocol. If we clearly state the expected behavior, then each side of the fence merely has to uphold it’s end of the protocol.
Yes, the separation probably means more tests. But, those tests will be smaller, faster to execute, and easier to write and maintain. When we do perform whole-system testing (which is always rarely relative to the automated protocol testing, because of the costs involved), it will catch use-cases of the interaction not already covered by the piece-wise tests. However, a record of the command/response sequence in each failing whole-system use-case, can be rolled backing into separate piece-wise automated tests, one for each side.
Ultimately, our goal is to catch bugs earlier by exercising the behavior protocol of each side separately. By working toward that goal in this way, we can also ensure that we meet our system requirements by encoding them into automated behavior tests exercised against both sides of the wall.
## My Career Forks
Over the past couple months, I’ve busied myself with finding offers of other employment. IMS (now Zodiac Inflight Innovations) has not given me the career growth that I initially anticipated. I find that, though they are getting better, management has been fairly schizophrenic. All the time chasing to put out fires and very little of the time investing in software quality practices that prevent such emergencies.
I interviewed with Google twice. Once last month at the Irvine Office for a position as Sr. Software Engineer. They decided not to hire at that time, because my performance during the interview was “on the edge.” However, their recruiters reached out later to have me interview for a Software Engineer in Test position. I subsequently read the book “How Google Tests Software” and was quite impressed with the specialist role. It’s more of a framework and tools builder for the other engineers, all with the goal of improving quality.
During the time between interviews, the kind fellows at JobSpring Partners, who helped me get hired by IMS (now Zodiac Inflight Innovations), followed up to discover that I was indeed unsatisfied with the career growth opportunities in my current position. They connected me with Fisker Automotive, which is rebuilding a team of software engineers so that they can rewrite the infotainment software that controls the Karma.
So, I stand at a cross-roads in my career. Do I choose a smaller company and pursue technical leadership, or choose the well-established and pursue technical skills growth? I did an analysis, to help myself decide. I would be comfortable with either choice in everything but the “Daily Work” category.
Daily work
fisker: system designer, software architecture
Skills improvement
– google: software engineering, how to program “at scale”
Personal Interactions
– fisker: upper management
Social Capital
– fisker: customer interaction
Technical Capital
– google: lots already in place, but must find a project to exploit it
– fisker: little, have to organize it all myself
Industry
– google: has wide variety of projects (incl. computational finance)
– fisker: automotive, embedded, gui design
On Paper:
– google: I have google on my resume, with crazy job title (they let you make one up)
– fisker: I put “declined an offer from google to work for fisker” on my resume
The Exit | 2018-10-19 22:59:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 11, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3011372983455658, "perplexity": 2808.4557973662145}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583512460.41/warc/CC-MAIN-20181019212313-20181019233813-00048.warc.gz"} |
http://swmath.org/software/13577 | # ComPASS
ComPASS: a tool for distributed parallel finite volume discretizations on general unstructured polyhedral meshes. The objective of the ComPASS project is to develop a parallel multiphase Darcy flow simulator adapted to general unstructured polyhedral meshes (in a general sense with possibly non planar faces) and to the parallelization of advanced finite volume discretizations with various choices of the degrees of freedom such as cell centres, vertices, or face centres. The main targeted applications are the simulation of CO$_2$ geological storage, nuclear waste repository and reservoir simulations. par The CEMRACS 2012 summer school devoted to high performance computing has been an ideal framework to start this collaborative project. This paper describes what has been achieved during the four weeks of the CEMRACS project which has been focusing on the implementation of basic features of the code such as the distributed unstructured polyhedral mesh, the synchronization of the degrees of freedom, and the connection to scientific libraries including the partitioner METIS, the visualization tool PARAVIEW, and the parallel linear solver library PETSc. The parallel efficiency of this first version of the ComPASS code has been validated on a toy parabolic problem using the Vertex Approximate Gradient finite volume spatial discretization with both cell and vertex degrees of freedom, combined with an Euler implicit time integration.
## References in zbMATH (referenced in 3 articles , 1 standard article )
Showing results 1 to 3 of 3.
Sorted by year (citations) | 2018-04-20 10:28:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.30630365014076233, "perplexity": 1219.5615319668657}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125937440.13/warc/CC-MAIN-20180420100911-20180420120911-00263.warc.gz"} |
https://www.physicsforums.com/threads/what-makes-light.733478/ | # What makes light?
1. ### rogerk8
220
Hi!
I wonder what makes light when it comes to excited and/or ionized atoms.
I have tried to study this phenomena both by looking up ionization and recombination on Wikipedia but the answer is missing.
Let's say that an atom is hit by a high energetic particle.
Two things, I believe, then might happen:
1) The outer electron escapes from the atom and the atom is thus ionized.
2) The outer electron is pushed into one of perhaps several outer shells but still resides in the atom and is thus excited.
Now,
In the case of 1)
The only reason for emitting light is if the state of energy is higher before it finally recombines with an electron.
In the case of 2)
Emission of light is made when the atom simply falls back to its original and lower energy state.
But all talk about ionizations making light makes me confused.
Which is true and how can an ionized (excited is ok due to "mgh") atom have a higher energy than a neutral one?
Much obliged for an answer on this one.
Best regards, Roger
2. ### DrClaude
2,327
I don't know where you got the idea that ionization of an atom is accompanied by emission of light, because it most cases it isn't. Exceptions to that might be if the electron being ionized is not the least-bound electron, but a core electron, in which case another electron from the atom will fill the gap, and hence emit a photon.
And why do you mention "mgh"? The binding of electrons in an atom is due to the Coulomb interaction, not gravity. Electrons can exist in different energy orbitals around the nucleus, and if there is a vacancy in a lower energy orbital, a higher energy electron can change state to occupy that lower energy orbital, and the difference in energy is compensated by the emission of a photon of the proper wavelenght.
3. ### rogerk8
220
This is a very good explanation. Thanks!
This is also a very good explanation. Thanks! I do however feel stupid now
Elaborating:
$$F=k\frac{q_1q_2}{r^2}[N]$$
$$W=\int_{r_1}^{r_2} Fdr=-kq_1q_2(\frac{1}{r_2}-\frac{1}{r_1})=-ke^2(\frac{1}{r_1}-\frac{1}{r_2})[Nm]$$
which means that for high r2 the work done by ionizing the atom is
$$W=-\frac{ke^2}{r_1}[Nm]$$
hence, the energy of the atom has increased(?)
Best regards, Roger
Last edited: Jan 18, 2014
4. ### DrClaude
2,327
It doesn't work like that. You have to use quantum mechanics, where electrons do not follow "orbits" at fixed ##r## around the nucleus.
5. ### rogerk8
220
Not even as a principle, you mean?
I do however know that
$$E=hf$$
where Planck's constant comes from quantum mechanics, right?
And E is the energy difference of the "orbits" or possible energy states, right?
Best regards, Roger
6. ### DrClaude
2,327
Not if you want the correct answer. It is possible to use the Bohr model to get a number that is correct for hydrogen, but this is not what an atom really looks like.
That equation is valid for photons. If ##E## is the energy lost by the electron due to the change in orbital, the formula will give you the frequency ##f## of the emitted photon.
7. ### rogerk8
220
A lazy question, what determines E?
Best regards, Roger
8. ### rogerk8
220
The electron's total energy at any radius:
$$E=\frac{1}{2}m_ev^2-\frac{Zk_ee^2}{r}=-\frac{Zk_ee^2}{2r}$$
This means that it takes energy to pull the orbiting electron away from the atom.
And when this energy is taken as in an ionization, the atom energy must lessen, right?
And an electron closer to the nucleus (and therefore higher energy) may fill that gap which then gives rise to light, right?
But what happens to that closer orbit and it's vacancy?
Best regards, Roger
9. ### DrClaude
2,327
You can see this basically as giving the energy to the electron to remove it from the atom. From that point of view, the energy of the atom left behind is unchanged. But we have to be careful, because you usually don't consider each independently, but take the ion + electron system as a whole.
That electron would have lower energy (there is a negative sign in the equation).
10. ### rogerk8
220
So you are giving energy to the electron (and not the atom) to remove it from the atom. And the energy of the atom is unchanged (and not increased while work is done to it or perhaps I should say the electron). Removing the electron far from the atom, according to the formula, however makes the electron energy "zero" as relative to the atom.
Recombination then would mean nothing to the atom. Yet, I think, the electron finding it's way back to the "shell" it came from would mean that the photon energy sent out is equivalent to the energy of that specific shell. Because that is the increasement of energy for that electron.
An electron closer to the nucleus obviously has higher (negative) energy.
I have a really hard time understanding this but let's look at it in layman terms. Positive energy gives "heat", negative energy then must give "cold". In other words, positive energy is an increasement, negative energy is a decreasement.
So if the negative energy is high, there is actually a decreasement of (electron) energy?
This would mean that an electron close to the nucleus would need a higher additional energy to escape the atom, right? All of this because it's energy is so negative which has to be overcome.
This actually sounds reasonable
Best regards, Roger
Last edited: Jan 19, 2014
11. ### DrClaude
2,327
I've been trying to keep it simple, which may have made things less clear than they should be!
We should take the system ion + electron and look at the energy of the system in total. Setting the zero of energy as the the energy when distance between the two is infinite, the energy is negative for the netral atom, as it is the more stable configuration.
Again, from the point of view of the ion + electron system: yes, if the ion captures the electron, the binding energy must be released, most probably through emission of a photon.
A bigger negative energy means lower energy. The zero of energy is arbitrary.
As I said above, where you put the zero is arbitrary, so there is no difference between negative and positive energy. All that is important is relative energy. We usually set the zero for the system at hand in order to easily make a distinction between different possiblities. In some cases, you set the zero for an infinite separation (as above). Sometimes, you will set the zero as the ground state of the atom.
Also, in this context you shouldn't think about "hot" or "cold". There is no temperature here, as we are working at fixed energy. Only when you have many particles does it make sense to use the language of thermodynamics.
Yes. The closer an eletron is to the nucleus, the greater the energy needed to remove it from the atom.
And just to be clear: the picture here is very classical. There are some subtleties due to quantum mechanics that need to be taken into account (such as the fact that an electron cannot be said to be at a given distance from the nucleus). Also, in many-electron atoms, electron-electron interactions also affect the picture.
12. ### rogerk8
220
You have obviously put lots of time and effort into describing this to me and I thank you for it!
Two things:
1) I didn't know that energy could be relative (and thus negative) but as you describe it it makes sense.
2) I do however still don't get how a higher negative energy could mean a lower energy because intuetively an electron closer to the atom ought to have higher energy simply to keep its distance from the atom's electromagnetic force.
This negative stuff obviously confuses me
Best regards, Roger
13. ### DrClaude
2,327
Most often, the exact total energy is not important, only the relative energy. For instance, when studying atoms, it is exceptional to have to take into account the "rest energy" ##mc^2## when calculating the energy of the atom. Where we chose to put to zero of energy is arbitrary.
It is the same as with gravity: the lowest energy is when the two bodies are closest together. If you want to move an electron away from the nucleus, or lift something off the ground, you have to put in energy into the system.
14. ### rogerk8
220
I totally understand now, thanks!
Best regards, Roger
15. ### rogerk8
220
I have understood that there are more ways to generate light. The mechanism I am thinking of is thermal radiation and reading about this in Wikipedia took me so far as to charge-acceleration and dipole oscillation.
While surfing around in Wikipedia I got to remember that light is a TEM-wave.
I now have an idea how thermal radiation makes light. Let's consider charge-acceleration first and please tell me if I'm wrong.
Being an electrical engineer, I began to think:
$$U=-L\frac{dI}{dt}=-jwLI$$
where
$$I=I_0e^{jwt}$$
which both means that jw is a derivation operator for any stationary signal and that the current and the voltage is 90 degrees out of phase in an inductor.
If we consider the Bohr Model and the Bohr Radius rB we have an orbiting moving charge (at a constant speed, preliminary) which by definition is equal to current.
Now, the magnetic flux through a coil of area S may be simplified to
$$\phi=BS=\mu_0HS=\mu_0NIS[Wb]$$
Where N=1 for our orbiting electron.
$$e=\oint Edl=-\frac{d\phi}{dt}=-jw\phi[V]$$
states the emf-induction due to magnetic flux change.
I have some stupid thoughts regarding how this emf may be measured. Perhaps simply on either side of the electron
Viewing these equations we do however have both B and E 90 degrees out of phase. And this is by definition a TEM-wave.
The speed of the (circulating) charge/current need however to be non-constant (otherwise no induction can be made) which means that we have to accelerate the charge by for instance heat.
For a while there I thought "what is acceleration of charge?". But then I stupidly realised that it just is change of velocity like any sinusoidal (f.i) signal.
My reasoning makes ordinary atoms radiate no light unless they are (being) heated up or excited. Strange. I must be wrong because almost everything has color. On the other hand, might vibration be the reason in that case?
The conclusion must be that every AC-driven coil (which might include dipole oscillation) or accelerated charge like above give rise to TEM (even though frequency differ enormously). Sounds a bit like radio
While the Bohr Model just is for simplification, a free accelerated or decelerated charge might suit even better for this reasoning. In this case it is more obvious that there are no spectral lines when it comes to thermal radiation and this is simply because of the "linear" speed states while adding kT.
How wrong am I?
Best regards, Roger
Last edited: Feb 1, 2014 | 2015-05-28 10:16:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8236660361289978, "perplexity": 669.2068574624323}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207929272.95/warc/CC-MAIN-20150521113209-00206-ip-10-180-206-219.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/414321/example-in-which-the-first-part-of-the-definition-of-ufd-fails-to-hold?answertab=active | Example in which the first part of the definition of UFD fails to hold
There is this example in which the first part of the definition of UFD (that is the existence of factorisation) fails to hold that I don't quite understand.
Let $R=\mathbb{R}[X_1,X_2,\dots]$, and let $I\triangleleft R$ be the ideal generated by the set $\{X_2^2-X_1,X_3^2-X_2,X_4^2-X_3,\dots\}\subset R$.
Then in $R/I$ the element $X_1+I$ has no factorisation as a product of irreducibles (and is not a unit): $$X_1+I=(X_2+I)(X_2+I)=(X_3+I)(X_3+I)(X_3+I)(X_3+I)=\dots$$
My questions are:
1. First of all, why is $X_1+I$ is an element in $R/I$? By using the definition of quotient ring, $X_1\in R$, right? But if we use the definition of polynomial ring $R=\mathbb{R}[X_1,X_2,\dots]=((\mathbb{R}[X_1])[X_2])[X_3]\dots$, how can we show $X_1\in R$? These multilayers seems a bit confusing for me..
2. I also have little clue how can we derive $X_1+I=(X_2+I)(X_2+I)$? I know that somehow we need to use the definition of $I$ as the generating set of $\{X_2^2-X_1,X_3^2-X_2,X_4^2-X_3,\dots\}$, but not entirely sure how to proceed.
It will be really appreciated if anyone could help me out on this.
Thanks!
-
$\bf R$ is the reals, and the number $1$ is in $\bf R$, so $X_1$ is in ${\bf R}[X_1]$, so I'm not sure where the difficulty is in seeing that it's in $R$.
For the second question, the way multiplication of cosets goes, $(X_2+I)(X_2+I)$ is $X_2^2+I$, the coset containing $X_2^2$. But $X_2^2-X_1$ is in $I$, so $X_2^2+I=X_1+I$.
Another example along the same lines is that the set of all algebraic numbers is not a UFD because $$2=(\sqrt2)^2=(\root4\of2)^4=(\root8\of2)^8=\dots$$
Thanks for your answer. For the first question, I know that $X_1\in\mathbb{R}[X_1]$ but don't know why $X_1\in R$. Here $R$ is different to $\mathbb{R}$, $\mathbb{R}$ is the reals whereas $R$ is the polynomial ring $\mathbb{R}[X_1,X_2,\dots]$, hopefully that clarifies my question. – user71346 Jun 8 '13 at 1:53
Do you understand why $x$ is in ${\bf R}[x,y]$? ${\bf R}[x,y]$ is polynomials in $x$ and $y$. If $f$ is a polynomial in $x$ then it is also a polynomial in $x$ and $y$. I'm having trouble understanding exactly what the difficulty is. – Gerry Myerson Jun 8 '13 at 3:36 | 2015-11-25 08:29:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9246324300765991, "perplexity": 59.502549536050616}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398445033.85/warc/CC-MAIN-20151124205405-00216-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://cran.rediris.es/web/packages/REddyProc/vignettes/aggUncertainty.html | # Aggregating uncertainty to daily and annual values
## Example setup
We start with half-hourly $$u_*$$-filtered and gapfilled NEE_f values. For simplicity this example uses data provided with the package and omits $$u_*$$ threshold detection but rather applies a user-specified threshold.
With option FillAll = TRUE, an uncertainty, specifically the standard deviation, of the flux is estimated for each record during gapfilling and stored in variable NEE_uStar_fsd.
library(REddyProc)
library(dplyr)
EddyDataWithPosix <- Example_DETha98 %>%
filterLongRuns("NEE") %>%
fConvertTimeToPosix('YDH',Year = 'Year',Day = 'DoY', Hour = 'Hour')
EProc <- sEddyProc$new( 'DE-Tha', EddyDataWithPosix, c('NEE','Rg','Tair','VPD', 'Ustar')) EProc$sMDSGapFillAfterUstar('NEE', uStarTh = 0.3, FillAll = TRUE)
results <- EProc$sExportResults() results_good <- results %>% filter(NEE_uStar_fqc <= 1) summary(results_good$NEE_uStar_fsd)
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.03535 1.69194 2.31331 2.70544 3.39922 17.19675
We can inspect, how the uncertainty scales with the flux magnitude.
results_good %>% slice(sample.int(nrow(results_good),400)) %>%
plot( NEE_uStar_fsd ~ NEE_uStar_fall, data = . )
## Omitting problematic records
REddyProc flags filled data with poor gap-filling by a quality flag in NEE_<uStar>_fqc > 0 but still reports the fluxes. For aggregation we recommend computing the mean including those gap-filled records, i.e. using NEE_<uStar>_f instead of NEE_orig. However, for estimating the uncertainty of the aggregated value, the the gap-filled records should not contribute to the reduction of uncertainty due to more replicates.
Hence, first we create a column similar NEE_orig_sd to NEE_<uStar>_fsd but where the estimated uncertainty is set to missing for the gap-filled records.
results <- EProc$sExportResults() %>% mutate( NEE_orig_sd = ifelse( is.finite(.data$NEE_uStar_orig), .data$NEE_uStar_fsd, NA), NEE_uStar_fgood = ifelse( is.finite(.data$NEE_uStar_fqc <= 1), .data$NEE_uStar_f, NA) ) If the aggregated mean should be computed excluding poor quality-gap-filled data, then its best to use a column with values set to missing for poor quality, e.g. using NEE_<uStar>_fgood instead of NEE_<uStar>_f. However, the bias in aggregated results can be larger when omitting records, e.g. consistently omitting more low night-time fluxes, than with using poor estimates of those fluxes. ## Random error For a given u* threshold, the aggregation across time uses many records. The random error in each record, i.e. NEE_fsd, is only partially correlated to the random error to records close by. Hence, the relative uncertainty of the aggregated value decreases compared to the average relative uncertainty of the individual observations. ### Wrong aggregation without correlations With neglecting correlations among records, the uncertainty of the mean annual flux is computed by adding the variances. The mean is computed by $$m = \sum{x_i}/n$$. And hence its standard deviation by $$sd(m) = \sqrt{Var(m)}= \sqrt{\sum{Var(x_i)}/n^2} = \sqrt{n \bar{\sigma^2}/n^2} = \bar{\sigma^2}/\sqrt{n}$$. This results in an approximate reduction of the average standard deviation $$\bar{\sigma^2}$$ by $$\sqrt{n}$$. results %>% summarise( nRec = sum(is.finite(NEE_orig_sd)) , NEEagg = mean(NEE_uStar_f) , varSum = sum(NEE_orig_sd^2, na.rm = TRUE) , seMean = sqrt(varSum) / nRec , seMeanApprox = mean(NEE_orig_sd, na.rm = TRUE) / sqrt(nRec) ) %>% select(NEEagg, nRec, seMean, seMeanApprox) ## NEEagg nRec seMean seMeanApprox ## 1 -1.657303 10901 0.02988839 0.02650074 Due to the large number of records, the estimated uncertainty is very low. ### Considering correlations When observations are not independent of each other, the formulas now become $$Var(m) = s^2/n_{eff}$$ where $$s^2 = \frac{n_{eff}}{n(n_{eff}-1)} \sum_{i=1}^n \sigma_i^2$$, and with the number of effective observations $$n_{eff}$$ decreasing with the autocorrelation among records (Bayley 1946, Zieba 2011). The average standard deviation $$\sqrt{\bar{\sigma^2_i}}$$ now approximately decreases only by about $$\sqrt{n_{eff}}$$: $Var(m) = \frac{s^2}{n_{eff}} = \frac{\frac{n_{eff}}{n(n_{eff}-1)} \sum_{i=1}^n \sigma_i^2}{n_{eff}} = \frac{1}{n(n_{eff}-1)} \sum_{i=1}^n \sigma_i^2 \\ = \frac{1}{n(n_{eff}-1)} n \bar{\sigma^2_i} = \frac{\bar{\sigma^2_i}}{(n_{eff}-1)}$ First we need to quantify the error terms, i.e. model-data residuals. For all the records of good quality, we have an original measured value NEE_uStar_orig and modelled value from MDS gapfilling, NEE_uStar_fall. For computing autocorrelation, equidistant time steps are important. Hence, instead of filtering, the residuals of bad-quality data are set to missing. results <- EProc$sExportResults() %>%
mutate(
resid = ifelse(NEE_uStar_fqc == 0, NEE_uStar_orig - NEE_uStar_fall, NA)
,NEE_orig_sd = ifelse(
is.finite(.data$NEE_uStar_orig), .data$NEE_uStar_fsd, NA)
)
Now we can inspect the the autocorrelation of the errors.
acf(results$resid, na.action = na.pass, main = "") The empirical autocorrelation function shows strong positive autocorrelation in residuals up to a lag of 10 records. Computation of effective number of observations is provided by function computeEffectiveNumObs from package lognorm based on the empirical autocorrelation function for given model-data residuals. Note that this function needs to be applied to the series including all records, i.e. not filtering quality flag before. autoCorr <- lognorm::computeEffectiveAutoCorr(results$resid)
nEff <- lognorm::computeEffectiveNumObs(results$resid, na.rm = TRUE) c(nEff = nEff, nObs = sum(is.finite(results$resid)))
## nEff nObs
## 4230.522 10901.000
We see that the effective number of observations is only about a third of the number of observations.
Now we can use the formulas for the sum and the mean of correlated normally distributed variables to compute the uncertainty of the mean.
resRand <- results %>% summarise(
nRec = sum(is.finite(NEE_orig_sd))
, NEEagg = mean(NEE_uStar_f, na.rm = TRUE)
, varMean = sum(NEE_orig_sd^2, na.rm = TRUE) / nRec / (!!nEff - 1)
, seMean = sqrt(varMean)
#, seMean2 = sqrt(mean(NEE_orig_sd^2, na.rm = TRUE)) / sqrt(!!nEff - 1)
, seMeanApprox = mean(NEE_orig_sd, na.rm = TRUE) / sqrt(!!nEff - 1)
) %>% select(NEEagg, seMean, seMeanApprox)
resRand
## NEEagg seMean seMeanApprox
## 1 -1.657303 0.04798329 0.04254471
The aggregated value is the same, but its uncertainty increased compared to the computation neglecting correlations.
Note, how we used NEE_uStar_f for computing the mean, but NEE_orig_sd instead of NEE_uStar_fsd for computing the uncertainty.
### Daily aggregation
When aggregating daily respiration, the same principles hold.
However, when computing the number of effective observations, we recommend using the empirical autocorrelation function estimated on longer time series of residuals (autoCorr computed above) in computeEffectiveNumObs instead of estimating them from the residuals of each day.
First, create a column DoY to subset records of each day.
results <- results %>% mutate(
DateTime = EddyDataWithPosix$DateTime # take time stamp form input data , DoY = as.POSIXlt(DateTime - 15*60)$yday # midnight belongs to the previous
)
Now the aggregation can be done on data grouped by DoY. The notation !! tells summarise to use the variable autoCorr instead of a column with that name.
aggDay <- results %>%
group_by(DoY) %>%
summarise(
DateTime = first(DateTime)
,nEff = lognorm::computeEffectiveNumObs(
resid, effAcf = !!autoCorr, na.rm = TRUE)
, nRec = sum(is.finite(NEE_orig_sd))
, NEE = mean(NEE_uStar_f, na.rm = TRUE)
, sdNEE = if (nEff <= 1) NA_real_ else sqrt(
mean(NEE_orig_sd^2, na.rm = TRUE) / (nEff - 1))
, sdNEEuncorr = if (nRec <= 1) NA_real_ else sqrt(
mean(NEE_orig_sd^2, na.rm = TRUE) / (nRec - 1))
, .groups = "drop_last"
)
aggDay
## # A tibble: 365 × 7
## DoY DateTime nEff nRec NEE sdNEE sdNEEuncorr
## <int> <dttm> <dbl> <int> <dbl> <dbl> <dbl>
## 1 0 1998-01-01 00:30:00 11.0 21 0.124 0.760 0.536
## 2 1 1998-01-02 00:30:00 3.66 7 0.00610 1.56 1.04
## 3 2 1998-01-03 00:30:00 0 0 0.0484 NA NA
## 4 3 1998-01-04 00:30:00 0 0 0.303 NA NA
## 5 4 1998-01-05 00:30:00 10.9 28 0.195 0.861 0.521
## 6 5 1998-01-06 00:30:00 18.0 48 0.926 0.615 0.370
## 7 6 1998-01-07 00:30:00 18.0 48 -0.337 0.566 0.340
## 8 7 1998-01-08 00:30:00 17.7 46 -0.139 0.525 0.320
## 9 8 1998-01-09 00:30:00 17.5 45 0.614 0.474 0.290
## 10 9 1998-01-10 00:30:00 15.4 36 0.242 0.641 0.411
## # … with 355 more rows
The confidence bounds (+-1.96 stdDev) computed with accounting for correlations in this case are about twice the ones computed with neglecting correlations.
## u* threshold uncertainty
There is also flux uncertainty due to uncertainty in u* threshold estimation. Since the same threshold is used for all times in a given uStar scenario, the relative uncertainty of this component does not decrease when aggregating across time.
The strategy is to
1. estimate distribution of u* threshold
2. compute time series of NEE (or other values of interest) for draws from this distribution, i.e. for many uStar-scenarios
3. compute each associated aggregated value
4. and then look at the distribution of the aggregated values.
Note that the entire processing down to the aggregated value has to be repeated for each uStar scenario. Hence, obtaining a good estimate of this uncertainty is computationally expensive.
1. First, we estimate many samples of the probability density of the unknown uStar threshold.
# for run-time of the vignette creation, here we use only few (3) uStar quantiles
# For real-world applications, a larger sample (> 30) is required.
nScen <- 3 # nScen <- 39
EddyDataWithPosix <- Example_DETha98 %>%
filterLongRuns("NEE") %>%
fConvertTimeToPosix('YDH',Year = 'Year',Day = 'DoY', Hour = 'Hour')
EProc <- sEddyProc$new( 'DE-Tha', EddyDataWithPosix, c('NEE','Rg','Tair','VPD', 'Ustar')) EProc$sEstimateUstarScenarios(
nSample = nScen*4, probs = seq(0.025,0.975,length.out = nScen) )
uStarSuffixes <- colnames(EProc$sGetUstarScenarios())[-1] uStarSuffixes # in real world should use > 30 ## [1] "uStar" "U2.5" "U50" "U97.5" 2. Produce time series of gapfilled NEE for each scenario. They are stored in columns distinguished by a suffix with the quantile. EProc$sMDSGapFillUStarScens('NEE')
3. Compute the annual mean for each scenario. Method sEddyProc_sApplyUStarScen calls a user-provided function that takes an argument suffix for each u*-threshold scenario. Here, we use it to create the corresponding NEE column name and compute mean across this column in the data exported from REddyProc.
computeMeanNEE <- function(ds, suffix){
column_name <- paste0("NEE_",suffix,"_f")
mean(ds[[column_name]])
}
FilledEddyData <- EProc$sExportResults() NEEagg <- unlist(EProc$sApplyUStarScen(computeMeanNEE, FilledEddyData))
NEEagg
## uStar U2.5 U50 U97.5
## -1.616926 -1.631296 -1.614860 -1.645656
4. compute uncertainty across aggregated values
sdNEEagg_ustar <- sd(NEEagg)
sdNEEagg_ustar
## [1] 0.01432019
## Combined aggregated uncertainty
Assuming that the uncertainty due to unknown u*threshold is independent from the random uncertainty, the variances add.
sdAnnual <- data.frame(
sdRand = resRand$seMean, sdUstar = sdNEEagg_ustar, sdComb = sqrt(resRand$seMean^2 + sdNEEagg_ustar^2)
)
sdAnnual
## sdRand sdUstar sdComb
## 1 0.04798329 0.01432019 0.05007459 | 2023-02-03 09:35:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5412058234214783, "perplexity": 12302.6035754815}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500044.66/warc/CC-MAIN-20230203091020-20230203121020-00401.warc.gz"} |
https://electronics.stackexchange.com/questions/349365/power-factor-improvment-c-k-ratio | # Power factor improvment c/k ratio
All power factor improvment controller has a setting c/k ratio. Their manual says it is the lowest step of capacitor bank. It controls switching threshold. But how what is its significance?
$C/K = \frac{Q }{I1/5A * U_{LL} * 1.73}$
$U_{LL}$ = line-to-line voltage in volts | 2019-07-23 05:25:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2654869556427002, "perplexity": 9816.956221626418}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195528869.90/warc/CC-MAIN-20190723043719-20190723065719-00360.warc.gz"} |
https://cs.stackexchange.com/questions/25804/understanding-expected-running-time-of-randomized-algorithms | # Understanding Expected Running Time of Randomized Algorithms
I want to understand the expected running time and the worse-case expected running time.
I got confused when I saw this figure (source),
where $I$ is the input and $S$ is the sequence of random numbers.
What I don't understand from the above equation is why the expected running time is given for one particular input $I$?
I always thought that for a problem $\pi$, $E(\pi) = \sum_{input \in Inputs}(Pr(input)*T(input))$ , isn't this correct?
So, let's assume Pr(x) is the uniform distribution, and we are to find the expected running time of the problem of searching an element in a $n$ element array using linear search.
Isn't the expected running time for linear search,
$$E(LinearSearch) = \frac{1}{n}\sum_1^ni$$
And what about the worst case expected running time, isn't it the time complexity of having the worst behavior? Like the figure below,
I would highly appreciate if someone can help me understand the two figures above.
There are two notions of expected running time here. Given a randomized algorithm, its running time depends on the random coin tosses. The expected running time is the expectation of the running time with respect to the coin tosses. This quantity depends on the input. For example, quicksort with a random pivot has expected running time $\Theta(n\log n)$. This quantity depends on the length of the input $n$.
Given either a randomized or a deterministic algorithm, one can also talk about its expected running time on a random input from some fixed distribution. For example, deterministic quicksort (with a fixed pivot) has expected running time $\Theta(n\log n)$ on a randomly distributed input of length $n$. This is known as average-case analysis. Your example of linear search fits this category. | 2020-07-06 05:37:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8463735580444336, "perplexity": 225.4846498594337}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655890105.39/warc/CC-MAIN-20200706042111-20200706072111-00557.warc.gz"} |
https://chemistry.stackexchange.com/questions/32921/dispersion-of-lithium-ions-in-nature-ratio-similar-to-isotopes?answertab=oldest | # Dispersion of Lithium Ions in Nature (Ratio Similar to Isotopes?)
I'm working on a project that identifies average characteristics of different elements. I'm stuck on lithium, I can't find a way to ratio lithium I and lithium II that symbolizes how it is in nature. I tried looking around on the internet but I haven't found any mention of ratios to ions only ratios to isotopes. Does anyone know of anything that might help?
• Doesn't really answer the question. If this means anything I could use the ratio in plasma conditions. – user4960003 Jun 15 '15 at 12:23
• Alright I'm very frustrated, not with this site but with my project, does anyone out there know or understand how lithium ions are dispersed in nature? – user4960003 Jun 16 '15 at 0:48
• I'm not entirely sure what you're after. Are you trying to find $\ce{Li+:Li2+}$? It's going to be nearly infinite (or nearly zero, if you reverse it.) Aside from in plasmas, lithium is almost always in the +1 oxidation state. – Jason Patterson Jun 16 '15 at 1:08
• The reference to isotopes in the question is very confusing if you really are asking about the ratio of lithium(I) ions to lithium(II) ions. Outside of high-energy plasmas, particle accelerators, or the vicinity of stellar cores, there aren't going to be many lithium(II) ions on Earth or in the universe. – Curt F. Jun 16 '15 at 2:14 | 2019-09-16 16:26:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4921824336051941, "perplexity": 721.682535358666}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514572879.28/warc/CC-MAIN-20190916155946-20190916181946-00011.warc.gz"} |
https://en.wikipedia.org/wiki/smooth_number?oldid=294190743 | # Smooth number
Jump to navigation Jump to search
In number theory, a positive integer is called B-smooth if none of its prime factors are greater than B. For example, 1,620 has prime factorization 22 × 34 × 5; therefore 1,620 is 5-smooth since none of its prime factors are greater than 5. 5-smooth numbers are also called regular numbers or Hamming numbers and arise in the study of Babylonian mathematics, music theory, and as a test problem for functional programming. 7-smooth numbers are sometimes called highly composite, although this conflicts with another meaning of that term.
Note that B does not have to be a prime factor. If the largest prime factor of a number is p then the number is B-smooth for any Bp. Usually B is given as a prime, but composite numbers work as well. A number is B-smooth if and only if it is p-smooth, where p is the largest prime less than or equal to B.
An important practical application of smooth numbers is for fast Fourier transform (FFT) algorithms such as the Cooley-Tukey FFT algorithm that operate by recursively breaking down a problem of a given size n into problems the size of its factors. By using B-smooth numbers, one ensures that the base cases of this recursion are small primes, for which efficient algorithms exist. (Large prime sizes require less-efficient algorithms such as Bluestein's FFT algorithm.)
## Powersmooth numbers
Further, m is called B-powersmooth if all prime powers ${\displaystyle \scriptstyle p_{i}^{n_{i}}}$ dividing m satisfy:
${\displaystyle p_{i}^{n_{i}}\leq B.\,}$
For example, 243251 is 16-powersmooth since its greatest prime factor power is 24 = 16. The number is also 17-powersmooth, 18-powersmooth, 19-powersmooth, etc.
B-smooth and B-powersmooth numbers have applications in number theory, such as in Pollard's p − 1 algorithm. Such applications are often said to work with "smooth numbers," with no B specified; this means the numbers involved must be B-smooth for some unspecified small number B; as B increases, the performance of the algorithm or method in question degrades rapidly. For example, the Pohlig-Hellman algorithm for computing discrete logarithms has a running time of O(B1/2) for groups of B-smooth order.
## Distribution
Let ${\displaystyle \scriptstyle \Psi (x,y)}$ denote the de Bruijn function, the number of y-smooth integers less than or equal to x.
If the smoothness bound B is fixed and small, there is a good estimate for ${\displaystyle \scriptstyle \psi (x,B)}$:
${\displaystyle \Psi (x,B)\sim {\frac {1}{\pi (B)!}}\prod _{p\leq B}{\frac {\log x}{\log p}}.}$
Otherwise, define the parameter u as u = log x / log y: that is, x = yu. Then we have
${\displaystyle \Psi (x,y)=x\cdot \rho (u)+O\left({\frac {x}{\log y}}\right)}$
where ${\displaystyle \scriptstyle \rho (u)}$ is the Dickman-de Bruijn function. | 2018-10-18 00:49:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 7, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8160120844841003, "perplexity": 589.9780546441771}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583511365.50/warc/CC-MAIN-20181018001031-20181018022531-00195.warc.gz"} |
http://timotheepoisot.fr/2015/04/02/do-not-enforce-R/ | Don't enforce R as a standard
Yesterday, I received the reviews for a paper of mine. It was rejected with an invitation to resubmit, so far, so good. In this paper, we present a lot of new measures to work on probabilistic networks, and it’s all in the preprint if you really want to read more about that. To do the paper, as in, to produce the figures and do the analysis, I wrote a package in Julia. I’m proud of this package. It’s fast, defensively programmed, well tested, already parallelized if you use several CPUs, and all that. It’s also released under the MIT “Expat” license, so you are free to do with it as you please.
It’s important to specify that this paper is not a software paper. It’s the description of methods, and it so happens that we decided to release the package alongside it. This is where things got complicated.
Reviewer 1 wants more practical details about the software (despite it not being the purpose of the paper), and that we have to justify that it is indeed usable. Some of this point is fair (I will write a short introduction, and the code used for the figures will be published alongside the paper). Reviewer 3 states that since Julia is not frequently used by ecologists, and therefore it’s dubious that the methods are usable, or even useful, and a R package would be better.
The three reviews were helpful and constructive, but these two comments infuriated me. Let me start with a bit of background. I’m not anti-R. I gave department-wide R training for graduate students and faculty. I wrote R packages to go with papers, and released them on GitHub, and I keep on fixing bugs and maintaining them. I use R on a daily basis. I contributed code to other people’s R packages. One of my lab machines runs RStudio server for colleagues that lack computational power. But it’s a tool. It’s neither a religion, nor a standard, nor a pre-requisite for getting your paper accepted in an ecological journal.
Let me put things in perspective.
First, a package written in anything is better than no package at all. And as far as I’m concerned, this should be the end of the argument. And especially if this is not a software paper (but even then), the language a software is written on has nothing to do with the scientific merit of the paper. Unless the choice of language means bad performance or a significant risk of errors, or the impossibility to run the code, it can be written in lisp or cobold for all I care. I cannot emphasize this point enough: any code organized in a software library and released under a FOSS License is better than no code at all.
Second, what to write a piece of software in, as the guy in charge of actually writing the code, is a conscious decision. And knowing the type of data, and use cases (because I’ve since used this package in a few projects), and the algorithm used, and all that, I decided that R was not the best choice I can make. Could I write a R version of it? Yes. But I won’t, because I don’t have that much time, it won’t be as efficient, and I’m not interested in doing it.
Third, there is this thing in the open-source crowd, where you don’t get to be entitled about a piece of software that is available for free. This is extremely important. That something is released does not imply that the maintainers will do free work for you. Especially when the message is “I don’t like this because this is not the way I do it, so can you start again?”. The whole point of FOSS is that if you don’t like it, you can fork it. But you don’t get to complain about something that is given away for free. This is just rude (see also, first point – it’s better than nothing). I don’t expect praise for writing code (except by my continuous integration engine, that sends me comforting Build passed emails). But if I go the extra mile of writing and organizing and optimizing code, it’s grating that people complain because it’s not matching their personal preferences.
Fourth, this points back to a larger issue – R is not the only computational tool we have. Every time these is a pushback about something on the basis that this is not R, we are losing opportunities. I’ve had discussion with people saying that they won’t even try to publish software, because it’s not in R and they don’t feel like fighting an uphill battle where most of their energy will go into their language choice. And because of this “R or bust” mentality that some people are so vocal about, cool things are not properly released. There is a cost.
Every time you complain about the language code is written in, Ethan White has to buy a new computer:
I know that accessibility is important – this is what picking FOSS platforms and languages is about, and this is why FOSS licenses for scientific code should be enforced. But beyond that, what should matter is, does it works as advertised?. Software is not something that happens magically out of thin air. People write it, people like me, and our choice of how to write takes precedence over your ideal language in which to run it. Replying to reviewers is an exercise in picking your battles – this one I’ll fight. | 2019-01-23 05:07:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.363267183303833, "perplexity": 738.5778201959943}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583897417.81/warc/CC-MAIN-20190123044447-20190123070447-00180.warc.gz"} |
https://texfaq.org/FAQ-mathml | Frequently Asked Question List for TeX
# Math on the Web
An earlier answer (“converting to HTML”) addresses the issue of converting existing (La)TeX documents for viewing on the Web as HTML.
Better font availability and the support for new Web standards means that there are now several possibilities for good rendering of mathematics on the web.
• Font technologies: Direct representation of mathematics in browsers was hampered by the limited range of symbols in the fonts that were available. However, all modern operating systems now include OpenType fonts with large collections of symbols and the availablity of web font technology means that page authors may specify fonts without relying on the reader having pre-installed suitable fonts.
The available OpenType math fonts are discussed in OpenType fonts
• Direct interpretaton of a subset of LaTeX math markup by Javascript. The speed of modern javaScript engines means that it is feasible to serve web pages that contain fragments of TeX markup that is converted in the reader’s browser. Two main systems are in common use:
• MathJax is the most widely used JavaScript Library for rendering mathematics. It supports several input syntaxes includedin a subset of LaTeX math syntax, and may be configured to render using several output forms, MathML, or SVG or (most commonly) HTML+CSS. While normally used as a JavaScript Library running in the reader’s browser it is also possible (using its Node.js interface) to do the conversion in advance,and serve the generated HTML pages.
• KaTeX Is a newer alternative JavasScript Library, its main aim is to be simpler and faster than MathJax. It has fewer input or output forms and covers a smaller range of LaTeX constructs, but is a viable alternative for pages that do not require the additional features of MathJax.
• Conversion of (La)TeX source to XML is already available (through TeX4ht at least), and work continues in that arena. The alternative, authoring in XML (thus producing documents that are immediately Web-friendly, if not ready) and using (La)TeX to typeset is also well advanced. One useful technique is transforming the XML to LaTeX, using an XSLT stylesheet or code for an XML library, and then simply using LaTeX; alternatively, one may typeset direct from the XML source.
• Direct representation of mathematics MathML is a standard for representing maths on the Web; Browser support for MathML is provided by firefox, and safari and other browsers using te same underlying html rendering libraries. At the current time it is not supported by Chrome or Edge browsers. MathML in the page may be rendered by MathJax, with an output identical to its TeX r.endering. MathJax uses a variant of MathML as its intermediate format)
The MathJax project’s site also allows you to download your own copy and install it on one of your servers. MathJax is open source software.
• An approach different from (La)TeX conversion is taken by the GELLMU Project. Its article XML document type, which has a markup vocabulary close to LaTeX that can be edited using LaTeX-like markup (even though it is not LaTeX — so far), comes with translators that make both PDF (via pdflatex) and XHTML+MathML. Such an approach avoids the inherent limitations of the “traditional” (La)TeX translation processes, which have traps that can be sprung by unfettered use of (La)TeX markup.
• Graphics SVG is a standard for graphics representation on the web. While the natural use is for converting existing figures, representations of formulas are also possible.
• An interesting alternative is MathTeX, which sits on your server as a CGI script, and you use it to include your TeX, in your HTML, as if it were an image:
<img src="/cgi-bin/mathtex.cgi?f(x)=\int\limits_{-\infty}^xe^{-t^2}dt">
(Mathtex supersedes the author’s earlier mimetex.)
FAQ ID: Q-mathml
Last updated: 2018-05-25 | 2021-06-24 05:41:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6363583207130432, "perplexity": 3646.9792141429284}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488551052.94/warc/CC-MAIN-20210624045834-20210624075834-00088.warc.gz"} |
https://www.physicsforums.com/threads/work-done-by-friction-by-wheels.841792/ | Work done by friction by wheels
Tags:
1. Nov 7, 2015
almarpa
Hello all.
I have a doubt about the work done by friction force on a wheels , in two diferent situations:
a) In Morin's mechanics book, chapter 5 (page 146) he considers the situation in which a car is braking without skidding. He claims that the friction from the ground on the tires causes the car to slow down. But then, he says that this force is not doing work on the car, because the force acts over zero distance. I do not really undertand why there is no work here.
b) Imagine the same situation, but now, the car slows down while skidding. In this situation, the wikipedia says that there is work done by friction (see https://en.wikipedia.org/wiki/Work_(physics), section "Moving in a straight line (skid to a stop)". I guess that friction does work only if the wheels of the car are skidding, but it does no work if the wheels are just rolling. I tink it is related to the point of application of the force, but I can't see the point obout it
Can anybody explain me why?
Best regards
2. Nov 7, 2015
PeroK
First, take a simple example. If you push a heavy object, but can't get it moving because of static friction, then no work is done. But, if you get it moving then work is done by kinetic friction.
In case b), you have an example of kinetic friction at work. The tyres will heat up etc.
In case a) the work is being done in the braking mechanism. The brakes will get hot. But, braking does rely on static friction on the ground. If the car were moving through the air, then applying the brakes would do nothing. To explain this, you can see that the tyre is not actually moving along the ground - it's rolling. One of the things about rolling (without slipping) is that the part of the tyre/wheel that touches the ground is instantaneously at rest. It grips using static friction, rather than kinetic friction. This is what allows rolling to be so efficient: each small area of the wheel touches the ground without slipping, grips momentarily using static friction, then moves off the ground. And the next small area of the wheel takes over. So, there is effectively no heat or energy loss to friction: either in accelerating, braking or moving with constant speed.
3. Nov 7, 2015
almarpa
Thanks a lot.
Trata helps clarify.
4. Dec 13, 2015
almarpa
Sorry, but after reading Kleppner - Kolenkow (2nd ediction) example 7.16, I have another doubt about work done by friction.
In this example, a uniform drum of radius b, mass M, weight W = Mg, and moment of inertia I = Mb2/2 is on a plane of angle β. The drum starts from rest and rolls without slipping, and we have to find the speed V of its center of mass after it has descended a height h.
In the solution, they calculate the total work done by all the forces while the certer of mass moves along the plane. For them, the forces doing work are the weight W (that is acting at the center of mass), and the friction f between the drum and the plane as well. So they consider that work done by friction is not zero, but -fL (being L the distance covered) (!!).
After that, they calculate the work done by the total torque on the drum. The only force providing a torque is friction (because weight is acting on the center of mass, and normal force is paralell to position vector). So this torque is responsible for the rolling of the drum, and accounts for the change of rotational kinetic energy of the drum. In adition, they get that work done by torque is fL, so it equals the work done by friction.
Finally, they conclude that work done by friction (-fL) is decreasing the the center of mass kinetic energy in exactly the same amount that torque exerted by friction (fL) is increasing the rotational energy. So here friction simply transforms mechanical energy from one mode to another.
Now, if it is true, the friction must be doing work, but previously you told me that friction is not doing work. I am really confused. What is really happening here??
Thanks for your replies.
5. Dec 13, 2015
PeroK
K & K sum up that example very well at the end by saying that in rolling without slipping friction transforms linear to rotational energy with no energy being dissipated as heat. Although, you could also say that friction transforms gravitational PE to rotational KE.
"Doing work" is putting energy in or taking energy out of a system. Transforming energy from one form of KE to another isn't "work".
In fact, if you look at the definition of work in K & K: secion 5.3.3. it gives $W_{ba} = K_b - K_a$. In this case, all the change in KE is due to gravity. Friction, by definition, does no work in this example.
6. Dec 13, 2015
A.T.
Just as they say: The linear and rotational work done by friction cancel each other, so friction is doing no work in total.
7. Dec 14, 2015
almarpa
But then, or Morin is wrong, or Kleppner and Kolenkow are wrong. Let me explain:
1) As I mentioned in the first contribution to this thread, in Morin's mechanics book, chapter 5 (page 146) he considers the situation in which a car is braking without skidding. He claims that the friction from the ground on the tires causes the car to slow down. But then, he says that this force is not doing work on the car, because the force acts over zero distance. So, for Morin, friction is nor doing work.
2) On the other hand, in Kleppner and Kolenkow mechanics book (2nd ediction) example 7.16, there is a drum rolling without slipping on a plane. The situation is the same than in the example in Morin's book. Nevertheless, they do not claim that work is zero, and compute the work that is being done by friction as the integral of f (the friction force) between x1 and x2, obtaining that this work equals to -fL (where L is the distance travelled between x1 and x2). So, for Kleppner and Kolenkow, work done by friction is not zero.
Evidently, one of them must be wrong, because the reasoning they follow is opposite, indepentdently that the torque due to the friction force is doing work or not.
What do you think about it?
PS: By the way, if friction does no work, how can Morin claim that the friction from the ground on the tires causes the car to slow down? By the W-E theorem, if friction does no work, the friction would not contributte to the loss of kinetic energy of the car...
As you can see, I am reallu puzzled about this issue...
8. Dec 14, 2015
A.T.
They both agree that friction is doing zero work. They just decompose that work into linear and rational parts differently, by choosing a different reference point.
9. Dec 14, 2015
almarpa
I see...
So in Morin, the reference point is the contact point between the wheel and the road. But in Kleppner- Kolenkow, the referece point is the center of mass of the drum. Am I right?
Nevertheless, when we change the reference point to the center of mass, is the friction force is still acting over zero distance, or not? This issue is not still clear to me...
Thank you so much.
10. Dec 14, 2015
PeroK
You're missing a subtlety in K&K's solution. They integrate $f$ with respect to the angle around the drum. From the drum's reference frame $f$ works it's way round the circumference and does work equal to $fb\theta$. And that's what causes the rotational motion.
Then, they take the force down the slope as $F-f$ where $f$ is doing negative work equal to the positive work it does above. Overalll, therefore, the work done is:
$(Fl - fl) + fl$
You can now interpret this two ways:
a) Friction is doing no overall work.
b) Friction is transforming linear KE into rotational KE by doing both positive and negative work at the same time.
The problem is perhaps clearer if you look at the overall energy. $E_{initial} = GPE = E_{final} = LKE + RKE$. With no energy lost to friction/heat. So that's clear. If you now reread K&K's final note, I think this is what they are trying to explain: friction in this case was not a dissipative force. They used it (rather cleverly) as a transformative force.
11. Dec 14, 2015
PeroK
The final point, which perhaps you may get from my post above, is that from the drum's perspective friction works its way round the circumference as a motive force.
12. Dec 14, 2015
almarpa
Mmmmm.
I think now I understand it all.
The key point is that Morin chooses as his reference point the point of contact between the wheel and the ground (which is instantaneously at rest); so for him, friction does no "linear" work nor "rotational" work at all. On the other hand, Kleppner and Kolenkow choose as their reference point the center of mass of the rolling drum (which, in the center of mass reference frame is at rest); so for them, friction does both "linear" and "rotational" work, which happen to be identical (but with opposite signs), and cancel each other, so the total work done by frction is zero.
I find this argumentation plausible, but I need to work it out to check if it is right.
Thanks A.T, and PeroK for your clever indications. If I had any other doubts, I would post them here as well.
Regards.
13. Dec 14, 2015
almarpa
OK, I have tried to solve Kleppner - Kolenkow example taking the point of contact between the drum and the ground as the origin of the reference frame.
That implies the new moment of inertia of the drum is I=ICM+Mb2=(3/2)Mb2. Now the problem is easier, because the W-E theorem for the rotational motion equation is enough to get the answer. Now the torque is only due to the weight, and not to the friction, so we get:
∫τ dθ=(1/2)Iω2
bWsenβ θ=(1/2)(3/2)Mb2ω2
, so:
V=√(4gh/3)
which is the same answer that Kleppner - Kolenkow obtain by using the W-E ecuations both for the lineal and rotation motion.
Now, I have a doubt. Altough it is not necesary for the problem, how should I use the W-E theorem for the traslational motion with this new reference point (the point of contact between the drum and the ground)??
∫F dr=T2-T1
Now the only force doing work is weight, because friction acts over zero distance, and normal force is perpendicular to the displacement. But, with this reference point, what is the kinetic energy?? I mean, what is the speed I shoud consider? The linear speed of the center of mass? The linear speed of that point? Both the rotational and the linear speed(in this case, what linear speed and why?)? Maybe the linear speed is zero (because this point is instantaneously at rest)?...
Regards.
14. Dec 14, 2015
Staff: Mentor
One must be careful about interpreting the W-E theorem. Despite the name, it's not really a statement about "real" work, but of pseudo-work (sometimes called "center of mass" work). It is an application of Newton's 2nd law:
$$F_{net}\Delta x_{cm}=\Delta (\frac{1}{2}m v_{cm}^2)$$
Note that you are multiplying the net force times the displacement of the center of mass, which allows you to calculate the resulting change in the kinetic energy of the center of mass. In Morin's example, the friction provides the net force and you can use the theorem to calculate the change in KE of the car. The friction causes the car to slow, but it does no work on the car. (The real work would involve friction times the displacement of the point of application--which is zero, since there is no slipping.)
The same analysis, of course, applies to an accelerating car. Friction causes the car to accelerate, but does no (real) work on the car (assuming no slipping). Nonetheless, the W-E theorem applies as before. The energy comes from within the car, not from the road. (But that external force from the road is certainly needed to create the motion of the car.)
15. Dec 20, 2015
Shreyas Samudra
Here there's nothing like choosing a reference , its all about choosing a system - centre of mass or
- Whole body
16. Dec 20, 2015
Shreyas Samudra
All this is explaining that change in internal energy of the system is manifesting in form of energy , cause there's friction inspiring that !!
17. Dec 20, 2015
almarpa
I will try to work it out again keeping this in mind.
What I still do not understand is the fact that, if friction is nor doing work, why it is sais that friction from the ground on the tires causes the car to slow down...
In my textbooks, nobody mention pseudo-work, they just tlak about work. So by W-E theorem, if frction does no work, it should not cause the car to slow down...
Thanks.
18. Dec 21, 2015
Shreyas Samudra
Look,
For a complicated system(where motion of all the parts of the system is not same), we can apply work energy theorem in 2 ways
1] WE theorem for COM, i.e you treate the whole body as a point particle, and find work done by all the forces on COM i.e integral of force times dx(COM)
So you will equate all this to change in KE of COM (here you will not include rotational KE of the body as, now your body is a point particle)
2]Apply WE theorem for whole body , here rotational KE will be considered and work done by those forces which are not causing displacement at the point of contact will be zero. | 2017-10-22 14:08:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7509325742721558, "perplexity": 490.16909494972447}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187825264.94/warc/CC-MAIN-20171022132026-20171022151939-00020.warc.gz"} |
https://mathoverflow.net/questions/346356/hitting-time-estimates | # Hitting time estimates
In a number of different contexts, I have wanted to estimate hitting times for a monotonic process $$(T_n)$$ taking values in the reals (or sometimes a process $$(T_n,X_n)$$ taking values in $$\mathbb R^2$$ where the first component is monotonic). I'm assuming the step size is small, and am interested in the first time, $$\tau_1$$, that $$T_n$$ exceeds some threshold $$a$$ (and in the 2d case, possibly the distribution of $$X_n$$ at this hitting time).
For some concreteness, assume $$a=1$$, $$T_0=0$$ and $$(T_n)$$ is Markov, where the jump size, $$T_{n+1}-T_n$$ is much smaller than 1. If necessary, we can assume that the jump sizes are i.i.d., although I'd ultimately prefer to have something more flexible where the distribution of $$T_{n+1}-T_n$$ is in some sense continuously dependent on $$T_n$$. I would like to obtain reasonably precise information on the distribution of $$\tau_1$$.
Is there an established machinery that can address questions of this type?
Here is a specific simple (made up) instance that I would be very interested to see a clean answer to (especially, as indicated above, an answer that might be generalizable away from the i.i.d. jump case): Let $$T_{n+1}-T_n\sim \exp(\text{Unif}[-2k,-k])$$ and $$T_0=0$$. What can be said about the distribution of $$\tau_1$$? How does the distribution of $$\tau_1$$ depend on $$k$$?
• I am not sure if I understood the question correctly. The distribution of $\tau_1$ is given by the formula $P(\tau_1>n)=P(T_n\le a)$ for all natural $n$. So, the problem reduces to that of the distribution of $T_n$. Already in the iid jump case, there are a million papers on that. – Iosif Pinelis Nov 18 at 22:34
• In the i.i.d. case, this is precisely what renewal theory studies. Its extension to the "Markovian" setting (although perhaps slightly different from what you wrote; I would have to check that) is called Markov renewal theory; Çinlar wrote a survey of this subject long ago. – Mateusz Kwaśnicki Nov 18 at 22:58
• Thanks @IosifPinelis. I think you have understood my question. I agree with your reduction, and this is a helpful observation. Do you know any papers where this is actually used to get explicit about the distribution on $\tau$? – Anthony Quas Nov 19 at 0:22
• @AnthonyQuas : I think I saw this reduction somewhere long ago, when I was a student. I don't remember where, we almost used no textbooks. Or maybe my memory is mistaken. I think the answer will depend on what properties of the distribution of $\tau_1$ you want to study. – Iosif Pinelis Nov 19 at 0:41
• I think this paper by Reinert and Yang begins with an up-to-date discussion of known quantitative bounds on the distribution of $\tau$. – Mateusz Kwaśnicki Nov 20 at 9:13
The simple key observation is that $$$$P(\tau_1\le n)=P(T_n>1). \tag{1}$$$$ Note that $$ET_n=n\de$$ and $$Var\,T_n=n\de^2\si^2$$.
So, by Chebyshev's inequality, for any fixed $$t\in(0,1)$$ and any $$n\le(1-t)/\de$$, $$$$P(\tau_1\le n)=P(T_n>1)\le\frac{n\de^2\si^2}{(1-n\de)^2} \le\frac{(1-t)\de\si^2}{t^2}\to0;$$$$ similarly, $$P(\tau_1\ge n)\to0$$ if $$n\sim(1+t)/\de$$ and hence if $$n\gtrsim(1+t)/\de$$. So, $$$$\de\tau_1\to1$$$$ in probability and hence without loss of generality we need consider only $$$$n\sim1/\de$$$$ in (1).
Next, by the Berry--Esseen inequality, $$$$P(\tau_1\le n)=P(T_n>1)=P\Big(Z>\frac{1-n\de}{\de\si\sqrt n}\Big)+R =P\Big(Z\le\frac{n-1/\de}{\si\sqrt n}\Big)+R,$$$$ where $$Z\sim N(0,1)$$, $$$$|R|\le C\frac\be{\si^3\sqrt n}\sim C\frac{\be\sqrt\de}{\si^3}\to0$$$$ by (0), and $$C$$ is a universal constant. Also, assuming $$|\frac{n-1/\de}{\si\sqrt n}|=O(1)$$, we have $$$$P\Big(Z\le\frac{n-1/\de}{\si\sqrt n}\Big) =P\Big(Z\le\frac{n-1/\de}{\si/\sqrt\de}\Big)+o(1).$$$$
So, $$\tau_1$$ is asymptotically normal with asymptotic mean $$1/\de$$ and asymptotic variance $$\si^2/\de$$. | 2019-12-15 15:03:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 47, "wp-katex-eq": 0, "align": 0, "equation": 8, "x-ck12": 0, "texerror": 0, "math_score": 0.9514965415000916, "perplexity": 273.8688909153546}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575541308604.91/warc/CC-MAIN-20191215145836-20191215173836-00463.warc.gz"} |
https://stuxnet999.github.io/volatility/2020/08/08/Writing-Plugins-Volatility-Part2.html | # Writing a simple Volatility plugin - Part 2
This post covers the basics of writing a simple plugin for the Volatility framework using Unified Output and using generator functions in python.
This post is a continuation of the last post that I’ve written on writing plugins for Volatility 2. In the previous post, we developed the function render_text() which was used to produce output to stdout.
In this article, I will be talking about why render_text() is less efficient and why an alternative is required to write plugins efficiently.
## Pre-requisites
• If you are here for the first time, you may want to read the part-1 of this blog series here.
• You must be familiar with the usage of common data types in python (lists, tuples etc…)
• Experience with using Volatility
As I had said in my previous post, the 3rd prerequisite is not required but its a plus if you have used volatility before.
## Quick recap
In this section, we’ll have a quick recap about the final code we wrote in the previous blog post. We will modify the same piece of code and build a more efficient and smarter code using simple python concepts.
import volatility.plugins.common as common
import volatility.utils as utils
import volatility.win32 as win32
class TestPlugin(common.AbstractWindowsCommand):
""" Works exactly like pslist """
def calculate(self):
def render_text(self, outfd, data):
outfd.write("{0}\t {1}\t {2}\n".format(PID, CreateTime, Process_name))
We created the render_text() function to list all the active processes in the system and we directed the output to stdout.
And it worked well too.
## Drawbacks of render_text()
Though our previous plugin was working fine, it was not ideal. Let us see why.
• Using render_text() limits us from producing the output of other common, important formats.
• If we go by this logic, we might have to write a render_X() function to generate every other format we may need.
• X → text, JSON, SQLite, xlsx, dot etc…
Volatility has a standard list of renderers which are most commonly used in the industry. So we have to write code which is compatible with their standards.
So let us dive in and try to build a better plugin which can render all the standard formats.
## Unified output & generator
def generator(self, data):
def unified_output(self,data):
tree = [
("PID", int),
("Create Tame", str),
("Process Name", str)
]
return TreeGrid(tree, self.generator(data))
### Understanding the code
You might be familiar with the some of the terminologies here like UniqueProcessId, ImageFileName, CreateTime etc. I have covered the detailed definition of these terminologies in the previous post and also introduced about the _EPROCESS structure
So we have 2 functions now, generator() & unified_output().
TreeGrid takes in a tuple the name and data type of each column which is produced.
The generator() function returns the value which is mapped to the corresponding item in the tree (In this case PID, Process Name, Create Time ).
Note: The data type of the corresponding fields in both the tree tuple-list and the yield in the generator() must be the same.
Also TreeGrid is imported from the volatility.renderers module.
Now we have everything that we require. Let us incorporate this in our previous code and test it.
## Testing output
So the final code that we have is,
import volatility.plugins.common as common
import volatility.utils as utils
import volatility.win32 as win32
from volatility.renderers import TreeGrid # Importing TreeGrid.
class TestPlugin(common.AbstractWindowsCommand):
""" Works exactly like pslist """
def calculate(self):
def generator(self, data):
def unified_output(self,data):
tree = [
("PID", int),
("Create Time", str),
("Process Name", str)
]
return TreeGrid(tree, self.generator(data))
We will test the output in some of the standard renderer types and see if our code works perfectly
### Testing TEXT
Let us proceed with the format most commonly used.
$volatility --plugins=testplugin/ -f memorydump.vmem --profile=Win7SP1x86 testplugin --output=text ### Testing SQLite $ volatility --plugins=testplugin/ -f memorydump.vmem --profile=Win7SP1x86 testplugin --output=sqlite --output-file=test.sqlite
I’m using DB browser to view the SQLite file.
### Testing DOT
\$ volatility --plugins=testplugin/ -f memorydump.vmem --profile=Win7SP1x86 testplugin --output=dot --output-file=test.dot
## Final thoughts
Now with this, I complete the series of writing simple volatility plugins. There can be more additions made to this code to make it a fully-fledged plugin, for example adding command-line arguments etc..
With learnings from this blog post, I hope you will be able to read and understand more of Volatility’s plugin codebase. | 2020-12-02 21:27:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.48021116852760315, "perplexity": 4324.242433631099}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141716970.77/warc/CC-MAIN-20201202205758-20201202235758-00623.warc.gz"} |
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# Star formation in the Sh 2-53 region influenced by accreting molecular filaments
Preprint
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There is no author summary for this article yet. Authors can add summaries to their articles on ScienceOpen to make them more accessible to a non-specialist audience.
### Abstract
We present a multi-wavelength analysis of a $$\sim$$30$$' \times$$30$$'$$ area around the Sh 2-53 region (hereafter S53 complex), which is associated with at least three H II regions, two mid-infrared bubbles (N21 and N22), and infrared dark clouds. The $$^{13}$$CO line data trace the molecular content of the S53 complex in a velocity range of 36--60 km s$$^{-1}$$, and show the presence of at least three molecular components within the selected area along this direction. Using the observed radio continuum flux of the H II regions, the derived spectral types of the ionizing sources agree well with the previously reported results. The S53 complex harbors clusters of young stellar objects (YSOs) that are identified using the photometric 2--24 $$\mu$$m magnitudes. It also hosts several massive condensations (3000-30000 $$M_\odot$$) which are traced in the {\em Herschel} column density map. The complex is found at the junction of at least five molecular filaments, and the flow of gas toward the junction is evident in the velocity space of the $$^{13}$$CO data. Together, the S53 complex is embedded in a very similar "hub-filament" system to those reported in Myers, and the active star formation is evident towards the central "hub" inferred by the presence of the clustering of YSOs.
### Author and article information
###### Journal
25 December 2017
###### Article
1712.09352 | 2020-10-22 13:10:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5554742813110352, "perplexity": 4437.537306053822}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107879537.28/warc/CC-MAIN-20201022111909-20201022141909-00568.warc.gz"} |
http://appliedmechanics.asmedigitalcollection.asme.org/issue.aspx?journalid=112&issueid=937874&direction=P | 0
IN THIS ISSUE
### Research Papers
J. Appl. Mech. 2019;86(5):051001-051001-9. doi:10.1115/1.4042576.
The kinetic energy of a mass moving horizontally can be completely converted into potential energy using a spring as an intermediary. The spring can be used to temporarily store some of the energy of the mass and change the direction of motion of the mass from horizontal to vertical. A nondimensional framework is used to study this problem for a point mass, first with a linear spring and then with a nonlinear spring that is an elastica. Solutions to the problems with the linear spring and elastica show many similarities and some dissimilarities. The dynamics of the point mass and elastica resemble the mechanics of a pole-vault; and therefore, a nonconservative external torque is introduced to parallel the muscle work done by vaulters. For the nonconservative system, the problem is solved for complete transformation of the kinetic energy of the mass and the work done by the external torque into potential energy of the mass. The initial velocities for the two cases, with and without the nonconservative force, are quite similar; and therefore, the maximum potential energy of the mass is higher in the presence of the nonconservative force. A realistic dimensional example is considered; the solution to the problem, despite several simplifying assumptions, is found to be similar to data of elite pole vaulters presented in the literature.
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2019;86(5):051002-051002-10. doi:10.1115/1.4042575.
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2019;86(5):051003-051003-13. doi:10.1115/1.4042573.
We theoretically study the electromechanical behaviors of a laminated thin-film piezoelectric semiconductor (PS) composite plate with flexural deformation. The nonlinear equations for drift currents of electrons and holes are linearized for a small carrier concentration perturbation. Following the structural theory systemized by R. D. Mindlin, a system of two-dimensional (2D) equations for the laminated thin-film PS plate, including the lowest order coupled extensional and flexural motion, are presented by expanding the displacement, potential, and the incremental concentration of electrons and holes as power series of the plate thickness. Based on the derived 2D equations, the analytical expressions of the electromechanical fields and distribution of electrons in the thin-film PS plate with an n-type ZnO layer subjected to a static bending are presented. The numerical results show that the electromechanical behaviors and piezotronic effects can be effectively controlled by the external applied force and initial concentration of carriers. The derived 2D equations and numerical results in this paper are helpful for developing piezotronic devices.
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2019;86(5):051004-051004-11. doi:10.1115/1.4042574.
Surface energy outside the contact zone, which is ignored in the classical Johnson–Kendall–Roberts (JKR) model, can play an essential role in adhesion mechanics of soft bodies. In this work, based on a simple elastic foundation model for a soft elastic half space with constant surface tension, an explicit expression for the change of surface energy outside the contact zone is proposed for a soft elastic substrate indented by a rigid sphere in terms of two JKR-type variables , a), where a is the radius of the contact zone and δ is the indentation depth of the rigid sphere. The derived expression is added to the classical JKR model to achieve two explicit equations for the determination of the two JKR variables , a). The results given by the present model are demonstrated with detailed comparison with known results reported in recent literature, which verified the validity and robust accuracy of the present method. In particular, the present model confirms that the change of surface energy of the substrate can play an essential role in micro/nanoscale contact of soft materials (defined by $W/(E*R)≥0.1$, where W is the adhesive energy, $E*$ is the substrate elasticity, and R is the rigid sphere radius). The present model offers a simpler analytical method for adhesion mechanics of a rigid sphere on a soft elastic substrate when compared with several existing methods proposed in recent literature that request more substantial numerical calculations.
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2019;86(5):051005-051005-7. doi:10.1115/1.4042577.
We consider the maximum value of the magnitude of transformation strain for an Eshelby inclusion set by the requirement of non-negative dissipation. The general formulation for a linear elastic solid shows that the dissipation associated with a strain transformation can be calculated as an integral over the transformed inclusion. Closed-form expressions are given for the maximum transformation strain magnitude in an isotropic linear elastic solid for both cylindrical and spherical inclusions that have undergone transformations corresponding to either a pure volume (or area) change or a pure shear. Most results presented are for transformations in an infinite solid and presume uniform material properties. Examples of the effect of a finite boundary and of differing material properties inside and outside the transformed inclusion are also given. The analytical results indicate that non-negative dissipation typically limits the transformation strain to being a constant of order unity times the critical stress at transformation divided by a relevant elastic modulus.
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2019;86(5):051006-051006-11. doi:10.1115/1.4042893.
Modeling the interface between two adherents in a co-cured composite joint for a delamination analysis is always a challenge since properties and thickness of the material forming the interface are not clearly defined or well characterized. In a conventional finite element (FE) analysis using virtual crack closure technique (VCCT) based on a linear elastic fracture mechanics (LEFM) theory, adherents are assigned to share the same common nodes along their intact interface. On the other hand, an FE analysis using cohesive elements or analytical methods based on an adhesive joint model for a delamination analysis of a co-cured joint will require modeling of the interface as well as the appropriate selection of its thickness and properties. The purpose of this paper is to establish the applicability and limitation of the adhesive joint model for a delamination analysis of a co-cured composite joint. In particular, it will show that when certain requirements are met, the strain energy release rates (SERR) become independent or nearly independent of the adhesive stiffness and thickness, and thus, SERR of an adhesive joint will be the same as that for a co-cured joint. These requirements are determined from a theoretical consideration, and they can be expressed explicitly in terms of joint characteristic (or load transfer) lengths and joint physical lengths. The established requirements are further validated by numerical results for various cracked joint geometries. Finally, implication of a mode ratio obtained by the proposed adhesive joint model for a corresponding delamination crack is discussed.
