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https://stats.stackexchange.com/questions/532623/how-should-evaluate-a-testing-set-using-a-pattern-learned-with-pca
|
# How should evaluate a Testing set using a pattern learned with PCA?
I tried to search for an answer about the evaluation and interpretation of a pattern learned with PCA, on data from a Testing Set, but I found no answer. Let me explain my situation:
My Data Mining task is to implement an instance of the KDD process in order to learn an accurate classifier of Windows applications.
I have a single "dataset.csv" file on which I do the initial split into Training Set and Testing Set (the entire learning & validation phases will focus exclusively on the Training Set).
My choice is to learn a classifier with the Gaussian Naive Bayes algorithm, so I decide to apply the transformation of my Training Set through the PCA, in order to ensure independence between the attributes (so, for example, on my Training Set with 50 features, I decide to transform it into a new Training Set with 10 Principal Components and learn & build from it my pattern with the Gaussian Naive Bayes and identifying the best configuration with a GridSearchCV).
Once learned my pattern, I have to perform the final evaluation on the whole Testing Set splitted at the start.
Here's my problem: The pattern was learned on a Training Set composed of 10 columns (i.e. 10 Principal Components), but my Testing Set is composed of 50 features (like my Training Set), and this conflicts with the learned pattern because the number of columns is different and I cannot make predictions. What should I do?
EDIT
To be correct, I specify the small details of my problem so that you can give me as complete an answer as possible.
I will describe very quickly what I did:
1. I split the starting dataset into Training Set and Testing Set using sklearn.model_selection.train_test_split with test_size=0.3 (i.e. 30%).
2. After splitting, I performed (on the Training Set) Data Cleaning (replacing any missing values with statistical.mode for each column) and Data Scaling using sklearn.preprocessing.MinMaxScaler.
3. After this little PreProcessing, I performed the PCA on the entire Training Set's indipendent variables, thus obtaining the new Training Set composed of 10 columns (i.e. the 10 Principal Components) and then I used sklearn.model_selection.GridSearchCV to learn the best configuration of sklearn.naive_bayes.GaussianNB (which is the estimator parameter), for the different var_smoothing values that I insert in a list and associate with the param_grid parameter, and setting cv=5 parameter for K-Fold Cross Validation used by GridSearchCV and finally fit on the entire Training Set.
This is how I learn my model, which now I have to test on the Testing Set.
I would like slightly more clarity on what should be done from this point on, so my doubt is: Would it be correct to perform the PCA exclusively on the Testing Set, calculating the same number of Principal Components and then evaluating?
• One would apply the PCA to all dataset before splitting to test/train. Jun 29 at 14:33
• That's totally wrong, the reason is that you should train your model only on the training data, without using any information regarding the testing data. If you apply PCA on the whole data (including the test data) before training the model (so, before the split), then you in fact use some information from the test data. Thus, you cannot really judge the behaviour of your model using the test data, because it is not an unseen data anymore. Jun 29 at 15:34
• Not really. PCA is "unsupervised". There is no peaking here. Test set is still unseen data, instances on the test set are not used in building the model. Jun 29 at 18:50
• @MehmetSuzen it causes data leakage, a somewhat related discussion: stats.stackexchange.com/questions/55718/… Jun 29 at 19:02
• @MehmetSuzen thanks for the article link. I couldn't exactly find parts supporting your idea (just skimmed). It's not only SO community, for example, the data leakage section in sklearn documentation exactly describes this kind of scenario: scikit-learn.org/stable/common_pitfalls.html . Simply, introducing information about the test set may introduce some trends in features that may not be readily available in training data. Thus, target or not, introduction of test set always carry a danger of leakage, despite one might get away with it on occasions. Jun 29 at 19:39
Just a minor correction: after PCA, you use the projections onto the principal components as features, not the PCs themselves. But, you'll have reduced set of features as you mentioned, say 10.
You'll set up a pipeline (e.g. you can utilize the Pipeline object in scikit-learn as I understand from your notation, you're using it) with steps PCA and GaussianNaiveBayes, and use grid search for hyper-parameter optimization (HPO).
This is different your proposed solution. In your second and third steps, you also introduce some leakage to the validation folds because you did PCA & data scaling beforehand. As I mentioned above, you should think all the operations you performed as a single model/pipeline and apply CV to it. This is harder to implement in code if you don't use pipelines, but it's the right thing to do.
Finally, with the best HPs selected, the final model (pipeline) will be fitted on the training set. This fitted model can predict the test set as well, because the pipeline has PCA step with PCs found for the training set, and there will be no dimension mismatch issue.
To reiterate, you won't fit PCA or scaling to the test set, you'll use fitted models/objects on the training set to be applied on the test set.
• I didn't know the Pipeline library from sklearn, so I didn't use it. To be correct, I will add a little more details to the question, so that you can give me a more precise answer. Jun 29 at 20:49
• @DavidZoy You don't have to use it, it's just very convenient for this kind of situations. I've amended my answer. Jun 29 at 21:25
• Use grid.best_estimator_ for the fitted model Jul 1 at 18:52
• You should have access to them in the best_estimator_ object. Jul 1 at 19:58
• I found them, thank you again! Jul 1 at 20:06
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2021-09-19 01:14:56
|
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|
https://www.tutorialspoint.com/construct-a-triangle-with-sides-5-cm-6-cm-and-7-cm-and-then-another-triangle-whose-sides-are-frac-7-5-of-the-corresponding-sides-of-the-first-tr
|
Construct a triangle with sides 5 cm, 6 cm and 7 cm and then another triangle whose sides are $\frac{7}{5}$ of the corresponding sides of the first triangle.
AcademicMathematicsNCERTClass 10
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Given:
A triangle with sides 5 cm, 6 cm and 7 cm.
To do:
We have to construct a triangle with sides 5 cm, 6 cm and 7 cm and then another triangle whose sides are $\frac{7}{5}$ times the corresponding sides of the first triangle.
Solution:
Steps of construction:
(i) Draw a $\triangle ABC$ with sides $AB = 5\ cm, BC = 7\ cm$ and $AC = 6\ cm$.
(ii) Draw an acute angle $CBX$ below $BC$ at point $B$.
(iii) Mark the ray $BX$ as $B_1, B_2, B_3, B_4, B_5, B_6$ and $B_7$ such that $BB_1= B_1B_2 = B_2B_3 = B_3B_4 = B_4B_5 = B_5B_6 = B_6B_7$.
(iv) Join $B_5$ and $C$.
(v) Draw $B_7C’$ parallel to $B_5C$, where $C’$ is a point on extended line $BC$.
(vi) Draw $A’C’ \| AC$, where $A’$ is a point on extended line $BA$.
Therefore,
$A’BC’$ is the required triangle.
Updated on 10-Oct-2022 13:23:52
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2022-11-26 13:39:35
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https://problemsolvingwithpython.com/06-Plotting-with-Matplotlib/06.03-Line-Plots/
|
# Line Plots
## Line Plots
Line plots can be created in Python with Matplotlib's pyplot library. To build a line plot, first import Matplotlib. It is a standard convention to import Matplotlib's pyplot library as plt. The plt alias will be familiar to other Python programmers.
If using a Jupyter notebook include the line %matplotlib inline after the imports. %matplotlib inline is a Jupyter notebook magic command which causes Matplotlib plots to display directly inside Jupyter notebook output cells.
To build our first plot, we will use NumPy, a numerical computing library for Python. NumPy is typically imported with the alias np.
In [1]:
import matplotlib.pyplot as plt
import numpy as np
# include if using a Jupyter notebook
%matplotlib inline
NumPy's np.arange() function creates an array of numbers with the parameters np.arange(start,stop,step). NumPy's np.sin() and np.pi functions do what you expect.
In [2]:
x = np.arange(0, 4 * np.pi, 0.1)
y = np.sin(x)
To create a line plot, pass an array or list of numbers as an argument to Matplotlib's plt.plot() function. The command plt.show() is needed at the end to show the plot. Make sure to include the double parenthesis () in plt.show().
In [3]:
plt.plot(x, y)
plt.show()
### Features of a Matplotlib plot
A variety of features on a Matplotlib plot can be specified. The following is a list of commonly defined features:
#### Line Color, Line Width, Line Style, Line Opacity and Marker Options
The color, width, and style of line in a plot can be specified. Line color, line width, and line style are included as extra arguments in the plt.plot() function call.on call.
plt.plot(<x-data>,<y-data>,
linewideth=<float or int>,
linestyle='<linestyle abbreviation>',
color='<color abbreviation>',
marker='<marker abbreviation>')
An example plt.plot() function call including line color, line width, and line style options is:
plt.plot(x, y,
linewidth=2.0,
linestyle='+',
color='b',
alpha=0.5,
marker='o')
Below is a list of linewidths (many other widths are also available).
linewidth=<float or int> Line Width
0.5 0.5 pixels wide
1 1 pixel wide
1.5 1.5 pixels wide
2 2 pixels wide
3 3 pixels wide
Below is a list of line styles.
linestyle='<style abbreviation>' Line Style
'-' or 'solid' solid line (default)
'--' or 'dashed' dashed line
'-.' or 'dashdot' dash-dot line
':' or 'dotted' dotted line
'None' or ' ' or '' no line
Below is a list of color abbreviations. Note 'b' is used for blue and 'k' is used for black.
color ='<color abbreviation>' Color Name
'b' Blue
'c' Cyan
'g' Green
'k' Black
'm' magenta
'r' Red
'w' White
'y' Yellow
Colors can also be specified in hexadecimal form surrounded by quotation marks like '#FF69B4' or in RGBA (red, green, blue, opacity) color surrounded by parenthesis like (255,182,193,0.5).
color ='<color abbreviation>' Color Format
(255,182,193,0.5) RGBA
Below is a list of alpha (opacity) values (any alpha value between 0.0 and 1.0 is possible).
alpha = <float or int> Opacity
0 transparent
0.5 Half transparent
1.0 Opaque
Below is a list of maker styles.
marker='<marker abbreviation>' Marker Style
'.' point
',' one pixel
'o' circle
'v' triangle_down
'^' triangle_up
'8' octagon
's' square
'p' pentagon
'*' star
'h' hexagon 1
'H' hexagon 2
'+' plus
'P' filled plus
'x' x
'X' filled x
'D' diamond
'd' thin diamond
In addition to marker='<marker style>', the color of the marker edge, the color of the marker face and the size of the marker can be specified with:
plt.plot( ....
markeredgecolor='<color abbreviation>',
markerfacecolor='<color abbreviation>',
markersize=<float or int>
....)
#### Title
The plot title will be shown above the plot. The plt.title() command accepts a string as an argument.
plt.title('My Title')
#### x-axis label
The x-axis label is shown below the x-axis. The plt.xlabel() command accepts a string as an argument.
plt.xlabel('My x-axis label')
#### y-axis label
The y-axis label is shown to the left of the y-axis. The plt.ylabel() command accepts a string as an argument.
plt.ylabel('My y-axis label')
#### Legend
You can use the plt.legend() command to insert a legend on a plot. The legend will appear within the plot area, in the upper right corner by default. The plt.legend() command accepts a list of strings and optionally accepts a loc= argument to position the legend in a different location
plt.legend(['entry1','entry2'], loc = 0)
The following are the location codes for the legend location. These numbers need to be placed after loc= in the plt.legend() call.
Legend Location loc = <number>
'best' 0
'upper right' 1
'upper left' 2
'lower left' 3
'lower right' 4
'right' 5
'center left' 6
'center right' 7
'lower center' 8
'upper center' 9
'center' 10
#### Grid
A grid can be shown on the plot using the plt.grid() command. By defaut, the grid is turned off. To turn on the grid use:
plt.grid(True)
The only valid options are plt.grid(True) and plt.grid(False). Note that True and False are capitalized and are not enclosed in quotations.
#### Tick Labels
Tick labels can be specified on the plot using plt.xticks() and plt.yticks(). To add tick labels use:
plt.xticks([locations list],[labels list])
plt.yticks([locations list],[labels list])
The [locations list] can be a Python list or NumPy array of tick locations . The [labels list] is a Python list or NumPy array of strings.
### Build a plot in five steps
Line plots with the specified features above are constructed in Python according to this general outline:
1. Imports
2. Define data
3. Plot Data including options
5. Show the plot
Details of each step is explained below.
#### 1. Imports
Import matplot.pyplot as plt, as well as any other modules needed to work with the data such as NumPy or Pandas. If using a Jupyter notebook include the line %matplotlib inline in the import section.
#### 2. Define data
The plot needs to contain data. Data is defined after the imports. Typically, data for plots is contained in Python lists, NumPy arrays or Pandas dataframes.
#### 3. Plot data including options
Use plt.plot() to plot the data you defined. Note the plt.plot() line needs to be called before any other plot details are specified. Otherwise, the details have no plot to apply to.
Besides data, the plt.plot() function can include keyword arguments such as:
• linewideth= <float or int>
• linestyle='<linestyle abbreviation>'
• color='<color abbreviation>'
• alpha= <float or int>
• marker='<marker abbreviation>'
• markeredgecolor='<color abbreviation>'
• markerfacecolor='<color abbreviation>'
• markersize=<float or int>
Add details such as a title, axis labels, legend, grid, and tick labels. Plot details to add include:
• plt.title('<title string>')
• plt.xlabel('<x-axis label string>')
• plt.ylabel('<y-axis label string>')
• plt.legend(['list','of','strings'])
• ptl.grid(<True or False>)
• plt.xticks([locations list or array], [labels list])
• plt.yticks([locations list or array], [labels list])
#### 5. Show the plot
Use the plt.show() command to show the plot. plt.show() causes the plot to display in a Jupyter notebook or pop out as a new window if the plot is constructed in a separate .py file. Note that plt.show() needs to be called after all of the plot specifications. The next code section follows this outline. The resulting plot is shown below.
In [4]:
# Imports
import numpy as np
import matplotlib.pyplot as plt
# include if using a Jupyter notebook. Remove if using a .py-file.
%matplotlib inline
# Define data x = np.arange(0, 4 * np.pi, 0.2) y = np.sin(x)
# Plot data including options plt.plot(x, y, linewidth=0.5, linestyle='--', color='r', marker='o', markersize=10, markerfacecolor=(1, 0, 0, 0.1))
# Add plot details plt.title('Plot of sin(x) vs x from 0 to 4 pi') plt.xlabel('x (0 to 4 pi)') plt.ylabel('sin(x)') plt.legend(['sin(x)']) plt.xticks( np.arange(0, 4*np.pi, np.pi/2), ['0','$\pi$/2','$\pi$','$3\pi/2$','2$\pi$','5$\pi$/2','3$\pi$','7$\pi$/2']) plt.grid(True)
# Show the plot plt.show()
|
2019-05-24 07:20:47
|
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|
http://blog.gmane.org/gmane.comp.tex.pgf.user/month=20080201
|
1 Feb 01:51 2008
### problem using a tikz matrix within beamer
Hello everybody,
I get some "Package pgfbasematrix Error: Single ampersand used with
wrong catcode." error message when using a tikz matrix within beamer.
Some simple example file is attached (uncomment the line to get the error).
I found a solution there :
that is: replace & by \pgfmatrixnextcell. But I was wondering if there
were a better way to fix it.
Thanks for any help
--
--
Sébastien Barthélemy
Attachment (beamer-tikz-ampersand.tex): application/x-tex, 571 bytes
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1 Feb 04:08 2008
### Re: problem using a tikz matrix within beamer
Dear Sébastien,
You can use another control sequence instead of \pgfmatrixnextcell by
using the "ampersand replacement" option.
\begin{tikzpicture}[auto]
\tikzstyle{frame} = [rectangle, draw=blue, thick, fill=blue!20];
\matrix [column sep=0.5cm,ampersand replacement=\&]
{
\node [frame] {$P_0$}; \& %<-- note the \ in front of the %
\node [frame] {$P_1$}; \&
\node [frame] {$P_2$}; \\
};
\end{tikzpicture}
I've also read that you can use the fragile option but I haven't
tried that.
Yours,
Matthew Leingang
On Jan 31, 2008, at 7:51 PM, Sébastien Barthélemy wrote:
> Hello everybody,
>
> I get some "Package pgfbasematrix Error: Single ampersand used with
> wrong catcode." error message when using a tikz matrix within beamer.
>
> Some simple example file is attached (uncomment the line to get the
1 Feb 09:58 2008
### Re: problem using a tikz matrix within beamer
Hi!
actually, just add the "fragile" option to beamer frames that contain
a matrix. That will fix the problem.
Till
Am 01.02.2008 um 04:08 schrieb Matthew Leingang:
> Dear Sébastien,
>
> You can use another control sequence instead of \pgfmatrixnextcell by
> using the "ampersand replacement" option.
>
> \begin{tikzpicture}[auto]
> \tikzstyle{frame} = [rectangle, draw=blue, thick, fill=blue!
> 20];
>
> \matrix [column sep=0.5cm,ampersand replacement=\&]
> {
> \node [frame] {$P_0$}; \& %<-- note the \ in front of the %
> \node [frame] {$P_1$}; \&
> \node [frame] {$P_2$}; \\
> };
> \end{tikzpicture}
>
> I've also read that you can use the fragile option but I haven't
> tried that.
>
> Yours,
1 Feb 11:30 2008
### Re: problem using a tikz matrix within beamer
2008/2/1, Till Tantau <tantau <at> tcs.uni-luebeck.de>:
> Hi!
>
> actually, just add the "fragile" option to beamer frames that contain
> a matrix. That will fix the problem.
Thanks to both of you
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1 Feb 11:47 2008
### Alignment/overlay of images and rectangles using TikZ in beamer
Hi there!
I'm - meanwhile rather desperately - trying to solve the following task:
I've defined an image to be 0.8\paperwidth. Now I want to display this image
on a slide and then overlay some rectangles for clarification.
One way I tried to accomplish this was:
\begin{tikzpicture}{remember picture,overlay}
\pgfuseimage{precrbias}
\end{tikzpicture}
\only <2> { \begin{tikzpicture}{remember picture,overlay}
\draw[red,overlay] (1,1) rectangle (1.5,1.5);
\draw[blue,overlay] (2.5,2.5) rectangle (1,1);
\draw[green,overlay] (5,5) rectangle (2.5,2.5);
\end{tikzpicture}}
(don't wonder about the coordinates, I'm just playing around so far)
With that I ended up having the image aligned at the center of the frame,
and the rectangles were where I wanted to have them.
In a second try, I put the image into a node, so "\node {
\pgfuseimage{precrbias} };". Now, the image was properly aligned, but the
rectangles show up next to it.
Putting the image outside any tikzpicture environment did not work either!
Any suggestions on what to do next?
1 Feb 12:04 2008
### Off-Topic: Problem with TeX expansion - help needed (capitalize words)
Hi,
I am sorry for abusing this list for a (difficult) TeX problem, but this is
the only list I know that is read by TeX wizards - please tell me if you know
a good place for such questions (that preferably does not require Usenet
access).
I am trying to solve a problem which I have had a lot of times already, and I
could absolutely not find any solution in the WWW so far: I am using
prettyref for styled references (i.e. \prettyref{eq:foo} gets expanded to
equation~\eqref{eq:foo}), which I want to capitalize at the beginning of
sentences.
After reading the TeXbook, I became brave and tried the following macro def:
\def\capitalize#1{\uppercase{#1}}
\capitalize simple text works.
With macros (or LaTeX commands), I could get it to work by adding
\expandafter:
\def\simplemacro{a simple macro}
\expandafter\capitalize \simplemacro{} works.
Question 1: Why does adding \expandafter to \capitalize's definition not work?
\def\capitalize#1{\expandafter\uppercase{#1}}
\capitalize \simplemacro{} does not work.
Obviously, this is because of the braces, but how can I solve this?
1 Feb 12:33 2008
### Re: Off-Topic: Problem with TeX expansion - help needed (capitalize words)
On Feb 1, 2008 12:04 PM, Hans Meine <meine <at> informatik.uni-hamburg.de> wrote:
> Hi,
>
> I am sorry for abusing this list for a (difficult) TeX problem, but this is
> the only list I know that is read by TeX wizards - please tell me if you know
> a good place for such questions (that preferably does not require Usenet
> access).
>
I recommend posting to comp.text.tex, which is the largest forum for
TeX users. It is a usenet group, but you can use Google groups to
both read and post to the group:
The only thing you'll need is a Google account. I think also you can
use Nabble to access usenet groups.
- Kjell Magne Fauske
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1 Feb 12:39 2008
### Re: Off-Topic: Problem with TeX expansion - help needed (capitalize words)
Hans,
Not sure how to solve your problem, but you might consider the fancyref
package.
It uses two formats \Fref and \fref. You can use the former at the
beginning of the sentence and latter anywhere else. Depending on the
label name (e.g. chap:, sec:, fig:, eq: and others) and language it
chooses the correct style for the reference. It also can use the
varioref package for automatic reference to the page too (e.g. figure
2.1 on page 21, or Equation 5.2 on the following page).
Regards,
Pascal Wolkotte
On Fri, 2008-02-01 at 12:04 +0100, Hans Meine wrote:
> Hi,
>
> I am sorry for abusing this list for a (difficult) TeX problem, but this is
> the only list I know that is read by TeX wizards - please tell me if you know
> a good place for such questions (that preferably does not require Usenet
> access).
>
> I am trying to solve a problem which I have had a lot of times already, and I
> could absolutely not find any solution in the WWW so far: I am using
> prettyref for styled references (i.e. \prettyref{eq:foo} gets expanded to
> equation~\eqref{eq:foo}), which I want to capitalize at the beginning of
> sentences.
>
> After reading the TeXbook, I became brave and tried the following macro def:
1 Feb 12:53 2008
### Re: Off-Topic: Problem with TeX expansion - help needed (capitalize words)
On Feb 1, 2008, at 6:04 AM, Hans Meine wrote:
> Hi,
>
> I am sorry for abusing this list for a (difficult) TeX problem, but
> this is
> the only list I know that is read by TeX wizards - please tell me
> if you know
> a good place for such questions (that preferably does not require
> Usenet
> access).
The TUG has a mailing list:
http://tug.org/mailman/listinfo/texhax
You have a good question and there are many TeXperts who can help.
Make sure you follow the etiquette of explaining what you wanted to
happen, what you tried, what actually happened, and minimal sample
code. There's a mini-flame war going on right now as to whether the
experts on the list are gentle enough. I think they are, but they
don't suffer fools gladly.
--Matt
--
Matthew Leingang
Preceptor in Mathematics
Harvard University
1 Feb 13:04 2008
### Re: Alignment/overlay of images and rectangles using TikZ in beamer
On Feb 1, 2008 11:47 AM, msu79 <martin <at> suklitsch.at> wrote:
>
> Hi there!
>
>
> I'm - meanwhile rather desperately - trying to solve the following task:
> I've defined an image to be 0.8\paperwidth. Now I want to display this image
> on a slide and then overlay some rectangles for clarification.
>
> One way I tried to accomplish this was:
> \begin{tikzpicture}{remember picture,overlay}
> \pgfuseimage{precrbias}
> \end{tikzpicture}
> \only <2> { \begin{tikzpicture}{remember picture,overlay}
> \draw[red,overlay] (1,1) rectangle (1.5,1.5);
> \draw[blue,overlay] (2.5,2.5) rectangle (1,1);
> \draw[green,overlay] (5,5) rectangle (2.5,2.5);
> \end{tikzpicture}}
>
> (don't wonder about the coordinates, I'm just playing around so far)
>
> With that I ended up having the image aligned at the center of the frame,
> and the rectangles were where I wanted to have them.
>
> In a second try, I put the image into a node, so "\node {
> \pgfuseimage{precrbias} };". Now, the image was properly aligned, but the
> rectangles show up next to it.
>
> Putting the image outside any tikzpicture environment did not work either!
>
|
2015-11-28 00:30:01
|
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|
https://gamedev.stackexchange.com/questions/153530/hlsl-mipmap-sampling-in-pixel-shader
|
# HLSL MipMap sampling in pixel shader
Using the Texture2D.Sample function, how does the pixel shader infer the mipmap level to use? Or does it use mipmap level 0? I am using the ps_4_0 shader model. And replacing Sample, with SampleLevel(state, pos, 0.0), does that give same results? In what cases does it give same results?
struct PS_INPUT
{
float4 pos : SV_POSITION;
float3 tex : TEXCOORD0; // texture coordinate
};
struct PS_OUTPUT
{
float4 color : SV_Target0;
};
Texture2D tex;
SamplerState state {
Filter = MIN_MAG_LINEAR_MIP_POINT;
};
PS_OUTPUT main(PS_INPUT input) : SV_Target
{
PS_OUTPUT output;
output.color = tex.Sample(state, input.pos);
return output;
}
Sample calculates the mip level to sample based on screen space derivatives of the texture coordinates provided. Derivatives of texture coordinates can be computed as follows:
float2 derivX = ddx(myTexCoord.xy);
float2 derivY = ddy(myTexCoord.xy);
But this is hidden inside the Sample function implementation. You can achieve the same result by using SampleGrad function and providing the derivatives yourself:
float4 textureColor = myTexture.SampleGrad(mySampler, myTexCoord.xy, derivX, derivY);
If you use SampleLevel, that expects the mip level instead of the derivatives, but you could compute a mip level from the derivatives yourself and use SampleLevel:
float delta_max_sqr = max(dot(derivX, derivX), dot(derivY, derivY));
float mip = 0.5 * log2(delta_max_sqr) * myTexture_NumberOfMIPLevels;
float4 textureColor = myTexture.SampleLevel(mySampler, myTexCoord.xy, mip);
However the SampleLevel might not give the exact results of the Sample method because Sample can use a different method based on the sampler descriptor's MipFilter and MipLODBias settings.
Use case:
Calculating derivatives using ddx and ddy needs information from neighbour pixels, so you can't use the functions inside dynamic branches or loops (which are maybe not running for neighboring pixels). In those cases the derivatives could be precomputed and you can use the SampleGrad or SampleLevel instructions inside the dynamic branches/loops.
• As long as you are comfortable requiring D3D_FEATURE_LEVEL_10_0 or higher, this works. – Chuck Walbourn Jan 24 '18 at 17:59
|
2019-11-22 02:29:31
|
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|
https://developer.mozilla.org/ja/docs/Web/MathML/Element/msup
|
# <msup>
MathML `の <msup>` 要素は，式に上付きを付けるために用います。
## 属性
class, id, style
Provided for use with stylesheets.
href
Used to set a hyperlink to a specified URI.
mathbackground
The background color. You can use `#rgb`, `#rrggbb` and HTML color names.
mathcolor
The text color. You can use `#rgb`, `#rrggbb` and HTML color names.
superscriptshift
## 例
Sample rendering:
Rendering in your browser: ${X}^{2}$
``````<math>
<msup>
<mi>X</mi>
<mn>2</mn>
</msup>
</math>
``````
## ブラウザ毎の互換性
BCD tables only load in the browser
### Gecko-specific notes
• Starting with Gecko 26.0 (Firefox 26 / Thunderbird 26 / SeaMonkey 2.23 / Firefox OS 1.2) it is no longer possible to use `<none />` as a child element. The rendering has been made more consistent with equivalent configurations of `<msub>` and `<mmultiscripts>` and a bug with an incorrect application of` the superscriptshift` attribute has been fixed (see バグ 827713 for details).
## 仕様
Specification Status Comment
MathML 3.0
msup の定義
MathML 2.0
msup の定義
• `<msub>` (Subscript)
• `<msubsup>` (Subscript-superscript pair)
• `<mmultiscripts>` (Prescripts and tensor indices)
|
2022-05-23 05:30:41
|
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|
https://www.irif.fr/~sighirea/sl-comp/18/index.html
|
# SL-COMP 2018
## Synopsis
SL-COMP is an initiative for collecting decision problems in Separation Logic and solvers dealing with such problems.
This initiative started in 2014 and it was inspired by the very popular initiatives on other theories, like first-order theories for SMT-COMP and Boolean constraints for SAT.
## Key Dates
• June 24: deadline for solver registration
• July 1: first round of the competition
• July 8: last round of the competition
• July 13: presentation at ADSL 2018; all participants are welcome to present their solver!
## Introduction
Separation Logic (SL) is an established and fairly popular Hoare logic for the imperative, heap-manipulating programs.
Its high expressivity, its ability to generate compact proofs and its support for local reasoning motivated the development of tools for automatic reasoning about programs using dynamic memory and whose properties are specified using SL.
These tools intensively use (semi-)decision procedures for checking satisfiability and entailment problems in SL. In the last ten years, several papers reported on the design and implementation of such solvers.
To improve the status of these solvers and to motivate the research in this area, we started in May 2014 a competition of solvers for Separation Logic problems. This competition was inspired by the very popular competitions on other logic theories, e.g., SMT-COMP and SAT. This first edition of SL-COMP has been held with the support of the StarExec platform and of SMT-COMP officers, mainly David Cok.
The input format of problems extended the format SMT-LIB with SL constructs. This format has been changed for the second edition of SL-COMP in 2018 to better exploit the new features of SMT-LIB like datatypes definition and mutually recursive functions.
## Specifications and rules
The input problems are specified using the format described here and commented in this post.
Problems to be solved are either satisfiability problems or entailment problems. An entailment problem consists in checking the validity of the formula $$A \models B$$ and it is coded by the formula $$A \land \lnot B$$.
The possible answers of a solver are: sat, unsat, unknown.
The solvers shall run on the StarExec platform. Solvers may use a pre-processor to transform the input file format to the solver's format. The input problems are not scrambled.
## Benchmarks and divisions
• SL-COMP 2018 collected a set of 800 problems. This set includes the one in the 2014 edition and nerly 200 new problems. The current set of problems is split into the following divisions:
• SL-COMP 2014 collected a set of 678 problems The problems collected have the following properties:
• 25% satisfiability problems versus 75% entailment problems
• 41% crafted problems versus 59% random generated problems
• The problems were split in four divisions:
• sll0a_sat: satisfiability problems for the symbolic heap fragment of Separation Logic with a unique inductive predicate specifying acyclic singly linked list.
• sll0a_entl: entailment problems for the above fragment.
• FDB_entl: entailment problems for the symbolic heap fragment of Separation Logic with linear inductive definitions specifying various kinds of lists in a restricted way.
• UDB_sat: satisfiability problems for the symbolic heap fragment of Separation Logic with general, user defined inductive definitions specifying lists, trees, etc.
## Participants
The final list of participants will be announces June 25th, 2018. Some participants already confimed their presence to SL-COMP'18:
• Asterix: divisions qf_shls_sat and qf_shls_entl
• COMP-SPEN: division qf_shlidlia_entl
• Cyclist: all divisions
• Harrsh: division qf_shls_sat
• Inductor: divisions qf_shid_entl
• Sleek: all divisions
• SongBird: all divisions
• SPEN: divisions qf_shls_sat, qf_shls_entl, qf_shlid_entl, and qf_shlidlia_entl
The list of participants at SL-COMP 2014 is provided here.
## Results
The results will be announces July 13th, 2018, at the ADSL workshop.
## Committee
The organisation committee of SL-COMP 2018 includes the organisers of the ADSL workshop, namely Nikos Gorogiannis, Radu Iosif and Mihaela Sighireanu.
The competition committee will include a member for each participating solver.
## Mailing list
Any question related to this competition shall be sent to the organisation committee and to the mailing list.
## Previous SL-COMPs
Created: 2018-06-18 Mon 11:09
Validate
|
2018-06-22 07:19:54
|
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|
https://www.hackmath.net/en/math-problem/1441
|
# Otto and Joachim
Otto and Joachim go through the woods. After some time Otto tire and make 15 minutes stop. Joachim meanwhile continues at 5 km/h. Otto when he set off again, first running speed of 7 km/h, but it keep only 30 sec and 1 minute must continue at 3 km/h. This keeps cycling. How long Otto Joachim catch up?
Correct result:
t = -1 min
#### Solution:
$s_1 = 5\cdot 15/60 = 1.25 \ km \ \\ s_J = 5\cdot 1.5/60 = 0.125 \ km \ \\ s_O = (7\cdot 0.5 + 3\cdot 1)/60 = 0.10833 \ km \ \\ \ \\ 0.10833 \ km < 0.125 \ km \ \\ s_O < s_J => t=\inf \ min$
We would be pleased if you find an error in the word problem, spelling mistakes, or inaccuracies and send it to us. Thank you!
Tips to related online calculators
Do you have a linear equation or system of equations and looking for its solution? Or do you have quadratic equation?
Do you want to convert velocity (speed) units?
Do you want to convert time units like minutes to seconds?
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|
2021-01-27 13:14:42
|
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|
https://xianblog.wordpress.com/tag/importance-sampling/
|
## nested sampling via SMC
Posted in Books, pictures, Statistics with tags , , , , , , , , , , , , on April 2, 2020 by xi'an
“We show that by implementing a special type of [sequential Monte Carlo] sampler that takes two im-portance sampling paths at each iteration, one obtains an analogous SMC method to [nested sampling] that resolves its main theoretical and practical issues.”
A paper by Queenslander Robert Salomone, Leah South, Chris Drovandi and Dirk Kroese that I had missed (and recovered by Grégoire after we discussed this possibility with our Master students). On using SMC in nested sampling. What are the difficulties mentioned in the above quote?
1. Dependence between the simulated samples, since only the offending particle is moved by one or several MCMC steps. (And MultiNest is not a foolproof solution.)
2. The error due to quadrature is hard to evaluate, with parallelised versions aggravating the error.
3. There is a truncation error due to the stopping rule when the exact maximum of the likelihood function is unknown.
Not mentioning the Monte Carlo error, of course, which should remain at the √n level.
“Nested Sampling is a special type of adaptive SMC algorithm, where weights are assigned in a suboptimal way.”
The above remark is somewhat obvious for a fixed sequence of likelihood levels and a set of particles at each (ring) level. moved by a Markov kernel with the right stationary target. Constrained to move within the ring, which may prove delicate in complex settings. Such a non-adaptive version is however not realistic and hence both the level sets and the stopping rule need be selected from the existing simulation, respectively as a quantile of the observed likelihood and as a failure to modify the evidence approximation, an adaptation that is a Catch 22! as we already found in the AMIS paper. (AMIS stands for adaptive mixture importance sampling.) To escape the quandary, the authors use both an auxiliary variable (to avoid atoms) and two importance sampling sequences (as in AMIS). And only a single particle with non-zero incremental weight for the (upper level) target. As the full details are a bit fuzzy to me, I hope I can experiment with my (quarantined) students on the full implementation of the method.
“Such cases asides, the question whether SMC is preferable using the TA or NS approach is really one of whether it is preferable to sample (relatively) easy distributions subject to a constraint or to sample potentially difficult distributions.”
A question (why not regular SMC?) I was indeed considering until coming to the conclusion section but did not find it treated in the paper. There is little discussion on the computing requirements either, as it seems the method is more time-consuming than a regular nested sample. (On the personal side, I appreciated very much their “special thanks to Christian Robert, whose many blog posts on NS helped influence this work, and played a large partin inspiring it.”)
## ensemble rejection sampling
Posted in Statistics with tags , , , on March 25, 2020 by xi'an
George Deligiannidis, Arnaud Doucet and Sylvain Rubenthaler have constructed a form of Rao-Blackwellised estimate out of a regular rejection sampler. Doubly surprisingly as turning importance sampling into regular sampling plus gaining over the standard accept-reject estimate. They call their approach ensemble rejection sampling. This is done by seeing the N-sample created from the proposal as an importance sampler, exploiting the importance weights towards estimating the (intractable) normalising constant of the target density, and creating an upper bound on this estimate Ẑ. That depends on the current value X from the N-sample under consideration for acceptance as
Z⁺=Ẑ+{max(w)-w(X)}/N
with a probability Ẑ/Z⁺ to accept X. The amazing result is that the X thus marginaly produced is distributed from the target! Meaning that this is a case for a self-normalised importance sampling distribution producing an exact simulation from the target. While this cannot produce an iid sample, it can be exploited to produce unbiased estimators of expectations under the target. Without even resampling and at a linear cost in the sample size N.
The method can be extended to the dynamic (state-space) case. At a cost of O(N²T) as first observed by Radford Neal. However, the importance sample seems to be distributed from a product of proposals that do not account for the previous particles. But maybe accounting for the observations. While the result involves upper bounds on the dynamic importance weights, the capacity to deliver exact simulations remains a major achievement, in my opinion.
## Hastings at 50, from a Metropolis
Posted in Kids, pictures, Running, Travel with tags , , , , , , , , , , , , , , , , , , , , , , on January 4, 2020 by xi'an
A weekend trip to the quaint seaside city of Le Touquet Paris-Plage, facing the city of Hastings on the other side of the Channel, 50 miles away (and invisible on the pictures!), during and after a storm that made for a fantastic watch from our beach-side rental, if less for running! The town is far from being a metropolis, actually, but it got its added surname “Paris-Plage” from British investors who wanted to attract their countrymen in the late 1800s. The writers H.G. Wells and P.G. Wodehouse lived there for a while. (Another type of tourist, William the Conqueror, left for Hastings in 1066 from a wee farther south, near Saint-Valéry-sur-Somme.)
And the coincidental on-line publication in Biometrika of a 50 year anniversary paper, The Hastings algorithm at fifty by David Dunson and James Johndrow. More of a celebration than a comprehensive review, with focus on scalable MCMC, gradient based algorithms, Hamiltonian Monte Carlo, nonreversible Markov chains, and interesting forays into approximate Bayes. Which makes for a great read for graduate students and seasoned researchers alike!
## sampling-importance-resampling is not equivalent to exact sampling [triste SIR]
Posted in Books, Kids, Statistics, University life with tags , , , , , , on December 16, 2019 by xi'an
Following an X validated question on the topic, I reassessed a previous impression I had that sampling-importance-resampling (SIR) is equivalent to direct sampling for a given sample size. (As suggested in the above fit between a N(2,½) target and a N(0,1) proposal.) Indeed, when one produces a sample
$x_1,\ldots,x_n \stackrel{\text{i.i.d.}}{\sim} g(x)$
and resamples with replacement from this sample using the importance weights
$f(x_1)g(x_1)^{-1},\ldots,f(x_n)g(x_n)^{-1}$
the resulting sample
$y_1,\ldots,y_n$
is neither “i.” nor “i.d.” since the resampling step involves a self-normalisation of the weights and hence a global bias in the evaluation of expectations. In particular, if the importance function g is a poor choice for the target f, meaning that the exploration of the whole support is imperfect, if possible (when both supports are equal), a given sample may well fail to reproduce the properties of an iid example ,as shown in the graph below where a Normal density is used for g while f is a Student t⁵ density:
## Mallows model with intractable constant
Posted in Books, pictures, Statistics with tags , , , , , , , , on November 21, 2019 by xi'an
The paper Probabilistic Preference Learning with the Mallows Rank Model by Vitelli et al. was published last year in JMLR which may be why I missed it. It brings yet another approach to the perpetual issue of intractable normalising constants. Here, the data is made of rankings of n objects by N experts, with an assumption of a latent ordering ρ acting as “mean” in the Mallows model. Along with a scale α, both to be estimated, and indeed involving an intractable normalising constant in the likelihood that only depends on the scale α because the distance is right-invariant. For instance the Hamming distance used in coding. There exists a simplification of the expression of the normalising constant due to the distance only taking a finite number of values, multiplied by the number of cases achieving a given value. Still this remains a formidable combinatoric problem. Running a Gibbs sampler is not an issue for the parameter ρ as the resulting Metropolis-Hastings-within-Gibbs step does not involve the missing constant. But it poses a challenge for the scale α, because the Mallows model cannot be exactly simulated for most distances. Making the use of pseudo-marginal and exchange algorithms presumably impossible. The authors use instead an importance sampling approximation to the normalising constant relying on a pseudo-likelihood version of Mallows model and a massive number (10⁶ to 10⁸) of simulations (in the humongous set of N-sampled permutations of 1,…,n). The interesting point in using this approximation is that the convergence result associated with pseudo-marginals no long applies and that the resulting MCMC algorithm converges to another limiting distribution. With the drawback that this limiting distribution is conditional to the importance sample. Various extensions are found in the paper, including a mixture of Mallows models. And an round of applications, including one on sushi preferences across Japan (fatty tuna coming almost always on top!). As the authors note, a very large number of items like n>10⁴ remains a challenge (or requires an alternative model).
## distilling importance
Posted in Books, Statistics, University life with tags , , , , , , , , , , on November 13, 2019 by xi'an
As I was about to leave Warwick at the end of last week, I noticed a new arXival by Dennis Prangle, distilling importance sampling. In connection with [our version of] population Monte Carlo, “each step of [Dennis’] distilled importance sampling method aims to reduce the Kullback Leibler (KL) divergence from the distilled density to the current tempered posterior.” (The introduction of the paper points out various connections with ABC, conditional density estimation, adaptive importance sampling, X entropy, &tc.)
“An advantage of [distilled importance sampling] over [likelihood-free] methods is that it performs inference on the full data, without losing information by using summary statistics.”
A notion used therein I had not heard before is the one of normalising flows, apparently more common in machine learning and in particular with GANs. (The slide below is from Shakir Mohamed and Danilo Rezende.) The notion is to represent an arbitrary variable as the bijective transform of a standard variate like a N(0,1) variable or a U(0,1) variable (calling the inverse cdf transform). The only link I can think of is perfect sampling where the representation of all simulations as a function of a white noise vector helps with coupling.
I read a blog entry by Eric Jang on the topic (who produced this slide among other things) but did not emerge much the wiser. As the text instantaneously moves from the Jacobian formula to TensorFlow code… In Dennis’ paper, it appears that the concept is appealing for quickly producing samples and providing a rich family of approximations, especially when neural networks are included as transforms. They are used to substitute for a tempered version of the posterior target, validated as importance functions and aiming at being the closest to this target in Kullback-Leibler divergence. With the importance function interpretation, unbiased estimators of the gradient [in the parameter of the normalising flow] can be derived, with potential variance reduction. What became clearer to me from reading the illustration section is that the prior x predictive joint can also be modeled this way towards producing reference tables for ABC (or GANs) much faster than with the exact model. (I came across several proposals of that kind in the past months.) However, I deem mileage should vary depending on the size and dimension of the data. I also wonder at the connection between the (final) distribution simulated by distilled importance [the least tempered target?] and the ABC equivalent.
## Why do we draw parameters to draw from a marginal distribution that does not contain the parameters?
Posted in Statistics with tags , , , , , , , on November 3, 2019 by xi'an
A revealing question on X validated of a simulation concept students (and others) have trouble gripping with. Namely using auxiliary variates to simulate from a marginal distribution, since these auxiliary variables are later dismissed and hence appear to them (students) of no use at all. Even after being exposed to the accept-reject algorithm. Or to multiple importance sampling. In the sense that a realisation of a random variable can be associated with a whole series of densities in an importance weight, all of them being valid (but some more equal than others!).
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2020-04-06 09:39:41
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https://math.stackexchange.com/questions/2711315/why-are-there-two-separate-branches-of-calculus
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# Why are there two separate branches of calculus?
Historically, there were two branches, differential and integral, right? Did they only form a unified calculus after the discovery of the Fundamental Theorem of Calculus?
Today, are the two branches still researched independently of each other?
• In a sense, yes, differentiation and integration are still studied separately. Modern integrals tend to involve more measure theory (most commonly, the Lebesgue integral). Certain Banach Spaces allow for a "Radon-Nikodym derivative" of one measure with respect to the other, and is defined purely in terms of the integral. Compare and contrast to an indefinite (Riemann) integral, which is defined purely in terms of the derivative! – Theo Bendit Mar 28 '18 at 5:27
• By mid-20th century it was clear that both the processes of integration and differentiation were (very important) examples of continuous linear maps (from one topological vector space of functions to a possibly different topological vector space of functions. Although some parts of the standard curriculum preserve an alleged separation between "integration" and "differentiation", this is mostly due to inertia rather than scientific content. – paul garrett Mar 28 '18 at 13:03
Calculus just means a way of calculating. There are many types of calculus, some of which are modern and currently actively studied (e.g. functional calculus, umbral calculus, difference calculus), while some are more or less fully understood. Differential and Integral calculus, i.e. the standard "calculus sequence", falls in the latter category. Note: mathematical analysis is a very active field of study, but it consists of generalizations and applications of the calculus sequence.
There are also abandoned types of calculus, like techniques for calculating square roots by hand example, which used to be taught like long division in school.
It depends on what you're referring to when you use the term "calculus".
Calculus itself comes from the Latin word meaning small pebble used for calculation, and is used to not only refer to differential and integral calculus as you describe. There are many other fields that have the name "calculus" to describe methods of calculation, such as propositional calculus, lambda calculus, etc. Before "standard" calculus, the term was widely used to refer to any field of mathematics.
But in the scope of differential and integral calculus, you are correct; the idea of infinitesimal change and infinitesimal summation predate Newton and Leibniz by centuries. The Fundamental Theorem of Calculus was only developed in the 17th and 18th centuries to connect these ideas. First introduced by Leibniz's and Newton's precursors, it was refined by these two. "Standard" calculus was then rigorized by later figures with the definition of limits such as used in derivatives, contrary to "infinitesimal ratios" presented by Leibniz.
historically, there were two branches right? differential and integral? Did they only form a unified calculus after the discovery of Fundamental Theorem of Calculus?
Are the two branches today still researched independently of each other?
in general there are two branches, diff and integral, but these are not two separate branches, not really. The derivative and integral are inverses of each other, like multiplication and division. Calculus had two independent discoverers, Newton and Leibniz. Newton is given credit for being the first to discover calculus. With Newton integral calculus came first, what he called fluxions, and differential came later. Newton used fluxions (what we call the fundamental theorem) to calculate the area under curves. Fluxions was initially derived from the (Newton's) binomial theorem. Leibniz, however, discovered differential calculus before integral, as best we know, as his treatise on differential calculus was published some time before his treatise on integral calculus. Newton is given credit for being the first to discover calculus, even though Leibniz published his findings before Newton did. Newton was very late in publishing, but his work predated Leibniz. We get our present day notation (dy/dx, etc) from Leibniz. We also get the integral sign from Leibniz. It comes from the Latin summa, meaning sum. Calculus itself is a Latin word meaning pebble, used for calculation in ancient times, as noted in a previous post.
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2019-10-21 02:06:34
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https://brilliant.org/discussions/thread/triangle-inequality-5/
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# Triangle Inequality
Suppose there is a triangle with sides $$a,b$$ and $$c$$. Prove that
$a^2 + b^2 + c^2 \geq \sqrt{48} \times A$
where $$A$$ denote the area of the triangle.