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2019;86(5):051007-051007-11. doi:10.1115/1.4042567.
The objective of the present work is to investigate the possibility of improving both stiffness and energy absorption in interlocking, architectured, brittle polymer blocks through hierarchical design. The interlocking mechanism allows load transfer between two different material blocks by means of contact at the mating surfaces. The contacting surfaces further act as weak interfaces that allow the polymer blocks to fail gradually under different loading conditions. Such controlled failure enhances the energy absorption of the polymer blocks but with a penalty in stiffness. Incorporating hierarchy in the form of another degree of interlocking at the weak interfaces improves stress transfer between contacting material blocks; thereby, improvement in terms of stiffness and energy absorption can be achieved. In the present work, the effects of hierarchy on the mechanical responses of a single interlocking geometry have been investigated systematically using finite element analysis (FEA) and results are validated with experiments. From finite element (FE) predictions and experiments, presence of two competing failure mechanisms have been observed in the interlock: the pullout of the interlock and brittle fracture of the polymer blocks. It is observed that the hierarchical interface improves the stiffness by restricting sliding between the contacting surfaces. However, such restriction can lead to premature fracture of the polymer blocks that eventually reduces energy absorption of the interlocking mechanism during pullout deformation. It is concluded that the combination of stiffness and energy absorption is optimal when fracture of the polymer blocks is delayed by allowing sufficient sliding at the interfaces.
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2019;86(5):051008-051008-11. doi:10.1115/1.4042894.
This work examines elastic wave propagation phenomena in open-cell foams with the use of the Bloch wave method and finite element analysis. Random foam topologies are generated with the Surface Evolver and subsequently meshed with Timoshenko beam elements, creating open-cell foam models. Convergence studies on band diagrams of different domain sizes indicate that a representative volume element (RVE) consists of at least 83 cells. Wave directionality and energy flow features are investigated by extracting phase and group velocity plots. Explicit dynamic simulations are performed on finite size domains of the considered foam structure to validate the RVE results. The effect of topological disorder is studied in detail, and excellent agreement is found between the wave behavior of the random foam and that of both the regular and perturbed Kelvin foams in the low-frequency regime. In higher modes and frequencies, however, as the wavelengths become smaller, disorder has a significant effect and the deviation between regular and random foam increases significantly.
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2019;86(5):051009-051009-6. doi:10.1115/1.4042919.
A rigid inclusion is embedded at a finite depth in a soft layer resting on a rigid substrate. A spherical indenter presses vertically onto the surface, deforming the matrix and displacing the inclusion. A subsurface inclusion initially near the indentation axis moves primarily downward, until an unstable lateral jump occurs to minimize the energy stored in the elastic medium. Such an instability is unique to soft materials undergoing large deformation. A two-dimensional plane-strain finite element analysis is used to simulate the 3D phenomenon.
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2019;86(5):051010-051010-10. doi:10.1115/1.4042570.
Emerging stretchable piezoelectric devices have added exciting sensing and energy harvesting capabilities to wearable and implantable soft electronics. As conventional piezoelectric materials are intrinsically stiff and some are even brittle, out-of-plane wrinkled or buckled structures and in-plane serpentine ribbons have been introduced to enhance their compliance and stretchability. Among those stretchable structures, in-plane piezoelectric serpentine ribbons (PSRs) are preferred on account of their manufacturability and low profiles. To elucidate the trade-off between compliance and sensitivity of PSRs of various shapes, we herein report a theoretical framework by combining the piezoelectric plate theory with our previously developed elasticity solutions for passive serpentine ribbons without piezoelectric property. The electric displacement field and the output voltage of a freestanding but nonbuckling PSR under uniaxial stretch can be analytically solved under linear assumptions. Our analytical solutions were validated by finite element modeling (FEM) and experiments using polyvinylidene fluoride (PVDF)-based PSR. In addition to freestanding PSRs, PSRs sandwiched by polymer layers were also investigated by FEM and experiments. We found that thicker and stiffer polymers reduce the stretchability but enhance the voltage output of PSRs. When the matrix is much softer than the piezoelectric material, our analytical solutions to a freestanding PSR are also applicable to the sandwiched ones.
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2019;86(5):051011-051011-8. doi:10.1115/1.4042920.
Laminated ribbons have been widely adopted for structures of flexible electronics to simultaneously achieve the electronic functions and mechanical performances. Their effective tensile stiffness and bending stiffness, which are extensively used as fundamental parameters in the mechanical analysis, are usually obtained by the plane-strain hypothesis for simplicity. However, it is found that the practical condition is usually closer to the traction free, even for the cases with a relatively large width. Here, a traction-free model is proposed to analytically obtain the effective tensile stiffness and bending stiffness of laminated ribbons, which can be used directly in the mechanical analysis of flexible electronics. The prediction of the traction-free model agrees very well with the precise result obtained by 3D finite element analysis (FEA) for the cases that are in the range of structure designs of flexible electronics. It is found that the tensile/bending stiffness of traction-free model is between the plane-stress model and plane-strain model, but is closer to the plane-stress model. The use of the plane-strain model sometimes may yield a considerable error in the mechanical analysis of flexible electronics. The parameter study shows that this model is very important for the problems with advanced materials, such as metamaterials with negative Poisson's ratio. This work provides a theoretical basis for the mechanical analysis of flexible electronics.
Commentary by Dr. Valentin Fuster
### Technical Brief
J. Appl. Mech. 2019;86(5):054501-054501-6. doi:10.1115/1.4042696.
Conventional wisdom would have it that moving mechanical systems that dissipate energy by Coulomb friction have no relationship between force and average speed. One could argue that the work done by friction is constant per unit of distance travelled, and if propulsion forces exceed friction, the net work is positive, and the system accumulates kinetic energy without bound. We present a minimalistic model for legged propulsion with slipping under Coulomb friction, scaled to parameters representative of single kilogram robots and animals. Our model, amenable to exact solutions, exhibits nearly linear (R2 > 0.96) relationships between actuator force and average speed over its entire range of parameters, and in both motion regimes, it supports. This suggests that the interactions inherent in multilegged locomotion may lead to governing equations more reminiscent of viscous friction than would be immediately obvious.
Commentary by Dr. Valentin Fuster | 2019-03-24 07:14:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4800703525543213, "perplexity": 1226.1328393065517}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912203378.92/warc/CC-MAIN-20190324063449-20190324085449-00010.warc.gz"} |
https://web.math.sinica.edu.tw/bulletin/archives_articlecontent16.jsp?bid=MjAwODMwOQ== | Archives
The logarithmic coefficients of close-to-convex functions
by Zhongqiu Ye
Vol. 3 No. 3 (2008) P.445~P.452
ABSTRACT
We prove that if $n \ge 2$ for each close-to-convex functions in S whose n-th logarithmic coefficients $\gamma_n$ satisfies $|\gamma_n| \le A$log $n/n$, where A is an absolute constant.
KEYWORDS
Close-to-convex functions, logarithmic coefficients, starlike functions
MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 30C45
MILESTONES | 2023-01-31 04:08:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9506898522377014, "perplexity": 11565.871228996915}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499842.81/warc/CC-MAIN-20230131023947-20230131053947-00859.warc.gz"} |
https://pos.sissa.it/350/065/ | Volume 350 - 7th Annual Conference on Large Hadron Collider Physics (LHCP2019) - Posters
Upgrade of the ATLAS Thin Gap Chamber Electronics for HL-LHC runs
H. Asada* and On behalf of the ATLAS Muon Collaboration
Full text: pdf
Pre-published on: August 29, 2019
Published on: December 04, 2019
Abstract
The High-Luminosity LHC (HL-LHC) is planned to start the operation in 2026 with an instantaneous luminosity of $7.5 \times 10^{34}~\rm{cm^{-2}s^{-1}}$. In order to cope with higher proton-proton collision rate, the trigger and readout electronics of ATLAS Thin Gap Chamber (TGC) needs to be replaced. All hit data will be transferred from the frontend to the backend boards, and a fast-tracking algorithm will be applied on these hits for the first-level muon triggering. The first prototype of the frontend board has been developed with full functions required for HL-LHC runs including the data transfer of 256 channels with a $16~\rm{Gbps}$ bandwidth and the control of the discriminator threshold. They were demonstrated at the CERN SPS beam facility. The rate of single event upsets in Kintex-7 FPGA integrated on the prototype board was measured in the ATLAS detector area, and automatic error correction was demonstrated. The fast-tracking algorithm was performed using a Monte-Carlo sample and data taken by ATLAS. The result indicates that the advanced trigger based on fast-tracking reduces the trigger rate by 30% while increasing the efficiency by a few percent. These studies provide essential ingredients in the development of ATLAS TGC electronics for HL-LHC.
DOI: https://doi.org/10.22323/1.350.0065
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http://weyl.math.toronto.edu/victor_ivrii/research/publications-to-download/ | ## Current Project: Sharp Spectral Asymptotics Reborn
2. Sharp Spectral Asymptotics for Operators with Irregular Coefficients.
Internat. Math. Res. Notices 2000, no. 22, 115–1166.
3. (with M. Bronstein) Sharp Spectral Asymptotics for Operators with Irregular Coefficients. Pushing the Limits
(final version as of Sep. 2002). Comm. Part. Diff. Equats., v. 28, no 1&2, pp. 99–123, (2003).
4. Sharp Spectral Asymptotics for Operators with Irregular Coefficients. Pushing the Limits II
(final version as of Sep. 2002). Comm. Part. Diff. Equats., v. 28, no 1&2 pp. 125–156, (2003).
5. Sharp Spectral Asymptotics for Magnetic Schrödinger Operator with Irregular Potential.
(Aug. 2, 2004) Russian Journal of Mathematical Physics, 11:4 (2004), 415–428.
6. Publications in Bibserver.
7. Publications in arXiv.
8. Personal page on MathSciNet.
## Talks
Do not open via web browser! Some of them are large files!
### TeX, LaTeX & Friends
1. e–Articles, e–Books and e–Talks too.
2. TeX Freak.
3. Beamer All the Way.
4. e–Articles, e–Books and e–Talks too (new version; you need Adobe Reader v. 8 or above; extract separate files from Collection (also known as Portfolio).
## Previous Projects
### Multiparticle Quantum Asymptotics
1. Around Scott correction terms.
(Proceedings of the Conference, Saint–Jean–de–Monts, France, June 1994)
2. Asymptotics of the ground state energies of large Coulomb systems
(with I.M.Sigal). Ann. Math., 138(1993), 243–335.
3. Semiclassical asymptotics for exchange energy.
Séminaire sur les Équations aux Dérivées Partielles, 1993–1994, Exp. No. XX, 12 pp., École Polytech., Palaiseau, 1994.
4. Holzhau94.
Semiclassical spectral asymptotics and multiparticle quantum theory (Proceedings of the Conference, Holzhau, Germany, July 1994)
5. Asymptotics of the ground state energy of heavy molecules in the strong magnetic field.
My talk in Minnesotta May 1995.
6. Asymptotics of the ground state energy of heavy molecules in the strong magnetic field. I.
(atoms with magnetic field B=o(N3) and molecules with B=0), Russian Journal of Mathematical Physics, 4:1 (1996), 29–74.
7. Asymptotics of the ground state energy of heavy molecules in the strong magnetic field. II
(molecules with magnetic field B=o(N3)), Russian Journal of Mathematical Physics, 5:3 (1997), 321–354.
8. Heavy molecules in the strong magnetic field.
(Short note: molecules with magnetic field B=o(N3): asymptotics of the ground state energy, ionization energy estimate, estimates for the excessive negative and positive charges) Russian Journal of Math. Phys., 4, (1996), N 1, 29–74.
9. Heavy molecules in the strong magnetic field. Estimates for ionization energy and excessive charge.
(molecules with magnetic field B=o(N3))
10. Heavy molecules in the strong magnetic field
My talk at conference at St.–Jean–de–Monts, 1997 (molecules with magnetic field B=o(N3)).
11. Heavy atoms in the superstrong magnetic field.
My talk at conference at M.Sh.Birman conference, Stockholm, Jan 1998. (Atoms in magnetic field B>cN3)
### Precise spectral asymptotics for Neumann Laplacian in domains with cusps
12. Precise spectral asymptotics for Neumann Laplacian in domains with cusps.
13. Eigenvalue asymptotics for Neumann Laplacian in domains with ultra–thin cusps.
(my talk in Ecole Polytechnique, Jan 1999) I deal with Laplacians and similar operators generated by generalized Maxwell system
### Miscellaneous
14. Accurate Spectral Asymptotics for Periodic Operators.
(Proceedings of the Conference, Saint–Jean–de–Monts, France, June 1999)
15. Semiclassical spectral asymptotics.
(Proceedings of the Conference, Nantes, France, June 1991)
## Old Papers Digitalized (djvu)
Check LizardTech for plug–ins/readers for Windows and MacOS and DjVuLibre for plug–ins/readers for many platforms.
How to digitalize.
### Really Old Book
16. Precise Spectral Asymptotics for Elliptic Operators Acting in Fiberings over Manifolds With Boundary
Lecture Notes in Mathematics, vol 1100, Springer–Verlag (January 1, 1985).
If you do not have djvu viewers you can still see it via Java applet.
### Some of Articles
17. Exponential decay of the solution of the wave equation outside an almost star-shaped region, Soviet. Math. Dokl., 10:6 (1989) 1527–1530.
18. (with V. Petkov) Necessary Conditions for the Cauchy Problem for Non–Strictly Hyperbolic Equations to be Well–Posed, Russian Math. Surveys, 29:5 (1974), 1–70; Russian
19. Well–posedness in Gevrey classes of the Cauchy problem for non–strict hyperbolic operators, Mat. Sbornik, 25:3 (1975), 390–413.
20. Sufficient Conditions for Regular and Completely Regular Hyperbolicity, Trudy Moskovskogo Matem. Obshchestva, 33 (1976), 1–65.
21. Wave fronts of solutions of certain pseudodifferential equations, Functional Analysis and Its Applications, 10:2 (1976), 141–142.
22. Cauchy problem conditions for hyperbolic operators with characteristics of variable multiplicity for Gevrey classes, Siberian Mathematical Journal, 17:6 (1976), 921–931.
23. Conditions for correctness in Gevrey classes of the Cauchy problem for weakly hyperbolic equations, Siberian Mathematical Journal, 17:3 (1976), 422–435.
24. The Well-Posedness of the Cauchy Problem For Nonstrictly Hyperbolic Operators. III. The Energy Integral, Trudy Moskovskogo Matem. Obshchestva, 34 (1977), 149–168.
25. Propagation of Singularities of Solutions of Symmetric Hyperbolic Systems, ICM 1978 (Helsinki), 771–776.
26. Wave Front Sets of Solutions of Certain Pseudodifferential Operators, Trudy Moskovskogo Matem. Obshchestva, 39 (1979), 49–86. Also in Russian.
27. Wave Front Sets of Solutions of Certain Hyperbolic Pseudodifferential Operators, Trudy Moskovskogo Matem. Obshchestva, 39 (1979), 87–119. Also in Russian.
28. Propagation of singularities of solutions of nonclassical boundary–value problems for the wave equation, Functional Analysis and Its Applications, 13:3 (1979), 226–227.
29. Wave fronts of solutions of boundary–value problems for symmetric hyperbolic systems, Siberian Mathematical Journal, 20:4 (1979), 516–524.
30. Wave fronts of solutions of boundary–value problems for symmetric hyperbolic systems II. Systems with characteristics of constant multiplicity, Siberian Mathematical Journal, 20:5 (1979), 722–734.
31. Wave fronts of solutions of symmetric pseudodifferential systems, Siberian Mathematical Journal, 20:3 (1979), 390–405.
32. Second term of the spectral asymptotic expansion of the Laplace – Beltrami operator on manifolds with boundary, Functional Analysis and Its Applications, 14:2 (1980), 98–106. Also in Russian.
33. Wave fronts for solutions of boundary–value problems for a class of symmetric hyperbolic systems, Siberian Mathematical Journal, 21:4 (1980), 527–534.
34. Wave fronts of solutions of boundary–value problems for symmetric hyperbolic systems. III. Systems with characteristics of variable multiplicity, Siberian Mathematical Journal, 21:1 (1980), 54–60.
35. The propagation of singularities of solutions of nonclassical boundary value problems for second order hyperbo1jc equations , Trudy Moskovskogo Matem. Obshchestva, 43 (1981), 87–99.
36. Exact spectral asymptotics for the Laplace – Beltrami operator in the case of general elliptic boundary conditions, Functional Analysis and Its Applications, 15:1 (1981), 59–60.
37. Accurate spectral asymptotics for elliptic operators that act in vector bundles, Functional Analysis and Its Applications, 16:2 (1982), 101–108. (Also in Russian)
38. Two papers with O.Zaitseva:
• Correctness of the Cauchy problem for some hyperbolic operators with characteristics of high variable multiplicity, Russian Math. Surveys, 37:3 (1982), 187–188;
• Strict and nonstrict inequalities in conditions for well-posedness of the Cauchy problem, Russian Math. Surveys, 40:2 (1985), 179–180.
39. Asymptotics of a spectral problem connected with the Laplace–Beltrami operator on a manifold with boundary, Functional Analysis and Its Applications, 17:1 (1983), 56–57.
40. Global and partially global operators. Propagation of singularities and spectral asymptotics. Microlocal analysis (Boulder, Colo., 1983), 119–125, Contemp. Math., 27, Amer. Math. Soc., Providence, RI, 1984.
41. Three spectral problems revised. In Hyperbolic equations and related topics (Katata/Kyoto, 1984), pages 85– 88. Academic Press, Boston, MA, 1986.
42. Precise eigenvalue asymptotics for transversally elliptic operators In Current topics in partial dirential equations, pages 55–62. Kinokuniya, Tokyo, 1986.
43. Asymptotics of the discrete spectrum for certain operators in Rd, Functional Analysis and Its Applications, 19:1 (1985), 61–62.
44. On the number of negative eigenvalues of Schrödinger operators with singular potentials Hyperbolic equations (Padua, 1985), 74–81, Pitman Res. Notes Math. Ser., 158, Longman Sci. Tech., Harlow, 1987.
45. (with S.Fedorova) Dilatation and the asymptotics of the eigenvalues of spectral problems with singularities, Functional Analysis and Its Applications, 20:4 (1986), 277–281. (also in Russian)
46. Estimates for a Number of Negative Eigenvalues of the Schrödinger Operator with Singular Potentials, ICM 1986, Berkeley, 1084–1093.
47. Precise spectral asymptotics for elliptic operators on manifolds with boundary, Siberian Mathematical Journal, 28:1 (1987), 80–86.
48. Linear hyperbolic equations (in Russian) Itogi nauli i tehniki. Ser. Sovr. problemy matematiki, Fundamental'nye napravleniya, v. 33, 157–247, 1988, VINITI.
49. Spectral asymptotics for the family of commuting operators. Operator calculus and spectral theory (Lambrecht, 1991), 139–148, Oper. Theory Adv. Appl., 57, Birkhauser, Basel, 1992.
50. (with C.Fefferman, L.Seco, M.Sigal) The energy asymptotics of large Coulomb systems, Schrödinger operators (Aarhus, 1991), Lecture Notes in Phys., 403, Springer, Berlin, 1992, 79–99.
## Some papers written under my supervision
• Olga Zaitseva, Six papers on weakly hyperbolic equations, Izvestiya Vysshih Uchebnyh Zavedenij, ser. Matematika, (1980–1987), [in Russian] and [in English].
• Mariya Zaretskaya, Four papers on quadratic quantum Hamiltonians, Izvestiya Vysshih Uchebnyh Zavedenij, ser. Matematika, (1983–1989), [in English]. Note: Translator definitely had no clue about mathematics. | 2017-04-30 20:31:39 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8038804531097412, "perplexity": 4079.5445664766494}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917125849.25/warc/CC-MAIN-20170423031205-00242-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://ceftravas.com.ng/cz2ve/directional-derivative-along-a-curve.php | # Directional derivative along a curve
Olivem 2020
The Hallstar Company
Given a curve x (), the requirement of constancy of a tensor T along this curve in flat space is simply = = 0. Let . In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. Lecture 7 Gradient and directional derivative (cont’d) In the previous lecture, we showed that the rate of change of a function f(x,y) in the direction of a vector u, called the directional derivative of f at a in the direction uˆ, is simply the dot product of the gradient vector ∇~ f(a) with the unit direction vector ˆu: D directional derivative operators along curves through p. The distance along the red line from K (Kearney, Nebraska) to S (Sioux City, Iowa) is 300 km. 3. 2 and 3. In this case the boundary curve C will be where the surface intersects the plane z=1 and so will be I. Call this new point q. For permissions beyond the scope of this license, please contact us. But in all other directions, the directional deriva-tive does not exist. Let f (x, y, z) f (x, y, z) be a differentiable function of three variables and let u = cos α i + cos β j + cos γ k u = cos α i + cos β j + cos γ k be a unit vector. Link to PDF Directional derivative. What are the units of the directional derivative? 2. Both are second-order derivative operators 2. Directional Derivatives section 12. Since the function is differenciable, by theorem, ?z(x) . When u = i, the directional derivative at P 0 is ¶ f /¶ x evaluated at (x 0, y 0). It therefore generalizes the notion of a partial derivative, in which directional derivative, divergence of a vector function, Curl of a vector of this surface the normal is along the vector. Therefore the value of the directional derivative of a function does not depend on the choice of the coordinate system, i. So the derivative of f along the vector (a,b) is zero [some call this the directional derivative, but the directional derivative is along a unit vector]. Since the function f does not change along level curve or surfaces, that is gives the rate of change along a line parallel to the as expected, this directional derivative is the ith The curve of steepest descent will be in the opposite. A connection allows you to define the concept of a "constant" vector along a curve, i. 29 Sep 2016 Solution: (a) The directional derivative of f in the direction of v at the point. This is analogous to walking along a path in the rolling meadow along which the elevation does not change. 1. Answer to: Find the directional derivative of F ( x , y , z ) = x y + 2 x z 2 y + z 2 at the point ( 1 , - 2 , 1 ) along the curve defined by for Teachers for Schools for Working Scholars for DERIVATIVES ALONG VECTORS AND DIRECTIONAL DERIVATIVES Math 225 Derivatives Along Vectors Suppose that f is a function of two variables, that is,f: R2 → R, or, if we are thinking without coordinates, f: E2 → R. ) If the function f is differentiable at , then the directional derivative exists along any unit vector and one has. 17 Oct 1996 Recall the definition of partial derivative. As we smoothly change t, we smoothly change , and so we smoothly change the arc length of: the directional derivative (D L)() = d dt L(+ t ) t!0 3 The directional derivative of a scalar function. . The directional derivative is greatest when its dot product with the gradient vector is greatest, which is exactly when the two vectors are pointing in the same direction! So to find the maximum value of a function, you travel in the direction of the gradient vectors of the function. Q moving along the curves C1 and C2 have position vectors 1( ) and 2( ) at time . In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. integration. directional derivatives in two directions, namely, along the x-axis the function is constantly 0, so the partial derivative df dx is 0; likewise along the y-axis, and df dy is 0. Just picture Richard Nixon. The idea behind the directional derivative is to reproduce the partial derivative, but for directions other than along a constant x or constant y value. v Directional derivative and gradient examples by Duane Q. along a unit vector. We will begin with a definition of geodesics, then present various method for finding Course Material. " It is a numerical value associated with the surface created with the surf command. Again, a has to be a unit vector, here a is not a unit vector. Claim. As before, we can rewrite this as directional derivative so that ruh1;yi= 0: By following how we solved the constant coe cient case, we know that the solution u(x;y) is constant along some curve, whose direction (tangent vector) is given by h1;yi. has a directional derivative along every direction at every point but f is not differentiable curve (explicitly, if the slope of the line is m , then the half-line is above Definition 265 The directional derivative of f at a point (xo,yo) in the direction . Hence, the direction of greatest increase of f is the same direction as the gradient vector. 4. 3). State the definitions of limit, directional limit and limit along a curve of a function of several variables; calculate these limits for simple examples; prove and apply the Rules for Limits to calculations for more complicated functions, A tangent line to this trace curve is displayed at the input point, and the value of the directional derivative of the function in the direction of the direction vector will be displayed in the green display above the 3D plot. the gradient Thus, the directional derivative is zero. Thus, for a function of two variables, the gradient is normal to the level curves, . If one defines to be all the functions that are differentiable at the point p, then one can interpret to be a linear functional such that and is a directional derivative of f in the direction of the curve . mit. is constant along the curve (see the figures in the reference [3]). 5. Review The Directional Derivative The Gradient Vector Three Dimensions Maximum Rate of Change Remember that ab = jajjbjcos(q) where q is the angle between a and b. 0 License. Find the angle between the Lecture Notes for Engineering Mathematics III Sai-Mang Pun1 Seventh week: 7 - 11 October 2019 1Department of Mathematics, Texas A&M University, College Station, TX, USA derivative is NOT a directional derivative since it does not satisfy the rst equation. A continuous function r : [a,b] ⊂ R → Rn is called a parametric curve in Rn. Math 324 G: 14. Solutions to Quizzes . Definition: The rate of change of a function That is, the limit is independent of choice path (In, Yn) → (a,b). One would suspect that the derivative in the direction of vec u at the point (1,1) would give the slope of the tangent line to the curve on the surface of Figure 2 at the point (1,1). The directional derivative of a scalar point function Φ(x, y, z) is the rate of change of the function Φ(x, y, z) at a particular point P(x, y, z) as measured in a specified direction. of a function Explanation of Normal Derivative Exam 2 Sample SOLUTIONS 1. Let’s start from a few basics and show this fact in$\mathbb{R}^2$ (for convenience). 6 Autumn 2017 4. The third type of derivative we discuss is a new, time-dependent version of the usual directional derivative along a curve used to define parallel transport [3– 5]. 15: Graph of a surface, its level curves and 2 gradient vectors. Mathematically, if v is in the direction of a contour line, . 2) The existence of this limit means its value is the same regardless of the path along which h ! 0 ; in particular, it is zero along the path t 7!t v for any v , 0 and t > 0. This is the In other words, it's not enough for the directional derivative to exist in the x and y directions in order to guarantee that the directional derivative will exist in every direction. The map Tγ is called the parallel transport along the path γ. The directional derivative For a function f of one variable x, the derivative expresses the rate of change of f(x) as x varies. }\) The directional derivative takes on its greatest positive value if theta=0. Hint: Find the distance D as a function of x; y where x is the distance of the family east from their home and y is the distance of the family north from their home. In higher dimension, we can ask how the function value f(x) changes as x varies along a particular direction. The slope of the tangent line to the slice curve at (x 0,y 0, f(x 0,y 0) is the directional derivative. But it's more than a mere storage device, it has several wonderful interpretations and many, many uses. 6. does not give a well-defined way to take directional derivatives of vector fields along curves. It is easier, however, Example 14. Also illustrate it using TEC 11. SOLUTIONS TO HOMEWORK ASSIGNMENT #4, MATH 253 1. represents the slope of a curve. Lecture 12: Directional and Partial Derivatives 12-4 Example If w= ln(x2 + y2 + z2), then @x@w= 2x x2 + y2 + z2 12. Estimate the value of the directional derivative of the pressure function at Kearney in the direction of Sioux City. I feel a better way to look at it is that partial derivatives actually tell us the " directional derivate" along the î vector (for x derivative) and j vector (for y derivative). I Directional derivative of functions of three variables. 5 Directional Derivatives and Gradient Vectors 3 Note. This visual is pictured in Ex. We finally demonstrate that is not continuous at by finding a curve approaching the origin along which the limit at the origin is not zero. 6 Directional Derivatives ¶ permalink. The vector u controls the direction along the surface; We consider the blue curve of intersection of the surface with the vertical plane containing the vector u. Both are rotationally invariant Aside: The fact that the second directional derivative along the gradient is rotationally invariant is Unformatted text preview: THE DIRECTIONAL DERIVATIVE We have studied the meaning of the partial derivatives of a function of two variables which are defined by the limits , lim , , and , lim , These correspond to the instantaneous rate of change in the function at some point , as we move parallel to the coordinate axes. Suppose is a function of 2 variables. We will now see that this notion can be generalized to any direction in R3. where the on the right denotes the gradient and is the dot product. Estimating directional derivatives from level curves . Definition: The rate of change of a function $f(x,y)$ in the direction of a unit vector $\vec{v}=\begin{bmatrix}a\\b\end{bmatrix}[/math Def. We have shown that a logarithmic ratio of two densities divided by the distance be-tween the two positions is approximately the directional derivative of the logarithmic The way the covariant derivative was presented to me was by first showing that a vector field can provide a directional derivative for smooth functions on a manifold. Proof: Suppose that (a,b) is any vector that is tangent to the level curve of f through (x0,y0). We now want to generalize this operator to ask how vectors, and other sorts of tensors, vary as we move along that same curve thought here. the point P. Directional derivative. But in general what about the rate of change in other directions? The one tangent to your path, namely the unit tangent vector \TT, so $${dh\over ds} = \grad{h} \cdot \TT \label{Directional}$$ Evaluating the dot product answers the question, without ever worrying about arclength. Change the function and repeat the previous steps. You know, maybe this here is like 0. And in that direction (since ), the value of the directional derivative is . Wolfram|Alpha is a great resource for determining the differentiability of a function, as well as calculating the derivatives of trigonometric, logarithmic, exponential, polynomial and many other types of mathematical expressions. looking at a level curve: f(x;y) = kfor some xed number k. Fluids – Lecture 12 Notes 1. When there are two independent variables, say w = f(x;y) is di erentiable and where both x and y are di erentiable functions The function in f is converted to ppform, and the directional derivative of its polynomial pieces is computed formally and in one vector operation, and put together again to form the ppform of the directional derivative of the function in f. The directional derivative in the direction of v is the rate of change along a We define a curve as a map α : I → R3, where I is an open interval of the real line. The directional derivative takes on its greatest negative value if theta=pi (or 180 degrees). The directional derivative immediately provides us with some additional information. How do I calculate this directional derivative? Calculate the directional derivtive df/ds of the function f(x,y,z) = x^2 + y^2 + z^2 along the tangent vector of the helix x = cos t, y = sin t, z = t at the point where t = pi/4 The directional derivative of a scalar point function at point in the direction of a vector point function is given by , where is unit vector along . Note: If is any scalar point function, then along the direction of , the directional derivative of is maximum, and also the maximum value of directional derivative of at point is given by . If the line lhas symmetric equations x 1 2 = y 3 = z+ 2 7; nd a vector equation for the line l 0such that l contains the pint (2,1,-3) and i. Directional derivatives tell you how a multivariable function changes as you move along some vector in its input space. 331 (3/23/08) Estimating directional derivatives from level curves We could find approximate values of directional derivatives from level curves by using the techniques of the last section to estimate the x- and y-derivatives and then applying Theorem 1. The gradient vector, let's call it g, we can find by taking the partial derivatives of f(x,y,z) in x, y, and z: Think of this as the plane z= f(x,y)= 1 with (x, 0, 1) and (0, y, 1), above x and y axes, lowered down to z= 0. DIRECTIONAL DERIVATIVE ALONG A CURVE. Parametric curve r : R → Rn. We translate a covector S along δ then δ′ 15 Apr 2017 It means, along the tangent vector to the curve. • The maximal directional derivative of the scalar field f(x,y,z) is in the direction of the gradient vector ∇f. The dot product has its maximum value when the vector a points in the same direction as b. Calculation. Consider a curved rectangle with an infinitesimal vector δ along one edge and δ′ along the other. We therefore define the covariant derivative along the path to be given by an operator We therefore define the covariant derivative along the path to be given by an operator SPACE CURVES, TANGENT VECTOR, PRINCIPAL NORMAL, BINORMAL, CURVATURE, TORSION, FRENET-SERRET FORMULAS, SPHERICAL INDICATRICES. The search, in this case, is for a curve along which the inverse edge indicator gets the smallest possible values. Or, the "directional derivative of Xµ in the direction Yν". 8 Convention By the directional derivative in the direction along a curve at the given point (a along the curve y=x2 1, The directional derivative of a scalar field along a vector , denoted , is the derivative of as one moves along a straight path in the direction. 1 Parameterized Lines Let L be the line through point p~ in direction ~v. Hence, the maximal directional derivative of the scalar field is in the direction of the gradient vector itself. 5 and 9. (2,2,1), which direction should one travel along the curve of is useful to know how / changes as its variables change along any path from a given point. Look at the curve where zfxy or ,() intersects the plane at (1,2,0). Consider a curve : [0;1] !Mand let _ = d =dtbe its velocity. r. Directional Derivative : Let f: R3! provided the limit exists, the directional derivative of f in the direction of u at c. of f at the point x0 is perpendicular to the tangent vector at x0 to any curve γ(t) that. Move a distance Δλ along the integral curve of V passing through p. • If a surface is given by f(x,y,z) = c where c is a constant, then So when we take the directional derivative of ##f## along the second curve, you can think of the function ##f## travelling twice as fast as it would along the first curve. Relation to Lie derivative. I want to measure how much the surface normal "twists" at p, when moving along c in the tangent direction v = c'. This operator can be interpreted as a tangent vector Analogies: Calc I/II concepts in comparison with analogous Calc III concepts Rob Donnelly From Murray State University’s Calculus III, Fall 2001 Math 211, Multivariable Calculus, Fall 2011 What is the directional derivative of f at (1;3) in the direction of the vector for any curve described by a order derivative along the curve. 5 Directional Derivatives and Gradient Vectors 1005 Directional Derivatives and Gradient Vectors If you look at the map (Figure 14. H. Covariant derivative explained. directional derivative along a curve. is known as the directional derivative of f along the direction PQ. Directional Derivatives The Question Suppose that you leave the point (a,b) moving with velocity ~v = hv 1,v 2i. The directional derivative of the function at the point along the direction of the vector is the slope of the tangent line to the previous curve at . One sees easily that the directional derivative formula above is precisely the appropriate limit of a Level Curve, Gradient And Directional Derivative: Letz=f(x,y)=x2-xy +y2+y, Consider The Level Curve F(x,y)-2, What Is The A. Use position vector and directional vector to come up with the line. I Properties of the the gradient vector. Nykamp is licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 4. If you type get(h1) at the Matlab prompt, you will get a list of the current properties and their values for the surface in Figure 1. A curve on the manifold is defined as a differentiable map . In a curved spacetime the Lie derivative of a function fis again its directional derivative, L uf= u r f: (7) If u is the 4-velocity of a fluid, generating the fluid trajectories in spacetime, L ufis commonly termed the convective derivative of f. The definition of a space curve is essentially an analytical implementation of this view. For a quadratic cost function in Rn, Newton’s method identifies a zero of the gradient in one step. 116 Partial and Directional derivatives, Differentiability. Find the position vector function of a particle that has an acceleration function a(t) = cos(t=2)i+ k; an initial velocity v(0) = 3j, and an initial position r(0) = 0. Directional derivative on a mountain shown as mesh plot. filling curve, the directional derivative along the curve provides a good measure ot the rate of change of the image intensities along the scanningdirection. When u = j, the directional derivative at P 0 is ¶ f /¶ y evaluated My answer for c is: She can traval along the level curve to be at a level path, which is that the change of the height is zero. parallel translation along a curve. The contour map shows the average maximum temperature for November 2004 (in ). Gradients and Normals Page 3 of 4 Any curve in space can be written as p (t) for a parameter , and if we require this curve to be on the isosurface along the gradient at (x;y) is the second derivative along this line. Directional derivative of functions of three variables. 2 MATH11007 NOTES 18: THE DIRECTIONAL DERIVATIVE. There are both 2D and 3D views, with the constraint curve laid out upon the graph of the Directional derivatives and the gradient For a function f(x,y), we have f_x\equiv slope in the x-direction, and f_y\equiv slope in the y-direction. Derivatives measure the rate of change along a curve with respect to a given real or complex variable. ; Use the gradient to find the tangent to a level curve of a given function. As in the scatterplots in Figures C. A pixel in an image is classified as a curve point if the first derivative along n(t) vanishes Directional derivatives: In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. The level curve is tangent to -\bfi + \bfj at the point (1,1). If not, we will prove it in this problem as follows: (a)Assume fis a di erentiable function of xand yand that fhas a directional derivative in the direction of any unit Line Integral of a Vector Field A line integral (sometimes called a path integral) is an integral where the function to be integrated is evaluated along a curve. Since the dot product is zero, the gradient is orthogonal to the tangent to the level curve as shown. Now, recall that the directional derivative requires that we approach along the line . Compute the slope of the line tangent to the level curve at P and verify that the tangent line is orthogonal to the gradient at that point. The rate of change of f in the direction of u is the slope of the tangent line to C at P. The directional derivative immediately provides us with some additional Since along contour lines the change in height is zero, this means the directional derivative along the contour is zero. If f is the temperature in a room and ~r(t) is a curve with velocity~r 0 (t), then rf(~r(t)) ~r 0 (t) is the temperature change, one measures on the point moving on a curve ~r(t) experiences: the chain rule told us that this is Then we notice that each curve through p defines an operator on this space, the directional derivative, which maps f df /d (at p). occur if w were dragged along by the flow generated by v. The orange vector is the projection of the gradient of onto the tangent line of the constraint curve; its direction is the direction of increase along the constraint and its magnitude is the slope (that is, the directional derivative) of in that direction. At the origin, the derivative in both x and y directions exist. To establish this idea we must demonstrate two things: (I) that the space of directional derivatives is a vector space; and (II) that it is the vector space we want (it has the same dimensionality as M, yields a natural idea of a vector pointing along a certain direction, and so on). In general if p2M let C1(p) be the functions de ned in some neighborhood of p2M, which are di erentiable at p. Next we consider the directional derivative of a scalar function f(x;y;z). Properties of the the gradient vector. find the gradient vector at a given point of a function. A covariant derivative introduces an extra geometric structure on a manifold which allows vectors in neighboring tangent spaces to be Tangent Lines to Level Curves In Exercises 25-28, sketch the curve f(x, y) = c together with of and the tangent line at the given point. In the case of a closed curve it is also called a contour integral. This is because the scalar product is zero, i. So we can’t use it as a directional derivative. Properties of the the gradient vector Remark: If θ is the angle between ∇ f and u, then holds D u f = ∇ f · u ⇒ D u f = |∇ f | cos(θ). curve f(x,y) = 10. d/dλ= (dyν/dλ)(d/dyν) Now: What is the particular form of the correction Γ? ∇ ν Xµ = ∂ ν X µ + Γµ νσ X s The derivative of Xµ in a curved space The derivative of Xµ in flat space Correction = + factors First Note: The gradient is a fancy word for derivative, or the rate of change of a function. When ∇f is not parallel to ∇g, we can see that we can travel along g(x,y) = k Student[MultivariateCalculus] DirectionalDerivative compute the directional derivative Calling Sequence Parameters Options Description Examples Calling 14 Sep 2016 Let's start from a few basics and show this fact in[math]\mathbb{R}^2$ (for convenience). We can generalize the partial derivatives to calculate the slope in any direction. The function to be integrated may be a scalar field or a vector field. . 31 Example 3. help you understand what is happening in the above level curve plot. The animation that I've created to help me, uses the function The animation shows: the surface a unit vector rotating about the point (1, 1, 0) Directional Derivative of a Scalar Point Function. The gradient vector and directional derivatives. But how do we compute this derivative? Note that z=f(x,y) and x and y are functions of t. This [5, 6], Dede et al. What are the units of the directional derivative? contour map shows the average maximum temperature for 2004 (in °C). edu/18-02SCF10 License: Creative Commons BY-NC-SA More in Chain Rule In the one variable case z = f(y) and y = g(x) then dz dx = dz dy dy dx. Figure 3. DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR 161 We can express the directional derivative in terms of the gradient. It is the rate of change of z in the direction of u r. It is computed by modulating the gradient vector grad of the image function , by the unit vector along the scanning direction definedbythe spacefilling curve. 2. Section 14. Lagrange Multiplier j is a curve in the domain D. 5, there are dotted lines to indicate the location of the peaks of the ideal curves. Thatis, grad Call this curve α λ (μ). A directional derivative along a curve (t) such that (0) = pis a linear functional on this space de ned by _(f) := d But, we can still ask, is there a derivative in every direction? And that's basically, yes, that's the directional derivative. The first step in taking a directional derivative, is to specify the direction. EXAMPLE. A bug living in the surface and following such a curve would perceive it to be straight. Introduction Hello it's a me again drifter1! Today, we will continue with our Mathematical Analysis series of Mathematics by getting into Directional Derivatives that are based on Partial Derivatives that we covered last time. is, at that instant you are moving along the level curve f(x, y) = f(a, b) . We shall now show that the space of directional derivatives along a curve on a di↵erential MATH W80 Daily Notes Directional Derivatives (Section II. Source. So one solution for y is (5, 20). It’s actually fairly simple to derive an equivalent formula for taking directional derivatives. e. The directional derivative is the slope of the tangent line to this curve in the direction of u r. 6) 1. Action of \directional derivatives" on Vectors: an ffi Connection As one moves on a manifold, along a curve with tangent vector ~u, we write the derivative, in that direction, of a scalar function, f 2F, as ~u(f) = u f; . Note that Directional derivatives help us find the slope if we move in a direction different from the one specified by the gradient. The parameterized curve α : R → R3 defined by α(t) = p~+t~v is a parameterization of L. Directional derivatives are derivatives of height functions over particular straight lines in the domain of a function. This establishes (a) and (b) for "most rapid increase", and similar reasoning gives the statements for "most rapid decrease". , along the path y(λ). However, in practice this can be a very difficult limit to compute so we need an easier way of taking directional derivatives. Observations regarding the Laplacian and the second directional derivative along the gradient: 1. That is because, along a level curve, the value of the function is CONSTANT, and therefore, 2. This leads us to the concept of the directional derivative of $$f$$ at a particular point $$\rr=\rr_0=\rr(u_0)$$ along the vector $$\vv\text{,}$$ which is traditionally defined as follows: 1 It is often assumed that Determine the directional derivative in a given direction for a function of two variables. Lecture 28 : Directional Derivatives, Gradient, Tangent Plane The partial derivative with respect to x at a point in R3 measures the rate of change of the function along the X-axis or say along the direction (1;0;0). 1 Calculus of variations. Observe the curve that results from the intersection of the surface of the function with the vertical plane corresponding to . Seeing . Suppose for a moment that we can parametrize such a curve by x, so that the curve is given by (x In mathematics, the directional derivative of a multivariate differentiable function along a given . We derive a closed-form, numerically stable and e cient algorithm to compute the gradient of a B ezier curve on manifolds with respect to its control points, expressed as a concatenation of so-called adjoint Jacobi elds. a vector in either of these directions is tangent to the level curve at that point. Directional derivatives, Definition and examples Then the directional derivative along p is. All assigned readings and exercises are from the textbook Objectives: Make certain that you can define, and use in context, the terms, concepts and formulas listed below: 1. Then, the directional derivative of f f in the direction of u u is given by Section 3: Directional Derivatives 10 We now state, without proof, two useful properties of the direc-tional derivative and gradient. We . Thus, at a curve point, the first derivative in the direction n(t) should vanish and the second directional derivative should be of large absolute value. 17. The second-order information on the cost function is incorporated through the directional derivative of the gradient. What about the rates of change in the other directions? Definition For any unit vector, u =〈u x,u y〉let If this limit exists, this is called the directional derivative of f at the , the variation of the curve as one displaces from r 0 in the direction of u^ is purely in the zdirection, and so it is natural to try to study the rate of change of zas one moves along the u^ direction by a small displacement h^u. Geometrically the number fy(x0,y0) is the slope of the tangent line at the point (x0,y0,z0) to a curve on If this limit exists, this is called the directional derivative of f at the point (a,b) in the direction of u. 2, page 791-792, Figure 5. Since the covariant derivative of a tensor field T at a point p depends only on value of the vector field X at p one can define the covariant derivative along a smooth curve γ(t) in a manifold: Note that the tensor field T only needs to be defined on the curve γ(t) for this definition to make sense. The directional derivative is computed by taking the dot product of the gradient of and a unit vector of "tiny nudges" representing the direction. For simplicity, I'll consider (c) in the case of a level curve. Since the value of the function is constant along the curve, the directional derivative in the direction tangent to the curve must be zero. derivative is compatible with the metric, but like the convective derivative it depends on gradients of the velocity. (2015) introduced the directional q-frame along a space curve to construct a tubular surface [13]. Geometrically, it is the slope of the line tangent to the graph of the function when the function is restricted to a parameterized curve in the domain, times the speed of travel along that curve. This thesis studies and its derivatives along a path which is normal to the object boundary -- moving along the gradient direction -- in order to create an opacity function. Let Φ(x, y, z) be a scalar point function possessing first partial derivatives throughout some region R of space. Section 2. 0 0 0 |u|=1 x P = (x , y ) u P y z f (x,y For the directional derivative, instead of slicing along the positive x- and y-directions, we slice the graph along a direction (cos θ, sin θ). Find the directional derivative of the function f(x, y) x2y3 4y at the point (2, 1) in the direction of the vector v 2 i 5 j. 13. But the latter depends only on the tangent direction of the curve at the given point, not on the detailed shape of the curve. If one defines to be all the functions that are differentiable at the point p, then one can interpret to be an operator such that and is a directional derivative of f in the direction of the curve . Given a vector field V(t) defined along , we can define the covariant derivative of V to be DV dt = r _V. Although the antisymmetry is trivial with this formulation, the independence upon local coordinates is not. We can think of a space curve as a path of a moving point. using a single quantity such as the derivative. 1 Directional Derivatives 1. Now find directional derivative of Q along A= 5i + 3j + 2k. We then asked how we could get the directional derivative for higher rank tensors. ) If you take a point (x 0;y 0) on this level curve and move in a direction tangent to the curve, the directional derivative is zero. If we imagine MˆRN then we literally take a tangent plane. To quote a famous editorial (Hugo Rossi, 1996): > In the fall of 1972 President Nixon announced that the rate of increase of inflation was decreasing. The directional derivative is a generalization of the partial derivatives. The partial derivatives of a function $$f$$ tell us the rate of change of $$f$$ in the direction of the coordinate axes. What Is The Equation Of The Normal Line At (1, 1) For F(x,y)-2? D. The equation x = f(x,y) represents a surface S in space. Space curve. Solutions for practice problems, Fall 2016 Qinfeng Li December 5, 2016 Problem 1. If f is the temperature in a room and r(t) is The figure below shows the level curves, defined by f(x,y)=c, of the surface. Directional Derivatives We know we can write The partial derivatives measure the rate of change of the function at a point in the direction of the x-axis or y-axis. And the one that I had graphed is x-squared plus y-squared, f of x, y, equals x-squared plus y-squared (a) shows the curve for first directional derivative versus data value; (b) the curve for second directional derivative versus data value. Given a vector field along M, Y : M → IR3, for p ∈ M,X ∈ T pM, the directional derivative of Y in the direction X, denoted ∇ XY, is defined as, ∇ XY = d dt Y σ(t)| t=0 Recall that a level curve is defined by a path in the $$xy$$-plane along which the $$z$$-values of a function do not change; the directional derivative in the direction of a level curve is 0. There is a very simple way to do this: substitute x,y,z with the coordinates of the curve, getting: F(t)=f(x(t),y(t) partial derivatives. Suppose I The vertical plane that passes through P and P 0 (x 0, y 0) parallel to u intersects S in a curve C. Instead, this rate of change is a vector quantity, called the gradient, denoted by rf. For z= f(x;y), the gradient rf(P) is perpendicular to the level curve of fthrough P. The vertical plane that passes through P and P0(x0,y0,0) parallel to u intersects S in a curve C. Suppose we have a curve c 1 with tangent vector V 1 and a curve c 2 with tangent vector V 2. The material here on slope of a linear function thus provides the precalculus foundation for the derivative in multi-dimensional calculus Since the derivative of f(x) at x0 is the slope of the curve y = f(x) at that point the expansion to flrst order represents a linear approximation to the curve at x = x0 using the tangent to the curve at x0. The first claim, that directional derivatives form a vector space, seems straightforward. the covariant. Partial derivatives and directional derivatives. the directional derivative of f along the Similarly, every smooth vector eld along is a \direction" along which we could vary to get a new curve +t . Now try your hand at the chain rule. We rst note that if is the angle between rf(x The rate of change of a scalar field f in an arbitrary direction S is designated by d d s [f] and called a directional derivative. At any point on the y-axis the derivative in the y-direction exists and is 0. 300 km. Let w = xyz+x3. ! along curve from Part a) at point (0, 1, 1) in direction of increasing x vector at xk as the vector along which the directional derivative of grad f is equal to −grad f(xk). Green’s Theorem relates the path integral of a vector field along an oriented, simple closed curve in the xy-plane to the double integral of its derivative over the region enclosed by the curve. 3 2. (1,2,3) . Each component of the gradient is the partial derivative of fwith respect to one of its independent variables, x, yor z. Solutions to Exercises. 6 Nov 2004 The directional derivative is a generalization of a partial derivative. The directional derivative of the function f : D ⊂ R2 → R at the point P0 = (x0,y0) . We draw the level curve with the tangent vector at $(1,1)$. If u is tangent to a level curve of the function f(x,y) (or a level surface of f(x,y,z)), then Level curves take their shape from the intersection of z = f (x,y) and z = c. f = 6 above it is ∆f = 6 − 5 = 1, and the distance between the level curves along the s-axis is ∆s ≈ 1. The covariant derivative is a generalization of the Euclidean directional derivative to the manifold setting. 20 May 2015 directional derivative operators along curves through p. f(t) x y z f(x,y) P = (x ,y ) 0 0 0 How's that possible? What's the directional derivative of the potential energy, in the direction of $\vec{\mathrm ds}\,,$ in the case of potential energy in three dimensions? And what is its physical meaning? Without calculation, find the directional derivative at $(1,1)$ in the direction $-\bfi+\bfj$. But it is enough provided that these directional derivatives happen to be continuous. You're not thinking of the actual vector actually taking a step along that, but you'd be thinking of taking a step along, say, h multiplied by that vector, and h might represent some really, really small numbers. i. Download this Page as a PDF: Note: Images are replaced by captions. This seems to be the component of the directional derivative of N perpendicular to v, ie Dt N . Prove that the following di erential equations are satis ed by the given functions: (a) @2u @x 2 @2u @y + @2u @z Here a surface is drawn, along with a dashed curve in the -plane. The directional derivative of f : Rn → R along the direction u at the point x is . ]. 24 Aug 2004 Directional Derivatives. Sometimes the covariant derivative along a curve is called absolute or intrinsic derivative. is the function defined by the limit (See other notations below. 32 Example 3 SOLUTION is Gateaux differentiable at (0, 0), with its derivative there being g(a, b) = 0 for all (a, b), which is a linear operator. $\begingroup$ @user29751: It is maybe worth adding that on an arbitrary pseudo-Riemannian manifold (i. – The chain rule measures the instantaneous rate of change of a func-tion with respect to a parameterization. The directional derivative of a vector field along M is defined in a manner similar to the directional derivative of a function defined on M. As indicated before, we wish to have a way to discuss the rate of change of the function f(x,y) in any direction, not just along the x or y axis. MATH 2530 NOTES Today, we are going to discuss the directional derivative. Koh ,b Krishna Ramaswamyc February 2004 ABSTRACT A large class of fixed income trading strategies focuses on opportunities offered by the tive and the largest absolute value in the second derivative. True or False, and explain: (a) There exists a function fwith continuous second partial derivatives such that f x(x;y) = x+ y2 f y= x y2 SOLUTION: False. Partial derivatives give us an understanding of how a surface changes when we move in the $$x$$ and $$y$$ directions. Section 12. Section 1-2 : Direction Fields. 2 Find a tangent vector to $z=x^2+y^2$ at $(1,2)$ in the direction of the vector $\langle 3,4\rangle$ and show that it is parallel to the tangent plane Gradient and directional derivative Instructor: Joel Lewis View the complete course: http://ocw. Indeed, the principle underlying W82’s Eq. 001. Tangent line to that curve, and we're wondering what its slope is, so, the reason that the directional derivative is gonna give us this slope, is because, another notation that might be kinda helpful for what this directional derivative is, some people will write partial f, and partial v. The rate of change of f in the direction u is the slope of the tangent to C at P. 6 Directional Derivatives and the Gradient Motivating Questions. This leads to the idea of the directional derivative: what is the rate of . The exterior derivative works as a generalization of differentiation which maps 1-forms to 2-forms, etc; the velocity curve works as a generalization of the derivative to curves; and finally, the tangent map is a generalization of the derivative to mappings. The gradient is going to be NORMAL/PERPEDICULAR to the level curve/surface. e one does not require the scalar products on the tangent spaces to be positive-definite) there are in general many different choices of a covariant derivative, but one which is very "natural": The Levi-Civita connection (where in this context, connection is synonymous for covariant derivative). Find the directional derivative of w along the curve at P. 1248. Since the value of the function is constant along the curve, the directional derivative in the tangent direction to the curve is zero. Then at every point on the manifold we have a unique tangent space where these vectors live. Intuitively, the directional derivative of f at a point x represents the rate of change of f with respect to time when moving past x at velocity v. in a moving frame of reference’’ (or the directional de-rivative) that contains no second-order derivative in the denominator (Petterssen 1956, sections 3. 1 What direction produces the greatest directional derivative? The smallest? . Call this curve α λ (μ). 6: Directional Derivatives and the Gradient Vector Recall that if f is a di erentiable function of x and y and z = f(x;y), then the partial Directional Derivative, Gradient and Level Set Liming Pang 1 Directional Derivative The partial derivatives of a multi-variable function f(x;y), @f @x and @f @y, tell us the rate of change of the function along the x-axis and y-axis respectively. is the function defined by the limit [1] (See other notations below. So, the directional derivative Du f(x0,y0) has its maximum when u points in the same can rewrite Expression 7 for the directional derivative as ; This expresses the directional derivative in the direction of u as the scalar projection of the gradient vector onto u. We started by listing a number of qualities we wanted our new derivative operator to have. 1 Integral curves In general, a (piecewise smooth) parameterised curve C ⊂ R2 can be viewed as Geodesics∗ (Com S 477/577 Notes) Yan-BinJia Nov2,2017 Geodesics are the curves in a surface that make turns just to stay on the surface and never move sideways. And by the way, if these conditions are met, we say that f is a continuously The derivative along a curve is also used to define the parallel transport along the curve. The result is called the directional derivative. Given a curve with tangent T, and a vector field Y defined along the curve, if the covariant derivative of Y in the direction of T is zero, then Y is parallel translated along the curve. So in the last video, I defined the gradient, but let me just take a function here. What Is The Equation Of The Tangent Line At (1, 1) For F(x,y) -2? C. The gradient In most cases, there is always one direction u where the directional derivative Duf(a) is the largest. A level curve is a curve . Hopefully you found the relationship in the last problem. Directional derivatives in the direction of the standard basis vectors will be of special importance. At any point on the x-axis the derivative in the x-direction exists (and is 0). Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. The function f could be the distance to some point or curve, the altitude function for some landscape, or temperature (assumed to be So, the definition of the directional derivative is very similar to the definition of partial derivatives. 9. where the on the right denotes the gradient and is the Euclidean inner product. 1. 2 . Find a) its maximum rate of increase at (1, 2, 1), and Implicit Function Theorem, Implicit Differentiation 6. This is called the directional derivative of the function f at the point (a, b) in the direction v. When we have a curve going through these points we can take the directional derivative at every point and have a vector field in coordinate basis at every point along the curve using the parameter of the curve. For instance, along the line y= xthe function is f(x;x) = jxj= p 2, which has no derivative at where is the angle between a and b, the directional derivative can be used to determine the direction along which fincreases most rapidly, decreases most rapidly, or does not change at all. The directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through … Use derivative of p(t) (which is actually v(t)) to find the directional (tangent) vector. This means the directional derivative in the direction of the the tangent to a level curve is 0. The ellipsoid x^2+4y^2+z^2 = 18 and the plane x+2y−z = 4 intersect in a curve Γ. Directional derivative is the projection of the gradient vector along the given vector. This curve is a geodesic. The Newtonian limit of u is the 4 The formula for the directional derivative gives us the following fact. I The gradient vector and directional derivatives. That is, we would like to find the curve C that minimizes this functional. What’s the curve that generates V 1 +fV 2? It is not clear what this curve Derivative along curve. What is the slope of the tangent line to the level curve at — 8)? Select the correct choice below and, if necessary, fill in the answer box in your choice. (3) can be seen from the definition of the directional derivative along a parameterized curve in the cylindrical co- The directional derivative of a scalar function. Proof. Example From our work above, if f(x,y) = 4 − 2x2 − y2 and u = − 1√ 2 (1,1), then D uf(1,1) = 3 √ 2. Suppose, for example, that you Use directional derivatives to nd the direction the family should drive to increase their distance from the gas leak as rapidly as possible. Note. 02 – Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors Rate of change of a function along a parameterized curve i pray to budha i pass this midterm, i played too much pokemon go now i sufferrr. the directional derivative in the direction of a vector that is TANGENT to the level curve at the point is 0. , α(0) = p~,α0(0) = ~v. Give the coordinates of a point with the property that the directional derivative of w at (2,−6,3) in the direction toward that point is as large as that is correct (but it is not analytical, so I'll need a numerical method), so what I want to do is take the directional derivative of that surface along another fitted 2D curve that is only dependent on only x and y. If the function f is differentiable at , then the directional derivative exists along any vector and one has. Theorem 274 If fis a di⁄erentiable function in xand y, then fhas a direc- Because a function has constant value along a level curve, the directional derivative is zero in the direction tangent to the level curve. 31 May 2018 In the section we introduce the concept of directional derivatives. as we move infinitesimally along the direction in which u(x0,y0) points. This topic is given its own section for a couple of reasons. However, the constant ratio of this curve is generally different from the constant ratios of the other curves. A vector field is called parallel if the covariant derivative Profiting from Mean-Reverting Yield Curve Trading Strategies* Choong Tze Chuaa, Winston T. One way to specify a direction is with a vector $\vc{u}=(u_1,u_2)$ that points in the direction in which we want to compute the slope. Gauss’ Divergence Theorem extends this result to closed surfaces and Stokes’ Theorem generalizes it to simple closed surfaces in space. To give you a brief idea of what you can expect to be able to do at the end of the course here are the Intended Learning Outcomes:. ; Explain the significance of the gradient vector with regard to direction of change along a surface. Directional Derivatives Definition: The directional derivative of f x,y at the point a,b andinthedirectionoftheunit vector u 〈u1,u2 , denoted as D u f a,b , is defined by Du f a,b lim h→0 f a hu1,b hu2 −f a,b h provided the limit exists. the directional derivative operator is a geometrical object. Motion along a curve; f$. is the function defined by the limit. Final Quiz. Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Directional Derivative of a Function of Three Variables. Note that the usual partial derivatives are special cases of the directional deriva-tive: Duf = fx with u = h1, 0i = i Duf = fy with u = h0, 1i = j Comments. choosing local maximal of second derivative along direction. Similarly, in multi-variable calculus, we will use the concept of the slope of a plane to define the directional derivative, which represents the slope of a surface. The height of a mountain ranged described by a function f(x,y) is shown as a mesh plot. y = 0 where y is a vector that's perpendicular to the gradien and y is the tangent line to the level curve. n. To produce a transported curve α* λ + Δλ (μ) passing through q, simply transport each point in α λ (μ) this same distance Δλ along the integral curve passing through α λ (μ). We will make the following claim: the tangent space T p can be identified with the space of directional derivative operators along curves through p. The directional derivative of a function at a point on a given curve or surface in the direction of the normal to the curve or surface. Example on Directional Derivative and Gradient Consider the scalar function, 2 2 f (x, y, z) x yz 4xz. If f is the temperature in a room and ~r(t) is a curve with velocity ~r ′(t), then ∇f(~r(t))·~r ′(t) is the temperature Section 12. Estimate the value of the directional derivative of this temperature function at Dubbo Example. For a differentiable function z = f(x,y), it is known that the directional derivative at (2,3) in the direction of (12,5) is 62 13, and the directional derivative at (2,3) in the direction of the point Q(4,4) is is the curve on the surface shown in Figure 2. Given a function f(x,y) whose surface is graphed below, consider the point f(x0,y0): Directly below f(x0,y0) is the point (x0,y0,0 Let c be the curve of intersection between S and P through a point p on the surface. So, what if we move in another direction, let's say, the direction of some unit vector, let's call it u . along a vector. So the gradient rf(x 0;y 0) is orthogonal to the level If the function f is differentiable at x, then the directional derivative exists along any vector v, and one has. Ok, first do I find the derivative of r(t) to get 3j +t(2i-2k) and then use 2i-2k as the direction vector to find the unit vector? Then the unit vector would be Home » Partial Differentiation » Directional Derivatives. Because u is a unit vector, the value of t is precisely the distance along the . Modifying the function g, different results can be obtained. ] On a smooth manifold M, to de ne the \directional derivative" of a vector eld Y along X, our rst candidate is the Lie derivative L XY = [X;Y]: Unfortunately it does not satisfy the rst equation in (2). ; Determine the gradient vector of a given real-valued function. OK, so these are derivatives in the direction of I hat or j hat, the vectors that go along the x or the y axis. Like the partial derivatives, it is a scalar. 25 Nov 2015 kg ≡ 0 everywhere along the curve c = P ∩Σ and hence c is a geodesic smooth function f : Rn → R, the directional derivative of f at p in the Definition: The Directional Derivative of f(x,y) at (a,b) in the direction u is .$ By computation, find the directional derivative at $(1,1)$ in the direction of $-\bfi + \bfj$. Hopefully the definition of the directional derivative makes sense: you calculate the di↵erence quotient using x and y that are allowed to move along the direction of u. directional derivatives will be C/m). 5. Dx f =fx (x, y ) is the rate of change of f in the x-direction. Learn with flashcards, games, and more — for free. That is, rf= h @f @x @f @y @f @z i: For example, the partial derivative of f with respect Looking for Normal Derivative? Find out information about Normal Derivative. Restricting to just the points on this circle gives the curve shown on the surface. 23) showing contours on the West Point Area along the Hudson River in New Y ork, you will notice that the tributary streams flow perpendicular to the contours. The question asks to find the directional derivative of f(x,y,z)=x^2+yz at the point (1,-3,2) in the direction of the path r(t)=t^2i + 3tj+(1-t^2)k. 5, Directional derivatives and gradient vectors p. Then write an equation for the tangent line. About Khan Academy: Khan Academy offers practice exercises, instructional derivatives along slice curves in vertical planes parallel to the x- and y- axes. We have to convert it to a unit vector, and it is very important that we make sure that we are working with unit vectors. Since the above limit exists, the result holds along any path along which , so it certainly holds along this path. The full range of these packages and some instructions,. 1261 (Not confident at all) I think you meant its the pushforward of: the local derivative of Y along X with the manifold "flowing along" X minus of the local derivative of X along Y with the manifold "flowing along" Y this time. Conceptually, optimization along a curve is easy: read f “as you go along the curve”;. 6 – set around 6 θ π. 6 1 Definition of the directional derivative Partial derivatives allow us to see how fast a function changes. But, we know that the dot product of the gradient and the direction is by definition the directional derivative, so we have Directional derivative and partial derivatives Remark: The directional derivative D uf P0 is the derivative of f along the line r(t) = hx 0,y 0i + u t. One of the most famous variational problems involves constraining a particle to travel along a curve (imagine that the particle slides along a frictionless track). In order to calculate second derivative and curve direction of the image, partial derivative of the input image r v,r w,r v v,r v Calculate the directional derivative at the point and in the direction indicated: a. Consider the curve: Consider the limit: Since the function is not continuous at , it cannot be differentiable and cannot have a gradient vector at . Stream Function First we note the geometric relation along the curve, taking the directional derivative of the potential along Covariant Derivative Directional derivative =rate of change along straight line More general setting: f: a differentiable function γ: a smooth parametric curve Question: How does f change as we move along γ? Definition The rate of change of f(γ(t)) with respect to t is called the covariant derivative of f along γ and is denoted by ∇γ′f . The most difficult idea to convey in the entire course, for me, is that of the directional derivative and the gradient vector. and a unit vector u = a, b 2, we define the directional derivative. Example 20 Feb 1999 To put this directional derivative in terms of the partials in the directions of Any curve in space can be written as p t for a parameter t, and if we . Def. 3 through C. Gradient Vf At (1, 1)? B. The idea of the method of characteristics is to reduce the pde to an ode by first finding the behaviour of φ along a curve defined by the flow of the vector field u. Hence, the directional derivative is the dot product of the gradient and the vector u. It’s a vector (a direction to move) that Points in the direction of greatest increase of a function (intuition on why) Is zero at a local maximum or local minimum (because there is no single direction of increase Limit and Derivative of Vector Function; Example of Position, Velocity and Acceleration in Three Space; Tangent Line to a Parametrized Curve; Angle of Intersection Between Two Curves; Unit Tangent and Normal Vectors for a Helix; Sketch/Area of Polar Curve r = sin(3O) Arc Length along Polar Curve r = e^{-O} Showing a Limit Does Not Exist The gradient stores all the partial derivative information of a multivariable function. The directional q-frame offers two key advantages over the Frenet frame [3, 8]: a) it is well defined even if the curve has vanishing second derivative [11], b) it avoid the unnecessary twist around the tangent. ) If the function f is differentiable at x, then the directional derivative exists along any unit vector u, and one has. We've done that in the next command here, along with turning the scaling off. Suppose further that the temperature at (x,y) is f(x,y). In fact, the directional derivative operator is linear, so you immediately have ##D_{2v}f = 2D_v f##, which shows that scaling ##v## just scales the directional derivative. Remark: The directional derivative D uf P0 is the derivative of f along the line r(t) = hx 0,y 0i + u t. We have that α(t) is the position of a point moving along L that is at p~ at t = 0 and has velocity vector ~v, i. By de nition the function does not change along a level curve. The slope of the tangent line to this curve (within the vertical plane) at the point C IS the directional derivative of the function at A in the direction of u. The directional derivative is the dot product of the gradient of the function and the direction vector. Obviously, as you move along this curve the function value doesn’t change (it’s always just k. Several examples illustrate the capabilites and validity of this The Gradient and Directional Derivative - (12. 3 Higher order partial derivatives If f: Rn!R, then any partial derivative of f is also a function from Rn to R. What is the directional derivative? In R3, D V 1+fV 2 (h) = [Dh] >[V 1 + fV 2] = [Dh]>V 1 + f[Dh]>V 2 What could go wrong? We used a curve to de ne a derivative. Compute the directional derivative of w at the point (2,−6,3) in the direction toward the origin. 14. Tangent vectors are directional derivatives along paths. Since x, y and z can be expressed as functions of the arc length s, measured along the curve S, we can write The Directional Derivative We now turn to the directional derivative. And the directional derivative is similar. Hint: consider the level curve at \$(1,1). derivative along the curve by a simple extension of equations (36) and (38) We have introduced the symbol ∇V for the directional derivative, i. So, the directional derivative of a, and the directional derivative of f in the direction of a at p is equal to the gradient of f at p dotted with a. Page 1 14. Directional derivatives and gradient vectors. The unit vector describes the proportions we want to move in 1. However, f is not continuous at (0, 0) (one can see by approaching the origin along the curve (t, t 3)) and therefore f cannot be Fréchet differentiable at the origin. Math 18. Letting approach along this path is found by setting , and the limit is now found by taking . Now imagine fitting a tangent line to the curve representing the cross . First, understanding direction fields and what they tell us about a differential equation and its solution is important and can be introduced without any knowledge of how to solve a differential equation and so can be done here before we get into solving them. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine The directional derivative of a scalar function. Also, the gradient vector gives the maximum rate of change of a function. With directional derivatives we can now ask how a function is changing if we Directional derivatives tell you how a multivariable function changes as you move along some vector in its input space. Directional derivatives of invariants: the goal here is to evaluate the derivative of a as functions of the arc lengths, measured along the curve S, we can write. Theorem EX 5 Graph gradient vectors and level curves for. - [Voiceover] So here I'd like to talk about what the gradient means in the context of the graph of a function. We made the comparison to standing in a rolling meadow and heading due east: the amount of rise/fall in doing so is comparable to \(f_x\text{. Lagrangian mechanics is based on the calculus of variations, which is the subject of optimization over a space of paths. If z0 = f(x0,y0), then the point P(x0,y0,z0) lies on S. A directional derivative is a derivative along a slice curve in a vertical plane which makes an angle θ with the horizontal. Tech. If the covariant derivative of T in the direction of the curve is zero, then the curve is a geodesic. The Gradient and Applications This unit is based on Sections 9. Some comments of explanation are in order: h1 contains a "handle. Directional derivative and partial derivatives. Sometimes authors write D v instead of . The derivative gives the instantaneous rate of change of with respect to . It is important to understand that defining parallel translation is an extra assumption or geometric structure added to the smooth manifold. Directional Derivatives — §11. 6 , Chapter 9. directional derivative along a curve
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https://docs.displayr.com/wiki/The_Relationship_Between_Cluster_Analysis,_Latent_Class_Analysis_and_Self-Organizing_Maps | # The Relationship Between Cluster Analysis, Latent Class Analysis and Self-Organizing Maps
There are four main types of algorithms in use for cluster-based segmentation:
1. Hierarchical cluster analysis.
2. K-Means Cluster Analysis.
3. Latent class analysis.
4. Self-organizing maps.
## Overall evaluation of the algorithms
Where the goal is to form segments, latent class analysis is almost always preferable to any of the other algorithms. Indeed, the other algorithms should generally be regarded as "plan B" algorithms, only used when latent class analysis cannot be used. This is because latent class analysis has important strengths relative to the other algorithms, whereas the other algorithms have no substantive advantages over latent class analysis. However, as ultimately segmentation is part art and part science, it is often the case that the other algorithms can lead to useful and even superior solutions to those obtained from latent class analysis, so the best approach is to use latent class analysis if in a rush but to consider multiple different segmentation where time permits.
### The strengths of latent class analysis
• It is able to accommodate lots of different types of data. For example, it can be used to create segments using combinations of categorical, numeric and other more exotic types of data, whereas most programs developed for the other algorithms can only accommodate numeric variables.
• It can deal with missing data in a sensible way, allocating people into segments based on their available data, whereas the standard implementations of the other algorithms only work with no missing data (technically, latent class analysis makes a missing at random assumption, whereas other methods make a less plausible assumption that any data is missing completely at random).
• It can accommodate weights. Implementations of the other algorithms generally ignore weights (and, there are no standard weighted versions of any of these other algorithms).
• It is a theoretically superior model. That is, latent class models are built upon many decades of statistical theory. By contrast, the other algorithms are all one-off algorithms which have no strong theoretical support. This is explained in a little more detail below.
• Latent class algorithms can be modified to incorporate lots of varied phenomena (e.g., predictor variables, complex sampling, response biases), all of which are not readily addressed with the alternative algorithms.
### The advantages of the other algorithms
As discussed below, k-means cluster analysis can be viewed as a variant of latent class analysis. Its only advantage over latent class analysis is that it is much faster to compute which means that with huge database k-means can be preferable.
Hierarchical cluster analysis can produce a dendrogram (i.e., a graph which shows the order with which segments are grouped together).
Self-organizing maps create clusters that are ordered on a two dimensional "map" and, where a large number of clusters are created, this can be beneficial from a presentation perspective.
While each of these advantages can be relevant in some circumstances they are, by and large, irrelevant in most segmentation studies which is why latent class is, in general, superior.
## The relationship between the algorithms
Each of latent class analysis, k-means cluster analysis and self-organizing map algorithms have an almost identical structure:[note 1]
Step 1: Initialization. Observations are assigned to a pre-determined number of clusters. Most commonly this is done randomly (either by randomly assigning observations to clusters or by randomly generating parameters). However, it can involve assigning respondents to pre-existing groups (e.g., by age) or using another algorithm to form the initial clusters (e.g., hierarchical cluster analysis to form some initial clusters prior to running k-means cluster analysis).
In the case of self-organizing maps, each cluster is assigned a location on a grid (e.g., if there are 12 clusters, the clusters may be assigned positions in a grid with three rows and four columns).
Step 2: Initial cluster description. A statistical summary is prepared of each cluster. With k-means and self-organizing maps this involves computing the mean value for each variable in each cluster. Latent class analysis also typically involves computation of the means, occasionally measures of variation (e.g., the standard deviation) as well as the sizes of the clusters.
Step 3: Computing the distance between each observation and each cluster. A measure of the distance between each observation and each cluster is computed. With latent class analysis, a probability of cluster membership is computed; this probability takes into account both the distance from of each observation from each cluster and the size of the cluster.
Step 4: Revising the cluster descriptions. Using the result of Step 3 the cluster descriptions are updated. This occurs in slightly different ways for each of the algorithms:
• Latent class analysis: A weighted analysis is undertaken for each cluster, computing the cluster description with the probability of cluster membership as the weight and computing the size of each cluster as the average of the probabilities.
• Cluster analysis: The mean for each cluster on each variable is computed as the average values of the variables for the observations that are most similar to the cluster's current description.[note 2] Note that this is basically the same process as with latent class analysis, except that the "weights" are all 1s and 0s (i.e., where a 1 indicates closest to a cluster).
• Self-organizing maps: The mean for each cluster on each variable is computed as the average values of the variables for the observations that are most similar to the cluster 's current description and observations in clusters that are located nearby on the grid.
Step 5: Iteration to convergence'. Step 3 and 4 are referred to jointly as an iteration. They are repeated in a continuous cycle until the descriptions stabilize (which is referred to as convergence). In the case of cluster analysis, this may take only a few iterations. In the case of latent class analysis and self-organizing maps, it can take hundreds or thousands of observations.
The latent class model is, in general, best from a theoretical perspective as the ways that it differs from the other two algorithms are, from a theoretical perspective, desirable:
1. When allocating observations to clusters it is necessarily the case that there will often be some uncertainty (e.g., if an observation has equally close to two clusters). This is taken into account by only the latent class algorithm .
2. It can be proved using probability theory that if allocating respondents to clusters it is appropriate to take into account the size of the clusters. For example, if a respondent is equally similar to two clusters, but one cluster is ten times the size of the other cluster, the respondent should be more likely to be allocated to the larger cluster (due to Bayes' Theorem). Only the latent class approach complies with this principle.
Note that these two advantages relate to the earlier discussion of the theoretical advantages of latent class analysis. As discussed above, there are other more pragmatic benefits to using latent class analysis.
### Comparing hierarchical cluster analysis with the other algorithms
Hierarchical cluster analysis forms clusters as follows:
Step 1: Each observation is assigned to a cluster (i.e., if there are 100 observations then there are 100 clusters).
Step 2: The distance is computed between all pairs of clusters.
Step 3: The two most similar clusters are merged.
Step 4: Repeat steps 2 and 3 until only a single cluster remains.
Numerous hierarchical cluster analysis algorithms have been developed and they differ in terms of how they conducted step 2 (e.g., do they compute the distance between the two most similar observations in a cluster, the two least similar, a the average, etc.).
A strength of hierarchical cluster analysis is that it always ensures that the most similar observations are in the same clusters. However, this strength also causes its great weakness: often there are situations where a cluster should be split and re-allocated to other clusters, but because no such step exists in hierarchical cluster analysis this never occurs. The net effect of this is that commonly clusters created by hierarchical cluster analysis are less homogeneous than clusters formed by the other algorithms.
The only situation where a hierarchical cluster analysis algorithm will better explain the variation in data is when there has been a problem with the other algorithms, such as with the random selection of initial clusters being poor. Historically this strength was a sufficient reason to consider using hierarchical cluster analysis, either as a standalone algorithm or as a stage before k-means cluster analysis. However, most k-means cluster analysis, latent class and self-organizing map programs can now compute lots of different segmentations, each using different start-points, making hierarchical cluster analysis a generally inferior method, except where there is an interest in the dendrogram (which is a tree showing the history of the merging of the different clusters).
## Notes
Template:Reflist
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https://e-maxx-eng.appspot.com/algebra/sieve-of-eratosthenes.html | # Sieve of Eratosthenes
Sieve of Eratosthenes is an algorithm for finding all the prime numbers in a segment $[1;n]$ using $O(n \log \log n)$ operations.
The idea is simple: at the beginning we write down a row of numbers and eliminate all numbers divisible by 2, except number 2 itself, then divisible by 3, except number 3 itself, next by 7, 11, and all the remaining prime numbers till $n$.
## Implementation
int n;
vector<char> prime (n+1, true);
prime[0] = prime[1] = false;
for (int i=2; i<=n; ++i)
if (prime[i])
if (i * 1ll * i <= n)
for (int j=i*i; j<=n; j+=i)
prime[j] = false;
This code first marks all numbers except zero and one as prime numbers, then it begins the process of sifting composite numbers. For this it goes through all numbers $2$ to $n$ in a cycle. If the current number $i$ is a prime number, it marks all numbers that are multiples of $i$ as composite numbers, starting from $i^2$ as all smaller numbers that are multiples of $i$ necessary have a prime factor which is less than $i$, so all of them were sifted earlier. (Since $i^2$ can easily overflow the type int, the additional verification is done using type long long before the second nested cycle).
Using such implementation the algorithm consumes $O(n)$ of the memory (obviously) and performs $O(n \log \log n)$ (this is proven in the next section).
## Asymptotic analysis
Let's prove that algorithm's running time is $O(n \log \log n)$. It will take $\frac n p$ actions for every prime $p \le n$ the inner cycle performs. Hence, we need to evaluate the next value:
$$\sum_{\substack{p \le n, \\ p\ is\ prime}} \frac n p = n \cdot \sum_{\substack{p \le n, \\ p\ is\ prime}} \frac 1 p.$$
Let's recall two known facts. First fact: the number of prime numbers that are less than or equal to $n$ approximately equals $\frac n {\ln n}$. Second fact: the $k$-th prime number approximately equals $k \ln k$ (that follows from the first fact). Then, we can write down the sum in a such way:
$$\sum_{\substack{p \le n, \\ p\ is\ prime}} \frac 1 p \approx \frac 1 2 + \sum_{k = 2}^{\frac n {\ln n}} \frac 1 {k \ln k}.$$
Here we separated the first prime number from the rest of the numbers in the sum, since $k = 1$ in approximation $k \ln k$ is $0$ and causes division by zero operation.
Now, let's evaluate this sum using the integral of a same function over $k$ from $2$ to $\frac n {\ln n}$ (we can make such approximation because, in fact, the sum is related to the integral as its approximation using rectangle method):
$$\sum_{k = 2}^{\frac n {\ln n}} \frac 1 {k \ln k} \approx \int_2^{\frac n {\ln n}} \frac 1 {k \ln k} dk.$$
The antiderivative for the integrand is $\ln \ln k$. Using a substitution and removing terms of lower order, we'll get the result:
$$\int_2^{\frac n {\ln n}} \frac 1 {k \ln k} dk = \ln \ln \frac n {\ln n} - \ln \ln 2 = \ln(\ln n - \ln \ln n) - \ln \ln 2 \approx \ln \ln n.$$
Now, returning to the original sum, we'll get its approximate evaluation:
$$\sum_{\substack{p \le n, \\ p\ is\ prime}} \frac n p \approx n \ln \ln n + o(n),$$
Q.E.D.
More strict proof (that gives more precise evaluation which is accurate within constant multipliers) you can find in the book authored by Hardy & Wright "An Introduction to the Theory of Numbers" (p. 349).
## Different optimizations of the Sieve of Eratosthenes
The biggest weakness of the algorithm is that it "walks" along the memory, constantly getting out of the cache memory limits. Because of that the constant which is concealed in $O(n \log \log n)$ is comparably big.
Besides, the consumed memory is the bottleneck for big $n$.
The methods presented below allow us to reduce the quantity of the performed operations, as well as to shorten the consumed memory noticeably.
### Sieving by the prime numbers till root
Obviously, to find all the prime numbers until $n$, it will be enough just to perform the sieving only by the prime numbers, which do not exceed the root of $n$.
Thus, the outer cycle of the algorithm will change:
for (int i=2; i*i<=n; ++i)
Such optimization doesn't affect the running time (indeed, by repeating the proof presented above we'll get the evaluation $n \ln \ln \sqrt n + o(n)$, which is asymptotically the same according to the properties of logarithm), though the number of operations will reduce noticeably.
### Sieving by the odd numbers only
Since all even numbers (except $2$) are composite, we can stop checking even numbers at all. Instead, we need to operate with odd numbers only.
First, it will allow us to half the needed memory. Second, it will reduce the number of operations performing by algorithm approximately in half.
### Reducing consumed memory
We should notice that algorithm of Eratosthenes operates with $n$ bits of memory. Hence, we can essentially reduce consumed memory by preserving not $n$ bytes, which are the variables of Boolean type, but $n$ bits, i.e. $\frac n 8$ bytes of memory.
However, such approach, which is called bit-level compression, will complicate the operations with these bits. Read or write operation on any bit will require several arithmetic operations and ultimately slow down the algorithm.
Thus, this approach is justified provided $n$ is so big that we cannot allocate $n$ bytes of the memory anymore. In this case we will trade saving memory ($8$ times less) with significant slowing down of the algorithm.
After all, it's worth mentioning the data structures that automatically do a bit-level compression, such as vector<bool> and bitset<>, have been already implemented in C++ language.
However, if speed is very important, it's better to implement a bit-level compression manually using bit operations. Still, the compilers cannot generate sufficiently fast code for today.
### Block sieving
It follows from the optimization "sieving by the prime numbers till root" that there is no need to keep the whole array $prime[1...n]$ all the time. For performing of sieving it's enough to keep just prime numbers until root of $n$, i.e. $prime[1... \sqrt n]$ and to build the remaining part of array in blocks. In doing so, we need to keep one block only at the present moment in time.
Let $s$ be a constant which determines the size of the block, then we have $\lceil {\frac n s} \rceil$ blocks altogether, and block $k$ ($k = 0 ... \lfloor {\frac n s} \rfloor$) contains numbers in a segment $[ks; ks + s - 1]$. We can work on blocks by turns, i.e. for every block $k$ we will go through all the prime numbers (from $1$ to $\sqrt n$) and perform sieving by them inside of a current block only. It is worth working on the first block accurately because of different reasons: first, all the prime numbers from $[1; \sqrt n]$ shouldn't remove themselves; second, the numbers $0$ and $1$ should be marked as non-prime numbers. While working on the last block it should not be forgotten that the last needed number $n$ is not necessary located in the end of the block.
Here we have the implementation of block sieving. The program reads the number $n$ and finds the number of prime numbers from $1$ to $n$.
const int SQRT_MAXN = 100000; // square root of maximum value of N
const int S = 10000;
bool nprime[SQRT_MAXN], bl[S];
int primes[SQRT_MAXN], cnt;
int main() {
int n;
cin >> n;
int nsqrt = (int) sqrt (n + .0);
for (int i=2; i<=nsqrt; ++i)
if (!nprime[i]) {
primes[cnt++] = i;
if (i * 1ll * i <= nsqrt)
for (int j=i*i; j<=nsqrt; j+=i)
nprime[j] = true;
}
int result = 0;
for (int k=0, maxk=n/S; k<=maxk; ++k) {
memset (bl, 0, sizeof bl);
int start = k * S;
for (int i=0; i<cnt; ++i) {
int start_idx = (start + primes[i] - 1) / primes[i];
int j = max(start_idx,2) * primes[i] - start;
for (; j<S; j+=primes[i])
bl[j] = true;
}
if (k == 0)
bl[0] = bl[1] = true;
for (int i=0; i<S && start+i<=n; ++i)
if (!bl[i])
++result;
}
cout << result;
}
The running time of block sieving is the same as for regular sieve of Eratosthenes (unless the size of the blocks is very small), but the needed memory will shorten to $O(\sqrt n + s)$ and "the random walking" on the memory will be reduced. On the other hand, there will be a division for each pair of a block and prime number from $[1; \sqrt n]$, and that will be far worse for smaller block sizes. Hence, it is necessary to keep balance when selecting constant $s$.
According to the performed experiments, we have the best speed of work when $s$ has a value approximately from $10^4$ to $10^5$.
## Advancement to the linear time complexity
We can convert the Eratosthenes algorithm into another algorithm that will have linear time complexity. Look at the article Sieve of Eratosthenes Having Linear Time Complexity. However, this algorithm has its own weaknesses. | 2018-05-24 15:18:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6499363780021667, "perplexity": 819.2852235360217}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794866511.32/warc/CC-MAIN-20180524151157-20180524171157-00468.warc.gz"} |
https://www.r-bloggers.com/how-to-be-bayesian-and-spare-yourself-a-dreadful-afternoon-with-your-stupid-football-team-losing-the-derby-2/ | # How to be Bayesian and spare yourself a dreadful afternoon with your stupid football team losing the derby
May 8, 2016
By
(This article was first published on R on Gianluca Baio, and kindly contributed to R-bloggers)
Yesterday was the second-last game of the Italian Serie A; I’ve been a Sampdoria supported since I was 12 $$-$$ at that time, they were starting to become one of the best clubs in Serie A (and that was back in the 80’s when Serie A was arguably the best league in the world), although they hadn’t won anything and didn’t have prospects for that season either. But they were a young, good side, playing nicely and so I kind of fell in love with them (and their shirt). Then they did become a very good side, winning the league and a few more trophies $$-$$ so good timing on my part! But also, then they reverted to some relative mediocrity $$-$$ of course, once you’ve decided you support a team, you’re stuck with them no matter what.
Anyway, this season has been rather crappy and yesterday it was a crucial game: we were playing the derby against local rival Genoa entering the game with 40 points and two games left in the campaign. Two teams couldn’t reach us any more (as they were trailing by over 6 points). But at least one between Carpi and Palermo could still overtake us if we lost our two remaining games and they won all of theirs. Also, Udinese was just one point behind us so they too could overtake us, technically. With three teams being relegated, we weren’t statistically safe yet.
So, that’s kind of nervous and earlier last week I thought about this a bit. I had a bad feeling about our game, because we’ve not been great lately (the previous game we were beaten by Palermo) and, clearly, Genoa would try really hard to mess it up for us… But, irrespective of the outcome of the derby, if at least one between Carpi, Palermo and Udinese failed to win their match we would be safe (as there wouldn’t be enough points left for them to catch us). Carpi played at home against Lazio, whose season hasn’t been great either, but they were already safe and with not much else to fight for, except a strong finish; Palermo were away at Fiorentina, who theoretically were still fighting for a Europa league qualification and so should have something to play for; Udinese were away at Atalanta, who much as Lazio were mathematically safe and with not much to play for.
Although one can make a much more complex model, I reasoned that instead of the actual result, what was only important was the chance that either of the three teams behind us would win and so I set up a model with $y_{} (_{})$, $$y_{\rm{Pal}} \sim \mbox{Bernoulli}(\theta_{\rm{Pal}})$$ and $$y_{\rm{Udi}} \sim \mbox{Bernoulli}(\theta_{\rm{Udi}})$$ where the “success” would in fact be the worst possible outcome, ie a win for them.
Then I set up some priors: I reasoned that because they were playing at home, Carpi may have a slightly higher chance of winning the game $$-$$ I figured something about 35%. Also, I thought (hoped) that Lazio wouldn’t be a walkover and so I assumed that 90% of the mass for the chance of Carpi winning their game was around 45%. These can be turned into an informative Beta(15.80107,28.4877) prior $$-$$ it’s fairly easy to work out the parameters of a Beta distribution given the mode (0.35, in this case) and some percentile (0.45 as the 90th percentile, in this case); Christensen et al (page 100) show some theory, while this is some relevant R code.
This is effectively the prior I was assuming:
and I thought it was just about reasonable (the dotted vertical lines indicate a rough estimate of the 95% prior credible interval). Then I did something similar to derive the priors for a Palermo and Udinese win $$-$$ because they were playing away, I figured they would have an average chance of winning of around 20% with 90% of the mass before 40%, which can be turned into a Beta(3.279775,10.1191) prior, looking like this:
Again, I was relatively happy with this and so used these priors in my model, which one could code in R as something like
p.car ~ rbeta(10000,15.80107,28.4877) # P(win) on average .35 and with 95% mass <= .45
p.pal ~ rbeta(10000,3.279775,10.1191) # P(win) on average .2 and with 95% mass <=.4
p.udi ~ rbeta(10000,3.279775,10.1191) # P(win) on average .2 and with 95% mass <=.4
p.safe <- 1-(p.car*p.pal*p.udi)
The most important variable in the model is the probability of Sampdoria being mathematically certain of avoiding relegation, p.safe, which is 1 minus the probability of the worst happening $$-$$ this assumes independence in the three games for Palermo, Carpi and Udinese; in general that’s probably not the best assumption, but in this case they kind of had to win to have a good shot at safety themselves and so I think it’s OK to assume independence. The results were kind of reassuring:
$$-$$ I got an estimated posterior average of 97.8% with a 95% credible interval of 93.8 to 99.7%.
I am not really one to stay at home on a Sunday just to watch the football game (so perhaps I’m not really a footaball fan?) and we’d planned to see some friends, but this reassured me that we shouldn’t be in too much trouble, even if we lost the derby. In the event, Kobi wasn’t great (possibly as a result of venturing an outing at the seaside on Saturday) and so we stayed at home $$-$$ but I decided not to bother with watching the game (again: a) a bold move for a real football fan, confident about his team; b) a cowardly move from a real football fan scared of what the outcome may be; c) not a real football fan).
We did lose the derby very badly, but Carpi, Palermo and Udinese all failed to win their games, which means we are safe. I’m glad I didn’t watch the game…
R-bloggers.com offers daily e-mail updates about R news and tutorials on topics such as: Data science, Big Data, R jobs, visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, git, hadoop, Web Scraping) statistics (regression, PCA, time series, trading) and more... | 2019-10-21 22:43:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.32187169790267944, "perplexity": 1862.5228736863316}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987795253.70/warc/CC-MAIN-20191021221245-20191022004745-00324.warc.gz"} |
https://www.physicsforums.com/threads/simple-quantum-exam-question.59776/ | # Simple quantum exam question
1. Jan 14, 2005
### chrismuktar
Hi,
This question appears in it's original form here:
http://www.chriscentral.com/physics/PC3101.PDF (question 2)
I have had a go and put some notes into the text below... I am stuck on parts c) (ii) and (iii).
Any help much appreciated! Thanks.
_______________________________________________
A quantum mechanical system is described by a two-dimensional state space spanned by the orthonormal vectors |1> and |2>. The Hamiltonian of the system can be defined by:
H|1> = a|1> + 2a|2>
H|2> = 2a|1> + a|2>
with "a" a positive constant.
(a) Write down the matrix representation of H in the basis {|1> |2>}.
Easy:
(a 2a)
(2a a )
(b) Find the energy eigenvalues of the systems and associated orthonormal eigenvectors.
I get {3, 1/root2*(1,-1)} and {-1,1/root2*(1,1)}, where this has been expressed in the form {eigenvalue,eigenvector}.
(c) If at time t = 0 the system is described by the state vector |2>:
(i) Find the mean value of the energy <H> at time t = 0.
I get <H> = a.
(ii) What is the probability that the system will be found having its highest possible energy at time t = 0?
(iii) If the highest energy is measured at time t = 0, calculate the state vector of the system at time t.
_________________________________
Any ideas? Thanks! Chris.
Last edited by a moderator: Apr 21, 2017
2. Jan 14, 2005
### vanesch
Staff Emeritus
Some hints:
For (ii), what is the state that corresponds to that highest energy ? If you have a given state |psi> and you have an observable A with an eigenstate |a> and eigenvalue a, remember that the probability to measure the eigenvalue a is given by |<a|psi>|^2 (Born's rule).
For (iii), we know that after we measured an eigenvalue a (no matter what was the probability to find that ; now we HAVE measured it), then the state immediately after that measurement is |a>, the corresponding eigenvector.
In this case, we have the hamiltonian ; what's special about the time evolution of the eigenstates of a hamiltonian ?
hope this helps...
cheers,
Patrick.
3. Jan 14, 2005
### chrismuktar
For ii) I have been attempting use Born's rule however I still find myself at a loss. The key things I ask myself here are:
The system is in state |2>, which is NOT an eigenstate of the hamiltonian, therefore there is uncertain energy (which we expect), so it is fair to say maybe:
$$H|1> = a_{1}|E_{1}> + a_{2}|E_{2}>$$
$$H|2> = a_{1}|E_{1}> - a_{2}|E_{2}>$$
where I think $$a_{n}$$ is the nth eigenvalue of the hamiltonian matrix.
The plus's and minus's come from the eigenvectors of the hamiltonian.
Still, I'm lost becaue I'm not sure what to do with it (or even if it's correct).
For iii) right after measurement the system is in a state $$|2>$$ which is our boundary condition. I imagine that our system's state vector is going to be something like:
$$|\psi>=|1>exp(iE_{1}t/\hbar) +|2>exp(iE_{2}t/\hbar)$$
but I'm not sure yet.
Thoughts?
Thanks for the help btw!
4. Jan 14, 2005
### Norman
For part (c)
You will need to write the vector |2> in terms of the eigenvectors you found in part (b), then you know everything! You know what H will give when it acts on that state. And you know the probabilities of measuring a given energy (in this case it will be the number in front of your eigenvectors squared).
By the way how did you get part i) without using the eigenvectors?
5. Jan 14, 2005
### chrismuktar
Hey Norman,
Surely the question gives me |2> as eigenvectors, right? I'm still a little miffed, I think someone is just going to have to post the answer and explain it. My lecturer didn't make me understand it fully either :-(
For part i):
$$<H> = <2|H|2> = <2|(2a|1>+a|2>)> = a$$
6. Jan 15, 2005
### vanesch
Staff Emeritus
No ! Not that way, you do not have to apply H in the left hand side.
You just write your actual state, |2> as a combination of the eigenvectors of H:
$$|2> = a_1 |E_1> + a_2 |E_2>$$
and then $$|a_2|^2$$ is simply the probability to find, after an energy mesurement, the energy state 2.
Note also that $$a_2 = <E_2 | 2>$$ so that's very easy to do !
No, it is not in state $$|2>$$ but it is in state $$|E_2>$$, because you measured the energy to be $$E_2$$. So because of that, the system is now in the corresponding eigenvector.
NO! In order to write the above, you need not to work with the vectors |1> and |2> but with the ENERGY EIGENSTATES.
And you ARE (after the measurement) in one single energy eigenstate....
cheers,
Patrick.
7. Jan 15, 2005
### chrismuktar
I'm going to go through the question now from the beginning and we'll see if it's all correct...
a) $$H = a\left(\begin{array}{cc}1&2\\2&1\end{array}\right)$$
b) $$\lambda_{1} = -1a, e_{1}=\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\-1\end{array}\right)$$
$$\lambda_{2} = 3a, e_{2}=\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\1\end{array}\right)$$
c)
i) $$<H> = a$$
ii)
$$|1> = \left(\begin{array}{c}\lambda_{1}\\\lambda_{2}\end{array}\right).e_{1} = \frac{1}{\sqrt{2}}(|-1a>-|3a>)$$
$$|2> = \left(\begin{array}{c}\lambda_{1}\\\lambda_{2}\end{array}\right).e_{2} = \frac{1}{\sqrt{2}}(|-1a>+|3a>)$$
We want the state
$$|< 3a | 2 >|^2 = \left|\frac{1}{\sqrt{2}} <3a | (|-1a>+|3a>)\right|^2 = \frac{1}{2}$$
iii)
The system is in eigenstate $$3a$$ at $$t=0$$. We want the time evolution of this state alone, which is
$$|3a(t)> = |3a>\exp^{\frac{i3at}{\hbar}}$$
Some things I want to check:
I were to swap around $$e_{1}$$ for $$e_{2}$$ and viceversa, in this case it produces the same answer, BUT, how should I be associating these eigenvectors states; does it matter which way around they are in general?
Thanks to all again,
Chris.
Last edited by a moderator: Apr 21, 2017 | 2018-02-21 16:02:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8328816294670105, "perplexity": 692.2079864563094}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891813626.7/warc/CC-MAIN-20180221143216-20180221163216-00692.warc.gz"} |
http://www.lastfm.es/user/EvgenyaPblcb/library/music/Cypress+Hill/_/Stoned+Is+The+Way+Of+The+Walk?setlang=es | # Colección
Música » Cypress Hill »
## Stoned Is the Way of the Walk
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Stoned Is the Way of the Walk 2:46 18 Sep 2013, 6:47
Stoned Is the Way of the Walk 2:46 17 Sep 2013, 10:56
Stoned Is the Way of the Walk 2:46 24 May 2013, 11:19
Stoned Is the Way of the Walk 2:46 14 Dic 2011, 6:36
Stoned Is the Way of the Walk 2:46 13 Dic 2011, 12:04
Stoned Is the Way of the Walk 2:46 13 Dic 2011, 10:39
Stoned Is the Way of the Walk 2:46 13 Dic 2011, 9:08
Stoned Is the Way of the Walk 2:46 13 Dic 2011, 7:07
Stoned Is the Way of the Walk 2:46 8 Sep 2011, 18:53
Stoned Is the Way of the Walk 2:46 20 Jun 2011, 5:56
Stoned Is the Way of the Walk 2:46 19 Jun 2011, 13:09
Stoned Is the Way of the Walk 2:46 17 Jun 2011, 20:38
Stoned Is the Way of the Walk 2:46 17 Jun 2011, 19:51
Stoned Is the Way of the Walk 2:46 17 Jun 2011, 19:05
Stoned Is the Way of the Walk 2:46 17 Jun 2011, 13:06
Stoned Is the Way of the Walk 2:46 17 Jun 2011, 11:27
Stoned Is the Way of the Walk 2:46 17 Jun 2011, 9:49
Stoned Is the Way of the Walk 2:46 17 Jun 2011, 8:10
Stoned Is the Way of the Walk 2:46 17 Jun 2011, 6:33
Stoned Is the Way of the Walk 2:46 16 Jun 2011, 21:03
Stoned Is the Way of the Walk 2:46 16 Jun 2011, 20:16
Stoned Is the Way of the Walk 2:46 16 Jun 2011, 19:30
Stoned Is the Way of the Walk 2:46 16 Jun 2011, 18:22 | 2015-02-27 08:00:17 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8716672658920288, "perplexity": 4413.43918937032}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936460577.67/warc/CC-MAIN-20150226074100-00146-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://techwhiff.com/learn/a-certain-project-has-an-average-accounting/393097 | # A certain project has an Average Accounting Return of .75 and incomes of 10,000, 50,000 and...
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A certain project has an Average Accounting Return of .75 and incomes of 10,000, 50,000 and 30,000. What is the amount of the initial investment?
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• 251.
Massachusetts General Hospital, Boston, MA.
Massachusetts General Hospital, Boston, MA. Umeå University, Faculty of Science and Technology, Department of Computing Science. Massachusetts General Hospital, Boston, MA.
Development of a Knee Phantom for the Evaluation of Methods for Measuring Knee Joint Kinematics2004In: Transactions of the Orthopaedics Research Society, ORS , 2004, Vol. 29, p. 1385-1385Conference paper (Other academic)
• 252.
Massachusetts General Hospital, Boston, MA.
Massachusetts General Hospital, Boston, MA. Massachusetts General Hospital, Boston, MA. Massachusetts General Hospital, Boston, MA. Sahlgrenska University Hospital. Umeå University, Faculty of Science and Technology, Department of Computing Science. Massachusetts General Hospital, Boston, MA. Massachusetts General Hospital, Boston, MA.
Experimental Assessment of Precision and Accuracy of Radiostereometric Analysis (RSA) for the Determination of Polyethylene Wear in a Total Hip Replacement Model2001In: Proceedings of the 11th annual meeting of AAKHS, AAKHS , 2001, p. 49-49Conference paper (Other academic)
• 253.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Pipelined Messaging Gateway2015Independent thesis Advanced level (degree of Master (One Year)), 20 credits / 30 HE creditsStudent thesis
The SMS gateway EMG uses separate threads for each connection. This requires a large number of locks to avoid data corruption, and causes problems when scaling to thousands of connections. The CPU load goes very high, and the amount of addressable memory per thread becomes low.
This report examines an alternative implementation, where each thread instead handles a separate part of the processing. The two architectures are compared based on throughput, resource utilisation, code complexity and more.
The results show what while throughput is about the same, the alternative implementation is better at keeping the CPU load within reasonable limits. However, it also requires more advanced data structure and algorithms to be efficient.
• 254.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Continuous Collision Detection for Wires with Adaptive Resolution2017Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
For interactive simulations using a physics engine, a fixed time step is often necessary in order to maintain real-time performance. Furthermore, collisions between the simulated geometric objects have to be detected by solving a computational problem called collision detection. In its discrete formulation, the geometric configurations of the bodies are evaluated at each simulation time step, whereas its continuous variant also considers the bodies’ motion in betweenthe time steps. A fixed simulation time step can lead to missed collisions if only discrete collision detection is performed. This problem arises especially when simulating thin objects such as wires, chains, or ropes for applications like heavy lifting or anchor handling. In order to be able to simulate wires interacting with each other in real-time simulations, continuous collision detection is therefore necessary.An existing simulation model for wires, chains, and ropes using adaptive wire resolution has been augmented using continuous collision detection. This addition has been integrated into the physics engine AGX Dynamics. Issues in existing methods for continuous collision detection of moving line segments caused by co-linearity and co-planarity have been identified, classified and addressed. Using this augmented approach to continuous collision detection allows for alarger fixed simulation step size compared to discrete collision detection, and thus decreases the total run time by up to 58.22% in relevant scenarios.
• 255.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
A design study on applications for mobile devices2012Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
The question of how we should develop mobile software is - with the proliferation of smartphones and HTML5 - an interesting and hot topic. This thesis introduces and compares the different approaches to mobile software development. It focuses on the basic web, native and hybrid approaches but also touches on cross platform tools. The aim of the thesis is to provide a solid understanding of the alternatives and help guide a potential choice between the techniques.
As a guide to making the decision this work is far from complete, it's a fast moving area with a lot of uncertainty and accounting for all (or even most) of the variables involved in the decision is next to impossible. The primary observation is that the native approach continues to dominate when it comes to user experience and functionality, but the other approaches have great potential and are improving at a fast pace. Costwise, native is often the inferior alternative, especially when targeting many platforms. As many existing resources, such as a website and web developers, can be used to make web or hybrid applications, the cost difference increases even more.
• 256. Breitgand, D.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Policy-Driven Service Placement Optimization in Federated Clouds2011Report (Other academic)
Efficient provisioning of elastic services constitutes a significant management challenge for cloud computing providers. We consider a federated cloud paradigm, where one cloud can subcontract workloads to partnering clouds to meet peaks in demand without costly over-provisioning. We propose a model for service placement in federated clouds to maximize profit while protecting Quality of Service (QoS) as specified in the Service Level Agreements (SLA) of the workloads. Our contributions include an Integer Linear Program (ILP) formulation of the generalized federated placement problem and application of this problem to load balancing and consolidation within a cloud, as well as for cost minimization for remote placement in partnering clouds. We also provide a 2-approximation algorithm based on a greedy rounding of a Linear Program (LP) relaxation of the problem. We implement our proposed approach in the context of the RESERVOIR architecture.
• 257.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Real-time DATMO Based on Deep 3D Point Cloud Features3D object detection in point cloud data2019Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
Perception for autonomous drive systems is the most essential task for safe and reliable driving. LiDAR sensors can be used for this and are vying for being crowned as an essentia lelement in this task.In this thesis, we present a novel real-time solution for detection and tracking of moving objects which utilizes deep learning based 3D object detection. On one hand, we present YOLO++, a 3D object detection network on point clouds only. A network that expands YOLOv3, the latest contribution to standard real-time object detector for three channel images. YOLO++ extracts the standard YOLO’s predictions plus an angle and a height from projected point clouds. Our unified architecture is fast. It processes images in 20 frames per second. Our experiments on the KITTI benchmark suite show that we achieve state-of-the-art efficiency but with a mediocre accuracy for car detection comparable to the result of Tiny-YOLOv3 on the COCO dataset.On the other hand, we present a multi-threaded object tracking solution that makes use o fthe detected objects by YOLO++. Each observation is associated to a thread with a novel concurrent data association process where each of the threads contain an Extended Kalman Filter that is used for predicting and estimating the object’s state over time. Futhermore, a LiDAR odometry algorithm is used to obtain absolute information about the movement, since the movement of objects are inherently relative to the sensor perceiving them. We obtain 33 state updates per second with an equal amount of threads to the number of cores inour main workstation.Even if the joint solution has not been tested on a system with enough computational power it is ready for deployment. We expect it to be runtime constrained by the slowest subsystem which happens to be the object detection system. This satisfies our real-time constraint of 10 frames per second of our final system by a large margin. Finally, we show that our system can take advantage of the predicted semantic information from the Kalman Filters in order to enhance the inference process in our YOLO++ architecture.
• 258.
Swedish Defence Research Agency (FOI).
Umeå University, Faculty of Science and Technology, Department of Computing Science. Swedish Defence Research Agency (FOI). Swedish Defence Research Agency (FOI). Swedish Defence Research Agency (FOI).
Abstraction techniques for social networks2010In: Proceedings of the 2010 International Conference on Advances in Social Networks Analysis and Mining, IEEE, 2010Conference paper (Refereed)
• 259.
University of Bath.
Umeå University, Faculty of Science and Technology, Department of Computing Science. University of Bath, Bath, UK.
How society can maintain human-centric artificial intelligence2019In: Human-centered digitalization and services / [ed] Marja Toivonen and Eveliina Saari, Springer, 2019, p. 305-323Chapter in book (Refereed)
Although not a goal universally held, maintaining human-centric artificial intelligence is necessary for society's long-term stability. Fortunately, the legal and technological problems of maintaining control are actually fairly well understood and amenable to engineering. The real problem is establishing the social and political will for assigning and maintaining accountability for artifacts when these artifacts are generated or used. In this chapter we review the necessity and tractability of maintaining human control and the mechanisms by which such control can be achieved. What makes the problem both most interesting and most threatening is that achieving consensus around any human-centered approach requires at least some measure of agreement on broad existential concerns.
The full text will be freely available from 2021-06-04 11:17
• 260.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Computing network centrality measures on fMRI data using fully weighted adjacency matrices2016Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
A lot of interesting research is currently being done in the field of neuroscience, a recent subject being the effort to analyse the the human brain connectome and its functional connectivity. One way this is done is by applying graph-theory based network analysis, such as centrality, on data from fMRI measurements. This involves creating a graph representation from a correlation matrix containing the correlations over time between all measured voxels. Since the input data can be very big, this results in computations that are too memory and time consuming for an ordinary computer. Researchers have used different techniques to work around this problem, examples include thresholding correlations when creating the adjacency matrix and using a smaller input data with lower resolution.This thesis proposes three ways to compute two different centrality measures, degree centrality and eigenvector centrality, on fully weighted adjacency matrices that are built from complete correlation matrices computed from high resolution input data. The first is reducing the problem by doing the calculations in optimal order and avoiding the construction of the large correlation matrix. The second solution is to distribute and do the computations in parallel on a large computer cluster using MPI. The third solution is to calculate as large sets as possible on an ordinary laptop using shared-memory parallelism with OpenMP. Algorithms are presented for the different solutions, and the effectiveness of the implementations of them is tested.
• 261.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Automated software testing for cross-platform systems2012Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
SILK is the preferred audio codec to use in a call between Skype clients. Everytime the source code has been changed there is a risk the code is no longer bit-exact between all the dffierent platforms. The main task for this thesis is to make it possible to test bit-exactness between platforms automatically to save resources for the company.
During this thesis a literature study about software testing has been carried out to find a good way of testing bit-exactness between different platforms. The advantages and disadvantages with the different testing techniques was examined during this study.
The result of the thesis is a framework for testing bit-exactness between several different platforms. Based on the conclusions from the literature study the framework is using a technique called data-driven testing to carry out the bit-exactness tests on SILK.
• 262.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
JavaScript and Web Integration of AgX Multiphysics Engine2011Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
JavaScript is the de facto language for client-side scripting on the web and has in recent years received a great boost in performance thanks to the `browser wars' where the different browser vendors compete to get higher market shares. AgX is a physics engine developed by Algoryx in Umeå, intended for use in simulations. This thesis explores possible methods of integrating JavaScript with AgX and in extension the possibility of integrating AgX into the web browser as a plugin, effectively enabling AgX for use in different web applications. Efforts were made to combine the embedded JavaScript API with the AgX browser plugin. This was found to be unfeasible for different reasons, although the AgX plugin as a separate piece of software was shown to work well.
• 263.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Supporting Communication and Collaboration in the Process Automation Industry2011Independent thesis Advanced level (degree of Master (One Year)), 20 credits / 30 HE creditsStudent thesis
This thesis shows new domains for social media applications. More specifically, it explores how communication and collaboration can be supported in the process automation industry.´A concept demonstrator was implemented using the Sencha Touch framework. The prototype is based on several identified use cases, and has been tested and evaluated with end users.The design and functionality is inspired from social media applications such as Facebook and Stack Overow. These kinds of popular social media platforms have developed an intuitive way of structuring and grouping information. This report shows that these information structures are indeed applicable in non traditional domains, such as the process automation industry.The concept answers to identified problem scenarios, e.g., communicating information between shifts and support of handling alarms. It also approaches personalization in order to support users focus and interest.
• 264.
Department of Geography, University of Zurich, Switzerland.
Umeå University, Faculty of Science and Technology, Department of Computing Science. Department of Geography, University of Zurich, Switzerland.
Distributing Attention Between Environment and Navigation System to Increase Spatial Knowledge Acquisition During Assisted Wayfinding2018In: Proceedings of Workshops and Posters at the 13th International Conference on Spatial Information Theory (COSIT 2017) / [ed] Fogliaroni P., Ballatore A., Clementini E., Springer, 2018, p. 19-22Conference paper (Refereed)
Travelers happily follow the route instructions of their devices when navigating in an unknown environment. Navigation systems focus on route instructions to allow the user to efficiently reach a destination, but their increased use also has negative consequences. We argue that the limitation for spatial knowledge acquisition is grounded in the system’s design, primarily aimed at increasing navigation efficiency. Therefore, we empirically investigate how navigation systems could guide users’ attention to support spatial knowledge acquisition during efficient route following tasks.
• 265.
University of Zurich, Switzerland.
Umeå University, Faculty of Science and Technology, Department of Computing Science. University of Zurich, Switzerland.
How does navigation system behavior influence human behavior?2019In: Cognitive Research: Principles and Implications, E-ISSN 2365-7464, Vol. 4, no 5Article in journal (Refereed)
• 266.
Department of Mathematics, Faculty of Science, University of Zagreb, Zagreb, Croatia.
Umeå University, Faculty of Science and Technology, Department of Computing Science. Institute of Mathematics, EPFL, Lausanne, Switzerland.
A Householder-Based Algorithm for Hessenberg-Triangular Reduction2018In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 39, no 3, p. 1270-1294Article in journal (Refereed)
The QZ algorithm for computing eigenvalues and eigenvectors of a matrix pencil $A - \lambda B$ requires that the matrices first be reduced to Hessenberg-triangular (HT) form. The current method of choice for HT reduction relies entirely on Givens rotations regrouped and accumulated into small dense matrices which are subsequently applied using matrix multiplication routines. A nonvanishing fraction of the total flop-count must nevertheless still be performed as sequences of overlapping Givens rotations alternately applied from the left and from the right. The many data dependencies associated with this computational pattern leads to inefficient use of the processor and poor scalability. In this paper, we therefore introduce a fundamentally different approach that relies entirely on (large) Householder reflectors partially accumulated into block reflectors, by using (compact) WY representations. Even though the new algorithm requires more floating point operations than the state-of-the-art algorithm, extensive experiments on both real and synthetic data indicate that it is still competitive, even in a sequential setting. The new algorithm is conjectured to have better parallel scalability, an idea which is partially supported by early small-scale experiments using multithreaded BLAS. The design and evaluation of a parallel formulation is future work.
• 267.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
A Multimodal Approach to Autonomous Document Categorization Using Convolutional Neural Networks2018Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
When international students apply for the Swedish educational system, they send documents to verify their merits. These documents are categorized and evaluated by administrators. This thesis approach the problem of document classification with a multimodal convolutional network. By looking at both image and text features together, it is examined if the classification is better than any of the sources alone. The best result for single source classification was when the input was text at 85.2% accuracy, this was topped by the multimodal approach with a accuracy of 88.4%.This thesis concludes that there is a gain in accuracy when using a multimodal approach.
• 268.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
How to provide a user friendly search interface based upon a libraries Open Public Access Catalogue2011Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
Today many libraries offer their services via Internet and reach billions of visitors in different ages. The problem is that most of these library web sites are not customized for the users and they don’t deliver a good user experience. In this thesis we have studied both the interface and the users of an Open Public Access Catalog called CS Library. We found out how the users use the current system and how they want to use it. To achieve this we gathered data with different user studies at the Ume°a city library. A new interface was developed from the outcome of our user study. We created a set of guidelines that were followed when a new user customized design was developed for CS Library. The new design offers a better user experience and is customized by the users needs, desires and thoughts. The new design has been implemented as an interactive prototype with HTML, CSS, Javascript and Actionscript to convey a better feeling of the interface.
• 269. Byers, R.
Umeå University, Faculty of Science and Technology, Departement of Computing Science.
Structured condition numbers for invariant subspaces2006In: Siam Journal on Matrix Analysis and Applications, Vol. 28, no 2, p. 326-347Article in journal (Refereed)
Invariant subspaces of structured matrices are sometimes better conditioned with respect to structured perturbations than with respect to general perturbations. Sometimes they are not. This paper proposes an appropriate condition number c(S), for invariant subspaces subject to structured perturbations. Several examples compare c(S) with the unstructured condition number. The examples include block cyclic, Hamiltonian, and orthogonal matrices. This approach extends naturally to structured generalized eigenvalue problems such as palindromic matrix pencils.
• 270.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Hybrid Clouds: Implementation and obstacles2016Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
Hybrid cloud is the approach companies want to adopt for its future in the cloud since hybrid cloud allows you to boost the capacity or the capability of a cloud service by aggregation, integration or customization with another cloud service. Those services can be both private or public. Implementing a hybrid cloud is a big process and companies have difficulties finding a good standard for it. In this thesis, the key points and obstacles in the implementation of the hybrid cloud are pinpointed. One obstacle, workflows are studied closer. Workflows are the result of cloud orchestration, arrangement and coordination of automated tasks. The thesis covers the implementation process of workflows. The result of the thesis is key findings and motivation for the implementation of hybrid cloud and how workflows should be implemented
• 271.
Dublin City University, Ireland.
Dublin City University, Ireland. Centre for Research and Technology Hellas, Greece. Centre for Research and Technology Hellas, Greece. Dublin City University, Ireland. Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N). Dublin City University, Ireland. Dublin City University, Ireland.
A review of cloud computing simulation platforms and related environments2017In: Proceedings of the 7th International Conference on Cloud Computing and Services Science: Volume 1: CLOSER, 2017, Vol. 1, p. 679-691Conference paper (Refereed)
Recent years have seen an increasing trend towards the development of Discrete Event Simulation (DES) platforms to support cloud computing related decision making and research. The complexity of cloud environments is increasing with scale and heterogeneity posing a challenge for the efficient management of cloud applications and data centre resources. The increasing ubiquity of social media, mobile and cloud computing combined with the Internet of Things and emerging paradigms such as Edge and Fog Computing is exacerbating this complexity. Given the scale, complexity and commercial sensitivity of hyperscale computing environments, the opportunity for experimentation is limited and requires substantial investment of resources both in terms of time and effort. DES provides a low risk technique for providing decision support for complex hyperscale computing scenarios. In recent years, there has been a significant increase in the development and extension of tools to support DES for cloud computing resulting in a wide range of tools which vary in terms of their utility and features. Through a review and analysis of available literature, this paper provides an overview and multi-level feature analysis of 33 DES tools for cloud computing environments. This review updates and extends existing reviews to include not only autonomous simulation platforms, but also on plugins and extensions for specific cloud computing use cases. This review identifies the emergence of CloudSim as a de facto base platform for simulation research and shows a lack of tool support for distributed execution (parallel execution on distributed memory systems).
• 272.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
EVALUATION OF MACHINE LEARNING ALGORITHMS FOR SMS SPAM FILTERING2019Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
The purpose of this thesis is to evaluate different machine learning algorithms and methods for text representation in order to determine what is best suited to use to distinguish between spam SMS and legitimate SMS. A data set that contains 5573 real SMS has been used to train the algorithms K-Nearest Neighbor, Support Vector Machine, Naive Bayes and Logistic Regression. The different methods that have been used to represent text are Bag of Words, Bigram and Word2Vec. In particular, it has been investigated if semantic text representations can improve the performance of classification. A total of 12 combinations have been evaluated with help of the metrics accuracy and F1-score.The results shows that Logistic Regression together with Bag of Words reach the highest accuracy and F1-score. Bigram as text representation seems to work worse then the others methods. Word2Vec can increase the performnce for K-Nearst Neigbor but not for the other algorithms.
• 273.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Collision Detection of TriangleMeshes using GPU2011Independent thesis Advanced level (degree of Master (One Year)), 20 credits / 30 HE creditsStudent thesis
Collision detection in physics engines often use primitives such as spheres and boxes since collisions between these objects are straightforward to compute. More complicated objects can then be modeled using compounds of these simpler primitives. However, in the pursuit of making it easier to construct and simulate complicated objects, triangle meshes are a good alternative since it is usually the format used by modeling tools. This thesis demonstrates how triangle meshes can be used directly as collision objects within a physics engine. The collision detection is done using triangle mesh models with tests accelerated using a tree-based bounding volume hierarchy structure. OpenCL is a new open industry framework for writing programs on heterogeneous platforms, including highly parallel platforms such as Graphics Processing Units(GPUs). Through the use of OpenCL, parallelization of triangle mesh collision detection is implemented for the GPU, then evaluated and compared to the CPU implementation
• 274.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Navigating to real life objects in indoor environments using an Augmented Reality headset2017Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
Augmented Reality (AR) headmounts are an rising technology with great chances to be a common gadget used by a lot of people in the near future. With the rise of this technology, new possibilities opens up on how to combine the interaction of the real world together with the virtual world. This thesis meets some of these upcoming interaction questions in terms of indoor navigation. The thesis introduces an approach for designing and implementing an AR-based system that is able to let users navigate around an indoor environment to find various real life objects while using an Augmented Reality headmount. The thesis also discusses how to personalize the navigation to different users in different environments. A proof-of concept was implemented and evaluated with several users inside different indoor environments, e.g., a real food store, where the results showed that users were more effective finding objects while using the AR-based system, compared to when not using the AR-basedsystem.
• 275.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Adaptive least squares matching as a non-linear least squares optimization problem2002In: Proceedings SSAB 2002: symposium on Image Analysis, 2002Conference paper (Other academic)
Adaptive Least Squares Matching (ALSM) is a powerful technique for precisely locating objects in digital images. The method was introduced to the photogrammetric community by Gruen in 1985 and has since been developed further. The purpose of this paper is to study the basic ALSM formulation from a least squares optimization point of view. It turns out that it is possible to describe the basic algorithm as a variation of the Gauss-Newton method for solving weighted non-linear least squares optimization problems. This opens the possibility of applying optimization theory on the ALSM problem. The line-search algorithm for obtaining global convergence is especially described and illustrated
• 276.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Comparison of resection: intersection algorithms and projection geometries in radiostereometry2002In: ISPRS journal of photogrammetry and remote sensing (Print), ISSN 0924-2716, E-ISSN 1872-8235, Vol. 56, no 5-6, p. 390-400Article in journal (Refereed)
Three resection-intersection algorithms were applied to simulated projections and clinical data from radiostereometric patients. On simulated data, the more advanced bundle-adjustment-based algorithms outperformed the classical Selvik algorithm, even if the error reductions were small for some parameters. On clinical data, the results were inconclusive. The two different projection geometries had a much larger influence on the error size and distribution. For the biplanar configuration, the position and motion errors were small and almost isotropic. For the uniplanar configuration, the position errors were comparably high and anisotropic, but still resulted in a high accuracy for some motion parameters at the expense of others. The simplified resection-intersection algorithm by Selvik may still be considered a good and robust algorithm for radiostereometry. More studies will have to be performed to find out how the theoretical advantages of the bundle methods can be utilized in clinical radiostereometry. (C) 2002 Elsevier Science B.V. All rights reserved.
• 277.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Improving the robustness of least squares template matching with a line-search algorithm2002In: Close-range imaging, long-range vision: proceedings of the Commission V symposium, ISPRS , 2002, Vol. 34, no 5, p. 7-11Conference paper (Other academic)
The Adaptive Least Squares Matching (ALSM) problem of Gruen is conventionally described as a statistical estimation problem. This paper shows that the ALSM problem may also be interpreted as a weighted non-linear least squares problem. This enables optimization theory to be applied to the ALSM problem. The ALSM algorithm may be interpreted as an instance of the well-known Gauss-Newton algorithm. A problem-independent termination criteria is introduces based on angles in high-dimensional vector spaces. The line-search modification of the Gauss-Newton method is explained and applied to the ALSM problem. The implications of the line-search modification is an increased robustness, reduced oscillations, and increased pull-in range. A potential drawback is the increased number of convergences toward side minima in images with repeating patterns.
• 278.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Metod och anordning för identifikation och orientering av stereoröntgenbilder2002Patent (Other (popular science, discussion, etc.))
• 279.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
UmRSA Digital Measure — A Measurement Program for Digital Radiostereometry1998In: Proceedings of the SSAB Symposium on Image Analysis, SSAB , 1998, p. 69-71Conference paper (Other academic)
• 280.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
INSA Strasbourg, France.
Bundle adjustment with and without damping2013In: Photogrammetric Record, ISSN 0031-868X, E-ISSN 1477-9730, Vol. 28, no 144, p. 396-415Article in journal (Refereed)
The least squares adjustment (LSA) method is studied as an optimisation problem and shown to be equivalent to the undamped Gauss-Newton (GN) optimisation method. Three problem-independent damping modifications of the GN method are presented: the line-search method of Armijo (GNA); the Levenberg-Marquardt algorithm (LM); and Levenberg-Marquardt-Powell (LMP). Furthermore, an additional problem-specific "veto" damping technique, based on the chirality condition, is suggested. In a perturbation study on a terrestrial bundle adjustment problem the GNA and LMP methods with veto damping can increase the size of the pull-in region compared to the undamped method; the LM method showed less improvement. The results suggest that damped methods can, in many cases, provide a solution where undamped methods fail and should be available in any LSA software package. Matlab code for the algorithms discussed is available from the authors.
• 281.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
INSA Strasbourg, France.
Camera Calibration using the Damped Bundle Adjustment Toolbox2014In: ISPRS Annals - Volume II-5, 2014: ISPRS Technical Commission V Symposium 23–25 June 2014, Riva del Garda, Italy / [ed] F. Remondino and F. Menna, Copernicus GmbH , 2014, Vol. II-5, p. 89-96Conference paper (Refereed)
Camera calibration is one of the fundamental photogrammetric tasks. The standard procedure is to apply an iterative adjustment to measurements of known control points. The iterative adjustment needs initial values of internal and external parameters. In this paper we investigate a procedure where only one parameter - the focal length is given a specific initial value. The procedure is validated using the freely available Damped Bundle Adjustment Toolbox on five calibration data sets using varying narrow- and wide-angle lenses. The results show that the Gauss-Newton-Armijo and Levenberg-Marquardt-Powell bundle adjustment methods implemented in the toolbox converge even if the initial values of the focal length are between 1/2 and 32 times the true focal length, even if the parameters are highly correlated. Standard statistical analysis methods in the toolbox enable manual selection of the lens distortion parameters to estimate, something not available in other camera calibration toolboxes. A standardised camera calibration procedure that does not require any information about the camera sensor or focal length is suggested based on the convergence results. The toolbox source and data sets used in this paper are available from the authors.
• 282.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
INSA Strasbourg, France.
Experiments with Metadata-derived Initial Values and Linesearch Bundle Adjustment in Architectural Photogrammetry2013Conference paper (Refereed)
According to the Waldhäusl and Ogleby (1994) "3x3 rules", a well-designed close-range architetural photogrammetric project should include a sketch of the project site with the approximate position and viewing direction of each image. This orientation metadata is important to determine which part of the object each image covers. In principle, the metadata could be used as initial values for the camera external orientation (EO) parameters. However, this has rarely been used, partly due to convergence problem for the bundle adjustment procedure.
In this paper we present a photogrammetric reconstruction pipeline based on classical methods and investigate if and how the linesearch bundle algorithms of Börlin et al. (2004) and/or metadata can be used to aid the reconstruction process in architectural photogrammetry when the classical methods fail. The primary initial values for the bundle are calculated by the five-point algorithm by Nistér (Stewénius et al., 2006). Should the bundle fail, initial values derived from metadata are calculated and used for a second bundle attempt.
The pipeline was evaluated on an image set of the INSA building in Strasbourg. The data set includes mixed convex and non-convex subnetworks and a combination of manual and automatic measurements.
The results show that, in general, the classical bundle algorithm with five-point initial values worked well. However, in cases where it did fail, linesearch bundle and/or metadata initial values did help. The presented approach is interesting for solving EO problems when the automatic orientation processes fail as well as to simplify keeping a link between the metadata containing the plan of how the project should have become and the actual reconstructed network as it turned out to be.
• 283.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
INSA Strasbourg, France.
External Verification of the Bundle Adjustment in Photogrammetric Software Using the Damped Bundle Adjustment Toolbox2016In: XXIII ISPRS Congress, Commission V: Volume XLI-B5 / [ed] L. Halounova, V. Šafář, F. Remondino, J. Hodač, K. Pavelka, M. Shortis, F. Rinaudo, M. Scaioni, J. Boehm, and D. Rieke-Zapp, International Society of Photogrammetry and Remote Sensing , 2016, Vol. XLI-B5, p. 7-14Conference paper (Refereed)
The aim of this paper is to investigate whether the Matlab-based Damped Bundle Adjustment Toolbox (DBAT) can be used to provide independent verification of the BA computation of two popular software—PhotoModeler (PM) and PhotoScan (PS).
For frame camera data sets with lens distortion, DBAT is able to reprocess and replicate subsets of PM results with high accuracy. For lens-distortion-free data sets, DBAT can furthermore provide comparative results between PM and PS. Data sets for the discussed projects are available from the authors.
The use of an external verification tool such as DBAT will enable users to get an independent verification of the computations of their software. In addition, DBAT can provide computation of quality parameters such as estimated standard deviations, correlation between parameters, etc., something that should be part of best practice for any photogrammetric software. Finally, as the code is free and open-source, users can add computations of their own.
• 284.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
National Institute of Applied Sciences of Strasbourg. Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, Department of Computing Science.
Pros and cons of constrained and unconstrained formulation of the bundle adjustment problem2004In: ISPRS Congress Istanbul 2004, Proceedings of Commission III, ISPRS , 2004, Vol. XXXV, no B3, p. 589-594Conference paper (Other academic)
Two implementations of the bundle adjustment problem were applied to a subset of the Zurich City Hall reference data set. One implementation used the standard Euler angle parameterisation of the rotation matrix. The second implementation used all nine elements of the rotation matrix as unknowns and six functional constraints. The second formulation was constructed to reduce the non-linearity of the optimisation problem. The hypothesis was that a lower degree of non-linearity would lead to faster convergence. Furthermore, each implementation could optionally use the line search damping technique known from optimisation theory. The algorithms were used to solve the relative orientation problem for a varying number of homologous points from 33 different camera pairs. The results show that the constrained formulation has marginally better convergence properties, with or without damping. However, damping alone halves the number of convergence failures at a minor computational cost. The conclusion is that except to avoid the singularities associated with the Euler angles, the preferred use of the constrained formulation remains an open question. However, the results strongly suggest that the line search damping technique should be included in standard implementations of the bundle adjustment algorithm.
• 285.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
3D measurements of buildings and environment for harbor simulators2009Report (Other academic)
Oryx Simulations develops and manufactures real-time physics simulators for training of harbor crane operator in several of the world’s major harbors. Currently, the modelling process is labor-intensive and a faster solution that can produce accurate, textured models of harbor scenes is desired. The accuracy requirements vary across the scene, and in some areas accuracy can be traded for speed. Due to the heavy equipment involved, reliable error estimates are important throughout the scene. This report surveys the scientific literature of 3D reconstruction algorithms from aerial and terrestrial imagery and laser scanner data. Furthermore, available software solutions are evaluated. The conclusion is that the most useful data source is terrestrial images, optionally complemented by terrestrial laser scanning. Although robust, automatic algorithms exist for several low-level subproblems, no automatic high-level 3D modelling algorithm exists that satisfy all the requirements. Instead, the most successful high-level methods are semi-automatic, and their respective success depend on how well user input is incorporated into an efficient workflow. Furthermore, the conclusion is that existing software cannot handle the full suite of varying requirements within the harbor reconstruction problem. Instead we suggest that a 3D reconstruction toolbox is implemented in a high-level language, Matlab. The toolbox should contain state-of-the-art low-level algorithms that can be used as “building blocks” in automatic or semi-automatic higher-level algorithms. All critical algorithms must produce reliable error estimates. The toolbox approach in Matlab will be able to simultaneously support basic research of core algorithms, evaluation of problem-specific high-level algorithms, and production of industry-grade solutions that can be ported to other programming languages and environments.
• 286.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
University of Western Australia. Massachusetts General Hospital, Boston, MA.
Validation of marker-based X-ray measurements of joint kinematics2006In: Proceedings SSBA 2006: symposium on image analysis / [ed] Fredrik Georgsson, Niclas Börlin, Umeå: Umeå University, Department of Computing Science, Umeå University , 2006, p. 113-116Conference paper (Other academic)
Radiostereometric Analysis (RSA) is an established method for measuring the motion of the skeleton. However, in order to measure dynamic joint kinematics, RSA requires expensive, custom-built hardware. Furthermore, the working volume is restricted to the region around where the beams intersect.
The Single-plane RSA Flouroscopy (SPRSAF) has the potential to overcome these limitations. This paper is the first validation of SPRSAF versus RSA on images with clinical image quality.
The results say that SPRSAF has a rotational error of (1.3,2.9, 11.6) degrees for rotation about the three primary axes. The corresponding translation results are (8.5, 1.0, 1.5) mm. This indicates that SPRSAF has the precision needed to be clinically useful in at least four of the six degrees of freedom.
• 287.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Sahlgrenska University Hospital.
Radiostereometry Based On Digitized Radiographs1997In: Proceedings of the 43rd Annual Meeting of The Orthopaedic Research Society, 1997, p. 626-626Conference paper (Other academic)
• 288.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Umeå University, Faculty of Medicine, Department of Odontology, Prosthetic Dentistry.
An Implant-oriented method for dental digital subtraction radiography1999In: Computer Methods in Biomechanics & Biomedical Engineering — 2 / [ed] Middleton, J., Gordon and Breach Science Publishers , 1999, p. 705-712Conference paper (Other academic)
• 289.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Umeå University, Faculty of Medicine, Department of Odontology, Prosthetic Dentistry.
The threaded dental implant as a reference object for image alignment2001In: Computer Methods in Biomechanics and Biomedical Engineering, ISSN 1025-5842, E-ISSN 1476-8259, Vol. 4, no 5, p. 421-431Article in journal (Refereed)
This paper presents a method that uses the threaded dental implant as a reference object for the inter-image alignment necessary for digital subtraction radiography. The implant is furthermore used to define a measurement coordinate system and to automate the placement of reference areas used for contrast correction. The method is intended for studies of diffuse bone density changes in the vicinity of the implant. The method is shown to be insensitive to large variations in exposure time and geometry, and is together with the contrast correction method of Ruttimann et al., able to detect clinically invisible simulated bone density changes.
• 290.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, Department of Computing Science.
A globally convergent gauss-newton algorithm for the bundle adjustment problem with functional constraints2003In: Optical 3-D measurement techniques: applications in GIS, mapping, manifactoring, quality control, robotics, navigation, mobile mapping, medical imaging, VR generation and animation / [ed] A. Gruen, H. Kahmen, Wichmann-Verlag , 2003, Vol. 2, p. 269-276Chapter in book (Other academic)
This paper describes a Gauss-Newton-based algorithm for the bundle adjustment problem with functional constraints (GNC). The GNC algorithm has superior theoretical convergence properties compared to the conventional bundle algorithm. Both algorithms were applied to simulated measurements of a sphere with 2-3 cameras and 4-9 points. For 2 cameras and 4-5 points, the GNC converged in substantially more cases. For the other configurations, the convergence properties were similar. The added cost for the GNC algorithm was less than 0.01 iterations on average. The GNC algorithm need to be evaluated on real-world problems, but the results suggest that the algorithm will be more reliable for minimum data problems and have a minimal overhead for easy problems.
• 291.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
INSA Strasbourg, France. INSA Strasbourg, France.
Implementing Functional Modularity for Processing of General Photogrammetric Data with the Damped Bundle Adjustment Toolbox (DBAT)2019In: Int. Arch. Photogramm. Remote Sens. Spatial Inf. Sci., XLII-2/W17, 69–75, 2019 / [ed] P. Grussenmeyer, A. Murtiyoso, H. Macher, and R. Assi, 2019, Vol. XLII-2/W17, p. 69-75Conference paper (Refereed)
The Damped Bundle Adjustment Toolbox (DBAT) is a free, open-source, toolbox for bundle adjustment. The purpose of DBAT is to provide an independent, open-source toolkit for statistically rigorous bundle adjustment computations. The capabilities include bundle adjustment, network analysis, point filtering, forward intersection, spatial intersection, plotting functions, and computations of quality indicators such as posterior covariance estimates and parameter correlations. DBAT is written in the high-level Matlab language and includes several processing example files. The input formats have so far been restricted to PhotoModeler export files and Photoscan (Metashape) native files. Fine-tuning of the processing has so far required knowledge of the Matlab language.
This paper describes the development of a scripting language based on the XML (eXtensible Markup Language) language that allow the user a fine-grained control over what operations are applied to the input data, while keeping the needed programming skills at a minimum. Furthermore, the scripting language allows a wide range of input formats. Additionally, the XML format allows simple extension of the script file format both in terms of adding new operations, file formats, or adding parameters to existing operations. Overall, the script files will in principle allow DBAT to process any kind of photogrammetric input and should extend the usability of DBAT as a scientific and teaching tool for photogrammetric computations.
• 292.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Photogrammetry and Geomatics Group, ICube Laboratory UMR 7357, INSA Strasbourg, France. Photogrammetry and Geomatics Group, ICube Laboratory UMR 7357, INSA Strasbourg, France. 3D Optical Metrology (3DOM) unit, Bruno Kessler Foundation (FBK), Trento, Italy; COMEX SA–Innovation Department, COMEX, Marseille, France. LIS, I&M Team, Aix-Marseille Université, Polytech Luminy, Marseille, France; Institute of Theoretical Physics, ETH Zurich, Zurich, Switzerland.
Flexible Photogrammetric Computations Using Modular Bundle Adjustment: The Chain Rule and the Collinearity Equations2019In: Photogrammetric Engineering and Remote Sensing, ISSN 0099-1112, Vol. 85, no 5, p. 361-368Article in journal (Refereed)
The main purpose of this article is to show that photogrammetric bundle-adjustment computations can be sequentially organized into modules. Furthermore, the chain rule can be used to simplify the computation of the analytical Jacobians needed for the adjustment. Novel projection models can be flexibly evaluated by inserting, modifying, or swapping the order of selected modules. As a proof of concept, two variants of the pinhole projection model with Brown lens distortion were implemented in the open-source Damped Bundle Adjustment Toolbox and applied to simulated and calibration data for a nonconventional lens system. The results show a significant difference for the simulated, error-free, data but not for the real calibration data. The current flexible implementation incurs a performance loss. However, in cases where flexibility is more important, the modular formulation should be a useful tool to investigate novel sensors, data-processing techniques, and refractive models.
• 293.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
INSA Strasbourg, France. INSA Strasbourg, France. 3D Optical Metrology (3DOM) unit, Bruno Kessler Foundation (FBK), Trento, Italy. 3D Optical Metrology (3DOM) unit, Bruno Kessler Foundation (FBK), Trento, Italy.
Modular Bundle Adjustment for Photogrammeric Computations2018In: ISPRS Technical Commission II Symposium 2018, ISPRS , 2018, Vol. XLII-2, p. 133-140Conference paper (Refereed)
In this paper we investigate how the residuals in bundle adjustmentcan be split into a composition of simple functions. According to thechain rule, the Jacobian (linearisation) of the residual can be formedas a product of the Jacobians of the individual steps. Whenimplemented, this enables a modularisation of the computation of thebundle adjustment residuals and Jacobians where each component haslimited responsibility. This enables simple replacement of componentsto e.g. implement different projection or rotation models byexchanging a module. The technique has previously been used toimplement bundle adjustment in the open-source package DBAT (Borlinand Grussenmeyer, ¨ 2013) based on the Photogrammetric and ComputerVision interpretations of Brown (1971) lens distortion model. In thispaper, we applied the technique to investigate how affine distortionscan be used to model the projection of a tilt-shift lens. Two extendeddistortion models were implemented to test the hypothesis that theordering of the affine and lens distortion steps can be changed toreduce the size of the residuals of a tilt-shift lens calibration.Results on synthetic data confirm that the ordering of the affine andlens distortion steps matter and is detectable by DBAT. However, whenapplied to a real camera calibration data set of a tilt-shift lens, nodifference between the extended models was seen. This suggests thatthe tested hypothesis is false and that other effects need to bemodelled to better explain the projection. The relatively lowimplementation effort that was needed to generate the models suggestthat the technique can be used to investigate other novel projectionmodels in photogrammetry, including modelling changes in the 3Dgeometry to better understand the tilt-shift lens.
• 294.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Katholieke Universiteit Nijmegen, Nijmegen, Holland. Sahlgrenska University Hospital, Göteborg, Sweden.
The precision of radiostereometric measurements: manual vs. digital measurements2002In: Journal of Biomechanics, ISSN 0021-9290, E-ISSN 1873-2380, Vol. 35, no 1, p. 69-79Article in journal (Refereed)
The precision of digital vs. manual radiostereometric measurements in total hip arthroplasty was evaluated using repeated stereoradiographic exposures with an interval of 10–15 min. Ten Lubinus SP2 stems cemented into bone specimens and 12 patients with the same stem design were used to evaluate the precision of stem translations and rotations. The precision of translations and rotations of the cup and femoral head penetration was studied in 12 patients with whole polyethylene cups.