Note by Harmanjot Singh
2 years, 5 months ago
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## Comments
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$A = \frac14 \sqrt{ (a^2 +b^2+c^2)^2 - 2(a^4+b^4+c^4) }$
So we have to prove that $$a^2 + b^2+ c^2 \geq \dfrac{\sqrt{48}}4 \sqrt{(a^2+ b^2+c^2)^2 - 2(a^4 + b^4 + c^4) }$$.
Equivalently, (squaring both sides), we want to prove that $$16(a^2 + b^2+c^2) \geq 48( (a^2+b^2+c^2)^2 - 2(a^4 + b^4 + c^4) )$$ or $$(a^2 + b^2 + c^2)^2 \leq 3(a^4 + b^4 + c^4 )$$ which is obviously true when we apply the power mean inequality on the numbers $$a^2 , b^2, c^2$$.
$\text{QM}(a^2, b^2,c^2) \geq \text{AM}(a^2,b^2,c^2) \Rightarrow \sqrt[4]{\frac{a^4 + b^4+c^4}{3} } \geq \sqrt{ \frac{a^2+b^2 +c^2}3 }$
Equality holds when the triangle is equilateral, or $$a=b=c$$.
- 2 years, 5 months ago
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2018-04-20 16:39:32
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https://wiki.contextgarden.net/Command/startproject
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\startproject
Syntax
\startproject ... ... \stopproject ... command ... text
Description
`\startproject ... \stopproject` starts a project. A project is a structure to produce multiple PDF files that use the same \environments. Each PDF file is defined by a \product; each product may (re)use several \components.
Example
```\startproject projectname
\startfrontmatter
\product titlepage %Titlepage for the book
\stopfrontmatter
\startbodymatter
\product issue1 % Local TOC
% Introduction
% Contents
\product issue2 % Local TOC
% Introduction
% Contents
\product issue3 % Local TOC
% Introduction
% Contents
\stopbodymatter
\startbackmatter
\product appendices % Appendix 1
% Appendix 2
% Impressum
\stopbackmatter
\nomorefiles
\stopproject
```
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2019-08-20 18:10:52
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https://math.stackexchange.com/questions/2603534/chain-homotopy-on-linear-chains-confusion-from-hatchers-book
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# Chain homotopy on linear chains: confusion from Hatcher's book
I am following the book of Hatcher, see page 121-122 for notations.
Define $T:LC_n(Y)\rightarrow LC_{n+1}(Y)$ as follows inductively: for $n=-1$ it is zero map. For $n\geq 0$,
$$T(\lambda):=b_{\lambda}(\lambda-T\partial\lambda).$$
Question. Consider $n=0$. If $\lambda:\Delta^0\rightarrow Y$, $\lambda(v_0)=w_0$ then $\lambda=[w_0]$ (by notation) and so $$T(\lambda)=b_{\lambda}(\lambda-T\partial\lambda)=b_{\lambda}([w_0]-T\partial[w_0])=b_{\lambda}([w_0]-T[\phi])=b_{\lambda}[w_0]=[w_0,w_0];$$ is this correct? If yes, move to $n=1$, for $\mu:\Delta^1\rightarrow Y$, $\mu(v_0)=w_0, \mu(v_1)=w_1$, we have $$T(\mu)=b_{\mu}(\mu-T\partial\mu).$$ Here $T\partial\mu=T([w_1]-[w_0])=T[w_1]-T[w_0]=[w_1,w_1]-[w_0,w_0]$ is this correct? Therefore $$T(\mu)=b_{\mu}([w_0,w_1]-[w_1,w_1]+[w_0,w_0])=[b,w_0,w_1]-[b,w_1,w_1]+[b,w_0,w_0].$$ Here $b$ is image of barycenter of $[v_0,v_1]$ under $\mu$ in $Y$. This last calculation I am not realizing well geometrically as Hatcher explains on page 122; I feel I am wrong somewhere, or misunderstanding something. Can one explain this?
Notations
• $\Delta^n=[v_1,v_1,\cdots,v_n]$ is the standard $n$-simplex of dimension $n$ in Euclidean space.
• $C_n(X)$ denote the free abelian group on all continuous maps $\Delta^n\rightarrow X$.
• $LC_n(X)$ denotes the subgroup of $C_n(X)$ generated by 'linear maps' $\Delta^n\rightarrow X$.
• A 'linear map' $\lambda:\Delta^n\rightarrow X$ is also denote by $[\lambda(v_0),\cdots, \lambda(v_n)]$ (repetition is possible in images).
• For $b\in Y$, define $b:LC_n(Y)\rightarrow LC_{n+1}(Y)$ by $b([w_0,\cdots,w_n])=[b,w_0,\cdots,w_n]$.
• For $\lambda:\Delta^n\rightarrow Y$, define $S(\lambda):=b_{\lambda}(S\partial \lambda)$ with $S([\phi)]=[\phi]$ (empty simplex).
• I am studying this now and getting the same answers you have. I have the same questions as you now. Did you end up figuring it out? – Juan Lanfranco Mar 6 at 23:56
• I think the singular simplices like $[b, w_i, w_i]$ are not intuitive since they are not embeddings of $2$-simplices. But if you draw out the $n=1$ case with the points $w_0, b_\lambda, w_1$ making an $L$-shape, you can see it more clearly and it is similar to Hatcher's $n=2$ case. – Juan Lanfranco Mar 7 at 0:04
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2019-10-22 09:33:42
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https://www.numerade.com/questions/determine-the-amplitude-period-and-phase-shift-for-the-given-function-graph-the-function-over-one-15/
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Our Discord hit 10K members! 🎉 Meet students and ask top educators your questions.Join Here!
Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the $x$ -intercepts and the coordinates of the highest and Iowest points on the graph.$$y=1-\cos \left(2 x-\frac{\pi}{3}\right)$$
$\pi / 6$
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so we have. Why is equal to one minus co. Sign off two X minus height over sea, and this can be figured in as y school to one minus co. Sign off two times X minus pi over six. So the altitude ISS won. The period is two pi over two, which is equal to pi. And there's a phase shift equal to pi over six. And there's also a word of The shift is equal to one, and there's only one X intercept, and Juan X X intercept is at X is equal to high or six. And then on the graph, you can see the location off the maximum point the quarters to power three, comma two and in accordance off the minimum point rpai or six comma zero.
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Heather Z.
Oregon State University
Kayleah T.
Harvey Mudd College
Caleb E.
Baylor University
Michael J.
Idaho State University
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2021-05-09 14:00:07
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https://bettersolution4u.wordpress.com/
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# SPSS Tutorial Class for M.Ed Course
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# How to Write Bengali Bold and Italic Font Text in English Document in LaTeX?
In the previous post we discussed how to write Bengali in LaTeX. Use polyglossia package for Bangla font and compile it with XeLaTeX command. In this post we will discuss how to use “Bangla Normal”, “Bangla Italic“, “Bangla Bold” style in LaTeX. However, the SolaimanLipi is used as Bengali font. The “Times New Roman” is used for English font.
The Bangla font and style files can run only with the help of AutoFakeBold=4.0 font option. If you also want to fake italics/slanted font try using AutoFakeSlant=0.4 where the number also indicates the strength of the slant.
Minimum working example :
\documentclass[12pt,a4paper]{article}
% For a bilingual document
\RequirePackage{fontspec}
\RequirePackage{polyglossia}
\setmainlanguage{english}
\defaultfontfeatures{Ligatures=TeX}
% Times New Roman used for English
\setmainfont[Mapping=tex-text, Ligatures=TeX]{Times New Roman}
\setmainlanguage[numerals=Devanagari]{bengali}
\setotherlanguage{english}
% Bengali
\newfontfamily\bengalifont[Script=Bengali,AutoFakeBold=4.0,AutoFakeSlant=0.4]{SolaimanLipi}
\newfontfamily\bengalifontbf[Script=Bengali,AutoFakeBold=4.0,AutoFakeSlant=0.4]{SolaimanLipi}
\newfontfamily\bengalifontsf[Script=Bengali,AutoFakeBold=4.0,AutoFakeSlant=0.4]{SolaimanLipi}
\title{LaTeX ইংলিশ ডকুমেন্টে বাংলা বোল্ড এবং ইটালিক ফন্ট লেখাটি কীভাবে লিখবেন?}
\author{মোঃ কুতুবউদ্দিন সরদার}
\begin{document}
\maketitle
সাধারন স্টাইল, \textbf{বোল্ড ফন্ট স্টাইল }, \textit{ইটালিক ফন্ট স্টাইল । }
\end{document}
Output:
# How to Write Bengali with Custom Font Package in LaTeX?
LaTeX uses fontspec and polyglosia packages to write multilingual documents. In this case, you need to use the XeLaTeX or LuaLaTeX compiler instead of the PDFLaTeX compiler.
There are a number of things to consider here
1. When loading fonts which are stored in the texmf-tree you should use file names and not font names, as font names don’t work on all OS out of the box. So use \newfontfamily\bengalifont[Script=Bengali,Scale=0.85]{AkaashNormal.ttf} instead of \newfontfamily\bengalifont[Script=Bengali,Scale=0.85]{AkaashNormal}.
\documentclass[12pt]{article}
\usepackage{polyglossia}
\setmainlanguage[numerals=Devanagari]{bengali}
\setotherlanguage{English}
\newfontfamily\englishfont[Scale=MatchLowercase]{AkaashNormal}
\newfontfamily\bengalifont[Script=Bengali]{Akaash}
\begin{document}
\author{মোঃ কুতুবউদ্দিন সরদার}
\title{ল্যাটেক্সে বাংলা লেখার সহজ উপায়}
\date{\today}
\maketitle
\section{ভূমিকা}
LaTeX এ বহুভাষিক document লেখার জন্য fontspec এবং polyglosia package গুলি ব্যবহার করা হয়। এই ক্ষেত্রে PDFLaTeX compiler এর পরিবর্তে XeLaTeX বা LuaLaTeX compiler ব্যবহার করতে হয়।
\end{document}
Output:
# Authorization Error : PDF to Image not allowed by ImageMagick
For some time ImageMagick adopted a security policy. It has become a little more rigorous with ImageMagick itself and Linux distributions included. Here is the Quick and easy solution. Run the following code in the terminal:
sudo mv /etc/ImageMagick-6/policy.xml /etc/ImageMagick-6/policy.xml.off
When done, you can restore the original with
sudo mv /etc/ImageMagick-6/policy.xml.off /etc/ImageMagick-6/policy.xml
# How to install sage on ubuntu 18.04 LTS?
From the official website, there are several methods to install sagemath.
The easiest one is to run the following commands:
sudo apt-get install build-essential m4 dpkg-dev
sudo apt-get install sagemath
For more details about the different installation methods, please visit the installation guide
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2021-08-02 23:50:09
|
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http://math.iisc.ac.in/seminars/2019/2019-04-30-raghavendra-tripathi.html
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#### MS Thesis colloquium
##### Venue: LH-1, Mathematics Department
The systematic study of determinantal processes began with the work of Macchi (1975), and since then they have appeared in different contexts like random matrix theory (eigenvalues of random matrices), combinatorics (random spanning tree, non-intersecting paths, measures on Young diagrams), and physics (fermions). A particularly interesting and well-known example of a discrete determinantal process is the Uniform spanning tree (UST) on finite graphs. We shall describe UST on complete graphs and complete bipartite graphs—in these cases it is possible to make explicit computations that yield some special cases of Aldous’ result on CRT.
The defining property of a determinantal process is that its joint intensities are given by determinants, which makes it amenable to explicit computations. One can associate a determinantal process with a finite rank projection on a Hilbert space of functions on a given set. Let $H$ and $K$ are two finite dimensional subspaces of a Hilbert space, and let $P$ and $Q$ be determinantal processes associated with projections on $H$ and $K$, respectively. Lyons showed that if $H$ is contained in $K$ then $P$ is stochastically dominated by $Q$. We will give a simpler proof of Lyons’ result which avoids the machinery of exterior algebra used in the original proof of Lyons and also provides a unified approach of proving the result in discrete as well as continuous case.
As an application of the above result, we will obtain the stochastic domination between the largest eigenvalue of Wishart matrix ensembles $W(N,N)$ and $W(N-1,N+1)$. It is well known that the largest eigenvalue of Wishart ensemble $W(M,N)$ has the same distribution as the directed last-passage time $G(M,N)$ on $Z^2$ with i.i.d. exponential weights. We, thus, obtain stochastic domination between $G(N,N)$ and $G(N-1,N+1)$ - answering a question of Riddhipratim Basu. Similar connections are also known between the largest eigenvalue of Meixner ensemble and directed last-passage time on $Z^2$ with i.i.d. geometric weights. We prove a stochastic domination result which combined with the Lyons’ result gives the stochastic domination between Meixner ensemble $M(N,N)$ and $M(N-1,N+1)$.
Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ; E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 17 Aug 2019
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2019-08-18 11:50:46
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https://www.physicsforums.com/threads/kirchhoffs-law-problem-resistors-in-series-and-parallel-and-a-switch.868279/
|
# Kirchhoff's Law Problem -- Resistors in Series and Parallel and a Switch
1. Apr 22, 2016
### DevonZA
1. The problem statement, all variables and given/known data
2. Relevant equations
Rparellel=1/R1+1/R2...
IR1=R2/R1+R2 x I
3. The attempt at a solution
I am unsure of how to answer d) and e) using KVL because I count 4 junctions?
Where should I start?
2. Apr 22, 2016
### BvU
Check your first equation ! It's not for Rpalrallel but for 1/=Rpalrallel !
And the 20 and 25 Ohm resistors are NOT parallel to e.g. the 12 Ohm !
3. Apr 22, 2016
### DevonZA
Yes sorry that was a typo.
The total resistance for the parallel resistors still equals 15.18ohms but yes I should have done them separately:
(1/12+1/10+1/16)^-1 = 4.07ohms
(1/20+1/25)^-1= 11.11ohms
Do my values look correct?
4. Apr 22, 2016
### BvU
I only know $${1\over 12} + {1\over 10}+ {1\over 16}+ {1\over 20}+ {1\over 25} \ne 15.82$$
5. Apr 22, 2016
### DevonZA
(1/12+1/10+1/16+1/20+1/25)^-1=15.18 ohms
Yes values are answers in the book
6. Apr 22, 2016
### BvU
Then why ask if the values are correct ?
7. Apr 22, 2016
### BvU
No. $${1\over 12} + {1\over 10}+ {1\over 16}+ {1\over 20}+ {1\over 25} = {1\over 2.98}$$However,$${1\over {1\over 12} + {1\over 10}+ {1\over 16} } + {1 \over {1\over 20}+ {1\over 25} } = 4.07 + 11.11 = 15.18$$
Last edited: Apr 22, 2016
8. Apr 22, 2016
### DevonZA
9. Apr 22, 2016
### SammyS
Staff Emeritus
What sort of device is R7 that its resistance changes depending on whether the switch is opened or closed
10. Apr 22, 2016
### BvU
The parts d and e are a bit weird, because basically you use KVL to calculate R7 in a and b.
And it wouldn't be proof, just showing.
I should think they want you to show that 1.2 A * (4.07 + 60 + 11.11 + 24.82) Ohm = 120 V
11. Apr 22, 2016
### CWatters
+1
Although I prefer to make the voltages around the loop sum to zero.
12. Apr 22, 2016
### SammyS
Staff Emeritus
@DevonZA ,
Are you saying that this problem is from a textbook?
What book? Who is the author ?
Last edited: Apr 22, 2016
13. Apr 23, 2016
### DevonZA
Can you show this?
14. Apr 23, 2016
### DevonZA
Hi Sammy this is from our varsity study guide. The author is most likely the lecturer who doesn't respond to me hence why I must ask these questions on here.
15. Apr 23, 2016
### CWatters
With reference to the equivalent circuit below... KVL says that going around a loop the voltages sum to zero.
If I arbitrarily choose to start at the -ve terminal of the battery and go around clockwise we have to prove that..
+120 + (-V1) + (-V2) + (-V3) + (-V4) = 0
V1 = 1.2 * 4.07 = 4.884V
V2 = 1.2 * 60 = 72V
V3 = 1.2 * 11.11 = 13.332V
V4 = 1.2 * 24.82 = 29.784V
Substitute..
+120 - 4.884 - 72 -13.332 - 29.784 = 0
16. Apr 23, 2016
### DevonZA
Thank you for the nice clear explanation CWatters
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2018-03-23 13:28:36
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https://rpg.stackexchange.com/questions/105935/does-disciple-of-life-work-with-spells-cast-from-a-staff-of-healing
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# Does Disciple of Life work with spells cast from a Staff of Healing? [duplicate]
If I have a Cleric who is Life domain and a Staff of Healing does Disciple of Life work with the spells cast from the Staff?
Staff of Healing: While holding it, you can use an action to expend 1 or more of its charges to cast one of the following spells from it, using your spell save DC and spellcasting ability modifier: [spells follow]
Disciple of Life: Also starting at 1st level, your healing spells are more effective. Whenever you use a spell of 1st level or higher to restore hit points to a creature, the creature regains additional hit points equal to 2 + the spell’s level
## marked as duplicate by Miniman dnd-5e StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Aug 26 '17 at 2:10
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
## 1 Answer
Yes, Disciple of Life applies to spells from the staff.
The staff ultimately allows you to cast the spells from it ("you can use an action to [...] cast one of the following spells from it"), and Disciple of Life applies to all spells you cast; it makes no restrictions along the lines of 'Whenever you use a cleric spell of 1st level or higher' that would prevent it from applying to the staff's spells.
• '...a Cleric spell...' but would it prevent that though? If Healing Word was on the Cleric Spell List and prepared (or not) isnt Healing Word cast by anything the Cleric is wielding or wearing a cleric spell for all intents and purposes? – Airatome Aug 26 '17 at 1:44
• @Airatome I don't believe so; my understanding is that something is considered a {class} spell when you have it known and/or prepared from one of your classes and when you cast it through a class feature. Even if you had it prepared that day, an instance cast from the staff wouldn't be a {class} spell since it's not a class feature that you're casting it through. That said, I'd be behind asking the question "What makes something considered to be a 'class' spell?", as that's just my understanding. – CTWind Aug 26 '17 at 1:50
• You may be right...I think this definitely warrants a new question. Feel free to ask it as I can not accurately typw a question on my phone :( . Im curious the answer(s) – Airatome Aug 26 '17 at 1:59
• @Airatome I've placed the question here. – CTWind Aug 30 '17 at 18:55
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2019-07-16 16:40:06
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https://getrevising.co.uk/revision-tests/distance-time-graphs
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# Distance-Time Graphs, Velocity and Acceleration
HideShow resource information
M E P Q B L X P N C O R G S G O X V W E K A J I H O Q A Q C S Q R A G G L C Q J W F I W S M B Y H Y A E R D V N P M W T Q N S K V S P N D W S B U A U B C F P E S Y T O X E I K S B J T K E W B H R P T J T R K E F H W F S R E K B Y I E I O Q L A N B O C K I H R H I K C G C G S H T T D L F D D X R L T P B R O O I F E J B L E B A G Q K J W P D L N T T M G B V D U E B Q N M V A X E D D B I X W F T Y S M H D E D X D V S N E T H Y F Q P L N Y J S P E E D V A U O B J M S J T R A C C E L E R A T I O N C U N K T A D C H W P A H M J G U I W Q L E T C P T J G V M V F S L O W E R D M B F J I G X K M H R V H A N L U O H E O W X B Q D P N G X S F J O M C
### Clues
• ? = change of velocity X time taken (12)
• Acceleration is the change of ? per second (8)
• An object moving at a constant speed travels at the same ? every second (8)
• Deceleration is the change of velocity per second when an object becomes ? (6)
• Measured in metre/seconds (5)
• Speed in metres/second = Distance travelled in meters divided by ? (4, 5)
• The steeper the ? of the line on a distance-time graph of a moving object, the greater the object's speed is (8)
• Velocity is speed in a given ? (9)
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2016-12-09 13:35:17
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https://testbook.com/question-answer/the-base-of-a-right-prism-is-a-regular-hexagon-tha--609a3c1fcef1e841511e3284
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# The base of a right prism is a regular hexagon that has side of 4 cm then, find the volume of the prism if its height is 12√3 cm.
1. 480 cm3
2. 600 cm3
3. 864 cm3
4. 840 cm3
## Answer (Detailed Solution Below)
Option 3 : 864 cm3
## Detailed Solution
Given:
The base of a right prism is a regular hexagon of side 4 cm
Its height is 12√3 cm
Formula used:
The volume of the prism = Area of base × Height
Area of regular hexagon = 6 × (√3/4)a2
Calculations:
According to the question, we have
The area of a regular Hexagon is
Area of regular Hexagon = 6 × (√3/4)a2
⇒ Area = 6 × (√3/4) × (4)2
⇒ Area = 6 × √3 × 4
⇒ Area = 24√3 cm2
Hence, The volume of the prism is
The volume of the prism = Area of base × Height
⇒ Volume = 24√3 × 12√3
⇒ Volume = 24 × 12 × 3
⇒ Volume = 864 cm3
∴ The Volume of the prism is 864 cm3.
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2021-10-26 21:49:32
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https://www.semanticscholar.org/paper/Global-bifurcation-for-corotating-and-vortex-pairs-Garc'ia-Haziot/d1574d32f71e220daf0a7411096c68ec749ca07f
|
• Corpus ID: 248377088
# Global bifurcation for corotating and counter-rotating vortex pairs
@inproceedings{Garcia2022GlobalBF,
title={Global bifurcation for corotating and counter-rotating vortex pairs},
author={Claudia Garc'ia and Susanna V. Haziot},
year={2022}
}
• Published 24 April 2022
• Mathematics
Abstract. The existence of a local curve of corotating and counter-rotating vortex pairs was proven by Hmidi and Mateu in [HM17] via a desingularization of a pair of point vortices. In this paper, we construct a global continuation of these local curves. That is, we consider solutions which are more than a mere perturbation of a trivial solution. Indeed, while the local analysis relies on the study of the linear equation at the trivial solution, the global analysis requires on a deeper…
2 Citations
Multipole vortex patch equilibria for active scalar equations
• Mathematics
• 2021
. We study how a general steady configuration of finitely-many point vortices, with Newtonian interaction or generalized surface quasi-geostrophic interactions, can be desingularized into a steady
Time periodic doubly connected solutions for the 3D quasi-geostrophic model
• Mathematics
• 2022
In this paper, we construct time periodic doubly connected solutions for the 3D quasi-geostrophic model in the patch setting. More specifically, we prove the existence of nontrivial m-fold doubly
## References
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Global Bifurcation of Rotating Vortex Patches
• Mathematics
Communications on Pure and Applied Mathematics
• 2019
We rigorously construct continuous curves of rotating vortex patch solutions to the two‐dimensional Euler equations. The curves are large in that, as the parameter tends to infinity, the minimum
Existence of Corotating and Counter-Rotating Vortex Pairs for Active Scalar Equations
• Mathematics, Physics
• 2016
In this paper, we study the existence of corotating and counter-rotating pairs of simply connected patches for Euler equations and the $${{\rm (SQG)}_{\alpha}}$$(SQG)α equations with {\alpha \in
Steady asymmetric vortex pairs for Euler equations
• Mathematics
Discrete & Continuous Dynamical Systems - A
• 2021
In this paper, we study the existence of co-rotating and counter-rotating unequal-sized pairs of simply connected patches for Euler equations. In particular, we prove the existence of curves of
Boundary effects on the emergence of quasi-periodic solutions for Euler equations
• Mathematics
• 2022
In this paper, we highlight the importance of the boundary effects on the construction of quasiperiodic vortex patches solutions close to Rankine vortices and whose existence is not known in the
A general theory for two-dimensional vortex interactions
A general theory for two-dimensional vortex interactions is developed from the observation that, under slowly changing external influences, an individual vortex evolves through a series of
A family of steady, translating vortex pairs with distributed vorticity
An efficient relaxation method is developed for computing the properties of a family of vortex pairs with distributed vorticity, propagating without change of shape through a homogeneous, inviscid
On the trivial solutions for the rotating patch model
In this paper, we study the clockwise simply connected rotating patches for Euler equations. By using the moving plane method, we prove that Rankine vortices are the only solutions to this problem in
Equilibrium shapes of a pair of equal uniform vortices
• Physics
• 1980
The shapes and properties of two equal corotating uniform vortices, rotating steadily about each other, are calculated. An integrodifferential equation for the bounding contour is solved numerically,
|
2022-07-04 06:34:25
|
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|
https://damitr.org/tag/golem/
|
# Explosives or Not
We have earlier seen some quotes from the book The Golem: What You Should Know About Science. There are two companion volumes to this book The Golem Unleashed: What You Should Know about Technology and Dr. Golem: How to think about Medicine. These series of books by Harry Collins and Trevor Pinch provide us with examples from these fields which most of the times are ‘uncontested’. For example in the first volume they discuss about the famous 1920 experimental confirmation of Einstein’s predictions in general relativity by Eddington. This experiment is told as a matter-of-fact anecdote in physics, where petty borders of nationalism could not stop physics and physicists. But in the book, as they show inspite of scanty or almost no positive evidence, Eddington “Concluded” that the predictions were true. This they term “experimenters’ regress”.
The experimenter’s regress occurs when scientists cannot decide what the outcome of an experiment should be and therefore cannot use the outcome as a criterion of whether the experiment worked or not.
The Golem Unleashed pp. 106
In The Golem Unleashed they present us with many examples of this from field of technology. One of the examples is from the Challenger accident which Feynman made famous by courtroom drama. In this case they call the “experimenter’s regress” as “technologist’s regress”.
Recently I read (all further quotes from the same link)an episode in India which would fit in very with these episodes. This is regarding baggage scanning machines installed at Indian airports. They were brought at 2 crore rupees per unit in 2010. But in August 2011 they failed the tests on tasks they were supposed to do.
The scanners are called in-line baggage inspection systems as they scan bags that go into the cargo hold of the aircraft after passengers check in and hand over their luggage to the airline. They use x-ray imaging and “automatic intelligence” to verify the contents of bags and determine whether they include explosives.
Now one would think that this would be as easy as it gets. Either the scanner detects whether the explosives are present in the baggage or they do not. But it is not as simple as it seems so. Now when the tests were done, the testers found the machines failed.
During the tests, security sources said that a technological specification committee of officials from the IB, RAW, SPG, NSG, BCAS and the civil aviation ministry passed bags containing 500 gm of six kinds of explosives, including PETN and ammonium nitrate, as well as IEDs through these systems. The scanners did not flag any of these bags as suspicious, the sources said.
So after this “failure” the companies which supplied these machines were asked to improve upon the machines or to share the software to recalibrate them. But the companies and interestingly Airport Authortiy of India AAI said that the testing methods were at fault. Now the explosives were passed and the machines did not detect them, then how can companies say that the testing methods were not working?
The machines work on the so called 70:30 principle.
“Though it works on a 70:30 principle, if there is an explosive in the 70 per cent, it will throw up the image of each and every bag that has dangerous substances. We would like to emphasise that the systems supplied and installed by our company at Indian airports are of state-of-the-art technology and are fully compliant with current standards.”
The 70:30 principle refers to the “automatic intelligence” used by Smiths Detection machines to clear 70 per cent of the baggage and reject the rest, according to the Airports Authority of India (AAI). “The machines reject 30 per cent of the baggage, the images of which are then sent to the screener. These systems have automatic intelligence capability and have been tested against a wide range of substances considered dangerous for aircraft. The details and specifications are never disclosed, or else terrorists would understand the software,”
But if anyway machines are doing the job, why not do it 100%? And the funny thing is that they are not sharing the software, which is the main agenda of the proprietary software companies. This is a case where people realize that they are just Users of the software under question. This argument that “or else terrorists would understand the software” does not hold. They don’t need to if the machine is going to reject a whole lot of bags And in anyway if there are bus/holes in the software, a thousand eyes repair them much faster than a few. And this is The companies further say that
“The technology or physics is that x-ray based system can’t detect explosives, it is only approximate detection of dangerous substances,”
Why is the AAI siding (they are rather defending the companies) with the companies is something worth pondering.
AAI people say “The problem could be due to the sheer ignorance of officers who lacked the skills to test for explosives,”
Still with no unanimity in the testing results, the case truly presents us with a “technologist’s regress.”
# The Golem at Large
Recently I completed reading of the second book in the Golem series, the complete being The Golem at Large: What you should know about technology by Harry Collins and Trevor Pinch. The book discusses cases from technology field in which there is a ‘regress’, in even expert people are not able to decide objectively what to make out of results of experiment, which at first sight seem to be so objective.
Some of the examples that they choose are well known, some are not. For example the much famed demonstration by Richard Feynman on O-Rings is brought out from its almost cult status. The demonstration by Feynman when looked at with all the background seems to be very naive. Similarly many other examples de-mythify different examples from different technologies.
Some of the quotes that I have liked are as under.
+ 4 It would, of course, be foolish to suggest that technology and
science are identical. Typically, technologies are more directly
influence than are sciences.
+ 6 But disputes are representative and illustrative of the roots of
knowledge; they show us knowledge in the making.
+ 10 It would be wrong to draw any conclusions for science and
technology in general from wartime statements; wartime claims
about the success of the missile reflect the demands of war rather
than the demands of truth.
+ 28 As always, if only we could fight the last war again we would
do it so much better.
+ 28 Just as military men dream of fighting a war in which there is
never any shortage of information or supplies, while the enemy
always does the expected, so experts have their dreams of
scientific measurement in which signal is signal and noise follows
the model given in the statistical textbooks. As the generals
dream of man- oeuvres, so the experts dream of the mythical model
of science.
+ 28 Even when we have unlimited access to laboratory conditions, the
process of measurement does not fit the dream; that was the point
of our earlier book ¡V the first volume of the Golem series.
+ 32 Skimp, save and cut corners, give too much decision-making
power to reckless managers and uncaring bureaucrats, ignore the
pleas of your best scientists and engineers, and you will be
punished.
+ 38 Whether two things are similar or different, Wittgenstein
noted, always involves a human judgement.
+ 40 The correct’ outcome can only be achieved if the experiments or
tests in question have been performed competently, but a competent
experiment can only be judged by its outcome.
+ 62 The treatment of the controversial aspects must be different to
the uncontroversial aspects. The same is true of what we loosely
refer to as experiments: one does not do experiments on the
uncontroversial, one engages in demonstrations.
+ 64 In an experiment, that would be cheating, but in a display, no
one would complain. A demonstration lies somewhere in the middle
of this scale. Classroom demonstrations, the first bits of science
we see, are a good case. Teachers often know that this or that
experiment’ will only work if the conditions are just so’, but
this information is not vouchsafed to the students.
+ 64 A demonstration or display is something that is properly set
before the lay public precisely because its appearance is meant
to convey an unambiguous message to the senses, the message that
we are told to take from it. But the significance of an experiment
can be assessed only be experts.
+ 71 Anything seen on television is controlled by the lens, the
director, the editor and the commentators. It is they who control
the conclusions that seem to follow from the direct evidence of
the senses’.
+ 74 The public were not served well, not because they necessarily
evidence needed to draw conclusions with the proper degree of
provisionality. There is no short cut through the contested
terrain which the golem must negotiate.
+ 77 A vast industry supported by national governments makes sure it
understands how oil is found, where it is found and who has the
rights to find it.
+ 82 In some ways it is easier to delve into the first few
nanoseconds of the universe than to reconstruct something buried
deep in the core of the earth.
+ 86 This is the experimenter’s regress’. If you believe that
microbiological activity exists at great depths then this is
evidence that a competently performed experiment has been carried
out. If you believe that microbiological activity is impossible or
extremely unlikely then the evidence of biological activity is
evidence for doubting the experiment. Experiment alone cannot
settle the matter.
+ 91 In short, Gold’s non-biological theory and its assessment are
intertwined with the politics and commerce of oil
exploration. There is no neutral place where a pure’ assessment
of the validity of his claims can be made.
+ 96 With several hundred equations to play with, this is an area
where theory’ and guesswork’ are not as far apart as
conventional ideas about science would encourage us to think.
+ 102 I think there are really two different approaches. One is to
say that this is a branch of science and that everything must be
based on objective criteria which people can understand. The other
is to say that is just too inflexible, and that there’s something
called judgement – intuition if you like – which has its place in
the sciences and that it’s the people who are intuitive who are
successful.
+ 104 It is also possible to argue that modellers who did not suffer from big
mistakes were lucky while some others were unlucky to have been wrong.
+ 106 Even if you believe that large errors are bound to prove you
wrong, you may still argue about the meaning of large’ and you
may still think that the difference between accuracy and
inaccuracy was not clever economics but luck. Finally, you may
always say that the economy changed radically.
+ 106 … it was not the model but the economy that was wrong.
+ 107 The experimenter’s regress occurs when scientists cannot
decide what the outcome of an experiment should be and therefore
cannot use the outcome as a criterion of whether the experiment
worked or not.
+ 107 Oh absolutely, that’s why it’s absolutely pointless to publish
these forecast error bands because they are extremely
large. . . . I’m all for publishing full and frank statements but
you see the difficulty [with] these standards errors is that
they’re huge.
+ … In fact, we could have done this at the National Institute in
the mid 70s, but we suppressed it on the grounds that the standard
errors were so large, that it would have been difficult for
non-specialists, you know people using the models, using the
forecasts, to appreciate. It would have discredited them.
+ 108 Science is often used as a way of avoiding responsibility;
some kinds of fascism can be seen as the substitution of
calculation for moral responsibility.
+ 110 That is, it selected those who were . . . willing to
subordinate their education to their careers’.
+ 111 The economists who build the models deserve credibility, but
their models do not; one should not use the same criteria to judge
expert advice as one uses to judge the coherence of a model.
+ 124 Flipping to and fro between science being all about certainty
and science being a political conspiracy is an undesirable state
of affairs.
+ 149 In effect, a group of lay people had managed to reframe the
scientific conduct of clinical research: they changed the way it
was conceived and practised.
+ 151 Feynman gives the impression that doubts can always be simply
resolved by a scientist who is smart enough.
+ 151 The danger is always that enchantment is the precursor of
disenchantment.
+ 153 Golem science and technology is a body of expertise, and
expertise must be respected. But we should not give unconditional
respect before we understand just what the expertise comprises and
whether it is relevant. To give unconditional respect is to make
science and technology a fetish.
|
2023-01-30 12:06:40
|
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|
https://mathsmartinthomas.wordpress.com/category/number-theory-combinatorics/
|
# STEP I-2009-1
A proper factor of an integer N is a positive integer, not 1 or N, that divides N.
Show that $3^2 \times 5^3$ has exactly 10 proper factors. Determine how many other integers of the form $3^m \times 5^n$ (where m and n are integers) have exactly 10 proper factors.
Let N be the smallest positive integer that has exactly 426 proper factors. Determine N, giving your answer in terms of its prime factors. Continue reading
# STEP III/2015/5
In the following argument to show that √2 is irrational, give proofs appropriate for steps 3, 5 and 6.
1. Assume that √2 is rational.
2. Define the set S to be the set of positive integers with the following property:
n is in S if and only if n√2 is an integer.
3. Show that the set S contains at least one positive integer.
4. Define the integer k to be the smallest positive integer in S
5. Show that (√2-1)k is in S.
6. Show that steps 4 and 5 are contradictory and hence that √2 is irrational.
Prove that $2^\frac{1}{3}$ is rational if and only if $2^\frac{2}{3}$ is rational.
Use an argument similar to that of part (i) to prove that $2^\frac{1}{3}$ and $2^\frac{2}{3}$ are irrational.
# STEP II/2005/2
For any positive integer N, the function f(N) is defined by
$f(N) = N\Big(1-\frac1{p_1}\Big)\Big(1-\frac1{p_2}\Big) \cdots\Big(1-\frac1{p_k}\Big)$
where $p_1, p_2, \dots , p_k$ are the only prime numbers that are factors of N.
Thus $f(80)=80(1-\frac{1}{2})(1-\frac{1}{5})\,$
(1) Evaluate f(12) and f(180)
(2) Show that f(N) is an integer for all N
(3) Prove, or disprove by means of a counterexample, each of the following:
(a) $f(m) f(n) = f(mn)$
(b) $f(p) f(q) = f(pq)$ if p and q are distinct prime numbers;
(c) $f(p) f(q) = f(pq)$ only if p and q are distinct prime numbers.
(4) Find a positive integer m and a prime number p such that
$f(p^m) = 146410$
# STEP II/2018/6
(i) Find all pairs of positive integers (n,p), where p is a prime number, that satisfy
$n!+ 5 = p$
(ii) In this part of the question you may use the following two theorems:
For $n\ge 7, 1! \times 3! \times \cdots \times (2n-1)! > (4n)!\,$
For every positive integer n, there is a prime number between 2n and 4n.
Find all pairs of positive integers (n,m) that satisfy
$1! \times 3! \times \cdots \times (2n-1)! = m!$
STEP I/2006/1 – Find the integer, n, that satisfies $n^{2} < 33\,127 < (n + 1)^2$. Find also a small integer m such that $(n + m)^2 - 33\,127$ is a perfect square. Hence express 33,127 in the form pq, where p and q are integers greater than 1.
By considering the possible factorisations of 33,127, show that there are exactly two values of m for which $(n + m)^2 - 33\,127$ is a perfect square, and find the other value.
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2019-10-13 20:37:59
|
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|
https://en.m.wikipedia.org/wiki/Higher_order_function
|
# Higher-order function
(Redirected from Higher order function)
In mathematics and computer science, a higher-order function is a function that does at least one of the following:
All other functions are first-order functions. In mathematics higher-order functions are also termed operators or functionals. The differential operator in calculus is a common example, since it maps a function to its derivative, also a function. Higher-order functions should not be confused with other uses of the word "functor" throughout mathematics, see Functor (disambiguation).
In the untyped lambda calculus, all functions are higher-order; in a typed lambda calculus, from which most functional programming languages are derived, higher-order functions that take one function as argument are values with types of the form ${\displaystyle (\tau _{1}\to \tau _{2})\to \tau _{3}}$.
## General examples
• map function, found in many functional programming languages, is one example of a higher-order function. It takes as arguments a function f and a list of elements, and as the result, returns a new list with f applied to each element from the list.
• Sorting functions, which take a comparison function as a parameter, allowing the programmer to separate the sorting algorithm from the comparisons of the items being sorted. The C standard function qsort is an example of this.
• fold
• Function composition
• Integration
• Callback
• Tree traversal
## Support in programming languages
### Direct support
The examples are not intended to compare and contrast programming languages, but to serve as examples of higher-order function syntax
In the following examples, the higher-order function twice takes a function, and applies the function to some value twice. If twice has to be applied several times for the same f it preferably should return a function rather than a value. This is in line with the "don't repeat yourself" principle.
#### OCAML
Explicitly
let add3 x = x + 3
let twice f x = f (f x)
print_int (twice add3 7) (* 13 *)
One-Line
print_int ((fun f x -> f (f x)) ((+)3) 7) (* 13 *)
#### APL
twice←{⍺⍺ ⍺⍺ ⍵}
plusthree←{⍵+3}
g←{plusthree twice ⍵}
g 7
13
Or in a tacit manner:
twice←⍣2
plusthree←+∘3
g←plusthree twice
g 7
13
#### J
Explicitly,
twice=. adverb : 'u u y'
plusthree=. verb : 'y + 3'
g=. plusthree twice
g 7
13
or tacitly,
twice=. ^:2
plusthree=. +&3
g=. plusthree twice
g 7
13
or point-free style,
+&3(^:2) 7
13
#### Python
>>> def twice(f):
... def result(a):
... return f(f(a))
... return result
>>> plusthree = lambda x: x+3
>>> g = twice(plusthree)
>>> g(7)
13
#### Clojure
(defn twice [function x]
(function (function x)))
(twice #(+ % 3) 7) ;13
In Clojure, "#" starts a lambda expression, and "%" refers to the next function argument.
#### Scheme
(define (add x y) (+ x y))
(define (f x)
(lambda (y) (+ x y)))
(display ((f 3) 7))
In this Scheme example, the higher-order function (f x) is used to implement currying. It takes a single argument and returns a function. The evaluation of the expression ((f 3) 7) first returns a function after evaluating (f 3). The returned function is (lambda (y) (+ 3 y)). Then, it evaluates the returned function with 7 as the argument, returning 10. This is equivalent to the expression (add 3 7), since (f x) is equivalent to the curried form of (add x y).
#### Erlang
or_else([], _) -> false;
or_else([F | Fs], X) -> or_else(Fs, X, F(X)).
or_else(Fs, X, false) -> or_else(Fs, X);
or_else(Fs, _, {false, Y}) -> or_else(Fs, Y);
or_else(_, _, R) -> R.
or_else([fun erlang:is_integer/1, fun erlang:is_atom/1, fun erlang:is_list/1],3.23).
In this Erlang example, the higher-order function or_else/2 takes a list of functions (Fs) and argument (X). It evaluates the function F with the argument X as argument. If the function F returns false then the next function in Fs will be evaluated. If the function F returns {false,Y} then the next function in Fs with argument Y will be evaluated. If the function F returns R the higher-order function or_else/2 will return R. Note that X, Y, and R can be functions. The example returns false.
#### Elixir
In Elixir, you can mix module definitions and anonymous functions
defmodule Hop do
def twice(f, v) do
f.(f.(v))
end
end
add3 = fn(v) -> 3 + v end
Alternatively, we can also compose using pure anonymous functions.
twice = fn(f, v) -> f.(f.(v)) end
add3 = fn(v) -> 3 + v end
#### JavaScript
const twice = (f, v) => f(f(v));
const add3 = v => v + 3;
#### Go
func twice(f func(int) int, v int) int {
return f(f(v))
}
func main() {
f := func(v int) int {
return v + 3
}
twice(f, 7) // returns 13
}
Notice a function literal can be defined either with an identifier (twice) or anonymously (assigned to variable f). Run full program on Go Playground!
#### Scala
def twice(f:Int=>Int) = f compose f
twice(_+3)(7) // 13
#### Java (1.8+)
Function<Function<Integer, Integer>, Function<Integer, Integer>> twice = f -> f.andThen(f);
twice.apply(x -> x + 3).apply(7); // 13
#### Kotlin
fun <T> twice(f: (T)->T): (T)->T = {f(f(it))}
fun f(x:Int) = x + 3
println(twice(::f)(7)) // 13
#### Lua
local twice = function(f,v)
return f(f(v))
end
local f = function(v)
return v + 3
end
print(twice(f,7)) -- 13
#### Swift
// generic function
func twice<T>(_ v: @escaping (T) -> T) -> (T) -> T {
return { v(v($0)) } } // inferred closure let f = {$0 + 3 }
twice(f)(10) // 16
#### Rust
// Take function f(x), return function f(f(x))
fn twice<A>(function: impl Fn(A) -> A) -> impl Fn(A) -> A
{
move |a| function(function(a))
}
// Return x + 3
fn f(x: i32) -> i32 {
x + 3
}
fn main() {
let g = twice(f);
println!("{}", g(7));
}
#### Ruby
def twice(f, x)
f.call f.call(x)
end
add3 = ->(x) { x + 3 }
#### C++
With generic lambdas provided by C++14:
#include <iostream>
auto twice = [](auto f, int v)
{
return f(f(v));
};
auto f = [](int i)
{
return i + 3;
};
int main()
{
std::cout << twice(f, 7) << std::endl;
}
Or, using std::function in C++11 :
#include <iostream>
#include <functional>
auto twice = [](const std::function<int(int)>& f, int v)
{
return f(f(v));
};
auto f = [](int i)
{
return i + 3;
};
int main()
{
std::cout << twice(f, 7) << std::endl;
}
#### D
import std.stdio : writeln;
alias twice = (f, i) => f(f(i));
alias f = (int i) => i + 3;
void main()
{
writeln(twice(f, 7));
}
#### ColdFusion Markup Language (CFML)
twice = function(f, v) {
return f(f(v));
};
f = function(v) {
return v + 3;
};
writeOutput(twice(f, 7)); // 13
#### PHP
$twice = function($f, $v) { return$f($f($v));
};
$f = function($v) {
return $v + 3; }; echo($twice($f, 7)); // 13 #### R twice <- function(func) { return(function(x) { func(func(x)) }) } f <- function(x) { return(x + 3) } g <- twice(f) > print(g(7)) [1] 13 #### Perl 6 sub twice(Callable:D$c) {
return sub { $c($c($^x)) }; } sub f(Int:D$x) {
return $x + 3; } my$g = twice(&f);
set f {{v} {return [expr $v + 3]}} # result: 13 puts [apply$twice $f 7] Tcl uses apply command to apply an anonymous function (since 8.6). #### XQuery declare function local:twice($f, $x) {$f($f($x))
};
declare function local:f($x) {$x + 3
};
local:twice(local:f#1, 7) (: 13 :)
### XACML
The XACML standard defines higher-order functions in the standard to apply a function to multiple values of attribute bags.
rule allowEntry{
permit
condition anyOfAny(function[stringEqual], citizenships, allowedCitizenships)
}
The list of higher-order functions is can be found here.
### Alternatives
#### Function pointers
Function pointers in languages such as C and C++ allow programmers to pass around references to functions. The following C code computes an approximation of the integral of an arbitrary function:
#include <stdio.h>
double square(double x)
{
return x * x;
}
double cube(double x)
{
return x * x * x;
}
/* Compute the integral of f() within the interval [a,b] */
double integral(double f(double x), double a, double b, int n)
{
int i;
double sum = 0;
double dt = (b - a) / n;
for (i = 0; i < n; ++i) {
sum += f(a + (i + 0.5) * dt);
}
return sum * dt;
}
int main()
{
printf("%g\n", integral(square, 0, 1, 100));
printf("%g\n", integral(cube, 0, 1, 100));
return 0;
}
The qsort function from the C standard library uses a function pointer to emulate the behavior of a higher-order function.
#### Macros
Macros can also be used to achieve some of the effects of higher order functions. However, macros cannot easily avoid the problem of variable capture; they may also result in large amounts of duplicated code, which can be more difficult for a compiler to optimize. Macros are generally not strongly typed, although they may produce strongly typed code.
#### Dynamic code evaluation
In other imperative programming languages, it is possible to achieve some of the same algorithmic results as are obtained via higher-order functions by dynamically executing code (sometimes called Eval or Execute operations) in the scope of evaluation. There can be significant drawbacks to this approach:
• The argument code to be executed is usually not statically typed; these languages generally rely on dynamic typing to determine the well-formedness and safety of the code to be executed.
• The argument is usually provided as a string, the value of which may not be known until run-time. This string must either be compiled during program execution (using just-in-time compilation) or evaluated by interpretation, causing some added overhead at run-time, and usually generating less efficient code.