The use of a measurement method based on digitised radiographs improved the precision for some of the motion parameters, whereas many of them did not change. A corresponding pattern was observed for both the intra- and interobserver error. Of the wear parameters, the most pronounced improvements were the 3D wear and in the proximal-distal direction, although the anterior-posterior precision was also improved. The mean errors of rigid body and elliptic fitting decreased in all evaluations but one, consistent with a more reproducible identification of the markers centres and the edge of the femoral head.
Increased precision of radiostereometric measurements may be used to increase the statistical power of future randomised studies and to study new fields in orthopaedics requiring higher precision than has been available with RSA based on manual measurements.
• 295. Börstler, Jurgen
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Beauty and the Beast: on the readability of object-oriented example programs2016In: Software quality journal, ISSN 0963-9314, E-ISSN 1573-1367, Vol. 24, no 2, p. 231-246Article in journal (Refereed)
Some solutions to a programming problem are more elegant or more simple than others and thus more understandable for students. We review desirable properties of example programs from a cognitive and a measurement point of view. Certain cognitive aspects of example programs are captured by common software measures, but they are not sufficient to capture a key aspect of understandability: readability. We propose and discuss a simple readability measure for software, SRES, and apply it to object-oriented textbook examples. Our results show that readability measures correlate well with human perceptions of quality. Compared with other readability measures, SRES is less sensitive to commenting and whitespace. These results also have implications for software maintainability measures.
• 296.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, Department of Computing Science.
Evaluating OO Example Programs for CS12008In: Proceedings of the 13th annual conference on Innovation and technology in computer science education, 2008, p. 47-52Conference paper (Refereed)
• 297.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Umeå University, Faculty of Science and Technology, Department of Computing Science. Glasgow Caledonian University, Scotland, UK.
On the quality of examples in introductory Java textbooksIn: Transactions on Computing EducationArticle in journal (Refereed)
Example programs play an important role in the teaching and learning of programming. Students as well as teachers rank examples as the most important resources for learning to program. Ex- ample programs work as role models and must therefore always be consistent with the principles and rules we are teaching.
However, it is difficult to find or develop examples that are fully faithful to all principles and guidelines of the object-oriented paradigm and also follow general pedagogical principles and practices. Unless students are able to engage with good examples, they will not be able to tell desirable from undesirable properties in their own and others’ programs.
In this paper we report on a study in which experienced educators evaluated the quality of object-oriented example programs for novices from popular Java textbooks. The evaluation was accomplished using an on-line checklist that elicited responses on the technical, object-oriented, and didactic quality of examples.
In total 25 reviewers contributed 215 reviews to our data set, based on 38 example programs from 13 common introductory programming textbooks. Results show that the evaluation instru- ment is reliable in terms of inter-rater agreement. Overall, example quality was not as good as one might expect from common textbooks, in particular regarding certain object-oriented properties.
We conclude that educators should be careful when taking examples straight out of a textbook.
• 298.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Classes or Objects? CRC-cards Considered Harmful: Extended Abstract2004In: First Workshop of the Scandinavian Pedagogy of Programming Network, 2004Conference paper (Refereed)
• 299.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
CRC-Cards and Roleplay Diagrams--Informal Tools to Teach OO Thinking2007In: 2nd Workshop on Computer Science Education, 2007Conference paper (Refereed)
• 300.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Improving CRC-Card Role-Play with Role-Play Diagrams2005In: OOPSLA'05 Addendum to the Proceedings (Educators' Syposium), 2005, p. 356-364Conference paper (Refereed)
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• rtf | 2020-01-21 00:39:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.32106712460517883, "perplexity": 2323.1228440090385}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250601040.47/warc/CC-MAIN-20200120224950-20200121013950-00442.warc.gz"} |
https://chemistry.stackexchange.com/questions/93773/is-1-2-hydride-shift-possible-in-ketonic-cations | # Is 1,2- Hydride shift possible in ketonic cations?
The question in my worksheet asks me to predict the result of the reaction between benezene and $\ce{(CH3CH2CH2CO)2O}$ in presence of anhydrous $\ce{AlCl3}$.
My attempt:
After complex formation with $\ce{AlCl3}$ we get this carbocation: $\ce{CH3CH2CH2CO+}$ that has two resonance structures.
But, if we consider hydride shift we get $\ce{CH3CH2C^+HC=O}$ which is not only stabilised by resonance but also by the $\text{+I}$ effect of ethyl group.
However, the answer is $\ce{PhCOCH2CH2CH3}$ which isn't formed by the second carbocation with hydride shift .
So I'd like to know if hydride shift is possible in ketonic carbocations or not? And if it is, is the answer wrong?
• If hydride transfer were to occur, the resultant carbonyl would be stabilizing a cation. A definite NO_NO! Read up on the the Friedel-Crafts reaction. Mar 21 '18 at 16:09
## 1 Answer
The carbocation doesn't rearrange since oxygen donates it's lone pair and the resulting carbocation is quite stable since the octet of all the atoms is complete. | 2022-01-27 08:44:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7063802480697632, "perplexity": 2922.2523296238396}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320305242.48/warc/CC-MAIN-20220127072916-20220127102916-00391.warc.gz"} |
https://stats.stackexchange.com/tags/hidden-markov-model/info | Tag Info
Hidden Markov Models are used for modelling systems that are assumed to be Markov processes with hidden (i.e. unobserved) states.
A hidden Markov model is a three-tuple $\left\langle\vec{\pi},A,B\right\rangle$ with $\vec{\pi}$ a probability vector over the $n$ hidden states, $A$ an $n\times n$ transition matrix and $B$ an $n\times m$ emission matrix. $\pi_i$ describes the probability of the system being in hidden state $i$ at time step $0$, $a_{ij}$ describes the probability of the system being in hidden state $j$ at time $t+1$ given it was in hidden state $i$ at time $t$. $b_{ik}$ describes the probability of observing $k$ given the system is in hidden state $i$.
A Markov model thus describes a Markov process, but where the state of the system is "hidden" and only observations can be seen. Such process can be trained for several applications (speech recognition, part-of-speech tagging and computational biology). It can be trained using the popular Baum-Welch algorithm and the most probable sequence of hidden states can be calculated using the Viterbi algorithm.
The standard HMM can be viewed graphically as follows:
Where $Z_t$ is the hidden state and $X_t$ is the observation at time $t$. We can use the conditional independence structure expressed by this graphical model to derive the algorithms above.
Extensions exist such that continuous output is supported as well. | 2020-06-02 22:08:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8894678950309753, "perplexity": 249.77488350237059}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347426801.75/warc/CC-MAIN-20200602193431-20200602223431-00008.warc.gz"} |
https://2021.help.altair.com/2021/hwsolvers/acusolve/topics/acusolve/training_manual/time_disc_r.htm | # Time Discretization
This section describes the various approaches used to discretize the governing equation in the temporal domain such as two step, multistep and multistage methods.
In order to determine a numerical solution of the governing differential equations the temporal domain must also be discretised apart from the spatial domain. The direction of influence of the time coordinate is only in the future. Therefore, all the solution methods for time dependent problems advance in time from a given initial data.
A vast majority of the methods used for temporal discretization are linear in nature. The time dependent variable is updated using a linear combination of the variable and its time derivatives. Linear approaches can be broadly categorized based on the number of steps, stages and derivatives used in the discretization.
Some of the widely used time discretization approaches are described below.
## Generalized Two Step Methods
Two step methods involve function values at two instances in time, generally considering the current time step at which the solution is known and the next time step at which the solution has to be computed.
Consider a first order ordinary differential equation for a dependent variable $\phi$ expressed as: (1)
$\frac{d\phi }{dt}=f\left(t,\phi \right)$
with an initial condition $\phi \left({t}_{0}\right)={\phi }^{0}$ .
A generalized scheme using a weighted average value for the approximation of the variable value at the n+1th time step is expressed as: (2)
${\phi }^{n+1}={\phi }^{n}+\Delta t\left[\theta {f}^{n+1}+\left(1-\theta \right){f}^{n}\right]$
where ${f}^{n}$ represents the function value at the nth time step and $\theta$ represents the weight.
The nature and stability of the temporal discretization scheme depends on the choice of the weight $\theta =0,\frac{1}{2}$ and $1$ represent the forward Euler, Crank-Nicholson and the backward Euler schemes, respectively.
## Multistep Methods
Multistep methods involve function values at more than two instances of time. These methods are generally derived by fitting a polynomial to the temporal derivative of the dependent variable, that is, $f\left(t,\phi \right)$ .
These methods include the Adams-Bashforth and Adams-Moutlton methods. The order of the method depends on the number of points in time at which the polynomial fitting it used. A third order accurate Adams-Moulton method is expressed as: (3)
${\phi }^{n+1}={\phi }^{n}+\frac{\Delta t}{2}\left[5{f}^{n+1}+8{f}^{n}-{f}^{n-1}\right]$
These methods require initial data at many steps, hence they are not self starting.
## Multistage Methods
Multistage methods involve computation of the function values multiple times at the same time step. They generally involve predictor and corrector steps to compute the values at the n+1th time step.
Numerical solution schemes are often referred to as being explicit or implicit. When a direct computation of the dependent variables can be made in terms of known quantities the computation is said to be explicit. When the dependent variables are defined by coupled set of equations, and either a matrix or iterative technique is needed to obtain the solution, the numerical method is said to be implicit.
Explicit methods are easy to program but are conditionally stable whereas implicit methods offer better stability but are computationally expensive. Predictor-Corrector methods offer a compromise between these choices. A variety of methods exist based on the choice of base method and the time instants used in the predictor and corrector steps.
The most popular methods in this category are the Runge-Kutta methods. A fourth Runge-Kutta method is constructed as follows:
• Explicit Euler Predictor: ${\phi }_{*}^{n+\frac{1}{2}}={\phi }^{n}+\frac{\Delta t}{2}{f}^{n}$
• Implicit Euler Corrector: ${\phi }_{**}^{n+\frac{1}{2}}={\phi }^{n}+\frac{\Delta t}{2}{f}_{*}^{n+\frac{1}{2}}$
• Mid-point rule Predictor: ${\phi }_{*}^{n+1}={\phi }^{n}+\Delta t{f}_{**}^{n+\frac{1}{2}}$
• Simpsons rule Corrector: ${\phi }^{n+1}={\phi }^{n}+\frac{\Delta t}{6}\left[{f}^{n}+2{f}_{*}^{n+\frac{1}{2}}+2{f}_{**}^{n+\frac{1}{2}}+{f}_{*}^{n+1}\right]$
## Generalized- $\alpha$ Method
The generalized $\alpha$ method is an implicit method of time integration which achieves high frequency numerical dissipation while at the same time minimizing unwanted low frequency dissipation and offers unconditional stability for linear problems. It is a variant of the generalized two step theta scheme discussed above where the first temporal derivatives are evaluated as variables.
For a linear system defined by: (4)
$\stackrel{˙}{\phi }=\frac{d\phi }{dt}=\lambda \phi$
The generalized $\alpha$ method for integration from time step ${t}_{n}$ to ${t}_{n+1}$ is constructed as follows:
• ${\stackrel{˙}{\phi }}_{n+{\alpha }_{m}}=\lambda {\phi }_{n+{\alpha }_{f}}$
• ${\phi }_{n+1}={\phi }_{n}+\Delta t{\stackrel{˙}{\phi }}_{n}+\Delta t\gamma \left({\stackrel{˙}{\phi }}_{n+1}-{\stackrel{˙}{\phi }}_{n}\right)$
• ${\stackrel{˙}{\phi }}_{n+{\alpha }_{m}}={\stackrel{˙}{\phi }}_{n}+{\alpha }_{m}\left({\stackrel{˙}{\phi }}_{n+1}-{\stackrel{˙}{\phi }}_{n}\right)$
• ${\phi }_{n+{\alpha }_{f}}={\phi }_{n}+{\alpha }_{n}\left({\phi }_{n+1}-{\phi }_{n}\right)$
where $\Delta t$ is the time step size $\left(\Delta t={t}_{n+1}-{t}_{n}\right)$ and ${\alpha }_{m},{\alpha }_{f},\gamma$ are free parameters.
The above four equations combine to yield the following system: (5)
${\varnothing }_{n+1}=c{\varnothing }_{n}$
where the solution vector ${\varnothing }_{n}$ at ${t}_{n}$ is defined as ${\varnothing }_{n}={\left\{{\phi }_{n},\Delta t{\stackrel{˙}{\phi }}_{n}\right\}}^{T}$ . | 2023-01-28 19:58:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 31, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6670903563499451, "perplexity": 355.5887273604986}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499654.54/warc/CC-MAIN-20230128184907-20230128214907-00655.warc.gz"} |
https://www.ias.ac.in/describe/article/pram/014/01/0057-0074 | • $${}_{\Lambda \Lambda }^{10} Be$$ in Faddeev-Yakubovsky formalismin Faddeev-Yakubovsky formalism
• # Fulltext
https://www.ias.ac.in/article/fulltext/pram/014/01/0057-0074
• # Keywords
Four-body equations; identical particles; two-body separable potentials; Bateman approximation; separation energy; Lambda particles in$${}_{\Lambda \Lambda }^{10} Be$$; Faddeev-Yakubovsky formalism
• # Abstract
The four-body dynamical equations for two distinct pairs of identical particles derived earlier are applied to investigate the system$${}_{\Lambda \Lambda }^{10} Be$$. The two-body potentials have been taken to be of the Yamaguchi form, and the Bateman approximation has been used for the other amplitudes. From the set of coupled integral equations, the separation energy, BΛΛ, for the two Λ particles in$${}_{\Lambda \Lambda }^{10} Be$$ is obtained as 43·97 MeV.
• # Author Affiliations
1. Department of Theoretical Physics, Indian Association for the Cultivation of Science, Jadavpur, Calcutta - 700 032
2. Department of Physics, Jadavpur University, Calcutta - 700 032
• # Pramana – Journal of Physics
Current Issue
Volume 93 | Issue 5
November 2019
• # Editorial Note on Continuous Article Publication
Posted on July 25, 2019 | 2019-09-19 06:38:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2633432149887085, "perplexity": 8895.117664366318}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573444.87/warc/CC-MAIN-20190919060532-20190919082532-00082.warc.gz"} |
https://codegolf.stackexchange.com/questions/105861/can-this-number-be-written-in-3x-1-format/105923 | # Can this number be written in (3^x) - 1 format?
Challenge:
Create a program that accepts a positive integer and checks if it can be written in the form of (3^x)-1, where X is another positive integer.
If it can, output X
If it can't, output -1 or a falsy statement.
Example inputs/outputs
Input:
2
It can be written as (3^1) - 1, so we output x which is 1
Output:
1
Input:
26
26 can be written as (3^3) - 1, so we output x (3)
Output:
3
Input:
1024
1024 can't be written in the form of (3^x) - 1, so we output -1
Output:
-1
This is so least amount of bytes wins
Related OEIS: A024023
• I ask to output X because I believe it's more challenging that way. Simply finding if it is of format 3^x - 1 would be too easy for a challenge, in my opinion. – P. Ktinos Jan 6 '17 at 14:56
• Unless if it's a falsy statement in your programming language, then no. – P. Ktinos Jan 6 '17 at 16:57
• May I want the number to be input in ternary? – John Dvorak Jan 6 '17 at 17:35
• having to handle non-negative intergers would make 0 3^0-1 a valid output and thus not useable as false, – Jasen Jan 7 '17 at 7:40
• anyone thinking of using log() in their answer should confirm it giives the correct answer 5 when 242 is input. – Jasen Jan 7 '17 at 9:57
## Pyke, 9 6 bytes
3m^Qh@
Try it here!
3m^ - map(3**i, range(input))
@ - V in ^
Qh - input + 1
Old 9 byte version:
b3'l}\2q*
Try it here!
## Pyth 8 bytes
xm^3dUQh
UQ # generate all values 1..Q (Q is the input)
m^3d # map 3^d over this ^ list
x h # find the input+1 (hQ) in the result of the last command
Try here
# R, 34 25 bytes
a=log(scan()+1,3);a*!a%%1
Calculate the base 3 logarithm of the input + 1. Test if the result is an integer, if it is it outputs it, if not it outputs 0 as falsey value. Thanks to @Billywob for the extra 9 bytes off!
Test cases:
> a=log(scan()+1,3);a*!a%%1
1: 1024
2:
[1] 0
> a=log(scan()+1,3);a*!a%%1
1: 26
2:
[1] 3
Old version at 34 bytes which outputs -1 as falsey value.
a=log(scan()+1,3);if(!a%%1,a,-1)
• If you do a*!a%%1 it will output a if true and 0 otherwise and you can skip the if thing. – Billywob Jan 6 '17 at 16:07
• The spec says "If it can't, output -1 or a falsy statement." and 0 is interpreted as FALSE in R so I would say it's valid. – Billywob Jan 6 '17 at 16:19
# C, 81 bytes
i,j,k;f(n){for(i=0;++i<n;){for(k=3,j=0;++j<i;k*=3);if(n==k-1)return i;}return-1;}
• I think you save bytes by using pow(3,i) instead of defining your own. Gcc complains about the missing #include <math.h> but compiles it anyway. I did have to cast to int. You also may be able to gain some by adding an r variable, initializing to -1, and then if(n==k-1)r=i;}return r;} – nmjcman101 Jan 6 '17 at 18:54
• @nmjcman101 Thanks, but pow() produces some incorrect results because of floating point inaccuracy. (When cast to int, 2.9999 will be 2, not 3). Adding a variable r sounds like a good idea, but it actually results in a 2 bytes longer code. – Steadybox Jan 6 '17 at 22:33
• yeah, log doesn't work, pow probably won't either. – Jasen Jan 7 '17 at 9:28
# Japt, 8 bytes
o m!³a°U
Try it online!
This code expands into the following:
Uo m!p3 a++U
Uo // Create the range [0...U).
m!p3 // Map each item X to 3**X.
a++U // Take the index of U+1. Returns -1 if it doesn't exist.
// Implicit: output result of last expression
• Very nice solution. – Oliver Jan 7 '17 at 6:04
# GolfScript, 11 bytes
~..3\?\)%!*
Try it online!
Uses the fact that 3n is divisible by n+1 if and only if n+1 itself is a power of 3. Outputs its input n if n+1 is a power of 3, otherwise outputs 0 (which is falsy in GolfScript).
De-golfed:
~ # eval the input, converting it from string to integer
.. # make two copies of the input number
3\? # raise 3 to the power of the input number
\)% # reduce the result modulo the input number plus one
! # boolean negate the result, mapping 0 to 1 and all other values to 0
* # multiply the input number with the result
Ps. Here's a simple test harness that runs the code above (minus the initial ~, which is not needed since the inputs are already numbers) on all integers from 0 to 9999 and prints those for which it returns a truthy result:
10000,{ ..3\?\)%!* },
The output of this program should be:
[2 8 26 80 242 728 2186 6560]
(The output doesn't include 0 because, even though the formula used does correctly detect it as one less than a power of 3, the result is still 0 × 1 = 0, and thus falsy. Fortunately, 0 is not a positive integer, and thus isn't a valid input for this challenge anyway.)
# Octave, 23 bytes
@(x)find(3.^(1:x)-1==x)
Verify all test cases!
Explanation:
This is an anonymous function that takes a positive integer x as input. .^ is element-wise power in Octave, so 3.^(1:x) is 3^1, 3^2, 3^3 .... Subtracting 1 gives 3^1-1, 3^2-1, 3^3-1 ... which can be compared to x.
find(a,b) takes a vector a as input, and attempts to find the scalar b in that vector and returns its index. If it's not found then it will output an empty matrix []. An empty matrix is a falsey value in Octave.
find(3.^(1:x)-1==x) searches for x in the vector 3^1-1, 3^2-1, 3^3-1 ... and attempts to return its index. If it's not in the vector then it returns an empty (falsey) matrix.
# C, 76 bytes
main(i,a,c){scanf("%d",&a);for(c=0,++a;i<a;i*=3,++c);printf("%d",(i==a)*c);}
# Sagemath, 45 bytes
This is simply @dfernan's solution repackaged as Sagemath (which is basically Python + some math libraries loaded by default and syntactic sugar).
In Sagemath, we can avoid the import math and we can use ^ for exponentiation, so we save a few chars.
def f(n):x=ceil(log(n,3));print((3^x-1==n)*x)
Test it online
c++ 60 bytes
int f(n){float o=log1p(n)/log(3);return o/floor(o)!=1?-1:o;}
explanation:
int f(n){
float o=log1p(n)/log(3); // eval for x using log3 function
return o/floor(o)!=1?-1:o; // if no remainder output X
}
• I think you don't have to use uint16_t. You can use normal int. – Roman Gräf Jan 8 '17 at 19:55
• I guess I could to save a few bytes but it seemed to me that if I wasn't safeguarding against negative integer input then I wasn't following the guidelines. – mreff555 Jan 14 '17 at 21:04
• this compile to me: int f(n){float o=log1p(n)/log(3);return o/floor(o)!=1?-1:o;} – user58988 Apr 29 '17 at 15:53
• @mreff555 You don't need to safeguard against negative input, you can assume positive input. – Erik the Outgolfer Apr 29 '17 at 16:05
# Python 2, 34 bytes
lambda n:(~n>>3**n%-~n*n)**4/80%80
Try it online!
Works for all Python ints, up to at least 2^100.
# Pip, 13 bytes
aTB:3MNa=2&#a
Try it online!
### Explanation
a is 1st command-line argument (implicit)
aTB:3 Convert a to base 3 and assign back to a
MNa=2 Does the min of a's digits equal 2?
& Logical-and
#a Length of a
If there are non-2 digits, we get the falsey value 0; otherwise, we get
the number of digits
• "26 can be written as (3^3) - 1" 124 can be written as (5^3)-1 but your code for 124 not print 5 print 0 – user58988 Feb 2 '18 at 15:48
• Ok I confuse exponent and base – user58988 Feb 2 '18 at 15:51
# 05AB1E, 6 bytes
>3.n.ï
Try it online!
> increments, 3 pushes a 3 to the stack, .n find the logarithm with base 3, .ï checks if it is equal to its integer part.
Returns 0 for falsy: If it can't, output -1 or a falsy statement.
• "26 can be written as (3^3) - 1" 124 can be written as (5^3)-1 but your code for 124 not print 5 print 0 – user58988 Feb 2 '18 at 15:47
• @RosLuP I believe the output of my program is correct, and other answers seem to agree – Mr. Xcoder Feb 2 '18 at 15:49
• Ok I confuse the exponent and the base – user58988 Feb 2 '18 at 15:50
# APL (Dyalog Unicode), 12 bytesSBCS
Anonymous tacit prefix function.
(⊢×⌊=⊢)3⍟1+⊢
Try it online!
1+⊢ increment
3⍟ log3
() apply the following function:
⌊=⊢ is the floor equal to the argument? (0 or 1)
⊢× multiply the argument by that
• "26 can be written as (3^3) - 1" 124 can be written as (5^3)-1 but your code for 124 not print 5 print 0 – user58988 Feb 2 '18 at 15:44
• Ok I confuse exponent and base – user58988 Feb 2 '18 at 15:51
# APL (Dyalog), 11 bytes
⊢|⊢⍳⍨¯1+3*⍳
Try it online!
Uses ⎕IO←0.
How?
⍳ - range of 0 to n-1.
3* - raise 3 to the power of each element.
¯1+ - decrement each by 1.
⊢⍳⍨ - search the index of n in that list (if not exists, this would return the maximum index plus 1 - which is n.
⊢| - modulo by n. This would keep the index, if found, and zero-out numbers not contained in the list that would produce n % n = 0.
# APL (Dyalog), 14 bytes
(∧/×≢)2=3⊥⍣¯1⊢
Try it online!
• "26 can be written as (3^3) - 1" 124 can be written as (5^3)-1 but your code for 124 not print 5 print 0 – user58988 Feb 2 '18 at 15:45
• Ok I confuse exponent and base – user58988 Feb 2 '18 at 15:52
• @RosLuP you mind deleting the comments? people use to DV without much thinking when seeing these – Uriel Feb 3 '18 at 16:17
# APL NARS, 14 chars or 28 bytes
{r×r=⌊r←3⍟1+⍵}
Test:
f←{r×r=⌊r←3⍟1+⍵}
f 2
1
f 26
3
f 1024
0
• This is 28 bytes in NARS, but exactly the same solution is only 14 bytes in Dyalog APL. Also, you can save two bytes by conversion to tradfn, r×r=⌊r←3⍟1+⎕, letting the program prompt for input. – Adám Dec 25 '17 at 16:15
• @Adám i prefer functions – user58988 Dec 25 '17 at 19:47
# Befunge-93, 26 bytes
&1+>:3%v
1+\^v-1_3/\
.@.$_ Try It Online Prints 0 as the falsey. ### How it Works &1+... Gets the input and adds one ...... ...... ...>:3%v Check if the number is divisible by 3 1+\^..._3/\ If not, divide the number by 3 and increment a counter ... Repeat until the number is not divisible by 3 ......... If the final number is a one, print the counter ....v-1_... Else pop the counter and print a 0 .@.$_ End the program
## JavaScript (ES6), 40 bytes
f=(n,p=0,k=1)=>n<k?n>k-2&&p:f(n,p+1,k*3)
Returns false or the power. A simple port of @Arnauld's ES7 answer would have taken 43 bytes.
• I think f=(n,p,k=1)=>n<k?n>k-2&&p:f(n,-~p,k*3) works and saves 2 bytes. – Arnauld Jan 7 '17 at 0:24
# PHP, 36 47 bytes
If log(input+1,3) differs from its integer value, print 0; else print the logarithm:
<?=(0|$x=log($argv[1]+1,3))-$x?0:$x; (36 bytes) fails for 242.
<?=strstr($x=log($argv[1]+1,3),".")?0:$x; and <?=(0|$x=log($argv[1]+1,3))-$x>1e-7?0:$x; (41 bytes) may fail for larger $x.
This version is safe:
for(;3**++$x<$n=1+$argv[1];);echo$n<3**$x?0:$x;
1.Loop $x up from 1 while 3^$x is smaller than argument+1.
2.Print 0 if the expression is larger than input+1, \$x else.
Takes input from command line argument. Run with -nr.
• gives wrong answer for 242 input – Jasen Jan 7 '17 at 9:22
• @Jasen: The downvote was ridiculous, but it´s fixed now. – Titus Jan 7 '17 at 11:36
# Java 7, 180 bytes
class A{public static void main(String[]q)throws Exception{int a,b=0;while((a=System.in.read()-48)>=0)b=b*10+(a);double k=Math.log(b+1)/Math.log(3);System.out.print(k%1==0?k:-1);}}
Really simple approach. Input, then add one, then log3 the number, and if it's a integer, print it; otherwise, print -1. Could use some work.
Only works up to (3^19)-1.
• Why is this non-competing? The Non-Competing status is only reserved for languages or language features that were added after the challenge was posted. – user41805 Jan 7 '17 at 11:10
• As I said, size doesn't matter, so this can compete. – P. Ktinos Jan 7 '17 at 20:55
• @KritixiLithos Oh okay. I wasn't aware of the exact meaning of Non-Competing. Thanks. Also, I'll update the title. – HyperNeutrino Jan 8 '17 at 3:19
• You can use interface A{...} and drop the public from main(). Why do you have (a) instead of a in the while loop. I think you can use float k=... instead of double k=.... Should be -5 bytes if I counted right. – Roman Gräf Jan 8 '17 at 20:04
• @RomanGräf I appreciate your suggestions; however, none of them are of any use for me, unfortunately. Your first suggestion only works in Java 8, and there is already a far better Java 8 solution out there. I have (a) in the while loop because I am doing a comparison of an assignment statement, and assignment has the lowest priority on the order of operations and thus requires a set of brackets around it. Finally, I have double because Math#log returns a double and casting would obviously be much slower. Regardless, thank you for the suggestions, but I will not be incorporating them. – HyperNeutrino Jan 9 '17 at 3:20
# Common Lisp (SBCL), 83 50 bytes
(let*((i(read))(r(log(1+ i)3)))(if(=(mod r 1)0)r))
Old version:
(let((i(read)))(if(find-if(lambda(x)(not(eq x #\2)))(format()"~3R"i))()(log(1+ i)3)))
Feedback encouraged!
# CJam, 11 bytes
ri3b_2-!\,*
Basically it determines if the number's trinary (base 3) representation contains only 2's, and if it does, outputs the number of 2's. It outputs 0 if the number is not only 2's.
Try it online!
Explanation
ri e# Get input as an integer
3b e# Convert to base 3 (an array of the digits in base 3)
_ e# Duplicate
2- e# Remove all 2's from the array
! e# Boolean negation. Yields 1 if the array contained only 2's (and is now
e# empty), 0 otherwise
\ e# Swap top 2 elements of stack
, e# Take the length of the base 3 digits array
* e# Multiply by the boolean value from before
• "Trinary" is more commonly referred to as Ternary. – FlipTack Jan 9 '17 at 18:23
## C 90 bytes
f(x){i=1;for(;i<x;i++){if((pow(3,i)-1)==x){printf("%d",i);break;}if(i==(x-1))puts("-1");}}
Ungolfed Version:
void f(int x)
{
int i=1;
for(;i<x;i++)
{
if((pow(3,i)-1)==x)
{
printf("%d",i);
break;
}
if(i==(x-1))
puts("-1");
}
}
• for x=1 what would return/print that above? – user58988 Apr 29 '17 at 15:43
# SmileBASIC, 32 bytes
INPUT X
X=LOG(X+1,3)?X*(X==X>>0)
Outputs 0 for false.
# Cardinal, 91 bytes
%
:
+
~
v~d<
0 /{<
+ A #-?+M?"-1"@
+ ! @.-<
+ M # M!/ <
>~ # ^ } \ +^%
According to the specifications of Cardinal, this shouldn't work for inputs above 255. However, due to the implementation in TIO accepting values over 255, it will work past 255 up to 3^34.
Try it online!
## Explanation
%
:
+
Input + 1
~
v~d<
0 /{<
+ A
+ !
+ M
>~ # ^
While divisible by 3
#-?+M?"-1"@
Output -1 if not divisible by 3 or equal to 1 and end program
.-<
# M!/ <
} \ +^%
Output counter for how many times the input + 1 has been divided by 3 before equalling 1
## QBIC, 18 bytes
:[a|c=3^b-1~c=a|?b
Explanation
: Get 'a' from the cmd-line
[a| FOR b=1; b<=a; b++ (this runs a lot longer than we need...)
c=3^b-1 Set c to be 3^b-1
~c=a IF this is the input given
|?b THEN print b
END IF and NEXT are added implicitly
This would print multiple bs if it would be possible to have multiple solutions. Right now, it runs on after finding a solution. We could substitute ?b for _Xb to quit after finding a solution, but that adds a byte. Also, the FOR-loop [a| could be initialised as [a/3| to save us a lot of iterations, but that adds another 2 bytes.
# Axiom,72 65 63 bytes
g(x:PI):INT==(z:=floor(y:=log(x+1.)/log(3.));y-z~=0=>-1;z::INT)
some test
(51) -> [[x,g(x)] for x in [1,2,3,4,5,6,7,8,9,10,26,27,79,80,242]]
(51)
[[1,- 1], [2,1], [3,- 1], [4,- 1], [5,- 1], [6,- 1], [7,- 1], [8,2],
[9,- 1], [10,- 1], [26,3], [27,- 1], [79,- 1], [80,4], [242,5]]
# TI-Basic (TI-84 Plus CE) 17 15 bytes
logBASE(Ans+1,3)
Ansnot(Ans-int(Ans
All tokens used are one-byte except logBASE(, which is two.
Explanation:
logBASE(Ans+1,3) # inverse of 3^n-1 is log3(x+1)
Ansnot(Ans-int(Ans # if X is an integer, X-int(x is 0, so not(X-int(X is 1, which is multiplied by X
# if X is not an integer, X-int(X is not 0, so not(X-int(X is zero, which is multiplied by X
# last value evaluated is implicitly returned
# J, 14 bytes
[:(*>.=])3^.>:
Try it online!
Here's a variant using a range of powers and indexing (@Dennis's method).
(~:*])>:i.~3^i.
Here's a variant using base conversion (@orlp's method).
[:(#*&(*/)2=])3&#.inv
# Factor, 30 bytes
[ 1 + 3 over [0,b) n^v index ]
Try it online!
Port of Dennis' Jelly answer. The index word conveniently returns f (the only falsy value in Factor) when the item is not found.
[ ! anonymous lambda
1 + 3 over ! ( n+1 3 n+1 )
[0,b) ! ( n+1 3 {0..n} )
n^v ! ( n+1 {3^0..3^n} )
index ! 0-based index of n+1 in {3^0..3^n}; false if not found
]
# Factor, 45 bytes
[ 3 >base dup [ 50 = ] all? swap length and ]
Try it online!
A base-conversion approach.
[ ! anonymous lambda, accepting a positive integer n
3 >base ! ( str ) convert to base-3 string
dup [ 50 = ] all? ! ( str ? ) test if it is all 2's
swap length and ! ( len/f ) if true, return the length of str
! otherwise return false
]
` | 2021-02-28 19:09:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2749224305152893, "perplexity": 4420.4529995792645}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178361723.15/warc/CC-MAIN-20210228175250-20210228205250-00354.warc.gz"} |
http://tex.stackexchange.com/questions/29438/prevent-the-file-path-of-an-image-appearing-above-the-image-when-using-beginfi | # Prevent the file path of an image appearing above the image when using \begin{figure} \includegraphics{filepath}
I have inserted an image in my LaTeX document and the image always displays with the file path of the image directly above it. How can I prevent this file path from displaying?
I am inserting the image like this:
\begin{figure}
\centering
\includegraphics{C:/Users/Name/Pictures/image1.png}
\caption{a caption}
\label{fig:reference}
\end{figure}
Here is a working example of document that produces this behaviour:
\documentclass{article}
\usepackage{graphicx}
\begin{document}
\begin{figure}
\includegraphics{C:/Users/Name/Pictures/image 1.png}
\end{figure}
\end{document}
The image was inserted using the TeXnicCenter Insert>Picture function.
When I generate the pdf document, directly above any image (I have inserted multiple images using this technique) the file path displays above the image, in the case above it would show: "file\path\of\image", how can I prevent this from displaying?
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Welcome to TeX.sx! Please add a minimal working example (MWE) that illustrates your problem. You're probably loading some package or using some option that causes this behaviour. – Jake Sep 25 '11 at 7:46
This is caused by the draft option, either for the graphicx package or the class. – Joseph Wright Sep 25 '11 at 7:50
Is file\path\of\image meant to be a normal path or really three macros? You need to use / not even under MS Windows, AFAIK. – Martin Scharrer Sep 25 '11 at 7:54
@Martin: Sorry it is just meant to be an imaginary file path. I will update the question to better reflect this. – Aesir Sep 25 '11 at 7:58
Can you post the log you get for your example: it is fine for me. By the way, it's not a good idea to provide the pdftex option for the graphicx package. Unless you are using dvipdfmx, graphicx will pick up automatically on the correct driver. – Joseph Wright Sep 25 '11 at 8:00
Avoid special characters (blank spaces, underscores, …) in the name and the path of the file. For extended file name processing try the grffile package from the oberdiek bundle.
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Thanks for all the help everybody. Loading the package grffile has sorted the problem! – Aesir Sep 25 '11 at 11:56
Normally you shouldn't use white spaces or special characters in path. To use them you can load the package grffile
Alternatively you can use:
\includegraphics[]{\string"path"}
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@diabonas: Thanks. – Marco Daniel Sep 25 '11 at 11:33
Another common reason for the same problem is one may use "draft" mode.
For example,
\documentclass[draft,a4paper,11pt]{report}
Simply removing the draft option will solve the problem.
- | 2015-12-01 14:47:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8846743106842041, "perplexity": 2057.240477075628}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398468233.50/warc/CC-MAIN-20151124205428-00265-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://cs184.eecs.berkeley.edu/sp19/lecture/4-40/transforms | Lecture 4: Transforms (40)
sheaconlon
This example seems to work like the image alignment homework problems that some of us might remember from 16A. The two vectors define three points and three points are needed to uniquely recover the transformation.
john-b-yang
Does it matter which vectors we assign to be "u" and "v"? In other words, if I switched the coordinates for the two so that the matrix looked like [[0, sqrt(2)/2, 1], [-2, sqrt(2)/2, 1]...], would that matter? My guess is that it does.
You must be enrolled in the course to comment | 2019-08-26 03:47:42 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8097900152206421, "perplexity": 1029.3903675926374}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027330962.67/warc/CC-MAIN-20190826022215-20190826044215-00201.warc.gz"} |
https://www.doubtnut.com/question-answer-physics/a-plate-of-area-2m2-is-made-to-move-horizontally-with-a-speed-of-2m-s-by-applying-a-horizontal-tange-643194363 | HomeEnglishClass 11PhysicsChapterFluid Mechanics
A plate of area 2m^(2) is made...
# A plate of area 2m^(2) is made to move horizontally with a speed of 2m//s by applying a horizontal tangential force over the free surface of a liquid. If the depth of the liquid is 1m and the liquid in contact with the bed is stationary. Coefficient of viscosity of liquid is 0.01 poise. Find the tangential force needed to move the plate.
Updated On: 17-04-2022
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Velocity gradient =(2-0)/(1-0)=2 s^(-1) <br> <img src="https://d10lpgp6xz60nq.cloudfront.net/physics_images/ARH_NEET_PHY_OBJ_V01_C13_S01_023_S01.png" width="80%"> <br> From Newton's law of viscous force, <br> |F| =eta A (Delta v)/(Delta y)=(0.01xx10^(-1))(2)(2)=4xx10^(-3)N <br> So, to keep the plate moving a force of 4xx10^(-3) N must be applied. | 2022-05-18 02:44:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20271529257297516, "perplexity": 4867.4502657425555}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662521041.0/warc/CC-MAIN-20220518021247-20220518051247-00698.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/153753-find-all-matrices-b-c-d-will-commute-every-matrix-set-s.html | # Math Help - Find all matrices {(a,b);(c,d)} that will commute with every matrix in set S
1. ## Find all matrices {(a,b);(c,d)} that will commute with every matrix in set S
where set S is the set of all 2x2 matrices
so i set an arbitrary matrix from the set S {(w,x);(y,z)}
so (sorry i dont know the matrix format :< )
[a b][w x] = [w x][a b]
[c d][y z] [y z] [c d]
and got
[(ax+bz) (ay+bw)] = [(xa+yc) (xb+yd)]
[(cx+dz) (cy+dw)] = [(za+wc) (zb+wd)]
and then equating them but im stuck here, lol xD
seems like i forgot algebra! XD any help please?
2. so your matrix must commute with $e_{11}=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ and $e_{12}=\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}.$ thus ...
3. hmm, why should they commute with that 2 specific matrices sir? sorry
4. Well, first you want it to commute with all matrices so you might as well chose some. The more important reason is that it is easy to work with these easy matrices. If you use all four of these matrices with one entry 1, you can build all other matrices as a linear combination of those four. If your matrix commutes with each of the four, then it should commute with all linear combinations of these four. | 2015-05-25 10:46:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8922843933105469, "perplexity": 1205.5256296888015}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207928479.19/warc/CC-MAIN-20150521113208-00086-ip-10-180-206-219.ec2.internal.warc.gz"} |
http://www.mathcaptain.com/number-sense/multiplying-decimals.html | Among four basic operations in arithmetic (Addition, Subtraction, Multiplication and Division), multiplication is one among them and it gives the product of two numbers. There is an inverse relationship between multiplication and division.
Consider the equation $\frac{54}{6}$ = 9 has inverse relationships:
6 $\times$ 9 = 54
9 $\times$ 6 = 54
We usually use an asterisk or cross ( * or x ) symbol to denote multiplication. Multiplication of decimals is normally done by ignoring the decimal points and then put the decimal point in the answer, it should have as many decimal places as the two original numbers combined.
How to Multiply Decimals?
When multiplying decimals follow the steps given below:
1. Multiply the numbers assuming they are whole numbers and do not place the decimal point.
2. As similar to whole numbers starting on the right multiply each digit in the top number by each digit in the bottom number and add the products.
3. Now put the decimal point in the answer and make sure it has as many decimal places as the two original numbers combined (sum).
Multiplying Decimals by Whole Numbers
Multiplying a decimal by a whole number will be same as multiplying two whole numbers:
If there is one digit after the decimal point in the question then there will be one digit after the decimal point in the answer.
If there are two digits after the decimal point in the question then there will be two digits after the decimal point in the answer.
So, in general if there are n decimal places in the decimal number being multiplied then place the decimal point n places from the right of the solution.
Examples of Multiplying Decimals
Solved Examples
Question 1: Multiply 0.04 $\times$ 0.111
Solution:
Ignore the decimal points
04 $\times$ 011
As zeroes add nothing to the value we multiply
4 $\times$ 11 = 44
The above two digits when combined gives 5 decimal digits so the product is 0.00044.
Question 2: Multiply 0.874 $\times$ 401
Solution:
Question 3: Calculate 7.563 $\times$ 8
Solution: | 2018-08-21 17:57:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7563305497169495, "perplexity": 380.46468066260854}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221218391.84/warc/CC-MAIN-20180821171406-20180821191406-00299.warc.gz"} |
https://socratic.org/questions/how-do-you-factor-2a-2-32#604515 | # How do you factor 2a^2-32?
Apr 30, 2018
$2 {a}^{2} - 32 = 2 \left(a - 4\right) \left(a + 4\right)$
#### Explanation:
$2 {a}^{2} - 32 = 2 \left({a}^{2} - 16\right)$ (factoring out 2)
$= 2 \left(a - 4\right) \left(a + 4\right)$
^This is an identity, ${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$
Apr 30, 2018
$2 \left(a + 4\right) \left(a - 4\right)$
#### Explanation:
To factor $2 {a}^{2} - 32$
Begin by factoring out 2 from each term.
$2 \left({a}^{2} - 16\right)$
${a}^{2} - 16$ is the difference of two squares and can be factored as ${a}^{2} - {b}^{2} = \left(a + b\right) \left(a - b\right)$
$2 \left(a + 4\right) \left(a - 4\right)$
Apr 30, 2018
$2 \left(a - 4\right) \left(a + 4\right)$
#### Explanation:
Factorize the expression $\left(2 {a}^{2} - 32\right)$ first, which will give us
$2 \left({a}^{2} - 16\right)$
But $\left({a}^{2} - 16\right)$ is a perfect square expression. Therefore it can further be factorized to
$\left({a}^{2} - 16\right)$
=${\left(a - 16\right)}^{2}$
=$\left(a - 4\right) \left(a + 4\right)$
Hence joining all them will sum up to
$\therefore 2 \left(a - 4\right) \left(a + 4\right)$ | 2023-03-23 02:12:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 18, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8299397826194763, "perplexity": 2478.1110662449487}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296944606.5/warc/CC-MAIN-20230323003026-20230323033026-00744.warc.gz"} |
https://www.acthreceptor.com/2020/10/29/8061/ | # Thiol agents can influence the equilibrium amongst these two states (Calero and Calvo, 2008). As
Thiol agents can influence the equilibrium amongst these two states (Calero and Calvo, 2008). As a result, within a related manner NO can react generating an Snitrosylation of thiol groups at Cysloop C177 and C191 and, in turn, this covalent modification induces protein Curdlan MedChemExpress structural rearrangements that impact on GABA binding and channel gating (Chang and Weiss, 2002). The leftward shift along with the concomitant improve in the maximal existing values, observed in D curves for GABA inside the presence of NO, are compatible with this hypothesis. This interpretation is also constant with all the effects of reducing agents that stop Cysloop formation and behave as GABAr1 receptor potentiators (Calero and Calvo, 2008). Interestingly, preceding research on NMDA receptors showed that redox modulation induced by each lowering thiol agents and NOinduced Snitrosylation is mediated through exactly the same extracellular cysteines (Lipton et al., 2002). In addition to NMDA receptors, ryanodine receptors, TRP channels and many other membranesignalling proteins are physiological targets for cysteine Snitrosylation (Eu et al., 2000; Lipton et al., 2002; Yoshida et al., 2006). Nonetheless, the modulation of Cysloop receptors by Snitrosylation was still not substantiated. It was shown that the redox modulation of Cysloop receptors, which includes the GABAC receptors, is ordinarily reversible (Amato et al., 1999; Pan et al., 2000; Calero and Calvo, 2008). Similarly, we found that NO modulation of GABAr1 receptors is conveniently reversible. As a result, the present final results also suggest that other redoxsensitive amino acid residues in the r1 subunits, including tryptophane, methionine and tyrosine, are not involved, primarily since these residues are typically modified by reactive nitrogen species in an irreversibly manner (e.g. by peroxynitrite, which might be developed by the reaction of NO with superoxide). Nitrosothiols are ordinarily incredibly labile within the presence of lowering reagents, but our experiments showed that NO effects on GABAr1 receptors can also be washed out inside the absence of reducing agents. A attainable explanation is that chemical modification from the extracellular redox internet site (the disulfide bond that types the Cysloop) produces a transient conformational adjust in the receptor that, in the absence of NO, rapidly relaxes to a lower energy state by excluding the NO group. This description is compatible together with the actions of MTSEA on GABAr1 receptors. Commonly, the effects of this cysteinespecific reagent call for the presence of decreasing agents in order to be washed out (Xu and Akabas, 1993; Choi et al., 2000). In contrast, we identified here that MTSEA applications produced a quick potentiation of the GABAr1 receptor responses that spontaneously disappeared in the course of bath perfusion with a typical Ringer’s answer.Pharmacological and physiological relevance from the modulation of GABAC receptors by NOGABAC receptors mediate several modes of inhibitory actions within the retina (Lukasiewicz et al., 2004). They are highly expressed in retinal bipolar cells (Koulen et al., 1998) and play an essential function in the control of axon terminal excitability by mediating reciprocal synapses with Aa861 Inhibitors targets amacrine cells (Matthews et al., 1994; Dong and Werblin, 1998; Hartveit, 1999).Nitric oxide and GABAC receptorsBJPGABAC receptors also mediate tonic inhibitory currents, which is usually persistently activated by low concentrations of ambient GABA, locally controlled by GABA transporters positioned on amacrine cells (Hull et al., 2006; Jones and Palmer,. | 2020-11-25 11:22:40 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8043150305747986, "perplexity": 7463.142636538132}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141182776.11/warc/CC-MAIN-20201125100409-20201125130409-00404.warc.gz"} |
https://www.physicsforums.com/threads/the-distance-between-2-spheres-in-pendulum.658658/ | # The distance between 2 spheres in pendulum
1. Dec 13, 2012
### Dr.Phy
Hi guys,first nice to meet you im new here.Im working on a physic homework and i.ve already done it but i dont know why the x=2lsinx ( x=distance between 2 spheres in pendulum).
(P.S sorry for my bad english,im from Germany). Is that from a trig identity? I think i've understand it but if someone can tell mi with a pic it will be very helpful.
Last edited: Dec 13, 2012
2. Dec 13, 2012
### WillemBouwer
just ad a pic of your problem and also your attemot at a solution and we can figure some solution out | 2018-02-23 12:42:50 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8078637719154358, "perplexity": 2694.9760094480794}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891814700.55/warc/CC-MAIN-20180223115053-20180223135053-00112.warc.gz"} |
https://www.groundai.com/project/charge-and-spin-transport-at-the-quantum-hall-edge-of-graphene/ | Charge and Spin Transport at the Quantum Hall Edge of Graphene
# Charge and Spin Transport at the Quantum Hall Edge of Graphene
Dmitry A. Abanin, Patrick A. Lee, Leonid S. Levitov Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139
###### Abstract
Landau level bending near the edge of graphene, described using 2d Dirac equation, provides a microscopic framework for understanding the quantum Hall Effect (QHE) in this material. We review properties of the QHE edge states in graphene, with emphasis on the novel phenomena that arise due to Dirac character of electronic states. A method of mapping out the dispersion of edge states using scanning tunneling probes is proposed. The Zeeman splitting of Landau levels is shown to create a particularly interesting situation around the Dirac point, where it gives rise to counter-circulating modes with opposite spin. These chiral spin modes lead to a rich variety of spin transport phenomena, including spin Hall effect, spin filtering and injection, and electric detection of spin current. The estimated Zeeman spin gap, enhanced by exchange, of a few hundred Kelvin, makes graphene an attractive system for spintronics. Comparison to recent transport measurements near is presented.
## I Introduction
Isolation and gating Novoselov04 () of graphene, a monolayer of graphite, has enabled the observation of interesting transport effects, resulting from Dirac fermion-like character of excitations. In particular, graphene hosts an integer quantum Hall effect (QHE) with unusual plateau structure Novoselov05 (); Zhang05 (). It was found that the QHE plateaus in monolayer are arranged symmetrically around the neutrality point, occurring at filling factors which are half-integer multiples of four, which is the combined spin and valley degeneracy of graphite. This form of QHE came to be known as the anomalous or half-integer QHE.
The simplest framework allowing to understand this behavior of QHE is provided by the structure of Landau levels (LL) of a 2d massless Dirac equation 00footnotemark: 0 which has the particle-hole symmetric spectrum 00footnotetext: The result (1) follows from the well-known representation of the Schrödinger-Pauli Hamiltonian as a square of a massless Dirac Hamiltonian, . The Dirac spectrum (1) is then obtained by taking a square root of the nonrelativistic Landau level spectrum.
En=±ε0√|n|,ε0=ℏv0√2eB/ℏc (1)
with the sign plus for positive , and minus for negative . Here is the magnetic field and is the velocity at the graphite Dirac point or . Due to spin and valley degeneracy (we shall discuss the role of Zeeman splitting below), each of the levels (1) contributes a step of to the quantized Hall conductivity. The particle-hole symmetry of the spectrum (1), with the level positioned at , suggests that the QHE plateaus must occur at , which is indeed what is observed in experiment Novoselov05 (); Zhang05 (). The anomalous QHE in graphene can be understood in a more fundamental way in terms of a quantum anomaly of the zeroth Landau level Gusynin05 (). The special character of the monolayer spectrum (1) is underscored by the difference between the QHE properties observed in graphene monolayer and bilayer systems Novoselov06 ().
One of the most dramatic consequences of the Dirac LL spectrum (1) is the appearance of the new energy scale . The square root dependence leads to a much larger level spacing than that for electrons with the quadratic dispersion conventional for semiconductors. For typical magnetic field of , the separation of the lowest LL () is quite large, , which enables QHE to persist up to room temperature GeimKimAPS2007 ().
To gain insight into the microscopic origin of the anomalous QHE, it is useful to develop the edge-states approach, which provides an intuitive and simple picture of the conventional QHE Halperin (). The edge states for graphene were studied using a numerical solution of the tight-binding model Peres () and also with the help of the Dirac equation Abanin06a (); Brey06 (). It was found that the energy levels (1), valley-degenerate in the graphene bulk, are split near the edge due to valley mixing at the boundary. Interestingly, it turns out that the structure and dispersion of the edge states depend on particular crystallographic orientation of the edge. For the so-called armchair edge, a simple particle-hole symmetric splitting was found Abanin06a (). Particle-like and hole-like states have different sign of energy dispersion, giving rise to counter-propagating modes with opposite chirality. A somewhat more complicated situation occurs near the zigzag edge, where dispersing edge states coexist with an additional dispersionless surface state Peres (). Despite these differences, the armchair, zigzag and other edges have same numbers of dispersing edge states of both chiralities. As we discuss below in Sec.IIII, this ensures the universal half-integer character of QHE in graphene.
One unique aspect of QHE in graphene is that its electronic states, owing to the monolayer character of this material, are fully exposed and, similar to the surface states of 3d materials, can be investigated by scanning tunneling microscopy (STM) probes with atomic resolution. Moreover, some useful information can even be obtained by imaging the top layer of 3d graphite, as demonstrated by recent STM studies of Dirac Landau levels Matsui05 (); Niimi06b () and of electron states near atomically thin edges Kobayashi06 (); Niimi06a (). In Sec.IIII we discuss new possibilities for scanning experiments that arise in graphene. The characteristic spatial scale of the states in a Landau level, given by the magnetic length , is about for the field of . Being large compared to the STM spatial resolution, it allows to image individual electronic states with sub- resolution and, in particular, to study Landau level bending near graphene edge. This bending in fact mimics the edge states momentum dispersion, due to the position-momentum duality of the Landau levels. As we shall see, the STM technique has sufficient resolution to map out the dispersion of QHE edge states.
Another novel feature of graphene is the simultaneous presence of the QHE edge modes of opposite chiralities, propagating in the opposite directions. Being particle-like and hole-like, they occur at the energies and , respectively. For electron density detuned from the neutrality point, , only one of the chiralities contributes to transport. However, as we shall see, near the states of both chiralities can participate in transport, leading to rather unusual transport properties. In particular, in the presence of Zeeman spin splitting of Landau levels, the state features an energy gap in the bulk and, simultaneously, a pair of edge states of opposite chirality and opposite spin polarization Abanin06a (); Fertig06 (). These states carry spin-up and spin-down electrons in the opposite directions along graphene edge, exhibiting quantized spin Hall effect but no charge Hall effect, owing to the particle-hole symmetry at . As we discuss in Sec.IIIIII, these spin-polarized chiral edge states exhibit interesting spin transport phenomena, such as spin filtering and spin injection, whereas the spin Hall effect provides a natural tool for the detection of spin current.
Interestingly, the counter-propagating edge states manifest themselves directly in charge transport. As we discuss in Sec.IVIV, near longitudinal resistivity remains finite, , while the Hall effect, which is absent at the particle-hole symmetry point , appears at nonzero due to conductivity in the bulk. This leads to Hall resistance changing sign at without exhibiting a clear plateau. The bulk conductivity short-circuits the edge transport and suppresses longitudinal resistivity, leading to a prominent peak in near and a plateau in These predictions, as well as the behavior of resistance fluctuations, which are enhanced near zero , are in agreement with experiment Zhang06 (); Abanin07 ().
Looking beyond graphene, interesting massless Dirac fermion states have been predicted a while ago Volkov85 () at interfaces of narrow-gapped HgTe and PbTe semiconductors. In Ref.Volkov85 (), which anticipated many of the features of electronic states in graphene, 2d Dirac states occur in a band-inverted heterojunction plane due to spin orbit interaction in 3d bulk, in the absence of magnetic field.
We note also that there have been interesting predictions of quantized spin Hall effect in certain 2d insulators KaneMele (); Bernevig06 (). In particular, the proposal of Ref.Bernevig06 () can be viewed as a 2d version of the 3d situation discussed in Ref.Volkov85 (). In these schemes an energy gap forms in the bulk due to spin-orbit even in the absence of magnetic field, similar to Ref.Volkov85 (). At the same time, counter-propagating modes carrying opposite spins appear on the edge, which are responsible for the quantized spin Hall current. As was emphasized in Ref.Fu06 (), there are general symmetry requirements, rooted in the time-reversal symmetry, protecting counter-propagating gapless excitations at an insulator boundary. In particular, certain invariants must exist, which are realized as spin component in the case of graphene KaneMele (); Abanin06a (), and were linked to supersymmetry for heterojunction systems Volkov85 ().
We note that the nature of the edge states of Refs.KaneMele (); Bernevig06 () is essentially identical to ours. The only difference, apart from different size of the energy gap, is that Rashba spin-orbit may cause backscattering in our case (see Sec.III) but not in the situation of Refs.Volkov85 (); KaneMele (); Bernevig06 (), while magnetic impurities will cause backscattering in both cases. In particular, our discussion in Sec.IV of and and our proposals to operate and detect spin current can serve as diagnostic tools should these schemes become realized experimentally.
The rest of the paper is organized as follows. In Sec.II we introduce the edge states using the Dirac equation framework. We focus on the two main edge types, armchair and zigzag, however we emphasize the generic aspects that are applicable to other edges. Then we discuss the anomalous QH effect and the possibility of imaging the edge states with scanning tunneling probes. In Sec.III we focus on spin-polarized chiral edge states and related spin transport phenomena. We also comment on spin relaxation mechanisms and present estimates of the spin relaxation time. In Sec.IV we introduce a transport model which accounts for both edge and bulk transport. This allows us to connect the chiral spin-polarized edge picture with recent transport measurements near the neutrality point Zhang06 (); Abanin07 ().
## Ii II. Dirac QHE edge states
Here we analyze electron states near zigzag and armchair edges (Fig.1a), the two most common graphene edge types, using massless Dirac model diVincenzo (). This exercise, which amounts to setting the boundary conditions for the Dirac spinor and solving an appropriate 1d eigenvalue problem, provides a fully microscopic picture of graphene QHE. Below we use this approach to illustrate the interplay between the QHE edge states and surface states for the zigzag edge, and discuss the possibility of imaging the edge states with scanning tunneling probes.
Let us first recall how low-energy graphene excitations are obtained in the tight-binding model diVincenzo () near the Dirac valleys and , located at the two non-equivalent Brillouin zone corners (see Fig.1(b)). There are two linearly independent zero-energy Bloch functions for each of the points , , each residing only on one sublattice ( or ) and vanishing on the other sublattice. Our choice of Bloch functions for the valley is shown in Fig.1(c). The Bloch functions for the valleys and are related by complex conjugation.
The wave function of low-lying excitations near and , is written as a superposition of these four zero-energy Bloch functions multiplied by slowly varying envelope functions , , , , with and being the wave function amplitudes on the sublattice and . (Our choice of the signs for the valley is convenient for treating an armchair boundary, as we shall see below.) The envelope functions , , and , describe excitations near and , respectively. The effective low-energy Hamiltonian, obtained by keeping only lowest-order gradients of and , takes the massless Dirac form 111We note that the Hamiltonian used in our paper Abanin06a () is related to the Hamiltonian (2) by variable transformation , .:
(2)
where , .
Landau levels in an external -field, described with the gauge , can be obtained for the states with the dependence from 1d Hamiltonians
HK,K′=iε0√2[0±∂y+(y−y∗)±∂y−(y−y∗)0], (3)
where and . Here and are measured in the units of and , respectively. The spectrum of , Eq.(3), yields the Dirac Landau levels, Eq.(1), where the eigenstates for the two valleys are given by
(uK,n,vK,n)=A(cnφn−1(y−y∗),φn(y−y∗)), (4) (uK′,n,vK′,n)=A(φn(y−y∗),cnφn−1(y−y∗)). (5)
Here are the eigenfunctions of the magnetic oscillator, , the normalization factor equals 1 for and for , with and . Note that zeroth LL states reside solely on sublattice for valley and on sublattice for valley 222 This property is specific for the zeroth LL, making the splitting of the LL due to Coulomb interaction distinctly different from that of other LLs (see Abanin06b (); Goerbig06 ())..
We now analyze how LL spectrum is modified near the armchair edge. We consider graphene sheet in the halfplane with an armchair edge parallel to the axis (see Fig.1(a)). Energy levels near the edge are determined from the Dirac eigenvalue equations , where , and are given by Eq.(3). To analyze this eigenvalue problem, we exclude components and consider eigenvalue equations with spectral parameter for components:
12(−∂2y+(y−y∗)2+1)uK=λuK,12(−∂2y+(y−y∗)2−1)uK′=λuK′, (6)
The boundary conditions for Eqs.(6) can be obtained from the tight-binding model, which is valid up to the very last row near , by setting the wave function equal zero at the boundary. Since the armchair edge has lattice sites of both and type (see Fig.1(a)), the wave function on both sublattices should vanish at the edge. In terms of the envelope functions , , taken at , this condition translates into
uK=uK′,vK=vK′. (7)
We obtain a pair of differential equations (6) on the semi-axis , coupled at the boundary via Eq.(7). To simplify this problem, let us map Eq.(6) for onto the positive semi-axis , by , while keeping on the negative semi-axis , and treat it as an eigenvalue problem in the domain with the wavefunction given by at negative and by at positive . The first boundary condition, Eq.(7), then means that the wavefunction is continuous at , while the second condition implies continuity of the derivative . (This can be seen by expressing in terms of using Eqs.(3).) Thus we obtain a 1d Schrödinger problem in the potential Abanin06a ()
V(y)=12(|y|+y∗)2−12sgn(y), (8)
defined on the entire axis. After finding the spectrum numerically, we obtain the energy levels of the Dirac fermions as
E(px)=±ε0√λ(y∗),y∗=−px, (9)
whereby the particle-hole symmetry is restored due to the two possible signs of . The dispersion (9) is illustrated in Fig.2a. We note that the LL double valley degeneracy in the bulk is lifted at the boundary.
The particle-hole symmetric edge states spectrum in Fig.2a instantly explains the “half-integer” Hall quantization in graphene. Indeed, for any electron density with integer in the bulk there is an odd number of the edge modes crossing the Fermi level, which means that the Hall conductivity is quantized as , where the factor two accounts for spin degeracy.
Spin degeracy of the Landau levels is lifted by the Zeeman interaction, which is substantial in graphene,
EZ=gμBB≈50K,g≈2, (10)
for (compare to in GaAs quantum wells). Zeeman-split edge states, depicted in Fig.2b, have interesting characteristics for electron density near neutrality, . At this density the state in the bulk is spin-polarized, with the Zeeman gap further enhanced by exchange (see below). The two branches of counterpropagating edge states near , carrying opposite spin, have interesting properties that will be discussed in more detail in Sec.III.
We now analyze the zigzag edge, which even at hosts a band of dispersionless zero-energy states bound to the edge Fujita (). We shall refer to these states as surface states, to distinguish them from the dispersing QHE edge states. As we shall see, the surface states contribute to the splitting of LL near the zigzag edge, in agreement with the tight-binding calculations Peres ().
We consider graphene sheet in the half-plane , with its first row consisting of atoms (see Fig.1a). For the states with the dependence , with , , from (2) we obtain 1d Hamiltonians
HK,K′=ε0√2[0∂x±(x−x∗)−∂x±(x−x∗)0], (11)
where . Similarly to the armchair case, the spectrum can be found from the eigenvalue equation , where , which should be supplemented with the boundary conditions. For our zigzag edge the wavefunction have must vanish on all sites at . For that both envelope functions have to vanish at the boundary,
uK=0,uK′=0. (12)
Excluding components, we obtain two separate eigenvalue problems for the spectral parameter ,
12(−∂2x+(x−x∗)2+1)uK=λuK,12(−∂2x+(x−x∗)2−1)uK′=λuK′, (13)
where both and satisfy the hard wall boundary conditions (12). The amplitudes on the sublattice can be expressed via ampliudes on the sublattice and eigenenergy ,
vK=(ε0/√2E)(−∂x+(x−x∗))uK,vK′=(ε0/√2E)(−∂x−(x−x∗))uK′. (14)
The eigenvalue problems (13) with the hard-wall boundary conditions (12) are familiar from the theory of edge states in the conventional QHE Halperin (), and their spectrum can be found numerically. The Dirac fermion energy dispersion is shown in Fig.3.
The behavior of LL’s is similar to the armchair case: there are two branches of the edge states, one for each valley, degenerate in the bulk, , which split near the edge. The zeroth LL, however, coexists with the surface state, which makes its behavior rather peculiar and different for the two valleys.
In the valley, which we discuss first, the zeroth LL states reside solely on the sublattice, see Eq.(4), and therefore automatically satisfy the boundary condition . Thus there are zero-energy states for arbitrary values of , of the form . Let us consider the states with far outside the graphene half-plane, . Not too far from the boundary, such states can be approximated by an exponential
vK(0
which is identical to surface state wave function Fujita (). Thus the zeroth LL for valley near the edge transforms into the surface mode. Being dispersionless, this mode does not contribute to the edge transport. The edge state spectrum for the valley is displayed in Fig.3a.
Now let us consider the zeroth LL for the valley . For , we approximate the ground state of the oscillator (13) with the hard-wall boundary condition as
uK′(x)=ψx∗(x)≈φ0(x−x∗)−φ0(x+x∗). (16)
The ground state energy is then approximated by
λ0(x∗)≈⟨h⟩, (17)
where is the effective Hamiltonian for component, Eq.(13), and denotes averaging over the normalized wave function (16).
Evaluating for , when the state (16) has unit norm with exponential accuracy, we obtain
λ0(x∗)≈x∗π−1/2e−x2∗. (18)
From the relation , we find the energies for the two branches of dispersing edge states
E±(x∗)≈±(2x∗)1/2π−1/4e−x2∗/2ε0. (19)
Plugging this expression in Eq.(14), we obtain the wave function on sublattice for these two branches,
vK′=±x1/2∗e−x∗x. (20)
which is again the surface state wave function (compare to Eq.(15)). We therefore conclude that for the valley the zeroth Landau level and the surface state mix giving rise to two dispersing edge modes. This is in agreement with the spectrum displayed in Fig.3b. We see that, although the Dirac model is applicable only in a small part of the Brilloin zone, near points and , it provides a description of the states at the zigzag edge, including the surface state, which is in agreement with the results of the tight-binding model of Ref.Peres (). The surface mode in the vicinity of and is given by Eqs.(15),(20).
Interestingly, the and sites contribute equally to the splitting of the zeroth LL, . This is somewhat counterintuitive, since this LL is solely on the sublattice in the bulk, while the surface mode is solely on the sublattice . This equal participation property can be understood as follows. The spinor states with , given by Eqs.(16),(20), are eigenstates of the Dirac Hamiltonian with the boundary condition (12), with the energies . Thus these states are orthogonal, which implies that the integrals of and are equal. We further note that the integral of the square of the component of our edge state wave function (20) over indeed equals one, in agreement with our choice of normalization on the sublattice in Eq.(16).
To sum up, for the zigzag edge, the zeroth LL gives rise to two dispersing edge states for one of the valleys, while for the other valley the zeroth LL morphs into the dispersionless surface mode which does not contribute to the edge current. Therefore, despite the presence of the surface mode, the number of dispersing QHE edge states with and for the zigzag boundary is the same as for the armchair boundary, giving rise to “half-integer” quantization of Hall current.
Finally, we briefly discuss how the edge states in graphene can be investigated using the STM technique Matsui05 (); Niimi06b (); Kobayashi06 (); Niimi06a (). Due to the Landau level momentum-position duality relation, , the edge state momentum dispersion shown in Figs.2,3 translates into the excitation energy dependence on the distance from the edge. The characteristic scale for the latter is set by the magnetic length , which for typical fields is about 50-80 times greater than the spatial resolution of STM instruments on graphite surface. This makes STM technique particularly convenient for this kind of studies.
A link between the edge states dispersion and the position-dependent tunneling spectroscopy can be established as follows. We shall use the solutions for the edge state wave function given above to calculate the local density of states (LDOS) near the zigzag edge (other edge types can be dealt with similarly). For each of the graphene sublattices LDOS is given by
ρA(E,x)=∑α|uα(x)|2δ(E−Eα),ρB(E,x)=∑α|vα(x)|2δ(E−Eα), (21)
where is the distance from the edge, and denotes the set of eigenstates of the and Hamiltonians (11) with the hard-wall boundary condition (12).
Using the eigenfunctions , and the energies found from Eqs.(13),(14) as discussed above, we obtain LDOS for the and sublattices which is displayed in Fig.4. We see that the position-independent Landau level bands, dominating LDOS far from the edge, bend away from near the edge. This bending mimics the edge states momentum dispersion shown in Fig.3. Note, however, that LDOS is nonzero only for , whereas the edge state momentum can be both positive and negative. The spatial width of the bending bands is determined by the width of the eigenfunctions , , which is of the magnetic length scale.
## Iii III. Spin-polarized chiral edge states and spin transport.
As we noted above, at the neutrality point graphene hosts gapless spin-polarized edge states (see Fig.2(b)). The Zeeman energy gap in the bulk, Eq.(10), is enhanced by the Coulomb interaction. A Hartree-Fock estimate of this enhancement Abanin06a () gives
Δ=π1/2e22κℏv0(1−α)ε0≈0.456⋅(1−α)ε0, (22)
where is RPA screening function, and the parameter describes relative strength of Coulomb and exchange correlations. Assuming , i.e. ignoring correlations of electrons with opposite spin, we obtain a spin gap for . Taking into account the substrate dielectric constant, , changes the result only slightly ( for ). This approximation, while pointing at a correct order of magnitude of a few hundred Kelvin, probably somewhat overestimates the spin gap since it ignores correlations and disorder effects.
The chiral spin-polarized edge states offer a unique setting to study spin transport. In particular, the spin-split state may be used to generate and detect spin-polarized currents. This spin transport regime seems attractive due to the large bulk gap and high stability of the edge states. Moreover, increased quality of samples should allow existence of spin polarized edge states even at relatively low magnetic fields.
Since the purpose of this section is mostly illustrative, we will keep our discussion as simple as possible. In particular, we shall ignore transport in the bulk, leaving the discussion of its role for Sec.IV. We also first neglect spin flip backscattering between edge states within one edge. Estimates of the spin flip rate wil be given below, Eq.(24). A general approach, based on the Landauer-Büttiker formalism Buttiker (), which can be used to calculate spin and charge currents at the edge for any configuration of current and voltage leads, was presented in Ref.Abanin06a ().
In this approach, transport is described by a scattering matrix Buttiker (), with the edge states playing the role of scattering channels, and the reservoirs supplying in-states and absorbing out-states. Current in each mode is described by the relation , where is the reservoir chemical potential for given spin projection. We consider the Hall bar geometry with four contacts 1-4 (see Fig.5), where the contacts 1 and 4 serve as current source and drain. For these two contacts we do not assume spin mixing, so that the injected and drained current may be spin polarized. The contacts 2, 3 are voltage probes, which means that they do not drain current from the system. Furthermore, we assume that the probes provide full spin mixing, i.e. chemical potentials of outgoing spin-up and spin-down electrons are equal.
The simplest situation arises when unpolarized current is injected through contact 1. Then the up- and down-spins spatially separate in a symmetric way, flowing along the opposite edges of the bar. This can be interpreted as circulating spin current, and described as spin-Hall effect with quantized spin conductance . No electric voltage will be induced between the voltage probes 2, 3 in this case (zero charge-Hall effect).
This device can be used as a detector of spin polarized current, made possible by the reciprocal of the spin Hall effect, in which the electric Hall voltage is directly related to spin rather than charge current. Suppose the up-spin and down-spin electrons, injected through contact 1, have unequal chemical potentials, . Then the currents flowing into the probes 2 and 3, , after equilibration and spin mixing in the probes, induce voltages . The resulting Hall voltage is directly proportional to spin current. At the same time, an unpolarized current (for which ) flows symmetrically in the upper and lower edges without generating Hall voltage.
Spin transport at also allows to realize spin filter. Suppose that the upper and lower edges of the device in Fig.5 are made asymmetric, which can be achieved, for example, simply by removing probe 3. Then we inject unpolarized current through contact 1. The injected current will be distributed equally between the upper and lower edges in cross section A. In cross section B, however, the net current will be spin polarized due to spin mixing in probe 2. The down-spin current reaching the drain in the upper edge equals while the up-spin current in the lower edge is . Therefore, the total drained current becomes spin polarized. The spin polarized current can be fed into another system (see Fig.5), where it can be detected using Hall probes 5 and 6 as discussed above.
More complicated circuits can be assembled which generate spin currents and detect them elsewhere. Note that the important principle is that as long as backscattering is not allowed, the edge current can travel long distances and the circuit is nonlocal, just as in the integer QHE Buttiker (). In this case the current-voltage relationship is obtained by solving the circuit equations as described in Ref.Abanin06a (). The spatial scale of nonlocality is controlled by spin relaxation which can be due to spin-orbit interaction or due to magnetic impurities near graphene edge.
For simplicity, here we limit the discussion to the effects of spin-orbit. There are two main spin-orbit terms in the graphene Hamiltonian KaneMele (); Min06 (), the so-called intrinsic and Rashba interaction, given by
HSO=λSOσzτzsz,HR=λR(σxτzsy−σysx), (23)
where Pauli matrices act in the sublattice space (Dirac spinor), while act in the valley space, and represent physical spin. Estimates from band calculations Min06 () give and a negligibly small .
To estimate the backscattering rate due to the spin-orbit interaction (23), we note that for an ideal atomically sharp edge the spin-orbit would couple the left and right states with the same momentum, opening a minigap at branch crossing: . However, this momentum-conserving interaction alone cannot backscatter edge states, and we need to take disorder into account. Edges of graphite monolayers have been imaged using STM probes Kobayashi06 (); Niimi06a (), where it was found that typically edge disorder can be viewed as patches of missing atoms of characteristic size . Taking into account the left-right branch mixing by spin-orbit , characterized by small mixing ratio of away from branch crossing, we obtain an estimate of the backscattering mean free path:
ℓ(ε)∼(ε/λR)2(ℓB/d)2d,|ε|≳λR, (24)
which gives for typical . The factor accounts for the magnetic field dependence of disorder matrix elements.
The quadratic energy dependence in (24), with spin flip rate having a sharp peak near branch crossing, suggests Abanin06a () the possibility to control backscattering using local gate. By tuning local chemical potential to and from the branch crossing, where the spin flip rate has a sharp peak, Eq.(24), we can induce or suppress backscattering in a controlled way. Spin filtering is achieved by controlling local gates on opposite sides of the Hall bar asymmetrically.
## Iv IV. Edge and bulk transport at ν=0.
Spin flip backscattering (24) can be incorporated in the edge transport model, described by coupled equations for particle density in the two spin-polarized modes:
∂tn1+∂xφ1=γ(φ2−φ1)∂tn2−∂xφ2=γ(φ1−φ2),ni=νiφi, (25)
where is the backscattering mean free path (24) taken for at the Fermi level, and are compressibilities of the modes. (For brevity, we use 1 and 2 instead of and .) In writing Eqs.(25) we implicitly assume that fast energy relaxation maintains local equilibrium of each of the modes, which is consistent with metallic temperature dependence of transport coefficients Abanin07 ().
In a stationary state, Eqs.(25) have an integral which expresses current conservation at the edge. [In this section we use the units of .] The general solution in the stationary current-carrying state is
φ1,2(x)=φ∗1,2−Ex,E=γ~I (26)
Taking into account that is the current in one edge, we calculate the total current as
I=2~I=2γE (27)
To describe the longitudinal resistance in the four-terminsl geometry, one must add potential drop on voltage probes Abanin07 (), which gives , where is the distance between the probes. Comparing to the data for at we estimate Abanin07 () . This mean free path value, which is relatively small on the scale predicted by Eq.(24), can be explained if spin flip processes are dominated by nonintrinsic effects, such as magnetic impurities localized near the edge.
It is crucial that the edge transport model (25) treats both edges of a Hall bar in an identical way, thus predicting zero Hall effect. In order to understand the observed density dependence of Hall coefficient Zhang06 (); Abanin07 (), which changes sign smoothly at without exhibiting a plateau, and of which has a sharp peak at , we need to incorporate transport in the bulk in our model. In the full edge+bulk model, the density dependence of transport coefficients arises from bulk currents short-circuiting edge currents away from . This explains, as we shall now see, the Hall effect, the peak of , the resistance fluctuations near , as well as the behavior of and .
We describe the transport problem in the bulk by the current-field relation, separately for each spin projection:
ji=−^σi∇ψi,^σi=(σ(i)xxσ(i)xy−σ(i)xyσ(i)xx),i=1,2,
where are electrochemical potentials for two spin states. We assume that the bulk conductivities , as a function of density , are peaked at the spin-split Landau levels. For simplicity, here we ignore possible valley splitting, in which case the spin up and down Landau levels occur at around the Dirac point. As a simplest model, below we use Gaussians
σ(1)xx(ν)=e−A(ν−1)2,σ(2)xx(ν)=e−A(ν+1)2 (28)
with the parameter describing the width of the levels. The Hall conductivities exhibit plateaus on either side of the peak in . The dependence of on can be modeled with the help of the semicircle relation which often provides a good description of conventional QHE systems semicircle_relation ().
The condition of charge continuity, , gives a 2d Laplace’s equation for the potentials, . This equation must be solved together with the boundary conditions phenomenologically describing bulk-edge coupling:
n.ji=g(ψi−φi) (29)
where is a normal vector to the boundary, and represents the edge-bulk leakage current density.
Although a general solution of this problem can be given with the help of Fourier method, here we consider only the case when the potentials are varying slowly on the scale of the bar width , which will suffice for our analysis of a homogeneous current flow. In this case, linearizing in the direction transverse to the bar, we can write Eqs.(29) for both edges of the bar as
−σxy∂xψi+σxx(ψi′−ψi)/w=g(ψi−φi)σxy∂xψi′+σxx(ψi−ψi′)/w=g(ψi′−φi′), (30)
, where the primed and unprimed quantities denote variables at opposite edges of the bar.
Equations for the edge variables are obtained by adding the bulk-edge leakage term to Eqs.(25), giving
∂xφ1=γ(φ2−φ1)+g(ψ1−φ1),−∂xφ2=γ(φ1−φ2)+g(ψ2−φ2), (31)
along with a similar pair of equations for , at the opposite edge.
The solution of these eight equations, describing uniform current, is of the form , , etc., with the same linear part for all quantities. Using the algebraic structure of this linear system and the symmetry between the edges, we reduce the number of equations from eight to two. First, it is convenient to express the parameters through using Eqs.(31), which gives
φ∗1=γ+g2γ+gψ∗1+γ2γ+gψ∗2+E2γ+gφ∗2=γ+g2γ+gψ∗2+γ2γ+gψ∗1−E2γ+g (32)
Writing similar equations for the variables at the opposite edge to express , through , , and substituting the result in Eqs.(30), we obtain four equations for and which have the form
−~σ(1)xywE=σ(1)xx(ψ∗1′−ψ∗1)+λ(ψ∗2−ψ∗1)~σ(1)xywE=σ(1)xx(ψ∗1−ψ∗1′)+λ(ψ∗2′−ψ∗1′)−~σ(2)xywE=σ(2)xx(ψ∗2′−ψ∗2)+λ(ψ∗1−ψ∗2)~σ(2)xywE=σ(2)xx(ψ∗2−ψ∗2′)+λ(ψ∗1′−ψ∗2′) (33)
where the coefficients in this linear system are defined as
~σ(1,2)xy=σ(1,2)xy±g2γ+g,λ=wγg2γ+g. (34)
The quantities represent the sum of the bulk and edge contributions to Hall conductivity for each spin.
Symmetry between the edges allows to further reduce the number of independent variables. For that we add the first two equations to obtain . Also we note that all potentials can be changed by the same constant that can be chosen so that the new quantities and satisfy . After that Eqs.(33) yield
~σ(1)xywE=2σ(1)xxψ∗1−λ(ψ∗2−ψ∗1)~σ(2)xywE=2σ(2)xxψ∗2−λ(ψ∗1−ψ∗2) (35)
These two equations can be solved to find .
Now we can find the current as a sum of the edge and bulk contributions, , where
Iedge=φ1−φ2+φ2′−φ1′=2(φ∗1−φ∗2)
and
Ibulk=σ(1)xy(ψ1−ψ1′)+σ(1)xxwE+σ(2)xy(ψ2−ψ2′)+σ(2)xxwE
After expressing through with the help of Eqs.(32) and using the solution of Eqs.(35), we obtain a relation , where
2~γ=42γ+g+wρ(1)xx+wρ(2)xx−λw(~σ(1)xy/σ(1)xx−~σ(2)xy/σ(2)xx)22+λ/σ(1)xx+λ/σ(2)xx. (36)
The quantities are defined as . The quantity , Eq.(36), replaces in Eq.(27). In the absence of bulk conductivity, , we recover the result for pure edge transport, .
The Hall voltage can be calculated from this solution as , where , are variables at opposite edges. We obtain , where
ξ=2w~σ(1)xy(λ+σ(2)xx)+~σ(2)xy(λ+σ(1)xx)2σ(1)xxσ(2)xx+λσ(2)xx+λσ(1)xx. (37)
This quantity vanishes at , since and at this point due to particle-hole symmetry.
Transport coefficients, obtained from this model for typical parameter values, are displayed in Fig.6 which reproduces many of the key features of the data (see Fig.1 in Ref.Abanin07 ()). In particular, the peak in is due to edge transport near . The suppression of at finite is due to the bulk conductivity short-circuiting the edge transport. The bulk and edge contributions to transport can be discerned from the double peak structure in in Fig.6. The peaks correspond to the bulk Landau level contributions, Eq.(28), whereas the part of between the peaks, exceeding the superposition of two Gaussians, Eq.(28), is the edge contribution. The Hall resistance is nonzero due to imbalance in for opposite spin polarizations away from . Interestingly, in Fig.6 exhibits no plateau, while calculated from and displays an under-developed plateau-like feature. Overall, this behavior resembles that of the experimentally measured transport coefficients Zhang06 (); Abanin07 ().
Another notable feature of the measured and is enhanced fluctuations near zero . These fluctuations are found to be strong in Ref.Zhang06 (), where changes sign several times near . They are also present, although are not as dramatic, in Ref.Abanin07 (). In the latter case, both and exhibit noisy behavior in the interval near comparable to the peak width. As Ref.Abanin07 () points out, this behavior is consistent with the edge transport model. In the absence of bulk transport, the distribution of potential along the edge depends on the local backscattering rate , whereby Eq.(26) is replaced by
φ1,2(x)=φ1,2(0)−~I∫x0γ(x′)dx′.
Fluctuations of arise due to its sensitivity to the local value of Fermi energy in the spin-orbit scattering model, Eq.(24), and, similarly, for the magnetic impurity scattering mechanism. Assuming that the random part of is of a white noise character, we obtain strong fluctuations along the edge of magnitude that scales as a square root of the edge length. These fluctuations will contribute equally to the longitudinal and transverse voltage, since they are uncorrelated on the opposite sides of the Hall bar. The absence of fluctuations away from can be understood as a result of bulk conductivity short-circuiting the edge current, which will equilibrate potentials on the opposite sides of the Hall bar.
The above discussion summarizes the results drawn from an attempt to model quantum Hall transport in graphene at by counter-circulating edge states. By taking into account backscattering within one edge as well as conduction in the bulk which short-circuits edge transport away from the neutrality point, this model accounts for the observed behavior of transport coefficients. Still, since no direct evidence for spin polarization has yet been found, more experimental and theoretical work will be needed to confirm the chiral spin-polarized edge picture of the state. If proven to exist in graphene, these states will provide a unique setting to study spin transport as well as other interesting phenomena.
This work is supported by NSF MRSEC Program (DMR 02132802), NSF-NIRT DMR-0304019 (DA, LL), and NSF grant DMR-0517222 (PAL).
## References
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• (4) V. P. Gusynin and S. G. Sharapov, Phys. Rev. Lett. 95, 146801 (2005).
• (5) K. S. Novoselov et al., Nature Physics 2, 177 (2006).
• (6) K. S. Novoselov et al., Science 315, 1379 (2007).
• (7) B. I. Halperin, Phys. Rev. B25, 2185 (1982).
• (8) N. M. R. Peres, F. Guinea, A. H. Castro Neto, Phys. Rev. B73, 125411 (2006).
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• (18) B. A. Volkov and O. A. Pankratov, Pis’ma Zh. Eksp. Teor. Fiz. 42, 145 (1985) [Engl. transl. JETP Lett. 42, 178 (1985)]; review in: O. A. Pankratov, Semicond. Sci. Technol. 5, S204-S209 (1990).
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https://www.dummies.com/education/math/pre-algebra/how-to-measure-in-three-dimensions/ | # How to Measure in Three Dimensions
Measuring in three dimensions is similar to measuring in two dimensions; however, in 3-D, the boundary of a solid is called its surface area (not its perimeter) and what’s inside a solid is called its volume (not its area).
The surface area of a solid is a measurement of the size of its surface, as measured in square units such as square inches (in.2), square feet (ft.2), square meters (m2), and so forth. The volume (V) of a solid is a measurement of the space it occupies, as measured in cubic units such as cubic inches (in.3), cubic feet (ft.3), cubic meters (m3), and so forth.
## Measuring spheres
The center of a sphere is a point that’s the same distance from any point on the sphere itself. This distance is called the radius (r) of the sphere. If you know the radius of a sphere, you can find out its volume using the following formula:
Because this formula includes p, using 3.14 as an approximate value for p gives you an approximation of the volume. For example, here’s how to figure out the approximate volume of a ball whose radius is 4 inches:
(Note: In the preceding problem, you use equal signs when a value is equal to whatever comes right before it and approximately-equal-to signs () when you round.)
## Measuring cubes
The main measurement of a cube is the length of its side (s). Using this measurement, you can find out the volume of a cube using the following formula:
V = s3
So if the side of a cube is 5 meters, here’s how you figure out its volume:
V = (5 m)3 = 5 m 5 m 5 m = 125 m3
You can read 125 m3 as 125 cubic meters or, less commonly, as 125 meters cubed.
## Measuring boxes (rectangular solids)
The three measurements of a box (or rectangular solid) are its length (l), width (w), and height (h). The box pictured in the figure below has the following measurements:
l = 4 m, w = 3 m, and h = 2 m.
You can find the volume of a box using the following formula:
V = l w h
So here’s how to find the volume of the box pictured above:
V = 4 m 3 m 2 m = 24 m3
## Measuring prisms
Finding the volume of a prism is easy if you have two measurements. One measurement is the height (h) of the prism. The second is the area of the base (Ab). The base is the polygon that extends vertically from the plane.
Here’s the formula for finding the volume of a prism:
V = Ab h
For example, suppose a prism has a base with an area of 5 square centimeters and a height of 3 centimeters. Here’s how you find its volume:
V = 5 cm2 3 cm = 15 cm3
Notice that the units of measurements (cm2 and cm) are also multiplied, giving you a result of cm3.
## Measuring cylinders
You find the volume of cylinders the same way you find the area of prisms — by multiplying the area of the base (Ab) by the cylinder’s height (h):
V = Ab h
Suppose you want to find the volume of a cylindrical can whose height is 4 inches and whose base is a circle with a radius of 2 inches. First, find the area of the base by using the formula for the area of a circle:
Ab = p r2
3.14 (2 in.)2
= 3.14 4 in.2
= 12.56 in.2
This area is approximate because you use 3.14 as an approximate value for p.
Now use this area to find the volume of the cylinder:
V 12.56 in.2 4 in. = 50.24 in.3
Notice how multiplying square inches (in.2) by inches gives a result in cubic inches (in.3).
## Measuring pyramids and cones
The two key measurements for pyramids and cones are the same as those for prisms and cylinders: the height (h) and the area of the base (Ab). Here’s the formula for the volume of a pyramid or a cone:
For example, suppose you want to find the volume of an ice cream cone whose height is 4 inches and whose base area is 3 square inches. Here’s how you do it:
Similarly, suppose you want to find the volume of a pyramid in Egypt whose height is 60 meters with a square base whose sides are each 50 meters. First, find the area of the base:
Ab = s2 = (50 m)2 = 2,500 m2
Now use this area to find the volume of the pyramid: | 2019-01-24 04:04:25 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8060203194618225, "perplexity": 518.8740986032025}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547584518983.95/warc/CC-MAIN-20190124035411-20190124061411-00315.warc.gz"} |
https://papers.nips.cc/paper/2010/hash/301ad0e3bd5cb1627a2044908a42fdc2-Abstract.html | #### Authors
Han Liu, Kathryn Roeder, Larry Wasserman
#### Abstract
A challenging problem in estimating high-dimensional graphical models is to choose the regularization parameter in a data-dependent way. The standard techniques include $K$-fold cross-validation ($K$-CV), Akaike information criterion (AIC), and Bayesian information criterion (BIC). Though these methods work well for low-dimensional problems, they are not suitable in high dimensional settings. In this paper, we present StARS: a new stability-based method for choosing the regularization parameter in high dimensional inference for undirected graphs. The method has a clear interpretation: we use the least amount of regularization that simultaneously makes a graph sparse and replicable under random sampling. This interpretation requires essentially no conditions. Under mild conditions, we show that StARS is partially sparsistent in terms of graph estimation: i.e. with high probability, all the true edges will be included in the selected model even when the graph size asymptotically increases with the sample size. Empirically, the performance of StARS is compared with the state-of-the-art model selection procedures, including $K$-CV, AIC, and BIC, on both synthetic data and a real microarray dataset. StARS outperforms all competing procedures. | 2022-07-01 04:31:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5894399881362915, "perplexity": 926.5080367581769}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103920118.49/warc/CC-MAIN-20220701034437-20220701064437-00071.warc.gz"} |
http://mathoverflow.net/questions/8924/diffeomorphism-of-3-manifolds/8951 | # Diffeomorphism of 3-manifolds
Surgery theory aims to measure the difference between simple homotopy types and diffeomorphism types. In 3 dimensions, geometrization achieves something much more nuanced than that. Still, I wonder whether the surgeons' key problem has been solved. Is every simple homotopy equivalence between smooth, closed 3-manifolds homotopic to a diffeomorphism?
In related vein, it follows from J.H.C. Whitehead's theorem that a map of closed, connected smooth 3-manifolds is a homotopy equivalence if it has degree $\pm 1$ and induces an isomorphism on $\pi_1$. Is there a reasonable criterion for such a homotopy equivalence to be simple? One could, for instance, ask about maps that preserve abelian torsion invariants (e.g. Turaev's).
-
Turaev defined a simple-homotopy invariant which is a complete invariant of homeomorphism type (originally assuming geometrization).
Here is the Springer link if you have a subscription: Towards the topological classification of geometric 3-manifolds
He claims in the paper that a map between closed 3-manifolds is a homotopy equivalence if and only if it is a simple homotopy equivalence, but he says that the proof of this result will appear in a later paper. I'm not sure if this has appeared though (I haven't searched through his later papers on torsion, and there's no MathScinet link).
-
The main issue is the connect-sum decomposition, no? Ie: oriented simple homotopy equivalent 3-manifolds have oriented simple-homotopy equivalent prime summands. Once you're past that JSJ + geometrization tools take over. – Ryan Budney Dec 15 '09 at 21:34
Agol gets the green box for pointing Turaev's paper which, alongside his 1988 paper "Homeomorphisms of geometric three-dimensional manifolds", MR0940851, apparently answers both questions affirmatively. However, Turaev's argument draws together threads that Paul, Daniel, John, algori, Henry and Ryan mentioned - in particular, Waldhausen's work on the Haken case. Thanks to all. – Tim Perutz Dec 16 '09 at 2:58
Agol, Tim -- I can't access Turaev's paper right now, but if he claims that a homotopy equivalence is a simple homotopy equivalence for 3-manifolds, this would seem a little strange, since for lens spaces this is false: L(7,1) and L(7,2) are homotopy equivalent, but have different simple homotopy types. – algori Dec 17 '09 at 16:21
@ algori: he's claiming that simple homotopy equivalent manifolds are homeomorphic. – Ian Agol Dec 18 '09 at 17:56
Waldhausen proved that homotopy equivalence is homotopic to homeo (and hence diffeo) for Haken 3-manifolds. Perelman extends that to irreducible/infinite pi_1.
It's an old conjecture that the Whitehead group of any torsion free group is trivial.
Irreducible 3-manifolds either have finite or torsion free pi_1, so given Perelman again only S^3/G have potentially non-simple homotopy equivalences.
-
Thanks! Didn't know any of that. When you say "Perelman extends that", do you mean that, in light of Perelman, Waldhausen's arguments apply to all irreducible 3-manifolds with infinite $\pi_1$? – Tim Perutz Dec 15 '09 at 5:08
yes, if I'm recalling correctly: if there exist incompressible tori (or surfaces) then Waldhausen applies. Perelman says that atoroidal manifolds are either hyperbolic, so Mostow applies, or (simple) Seifert-fibered (and hence classified), I assume it is known precisely what non-Haken SF manifolds admit h.e. that aren't homotopic to diffeos, presumably just lens spaces. – Paul Kirk Dec 15 '09 at 23:46
This doesn't actually answer the question, but concerns Tim's comment:-
"The core of the question - I think! - is whether group theory, plus a bit of extra topological input, recognizes the geometric pieces of a 3-manifold."
It's certainly true that the fundamental group sees a lot of the geometry. Scott and Swarup proved that you can reconstruct the JSJ (torus) decomposition of an irreducible 3-manifold from the fundamental group. If your manifold isn't irreducible then the Kneser--Milnor decomposition corresponds exactly to the Grushko decomposition of π1. And π1 also determines the geometry of geometric pieces---Seifert-fibered pieces have normal cyclic subgroups etc.
(Of course, you need the Poincare Conjecture to know that you didn't connect sum with a fake 3-sphere!)
-
About the elliptic case $S^3/G$: elliptic 3-manifolds are classified up to homeomorphism by their $\pi_1$'s, except for the lens spaces. For the lens spaces the simple homotopy type classification is equivalent to the homeomorphism classification (see e.g Milnor, Whitehead torsion, Bulletin AMS, 72) but differs from the homotopy classification.
-
Regarding your first question, in 1953 Moise proved the (manifold) Hauptvermutung for 3-manifolds (Ann. of Math. 58, pp. 458-480). One way to state his result is that every homeomorphism (diffeomorphism) between compact 3-manifolds is homotopic to a PL homeomorphism.
-
I don't know in general, so I'll just the more obvious cases. For hyperbolic 3-manifolds, this is implied by Mostow's rigidity theorem, which states that a homotopy equivalence of hyperbolic manifolds $n$-manifolds is homotopic to an isometry. It's also true for $S^3$, since both $Diff(S^3)$ and $Aut^h(S^3)$ have two components.
-
Right - indeed, MR is stronger still, in that it only requires an abstract isomorphism of $\pi_1$. The core of the question - I think! - is whether group theory, plus a bit of extra topological input, recognizes the geometric pieces of a 3-manifold. – Tim Perutz Dec 15 '09 at 4:47
The $\pi_1$ issue isn't really part of Mostow rigidity proper. Hyperbolic manifolds are $K(\pi,1)$'s so an isomorphism of the fundamental group is the same thing as a homotopy-equivalence of the spaces. The technique falls under basic obstruction theory, which in this case is pretty much "follow your nose". – Ryan Budney Dec 15 '09 at 21:59 | 2014-09-01 11:38:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8682424426078796, "perplexity": 864.659405297917}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1409535917663.12/warc/CC-MAIN-20140901014517-00411-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/oscillation-questions.672474/ | # Oscillation Questions
1. Feb 17, 2013
### Arkuski
Suppose that $f$ is bounded by $M$. Prove that $ω(f^2,[a,b])≤2Mω(f,[a,b])$.
I can show that $ω(f,[a,b])≤2M$ and that $ω(f^2,[a,b])≤M^2$ but this procedure is getting me nowhere. I also have a similar problem that likely calls for the same approach:
Suppose that $f$ is bounded below by $m$ and that $m$ is a positive number. Prove that $ω(1/f,[a,b])≤ω(f,[a,b])/m^2$.
This one I think I have right but my instructor is telling me that it's wrong. Since all values are positive, by the nature of $\frac{1}{x}$, $\displaystyle\sup f = \frac{1}{\displaystyle\inf f}$ and $\displaystyle\inf f = \frac{1}{\displaystyle\sup f}$. We can now analyze the oscilation as follows:
$ω(1/f,[a,b])=\frac{1}{\displaystyle\inf f}-\frac{1}{\displaystyle\sup f}=\frac{ω(f,[a,b])}{(\displaystyle\inf f)(\displaystyle\sup f)}≤\frac{ω(f,[a,b])}{m^2}$
2. Feb 17, 2013
### haruspex
It's probably not the most elegant, but you could try breaking it into separate cases according to the signs of sup f and inf f.
I think you mean $\displaystyle\sup \frac{1}{f} = \frac{1}{\displaystyle\inf f}$ etc. Other than that, your proof looks fine. | 2017-08-22 09:54:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9062156677246094, "perplexity": 163.54403755642542}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886110573.77/warc/CC-MAIN-20170822085147-20170822105147-00268.warc.gz"} |
https://www.gamedev.net/forums/topic/288733-iterating-through-vectors/ | # iterating through vectors
This topic is 5081 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
I do it a lot. Forward and back. No only do I do it a lot, I examine a variable in each item in the vector for one reason or another. How slow and stupid is this? ex:
std::vector<CClass*> Vector;
std::vector<CClass*>::iterator my_iter;
for(my_iter = Vector.begin(); my_iter != Vector.end(); my_iter++)
{
if((*my_iter)->someVar == someOtherVar)
{
DoSomeWork();
}
}
What do you think?
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Might be worth looking at functors, for a number of reasons (Meyers Effective STL has the best section on this that I've read). Then you'd end up with something like this (if I haven't got any code wrong).
// Assuming checking ints, change as required (even template if you want)class functor{ functor(int checkVar) : checkVar_(checkVar) {} void operator() (const CClass* &in) { if(in->someVar == checkVar_) DoSomeWork(); } private: int checkVar_;};// Need to include <algorithm>std::for_each(Vector.begin(), Vector.end(), functor(someOtherVar));
Jim.
Edit : Digging out the Meyer book - he cites 3 reasons for using algorithms :
Efficiency (for example, in your example you calculate Vector.end() every time you run through the loop, as opposed to once only)
Correctness (less likely to get errors with a library function)
Maintainability (working from a known vocabulary)
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Quote:
Original post by JimPriceMight be worth looking at functors, for a number of reasons (Meyers Effective STL has the best section on this that I've read). Then you'd end up with something like this (if I haven't got any code wrong).*** Source Snippet Removed ***Jim.Edit : Digging out the Meyer book - he cites 3 reasons for using algorithms :Efficiency (for example, in your example you calculate Vector.end() every time you run through the loop, as opposed to once only)Correctness (less likely to get errors with a library function)Maintainability (working from a known vocabulary)
I'll have to look into that.
But how bad is iterating through a vector? Really bad? the ones I use arent that huge (yet).
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I'll be honest, I can't personally quantify what improvements you might make - and even Meyer, in the same article, pretty much says that the use of algorithms versus hand-crafted loops is situationally dependent - and in your example, it sounds like he'd probably hand-craft (to avoid the creation of a functor that ostensibly does very little).
The other efficiency considerations (and these are the ones that are noted as more important than just re-calculating Vector.end() each time) I suspect are more relevant to specific algoithms, as opposed to for_each. The interesting example is where he talks about how the STL algorithms are likely to be optimised for the container you are using - for example, a deque is likely to store it's data in one or more fixed size arrays - and therefore the STL can use this knowledge to use pointer-based traversal instead of iterator-based traversal. He quotes some (container-specific) implementations as being 20% faster than "normal" traversal.
Finally, I thought I'd add some links to Guru of the Week that deal with this. Firstly, a general examination of creation of temporaries using hand-crafted loops: GotW #3. And finally my favorite : creating mastermind using algorithms: GotW #41.
Jim.
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What about the find() method? Does that work on vectors? I was under the impression it didn't. I would think that would be a better way to do what I want.
Basically, I've got two vectors full of pointers to one of my own classes. I want to search one vector for an object that meets a certain criteria. Once I find that object, I want to add it to the second vector (I've just been using push_back()) and then remove it from the first. I'm using vectors instead of lists because I need the my_list[] syntax. I use an int to hold my current position in the list (for various reasons).
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Morning,
Vector has no find member function, but the algorithm functions work with all iterators. So you'd use something like this (off the top of my head, so no guarantees it will work)
#include <vector>#include <algorithm>// You said one of your own classes, so taking it literally // - although the extension to polymorphism is obviousclass Base{//stuff};bool FindCorrectPointer(const Base* &in){ if(in->YesPlease) return true; return false;}int main(){ typedef std::vector<Base*> MyTypedef; typedef MyTypedef::iterator MyIterator; MyTypedef myContainer1; MyTypedef myContainer2; MyIterator myIterator;// This line becomes more complicated if you want to pass parameters to the search function// Need to look up function adapters - like bind2nd// Also note find_if only returns first occurrence // - but you can get around this using a loop containing find_if that exits when it hits myContainer1.end() myIterator = std::find_if(myContainer1, myContainer2, FindCorrectPointer());// Check to see if this element existed if(myIterator != myContainer1.end()) { myContainer2.push_back(*myIterator);// The erase function now points at the next element in the container myIterator = myContainer1.erase(myIterator); }};
Benefits - if later on you decide to change to storage container, you can now do it just be changing the typedef statements at the beginning (no need to worry about [] and storing ints). The one thing you might want to change is the use of find - member functions (when present) are always preferred to algorithms, for the same reasons as mentioned previously (ie they are tailored to the underlying structure of the container).
HTH,
Jim.
Edit : loads of silly typos
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The main problem with using STL algorithms for what you want to do is that algorithms work over a range of items, completely. You want to find one value, copy it, remove it, and stop (right?).
See, the really nice thing about using STL algos instead of for loops is that they're more predictable... i.e. when you see a for loop, you have to go scan through its contents to see if there are any 'break's or any statements that affect the iterating variable.
OTOH, when you need those things you have to write a for loop.
Anyway, efficiency is not really your main worry, because erasing from vectors is damn slow... and how does that affect iterators? I know some operations invalidate iterators, but I'm not sure when or how. You should be fine though, since you don't use the iterator after you erase it.
If you gave a more specific description of what you're trying to accomplish overall, STL might be able to help you do it differently (i.e. if you maintained sorted vectors, you could use lower_bound(), or maybe [] syntax isn't so important after all and you can use std::set).
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Quote:
The main problem with using STL algorithms for what you want to do is that algorithms work over a range of items, completely. You want to find one value, copy it, remove it, and stop (right?).
Er - I don't think they do. The find and find_if algorithms stop when they hit first object found - otherwise they wouldn't be able to return an iterator (I suppose they could return a std::vector of iterators, for example, but I'm guessing they don't do this for exactly the reason you point out). That's why you have to write code like this:
std::vector<int> Vec;std::vector<int>::iterator It;// Possibly populate Vecwhile(It != Vec.end()){ It = std::find(Vec.begin(), Vec.end(), value); if(It != Vec.end()) { It->DoSomething;// And you could use erase here, for example, because it returns an iterator to the next element of Vec }}
Here's a comparison of hand-crafted, algorithm and member function codes (pretty much taken from effective STL). Take this code:
set<int> s;// Populate s with 1,000,000 ints//Method 1:set<int>::iterator i; // Have to define here so I can use i outside the loopfor(i = s.begin(); i != s.end(); ++i){ if(*i == 727) break;}//Method 2:set<int>::iterator i = std::find(s.begin(), s.end(), 727);//Method 3:set<int>::iterator i = s.find(727);
Method 1 : stops when it hits first 727 - so could loop once or a million times.
Method 2 : stops when it hits first 727 - so could loop once or a million times.
Method 3 : stops when it hits first 727 - but set is stored as a red-black tree, so at most loops about 40 times.
This is why member functions are preferred to algorithms - an algorithm has to be generic so it can work with any iterator-providing container, but a member function can be written to take advantage of the structure of the container.
OK - so this doesn't make a big difference with std::vector in these simple examples (unless the underlying implementation uses pointer-based traversal, as vectors are typically stored as a big array), but if there is a change in container type then this becomes relevant.
Quote:
OTOH, when you need those things you have to write a for loop.
No you don't - for the reasons mentioned above. But I do agree that choice of hand-crafted versus algorithm is situationally dependent - but it's not as clear cut as this.
Quote:
Anyway, efficiency is not really your main worry, because erasing from vectors is damn slow
This is a good point and is something I'd overlooked. Depending upon how frequently you're doing it a std::list might be more efficient.
Quote:
I know some operations invalidate iterators, but I'm not sure when or how.
Another reason for using algorithms, because they do know what's happening to the iterator.
Jim.
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it looks to me like you are using the wrong container for the job. if you are looping thruogh the container looking for a specific value frequently, you should be using either a std::set or a std::map. this is what these containers were designed for! each std::container has its own place and purpose, don't restrict yourself to only using vectors. i think i use almost all of the containers in my current project (deque,set,map,vector,list..) your code would turn to (something like) this:
std::set<int>::iterator it = my_set.find(_checkVar);if(it != my_set.end()){ DoSomeWork();}
not only does this make your code cleaner and easier to write, but it also makes it faster. sorry if this was already suggested, im at work and could only skim through the thread.
[Edited by - graveyard filla on December 16, 2004 1:14:15 PM]
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Original post by benutneI do it a lot. Forward and back. No only do I do it a lot, I examine a variable in each item in the vector for one reason or another.How slow and stupid is this?ex:*** Source Snippet Removed ***What do you think?
Prefer ++iter to iter++, iter++ creates a temporary.
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× | 2018-11-15 03:52:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25173065066337585, "perplexity": 1904.254994191463}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039742483.3/warc/CC-MAIN-20181115033911-20181115055911-00220.warc.gz"} |
https://tex.stackexchange.com/questions/516754/how-to-pass-contents-of-environment-declared-with-newenvironment-in-latex2e-a | # How to pass contents of environment (declared with \newenvironment in LaTeX2e) as argument to that environment?
I know there's an option in expl3, but how can I do the same in LaTeX2e (or TeX)?
\newenvironment*{myEnv}[1][]{}{}
\begin{myEnv}
I first need to pass this to this environment as an argument.
\end{myEnv}
• This is what the environ package does, the body of the environment is then stored in \BODY. – user194703 Nov 17 '19 at 7:31
• You can do everything what packages do by copying the contents of these packages in the preamble (sandwiched between \makeatletter and \makeatother, of course). I do not know what defines an "official" package, but environ is definitely a very nice and robust package that gets widely used. The source of the package is not very long, so you can have a look at it. – user194703 Nov 17 '19 at 7:38
• Example: \documentclass{article} \usepackage{environ} \NewEnviron{myEnv}[1][]{\underline{\BODY}} \begin{document} \begin{myEnv} I first need to pass this to this environment as an argument. \end{myEnv} \end{document} – user194703 Nov 17 '19 at 7:40
• @Schrödinger'scat, for some reason my custom macro can't break down this \BODY parameter, but can break down regular text into letters. In other words \BODY doesn't behave like regular text. – bp2017 Nov 17 '19 at 7:51
• Well, hard to tell what is going on without seeing your custom macro. You may have to replace \MyCustomMacro{\BODY} by \edef\temp{\noexpand\MyCustomMacro{\BODY}}\temp, but really hard to say. – user194703 Nov 17 '19 at 7:55
The idea of 'collecting the body' is nowadays available in xparse
\documentclass{article}
\usepackage{xparse}
\begin{document}
\NewDocumentEnvironment{myEnv}{O{}+b}{The argument was '#2'}{}
\begin{myEnv}
I first need to pass this to this environment as an argument.
\end{myEnv}
\end{document}
Here, #2 is the second argument (the body).
As pointed out in comments, the long-standing environ package does the same but provides the result as \BODY.
• @bp2017 You need a more sophisticated loop to deal with spaces: there are several approaches (and examples on the site), depending on the other requirements (working by expansion, treatment of braces, ...). I'll adjust my answer to take \par tokens. – Joseph Wright Nov 17 '19 at 8:18
• When I pass \BODY to \zStart as \zStart{\BODY} I don't get processed text, it's as if \zStart didn't work. – bp2017 Nov 17 '19 at 8:20
• @bp2017 Well no, because there is only one token, \BODY. You'll need to expand it once: \expandafter\zStart\expandafter{\BODY}. TeX macros receive the tokens you pass as written, not the expansion of the tokens. I think that was mentioned in comments on the question. – Joseph Wright Nov 17 '19 at 8:23
• – Joseph Wright Nov 17 '19 at 8:39 | 2021-04-13 17:23:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8989792466163635, "perplexity": 1916.7270852289191}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038073437.35/warc/CC-MAIN-20210413152520-20210413182520-00266.warc.gz"} |
https://quantnet.com/threads/anyone-got-columbia-mfe-offer-1st-deposite-date-request-a-expedite-decision-on-nyu-or-baruch.19651/page-2 | Columbia MFEanyone got columbia MFE offer (1st deposite date?) request a expedite decision on NYU or Baruch...
yo yue
Member
C++ Student
its personal preference and based on couple of data points i have collected...
biased in a way...
one is Baruch is very hard to get in ... their candidates are relatively strong among the peers... second the program is more affordable...
implol
Member
its personal preference and based on couple of data points i have collected...
biased in a way...
one is Baruch is very hard to get in ... their candidates are relatively strong among the peers... second the program is more affordable...
ah i see. you are right about baruch for these two points. but it will be harder for you to network with baruch degree. i currently go to an undergrad that is super affordable and difficult to get in (below 8% acceptance rate). but students have hard time to land internships at top companies.
p.s. : this is not meant to start a war nor to argue which program is better. just point out the pros and cons so that you can make the best decision for you.
Lolo_D
New Member
Hello,
For those who have already accepted the offer,
Could you tell us how you have paid the 4000$deposit? I am wondering that because I have not accepted the offer yet, but as 4000$ is a huge amount, I am not sure that I can pay by card.
Do I have to ask my bank to do the transfert payement?
Thank you
yo yue
Member
C++ Student
I sent an email to Baruch, they are super quick. Get my first interview invite the next business day. Then get my second interview invite on the same day I did my first interview. NYU on the other hand essentially replied me "keep waiting"
CMU is gonna be tough. R3 decision is in May. They have not sent out R2 interview yet. There is just no way you can avoid paying for the deposit for other schools.
Who will you choose between Baruch and Columbia?
I sent email to Baruch this morning... no one replied yet... which email you send to... hope its not the case that they dont want give me an interview at all...
cxiao
Active Member
C++ Student
I sent email to Baruch this morning... no one replied yet... which email you send to... hope its not the case that they dont want give me an interview at all...
I take a look at your tracker, you applied 2 weeks ago? I submitted my application back in December. I think that's a major reason why they are so quick. In your case, I don't think they will have enough time to review your application.
yo yue
Member
C++ Student
I take a look at your tracker, you applied 2 weeks ago? I submitted my application back in December. I think that's a major reason why they are so quick. In your case, I don't think they will have enough time to review your application.
I thought they are doing stuff on a rolling basis... once your package complete then it heads to review stage... coz baruch doesnt have a fixed seat cap then I thought they will be highly efficient in this way... Hope I dont screw up my deadlines...
Did you have your second interview yet...
cxiao
Active Member
C++ Student
I thought they are doing stuff on a rolling basis... once your package complete then it heads to review stage... coz baruch doesnt have a fixed seat cap then I thought they will be highly efficient in this way... Hope I dont screw up my deadlines...
Did you have your second interview yet...
They are on a rolling basis. But I don't think they are that quick.
I did my second interview. Still waiting for the result.
yo yue
Member
C++ Student
They are on a rolling basis. But I don't think they are that quick.
I did my second interview. Still waiting for the result.
best of luck and fingers crossed
I did checked prior year and the one before... there is case where the interview is 8 days after application submission - submitted right b4 deadlines. i guess if the profile is compelling enough then they will make a case... profile itself doesnt matter taht much as long as one get interview actually, they will test u through the interview anyways...
yo yue
Member
C++ Student
They are on a rolling basis. But I don't think they are that quick.
I did my second interview. Still waiting for the result.
congrats... I think the one got adimtted is you rite? finalized your decision yet?
cxiao
Active Member
C++ Student
congrats... I think the one got adimtted is you rite? finalized your decision yet?
wink wink. Not yet. Still thinking.
Pavlos Sakoglou
Well-Known Member
C++ Student
Columbia vs Baruch: The battle rages on!!!
Columbia is more famous and that's why they are in the top of the list. Their classes have many students (fin math has 60-70 students) and teaching gets less personal. Don't get me wrong, it is probably one of the best in the country, but in a specialist master you need more efficiency than fame.
Baruch is half price than Columbia, guarantees you paid intership between 2nd and 3rd semester, and a high paying job right after your graduation. The faculty is excellent and care about your success, you get to have a mentor during your MFE, alumni and financial professionals, and the small size of the class makes it easier to comprehent the material. They are committed and provide you with expert knowledge and a valuable network to get employed right away.
Of course, there is still the argument that everyone knows Columbia outside of New York and Columbia will help you with any job application later, while Baruch only guarantees you the first job in some major cities, then it is up to you. But most of the times is up to you.
Personally, I choose Baruch over Columbia. I am recently doing the Pre-MFEs in Baruch and I am extremely satisfied. They are super intelligent and with great sense of humor. Great environment, makes you feel part of the group.
Last edited:
yo yue
Member
C++ Student
Columbia vs Baruch: The battle rages on!!!
Columbia is more famous but one of the main reasons that it is in the top of the list is its reputation. Their classes have many students (fin math has 60-70 students) and teaching gets less personal. Don't get me wrong, it is probably one of the best in the country, but in a specialist master you need more efficiency than fame.
Baruch is half price than Columbia, guarantees you paid intership between 2nd and 3rd semester, and a high paying job right after your graduation. The faculty is excellent and care about your success, you get to have a mentor during your MFE, alumni and financial professionals, and the small size of the class makes it easier to comprehent the material. They are committed and provide you with expert knowledge and a valuable network to get employed right away.
Of course, there is still the argument that everyone knows Columbia outside of New York and Columbia will help you with any job application later, while Baruch only guarantees you the first job in some major cities, then it is up to you. But most of the times is up to you.
Personally, I choose Baruch over Columbia. I am recently doing the Pre-MFEs in Baruch and I am extremely satisfied. They are super intelligent and with great sense of humor. Great environment, makes you feel part of the group.
Thanks so much for sharing this... I found Baruch is more and more attractive and compelling to me based on so many excellent reviews... Dan must have done an outstanding job to make every graduate succeeding... It seems fair to say, there is no weak graduate from Baruch... for Columbia and NYU I dont have enough data points... I would imagine it will be difficult to keep with Baruch given their interview process so rigorous and selective...
curaider
New Member
C++ Student
Thanks so much for sharing this... I found Baruch is more and more attractive and compelling to me based on so many excellent reviews... Dan must have done an outstanding job to make every graduate succeeding... It seems fair to say, there is no weak graduate from Baruch... for Columbia and NYU I dont have enough data points... I would imagine it will be difficult to keep with Baruch given their interview process so rigorous and selective...
Why should the rigorous and selective admission process of Baruch concern you once you are admitted? Surely Baruch MFE is a successful program, but if NYU MathFin and Columbia MFE are equally successful with "so so" students, it probably means their added values are higher.
With no offense to anyone, you are certainly aware of the relationship between QuantNet and Baruch right?
Pavlos Sakoglou
Well-Known Member
C++ Student
Why should the rigorous and selective admission process of Baruch concern you once you are admitted? Surely Baruch MFE is a successful program, but if NYU MathFin and Columbia MFE are equally successful with "so so" students, it probably means their added values are higher.
With no offense to anyone, you are certainly aware of the relationship between QuantNet and Baruch right?
I agree. I think any of these programs are equally great and probably the best in the world.
Though, looking at the samilar quality they all provide and the prices they have, I will target Baruch and apply there, even though Columbia and NYU will give you a great advantage later as they are more prestigious. I would choose them for a PhD any time, but for MFE I think Baruch wins. Unless, you know, it is your last degree or something and the school reputation counts more.
implol
Member
I think fits matter more when you talk about top programs. No need to fight over. Everybody wants different things from the education. Prestige vs best value. People have different preferences.
yo yue
Member
C++ Student
Why should the rigorous and selective admission process of Baruch concern you once you are admitted? Surely Baruch MFE is a successful program, but if NYU MathFin and Columbia MFE are equally successful with "so so" students, it probably means their added values are higher.
With no offense to anyone, you are certainly aware of the relationship between QuantNet and Baruch right?
yeah... I have to admit I am biased in a way...
for NYU, I am expecting more from student's resume... they might just not put the fancy stuff on it...
Pavlos Sakoglou
Well-Known Member
C++ Student
I think fits matter more when you talk about top programs. No need to fight over. Everybody wants different things from the education. Prestige vs best value. People have different preferences.
agreed.
yo yue
Member
C++ Student
Why should the rigorous and selective admission process of Baruch concern you once you are admitted? Surely Baruch MFE is a successful program, but if NYU MathFin and Columbia MFE are equally successful with "so so" students, it probably means their added values are higher.
With no offense to anyone, you are certainly aware of the relationship between QuantNet and Baruch right?
hey, for NYU... if you go thru their resume book - online you can check to see if they get employed or not... someone did so and find out 8 out of 45 havent got job after graduate 2 month. A good school no matter how good their program is, they shouldnt be afraid to publish their placement rate and salary range like baruch and Berkeley... if it is not transparent, I dont think that is good indicator... at the end of the day, people judge the success of the program by placement, salary...
NYU top applied math department - no doubt about it, people graduate from this go to the top destination but keeps in mind there are people only look good on the paper... or even fake some experience of any sort... with this said, schools have interviews usually do a better job than the ones that dont by filtering the quality of students...
just my personal idea... hope everyone get in to the school they really want
Last edited: | 2019-07-17 01:28:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22472622990608215, "perplexity": 2425.7167509385913}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195525004.24/warc/CC-MAIN-20190717001433-20190717023433-00006.warc.gz"} |
https://physics.stackexchange.com/questions/536714/is-there-any-proof-of-why-a-charge-is-scalar/536720 | Is there any proof of why a charge is scalar?
I tried to prove it by contradiction as, if it was a vector then the isotropic space property that nature currently has now would not hold; But that proof eventually broke down. So, I was wondering if a proof actually exists or it is just experimentally told as a property?
• Hard to say, but it has something to do that elementary charge is space and time invariant Mar 17, 2020 at 9:19
• Does this answer your question? How can we prove charge invariance under Lorentz Transformation? Mar 17, 2020 at 14:25
• You cannot mathematically prove something about the physical universe. All the mathematics does is rearrange where you are making assumptions. But in the end, you are still making assumptions. The actual "proof" is the experimental evidence accrued supporting those assumptions. (And the reason I quoted "proof" is because despite using the same word, the mathematical and scientific concepts of "proof" are significantly different.) Mar 17, 2020 at 16:43
• Can you clarify what you intend by “scalar”? Do you just mean something that is not a three dimensional vector or do you mean the technical meaning in terms of four-vectors and tensors?
– Dale
Mar 17, 2020 at 19:19
The point-like sources of electromagnetic field don't have to be scalar like. We can have dipole sources, quadrupole sources and so on. In general, any feld generated by a point-like source can be decomposed using so-called multipole expansion.
$$V(r,\theta,\varphi) = \sum_{l=0}^\infty \sum_{m=-l}^l C^m_l(r) Y^m_l(\theta, \varphi)$$
where functions $$Y^m_l$$ are specific functions called spherical harmonics.
The electric charge is the property of the source that generates the spherically isotropic field (l=0,m=0), and that means it is a scalar by definition.
That is, if we consider only the transformations of the space. If we consider space-time transformation, then charge density is not a scalar density, but just the time-component of a four-vector called the four-current: $$(\rho, \vec j)$$.
If the question is whether the elementary particles possess only scalar charges, then it depends on how deep do you go. The truly fundamental particles like electrons or quarks seem to have only scalar charges, but composite particles may have for example the dipole moment, which is a vector. This is the case of a neutron, which despite having no net electric charge, still possesses a small electric dipole moment. Even fundamental particles may theoretically have a dipole moment, and there are theoretical predictions to detect the electron's dipole moment, but the predicted value is too small to be observed experimentally. The same for other fundamental particles. For all practical uses, the fundamental particles can be treated as possessing only scalar charge.
• I've understood that OP is asking about elementary charges, not multipoles Mar 17, 2020 at 9:15
• @AgniusVasiliauskas Introducing multipole expansion however lets you define the charge as the monopole moment, and then it's scalar by definition. Mar 17, 2020 at 9:22
• @AdamLatosiński . So BY DEFINATION means there’s no proof? Mar 17, 2020 at 16:33
• Small nitpick: technically we don't know that the neutron electric dipole moment is nonzero. We certainly expect it to be nonzero, but we have only measured null results at current sensitivity. Mar 17, 2020 at 17:21
• @AshishKumar I'm confused. If you define "two" as the next natural number after "one", do you require a proof that "two" is the next natural number after "one"? If you define electric charge as the scalar (i.e. monopole or isotropic) source of electric field, do you need to prove that the electric charge is a scalar? I mean, if you use different definition of electric charge, then maybe you do need a proff. But then please tell us what definition you use. Mar 17, 2020 at 20:45
Yes, the proof that charge is a scalar is quite simple. Start with Gauss' law: $$\nabla \cdot \vec E = \frac{\rho}{\epsilon_0}$$ The divergence of a vector is a scalar, so the quantity on the right is a scalar. Then define $$Q = \iiint \rho \ dx \ dy \ dz$$ since $$\rho$$ is a scalar the integral of $$\rho$$ over a volume is also a scalar, so $$Q$$ is a scalar.
• $\nabla \cdot \vec E$ is not a scalar but the time component of a fourvector density. Mar 17, 2020 at 12:56
• I understood that the OP was asking about “scalars” in the vernacular meaning simply not a (three-)vector”. I do not believe that they were asking about scalars in the technical meaning of the term as a pseudo-Riemannian scalar such as the contraction of two four-vectors.
– Dale
Mar 17, 2020 at 18:49
The properties of a particles cannot be a vector quantity. I believe physically there's no need to define the charge as a vector quantity. Because the important thing is its magnitude. The vector is a mathematical concept that can be used in physics to describe nature. The charge of a particle does not need this kind of description.
• The properties of a substance cannot be a vector quantity Wrong. Take a look at complex refractive index of material, which is defined in complex plane as : $$\begin{bmatrix}n\\i\kappa\end{bmatrix}$$. Or take a look into second order Cauchy stress tensor $\sigma$ of material in continuum mechanics: $$\left[{\begin{matrix}\sigma _{11}\,\sigma _{12}\,\sigma _{13}\\\sigma _{21}\,\sigma _{22}\,\sigma _{23}\\\sigma _{31}\,\sigma _{32}\,\sigma _{33}\\\end{matrix}}\right]$$ .So in principle, material property can be described by tensor of any order Mar 17, 2020 at 19:06
• @AgniusVasiliauskas By substance I mean leptons and quarks. Not normal material. Mar 17, 2020 at 22:37
• Spin angular momentum or magnetic dipole moment of particle IS vector quantity Mar 18, 2020 at 10:02
• @AgniusVasiliauskas "In some ways, spin is like a vector quantity; it has a definite magnitude, and it has a "direction" (but quantization makes this "direction" different from the direction of an ordinary vector)". So its not a vector quantity. Mar 18, 2020 at 10:34
• No, it is. Quantized vector is still a vector. Mar 18, 2020 at 11:54 | 2022-07-05 13:01:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 8, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7644820213317871, "perplexity": 450.24521602785836}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104576719.83/warc/CC-MAIN-20220705113756-20220705143756-00642.warc.gz"} |
http://math.stackexchange.com/questions/27712/alternative-proof-of-the-cardinality-of-the-set-of-all-mappings | # Alternative proof of the cardinality of the set of all mappings
Can anyone please tell me if there is any other proof for the cardinality of all mappings, that is not by induction, i.e., not this one (http://www.proofwiki.org/wiki/Cardinality_of_Set_of_All_Mappings) ?
Thanks
-
You could just define $\vert T\vert^{\vert S\vert}$ to be $\left\vert T^S\right\vert$ and you're done! Actually for the non-finite case, that's probably the best way. – George Lowther Mar 18 '11 at 1:50
If $A=\{a_1, a_2, \ldots, a_n\}$ has $n$ elements and $B$ has $m$ elements, a mapping $f$ from $A$ to $B$ is defined uniquely by choosing $f(a_1)$ ($m$ options), then $f(a_2)$ ($m$ options again), and so on until you choose $f(a_n)$ (as always, $m$ options). By the product rule, there are $m \times m \times \cdots \times m$ ($n$ factors) overall, namely, $m^n$. | 2014-04-25 01:44:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8805884718894958, "perplexity": 223.77177303658704}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00206-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://tailieu.vn/doc/lecture-methods-of-electric-power-systems-analysis-lesson-6-power-operations-power-flow-2529075.html | # Lecture Methods of Electric power systems analysis - Lesson 6: Power operations, power flow
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Lecture Methods of Electric power systems analysis - Lesson 6: Power operations, power flow provide students with knowledge about three bus powerworld simulator case; basic power control; power flow in transmission line is limited by heating considerations; overloaded transmission line; automatic generation control;...
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## Nội dung Text: Lecture Methods of Electric power systems analysis - Lesson 6: Power operations, power flow
1. ECEN 615 Methods of Electric Power Systems Analysis Lecture 6: Power Operations, Power Flow Prof. Tom Overbye Dept. of Electrical and Computer Engineering Texas A&M University overbye@tamu.edu
2. Announcements • Read Chapter 6 from the book • The book formulates the power flow using the polar form for the Ybus elements • Homework 2 is due on Thursday September 17 1
3. Three Bus PowerWorld Simulator Case PowerWorld Case Name: B3Slow Load with Area Name: Home ACE: -15.5 MW green Home Area 25.4 MW MW Load: 316.2 MW MW Gen: 301.0 MW 25.5 MW MW Losses: 0.28 MW arrows Bus 2 5.3 Mvar A -4.9 Mvar A Bus 1 1.00 pu indicating 210.8 MW 105.4 Mvar MVA MVA slack 1.00 pu amount A A 115.4 MW -1.9 Mvar of MW 151.0 MW AGC OFF 34.3 MW MVA A A MVA 10.1 MW 100 MW AVR ON 10.6 Mvar 3.1 Mvar flow 121.3 Mvar 34.5 MW MVA MVA 10.1 MW -3.0 Mvar Other Area -10.0 Mvar Scheduled Transactions 1.00 pu 0.0 MW Bus 3 105.4 MW Note the Off AGC 150.0 MW 52.7 Mvar power Used 39.7 Mvar AVR ON AGC ON balance at to control each bus output of Direction of green arrow is used to indicate generator direction of real power (MW) flow; the blue arrows show the reactive power 2
4. Basic Power Control • Opening a circuit breaker causes the power flow to instantaneously (nearly) change. • No other way to directly control power flow in a transmission line. • By changing generation we can indirectly change this flow. • Power flow in transmission line is limited by heating considerations • Losses (I^2 R) can heat up the line, causing it to sag. 3
5. Transmission Line Limits • Power flow in transmission line is limited by heating considerations. • Losses (I2 R) can heat up the line, causing it to sag. • Each line has a limit; many utilities use winter/summer limits. 4
6. Overloaded Transmission Line Area Name: Home ACE: -263.1 MW Home Area MW Load: 559.2 MW -162.5 MW MW Gen: 301.0 MW 165.3 MW Bus 2 39.8 Mvar MW Losses: 4.91 MW -25.8 Mvar Bus 1 A A 1.000 pu 112% 112% 372.8 MW MVA MVA 186.4 Mvar slack 1.000 pu A A 363.0 MW MVA -52.3 Mvar MVA -59.2 MW 151.0 MW AGC OFF A A 97.7 MW 18.8 Mvar 100.0 MW AVR ON -26.5 Mvar 245.0 Mvar MVA MVA 59.8 MW -96.2 MW Other Area -16.9 Mvar 31.6 Mvar Scheduled Transactions Bus 3 1.000 pu 0.0 MW 186.4 MW Off AGC 93.2 Mvar 150.0 MW AGC OFF 107.9 Mvar AVR ON 5
7. Interconnected Operation Balancing Authority (BA) Areas • North American Eastern and Western grids are divided into balancing authority areas (BA) – Often just called an area • Transmission lines that join two areas are known as tie-lines. • The net power out of an area is the sum of the flow on its tie-lines. • The flow out of an area is equal to total gen - total load - total losses = tie-flow 6
8. US Balancing Authorities 7
9. Area Control Error (ACE) • The area control error is the difference between the actual flow out of an area, and the scheduled flow – ACE also includes a frequency component that we will probably consider later in the semester • Ideally the ACE should always be zero • Because the load is constantly changing, each utility (or ISO) must constantly change its generation to “chase” the ACE • ACE was originally computed by utilities; increasingly it is computed by larger organizations such as ISOs 8
10. Automatic Generation Control • Most utilities (ISOs) use automatic generation control (AGC) to automatically change their generation to keep their ACE close to zero. • Usually the control center calculates ACE based upon tie-line flows; then the AGC module sends control signals out to the generators every couple seconds. 9
11. Three Bus Case on AGC Area Name: Home ACE: -0.0 MW Home Area MW Load: 330.2 MW MW Gen: 330.6 MW -21 MW 21 MW MW Losses: 0.40 MW Bus 2 4 Mvar -4 Mvar A A Bus 1 MVA 220 MW MVA 110 Mvar slack A A 100 MW 158 MW MVA 2 Mvar 127 Mvar MVA -41 MW -21 MW AGC ON 13 Mvar A A 6 Mvar 100 MW 0 Mvar 41 MW MVA MVA 21 MW Scheduled Transactions -12 Mvar Bus 3 -6 Mvar Other Area 0.0 MW Area AGC Status: Part. AGC 173 MW 110 MW 37 Mvar 55 Mvar AGC ON Generation Net tie flow is is automatically close to zero changed to match change in load 10
12. Generator Costs • There are many fixed and variable costs associated with power system operation • The major variable cost is associated with generation. • Cost to generate a MWh can vary widely • For some types of units (such as hydro and nuclear) it is difficult to quantify • More others such as wind and solar the marginal cost of energy is essentially zero (actually negative for wind!) • For thermal units it is straightforward to determine • Many markets have moved from cost-based to price- based generator costs 11
13. Economic Dispatch • Economic dispatch (ED) determines the least cost dispatch of generation for an area. • For a lossless system, the ED occurs when all the generators have equal marginal costs. IC1(PG,1) = IC2(PG,2) = … = ICm(PG,m) 12
14. Power Transactions • Power transactions are contracts between areas to do power transactions. • Contracts can be for any amount of time at any price for any amount of power. • Scheduled power transactions are implemented by modifying the area ACE: ACE = Pactual,tie-flow - Psched 13
15. 100 MW Transaction Area Name: Home ACE: -0.0 MW Home Area MW Load: 335.8 MW MW Gen: 436.8 MW 35 MW -34 MW MW Losses: 1.01 MW Bus 2 -7 Mvar 7 Mvar A A Bus 1 MVA 224 MW MVA 112 Mvar slack A A 0 MW 226 MW MVA 28 Mvar 116 Mvar MVA -33 MW -66 MW AGC ON 10 Mvar A A 21 Mvar 100 MW 0 Mvar 33 MW MVA MVA 66 MW Scheduled Transactions -10 Mvar Bus 3 -19 Mvar Other Area 100.0 MW Area AGC Status: Part. AGC 211 MW 112 MW 28 Mvar 56 Mvar AGC ON Net tie-line Scheduled 100 MW transaction flow is now from the Home Area to the 100 MW Other Area 14
16. Security Constrained ED • Transmission constraints often limit system economic operation. • Such limits required a constrained dispatch in order to maintain system security. • In the three bus case the generation at bus 3 must be constrained to avoid overloading the line from bus 2 to bus 3. 15
17. Security Constrained Dispatch Area Name: Home ACE: 0.1 MW Home Area MW Load: 580.0 MW MW Gen: 685.9 MW -22 MW 22 MW MW Losses: 5.90 MW Bus 2 4 Mvar -4 Mvar A A Bus 1 MVA 387 MW MVA 193 Mvar slack A A -0 MW 223 MW 100% MVA 37 Mvar 246 Mvar MVA -142 MW -122 MW AGC ON A 49 Mvar A 41 Mvar 100 MW 100% 0 Mvar 145 MW MVA MVA 124 MW Scheduled Transactions -37 Mvar Bus 3 -33 Mvar Other Area 100.0 MW Area AGC Status: OPF 463 MW 193 MW 26 Mvar 97 Mvar AGC ON Dispatch is no longer optimal due to need to keep the line from bus 2 to bus 3 from overloading 16
18. Multi-Area Operation • If areas have direct interconnections then they may directly transact, up to the capacity of their tie-lines. • Actual power flows through the entire network according to the impedance of the transmission lines. • Flow through other areas is known as “parallel path” or “loop flow.” 17
19. Seven Bus Case: One-line Area Top System has three areas has five 44 MW A 42 MW 31 MW A 31 MW 80 MW 30 Mvar buses 1.05 pu Bus 1 MVA Bus 3 0.99 pu MVA Bus 4 1.00 pu 61 MW 105 MW 37 MW 110 MW 32 MW AGC ON 40 Mvar A A A A 93 MW MVA MVA Case Hourly Cost AGC ON MVA 38 MW MVA 16933 $/h A 14 MW 60 MW 33 MW 1.04 pu 79 MW MVA 77 MW 1.01 pu Bus 2 Top Area Cost Bus 5 8030$/h 40 MW 39 MW 130 MW 40 MW 40 Mvar 20 Mvar A A 170 MW AGC ON MVA MVA 40 MW 40 MW 20 MW A 20 MW 1.04 pu 1.04 pu MVA Bus 6 20 MW A 20 MW Bus 7 200 MW Left Area Cost MVA Right Area Cost slack 200 MW 0 Mvar 4189 $/h 4714$/h 0 Mvar 200 MW AGC ON 201 MW AGC ON Area Left Area has one Right has bus one bus PowerWorld Case: B7Flat 18
20. Seven Bus Case: Area View Actual Top Area Losses 7.1 MW flow System has 40.2 MW between -40.2 MW 40 MW of 0.0 MW 0.0 MW areas “Loop Flow” Scheduled Left Right Area Losses 40.2 MW Area Losses flow 0.3 MW 0.0 MW 0.7 MW 19 | 2022-06-30 16:53:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3065682649612427, "perplexity": 12586.76187678727}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103850139.45/warc/CC-MAIN-20220630153307-20220630183307-00508.warc.gz"} |
https://nrich.maths.org/11017/solution | ### Kissing Triangles
Determine the total shaded area of the 'kissing triangles'.
### Isosceles
Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.
### Linkage
Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?
# Triangles in a Square
##### Age 11 to 14 Challenge Level:
We had lots of great solutions to this problem. Thank you to all of you who wrote in!
Julian from the British School Manila, Ahrus and Ben from Dixons Trinity Academy, and Kira from Wycombe High School all managed to find the area of each of the coloured triangles on the $5 \times 5$ grid. Here's what Kira did:
How to calculate the area of each triangle:
When the triangles are drawn on the dotty paper they are surrounded by $3$
right-angled triangles. The total area of all three right-angled triangles
must be subtracted from the area of the square ($25$ squares) to give the
area of the triangle.
Red triangle is $10.5$ squares
Yellow triangle is $10.5$ squares
Blue triangle is $11.5$ squares
Green triangle is $11$ squares
Purple triangle is $10.5$ squares
Therefore the blue triangle has the largest area.
Jurmana, Haidi, and Nour from the Continental School Cairo all found the largest and smallest triangles they could make on a $5 \times 5$ grid:
The largest area I found was $12.5$ cm$^2$ when the corners of the triangle are at $(0,0)$, $(0,5)$ and $(5,5)$.
The smallest area I found was $2.5$ cm$^2$.
Here's how Julian found a general formula for the area of a triangle with vertices at $(5,5)$, $(0,y)$, and $(x,0)$:
Area of the top left white triangle:
base: $5$
height: $5-y$
area: $\frac{5(5-y)}{2}$
Area of the bottom left white triangle:
base: $x$
height: $y$
area: $\frac{xy}{2}$
Area of the bottom right white triangle:
base: $5-x$
height: $5$
area: $\frac{5(5-x)}{2}$
Since the area of the colored triangle is $25$-(the area of the three
triangles covering the white space), we can add the three areas and
subtract it from $25$.
This will give us an area of $\frac{5x+5y-xy}{2}$.
Victor from Dulwich College Seoul found this formula and used it to find all the possible areas of triangles on a $5 \times 5$ grid:
The possible areas are $2.5cm^2$, $4.5cm^2$, $5cm^2$, $6.5cm^2$, $7.5cm^2$, $8cm^2$, $8.5cm^2$, $9.5cm^2$,$10cm^2$, $10.5cm^2$, $11cm^2$, $11.5cm^2$, $12cm^2$, $12.5cm^2$ which is a total of $14$ areas.
Kira thought about triangles on bigger grids:
When a triangle is drawn in a $6$ by $6$ grid the area can be represented by
$3x + 3y – 0.5xy$.
When a triangle is drawn in a $7$ by $7$ grid the area can be represented by
$3.5x + 3.5y – 0.5xy$.
This shows that there is a pattern for finding the area of the triangle. $x$
and $y$ are multiplied by half of the number of squares up or down.
Let $n$ represent size of grid eg $7$ by $7$ grid is when $n=7$:
Area of triangle in square = $0.5nx + 0.5 ny - 0.5xy$ squares. | 2020-09-18 08:39:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6605156660079956, "perplexity": 758.5861038340435}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400187354.1/warc/CC-MAIN-20200918061627-20200918091627-00412.warc.gz"} |
http://www.physicsforums.com/showthread.php?p=4160631 | # Differential equation resembling to cycloid
by tom-73
Tags: cycloid, differential, equation, resembling
P: 2 What is the function corresponding to this ODE: In complex notation it obviously shows up like this: a * z''(t) + b * |z'(t)| * z'(t) + c = 0; The numerical solution shows a graph resembling to a cycloid. Thanks for any help! Tom
Homework
HW Helper
Thanks
P: 8,912
Quote by tom-73 What is the function corresponding to this ODE: In complex notation it obviously shows up like this: a * z''(t) + b * |z'(t)| * z'(t) + c = 0;
It does? I don't see how.
If you divide through by the surd and subtract the 1st eqn from the second, I believe you get something integrable.
P: 2 Thank you for your comment. I tried to devide and subtract. The problem is the term in the middle: (y' - eps*x') vs. (x' + eps*y') It makes the situation even worse - I did not succeed in finding a simplified pattern. The complex notation in my first post has been derived by simply multiplying the 2nd equation by i and adding the result to the first equation. What I investigated in the meanwhile: The cycloid ODE in complex notation should be a * z''(t) + b * z'(t) + c = 0; The only difference is the multiplication with |z'| in the middle which in fact produces a value near 1 for curtate cycloids with r1 << r0 (the point tracing out the curve is inside the circle, which rolls on a line AND it is close to the center). The ODEs in my first posts describe a phugoid, a more general form of the cycloid I suppose. It seems that the phugoid has no analytic solution. Any suggestions? Tom
Homework
HW Helper
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P: 8,912
## Differential equation resembling to cycloid
Sorry, I overlooked what happens to the RHS. My original suggestion was nonsense.
The complex notation in my first post has been derived by simply multiplying the 2nd equation by i and adding the result to the first equation.
Ah yes, I see it now. Sorry for the noise.
Related Discussions Calculus & Beyond Homework 1 Differential Equations 5 Differential Equations 7 General Physics 4 Introductory Physics Homework 2 | 2014-03-12 07:03:45 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8053750991821289, "perplexity": 724.9199063715347}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394021446249/warc/CC-MAIN-20140305121046-00081-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://www.zbmath.org/?q=an%3A0687.35071 | # zbMATH — the first resource for mathematics
Navier-Stokes equations. (English) Zbl 0687.35071
Chicago Lectures in Mathematics. Chicago, IL etc.: University of Chicago Press. ix, 190 p. $34.95/hbk;$ 14.95/pbk; £27.95/hbk; £11.95/pbk (1988).