#### Objects
In object-oriented programming languages that do not support higher-order functions, objects can be an effective substitute. An object's methods act in essence like functions, and a method may accept objects as parameters and produce objects as return values. Objects often carry added run-time overhead compared to pure functions, however, and added boilerplate code for defining and instantiating an object and its method(s). Languages that permit stack-based (versus heap-based) objects or structs can provide more flexibility with this method.
An example of using a simple stack based record in Free Pascal with a function that returns a function:
program example;
type
int = integer;
Txy = record x, y: int; end;
Tf = function (xy: Txy): int;
function f(xy: Txy): int;
begin
Result := xy.y + xy.x;
end;
function g(func: Tf): Tf;
begin
result := func;
end;
var
a: Tf;
xy: Txy = (x: 3; y: 7);
begin
a := g(@f); // return a function to "a"
writeln(a(xy)); // prints 10
end.
The function a() takes a Txy record as input and returns the integer value of the sum of the record's x and y fields (3 + 7).
#### Defunctionalization
Defunctionalization can be used to implement higher-order functions in languages that lack first-class functions:
// Defunctionalized function data structures
template<typename T> struct Add { T value; };
template<typename T> struct DivBy { T value; };
template<typename F, typename G> struct Composition { F f; G g; };
// Defunctionalized function application implementations
template<typename F, typename G, typename X>
auto apply(Composition<F, G> f, X arg) {
return apply(f.f, apply(f.g, arg));
}
template<typename T, typename X>
auto apply(Add<T> f, X arg) {
return arg + f.value;
}
template<typename T, typename X>
auto apply(DivBy<T> f, X arg) {
return arg / f.value;
}
// Higher-order compose function
template<typename F, typename G>
Composition<F, G> compose(F f, G g) {
return Composition<F, G> {f, g};
}
int main(int argc, const char* argv[]) {
auto f = compose(DivBy<float>{ 2.0f }, Add<int>{ 5 });
apply(f, 3); // 4.0f
apply(f, 9); // 7.0f
return 0;
}
In this case, different types are used to trigger different functions via function overloading. The overloaded function in this example has the signature auto apply.
|
2019-06-15 23:00:02
|
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|
https://nigerianscholars.com/past-questions/general-paper/question/294831/
|
Home » » A principle that advocates total equality of members of a society is called ____...
# A principle that advocates total equality of members of a society is called ____...
### Question
A principle that advocates total equality of members of a society is called ________.
### Options
A) Communalismm
B) egalitarianism
C) totalitarianism
D) oligarchy
|
2022-01-20 20:42:29
|
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|
http://mathhelpforum.com/calculus/124198-i-don-t-know-what-i-did-wrong-integral.html
|
# Thread: I don't know what I did wrong on this integral..
1. ## I don't know what I did wrong on this integral..
Question is: Evaluate $\int_{25}^{36}\frac{ln(y)}{\sqrt(y)}dy$
Integrating by parts,
$u=ln(y), du=\frac{dy}{y}$
$dv=\frac{1}{\sqrt{y}}dy, v = 2\sqrt{y}$
$=[2\sqrt{y}ln(y) - 2\int\frac{1}{\sqrt{y}}dy]_{25}^{36}$
$= [2\sqrt{y}ln(y) - 4\sqrt{y}]_{25}^{36}$
$=(12ln(36)-24) - (10log25-20)$
$=12log36-10log25-4$
Strangely this is the same answer that Wolfram Alpha gives, but the decimal value is different. When evaluated on my calculator I'm getting 0.696, the correct answer is 6.813 can someone point out my error? Thanks!
2. You changed the "ln" to a "log".
$ln=log_{e}$
$log=log_{10}$
Big difference.
3. you are using log when you should be using ln (natural log) yes the base matters.... =) no mistakes with calculus, just logarithms
4. That's the second time in a row I made that same mistake! Maybe it's my lack of sleep...haha thank you guys!
|
2017-03-30 05:49:30
|
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|
http://www.open3d.org/docs/release/python_api/open3d.geometry.TriangleMesh.html
|
# open3d.geometry.TriangleMesh¶
class open3d.geometry.TriangleMesh
TriangleMesh class. Triangle mesh contains vertices and triangles represented by the indices to the vertices. Optionally, the mesh may also contain triangle normals, vertex normals and vertex colors.
class Type
Enum class for Geometry types.
HalfEdgeTriangleMesh = Type.HalfEdgeTriangleMesh
Image = Type.Image
LineSet = Type.LineSet
PointCloud = Type.PointCloud
RGBDImage = Type.RGBDImage
TetraMesh = Type.TetraMesh
TriangleMesh = Type.TriangleMesh
Unspecified = Type.Unspecified
VoxelGrid = Type.VoxelGrid
__init__(*args, **kwargs)
1. __init__(self: open3d.cpu.pybind.geometry.TriangleMesh) -> None
Default constructor
1. __init__(self: open3d.cpu.pybind.geometry.TriangleMesh, arg0: open3d.cpu.pybind.geometry.TriangleMesh) -> None
Copy constructor
1. __init__(self: open3d.cpu.pybind.geometry.TriangleMesh, vertices: open3d.cpu.pybind.utility.Vector3dVector, triangles: open3d.cpu.pybind.utility.Vector3iVector) -> None
Create a triangle mesh from vertices and triangle indices
clear(self)
Clear all elements in the geometry.
Returns
open3d.cpu.pybind.geometry.Geometry
cluster_connected_triangles(self)
Function that clusters connected triangles, i.e., triangles that are connected via edges are assigned the same cluster index. This function returns an array that contains the cluster index per triangle, a second array contains the number of triangles per cluster, and a third vector contains the surface area per cluster.
Returns
Tuple[open3d.cpu.pybind.utility.IntVector, List[int], open3d.cpu.pybind.utility.DoubleVector]
compute_adjacency_list(self)
Returns
open3d.cpu.pybind.geometry.TriangleMesh
compute_convex_hull(self)
Computes the convex hull of the triangle mesh.
Returns
Tuple[open3d.cpu.pybind.geometry.TriangleMesh, List[int]]
compute_triangle_normals(self, normalized=True)
Function to compute triangle normals, usually called before rendering
Parameters
normalized (bool, optional, default=True) –
Returns
open3d.cpu.pybind.geometry.TriangleMesh
compute_vertex_normals(self, normalized=True)
Function to compute vertex normals, usually called before rendering
Parameters
normalized (bool, optional, default=True) –
Returns
open3d.cpu.pybind.geometry.TriangleMesh
static create_arrow(cylinder_radius=1.0, cone_radius=1.5, cylinder_height=5.0, cone_height=4.0, resolution=20, cylinder_split=4, cone_split=1)
Factory function to create an arrow mesh
Parameters
• cylinder_height (float, optional, default=5.0) – The height of the cylinder. The cylinder is from (0, 0, 0) to (0, 0, cylinder_height)
• cone_height (float, optional, default=4.0) – The height of the cone. The axis of the cone will be from (0, 0, cylinder_height) to (0, 0, cylinder_height + cone_height)
• resolution (int, optional, default=20) – The cone will be split into resolution segments.
• cylinder_split (int, optional, default=4) – The cylinder_height will be split into cylinder_split segments.
• cone_split (int, optional, default=1) – The cone_height will be split into cone_split segments.
Returns
open3d.cpu.pybind.geometry.TriangleMesh
static create_box(width=1.0, height=1.0, depth=1.0)
Factory function to create a box. The left bottom corner on the front will be placed at (0, 0, 0).
Parameters
• width (float, optional, default=1.0) – x-directional length.
• height (float, optional, default=1.0) – y-directional length.
• depth (float, optional, default=1.0) – z-directional length.
Returns
open3d.cpu.pybind.geometry.TriangleMesh
static create_cone(radius=1.0, height=2.0, resolution=20, split=1)
Factory function to create a cone mesh.
Parameters
• height (float, optional, default=2.0) – The height of the cone. The axis of the cone will be from (0, 0, 0) to (0, 0, height).
• resolution (int, optional, default=20) – The circle will be split into resolution segments
• split (int, optional, default=1) – The height will be split into split segments.
Returns
open3d.cpu.pybind.geometry.TriangleMesh
static create_coordinate_frame(size=1.0, origin=array([0.0, 0.0, 0.0]))
Factory function to create a coordinate frame mesh. The coordinate frame will be centered at origin. The x, y, z axis will be rendered as red, green, and blue arrows respectively.
Parameters
• size (float, optional, default=1.0) – The size of the coordinate frame.
• origin (numpy.ndarray[float64[3, 1]], optional, default=array([0., 0., 0.])) – The origin of the cooridnate frame.
Returns
open3d.cpu.pybind.geometry.TriangleMesh
static create_cylinder(radius=1.0, height=2.0, resolution=20, split=4)
Factory function to create a cylinder mesh.
Parameters
• height (float, optional, default=2.0) – The height of the cylinder. The axis of the cylinder will be from (0, 0, -height/2) to (0, 0, height/2).
• resolution (int, optional, default=20) – The circle will be split into resolution segments
• split (int, optional, default=4) – The height will be split into split segments.
Returns
open3d.cpu.pybind.geometry.TriangleMesh
TriangleMesh.create_from_point_cloud_alpha_shape(alpha, alpha, tetra_mesh, pt_map)
Alpha shapes are a generalization of the convex hull. With decreasing alpha value the shape schrinks and creates cavities. See Edelsbrunner and Muecke, “Three-Dimensional Alpha Shapes”, 1994.
Parameters
• alpha (float) – Parameter to control the shape. A very big value will give a shape close to the convex hull.
• alpha – Parameter to control the shape. A very big value will give a shape close to the convex hull.
• tetra_mesh (open3d.geometry.TetraMesh) – If not None, than uses this to construct the alpha shape. Otherwise, TetraMesh is computed from pcd.
• pt_map (List[int]) – Optional map from tetra_mesh vertex indices to pcd points.
Returns
open3d.cpu.pybind.geometry.TriangleMesh
static create_from_point_cloud_ball_pivoting(pcd, radii)
Function that computes a triangle mesh from a oriented PointCloud. This implements the Ball Pivoting algorithm proposed in F. Bernardini et al., “The ball-pivoting algorithm for surface reconstruction”, 1999. The implementation is also based on the algorithms outlined in Digne, “An Analysis and Implementation of a Parallel Ball Pivoting Algorithm”, 2014. The surface reconstruction is done by rolling a ball with a given radius over the point cloud, whenever the ball touches three points a triangle is created.
Parameters
• pcd (open3d.cpu.pybind.geometry.PointCloud) – PointCloud from which the TriangleMesh surface is reconstructed. Has to contain normals.
• radii (open3d.cpu.pybind.utility.DoubleVector) – The radii of the ball that are used for the surface reconstruction.
Returns
open3d.cpu.pybind.geometry.TriangleMesh
static create_from_point_cloud_poisson(pcd, depth=8, width=0, scale=1.1, linear_fit=False, n_threads=- 1)
Function that computes a triangle mesh from a oriented PointCloud pcd. This implements the Screened Poisson Reconstruction proposed in Kazhdan and Hoppe, “Screened Poisson Surface Reconstruction”, 2013. This function uses the original implementation by Kazhdan. See https://github.com/mkazhdan/PoissonRecon
Parameters
• pcd (open3d.cpu.pybind.geometry.PointCloud) – PointCloud from which the TriangleMesh surface is reconstructed. Has to contain normals.
• depth (int, optional, default=8) – Maximum depth of the tree that will be used for surface reconstruction. Running at depth d corresponds to solving on a grid whose resolution is no larger than 2^d x 2^d x 2^d. Note that since the reconstructor adapts the octree to the sampling density, the specified reconstruction depth is only an upper bound.
• width (int, optional, default=0) – Specifies the target width of the finest level octree cells. This parameter is ignored if depth is specified
• scale (float, optional, default=1.1) – Specifies the ratio between the diameter of the cube used for reconstruction and the diameter of the samples’ bounding cube.
• linear_fit (bool, optional, default=False) – If true, the reconstructor will use linear interpolation to estimate the positions of iso-vertices.
• n_threads (int, optional, default=-1) – Number of threads used for reconstruction. Set to -1 to automatically determine it.
Returns
Tuple[open3d.cpu.pybind.geometry.TriangleMesh, open3d.cpu.pybind.utility.DoubleVector]
static create_icosahedron(radius=1.0)
Factory function to create a icosahedron. The centroid of the mesh will be placed at (0, 0, 0) and the vertices have a distance of radius to the center.
Parameters
radius (float, optional, default=1.0) – Distance from centroid to mesh vetices.
Returns
open3d.cpu.pybind.geometry.TriangleMesh
static create_moebius(length_split=70, width_split=15, twists=1, raidus=1, flatness=1, width=1, scale=1)
Factory function to create a Moebius strip.
Parameters
• length_split (int, optional, default=70) – The number of segments along the Moebius strip.
• width_split (int, optional, default=15) – The number of segments along the width of the Moebius strip.
• twists (int, optional, default=1) – Number of twists of the Moebius strip.
• raidus (float, optional, default=1) –
• flatness (float, optional, default=1) – Controls the flatness/height of the Moebius strip.
• width (float, optional, default=1) – Width of the Moebius strip.
• scale (float, optional, default=1) – Scale the complete Moebius strip.
Returns
open3d.cpu.pybind.geometry.TriangleMesh
static create_octahedron(radius=1.0)
Factory function to create a octahedron. The centroid of the mesh will be placed at (0, 0, 0) and the vertices have a distance of radius to the center.
Parameters
radius (float, optional, default=1.0) – Distance from centroid to mesh vetices.
Returns
open3d.cpu.pybind.geometry.TriangleMesh
static create_sphere(radius=1.0, resolution=20)
Factory function to create a sphere mesh centered at (0, 0, 0).
Parameters
• resolution (int, optional, default=20) – The resolution of the sphere. The longitues will be split into resolution segments (i.e. there are resolution + 1 latitude lines including the north and south pole). The latitudes will be split into 2 * resolution segments (i.e. there are 2 * resolution longitude lines.)
Returns
open3d.cpu.pybind.geometry.TriangleMesh
static create_tetrahedron(radius=1.0)
Factory function to create a tetrahedron. The centroid of the mesh will be placed at (0, 0, 0) and the vertices have a distance of radius to the center.
Parameters
radius (float, optional, default=1.0) – Distance from centroid to mesh vetices.
Returns
open3d.cpu.pybind.geometry.TriangleMesh
static create_torus(torus_radius=1.0, tube_radius=0.5, radial_resolution=30, tubular_resolution=20)
Factory function to create a torus mesh.
Parameters
• torus_radius (float, optional, default=1.0) – The radius from the center of the torus to the center of the tube.
• tube_radius (float, optional, default=0.5) – The radius of the torus tube.
• radial_resolution (int, optional, default=30) – The number of segments along the radial direction.
• tubular_resolution (int, optional, default=20) – The number of segments along the tubular direction.
Returns
open3d.cpu.pybind.geometry.TriangleMesh
TriangleMesh.crop(bounding_box, bounding_box)
Function to crop input TriangleMesh into output TriangleMesh
Parameters
Returns
open3d.cpu.pybind.geometry.TriangleMesh
deform_as_rigid_as_possible(self, constraint_vertex_indices, constraint_vertex_positions, max_iter, energy=DeformAsRigidAsPossibleEnergy.Spokes, smoothed_alpha=0.01)
This function deforms the mesh using the method by Sorkine and Alexa, ‘As-Rigid-As-Possible Surface Modeling’, 2007
Parameters
• constraint_vertex_indices (open3d.cpu.pybind.utility.IntVector) – Indices of the triangle vertices that should be constrained by the vertex positions in constraint_vertex_positions.
• constraint_vertex_positions (open3d.cpu.pybind.utility.Vector3dVector) – Vertex positions used for the constraints.
• max_iter (int) – Maximum number of iterations to minimize energy functional.
• energy (open3d.cpu.pybind.geometry.DeformAsRigidAsPossibleEnergy, optional, default=DeformAsRigidAsPossibleEnergy.Spokes) – Energy model that is minimized in the deformation process
• smoothed_alpha (float, optional, default=0.01) – trade-off parameter for the smoothed energy functional for the regularization term.
Returns
open3d.cpu.pybind.geometry.TriangleMesh
dimension(self)
Returns whether the geometry is 2D or 3D.
Returns
int
euler_poincare_characteristic(self)
Function that computes the Euler-Poincaré characteristic, i.e., V + F - E, where V is the number of vertices, F is the number of triangles, and E is the number of edges.
Returns
int
filter_sharpen(self, number_of_iterations=1, strength=1, filter_scope=FilterScope.All)
Function to sharpen triangle mesh. The output value ($$v_o$$) is the input value ($$v_i$$) plus strength times the input value minus he sum of he adjacent values. $$v_o = v_i x strength (v_i * |N| - \sum_{n \in N} v_n)$$
Parameters
• number_of_iterations (int, optional, default=1) – Number of repetitions of this operation
• strength (float, optional, default=1) –
• filter_scope (open3d.cpu.pybind.geometry.FilterScope, optional, default=FilterScope.All) –
Returns
open3d.cpu.pybind.geometry.TriangleMesh
filter_smooth_laplacian(self, number_of_iterations=1, lambda=0.5, filter_scope=FilterScope.All)
Function to smooth triangle mesh using Laplacian. $$v_o = v_i \cdot \lambda (sum_{n \in N} w_n v_n - v_i)$$, with $$v_i$$ being the input value, $$v_o$$ the output value, $$N$$ is the set of adjacent neighbours, $$w_n$$ is the weighting of the neighbour based on the inverse distance (closer neighbours have higher weight), and lambda is the smoothing parameter.
Parameters
• number_of_iterations (int, optional, default=1) – Number of repetitions of this operation
• lambda (float, optional, default=0.5) – Filter parameter.
• filter_scope (open3d.cpu.pybind.geometry.FilterScope, optional, default=FilterScope.All) –
Returns
open3d.cpu.pybind.geometry.TriangleMesh
filter_smooth_simple(self, number_of_iterations=1, filter_scope=FilterScope.All)
Function to smooth triangle mesh with simple neighbour average. $$v_o = \frac{v_i + \sum_{n \in N} v_n)}{|N| + 1}$$, with $$v_i$$ being the input value, $$v_o$$ the output value, and $$N$$ is the set of adjacent neighbours.
Parameters
• number_of_iterations (int, optional, default=1) – Number of repetitions of this operation
• filter_scope (open3d.cpu.pybind.geometry.FilterScope, optional, default=FilterScope.All) –
Returns
open3d.cpu.pybind.geometry.TriangleMesh
filter_smooth_taubin(self, number_of_iterations=1, lambda=0.5, mu=-0.53, filter_scope=FilterScope.All)
Function to smooth triangle mesh using method of Taubin, “Curve and Surface Smoothing Without Shrinkage”, 1995. Applies in each iteration two times filter_smooth_laplacian, first with filter parameter lambda and second with filter parameter mu as smoothing parameter. This method avoids shrinkage of the triangle mesh.
Parameters
• number_of_iterations (int, optional, default=1) – Number of repetitions of this operation
• lambda (float, optional, default=0.5) – Filter parameter.
• mu (float, optional, default=-0.53) – Filter parameter.
• filter_scope (open3d.cpu.pybind.geometry.FilterScope, optional, default=FilterScope.All) –
Returns
open3d.cpu.pybind.geometry.TriangleMesh
get_axis_aligned_bounding_box(self)
Returns an axis-aligned bounding box of the geometry.
Returns
open3d.geometry.AxisAlignedBoundingBox
get_center(self)
Returns the center of the geometry coordinates.
Returns
numpy.ndarray[float64[3, 1]]
get_geometry_type(self)
Returns one of registered geometry types.
Returns
open3d.geometry.Geometry.GeometryType
get_max_bound(self)
Returns max bounds for geometry coordinates.
Returns
numpy.ndarray[float64[3, 1]]
get_min_bound(self)
Returns min bounds for geometry coordinates.
Returns
numpy.ndarray[float64[3, 1]]
get_non_manifold_edges(self, allow_boundary_edges=True)
Get list of non-manifold edges.
Parameters
allow_boundary_edges (bool, optional, default=True) – If true, than non-manifold edges are defined as edges with more than two adjacent triangles, otherwise each edge that is not adjacent to two triangles is defined as non-manifold.
Returns
open3d.cpu.pybind.utility.Vector2iVector
get_non_manifold_vertices(self)
Returns a list of indices to non-manifold vertices.
Returns
open3d.cpu.pybind.utility.IntVector
get_oriented_bounding_box(self)
Returns an oriented bounding box of the geometry.
Returns
open3d.geometry.OrientedBoundingBox
static get_rotation_matrix_from_axis_angle(rotation: numpy.ndarray[float64[3, 1]]) → numpy.ndarray[float64[3, 3]]
static get_rotation_matrix_from_quaternion(rotation: numpy.ndarray[float64[4, 1]]) → numpy.ndarray[float64[3, 3]]
static get_rotation_matrix_from_xyz(rotation: numpy.ndarray[float64[3, 1]]) → numpy.ndarray[float64[3, 3]]
static get_rotation_matrix_from_xzy(rotation: numpy.ndarray[float64[3, 1]]) → numpy.ndarray[float64[3, 3]]
static get_rotation_matrix_from_yxz(rotation: numpy.ndarray[float64[3, 1]]) → numpy.ndarray[float64[3, 3]]
static get_rotation_matrix_from_yzx(rotation: numpy.ndarray[float64[3, 1]]) → numpy.ndarray[float64[3, 3]]
static get_rotation_matrix_from_zxy(rotation: numpy.ndarray[float64[3, 1]]) → numpy.ndarray[float64[3, 3]]
static get_rotation_matrix_from_zyx(rotation: numpy.ndarray[float64[3, 1]]) → numpy.ndarray[float64[3, 3]]
get_self_intersecting_triangles(self)
Returns a list of indices to triangles that intersect the mesh.
Returns
open3d.cpu.pybind.utility.Vector2iVector
get_surface_area(self: open3d.cpu.pybind.geometry.TriangleMesh) → float
Function that computes the surface area of the mesh, i.e. the sum of the individual triangle surfaces.
get_volume(self: open3d.cpu.pybind.geometry.TriangleMesh) → float
Function that computes the volume of the mesh, under the condition that it is watertight and orientable.
has_adjacency_list(self)
Returns True if the mesh contains adjacency normals.
Returns
bool
has_textures(self)
Returns True if the mesh contains a texture image.
Returns
bool
has_triangle_material_ids(self)
Returns True if the mesh contains material ids.
Returns
bool
has_triangle_normals(self)
Returns True if the mesh contains triangle normals.
Returns
bool
has_triangle_uvs(self)
Returns True if the mesh contains uv coordinates.
Returns
bool
has_triangles(self)
Returns True if the mesh contains triangles.
Returns
bool
has_vertex_colors(self)
Returns True if the mesh contains vertex colors.
Returns
bool
has_vertex_normals(self)
Returns True if the mesh contains vertex normals.
Returns
bool
has_vertices(self)
Returns True if the mesh contains vertices.
Returns
bool
is_edge_manifold(self, allow_boundary_edges=True)
Tests if the triangle mesh is edge manifold.
Parameters
allow_boundary_edges (bool, optional, default=True) – If true, than non-manifold edges are defined as edges with more than two adjacent triangles, otherwise each edge that is not adjacent to two triangles is defined as non-manifold.
Returns
bool
is_empty(self)
Returns True iff the geometry is empty.
Returns
bool
is_intersecting(self, arg0)
Tests if the triangle mesh is intersecting the other triangle mesh.
Parameters
arg0 (open3d.cpu.pybind.geometry.TriangleMesh) –
Returns
bool
is_orientable(self)
Tests if the triangle mesh is orientable.
Returns
bool
is_self_intersecting(self)
Tests if the triangle mesh is self-intersecting.
Returns
bool
is_vertex_manifold(self)
Tests if all vertices of the triangle mesh are manifold.
Returns
bool
is_watertight(self)
Tests if the triangle mesh is watertight.
Returns
bool
merge_close_vertices(self, eps)
Function that will merge close by vertices to a single one. The vertex position, normal and color will be the average of the vertices. The parameter eps defines the maximum distance of close by vertices. This function might help to close triangle soups.
Parameters
eps (float) – Parameter that defines the distance between close vertices.
Returns
open3d.cpu.pybind.geometry.TriangleMesh
normalize_normals(self)
Normalize both triangle normals and vertex normals to length 1.
Returns
open3d.cpu.pybind.geometry.TriangleMesh
orient_triangles(self)
If the mesh is orientable this function orients all triangles such that all normals point towards the same direction.
Returns
bool
paint_uniform_color(self, arg0)
Assigns each vertex in the TriangleMesh the same color.
Parameters
arg0 (numpy.ndarray[float64[3, 1]]) –
Returns
open3d.cpu.pybind.geometry.MeshBase
remove_degenerate_triangles(self)
Function that removes degenerate triangles, i.e., triangles that references a single vertex multiple times in a single triangle. They are usually the product of removing duplicated vertices.
Returns
open3d.cpu.pybind.geometry.TriangleMesh
remove_duplicated_triangles(self)
Function that removes duplicated triangles, i.e., removes triangles that reference the same three vertices, independent of their order.
Returns
open3d.cpu.pybind.geometry.TriangleMesh
remove_duplicated_vertices(self)
Function that removes duplicated verties, i.e., vertices that have identical coordinates.
Returns
open3d.cpu.pybind.geometry.TriangleMesh
remove_non_manifold_edges(self)
Function that removes all non-manifold edges, by successively deleting triangles with the smallest surface area adjacent to the non-manifold edge until the number of adjacent triangles to the edge is <= 2.
Returns
open3d.cpu.pybind.geometry.TriangleMesh
remove_triangles_by_index(self, triangle_indices)
This function removes the triangles with index in triangle_indices. Call remove_unreferenced_vertices to clean up vertices afterwards.
Parameters
triangle_indices (List[int]) – 1D array of triangle indices that should be removed from the TriangleMesh.
Returns
None
remove_triangles_by_mask(self, triangle_mask)
This function removes the triangles where triangle_mask is set to true. Call remove_unreferenced_vertices to clean up vertices afterwards.
Parameters
triangle_mask (List[bool]) – 1D bool array, True values indicate triangles that should be removed.
Returns
None
remove_unreferenced_vertices(self)
This function removes vertices from the triangle mesh that are not referenced in any triangle of the mesh.
Returns
open3d.cpu.pybind.geometry.TriangleMesh
remove_vertices_by_index(self, vertex_indices)
This function removes the vertices with index in vertex_indices. Note that also all triangles associated with the vertices are removed.
Parameters
vertex_indices (List[int]) – 1D array of vertex indices that should be removed from the TriangleMesh.
Returns
None
remove_vertices_by_mask(self, vertex_mask)
This function removes the vertices that are masked in vertex_mask. Note that also all triangles associated with the vertices are removed.
Parameters
vertex_mask (List[bool]) – 1D bool array, True values indicate vertices that should be removed.
Returns
None
TriangleMesh.rotate(R, R, center)
Apply rotation to the geometry coordinates and normals.
Parameters
• R (numpy.ndarray[float64[3, 3]]) – The rotation matrix
• R – The rotation matrix
• center (numpy.ndarray[float64[3, 1]]) – Rotation center used for transformation
Returns
open3d.cpu.pybind.geometry.Geometry3D
sample_points_poisson_disk(self, number_of_points, init_factor=5, pcl=None, use_triangle_normal=False, seed=- 1)
Function to sample points from the mesh, where each point has approximately the same distance to the neighbouring points (blue noise). Method is based on Yuksel, “Sample Elimination for Generating Poisson Disk Sample Sets”, EUROGRAPHICS, 2015.
Parameters
• number_of_points (int) – Number of points that should be sampled.
• init_factor (float, optional, default=5) – Factor for the initial uniformly sampled PointCloud. This init PointCloud is used for sample elimination.
• pcl (open3d.cpu.pybind.geometry.PointCloud, optional, default=None) – Initial PointCloud that is used for sample elimination. If this parameter is provided the init_factor is ignored.
• use_triangle_normal (bool, optional, default=False) – If True assigns the triangle normals instead of the interpolated vertex normals to the returned points. The triangle normals will be computed and added to the mesh if necessary.
• seed (int, optional, default=-1) – Seed value used in the random generator, set to -1 to use a random seed value with each function call.
Returns
open3d.cpu.pybind.geometry.PointCloud
sample_points_uniformly(self, number_of_points=100, use_triangle_normal=False, seed=- 1)
Function to uniformly sample points from the mesh.
Parameters
• number_of_points (int, optional, default=100) – Number of points that should be uniformly sampled.
• use_triangle_normal (bool, optional, default=False) – If True assigns the triangle normals instead of the interpolated vertex normals to the returned points. The triangle normals will be computed and added to the mesh if necessary.
• seed (int, optional, default=-1) – Seed value used in the random generator, set to -1 to use a random seed value with each function call.
Returns
open3d.cpu.pybind.geometry.PointCloud
TriangleMesh.scale(scale, center, scale, center)
Apply scaling to the geometry coordinates.
Parameters
• scale (float) – The scale parameter that is multiplied to the points/vertices of the geometry
• center (numpy.ndarray[float64[3, 1]]) – Scale center used for transformation
• scale – The scale parameter that is multiplied to the points/vertices of the geometry
• center – Scale center used for transformation
Returns
open3d.cpu.pybind.geometry.Geometry3D
select_by_index(self, indices, cleanup=True)
Function to select mesh from input triangle mesh into output triangle mesh. input: The input triangle mesh. indices: Indices of vertices to be selected.
Parameters
• indices (List[int]) – Indices of vertices to be selected.
• cleanup (bool, optional, default=True) – If true calls number of mesh cleanup functions to remove unreferenced vertices and degenerate triangles
Returns
open3d.cpu.pybind.geometry.TriangleMesh
simplify_quadric_decimation(self, target_number_of_triangles, maximum_error=inf, boundary_weight=1.0)
Function to simplify mesh using Quadric Error Metric Decimation by Garland and Heckbert
Parameters
• target_number_of_triangles (int) – The number of triangles that the simplified mesh should have. It is not guaranteed that this number will be reached.
• maximum_error (float, optional, default=inf) – The maximum error where a vertex is allowed to be merged
• boundary_weight (float, optional, default=1.0) – A weight applied to edge vertices used to preserve boundaries
Returns
open3d.cpu.pybind.geometry.TriangleMesh
simplify_vertex_clustering(self, voxel_size, contraction=SimplificationContraction.Average)
Function to simplify mesh using vertex clustering.
Parameters
• voxel_size (float) – The size of the voxel within vertices are pooled.
• contraction (open3d.cpu.pybind.geometry.SimplificationContraction, optional, default=SimplificationContraction.Average) – Method to aggregate vertex information. Average computes a simple average, Quadric minimizes the distance to the adjacent planes.
Returns
open3d.cpu.pybind.geometry.TriangleMesh
subdivide_loop(self, number_of_iterations=1)
Function subdivide mesh using Loop’s algorithm. Loop, “Smooth subdivision surfaces based on triangles”, 1987.
Parameters
number_of_iterations (int, optional, default=1) – Number of iterations. A single iteration splits each triangle into four triangles.
Returns
open3d.cpu.pybind.geometry.TriangleMesh
subdivide_midpoint(self, number_of_iterations=1)
Function subdivide mesh using midpoint algorithm.
Parameters
number_of_iterations (int, optional, default=1) – Number of iterations. A single iteration splits each triangle into four triangles that cover the same surface.
Returns
open3d.cpu.pybind.geometry.TriangleMesh
transform(self, arg0)
Apply transformation (4x4 matrix) to the geometry coordinates.
Parameters
arg0 (numpy.ndarray[float64[4, 4]]) –
Returns
open3d.cpu.pybind.geometry.Geometry3D
translate(self, translation, relative=True)
Apply translation to the geometry coordinates.
Parameters
• translation (numpy.ndarray[float64[3, 1]]) – A 3D vector to transform the geometry
• relative (bool, optional, default=True) – If true, the translation vector is directly added to the geometry coordinates. Otherwise, the center is moved to the translation vector.
Returns
open3d.cpu.pybind.geometry.Geometry3D
HalfEdgeTriangleMesh = Type.HalfEdgeTriangleMesh
Image = Type.Image
LineSet = Type.LineSet
PointCloud = Type.PointCloud
RGBDImage = Type.RGBDImage
TetraMesh = Type.TetraMesh
TriangleMesh = Type.TriangleMesh
Unspecified = Type.Unspecified
VoxelGrid = Type.VoxelGrid
property adjacency_list
The set adjacency_list[i] contains the indices of adjacent vertices of vertex i.
Type
List of Sets
property textures
The texture images.
Type
open3d.geometry.Image
property triangle_material_ids
material index associated with each triangle
Type
int array of shape (num_trianges, 1), use numpy.asarray() to access data
property triangle_normals
Triangle normals.
Type
float64 array of shape (num_triangles, 3), use numpy.asarray() to access data
property triangle_uvs
List of uvs denoted by the index of points forming the triangle.
Type
float64 array of shape (3 * num_triangles, 2), use numpy.asarray() to access data
property triangles
List of triangles denoted by the index of points forming the triangle.
Type
int array of shape (num_triangles, 3), use numpy.asarray() to access data
property vertex_colors
RGB colors of vertices.
Type
float64 array of shape (num_vertices, 3), range [0, 1] , use numpy.asarray() to access data
property vertex_normals
Vertex normals.
Type
float64 array of shape (num_vertices, 3), use numpy.asarray() to access data
property vertices
Vertex coordinates.
Type
float64 array of shape (num_vertices, 3), use numpy.asarray()` to access data
|
2021-01-20 22:20:36
|
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|
https://4coder.handmade.network/forums/t/1728-capslock_-_rebind#9590
|
44 posts
CapsLock - rebind
Is it currently possible to rebind the CapsLock key?
For example I am using Casey's custom configuration and I am trying to toggle the edit/entry mode using the CapsLock key.
Allen Webster
474 posts / 5 projects
Heyo
CapsLock - rebind
There is no way to rebind capslock in the 4coder customization.
Aidan
12 posts
CapsLock - rebind
I assume remapping caps lock is harder to do than any of the other keys as Vim doesn't let you remap it either. When I'm using Vim on Windows I use AutoHotKey to map caps lock to esc with this script.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 classname = "" keystate = "" *Capslock:: WinGetClass, classname, A if (classname = "Vim") { SetCapsLockState, Off Send, {ESC} } else { GetKeyState, keystate, CapsLock, T if (keystate = "D") SetCapsLockState, Off else SetCapsLockState, On return } return
Its a bit of a bulky solution but you should be able to replace
1 if (classname = "Vim")
with
1 if (classname = "4coder-win32-wndclass")
to get it to work with 4coder. I assume this is a feature Allen is planning to add in the future so something like this wouldn't be needed.
44 posts
CapsLock - rebind
Thanks for the tip. I know a lot of people swap the ctrl and capsLock key on emacs, turns out they are mostly using external programs like you mentioned. AutoHotKey was also mentioned for Windows. I am going to install it and test your script out.
Andrew Reece
51 posts / 3 projects
Maker of things
CapsLock - rebind
Edited by Andrew Reece on
I remapped my caps lock key (along with the rest of my keyboard) with the Microsoft Keyboard Layout Creator, which was ok - you can switch back to normal QWERTY/layout with a button shortcut, or localise changes to individual processes/applications. Getting it set up took some faffing about, but it was quite a while ago now and I don't really remember why.
Lifehacker suggests KeyTweak, for whatever that's worth.
|
2023-01-30 14:12:58
|
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|
http://openstudy.com/updates/503679c9e4b04ce0997d31a3
|
## anonymous 3 years ago Is the vector cross product only defined for 3D? Wikipedia formally introduces it as, for vectors $$\vec a$$ and $$\vec b$$,$\vec a\times \vec b=(||\vec a|| ||\vec b|| sin \Theta )\vec n$Where $$\vec n$$ is the unit normal vector. It also mentions that n is vector normal to the plane made by a and b, implying that a and b are 3D vectors. Wikipedia mentions something about a 7D cross product, but I'm not going to pretend I understand that. However, working off a definition I more or less invented (so I cannot tell how accurate it is), I can theorize a "cross product" that spans over any number
1. anonymous
(cont.) of dimensions: Let the cross product operator be C, such that when C is applied to a set of vectors S, we get the result v, which is orthogonal (as defined by cross product) to each vector in S. Assuming I have the right idea, $$C(S) = \vec v$$ implies that actually you need S to be of length $$k-1$$ if it contains vectors of $$k$$ dimensions, and that $$\vec v$$ contains $$k$$ dimensions. Am I onto a something or is this completely useless, and cross product was only ever meant for 3D (and apparently 7D)?
2. anonymous
@experimentX @mukushla sorry but you guys are the only ones who answer my questions anymore
3. anonymous
oh man im poor in this topic ...sorry.. :(
4. anonymous
It's ok. I'll go to SE if no one here helps...
5. anonymous
I will just mind map down what I carry around in my head conceding the vector product. If you have a cross product of $$n$$ vectors, then its cross product is defined in $$n+1$$ space. I believe we agree on that. You can reverse that statement and obtain the same as you have mentioned above. To obtain a cross product in three dimensions you need two vectors. In 4 Dimensions you need 3 Vectors.
6. anonymous
Agreed. I'll ask in SE and see what they say.
7. anonymous
in higher dimensions it's called wedge product I believe, no longer cross product.
8. anonymous
Well the wedge product is present in lower dimensions, too. I think it's something that's different but connected. I recall some seemingly simple formula from Wikipedia that obviously has layers of depth I can't fathom. It went like this: For a set of vectors V, $C(V)=\frac{\Lambda(V)}{E(V)}$Where the lambda is the wedge product and the E is the elementary algebra of the system, whatever that means.
9. anonymous
Also, before I saw that formula, I thought wedge products can only be taken on differentially sized vectors, but I guess the fact that their magnitude is infinitesimal doesn't mean anything...
10. anonymous
Hey I'd love to continue chatting but I g2g. Here's the link to the SE page if they say anything interesting: http://math.stackexchange.com/questions/185991/is-the-vector-cross-product-only-defined-for-3d
11. experimentX
sorry ... i don't find anything beyond 3d quite intuitive.
12. anonymous
@experimentX Well I did some playing around on MMA and it turns out that, at least the way I did it with the matrices, the magnitude of a 4D cross product gives the volume drawn out by the first three 4D vectors if their 4th component is zero (like the 3D cross product gives the area of the parallelogram of two 3D vectors with a third zero component)
13. experimentX
well that is one way to picture it out ... except that you need hyperspace to visualize :((
14. anonymous
Well yeah i think of hyperspace like an "internal shadow" where a higher value is a longer shadow, if that helps. Or you can do mass.
15. experimentX
yeah i watched a series of video on it. it was very nice ... but can't find it anywhere online.
|
2016-07-01 10:01:06
|
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|
https://courses.lumenlearning.com/wm-prealgebra/chapter/decimal-operations/
|
What you’ll learn to do: Use various operations to solve applications involving decimals
How many candy bars can Greg buy with the change in his piggy bank?
Greg is hosting a movie night for his friends, and he’s buying candy to share. Candy bars cost $0.79 each, and Greg plans to buy as many as possible with the change from his piggy bank. If he has$5.53, how many candy bars can he buy? To maximize his candy-buying power, Greg will need to use mathematical operations on decimals. In this section, you’ll learn how to perform familiar operations including addition, subtraction, multiplication, and division when working with decimals.
Before you get started, take this readiness quiz.
|
2021-10-27 02:59:55
|
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|
https://www.cell.com/cell-reports/fulltext/S2211-1247(12)00068-X?_returnURL=http%3A%2F%2Flinkinghub.elsevier.com%2Fretrieve%2Fpii%2FS221112471200068X%3Fshowall%3Dtrue
|
Resource| Volume 1, ISSUE 3, P265-276, March 29, 2012
# FoldEco: A Model for Proteostasis in E. coli
Open AccessPublished:March 15, 2012
## Summary
To gain insight into the interplay of processes and species that maintain a correctly folded, functional proteome, we have developed a computational model called FoldEco. FoldEco models the cellular proteostasis network of the E. coli cytoplasm, including protein synthesis, degradation, aggregation, chaperone systems, and the folding characteristics of protein clients. We focused on E. coli because much of the needed input information—including mechanisms, rate parameters, and equilibrium coefficients—is available, largely from in vitro experiments; however, FoldEco will shed light on proteostasis in other organisms. FoldEco can generate hypotheses to guide the design of new experiments. Hypothesis generation leads to system-wide questions and shows how to convert these questions to experimentally measurable quantities, such as changes in protein concentrations with chaperone or protease levels, which can then be used to improve our current understanding of proteostasis and refine the model. A web version of FoldEco is available at http://foldeco.scripps.edu.
## Highlights
• A computational model called FoldEco for proteostasis in E. coli was created
• FoldEco was used to generate testable hypotheses about proteostasis
• A web version of FoldEco is available at http://foldeco.scripps.edu
## Introduction
As they fold to their native states, most proteins sample marginally stable misfolded states that are susceptible to self-association (
• Chiti F.
• Dobson C.M.
Protein misfolding, functional amyloid, and human disease.
), which can turn the marginally stable misfolded monomers into stable aggregates, especially at high protein concentrations (
• Oosawa F.
• Asakura S.
Thermodynamics of the polymerization of protein.
). To minimize protein misfolding and aggregation, cellular pathways have evolved that either assist protein folding or degrade proteins that have failed to fold (
• Balch W.E.
• Morimoto R.I.
• Dillin A.
• Kelly J.W.
,
• Hartl F.U.
• Bracher A.
• Hayer-Hartl M.
Molecular chaperones in protein folding and proteostasis.
,
• Powers E.T.
• Morimoto R.I.
• Dillin A.
• Kelly J.W.
• Balch W.E.
Biological and chemical approaches to diseases of proteostasis deficiency.
). The assisted folding pathways feature chaperones; the degradation pathways feature proteases.
Biologically assisted folding and degradation enable organisms to maintain protein homeostasis, or proteostasis. Proteostasis is critical for the health of an organism. The truth of this statement is demonstrated by the consequences of failures of proteostasis. Deleting chaperone genes in E. coli results in, at best, sensitivity to stresses that affect protein folding (e.g., heat), and at worst, organisms that are not viable (
• Deuerling E.
• Bukau B.
Chaperone-assisted folding of newly synthesized proteins in the cytosol.
). In higher organisms, failure of proteostasis leads to loss-of-function diseases, such as cystic fibrosis, when too little native protein is produced, or gain-of-toxicity diseases, such as Alzheimer disease, when too much aggregated protein is produced (
• Balch W.E.
• Morimoto R.I.
• Dillin A.
• Kelly J.W.
,
• Chiti F.
• Dobson C.M.
Protein misfolding, functional amyloid, and human disease.
,
• Powers E.T.
• Morimoto R.I.
• Dillin A.
• Kelly J.W.
• Balch W.E.
Biological and chemical approaches to diseases of proteostasis deficiency.
).
Understanding proteostasis requires two steps. The first step is reductionist: the mechanisms of assisted folding pathways must be understood at a molecular level. The second step is holistic: the cooperation and competition among the assisted folding pathways, the “proteostasis network” (
• Balch W.E.
• Morimoto R.I.
• Dillin A.
• Kelly J.W.
,
• Powers E.T.
• Morimoto R.I.
• Dillin A.
• Kelly J.W.
• Balch W.E.
Biological and chemical approaches to diseases of proteostasis deficiency.
), must be understood when they are present and operating simultaneously. Neither of these steps can be accomplished by using intuition alone; mathematical models are necessary to make sense of experimental data and to design incisive experiments (
• Hu B.
• Mayer M.P.
• Tomita M.
Modeling Hsp70-mediated protein folding.
,
• Jewett A.I.
• Shea J.E.
Do chaperonins boost protein yields by accelerating folding or preventing aggregation?.
,
• Tehver R.
• Thirumalai D.
Kinetic model for the coupling between allosteric transitions in GroEL and substrate protein folding and aggregation.
,
• Wiseman R.L.
• Powers E.T.
• Buxbaum J.N.
• Kelly J.W.
• Balch W.E.
An adaptable standard for protein export from the endoplasmic reticulum.
).
To create a comprehensive, yet detailed, model for proteostasis, we focused on a simple organism: E. coli. E. coli have only one major Hsp70-based system, consisting of DnaK, the Hsp40 cochaperone DnaJ, and the nucleotide exchange factor GrpE (
• Genevaux P.
• Georgopoulos C.
• Kelley W.L.
The Hsp70 chaperone machines of Escherichia coli: a paradigm for the repartition of chaperone functions.
). They also have only one chaperonin system, consisting of GroEL and its cochaperone GroES (
• Horwich A.L.
• Farr G.W.
• Fenton W.A.
GroEL-GroES-mediated protein folding.
), and one disaggregation system, in which the Hsp104-type chaperone ClpB cooperates with the DnaK/DnaJ/GrpE system (
• Mogk A.
• Deuerling E.
• Vorderwülbecke S.
• Vierling E.
• Bukau B.
Small heat shock proteins, ClpB and the DnaK system form a functional triade in reversing protein aggregation.
). A wealth of available information on the biochemistry of these chaperones enabled us to build a model for proteostasis in E. coli. Here, we describe the resulting model, called FoldEco, and illustrate how it can be used to generate hypotheses about proteostasis in E. coli. We also introduce a publically accessible, web-based version of FoldEco at http://foldeco.scripps.edu. We offer this model to the scientific community as a general paradigm for the interplay of species and processes in proteostasis networks.
## Results and Discussion
### The FoldEco Model
FoldEco (Figures 1, S1, S2, S3, S4, and S5) tracks what happens to soluble proteins of interest (“client proteins”) as they are produced in the E. coli cytosol (membrane and periplasmic proteins are currently not included in FoldEco; see Extended Experimental Procedures). FoldEco has five systems:
#### 1. Protein Synthesis and Folding
The synthesis of a client protein (protein “i”) in FoldEco is represented as a simplified three-step process (see Extended Experimental Procedures). It begins with ribosomes (R) becoming translationally active (Ra,i). The protein then emerges from the ribosomal exit tunnel and the ribosome:nascent protein complex is formed (Ra:Ui). Finally, the ribosome releases the unfolded client protein (Ui) when synthesis is complete. A trigger factor monomer, T, can bind to the ribosome or ribosome:nascent protein complex yielding the species R:T, Ra,i:T, or Ra:Ui:T (
• Deuerling E.
• Bukau B.
Chaperone-assisted folding of newly synthesized proteins in the cytosol.
). The ribosome releases the unfolded client protein free (Ui) or in complex with trigger factor (T:Ui). The unfolded client protein reversibly folds to the native state (Ni) or reversibly misfolds to form an off-pathway intermediate (Mi). Mi kinetically partitions between self-association by nucleated polymerization (
• Oosawa F.
• Asakura S.