These lecture notes are about the exciting subject of the relation between the solutions of the Navier-Stokes equations and finite-dimensional phenomena. The authors describe the results in this field, results which have been obtained recently and which for a large part are due to the authors and a.o. to Roger Temam.
Although the text is not intended as a complete course on the Navier-Stokes equations, it presents a lot of general material. We mention: existence and uniqueness of weak solutions, regularity (an adaptation of the classical $$L^ 2$$-theory), inequalities, in fact the whole technical machinery one needs to study these problems. This also includes vanishing viscosity limits, analyticity and backward uniqueness.
The heart of the matter lies in the last 3 chapters: exponential decay of volume elements, global Lyapunov exponents, Hausdorff and fractal dimension of the universal attractor, inertial manifolds.
There has been recent progress in lowering the bounds for the dimension of the universal attractor for 2D Navier-Stokes equations. This is discussed in chapter 14 together with upper bounds for bounded invariant sets in the 3D case.
This is an interesting and well-written text on a fascinating subject.
Reviewer: F.Verhulst
##### MSC:
35Q30 Navier-Stokes equations 35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations 35B41 Attractors 35Dxx Generalized solutions to partial differential equations 35B45 A priori estimates in context of PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids | 2021-03-05 12:58:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.552769124507904, "perplexity": 775.9031470803133}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178372367.74/warc/CC-MAIN-20210305122143-20210305152143-00028.warc.gz"} |
https://socratic.org/questions/how-do-you-integrate-int-x-3-cot-x-dx-using-integration-by-parts | # How do you integrate int x^3 cot x dx using integration by parts?
Nov 28, 2016
You can't.
#### Explanation:
The antiderivative of ${x}^{3} \cot x$ involves the Polylogarithm Function evaluated at imaginary values.
$I = \int {x}^{3} \cot x \mathrm{dx}$
Let $u = {x}^{3}$ so $\mathrm{du} = 3 {x}^{2} \mathrm{dx}$ and
let $\mathrm{dv} = \cot x \mathrm{dx}$ so $v = \int \cot x \mathrm{dx} = \ln \left\mid \sin \right\mid x$.
$I = {x}^{3} \ln \left\mid \sin \right\mid x - 3 \int {x}^{2} \ln \left\mid \sin \right\mid x \mathrm{dx}$
Let $u = {x}^{2}$ so $\mathrm{du} = 2 x \mathrm{dx}$
let $\mathrm{dv} = \ln \left\mid \sin \right\mid x \mathrm{dx}$ so $v = \frac{1}{2} i \left({x}^{2} + L {i}_{2} \left({e}^{2 i x}\right)\right) - x \ln \left(1 - {e}^{2 i x}\right) + x \ln \left\mid \sin \right\mid x$
Where $L {i}_{2} \left(x\right) = {\sum}_{k = 1}^{\infty} {x}^{k} / {k}^{2}$ $\text{ }$ (known as the polylogarithm or Jonquiere's function.)
At this point I'll let you finish yourself. (Because I'm out of my depth.) | 2020-12-04 14:14:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 13, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9960372447967529, "perplexity": 1824.3159441203559}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141737946.86/warc/CC-MAIN-20201204131750-20201204161750-00098.warc.gz"} |
http://openstudy.com/updates/50a9b5cee4b064039cbd18d9 | anonymous 3 years ago Alright guys I'm really stuck on this one.. I'm trying to solve a Log problem but I cant figure out what step is next... 3Ln(X + 4) -5 = 3 First step: add five to both sides 3Ln (X + 4) -5 = 3 +5 +5 3Ln (x + 4) = 8 What do I do next???
• This Question is Open
1. anonymous
Divide by 3 on both sides
2. anonymous
thn carry on
3. anonymous
so it should look something like this then, right... 3 Ln ( X + 4) = 8 ---- --- 3 3 Can you walk me through each step, please? I would really appreciate it
4. anonymous
Yes.. I will assist you... now what will be the equation?
5. anonymous
My question is to walk me through the next couple of steps after I divide by 3 on both sides. Can you show me what it should look like? Thanks..
6. anonymous
mmm.. you mean I have to write each steps??
7. anonymous
well if you could just briefly explain to me what to do after I divide by 3 that would be helpful.... I'm just completely lost on this problem. I pretty much only know how to do the first step
8. anonymous
Ok.. afet dividing by 3 you have to take inverse logarithm
9. anonymous
then you will be abl to find the value of x
10. anonymous
so what might that look like... could you write out the steps by placing the numbers in the correct spots. I know I'm asking a lot but it would really help me understand the problem better... :/
11. anonymous
actually writting step by step or direct answr is against the CoC... So I will write an example
12. zepdrix
Exponentiation is the inverse of the logarithmic function. Here's a quick example. $\large \ln x = 2$ This is a log with base e, we'll exponentiate both sides, rewrite both sides as exponents with base e. $\huge e^{\ln x}=e^2$ Since the exponential and the logarithm are inverse operations of one another, they essentially "cancel out". $\huge x=e^2$
13. anonymous
well examole by @zepdrix is sufficient , I guess... You have to work it out
14. anonymous
Ohhh ok, so natural Log (Ln) and (e) cancel out. ok. Well that does help out a little bit. Thank you for your time.
15. anonymous
welcome......
16. zepdrix
Yah performing the inverse operation of the logarithm is a bit tricky to get used to :) If you're still confused on how to perform that on your particular problem, just let us know.
17. anonymous
Thank you, I'm sure i'll figure it out. Your explanation did help. Yes, Inverse operation of Logarithm is tricky.
18. anonymous
all the best.... | 2016-10-21 18:45:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7085881233215332, "perplexity": 805.2351702455861}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988718296.19/warc/CC-MAIN-20161020183838-00077-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://www.shaalaa.com/question-bank-solutions/what-will-be-the-additive-inverse-of-39-additive-inverse-of-rational-number_159202 | # What will be the additive inverse of -39? - Mathematics
Sum
What will be the additive inverse of (-3)/9?
#### Solution
(-3)/9 "as the additive inverse of" 3/9
Is there an error in this question or solution? | 2021-04-16 00:27:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.40935036540031433, "perplexity": 1634.047574021506}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038088264.43/warc/CC-MAIN-20210415222106-20210416012106-00596.warc.gz"} |
https://math.stackexchange.com/questions/1000477/in-metric-space-x-is-connected-and-x-subset-y-subset-bar-x-prove-that-y/1000633 | In metric space, $X$ is connected and $X\subset Y \subset \bar X$, prove that $Y$ is connected.
Question: $X$ is connected and $X\subset Y\subset\bar X$, prove that $Y$ is connected.
This is one of my midterm questions this morning. I couldn't figure it out. But now I came up with this proof, so if anyone could check and see if it makes sense, I would really appreciate it (although that's not gonna change my grade).
Proof: For the purpose of arguing by contradiction, we assume that $Y$ is disconnected. Then we have nonempty, open sets $U,V$ such that $U \cup V=Y$ and $U \cap V=\emptyset$.
Since $X \subset Y$, then it's obvious that we either have $X \subset U$, or $X \subset V$. Without losing generality, we assume that $X\subset U$. Then $\bar X \subset \bar U$, then $Y \subset \bar U$, then $V \subset \bar U$.
Consider some $v \in V$. There exits some $\epsilon$ such that $B_\epsilon (x)\subset V$. At the same time, since $x\in V \subset \bar U$, $B_\epsilon(x) \cap U \neq \emptyset$. This is a contradiction since $U \cap V =\emptyset$. So $Y$ is connected.
• You seem to assume that the space is a metric space? – Asaf Karagila Oct 31 '14 at 21:59
• @AsafKaragila Oh yes. – 3x89g2 Oct 31 '14 at 22:00
We assume that $Y$ is disconnected. Then we have open sets U, V with $Y \subseteq U \cup V$, $U \cap V = \emptyset$, $Y \cap U \neq \emptyset \neq Y \cap V$. On the other hand, since $X \subseteq Y$ and X is connected, we can assume wlog that $X \subseteq U$. Now, since $Y \cap V \neq \emptyset$, there is a $z \in Y \cap V$. Since $Y \subseteq \bar X$, we have $z \in \bar X$. On the other hand, from $X \subseteq U$ and $z \in V$ and $U \cap V = \emptyset$, we get that $V$ is an open neighborhood of $z$ which doesn't contain any points of $X$. This means $z \notin \bar X$. Contradiction. So $Y$ must be connected.
The best proof i know is by using this equivalent characterization of a connected space: a space $X$ is connected if and only if every continuous function $X \rightarrow \{0,1\}$ is constant. If $X\subset Y\subset \overline{X}$ and if $Y \rightarrow \{0,1\}$ is continuous, then it restriction to $X$ is constant, and since $X$ is dense in $Y$, it follows that the function is constant, so $Y$ is connected.
• Why $X$ is dense in $Y$ ? – Empty Jan 28 at 16:24 | 2019-09-19 08:43:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9815497994422913, "perplexity": 55.3572112926233}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573465.18/warc/CC-MAIN-20190919081032-20190919103032-00277.warc.gz"} |
http://www.physicsforums.com/showpost.php?p=4267368&postcount=2 | View Single Post
Sci Advisor Thanks PF Gold P: 12,203 The (differential) equation of motion for a particle of mass m on a spring with spring constant k and a displacement x from the equilibrium position is m d2x/dt2 = -kx When you solve this, you find a solution of the form x= A sin(ωt - ∅) ∅ is an arbitrary value for the phase of the oscillation. That's where the variable ω comes from. Trig functions involve angles so ω has the dimension of an angle divided by time. Hence it's referred to as an angular frequency. There is one other point and that is that ω is in radians (as all good angles are) so is 2∏f, where f is the frequency in cycles per second (Hz). | 2014-09-03 00:10:17 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8969895243644714, "perplexity": 464.89649220669946}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1409535923940.4/warc/CC-MAIN-20140909040417-00422-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://tex.stackexchange.com/questions/616732/symbol-distance-of-double-triple-etc-integrals-of-cm | # Symbol distance of double, triple, etc. integrals of CM
Just a curiosity:
why in Computer Modern (and I think in other some fonts) the double integral signs have a high distance of default?
\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\begin{document}
$\iint_D f(x,y)\, dxdy$
\end{document}
With amsmath, the multiple integral signs are spaced using \intkern@, whose definition is
% amsmath.sty, line 654:
\def\intkern@{\mkern-6mu\mathchoice{\mkern-3mu}{}{}{}}
which means that there is a negative kern of 6mu, which is increased to 9mu in display style.
You can modify this default. Here's an example, but of course the redefinition should go in the preamble.
\documentclass[12pt]{article}
\usepackage{amsmath}
\begin{document}
\begin{center}
$\iint_D f(x,y)\,dx\,dy$
\end{center}
$\iint_D f(x,y)\, dx\,dy$
\makeatletter
\def\intkern@{\mkern-9mu\mathchoice{\mkern-4mu}{}{}{}}
\makeatother
\begin{center}
$\iint_D f(x,y)\,dx\,dy$
\end{center}
$\iint_D f(x,y)\, dx\,dy$
\end{document}
• Thank you immensely, immensely like the title of a song by Umberto Tozzi my favorite singer. Now I ask myself? But isn't the spacing \def\intkern@{\mkern-9mu\mathchoice{\mkern-4mu}{} better? Sep 25 at 22:44
• @Sebastiano It’s a question of personal preferences Sep 26 at 7:18
• I remember (but may be wrong), that the amsmath package has been updated recently. Many years ago I didn't remember that the two integral symbols had this large distance. Sep 26 at 19:11
• @Sebastiano Version 2.13, released 2000/07/18, has the same values. Sep 26 at 19:50 | 2021-10-17 18:28:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.8711041808128357, "perplexity": 3002.558444184785}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585181.6/warc/CC-MAIN-20211017175237-20211017205237-00069.warc.gz"} |
https://dsp.stackexchange.com/questions/52413/image-processing-why-is-sum-of-values-of-a-blurring-filter-1 | # Image processing - Why is sum of values of a blurring filter = 1?
Usually, blurring filters have the sum of all the values in the filter matrix equal to $$1$$.
Why is it so?
• The question contradicts the example, if the matrix that is provided is supposed to be a 3x3 convolution matrix. Can you please clarify? – A_A Oct 5 '18 at 12:25
## 2 Answers
Blurring an image means reducing its high frequencies while retaining its low frequencies.
Ususally this means a lowpass filter with a cutoff frequency of $$\omega_c$$. A standard low pass filter would have a DC response of $$1$$; i.e., $$H(0,0) = 1$$
This translates into time domain using the DTFT as: $$H(0,0) = 1 \implies \sum_n \sum_m h[n,m]e^{-j0n}e^{-j0n} =\sum_n \sum_m h[n,m] = 1$$
Which indicates that the sum of the impulse response samples equates to $$1$$.
For constant images to remain constant after blurring.
The most blurred images are flat-valued with any constant $$c$$. They are left invariant by traditional smoothers, low-pass or blur filters, hence $$\sum h_{k,l}c=c$$, for all $$c\neq 0$$ when $$\sum h_{k,l}=1$$, | 2019-04-22 18:55:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 10, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7818617820739746, "perplexity": 725.0039285051013}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578577686.60/warc/CC-MAIN-20190422175312-20190422201312-00275.warc.gz"} |
https://www.shaalaa.com/question-bank-solutions/can-the-following-groups-of-elements-be-classified-as-dobereiner-s-triad-na-si-cl-atomic-mass-of-be-9-na-23-mg-24-si-28-cl-35-ca-40-justify-your-answer-dobereiner-s-triads_102549 | # Can the Following Groups of Elements Be Classified as Dobereiner'S Triad: Na, Si, Cl Atomic Mass of Be - 9, Na - 23, Mg - 24, Si - 28, Cl - 35, Ca - 40. Justify Your Answer. - Science
Numerical
Can the following groups of elements be classified as Dobereiner's triad:
Na, Si, Cl
Atomic mass of Na - 23, Si - 28, Cl - 35.
#### Solution
According to Dobereiner's triad, the atomic mass of the second element is nearly equal to the average of the atomic masses of the first and the third elements provided the three elements are arranged in the increasing order of their atomic masses.
Na, Si, Cl:
Atomic mass of Na - 23, Si - 28, Cl - 35.
Average of atomic masses of Na and Cl = (23 + 35)/(2) = 29 which is nearly equal to the atomic mass of Si i.e. 28.
Thus, it can be considered as Dobereiner's triad.
Is there an error in this question or solution? | 2022-06-24 22:46:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.48996591567993164, "perplexity": 1766.1286223739758}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103033816.0/warc/CC-MAIN-20220624213908-20220625003908-00295.warc.gz"} |
http://physics.stackexchange.com/questions/43260/in-which-of-the-above-figures-will-the-light-bulb-be-glowing | # In which of the above figures will the light bulb be glowing?
What is principle of solution behind this induction problem (problem 29)? The problem can be find in here (problem 29).
Problem is: The five separate figures below involve a cylindrical magnet and a tiny light bulb connected to the ends of a loop of copper wire. These figures are to be used in following question. The plane of the wire loop is perpendicular to the reference axis. The states of motion of the magnet and of the loop of wire are indicated in the diagram. Speed will be represented by v and CCV represent counter clockwise. (look at the picture in here( problem 29))
29) In which of the above figures will the light bulb be glowing?
(a) I, III, IV (b) I, IV (c) I, II, IV (d) IV (e) None of these.
-
Let me ask that how do you post pictures to the questions in this website? – alvoutila Nov 2 '12 at 14:13 If you click on the little square on the question tool bar it asks you for the url of the image to upload either from your computer or from the web. – anna v Nov 2 '12 at 14:28 I tried but it failed. It takes wrong link like; [3]: i.stack.imgur.com/2cO6y.png – alvoutila Nov 2 '12 at 14:49 Maybe you haven't noticed. It displays the image right now. And, that's not a wrong link. That's where you've uploaded the image through the link you've provided :-) – Ϛѓăʑɏ βµԂԃϔ Nov 2 '12 at 15:20
Do you use formula " Faraday's law of induction states the induced electromotive force (EMF) in the wire is $E= - \frac{d \Phi_B}{dt}$ "? – alvoutila Nov 3 '12 at 11:10 @alvoutila: Yes, but carefully, because while the formula is true, it is due to two different effects (which are ultimately related by relativity)--- the Lorentz force law and the Faraday induction law, depending on whether it's the magnet or the wire that is moving. – Ron Maimon Nov 3 '12 at 13:45 | 2013-05-24 20:28:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5946558713912964, "perplexity": 481.505601274819}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368705043997/warc/CC-MAIN-20130516115043-00094-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://nag.com/numeric/MB/manual64_24_1/html/C09/c09eaf.html | Integer type: int32 int64 nag_int show int32 show int32 show int64 show int64 show nag_int show nag_int
Chapter Contents
Chapter Introduction
NAG Toolbox
NAG Toolbox: nag_wav_2d_sngl_fwd (c09ea)
Purpose
nag_wav_2d_sngl_fwd (c09ea) computes the two-dimensional discrete wavelet transform (DWT) at a single level. The initialization function nag_wav_2d_init (c09ab) must be called first to set up the DWT options.
Syntax
[ca, ch, cv, cd, ifail] = c09ea(a, icomm, 'm', m, 'n', n)
[ca, ch, cv, cd, ifail] = nag_wav_2d_sngl_fwd(a, icomm, 'm', m, 'n', n)
Description
nag_wav_2d_sngl_fwd (c09ea) computes the two-dimensional DWT of a given input data array, considered as a matrix A$A$, at a single level. For a chosen wavelet filter pair, the output coefficients are obtained by applying convolution and downsampling by two to the input, A$A$, first over columns and then to the result over rows. The matrix of approximation (or smooth) coefficients, Ca${C}_{a}$, is produced by the low pass filter over columns and rows; the matrix of horizontal coefficients, Ch${C}_{h}$, is produced by the high pass filter over columns and the low pass filter over rows; the matrix of vertical coefficients, Cv${C}_{v}$, is produced by the low pass filter over columns and the high pass filter over rows; and the matrix of diagonal coefficients, Cd${C}_{d}$, is produced by the high pass filter over columns and rows. To reduce distortion effects at the ends of the data array, several end extension methods are commonly used. Those provided are: periodic or circular convolution end extension, half-point symmetric end extension, whole-point symmetric end extension and zero end extension. The total number, nct${n}_{\mathrm{ct}}$, of coefficients computed for Ca${C}_{a}$, Ch${C}_{h}$, Cv${C}_{v}$, and Cd${C}_{d}$ together and the number of columns of each coefficients matrix, ncn${n}_{\mathrm{cn}}$, are returned by the initialization function nag_wav_2d_init (c09ab). These values can be used to calculate the number of rows of each coefficients matrix, ncm${n}_{\mathrm{cm}}$, using the formula ncm = nct / (4ncn)${n}_{\mathrm{cm}}={n}_{\mathrm{ct}}/\left(4{n}_{\mathrm{cn}}\right)$.
References
Daubechies I (1992) Ten Lectures on Wavelets SIAM, Philadelphia
Parameters
Compulsory Input Parameters
1: a(lda,n) – double array
lda, the first dimension of the array, must satisfy the constraint ldam$\mathit{lda}\ge {\mathbf{m}}$.
The m$m$ by n$n$ data matrix A$A$.
2: icomm(180$180$) – int64int32nag_int array
Contains details of the discrete wavelet transform and the problem dimension as setup in the call to the initialization function nag_wav_2d_init (c09ab).
Optional Input Parameters
1: m – int64int32nag_int scalar
Default: The first dimension of the array a.
Number of rows, m$m$, of data matrix A$A$.
Constraint: this must be the same as the value m passed to the initialization function nag_wav_2d_init (c09ab).
2: n – int64int32nag_int scalar
Default: The second dimension of the array a.
Number of columns, n$n$, of data matrix A$A$.
Constraint: this must be the same as the value n passed to the initialization function nag_wav_2d_init (c09ab).
Input Parameters Omitted from the MATLAB Interface
lda ldca ldch ldcv ldcd
Output Parameters
1: ca(ldca, : $:$) – double array
The first dimension of the array ca will be ncm${n}_{\mathrm{cm}}$ where ncm = nct / (4ncn)${n}_{\mathrm{cm}}={n}_{\mathrm{ct}}/\left(4{n}_{\mathrm{cn}}\right)$ and ncn${n}_{\mathrm{cn}}$, nct${n}_{\mathrm{ct}}$ are returned by the initialization function nag_wav_2d_init (c09ab)
The second dimension of the array will be ncn$\mathit{ncn}$ where ncn${n}_{\mathrm{cn}}$ is the parameter nwcn returned by function nag_wav_2d_init (c09ab)
ldcancm$\mathit{ldca}\ge {n}_{\mathrm{cm}}$ where ncm = nct / (4ncn)${n}_{\mathrm{cm}}={n}_{\mathrm{ct}}/\left(4{n}_{\mathrm{cn}}\right)$ and ncn${n}_{\mathrm{cn}}$, nct${n}_{\mathrm{ct}}$ are returned by the initialization function nag_wav_2d_init (c09ab).
Contains the ncm${n}_{\mathrm{cm}}$ by ncn${n}_{\mathrm{cn}}$ matrix of approximation coefficients, Ca${C}_{a}$.
2: ch(ldch, : $:$) – double array
The first dimension of the array ch will be ncm${n}_{\mathrm{cm}}$ where ncm = nct / (4ncn)${n}_{\mathrm{cm}}={n}_{\mathrm{ct}}/\left(4{n}_{\mathrm{cn}}\right)$ and ncn${n}_{\mathrm{cn}}$, nct${n}_{\mathrm{ct}}$ are returned by the initialization function nag_wav_2d_init (c09ab)
The second dimension of the array will be ncn$\mathit{ncn}$ where ncn${n}_{\mathrm{cn}}$ is the parameter nwcn returned by function nag_wav_2d_init (c09ab)
ldchncm$\mathit{ldch}\ge {n}_{\mathrm{cm}}$ where ncm = nct / (4ncn)${n}_{\mathrm{cm}}={n}_{\mathrm{ct}}/\left(4{n}_{\mathrm{cn}}\right)$ and ncn${n}_{\mathrm{cn}}$, nct${n}_{\mathrm{ct}}$ are returned by the initialization function nag_wav_2d_init (c09ab).
Contains the ncm${n}_{\mathrm{cm}}$ by ncn${n}_{\mathrm{cn}}$ matrix of horizontal coefficients, Ch${C}_{h}$.
3: cv(ldcv, : $:$) – double array
The first dimension of the array cv will be ncm${n}_{\mathrm{cm}}$ where ncm = nct / (4ncn)${n}_{\mathrm{cm}}={n}_{\mathrm{ct}}/\left(4{n}_{\mathrm{cn}}\right)$ and ncn${n}_{\mathrm{cn}}$, nct${n}_{\mathrm{ct}}$ are returned by the initialization function nag_wav_2d_init (c09ab)
The second dimension of the array will be ncn$\mathit{ncn}$ where ncn${n}_{\mathrm{cn}}$ is the parameter nwcn returned by function nag_wav_2d_init (c09ab)
ldcvncm$\mathit{ldcv}\ge {n}_{\mathrm{cm}}$ where ncm = nct / (4ncn)${n}_{\mathrm{cm}}={n}_{\mathrm{ct}}/\left(4{n}_{\mathrm{cn}}\right)$ and ncn${n}_{\mathrm{cn}}$, nct${n}_{\mathrm{ct}}$ are returned by the initialization function nag_wav_2d_init (c09ab).
Contains the ncm${n}_{\mathrm{cm}}$ by ncn${n}_{\mathrm{cn}}$ matrix of vertical coefficients, Cv${C}_{v}$.
4: cd(ldcd, : $:$) – double array
The first dimension of the array cd will be ncm${n}_{\mathrm{cm}}$ where ncm = nct / (4ncn)${n}_{\mathrm{cm}}={n}_{\mathrm{ct}}/\left(4{n}_{\mathrm{cn}}\right)$ and ncn${n}_{\mathrm{cn}}$, nct${n}_{\mathrm{ct}}$ are returned by the initialization function nag_wav_2d_init (c09ab)
The second dimension of the array will be ncn$\mathit{ncn}$ where ncn${n}_{\mathrm{cn}}$ is the parameter nwcn returned by function nag_wav_2d_init (c09ab)
ldcdncm$\mathit{ldcd}\ge {n}_{\mathrm{cm}}$ where ncm = nct / (4ncn)${n}_{\mathrm{cm}}={n}_{\mathrm{ct}}/\left(4{n}_{\mathrm{cn}}\right)$ and ncn${n}_{\mathrm{cn}}$, nct${n}_{\mathrm{ct}}$ are returned by the initialization function nag_wav_2d_init (c09ab).
Contains the ncm${n}_{\mathrm{cm}}$ by ncn${n}_{\mathrm{cn}}$ matrix of diagonal coefficients, Cd${C}_{d}$.
5: ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).
Error Indicators and Warnings
Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
Constraint: m = m${\mathbf{m}}=m$, the value of m on initialization (see nag_wav_2d_init (c09ab)).
Constraint: n = n${\mathbf{n}}=n$, the value of n on initialization (see nag_wav_2d_init (c09ab)).
ifail = 2${\mathbf{ifail}}=2$
Constraint: ldam$\mathit{lda}\ge {\mathbf{m}}$.
ifail = 3${\mathbf{ifail}}=3$
ldca is too small, the number of wavelet coefficients in the first dimension.
ldcd is too small, the number of wavelet coefficients in the first dimension.
ldch is too small, the number of wavelet coefficients in the first dimension.
ldcv is too small, the number of wavelet coefficients in the first dimension.
ifail = 6${\mathbf{ifail}}=6$
Either the initialization function has not been called first or icomm has been corrupted.
Either the initialization function was called with wtrans = 'M'${\mathbf{wtrans}}=\text{'M'}$ or icomm has been corrupted.
ifail = 999${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
Accuracy
The accuracy of the wavelet transform depends only on the floating point operations used in the convolution and downsampling and should thus be close to machine precision.
None.
Example
```function nag_wav_2d_sngl_fwd_example
m = int64(6);
n = int64(6);
wavnam = 'DB4';
mode = 'Half';
wtrans = 'Single level';
a = [8, 7, 3, 3, 1, 1;
4, 6, 1, 5, 2, 9;
8, 1, 4, 9, 3, 7;
9, 3, 8, 2, 4, 3;
1, 3, 7, 1, 5, 2;
4, 3, 7, 7, 6, 1];
fprintf('\nInput data a:\n');
disp(a);
[nwl, nf, nwct, nwcn, icomm, ifail] = nag_wav_2d_init(wavnam, wtrans, mode, m, n);
nwcm = double(nwct)/(4*double(nwcn));
% Perform Discrete Wavelet transform
[ca, ch, cv, cd, ifail] = nag_wav_2d_sngl_fwd(a, icomm);
fprintf('Approximation coefficients CA:\n');
disp(ca);
fprintf('Diagonal coefficients CD:\n');
disp(cd);
fprintf('Horizontal coefficients CH:\n');
disp(ch);
fprintf('Vertical coefficients CV:\n');
disp(cv);
% Reconstruct original data
[b, ifail] = nag_wav_2d_sngl_inv(m, n, ca, ch, cv, cd, icomm);
fprintf('Reconstruction b:\n');
disp(b);
```
```
Input data a:
8 7 3 3 1 1
4 6 1 5 2 9
8 1 4 9 3 7
9 3 8 2 4 3
1 3 7 1 5 2
4 3 7 7 6 1
Approximation coefficients CA:
6.3591 10.3477 8.0995 10.3210 8.7587 3.5783
11.5754 6.3762 12.1704 7.4521 8.6977 14.8535
2.0630 8.4499 15.4726 12.1764 3.8920 2.7112
10.2143 6.2445 13.8571 8.1060 7.7701 13.2127
6.3353 8.7805 10.2727 10.0472 6.8614 7.5814
11.7141 11.1018 5.2923 8.1272 14.5540 2.5729
Diagonal coefficients CD:
0.4777 1.0230 -0.3147 0.0625 0.0831 -1.3316
1.0689 1.5671 -2.1422 0.5565 1.7593 -2.8097
-0.9555 -1.9276 0.9195 -0.2228 -0.5125 2.6989
0.2899 0.4453 -0.5695 0.1541 0.4749 -0.7946
0.4944 1.4145 0.3488 -0.1187 -0.6212 -1.5177
-1.3753 -2.5224 1.7581 -0.4316 -1.1835 3.7547
Horizontal coefficients CH:
0.4100 -0.1827 1.5354 0.0784 0.8101 -1.3594
2.3496 -0.9422 2.3780 -1.0540 2.7743 -2.2648
-1.2690 0.0152 -6.9338 -1.7435 -1.6917 1.2388
0.6317 -0.0969 2.3300 0.4637 0.6365 -0.1162
-0.2343 0.3923 5.5457 2.1818 0.2103 -0.8573
-1.8880 0.8142 -4.8552 0.0736 -2.7395 3.3590
Vertical coefficients CV:
1.5365 5.9678 3.4309 -1.0585 -5.0275 -4.8492
0.6779 -0.0294 -5.3274 1.6483 4.8689 -1.8383
-1.1065 -2.8791 0.1535 0.0982 0.8417 2.8923
-0.1359 -2.6633 -5.8549 1.8440 6.2403 0.5697
1.4244 5.2140 1.6410 -0.4669 -3.2369 -4.5757
1.0288 2.2521 0.0574 -0.1359 -0.5170 -2.6854
Reconstruction b:
8.0000 7.0000 3.0000 3.0000 1.0000 1.0000
4.0000 6.0000 1.0000 5.0000 2.0000 9.0000
8.0000 1.0000 4.0000 9.0000 3.0000 7.0000
9.0000 3.0000 8.0000 2.0000 4.0000 3.0000
1.0000 3.0000 7.0000 1.0000 5.0000 2.0000
4.0000 3.0000 7.0000 7.0000 6.0000 1.0000
```
```function c09ea_example
m = int64(6);
n = int64(6);
wavnam = 'DB4';
mode = 'Half';
wtrans = 'Single level';
a = [8, 7, 3, 3, 1, 1;
4, 6, 1, 5, 2, 9;
8, 1, 4, 9, 3, 7;
9, 3, 8, 2, 4, 3;
1, 3, 7, 1, 5, 2;
4, 3, 7, 7, 6, 1];
fprintf('\nInput data a:\n');
disp(a);
[nwl, nf, nwct, nwcn, icomm, ifail] = c09ab(wavnam, wtrans, mode, m, n);
nwcm = double(nwct)/(4*double(nwcn));
% Perform Discrete Wavelet transform
[ca, ch, cv, cd, ifail] = c09ea(a, icomm);
fprintf('Approximation coefficients CA:\n');
disp(ca);
fprintf('Diagonal coefficients CD:\n');
disp(cd);
fprintf('Horizontal coefficients CH:\n');
disp(ch);
fprintf('Vertical coefficients CV:\n');
disp(cv);
% Reconstruct original data
[b, ifail] = c09eb(m, n, ca, ch, cv, cd, icomm);
fprintf('Reconstruction b:\n');
disp(b);
```
```
Input data a:
8 7 3 3 1 1
4 6 1 5 2 9
8 1 4 9 3 7
9 3 8 2 4 3
1 3 7 1 5 2
4 3 7 7 6 1
Approximation coefficients CA:
6.3591 10.3477 8.0995 10.3210 8.7587 3.5783
11.5754 6.3762 12.1704 7.4521 8.6977 14.8535
2.0630 8.4499 15.4726 12.1764 3.8920 2.7112
10.2143 6.2445 13.8571 8.1060 7.7701 13.2127
6.3353 8.7805 10.2727 10.0472 6.8614 7.5814
11.7141 11.1018 5.2923 8.1272 14.5540 2.5729
Diagonal coefficients CD:
0.4777 1.0230 -0.3147 0.0625 0.0831 -1.3316
1.0689 1.5671 -2.1422 0.5565 1.7593 -2.8097
-0.9555 -1.9276 0.9195 -0.2228 -0.5125 2.6989
0.2899 0.4453 -0.5695 0.1541 0.4749 -0.7946
0.4944 1.4145 0.3488 -0.1187 -0.6212 -1.5177
-1.3753 -2.5224 1.7581 -0.4316 -1.1835 3.7547
Horizontal coefficients CH:
0.4100 -0.1827 1.5354 0.0784 0.8101 -1.3594
2.3496 -0.9422 2.3780 -1.0540 2.7743 -2.2648
-1.2690 0.0152 -6.9338 -1.7435 -1.6917 1.2388
0.6317 -0.0969 2.3300 0.4637 0.6365 -0.1162
-0.2343 0.3923 5.5457 2.1818 0.2103 -0.8573
-1.8880 0.8142 -4.8552 0.0736 -2.7395 3.3590
Vertical coefficients CV:
1.5365 5.9678 3.4309 -1.0585 -5.0275 -4.8492
0.6779 -0.0294 -5.3274 1.6483 4.8689 -1.8383
-1.1065 -2.8791 0.1535 0.0982 0.8417 2.8923
-0.1359 -2.6633 -5.8549 1.8440 6.2403 0.5697
1.4244 5.2140 1.6410 -0.4669 -3.2369 -4.5757
1.0288 2.2521 0.0574 -0.1359 -0.5170 -2.6854
Reconstruction b:
8.0000 7.0000 3.0000 3.0000 1.0000 1.0000
4.0000 6.0000 1.0000 5.0000 2.0000 9.0000
8.0000 1.0000 4.0000 9.0000 3.0000 7.0000
9.0000 3.0000 8.0000 2.0000 4.0000 3.0000
1.0000 3.0000 7.0000 1.0000 5.0000 2.0000
4.0000 3.0000 7.0000 7.0000 6.0000 1.0000
```
Chapter Contents
Chapter Introduction
NAG Toolbox
© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2013 | 2015-07-30 18:25:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 89, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8599368929862976, "perplexity": 6364.917230797905}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042987552.57/warc/CC-MAIN-20150728002307-00271-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://www.jobilize.com/physics3/section/problems-blackbody-radiation-by-openstax?qcr=www.quizover.com | # 6.1 Blackbody radiation (Page 6/15)
Page 6 / 15
Check Your Understanding A molecule is vibrating at a frequency of $5.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{14}\text{Hz}.$ What is the smallest spacing between its vibrational energy levels?
$3.3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-19}\text{J}$
## Quantum theory applied to a classical oscillator
A 1.0-kg mass oscillates at the end of a spring with a spring constant of 1000 N/m. The amplitude of these oscillations is 0.10 m. Use the concept of quantization to find the energy spacing for this classical oscillator. Is the energy quantization significant for macroscopic systems, such as this oscillator?
## Strategy
We use [link] as though the system were a quantum oscillator, but with the frequency f of the mass vibrating on a spring. To evaluate whether or not quantization has a significant effect, we compare the quantum energy spacing with the macroscopic total energy of this classical oscillator.
## Solution
For the spring constant, $k=1.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\text{N/m},$ the frequency f of the mass, $m=1.0\text{kg},$ is
$f=\frac{1}{2\pi }\sqrt{\frac{k}{m}}=\frac{1}{2\pi }\sqrt{\frac{1.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\text{N/m}}{1.0\text{kg}}}\simeq 5.0\phantom{\rule{0.2em}{0ex}}\text{Hz}$
The energy quantum that corresponds to this frequency is
$\text{Δ}E=hf=\left(6.626\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-34}\text{J}·\text{s}\right)\left(5.0\text{Hz}\right)=3.3\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-33}\text{J}$
When vibrations have amplitude $A=0.10\text{m},$ the energy of oscillations is
$E=\frac{1}{2}k{A}^{2}=\frac{1}{2}\left(1000\text{N/m}\right){\left(0.1\text{m}\right)}^{2}=5.0\text{J}$
## Significance
Thus, for a classical oscillator, we have $\text{Δ}E\phantom{\rule{0.1em}{0ex}}\text{/}\phantom{\rule{0.1em}{0ex}}E\approx {10}^{-34}.$ We see that the separation of the energy levels is immeasurably small. Therefore, for all practical purposes, the energy of a classical oscillator takes on continuous values. This is why classical principles may be applied to macroscopic systems encountered in everyday life without loss of accuracy.
Check Your Understanding Would the result in [link] be different if the mass were not 1.0 kg but a tiny mass of 1.0 µ g, and the amplitude of vibrations were 0.10 µ m?
No, because then $\text{Δ}E\phantom{\rule{0.1em}{0ex}}\text{/}\phantom{\rule{0.1em}{0ex}}E\approx {10}^{-21}$
When Planck first published his result, the hypothesis of energy quanta was not taken seriously by the physics community because it did not follow from any established physics theory at that time. It was perceived, even by Planck himself, as a useful mathematical trick that led to a good theoretical “fit” to the experimental curve. This perception was changed in 1905 when Einstein published his explanation of the photoelectric effect, in which he gave Planck’s energy quantum a new meaning: that of a particle of light.
## Summary
• All bodies radiate energy. The amount of radiation a body emits depends on its temperature. The experimental Wien’s displacement law states that the hotter the body, the shorter the wavelength corresponding to the emission peak in the radiation curve. The experimental Stefan’s law states that the total power of radiation emitted across the entire spectrum of wavelengths at a given temperature is proportional to the fourth power of the Kelvin temperature of the radiating body.
• Absorption and emission of radiation are studied within the model of a blackbody. In the classical approach, the exchange of energy between radiation and cavity walls is continuous. The classical approach does not explain the blackbody radiation curve.
• To explain the blackbody radiation curve, Planck assumed that the exchange of energy between radiation and cavity walls takes place only in discrete quanta of energy. Planck’s hypothesis of energy quanta led to the theoretical Planck’s radiation law, which agrees with the experimental blackbody radiation curve; it also explains Wien’s and Stefan’s laws.
## Conceptual questions
Which surface has a higher temperature – the surface of a yellow star or that of a red star?
yellow
Describe what you would see when looking at a body whose temperature is increased from 1000 K to 1,000,000 K.
Explain the color changes in a hot body as its temperature is increased.
goes from red to violet through the rainbow of colors
Speculate as to why UV light causes sunburn, whereas visible light does not.
Two cavity radiators are constructed with walls made of different metals. At the same temperature, how would their radiation spectra differ?
would not differ
Discuss why some bodies appear black, other bodies appear red, and still other bodies appear white.
If everything radiates electromagnetic energy, why can we not see objects at room temperature in a dark room?
human eye does not see IR radiation
How much does the power radiated by a blackbody increase when its temperature (in K) is tripled?
## Problems
A 200-W heater emits a 1.5-µm radiation. (a) What value of the energy quantum does it emit? (b) Assuming that the specific heat of a 4.0-kg body is $0.83\text{kcal}\phantom{\rule{0.1em}{0ex}}\text{/}\phantom{\rule{0.1em}{0ex}}\text{kg}·\text{K},$ how many of these photons must be absorbed by the body to increase its temperature by 2 K? (c) How long does the heating process in (b) take, assuming that all radiation emitted by the heater gets absorbed by the body?
a. 0.81 eV; b. $2.1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{23};$ c. 2 min 20 sec
A 900-W microwave generator in an oven generates energy quanta of frequency 2560 MHz. (a) How many energy quanta does it emit per second? (b) How many energy quanta must be absorbed by a pasta dish placed in the radiation cavity to increase its temperature by 45.0 K? Assume that the dish has a mass of 0.5 kg and that its specific heat is $0.9\phantom{\rule{0.2em}{0ex}}\text{kcal}\phantom{\rule{0.1em}{0ex}}\text{/}\phantom{\rule{0.1em}{0ex}}\text{kg}·\text{K}.$ (c) Assume that all energy quanta emitted by the generator are absorbed by the pasta dish. How long must we wait until the dish in (b) is ready?
(a) For what temperature is the peak of blackbody radiation spectrum at 400 nm? (b) If the temperature of a blackbody is 800 K, at what wavelength does it radiate the most energy?
a. 7245 K; b. 3.62 μm
The tungsten elements of incandescent light bulbs operate at 3200 K. At what frequency does the filament radiate maximum energy?
Interstellar space is filled with radiation of wavelength $970\text{μ}\text{m.}$ This radiation is considered to be a remnant of the “big bang.” What is the corresponding blackbody temperature of this radiation?
The radiant energy from the sun reaches its maximum at a wavelength of about 500.0 nm. What is the approximate temperature of the sun’s surface?
what is cathodic protection
its just a technique used for the protection of a metal from corrosion by making it cathode of an electrochemical cell.
akif
what is interferometer
Show that n1Sino1=n2Sino2
what's propagation
is it in context of waves?
Edgar
It is the manner of motion of the energy whether mechanical(requiring elastic medium)or electromagnetic(non interference with medium)
Edgar
determine displacement cat any time t for a body of mass 2kg under a time varrying force ft=bt³+csinkt
A round diaphragm S with diameter of d = 0.05 is used as light source in Michelson interferometer shown on the picture. The diaphragm is illuminated by parallel beam of monochromatic light with wavelength of λ = 0.6 μm. The distances are A B = 30, A C = 10 . The interference picture is in the form of concentric circles and is observed on the screen placed in the focal plane of the lens. Estimate the number of interference rings m observed near the main diffractive maximum.
A Pb wire wound in a tight solenoid of diameter of 4.0 mm is cooled to a temperature of 5.0 K. The wire is connected in series with a 50-Ωresistor and a variable source of emf. As the emf is increased, what value does it have when the superconductivity of the wire is destroyed?
how does colour appear in thin films
hii
Sonu
hao
Naorem
hello
Naorem
hiiiiii
ram
🎓📖
Deepika
yaaa ☺
Deepika
ok
Naorem
hii
PALAK
in the wave equation y=Asin(kx-wt+¢) what does k and w stand for.
derivation of lateral shieft
hi
Imran
total binding energy of ionic crystal at equilibrium is
How does, ray of light coming form focus, behaves in concave mirror after refraction?
Sushant
What is motion
Anything which changes itself with respect to time or surrounding
Sushant
good
Chemist
and what's time? is time everywhere same
Chemist
No
Sushant
how can u say that
Chemist
do u know about black hole
Chemist
Not so more
Sushant
DHEERAJ
Sushant
But ask anything changes itself with respect to time or surrounding A Not any harmful radiation
DHEERAJ
explain cavendish experiment to determine the value of gravitational concept.
Cavendish Experiment to Measure Gravitational Constant. ... This experiment used a torsion balance device to attract lead balls together, measuring the torque on a wire and equating it to the gravitational force between the balls. Then by a complex derivation, the value of G was determined.
Triio
For the question about the scuba instructor's head above the pool, how did you arrive at this answer? What is the process? | 2019-10-16 21:57:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 14, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5743131637573242, "perplexity": 814.1013425041809}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986670928.29/warc/CC-MAIN-20191016213112-20191017000612-00284.warc.gz"} |
https://www.albert.io/learn/ap-physics-1-and-2/question/equivalent-resistance-parallel-circuit | Limited access
Five identical $20\ \Omega$ resistors are connected in parallel.
What is the equivalent resistance of the group of resistors?
A
$4 \space\Omega$
B
$20 \space\Omega$
C
$25 \space\Omega$
D
$100 \space\Omega$ | 2017-03-29 07:24:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.40652117133140564, "perplexity": 1863.3607554978882}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218190234.0/warc/CC-MAIN-20170322212950-00625-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://gmw.globalmathproject.org/station/I8S8B | 05 820 897
### Question 1
Performing addition and subtraction with decimals in a $1 \leftarrow 10$ machine is no different from performing addition and subtraction in a $1 \leftarrow 10$ machine without decimals.
For example, here is a picture of $22.37+5.841$.
We see the answer $2|7.11|11|1$. Just add left to right and don’t worry about explosions!
Now with explosions we obtain an answer that society understand. We get $28.211$. (Check this!)
And here is $10.23-4.57$.
We see the answer, after some annihilations, as $1|-4.-3|-4$. (Do you see this?)
Now with some unexplosions, we can fix up this answer for society to read. We get $10.23-4.57=5.66$.
What is $0.05+0.006$?
What is $0.05-0.006$?
Can you see the answer with dots-and-boxes?
### Question 2
Agatha says that computing $0.0348+0.0057$ is essentially the same work as computing $348+57$ in whole numbers. Do you agree?
Percy says that computing $0.0852+0.037$ is essentially the same work as computing $852+37$ in whole numbers. Do you agree?
You can either play with some of the optional stations below or go to the next island!
Register NOW and unlock all islands! | 2021-10-21 11:03:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 14, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8294942378997803, "perplexity": 808.4344684208061}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585405.74/warc/CC-MAIN-20211021102435-20211021132435-00299.warc.gz"} |
https://socratic.org/questions/how-do-you-find-three-consecutive-odd-integers-whose-sum-is-69 | # How do you find three consecutive odd integers whose sum is 69?
Jun 5, 2015
Let the smallest of the three consecutive odd numbers be $x$
The next two consecutive odd integers can be written as $\left(x + 2\right) , \left(x + 4\right)$
$\left(x\right) + \left(x + 2\right) + \left(x + 4\right) = 69$
$3 x + 6 = 69$
$3 x = 63$
color(purple)(x = 63/ 3 = 21
so ,$x + 2 = 23 , x + 4 = 25$
The numbers are color(purple)(21,23,25 | 2019-07-16 02:20:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 8, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5137125253677368, "perplexity": 773.0085044008871}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195524475.48/warc/CC-MAIN-20190716015213-20190716041213-00323.warc.gz"} |
https://www.jobilize.com/physics1/section/average-acceleration-average-and-instantaneous-acceleration-by-opensta?qcr=www.quizover.com | # 3.3 Average and instantaneous acceleration
Page 1 / 7
By the end of this section, you will be able to:
• Calculate the average acceleration between two points in time.
• Calculate the instantaneous acceleration given the functional form of velocity.
• Explain the vector nature of instantaneous acceleration and velocity.
• Explain the difference between average acceleration and instantaneous acceleration.
• Find instantaneous acceleration at a specified time on a graph of velocity versus time.
The importance of understanding acceleration spans our day-to-day experience, as well as the vast reaches of outer space and the tiny world of subatomic physics. In everyday conversation, to accelerate means to speed up; applying the brake pedal causes a vehicle to slow down. We are familiar with the acceleration of our car, for example. The greater the acceleration, the greater the change in velocity over a given time. Acceleration is widely seen in experimental physics. In linear particle accelerator experiments, for example, subatomic particles are accelerated to very high velocities in collision experiments, which tell us information about the structure of the subatomic world as well as the origin of the universe. In space, cosmic rays are subatomic particles that have been accelerated to very high energies in supernovas (exploding massive stars) and active galactic nuclei. It is important to understand the processes that accelerate cosmic rays because these rays contain highly penetrating radiation that can damage electronics flown on spacecraft, for example.
## Average acceleration
The formal definition of acceleration is consistent with these notions just described, but is more inclusive.
## Average acceleration
Average acceleration is the rate at which velocity changes:
$\stackrel{\text{–}}{a}=\frac{\text{Δ}v}{\text{Δ}t}=\frac{{v}_{\text{f}}-{v}_{0}}{{t}_{\text{f}}-{t}_{0}},$
where $\stackrel{\text{−}}{a}$ is average acceleration , v is velocity, and t is time. (The bar over the a means average acceleration.)
Because acceleration is velocity in meters divided by time in seconds, the SI units for acceleration are often abbreviated m/s 2 —that is, meters per second squared or meters per second per second. This literally means by how many meters per second the velocity changes every second. Recall that velocity is a vector—it has both magnitude and direction—which means that a change in velocity can be a change in magnitude (or speed), but it can also be a change in direction. For example, if a runner traveling at 10 km/h due east slows to a stop, reverses direction, continues her run at 10 km/h due west, her velocity has changed as a result of the change in direction, although the magnitude of the velocity is the same in both directions. Thus, acceleration occurs when velocity changes in magnitude (an increase or decrease in speed) or in direction, or both.
## Acceleration as a vector
Acceleration is a vector in the same direction as the change in velocity, $\text{Δ}v$ . Since velocity is a vector, it can change in magnitude or in direction, or both. Acceleration is, therefore, a change in speed or direction, or both.
Keep in mind that although acceleration is in the direction of the change in velocity, it is not always in the direction of motion. When an object slows down, its acceleration is opposite to the direction of its motion. Although this is commonly referred to as deceleration [link] , we say the train is accelerating in a direction opposite to its direction of motion.
#### Questions & Answers
what is a wave?
DAVID Reply
show that coefficient of friction of solid block inclined at an angle is equivalent to trignometric tangent of angle
DAVID
thanks for that definition.
Dodou Reply
Hi everyone please can dere be motion without force?
Lafon
no...
Enyia
Thanks
Lafon
hi
Omomaro
whats is schrodinger equation
Omomaro
l went spiral spring
Xalat
what is position?
Adhar Reply
position is simply where you are or where you were
Shii
position is the location of an object with respect to a two or three dimensional axes or space.
Bamidele
Can dere be motion without force?
Lafon
what is the law of homogeinity?
auson Reply
two electric lines of force never interested each other. why?
Sujit Reply
proof that for BBC lattice structure 4r\root 5 and find Apf for the BBC structure
Eric Reply
what is physics?
Abdulaziz Reply
physics is deine as the specific measrument of of volume, area,nd distances...
Olakojo
if a string of 2m is suspended an an extended 3m elasticity is been applied.... is hooks law obeyed?
Enyia
if a string of 2m is suspended an an extended 3m elasticity is been applied.... is hooks law obeyed?
Enyia
yes
Alex
proof that for a BBC lattice structure a= 4r/ root 5 find the APF for the BBC structure
Eric
if a string of 2m is suspended an an extended 3m elasticity is been applied.... is hooks law obeyed?
Enyia Reply
tell me conceptual quetions of mechanics
Syeda Reply
I want to solve a physical question
ahmed
ok
PUBG
a displacement vector has a magnitude of 1.62km and point due north . another displacement vector B has a magnitude of 2.48 km and points due east.determine the magnitude and direction of (a) a+ b and (b) a_ b
Kou Reply
quantum
George
a+b=2.9
SUNJO
a+b
Yekeen
use Pythogorous
Dhritwan
A student opens a 12kgs door by applying a constant force of 40N at a perpendicular distance of 0.9m from the hinges. if the door is 2.0m high and 1.0m wide determine the magnitude of the angular acceleration of the door. ( assume that the door rotates freely on its hinges.) please assist me to d
Mike
what is conditions met to produce shm
Enocy Reply
what is shm
Manzoor
shm?
Grant
Why is Maxwell saying that light is an electromagnetic wave?
Bong
1st condition; It(th e BBC's system) must have some inertia which will enable it to possess Kinetic energy 2. must be able to store potential energy
Calleb
I meant "the system" not the BBC'S....."
Calleb
what a answer bro
Manzoor
kindly tell us the name of your university
Manzoor
GUlam Ishaq Khan INSTITUTE of engineering science
ali
Department of Environment Ionian University Zante Greece
why light wave travel faster than sounds
ALI Reply
Why light travel faster than sounds?
ALI
Light travel faster than sound because it does not need any medium to travel through.
alhassan
when an aeroplane flies....why it does not fall on the earth?
Frazali
As an aeroplane moves, it hits a wind,we have the wind flowing at the upper and lower zone of the aeroplane, the one that is moving on the upper zone moves at a greater speed than that of the lower zone, this creates a low pressure on the upper zone and a greater pressure at the lower zone.
Kipkoech
which thing of aeroplane moves it upward?
Frazali
good question
Manzoor
about force
Barataa
am pleased to join the group
Nesru
yea
caleb
It a privilege to be here
olajire
hi
Awode
hello
Manzoor
Light speed is more than sound speed. C=3×10*8m/s V=320-340 m/s
siva
A body of mass 2kg slides down a rough plane inclined to horizontal at 30degrees. find the energy that is wasted as a result of friction if the co-efficient of kinetic f
official Reply
ten applications of Newton's second law of motion
Alale Reply
Calculate the volume at S.T.P of a gas whose volume at -5° and 746 mmHg
Mlungisi Reply
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By Mldelatte By Rhodes | 2019-11-20 09:18:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 3, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5685712695121765, "perplexity": 1325.1277101565704}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670535.9/warc/CC-MAIN-20191120083921-20191120111921-00270.warc.gz"} |
https://cs.stackexchange.com/questions/57541/understanding-dpa | # Understanding DPA
DPA denotes “Deterministic Pushdown Automata”.
Is the following (*) true regarding DPA:
After all symbols of the input sequence are consumed, the input sequence is accepted either (1) if the stack is empty or (2) if the current state of the DPA is an accepting one, or both.
My book is fairly vague on the subject (it doesn't mention it) and Wikipedia, in my opinion, doesn't offer a better explanation either:
There are two possible acceptance criteria: acceptance by empty stack and acceptance by final state. The two are not equivalent for the deterministic pushdown automaton (although they are for the non-deterministic pushdown automaton). The languages accepted by empty stack are the languages that are accepted by final state, as well as have no word in the language that is the prefix of another word in the language.
(What I don't understand is the equivalence they mention, and how does that influence the acceptance of a word.)
If my statement (*) doesn't hold, how would I determine, given the definition of a DPA, which condition are input sequences supposed to fulfill? Would I assume one or the other as the preferred one?
• What is your statement? – Auberon May 16 '16 at 21:06
• @Auberon I hope it's clear now. – C. White May 16 '16 at 21:09
You can define the language of a PDA in two different ways:
1. For a PDA $M$, $L_1(M)$ is the set of all words which can be processed by $M$ in such a way that at the end, the stack is empty.
2. For a PDA $M$, $L_2(M)$ is the set of all words which can be processed by $M$ in such a way that at the end, $M$ is at an accepting state.
A basic theorem shows that both acceptance criteria are equivalent:
For each language $L$ the following holds: there exists a PDA $M_1$ such that $L_1(M_1) = L$ iff there exists a PDA $M_2$ such that $L_2(M_2) = L$.
A language $L$ is context-free if there exists a PDA $M$ such that $L_1(M) = L$. Equivalently, a language $L$ is context-free if there exists a PDA $M$ such that $L_2(M) = L$. The two definitions are equivalent due to the theorem.
Some PDAs are deterministic – call those DPAs. The basic theorem doesn't hold for DPAs. According to Wikipedia, there exists a DPA $M_1$ such that $L_1(M_1) = L$ iff $L$ is prefix-free (no word is a prefix of another word) and there exists a DPA $M_2$ such that $L_2(M_2) = L$.
A language $L$ is deterministic context-free if there exists a DPA $M$ such that $L_2(M) = L$.
If $L_1(M) = L$ for some DPA $M$ then $L$ is deterministic context-free, but $L$ could be deterministic context-free without there existing a DPA $M$ such that $L_1(M) = L$.
• I'd to like to add that both conditions may be used in a PDA and that a PDA may push/pop multiple symbols to/from the stack. – Auberon May 16 '16 at 21:20
• Accepting "at the end" can be confusing terminology: the PDA might do some epsilon-moves before accepting after having read the complete input. – Hendrik Jan May 17 '16 at 0:06 | 2020-07-08 08:27:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6294278502464294, "perplexity": 471.5630792447652}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655896905.46/warc/CC-MAIN-20200708062424-20200708092424-00084.warc.gz"} |
http://randomthoughtsonjavaprogramming.blogspot.com/2017/ | ## Thursday, 22 June 2017
I got the following (unhelpful) message in my server log, when I changed some of my Java classes that are translated to JSON (and vice versa).
Turns out that I added a specific constructor to one of my Java classes, effectively removing the unspecified Default Constructor that Java always adds.
This default constructor is however essential to the proper working of JSON-Java mapping.
## Thursday, 15 June 2017
I am using EJBs as REST Services. It works pretty well. I added security on the EJB, by means of security definitions in the web.xml file and appropriate annotations on the EJB (@DeclareRoles and @RolesAllowed).
Unfortunately, when I try to access the methods in the EJB without being properly authorized, I received a 500 BadRequest. Instead I would really like to have a 401 Unauthorized.
I posted a question on StackOverflow1, but I have found the solution2 in the mean time, which I also posted, and will repost here.
It is possible to add an ExceptionMapper to your Application, which can map between an Exception and an appropriate HTTP Response.
# Note
My ApplicationConfig has now been expanded with a
.
# References
[2] RESTful Java with JAX-RS 2.0 (Second Edition) - Exception Handling
https://dennis-xlc.gitbooks.io/restful-java-with-jax-rs-2-0-2rd-edition/en/part1/chapter7/exception_handling.html
StackOverflow -
https://stackoverflow.com/questions/3297048/403-forbidden-vs-401-unauthorized-http-responses
## Thursday, 8 June 2017
### Casting JSON Object to TypeScript Class
I have implemented some HTTP service for my Angular App using the explanation at [1]. Now in resource [2] it is mentioned that it is important to provide the JSON Object received from the HTTP Service in the constructor of the data model.
I thought I had found a shortcut. I thought that as long as the JSON object received resembled the structure of the TypeScript class, that I could just cast it to the TypeScript class.
This worked fine, until it didn't, and then I got this huge error in my face.
# The problem
The problem started appearing when I defined a method in my TypeScript class. Naturally, this method is not available in the JSON Object, and no manner of Casting is going to make it magically appear there.
You get something like:
ERROR TypeError: item.getItemPriceAsInteger is not a function
at ItemService.webpackJsonp.71.ItemService.updateItem (http://localhost.com/main.bundle.js:811:67)
at ItemSettingsComponent.webpackJsonp.183.ItemSettingsComponent.update (http://localhost.com/main.bundle.js:508:28)
at ItemSettingsComponent.webpackJsonp.183.ItemSettingsComponent.saveItem (http://localhost.com/main.bundle.js:480:14)
at Object.eval [as handleEvent] (ng:///AppModule/ItemSettingsComponent.ngfactory.js:1663:24)
at handleEvent (http://localhost.com/vendor.bundle.js:13600:138)
at callWithDebugContext (http://localhost.com/vendor.bundle.js:14892:42)
at Object.debugHandleEvent [as handleEvent] (http://localhost.com/vendor.bundle.js:14480:12)
at dispatchEvent (http://localhost.com/vendor.bundle.js:10500:21)
at http://localhost.com/vendor.bundle.js:12428:20
at SafeSubscriber.schedulerFn [as _next] (http://localhost.com/vendor.bundle.js:5549:36)
# Solutions
There are several solutions available as described in [3, 4, 5].
# Chosen solution
I like the one provided in [6]. It uses TypeScript Decorators7. It can be installed as an npm package, according to [8].
To anyone using Java, the solution provided has an uncanny resemblance to JPA annotated Entities or JAXB annotated classes.
I am going to go ahead and try this one out, and see how it works.
I'll provide an update, once I get some results.
# References
[1] Angular Docs - HTTP Client
https://angular.io/docs/ts/latest/guide/server-communication.html
[2] Writing a Search Result
ng-book 2 - The Complete Book on Angular Nate Murray, Felipe Coury, Ari Lerner, Carlos Taborda
[3] StackOverflow - How do I cast a JSON object to a typescript class
https://stackoverflow.com/questions/22875636/how-do-i-cast-a-json-object-to-a-typescript-class
[4] StackOverflow - Angular2 cast a json result to an interface
https://stackoverflow.com/questions/34516332/angular2-cast-a-json-result-to-an-interface
[5] Angular2 HTTP GET - Cast response into full object
https://stackoverflow.com/questions/36014161/angular2-http-get-cast-response-into-full-object
[6] Mark Galae - TypeScript Json Mapper
http://cloudmark.github.io/Json-Mapping/
[7] TypeScript - Decorators
https://www.typescriptlang.org/docs/handbook/decorators.html
Ninja Tips 2 - Make your JSON typed with TypeScript
[8] npm - json-typescript-mapper
https://www.npmjs.com/package/json-typescript-mapper
## Thursday, 1 June 2017
### Bower
Wow. On the website for bower1, they mention the following quote:
“ ...psst! While Bower is maintained, we recommend yarn and webpack for new front-end projects!”2 3
Damn, it's hard to keep up with the advancements in Front-end Land!
# References
[1] Bower - A package manager for the web
https://bower.io/
Yarn - Fast, reliable, and secure dependency management.
https://yarnpkg.com/en/
webpack MODULE BUNDLER
https://webpack.github.io/
### flexibleJDBCRealm
I have recently changed my security realm settings, and I thought I'd document them here.
I'm still using the flexibleJDBCRealm1 as I've documented in previous blogs2,3.
In the Glassfish administration console, under Configurations -> server-config -> Security -> Realms -> myRealm, the settings are now as follows.
NameValueDescription
datasource.jndijdbc/mydbthe data source to my database
jaas.contextflexibleJdbcRealm
sql.groupsselect groupid from mmv_groups where name in (?)using a database view, makes it easier to change table layout without effecting the securityrealm
# Note
The SHA-512 encoding always creates 128 characters as the hash.
However, in the source code of the flexibleJDBCRealm, this hash is converted from a byte[] into a hexadecimal string by means of a call "new BigInteger(1, aData).toString(16);".
This effectively means that if the byte[] starts with one or more "0"s, these are removed in the BigInteger call leaving you with a hash that is less than 128 characters.
This is why I need to use "HEX:128", instead of just "HEX".
The values are easily verifiable in the database.
I can just do a
SELECT SHA2(usertable.password, 512) from usertable where user='mrbear';
It should yield the exact same result as the encryption function of the flexibleJDBCRealm.
# References
[1] FlexibleJDBCRealm
http://flexiblejdbcrealm.wamblee.org/site/
[2] Security Realms in Glassfish
http://randomthoughtsonjavaprogramming.blogspot.nl/2016/04/security-realms-in-glassfish.html
[3] Glassfish Security Realms
http://randomthoughtsonjavaprogramming.blogspot.nl/2014/10/glassfish-security-realms.html
[4] Installation instructions
http://flexiblejdbcrealm.wamblee.org/site/documentation/snapshot/installation.html
## Thursday, 25 May 2017
### "this" in JavaScript/TypeScript
I have been struggling with using "this" in JavaScript, ever since I got into that area of programming.
There are lots of warnings on the web, where programmers who are used to a certain behaviour regarding "this" (Like me) can fall into this trap.
I recently found some really good resources that explain it.
There's one1 that explains it a little regarding "this" in JavaScript.
But as I have been writing in TypeScript, I was looking for an explanation that focuses on TypeScript and helps me find the best solution to deal with this. I found that one in [2].
# For example
So I've got some code that could use a bit of a look-over.
Here's the troublesome bit.
TypeScript has an excellent Tutorial, which I've used time and again to write my things. One of the pages I've used is the explanation regarding HTTP which you can find at [3].
In it they mention a "handleError" method, which can handle HTTP errors of the PlayerService. Convenient, so I used it. It works.
Next, I wished for the handleError method in the PlayerService that takes care of HTTP connections to notify the ErrorsService. So naturally, I inject the ErrorsService into the PlayerService.
Unfortunately, in the handleError, the ErrorsService is 'undefined'. (See line 30 in the gist below)
It is explained in reference [2] why this is, but I like the following quote:
“The biggest red flag you can keep in mind is the use of a class method without immediately invoking it. Any time you see a class method being referenced without being invoked as part of that same expression, this might be incorrect.”
Now there are several solutions for this described in [2].
The solution below is what I came up with on my own, and I don't really like it, but it works.
# Local Fat Arrow
I prefer the solution called the "Local Fat Arrow", which looks like this:
I love it!
# References
[1] Mozilla Developer Network - javascript:this
https://developer.mozilla.org/en/docs/Web/JavaScript/Reference/Operators/this
[2] Github/Microsoft/TypeScript - 'this'in TypeScript
https://github.com/Microsoft/TypeScript/wiki/'this'-in-TypeScript
[3] ts GUIDE - HTTP CLIENT
https://angular.io/docs/ts/latest/guide/server-communication.html
## Thursday, 18 May 2017
Small followup of From Hibernate to Eclipselink1 post.
I am not entirely satisfied about the AdditionalCriteria4 thingy. I find it a chore to have to set a parameter on the EntityManager all the time to enable/disable it.
Biggest issue for me is that parameters set on the EntityManager are required. If they are omitted, an exception is thrown when querying.
Current solution in my software:
Turn the AdditionalCriteria on or off by means of a parameter that needs to be set on the EntityManager.
Looks like this:
Setting the parameter activePersonFilter can be done on the EntityManager as follows:
@PersistenceContext(properties =
{
@PersistenceProperty(name = "activePersonFilter", value = "0"),
@PersistenceProperty(name = "sundaydateFilter", value = "")
})
private EntityManager em;
Or
entityManager.setProperty("activePersonFilter", 0);
# Other solutions
There are some other solutions.
1. You can remove the additionalCriteria (set it to "") in a subclass, and use the subclass specifically. See [2].
2. You can customize any mapping in EclipseLink and add the requirements/conditions that you need. See [3].
3. I could just decide to create a view on the offending database table. Then create two entities. Sounds very similar to the first option.
4. I could solve the problem in software. Just have EclipseLink not filter anything. (Which is silly, I don't wish for my ORM to get the 1000 persons in the room from the database, if there are say only three persons active.)
5. I could remove the collection entirely, and retrieve the required Persons using a NamedQuery. (Which is bogus. I like the ORM to deal with this for me, instead of having to do it myself. It's what the ORM is for.)
# Customizing a Mapping
I have recently decided to try to customize the mapping specifically in Entities that have collections containing instances of Person class. That way I have more control. See reference [3] on how this works.
It requires a @Customizer annotation.
For instance, in a Room I only wish to see the active persons.
This requires me to define the PersonsFilterForRoom as follows.
"persons"
the name of the field that contains the collection
"room"
the name of the field in the Entity of the collection
"id"
the name of the field in the Room entity that identifies it
It works pretty good.
# Note
I also noticed that this way I could have two (Lazy! That's the important bit!) Collections in the same Entity at the same time referring to the same Person. One will contain all Persons and one will contain only the Active Persons.
This is ideal, for instance for Guilds.
Like so:
This way the customizer PersonsFilterForGuild is designed to only work on the activeMembers collection.
I like it!
# References
[2] StackOverflow - Disable additional criteria only in some entity relations
[3] Mapping Selection Criteria
## Friday, 12 May 2017
A small blog this time.
At work we sometimes have serious problems with non-deterministic tests.
Martin Fowler mentioned how this can be prevented or dealt with.1
I also noticed that these non-deterministic tests are (almost...) always in the end-to-end tests (or the functional tests, or however you wish to call them).
Martin Fowler also has something to say about those2
# References
[1] MartinFowler - Eradicating Non-Determinism in Tests
https://martinfowler.com/articles/nonDeterminism.html
[2] MartinFowler - TestPyramid
https://martinfowler.com/bliki/TestPyramid.html
## Thursday, 4 May 2017
### REST-assured
I am a card-carrying member of the NLJUG0, which provides Java Magazine (not the Oracle one) six times per year.
One of the issues contained an article about REST-assured1.
I have been using SoapUI5 to test my REST services, and that works fine. It's a nice graphical userinterface for me to fiddle with parameters and urls and HTTP requests and even write tests.
I am aware that it is probably possible to integrate SoapUI into my Build Pipeline, but I was really looking for something different. Something more in the line of programming, which is of course my forte. Something I could use in my unit-tests.
REST-assured was exactly what I needed and let me tell you, it's great!
# Usage
I will provide an example of how I use it.
As you can see, REST-assured is a very nice DSL (Domain Specific Language) and reads easily.
Some explanation of the above:
log().ifValidationFails()
I wish to log stuff, if the validation/test fails, so I can find out what is wrong. The output looks like
param(name, value)
for setting parameters at the end of the url, like ?stuff=value
pathParam(tag, value)
replaces {tag} in your url with the values. Convenient!
request methods
in the example above, we are using the PUT HTTP Request.
As it is used for testing, it is possible to verify the values afterwards. In the above this is visible as we expect to receive a 204 (NO_CONTENT).
We can extract the response, as is done above, to verify for example the json payload (if there is one) or get cookie values.
In the above example it is essential for the followup calls that we get the JSESSIONID cookie out of the request.
In subsequent REST calls, it is obvious that we need to send along the same JSESSIONID cookie.
# Some notes
I tried to send parameters, but a POST defaults to FORM parameters in the body, but I already have a BODY. But using "queryParam" instead of "param" fixes this problem.
I do enjoy using the "prettyPrint" method on a Response, to properly format a JSON body and dump it to standard output and see what I get. It's nice.
Getting some values out of your JSON formatted response body does require some serious work, though. Needs more research.
I am not entirely sure, I do not enjoy using http status codes like 200 or 204. I prefer something more readable like "NO_CONTENT", but I suppose I can deal with it myself. No biggy.
Update 14/05/2017: I'm also slightly sorry to find out that rest-assured includes Hamcrest. I prefer AssertJ at the moment myself.
# Postscriptum
The article in Java Magazine also mentioned WireMock3.
Though I do not use it, it seems excellent for testing the other side of the communications, if you need to test a client that communicates with a server via rest calls.
# References
[0] NLJUG
http://www.nljug.org/
[1] REST-assured
Teije van Sloten Java Magazine | 01 2017
[2] GitHub - Java DSL for easy testing of REST services
https://github.com/rest-assured/rest-assured
[3] WireMock
http://wiremock.org/
[4] GitHub - RestAssured Usage
https://github.com/rest-assured/rest-assured/wiki/usage
[5] SoapUI
https://www.soapui.org/
Testing REST Endpoints Using REST Assured
https://semaphoreci.com/community/tutorials/testing-rest-endpoints-using-rest-assured
RFC2616 - HTTP status codes
https://www.w3.org/Protocols/rfc2616/rfc2616-sec10.html
## Thursday, 27 April 2017
### Cucumber @After en @Before Hooks
We're using Cucumber at work to write tests, end-to-end-tests that access the user interface of the web application using Selenium.