Thermodynamics of the polymerization of protein.
) to form aggregates (Ai,j, where there are j monomers in the aggregate) or chaperone binding. We have not incorporated the possibility of cotranslational folding or misfolding (see Extended Experimental Procedures).
#### 2. The DnaK/DnaJ/GrpE, or KJE, System
The role of the KJE system in proteostasis is to bind misfolded proteins and unfold them, thereby giving them another chance to fold (
• Mayer M.P.
• Bukau B.
Hsp70 chaperones: cellular functions and molecular mechanism.
). In FoldEco, the mechanism of the KJE system begins with unfolded or misfolded protein binding to dimeric DnaJ (J2) or ATP-bound DnaK (KT), forming binary complexes (J2:Ui/Mi or KT:Ui/Mi, where the “/” indicates “either-or”). J2:Ui/Mi or KT:Ui/Mi can then add KT or J2, respectively, to form the ternary complex KT:Ui/Mi:J2. However, KT binds weakly to substrates (
• Gisler S.M.
• Pierpaoli E.V.
• Christen P.
Catapult mechanism renders the chaperone action of Hsp70 unidirectional.
,
• Mayer M.P.
• Schröder H.
• Rüdiger S.
• Paal K.
• Laufen T.
• Bukau B.
Multistep mechanism of substrate binding determines chaperone activity of Hsp70.
), so binding to J2 is preferred. J2 binding accelerates ATP hydrolysis by DnaK in the KT:Ui/Mi:J2 complex, which causes a conformational change within DnaK. This conformational change is propagated to the bound client protein, causing it to unfold. Thus, both KT:Ui:J2 and KT:Mi:J2 produce KD:Ui:J2 upon ATP hydrolysis (KD = ADP-bound DnaK). KD:Ui:J2 then releases J2 to give KD:Ui (KD:Ui can also be produced by slow ATP hydrolysis in KT:Ui/Mi). Nucleotide exchange in DnaK is induced by binding to dimeric GrpE (E2), producing KT:Ui:E2. Dissociation of E2 yields KT:Ui, which can dissociate to yield free unfolded protein, Ui. Note that because either U or M can enter the KJE cycle, and U is released, this chaperone team provides a conduit from M to U. Note also that KT:Ui can also unproductively reenter the KJE cycle by binding to J2, depending on the relative rates of substrate dissociation and J2 binding. The total number of KJE cycles that a particular client protein experiences before it folds is determined by the relative rates of entry into the KJE cycle and folding. Our model for the KJE cycle is in part based on that of
• Hu B.
• Mayer M.P.
• Tomita M.
Modeling Hsp70-mediated protein folding.
.
#### 3. The GroEL/GroES, or GroELS, System
The role of the GroELS system in proteostasis is to encapsulate client proteins, thereby providing them with an isolated environment in which to fold (
• Horwich A.L.
• Farr G.W.
• Fenton W.A.
GroEL-GroES-mediated protein folding.
). In FoldEco, the GroELS cycle begins when an unfolded or misfolded client protein binds to the cis ring of GroEL, which is either ATP-bound (GrLT) or not (GrL), thus yielding GrL:Ui/Mi or GrLT:Ui/Mi. Therefore, client proteins can bind to either the ATP-free or ATP-bound state of GroEL, but there should be very little of the former at typical in vivo ATP concentrations (
• Tyagi N.K.
• Fenton W.A.
• Horwich A.L.
GroEL/GroES cycling: ATP binds to an open ring before substrate protein favoring protein binding and production of the native state.
). GrL:Ui and GrL:Mi are both converted to GrLT:Ui by ATP-binding-induced forced unfolding of the bound client protein (
• Lin Z.
• Rye H.S.
GroEL stimulates protein folding through forced unfolding.
). GrLT:Ui/Mi is then capped by GroES (GrS), which releases the client protein into the cis cavity (GrLT:Ui/Mi:GrS), where it can fold or, in principle, misfold, to give GrLT:Ui/Mi/Ni:GrS. In FoldEco, the rate of folding of a client protein in the GroEL cavity can be the same as it is in solution or can be different, as desired by the user; both circumstances have been reported (
• Apetri A.C.
• Horwich A.L.
Chaperonin chamber accelerates protein folding through passive action of preventing aggregation.
,
• Chakraborty K.
• Chatila M.
• Sinha J.
• Shi Q.
• Poschner B.C.
• Sikor M.
• Jiang G.
• Lamb D.C.
• Hartl F.U.
• Hayer-Hartl M.
Chaperonin-catalyzed rescue of kinetically trapped states in protein folding.
), and both are therefore allowed. ATP hydrolysis in the cis ring yields GrLD:Ui/Mi/Ni:GrS, which can follow one of two pathways. In the first, ATP binding in the trans ring results in GrLD:Ui/Mi/Ni:GrS::GrLT and release of ADP, GrS, and the client protein from the cis ring. In the second pathway, binding of unfolded or misfolded client protein (Uk or Mk; the subscript k indicates that it could be a different type of client protein from the one in the cis cavity) in the trans ring results in GrLD:Ui/Mi/Ni:Grs::GrL:Uk/Mk, which then binds ATP in the trans ring, stimulating release of ADP, GrS, and client protein from the cis ring. The trans ring is left in the GrLT:Uk state, ready to reenter the cycle. The partitioning between the first and second pathways depends on the rates of binding of ATP versus substrate. As with the KJE cycle, the total number of GroELS cycles that a particular client protein experiences before it folds is determined by the relative rates of entry into the GroELS cycle and folding. Our model of the GroELS cycle is based in part on those of
• Tehver R.
• Thirumalai D.
Kinetic model for the coupling between allosteric transitions in GroEL and substrate protein folding and aggregation.
and
• Jewett A.I.
• Shea J.E.
Do chaperonins boost protein yields by accelerating folding or preventing aggregation?.
.
#### 4. The ClpB/DnaK/DnaJ/GrpE, or B+KJE, Disaggregation System
Disaggregation is critical for the recovery of E. coli from heat shock (
• Mogk A.
• Deuerling E.
• Vorderwülbecke S.
• Vierling E.
• Bukau B.
Small heat shock proteins, ClpB and the DnaK system form a functional triade in reversing protein aggregation.
). This function can be performed for small aggregates by the KJE cycle (
• Diamant S.
• Ben-Zvi A.P.
• Bukau B.
• Goloubinoff P.
Size-dependent disaggregation of stable protein aggregates by the DnaK chaperone machinery.
), but larger aggregates require the AAA+ ATPase ClpB, “B” for short (
• Mogk A.
• Deuerling E.
• Vorderwülbecke S.
• Vierling E.
• Bukau B.
Small heat shock proteins, ClpB and the DnaK system form a functional triade in reversing protein aggregation.
,
• Weibezahn J.
• Tessarz P.
• Schlieker C.
• Zahn R.
• Maglica Z.
• Lee S.
• Zentgraf H.
• Weber-Ban E.U.
• Dougan D.A.
• Tsai F.T.
• et al.
Thermotolerance requires refolding of aggregated proteins by substrate translocation through the central pore of ClpB.
). We based the mechanism of the B+KJE cycle in FoldEco on several studies that suggest that the KJE system prepares aggregates for B (
• Acebrón S.P.
• Fernández-Sáiz V.
• Taneva S.G.
• Moro F.
• Muga A.
DnaJ recruits DnaK to protein aggregates.
,
• Acebrón S.P.
• Martín I.
• del Castillo U.
• Moro F.
• Muga A.
DnaK-mediated association of ClpB to protein aggregates. A bichaperone network at the aggregate surface.
,
• Doyle S.M.
• Hoskins J.R.
• Wickner S.
Collaboration between the ClpB AAA+ remodeling protein and the DnaK chaperone system.
), which then removes monomers from aggregates. Thus, client protein aggregates of size j (Ai,j) bind first to J2 or KT (we have assumed a 1:1 stoichiometry of chaperone:aggregate in these complexes; see Extended Experimental Procedures). The resulting complexes (J2:Ai,j or KT:Ai,j) then bind to KT or J2, respectively, to form the ternary complex KT:Ai,j:J2. ATP hydrolysis produces KD:Ai,j:J2, where indicates that the aggregate is now prepared for ClpB binding. Release of J2 yields KD:Ai,j, which has two possible fates. First, it can bind to E2, which induces nucleotide exchange to give KT:Ai,j:E2. For small oligomers (j ≤ 4 in FoldEco), a monomer is lost and E2 dissociates to give KT:Ai,j-1 and a free monomer, Ui (
• Diamant S.
• Ben-Zvi A.P.
• Bukau B.
• Goloubinoff P.
Size-dependent disaggregation of stable protein aggregates by the DnaK chaperone machinery.
). For larger aggregates, E2 dissociates and KT:Ai,j is produced without monomer loss, consistent with the observation that the KJE system on its own cannot disperse large aggregates (
• Diamant S.
• Ben-Zvi A.P.
• Bukau B.
• Goloubinoff P.
Size-dependent disaggregation of stable protein aggregates by the DnaK chaperone machinery.
). KT:Ai,j-1 and KT:Ai,j can then either reenter the cycle or dissociate to give a free aggregate and KT. Second, KD:Ai,j can bind to B to form KD:Ai,j:B. E2 binding followed by nucleotide exchange gives E2:KT:Ai,j:B, from which E2 and KT dissociate to give Ai,j:B. Finally, a monomer is translocated through the central pore of B (
• Weibezahn J.
• Tessarz P.
• Schlieker C.
• Zahn R.
• Maglica Z.
• Lee S.
• Zentgraf H.
• Weber-Ban E.U.
• Dougan D.A.
• Tsai F.T.
• et al.
Thermotolerance requires refolding of aggregated proteins by substrate translocation through the central pore of ClpB.
) and the complex dissociates, giving Ai,j-1, Ui, and B.
The most important proteases for proteostasis in E. coli are the energy-dependent proteases (
• Gottesman S.
Proteolysis in bacterial regulatory circuits.
). FoldEco incorporates two degradation pathways. The first features Lon, a protease that degrades unfolded and misfolded client proteins (
• Gottesman S.
Proteolysis in bacterial regulatory circuits.
). This pathway begins with Lon and Ui or Mi associating to form the reversible complex Lon:Ui/Mi. The bound substrate is then transferred to the proteolytic chamber (with concurrent forced unfolding, if the substrate is misfolded), to form Lon:Ui, where indicates that the substrate is now committed to degradation. Degradation is processive, with the ATPase domains of Lon feeding the substrate into the proteolytic chamber. In the second pathway, ClpAP-type proteases (Dn) degrade native proteins (Ni) that have been tagged with a degradation signal, or degron (
• Gottesman S.
Proteolysis in bacterial regulatory circuits.
,
• Varshavsky A.
The N-end rule pathway and regulation by proteolysis.
). This pathway begins with Dn binding to Ni, perhaps after delivery by an adaptor protein such as ClpS (
• Erbse A.
• Schmidt R.
• Bornemann T.
• Schneider-Mergener J.
• Mogk A.
• Zahn R.
• Dougan D.A.
• Bukau B.
ClpS is an essential component of the N-end rule pathway in Escherichia coli.
), to form the reversible complex Dn:Ni. The bound client protein is then forcibly unfolded and transferred to the proteolytic chamber, yielding Dn:Ui. The substrate is degraded processively as described above. Note that degron installation, adaptor protein binding, and delivery of the substrate to the protease are subsumed into the binding step in FoldEco (see Extended Experimental Procedures).
### Implementing and Parametrizing FoldEco
FoldEco is implemented by writing the differential equations that describe the time-dependent concentrations of each species in the model. Solving this system of equations requires the model parameters (the rate constants and initial concentrations) to have numerical values. Fortunately, the rate constants for virtually every step in FoldEco can be estimated from the literature (Figures S1, S2, S3, S4, and S5). Note that parameters such as chaperone-substrate binding affinity and on/off rates are protein-specific. The default parameters that we have used here and on the FoldEco website were measured for model proteins and can be changed by users on the input page of the FoldEco website. Literature estimates of the initial concentrations of the proteostasis network components are also available (Table S1). Parameters for the folding of a particular client protein can be estimated from literature data, experiments, or models; for example, that of Ghosh and Dill for protein stability (
• Ghosh K.
• Dill K.
Cellular proteomes have broad distributions of protein stability.
,
• Ghosh K.
• Dill K.A.
Computing protein stabilities from their chain lengths.
) or those of
• Plaxco K.W.
• Simons K.T.
• Baker D.
Contact order, transition state placement and the refolding rates of single domain proteins.
or
• Ouyang Z.
• Liang J.
Predicting protein folding rates from geometric contact and amino acid sequence.
for folding rates. The differential equations are then solved numerically, and the time-dependent concentrations of all species are the output.
Any model of biological networks must by necessity contain some approximations. Thus, FoldEco in its present form does not account for the effect of bacterial growth on proteostasis, stress responses, or possible changes in ATP levels. These and other approximations and their potential effects on the model are discussed in the Extended Experimental Procedures.
Models such as FoldEco are powerful tools for rationalizing experimental observations and generating hypotheses. We describe in the following sections some illustrative examples of findings and predictions derived from our own exploration of FoldEco. We have deployed a web version of FoldEco (described in the final section) to allow others to use it for their own purposes.
### The Importance of Synthesis and Degradation Rates to Protein Aggregation
An important use for any model of proteostasis is to characterize the circumstances under which a client protein aggregates. Aggregation is possible when the client protein's aggregation-prone intermediate (in FoldEco, the Mi state) has a concentration higher than its critical concentration for aggregation, or Ccrit (
• Oosawa F.
• Asakura S.
Thermodynamics of the polymerization of protein.
). The concentration of Mi, in turn, is determined by the balance between the rates of protein synthesis and of degradation by Lon, the folding and misfolding energetics, and of course chaperones (see Extended Experimental Procedures).
To illustrate the effect of synthesis rate and Lon concentration on protein aggregation, we used FoldEco to examine the behavior of a model client protein in the nonphysiological, but nevertheless instructive, situation of being in the absence of any chaperones. The model protein was assigned a moderately aggregation-prone “biophysical profile,” which includes the following (see Figures 1 and S1): the folding rate and equilibrium constants (kf and Kf), the misfolding rate and equilibrium constants (km and Km), the monomer-aggregate association rate constant (ka), and the critical concentration for aggregation (Ccrit). Thus, the model protein was given moderately favorable misfolding parameters (km and Km = 1 s−1 and 10, respectively), more favorable folding parameters (kf and Kf = 1 s−1 and 1000, respectively), moderately fast aggregation kinetics (ka = 0.1 μM−1 s−1), and a strong aggregation propensity (Ccrit = 0.1 μM). The chosen value of Ccrit is comparable to the in vitro critical concentration of Aβ42, the 42-residue-long and most aggregation-prone of the major isoforms of the Aβ peptide, the aggregation of which is implicated in Alzheimer disease (
• Usui K.
• Hulleman J.D.
• Paulsson J.F.
• Siegel S.J.
• Powers E.T.
• Kelly J.W.
Site-specific modification of Alzheimer's peptides by cholesterol oxidation products enhances aggregation energetics and neurotoxicity.
). The other parameters used in these simulations are listed in Table S2. FoldEco was solved for this model protein at synthesis rates varying from 0.01 μM s−1 to 1 μM s−1. The upper end of this range is 70% of the maximal protein synthesis rate given the parameters used in this simulation (see Extended Experimental Procedures). At each synthesis rate, we determined the minimum concentration of Lon hexamer that would suppress aggregation to below 5% of total protein after 10,000 s of simulation time (about 2.75 hr; Figure 2A , black curve). This length of simulation time was arbitrarily chosen, but qualitatively similar results are obtained at different times and at steady state (Figure S6A ). The time required to reach steady state appears to depend primarily on the folding energetics (kf and Kf) and degradation rates (see Extended Experimental Procedures).
These simulations show the importance of synthesis and degradation rates to protein aggregation. The typical concentration of Lon hexamers in E. coli is about 0.3 μM (Table S1), which is the minimum concentration needed to suppress aggregation at the relatively slow synthesis rate of 0.019 μM s−1 (Figure 2A). Higher synthesis rates for this aggregation-prone model protein require higher concentrations of Lon (or the introduction of chaperones; see below). Heat shock can increase the concentration of Lon by up to four-fold (
• Zhao K.
• Liu M.
• Burgess R.R.
The global transcriptional response of Escherichia coli to induced σ32 protein involves σ32 regulon activation followed by inactivation and degradation of σ32 in vivo.
); a four-fold increase in Lon concentration would increase the synthesis rate that can be tolerated to 0.074 μM s−1 (Figure 2A). Expression from a plasmid can give even higher concentrations of Lon, which would enable higher synthesis rates.
Higher synthesis and degradation rates may be desirable for protein production because they lead to higher concentrations of native protein (Figure 2A, blue numbers). For example, at a synthesis rate of 0.01 μM s−1 and Lon hexamer concentration of 0.12 μM (the minimum required to suppress aggregation), the native state concentration of our model protein at 10,000 s is 41 μM. Increasing the synthesis rate 10-fold to 0.1 μM s−1 and the Lon hexamer concentration to 1.6 μM gives a native protein concentration of 112 μM at 10,000 s. Analyzing the kinetics of protein accumulation (Figure S6B) and a simplified model for protein synthesis and degradation (Figure S6C) reveals that this increase in the native protein concentration arises not just from the higher synthesis rate, but rather from the suppression of misfolding by degradation, which increases the concentration of the unfolded state during protein synthesis, and thereby increases the rate of production of the native state (see Extended Experimental Procedures).
The strategy of jointly increasing the synthesis rate and Lon concentration to increase native protein concentration comes at a cost to the cell. Our simulations suggest that although the native protein concentration increases as the synthesis and degradation rates increase, the fraction of protein that is degraded increases as well (Figure 2A, red numbers). Thus, while our model protein can be kept soluble when the synthesis rate is as high as 1 μM s−1, very high Lon concentrations (13 μM; Figure 2A) are required to do so. In these circumstances, 94% of the total synthesized protein becomes degraded. The enormous metabolic cost of so much wasted protein synthesis would be acceptable to an experimenter expressing a protein in E. coli, but would be a hindrance to an organism in its natural setting (
• Stoebel D.M.
• Dean A.M.
• Dykhuizen D.E.
The cost of expression of Escherichia coli lac operon proteins is in the process, not in the products.
).
Proteostasis can be efficiently maintained at considerably lower metabolic cost by using chaperones to minimize aggregation. To demonstrate this notion, we repeated the simulations described above with a full complement of chaperones (20 μM trigger factor, 30 μM DnaK, 1 μM DnaJ, 15 μM GrpE, 3 μM GroEL tetradecamers, 5 μM GroES heptamers, 0.3 μM ClpB hexamers; Table S1). At physiological concentrations of Lon (0.3 μM), chaperones permit synthesis rates up to almost 0.3 μM s−1, about 15-fold higher than in the absence of chaperones, and yield much higher concentrations of native protein (Figure 2B). Higher synthesis rates, however, require sharply increasing concentrations of Lon to suppress aggregation. Thus, chaperones do not entirely obviate the need for degradation during the expression of aggregation-prone proteins.
### Misfolding-Prone Proteins Benefit the Most from the KJE System
In addition to enhancing our understanding of in vivo protein aggregation, models for proteostasis should also be useful for revealing how client proteins with particular biophysical profiles benefit from particular chaperoning mechanisms. To find client proteins that benefit strongly from the KJE system, we generated 4,000 random biophysical profiles, input each of them into FoldEco, and solved the model with a modest synthesis rate (0.02 μM s−1), typical concentrations of Lon (0.3 μM hexamer) and trigger factor (20 μM; Table S1), and with or without the KJE system (30 μM DnaK, 1 μM DnaJ, and 15 μM GrpE; Table S1). The GroELS system and ClpB were excluded from the simulations in order to yield a focused view of profiles for which the KJE system alone influenced client outcome. Although such conditions cannot exist in reality, because GroEL is an essential protein in E. coli, isolating the KJE system like this is the best way to understand its client preferences and overall capacity to maintain proteostasis.
To generate the random biophysical profiles, kf and km were varied between 10−3 and 103 s−1; ka was varied between 10−3 and 103 μM−1 s−1; Kf was varied between 102 and 108; Km varied between 10−3 and 103; and Ccrit was varied between 1 mM and 1 nM (Table S3). Where possible, the bounds on these variables were based on available protein-folding data (see Extended Experimental Procedures). The parameters used for the association between client proteins and DnaK and DnaJ2 were those measured for the heat shock transcription factor σ32, a known substrate of the KJE system (
• Gamer J.
• Multhaup G.
• Tomoyasu T.
• McCarty J.S.
• Rüdiger S.
• Schönfeld H.J.
• Schirra C.
• Bujard H.
• Bukau B.
A cycle of binding and release of the DnaK, DnaJ and GrpE chaperones regulates activity of the Escherichia coli heat shock transcription factor σ32.
,
• Mayer M.P.
• Schröder H.
• Rüdiger S.
• Paal K.
• Laufen T.
• Bukau B.
Multistep mechanism of substrate binding determines chaperone activity of Hsp70.
). The other parameters used in these simulations are listed in Table S4.
We examined the biophysical profiles of the client proteins with the largest (top 5%) increases in the concentration of the native state at an arbitrarily chosen time point (10,000 s) upon adding the KJE system (Table S3). The greatest increases varied from about +50 to +139 μM, which are substantial enhancements given that 200 μM of protein was synthesized. The outstanding feature of the biophysical profiles of the top KJE substrates is their uniformly high misfolding propensities (Figure 3A ). This feature shows that client proteins that benefit most from the KJE system have a strong tendency to misfold. By pumping client proteins from the Mi state, which cannot fold, to the Ui state, which can, the KJE system promotes folding and rescues client proteins from aggregation or degradation. Which of these two fates a client protein is rescued from depends on its aggregation propensity, as measured by Ccrit. At a synthesis rate of 0.02 μM s−1, client proteins need an exceptionally strong aggregation propensity (low Ccrit value) to aggregate because so little protein is produced: the median Ccrit value for proteins that aggregated more than 5% (total aggregate concentration > 10 μM) in the absence of the KJE system was 7.4 nM (Figure S7A ). Thus, only 25% of the top KJE substrates are rescued primarily from aggregation (Table S3). The rest are rescued from degradation.
### Slow Folding Proteins Benefit the Most from the GroELS System
To find the biophysical profiles that benefit the most from the GroELS system, we repeated the simulation experiment described above with or without the GroELS system (3 μM GroEL tetradecamers, 5 μM GroES heptamers; Table S1). The KJE system and ClpB were excluded in order to focus on the impact of the GroELS system, and the folding and misfolding parameters were assumed to be the same inside and outside the GroEL cavity. Binding parameters for typical GroELS system substrates were used (Figure S3, Table S4). At a synthesis rate of 0.02 μM s−1, the top 5% of increases in the concentration of the native state upon adding the GroELS system varied from +102 to +153 μM (Table S3). In addition, the native state concentration increased by more than 10 μM for 2,223 of the 4,000 randomly generated client proteins when the GroELS system was added, but for only 737 client proteins when the KJE system was added (Table S3). This result demonstrates that the GroELS system is beneficial to a broader range of our randomly generated client proteins than the KJE system at this synthesis rate. The efficiency of the GroELS system is especially remarkable considering that we assumed that the folding and misfolding rates in the GroEL cavity were the same as they are in solution. If faster folding rates inside the cavity had been used, the effect of the GroELS system on native state concentration would presumably have been even larger.
A striking feature of the top GroELS substrates is their low and narrowly distributed folding rate constants (Figure 3B). This rate selectivity arises from two factors: fast-folding client proteins (kf > 0.1 s−1) tend not to need chaperone assistance, while very slow-folding client proteins (kf < 0.01 s−1) fold too slowly for the effect of the GroELS system to take hold on the time scale of the simulation. The optimal folding rate for GroEL substrates is determined by the rate constant for ATP hydrolysis by GroEL, which determines how long substrates remain encapsulated. In FoldEco, the rate constant for this process is set to 0.1 s−1 (corresponding to a half-life of about 7 s), as generally found experimentally (
• Ranson N.A.
• Burston S.G.
• Clarke A.R.
Binding, encapsulation and ejection: substrate dynamics during a chaperonin-assisted folding reaction.
,
• Rye H.S.
• Roseman A.M.
• Chen S.
• Furtak K.
• Fenton W.A.
• Saibil H.R.
• Horwich A.L.
GroEL-GroES cycling: ATP and nonnative polypeptide direct alternation of folding-active rings.
). In principle, the optimal folding rate constant for GroELS substrates could be changed by changing the ATP hydrolysis rate constant. However, if this rate constant is too low, the GroELS system retains substrates for so long that it quickly becomes saturated; if this rate constant is too high, the GroELS system does not retain slow-folding client proteins long enough for them to fold appreciably and, as noted above, fast-folding substrates have no need of the GroELS system. The ATP hydrolysis rate constant of GroEL appears to have evolved to an optimum.
The other noteworthy feature of the biophysical profiles of the top GroELS substrates is their low misfolding propensities (Figure 3B). In fact, only 12% of the top GroELS substrates are rescued from aggregation, compared with 25% of the top KJE substrates (Table S3). The GroELS system is apparently not optimized to correct misfolding. It is important to note, however, that while misfolding-prone proteins may not be among the top GroELS substrates, they nevertheless benefit substantially from the GroELS system. For example, of the 737 proteins whose native state concentration increases by > 10 μM upon addition of the KJE system, 657 (89%) benefit similarly upon addition of the GroELS system. This observation suggests, and experiments have shown (
• Vorderwülbecke S.
• Kramer G.
• Merz F.
• Kurz T.A.
• Rauch T.
• Zachmann-Brand B.
• Bukau B.
• Deuerling E.
Low temperature or GroEL/ES overproduction permits growth of Escherichia coli cells lacking trigger factor and DnaK.
), that the GroELS system can complement the loss of the KJE system, at least partially.
### Cooperation between the KJE and GroELS Systems
To determine whether the biophysical profiles that benefit the most from the KJE or GroELS systems are different when one system is introduced in the presence of the other than when they are introduced individually, we repeated the simulation experiment described above with both systems present. The results from these simulations were similar to those from the previous simulations. The top 5% of the increases in the concentration of the native state upon addition of the KJE system in the presence of the GroELS system ranged from +36 to +122 μM, and the native state concentration increased by more than 10 μM for 572 of the 4,000 client proteins. The top 5% of the increases in the concentration of the native state upon addition of the GroELS system in the presence of the KJE system ranged from +101 to +156 μM, and the native state concentration increased by more than 10 μM for 2,013 of the 4,000 client proteins. The biophysical profiles of the top KJE and GroELS substrates were nearly identical in the presence (Figures S7B and S7C, respectively) and absence (Figures 3A and 3B, respectively) of the other system.
The observations above suggest that the activities of the KJE and GroELS systems are independent and complementary. The extent to which this idea is true is demonstrated by Figure 3C. As noted above, 657 of the 4,000 biophysical profiles studied had increases in native state concentration upon addition of the KJE system or the GroELS system that were >10 μM. For this subset, we have plotted in Figure 3C the increase in native state concentration due to the simultaneous presence of the KJE and GroELS systems (the “combined effect”; y axis) against the sum of the increases due to the individual presence of either the KJE or the GroELS system (the “summed effects”; x axis). This plot can be split into three regions: region 1 contains the 415 cases for which the combined effects are less than the summed effects, region 2 contains the 161 cases for which the combined and summed effects are roughly equal, and region 3 contains the 81 cases for which the combined effects are greater than the summed effects.
The cases in region 2 are perhaps the easiest to understand: the effects of the KJE and GroELS systems are simply independent and additive. In contrast, the KJE and GroELS systems appear to interfere with each other to some extent for the cases in region 1. This is partly due to an artifact, as 247 of the cases in region 1 (60%) are there because the KJE or GroELS systems by themselves completely rescue the client protein (red open circles). In these cases, a second chaperone system will always be superfluous. For the remaining 168 cases in region 1, the KJE and GroELS systems are partially redundant (blue open circles).
For the cases in region 3, the KJE and GroELS systems are synergistic. The biophysical profiles of these cases (Figure 3D) show that they combine the features of the top KJE and GroELS substrates: they have high propensities to misfold (high Km) and slow folding rates (low kf). For such substrates, the ability of the KJE system to recover protein from the misfolded state apparently increases the input into the GroELS system, which then increases the flux of protein into the native state. The cases in regions 1 (Figure S7D) and 2 (Figure S7E) have lower misfolding propensities and faster folding rates, so that the KJE and GroELS systems are less able to amplify each other's effects.
### The Effect of Increasing the Synthesis Rate on the Biophysical Profiles of the Top KJE and GroELS Substrates
We probed the effect of a higher synthesis rate on the biophysical profiles of the top KJE and GroELS substrates by repeating the simulation experiments described above with a 10-fold higher synthesis rate (0.2 μM s−1). The top 5% of the increases in the concentration of the native state at 10,000 s upon addition of the KJE system by itself or in the presence of the GroELS system are proportionately higher than they were at the lower synthesis rate, ranging from +643 to +1526 μM without the GroELS system and from +512 to +1,507 μM with the GroELS system (2,000 μM of protein was synthesized; Table S5). The biophysical profiles of the top KJE substrates at this synthesis rate (Figure 4A ) are nearly identical with and without GroELS (Figure 4A and Figure S8A ) and are both similar to those of the top KJE substrates at the lower synthesis rate, but with a new feature. The misfolding propensities of the top KJE substrates are still high, but now their aggregation propensities are high as well (Figure 4A). Thus, aggregation appears to be more important at this synthesis rate than at the lower synthesis rate. Indeed, all of the top KJE substrates are rescued from aggregation rather than degradation at this synthesis rate. Rescue from aggregation is not guaranteed by the presence of the KJE system, however. The aggregation rate constants for the top KJE substrates tend to be low (Figure 4A), suggesting that fast aggregation interferes with the KJE system.
At the higher synthesis rate, the top 5% of the increases in the concentration of the free native state upon addition of the GroELS system ranged from +279 to +1050 μM without the KJE system and +135 to +1,119 with the KJE system (Table S5). Unlike the situation with the KJE system, the biophysical profiles of the top GroELS substrates in the absence (Figure 4B) or presence (Figure S8B) of the KJE system are different at this synthesis rate than at the lower synthesis rate. The folding rate constants are not as low as they were before for the top GroELS substrates. Instead, all of the parameters associated with misfolding and aggregation (km, Km, ka, and Ccrit) are high, indicating that the top beneficiaries of the GroELS system at higher synthesis rates are misfolding and aggregation prone rather than slow folding. Consistent with this observation, all of the top GroELS substrates are rescued primarily from aggregation.
These observations suggest that aggregation rather than degradation becomes the dominant problem in proteostasis as the synthesis rate increases, and the KJE and GroELS systems are comparably able of rescuing proteins from misfolding and aggregation. This assertion is demonstrated by the concentration of the native state increasing by more than 100 μM for 700 of the 4,000 randomly generated client proteins upon addition of the KJE system by itself (and for 554 client proteins when it is added in the presence of the GroELS system) and for 545 client proteins upon addition of the GroELS system by itself (and for 288 client protein when it is added in the presence of the KJE system).
The ability of the KJE and GroELS systems to cooperate is illustrated in Figure 4C, which is a plot of the combined versus summed effects of the KJE and GroELS systems for the 433 cases for which the KJE and GroELS systems each individually increase the concentration of the native state by >100 μM. We split this plot into three regions, as in Figure 3C. Also as in Figure 3C, many of the cases in region 1 (142 out of 273; red open circles) are there because they are almost completely rescued by either the KJE or GroELS system by itself. Perhaps the most notable feature of Figure 4C is that region 3 is almost completely unpopulated. The effects of the KJE and GroELS systems are at best additive at this synthesis rate, probably because the similarity of the biophysical profiles of their top substrates leaves little room for them to complement each other's activity.
### Testing Predictions from FoldEco
Here, we examine the hypotheses suggested by the simulations in the preceding sections in the light of experimental data from the literature. The first is that aggregation can be suppressed by decreasing the synthesis rate or by increasing the concentration of Lon. Siller et al. recently examined the effect of the synthesis rate on the in vivo folding of firefly luciferase (Luc) (
• Siller E.
• DeZwaan D.C.
• Anderson J.F.
• Freeman B.C.
• Barral J.M.
Slowing bacterial translation speed enhances eukaryotic protein folding efficiency.
). They found that slowing the rate of ribosomal translation decreased the extent of aggregation and increased the specific activity of Luc (see Extended Experimental Procedures for details). Siller et al. provide data for the relative specific activities of Luc after 15 min of expression at slow and fast translation rates (
• Siller E.
• DeZwaan D.C.
• Anderson J.F.
• Freeman B.C.
• Barral J.M.
Slowing bacterial translation speed enhances eukaryotic protein folding efficiency.
) and data that can be used to estimate the extent of aggregation (see Figure S9A and Extended Experimental Procedures). We attempted to reproduce their results by using FoldEco.
We constrained our simulations by using estimates of the folding (kf) and misfolding (km) rates for Luc folding based on in vitro studies of Luc refolding in the presence (
• Szabo A.
• Langer T.
• Schröder H.
• Flanagan J.
• Bukau B.
• Hartl F.U.
The ATP hydrolysis-dependent reaction cycle of the Escherichia coli Hsp70 system DnaK, DnaJ, and GrpE.
) and absence (
• Herbst R.
• Gast K.
• Seckler R.
Folding of firefly (Photinus pyralis) luciferase: aggregation and reactivation of unfolding intermediates.
) of the KJE system (see Extended Experimental Procedures for details on this and other constraints; see also Figure S9B). The other folding parameters had little effect on the results of the simulations. For the initial chaperone concentrations, we used the chaperone concentrations at the top of the confidence interval reported in Table S1 for Luc produced at both slow and fast translation rates (40 μM trigger factor, 50 μM DnaK, 2 μM DnaJ, 20 μM GrpE, 5 μM GroEL tetradecamer, 10 μM GroES heptamer, 0.5 μM Lon hexamer, 0.5 μM ClpB hexamer), because heterologous expression of aggregation-prone proteins such Luc is known to induce the heat-shock response (
• Hoffmann F.
• Rinas U.
Stress induced by recombinant protein production in Escherichia coli.
). Moreover, we note that these chaperone concentrations are similar to some experimental measurements of chaperone concentrations in nonstressed E. coli (see Table S1).
The results of our simulations at these chaperone concentrations match the experimental results well. They reproduce the increase in aggregation as well as the decrease in specific activity (Figures 5A and 5B , respectively; see also Table S6 and Extended Experimental Procedures) observed experimentally upon increasing the translation rate (
• Siller E.
• DeZwaan D.C.
• Anderson J.F.
• Freeman B.C.
• Barral J.M.
Slowing bacterial translation speed enhances eukaryotic protein folding efficiency.
). FoldEco also reproduces the trends in the data at lower chaperone concentrations, but the extent of aggregation is generally overestimated (Figures S9C and S9D; Table S7). Thus, FoldEco currently encompasses most of the aspects of in vivo proteostasis needed to understand the effect of protein synthesis rate on the partitioning of protein between aggregates and native, functional protein.
Despite this success, one observation made by Siller et al. was not reproduced in these simulations: that the absolute activity (not just the specific activity) of Luc is higher in their expression system when translation is slower, despite less protein being produced per cell. Siller et al. attribute this effect to cotranslational folding at lower translation rates (
• Siller E.
• DeZwaan D.C.
• Anderson J.F.
• Freeman B.C.
• Barral J.M.
Slowing bacterial translation speed enhances eukaryotic protein folding efficiency.
). The current version of FoldEco does not explicitly account for cotranslational folding, but we can mimic it by increasing the folding rate constant (kf) by a factor of about 2 in the simulations at the slower translation rate, but not at the faster translation rate (Figure S9E). This adjustment enables us to reproduce their observation (see Extended Experimental Procedures) and suggests that cotranslational folding doubles the effective folding rate of Luc.
This exercise demonstrates one of the most important uses of models for complicated systems: they enable concrete tests of whether a given set of processes is sufficient to explain a given set of experimental results. When they are not, it is clear that at least one important process have been left out of the model. In this case, cotranslational folding is not necessary to explain the decrease in aggregation of Luc when it is synthesized more slowly, but the importance of cotranslational folding to the production of active Luc is nevertheless unambiguous. The current version of FoldEco can be adapted to account for cotranslational folding by adjusting kf, but future versions will include it explicitly.
The other hypotheses that are suggested by our results with FoldEco have to do with the characteristics of the substrates for the KJE and GroELS systems. For example, on the basis of Figures 3A and 4A, we expect that DnaK substrates in E. coli should be aggregation prone. This prediction is consistent with the results from a recent study of DnaK interactors. Calloni et al. found that proteins that are highly enriched on DnaK (
• Calloni G.
• Chen T.
• Schermann S.M.
• Change H.-C.
• Genevaux P.
• Agostini F.
• Tartaglia G.G.
• Hayer-Hartl M.
• Hartl F.U.
DnaK functions as a central hub in the E. coli chaperone network.
) tend to be less soluble than the average proteins of soluble cell lysate (where solubility measurements were taken from a proteome-wide measurement of E. coli protein solubilities) (
• Niwa T.
• Ying B.W.
• Saito K.
• Jin W.
• Ueda T.
• Taguchi H.
Bimodal protein solubility distribution revealed by an aggregation analysis of the entire ensemble of Escherichia coli proteins.
). On the basis of Figures 3B and 4B, we expect that GroEL substrates in E. coli should be slow folding, aggregation prone, or both. The latter prediction is consistent with experimental results (
• Chapman E.
• Farr G.W.
• Usaite R.
• Furtak K.
• Fenton W.A.
• Chaudhuri T.K.
• Hondorp E.R.
• Matthews R.G.
• Wolf S.G.
• Yates J.R.
• et al.
Global aggregation of newly translated proteins in an Escherichia coli strain deficient of the chaperonin GroEL.
,
• Fujiwara K.
• Ishihama Y.
• Nakahigashi K.
• Soga T.
• Taguchi H.
A systematic survey of in vivo obligate chaperonin-dependent substrates.
,
• Kerner M.J.
• Naylor D.J.
• Ishihama Y.
• Maier T.
• Chang H.C.
• Stines A.P.
• Georgopoulos C.
• Frishman D.
• Hayer-Hartl M.
• Mann M.
• Hartl F.U.
Proteome-wide analysis of chaperonin-dependent protein folding in Escherichia coli.
). The former prediction must await proteome-wide determination of folding rate constants to be evaluated, but it is consistent with the finding that obligate GroEL substrates tend to have topologically complicated structures (
• Fujiwara K.
• Ishihama Y.
• Nakahigashi K.
• Soga T.
• Taguchi H.
A systematic survey of in vivo obligate chaperonin-dependent substrates.
,
• Kerner M.J.
• Naylor D.J.
• Ishihama Y.
• Maier T.
• Chang H.C.
• Stines A.P.
• Georgopoulos C.
• Frishman D.
• Hayer-Hartl M.
• Mann M.
• Hartl F.U.
Proteome-wide analysis of chaperonin-dependent protein folding in Escherichia coli.
), because proteins with such structures generally fold slowly (
• Plaxco K.W.
• Simons K.T.
• Baker D.
Contact order, transition state placement and the refolding rates of single domain proteins.
).
### FoldEco as a Web Resource
To enable those interested to perform their own experiments with FoldEco, we have created a web version of FoldEco, the home page of which is http://foldeco.scripps.edu. This website has an overview of the model, descriptions of its components and subsystems, and basic and advanced interactive pages into which parameters can be input and then sent to a server. The server solves FoldEco with the user-defined parameters and returns the output. There are several options for the output: fully graphical depictions of the simulation results (including static images, movies, and interactive animation), concentration versus time plots for a selection of the species in the model, and downloadable tables with concentration versus time data for some or all of the species in the model. Complete guides to the input parameters and the output data are provided on the website. We expect this publicly available resource to prove useful as a means to both generate hypotheses and rationalize existing data about proteostasis.
## Experimental Procedures
### General
The differential equations that make up FoldEco were solved numerically with the use of Mathematica 8.0.1 (Wolfram Research) on a Dell Precision T7500 Workstation running Windows 7 Professional (64 bit) with a 3.33 GHz Intel Xeon W5590 Quadcore CPU and 24 GB of RAM. FoldEco is implemented on the web with the use of Wolfram webMathematica 3.0 running on a VMware virtual server running CentOS installed on an Oracle Sun Blade server. Details about setting up and solving FoldEco can be found in the Extended Experimental Procedures. A Mathematica application file with the code for FoldEco and a Mathematica notebook file demonstrating how to use FoldEco are provided in Folder S1.
### Setting Up and Solving FoldEco
The differential equations that make up FoldEco can be derived from Figure 1 (or Figures S1, S2, S3, S4, and S5). The net rate of change of each species' concentration is equal to the sum of the rates of the processes that produce it, less the sum of the rates of the processes that consume it (where the rate of a process is its rate constant multiplied by the concentrations of the species that the process consumes). For example, free inactive ribosomes (R in Figure 1) are consumed when they become translationally active (forming Ra,i) or when they bind trigger factor (T). R is produced by dissociation of R:T, and by dissociation of Ra:Ui and Ra:Ui:T after translation is complete. The rate of change of the concentration of R is therefore
$d[R]dt=−s0,1[R][T]−si,3[R]+s0,2[R:T]+si,11[Ra:Ui]+si,12[Ra:Ui:T]$
(1)
where the square brackets indicate concentrations, d[R]/dt is the time derivative of [R], and s0,1, si,3, s0,2, si,11, and si,12 are rate constants (the first subscript of these rate constants is 0 if the process does not involve a client protein and i if it does; the second subscript indicates the reaction number in the synthesis system of FoldEco; see Figure S1). FoldEco is solved by using the numerical differential equation solver in Mathematica, which adaptively chooses its integration methods and step sizes depending on the properties of the system of differential equations (e.g., smoothness and stiffness of the solutions).
### Assumptions and Simplifications in FoldEco
Models like FoldEco that are intended to capture the complexity of biological processes occurring in vivo inevitably contain some assumptions and simplifications; without them, it is sometimes impossible to make progress. Here we present a list of the most important assumptions and simplifications in the current version of FoldEco.
### FoldEco Is a Deterministic, Not a Stochastic, Model
FoldEco consists of a set of ordinary differential equations that are numerically integrated to produce a solution that consists of the time-dependent concentrations of the species in the model. As such, FoldEco is valid for examining the average state of proteostasis in populations of E. coli under a given set of conditions. Focusing on population averages means that FoldEco cannot capture cell-to-cell variability, but it has the advantage that measuring population averages for variables of interest is often experimentally easier than measuring the same variables in individual cells.
### FoldEco Assumes Free Mixing of Components
FoldEco treats E. coli as single compartments (the cytoplasm) that are freely mixed; that is, we assume that the concentrations of the species in the model do not vary across the cell. Many factors could cause concentration heterogeneities within cells. However, the data to parameterize the spatial dependence of in vivo protein concentrations is only now becoming available (
Beck, M., Topf, M., Frazier, Z., Tjong, H., Xu, M., Zhang, S., and Alber, F. (2011). Exploring the spatial and temporal organization of a cell's proteome. J. Struct. Biol. 173, 483–496.
), and some of it suggests that FoldEco components like GroEL (
Charbon, G., Wang, J., Brustad, E., Schultz, P.G., Horwich, A.L., Jacobs-Wagner, C., and Chapman, E. (2011). Localization of GroEL determined by in vivo incorporation of a fluorescent amino acid. Bioorg. Med. Chem. Lett. 21, 6067–6070.
) and DnaK and DnaJ (
Taniguchi, Y., Choi, P.J., Li, G.W., Chen, H., Babu, M., Hearn, J., Emili, A., and Xie, X.S. (2010). Quantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells. Science 329, 533–538.
) are in fact uniformly distributed. Thus, rather than complicating our model by adding the capacity to account for spatially heterogeneous concentrations of FoldEco components as well as introducing parameters for which we do not have estimates, we use the free mixing approximation.
### Parameters Measured In Vitro Are Used for Processes Occurring In Vivo
The default parameters used in FoldEco were all measured in vitro. However, FoldEco is meant to simulate processes in vivo. Some features of the in vivo environment are easy to mimic in vitro (pH, ionic strength), but some are not; in particular, the presence of the cellular proteome is expected to affect protein folding and protein association energetics (
Ebbinghaus, S., and Gruebele, M. (2011). Protein Folding Landscapes in the Living Cell. J. Phys. Chem. Lett. 2, 314–319.
,
Gershenson, A., and Gierasch, L.M. (2011). Protein folding in the cell: challenges and progress. Curr. Opin. Struct. Biol. 21, 32–41.
). Measurements of protein stabilities in vivo have suggested that they are often similar to in vitro stabilities (
Ghaemmaghami, S., and Oas, T.G. (2001). Quantitative protein stability measurement in vivo. Nat. Struct. Biol. 8, 879–882.
,
Ignatova, Z., and Gierasch, L.M. (2004). Monitoring protein stability and aggregation in vivo by real-time fluorescent labeling. Proc. Natl. Acad. Sci. USA 101, 523–528.
), but they are sometimes higher (
Dhar, A., Girdhar, K., Singh, D., Gelman, H., Ebbinghaus, S., and Gruebele, M. (2011). Protein stability and folding kinetics in the nucleus and endoplasmic reticulum of eucaryotic cells. Biophys. J. 101, 421–430.
,
Ebbinghaus, S., Dhar, A., McDonald, J.D., and Gruebele, M. (2010). Protein folding stability and dynamics imaged in a living cell. Nat. Methods 7, 319–323.
), and high protein concentrations can also lower protein stability (
Miklos, A.C., Sarkar, M., Wang, Y., and Pielak, G.J. (2011). Protein crowding tunes protein stability. J. Am. Chem. Soc. 133, 7116–7120.
). Given this uncertainty, we have taken a conservative approach and chosen to use the in-vitro-measured parameters without correction. These parameters will be replaced in the model as specific information about rate or equilibrium constants for the processes in FoldEco becomes available.
### FoldEco Does Not Account for Bacterial Growth
In its present form, FoldEco does not account for bacterial growth, which in effect dilutes proteins as they are being synthesized. Thus, FoldEco best represents bacteria that are not growing. This condition is met by bacteria that are overexpressing heterologous proteins, for which growth is strongly inhibited (
Dong, H., Nilsson, L., and Kurland, C.G. (1995). Gratuitous overexpression of genes in Escherichia coli leads to growth inhibition and ribosome destruction. J. Bacteriol. 177, 1497–1504.
,
Kurland, C.G., and Dong, H. (1996). Bacterial growth inhibition by overproduction of protein. Mol. Microbiol. 21, 1–4.