I recently added an @After hook to a class that contained my StepDefinitions.
However, this @After hook was also called by all other scenarios1, which was not my intention.
As a matter of fact, that @After I added was executing similar code as an @After in another StepDefinition class. I verified that both @After annotated methods were executed for each and every scenario, and they were.
So I decided to move all @After annotated methods into a "GlobalStepDefinition" class, and collaps all of them into one method.
Incidentally, reference [3] shows why we should not have many of these end-to-end tests.
# References
[1] GitHub Issues - Before and After methods invoked for unused step definition classes #1005
https://github.com/cucumber/cucumber-jvm/issues/1005
[2] Cucumber - Polymorphic Step Definitions
https://cucumber.io/blog/2015/07/08/polymorphic-step-definitions
[3] MartinFowler.com - TestPyramid
https://martinfowler.com/bliki/TestPyramid.html
## Sunday, 23 April 2017
### Problems with Resolution and My Monitor in Fedora Core 25
Well, my monitor always has been a bit of a problem child, but it worked, so I didn't mind.
I let it bounce once on the floor, but besides some slight discolouring in the lower-right corner, it was fine.
It reports EDID settings that are completely crap, but I got used to ignoring those, using xrandr.
# XRandr settings that work for me
The following settings work:
xrandr --newmode "1920x1440" 339.50 1920 2072 2280 2640 1440 1443 1447 1514 -
xrandr --newmode "1600x1200" 235.00 1600 1728 1896 2192 1200 1203 1207 1262 -
xrandr --newmode "1280x1024" 159.50 1280 1376 1512 1744 1024 1027 1034 1078
xrandr --output VGA-0 --mode 1920x1440
# Problem
Then I upgraded to Fedora Core 25, and my monitor showed me a handsome 1024x768, which was a disappointment to say the least. (I'm used to 1920x1440.)
Using xrandr gave me the cryptic error message:
bash-4.3$xrandr --output XWAYLAND0 --mode "1920x1440" xrandr: Configure crtc 0 failed After some research I noticed that Fedora Core 25 is the first one to use Wayland1 as the default. # Solution Switching back to the old Xorg2 fixed my problem. # Checking graphics card bash-4.3$ lspci -nnk |grep -A 3 -i vga
01:00.0 VGA compatible controller [0300]: Advanced Micro Devices, Inc. [AMD/ATI] Juniper XT [Radeon HD 5770] [1002:68b8]
Subsystem: ASUSTeK Computer Inc. Device [1043:0344]
# References
[1] Wayland Desktop
https://wayland.freedesktop.org/
[2] Fedora Project - Switching back to Xorg
https://fedoraproject.org/wiki/Changes/WaylandByDefault
Fedoraforum.org - how to install amd/ati driver on fedora 25?
AskFedora - How to add a custom resolution to Weyland Fedora 25?
ArchLinux - Forcing modes and EDID
https://wiki.archlinux.org/index.php/Kernel_mode_setting#Forcing_modes_and_EDID
Bugzilla Redhat - My Bugreport
https://bugzilla.redhat.com/show_bug.cgi?id=1443761
## Saturday, 15 April 2017
### Keyset pagination
In the past I have used the MySQL equivalent of pagination. In other words, the splitting up of a ResultSet into pages of a fixed number of entries, by means of using SQL1.
It looks like the following:
SELECT * FROM tbl LIMIT 5,10; # Retrieve rows 6-15
For compatibility with PostgreSQL, MySQL also supports the LIMIT row_count OFFSET offset syntax, which I've used in the past.
# Performance
Performance is a key point here, as MySQL requires the retrieval of the results in order to determine where the offset starts.
If the table is large, retrieval of pages at the end of the table are going to be extremely slow.
# Solution
A better way to deal with this, is to not use an offset, but use the key of the last row of the previous page, and use that in the query for the next page.
Obviously this only works if the resultset is sorted.
For more references that explain this a lot better, see [2] and [3].
# References
[1] MySQL 5.7 - 14.2.9. SELECT Syntax
https://dev.mysql.com/doc/refman/5.7/en/select.html
[2] Use the Index, Luke! - We need tool support for keyset pagination
http://use-the-index-luke.com/no-offset
[3] Use the Index, Luke! - Paging Through Results
http://use-the-index-luke.com/sql/partial-results/fetch-next-page
## Thursday, 6 April 2017
### Try Git
To anyone who is absolutely new to the exciting new world of Git1.
There seems to be a little website where you can try Git2, working in a (very very) limited sandbox environment.
# What is Git?
If you wish to know what Git is, there are loads of interesting articles on teh interwebs that explain it very well.
But I did find the following explanation in the README provided with the source tar-ball:
The name "git" was given by Linus Torvalds when he wrote the very
first version. He described the tool as "the stupid content tracker"
and the name as (depending on your mood):
- random three-letter combination that is pronounceable, and not
actually used by any common UNIX command. The fact that it is a
mispronunciation of "get" may or may not be relevant.
- stupid. contemptible and despicable. simple. Take your pick from the
dictionary of slang.
- "global information tracker": you're in a good mood, and it actually
works for you. Angels sing, and a light suddenly fills the room.
- "goddamn idiotic truckload of sh*t": when it breaks
# References
[1] Git --distributed-is-the-new-centralized
https://git-scm.com/
[2] Try Git
https://try.github.io/
## Thursday, 30 March 2017
### Setting session timeout in Glassfish
People complained that their sessions timed-out too quickly in Glassfish.
I checked and it is set to 30 minutes (default 1800 seconds), just a tad too little.
Increased it to 2 hours (7200 seconds).
Just went to Configurations - Web Container - Session Properties - Session Timeout.
It changes the domain.xml:
<session-properties timeout-in-seconds="7200"></session-properties>
# Problem
Of course, this completely and utterly failed to work in my case.
It turns out I already had a session timeout specified in the web.xml.
<session-config>
<session-timeout>
30
</session-timeout>
</session-config>
The session timeout in the web.xml is specified in minutes.
You can also specify it in the glassfish-web.xml file.1
<session-config>
<session-properties>
<property name="timeoutSeconds" value="600"/>
</session-properties>
</session-config>
# Precedence
You do need to check which setting takes precedence in your application. It's not clear from the documentation.
# References
[1] Glassfish 4.0 Application Deployment Guide
https://glassfish.java.net/docs/4.0/application-deployment-guide.pdf
iT Geek Help - Glassfish web container tuning settings
http://itgeekhelp.blogspot.nl/2009/03/glassfish-web-container-tuning-settings.html
StackOverflow - How to set session timeout in glassfish-web.xml configuration file?
http://stackoverflow.com/questions/33067985/how-to-set-session-timeout-in-glassfish-web-app-glassfish-web-xml-configurat
## Thursday, 23 March 2017
### AssertJ vs. Hamcrest
I recently came across a piece of code that used a Stack1. The Stack seems to inherit from Vector. The JavaDoc indicated (and so did my IDE, I think) that I should be using the Deque2 interface instead. To be precise:
“A more complete and consistent set of LIFO stack operations is provided by the Deque interface and its implementations, which should be used in preference to this class.”
Dequeue basically seems to be a specialized Queue3, that supports element insertion and removal at both ends4.
In order to get to grips with Deque, I decided to write some simple tests. These are JUnit Tests (version 4.12) and in one I used Hamcrest5 and in the other I went for AssertJ6.
Let's see what happens.
# A simple compare
Hamcrest:
assertThat(actual, equalTo(testdata2));
AssertJ:
assertThat(actual).isEqualTo(testdata2);
# Collections
Hamcrest:
assertThat(transmittedTestdata, hasSize(2));
AssertJ:
assertThat(transmittedTestdata).size().isEqualTo(2);
# Null Values
Hamcrest:
assertThat(actual, not(nullValue()));
AssertJ:
assertThat(actual).isNotNull();
# Exceptions
Hamcrest:
@Test(expected = NoSuchElementException.class)
public void testEmptyDequeueException()
{
Testdata pop = transmittedTestdata.pop();
}
AssertJ:
assertThatThrownBy(transmittedTestdata::pop).isInstanceOf(NoSuchElementException.class);
# Imports
A comparison between the required imports of Hamcrest and Assertj is interesting:
Hamcrest:
import java.util.Deque;
import java.util.NoSuchElementException;
import static org.hamcrest.CoreMatchers.equalTo;
import static org.hamcrest.CoreMatchers.not;
import static org.hamcrest.CoreMatchers.nullValue;
import static org.hamcrest.MatcherAssert.assertThat;
import static org.hamcrest.Matchers.empty;
import static org.hamcrest.Matchers.hasSize;
import org.junit.After;
import org.junit.AfterClass;
import org.junit.Before;
import org.junit.BeforeClass;
import org.junit.Test;
AssertJ:
import java.util.Deque;
import java.util.NoSuchElementException;
import static org.assertj.core.api.Assertions.assertThat;
import static org.assertj.core.api.Assertions.assertThatThrownBy;
import org.junit.After;
import org.junit.AfterClass;
import org.junit.Before;
import org.junit.BeforeClass;
import org.junit.Test;
# Notes
• I really like the AssertJ fluent API. It feels more natural to me than the Hamcrest one.
• It is way easier to find the appropriate matchers in AssertJ. I get the full benefit of my IDE code completion.
• Adding the appropriate import is way easier. Using Hamcrest, I always get a choice of five different imports for the same matcher.
• I need fewer imports anyways.
So far, I like AssertJ a lot.
I need to work with AssertJ a lot more, to see some of the interesting stuff.
# References
[1] Java 7 JavaDoc - Stack
https://docs.oracle.com/javase/7/docs/api/java/util/Stack.html
[2] Java 7 JavaDoc - Deque
https://docs.oracle.com/javase/7/docs/api/java/util/Deque.html
[3] Java 7 JavaDoc - Queue
https://docs.oracle.com/javase/7/docs/api/java/util/Queue.html
[4] Wikipedia - Double-ended queue
https://en.wikipedia.org/wiki/Double-ended_queue
[5] Hamcrest - Matchers that can be combined to create flexible expressions of intent
http://hamcrest.org/
[6] AssertJ - Quick start
http://joel-costigliola.github.io/assertj/assertj-core-quick-start.html
## Thursday, 16 March 2017
### reveal.js
Our architect recently put together a presentation regarding our new framework using reveal.js1.
I had never heard of reveal.js and I was intrigued. It seems to be a presentation framework that runs in your webbrowser, using npm2 and grunt3 and javascript and MarkDown4 and all that.
I figured I'd give it a try for my next presentation.
I downloaded a release6 and used the very clear instructions on how it works on GitHub5.
Installing a new release, seems to be nothing more than:
- unzip
- edit index.html
- browse to index.html
Luckily, I had the changes our architect made to bring it into line with the company layout guidelines. It was nothing more than a different css file that is based on the "white"-theme (which is also a css file). The default theme when you get a release is the "black"-theme, similar to the one visible at [1].
You can decide to just browse to the file index.html locally to display the presentation, but if you do a "grunt serve" a small webserver is started that serves the webpage and related resources. The latter option provides more functionality.
# MarkDown
<div class="reveal">
<div class="slides">
<section data-markdown="slides.md"
data-separator="^\n\n\n"
data-separator-vertical="^\n\n"
data-separator-notes="^Note:"
data-charset="utf-8">
</section>
</div>
</div>
As you can see above, you can specify how the sheets are divided. What exactly the sequence is for detecting a division.
Initializing the presentation is done using:
// More info https://github.com/hakimel/reveal.js#configuration
Reveal.initialize({
width : 1280,
height : 1024,
slideNumber: 'c/t',
showNotes: true,
history: true,
// More info https://github.com/hakimel/reveal.js#dependencies
dependencies: [
{ src: 'plugin/markdown/marked.js' },
{ src: 'plugin/markdown/markdown.js' },
{ src: 'plugin/notes/notes.js', async: true },
{ src: 'plugin/zoom-js/zoom.js', async: true } }
]
});
I set the "snowNotes" to true, because I wished to print out the sheets including the notes. See Printing below.
# CSS
It is easy to add custom CSS to individual slides. For example:
<!-- .slide: data-background="#ffffff" data-background-image="images/background_subtitle.png" data-background-size="auto 100%" data-background-repeat="no-repeat" -->
A common one used to change the font of the previous element (useful for source code):
<!-- .element: class="small" -->
# Printing
Printing your sheets seems to be as simple as surfing to the url:
http://localhost:8080/?print-pdf#/
It generated in Chrome browser a PDF file that you can simply print to file in the browser.
# Conclusions
It is very nice, if you are a programmer or web guy and you do not wish to fire up Microsoft Powerpoint.
An advantage is of course that MarkDown files can easily be added to your version control system.
Another advantage is that you can refer to images on the Internet/Intranet. I managed to do just that, by referring to images already on our Intranet Confluence pages. At least the images will always be up to date.
(p.s. It also means that in order to view my presentation properly, one has to be logged into Confluence. I found that out rather quickly, when trying my presentation out in one of our conference rooms.)
I don't really like the markdown setting displayed above, as it is too easy to add one line or remove one line to many.
I also had a problem where I must have made a grammatical mistake, and in my FireFox browser the presentation managed to hang and after several seconds I'd get a "Script is running too long. Do something about it?" message.
There are several keyboard shortcuts for navigation through the sheets during the presentation, which is nice, as the mouse isn't all that handy.
I don't much like the "sheet notes", which are displayed in a separate browser window. I usually have them turned off.
# References
[1] Reveal.ks - the HTML Presentation Framework
http://lab.hakim.se/reveal-js/
[2] NPM
https://www.npmjs.com/
[3] Grunt
https://gruntjs.com/
[4] Wikipedia - MarkDown
https://en.wikipedia.org/wiki/Markdown
[5] GitHub - reveal.js
https://github.com/hakimel/reveal.js
[6] reveal.js releases
https://github.com/hakimel/reveal.js/releases
GitHub- Basic Writing and Formatting Syntax
https://help.github.com/articles/basic-writing-and-formatting-syntax/
## Thursday, 9 March 2017
### Git Stash
This little blog post is just for me to remember my favorite "git stash" commands. It took me a little while to actually use the stash, but that is because IntelliJ provides a similar functionality called "shelving", which I had used all this time.
I use branches a lot when using Git, and the problem there is that Git usually complains if I wish to change branches, while I still have uncommitted changes in my current branch. Therefore the "stash" command is for me very valuable.
$git stash Get your uncommitted changes back from the stash:$ git stash apply
Get a list of your current stashes:
$git stash list stash@{0}: WIP on master: 049d078 added the index file stash@{1}: WIP on master: c264051 Revert "added file_size" stash@{2}: WIP on master: 21d80a5 added number to log Remove a no longer needed stash:$ git stash drop stash@{0}
Dropped stash@{0} (364e91f3f268f0900bc3ee613f9f733e82aaed43)
One command I particularly like is this one that does both an apply of your stash and once done automatically removes it from the list of stashes:
git stash pop
The stash has a lot of similarities to your standard Stack implementation (or Dequeue, depending on your point of view.)
I notice that if I do not clean up the place or use the "pop" subcommand, that my list of stashes tends to grow quite long unobtrusively.
# References
6.3 Git Tools - Stashing
https://git-scm.com/book/en/v1/Git-Tools-Stashing
Atlassian Tutorials - Git Stash
https://www.atlassian.com/git/tutorials/git-stash
Ariejan De Vroom - GIT: Using the stash
https://ariejan.net/2008/04/23/git-using-the-stash/
Git Stash - Man Page
https://git-scm.com/docs/git-stash
## Thursday, 2 March 2017
### Maven and the Dangers of Snapshots
Recently we've been causing problems in the regular builds of branches of our software.
Basically the problem is our own fault and is related to Maven Snapshots.
According to the guide1, a Snapshot is a library that is still under development, and may change rapidly as new versions of the Snapshot are pushed to the Nexus regularly.
If a dependency on a Snapshot is defined in your pom.xml, then Maven, as it should, always picks the latest Snapshot.
This is fine and dandy if you are currently developing your software, and you want the newest of the new of the libraries that your other software teams are developing.
# The Problem
It means that once you create a stable release of your software (and the appropriate Git branch for it to live in as well, of course) it is important to replace the Snapshot in the pom.xml with the appropriate released version.
We neglected to do just that.
# The Consequence
Our branch containing the release version of our software suddenly bombed with compile errors in the Deployment Pipeline.
This caused the maintenance people a headache, as the Git revision of the branch had not changed, between the previous build (which compiled just fine) and the new build (which bombed).
Despite the build being pulled from Git with the exact same revision, it was technically different from the previous build.
All because we kept developing the Snapshot and pushing it into the Nexus.
# What we should have done
• create a proper release of the library
• change the pom.xml in the branch to refer to this release.
• create a new snapshot of the library
• use the new snapshot in the pom.xml of the master branch (which is used for development)
Now the build of both the branch as well as the master should compile again.
# References
[1] Apache Maven - Getting Started
https://maven.apache.org/guides/getting-started/
Continuous Releasing of Maven Artifacts
https://dzone.com/articles/continuous-releasing-maven
## Thursday, 23 February 2017
### A Natural Progression Towards Lambda
At my work, in order to deal with a grid1 in the frontend and a list at the backend, we use a DataModel at the backend.
It seems simple enough, and used to work as follows:
private List<Person> list = Arrays.asList(new Person("Jim"), new Person("Jack"));
private ListDataModel<Person> dataModel = new ListDataModel<>(list);
This had some shortcomings when for example the user decided to select a different department, executing this code:
list = findPersonsByDepartment(department);
This seems to work just fine. A person selects a different department, and the employees data model updates itself. Or so one would think.
What happens is that the ListDataModel retains the old list. So, the frontend is never updated.
# Reusing the same list
Because of this little problem, our code retains a lot of the following statements, to make sure the same list is used over and over again:
list.clear();
It seems a slightly convoluted way to doing things.
# Anonymous inner classes
We soon found out that anonymous inner classes would solve this problem better, and in fact there are more anonymous inner classes than there are named DataModels in our current code base.
It looks like the following:
private ListDataModel<Person> dataModel = new ListDataModel<Person>()
{
@Override
public List<Person> getList()
{
return findPersonsByDepartment(department);
}
};
There now, any time the contents of the ListDataModel is requested in the frontend, a new and accurate List containing the department employees is returned.
# Passing code
Instead of creating an entire new anonymous inner subclass of a ListDataModel, it might be more elegant to create an interface especially for this purpose, call it the ListProvider interface.
As follows:
public interface ListProvider<T>
{
List<T> getList();
}
private ListDataModel<Person> dataModel = new ListDataModel<Person>(new ListProvider<>()
{
@Override
public List<Person> getList()
{
return findPersonsByDepartment(department);
}
});
# Using lambdas
The good part is that now with Java 8 we can start using Lambdas.
And in this case, we have an interface containing just one method. This is in essence the definition of a lambda.
So now the proper way to write this would be the following:
public interface ListProvider<T>
{
List<T> getList();
}
private ListDataModel<Person> dataModel =
new ListDataModel<Person>(() -> findPersonsByDepartment(department));
Convenient, isn't it?
In this case, the lambda is called a Supplier2 .
# References
[1] Welcome to the SlickGrid! (outdated sadly)
https://github.com/mleibman/SlickGrid/wiki
[2] Supplier (Java Platform SE 8)
https://docs.oracle.com/javase/8/docs/api/java/util/function/Supplier.html
## Thursday, 16 February 2017
### Angular - Semantic Versioning
Angular1 has switched to Semantic Versioning2
So, the brand new thing that is totally hot right now is Angular 4.0.
The versions released, and to be released are available here3.
Contrary to the image in the blog, the word for referring to all this is "Angular".
Looks like version 5 of Angular will be released later in the year.
I hope I can keep up.
# References
[1] Ok... let me explain: it's going to be Angular 4.0, or just Angular
http://angularjs.blogspot.nl/2016/12/ok-let-me-explain-its-going-to-be.html?inf_contact_key=8b9c809bd7a11da8e78370dff6483f15f2782c6760b6b1b77f6b008bc3804655&m=1
[2] Semantic Versioning 2.0.0
http://semver.org/
[3] Versioning and Releasing Angular
http://angularjs.blogspot.nl/2016/10/versioning-and-releasing-angular.html#Timebased_release_cycles_18
## Thursday, 9 February 2017
### Group by problem with Hibernate
Recently had a small problem that the group by function didn't work, if I added a subtable to the query. The GROUP BY expression did not match any longer.
Seems a long standing problem with Hibernate.
# References
Java Persistence with Hibernate, page 392
Christian Bauer, Gavin King, Gary Gregoy
HH-1615 - GROUP BY entity does not work
https://hibernate.atlassian.net/browse/HHH-1615
HH2436 - Allow grouping by entity reference (per JPA spec)
https://hibernate.atlassian.net/browse/HHH-2436
## Wednesday, 1 February 2017
### Extending SSL Certificate in Glassfish
This is a followup of the blog post SSL Certificates in Glassfish.
The reason for this followup, is that signing of websites and code seems to be a very error prone and manual process, that is done infrequently enough for all of us to forget afterwards.
It basically follows the same path as the previous blog post, but I find it convenient to write stuff down, in case I forget.
Now my certificate on my website had expired, and it took me a while, before I found the time and the motivation to extend the certificate.
I'm still with GoDaddy.com4. Thankfully, the CSR was already transmitted last year, and I can just reuse that one.
Once I submit the CSR, I am required to verify that I am the owner of the Domain. This time, thank goodness, it requires nothing more than the clicking of a link sent to the email address that is stored in the WHOIS information.
Nothing like putting a file in the rootmap of the webserver or some such, like the first time.
Once that is done, I need to download the new certificates from godaddy.com. They ask for the type of web server that they need to generate the certificates for. Glassfish is not mentioned anywhere, so I select "Other".
The zip file I then receive, contains the same files as mentioned in my previous blogpost1.
As I already installed all the root certificates, I choose to ignore the gd_bundle-g2-g1.crt file.
The more interesting file is the 2375839yrghfs5e7f.crt file.
## Replace the original self-signed certificate with the certificate you obtained from the CA
[glassfish@server config]$keytool -import -v -trustcacerts -alias s1as -file /home/glassfish/junk/2375839yrghfs5e7f.crt -keystore keystore.jks -storepass changeit Certificate reply was installed in keystore [Storing keystore.jks] ## Verifying the keystore.jks You can verify that all is well, by using the above command to check the keystore. You will see something like the following: Alias name: s1as Creation date: Feb 1, 2017 Entry type: PrivateKeyEntry Certificate chain length: 4 Certificate[1]: Owner: CN=www.server.org, OU=Domain Control Validated Issuer: CN=Go Daddy Secure Certificate Authority - G2, OU=http://certs.godaddy.com/repository/, O="GoDaddy.com, Inc.", L=Scottsdale, ST=Arizona, C=US Serial number: 8446c5db57d376ed Valid from: Wed Feb 01 14:27:00 CET 2017 until: Thu Feb 01 14:27:00 CET 2018 Certificate fingerprints: MD5: 75:7a:73:67:72:6a:6b:73:65:72:6e:79:20:62:61:77 SHA1: 75:7a:73:67:72:6a:6b:73:65:72:6e:79:20:62:61:77:79:20:72:67 SHA256: 75:7a:73:67:72:6a:6b:73:65:72:6e:79:20:62:61:77:79:20:72:67:68:20:61:77:65:72:3c:6f:3b:20:59:38 Signature algorithm name: SHA256withRSA Version: 3 Which shows that as of today, the keystore has a valid certificate that is exactly valid for one year. To apply your changes, restart GlassFish Server, according to chapter "To Sign a Certificate by Using keytool2". ## Verifying after reboot Earlier, when issuing the openssl command: openssl s_client -connect www.server.org:4848 The result was: SSL handshake has read 15360 bytes and written 339 bytes --- New, TLSv1/SSLv3, Cipher is ECDHE-RSA-AES256-GCM-SHA384 Server public key is 2048 bit Secure Renegotiation IS supported Compression: NONE Expansion: NONE No ALPN negotiated SSL-Session: Protocol : TLSv1.2 Cipher : ECDHE-RSA-AES256-GCM-SHA384 Session-ID: 5891E20F7C4FA7CBFA6ABF7E0EC6EC2D40C2CB4A148EFCEAE7F3179F5F80763F Session-ID-ctx: Master-Key: B8C7BA7AC15244DC581749AC9702609F8EB1BCE03F5B0CD53ECEE382D93877EBF6D5E3FE9F603D6D8253521A29EEB494 Key-Arg : None Krb5 Principal: None PSK identity: None PSK identity hint: None Start Time: 1485956532 Timeout : 300 (sec) Verify return code: 10 (certificate has expired) --- Notice especially that last bit. Once the glassfish was rebooted, the same command yields: SSL handshake has read 15370 bytes and written 339 bytes --- New, TLSv1/SSLv3, Cipher is ECDHE-RSA-AES256-GCM-SHA384 Server public key is 2048 bit Secure Renegotiation IS supported Compression: NONE Expansion: NONE No ALPN negotiated SSL-Session: Protocol : TLSv1.2 Cipher : ECDHE-RSA-AES256-GCM-SHA384 Session-ID: 5891E99B097CCC082475F5949A55ABD71C7AED902725AA6E98E77EAA3FC7BF01 Session-ID-ctx: Master-Key: 9465D76CDC8D4CA19E46B2367ECD35382BA8049707BBF1D4D06E0389E85F724BA646F3C2C9FD45CF256C12ED9A0714F0 Key-Arg : None Krb5 Principal: None PSK identity: None PSK identity hint: None Start Time: 1485958464 Timeout : 300 (sec) Verify return code: 0 (ok) --- Again, I would like to draw your attention to the last line. And that's it for now! # References [1] SSL Certificates in Glassfish http://randomthoughtsonjavaprogramming.blogspot.nl/2015/10/ssl-certificates-in-glassfish.html [2] GlassFish Server Open Source Edition Security Guide Release 4.0 https://glassfish.java.net/docs/4.0/security-guide.pdf [3] GlassFish Server Open Source Edition Administration Guide Release 4.0 https://glassfish.java.net/docs/4.0/administration-guide.pdf [4] GoDaddy: Hosting, domainregistration, websites and more... http://www.godaddy.com SSLShopper - most common java keytool keystore commands https://www.sslshopper.com/article-most-common-java-keytool-keystore-commands.html SSLShopper - SSL Certificate Verification https://www.sslshopper.com/ssl-checker.html ## Saturday, 28 January 2017 ### Race Condition in JavaScript Promises I recently encountered a race condition in the use of my promises. It seems trivial, but these little mistakes have a tendency to cause a lot of debugging time. getUrls(): ng.IPromise<any> { if (this.urls != null) { var urls = this.urls; return this.$q(function (resolve, reject) { resolve(urls); });
}
this.urls = {};
var urls: any = this.urls;
return this.$http.get("conf/urls.json").then(function (response: any) { urls.productionUrl = response.data.productionUrl; urls.testingUrl = response.data.testingUrl; return response.data; }); } Can you spot the problem? The problem occurs when two threads access the same method at the same time. Click here for the answer. I always have problems wrapping my mind around JavaScript promises. ## Thursday, 19 January 2017 ### Rewriting History with Git Well, this is basically a followup of my previous blogpost about git. A note of warning: rewriting history can be tricky, and you should perform this only on your local Git repository on things you haven't yet pushed to a remote (public) repository. For more detailed information on how you can work with it, see the References. Right here, right now, I'm going to provide the way I've used it in my current work. # Logs Rewriting history is done by "rebasing" your current checkins. It helps if you can easily retrieve your current checkins from the logs. The following shows my last 5 checkins in my local branch. [mrbear@localhost project]$ git log --pretty=format:"%h %s" HEAD~5..HEAD
75d7620 BUGS-0010 Make it visible whether user is using the test or the production version.
ca9217e BUGS-0010 Cache test/prod urls.
b1f93c1 BUGS-0010 Replace hardcoded urls with configuration.
46908ce BUGS-0010 Implement configuration using Gulp.
49115c3 BUGS-0010 Two ways: "gulp test" or "gulp production".
Bear in mind that the log shows the checkins from most recent (on top) to the least recent (last). Rebasing takes the checkins in the opposite order.
# Squashing checkins
Let us try to squash some commits that we've made together into one single commit.
[mrbear@localhost project]\$ git rebase -i HEAD~5
49115c3 BUGS-0010 Two ways: "gulp test" or "gulp production".
46908ce BUGS-0010 Implement configuration using Gulp.
b1f93c1 BUGS-0010 Replace hardcoded urls with configuration.
ca9217e BUGS-0010 Cache test/prod urls.
75d7620 BUGS-0010 Make it visible whether user is using the test or the production version.
# Rebase d5defcb..75d7620 onto d5defcb (5 command(s))
#
# Commands:
# p, pick = use commit
# r, reword = use commit, but edit the commit message
# e, edit = use commit, but stop for amending
# s, squash = use commit, but meld into previous commit
# f, fixup = like "squash", but discard this commit's log message
# x, exec = run command (the rest of the line) using shell
#
# These lines can be re-ordered; they are executed from top to bottom.
#
# If you remove a line here THAT COMMIT WILL BE LOST.
#
# However, if you remove everything, the rebase will be aborted.
#
# Note that empty commits are commented out
I've decided to squash 46908ce, so that this commit will be combined with its previous commit, the 49115c3.
I can even change the commit message and combine the two commit messages!
# This is a combination of 2 commits.
# The first commit's message is:
BUGS-0010 Implement configuration using Gulp.
# This is the 2nd commit message:
BUGS-0010 Two ways: "gulp test" or "gulp production".
# Please enter the commit message for your changes. Lines starting
# with '#' will be ignored, and an empty message aborts the commit.
#
# Date: Thu Jan 19 14:13:50 2017 +0100
#
# rebase in progress; onto 0a98e03
# You are currently editing a commit while rebasing branch 'BUGS-0010' on '0a98e03'.
#
# Changes to be committed:
# modified: config.xml
# modified: lang.json
# Untracked files:
# project/config.xml~
# project/res/
# project/www/conf/
# project/www/i18n/
#
The output will look something like this:
[detached HEAD 5b2f880] BUGS-0010 Configuration using gulp in two ways: "gulp test" or "gulp production".
Date: Mon Jan 9 13:20:45 2017 +0100
8 files changed, 36 insertions(+), 6 deletions(-)
rename project/{ => conf}/config.xml (94%)
rename project/{www/i18n => conf}/lang.json (99%)
# Reordering checkins
Reordering the checkins, is as simple as cut&pasting the appropriate lines into a different sequence.
# Removing checkins
Removing a checkin, can be done by just removing a line from the sequence. Use with caution.
# References
Atlassian Git Tutorial - Rewriting history
https://www.atlassian.com/git/tutorials/rewriting-history/git-rebase-i
Git - 7.6 Git Tools - Rewriting History
https://git-scm.com/book/en/v2/Git-Tools-Rewriting-History
## Thursday, 12 January 2017
### Base 64 Encoding and URLs
I recently had some issues with base64 encoding of images and documents, prior to sending them over HTTP to the frontend of my app, and there decoding it again.
The issues were manifold.
Let me try and indicate the problems I encountered.
In the backend I use the BaseEncoding class provided in the Core of Google Guava. (com.google.common.io.BaseEncoding) The method "base64()"speaks for itself.
At the frontend, using Javascript, I used the "atob()" method1 to get the whole base64 encoded string back into it's original shape.
# Problem 1 - String contains an invalid character
The "atob()" method threw an error, when attempting to decode String. After some research, involving comparing the string that is sent by the Backend, with the string received by the Frontend, I did notice two differences.
Apparently, the backend is sending a url-safe encoded string2, despite me not having specified that this is
what I want.
Well, the differences aren't major and a simple solution does the trick:
atob(contents.replace(/-/g, "+").replace(/_/g, "/"));
And voila, atob() no longer complains about invalid characters.
# Problem 2 - Decoded document does not match original document
My app stored the received document, after decoding, into a file locally, so it can be opened. My App is created using Cordova (and Ionic and some other stuff) and I use the Cordova File Plugin3 to write the file.
The PDF document that I was using as a test, seemed to be transferred just fine, but upon opening it on the Tablet, an empty PDF document was shown.
There were vast differences between the original document and the decoded document. The differences seem to focus on the decidedly "weird" characters. The alphabet seemed just fine.
Being at a loss for the moment, I decided to use a library named js-base644. That didn't help in the slightest. Using runkit5 to test it, I found that it actually decodes base64 encoding into UTF-8 badly.
There were a number of bugs reported with it in GitHub.
# Problem 3 - Cordova File Plugin
After switching back to the method "atob()", and comparing the output of "atob()" with the original document, I found them to be identical.
However, the file stored on the Tablet was still suffering from the exact same symptoms. Clearly something was going wrong with the Cordova File Plugin.
After looking at the documentation3, I found that the File Plugin will also output UTF-8, similar to the js-base64 library.
In the end, I found out that it only outputs UTF-8, if I write a string in a Blob to the File. If I change what I write into a JavaScript ArrayBuffer in a Blob, things work as they should.
And I finally got a nice PDF in the standard PDF Viewer of the Tablet.
# References
[1] MDN - Base64 encoding and decoding
https://developer.mozilla.org/en/docs/Web/API/WindowBase64/Base64_encoding_and_decoding
[2] RFC4648 The Base16, Base32, and Base64 Data Encodings - Section 5
https://tools.ietf.org/html/rfc4648#section-5
[3] File - Apache Cordova - cordova-plugin-file
https://cordova.apache.org/docs/en/latest/reference/cordova-plugin-file/
[4] js-base64 - Yet another Base64 transcoder in pure JS
https://www.npmjs.com/package/js-base64
[5] Runkit
https://runkit.com/npm/js-base64
StackOverflow - Using Javascript's atob to decode base64 doesn't properly decode utf-8 strings
http://stackoverflow.com/questions/30106476/using-javascripts-atob-to-decode-base64-doesnt-properly-decode-utf-8-strings
C#411 - Convert Binary to Base64 String
http://www.csharp411.com/convert-binary-to-base64-string/
## Thursday, 5 January 2017
### Inkscape
Inkscape is a vector graphics drawing tool for Linux, which I've used in the past.
Fedora magazine has a series of articles on it, which I link to here.
Part 1 - Getting started with Inkscape on Fedora
https://fedoramagazine.org/getting-started-inkscape-fedora/
Part 2 - Inkscape: Adding some colour | 2017-06-28 20:52:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2556483745574951, "perplexity": 1324.7842658419988}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128323801.5/warc/CC-MAIN-20170628204133-20170628224133-00551.warc.gz"} |
https://www.ni.com/documentation/en/labview-comms/latest/analysis-node-ref/complementary-incomplete-gamma-function/ | Complementary Incomplete Gamma Function (G Dataflow)
Computes the regularized complementary incomplete gamma function.
x
Input argument. If x is negative, the node uses the absolute value of x.
Default: 0
a
The lower limit of the regularized complementary incomplete gamma integral.
Default: Positive infinity
error in
Error conditions that occur before this node runs.
The node responds to this input according to standard error behavior.
Standard Error Behavior
Many nodes provide an error in input and an error out output so that the node can respond to and communicate errors that occur while code is running. The value of error in specifies whether an error occurred before the node runs. Most nodes respond to values of error in in a standard, predictable way.
error in does not contain an error error in contains an error
If no error occurred before the node runs, the node begins execution normally.
If no error occurs while the node runs, it returns no error. If an error does occur while the node runs, it returns that error information as error out.
If an error occurred before the node runs, the node does not execute. Instead, it returns the error in value as error out.
Default: No error
1 - g(x, a)
Value of the regularized complementary incomplete gamma function.
error out
Error information.
The node produces this output according to standard error behavior.
Standard Error Behavior
Many nodes provide an error in input and an error out output so that the node can respond to and communicate errors that occur while code is running. The value of error in specifies whether an error occurred before the node runs. Most nodes respond to values of error in in a standard, predictable way.
error in does not contain an error error in contains an error
If no error occurred before the node runs, the node begins execution normally.
If no error occurs while the node runs, it returns no error. If an error does occur while the node runs, it returns that error information as error out.
If an error occurred before the node runs, the node does not execute. Instead, it returns the error in value as error out.
Algorithm for Computing the Regularized Complementary Incomplete Gamma Function
The following equation defines the complement of the regularized incomplete gamma function.
${\mathrm{\Gamma }}_{c}\left(x,a\right)=\frac{1}{\mathrm{\Gamma }\left(x\right)}{\int }_{a}^{\infty }{t}^{x-1}{e}^{-t}dt$
The complement of the regularized incomplete gamma function is related to the regularized incomplete gamma function by the following identity.
$\mathrm{\Gamma }\left(x,a\right)+{\mathrm{\Gamma }}_{c}\left(x,a\right)=1$
The regularized incomplete gamma function is defined according to the following intervals for the input values.
$x\in \left[0,\infty \right),a\in \left(0,\infty \right)$
For any positive real value of lower limit a, the regularized incomplete gamma function is defined for nonnegative real values of x.
Where This Node Can Run:
Desktop OS: Windows
FPGA: Not supported
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Urbanization favors high wage earners
Abstract
As cities increase in size, total wages grow superlinearly, meaning that average wages are higher in larger cities. This phenomenon, known as the urban wage premium, supports the notion that urbanization and the growth of cities contribute positively to human well-being. However, it remains unclear how the distribution of wages changes as cities grow. Here we segment the populations of U.S. cities into wage deciles and determine the scaling coefficient of each decile’s aggregate wages versus city size. We find that, while total wages of all deciles grow superlinearly with city size, the effect is uneven, with total wages of the highest wage earners growing faster than all other deciles. We show that this corresponds with the predominance of high-wage jobs in larger cities. Thus, the effects of urbanization are mixed -- it is associated with higher average wages but with increasing inequality, thus inhibiting prospects for long-term sustainability.
Introduction
Urban scholars have long known that workers in larger cities earn higher average wages than workers in smaller cities, a phenomenon known as the urban wage premium1,2. Over a century ago, Alfred Marshall3 referred to this phenomenon as economies of agglomeration or scale, while the biologist Haldane, amongst others, defined this effect in both animals and humans as positive allometry4. Among cities, researchers have shown that both wages and incomes increase superlinearly as a power function of population size5,6,7,8,9, meaning that average incomes and wages are higher in larger cities. Findings such as these have contributed to the view that urbanization can be a positive contributor to human well-being10,11,12. Yet, in both human and natural systems, inequality of resource distribution is known to increase with time and development13,14 suggesting that while larger cities may bring higher average wages, they may also exhibit higher levels of inequality.
Such inequality is not only “a threat to economic progress, social cohesion and political stability”15, it also negatively impacts a city’s prospects of becoming sustainable16. However, the relationship between inequality and sustainability is complex17,18,19,20. The relationship is seemingly mediated through levels of social trust, with higher levels of inequality decreasing social trust, which in turn decreases sustainable behaviors. Yet social trust is also known to increase with higher levels of aggregate income. Thus, growth may offset some of the negative impacts of increased inequality21. Furthermore, increased inequality is correlated with higher levels of innovation22, which is an important contributor to future sustainability. This further complicates the relationship between inequality, growth, and sustainability.
Our goal in this study is to better understand the nature of this complex relationship between city size and inequality. While studies have shown that inequality of wages, incomes, and other indicators of prosperity exists between cities of different sizes5,15,23,24,25, we seek to understand how inequality within cities varies as a function of city size. We do this by examining how the distribution of wages within cities changes with size.
Researchers have previously addressed this question by analyzing descriptive metrics of distributions, such as GINI coefficients6,14 or the ratios of different distribution percentiles26,27. Here, we use power-law scaling analysis to search for systematic regularities of wage distributions as a function of city size. Power-law scaling has been used to characterize the relationship between city size and aggregate urban measures such as wages, income, housing prices, infrastructure, crime, innovation, information networks, professional diversity, etc.5,6,7,8,9,28,29,30,31,32. In this study, we focus on wages.
While others have examined the power-law scaling behavior of aggregate wages5,33, it remains unclear how the distribution of wages changes with increasing city size. This distribution can give important clues about the relationship between city size and wage inequality. To do so we divide the populations of 382 U.S. metropolitan statistical areas (MSAs) m into wage deciles d so that the lowest 10% of wage earners are in decile one and the highest 10% of wage earners are in decile ten. We then calculate the power-law scaling coefficient of a decile’s total wages Wm,d versus the MSA’s population Pm
$$W_{m,d} = \alpha P_m^\beta .$$
(1)
We take β to be the magnitude of a decile’s urban wage premium and compare how the wage premium is realized across different classes of wage earners.
Previous studies have used a similar method to examine how distributions of personal income changes with city size7,28. These studies divide urban populations into fixed bins based on an individual’s personal income and then fit the number of workers in each income bin against city size as a power law. Thus, while this method compares the number of workers per income bin versus population, our method compares total wages per population decile versus population. This is a critical distinction as the use of fixed bins, for either wages or income, is significantly affected by differences in cost of living among cities (see Supplementary Discussion and Supplementary Fig. 1 for further discussion of this method).
Results and discussion
We find that power-law scaling coefficient of decile wages versus population size varies significantly across wage deciles (Fig. 1). In all years studied the wage premium for the tenth decile of wage earners is significantly higher than any other decile and decreases monotonically across deciles. In 2019 the tenth decile exhibited β = 1.16, while the aggregate wage premium across all wage earners was β = 1.11.
The pattern across deciles remained remarkably stable over our study period of 2005–2019. The scaling coefficient of the tenth decile did increase slightly from 1.142 in 2005 to 1.156 in 2019. Similar increases were exhibited in deciles 5 through 9, while βs for deciles 1 and 2 decreased slightly.
Examining other decile characteristics, we find the proportion of workers with a college degree increases substantially with higher wage deciles while the proportion of female workers decreases from over half of workers in the lowest wage decile to about one fourth of workers in the highest decile (Table 1).
Scaling exponents of deciles
Despite the adverse effects of increased inequality15 and the UN’s Sustainable Development Goal to “reduce inequality within and among countries”34, inequality has continued to increase globally over the past 25 years23,35. Phenomena cited as drivers of this inequality include globalization, technology change, and the decline of labor unions36. Our study suggests that these various drivers manifest disproportionately in larger cities. This complicates the role of urbanization in enhancing human well-being as it creates a trade-off between the opportunity for higher wages that come with larger cities and increasing inequality of those wages.
Concentration of high-wage jobs in larger cities
To explain results we focus on the fact that cities require the presence of particular economic building blocks before they can grow more complex, and typically more lucrative, industries29. The degree to which those industries require others to build upon is reflected in the power-law scaling coefficient β′ of an MSA’s employment in an industry Em,i versus the MSA’s size Pm
$$E_{m,i} = \alpha P_m^{\beta ^\prime }.$$
(2)
Superlinear scaling (i.e. β > 1) indicates that industries tend to appear more in cities that have reached a population size capable of supporting both the emerging industry and its prerequisites. When calculating 2019 scaling coefficients for each 2-digit industry in the PUMS data, our results echoed those of Youn et al.29 which showed that management, professional, and scientific consulting services had the highest scaling coefficients (See Supplementary Table 1 for industry scaling results).
We apply that same technique to occupations using the U.S. Census Bureau’s 2-digit occupational groups. Growth in some occupations requires the existence of prerequisites, such as other occupations or a minimum city size, forming a hierarchy of increasingly complex occupations37,38. As with industries, the degree to which occupations depend on prerequisites is reflected in the scaling exponent β” of an MSA’s employment in an occupational Em,o versus the MSA’s size Pm
$$E_{m,o} = \alpha P_m^{\beta ^{\prime \prime }}.$$
(3)
The larger an occupation’s scaling coefficient, the larger the population required to support that occupation. Table 2 presents 2019 power-law scaling coefficients of all 2-digit occupation groups across MSAs.
The pattern shown in Fig. 1 suggests that those occupations emerging in larger cities also bring higher wages relative to occupations that do not require larger city size. This is supported by a positive correlation between an occupation’s scaling exponent and its average wages across MSAs (R2 = 0.46, excluding military occupations).
Higher scaling coefficients in the top wage deciles are also partly a result of the fact that mean wages of some occupation groups increase with city size25. In other words, not only do higher wage occupations concentrate in large cities, but workers in the same occupation group earn higher wages in larger cities. We quantify this effect by calculating the scaling coefficient β′′′ of total occupational wages Wm,o in an MSA versus total occupational employment Em,o in an MSA for each 2-digit occupation group o
$$W_{m,o} = \alpha E_{m,{o}}^{\quad\,\,\beta ^{\prime \prime \prime }}.$$
(4)
Results show that this effect is particularly pronounced in legal (β‴ = 1.14), sales (β‴ = 1.09), management (β‴ = 1.09), and arts (β‴ = 1.08) occupations. Again, the relationship between an occupation group’s power-law coefficient and the group’s mean wages is positive (R2 = 0.60, excluding military occupations). (See Supplementary Table 2 for full occupation results and Supplementary Table 3 for the same wage scaling analysis applied to industries).
There are likely many factors that contribute to the pattern of inequality shown in Fig. 1. We show that increasing wage inequality is related to the fact that more lucrative occupations both occur more frequently and pay higher wages in larger cities. Overall, our results use individual-level microdata to confirm a general trend identified in many other studies—that wages are higher in larger cities, that those higher wages are concentrated among the highest wage earners, and that this increases wage inequality in larger cities.
The impact of education level
While we have shown that higher inequality is related to the asymmetric concentration of high wage jobs in larger cities, this creates a new question of why high wage jobs concentrate in larger cities. Some have suggested that workers earn a premium in larger cities because they acquire valuable skills through knowledge spillovers that they could not acquire in smaller cities25. Others have suggested that larger cities offer more opportunities for high skilled workers to be matched with a job requiring high skill levels and that once a city has established a large concentration of high-skilled workers, they will continue to attract high-skilled workers away from smaller cities39. This latter assertion is supported by the fact that workers with a college education not only concentrate more in larger cities40,41, but that they have continued to do so through multiple generations14,24.
Because education level is positively correlated with wages40,41, we analyzed the 2019 power-law scaling coefficient of workers by educational attainment level to examine whether college education also plays a role in higher inequality levels in larger cities. Results in Table 3 show that not only do the numbers of college educated workers scale superlinearly with MSA size, but the scaling coefficient increases with the typical time required to complete each degree type. Thus, β is approximately linear for workers with some college or an associate degree, β = 1.15 for workers with a bachelor’s degree, and β = 1.26 for workers with a professional degree. Thus, not only do high wage jobs concentrate in larger cities, so too do workers with advanced education levels.
Furthermore, the number of workers with a high school education or less scales sublinearly with MSA size suggesting that not only do larger cities offer more opportunities work high-skilled workers, but they also offer diminishing opportunities for less skilled workers. This phenomenon likely exacerbates the appearance of higher levels of inequality in larger cities.
Stability of our findings over time
We conducted our analysis for every year from 2005 to 2019, showing three years of results graphically in Fig. 1. During this period the pattern of inequality across deciles remained qualitatively fixed and scaling exponents of individual deciles changed little over time. However, our study period is too short to conclude that the pattern we find among wage deciles holds over longer periods of time.
On the contrary, a large body of literature examines a broad transition starting in the late 1970s and continuing through the 1980s in which larger cities began to diverge from smaller cities in several characteristics, including interregional inequality23,26,42,43,44. This transition has also appeared in a recent long-term study of aggregate wage scaling of urban areas, with small cities having a higher wage premium than large cities prior to the mid-1990s45. Thus, there is ample indication that the trend we observe among wage deciles has looked qualitatively different in the past.
Future directions
This study lays the groundwork for several tantalizing future research directions. First, we note that our study excludes part-time and partial-year workers. Our goal was to do an initial study without the complications and possible distortions of these categories of workers. Yet, these workers, by earning less than their full-time counterparts, likely exacerbate effects of inequality and could be integrated in future studies.
Second, we briefly discussed the impact of differences in prices among cities on different scaling calculations. While we show in Supplementary Fig. 2 that prices do not affect the patterns uncovered in our analysis, it does change the magnitude of power-law scaling coefficients in our study and can significantly alter the results of other methods used to analyze wage and income distributions. Thus, further research is needed to better understand the impact of regional price differences on measures of inequality.
Finally, we propose that power-law scaling analysis of occupational and industrial employment versus city size may be supplemented by a similar analysis of skills. Do particular skills concentrate in larger cities and are those skills associated with higher wages? Addressing such questions will likely enhance our understanding of the drivers of inequality both between and within cities.
Methods
Data
We take wage and population data from the American Community Survey, Public Use Microdata Set (PUMS) published by the U.S. Census Bureau46. These data include approximately 400 individual-level attributes collected annually from 1% of the U.S. population. From these data we extracted each individual’s employment status, annual wages, sex, and educational attainment for the years 2005–2019.
Individuals represented in the PUMS dataset are assigned to spatial units called Public Use Microdata Areas (PUMAs), which are designed to encompass approximately 100,000 residents and which correspond to no other generally used geographical unit. Therefore, we use the mapping of PUMAs to U.S. metropolitan statistical areas (MSA) from iPums.org47 to assign individuals to MSAs and to calculate the total population of each MSA.
Synthesizing complete MSA populations
Each individual in the PUMS dataset is assigned an expansion factor, or weight, which estimates the total number of people in the individual’s PUMA having the same attributes, including wages. Those weights were adjusted by the PUMA-to-MSA crosswalk factor to estimate a total number of people per MSA at each wage value, occupation, industry, and educational attainment level. We then used the weighted n-tiles algorithm of the R package hutils48 to expand our sample and place every individual of an MSA into a population decile ordered by wages.
From this expanded dataset we extracted only individuals in the workforce that worked 50–52 weeks in the previous 12 months and that ordinarily worked 35 or more hours per week. This avoids distortions, particularly in lower wage deciles, due to part-time and partial-year employment. Note that PUMS data does not designate anyone under age 16 as employed even if they are employed.
With the resulting dataset, we calculated total wages by decile for each MSA, the log of which was regressed against the log of the MSA’s total population to calculate a decile’s power-law scaling coefficient β.
To better understand our main result, we further calculated the power-law scaling coefficient of number of MSA workers in each 2-digit occupation, in each 2-digit industry, and in each educational attainment category versus MSA population. In each case we used the US Census Bureau’s categories for occupations, industries, and educational attainment levels.
Data Availability
All data used in this study are publicly available from (1) The US Census Bureau at: https://www.census.gov/programs-surveys/acs/microdata/access.html, (2005–2019 American Community Survey, 1-year Public Use Microdata Samples); and (2) iPUMS at: https://usa.ipums.org/usa/resources/volii/MSA2013\_PUMA2010\_crosswalk.xls (Crosswalk Between 2013 MSAs and 2010 PUMAs).
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Acknowledgements
S.T.S. and J.M.A. received funding from the ASU Knowledge Exchange for Resilience which is supported by Virginia G. Piper Charitable Trust. The conclusions, views, and opinions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of the Virginia G. Piper Charitable Trust.
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Authors
Contributions
S.T.S. and J.M.A. conceived of the project, acquired the data, and performed the analysis. S.T.S., J.M.A., E.W., and M.B. wrote and revised the paper.
Corresponding author
Correspondence to Shade T. Shutters.
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Shutters, S.T., Applegate, J.M., Wentz, E. et al. Urbanization favors high wage earners. npj Urban Sustain 2, 6 (2022). https://doi.org/10.1038/s42949-022-00049-x
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• DOI: https://doi.org/10.1038/s42949-022-00049-x | 2022-06-27 01:02:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.39480578899383545, "perplexity": 6867.435569199796}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103322581.16/warc/CC-MAIN-20220626222503-20220627012503-00237.warc.gz"} |
https://tex.stackexchange.com/questions/538891/long-chemfig-reaction-scheme-in-two-lines-above-each-other | Long Chemfig reaction scheme in two lines above each other
The reaction scheme shown below is to long to fit in one line. I would like to place the two left compounds atop one another, so the + is in the middle between them, vertically centered with the arrow.
MWE:
\documentclass[10pt,a4paper]{article}
\usepackage[left=32.5mm, right=25mm, showframe]{geometry}
\usepackage[version=4]{mhchem}
\usepackage{chemfig}
\setchemfig{fixed length=true, atom sep=1.5em, arrow offset=6pt}
\def\x{\vphantom{C}}
\begin{document}
\setchemfig{arrow coeff=1.65}
\centering\small
\schemestart
\chemfig{\x-[@{li}]N(-[2]H)-\x|{(}CH_2{)}_{11}|\x-C(=[6]O)-[@{re}]}
\polymerdelim[delimiters={[]},height=25pt,depth=29pt]{li}{re}
\+
\chemfig{\x-[@{li}]N(-[2]H)-\x|{(}CH_2{)}_{11}|\x-C(=[6]O)-[@{re}]}
\polymerdelim[delimiters={[]},height=25pt,depth=29pt,indice=m]{li}{re}
\arrow{<->>[$T$, $t$, $\Delta p_{\ce{H2O}}$][]}
\chemfig{\x-[@{li}]N(-[2]H)-\x|{(}CH_2{)}_{11}|\x-C(=[6]O)-[@{re}]}
\polymerdelim[delimiters={[]},height=25pt,depth=29pt,indice=n+m]{li}{re}
\+
\chemfig{H_2O} \
\schemestop
\end{document}
EDIT: As @leandriis pointed out, simply putting the arrow text in a \parbox, such as in this question, solves the issue.
EDIT 2: I forgot to add a H and a OH group at the sides on all molecules, making them even longer. Thus, the original issue still exists and is even worse.
MWE 2:
\setchemfig{arrow label sep=5pt}
\centering\small
\schemestart
\chemfig{\x H-[@{li}]N(-[2]H)-\x|{(}CH_2{)}_{11}|\x-C(=[6]O)-[@{re}]OH}
\polymerdelim[delimiters={[]},height=25pt,depth=29pt]{li}{re}
\+
\chemfig{\x H-[@{li}]N(-[2]H)-\x|{(}CH_2{)}_{11}|\x-C(=[6]O)-[@{re}]OH}
\polymerdelim[delimiters={[]},height=25pt,depth=29pt,indice=m]{li}{re}
\arrow{<->>[\parbox{5cm}{\centering $T$, $t$,\\$\Delta p_{\ce{H2O}}$}][]}
\chemfig{\x H-[@{li}]N(-[2]H)-\x|{(}CH_2{)}_{11}|\x-C(=[6]O)-[@{re}]OH}
\polymerdelim[delimiters={[]},height=25pt,depth=29pt,indice=n+m]{li}{re}
\+
\chemfig{H_2O}
\schemestop
Using a vertical \subscheme and invisible arrows it is rather easy:
\documentclass[10pt,a4paper]{article}
\usepackage[left=32.5mm, right=25mm, showframe]{geometry}
\usepackage[version=4]{mhchem}
\usepackage{chemfig}
\setchemfig{fixed length=true, atom sep=1.5em, arrow offset=6pt}
\begin{document}
\begin{center}
\def\x{\vphantom{C}}
\setchemfig{arrow label sep=5pt}
\small
\schemestart
\subscheme[-90]{%
\chemfig{\x H-[@{li}]N(-[2]H)-\x|{(}CH_2{)}_{11}|\x-C(=[6]O)-[@{re}]OH}
\polymerdelim[delimiters={[]},height=25pt,depth=29pt]{li}{re}
\arrow{0}[,.3]
\+{0pt,1em}
\arrow{0}[,.3]
\chemfig{\x H-[@{li}]N(-[2]H)-\x|{(}CH_2{)}_{11}|\x-C(=[6]O)-[@{re}]OH}
\polymerdelim[delimiters={[]},height=25pt,depth=29pt,indice=m]{li}{re}
}
\arrow{<->>[{\begin{tabular}{c}$T$, $t$,\\$\Delta p_{\ce{H2O}}$\end{tabular}}][]}
\chemfig{\x H-[@{li}]N(-[2]H)-\x|{(}CH_2{)}_{11}|\x-C(=[6]O)-[@{re}]OH}
\polymerdelim[delimiters={[]},height=25pt,depth=29pt,indice=n+m]{li}{re}
\+
\chemfig{H_2O}
\schemestop
\end{center}
\end{document} | 2020-10-28 17:23:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9419311285018921, "perplexity": 4206.025165810909}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107900200.97/warc/CC-MAIN-20201028162226-20201028192226-00559.warc.gz"} |
https://www.gradesaver.com/textbooks/science/physics/physics-10th-edition/chapter-7-impulse-and-momentum-problems-page-193/28 | Physics (10th Edition)
The two slide together at a speed of $3.51m/s$.
We have Ashley, with mass $M=71kg$ and $V=+2.7m/s$, and Miranda, with mass $m=58kg$ at $v=+4.5m/s$. After Miranda jumps into the tube, the system has mass $M+m=129kg$ and runs at velocity $V_f$, which needs to be found. Since friction is ignored, the total linear momentum is conserved. Therefore, $$M\vec{V}+m\vec{v}=(M+m)\vec{V}_f$$ $$\vec{V}_f=\frac{M\vec{V}+m\vec{v}}{M+m}=+3.51m/s$$ So the two slide together at a speed of $3.51m/s$. | 2021-08-01 18:18:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9189090132713318, "perplexity": 349.86529366661324}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154214.63/warc/CC-MAIN-20210801154943-20210801184943-00183.warc.gz"} |
https://codereview.stackexchange.com/questions/2855/selection-sort-using-recursion | # Selection sort using recursion
How does this recursive selection sort look to everyone here? Am I missing anything 'pythonic' about the way I have done it?
def selection_sort(li, out=None):
if out is None:
out = []
li = li[:]
if len(li) == 0:
return out
small = min(li)
li.remove(small)
out.append(small)
return selection_sort(li, out)
def selection_sort(li, out=None):
I dislike the name "li," I think abbreviations are bad.
if out is None:
out = []
li = li[:]
Rather then using the out parameter to do this, I suggest creating a seperate internal function which is called. Otherwise, it looks like the caller to the function might reasonable pass out = something, when its only meant to be used internally.
if len(li) == 0:
return out
Better to use if not li:
small = min(li)
li.remove(small)
out.append(small)
return selection_sort(li, out)
Its a little hard to gauge this code because you are doing two things you shouldn't, using recursion when its not necessary and implementing your own sort routine. But if you are doing so for learning purposes that's fine.
EDIT
Iterative solution:
def selection_sort(li):
li = li[:]
out = []
while li:
smallest = min(li)
li.remove(smallest)
out.append(smallest)
return out
But why is this better than the recursive solution:
1. There is a limit to your stack space. If you try to sort a large list using this function you'll get a RuntimeError
2. Calling a function has extra overhead that iterating in a loop does not, the iterative version will be faster
3. A loop is usually easier to read then recursion. The while loop makes it easy to see whats doing on, whereas the recursive version requires some thought about the code to see the logic.
• Aye learning, and why not recursion for this situation? – Jakob Bowyer Jun 7 '11 at 19:06
• @Jakob, see edit – Winston Ewert Jun 7 '11 at 19:18
• +1: amplifying "don't have output parameters". Given Python's ability to return arbitrary tuples, there is never a reason to use an output parameter. For example: return (a_list, a_dict_of_dicts, status) is valid and idiomatic. Also pay attention to the difference between list.sort() and list.sorted() in the standard library (hint: one returns a value, the other doesn't, why?) – msw Jun 17 '11 at 1:09
• @msw, I wouldn't quite say "never." For example, numpy has reasonable use of out parameters for efficiency reasons. – Winston Ewert Jun 17 '11 at 23:14
• Fair 'nuff. How about "unless you know whyfor"? – msw Jun 18 '11 at 10:04 | 2019-05-23 08:05:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.35535576939582825, "perplexity": 3434.1454801106297}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232257156.50/warc/CC-MAIN-20190523063645-20190523085645-00024.warc.gz"} |
http://planetmath.org/FunctionContinuousAtOnlyOnePoint | # function continuous at only one point
Let us show that the function $f\colon\mathbbmss{R}\to\mathbbmss{R}$,
$f(x)=\begin{cases}x,&\mbox{when x is rational},\\ -x,&\mbox{when x is irrational},\end{cases}$
is continuous at $x=0$, but discontinuous for all $x\in\mathbbmss{R}\setminus\{0\}$ [1].
We shall use the following characterization of continuity for $f$: $f$ is continuous at $a\in\mathbbmss{R}$ if and only if $\lim_{k\to\infty}f(x_{k})=f(a)$ for all sequences $(x_{k})\subset\mathbbmss{R}$ such that $\lim_{k\to\infty}x_{k}=a$.
It is not difficult to see that $f$ is continuous at $x=0$. Indeed, if $x_{k}$ is a sequence converging to $0$. Then
$\displaystyle\lim_{k\to\infty}|f(x_{k})|$ $\displaystyle=$ $\displaystyle\lim_{k\to\infty}|f(x_{k})|$ $\displaystyle=$ $\displaystyle\lim_{k\to\infty}|x_{k}|$ $\displaystyle=$ $\displaystyle 0.$
Suppose $a\neq 0$. Then there exists a sequence of irrational numbers $x_{1},x_{2},\ldots$ converging to $a$. For instance, if $a$ is irrational, we can take $x_{k}=a+1/k$, and if $a$ is rational, we can take $x_{k}=a+\sqrt{2}/k$. For this sequence we have
$\displaystyle\lim_{k\to\infty}f(x_{k})$ $\displaystyle=$ $\displaystyle-\lim_{k\to\infty}x_{k}$ $\displaystyle=$ $\displaystyle-a.$
On the other hand, we can also construct a sequence of rational numbers $y_{1},y_{2},\ldots$ converging to $a$. For example, if $a$ is irrational, this follows as the rational numbers are dense in $\mathbbmss{R}$, and if $a$ is rational, we can set $y_{k}=x_{k}+1/k$. For this sequence we have
$\displaystyle\lim_{k\to\infty}f(y_{k})$ $\displaystyle=$ $\displaystyle\lim_{k\to\infty}y_{k}$ $\displaystyle=$ $\displaystyle a.$
In conclusion $f$ is not continuous at $a$.
## References
• 1 Homepage of Thomas Vogel, http://www.math.tamu.edu/ tom.vogel/gallery/node3.htmlA function which is continuous at only one point.
Title function continuous at only one point FunctionContinuousAtOnlyOnePoint 2013-03-22 14:56:19 2013-03-22 14:56:19 Andrea Ambrosio (7332) Andrea Ambrosio (7332) 7 Andrea Ambrosio (7332) Example msc 26A15 msc 54C05 DirichletsFunction FunctionDifferentiableAtOnlyOnePoint | 2018-03-18 09:47:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 46, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9839614629745483, "perplexity": 111.34023279603052}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257645604.18/warc/CC-MAIN-20180318091225-20180318111225-00664.warc.gz"} |
http://www.mychemistry.eu/2013/05/about-footnotes/ | You all know how easy it is to add footnotes in LaTeX. Most times it is as easy as calling \footnote{This is the footnote.}. Typographically the result often is not very pleasing, though. Let’s look at the following small example:
\documentclass{article}
\renewcommand\thempfootnote{\arabic{mpfootnote}}
\begin{document}
\begin{minipage}{.5\linewidth}
\noindent The three little pigs built their houses
out of straw\footnote{not to be confused with hay},
sticks\footnote{or lumber according to some sources}
and bricks\footnote{probably fired clay bricks}.
\end{minipage}
\end{document
}
This looks as follows:
The footnote numbers are huge and the gaps between them and the punctuation marks are not nice, either. With my favourite font it looks a little bit better although actually nothing happened.
\documentclass{article}
\usepackage{libertine}
\renewcommand\thempfootnote{\arabic{mpfootnote}}
\begin{document}
\begin{minipage}{.5\linewidth}
\noindent The three little pigs built their houses
out of straw\footnote{not to be confused with hay},
sticks\footnote{or lumber according to some sources}
and bricks\footnote{probably fired clay bricks}.
\end{minipage}
\end{document
}
The footnote numbers really should be typeset with superior figures. Not every font has them but Linux Libertine O does. Using them is quite easy, actually, since Michael Sharpe published his superiors package, which needs to be loaded before libertine:
\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[supstfm=libertinesups]{superiors}
\usepackage{libertine}
\renewcommand\thempfootnote{\arabic{mpfootnote}}
\begin{document}
\begin{minipage}{.5\linewidth}
\noindent The three little pigs built their houses
out of straw\footnote{not to be confused with hay},
sticks\footnote{or lumber according to some sources}
and bricks\footnote{probably fired clay bricks}.
\end{minipage}
\end{document
}
The result looks a lot better now! The only thing remaining is the gaps between footnote marks and punctuation marks. But thanks to Christian and his question on TeX.sx there is also an easy solution for that: my fnpct package.
\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[supstfm=libertinesups]{superiors}
\usepackage{libertine}
\usepackage{fnpct}
\renewcommand\thempfootnote{\arabic{mpfootnote}}
\begin{document}
\begin{minipage}{.5\linewidth}
\noindent The three little pigs built their houses
out of straw\footnote{not to be confused with hay},
sticks\footnote{or lumber according to some sources}
and bricks\footnote{probably fired clay bricks}.
\end{minipage}
\end{document
}
As you can see footnote marks and punctuation marks switch positions and moved closer together. The package lets you customize if you want the switching or not and also let’s you set the amount of kerning for each punctuation mark. What you see in the picture are just the default settings.
That is why fnpct offers an easy way out: \footnote gets an optional star that prevents switching with punctuation or — if switching was disabled through the package option — enables it. | 2020-09-26 14:05:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7738385796546936, "perplexity": 3989.560957632546}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400244231.61/warc/CC-MAIN-20200926134026-20200926164026-00280.warc.gz"} |
https://tex2e.github.io/blog/math/math-pronounce | # 晴耕雨読
## working in the fields on fine days and reading books on rainy days
The pronunciations of mathematical symbols are given in the list below.
Math Pronunciation LaTeX
$\dfrac{1}{2}$ one over two \dfrac
$1.2$ one point two .
$\therefore$ therefore \therefore
$\because$ because \because
$\forall x$ for all $x$ \forall
$\exists x$ for some $x$ \exists
$p \wedge q$ $p$ and $q$ \wedge
$p \vee q$ $p$ or $q$ \vee
$\neg p$ not $p$ \neg
$p \rightarrow q$ $p$ implies $q$ \rightarrow
$p \Rightarrow q$ if $p$ then $q$ \Rightarrow
$p \leftrightarrow q$ $p$ is equivalent to $q$ \leftrightarrow
$p \Leftrightarrow q$ $p$ if and only if [iff] $q$ \Leftrightarrow
$a’$ $a$ prime
$a’’$ $a$ double prime ’’
$\bar{a}$ $a$ bar \var
$\widetilde{a}$ $a$ tilde \widetilde
$\hat{a}$ $a$ hat \hat
$a^*$ $a$ star ^*
$\pm a$ plus or minus $a$ \pm
$\lvert a\rvert$ absolute value of $a$ \lvert \rvert
$a^2$ $a$ square ^2
$a^3$ $a$ cube ^3
$a^n$ $a$ to the $n$ [$n$th (power)] ^n
$a_n$ $a$ sub $n$ _n
$\sqrt{a}$ square root of $a$ \sqrt
$\sqrt[3]{a}$ cube root of $a$ \sqrt[3]
$\sqrt[n]{a}$ $n$th root of $a$ \sqrt[n]
$a \approx b$ $a$ is approximately equal to $b$ \approx
$a \le b$ $a$ is greater then or equal to $b$ \le
$a \ge b$ $a$ is less then or equal to $b$ \ge
$a \times b$ $a$ times $b$ \times
$a \cdot b$ $a$ times $b$ \cdot
$a \div b$ $a$ by $b$ \div
$a \,/\, b$ $a$ by $b$ /
$a:b$ the ratio of $a$ to $b$ :
$(a + b)c$ $a$ plus $b$ in parentheses times $c$
$(a + b)(a - b)$ $a$ plus $b$ times $a$ minus $b$
$n!$ $n$ factorial !