), and bacteria in the lag and stationary phases. Future versions of FoldEco will account for bacterial growth.
### FoldEco Does Not Account for the Presence of the Background Proteome
The FoldEco simulations reported herein track the synthesis and folding of a single client protein. However, no protein is produced in isolation: there is a background proteome that in principle could compete for proteostasis network components. This problem is acute during bacterial growth, as many, if not most, newly synthesized proteins interact with the proteostasis network (
• Calloni G.
• Chen T.
• Schermann S.M.
• Change H.-C.
• Genevaux P.
• Agostini F.
• Tartaglia G.G.
• Hayer-Hartl M.
• Hartl F.U.
DnaK functions as a central hub in the E. coli chaperone network.
,
• Chapman E.
• Farr G.W.
• Usaite R.
• Furtak K.
• Fenton W.A.
• Chaudhuri T.K.
• Hondorp E.R.
• Matthews R.G.
• Wolf S.G.
• Yates J.R.
• et al.
Global aggregation of newly translated proteins in an Escherichia coli strain deficient of the chaperonin GroEL.
,
• Fujiwara K.
• Ishihama Y.
• Nakahigashi K.
• Soga T.
• Taguchi H.
A systematic survey of in vivo obligate chaperonin-dependent substrates.
,
• Kerner M.J.
• Naylor D.J.
• Ishihama Y.
• Maier T.
• Chang H.C.
• Stines A.P.
• Georgopoulos C.
• Frishman D.
• Hayer-Hartl M.
• Mann M.
• Hartl F.U.
Proteome-wide analysis of chaperonin-dependent protein folding in Escherichia coli.
) and would therefore compete with a client protein of interest. The problem is much reduced when bacteria are not growing, as the proteostasis network does not interact to nearly the same extent with pre-existing proteins (
• Calloni G.
• Chen T.
• Schermann S.M.
• Change H.-C.
• Genevaux P.
• Agostini F.
• Tartaglia G.G.
• Hayer-Hartl M.
• Hartl F.U.
DnaK functions as a central hub in the E. coli chaperone network.
). Thus, again, FoldEco is most appropriate for use in situations like the overexpression of heterologous proteins, or the lag and stationary phases, where E. coli are producing few proteins other than the one of interest. In ongoing work, we are extending FoldEco to account for the proteome, using a realistic parameterization and coarse graining methods.
### FoldEco Does Not Have the Proteostasis Network Components for Membrane or Periplasmic Proteins
Membrane proteins have specialized systems for their folding and quality control (
Dalbey, R.E., Wang, P., and Kuhn, A. (2011). Assembly of bacterial inner membrane proteins. Annu. Rev. Biochem. 80, 161–187.
,
Knowles, T.J., Scott-Tucker, A., Overduin, M., and Henderson, I.R. (2009). Membrane protein architects: the role of the BAM complex in outer membrane protein assembly. Nat. Rev. Microbiol. 7, 206–214.
), as do periplasmic proteins (
Allen, W.J., Phan, G., and Waksman, G. (2009). Structural biology of periplasmic chaperones. Adv Protein Chem Struct Biol 78, 51–97.
). These systems are not included in the current version of FoldEco, which is intended for use with cytoplasmic proteins, but will be included in future versions.
### FoldEco Does Not Account for Stress Responses
FoldEco does not yet account for stress responses, like the heat shock response (
Guisbert, E., Yura, T., Rhodius, V.A., and Gross, C.A. (2008). Convergence of molecular, modeling, and systems approaches for an understanding of the Escherichia coli heat shock response. Microbiol. Mol. Biol. Rev. 72, 545–554.
), which can upregulate the levels of proteostasis network components in response to environmental changes that affect protein folding. However, such stress responses can be roughly approximated by increasing the initial concentrations of the relevant components and accounting for the effect of the stress on the protein folding parameters (for example, increasing the temperature would be expected to destabilize a protein). Future versions of FoldEco will incorporate stress responses.
### FoldEco Does Not account for Fluctuations in ATP Concentration
In this version of FoldEco, we have assumed that the concentration of ATP is in the millimolar range and is constant, so that ATP binding is never rate limiting. However, FoldEco can be used to track ATP use by the proteostasis network, since ATP consumption rates are known for the systems in FoldEco (Figures S1, S2, S3, S4, and S5).
### Concentrations of mRNA Are Assumed to Be at Steady State in FoldEco
The rate constant for translation initiation, si,3, is assumed to be constant throughout FoldEco simulations. This is equivalent to assuming that the concentration of mRNA for the client protein of interest is at steady state at the beginning of the simulation. This assumption is consistent with observations that transcription is not rate limiting in protein production (
Sandén, A.M., Prytz, I., Tubulekas, I., Förberg, C., Le, H., Hektor, A., Neubauer, P., Pragai, Z., Harwood, C., Ward, A., et al. (2003). Limiting factors in Escherichia coli fed-batch production of recombinant proteins. Biotechnol. Bioeng. 81, 158–166.
) that protein synthesis lags mRNA synthesis (
Golding, I., Paulsson, J., Zawilski, S.M., and Cox, E.C. (2005). Real-time kinetics of gene activity in individual bacteria. Cell 123, 1025–1036.
), and that protein translation can occur co-transcriptionally (
Miller, O.L., Jr., Hamkalo, B.A., and Thomas, C.A., Jr. (1970). Visualization of bacterial genes in action. Science 169, 392–395.
). However, given this assumption, one must be judicious in choosing the t = 0 time point when simulating experimental data. The t = 0 time point should be assigned to the time at which the client protein mRNA (and therefore the rate of protein synthesis) reaches steady state, not necessarily the time when expression is induced. When mRNA levels reach steady state rapidly compared to the time scale of the simulation (say, within 15 min for a 3 hr simulation), the time when expression is induced can be assigned to be t = 0. This condition should be satisfied most of the time, given that the approach of the concentration of a given mRNA to its steady state value is determined by its degradation rate, and the half-lives of mRNA in E. coli are generally 3-8 min (
Bernstein, J.A., Khodursky, A.B., Lin, P.H., Lin-Chao, S., and Cohen, S.N. (2002). Global analysis of mRNA decay and abundance in Escherichia coli at single-gene resolution using two-color fluorescent DNA microarrays. Proc. Natl. Acad. Sci. USA 99, 9697–9702.
). When this condition is not met, the time point at which the protein synthesis rate reaches steady state should be assigned to be t = 0.
### FoldEco Uses a Simplified Model of Translation
In FoldEco, protein synthesis is represented as a simplified, three-step process. The synthesis of a client protein (protein “i”) begins with the initiation of translation. This process involves several steps (
Laursen, B.S., Sørensen, H.P., Mortensen, K.K., and Sperling-Petersen, H.U. (2005). Initiation of protein synthesis in bacteria. Microbiol. Mol. Biol. Rev. 69, 101–123.
), but is simplified to a single step in FoldEco. This simplification is appropriate since there is a single rate limiting step in translation initiation: the rearrangement to the stable 30S initiation complex of the metastable ternary complex between the 30S ribosome subunit, mRNA, and the formyl-Met-charged tRNA (
Gualerzi, C., Risuleo, G., and Pon, C.L. (1977). Initial rate kinetic analysis of the mechanism of initiation complex formation and the role of initiation factor IF-3. Biochemistry 16, 1684–1689.
,
Laursen, B.S., Sørensen, H.P., Mortensen, K.K., and Sperling-Petersen, H.U. (2005). Initiation of protein synthesis in bacteria. Microbiol. Mol. Biol. Rev. 69, 101–123.
). The next step is mRNA translation up to the point that the nascent protein is long enough to emerge from the ribosomal exit tunnel, and the final step is translation of the remainder of the mRNA. We treat both of these processes as single steps, rather than treating the translation of individual codons separately, as has been done in some models of translation (
Zouridis, H., and Hatzimanikatis, V. (2007). A model for protein translation: polysome self-organization leads to maximum protein synthesis rates. Biophys. J. 92, 717–730.
). This approach is justified here, because the rate of protein synthesis rapidly comes to steady state (within ∼1 min for all of the simulations described in this work) on the time scales of interest to us (15 min to 2.75 hr). Using a finer grained model of translation is therefore not necessary.
### FoldEco Does Not Explicitly Account for Co-translational Folding and Misfolding
Many proteins are known to begin folding co-translationally, especially multi-domain proteins (
Eichmann, C., Preissler, S., Riek, R., and Deuerling, E. (2010). Cotranslational structure acquisition of nascent polypeptides monitored by NMR spectroscopy. Proc. Natl. Acad. Sci. USA 107, 9111–9116.
,
Evans, M.S., Sander, I.M., and Clark, P.L. (2008). Cotranslational folding promotes beta-helix formation and avoids aggregation in vivo. J. Mol. Biol. 383, 683–692.
,
Fedorov, A.N., and Baldwin, T.O. (1995). Contribution of cotranslational folding to the rate of formation of native protein structure. Proc. Natl. Acad. Sci. USA 92, 1227–1231.
,
Frydman, J., Erdjument-Bromage, H., Tempst, P., and Hartl, F.U. (1999). Co-translational domain folding as the structural basis for the rapid de novo folding of firefly luciferase. Nat. Struct. Biol. 6, 697–705.
,
Kelkar, D.A., Khushoo, A., Yang, Z., and Skach, W.R. (2012). Kinetic analysis of ribosome-bound fluorescent proteins reveals an early, stable, cotranslational folding intermediate. J. Biol. Chem. 287, 2568–2578.
,
• Siller E.
• DeZwaan D.C.
• Anderson J.F.
• Freeman B.C.
• Barral J.M.
Slowing bacterial translation speed enhances eukaryotic protein folding efficiency.
). The role of co-translational folding in E. coli, however, is diminished by two factors. First, bacterial ribosomes translate proteins much faster than eukaryotic ribosomes, leaving less time for co-translational folding (
• Siller E.
• DeZwaan D.C.
• Anderson J.F.
• Freeman B.C.
• Barral J.M.
Slowing bacterial translation speed enhances eukaryotic protein folding efficiency.
). Second, trigger factor and/or DnaK bind to most proteins as they emerge from the ribosomal exit tunnel (
Deuerling, E., Schulze-Specking, A., Tomoyasu, T., Mogk, A., and Bukau, B. (1999). Trigger factor and DnaK cooperate in folding of newly synthesized proteins. Nature 400, 693–696.
,
Teter, S.A., Houry, W.A., Ang, D., Tradler, T., Rockabrand, D., Fischer, G., Blum, P., Georgopoulos, C., and Hartl, F.U. (1999). Polypeptide flux through bacterial Hsp70: DnaK cooperates with trigger factor in chaperoning nascent chains. Cell 97, 755–765.
,
Wegrzyn, R.D., and Deuerling, E. (2005). Molecular guardians for newborn proteins: ribosome-associated chaperones and their role in protein folding. Cell. Mol. Life Sci. 62, 2727–2738.
). This role of trigger factor and DnaK is incorporated implicitly into the model by not including a pathway for co-translational folding or misfolding. This approach should yield acceptable results, except in the case of double deletions of trigger factor and DnaK. Future versions of FoldEco will explicitly include the possibility of co-translational folding and misfolding.
### Aggregation in FoldEco
Aggregates composed of two, three, etc. monomers and all of their complexes with chaperones (of which there are nine) count as individual species in FoldEco. The size of the system of equations in FoldEco therefore depends on the maximum aggregate size allowed. We have found that artifacts arise when the cut-off size is too small, while computation times are too long when the cut-off size is too large (computation time scales quadratically with the number of equations being solved: with a maximum aggregate size of 100, FoldEco can be solved in ∼1 s, but with a maximum size of 1000, 20-40 s are needed). We find that a maximum aggregate size between 50 and 200 is a good compromise between these considerations. An aggregate consisting of 200 monomers is much smaller than typical inclusion bodies (
Taylor, G., Hoare, M., Gray, D.R., and Marston, F.A.O. (1986). Size and density of protein inclusion bodies. Biotechnology 4, 553–557.
), but increasing the size cut-off does not change the behavior of the model substantially (see below for more information on the interpretation of the maximum fibril size setting).
### The Stoichiometry of B+KJE System Components Binding to Aggregates Is Assumed to Be 1:1
In principle, it should be possible for multiple ClpB molecules to be working in parallel on a single aggregate (provided that the aggregate is large enough). However, in the mechanism for the B+KJE disaggregation system (Figure S4), a 1:1 stoichiometry is assumed between disaggregation system components and aggregates. Unfortunately, this is a computational necessity. Each aggregate size is tracked individually in the model; thus, if aggregates composed of up to n monomers can each have m ClpB molecules bound to them, then n × m equations would be needed to track all of the aggregate:ClpB species, which would greatly increase computation times. This problem is mitigated, however, upon consideration of how FoldEco represents aggregation. The most intuitive way to view the aggregated states A2, A3,…, An is as individual oligomers. In a rough approximation, they also could be viewed as reactive sites on a single large aggregate instead, in which case the number concentration of aggregates $(∑i=2nAi)$ is equivalent to the number of reactive sites on the single aggregate, the size-weighted concentration of aggregates $(∑i=2ni×Ai)$ is equivalent to the total number of monomers in the single aggregate, and n corresponds to the maximum number of monomers in the aggregate per reactive site. In this scenario, ClpB molecules can attach to each reactive site, and the density of reactive sites is controlled by adjusting n, with the number of reactive sites decreasing as n increases.
### Proteostasis Network Components Not Currently in FoldEco
Some chaperoning systems are not yet included in FoldEco, like the bacterial small heat shock proteins, IbpA and IbpB, and the bacterial Hsp90, HtpG. However, the effects of these systems on proteostasis appear to be subtle (
Bardwell, J.C., and Craig, E.A. (1988). Ancient heat shock gene is dispensable. J. Bacteriol. 170, 2977–2983.
,
Thomas, J.G., and Baneyx, F. (1998). Roles of the Escherichia coli small heat shock proteins IbpA and IbpB in thermal stress management: comparison with ClpA, ClpB, and HtpG In vivo. J. Bacteriol. 180, 5165–5172.
). Nevertheless, future versions of FoldEco will incorporate these proteostasis network components as the necessary mechanistic information becomes available.
### Synthesis and Degradation Rate Simulation Experiments
We determined how the balance between synthesis and degradation affects aggregation in the absence of chaperones in FoldEco by seeking conditions that would allow a model client protein with kf = 1 s−1, Kf = 1000, km = 1 s−1, Km = 10, ka = 0.1 μM−1 s−1, and Ccrit = 0.1 μM (the other parameters used are listed Table S2) to be less than 5% aggregated after 10,000 s of simulation time. To do this, the chaperone concentrations were set to 0 and FoldEco was solved for the model protein at a constant synthesis rate while the Lon concentration was varied (using a quasi-Newton method as implemented by the Mathematica function FindMinimum) until the amount of aggregates at 10,000 s was 5% of total protein. Above this Lon concentration, in the unshaded region of Figure 2A, aggregates are < 5% of total protein; below it, in the shaded region of Figure 2A, aggregates are > 5%. This was repeated at several synthesis rates, ranging from 0.01 μM s−1 to 1 μM s−1, to give the points that, when joined, give the black curve in Figure 2A. FoldEco was solved again with the synthesis rates and Lon concentrations found above to determine the native protein concentration and the fraction of synthesized protein that has been degraded at 10,000 s. The synthesis rate was varied by varying si,3 and si,4 between 0.000504 s−1 and 0.375 s−1 with si,7 = si,8 = 0.25 s−1, si,11 = si,12 = 0.1 s−1 (appropriate for synthesis of a 200-residue protein at 20 residues s−1; Figure S1), and a ribosome concentration of 20 μM. This procedure was repeated with the default chaperone concentrations (Table S1) to determine how including chaperones modulated the effect of the balance between synthesis and degradation on the aggregation of the model client protein described above.
We solved FoldEco for the test protein described in the “Synthesis and Degradation Rate Simulation Experiments,” which has the following set of folding parameters: kf = 1 s−1, Kf = 1000, km = 1 s−1, Km = 10, ka = 0.1 μM−1 s−1, and Ccrit = 0.1 μM, and an initial Lon concentration of 0.3 μM. The other parameters used are listed in Table S2. The solutions for unfolded, misfolded and native protein are shown in Figure S6B. These plots show that protein accumulation occurs on two time scales. It begins rapidly, with the unfolded and misfolded states reaching a pseudo-steady state within 50 s. The concentrations of these species slowly increase after that until they reach their true steady state at 15,000-20,000 s.
### Analysis of a Simplified Model for Protein Synthesis, Folding, and Degradation
We analyzed the behavior of FoldEco on the two time scales described above using a simplified model that includes only synthesis, folding, and degradation (Figure S6C). Synthesis is condensed to a single step with rate σ. Degradation is simplified to a Briggs-Haldane scheme, where D is a Lon-type protease that degrades U and M. The rate constants for substrate binding, release, and degradation (δ1, δ2, δ3) are assumed to be the same for U and M. The folding and unfolding rate constants are φ1 and φ2, while those for misfolding and un-misfolding are μ1 and μ2. The rate equations for the system are
$d[U]dt=σ−μ1[U]+μ2[M]−φ1[U]+φ2[N]−δ1[D][U]+δ2[D:U]$
(S1)
$d[M]dt=μ1[U]−μ2[M]−δ1[D][M]+δ2[D:M]$
(S2)
$d[N]dt=φ1[U]−φ2[N]$
(S3)
$d[D:U]dt=δ1[D][U]−δ2[D:U]−δ3[D:U]$
(S4)
$d[D:M]dt=δ1[D][M]−δ2[D:U]−δ3[D:U]$
(S5)
$d[D]dt=−δ1[D][U]−δ1[D][M]+δ2[D:U]+δ2[D:M]+δ3[D:U]+δ3[D:M]$
(S6)
There is also one conservation equation:
$[D]tot=[D]+[D:U]+[D:M]$
(S7)
We begin by examining the true steady state, where all of the rate equations above are equal to 0. Setting equations S4 and S5 equal to 0 and rearranging them gives
$[D:U]s=δ1[U]s[D]sδ2+δ3$
(S8)
$[D:M]s=δ1[M]s[D]sδ2+δ3$
(S9)
where the subscript “s” indicates steady state. Summing equations (S1) and (S2), and equations (S4) and (S5) gives
$[d[U]dt+d[M]dt]s=0=σ−φ1[U]s+φ2[N]s−δ1[D]s([U]s+[M]s)+δ2([D:U]s+[D:M]s)$
(S10)
$[d[D:U]dt+d[D:M]dt]s=0=δ1[D]s([U]s+[M]s)−(δ2+δ3)([D:U]s+[D:M]s)$
(S11)
And from equation (S3) we have
$0=φ1[U]s−φ2[N]s$
(S12)
Summing equations (S10), (S11), and (S12) gives
$0=σ−δ3([D:U]s+[D:M]s)$
or
$[D:U]s+[D:M]s=σδ3$
(S13)
Inserting this into equation (S8) for the conservation of D and rearranging gives
$[D]s=[D]tot−σδ3$
(S14)
So, the concentration of free protease can be written simply in terms of the total protease concentration ([D]tot), the synthesis rate (σ), and the equivalent of kcat for the protease (δ3). In addition, combining equations (S10), (S12), (S13), and (S14) and rearranging gives
$[U]s+[M]s=σ(1+δ2/δ3)δ1([D]tot−σδ3)=σ(δ2+δ3)δ1δ3([D]tot−σδ3)$
(S15)
For convenience we define
$Δs=δ1([D]tot−σδ3)(1+δ2/δ3)=δ1δ3([D]tot−σδ3)δ2+δ3=δ1δ3[D]sδ2+δ3$
(S16)
Δs is the effective degradation rate constant. Equation (S15) can now be written as
$[U]s+[M]s=σΔs$
(S17)
We note that if σ/δ3 > [D]tot, then [D]s and Δs would be negative, which is not physically reasonable. In this situation, synthesis and degradation cannot be balanced, and there is no steady state. The amount of protein in the system simply increases without bound.
Equation (S17) shows that the sum of the concentrations of U and M at true steady state can be written in terms of the synthesis and degradation parameters, independent of the folding parameters. The ratio of [U]s and [M]s can be obtained by inserting equation (S9) into equation (S2) to give
$[d[M]dt]s=0=μ1[U]s−μ2[M]s−δ1[D]s[M]s+δ2(δ1[M]s[D]sδ2+δ3)$
(S18)
or
$0=μ1[U]s−μ2[M]s−[M]sδ1δ3([D]tot−σδ3)δ2+δ3=μ1[U]s−μ2[M]s−Δs[M]s$
(S19)
The ratio of [M] and [U] at steady state is then
$[M]s[U]s=μ1μ2+Δs$
(S20)
At equilibrium, in the absence of synthesis and degradation, this ratio would be
$[M]eq[U]eq=μ1μ2$
Since Δs has to be positive if steady state is possible, the steady state ratio of [M] and [U] has to be lower than their equilibrium ratio. In other words, the degradation system decreases the apparent steady-state stability of the misfolded state. Finally, the steady state ratio of [N] and [U] is
$[N]s[U]s=φ1φ2$
(S21)
The steady state ratio of [N] and [U] is the same as it is at equilibrium.
We analyze the earlier pseudo-steady state by assuming that the rate equations for all of the species in the system except N are approximately equal to 0, and that φ1[U] ≫ φ2[N] (i.e., unfolding is negligible compared to folding). The equations for processes that do not involve interconversion or interaction with the native state retain their form, so that equations (S8) and (S9) become
$[D:U]ps=δ1[U]ps[D]psδ2+δ3$
(S22)
$[D:M]ps=δ1[M]ps[D]psδ2+δ3$
(S23)
where the subscript “ps” indicates pseudo-steady state. We begin our analysis of the pseudo-steady state by summing equations (S1) and (S4), and equations (S2) and (S5) to get
$[d[U]dt+d[D:U]dt]ps=0=σ−(φ1+μ1)[U]ps+μ2[M]ps−δ3[D:U]ps$
(S24)
$[d[M]dt+d[D:M]dt]ps=0=μ1[U]ps−μ2[M]ps−δ3[D:M]ps$
(S25)
Inserting the expressions for [D:U] and [D:M] into equations (S24) and (S25) gives
$[d[U]dt+d[D:U]dt]ps=0=σ−(φ1+μ1)[U]ps+μ2[M]ps−[U]psδ1δ3[D]psδ2+δ3$
(S26)
$[d[M]dt+d[D:M]dt]ps=0=μ1[U]ps−μ2[M]ps−[M]psδ1δ3[D]psδ2+δ3$
(S27)
The coefficients in the last terms in these equations are similar to the expression for Δs. We therefore define
$Δps=δ1δ3[D]psδ2+δ3=δ1[D]ps1+δ2/δ3$
(S28)
where [D]ps is the pseudo-steady state concentration of D. Equations (S26) and (S27) can be rearranged and written in matrix form
$[(φ1+μ1+Δps)−μ2−μ1μ2+Δps][[U]ps[M]ps]=Rps[[U]ps[M]ps]=[σ0]$
(S29)
The determinant of the rate constant matrix Rps is
$det(Rps)=Δps(μ1+μ2+Δps)+φ1(μ2+Δps)$
(S30)
The solution to equation (S29) is
$[[U]ps[M]ps]=σdet(Rps)[μ2+Δpsμ1]$
(S31)
From this solution, the sum of [U] and [M] at pseudo-steady state is
$[U]ps+[M]ps=σ(μ1+μ2+Δps)det(Rps)=σΔps+φ1(11+μ1μ2+Δps)$
(S32)
This is similar in form to the expression at steady state in equation (S17), except that there is a second term in the denominator, and it depends explicitly on folding parameters: the folding rate φ1 and the quantity μ1/(μ2 + Δps). This latter quantity is the pseudo-steady state ratio of [M] to [U], as is evident from equation (S31):
$[M]ps[U]ps=μ1μ2+Δps$
(S33)
Again, Δps has to be positive, so the ratio of [M] and [U] at pseudo-steady state is less than it is at equilibrium, just as it is at true steady state. Thus, degradation decreases the apparent steady-state stability of the misfolded state throughout the time course of protein synthesis, and not just at steady state.
The effect of this apparent destabilization of the misfolded state on the concentration of the native state is mediated by its effect on the concentration of the unfolded state. The concentration of native protein at steady state is proportional to the concentration of unfolded protein, as illustrated by rearranging equation (S21).
$[N]s=φ1φ2[U]s$
(S34)
The rate of accumulation of native protein at pseudo-steady state is also proportional to the concentration of unfolded protein. Given the pseudo-steady state assumptions, the rate equation for [N] becomes
$[d[N]dt]ps=φ1[U]ps$
(S35)
Because degradation destabilizes the misfolded state relative to the unfolded state at both pseudo- and true steady state, the concentration of unfolded protein is higher than one would expect based on its equilibrium energetics under these conditions. As a consequence, both the rate of accumulation of natively folded protein and its final concentration are higher because of degradation. This result explains the observation that jointly increasing the synthesis rate and protease concentration can result in an increase in the native state concentration throughout the time course of a FoldEco simulation.
As noted above and as apparent from Figure S6B, protein accumulation in FoldEco happens in two stages. The pseudo-steady state is reached rapidly, and is followed by a much slower evolution to true steady state. The rate of approach to true steady state cannot be determined analytically, even for this simplified system. We can, however, get some information by examining the asymptotic rate of approach to steady state using perturbation methods. To do this, we first observe that the equations for [D:U] and [D:M] have the same form at pseudo- and true steady state, and so it is reasonable to assume that they retain this form throughout the approach to steady state. Thus,
$[D:U]t=δ1[U]t[D]tδ2+δ3$
(S36)
$[D:M]t=δ1[M]t[D]tδ2+δ3$
(S37)
where the subscript “t” indicates that the equations are valid between pseudo- and true steady state. Combining equations (S36) and (S37) with equation (S7) yields
$[D]t=[D]tot1+δ1([U]t+[M]t)δ2+δ3$
(S38)
Inserting equations (S36) and (S37) into equations (S1), (S2), and (S3) gives
$[d[U]dt]t=σ−μ1[U]t+μ2[M]t−φ1[U]t+φ2[N]t−δ1[D]t[U]t+δ2δ1[U]t[D]tδ2+δ3$
(S39)
$[d[M]dt]t=μ1[U]t−μ2[M]t−δ1[D]t[M]t+δ2δ1[M]t[D]tδ2+δ3$
(S40)
$[d[N]dt]t=φ1[U]t−φ2[N]t$
(S41)
Close to steady state, [U]t differs from [U]s by only a small quantity u:
$[U]t=[U]s+u$
(S42)
Similarly, for [M]t, [D]t, and [N]t:
$[M]t=[M]s+m$
(S43)
$[D]t=[D]s+d$
(S44)
$[N]t=[N]s+n$
(S45)
The value of d can be approximated in terms of u and m using equation (S38) and noting that [D]t = [D]s at steady state
$d=∂[D]s∂[U]su+∂[D]s∂[M]sm=−[D]tot(δ1δ2+δ3)(u+m)(1+δ1([U]s+[M]s)δ2+δ3)2$
(S46)
Using equations (S14) and (S15) to substitute for [U]s + [M]s enables us to simplify equation (S46) to
$d=−δ1[D]sδ2+δ3[D]s[D]tot(u+m)$
(S47)
We note that the last two terms in equation (S39) can be combined to give
$−δ1[D]t[U]t+δ2δ1[U]t[D]tδ2+δ3=−δ1δ3[U]t[D]tδ2+δ3$
(S48)
Using equations (S42) and (S43),
$δ1δ3[U]t[D]tδ2+δ3=δ1δ3([U]s+u)([D]s+d)δ2+δ3=δ1δ3[U]s[D]sδ2+δ3+δ1δ3[D]suδ2+δ3+δ1δ3[U]sdδ2+δ3+δ1δ3udδ2+δ3$
(S49)
Neglecting the last term in which the small quantities u and d are multiplied, and using equation (S47) to replace d gives
$δ1δ3[U]t[D]tδ2+δ3=δ1δ3[U]s[D]sδ2+δ3+δ1δ3[D]suδ2+δ3+δ1δ3[U]sδ2+δ3(−δ1[D]sδ2+δ3[D]s[D]tot(u+m))$
(S50)
Inserting equation (S16) into the above gives
$δ1δ3[U]t[D]tδ2+δ3=Δs[U]s+Δsu−Δs2[U]s(u+m)δ3[D]tot=Δs[U]s+Δs(1−Δs[U]sδ3[D]tot)u−Δs2[U]sδ3[D]totm$
(S51)
Similarly,
$δ1δ3[M]t[D]tδ2+δ3=Δs[M]s+Δs(1−Δs[M]sδ3[D]tot)m−Δs2[M]sδ3[D]totu$
(S52)
With this groundwork, we can write equations for du/dt, dm/dt, and dn/dt for the asymptotic rate of approach of the system to steady state. Inserting equations (S42)-(S45) into equations (S39)-(S41) and then simplifying with equations (S51) and (S52) gives the following:
$[d[U]dt]t=[d[U]dt]s+dudt=dudt=−μ1u+μ2m−φ1u+φ2n−Δs(1−Δs[U]sδ3[D]tot)u+Δs2[U]sδ3[D]totm$
(S53)
$[d[M]dt]t=[d[M]dt]s+dmdt=dmdt=μ1u−μ2m−Δs(1−Δs[M]sδ3[D]tot)m+Δs2[M]sδ3[D]totu$
(S54)
$[d[N]dt]t=[d[N]dt]s+dndt=dndt=φ1u−φ2n$
(S55)
In matrix form,
$ddt[umn]=[−(μ1+φ1+Δs(1−Δs[U]sδ3[D]tot))μ2+Δs2[U]sδ3[D]totφ2μ1+Δs2[M]sδ3[D]tot−(μ2+Δs(1−Δs[M]sδ3[D]tot))0φ10−φ2][umn]$
(S56)
Let the matrix of rate constants above be denoted Rp, where the subscript “p” indicates that it is for the rate equations of the perturbations u, m, and n. This system of equations is first-order, homogeneous, linear, and has constant coefficients. Its solutions are therefore sums of exponentials, the coefficients and exponents of which are derived from the eigenvectors and eigenvalues of Rp. The eigenvalue of Rp with the smallest magnitude is of particular interest because it determines the rate of approach of the system to steady state. Unfortunately, the eigenvalues of a 3 × 3 matrix are the roots of a third-order polynomial, and are therefore difficult to analyze. A lower bound on these eigenvalues, however, can be determined using the Gershgorin circle theorem (
Marcus, M., and Minc, H. (1964). A survey of matrix theory and matrix inequalities (New York: Dover Publications).
). For an n × n matrix A with elements aij, we can define
$ri=∑j=1,j≠in|aji|$
(S57)
The Gershgorin circle theorem states that each eigenvalue of A is in at least one of n circles in the complex plane centered at aii and with radius ri:
${z:|z−aii|≤ri}$
(S58)
For Rp, a 3 × 3 matrix, there are 3 circles. Since the diagonal elements of Rp are real, these circles are centered on the real number axis. Furthermore, Rp is diagonally similar to a symmetric matrix, so its eigenvalues are real as well. The bounds on the eigenvalues are therefore the points where the Gershgorin circles intersect the real number axis. To shrink the circles we can use a diagonal similarity transformation, wherein Rp is multiplied by a diagonal matrix P and its inverse P−1 on the right and left, respectively.
$P−1RpP$
(S59)
This operation leaves the eigenvalues of Rp unchanged, but changes the radii of the Gershgorin circles, so that the bounds can be improved. If we take
$P=[10001000c]$
(S60)
then
$P−1=[10001000c−1]$
(S61)
and
$P−1RpP=[−(μ1+φ1+Δs(1−Δs[U]sδ3[D]tot))μ2+Δs2[U]sδ3[D]totcφ2μ1+Δs2[M]sδ3[D]tot−(μ2+Δs(1−Δs[M]sδ3[D]tot))0φ1/c0−φ2]$
(S62)
where c is a positive constant. The value of c can be chosen so that all of the Gershgorin circles lie entirely in the left half of the complex plane. With this choice, we can now concern ourselves exclusively with the lowest-magnitude (least negative) bound on the eigenvalues, that is, the intersection of the Gershgorin circles with the real number axis that is closest to 0. The intersections of the Gershgorin circle with the real number axis are given by the center ± the radius of the circles. Since the circles are entirely in the left-half of the complex plane, the intersection for which the center and radius are added is closest to 0. The sum of the centers and radii of the Gershgorin circles happens to be equal to the column sums of the matrix. These column sums are
$Column1:−φ1(1−1c)−Δs+Δs2([U]s+[M]s)δ3[D]tot=−Δs[D]s[D]tot−φ1(1−1c)$
(S63)
$Column2:−Δs[D]s[D]tot$
(S64)
$Column3:−φ2(1−c)$
(S65)
Here we have used equations (S14) and (S17) to simplify these column sums. Since c can be chosen arbitrarily, we can use it to ensure that the column 1 and column 3 sums are equal:
$−Δs[D]s[D]tot−φ1(1−1c)=−φ2(1−c)$
(S66)
Rearranging equation (S66) shows that
$φ1+cφ2=Δs[D]s[D]totc(1−c)$
(S67)
The left-hand side of equation (S67) is equal to φ1 when c = 0 and increases to φ1 + φ2 when c = 1. The right-hand side is equal to 0 when c = 0 and tends to infinity when c → 1−. Thus, the left- and right-hand sides of equation (S67) have to be equal somewhere in the range 0 ≤ c ≤ 1. With this information, we can rearrange equation (S66) as follows:
$Δs[D]s[D]tot+φ1(c−1c)=φ2(1−c)$
(S68)
In equation (S68), the second term has to be negative, because it was shown above that 0 ≤ c ≤ 1. This observation means that
$Δs[D]s[D]tot>φ2(1−c)$
(S69)
In this inequality, the left-hand side is the bound on the eigenvalue from column 2 of Rp. The right-hand side is the bound on the eigenvalue from column 3 of Rp (which is equal to the bound from column 1 by virtue of our choice of c). Inequality (S69) shows that the lowest magnitude bound on the eigenvalues of Rp, which we will denote βmin, is therefore the one derived from columns 1 and 3, that is:
$βmin=φ2(1−c)$
(S70)
The rate of approach of the system to steady state clearly depends on φ2, the rate constant for unfolding of the native state. It also depends on the value of c. An explicit expression for c can be obtained by rearranging equation (S66):
$φ2c2+(φ1−φ2+Δs[D]s[D]tot)c−φ1=0$
(S71)
This equation is quadratic in c. The positive root is:
$c=−(φ1−φ2+Δs[D]s[D]tot)+(φ1−φ2+Δs[D]s[D]tot)2+4φ1φ22φ2$
(S72)
This root can be simplified by noting that φ1 ≫ φ2 for proteins with stable native states (recall that the equilibrium constant for folding, Kf, is equal to φ12 and is usually ≫ 1). Since 0 ≤ c ≤ 1, the term φ2c2 in equation (S71) can be neglected, and the root simplifies to
$c≅φ1φ1+Δs[D]s[D]tot$
(S73)
Inserting this into equation (S70) gives
$βmin≅−φ2(1−φ1φ1+Δs[D]s[D]tot)=−φ2Δs[D]s[D]totφ1+Δs[D]s[D]tot$
(S74)
Equation (S74) shows that when Δs[D]s/[D]tot ≫ φ1, βmin is
$βmin≅−φ2$
(S75)
which happens when degradation is much faster than folding. On the other hand, when φ1 ≫ Δs[D]s/[D]tot,
$βmin≅−φ2Δs[D]s[D]totφ1=−1KfΔs[D]s[D]tot$
(S76)
which happens when folding is much faster than degradation. In summary, the rate of approach of the system to steady state is governed by the smallest eigenvalue of Rp, which is the apparent rate constant for the approach of the system to steady state. The bound on this eigenvalue is βmin, which is a function of the folding, unfolding, and degradation rates, but is never larger than the unfolding rate constant of the client protein. In FoldEco, the unfolding rate constant is given by kf / Kf, the ratio of the folding rate and equilibrium constants.
### Examination of KJE and GroELS Substrates
Four thousand random biophysical profiles were generated, comprising folding rate and equilibrium constants (kf and Kf), misfolding rate and equilibrium constants (km and Km), monomer-aggregate association rate constants (ka), and critical concentrations for aggregation (Ccrit). The values of kf and km varied between 10−3 and 103 s−1; ka varied between 10−3 and 103 μM−1 s−1; Kf varied between 102 and 108; Km varied between 10−3 and 103; and Ccrit varied between 1 mM and 1 nM. In all cases, the parameters were log-uniformly distributed (i.e., their logarithms had a uniform distribution) between the indicated bounds.
The bounds for kf are taken from a set of reported folding rates of single domain proteins (
• Ouyang Z.
• Liang J.
Predicting protein folding rates from geometric contact and amino acid sequence.
,
• Plaxco K.W.
• Simons K.T.
• Baker D.
Contact order, transition state placement and the refolding rates of single domain proteins.
). The same bounds are used for km, since misfolding and folding should be kinematically similar, despite having different endpoints. The bounds for ka correspond to nonspecific bimolecular associations at the low end to the diffusion limit at the high end (
Northrup, S.H., and Erickson, H.P. (1992). Kinetics of protein-protein association explained by Brownian dynamics computer simulation. Proc. Natl. Acad. Sci. USA 89, 3338–3342.
). Values of Kf from 102 to 108 correspond to folding free energies from about −3 to −11 kcal mol−1, which are typical of single domain proteins (although the very low end of stabilities would probably only be found in proteins with destabilizing mutations) (
• Ghosh K.
• Dill K.
Cellular proteomes have broad distributions of protein stability.
,
• Ghosh K.
• Dill K.A.
Computing protein stabilities from their chain lengths.
). Values of Km from 10−3 to 103 correspond to misfolding free energies from about +4 to −4 kcal mol−1. This range reflects our expectation that the misfolded state should be unstable for well-behaved proteins, but usually less stable than the native state even for poorly-behaved proteins. The range for Ccrit (which is also equal to the monomer-aggregate dissociation constant) corresponds to weak, nonspecific protein-protein associations at the low end to moderately strong protein-protein associations at the high end (
Kastritis, P.L., Moal, I.H., Hwang, H., Weng, Z., Bates, P.A., Bonvin, A.M., and Janin, J. (2011). A structure-based benchmark for protein-protein binding affinity. Protein Sci. 20, 482–491.
).
While using the bounds described above ensures that individual parameters will have realistic values, it is possible that combinations of parameters that do not occur in nature will be included in our data set. Such parameter combinations, however, can still be instructive about trends in chaperone behavior. Moreover, we are unaware of experimental data on joint parameter distributions that could be used to justify blocking out regions of parameter space. We therefore uniformly sampled the entire six-dimensional parameter space.
FoldEco was solved at two synthesis rates (0.02 μM s−1 and 0.2 μM s−1) with each of the random biophysical profiles under three conditions: with neither the KJE nor the GroELS system, with the KJE system (30 μM DnaK, 1 μM DnaJ, 15 μM GrpE), or with the GroELS system (3 μM GroEL tetradecamer, 5 μM GroES hexamer). The desired synthesis rates were obtained by setting si,3 and si,4 to 0.001014 s−1 or 0.011628 s−1, with si,7 = si,8 = 0.25 s−1, si,11 = si,12 = 0.1 s−1 (appropriate for synthesis of a 200-residue protein at 20 residues s−1; Figure S1). In all cases, Lon (0.3 μM hexamer) and trigger factor (20 μM) were included, but ClpB and native state proteases were excluded. The same association rate and equilibrium constants with DnaK, DnaJ, and GroEL (those given in Figures S1, S2, S3, S4, and S5) were used for all of the random client proteins, and the folding and misfolding rates were assumed to be the same inside the GroEL cavity as in solution. We provide a complete list of the parameters used in Table S4 (other than kf, Kf, km, Km, ka, and Ccrit which are in Tables S3 and S5). Of the 4,000 solutions at the low synthesis rate, 13 (0.325%) failed error tests (i.e., the solutions had insufficient accuracy or precision) at time steps before 10,000 s. At the high synthesis rate, 11 (0.275%) solutions failed error tests. These problems arise almost exclusively in the rare cases in which a very fast aggregation rate (ka > 102.5 μM−1 s−1) is coupled with low fibril stability (Ccrit > 100 μM), and tendencies toward high misfolded state stability (Km > 1) and a low folding rate (kf < 1 s−1). For this small region of parameter space, the solutions to FoldEco are too stiff to integrate up to 10,000 s with the default accuracy and precision goals, even with Mathematica's powerful numerical differential equation solving algorithms. This problem could be addressed by relaxing the accuracy and/or precision goals, but since such a small portion of the parameter space was affected, we simply discarded the cases in question. The solutions were evaluated at 10,000 s and the top 5% of the increases in the concentration of the native state upon adding the KJE system or the GroELS system (top 200) were identified (Tables S3 and S5).
### Testing FoldEco against Experimental Data on the Effect of Translation Rate on Luciferase Folding
Siller et al. examined the effect of translation rate on the in vivo folding efficiency of firefly luciferase (Luc), an aggregation-prone protein (
• Siller E.
• DeZwaan D.C.
• Anderson J.F.
• Freeman B.C.
• Barral J.M.
Slowing bacterial translation speed enhances eukaryotic protein folding efficiency.
). They controlled the translation rate using streptomycin pseudo-dependent (SmP) ribosomes of E. coli, which have a translation rate of 5 amino acids per second (AA s−1) under normal conditions, but an approximately two-fold higher translation rate in the presence of streptomycin (
Ruusala, T., Andersson, D., Ehrenberg, M., and Kurland, C.G. (1984). Hyper-accurate ribosomes inhibit growth. EMBO J. 3, 2575–2580.
,
• Siller E.
• DeZwaan D.C.
• Anderson J.F.
• Freeman B.C.
• Barral J.M.
Slowing bacterial translation speed enhances eukaryotic protein folding efficiency.
,
Zengel, J.M., Young, R., Dennis, P.P., and Nomura, M. (1977). Role of ribosomal protein S12 in peptide chain elongation: analysis of pleiotropic, streptomycin-resistant mutants of Escherichia coli. J. Bacteriol. 129, 1320–1329.
). They found that the aggregation of Luc decreased at the slower translation rate (in the absence of streptomycin), while the specific and absolute activities of Luc increased, after both 15 min and 1 hr of Luc expression. Our goal in this experiment was to determine to what extent these experimental results can be reproduced in simulations with FoldEco.
To do this, we first needed quantitative information on all of these effects. The extents of aggregation in equivalent amounts of Luc produced at slower or faster translation rates (here, “slower” and “faster” translation rates refers to the translation rates of SmP ribosomes in the absence or presence of streptomycin, or about 5 or 10 AA s−1, respectively) after 15 min or 1 hr of expression are illustrated by a Western blot (15 min; their Figure 2a) or a Coomassie-stained gel (1 hr; their Figure 2b). These gels qualitatively show that there is less aggregation with slower translation of Luc than with faster translation. To quantify these results, we copied the images of the gels in the pdf file of the Siller et al. paper using Adobe Acrobat, pasted the image into Adobe Photoshop, and saved them as grayscale tif files. These files, which each consisted of a set of numbers between 0 and 1 that represent the gray level of the image (0 = black, 1 = white), were imported into Mathematica and inverted (so that 1 = black and 0 = white). The result for the gel from the 15 min time point is shown as a three-dimensional plot in Figure S9A, with the original gel shown as an inset. The bands representing Luc in the lanes with soluble and pelleted protein were integrated using Mathematica to determine the relative amounts of Luc in each of these bands. The resulting integrated intensities (in arbitrary units) for the Luc bands in the gel from the 15 min time point after background subtraction were: soluble Luc, slow translation = 188.9; pelleted Luc, slow translation = 64.2; soluble Luc, fast translation = 96.4; pelleted luc, fast translation = 86.6. Thus, the fraction of Luc that remained soluble at the slow translation rate (fsol) was 0.75. This fraction decreased to 0.53 at the faster translation rate. Unfortunately, the Luc bands were not sufficiently resolved from background bands in the gel from the 1 hr time point to reliably integrate them, so from here on we focused on the data from the 15 min time point.
The relative specific activity (rSA) of Luc at 15 min was determined directly from the bar graph in Figure 2a in Siller et al. (
• Siller E.
• DeZwaan D.C.
• Anderson J.F.
• Freeman B.C.
• Barral J.M.
Slowing bacterial translation speed enhances eukaryotic protein folding efficiency.
). This graph shows the relative Luc activity in a given amount of total protein and the Luc activity present in just the soluble fraction when Luc is expressed at slow or fast translation rates. We focused on the relative activities of the soluble fractions, since they had virtually all of the Luc activity. Measuring directly from the bar graph in Figure 2a in Siller et al., we found that the ratio of the relative activities in the soluble fractions from samples of Luc expressed at slower and faster translation rates was rSA = 1.7.
To simulate these experimental results using FoldEco, we first had to constrain the input parameters to the extent possible. It has been shown that Luc folds in vitro via a mechanism that involves an initial fast equilibration between the unfolded state and a pair of aggregation-prone intermediates, which we consider together to make up the misfolded state (M) in FoldEco. This first event happens within the dead time of manual mixing experiments. Based on this observation, we assigned km = 1 s−1 (
• Herbst R.
• Gast K.
• Seckler R.
Folding of firefly (Photinus pyralis) luciferase: aggregation and reactivation of unfolding intermediates.
) The stability of these two intermediates collectively was measured to be ∼4 kcal mol−1, which corresponds to Km = 800 (
Herbst, R., Schäfer, U., and Seckler, R. (1997). Equilibrium intermediates in the reversible unfolding of firefly (Photinus pyralis) luciferase. J. Biol. Chem. 272, 7099–7105.
). The overall stability of Luc was measured to be ∼12 kcal mol−1, which corresponds to Kf = 5 × 108 (
Herbst, R., Schäfer, U., and Seckler, R. (1997). Equilibrium intermediates in the reversible unfolding of firefly (Photinus pyralis) luciferase. J. Biol. Chem. 272, 7099–7105.
). Luc aggregates at concentrations below 100 nM, so we assigned Ccrit = 10 nM; however, this process is not fast at sub-micromolar concentrations, so we assigned ka = 0.1 μM s−1 (
• Herbst R.
• Gast K.
• Seckler R.
Folding of firefly (Photinus pyralis) luciferase: aggregation and reactivation of unfolding intermediates.
,
Herbst, R., Schäfer, U., and Seckler, R. (1997). Equilibrium intermediates in the reversible unfolding of firefly (Photinus pyralis) luciferase. J. Biol. Chem. 272, 7099–7105.
). The aggregation of Luc shows some signs of behaving like a downhill polymerization—the concentration of soluble aggregation-prone intermediate in equilibrium with aggregates has a weak tendency to increase in concentration as a function of total protein (
• Herbst R.
• Gast K.
• Seckler R.
Folding of firefly (Photinus pyralis) luciferase: aggregation and reactivation of unfolding intermediates.
)—so the supercritical concentration (see Figure S1) was set equal to Ccrit.
To constrain kf, we took advantage of literature data for the in vitro KJE-system-mediated refolding of Luc. Szabo et al. measured the kinetics of reactivation of Luc (0.25 μM) in the presence of 1.25 μM DnaK, 0.25 μM DnaJ, and 1.25 μM GrpE (
• Szabo A.
• Langer T.
• Schröder H.
• Flanagan J.
• Bukau B.
• Hartl F.U.
The ATP hydrolysis-dependent reaction cycle of the Escherichia coli Hsp70 system DnaK, DnaJ, and GrpE.
). Their data (fraction of active Luc versus time) were read off of their Figure 1 and are reproduced in Figure S9B. FoldEco was then used to fit this data set by setting the concentrations of the KJE components to their experimental values, and setting the concentrations of the ribosome, proteases, and the other chaperones equal to 0. With these settings, all of the systems in FoldEco are switched off except the KJE cycle, and FoldEco simplifies to a model of KJE-system-assisted in vitro folding. The folding parameters were set to the values noted above for Luc, and the parameters in Table S2 were used for the steps in the KJE cycle. FoldEco was then fit to the experimental data using the Mathematica function NonlinearModelFit with kf as the only adjustable parameter. The fit to the data was good (R2 = 0.98; Figure S9B), with the best-fit value of kf being 0.015 s−1 (95% confidence interval: 0.010 s−1 to 0.023 s−1). This value of kf completed the biophysical profile of Luc.
The parameters for protein synthesis were constrained as follows. Siller et al. reported that the rate of protein synthesis in their system was proportional to the rate of polypeptide elongation (steps si,7 and si,8, and si,11 and si,12 in Figure S1). Protein synthesis was therefore rate limited by this process rather than translation initiation. The rate of elongation was set to 5 AA s−1 for the slower translating bacteria and 10 AA s−1 for the faster translating bacteria (
Ruusala, T., Andersson, D., Ehrenberg, M., and Kurland, C.G. (1984). Hyper-accurate ribosomes inhibit growth. EMBO J. 3, 2575–2580.
). According to Figure S1, and given that Luc is 550 residues long, this makes si,7 = si,8 = 0.125 s−1 and si,11 = si,12 = 0.0098 s−1 for the slower translating bacteria, and si,7 = si,8 = 0.25 s−1 and si,11 = si,12 = 0.0196 s−1 for the faster translating bacteria. The rate constant for the initiation of translation (si,3) was set to 0.1 s−1, so that initiation was much faster than elongation. Figure 1A in Siller et al. suggests that their bacteria double in ∼1 hr during log-phase growth for the faster translating bacteria (
• Siller E.
• DeZwaan D.C.
• Anderson J.F.
• Freeman B.C.
• Barral J.M.
Slowing bacterial translation speed enhances eukaryotic protein folding efficiency.
), so the concentration of ribosomes (which correlates with the growth rate) for these bacteria was set to 20 μM (
Bremer, H., and Dennis, P.P. (1987). Modulation of chemical composition and other parameters of the cell by growth rate. In Escherichia coli and Salmonella typhimurium: Cellular and molecular biology, F.C. Neidhardt, ed. (Washington, D. C.: American Society for Microbiology), pp. 1527–1542.
). The concentration of ribosomes was kept the same for the slower translating bacteria, since their slow growth rate was assumed to be due to their translation rate, not the concentration of their ribosomes.
The concentrations of the other proteostasis network components were set using the information in Table S1. This table shows that reported measurements of chaperone concentrations vary widely, and any set of concentrations within the 50% confidence intervals in Table S1 can be supported with experimental data. Because the expression of aggregation-prone proteins like Luc is known to induce the heat-shock response (
• Hoffmann F.
• Rinas U.
Stress induced by recombinant protein production in Escherichia coli.
), we used concentrations from the tops of these confidence intervals, which are comparable to the concentrations reported by Ishihama et al. and Lopez-Campistrous et al. (
Ishihama, Y., Schmidt, T., Rappsilber, J., Mann, M., Hartl, F.U., Kerner, M.J., and Frishman, D. (2008). Protein abundance profiling of the Escherichia coli cytosol. BMC Genomics 9, 102.
,
Lopez-Campistrous, A., Semchuk, P., Burke, L., Palmer-Stone, T., Brokx, S.J., Broderick, G., Bottorff, D., Bolch, S., Weiner, J.H., and Ellison, M.J. (2005). Localization, annotation, and comparison of the Escherichia coli K-12 proteome under two states of growth. Mol. Cell. Proteomics 4, 1205–1209.
). Thus we set the chaperone concentrations as follows: [trigger factor] = 40 μM, [DnaK] = 50 μM, [DnaJ] = 2 μM, [GrpE] = 20 μM, [GroEL tetradecamer] = 5 μM, [GroES heptamer] = 10 μM, [Lon hexamer] = 0.5 μM, [ClpB hexamer] = 0.5 μM. Siller et al. found no difference in chaperone concentrations between the slower and faster translating bacteria (
• Siller E.
• DeZwaan D.C.
• Anderson J.F.
• Freeman B.C.
• Barral J.M.
Slowing bacterial translation speed enhances eukaryotic protein folding efficiency.
), so we used the above chaperone concentrations in our simulations of both conditions.
Using the parameters noted above and the default values in Table S2 for the remaining parameters, we simulated the expression of Luc in the slower and faster translating bacteria. We took the fraction of soluble Luc (fsol) to be
$fsol=1−concentration of aggregated Luctotal Luc concentration$
A substantial amount of non-native chaperone-bound Luc was included in the soluble fraction. Siller et al. do not report experiments that can be used to judge whether their soluble protein in fact included chaperone-bound, non-native Luc (
• Siller E.
• DeZwaan D.C.
• Anderson J.F.
• Freeman B.C.
• Barral J.M.
Slowing bacterial translation speed enhances eukaryotic protein folding efficiency.
). The ratio of specific activities of Luc in the slower and faster translating bacteria (rSA) was taken to be
$rSA=concentration of native Luc in slower translating bacteriatotal Luc concentration in slower translating bacteriaconcentration of native Luc in faster translating bacteriatotal Luc concentration in faster translating bacteria$
The results of our simulation at the 15 min time point are summarized in Table S6, according to which the values of fsol for the slower and faster translating bacteria are 0.62 and 0.35, and the value of rSA is 1.4. These values are compared to the experimental results in Figure 5.
To determine the effect of chaperone concentrations on our results, the simulations were repeated with the default chaperone concentrations from Table S1 ([trigger factor] = 20 μM, [DnaK] = 30 μM, [DnaJ] = 1 μM, [GrpE] = 15 μM, [GroEL tetradecamer] = 3 μM, [GroES heptamer] = 5 μM, [Lon hexamer] = 0.3 μM, [ClpB hexamer] = 0.3 μM). The results of these simulations at lower chaperone concentrations are summarized in Table S7, according to which the values of fsol for the slower and faster translating bacteria are 0.32 and 0.18, and the value of rSA is 1.6. These values are compared to the experimental results in Figures S9C and S9D. As stated in the main text, the simulations reproduce the trends in the experimental data, but they overemphasize aggregation.
Of the proteostasis network components in FoldEco, the results of the Luc expression simulations are most sensitive to the concentrations of the KJE system components (data not shown). The concentration of GroEL does not strongly affect our results, consistent with the experimentally observed inability of GroEL to facilitate Luc refolding in vitro (
Buchberger, A., Schröder, H., Hesterkamp, T., Schönfeld, H.J., and Bukau, B. (1996). Substrate shuttling between the DnaK and GroEL systems indicates a chaperone network promoting protein folding. J. Mol. Biol. 261, 328–333.
).
Siller et al. observe that not only the specific activity, but the absolute activity of Luc increases in the slower translating bacteria. According to their Figure 2c, the Luc produced by the slower translating bacteria is about 1.5-fold more active than that produced by the faster translating bacteria after 1 hr (
• Siller E.
• DeZwaan D.C.
• Anderson J.F.
• Freeman B.C.
• Barral J.M.
Slowing bacterial translation speed enhances eukaryotic protein folding efficiency.
). They attribute this effect to improved cotranslational folding when translation is slower. Since FoldEco does not explicitly account for cotranslational folding (see above), this effect is not reproduced in our simulations. We can, however, roughly approximate cotranslational folding by increasing the kf in the simulation with the slower translation rate. We find that increasing kf by a factor of about 1.9 reproduces the experimentally observed 1.5-fold difference in the absolute activity of Luc produced in the slower translating bacteria (Figure S9E). Thus, co-translational folding increases the effective folding rate of Luc by 1.9 fold.
## Acknowledgments
We acknowledge with gratitude the contribution of Daniel W. Farrell (Stony Brook University) to the implementation of the FoldEco program. We thank Jeffery W. Kelly, R. Luke Wiseman, Amber N. Murray (The Scripps Research Institute), Ken A. Dill (Stony Brook University), and members of the Gierasch group (University of Massachusetts-Amherst) for helpful discussions. We thank F.U. Hartl and M. Hayer-Hartl for giving us early access to their DnaK interactome results. This work was funded by National Institutes of Health grants OD000945 and GM027616.
## Supplemental Information
• Document S1. Tables S1, S2, S4, S6, and S7
• Table S3. List of Biophysical Profiles and Simulation Results for the Examination of KJE and GroELS System Substrates with a Synthesis Rate of 0.02 μM s−1, Related to Figure 3
• Table S5. List of Biophysical Profiles and Simulation Results for the Examination of KJE and GroELS System Substrates with a Synthesis Rate of 0.2 μM s−1, Related to Figure 5
• Folder S1. A Mathematica Application File with the Code for FoldEco and a Mathematica Notebook File Demonstrating How to Use FoldEco
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2022-12-06 08:03:18
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https://www.tes.com/teaching-resource/times-table-assessment-includes-both-x-and-for-6-7-8-9-11-and-12-and-answers-11781593
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# Times table assessment - Includes both x and ÷ for 6,7,8,9,11 and 12 and answers
Created by
lyndacatherine
A 6-page assessment document for use with KS2 to assess knowledge of times tables x6, x7, x8, x9, x11 and x12 as well as the related division facts. Pages include: 1) a grid of times tables questions (in columns per x table but in random order down the column) for children to complete; 2) answers to that; 3) a grid of division questions for each of the above x tables (also in columns for each times table but in random order down the column); 4) answers to that; 5) a blank grid with question numbers and column headings in case you want to call out the x tables questions orally for children to write the answer; 6) a blank division facts grid for the same purpose.
\$2.61
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### Info
Created: Nov 24, 2017
Updated: Feb 10, 2018
docx, 35 KB
Times-Tables-Assessment
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2020-02-29 14:26:13
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https://socratic.org/questions/how-do-you-solve-4-x-2-2-20
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# How do you solve -4(x+2)^2=-20?
$x = - 2 \pm \sqrt{5}$
Divide both sides by -4 to give ${\left(x + 2\right)}^{2} = 5$
Take the square root $\left(x + 2\right) = \pm \sqrt{5}$
Subtract 2 from both sides $x = - 2 \pm \sqrt{5}$
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2020-07-08 14:46:22
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https://www.semanticscholar.org/paper/Dissipation-in-the-effective-field-theory-for-First-Endlich-Nicolis/f7cfa86f36440e2dd968d2087ff8231b4bb8a62d
|
# Dissipation in the effective field theory for hydrodynamics: First order effects
@article{Endlich2013DissipationIT,
title={Dissipation in the effective field theory for hydrodynamics: First order effects},
author={Solomon Endlich and Alberto Nicolis and Rafael A. Porto and Junpu Wang},
journal={Physical Review D},
year={2013},
volume={88},
pages={105001}
}
• Published 27 November 2012
• Physics
• Physical Review D
We introduce dissipative effects in the effective field theory of hydrodynamics. We do this in a model-independent fashion by coupling the long-distance degrees of freedom explicitly kept in the effective field theory to a generic sector that “lives in the fluid,” which corresponds physically to the microscopic constituents of the fluid. At linear order in perturbations, the symmetries, the derivative expansion, and the assumption that this microscopic sector is thermalized allow us to…
70 Citations
Effective field theory of dissipative fluids
• Physics
• 2015
A bstractWe develop an effective field theory for dissipative fluids which governs the dynamics of long-lived gapless modes associated with conserved quantities. The resulting theory gives a path
On thermal fluctuations and the generating functional in relativistic hydrodynamics
• Physics
• 2015
A bstractWe discuss a real-time generating functional for correlation functions in dis-sipative relativistic hydrodynamics which takes into account thermal fluctuations of thehydrodynamic variables.
Effective action for relativistic hydrodynamics: fluctuations, dissipation, and entropy inflow
• Physics
Journal of High Energy Physics
• 2018
A bstractWe present a detailed and self-contained analysis of the universal SchwingerKeldysh effective field theory which describes macroscopic thermal fluctuations of a relativistic field theory,
Constructing higher-order hydrodynamics: The third order
• Physics
• 2016
Hydrodynamics can be formulated as the gradient expansion of conserved currents in terms of the fundamental fields describing the near-equilibrium fluid flow. In the relativistic case, the
Generalized global symmetries and dissipative magnetohydrodynamics
• Physics
• 2017
The conserved magnetic flux of U ( 1 ) electrodynamics coupled to matter in four dimensions is associated with a generalized global symmetry. We study the realization of such a symmetry at
An action principle for dissipative fluid dynamics
Fluid dynamics is the universal theory of low-energy excitations around equilibrium states, governing the physics of long-lived modes associated with conserved charges. Historically, fluid dynamics
Effective field theory of dissipative fluids (II): classical limit, dynamical KMS symmetry and entropy current
• Physics
• 2017
A bstractIn this paper we further develop the fluctuating hydrodynamics proposed in [1] in a number of ways. We first work out in detail the classical limit of the hydrodynamical action, which
The Effective Field Theory Approach to Fluid Dynamics
The Effective Field Theory Approach to Fluid Dynamics Solomon G. S. O. Endlich In this thesis we initiate a systematic study of fluid dynamics using the effective field theory (EFT) program. We
Theory of Diffusive Fluctuations.
• Physics
Physical review letters
• 2019
The recently developed effective field theory of fluctuations around thermal equilibrium is used to compute late-time correlation functions of conserved densities and it is found that the diffusive pole is shifted in the presence of nonlinear hydrodynamic self-interactions.
## References
SHOWING 1-10 OF 33 REFERENCES
Effective field theory for hydrodynamics: Thermodynamics, and the derivative expansion
• Physics
• 2012
We consider the low-energy effective field theory describing the infrared dynamics of non-dissipative fluids. We extend previous work to accommodate conserved charges, and we clarify the matching
Hydrodynamic transport coefficients in relativistic scalar field theory.
• Jeon
• Physics
Physical review. D, Particles and fields
• 1995
The effective Boltzmann equation is valid even at very high temperature where the thermal lifetime and mean free path are short compared to the Compton wavelength of the fundamental particles.
Effective field theory for hydrodynamics: Wess-Zumino term and anomalies in two spacetime dimensions
• Physics
• 2011
We develop the formalism that incorporates quantum anomalies in the effective field theory of non-dissipative fluids. We consider the effect of adding a Wess-Zumino-like term to the low-energy
Hall viscosity from effective field theory
• Physics
• 2011
For two-dimensional non-dissipative fluids with broken parity, we show via effective field theory methods that the infrared dynamics generically exhibit Hall viscosity--a conservative form of
Low-energy effective field theory for finite-temperature relativistic superfluids
We derive the low-energy effective action governing the infrared dynamics of relativistic superfluids at finite temperature. We organize our derivation in an effective field theory fashion-purely in
Hydrodynamics with triangle anomalies.
• Physics
Physical review letters
• 2009
It is shown that a hitherto discarded term in the conserved current is not only allowed by symmetries, but is in fact required by triangle anomalies and the second law of thermodynamics, which leads to a number of new effects, one of which is chiral separation in a rotating fluid at nonzero chemical potential.
Dissipative effects in the effective field theory of inflation
• Mathematics
• 2011
A bstractWe generalize the effective field theory of single clock inflation to include dissipative effects. Working in unitary gauge we couple a set of composite operators, {\mathcal{O}_{{\mu \nu
Dissipative effects in the worldline approach to black hole dynamics
• Physics
• 2006
We derive a long wavelength effective point-particle description of four-dimensional Schwarzschild black holes. In this effective theory, absorptive effects are incorporated by introducing degrees of
The quantum mechanics of perfect fluids
• Physics
• 2010
We consider the canonical quantization of an ordinary fluid. The resulting long-distance effective field theory is derivatively coupled, and therefore strongly coupled in the UV. The system however
Dissipative phenomena in quark-gluon plasmas.
• Physics
Physical review. D, Particles and fields
• 1985
Transport coefficients of small-chemical-potential quark-gluon plasmas are estimated and dissipative corrections to the scaling hydrodynamic equations for ultrarelativistic nuclear collisions are
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2022-06-30 14:16:22
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|
https://www.macdrifter.com/2013/12/bookmarker-macros-for-editorial.html
|
Bookmarker Macros for Editorial
This pair of macros for Editorial brings a whole new level of bookmarking to text files on the iPad. Ever want to bookmark a specific spot in a text file or remember where you left off editing a text file in Editorial? Me too. So, I created this pair of fairly simple macros.1 Here’s a little demo:
The first macro saves the current text selection to a text file named EditorialBookmarks in the Dropbox file storage. I chose this design for very specific reasons. I wanted my bookmarks to be transportable from one iPad to another if I get a new device. I also wanted to be able to see exactly what was in my bookmarks file by just looking at the plain text.
Editorial makes this easy with a single action named Get Bookmark URL which generates a string that represents the current document and text selection:
:::text
editorial://open/NameOfCurrentDocument.txt?root=dropbox&selection=673-704
Adding a bit more usefulness to this, the Save Bookmark macro also saves a name for the bookmark and appends it all to a bookmark text file.
The partner macro presents a list of the bookmarks in the bookmark text file. Tapping one opens the bookmark and selects the previous range.
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2023-02-02 15:51:32
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https://www.math.u-psud.fr/Titre-a-preciser-50256?lang=fr
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## $L^2$ Solvability of boundary value problems for divergence form parabolic equations with complex coefficients
### Mardi 19 avril 2016 14:00-15:00 - Kaj Nyström - Université d'Uppsala
Résumé : We consider parabolic operators of the form
$$\partial_t+\mathcalL,\ \mathcalL=-\mboxdiv\, A(X,t)\nabla,$$
in
$\mathbb R_+^n+2 :={(X,t)=(x,x_n+1,t)\in \mathbb R^n\times \mathbb R\times \mathbb R :\ x_n+1>0}$, $n\geq 1$.
We assume that $A$ is a $(n+1)\times (n+1)$-dimensional matrix which is bounded, measurable, uniformly elliptic and complex, and we assume, in addition, that the entries of A are independent of the spatial coordinate $x_n+1$ as well as of the time coordinate $t$. For such operators we prove that the boundedness and invertibility of the corresponding layer potential operators are stable on $L^2(\mathbb R^n+1,\mathbb C)=L^2(\partial\mathbb R^n+2_+,\mathbb C)$ under complex, $L^\infty$ perturbations of the coefficient matrix. Subsequently, using this general result, we establish solvability of the Dirichlet, Neumann and Regularity problems for $\partial_t+\mathcalL$, by way of layer potentials and with data in $L^2$, assuming that the coefficient matrix is a small complex perturbation of either a constant matrix or of a real and symmetric matrix.
Lieu : Salle 113-115 (Bâtiment 425)
$L^2$ Solvability of boundary value problems for divergence form parabolic equations with complex coefficients
décembre 2019 :
Département de Mathématiques Bâtiment 307 Faculté des Sciences d'Orsay Université Paris-Sud F-91405 Orsay Cedex Tél. : +33 (0) 1-69-15-79-56 Département Fermeture du département Laboratoire Formation
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2019-12-10 13:30:20
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https://www.tutorialspoint.com/parse-and-balance-angle-brackets-problem-in-javascript
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# Parse and balance angle brackets problem in JavaScript
We are given a string of angle brackets, and we are required to write a function that add brackets at the beginning and end of the string to make all brackets match.
The angle brackets match if for every < there is a corresponding > and for every > there is a corresponding <.
For example − If the input string is −
const str = '><<><';
## Output
Then the output should be −
const output = '<><<><>>';
Here, we added, '<' at the beginning and '>>' at the end to balance the string.
We will use a number that will keep count of the number of open '<' tags so far. And then, when we encounter a '>' tag, if there are no current open tags, we will add '<' to the beginning of the string (while keeping the open tag count at 0).
Then, at the end, add a number of '>'s matching the number of currently open tags.
## Example
The code for this will be −
const str = '><<><';
const buildPair = (str = '') => {
let count = 0;
let extras = 0;
for (const char of str) {
if (char === '>') {
if (count === 0) {
extras++;
} else {
count−−;
};
} else {
count++;
};
};
console.log(buildPair(str));
><<><>>
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2023-03-29 23:59:44
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https://plainmath.net/8908/solve-6%E2%88%928x-equal-7x%E2%88%9210x-plus-21-the-value-of-x-is-%E2%96%A1
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# Solve. 6−8x=7x−10x+21 The value of x is □.
Question
Equations and inequalities
Solve.
6−8x=7x−10x+21
The value of x is □.
2021-03-08
6−8x=7x−10x+21
6−8x=−3x+21
6−8x+8x=−3x+21+8x
6=5x+21
6−21=5x+21−21
-15=5x
$$\displaystyle{\frac{{{15}}}{{{5}}}}={\frac{{{5}{x}}}{{{5}}}}$$
-3=x
### Relevant Questions
Solve the equations and inequalities. Find the solution sets to the inequalities in interval notation. $$\displaystyle{3}{x}{\left({x}-{1}\right)}={x}+{6}$$
Solve the equations and inequalities. Write the solution sets to the inequalities in interval notation. $$\displaystyle{\left({x}^{{2}}-{9}\right)}^{{2}}-{2}{\left({x}^{{2}}-{9}\right)}-{35}={0}$$
Solve the equation $$\displaystyle{x}^{{{2}}}+{x}-{20}$$
Solve the equation $$\displaystyle{\left({x}-{1}\right)}^{{{2}}}={9}$$
Solve the equation $$\displaystyle{x}^{{{2}}}-{25}$$
Solve the equation $$\displaystyle{\frac{{{3}}}{{{2}}}}{x}={\frac{{{5}}}{{{6}}}}{x}+{2}$$
$$\displaystyle{a}^{{2}}+{12}{a}+{36}\leq{0}$$
$$\displaystyle{a}^{{2}}+{12}+{36}={0}$$
Solve the equations and inequalities. Find the solution sets to the inequalities in interval notation. $$\displaystyle\sqrt{{{t}+{3}}}+{4}={t}+{1}$$
Solve the equations and inequalities. Write the solution sets to the inequalities in interval notation. $$\displaystyle{9}^{{{2}{m}-{3}}}={27}^{{{m}+{1}}}$$
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2021-06-15 10:56:56
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http://patricktalkstech.com/burst-error/burst-error-correction.html
|
Home > Burst Error > Burst Error Correction
# Burst Error Correction
## Contents
An interleaver accepts a sequence of symbols and permutes them; a deinterleaver in return, permutes the symbols back into the original order. Therefore, the interleaved ( λ n , λ k ) {\displaystyle (\lambda n,\lambda k)} code can correct the burst of length h {\displaystyle h} . We define a a burst description to be a tuple where is the pattern of the error (that is the string of symbols beginning with the first nonzero entry in the The interleaver will just reorganize the input symbols at the output. http://patricktalkstech.com/burst-error/burst-error-correction-example.html
These redundant bits are added by the sender and removed by the receiver. Let be an irreducible polynomial of degree over , and let be the period of . I have prepared this report with my utmost earnestness and sincere effort. Following are typical parameters that a burst can have 1.
## Burst Error Correction Using Hamming Code
By the induction hypothesis, p | k − p {\displaystyle p|k-p} , then p | k {\displaystyle p|k} . By the induction hypothesis, , then . Now, Hamming code cannot correct 3 errors. Since must be an integer, we have .
• Therefore, M ( 2 ℓ − 1 + 1 ) ⩽ 2 n {\displaystyle M(2^{\ell -1}+1)\leqslant 2^{n}} implies M ⩽ 2 n / ( n 2 ℓ − 1 + 1
• Recommended Foundations of Programming: Databases Foundations of Programming: Object-Oriented Design Foundations of Programming: Fundamentals Error Detection And Correction Renu Kewalramani Parity check(Error Detecting Codes) Imesha Perera Computer Networks - Error Detection
• The error can then be corrected through its syndrome.
• Thanks.
But most importantly, we notice that each zero run is disjoint. Burst Error Definition But instead of repeating the entire data stream, a shorter group of bits may be appended to the end of each unit. Figure 8 shows the process of using redundant bits to check the accuracy of a data unit. Hence I will be copying/donating the same text to Wikipedia too.
Upon receiving it, we can tell that this is c 1 {\displaystyle \mathbf γ 4 _ γ 3} with a burst b . {\displaystyle \mathbf γ 0 .} By the above Burst Error Detection And Correction See our User Agreement and Privacy Policy. Consider a code operating on F 2 m {\displaystyle \mathbb {F} _{2^{m}}} . Abramson Bound(s) It is natural to consider bounds on the rate, block-length, and number of codewords in a burst-error-correcting code.
## Burst Error Definition
Export You have selected 1 citation for export. The reason is that even if they differ in all the other ℓ {\displaystyle \ell } symbols, they are still going to be different by a burst of length ℓ . Burst Error Correction Using Hamming Code But is irreducible, therefore it must divide both and ; thus, it also divides the difference of the last two polynomials, . Burst Error Correcting Codes Ppt Thus, g ( x ) = ( x 9 + 1 ) ( 1 + x 2 + x 5 ) = 1 + x 2 + x 5 + x
Hence, if we receive e1, we can decode it either to 0 or c. Check This Out Technol. It is capable of correcting any single burst of length . Therefore, a ( x ) + x b b ( x ) {\displaystyle a(x)+x^{b}b(x)} is either divisible by x 2 ℓ − 1 + 1 {\displaystyle x^{2\ell -1}+1} or is 0 Burst Error Correcting Convolutional Codes
Encoded message using random block interleaver 9. Inst. We can further revise our division of by to reflect , that is . Source Each of the M {\displaystyle M} words must be distinct, otherwise the code would have distance < 1 {\displaystyle <1} .
Generated Fri, 18 Nov 2016 09:50:21 GMT by s_mf18 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.10/ Connection Burst Error Correcting Convolutional Codes Pdf Finally, it also divides: . This effectively creates a random channel, for any burst that occurred is now (likely) scattered across the length of the received codeword.
## Thus it follows that no nonzero burst of length 2l or less can be a codeword Rieger Bound If l is the burst error correcting ability of an (n, k) linear
When 6. Let d ( x ) {\displaystyle d(x)} be the greatest common divisor of the two polynomials. Therefore, the detection failure probability is very small ( 2 − r {\displaystyle 2^{-r}} ) assuming a uniform distribution over all bursts of length ℓ {\displaystyle \ell } . Random Error Correcting Codes Generate message depending on loop invariant 5.
For example, the previously considered error vector E = ( 010000110 ) {\displaystyle E=(010000110)} , is a cyclic burst of length ℓ = 5 {\displaystyle \ell =5} , since we consider If it had a burst of length ⩽ 2 ℓ {\displaystyle \leqslant 2\ell } as a codeword, then a burst of length ℓ {\displaystyle \ell } could change the codeword to Thus, we can formulate as Drawbacks of Block Interleaver : As it is clear from the figure, the columns are read sequentially, the receiver can interpret single row only after it http://patricktalkstech.com/burst-error/burst-error-correction-technique.html The data unit, now enlarged by several hits, travels over the link to the receiver.
First we observe that a code can detect all bursts of length ⩽ ℓ {\displaystyle \leqslant \ell } if and only if no two codewords differ by a burst of length Reliable communication is assured if the hamming distance between the transmitter and receiver is less than or equal to one. With these requirements in mind, consider the irreducible polynomial , and let . Notice that such description is not unique, because D ′ = ( 11001 , 6 ) {\displaystyle D'=(11001,6)} describes the same burst error.
Then the number of errors that deinterleaved output may contain is For error correction capacity upto t, maximum burst length allowed = (nd+1)(t-1) For burst length of (nd+1)(t-1)+1,decoder may fail. Let divide . Following are typical parameters that a burst can have 1. By our assumption, is a valid codeword, and thus, must be a multiple of .
Generated Fri, 18 Nov 2016 09:50:21 GMT by s_mf18 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.9/ Connection We know that p ( x ) {\displaystyle p(x)} divides both (since it has period p {\displaystyle p} ) x p − 1 = ( x − 1 ) ( 1 Thus, the main function done by interleaver at transmitter is to alter the input symbol sequence. Interleaver Efficiency [4] A particularly useful definition for an interleaver is its efficiency.
For example, the efficiency of the block-interleaver mentioned above is . In this report the concept of Hamming Code, Burst Error, and how to detect & correct it are discussed first.
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2018-04-22 01:09:02
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https://arbourj.wordpress.com/2012/05/03/field-extensions/
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Recall that since a field does not have a proper ideal, a field morphism is either injective or the zero morphism (the kernel of the morphism is an ideal so it must be $\{ 0 \}$ or the whole field). Recall also that for any field $K,$ there is a unique ring homomorphism $\iota :\textbf{Z} \to K,$ and we define the characteristic of K, $\text{char } K,$ to be the integer $p \in \textbf{N}$ such that $p$ generates the principal ideal $\ker \iota.$
Note that since $\iota(\text{Z})$ is an integral domain (fields are integral domains) $\ker \iota$ must be a prime ideal hence either $\text{char } K = 0$ (if $\iota$ is injective) or $\text{char } K$ is a prime number.
Now, recall the construction of the quotient field of a ring, which is, in a universal way, the smallest field containing the ring. Since the quotient field of $\textbf{Z}$ is $\textbf{Q}$ and $\textbf{Z}/p\textbf{Z}$ is a field, a field $K_0$ of characteristic 0 (resp. $K_p$ of characteristic $p$) contains a copy of $\textbf{Q}$ (resp. a copy of $\textbf{Z}/p\textbf{Z}$). Moreover, these are their respective smallest subfield, called their prime subfield.
Since a field morphism is injective, for every morphism $K \to F,$ we can identify $K$ as a subfield of $F,$ it is then said that $F$ is a field extension of $K,$ noted $K \subseteq F$ or $F/K.$ So every field is a field extension of its prime subfield, this is why theory of fields is essentially about the study of field extensions.
Note that for $F/K$ a field extension, we have $\text{char }K = \text{char } F$ so fields of different characteristic do not interact with each others.
If $F/K$ is a field extension, then $F$ is a $K$-algebra hence a vector space over $K.$ A field extension $F/K$ is finite of degree $n$ if the dimension of $F$ over $K$ as a vector space is $\dim F = n \in \textbf{N}.$ The degree of a field extension $F/K$ is noted $[F : K]$ or simply $(F/K).$
We have already seen an example of a finite extension in last post where I talked about adjoining a root of a polynomial in a ring. There we saw that for $R$ a ring, the ring $R[x]/(f)$ is a minimal ring with respect to rings containing $R$ and a root for $f(x).$ In fact, if $R = K$ is a field, then $K[x]$ is a principal ideal domain (PID) so if $f(x)$ is irreducible, the ideal $(f)$ will be maximal and $K[x]/(f)$ will be a field extension of $K,$ minimal with respect to field extensions containing a root for $f(x).$ Although, as we have seen in the theorem of last post, this extension is minimal but not universal: if $F$ is another field extension of $K$ having a root for $f(x),$ we certainly have morphisms
$K \hookrightarrow K[x]/(f) \hookrightarrow F$
but the morphism from $K[x]/(f)$ to $F$ may not be unique.
Let $F/K$ be a field extension and $\alpha \in F.$ Then the smallest subfield of $F$ containing both $K$ and $\alpha$ is is denoted $K(\alpha).$ A field extension $F/K$ is called a simple extension if there is an element $\alpha \in F$ such that $F = K(\alpha).$
For example, if $F = K[x]/(f),$ for $f(x)$ an irreducible monic polynomial of degree d, the extension $F/K$ is simple of degree d. Indeed, if $\alpha$ is the coset of $x$ in $F,$ any subfield of $F$ containing $\alpha$ and $K$ must contain all polynomials in $\alpha$ over $K,$ since $K[x]/(f) \simeq K[\alpha],$ we must have $K(\alpha) = F.$
In fact, we will see that this is characteristic of finite simple extensions:
Theorem: Let $K(\alpha)/K$ be a simple extension and consider the evaluation morphism $\text{ev}_{\alpha} : K[x] \to K(\alpha)$ defined by $f(x) \mapsto f(\alpha).$ Then
1. $\text{ev}_{\alpha}$ is injective iff $K(\alpha)/K$ is infinite. In this case, $K(\alpha) \simeq K(x),$ the field of rational functions over $K.$
2. $\text{ev}_{\alpha}$ is not injective iff $K(\alpha)/K$ is finite. In this case there is a unique monic irreducible polynomial $f(x) \in K[x]$ of degree $[K(\alpha) : K]$ such that $K(\alpha) \simeq K[x] / (f).$
Proof: Recall that the quotient field of a ring is the smallest field containing that ring. So if $\text{ev}_{\alpha}$ is injective, then it extends uniquely to a field homomorphism $\phi$ from the quotient field of $K[x],$ namely $K(x),$ to the field $K(\alpha).$ But then the image of $\phi$ contains $K$ and $\alpha$ so since $K(\alpha)$ is minimal with this property, we must have $\phi(K(x)) = K(\alpha),$ hence the isomorphism between $K(x)$ and $K(\alpha).$ Moreover, since $\text{ev}_{\alpha}$ is injective, the image of the powers of $x;$ $(1, \alpha, \alpha^2, \ldots )$ are linearly independent so the extension $K(\alpha)/K$ is infinite.
On the other hand, if $\text{ev}_{\alpha}$ is not injective, then as seen in my last post, $\ker \text{ev}_{\alpha}$ is prime and generated by a unique monic irreducible non-constant polynomial. The first isomorphism theorem then gives us a homomorphism $\phi : K[x] / (f) \to K(\alpha)$ and since the image of $\phi$ is a subfield containing both $K$ and $\alpha,$ it is the whole $K(\alpha).$ As seen in my last post, we then have $[K(\alpha) : K] = \deg f.$
Finally, since either $\text{ev}_{\alpha}$ is injective or it isn’t, the converse of both points of the theorem is proved.
QED
In the first case of this theorem, $\alpha$ is said to be transcendantal over $K$ and in the second it is said to be algebraic.
Reference: P. Aluffi, Algebra : Chapter 0
M. Artin, Algebra, 2nd edition
E. Artin, Galois Theory.
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2023-01-30 04:11:49
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https://spinor.info/weblog/?m=201108
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I never thought Apple computers were hip. Every so often, I thought about buying Apple hardware, but if I did so, I’d want a development system, so my shopping cart at apple.ca rapidly ballooned to some 2,000 dollars… by which time I inevitably realized that I’d be buying expensive toys that would become obsolete long before I’d find the time needed to become proficient with Apple’s development tools.
And now here is an interesting article, from the Ottawa Citizen no less, elaborating on something that I felt all along: that despite its hip image, what Apple sold to the masses all along was really mediocrity.
Of course this probably means that I am not one of the cool kids, but if that is the case, so be it… life is way too short to worry about coolness.
I am not usually in the business of recommending software or hardware products, and it’s certainly not something anyone pays me to do… but recently, I began using two products, both of which have exceptional value, even though one came free of charge and the other cost only 150 dollars.
The free product is Secunia’s Personal Software Inspector (PSI), a software application that turned from something I never heard about into something I cannot live without virtually overnight. It is an application that keeps tabs on all the software installed on your computer and lets you know if any of them are out of date and require updates. Like antivirus software, PSI sits quietly in the background most of the time, but it pops up an unobtrusive warning whenever a new update becomes available, and even offers a direct link to the manufacturer’s download site. It is nice, incredibly useful, it recognizes hundreds of installed applications, and, well, it works as it is supposed to and doesn’t cost a penny.
The product I paid money for is a Cisco RV042 small business router. It does what small business routers do, connects your internal network to an external (DSL, cable, etc.) Internet connection. What makes it special is that it allows your internal network to be connected to two external connections at the same time, and it performs dynamic load balancing and failover functions between the two. I now set up my network architecture to take full advantage of it… and in the coming days, it will be working overtime, as I am planning a major change to my DSL service which will likely involve some unpredictable downtime. The router has other useful functions, too, not the least of which is that it can act as a VPN server, allowing a remote computer to connect to the internal network. The best part is that, like Secunia’s software, it simply works as advertised.
Back when I was learning the elementary basics of FORTRAN programming in Hungary in the 1970s, I frequently heard an urban legend according to which the sorry state of computer science in the East Bloc was a result of Stalin’s suspicion towards cybernetics, which he considered a kind of intellectual swindlery from the decadent West. It seemed to make sense, neglecting of course the fact that the technological gap between East and West was widening, and that back in the 1950s, Soviet computers compared favorably to Western machines; and that it was only in the 1960s that a slow, painful decline began, as the Soviets began to rely increasingly on stolen Western technology.
Nonetheless, it appears that Stalin was right after all, insofar as cybernetics is concerned. I always thought that cybernetics was more or less synonymous with computer science, although I really have not given it much thought lately, as the term largely fell into disuse anyway. But now, I am reading an intriguing book titled “The Cybernetic Brain: Sketches of Another Future” by Andrew Pickering, and I am amazed. For instance, until now I never heard of Project Cybersyn, a project conceived by British cyberneticists to create the ultimate centrally planned economy for socialist Chile in the early 1970s, complete with a futuristic control room. No wonder Allende’s regime failed miserably! The only thing I cannot decide is which was greater: the arrogance or dishonesty of those intellectuals who created this project. A project that, incidentally, also carried a considerably potential for misuse, as evidenced by the fact that its creators received invitations from other repressive regimes to implement similar systems.
Stalin may have been one of the most prolific mass murderers in history, but he wasn’t stupid. His suspicions concerning cybernetics may have been right on the money.
I am reading a very interesting paper by Mishra and Singh. In it, they claim that simply accounting for the gravitational quadrupole moment in a matter-filled universe would naturally produce the same gravitational equations of motion that we have been investigating with John Moffat these past few years. If true, this work would imply that that our Scalar-Tensor-Vector Gravity (STVG) is in fact an effective theory (which is not necessarily surprising). Its vector and scalar degrees of freedom may arise as a result of an averaging process. The fact that they not only recover the STVG acceleration law but the correct numerical value of at least one of the STVG constants, too, suggests that this may be more than a mere coincidence. Needless to say, I am intrigued.
As I’ve been asked about this more than once before, I thought I’d write down an answer to a simple question concerning the Pioneer spacecraft: if the “thermal hypothesis”, namely that the spacecraft are decelerating due to the heat they radiate, is true, how come this deceleration diminishes more rapidly, with a half-life of 20-odd years, than the primary heat source on board, which is plutonium-238 fuel with a half-life of 87.74 years?
The answer is simple: there are other half-lives on board. Notably, the half-life of the efficiency of the thermocouples that convert the heat of plutonium into electricity.
Now most of that heat from plutonium is simply wasted; it is radiated away, and while it may produce a recoil force, it does so with very low efficiency, say, 1%. The thermocouples convert about 6% of heat into electricity, but as the plutonium fuel cools and the thermocouples age, their efficiency decreases (this is in fact measurable, as telemetry tells us exactly how much electricity was generated on board at any given moment.) All that electrical energy has to go somewhere… and indeed it does, powering all on-board instrumentation that, like a home computer, ultimately turn all the energy they consume into heat. This heat is radiated away, and it is in fact converted into a recoil force with an efficiency of about 40%.
These are all the numbers we need. The recoil force, then, will be proportional to 1% of 100% − 6% = 94% plus 40% of 6% of the total thermal power (say, 2500 W at the beginning). The total power will decrease at a rate of $$2^{-T/87.74}$$, so after $$T$$ number of years, it will be $$2500\times 2^{-T/87.74}$$ W. As to the thermocouple efficiency, its half-life may be around 30 years; so the electrical conversion efficiency goes from 6% to $$6\times 2^{-T/30.0}$$ % after $$T$$ years.
So the overall recoil force can be calculated as being proportional to
$$P(T)=2500\times 2^{-T/87.74}\times\left\{\left[1-0.06\times 2^{-T/30.0}\right]\times 0.01+0.06\times 2^{-T/30.0}\times 0.4\right\}.$$
(This actually gives a result in watts. To convert it into an actual force, we need to divide by the speed of light, 300,000,000 m/s.) With a bit of simple algebra, this formula can be simplified to
$$P(T)=25.0\times 2^{-T/87.74}+58.5\times 2^{-T/22.36}.$$
The most curious thing about this result is that the recoil force is dominated by a term that has a half-life of only 22.36 years… which is less than the half-life of either the plutonium fuel or the thermocouple efficiency.
The numbers I used are not the actual numbers from telemetry (though they are not too far from reality) but this calculation still demonstrates the fallacy of the argument that just because the power source has a specific half-life, the thermal recoil force must have the same half-life.
I am catching up with my reading of recent issues of New Scientist, which arrived all at once after our recent postal strike.
Cephalopods are smart. So smart in fact that they are tool users, the only invertebrates we know about that have this ability. Yet they evolved entirely differently from us, having split from us some half a billion years ago on the evolutionary tree. Some argue that cephalopods deserve extra protection; on the other hand, we don’t even know how to anesthetize them properly.
I also wonder if the SETI folks are taking notice. We think we are so smart that we can talk to aliens? How about learning first how to communicate with a giant squid. Compared to aliens, these guys are our cousins after all.
Finally, a voice of reason.
I just read an opinion piece in New Scientist by Erle Ellis. His message is simple: Welcome to the Anthropocene. Ellis believes that the geological epoch called the Holocene is over; the landscape of the Earth has been altered irreversibly by humans, but not all such change is bad or unwelcome. In any case, there is no turning back. The question is not how to undo what we have done, but how to create a better, more sustainable Anthropocene, as we have become the creators, engineers, and stewards of this world.
This has also been my opinion for a long time. Humans are no less “natural” than apes, ants, whales, or trees. By extension, a skyscraper or a factory are no less natural than an anthill or a bird’s nest. However, it has happened in the past that a species overwhelmed and destroyed the environment in which it once thrived. Humans can suffer the same fate… except that we do possess oversize brains and the ability to plan ahead in the long term. What we need is not some romantic notion of a “pristine planet”, but to learn how to manage a planet of finite resources that is dominated, and irreversibly altered, by our presence.
First the first time in seven years (!), my main Internet connection is down, and will likely stay down until at least Tuesday. This being a long weekend, no telco technician is available until then, and they determined that the fault is likely a partial short in the physical circuit. Bloody hell.
Now I am scrambling to reroute everything to a backup server, provided courtesy of a good friend of mine. I asked him to only move around on tiptoes until Tuesday, and I am begging Murphy not to strike again until then…
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2022-07-03 23:35:41
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https://uo2.stratics.com/category/7-collectors-corner/posts/category/atlantic-news/posts/category/4-qa5/page/2/
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Posted by Tim Chappell | 2012 Nov 05 20:49 -0500 GMT
UO.stratics
Has there been any thought as to letting us use any of the boats in the huge lake in Malas near Umbra, or sail in any of the waters in Ilshenar? (Bazer – Apr 2011) Could rowboats only be allowed in Dungeons, Ilshenar, Ter Mur and Malas? (Basara – October 2012) Will you allow us to use rowboats in dungeons please? There is so much wasted water in like ice dungeon and I need to catch Winter Dragonfish and Blue lobsters so….thats all (Lord Gareth – June 2011) Maybe it would be nice to allow rowboat placement in dungeons? There some dungeons where a rowboat can be handy to fish. Otherwise rowboats don’t realy have a good use cause they are slow and have no hold.(Frarc -Mar 2011)I wouldn´t mind rowboats in Ilshenar, Gravewater Lake etc either …. (Ertai Vodalion – Mar 2011)
Kyronix: Using Rowboats in dungeons and Malas/Ilshenar is something that isn’t as simple as flipping the “rowboats on” switch. As far as dungeons go, all the areas where you’d presumably place a rowboat would need to be checked given the new level of accessibility, basically the dungeons weren’t built to be sailed, the same, albeit to a lesser degree, goes for Ilshenar. Malas however, that’s a pretty straightforward body of water and we’ll discuss the matter further. As far as fishing goes we’ve made some passes at various aspects of fishing that will be visible in the Pub 80 bug squashing extravaganza.
Essentially back when they increased the chances for lower end creatures to give relic frag generating loot they broke the loot generators ability to drop single mod items like 20DCI/HCI, 4MR, Stats, 150 luck, 35 EP, etc. Can they somehow undo what they did or give the loot generator better chances of dropping single mod items with max intensity? (DEVs can you please take a look into the loot generator?) (silent)
Phoenix: Such items are currently produced by low-end runic tools. We do not have plans for the near future to revisit monster loot generation.
(I rewrote the question into a more concise form – MMoore) Will you take a look at the contents of the holds of pirate ships, the loot of pirates, and the rewards for capturing pirates? The time investment, cost, and risk are not commensurate with the reward.
Phoenix: Yes, we will have a good look at these in a near future publish.
In the SA Dungeon, to the right of where the Slasher of Veils spawns at the Destroyed Monument/Statue in center of other buildings. When fighting the Slasher Of Veils in this area and a hell hound is on the statue area, provoking the hell hound on the Slasher can cause the Slasher to teleport on top of monument base then teleport players up also, killing them, leaving them with no way down and unable to retrieve items from their corpses. Will this be looked into in the near future? (From Trokip – submitted via PM) Also (asamdan, Ask the Devs Jul 2010)
Kyronix: I will submit an unconfirmed bug into the system, and if it checks out (which it sounds like it will) will look to altering the world building in that area so this no longer occurs.