$_nP_r$ permutation of $n$ things (taken) $r$ at a time _nP_r
$_nC_r$ combination of $n$ things (taken) $r$ at a time _nC_r
$a \equiv b \pmod{n}$ $a$ is congruent to $b$ modulo $n$ \equiv \pmod
$a|b$ $a$ is a divisor of $b$ |
$\vec{a}$ vector $a$ \vec
$% $ $m$ times $n$ matrix $a$ one one to $a$ $m$ $n$ \begin{pmatrix} \cdots \vdots \ddots
$\left( a_{ij} \right)$ matrix $a$ $i$ $j$ _{ij}
$\det A$ determinant of $A$ \det
$\lvert A\rvert$ determinant of $A$ \lvert \rvert
$A^{-1}$ inverse of $A$ ^{-1}
$A^t$ transpose of $A$ ^t
$\mathrm{adj}\, A$ adjoint of $A$ \mathrm{adj}\,
$\mathrm{tr}\, A$ trace of $A$ \mathrm{tr}\,
$A \bigotimes B$ Kronecker product (tensor product) of $A$ $B$ \bigotimes
$\{x\,|\,C\}$ set of $x$ satisfying $C$ { \,|\, }
$\{x_n\}$ sequence $x$ sub $n$ {}
$x_1, x_2, …$ $x$ one $x$ two and so on
$x \in \mathbb{Z}$ $x$ is an element of $\mathbb{Z}$ \in \mathbb
$\mathbb{Z} \ni x$ $\mathbb{Z}$ contains $x$ \mathbb \ni
$A \subset B$ $A$ is contained in $B$, $A$ is a subset of $B$ \subset
$A \supset B$ $A$ contains $B$ \supset
$A \varsubsetneqq B$ $A$ is properly contained in $B$ \varsubsetneqq
$A \cup B$ $A$ union $B$ \cup
$A \cap B$ $A$ intersection $B$ \cap
$\emptyset$ empty set \emptyset
$\dim A$ dimension of $A$ \dim
$\ln x$ natural logarithm of $x$ \ln
$\Re z$ real part of $z$ \Re
$\Im z$ imaginary part of $z$ \Im
$\lim_{n\to\infty} x_n$ limit of $x_n$ as $n$ tends to [approaches] infinity \lim \to \infty
$x_n \to -\infty \;(n\to\infty)$ $x_n$ tends to minus infinity as $n$ tends to [approaches] infinity \to \infty
$x_1 + \cdots + x_n$ the sum of the x’s from sub one to sub $n$ \cdots
$\sum_{k=1}^k x_k$ sum of $x_k$ k equals one to $n$ \sum
$\prod_{k=1}^k x_k$ product of $x_k$ k equals one to $n$ \prod
$f: X \to Y$ function $f$ mapping $X$ into $Y$ \to
$f: x \mapsto x$ function $f$ mapping $x$ to $y$ \mapsto
$f | E$ $f$ restricted on $E$ |
$\mathrm{Im}\,f$ image of $f$ \mathrm{Im}\,
$\mathrm{Ker}\,f$ kernel of $f$ \mathrm{Ker}\,
$f \circ g$ $f$ composition $g$ \circ
$f * g$ $f$ convolution $g$ *
$f \sim g$ $f$ is equivalent to $g$ \sim
$\lVert x\rVert$ norm of $x$ \lVert \rVert
$\dfrac{dy}{dx}$ $d$ $y$ over [by] $d$ $x$ d
$\dfrac{d^2y}{dx^2}$ $d$ two $y$ over [by] $d$ $x$ square d^2
$\dfrac{\partial z}{\partial x}$ round $d$ $z$ over round $d$ $x$ \partial
$\varphi(x)$ phi of $x$ \varphi
$\nabla \cdot f$ nabla dot $f$ \nabla \cdot
$\Delta f$ Laplacian of $f$ \Delta
$\int y \,dx$ integral $y$ $d$ $x$ \int \,dx
$\int_a^b y \,dx$ integral from $a$ to $b$ of $y$ $d$ $x$ \int \,dx
$\iint f(x,y) \,dxdy$ double integral of $f$ of $x$ $y$ \iint \,dxdy
$\iint_E f(x,y) \,dxdy$ double integral over $E$ of $f$ of $x$ $y$ \iint \,dxdy
$f \bot g$ $f$ is orthogonal to $g$ \bot
$P(E)$ probability of $E$ P()
$P(E|F)$ probability of $E$ under the condition [when] $F$ P(|)
$\bigcup\limits_{k=1}^{n} E_k$ union of $E_k$ $k$ running from one to $n$ \bigcup\limits
$\bigcap\limits_{k=1}^{n} E_k$ intersection of $E_k$ $k$ running from one to $n$ \bigcap\limits | 2020-02-21 14:27:30 | {"extraction_info": {"found_math": true, "script_math_tex": 8, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9985994696617126, "perplexity": 2279.7569821884995}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145533.1/warc/CC-MAIN-20200221142006-20200221172006-00112.warc.gz"} |
https://www.gradesaver.com/textbooks/math/trigonometry/trigonometry-7th-edition/chapter-3-section-3-3-definition-iii-circular-functions-3-3-problem-set-page-143/8 | ## Trigonometry 7th Edition
$\cos{225^o} = x = -\dfrac{\sqrt2}{2}$
RECALL: In a unit circle $\cos {t}=x \\\sin{t}=y \\\tan{t} = \dfrac{y}{x}, x\ne0 \\\cot{t} = \dfrac{x}{y}, y \ne 0 \\\sec{t} = \dfrac{1}{x}, x\ne0 \\\csc{t} = \dfrac{1}{y}, y \ne0$ Using the definition above and the unit circle in Figure 5 on page 137 of this book, then: Point on the unit circle: $\left(-\dfrac{\sqrt2}{2}, -\dfrac{\sqrt2}{2}\right)$ Thus, $\cos{225^o} = x = -\dfrac{\sqrt2}{2}$ | 2020-10-22 12:45:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8448816537857056, "perplexity": 230.6910723759871}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107879537.28/warc/CC-MAIN-20201022111909-20201022141909-00255.warc.gz"} |
http://www.ijfis.org/journal/view.html?uid=813&&vmd=Full | Title Author Keyword ::: Volume ::: Vol. 18Vol. 17Vol. 16Vol. 15Vol. 14Vol. 13Vol. 12Vol. 11Vol. 10Vol. 9Vol. 8Vol. 7Vol. 6Vol. 5Vol. 4Vol. 3Vol. 2Vol. 1 ::: Issue ::: No. 4No. 3No. 2No. 1
Improved Neighborhood Search for Collaborative Filtering
Yeounoh Chung1, Noo-ri Kim2, Chang-yong Park3, and Jee-Hyong Lee2
1Department of Computer Science, Brown University, Rhode Island, USA, 2Department of Electrical and Computer Engineering, Sungkyunkwan University, Suwon, Korea, 3LG Electronics, Seoul, Korea
Correspondence to: Jee-Hyong Lee (john@skku.edu)
Received February 1, 2018; Revised March 10, 2018; Accepted March 21, 2018.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
k-Nearest Neighbor (k-NN) and other user-based collaborative filtering (CF) algorithms have gained popularity because of the simplicity of their algorithms and performance. As the performance of such algorithms largely depends on neighborhood selection, it is important to select the most suitable neighborhood for each active user. Previous user-based CF simply relies on similar users or common experts in this regard; however, because users have different tastes as well as different expectations for expert advice, similar users or common experts may not always be the best neighborhood for CF. In search of a more suitable neighborhood, so-called personalized experts develop personalized expert features. Through experimentation, we show that personalized experts are different from similar users, common experts, or similar common experts. The personalized, expert-based CF algorithm outperforms k-NN and other user-based CF algorithms.
Keywords : Personalized experts, Recommender system, Collaborative filtering, Support vector machine
1. Introduction
With the success of many e-commerce services (e.g., Amazon, Netflix, Last.fm), recommender systems have gained significant interest and popularity in recent years, and significant effort has been dedicated to researching and building better recommender systems and algorithms [1]. One of the most popular algorithms for recommender systems is collaborative filtering (CF), which simply finds patterns among similar users or items [2]. CF achieved widespread success because of its simplicity and efficiency, despite several drawbacks (e.g., the sparsity problem) [37].
Typical CF (neighborhood or user-based CF) develops user profiles based on the item-consumption profiles of those users and provides personalized recommendations to active (or target) users based on a combination of similar user profiles. CF is based on the simple assumption that users with tastes that are similar to the active user may give more useful information, which may lead to a better recommendation. However, in some cases, users with similar tastes may not give useful information because the active user may have already consumed what the similar users have consumed. Ideally, to obtain the best performance, CF algorithms require users who can provide useful information for recommendations, and they may not necessarily be similar users.
In parallel to similar user-based CF, expert-based CF and recommender systems have been proposed. General users who lack domain knowledge often trust more reliable and knowledgeable experts when making decisions to purchase items. A study conducted in the field of retail and marketing shows that consumers regard expert opinions as more reliable [8]. In agreement with this observation, several recent studies have exploited the knowledge of experts [918]. Those approaches are based on the assumption that users with more expertise may give more useful information, which will lead to more accurate recommendations. Expert-based CF can be more robust for situations where there are not enough item-consumption histories available from which to draw similarities between users (i.e., the sparsity problem) than similar user-based CF [9, 12]. However, expert-based CF is limited in that the experts can only recommend items that are generally popular. In other words, the recommendations are less customized.
In this work, we seek to find a better neighborhood for user-based CF and to combine the merits of both user-based and expert-based approaches. The notion of personalized experts as the better neighborhood from which to provide useful information was first proposed in our previous works [19, 20]]. However, personalized expertise was expressed in crudely developed features for support vector machine (SVM) model training and yielded less accurate recommendations than k-Nearest Neighbor (k-NN). Here, we examine the notion of personalized expertise in various aspects and carefully design new expertise features to identify personalized experts for users with various profiles and preferences. Notably, our new personalized expert-based recommender system outperforms k-NN in terms of prediction accuracy. Furthermore, we present a better learning process for a single global SVM model to find customized expert groups for each user, without any given expert labels or explicit user feedback. The key idea is to train an SVM model to learn the mapping between different user profiles and the most beneficial groups of neighbors. In [19], we proposed to search personalized experts among similar users. This reduced the cost of training, but also bounded the personalized experts to similar users (what if a user does not want any suggestions from similar neighbors?). Instead, we refined the expert pool to be users with any expertise characteristics (e.g., early adopter, heavy access, niche-item access) and select more diversified personalized experts from them.
Our approach is expert-based, but unlike previous expert-based approaches, different experts are chosen for each active user to accommodate different needs. Some users prefer similar users; others prefer early adopters or even users with very eccentric tastes. Furthermore, this personalized expert identification problem is more thoroughly studied to yield a machine learning solution by using a SVM. The resulting recommendations from personalized expert-based CF prove to be more accurate than k-NN and expert-based CF systems and more customized than expert-based CF.
The rest of this study is organized as follows. In Section 2, we briefly discuss previous user-based CF algorithms. In Section 3, we describe the personalized expert identification problem along with the personalized expertise measures in detail. In Section 4, we present the experimental results and analysis. In Section 5, we further discuss the robustness of the proposed recommender system considering the sparsity problem. Finally, we conclude in Section 6.
2. RelatedWork
A recommender system based on a k-NN CF algorithm relies on collaborative opinions of a neighborhood with similar user profiles computed from item consumption histories. Because recommendations are generated based on user profiles alone, similar user-based recommender systems result in accurate recommendations for various users. However, the recommendations may be inaccurate if the item-consumption histories are not sufficient to build rich user profiles [46]. This lack of information in item-consumption histories is referred to as the sparsity problem, and it is one of the most limiting factors for its performance in practice. Many techniques, ranging from dimension reduction to sparse data smoothing, were proposed to address this issue [4, 2124].
To alleviate the sparsity problem and build a better recommender system, several researchers have suggested expert-based CF. Papagelis et al. [6] shows that expert profiles from a movie review website can be used to model user profiles of a much larger user group. By CF of the opinions of similar external experts, the authors were able to produce comparable recommendations to k-NN. Similarly, other external expert-based CF algorithms used external expert knowledge identified from web blogs or real human participants who can provide dynamic feedback for recommendations [11, 16]. This type of external expert-based CF is robust to the sparsity problem; however, it is very expensive to source expert knowledge in most cases, which may limit the scalability of the applications.
Instead of using external expert knowledge, other researchers focused on identifying experts among active users. As the performance of CF algorithms largely depends on neighbor selection (i.e., the source of collective opinions in CF), defining and identifying appropriate experts is important for successful expert-based recommender systems [1015, 17, 18]. The expert groups used in those works are early adopters, personal innovators, and users with highly common expert measures.
Song et al. [14] proposes three common expert measures and identifies a set of common experts from an active user group. Because the same common experts are used in CF for all active users, the resulting recommendations are less personalized. Similarly, Lee and Lee [12] identified common experts per similar item group in their recent work. Their approach suggests different expert groups for different item groups, but recommendations are still not personalized with respect to the active users.
3. Personalized Expert Search
Instead of simply choosing similar users, our approach chooses different experts for each active user who can better accommodate various needs and expectations. We define personalized experts per each active user as neighbors who are the most resourceful for CF-based recommendations. To efficiently determine whether a neighboring user is a personalized expert or not, we train a single global SVM model that learns the matching pattern between personalized experts and active users. Because the task is not just finding similar user profiles, the matching pattern can be complicated, and generating an accurate SVM learner to solve this personalized expert identification problem is challenging. In the following subsections, we discuss three challenges and the solutions for them.
### 3.1 How to Label Training Data?
Training an accurate SVM learner to find personalized experts for active users requires training data with labels–these labels should identify which experts belong to whom. Because such labels are not available (i.e., it is very expensive to receive explicit feedbacks from users), we approximate the labels with a random search.
First, we define a personalized expert group for an active user as a set of users who give the most accurate recommendations. With this definition, and by only using the training data, we select a group of users of a fixed size, called Vui, at random for each active user, ui, to carry out CF with the group and evaluate performance increases. For each iteration, we randomly switch one user in Vui with one user not in Vui. If the new Vui yields better recommendation accuracy, the new Vui is accepted.
This random search procedure repeats for a fixed number of iterations, and the final Vui is used as an approximated personalized expert group for ui. However, this technique is too costly from a computational perspective. To reduce the computational complexity, we assume that the personalized experts exhibit several degrees of common expertise that are accepted by the general population; in other words, we reduce the search space to a handful of users with a higher expertise. The expertise measures are defined in the next subsection. This generic random search algorithm is simple and yet very useful for obtaining a near-optimal solution. In solving ill-structured global optimization problems with many potential stationary points, a random search ensures convergence to a global optimum in terms of probability. Essentially, if the random selection does not ignore any part of the search space, then the algorithm is guaranteed to converge with a probability one [25]. As it follows a geometric distribution, the number of expected iterations until near-optimal convergence (within distance from the optimum) is as follows:
$E [N(Vui*+ɛ)]=1p(Vui*+ɛ).$
Finding the optimum is still very expensive for a practical recommender system, even with the search space reduction. In this work, we limit the number of iterations for finding personalized experts to 1,000, which is empirically shown to be sufficient.
### 3.2 How to Describe Personalized Expertise?
To extract a meaningful matching pattern, we carefully develop features to represent the relationship between any pair of users. The personalized expertise feature vector, Xij, indicates how an active user, ui, views a neighbor, uj . We measure such a pair-wise view with absolute and relative measures. The absolute expertise measures describe how much a neighbor uj is generally accepted as an expert, and the relative measures are used to represent the information of uj with respect to an active user, ui.
We express absolute expertise measures with four features: Early Adopter, Heavy Access, Niche-Item Access and Eccentricity. Early Adopter, Heavy Access and Niche-Item Access are common expertise measures [14] and Eccentricity indicates how eccentric and unique a user is. However, relative expertise measures are defined between a pair of users; namely, an active user and a neighbor. They are expressed in three features: Similarity, Common-Item Access, Unknown-Item Access.
In the following expression, we define the expertise measures used to express different notions of neighborhood expertise:
$X→ij=.$
Early Adopter (EA(ui)) uses new items before others and their opinion can have influence. It measures how long it takes for ui to access newly released items on average. Given reference time (TR), item released time of m (Tm), item rated time of ui (Tui,m), the list of items that ui accessed (I(ui)), we compute EA(ui) as follows:
$EA(ui)=∑m∈I(ui)TR-Tui,m|I(ui)|.$
Heavy Access (HA(ui)) measures how many items a user accessed. In general, more experience means more expertise:
$HA(ui)=log(|I(ui)|+1).$
Niche-Item Access (NA(ui)) measures the average unpopularity of accessed items. In a sense, users who find hidden items that are not popular are ad hoc experts. Given the list of users who accessed item m, U(m), we compute NA(ui) as follows:
$NA(ui)=∑m∈I(ui)log 2log (|U(m)|+1)/∣I(ui)∣.$
Eccentricity (EC(ui)) measures the average item preference deviation from the popular beliefs or the population mean. Some believe that experts must have different and more eccentric views on matters than the rest of the world. Given the average rating on m(m), the actual rating of ui on m(Rui,m), the upper bound of rating values (Rmax), the lower bound of rating values (Rmin), we compute EC(ui) as follows:
$EC(ui)=∑m∈I(ui)log (∣R¯m-Rui,m∣+1)log (Rmax-Rmin)/∣I(ui)∣.$
Similarity (Sim(ui, uj)) measures the similarity between two user profiles. It is measured with the Pearson correlation coefficient of the ratings of the two users. Users with similar item preferences may be more helpful; but some users may prefer users with very different item preferences, so we consider PR(ui, uj) in our neighbor search:
$Sim(ui,uj)=∑m∈I(ui)∩I(uj)(Rui,m-R¯ui)(Ruj,m-R¯uj)∑m∈I(ui)∩I(uj)(Rui,m-R¯ui)2×∑m∈I(ui)∩I(uj)(Ruj,m-R¯uj)2.$
Common-Item Access (CA(ui, uj)) is different from similarity. A user may trust other users with the same item experiences. If two users consume exactly the same set of items and both users like or dislike the same item, the similarity will be high. However, CA(ui, uj) will be high if the number of commonly accessed items is large:
$CA(ui,uj)=log(∣I(ui)∩I(uj)∣+1).$
Unknown-Item Access (UA(ui, uj)) measures how many new items uj has accessed, of which ui has no prior knowledge. ui may prefer neighbors with more experience with new items:
$UA(ui,uj)=log (∣I(ui)-I(uj)∣+1).$
### 3.3 How to Train SVM on Class-Imbalanced Data?
The performance of personalized expert-based CF largely depends on the qualifications of personalized experts; thus, the classification accuracy of the SVM learner is very important. One of the biggest concerns in approximating expert labels is that the number of personalized experts for ui is very small compared to the number of the entire user group. As a result, the accuracy of an SVM learner trained on such an imbalanced training data is degraded [26]. To cope with this, we use the cost sensitive support vector machine (C-SVM) learner [20], which assigns different training error penalties to different classes to effectively learn from imbalanced data [27]. The personalized expert identification problem transformed into an SVM optimization problem is as follows:
$minimizeW→12W→·W→+(C++C-)·∑i∑jɛijsubject to yij(W→·X→ij+b)≥1-ɛij, ɛij≥0,$
where C+ and C control the trade-off between training errors and margin maximization for positive and negative examples, respectively. By tuning the cost factor, C+/C, one can more effectively learn from class imbalanced data.
4. Experiment
In this section, we present experimental results to show that personalized expert-based CF can produce better recommendations than similar user- or common expert-based CF recommender systems. We use MovieLens data sets to accomplish this. The data sets are widely used in recommender systems and CF studies, and they are compiled and collected over various periods of time [4]. Specifically, we use MovieLens 100k data set (ML-100k), which contains 100,000 ratings from 943 users and 1,682 items. We divide the data set into five folds for cross-validation.
### 4.1 Evaluation Metrics
Different CF algorithms and recommender systems exhibit different performance characteristics, and several properties of recommender systems are traded-off at the expense of the other properties. Therefore, various performance metrics must be used to evaluate CF algorithms [6]. In this work, we consider both prediction accuracy evaluation and recommendation list evaluation.
The prediction accuracy is by far the most common and important performance evaluation metric in recommender system evaluation. To evaluate any CF based recommender systems in prediction accuracy, we use the Mean Absolute Error.
Mean Absolute Error (MAE) measures the average difference between the predicted ratings and the actual ratings. MAE for ui is calculated as follows:
$MAE(ui)=∑m∈I(ui)test∣R^ui,m-Rui,m∣∣I(ui)test∣.$
Here, ui,m is the predicted rating of ui on m, and I(ui)test is the accessed item lists of ui for the items in the test data. MAE(ui) of all users are then averaged to evaluate the MAE of recommender systems.
Recommendation list evaluation is important for studying various properties of recommender systems. In this domain, we consider Item Coverage, User Coverage, Diversity, Precision and Recall of returned recommendations.
Item Coverage (Covitem) measures the proportion of items that a recommender system can recommend from the entire item space:
$Covitem=∑m∈Itemtest|U(m)|·δ(m,Rec(U sertest))∑m∈Itemtest|U(m)|.$
δ(m,Rec(Usertest)) = 1, only if item m appears in any recommendation lists for a given test data; U(m) is the list of users who accessed item. A list of a fixed number of recommendations, Rec, is produced for each active user ui, and we define recommendable items as items with predicted ratings greater than the average rating of ui. Rec contains items with the highest predicted ratings.
Diversity (Div) measures how diverse recommendation lists are. The pairwise diversity for two users are computed by the following formula:
$Div(ui,uj)=∣Rec(Ui)∩Rec(uj)∣∣Rec∣.$
Div(ui, uj) for all pairs of users are then averaged to evaluate the Diversity of the recommender systems. This measure is of particular interest, if one is interested in the customization of recommendations given to each individual.
Precision (Prec) measures the proportion of the successful recommendations among all recommendations. Precision indicates the quality of the produced recommendations with an emphasis on recommendation successes, rather than recommendation failures. Precision is calculated by the following formula:
$Prec=|tp||tp|+|fp|.$
tp and fp are the numbers of true-positive and false-positive recommendation results, respectively. All possible recommendation results are shown in
Recall (Rec) measures the proportion of the successful recommendations with the respect to the items that users actually liked. Recall indicates the quality of the produced recommendations with an emphasis on recommendation failures, rather than recommendation successes. Recall is calculated by the following formula:
$Rec=|tp||tp|+|fn|.$
Because each active user has a different watch history and access counts for the items in the testing data, it is impossible to generate the same fixed size recommendation lists for all users. Therefore, Precision and Recall are measured on all recommendations that can be validated with true ratings in the testing data; both metrics measure the quality of recommendations in different aspects. Precision increases with more recommendation successes, while Recall increases with less missed successful recommendation opportunities.
### 4.2 Baseline
We compare the proposed recommender system with three different types of CF recommender systems: similar user-based recommender system (SU), common expert-based recommender system (CE) and similar common expert-based recommender system (SCE). SU computes pairwise similarities for every pair of users based on their previous rating histories; then, a number of similar neighboring users are selected. Finally, CF is used to predict the ratings or produce recommendations for each user. CE chooses a fixed number of experts considering three absolute expertise measures (Early Adopter, Heavy Access, Niche-Item Access) and then uses the chosen experts as the neighbors for all the users. The last baseline is SCE. It first creates a pool of common experts by considering three and chooses neighbors for each active user by similarity. Thus, it is also expected to strike a good balance between recommendation accuracy and customization.
In tuning the recommender systems, the neighborhood size, k, can be chosen using a validation data set; however, previous works using the MovieLens data set [28, 29] reported the same result when using a fixed size k for recommendations. In this work, we set k to be 50 to compare the performance of different neighborhoods.
To predict user preference (i.e., ratings), we use the following CF algorithm:
1. Select k users as a neighborhood for the given active user.
2. Assign a user weight to the selected users.
3. Compute a rating prediction of the active user ui on an item as weighted average rating of the neighborhood.
In SU, the Pearson correlation (i.e., Similarity) is used not only as the similarity measure between users but also as the weights of the selected users (w(ui, uj)). CE uses the expertise of users to choose a neighborhood in step 1 and the Pearson correlation to determine user weights in step 2. In step 3, the weighted average of the ratings of the selected neighborhood is computed using the following formula:
$R^ui,m=R¯ui+∑uj∈N(ui)w(ui,uj)·(Rui,m-R¯(uj))∑uj∈N(ui)w(ui,uj).$
We strictly follow the traditional CF algorithm to compare and focus on the qualities of different types of neighborhoods, and if none of the selected neighborhood has used the item, then the system predicts the mean user rating (R(ui)).
### 4.3 Results
We first compare the prediction accuracy in the MAE of different recommender systems. Table 2 shows comparison results and the proposed approach (PE) yields more accurate results than the baselines. It shows an 11.9% improvement over SU, 18.4% over CE, and 4.8% over SCE.
CE yields the least accurate results. It is interesting that SCE is the second best. Both PE and SCE are basically personalized expert-based approaches. SCE first identifies common experts and simply chooses neighbors from the common experts based on similarity to the active user; however, PE first learns the patterns of neighbor selection of each user by SVM considering absolute and relative expertise. Thus, PE can identify more personalized neighbors who can better serve users with different needs and expectations.
To examine various properties of the proposed recommender system, we evaluate recommendations produced by the system. Table 3 shows Item Coverage of recommendation lists produced by different recommender systems. Item Coverage measures the proportion of items that a recommender system can recommend, and the measure increases as the size of the recommendation list increases. In this respect, SU with similar movie tastes generates recommendation lists with higher Item Coverage, while both PE and CE give recommendations that are more widely acceptable, based on their expert knowledge. PE covers slightly more items than CE (2% increase from 0.3837 to 0.3917 at |Rec| = 20), and SCE sits in between SU and PE.
For some applications, it is more important to recommend a variety of items. The seller also needs to sell unknown and unpopular items in stock, in addition to the popular items. Table 4 shows the Diversity of the recommendation lists produced by different recommender systems. Diversity decreases as the recommendation list size increases, as more common items are included to recommendation lists to active users. Higher Diversity means that the more diverse recommendation lists are given to different active users. Similar to Item Coverage, SU yields the most diverse recommendation lists, and then SCE, PE and CE follow. The results indicate that SU provides more diverse recommendations that possibly better serve diverse preferences of users; however, recommendation lists with high Item Coverage and Diversity are not necessarily accurate, as shown in Table 2. In this work, we define personalized experts as neighbors who can help to generate the more accurate recommendations for an active user; hence, PE puts more importance on accuracy over recommendation list customization. If we want PE to generate more customized and diverse recommendation lists, we can accomplish that by searching for personalized experts who can provide diverse recommendations.
Table 5 shows the Precision and Recall of recommendations. The high Precision and low Recall of SU indicates that SU provides only a few recommendations, but with high confidence. However, CE recommends more items with fewer successes, resulting in low Precision and high Recall. PE and SCE achieve both high Precision and high Recall, which implies good recommendation quality. PE yields better recommendations than SCE because there is no significant difference in Precision and the Recall of PE is higher at 0.7357 (2.6% improvement over SCE at 0.7171). Taking the opinions of similar experts with simply high similarity and high common expertise results in good quality recommendations; however, because users need different levels of expert assistance (high or low measures), customizing the neighborhood in terms of various expertise measures including similarity further improves recommendation quality.
The results indicate that PE generates recommendations that are more accurate in terms of lower MAE and higher Precision and Recall than other recommender systems. In this work, we define personalized experts as neighbors who can help generate more accurate recommendations for an active user; hence, PE places more importance on accuracy over recommendation list customization and selects neighbors who can give the most accurate recommendations to each active user. If we want PE to generate more customized and diverse recommendation lists, we can facilitate that by searching for personalized experts who can give diverse recommendations, as opposed to the accurate recommendations discussed in this work.
5. Discussion
### 5.1 The Sparsity Problem
CF performance suffers when there is insufficient information, which is also known as the sparsity problem. A typical SU can generate accurate recommendations, but it is not robust to the sparsity problem. In this section, we compare the performance of different recommender systems with varying sparsity levels.
Table 6 shows different sparsity levels as we introduce more sparseness into the training data. The original data set is not very sparse (1–100000/(943·1682) = 0.9369) before splitting into training and testing data. To introduce more sparseness into the training data, we removed the rating information received during the last 1-month or 2-month period.
Table 7 illustrates the sensitivity of different recommender systems in relation to varying sparsity levels. The prediction accuracy of the user-based CF algorithm decreases as the training data sparseness increases. Among the four neighborhoods in comparison, PE yields the best prediction accuracy with the lowest MAE at all sparsity levels. The prediction accuracy of CE drops 25.9% from 1.3710–1.7260, the accuracy of SCE drops 25.0% from 1.3383–1.6733, the accuracy of k-NN drops 13.0% from 1.2829–1.4502, and the accuracy of PE drops 15.0% from 1.2000–1.3803 as the sparsity level increases from 95.8%–96.9%. At all sparsity levels, PE yields the most accurate prediction results.
The quality of the recommendation also degrades with increasing data sparseness. The Precision and Recall values from Tables 8 and 9 indicate that, with sparser data (95.8% and 96.9%), SU yields high Precision and low Recall recommendations, CE and SCE yield low Precision and low Recall recommendations, and PE yields high Precision and high Recall recommendations. At all sparsity levels, PE yields the best quality recommendations.
Neighborhoods are selected with the sparse training data, hence the lack of accurate information results in an inaccurate neighborhood selection; furthermore, it is more likely that none of the selected neighborhoods has watched the item in question. In such a case, a recommendation opportunity is missed as the CF algorithm in (16) predicts the active user average rating and the item is not recommended. To provide more accurate recommendations, it may be beneficial to only recommend a few items with confidence; however, many opportunities are missed using this method, as evidenced by the passive recommender system result in low Recall. Table 10 shows the recommendation miss rate of different neighborhoods.
The recommendation miss rate increases as training data sparsity level increases for all types of neighborhoods. An appropriate neighborhood should be able to provide answers to the request of each active user. Although, personalized experts are selected to maximize the prediction accuracy of the CF algorithm, PE can provide recommendations in most opportunities at a sparsity level of 94.9% (original training data), and even at 96.9% with sparser data. As seen in Table 10, SU provides more accurate predictions and recommendations than CE, while the recommendation miss rates of SU are higher than those of CE in most cases (at the sparsity level of 94.9% and 96.9%): CE recommends items more carelessly than SU, and it yields more recommendation failures than recommendation misses.
### 5.2 Neighborhood Study
In this subsection, we discuss how different neighborhood characteristics result in performance differences in different recommender systems. As seen in Section 4, different neighborhood-based CF algorithms exhibit different performance characteristics. For instance, SU results in recommendations with higher Diversity than other recommender systems. This is because SU recommends items that each active user likes; consequently, the overall recommendations for all users are more diverse. However, CE recommends items that the common experts like; this results in the overall recommendations with lower Diversity. We want to customize the neighborhood for each user to obtain the best recommendation result; we argue that neighborhoods for users should be different in terms of the degrees of various expertise measures from Section 3.2.
Figure 1 shows different neighborhood characteristics for three different users (User ID: 123, 456). The neighborhood size is 50, and the standardized expertise measures for all members within each neighborhood are averaged to define the characteristics of the neighborhood. SU, CE, SIMCE and PE show very different characteristics, but the patterns are similar among different users. The SU neighborhood consists of neighbors with the highest similarity only, and its radial graphs peak toward Sim measure. The CE neighborhood consists of neighbors with the highest common expertise measure (||〈EA,NA,HA〉||), and its radial graphs expand toward EA, NA, and EC, with a peak at EA. Note that CE consists of the same common experts for all users and the absolute measures (EA, NA, HA, EC) are constant, whereas relative measures vary by active user. SIMCE stands in-between SU and CE as its neighborhood consists of neighbors with high similarity and high common expertise. Lastly, the PE neighborhood consists of personalized experts; the neighborhood characteristic of PE is very different from the others and expands toward CA, UA, NA and HA. This confirms that personalized experts are not just similar users or common experts; PE provides more accurate recommendations to users as seen in Section 4.
Having shown that a personalized expert is a better alternative to similar users, we now examine how well personalized each expert group is for each active user. By using the Jaccard Index, we measure group similarity among different personalized expert groups. The Jaccard Index is one if two clusters are identical and it is zero if two clusters have no common elements. Given two groups, N1 and N2, the Jaccard Index is defined as follows:
$J(N1,N2)=|W1∩W2||W1∪W2|.$
Table 11 shows the neighborhood similarity averages for different types of neighborhoods. Given three different users (User ID: 15, 123, 456), we measure the Jaccard Index for every pair and average the pairwise values for each neighborhood type. As expected CE has neighborhood similarity of one, as the same common experts are suggested for all users; the neighborhoods for SU and SIMCE tend to be more diverse because they are more likely to select neighbors based on Sim and users have diverse preferences. We originally expected personalized expert groups for users to be more diverse than what we see here; however, the personalized expert groups overlap significantly and exhibit very similar neighborhood characteristics (high CA, UA, NA, HA), which are also obvious characteristics of heavy access users who access most of the items. In fact, 42 of the most heavy access users (top 5% in HA) are included in each personalized expert group. From this finding and our analysis of personalized expert groups, we conclude that our personalized expert search correctly identifies the most effective neighborhood for a given data set.
k-NN and other user-based CF algorithms gained much popularity for the simplicity of the algorithms and their performance. As the performance of such algorithms largely depends on the neighborhood selection, it is important to select the most suitable neighborhood for each active user. In this work, we customize the neighborhood for each active user and call such neighborhoods personalized experts; the proposed personalized expert-based recommender system serves users with more accurate recommendations. Furthermore, the proposed neighborhood-based recommender system is more robust to sparse data.
In the neighborhood study, we show that personalized experts are significantly different from similar users, common experts, or similar common experts, and the novel neighborhood (PE) is customized for each active user. We have shown a way to build a global model to find a personalized neighborhood for each active user, but building such a global model can be impractically costly (see Section 3.1) and limits the scalability of the system. In this regard, we plan to explore unsupervised or reinforcement learning algorithms in the future.
Acknowledgements
This research was supported by Next-Generation Information Computing Development Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2014M3C4A7030503). Also, this work was supported by the NRF grant funded by the Korea government (MSIP) (No. NRF-2016R1A2B4015820).
Conflict of Interest
Figures
Fig. 1.
Expertise measures (standardized) of different neighborhood types: similar users (k-NN), common experts (CE), similar common experts (SIMCE), personalized experts (PE). (a) User ID: 123, (b) User ID: 456.
TABLES
### Table 1
Recommendation results classification
RecommendedNot recommended
### Table 2
Prediction accuracy of recommender systems (MAE)
SUCESCEPR
MAE0.87090.94660.81110.7723
### Table 3
Item coverage of different recommender systems
|Rec|SUCESCEPR
100.86080.29730.36200.2986
200.92020.38370.51330.3917
300.94090.46020.61030.4803
400.95310.51000.67200.5611
500.95950.56140.72160.6368
### Table 4
Diversity of different recommender systems
|Rec|SUCESCEPR
100.92900.64050.88320.6914
200.91650.63930.84960.6726
300.90170.63330.82010.6663
400.88620.63170.79400.6672
500.87010.62870.76990.6669
### Table 5
Precision and recall of recommendations of recommender systems
SuCESCEPR
Precision0.65330.59850.64850.6433
Recall0.34120.63280.71710.7357
### Table 6
Training data sparsity levels
All data−1 month−2 month
ML-100K94.9%95.8%96.9%
### Table 7
Prediction accuracy by different sparsity levels (MAE)
Sparsity level (%)SUCESCEPE
94.90.88030.95000.81110.7762
95.81.28291.37101.33831.2000
96.91.45021.72601.67331.3803
### Table 8
Precision by different sparsity levels
Sparsity level (%)SUCESCEPE
94.90.65330.59850.64850.6433
95.80.65210.52910.54730.6521
96.90.64900.56820.58340.6490
### Table 9
Recall by different sparsity levels
Sparsity level (%)SUCESCEPE
94.90.34130.63290.71710.7358
95.80.29490.13750.11430.5490
96.90.27820.28180.25540.4790
### Table 10
Recommendation miss rate by different sparsity levels
Sparsity level (%)SUCESCEPE
94.90.50520.02610.04390.0072
95.80.52490.69980.78560.1578
96.90.52870.42300.53330.2248
### Table 11
Jaccard index of different recommender systems
SUCESCEPE
Jaccard0.07921.00000.22000.8363
References
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2. Deivendran, P, Mala, T, and Shanmugasundaram, B (2011). Content based recommender systems. International Journal of Computer Science & Emerging Technologies. 2, 148-152.
3. Formoso, V, Cacheda, F, and Carneiro, V (2008). Algorithms for efficient collaborative filtering. Efficiency Issues in Information Retrieval Workshop. Heidelberg: Springer, pp. 17-28
4. Herlocker, JL, Konstan, JA, Borchers, A, and Riedl, J 1999. An algorithmic framework for performing collaborative filtering., Proceedings of the 22nd Annual International ACM SIGIR Conference on Research and Development in Information Retrieval, Berkeley, CA, Array, pp.230-237.
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10. Kawamae, N 2010. Serendipitous recommendations via innovators., Proceedings of the 33rd International ACM SIGIR Conference on Research and Development in Information Retrieval, Geneva, Switzerland, Array, pp.218-225.
11. Kumar, A, and Bhatia, M (2012). Community expert based recommendation for solving first rater problem. International Journal of Computer Applications. 37, 7-13.
12. Lee, K, and Lee, K (2013). Using experts among users for novel movie recommendations. Journal of Computing Science and Engineering. 7, 21-29.
13. Rusmevichientong, P, Zhu, S, and Selinger, D 2004. Identifying early buyers from purchase data., Proceedings of the 10th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Seattle, WA, Array, pp.671-677.
14. Song, SI, Lee, S, Park, S, and Lee, SG 2012. Determining user expertise for improving recommendation performance., Proceedings of the 6th International Conference on Ubiquitous Information Management and Communication, Kuala Lumpur, Malaysia, Array.
15. Tyler, SK, Zhu, S, Chi, Y, and Zhang, Y 2009. Ordering innovators and laggards for product categorization and recommendation., Proceedings of the 3rd ACM Conference on Recommender Systems, New York, NY, Array, pp.29-36.
16. Kim, SW, Chung, CW, and Kim, D (2009). An opinion-based decision model for recommender systems. Online Information Review. 33, 584-602.
17. Cheng, L, Fan, Y, Yu, C, and Du, Y 2016. An improved trust-aware recommender system for personalized user recommendation in Tmall., Proceedings of the 2nd International Conference on Mechanical, electronic and Information Technology Engineering, Chongqing, China, Array, pp.60-63.
18. Huang, J, Zhu, K, and Zhong, N (2016). A probabilistic inference model for recommender systems. Applied Intelligence. 45, 686-694.
19. Chung, Y, Jung, HW, Kim, J, and Lee, JH (2013). Personalized expert-based recommender system: training C-SVM for personalized expert identification. Machine Learning and Data Mining in Pattern Recognition. Heidelberg: Springer, pp. 434-441
20. Chung, Y, Lee, SW, and Lee, JH (2013). Personalized expert-based recommendation. Journal of Korean Institute of Intelligent Systems. 23, 7-11.
21. Allison, B, Guthrie, D, and Guthrie, L (2006). Another look at the data sparsity problem. Text, Speech and Dialogue. Heidelberg: Springer, pp. 327-334
22. Billsus, D, and Pazzani, MJ 1998. Learning collaborative information filters., Proceedings of the 15th International Conference on Machine Learning, Madison, WI, pp.46-54.
23. Sarwar, B, Karypis, G, Konstan, J, and Riedl, J 2001. Item-based collaborative filtering recommendation algorithms., Proceedings of the 10th International Conference on World Wide Web, Hong Kong, China, Array, pp.285-295.
24. Sun, M, Lebanon, G, and Kidwell, P (2012). Estimating probabilities in recommendation systems. Journal of the Royal Statistical Society Series C (Applied Statistics). 61, 471-492.
25. Zabinsky, ZB (2009). Random search algorithms. Wiley Encyclopedia of Operations Research and Management Science. Chichester: John Wiley & Sons
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Biographies
Yeounoh Chung received his B.S. in Electrical and Computer Engineering and his M.S. in Computer Science from Cornell University, Ithaca, USA, in 2008 and 2009, respectively. He is currently pursuing Ph.D. in Computer Science at Brown University, Providence, RI, USA. His current research interests focus on big data management and data mining.
E-mail: yeounoh chung@brown.edu
Noo-ri Kim received the B.S. in computer engineering from Sungkyunkwan University, Suwon, Korea in 2013. He is currently pursuing his M.S.-Ph.D. in Computer Engineering at Sungkyunkwan University. His research interests include recommender systems, text mining, and machine learning.
E-mail: pd99j@skku.edu
Chang-yong Park received his B.S. in Computer Engineering from Dongguk University, Korea, in 2010, and his M.S. in Computer Engineering from Sungkyunkwan University in 2014. Now he works at LG Electronics as a software engineer. His research interests include software engineering, context-aware recommender system, and intelligent agents.
E-mail: changyong1.park@lge.com
Jee-Hyong Lee received his B.S., M.S., and Ph.D. in computer science from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 1993, 1995, and 1999, respectively. From 2000 to 2002, he was an international fellow at SRI International, USA. He joined Sungkyunkwan University, Suwon, Korea, as a faculty member in 2002. His research interests include recommender systems, intelligent systems, and machine learning.
E-mail: john@skku.edu
June 2018, 18 (2) | 2018-09-25 13:52:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 17, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5118153095245361, "perplexity": 2469.105925447443}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267161638.66/warc/CC-MAIN-20180925123211-20180925143611-00503.warc.gz"} |
https://tex.stackexchange.com/questions/373157/a-difficulty-to-understand-a-latex-error-message | # A difficulty to understand a latex error message
I have this error message that I don't understand.
Please if you can help me
• Welcome to LaTeX! Welcome to TeX.SX! Please help us help you and add a minimal working example (MWE) that illustrates your problem. Reproducing the problem and finding out what the issue is will be much easier when we see compilable code, starting with \documentclass{...} and ending with \end{document}. – Moriambar Jun 4 '17 at 7:21
• error message starts a bit earlier this is not enough to decode the problem – percusse Jun 4 '17 at 7:38
• latex is trying to input a file it can not find but you have only shown a small fragment of the error message so hard to help. please show the full error from the log from ! in a code block {} not quote so line endings are preserved. – David Carlisle Jun 4 '17 at 8:12
• Just a guess: pst-arrow is a recent package, which was a part of pstricks-add previously. Try 1) updating your installation; 2) installing pst-arrow. – Bernard Jun 4 '17 at 9:09
• It looks like you've got two separate accounts, which means you cannot edit your original post or leave comments. The Stack Exchange staff can merge them together for you. – Dai Bowen Jun 4 '17 at 13:22
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Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count). | 2019-07-23 17:57:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.685215413570404, "perplexity": 1957.7085920590337}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195529481.73/warc/CC-MAIN-20190723172209-20190723194209-00521.warc.gz"} |
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# RD Sharma solutions for Class 12 Maths chapter 16 - Tangents and Normals [Latest edition]
## Chapter 16: Tangents and Normals
Exercise 16.1Exercise 16.2Exercise 16.3Others
Exercise 16.1 [Pages 10 - 11]
### RD Sharma solutions for Class 12 Maths Chapter 16 Tangents and Normals Exercise 16.1 [Pages 10 - 11]
Exercise 16.1 | Q 1.01 | Page 10
Find the slope of the tangent and the normal to the following curve at the indicted point $y = \sqrt{x^3} \text { at } x = 4$ ?
Exercise 16.1 | Q 1.02 | Page 10
Find the slope of the tangent and the normal to the following curve at the indicted point $y = \sqrt{x} \text { at }x = 9$ ?
Exercise 16.1 | Q 1.03 | Page 10
Find the slope of the tangent and the normal to the following curve at the indicted point y = x3 − x at x = 2 ?
Exercise 16.1 | Q 1.04 | Page 10
Find the slope of the tangent and the normal to the following curve at the indicted point y = 2x2 + 3 sin x at x = 0 ?
Exercise 16.1 | Q 1.05 | Page 10
Find the slope of the tangent and the normal to the following curve at the indicted point x = a (θ − sin θ), y = a(1 − cos θ) at θ = −π/2 ?
Exercise 16.1 | Q 1.06 | Page 10
Find the slope of the tangent and the normal to the following curve at the indicted point x = a cos3 θ, y = a sin3 θ at θ = π/4 ?
Exercise 16.1 | Q 1.07 | Page 10
Find the slope of the tangent and the normal to the following curve at the indicted point x = a (θ − sin θ), y = a(1 − cos θ) at θ = π/2 ?
Exercise 16.1 | Q 1.08 | Page 10
Find the slope of the tangent and the normal to the following curve at the indicted point y = (sin 2x + cot x + 2)2 at x = π/2 ?
Exercise 16.1 | Q 1.09 | Page 10
Find the slope of the tangent and the normal to the following curve at the indicted point x2 + 3y + y2 = 5 at (1, 1) ?
Exercise 16.1 | Q 1.1 | Page 10
Find the slope of the tangent and the normal to the following curve at the indicted point xy = 6 at (1, 6) ?
Exercise 16.1 | Q 2 | Page 10
Find the values of a and b if the slope of the tangent to the curve xy + ax + by = 2 at (1, 1) is 2 ?
Exercise 16.1 | Q 3 | Page 10
If the tangent to the curve y = x3 + ax + b at (1, − 6) is parallel to the line x − y + 5 = 0, find a and b ?
Exercise 16.1 | Q 4 | Page 10
Find a point on the curve y = x3 − 3x where the tangent is parallel to the chord joining (1, −2) and (2, 2) ?
Exercise 16.1 | Q 5 | Page 10
Find the points on the curve y = x3 − 2x2 − 2x at which the tangent lines are parallel to the line y = 2x− 3 ?
Exercise 16.1 | Q 6 | Page 10
Find the points on the curve y2 = 2x3 at which the slope of the tangent is 3 ?
Exercise 16.1 | Q 7 | Page 10
Find the points on the curve xy + 4 = 0 at which the tangents are inclined at an angle of 45° with the x-axis ?
Exercise 16.1 | Q 8 | Page 10
Find the point on the curve y = x2 where the slope of the tangent is equal to the x-coordinate of the point ?
Exercise 16.1 | Q 9 | Page 10
At what points on the circle x2 + y2 − 2x − 4y + 1 = 0, the tangent is parallel to x-axis?
Exercise 16.1 | Q 10 | Page 10
At what point of the curve y = x2 does the tangent make an angle of 45° with the x-axis?
Exercise 16.1 | Q 11 | Page 10
Find the points on the curve y = 3x2 − 9x + 8 at which the tangents are equally inclined with the axes ?
Exercise 16.1 | Q 12 | Page 10
At what points on the curve y = 2x2 − x + 1 is the tangent parallel to the line y = 3x + 4?
Exercise 16.1 | Q 13 | Page 10
Find the point on the curve y = 3x2 + 4 at which the tangent is perpendicular to the line whose slop is $- \frac{1}{6}$ ?
Exercise 16.1 | Q 14 | Page 11
Find the points on the curve x2 + y2 = 13, the tangent at each one of which is parallel to the line 2x + 3y = 7 ?
Exercise 16.1 | Q 15 | Page 11
Find the points on the curve 2a2y = x3 − 3ax2 where the tangent is parallel to x-axis ?
Exercise 16.1 | Q 16 | Page 11
At what points on the curve y = x2 − 4x + 5 is the tangent perpendicular to the line 2y + x = 7?
Exercise 16.1 | Q 17.1 | Page 11
Find the points on the curve $\frac{x^2}{4} + \frac{y^2}{25} = 1$ at which the tangent is parallel to the x-axis ?
Exercise 16.1 | Q 17.2 | Page 11
Find the points on the curve$\frac{x^2}{4} + \frac{y^2}{25} = 1$ at which the tangent is parallel to the y-axis ?
Exercise 16.1 | Q 18 | Page 11
Find the points on the curve x2 + y2 − 2x − 3 = 0 at which the tangents are parallel to the x-axis ?
Exercise 16.1 | Q 19.1 | Page 11
Find the points on the curve $\frac{x^2}{9} + \frac{y^2}{16} = 1$ at which the tangent is parallel to x-axis ?
Exercise 16.1 | Q 19.2 | Page 11
Find the points on the curve $\frac{x^2}{9} + \frac{y^2}{16} = 1$ at which the tangent is parallel to y-axis ?
Exercise 16.1 | Q 20 | Page 11
Who that the tangents to the curve y = 7x3 + 11 at the points x = 2 and x = −2 are parallel ?
Exercise 16.1 | Q 21 | Page 11
Find the points on the curve y = x3 where the slope of the tangent is equal to the x-coordinate of the point ?
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Exercise 16.2 [Pages 27 - 29]
### RD Sharma solutions for Class 12 Maths Chapter 16 Tangents and Normals Exercise 16.2 [Pages 27 - 29]
Exercise 16.2 | Q 1 | Page 27
Find the equation of the tangent to the curve $\sqrt{x} + \sqrt{y} = a$ at the point $\left( \frac{a^2}{4}, \frac{a^2}{4} \right)$ ?
Exercise 16.2 | Q 2 | Page 27
Find the equation of the normal to y = 2x3 − x2 + 3 at (1, 4) ?
Exercise 16.2 | Q 3.01 | Page 27
Find the equation of the tangent and the normal to the following curve at the indicated point x4 − bx3 + 13x2 − 10x + 5 at (0, 5) ?
Exercise 16.2 | Q 3.02 | Page 27
Find the equation of the tangent and the normal to the following curve at the indicated point y = x4 − 6x3 + 13x2 − 10x + 5 at x = 1?
Exercise 16.2 | Q 3.03 | Page 27
Find the equation of the tangent and the normal to the following curve at the indicated point y = x2 at (0, 0) ?
Exercise 16.2 | Q 3.04 | Page 27
Find the equation of the tangent and the normal to the following curve at the indicated point y = 2x2 − 3x − 1 at (1, −2) ?
Exercise 16.2 | Q 3.05 | Page 27
Find the equation of the tangent and the normal to the following curve at the indicated point $y^2 = \frac{x^3}{4 - x}at \left( 2, - 2 \right)$ ?
Exercise 16.2 | Q 3.06 | Page 27
Find the equation of the tangent and the normal to the following curve at the indicated point y = x2 + 4x + 1 at x = 3 ?
Exercise 16.2 | Q 3.07 | Page 27
Find the equation of the tangent and the normal to the following curve at the indicated point$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text{ at }\left( a\cos\theta, b\sin\theta \right)$ ?
Exercise 16.2 | Q 3.08 | Page 27
Find the equation of the tangent and the normal to the following curve at the indicated point $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( a\sec\theta, b\tan\theta \right)$ ?
Exercise 16.2 | Q 3.09 | Page 27
Find the equation of the tangent and the normal to the following curve at the indicated point y2 = 4ax at $\left( \frac{a}{m^2}, \frac{2a}{m} \right)$ ?
Exercise 16.2 | Q 3.1 | Page 27
Find the equation of the tangent and the normal to the following curve at the indicated point $c^2 \left( x^2 + y^2 \right) = x^2 y^2 \text { at }\left( \frac{c}{\cos\theta}, \frac{c}{\sin\theta} \right)$ ?
Exercise 16.2 | Q 3.11 | Page 27
Find the equation of the tangent and the normal to the following curve at the indicated point xy = c2 at $\left( ct, \frac{c}{t} \right)$ ?
Exercise 16.2 | Q 3.12 | Page 27
Find the equation of the tangent and the normal to the following curve at the indicated point $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { at } \left( x_1 , y_1 \right)$ ?
Exercise 16.2 | Q 3.13 | Page 27
Find the equation of the tangent and the normal to the following curve at the indicated point $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( x_0 , y_0 \right)$ ?
Exercise 16.2 | Q 3.14 | Page 27
Find the equation of the tangent and the normal to the following curve at the indicated point $x^\frac{2}{3} + y^\frac{2}{3}$ = 2 at (1, 1) ?
Exercise 16.2 | Q 3.15 | Page 27
Find the equation of the tangent and the normal to the following curve at the indicated point x2 = 4y at (2, 1) ?
Exercise 16.2 | Q 3.16 | Page 27
Find the equation of the tangent and the normal to the following curve at the indicated point y2 = 4x at (1, 2) ?
Exercise 16.2 | Q 3.17 | Page 27
Find the equation of the tangent and the normal to the following curve at the indicated point 4x2 + 9y2 = 36 at (3cosθ, 2sinθ) ?
Exercise 16.2 | Q 3.18 | Page 27
Find the equation of the tangent and the normal to the following curve at the indicated point y2 = 4ax at (x1, y1)?
Exercise 16.2 | Q 3.19 | Page 27
Find the equation of the tangent and the normal to the following curve at the indicated point $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( \sqrt{2}a, b \right)$ ?
Exercise 16.2 | Q 4 | Page 27
Find the equation of the tangent to the curve x = θ + sin θ, y = 1 + cos θ at θ = π/4 ?
Exercise 16.2 | Q 5.1 | Page 28
Find the equation of the tangent and the normal to the following curve at the indicated points x = θ + sinθ, y = 1 + cosθ at θ = $\frac{\pi}{2}$ ?
Exercise 16.2 | Q 5.2 | Page 28
Find the equation of the tangent and the normal to the following curve at the indicated points $x = \frac{2 a t^2}{1 + t^2}, y = \frac{2 a t^3}{1 + t^2}\text { at } t = \frac{1}{2}$ ?
Exercise 16.2 | Q 5.3 | Page 28
Find the equation of the tangent and the normal to the following curve at the indicated points x = at2, y = 2at at t = 1 ?
Exercise 16.2 | Q 5.4 | Page 28
Find the equation of the tangent and the normal to the following curve at the indicated points x = asect, y = btant at t ?
Exercise 16.2 | Q 5.5 | Page 28
Find the equation of the tangent and the normal to the following curve at the indicated points x = a(θ + sinθ), y = a(1 − cosθ) at θ ?
Exercise 16.2 | Q 5.6 | Page 28
Find the equation of the tangent and the normal to the following curve at the indicated points x = 3cosθ − cos3θ, y = 3sinθ − sin3θ
Exercise 16.2 | Q 6 | Page 28
Find the equation of the normal to the curve x2 + 2y2 − 4x − 6y + 8 = 0 at the point whose abscissa is 2 ?
Exercise 16.2 | Q 7 | Page 28
Find the equation of the normal to the curve ay2 = x3 at the point (am2, am3) ?
Exercise 16.2 | Q 8 | Page 28
The equation of the tangent at (2, 3) on the curve y2 = ax3 + b is y = 4x − 5. Find the values of a and b ?
Exercise 16.2 | Q 9 | Page 28
Find the equation of the tangent line to the curve y = x2 + 4x − 16 which is parallel to the line 3x − y + 1 = 0 ?
Exercise 16.2 | Q 10 | Page 28
Find an equation of normal line to the curve y = x3 + 2x + 6 which is parallel to the line x+ 14y + 4 = 0 ?
Exercise 16.2 | Q 11 | Page 28
Determine the equation(s) of tangent (s) line to the curve y = 4x3 − 3x + 5 which are perpendicular to the line 9y + x + 3 = 0 ?
Exercise 16.2 | Q 12 | Page 28
Find the equation of a normal to the curve y = x loge x which is parallel to the line 2x − 2y + 3 = 0 ?
Exercise 16.2 | Q 13.1 | Page 28
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is parallel to the line 2x − y + 9 = 0 ?
Exercise 16.2 | Q 13.2 | Page 28
Find the equation of the tangent line to the curve y = x2 − 2x + 7 which perpendicular to the line 5y − 15x = 13. ?
Exercise 16.2 | Q 14 | Page 28
Find the equations of all lines having slope 2 and that are tangent to the curve $y = \frac{1}{x - 3}, x \neq 3$ ?
Exercise 16.2 | Q 15 | Page 28
Find the equations of all lines of slope zero and that are tangent to the curve $y = \frac{1}{x^2 - 2x + 3}$ ?
Exercise 16.2 | Q 16 | Page 28
Find the equation of the tangent to the curve $y = \sqrt{3x - 2}$ which is parallel to the 4x − 2y + 5 = 0 ?
Exercise 16.2 | Q 17 | Page 28
Find the equation of the tangent to the curve x2 + 3y − 3 = 0, which is parallel to the line y= 4x − 5 ?
Exercise 16.2 | Q 18 | Page 29
Prove that $\left( \frac{x}{a} \right)^n + \left( \frac{y}{b} \right)^n = 2$ touches the straight line $\frac{x}{a} + \frac{y}{b} = 2$ for all n ∈ N, at the point (a, b) ?
Exercise 16.2 | Q 19 | Page 29
Find the equation of the tangent to the curve x = sin 3ty = cos 2t at
$t = \frac{\pi}{4}$ ?
Exercise 16.2 | Q 20 | Page 29
At what points will be tangents to the curve y = 2x3 − 15x2 + 36x − 21 be parallel to x-axis ? Also, find the equations of the tangents to the curve at these points ?
Exercise 16.2 | Q 21 | Page 29
Find the equation of the tangents to the curve 3x2 – y2 = 8, which passes through the point (4/3, 0) ?
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Exercise 16.3 [Pages 40 - 41]
### RD Sharma solutions for Class 12 Maths Chapter 16 Tangents and Normals Exercise 16.3 [Pages 40 - 41]
Exercise 16.3 | Q 1.1 | Page 40
Find the angle of intersection of the following curve y2 = x and x2 = y ?
Exercise 16.3 | Q 1.2 | Page 40
Find the angle of intersection of the following curve y = x2 and x2 + y2 = 20 ?
Exercise 16.3 | Q 1.3 | Page 40
Find the angle of intersection of the following curve 2y2 = x3 and y2 = 32x ?
Exercise 16.3 | Q 1.4 | Page 40
Find the angle of intersection of the following curve x2 + y2 − 4x − 1 = 0 and x2 + y2 − 2y − 9 = 0 ?
Exercise 16.3 | Q 1.5 | Page 40
Find the angle of intersection of the following curve $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and x2 + y2 = ab ?
Exercise 16.3 | Q 1.6 | Page 40
Find the angle of intersection of the following curve x2 + 4y2 = 8 and x2 − 2y2 = 2 ?
Exercise 16.3 | Q 1.7 | Page 40
Find the angle of intersection of the following curve x2 = 27y and y2 = 8x ?
Exercise 16.3 | Q 1.8 | Page 40
Find the angle of intersection of the following curve x2 + y2 = 2x and y2 = x ?
Exercise 16.3 | Q 1.9 | Page 40
Find the angle of intersection of the following curve y = 4 − x2 and y = x2 ?
Exercise 16.3 | Q 2.1 | Page 40
Show that the following set of curve intersect orthogonally y = x3 and 6y = 7 − x?
Exercise 16.3 | Q 2.2 | Page 40
Show that the following set of curve intersect orthogonally x3 − 3xy2 = −2 and 3x2y − y3 = 2 ?
Exercise 16.3 | Q 2.3 | Page 40
Show that the following set of curve intersect orthogonally x2 + 4y2 = 8 and x2 − 2y2 = 4 ?
Exercise 16.3 | Q 3.1 | Page 40
Show that the following curve intersect orthogonally at the indicated point x2 = 4y and 4y + x2 = 8 at (2, 1) ?
Exercise 16.3 | Q 3.2 | Page 40
Show that the following curve intersect orthogonally at the indicated point x2 = y and x3 + 6y = 7 at (1, 1) ?
Exercise 16.3 | Q 3.3 | Page 40
Show that the following curve intersect orthogonally at the indicated point y2 = 8x and 2x2 + y2 = 10 at $\left( 1, 2\sqrt{2} \right)$ ?
Exercise 16.3 | Q 4 | Page 40
Show that the curves 4x = y2 and 4xy = k cut at right angles, if k2 = 512 ?
Exercise 16.3 | Q 5 | Page 40
Show that the curves 2x = y2 and 2xy = k cut at right angles, if k2 = 8 ?
Exercise 16.3 | Q 6 | Page 40
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other ?
Exercise 16.3 | Q 7 | Page 40
Prove that the curves y2 = 4x and x2 + y2 - 6x + 1 = 0 touch each other at the point (1, 2) ?
Exercise 16.3 | Q 8.1 | Page 41
Find the condition for the following set of curve to intersect orthogonally $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { and } xy = c^2$ ?
Exercise 16.3 | Q 8.2 | Page 41
Find the condition for the following set of curve to intersect orthogonally $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { and } \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1$ ?
Exercise 16.3 | Q 9 | Page 41
Show that the curves $\frac{x^2}{a^2 + \lambda_1} + \frac{y^2}{b^2 + \lambda_1} = 1 \text { and } \frac{x^2}{a^2 + \lambda_2} + \frac{y^2}{b^2 + \lambda_2} = 1$ intersect at right angles ?
Exercise 16.3 | Q 10 | Page 41
If the straight line xcos $\alpha$ +y sin $\alpha$ = p touches the curve $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ then prove that a2cos2 $\alpha$ $-$ b2sin$\alpha$ = p?
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[Pages 41 - 42]
### RD Sharma solutions for Class 12 Maths Chapter 16 Tangents and Normals [Pages 41 - 42]
Q 1 | Page 41
Find the point on the curve y = x2 − 2x + 3, where the tangent is parallel to x-axis ?
Q 2 | Page 41
Find the slope of the tangent to the curve x = t2 + 3t − 8, y = 2t2 − 2t − 5 at t = 2 ?
Q 3 | Page 41
If the tangent line at a point (x, y) on the curve y = f(x) is parallel to x-axis, then write the value of $\frac{dy}{dx}$ ?
Q 4 | Page 41
Write the value of $\frac{dy}{dx}$ , if the normal to the curve y = f(x) at (x, y) is parallel to y-axis ?
Q 5 | Page 41
If the tangent to a curve at a point (xy) is equally inclined to the coordinates axes then write the value of $\frac{dy}{dx}$ ?
Q 6 | Page 41
If the tangent line at a point (x, y) on the curve y = f(x) is parallel to y-axis, find the value of $\frac{dx}{dy}$ ?
Q 7 | Page 41
Find the slope of the normal at the point 't' on the curve $x = \frac{1}{t}, y = t$ ?
Q 8 | Page 41
Write the coordinates of the point on the curve y2 = x where the tangent line makes an angle $\frac{\pi}{4}$ with x-axis ?
Q 9 | Page 41
Write the angle made by the tangent to the curve x = et cos t, y = et sin t at $t = \frac{\pi}{4}$ with the x-axis ?
Q 10 | Page 42
Write the equation of the normal to the curve y = x + sin x cos x at $x = \frac{\pi}{2}$ ?
Q 11 | Page 42
Find the coordinates of the point on the curve y2 = 3 − 4x where tangent is parallel to the line 2x + y− 2 = 0 ?
Q 12 | Page 42
Write the equation on the tangent to the curve y = x2 − x + 2 at the point where it crosses the y-axis ?
Q 13 | Page 42
Write the angle between the curves y2 = 4x and x2 = 2y − 3 at the point (1, 2) ?
Q 14 | Page 42
Write the angle between the curves y = e−x and y = ex at their point of intersections ?
Q 15 | Page 42
Write the slope of the normal to the curve $y = \frac{1}{x}$ at the point $\left( 3, \frac{1}{3} \right)$ ?
Q 16 | Page 42
Write the coordinates of the point at which the tangent to the curve y = 2x2 − x + 1 is parallel to the line y = 3x + 9 ?
Q 17 | Page 42
Write the equation of the normal to the curve y = cos x at (0, 1) ?
Q 18 | Page 42
Write the equation of the tangent drawn to the curve $y = \sin x$ at the point (0,0) ?
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[Pages 42 - 44]
### RD Sharma solutions for Class 12 Maths Chapter 16 Tangents and Normals [Pages 42 - 44]
Q 1 | Page 42
The equation to the normal to the curve y = sin x at (0, 0) is ___________ .
• x = 0
• y = 0
• x + y = 0
• x − y = 0
Q 2 | Page 42
The equation of the normal to the curve y = x + sin x cos x at x = π/2 is ___________ .
• = 2
• x = π
• x + π = 0
• 2x = π
Q 3 | Page 42
The equation of the normal to the curve y = x(2 − x) at the point (2, 0) is ________________ .
• x − 2y = 2
• x − 2y + 2 = 0
• 2x + y = 4
• 2x + y − 4 = 0
Q 4 | Page 42
The point on the curve y2 = x where tangent makes 45° angle with x-axis is ______________ .
• (1/2, 1/4)
• (1/4, 1/2)
• (4, 2)
• (1, 1)
Q 5 | Page 42
If the tangent to the curve x = a t2, y = 2 at is perpendicular to x-axis, then its point of contact is _____________ .
• (a, a)
• (0, a)
• (0, 0)
• (a, 0)
Q 6 | Page 42
The point on the curve y = x2 − 3x + 2 where tangent is perpendicular to y = x is ________________ .
• (0, 2)
• (1, 0)
• (−1, 6)
• (2, −2)
Q 7 | Page 42
The point on the curve y2 = x where tangent makes 45° angle with x-axis is ____________________ .
• (1/2, 1/4)
• (1/4, 1/2)
• (4, 2)
• (1, 1)
Q 8 | Page 42
The point at the curve y = 12x − x2 where the slope of the tangent is zero will be _____________ .
• (0, 0)
• (2, 16)
• (3, 9)
• none of these
Q 9 | Page 42
The angle between the curves y2 = x and x2 = y at (1, 1) is ______________ .
• $\tan^{- 1} \frac{4}{3}$
• $\tan^{- 1} \frac{3}{4}$
• 90°
• 45°
Q 10 | Page 43
The equation of the normal to the curve 3x2 − y2 = 8 which is parallel to x + 3y = 8 is ____________ .
• x + 3y = 8
• x + 3y + 8 = 0
• x + 3y ± 8 = 0
• x + 3y = 0
Q 11 | Page 43
The equations of tangent at those points where the curve y = x2 − 3x + 2 meets x-axis are _______________ .
• x − y + 2 = 0 = x − y − 1
• x + y − 1 = 0 = x − y − 2
• x − y − 1 = 0 = x − y
• x − y = 0 = x + y
Q 12 | Page 43
The slope of the tangent to the curve x = t2 + 3 t − 8, y = 2t2 − 2t − 5 at point (2, −1) is ________________ .
• 22/7
• 6/7
• -6
• none of these
Q 13 | Page 43
At what point the slope of the tangent to the curve x2 + y2 − 2x − 3 = 0 is zero
• (3, 0), (−1, 0)
• (3, 0), (1, 2)
• (−1, 0), (1, 2)
• (1, 2), (1, −2)
Q 14 | Page 43
The angle of intersection of the curves xy = a2 and x2 − y2 = 2a2 is ______________ .
• 45°
• 90°
• none of these
Q 15 | Page 43
If the curve ay + x2 = 7 and x3 = y cut orthogonally at (1, 1), then a is equal to _____________ .
• 1
• -6
• 6
• 0
Q 16 | Page 43
If the line y = x touches the curve y = x2 + bx + c at a point (1, 1) then _____________ .
• b = 1, c = 2
• b = −1, c = 1
• b = 2, c = 1
• b = −2, c = 1
Q 17 | Page 43
The slope of the tangent to the curve x = 3t2 + 1, y = t3 −1 at x = 1 is ___________ .
• 1/2
• 0
• -2
Q 18 | Page 43
The curves y = aex and y = be−x cut orthogonally, if ___________ .
• a = b
• a = −b
• ab = 1
• ab = 2
Q 19 | Page 43
The equation of the normal to the curve x = a cos3 θ, y = a sin3 θ at the point θ = π/4 is __________ .
• x = 0
• y = 0
• x = y
• x + y = a
Q 20 | Page 43
If the curves y = 2 ex and y = ae−x intersect orthogonally, then a = _____________ .
• 1/2
• −1/2
• 2
• 2e2
Q 21 | Page 43
The point on the curve y = 6x − x2 at which the tangent to the curve is inclined at π/4 to the line x + y= 0 is __________ .
• (−3, −27)
• (3, 9)
• (7/2, 35/4)
• (0, 0)
Q 22 | Page 43
The angle of intersection of the parabolas y2 = 4 ax and x2 = 4ay at the origin is ____________ .
• π/6
• π/3
• π/2
• π/4
Q 23 | Page 43
The angle of intersection of the curves y = 2 sin2 x and y = cos 2 x at $x = \frac{\pi}{6}$ is ____________ .
• π/4
• π/2
• π/3
• none of these
Q 24 | Page 43
Any tangent to the curve y = 2x7 + 3x + 5 __________________ .
• is parallel to x-axis
• is parallel to y-axis
• makes an acute angle with x-axis
• makes an obtuse angle with x-axis
Q 25 | Page 43
The point on the curve 9y2 = x3, where the normal to the curve makes equal intercepts with the axes is
(a) $\left( 4, \frac{8}{3} \right)$
(b) $\left( - 4, \frac{8}{3} \right)$
(c) $\left( 4, - \frac{8}{3} \right)$
(d) none of these
Q 26 | Page 43
The slope of the tangent to the curve x = t2 + 3t − 8, y = 2t2 − 2t − 5 at the point (2, −1) is _____________ .
• $\frac{22}{7}$
• $\frac{6}{7}$
• $\frac{7}{6}$
• $- \frac{6}{7}$
Q 27 | Page 43
The line y = mx + 1 is a tangent to the curve y2 = 4x, if the value of m is ________________ .
• 1
• 2
• 3
• 1/2
Q 28 | Page 44
The normal at the point (1, 1) on the curve 2y + x2 = 3 is _____________ .
• x + y = 0
• x − y = 0
• x + y + 1 = 0
• x − y = 1
Q 29 | Page 44
The normal to the curve x2 = 4y passing through (1, 2) is _____________ .
• x + y = 3
• x − y = 3
• x + y = 1
• x − y = 1
• none of these
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## Chapter 16: Tangents and Normals
Exercise 16.1Exercise 16.2Exercise 16.3Others
## RD Sharma solutions for Class 12 Maths chapter 16 - Tangents and Normals
RD Sharma solutions for Class 12 Maths chapter 16 (Tangents and Normals) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 12 Maths solutions in a manner that help students grasp basic concepts better and faster.
Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.
Concepts covered in Class 12 Maths chapter 16 Tangents and Normals are Maximum and Minimum Values of a Function in a Closed Interval, Maxima and Minima, Simple Problems on Applications of Derivatives, Graph of Maxima and Minima, Approximations, Tangents and Normals, Increasing and Decreasing Functions, Rate of Change of Bodies Or Quantities, Introduction to Applications of Derivatives.
Using RD Sharma Class 12 solutions Tangents and Normals exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer RD Sharma Textbook Solutions to score more in exam.
Get the free view of chapter 16 Tangents and Normals Class 12 extra questions for Class 12 Maths and can use Shaalaa.com to keep it handy for your exam preparation
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