Will we ever be able to use soulstones that we find from other people for something besides deco? Like maybe using them in the creation of fragments or being able to wipe the stone empty to use them again, maybe with limited charges? (PowerFullPete, Jan, 2010) We all have a few odd unowned Soulstone’s gotten from housefalls or found in luna. These unowned stones are good for deco but Id like to see perhaps a trade in. Red ,Blue an green ones litter uo, tons of dye tubs,statues stc.. so to clean them up Id like to see something like the Holloween mask lady set up perhaps in luna to except the rewards trade in. Old for new choice. Would go along way to clean up unwanted rewards and get something we can use. (Lady Storm, Jun, 2010) Will we ever be able to re-link full soulstones or soulstone fragments to different accounts similarly to what is possible with teleport tiles? (Forte~, Sep, 2010) Can there be a way to use old soulstones? Especially when the players who originally owned them (and paid for them, real money here) have quit the game? They are already paid for? Why can’t we re-use them? I understand they are account bound. But why can’t the skills be re-moved and the stone be re-newed? It seems pointless to have hundreds of soulstones in game, that we ALREADY PAID FOR, be completely and utterly unusable. It’s real money floating around in a game, and being worthless. EA (as a last resort) could even sell us a tool that would un-bind them and wipe the skills from them for a couple of bucks. So that at least it could be used again. It costed someone money to purchase in the first place. Why can’t they be re-used if the player who bought it has quit the game entirely? (Templar Asarhi, Jun, 2011 )
Phoenix: We are currently evaluating ideas for what to do with these, including implementing an exchange program that will allow multiple soulstones to be turned in and exchanged for new soulstones.Siege Shard General Chat
Is it possible to get a better explanation of ‘drift speed’ and ‘turn delay’ on large ships? In other words, which ship type is fastest, and which is slowest? Also could we get these figures, plus ‘durability’ and ‘cannon damage’ figures for the Britannia Boat?
Phoenix: All ships move at the same speed, except rowboats.
The Tokuno and Orc ships have 100,000 hit points. The Gargish ship has 140,000 hit points. The Britannian ship has 200,000 hit points.
UO Auctions
Question: Has the Dev team ever considered adding further higher levels of ingot and stone that could be extremely rare but give material bonuses similar to that of heartwood? this may open up slightly more deversity on how you can kit up a human/gargoyle melee fighter, as the material bonuses of wooden armor often outshine metal/stone? (Treasure Seeker)
Phoenix: We do not have plans to add new types of special materials at this time.
Question: Have the Dev team thought about ways to bring back dragon scales as a recourse now that the armor is obsolete? i think perhaps an item crafted by tinkers that adds +1 resist or +5luck to a crafted armor piece would be a good way of bringing back the scales and also making balancing crafted sets a bit easier . (Treasure Seeker)
Bleak: We are currently working to bring back value to all types of armor including Dragon ArmorUO Mania
could you let us use the Retouching tool on an ethereal mount statuette in the trade window, so that other players don’t need to give us their ethereal mounts to make them transparent? (clorenz)
Phoenix: We will take this idea into consideration.
If you have something you want to know about Ultima Online and the team, post your questions on your fansite. All interesting questions will be collected and send to our team, so we can continue to answer them.
Posted by Kai Schober | 2012 Oct 31 20:43 -0400 GMT
UOMania
Can you add the timer for the spawn of the next token to the item property display of the veteran reward Shard Transfer Shields? Right now I need to write down the date to remember when the next token will come out. (clorenz) uo.stratics Also the number currently held on the shield (Tazar) Kyronix: This is something we can easily do and will add to the backlog, thanks!
Currently we can only get Bulk Order Book Covers buying the full 11th anniversary collection pack from the origin store. Can we get them in some other way, for example with a separate purchase? (Zangar) Mesanna: This is something we can add in the future.
Can we get the ability to remove and recover powerscrolls and recipes from a character, to apply them on another character? (clorenz) Mesanna: We as a team all feel this would not be a good system to put into UO due to the fact it cuts out the need to do champ spawns.
Can you make all items stackable? There’s still a lot of stuff that requires infinite storage space and is unstackable. (clorenz) uo.stratics We already have clockwork assemblies and arcane gems that are stackable. Can we get the power crystals to stack as well? (sablestorm) I can remember when clockwork assemblies weren’t stackable and they were fixed at some point in time. It would be great to be able to stack power crystals too. (Trixrnt4kids) Tasty treats – We are getting a lot of these now from the new Treasure Chests, and just wondering why they don’t stack? Thank you (Old Man of UO) Acid slug, Tasty treats and “scared fire ant goo” All 3 doens’t stack. Please, fix it because there are too many items (Ivory Norwind) Monster Stealables are barely used in game (from my experience talking to people) because they take so many lockdowns in a house to store due to the fact that they are unstackable. Please consider making them stack. Thanks (Nero Blackraven) Kyronix: There is a number of bugs related to stackable items in the system that we will be squashing during the Publish 80 bug sprint.
I’d like an item to open my bankbox from home. Similar to the chest of sending, but when you double-click it it opens the bankbox. (clorenz) Kyronix: This is certainly something we can discuss, maybe it’s time for a Britannian heat wave?
Can we get the ability to transfer money from a character on one shard to a character on another shard? Maybe with a “gold transfer token”. (clorenz) Mesanna: I guess I am not understanding the question, are you asking to be able to just transfer money? I guess my confusion is why not just do it via character transfer?
Red & Purple Pixies
Has the staff thought about making the seasonal change that removes the green from the trees and sets the ground in snow, something we can turn off by option? Frankly… I’m sick to death of it in real life and would rather not see that landscape in the game. Thus the option, so people can make a choice. (Lady Frany Flame) Bleak: This is a good suggestion and one we will keep in mind for both clients going forward.
UO Auctions
I find that the game can occasionally seem a little linear as far as dress codes go for the player hoping to achieve best kit. For example everyone ends up wearing sunglasses from library donations. Have the Devs ever considered adding any artifacts to the game that are actually random in which art they drop as. Ie a helm that could compete with the goggles, but can appear as a norsehelm/closed helm/samurai helm/dragon helm and a caster hat that could appear as any of the cloth hat art. etc i feel applying this sort of idea to some new drops might aide diversity in the way players end up dressed and make use of great in game art that ends up somewhat redundant. (Treasure Seeker) Bleak: We have considered doing this but there are several factors that need to taken into consideration for example the actual name of the artifact that may need to be changed based on the selected art or some are may need to be eliminated as a possibility due to meditation restrictions. This is something that we will keep in mind as we go forward in creating new artifacts.
UO.stratics
I know it was stated a long time ago that we would be able to un-alter items. Is this still in the works? Also, could we get the option to alter any similar item to any race? For example, rings and bracelets could be altered, but necklaces and earrings could not (unless there’s a similar artifact for the opposite race) (yadiman) Mesanna: After further research into this matter we are unable to put in the ability to un-alter items. There just isn’t a way to keep track of the original item.
If you have something you want to know about Ultima Online and the team, post your questions on your fansite. All interesting questions will be collected and send to our team, so we can continue to answer them.
Posted by Tim Chappell | 2012 Oct 23 15:12 -0400 GMT
The Ultima Online team has received another set of questions last week. Here are the answers.
Ultima Online Bibliothek Dear Devs, UO has been in our lives for 15 years, how about new merchandise to celebrate? The red Dragon with the 15years font would be a perfect and very cool object for (coloured!) T-Shirts, Hoodies, Lanyards and Stickers! (Medea, ) Mesanna: There is a new Tshirt on the bioware store for UO if you go to this link.
Ultima Online Bibliothek Hello Devs, is it possible to get rid of the timer on the green thorns? I think there is no reason anymore to not get rid of it, because to hunt for Vines or Vorpal Bunnies is nowadays no real issue anymore… thank you! (Medea) Mesanna: Currently there is a 3 minute delay on the thorns that the team have honestly not thought about removing . We will do some research and talk it over.
Ultima Online Bibliothek Hallo UO Team, what happened to the plans about New Naven, meaning replacing New Haven with the old Haven or Occlo? (Waldschrat) Mesanna: We have not made any plans to go forward with this change as of yet.
Ultima Online Bibliothek Dear UO Dev Team, how about a nice and somehow useful list about what action will bring how much points for Town Loyalty? Or even a formula on how the different options are calculated? I am a little confused about all the things we can do. For example: I hunted on Champion Spawns for hours for my tamer’s standing, but there was very little or not much standing earned. But my tailor has no problems maintaining and even get more standing with ease when she gives her BODs to the local workers! (And yes, I login daily to get the loyalty points refreshed). Thanks! Thanks for listening! Kyronix: As I am sure you realize its combination of Love, Hatred and Neutrality. Given the past loyalty decisions of the character it might take longer to build the relationship with the city you desire.
I’d love to know from the team if they plan on releasing these thrones in the spring cleaning or something before anyone makes a poor choice of purchase thinking it’ll be the only release. (Assia Penryn) Mesanna: I don’t have issue adding the throne to Spring Cleaning.
Japanese Players Question 1 Question: Are there any plans to revamp the aquarium? Nickname: Ivy Mesanna: We are looking to add new items not necessarily revamp it.
Knuckleheads.dk In the recent storyline Dupre and Lord Blackthorn appeared ingame. Why didn’t you/the EMs not use their special armors/skins (can be seen here and here)? Mesanna: The old armor was not used because it was not visible in the EC client so we had to remake Blackthorn.
Over the last few months, the size of my guild has more than doubled. Admin, things like promoting folks or doing guild titles, was ok while we were small, but now we’re so large, and so many folks have a alts with us too, it’s becoming a true pain in the ass. For example; scroll through 6 pages to find the char that needs a status altering, as soon as you’re done, it drops you out of the guild roster page. To get to the next char to alter, you have to go back in scroll through 6 pages, etc etc. Is there any way of making this simpler? What I would like to see, ideally, is 1) If I click on a char name, I get all the chars that one person has in the guild. 2) I can work on those chars, from that list, without being dumped out of the guild admin section and having to navigate back through. 3) Maybe some way of working on the admin bits from the UO site, available to me as a guild leader, with a choice of giving access to emissaries? Also, guild titles. What’s with me not being able to give someone the title ‘Dragon Master’ or ‘Hunt Master’ or similar? Those aren’t offensive in any way, what’s the problem? I know it fits, but it tells me it’s disallowed. And if you look at that, maybe you could fix the problem that’s causing my chest of glassblowing supplies to show up on engraving as ‘glablowing supplies’!(Cailleach, ask the devs) Mesanna: We would like to redo the guild system as we are aware of a lot of issues we would like to change it in. I know this is not the answer you want but it is on our list. As far as the title Dragon Master or Hunt Master those were disallowed due to the fact that you can not name yourself after a NPC in the game or a skill in the game. I have no problem at all for allowing glassblowing to show up as it should.
Posted by Tim Chappell | 2012 Oct 15 14:45 -0400 GMT
Its Monday again and we have another set of questions that were answered by the development team. There were even a couple questions answered by one of our GMs.
Can we get an option added to crystal portals to hop over to the Heartwood Gate in Yew? It would making visiting Heartwood that much more convenient! (sablestorm)
Mesanna: Sorry this is not possible to do the fact that Heartwood is on a Dungeon Server
UOForums – There’s several different types of chats so Im wondering if there is a personal message system in the works? Though I kinda enjoy the classic UO “drop me a little book in my mailbox” thing, I think we’ve evolved past that now. Thanks! (Quintus Batiatus)
Mesanna: The team has talked about different ways to improve on the existing mailing system but nothing concrete as of yet.
UOForums – I am looking forward to the Vendor Search option. Is there any details that can be given on this yet? Will you get coordinates to the vendors location, or be teleported to the vendors location, or be shown a map of where the vendor is? Thanks!(Quintus Batiatus)
Mesanna: How about all of the above? We want to give you options and make it useable by all.
UOForums – Im a veteran player since 1997. Ive come and gone a few times now. Though I enjoy the game as it is currently and find it fun to discover new things thats been added in recent years, I sometimes long for the days of the original release. Or maybe, the post Trammel release. The days when GM smiths sat outside the Brit Blacksmith shops, selling their wares. You know, the “stand outside the bank repeating the same items your trying to sell” days. My question is this, would it be possible or have a shard just for classic game play? Thanks! (Quintus Batiatus)
Mesanna: Not to give you a short answer on this one but it’s been hashed many times and the answer is still no.
‘Ello all! We have this fisherman in the SRC who is constantly frustrated by using his bait. He moans, that having to attach them individually everytime you cast is so fiddely for his clumsy hands, he doesn’t even bother with it anymore. Is there a way you can make it more manageable even for slightly inapt fisherman? Maybe create a hook which can be charged with bait and then work like the lava ones? I am sure fishermen (and women of course) around the realms would be glad for it …. and stop prodding me Vaughan, I did ask them now …. so stop it! (John II, A&A4 Comments)(also asked by Frarc, Ask the Devs)
Mesanna: Our stealing skill has just increased by .05 cause we are going to steal this awesome idea =)
If a house falls next to you and you resize a house next to it. Will you be able to place on the idoc spot? Or will you not be able to place because of the Idoc placement timer?(Lord Gareth)Mesanna: Gareth *shakes head* you should know this! There is a delay and the time delays is from 30 minutes to 2 hours it’s totally random.
UOMania clorenz – after paging a GM with one character, if I log off and switch to another character on the same account will I stay in queue and still receive an answer?
GM Inarea: Yes as long as they are on the same account.
UOMania clorenz – could we get also GM support via e-mail? It would be easier to explain the problem (especially for non native English speakers!)
GM Inarea: So if English is not your native language, when you page state in the beginning that you would like to be contacted via email instead and give a brief description of the issue.
UOMania clorenz – I left my Personal Bless Deed on Europa when I transferred my character to another shard. When I transferred the character back to Europa, the Personal Bless Deed was not working for him anymore. How do I get it fixed?
Mesanna: Short answer a GM can fix it for you, long answer is we have this on our list to fix in the big bug push next publish.
UOMania Sir Bolo – I’ve heard that under Lord Blackthorn’s rule the players will be able to run for office in the cities of Britannia. Is this just for roleplaying or will the player-run city councils also gain in-game powers? Can we have an example of the things they could do?
Kyronix: If you mean in terms of having the ability to add or change your city? To give your city random buffs or a facelift or add a soda jerk or additions like this? Maybe. All I can say right now is that it is roleplaying with a twist. All the details are not worked out, nothing is in stone . As soon as we have more details we will be happy to share them with you.
Posted by Tim Chappell | 2012 Oct 09 23:49 -0400 GMT
It is time for another Ask and Answer and this week we answer a lot of great questions from UO Mania.
UO Mania clorenz – Are there any plans to add books to store Scrolls of Alacrity or Powerscrolls similar to the books for Scroll of Transcendence?
Mesanna: When we did the SOT book we had planned to add these other ways to store items. There are just so many things we want to get done we have to take them one at a time. Please be patient they are coming.
UO Mania clorenz – Are there any plans to add jewelry boxes to store rings and bracelets similar to the seed boxes?
Mesanna: This is something the team as talked about several times, but it’s not as simple as the seed boxes. As soon as we can figure out a way to do this correctly to everyone’s satisfaction we will give you guys a jewelry box. The best part about this is I found a wonderful antique jewelry box that would be fitting for the art.
UO Mania clorenz – Can we have a preview of the 15th year Veteran Rewards? Are they useful or decorative?
Mesanna: All the 15th year vet rewards are useful except for the new statues we have added this year. (But the statues are pretty sweet) We will post some pictures close to the publish date for everyone.
UO Mania Bl4ckFir3 – Why do we still have to pay a monthly subscription, while many games have moved to a free-to-play model? Have you considered going Free-to-play with extra features and expansions for sale and the possibility to earn currency in-game to unlock them?
Mesanna: The option to take UO free to play is not up for discussion at this time.
UO Mania Wanderer – are you planning to provide some improvements for thieves in PvP?
Mesanna: Honestly no we don’t have any specific plans, but we do want to start the Topic specific HOC’s soon, PVP will be a topic we are going to want to talk to you guys about. I will say that the team wants to give thieves things to do in and out of PVP.
UO Mania Sir Bolo – the Singing Balls and Secret Chests are only available on the UO store in Japan at the moment. Are you going to make them available for sale also on origin.com in the US and in Europe?
Mesanna: I can add the Secret Chest in the future when I have other things to add to the origin store. I need everyone to understand this is a normal chest that has the ability to have a combo lock on it thus a great role playing tool.
UO Mania Sir Bolo – codes for the 15th Anniversary Commemorative Robe were handed out to players at the anniversary party in Virginia, and to the winners of the Memorable Moments contest. Are we going to get other opportunities to win the promotional codes for this robe?
Mesanna: Not at this time, we do not have any plans outside of the contest and the party to distribute the robe.
Recently, the veteran rewards are overflowing with Sosaria.
How about “Re-activation of vet rewards”? I mean, a reduction point is decided according to years of rewards and it returns to the acquisition point for a new rewards. For example, “1 year vet rewards” is 0.1point and 10 “1 year vet rewards” will be point for “Getting one 1 years vet rewards”.
Thank you for your time. (Peil)
Mesanna: I can see pro’s and con’s to this. This is not a decision that can be made lightly but we will bring this up at a later date for discussion.
Do the devs have any plans to make Gargoyle Clothing (Gargish Robes, Gargish Wing Armor, Gargish Talons) able to be used with a Arcane Gem to create Arcane Clothing for Gargoyles? The only Arcane Clothing that a Gargoyle can use right now, is the Arcane Robes, which there’s quite a few Robe slot items nowadays which are better (Conjurer’s Garb, Shroud of the Condemned, etc). (PlayerSkillFTW)
Phoenix: : Right now we don’t have plans to create Gargoyle versions of Arcane Clothing, because there are alternatives on equipment that make all spellcasting reagent-free.
Is it possible to change the color of the font on the menu for the peculiar seeds? The font is dark and the background is dark. Its hard for my old eyes to see how many resources there are on the plant. On the plain plants the font is white with the black background. Can the pecular seeds be changed to what the plain plants are. Thank you. (Sara Dale)
Kyronix: This is something that can be adjusted, I’ll go ahead and submit an unconfirmed bug report into the system so it can get added.
Kai Schober
1 Oct 2012 16:43:28 EST
It is Monday again. Although the Dev team already answered a ton of questions during the Anniversary Party on Saturday (a recording of this will be eleased later this week), they took the time to address these questions gathered on the different fan sites, too.
Slayer properties on instruments:
Was wondering if we could ever get a way to craft slayer musical instruments. Either thru runic saws or possibly new recipes for carpenters to make? (Xalan Dementia)
Could maybe someday imbuing move to instruments? (JamesC )
I’d like to be able to imbue GM instruments with slayer properties using the same ingredients as slayer weapons and to employ the currently unused ‘essence of persistence’ to either a) raise the number of charges to 1600, in line with the minor Ilshenar artifacts, Iolo’s Lute and Gwenno’s harp or b) add the ‘renewable’ property found on the minor Tokuno artifact flutes.
Kyronix: We are constantly talking about new ways to revitalize crafters, and this is another great series of suggestions to give some love to various crafter classes. We will add it to our backlog list.
When the punishment for murderers, banishment from Trammel, was introduced it actually did not deny access to content, because all that was in Trammel was also in Felucca. However as more and more content has been added over the years, access to content has been denied, making the punishment, in effect, more and more severe. Is the punishment for murderers therefore now too severe? Should they be allowed access to the other facets? Forged pardons exist in game, but is there any possibility that ‘full pardons’ might become available, possibly from Origin Store or clean up points, to allow long established red characters to ‘repent’ and become good citizens?
Kyronix: Do truly red players want to turn blue? I don’t know the plausibility of releasing an item that would erase all murder counts, but adding new ways to make Felucca a more attractive area to play is something we’ve talked about.
Can you fix the language filter? There should be a way to toggle the language filter on or off, besides just the chat filter. While marking runes for a rune library, my wife and I have come across many words that cannot be put onto runes. The filter denies the name as inappropriate. Here is a small list of words that are inappropriate: basement, frozen, passage, companion, titan, assassin, compassion, counselor and I am sure that we will come across others as we mark runes. With all of the character names, guild names, pet names that people can make which are much worse, you would think that we would not have these problems while making a rune library.
Phoenix: We do believe the obscenity filter is a bit too harsh, because it filters sub-strings that contain only slightly naughty words like “ass” and “tit”. We are considering how to adjust the filter and believe that relaxing it a bit would be the right thing to do.
Although we’re now approaching the 15th Aniversary: I would like to ask if the 14 yr. vet reward shard shield will be offered for Siege’s 14 yr. vets even if as deco, or are ye all thinking of something else perhaps that the dev. team or artists could create for Siege’s 14 yr. vets to claim as their 14 yr. customer vet reward? Perhaps if the shard shields just can not be offered even as deco only for us, maybe the art team could create a Siege shard RUG ? That would be awesome looking, like the dolphin rose or skull rugs for the 10 yr vet reward.
Hopefully yall will yet come up with, a 14 yr. vet reward for ye Siege customer base. Thank ye kindly! (Queen Zen)
Mesanna: We can provide deco shields for the 14th for Siege only.
The Brit Boat since this big monolith of a boat costs so much damn money, what say we make it so that either it doesn’t rot and die or that its stages are 20x more then normal boats. I rarely if ever use the ones i bought cause im afraid ill get busy at work and forget to refresh them and loose them. Which makes them useless if they are not being used no? (poo)
Kyronix: I have probably spent a small fortune in boats only to have them decay on me because I forget to refresh them and they go poof. While making the Brit Ship being completely nodecay is unlikely finding a middle ground is doable.
Now where garg necless and earrings are a part of a garg suit, I have to trash alot non gm jewelry as I can’t smelt them back to ingots. Will we see a fix for that? It’s really a pain when making suits to the Gargs. (FrejaSP)
Misk: Providing some sort of salvaging for tinkering has been discussed by the team. Up to this point my solution has been to craft these items near the Ter Mur Jeweler so that I can simply sell the ones that I did not need. Hey, 1 gold is 1 gold after all.
Did not see this posted, so was curious about it. Has there been any consideration to adding the mystic and necro equivalents to mage weapons? Some templates use magery as a weapon skill with -0 to -20 mage weapon. I think there would be definite interest in a -20 mysticism or -20 nercomancy weapons. Might breathe some life into some older templates and add variety. Could this be considered?(Shadowdark)
Phoenix: This suggestion has merit, and we are always looking for ideas for new and interesting item properties to help keep things fresh.
You can, as a rule, only get a group of Spellweavers together for a Focus in the busy evening hours which means Spellweaving is only useful between mid-evening and around 4 0r 5 in the morning because the Spellweaving timer runs out real time. Can it be switched to character time in game so that we can get the Focus when it’s available in the evening and then log out the character until we want to play it as a Spellweaver in the morning or afternoon? (Tanivar)
Phoenix: This is a good suggestion and we will consider a solution like is proposed here.
Certain item\item groups were removed from Cleanup turn in list. Would items listed below be returned as they were once where worth significant points?
– Books (mainly books gotten from fishing and quest reward from turning in Ancient Tome) turned off due to bug fix (old point value 100-250 points)
– Old Holiday Wands (100 charge and 500 charge ones) (Old point value 10000)
Also would the following items ever be added?
– Old charged weapons (hit curse\febleminded\Clumsy\ect charges) Excluding Hit MM\Fireball\Lightning\Heal Greater Heal (point value suggested 5000 same as other charged items currently can be turned in)(NBG)
Kyronix: Our goal is to keep a consistent rotation of items in the Cleanup list so that both the rewards and the turn-in items are fresh. Thanks for the suggestions!
f you have something you want to know about Ultima Online and the team, post your questions on your fansite. All interesting questions will be collected and send to our team, so we can continue to answer them.
Kai Schober
24 Sep 2012 12:36:20 EST
The Ultima Online team has received another set of questions last week. Here are the answers.
The Rustic and Gothic theme packs can not be bought together, as they could from uogamecodes.com, are there any plans to make that possible? (Snyder330)
When the change was made to Origin Store that was left out, we will add it back in next time we go to put something up on the store.
Some Origin stores outside of the US don’t all have the full range of items – when the High Seas booster will be available in EMEA region? (Sir Z), – Stygian Abyss is still not available in Brazil store (Bealank). Will the missing items become available?
We have reported this to my contact, also inquiring when the items from the Korean store will be back, when we hear something definite we will let you guys know.
Is there a reason to keep town loyalty up other than for the banners and the town titles? (Ingame Question)
Mesanna: Yes there is, we have another reason that will be revealed very soon. Sorry don’t want to spoil anything.
Will there be more and new casino games added in the future? (Ingame Question)
Phoenix: Yes! We are planning another dice game, plus a card game. Both will be in the style of the casino table games.
While I appresiate the good new new player guide you made for the “Playguide” section on herald.. is there any particular reason you removed alot of valid pages that people frequently used as a look up guide, such as the material bonus page? And why is the new new player guide in PDF? It is terrible to navigate for specific info. (Knuckleheads.dk)
Mesanna: We are moving over to a new site named UltimaOnline.com and it is in a html format, no more PDF.
Here’s my question: With everyone asking for new content and revamps for old dungeons like Doom, etc, would the DEVs consider re-instating imbuing artifacts/replicas as long as the item caps remain the same for non-exceptional items, meaning all artifacts/replicas would be imbueable to 450 property cap? I mean its a huge undertaking to come up with new artifacts/properties, think about the balance required, and most of the stuff will probably be a rehash of older items anyhow. Why not let us have some fun with it? I think it would really invigorate older lesser/minor artifacts without making anything overpowered, considering how some reforged/new shame loot is insanely good now. (Arcades)
Kyronix: This is a really interesting idea, but one that would need to be tread upon very carefully. What we’ve talked about is trying to incorporate allowing players to revitalize outdated artifacts via traditional crafting means and at the same time revitalizing the crafter class.
Hello Mesanna and Dev Team, because of the all-time discussion about the UO graphics I like to ask if there are any plans for the 2D/Classic Client to get better a screen resolution? We still have 800×600 max, this is quite outdated for today’s games. (Medea, Ultima Online Bibliothek Forum)
Kyronix: There are draw issues associated with the world building when increasing the size of the client window in the 2D/Classic client. We’ve experimented with it, but unfortunately the results were not aesthetically pleasing. You can always come over to the dark side and switch to the EC….*evil grin*
Hallo Mesanna and UO Dev Team, are there any plans to have the small, round Radar Map in the 2D/Classic Client replaced by a zoomable/bigger version of it or even a real map like the Enhanced Client has now? (Ultima Mapper works not on my system so I have no really usable solution at the moment.) And if a new map would be built in, is there a possibility to add Towns, Dungeons and important points too? (Waldschrat and MeneTekel, Ultima Online Bibliothek Forum)
Phoenix: Currently we are not very likely to make any significant changes to the Classic client. The Enhanced client has a much-improved map. Since that client’s UI can be completely customized, player-created user interfaces for the EC offer greater improvements still.
Dear UO Team, what about completing the house- and roof house building tiles? There are some of them and it would be fantastic to have with every set the complete (small) walls, windows, archs and for the roofs all directions of them. (Medea, Ultima Online Bibiothek Forum)
Mesanna: We do know there are missing directions of certain tiles but if you could email me the wall tiles and roof tiles you would like to see in the customization gump that would help us with adding the tiles you guys want.
Dear Dev Team, I am thinking about new instruments for bards – what about a bagpipe? It would fit quite perfectly for some of the races/classes! (Raknar Skarsol, Ultima Online Bibliothek Forum)
Mesanna: First off before I answer this question, are you JP in disguise? *grins* I think bagpipes would be different, but since the instruments are not equippable some of the magic would be lost don’t you think? Of course if you just want some for display and sound that would not be as difficult.
If you have something you want to know about Ultima Online and the team, post your questions on your fansite. All interesting questions will be collected and send to our team, so we can continue to answer them.
Discuss this issue.
Kai Schober
17 Sep 2012 12:13:34 EST
The Ultima Online team has received another set of questions last week. Here are the answers.
Concerns
People using high Seas boats, moored along side each other, to ‘house hide’ in PvP.
Mesanna: We have written this up as a bug and hope to have it fixed soon.
Questions
Are there any plans to make gargoyle horn dye and/or restyle available on hairdressers? Daslo, Neverplay UO, Cerwin Vega
Mesanna: After speaking with the team we think Horn Dressers is a good idea…Thanks for the suggestion
Are there any plans to revamp Doom? By this I mean changing or evolving the bosses, changing the loot, artifact pool or drop mechanics. There are several problems, for one an increase in players causes more bosses to spawn, however there is no obligation to complete a full round (which would be my definition of running a gauntlet), so people do the 5 rooms then leave before the Dark Fathers, usually leaving one skilled soloer to spend 30 mins plus killing each of them. Now we have gargoyles some gargoyle drops should probably also be included. (CKTC, ask the devs)
Mesanna: CKTC, Doom is on our list, it needs updated as much as champ spawns, dungeons and peerless do.
So awhile ago, I inquired about this and someone said the refer a friend program would be making a return. I recall someone saying the program would be back, but now I can’t find where this was said. Is there any chance the program will be making a return soon? I am asking because there are many players who gave their accounts away or sold them. Some of them that I have pestered mean to return via the trial so it would be really great to get referred by them if the referral program returns. (Sablestorm, Stratics)
Mesanna: Sablestorm, at this time we are not starting up the Buddy Referral program. With all the account management changes this is something that was not moved into the new system and would require.
A while back I read about EA’s plan for WAR/UO/DAoC: They planned to switch from ‘continuing content patches’ to so called ‘booster packs’. For UO that resulted in the ‘High Seas booster’ and the ‘Gothic’ and ‘Rustic’ theme packs. My question is: Will you continue to add boosters? And if so, will they be more in the line with the theme packs or the High Seas expansion? What is the timeframe in which you will add new content (other then differently colored items… . (from Knuckleheads.dk)
Mesanna: No UO is not going to be working on a booster pack or a theme pack for a while. We want to take the time to do a big bug push. We want to put in an in game Vendor Search, we want to be able to update the outdated armor in the game, we want to improve the fishing rewards, we want to finish virtues, and we want to finish the high res art. So at this time we don’t have a booster pack or expansion in the works. I am not saying we never well but just not at this time.
When will the Virtue system be completed? (from Knuckleheads.dk)
Mesanna: We have come up with several ideas but just don’t feel they are worthy of the last two virtues. We have read some of your ideas on stratics but would love to hear more.
Will the “brainstormed” Order/Chaos involve Warfare in usual None-PvP areas, Such as trammel? (from Knuckleheads.dk)
Mesanna: We have not gotten that far in the discussions with this, hopefully we can talk to everyone during the 15th Anniversary Party and get a good discussion going. Since a lot of you are not able join us we will also start a thread when we are at the point of go or no go on the project.
Non-medable armor is underpowered compared to medable armor. Is there any plans to boosting non-medable armor without ruining the usefulness of leather? (from UOForums.com)
Mesanna: We are in discussion regarding updating armor, making all armor useful again. We are going to do our best not to mess up leather but still make the other materials a viable option.
It is very bad impression goldspammers leave on newbies and returners, with the default channel being help, trials unable to change, and returners often not know they can switch channels. Could you at least make the default channel “General” if the account privileges grants access to it? (from UOForums.com)
Stephen: If you account is not a trial account we will make the change to have the default as general instead of help, this will be made in one of the upcoming publishes. Trail accounts will still be placed in the help channel. Please keep in mind the gold spammers are trial accounts those have to stay in the help channel.
Is it possible to see if your character is friended to the Ants? If not wish they could make something on the character sheet.(Ingame Question from Drachenfels)
Mesanna: Currently there is not a way to tell if your friended to the ants other than not getting attacked. It is something we can add in the future so I will add it to the backlog.
Still I must ask, Will you please fix houses to allow more than 10 Co-owners? Many of us have multiple accounts and are sick and tired of being nothing but a Friend in our own homes… I know I’m not alone and I know I’ve asked time and time again but PLEASE can we have at least 70 co-owner slots on our house. From MalagAste, Uhall. Risso, Ask the Devs (Clarification: Only the owning account characters are all counted as owning. Subsequent account characters are treated individually. Therefore it is not possible to co-own more than 8 characters from other accounts, any above that number can only be friended.)
Mesanna: Everyone will be happy to hear (or I hope they will be) that we as a team have talked about it and want to make a change to the housing co owners. Instead of 10 co owners we want to change it to 10 accounts which is 70 players total.
If you have something you want to know about Ultima Online and the team, post your questions on your fansite. All interesting questions will be collected and send to our team, so we can continue to answer them.
Kai Schober
10 Sep 2012 10:35:40 EST
You ask and we answer. In the last days our team took the time to address some of the questions that keep coming in. If you have something you want to know about Ultima Online and the team, post your questions on your fansite. All interesting questions will be collected and send to our team, so we can continue to answer them.
Here are the first seven:
1. Would it be possible to convert the older quests to the new Quest system? The older system only allows one quest to be active at a time. (Drachenfels, General Chat)
Mesanna: Are you speaking about the context questions vs the paperdoll quests? I know this is not the answer you want to hear but without looking deeper into the code we can not give a definite answer for this question.
2. Are Harpsichord rolls a permanent or time limited drop? If limited, when will they stop? (popps, Stratics)
Mesanna: This is a permanent addition to the game, hopefully in the future we can add additional music rolls.
3. Can you mention Fansites by URL in game…. there has been talk in the past you can and then some say you cannot (Rupert Avery, UO Auctions)
Mesanna: Yes, you can advertise any URL in game on your profile as long as it DOES NOT sell any items.
4. There’s a number of items that were introduced with SA that resemble & relate to the other Imbuing crafting ingredients. But they have no current use. Fey Wings, Fur, Horn of Abyssal Inferno, Kepetch Wax, Lodestone, Primeval Lich Dust, Slith Eye, Vile Tentacles, Void Core. Are there plans to make all those items, or some of them usable at one point? (Nystul, Ask the Devs)
Kyronix: We’re always looking for new and creative uses for underutilized ingredients and are currently working on adding some to upcoming content.
5. Hello, I just want to ask a question about Distilling. Are there any plans to expand it so that wines, beers, ales and sake can be made? Thanks (Edward Striker, Ask the Devs)
Mesanna: I will add this to our list of things to do, good suggestion Edward.
6. Is there any plans to make Golems semi “bondable” at some point in time? The big problem with Golems nowadays is that not only are they mostly underpowered, but also that any training put into them is undone when they can die so easily. Another part of the solution, would of course be to lessen the cooldown incurred upon healing the Golem. The cooldown is currently like 30 seconds, which is WAY too long. It should be shortened to something like 10 seconds. (PlayerSkillFTW, Ask the Devs)
Mesanna: We have talked about adding new golens to advanced Tinkering, but we are not going to add another golem to be a training punching bag for Luna.
7. Are there any plans to review getting to and from certain areas of Siege? There are many places on the shard that you cannot gate to from the Ter Mur facet and also from many places in the Tokuno facet, for example to areas around Luna and also areas around Yew and Skara Brae. (When you try, you get a very unhelpful message that says your spell doesn’t seem to work.) The inconsistencies make no sense at all and cause a lot of frustration for newcomers to the shard, especially when they work hard to make enough gold to place a house in an area that looks ideal for housing (because it’s sooooo empty) and then find out that the reason for the emptiness is just plain old bugginess. (Tina Small, Ask the Devs)
Mesanna: Tina we agree, and we are entering a problem report on this today.
That’s all for today, but we are sure you have more questions.
Keep them coming and we’ll see you in the game.
## QA5 – Stephen “Bleak” Brown – Software Engineer
A Few Questions and Answers to give an insight to:
Stephen “Bleak” Brown
Software Engineer
For those of us who don’t know, what has your “career track” been with the gaming industry?
My “career track” track begins with Ultima Online, I joined the team at the beginning of 2008. I got my feet wet with Publish 53 it seems like it was just yesterday Being an engineer I have worked on pretty much all of the systems UO has to offer.
What project that you were involved in are you the proudest of?
Where do I begin…I can’t choose just one: NPC Ship AI, Gargoyle throwing fixes, Item Insurance Gump, Random Treasure Map System, and Focus Skill Spec just to name a few.
I think most of us understand and respect Dev’s having to remain tight lipped about upcoming content etc. However, when it comes to past content only, what was the hardest thing for you to personally remain tight lipped about?
I’m not one to spoil a surprise but I would have to say the hardest thing to remain tight lipped about was the most recent patch to the Enhanced Client which fixed some serious client crashes.
When it comes to the current team, who is the practical joker of the bunch and have they gotten you? OR What was your best retaliatory strike against “the joker”?
Why So Serious? It would be a tie between Misk and me. As for retaliatory strikes, most just end with nerf gun wars.
When you are not chained to the desk working, what do you enjoy doing?
When I’m not chained to my desk I enjoy reading, playing MOBA games, watching football, duking it out on test center, and constantly checking the boards.
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2021-11-28 18:39:44
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https://wikimili.com/en/Linear_response_function
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Linear response function
Last updated
A linear response function describes the input-output relationship of a signal transducer such as a radio turning electromagnetic waves into music or a neuron turning synaptic input into a response. Because of its many applications in information theory, physics and engineering there exist alternative names for specific linear response functions such as susceptibility, impulse response or impedance, see also transfer function. The concept of a Green's function or fundamental solution of an ordinary differential equation is closely related.
A neuron, also known as a neurone and nerve cell, is an electrically excitable cell that communicates with other cells via specialized connections called synapses. All animals except sponges and placozoans have neurons, but other multicellular organisms such as plants do not. A neuron is the main component of nervous tissue.
In the nervous system, a synapse is a structure that permits a neuron to pass an electrical or chemical signal to another neuron or to the target effector cell.
Information theory studies the quantification, storage, and communication of information. It was originally proposed by Claude Shannon in 1948 to find fundamental limits on signal processing and communication operations such as data compression, in a landmark paper entitled "A Mathematical Theory of Communication". Applications of fundamental topics of information theory include lossless data compression, lossy data compression, and channel coding. Its impact has been crucial to the success of the Voyager missions to deep space, the invention of the compact disc, the feasibility of mobile phones, the development of the Internet, the study of linguistics and of human perception, the understanding of black holes, and numerous other fields.
Mathematical definition
Denote the input of a system by ${\displaystyle h(t)}$ (e.g. a force), and the response of the system by ${\displaystyle x(t)}$ (e.g. a position). Generally, the value of ${\displaystyle x(t)}$ will depend not only on the present value of ${\displaystyle h(t)}$, but also on past values. Approximately ${\displaystyle x(t)}$ is a weighted sum of the previous values of ${\displaystyle h(t')}$, with the weights given by the linear response function ${\displaystyle \chi (t-t')}$:
In physics, a force is any interaction that, when unopposed, will change the motion of an object. A force can cause an object with mass to change its velocity, i.e., to accelerate. Force can also be described intuitively as a push or a pull. A force has both magnitude and direction, making it a vector quantity. It is measured in the SI unit of newtons and represented by the symbol F.
${\displaystyle x(t)=\int _{-\infty }^{t}dt'\,\chi (t-t')h(t')+\dots \,.}$
The explicit term on the right-hand side is the leading order term of a Volterra expansion for the full nonlinear response. If the system in question is highly non-linear, higher order terms in the expansion, denoted by the dots, become important and the signal transducer cannot adequately be described just by its linear response function.
The Volterra series is a model for non-linear behavior similar to the Taylor series. It differs from the Taylor series in its ability to capture 'memory' effects. The Taylor series can be used for approximating the response of a nonlinear system to a given input if the output of this system depends strictly on the input at that particular time. In the Volterra series the output of the nonlinear system depends on the input to the system at all other times. This provides the ability to capture the 'memory' effect of devices like capacitors and inductors.
The complex-valued Fourier transform ${\displaystyle {\tilde {\chi }}(\omega )}$ of the linear response function is very useful as it describes the output of the system if the input is a sine wave ${\displaystyle h(t)=h_{0}\cdot \sin(\omega t)}$ with frequency ${\displaystyle \omega }$. The output reads
The Fourier transform (FT) decomposes a function of time into its constituent frequencies. This is similar to the way a musical chord can be expressed in terms of the volumes and frequencies of its constituent notes. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform of a function of time is itself a complex-valued function of frequency, whose magnitude component represents the amount of that frequency present in the original function, and whose complex argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the time domain. There is also an inverse Fourier transform that mathematically synthesizes the original function from its frequency domain representation.
${\displaystyle x(t)=|{\tilde {\chi }}(\omega )|\cdot h_{0}\cdot \sin(\omega t+\arg {\tilde {\chi }}(\omega ))\,,}$
with amplitude gain ${\displaystyle |{\tilde {\chi }}(\omega )|}$ and phase shift ${\displaystyle \arg {\tilde {\chi }}(\omega )}$.
An amplifier, electronic amplifier or (informally) amp is an electronic device that can increase the power of a signal. It is a two-port electronic circuit that uses electric power from a power supply to increase the amplitude of a signal applied to its input terminals, producing a proportionally greater amplitude signal at its output. The amount of amplification provided by an amplifier is measured by its gain: the ratio of output voltage, current, or power to input. An amplifier is a circuit that has a power gain greater than one.
Example
Consider a damped harmonic oscillator with input given by an external driving force ${\displaystyle h(t)}$,
${\displaystyle {\ddot {x}}(t)+\gamma {\dot {x}}(t)+\omega _{0}^{2}x(t)=h(t).\,}$
The complex-valued Fourier transform of the linear response function is given by
${\displaystyle {\tilde {\chi }}(\omega )={\frac {{\tilde {x}}(\omega )}{{\tilde {h}}(\omega )}}={\frac {1}{\omega _{0}^{2}-\omega ^{2}+i\gamma \omega }}.\,}$
The amplitude gain is given by the magnitude of the complex number ${\displaystyle {\tilde {\chi }}(\omega ),}$ and the phase shift by the arctan of the imaginary part of the function, divided by the real one.
From this representation, we see that for small ${\displaystyle \gamma }$ the Fourier transform ${\displaystyle {\tilde {\chi }}(\omega )}$ of the linear response function yields a pronounced maximum ("Resonance") at the frequency ${\displaystyle \omega \approx \omega _{0}}$. The linear response function for a harmonic oscillator is mathematically identical to that of an RLC circuit. The width of the maximum ${\displaystyle ,\Delta \omega ,}$ typically is much smaller than ${\displaystyle \omega _{0},}$ so that the Quality factor ${\displaystyle S:=\omega _{0}/\Delta \omega }$ can be extremely large.
Kubo formula
The exposition of linear response theory, in the context of quantum statistics, can be found in a paper by Ryogo Kubo. [1] This defines particularly the Kubo formula, which considers the general case that the "force" h(t) is a perturbation of the basic operator of the system, the Hamiltonian, ${\displaystyle {\hat {H}}_{0}\to {\hat {H}}_{0}-h(t'){\hat {B}}(t')\,}$ where ${\displaystyle {\hat {B}}}$ corresponds to a measurable quantity as input, while the output x(t) is the perturbation of the thermal expectation of another measurable quantity ${\displaystyle {\hat {A}}(t)}$. The Kubo formula then defines the quantum-statistical calculation of the susceptibility ${\displaystyle \chi (t-t')}$ by a general formula involving only the mentioned operators.
As a consequence of the principle of causality the complex-valued function ${\displaystyle {\tilde {\chi }}(\omega )}$ has poles only in the lower half-plane. This leads to the Kramers–Kronig relations, which relates the real and the imaginary parts of ${\displaystyle {\tilde {\chi }}(\omega )}$ by integration. The simplest example is once more the damped harmonic oscillator. [2]
Related Research Articles
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x:
In engineering, a transfer function of an electronic or control system component is a mathematical function which theoretically models the device's output for each possible input. In its simplest form, this function is a two-dimensional graph of an independent scalar input versus the dependent scalar output, called a transfer curve or characteristic curve. Transfer functions for components are used to design and analyze systems assembled from components, particularly using the block diagram technique, in electronics and control theory.
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be readily derived by integrating the product rule of differentiation.
In electrical engineering and control theory, a Bode plot is a graph of the frequency response of a system. It is usually a combination of a Bode magnitude plot, expressing the magnitude of the frequency response, and a Bode phase plot, expressing the phase shift.
Fourier optics is the study of classical optics using Fourier transforms (FTs), in which the waveform being considered is regarded as made up of a combination, or superposition, of plane waves. It has some parallels to the Huygens–Fresnel principle, in which the wavefront is regarded as being made up of a combination of spherical wavefronts whose sum is the wavefront being studied. A key difference is that Fourier optics considers the plane waves to be natural modes of the propagation medium, as opposed to Huygens–Fresnel, where the spherical waves originate in the physical medium.
In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response is of finite duration, because it settles to zero in finite time. This is in contrast to infinite impulse response (IIR) filters, which may have internal feedback and may continue to respond indefinitely.
In control theory and signal processing, a linear, time-invariant system is said to be minimum-phase if the system and its inverse are causal and stable.
The fluctuation–dissipation theorem (FDT) or fluctuation–dissipation relation (FDR) is a powerful tool in statistical physics for predicting the behavior of systems that obey detailed balance. Given that a system obeys detailed balance, the theorem is a general proof that thermodynamic fluctuations in a physical variable predict the response quantified by the admittance or impedance of the same physical variable, and vice versa. The fluctuation–dissipation theorem applies both to classical and quantum mechanical systems.
In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). This linear operator is given by convolution with the function :
Linear phase is a property of a filter, where the phase response of the filter is a linear function of frequency. The result is that all frequency components of the input signal are shifted in time by the same constant amount, which is referred to as the group delay. And consequently, there is no phase distortion due to the time delay of frequencies relative to one another.
Linear time-invariant theory, commonly known as LTI system theory, investigates the response of a linear and time-invariant system to an arbitrary input signal. Trajectories of these systems are commonly measured and tracked as they move through time, but in applications like image processing and field theory, the LTI systems also have trajectories in spatial dimensions. Thus, these systems are also called linear translation-invariant to give the theory the most general reach. In the case of generic discrete-time systems, linear shift-invariant is the corresponding term. A good example of LTI systems are electrical circuits that can be made up of resistors, capacitors, and inductors.. It has been used in applied mathematics and has direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas.
Nondimensionalization is the partial or full removal of units from an equation involving physical quantities by a suitable substitution of variables. This technique can simplify and parameterize problems where measured units are involved. It is closely related to dimensional analysis. In some physical systems, the term scaling is used interchangeably with nondimensionalization, in order to suggest that certain quantities are better measured relative to some appropriate unit. These units refer to quantities intrinsic to the system, rather than units such as SI units. Nondimensionalization is not the same as converting extensive quantities in an equation to intensive quantities, since the latter procedure results in variables that still carry units.
The Kramers–Kronig relations are bidirectional mathematical relations, connecting the real and imaginary parts of any complex function that is analytic in the upper half-plane. These relations are often used to calculate the real part from the imaginary part of response functions in physical systems, because for stable systems, causality implies the analyticity condition, and conversely, analyticity implies causality of the corresponding stable physical system. The relation is named in honor of Ralph Kronig and Hans Kramers. In mathematics these relations are known under the names Sokhotski–Plemelj theorem and Hilbert transform.
In signal processing, a causal filter is a linear and time-invariant causal system. The word causal indicates that the filter output depends only on past and present inputs. A filter whose output also depends on future inputs is non-causal, whereas a filter whose output depends only on future inputs is anti-causal. Systems that are realizable must be causal because such systems cannot act on a future input. In effect that means the output sample that best represents the input at time comes out slightly later. A common design practice for digital filters is to create a realizable filter by shortening and/or time-shifting a non-causal impulse response. If shortening is necessary, it is often accomplished as the product of the impulse-response with a window function.
In numerical analysis, the split-step (Fourier) method is a pseudo-spectral numerical method used to solve nonlinear partial differential equations like the nonlinear Schrödinger equation. The name arises for two reasons. First, the method relies on computing the solution in small steps, and treating the linear and the nonlinear steps separately. Second, it is necessary to Fourier transform back and forth because the linear step is made in the frequency domain while the nonlinear step is made in the time domain.
In electronics, complex gain is the effect that circuitry has on the amplitude and phase of a sine wave signal. The term complex is used because mathematically this effect can be expressed as a complex number.
In physics, nonlinear resonance is the occurrence of resonance in a nonlinear system. In nonlinear resonance the system behaviour – resonance frequencies and modes – depends on the amplitude of the oscillations, while for linear systems this is independent of amplitude.
Phase stretch transform (PST) is a computational approach to signal and image processing. One of its utilities is for feature detection and classification. PST is related to time stretch dispersive Fourier transform. It transforms the image by emulating propagation through a diffractive medium with engineered 3D dispersive property. The operation relies on symmetry of the dispersion profile and can be understood in terms of dispersive eigenfunctions or stretch modes. PST performs similar functionality as phase-contrast microscopy, but on digital images. PST can be applied to digital images and temporal data.
In mathematics, the exponential response formula (ERF), also known as exponential response and complex replacement, is a method used to find a particular solution of a non-homogeneous linear ordinary differential equation of any order. The exponential response formula is applicable to non-homogeneous linear ordinary differential equations with constant coefficients if the function is polynomial, sinusoidal, exponential or the combination of the three. The general solution of a non-homogeneous linear ordinary differential equation is a superposition of the general solution of the associated homogeneous ODE and a particular solution to the non-homogeneous ODE. Alternative methods for solving ordinary differential equations of higher order are method of undetermined coefficients and method of variation of parameters.
References
1. Kubo, R., Statistical Mechanical Theory of Irreversible Processes I, Journal of the Physical Society of Japan, vol. 12, pp. 570–586 (1957).
2. De Clozeaux,Linear Response Theory, in: E. Antončik et al., Theory of condensed matter, IAEA Vienna, 1968
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2019-10-20 13:32:46
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https://tex.stackexchange.com/questions/257469/setmainfont-error-with-most-recent-fontspec-version?noredirect=1
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# \setmainfont error with most recent fontspec version [duplicate]
I get a mysterious error running the following MWE (with both XeLaTeX and LuaLaTeX):
\documentclass{article}
\usepackage{fontspec}
\setmainfont{texgyrepagella-regular.otf}
\begin{document}
Hello World!
\end{document}
The error description in the .log file reads:
...
LaTeX Info: Redefining \upshape on input line 2245.
("C:\Program Files\MiKTeX 2.9\tex\latex\fontspec\fontspec.cfg")))
! Undefined control sequence.
l.5 \setmainfont{texgyrepagella-regular.otf}
The control sequence at the end of the top line
...
And the document that's produced looks like this:
I'm using MiKTeX 2.9 on Windows 7 with a very recent version of fontspec, which according to MiKTeX's Update Manager was packaged only four days ago (2015-07-214).
Is this a bug?
• @JosephWright: That here is the other way round: fontspec is too new and \str_case:nnF is unknown. Einbandi: Update the l3-packages. – Ulrike Fischer Jul 28 '15 at 13:00
• @UlrikeFischer: I just updated all the l3-related packages I could find but unfortunately the problem persits. – einbandi Jul 28 '15 at 13:13
• Did you run the update manager in admin and user mode? – Ulrike Fischer Jul 28 '15 at 13:15
• So far only in admin mode. I'm running in user mode right now and will get back to you. – einbandi Jul 28 '15 at 13:16
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2020-02-21 13:33:08
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https://forum.allaboutcircuits.com/ubs/oven-temperature-controller-makewithmaxim-contest.1359/
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# Oven Temperature Controller (#MakeWithMaxim Contest)
Introduction
I'm using the MAX31856EVSYS to build a oven temperature controller. Recipes tend to use different units of temperature: centigrade, fahrenheit & gas mark. It is not difficult to convert between these (many ovens also show both on the temperature dial), but I thought of building a system which allows the user to set the temperature (in any unit) and the baking time.
For now, the system will only monitor temperature and alert the user if it exceeds a narrow limit (if the oven's inbuilt temperature control system isn't working correctly or isn't calibrated well enough) or if the baking time is up.
My plan is to incorporate a closed loop temperature control system that lets the user set a desired temperature profile (eg. 200C for 15 minutes, 300C for 10 minutes). The system will then automatically sense the temperature and increase/decrease the temperature by controlling the power input to the oven (using a SSR for electric ovens, and a servo motor that controls the knob position for gas fired ovens).
I initially planned on using an ESP8266 for this project, but after taking a closer look, I realised that it would be easier to get it working with a Raspberry Pi (I was short on time since I received my board very late). The Pi is overkill for this, so I'll be porting it over to the ESP8266 eventually - using the ESP8266 lets me generate alerts on my phone since its got WiFi.
For now, I'm building a control interface for the temperature controller using the Raspberry Pi 3. I'm using a Python application to display a GUI on the Pi's 2.4" touchscreen HAT that lets the user view the status of the system. The user will be able to set & adjust the temperature using a rotary encoder.
BOM
For Sensing:
Type K Thermocouple
MAXIM MAX31856
Raspberry Pi
Rotary encoder
For control (to be implemented):
SSR/ relay for electric oven, servo for gas.
Schematics
View attachment 189070
Instructions
Using the MAX31856 with the Raspberry Pi is very easy: there are a couple of Python libraries available, which greatly simplifies the software side of things. Using the board without an existing library isn't a problem either: the datasheet contains all the details of the registers that need for be read.
I used Stephen Smith's Raspberry Pi Python Library for the MAX31856 which uses a software implementation of SPI which allows you to use any GPIO pins for SPI. The USB2PMB1 (PMOD to USB board) wasn't needed for this, but I did try it out: the evaluation software lets you play around with a range of configuration options for the MAX31856.
View attachment 189093
Use jumpers to connect the MAX31856PMB1 module (or any breakout board with the MAX31856) to the Raspberry Pi.
6 connections need to be made:
• VCC (+3.3V)
• GND
• SCK (SCLK or SPI Clock)
• CS (Chip select)
• SDI (Serial Data Input - MOSI on the Pi)
• SDO(Serial Data Output - MISO on the Pi)
View attachment 189092
Also connect the Type K thermocouple to the terminal block, with the yellow wire on the plus side.
View attachment 189087
View attachment 189091
Source Code
I used this script to read data temperature from the MAX31856 and print it to the Python terminal.
Code:
import time, math
import max31856
csPin=19
misoPin = 26
mosiPin = 16
clkPin =20
while csPin == 19: #run an infinite loop
#all the following code is in the loop, so indent it
max = max31856.max31856(csPin,misoPin,mosiPin,clkPin)
print( "Thermocouple Temp: %f degC" % thermoTempC)
print ("Cold Junction Temp: %f degF" % juncTempF)
print()
time.sleep(0.5)
I'm also did a little work on the GUI for the Python app that I'll be displaying on the Pi Display HAT.
It's far from complete, but this is what I have in mind.
I'll be adding options to set the temperature (in C, F or Gas Mark), and an option for a timer.
Once the user sets all the parameters, I'm thinking of displaying another page which will display a graph of temperature vs time for the duration of that particular session. At the end, it automatically turns off the oven (I'm sticking with an electric oven for now) and rings an alarm.
This code is only for the GUI of the first page, I''ll be adding including the MAX31856 library so that it can read temperature, and I've got to add functions to read the rotary encoder which will be connected to the GPIO pins.
This is a screenshot of the Python GUI program (taken on a Windows PC, but all I need to do is run it on Pi - porting Python is very easy!)
View attachment 189088
Code:
from tkinter import Tk, Label, Button, Entry, StringVar
LARGE_FONT = ("Verdana", 12)
class GUI:
def __init__(self, master):
self.master = master
master.title("Oven Temperature Controller")
self.label = Label(master, text="Oven Temperature Controller", font=LARGE_FONT)
self.label.place(x=60,y=0)
self.label2 = Label(master, text="Current Temperature: ")
self.label2.place(x=0, y=40)
self.templabel = Label(master, text="234.9")
self.templabel.place(x=120, y=40)
self.templabel1 = Label(master, text="°C")
self.templabel1.place(x=150, y=40)
self.templabel2 = Label(master, text="234.9")
self.templabel2.place(x=120, y=60)
self.templabel21 = Label(master, text="°F")
self.templabel21.place(x=150, y=60)
self.templabel3 = Label(master, text="Gas Mark")
self.templabel3.place(x=120, y=80)
self.templabel31 = Label(master, text="9")
self.templabel31.place(x=172, y=80)
self.label2 = Label(master, text="Set New Temperature: ")
self.label2.place(x=0, y=120)
#v=StringVar()
self.enter = Entry(master,width=6)
self.enter.place(x=130,y=120)
self.c_button = Button(master, text="Set New Temp")
self.c_button.place(x=180, y=116)
def greet(self):
print("Greetings!")
#temperature conversion units
def CtoF(C):
return (C*(9/5) + 32)
def FtoC(F):
return ((F-32)*(5/9))
#gas mark only works for ranges >1
def GasMarktoF(gasmark):
return (275+(gasmark-1)*25)
def GasMarktoC(gasmark):
return (((275+(gasmark-1)*25)-32)*(5/9))
root = Tk()
my_gui = GUI(root)
root.title("Oven Temperature Controller")
root.geometry("360x220")
root.mainloop()
Author
ck_007
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181
Last update
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2021-04-21 11:31:55
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https://tex.stackexchange.com/questions/308767/subfigure-inside-tabular-multirow
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# subfigure inside tabular multirow
I'm trying to align 3 subfigures in a 2-by-2 table, where the second column is a multirow spanning the two rows.
Here's where I am:
Usually \multirow vertically centers its contents, but that's not happening with the subfigure.
Here's the code to generate the previous image.
\documentclass[]{article}
\usepackage[margin=1in]{geometry}
\usepackage[subrefformat=parens,labelformat=parens]{subcaption} %% provides subfigure
\usepackage{tabularx} %% tabularx and multirow so that we can do nice subfigure layouts
\usepackage{multirow}
\usepackage[demo]{graphicx}
\begin{document}
\begin{figure}
\begin{tabular}{c|c}
\begin{subfigure}{0.3\linewidth}
\begin{center}
\includegraphics[width=1.9in, height=1.9in]{logo}
\caption{}
\end{center}
\end{subfigure} & \multirow{2}{.5\textwidth}{
\begin{subfigure}{0.3\linewidth}
\begin{center}
\includegraphics[width=3.9in, height=3.9in]{logo}
\caption{}
\end{center}
\end{subfigure}
} \\
\cline{1-1}
\begin{subfigure}{0.3\linewidth}
\begin{center}
\includegraphics[width=1.9in, height=1.9in]{logo}
\caption{}
\end{center}
\end{subfigure} & \\
\end{tabular}
\end{figure}
\end{document}
That-a-way? You can reference the subfigures adding a label in the first argument of \subcaptionbox which is to contain the caption text and the label: \documentclass[]{article} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage[margin=1in, showframe]{geometry} \usepackage{caption} \usepackage[subrefformat=parens,labelformat=parens]{subcaption} %% provides subfigure \usepackage{multirow} \usepackage{cleveref} \usepackage[demo]{graphicx}
\begin{document}
\begin{figure}
\centering\setlength\tabcolsep{1.25em}
\begin{tabular}{c|c}
\subcaptionbox{\vspace*{2ex}\label{1st-fig}}{\includegraphics[width=1.9in, height=1.9in]{logo}} & %
\multirow{2}{*}[\dimexpr1.9in-\baselineskip-0.5\abovecaptionskip-0.8ex\relax]{\subcaptionbox{label{3rd-fig}}{\includegraphics[width=3.9in, height=3.9in]{logo}}}
\\
\cline{1-1}
\subcaptionbox{\label{2nd-fig}}{\includegraphics[width=1.9in, height=1.9in]{logo}}
&
\end{tabular}
\caption{Some figures}
\end{figure}
We see in \cref{1st-fig} …
\end{document}
• Thank you! Can you explain why my approach didn't work? Also, where can I place a \label so that I can reference the subfigures? I stripped that from my minimal example, but I had it after \caption{} – Gus May 9 '16 at 22:54
• @Gus: The label has to go in the first argument of \subcaptionbox (see my edit). Why your approach doesn't work is hard to say. My first idea is that multirow isn't really done for that, as multirow{2} means the cell will use 2 lines, not 2 rows. So you should use the optional argument to make a manual adjustment. – Bernard May 11 '16 at 0:02
• I don't believe you edited the code to correspond to the new figure. – Gus May 24 '16 at 1:51
• @Gus: Oh! yes. I messed up with different versions of the code. It's corrected now. Thanks for pointing it. – Bernard May 24 '16 at 9:02
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2020-07-08 02:17:02
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https://www.nathantsoi.com/blog/ebay-hc-05-bluetooth-setup-guide/index.html
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# Ebay HC-05 Bluetooth Setup Guide
• First, we'll change the baud rate as described by Oscar. Use this link if you're using an Arduino to change the baud.
• Using my cool CP102 USB to UART adapter, I tried following the directions from painless360, summarized below, but I couldn't change my com port. Turns out you need to run Arduino as an administrator to change the COM port. Also those commands are for the new HC-06.
• Here is another great guide I used, specific to the HC-05 I ordered.
• The HC-05 is different from the HC-06 in 2 respects, it requires the button be depressed when powering on to go into AT mode and it wants both NL and CR sent with the command. Also, my UART to USB adapter has the option of 3 or 5v. 3v didn't work, so I had to use 5v.
• Plug it in
• Open the Arduino Studio
• Open the serial Monitor
• Pick "Both NL & CR"
• Set serial monitor BAUD to 9600
• NOTE: Not sure why this is, but my board wouldnt work unless I set the BAUD to 38400. If the following doesnt work, try the AT command with every BAUD until you find the right one.
• Also note, the AT+UART command, which returns the BAUD, showed that it was running 9600,0,0. Edit: This is for non-AT mode.
• Type AT
• It should say "OK"
• Change BAUD enter AT+UART=115200,1,0 where 1 is the stop bit and 0 is the parity. On the HC-06 the command is just AT+BAUD8 More documentation for the HC-05 is available at iteadsstudio.com
• It should say OK on the HC-05, OK115200 on the HC-06
• This will change the BAUD in bluetooth mode, not the baud in AT mode, so your connection should remain open
• Change the name, on the HC-05 AT+NAME? doesnt seem to work
• For the HC-06: AT+NAMEputhtenamehere, e.g. AT+NAMESuperCopter
• Check the pin with AT+ PSWD? and set it with AT+ PSWD=0000 where 0000 is your pin
• Use AT+PIN1234 for the HC-06
• If you're using a cc3d like me, here's how you wire it onto your copter
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2021-01-27 07:59:24
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http://mathhelpforum.com/advanced-statistics/184502-queuing-theory-m-m-2-queue-twist.html
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## Queuing theory M/M/2 queue with twist
Hello!
I've gotten stuck on a problem that is stated as follows:
Given a queueing system with possible an possible throughput with 2 servers. Server 1 and server 2 have an handling time that are distributed exponentially with parameter μ1 and μ2 respectively. Customers arrive with an arrive with an poisson distribution of rate λ, and will prioritise going to an free server (a server not handling another customer). If both are free customers will prioritise going to server 1. For this case, answer the following questions:
(1) Find the balance equation expressed with p[n] for this system when there are n customers in the system. However, when n=1 only one of the servers are working, to differ between these states denote them with a and b.
(2) Solve the equation you got in (1)
(3) Find the average number of customers in the system.
My attempt at an sollution:
Since writing my state graph here is kind of a lot of work I'll skip that step and try to explain my thinking in words.
For 0 customers: Customer arrives from states 1a and 1b with parameter μ1 and μ2 respectively. The system goes to state 1a if an customer arrive as stated in the excercise problem.
For 1 customer being served by register 1: This state is entered either if a new customer comes (parameter λ) or if there were 2 customers but the other registered finished serving their customer (parameter μ2). It leaves to state 0 (parameter μ1) and state 2 (parameter λ).
For 1 customer being served by register 2: This state is entered if there were 2 customers but the other register finished serving their customer (parameter μ1). It leaves to state 0 (parameter μ2) and state 2 (parameter λ).
For 2 customers being served: Arrival with parameter λ from both state 1a and 1b. Also entered from state 3 with parameter μ1 + μ2. Leaves to state 1a with parameter μ1, 1b with parameter μ2 and 3 with parameter λ.
For n>=3 customers being served: Incomming parameter λ from previous state and parameter μ1 + μ2 from the next one. Reveresely, outgoing to the prevoiusl state with parameter μ1 + μ2 and next with parameter λ.
Since the events "Customer stays" and "Customer arrives" are independent, with above reasoning, we get an balance equation as following:
State : Equation : "Equation number"
0 : p[0]*λ = μ[1]*p[1a] + μ2*p[1b] : (1)
1a : p[1a]*(μ1 + λ) = λ*p[0] + p[2]*μ2 : (2)
1b : p[1b]*(μ2 + λ) = p[2]*μ1 : (3)
2 : p[2]*(μ1 + μ2 + λ) = λ*(p[1a] + p[1b]) + p[3]*(μ1 + μ2) : (4)
n>=3 : p[n]*(μ1 + μ2 + λ) = p[n-1]*λ + p[n+1]*(μ1 + μ2) : (5)
Also, $\sum_{i=0}^{\infty} p_{i} = 1$ with $p_{1}=p_{1a} + p_{1b}$.
Now, if something here is wrong I've probably misunderstood something quite basic, but assuming it's right I proceeded to (2).
(2) This is where I get really screwed. The equations get ridiculously long and I'm not gonna write them here. The way I tried solving it was:
Step 1: Express p[2] with p[1a] and p[1b] respectively using equation (2) and (3).
Step 2: Put these equations in equation (1) and calculate the relationship between p[0] and p[2].
Step 3: Put my expressions from Step 1 and 2 into equation (4), thus giving me the relation between p[0], p[2], p[3] (this is ridiculously long though...)
Step 4: Use induction to prove that $p_{n} = (\frac{\lambda}{\mu_{1} + \mu_{2}})^{n-2} p_{2}$. With $\sum_{i=0}^{\infty} p_{i} = 1$ I'll also be able to calculate p[0].
Now, anyone can provide me with help or feedback? Any hint or comment appreciated. I think I've got a good way to calculate part (3) of the problem, but that's for when I've solved the first parts.
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2017-06-26 19:24:41
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http://clay6.com/qa/47875/the-plates-of-a-parallel-capacitor-have-an-area-of-100-cm-2-each-and-are-se
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# The plates of a parallel capacitor have an area of $100 \;cm^2$ each and are separated by $2.0\; mm$. The capacitor is charged by connecting it to $200\; V$ supply. How much electrostatic energy is stored in the capacitor?
Can you answer this question?
## 1 Answer
0 votes
$(A) 8.85 \times 10^{-7} J$
Hence A is the correct answer.
answered Jun 22, 2014 by
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2016-10-23 12:05:53
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https://www.elevri.com/courses/linear-algebra/matrix-arithmetics/matrix-multiplication
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## Learn matrix multiplication, scalar multiplication, addition and subtraction
### Definition of a matrix
A matrix is a rectangular list of numbers, called elements. Each matrix has rows and columns and its size is called (read "m times n"). Here are a few examples:
Furthermore, matrices are usually noted as integers (A, B, C, etc.). Let the matrix A be a -matrix. Each element and its i.d. (position in the matrix) are usually noted as in the following way:
Summation of matrices can only take place elementally and if the matricies have the same dimensions. Let both and be a pair of -matrices, and that and that then it applies that;
Here we have two examples:
### Scalar multiplication
Let be an -matrix. Then the following applies to all vectors and in and each scalar :
Scalar multiplication with a matrix works intuitively. Let be a -matrix summed times. Then it applies that:
and for each element in it applies that:
### Matrix multiplication
In order for the multiplication between two matrices to be defined, it is required that the number of columns of the left matrix must correspond to the number of rows of the right matrix. That is, the dimensions of the result matrix are the number of rows of the left matrix times the number of columns of the right matrix. In other words:
But what will be the result matrix of ? We show the simplest multiplication between matrix and vector :
Let's take an example:
We know from the above that the dimensions of the result of become and the result is:
Let's take another example of matrix multiplication where has an additional column and thus is noted as the matrix :
We know from the above that the dimensions of the result of become . The result is:
On a general basis, we conclude that the product of two matrices, and , is calculated by multiplying the rows of with the columns of . So the result is:
where the elements of the matrix become:
### Inner and outer products
In linear algebra, we talk about inner and outer products between two vectors of the same dimension, and . These two are defined as follows:
• inner product: , i.e. a scalar
• outer product: , a matrix
Take the following example, let:
Then it applies that the inner and outer product are:
### Identity, inverse and transpose
The student needs to be aware of the following matrices that we deal with in this section:
• (the identity matrix)
• (the inverse of )
• (the transpose of )
#### The identity matrix
above refers to the identity matrix, which can be seen as a multidimensional one, an -matrix where all elements are 0 except the diagonal elements, which are all 1.
The identity matrix functions as 1 being the identity operator for all numbers:
namely that:
Note here that multiplication by is commutative.
#### Inverse
The identity matrix also causes the existence of an inverse matrix, noted as . The following property applies:
If is a multidimensional one, then can be seen as , even if that operation is mathematically illegal.
#### Transpose
Last but not least, we have the transpose of , which is noted . It can be seen as a rotation of , where its rows become columns, as follows:
### The laws of matrix arithmetic
The following laws apply to matrix addition;
• (commutative law)
• (associative law)
#### Laws of multiplication
Note that the commutative law does not apply to matrix multiplication, i.e.:
However, the following laws do apply:
• (associative law)
• (distributive law)
• (identity)
• (linearity)
• (linearity)
## Good outline for linear algebra and short to-do list
We work hard to provide you with short, concise and educational knowledge. Contrary to what many books do.
## Get exam problems for old linear algebra exams divided into chapters
The trick is to both learn the theory and practice on exam problems. We have categorized them to make it extra easy.
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2023-03-24 02:24:13
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https://socratic.org/questions/how-do-you-solve-x-1-5-1-2#406752
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# How do you solve x-1/5=-1/2?
Apr 14, 2017
See below.
#### Explanation:
Let's start by adding $\frac{1}{5}$ to both sides.
$x - \frac{1}{5} = - \frac{1}{2}$
$x = - \frac{1}{2} + \frac{1}{5}$
Now, we have to make common denominators (which is $10$, as $\lcm \left(2 , 5\right) = 10$)
$x = - \frac{5}{10} + \frac{2}{10}$
$x = - \frac{3}{10}$.
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2022-12-06 10:39:53
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https://www.ryantolsma.com/physics/2021/04/06/speedrun.html
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In this post, I will attempt to derive and explain some consequences of special relativity as fast as possible, starting only with the invariance of physics in inertial frames and the constancy of the speed of light. I will assume basic familiarity of Lagrangian mechanics and Einstein summation convention.
## Basics
Consider a particle moving at the speed of light along the $$x$$-axis, such that $$x = \pm ct$$ and the other coordinates are constant. Note that $$c^2t^2 -x^2=0$$. Since this must be invariant under all changes of reference frame $$x\to x'$$, it follows that $$c^2t'^2 - x'^2 = 0$$ as well. Using natural units, more generally this implies that any changes of an intertial reference frame $$(t,x, y, z) \to (t', x', y', z')$$ must preserve the equality
$t^2 - x^2 - y^2 - z^2 = t'^2 - x'^2 -y'^2 - z'^2$
which as an invariant (squared) norm gives what’s known as the Minkowski metric. Any transformation respecting this metric is a symmetry of relativistic physics, and the group of all these symmetry preserving transformations is known as the Lorentz Group. From here on out, for a four vector $$x^\mu = (t,x, y, z)$$ we take $$x^\mu x_\mu = t^2 - x^2-y^2-z^2$$ to be its squared norm under this metric.
## Symmetries
Now that we have an equation which enables us to identify if a transformation is a symmetry, we want to be able to classify them. First, note that for a 3D rotation acting on the $$x,y,z$$ components, given by a matrix $$R$$ satisfying $$RR^T = I$$, we can easily see that
$(Rx)^\mu (Rx)_\mu = t^2 - \vert\vert(Rx)_i\vert\vert_2^2 = t^2 - x^2 - y^2 - z^2 = x^\mu x_\mu$
preserves the metric and is thus a symmetry as rotations preserve the Euclidean norm. By a similar process, you can try convincing yourself that time reversal and space inversion also qualify as symmetries. Next, we want to understand how coordinates should transform over shifts in velocity. To do this, consider coordinates $$x^+ = t+x$$ and $$x^- = t-x$$ (leaving the other components constant). Then by construction $$(x^+)(x^-)$$ is invariant. An obvious invariant preserving transformation is to then take $$x'^+ = \lambda x^+$$ for some constant $$\lambda$$ and $$x'^- = \frac{1}{\lambda}x^-$$ such that $$(x^+)(x^-) = (x'^+)(x'^-)$$. Explicitly expanding out and solving for $$x' = \frac{\lambda}{2}x^+ - \frac{1}{2\lambda}x^-$$ in terms of $$(t,x)$$ and similarly for $$t'$$, we find that
\begin{align*} x' &= \frac{\lambda^2 +1}{2\lambda}x + \frac{\lambda^2-1}{2\lambda}t \\ t' &= \frac{\lambda^2-1}{2\lambda}x + \frac{\lambda^2+1}{2\lambda}t \end{align*}
In particular, choosing a frame with $$x'=0$$ implies that $$x = \frac{1-\lambda^2}{1+\lambda^2}t$$ and thus $$x = vt$$ with $$v = \frac{1-\lambda^2}{1+\lambda^2}$$. Using this definition of $$v$$ and plugging in to solve for $$(t', x')$$, it becomes apparent that
\begin{align*} t' &= \frac{t-vx}{\sqrt{1-v^2}} &= \gamma(t - vx) \\ x' &= \frac{x -vt}{\sqrt{1-v^2}} &= \gamma(x-vt) \\ y' &= y \\ z' &= z \end{align*}
where we adopt the standard notation for the Lorentz Factor $$\gamma = \frac{1}{\sqrt{1-v^2}}$$. The derived equation is often called the Lorentz Boost applied along the $$x$$-axis and intuitively corresponds to shifting to another reference frame with a constant velocity $$v$$. To boost along an arbitrary axis $$w$$, one simply rotates $$w$$ to the $$x$$-axis, applies the standard Lorentz Boost, and rotates the frame back. Although I won’t show it here, every element of the the Lorentz Group can be broken down into some composition of rotations, boosts, time reversal, and spatial inversion. These completely describe the symmetry transformations of the Minkowski Space special relativity resides in.
For some fun, I’ll show a couple principles underlying some classic “paradoxes” you may have seen before. First, we illustrate the effects of Lorentz Contraction. Consider a pole of length $$L$$ sitting still at the origin along the $$x$$-axis in the $$(t, x)$$ frame. Now consider a frame $$(t', x')$$ constructed by applying a boost of velocity $$v$$. At $$t'=0$$, by the Lorentz Boost formulas we know that $$t = vx$$, and so at the tip of the pole where $$x=L$$ we find that $$t=vL$$. Now, applying the invariance of norm, we see that
\begin{align*} x'^2 - t'^2 &= x^2 - t^2 \\ x'^2 &= L^2 - v^2L^2 \\ x' &= L\sqrt{1 - v^2} = \frac{L}{\gamma} \end{align*}
and thus the length of the pole in the moving $$(t', x')$$ frame actually decreases by a factor of $$\frac{1}{\gamma}$$.
Next, to illustrate time dilation we consider the same setting, but now note what happens at $$x'= 0$$. Plugging into our equations from teh Lorentz Boost, we see that this directly implies $$x = vt$$, and thus applying invariance of the norm again:
\begin{align*} t'^2 - x'^2 &= t^2 - x^2 \\ t'^2 &= t^2 - v^2t^2 \\ t' &= t\sqrt{1-v^2} = \frac{t}{\gamma} \end{align*}
which shows that clocks in this moving frame actually appear to be running slower by a factor of $$\frac{1}{\gamma}$$ compared to the stationary frame. Together, these two concepts can be applied to show that simultaneity is not a universal concept, and that simultaneous events in one frame can appear time-separated in another. I highly encourage the ambitious reader to try deriving this themselves using these tools.
## Proper Time and Action
Proper time is an incredibly helpful concept and physically describes the amount of time an object experiences with respect to its own frame. For example, suppose that someone travels along a path from point $$a$$ to point $$b$$ with their own reference identified as $$(\tau, x')$$. At each instant, we can apply our good friend invariance of the norm, to see that $$d\tau^2 - dx'^idx'^i = dt^2 - dx^idx^i$$ for the travelling and stationary observer frames. However, for the traveller, in their frame of reference, they always have $$dx^i = 0$$ as their own position defines the origin of their frame. Thus, it follows that
$d\tau = \sqrt{dt^2 - dx^i dx^i}$
To compute the total proper time over their journey, we can integrate, yielding
\begin{align*} \tau_{a,b} &= \int_a^b d\tau \\ &= \int_a^b \sqrt{dt^2 - dx^idx^i} \\ &= \int_a^b dt\sqrt{1 - \frac{dx^i}{dt}\frac{dx^i}{dt}} \\ &= \int_a^b \sqrt{1-v^2}dt \end{align*}
Note how our derivation is actually independent of which path our friend takes as long as the endpoints are fixed, and thus our definition of proper time is actually a path invariant! From Lagrangian mechanics, you may recall the Principle of Least Action in which a quantity which is stationary to path variation is used to derive equations of motion. By inspection, our definition of proper time seems like a natural invariant to use to define such an action principle:
\begin{align*} S &= -m \tau_{a,b} \\ &= -m \int_a^b \sqrt{1-v^2}dt \end{align*}
where we judiciously introduce a factor of $$-m$$ as the invariance of action is maintained under constant scaling.
## Energy
Now that we have an action, we’re almost there. Recalling the definition of the Lagrangian from action, $$S = \int \mathcal{L}$$, we immediately can identify that here we have $$\mathcal{L} = -m \sqrt{1 - v^2}$$. As this may still look unfamiliar, we can reintroduce units using the fact that $$\mathcal{L}$$ has units of energy, to see that
$\mathcal{L} = -mc^2\sqrt{1 - \frac{v^2}{c^2}}$
Taking a first order Taylor expansion and assuming that $$\frac{v}{c} << 1$$ in the classical mechanics regime, we see that
$\mathcal{L} \approx -mc^2(1 - \frac{v^2}{2c^2}) = -mc^2 + \frac{mv^2}{2}$
Now, taking the Legendre Transform to derive the Hamiltonian we see that
$\mathcal{H} = mc^2 + \frac{mv^2}{2}$
This should look familiar! The Hamiltonian $$\mathcal{H}$$ represents the energy of a system, and what we’ve now derived consists of a classical kinematic energy term $$\frac{mv^2}{2}$$ and a constant factor $$mc^2$$. Setting $$v = 0$$, we see that $$E = mc^2$$ is the resting energy of our system and is exactly the celebrated mass-energy equivalence formula Einstein derived a century ago!
#### Remarks
I hope you were able to enjoy reading this and found it at least somewhat insightful! If you’re interested in these sorts of things, the book Spacetime Physics by Taylor and Wheeler is a classic and has plenty of fun paradox brainteasers to play around with.
|
2021-04-19 21:02:27
|
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|
https://study.com/learn/lesson/multiplying-exponenetial-expressions.html
|
# Multiplying Expressions with Exponents
Stephen Sacchetti, Jennifer Beddoe
• Author
Stephen Sacchetti
Stephen graduated from Haverford College with a B.S. in Mathematics in 2011. For the past ten years, he has been teaching high school math and coaching teachers on best practices. In 2015, Stephen earned an M.S. Ed from the University of Pennsylvania where he currently works as an adjunct professor.
• Instructor
Jennifer Beddoe
Jennifer has an MS in Chemistry and a BS in Biological Sciences.
Learn about multiplying exponents with different bases, as well as multiplying exponents with the same base. Explore multiplying exponents rules, how to multiply exponents with different bases, and what happens when you multiply exponents. Updated: 11/08/2021
Show
## How Do You Multiply Exponents?
This lesson will focus on operations on numbers that have exponents. In particular, the main goal of this lesson will be to answer the question of when and how do you multiply exponents. After establishing the when and how there will be an exploration of multiplying exponents rules that outline how to combine exponents in a variety of situations.
### What are Exponents?
The four basic operations of arithmetic are addition, subtraction, multiplication, and division. Recall that multiplication is simply a shorter way to write repeated addition. For example, {eq}3 + 3 + 3 + 3 + 3 + 3 = 6 \times 3 {/eq}. Similarly, an exponent is an operation that is a shorter way to write repeated multiplication. For example, {eq}3 \times 3 \times 3 \times 3 \times 3 \times 3 = 3^6 {/eq}. In this case, 6 is called the exponent and 3 is called the base. At times, the word power is also used to describe an exponent. For instance, the expression in Fig. 1 might be called "two to the power of five."
More generally, any exponential expression can be written as {eq}b^a {/eq} where a is the exponent and b is the base.
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## Multiplying Exponents with the Same Base: Do You Add Exponents When Multiplying?
What do you do when multiplying exponents? Since exponents mean repeated multiplication, a natural question of what happens when you multiply exponents might arise. In this section, the idea of multiplying two bases raised to exponents will be explored. For example, consider the expression {eq}2^5 \times 2^3 {/eq}. It may be tempting here to assume that the 2's multiply together or that the 5 and 3 multiply together, but in fact, neither of these is correct. Instead, lean on the definition of an exponent to rewrite this expression before combining any terms:
{eq}2^5 \times 2^3 = (2 \times 2 \times 2 \times 2 \times 2) \times (2 \times 2 \times 2) {/eq}
Noticing that there are eight total 2's here, this expression can be rewritten as {eq}2^8 {/eq}. This idea can be generalized to show that you add exponents when multiplying like bases: {eq}b^x \times b^y = b^{x + y} {/eq}
## Multiplying Exponents With Different Bases
But what happens when multiplying exponents with different bases? This cannot be done by adding the exponents as in the previous section because it is unclear what to do with the base. This section will show how to multiply exponents with different bases using two different methods of simplification.
### Exponential Multiplication: Simplification Method
The first method is the simplification method. Essentially, this method is used to write out the exponential expression as a long string of multiplication and then combine all of the terms. Consider the example: {eq}3^4 \times 4^3 {/eq}. Writing this out gives: {eq}3 \times 3 \times 3 \times 3 \times 4 \times 4 \times 4 {/eq} and none of the 3's can be combined with the 4's other than by multiplying out this product. Doing so gives an answer of 5184.
### Exponential Multiplication: Grouping Method
The second method of multiplying exponential expressions with different bases is grouping. The grouping method essentially says to rearrange the terms so that the multiplication can be done a bit more easily. Recall the example above of {eq}3^4 \times 4^3 {/eq} Notice that there is one more 3 than there are 4's. So we can combine the same number of 3's and 4's and have one 3 left over. For instance:
{eq}3 \times 3 \times 3 \times 3 \times 4 \times 4 \times 4 = 3 \times 4 \times 3 \times 4 \times 3 \times 4 \times 3 = 12 \times 12 \times 12 \times 3 = 12^3 \times 3 {/eq}
So it is possible to rewrite the expression using exponents: {eq}3^4 \times 4^3 = 3^3 \times 4^3 \times 3 = 12^3 \times 3 {/eq}.
More generally, it can be said that {eq}a^x \times b^x = (ab)^x {/eq} or, in words, multiplying different bases raised to the same power means that the bases can be multiplied together first and then raised to that power.
Notice, though, that this grouping method requires having the same exponent on each base. For instance, to group {eq}x^5 \times y^7 {/eq}, first rewrite it as {eq}x^5 \times y^5 \times y^2 {/eq} and then group the terms with the same exponents: {eq}(xy)^5 \times y^2 {/eq}
## Multiplying Exponents Rules: All Exponent Rules
There are seven rules to remember for multiplying bases with exponents. Each multiplying exponent rule will be shown below with an example to help illustrate it. Some of these rules do not involve multiplication but will be useful in solving more complicated problems involving multiplying exponents.
### Rule 1: Product of Powers
If multiplying exponential expressions with the same base, then add the exponents together: {eq}2^4 \times 2^7 = 2^{11} {/eq}
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|
2022-01-19 03:44:17
|
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|
https://newproxylists.com/tag/parameters/
|
## tables – How to better display a long list of parameters: valuable items?
I would like to ask if there is a better way to display a number of categories containing long lists of items "parameter: value" than what I have for the moment :
Category 1
parameter1: value1
parameter2: value2
Category2
parameter1: value1
parameter2: value2
There are 4 categories. The number of parameters in a list can contain up to 15 elements. The value can be a word as well as several sentences. All this information must be visible simultaneously.
Thank you
## plugins – How to pass multiple custom fields as shortcode parameters
I'm learning to pass parameters in short code, I know the basic step after being read by WP Codex.
But for now, the plugin comes with custom fields and as a title, I want to know how to do it.
Here is the code of the plugin that adds custom fields
``````\$prefix = '_al_listing_';
\$fields = ();
\$fields() = (
'name' => __( 'Price', 'auto-listings' ),
'id' => \$prefix . 'price',
'type' => 'number',
'min' => 0,
'step' => '0.01',
);
\$fields() = (
'name' => __( 'Suffix', 'auto-listings' ),
'desc' => __( 'Optional text after price.', 'auto-listings' ),
'id' => \$prefix . 'price_suffix',
'type' => 'text',
);
\$fields = apply_filters( 'auto_listings_metabox_details', \$fields );
ksort( \$fields );
return (
'id' => \$prefix . 'details',
'title' => __( 'Details', 'auto-listings' ),
'post_types' => 'auto-listing',
'fields' => \$fields,
'context' => 'side',
);
``````
And here is the code I've built to call a shortcode with parameters, but I do not know what to pass 1 custom field at a time.
``````public function listings( \$atts ) {
\$atts = shortcode_atts(
(
'orderby' => 'date',
'order' => 'asc',
'number' => '20',
'price' => ''
),
\$atts
);
\$query_args = (
'post_type' => \$post-type,
'post_status' => 'publish',
'meta_key' => '_al_listing_price',
'orderby' => 'meta_value',
'meta_value' => \$atts('price'),
'order' => \$atts('order'),
'posts_per_page' => \$atts('number'),
);
return \$this->listing_loop( \$query_args, \$atts, 'listings' );
}
``````
Any help is appreciated.
## C # to C ++ (dependency injection, passing parameters)
I am really new to C ++ and I am trying to understand some C # conversions in C ++. I have trouble understanding when to use a parameter like reference / pointer / smart pointer or what to return.
I've read that it's best to use smart pointers as much as possible instead of returning a pointer.
I made an example in C # and converted it to C ++. Did I do something wrong or what would be an improvement of my code?
Greetings
C #:
``````public struct Container
{
public int Index { get; set; }
}
public interface IA
{
Container() CreateContainers();
}
public class A : IA
{
public Container() CreateContainers()
{
return new Container()
{
new Container() { Index = 1 },
new Container() { Index = 2 }
};
}
}
public interface IB
{
void DoSomething();
void DoOtherStuff(List someContainer);
}
public class B : IB
{
public B(IA a)
{
_a = a;
}
public void DoSomething()
{
var containerList = _a.CreateContainers();
for (var i = 0; i < containerList.Length; i++)
{
//do something
}
}
public void DoOtherStuff(List someContainer)
{
someContainer.Add(new Container() { Index = 4 });
someContainer.Add(new Container() { Index = 5 });
someContainer.Add(new Container() { Index = 6 });
}
}
public class Program
{
public void Main()
{
IA a = new A();
IB b = new B(a);
b.DoSomething();
var list = new List();
b.DoOtherStuff(list);
}
}
``````
C ++:
``````//Container.h
struct Container
{
public:
Container() noexcept : _index(0) {}
int GetIndex() const { return _index; }
void SetIndex(const int value) { _index = value; }
private:
int _index;
};
//IA.h
class IA
{
public:
virtual ~IA() noexcept = default;
virtual std::unique_ptr> CreateContainers() = 0;
};
//A.h
class A : public IA
{
public:
std::unique_ptr> CreateContainers() override;
};
//A.cpp
std::unique_ptr> A::CreateContainers()
{
Container c1;
c1.SetIndex(1);
Container c2;
c2.SetIndex(2);
auto result = std::make_unique>(2);
result->push_back(c1);
result->push_back(c2);
return result;
};
//IB.h
class IB
{
public:
virtual ~IB() noexcept = default;
virtual void DoSomething() = 0;
virtual void DoOtherStuff(std::vector& someContainer) = 0;
};
//B.h
class B : public IB
{
public:
explicit B(IA& a) noexcept;
void DoSomething() override;
void DoOtherStuff(std::vector& someContainer) override;
private:
IA& _a;
};
//B.cpp
B::B(IA& a) noexcept :
_a(a)
{
}
void B::DoSomething()
{
const auto containerList = _a.CreateContainers();
for (auto i = 0; i < containerList->size(); i++)
{
//do something
}
}
void B::DoOtherStuff(std::vector& someContainer)
{
Container c1;
c1.SetIndex(4);
Container c2;
c2.SetIndex(5);
Container c3;
c3.SetIndex(6);
someContainer.push_back(c1);
someContainer.push_back(c2);
someContainer.push_back(c3);
}
//main.cpp
int main()
{
const std::unique_ptr a = std::make_unique();
std::unique_ptr b = std::make_unique(*a);
b->DoSomething();
auto list = std::make_unique>();
b->DoOtherStuff(*list);
}
$$```$$
``````
## adjustment – Combination of adjusted parameters to create a variable and tell me the error it contains
Assuming you are using mathematica 12 (or higher),
it's pretty simple with the use `Around`, `VectorAround`, `AroundReplace`, etc.
Mathematica, in these versions, has an integrated error propagation:
So, you extract the covariance matrix and use `AroundReplace`
(he even uses the correlated error propagation)
So let's start: assuming your adjustment is in the variable `fit`:
``````covMat = fit("CovarianceMatrix");
bestParams = fit("BestFitParameters");
vecErr = bestParams((All, 1)) ->
VectorAround(bestParams((All, 2)), 0.5*(covMat + Transpose(covMat)))
AroundReplace(ArcTan(-b/a), vecErr)
``````
the `0.5*(covMat + Transpose(covMat))` is used because the covariance matrix is computed by a matrix inversion that is sometimes insufficiently stable to produce an absolute symmetric matrix that `VectorAround` will complain about. We therefore symmetry with force by hand.
He will even tell you what is the formula of your error if you plug symbols:
``````FullSimplify(
AroundReplace(
ArcTan(-b/a), {{a, b} ->
VectorAround({A, B}, {{C(A), C(A, B)}, {C(A, B), C(B)}})}),
A (Element) Reals && B (Element) Reals)
``````
## blockchain – Understanding the parameters of the API iquidus
I'm trying to understand the values that are asked of me in the settings.json file to install the IQs explorer. I know how to get the genesis block because I hard-coded it when I created a fork to create a new altcoin. However, I do not know how to get the genesis_tx. I know you can convert it from the genesis block but I do not know how?
"Genesis": {
"genesis_tx": "",
"genesis_block": ""}
In terms of API parameters, they ask for an address, blockindex, blockhash and txhash. The blockindex can be any value in the height of your block … so, do I select any index value with the corresponding address in the list of transactions? Also, how can I get txhash and blockhash? In my opinion, these values are subject to change and I know that they are stored in the blockchain … Does any one know how to retrieve this information?
"api": {
"blockindex": "",
"blockhash": "",
"txhash": "",
}
## seo – Does the blocking of URL parameters in the Google search console remove pages from the index?
In GSC, it is possible to tell Google how to handle the URL parameters. Google has suddenly started indexing more than 600,000 URLs with settings on our website. I've set the parameter in GSC indefinable and about 400,000 pages disappear from the index. So I thought everything was working as expected and that the rest of the pages would also be deleted. After one month, 300,000 pages are indexed again.
Should this work and should I wait or should I set pages with noindex?
## 8 – Need to document the AJAX recall parameters?
Considering the `PHPCS` Warning:
phpcs: hook implementations should not duplicate @param documentation
Is it possible to say PHPCS that the function is an AJAX callback (for such warnings to be generated)?
Or, should we document all AJAX recall settings, with `@param` PHPDoc?
## Context:
Our company has been discussing the AJAX booster function below:
``````/**
* Prevents popup's redirection from dashboard to another link.
*
* Users were redirected to another page as soon as the Event-creation
* popup form was submitted, just added an AJAX handler to submit button
* which prevents the redirect using ajax,
* and when there is any error on the form, the message is sent to the popup.
*/
function _MYMODULE_popup_submit(array &\$form, FormStateInterface \$formState) {
// ...
}
``````
Where a developer insisted to add `@param` PHPDoc comments on each AJAX reminder as:
``````/**
* Prevents popup's redirection from dashboard to another link.
*
* Users were redirected to another page as soon as the Event-creation
* popup form was submitted, just added an AJAX handler to submit button
* which prevents the redirection using ajax,
* and when there is any error on the form, the message is sent to the popup.
*
* @param array &\$form
* All data of the form.
* @param DrupalCoreFormFormStateInterface \$formState
* State of the form.
*
* @return DrupalCoreAjaxAjaxResponse
* Returns the data back as AjaxResponse object.
*/
function _amt_dayview_popup_submit(array &\$form, FormStateInterface \$formState) {
// ...
}
``````
And the other considered it as understandable and that it should not be documented:
… remember that PHPCS warn you whenever we use `@param` on any hook, like "`phpcs: Hook implementations should not duplicate @param documentation`"and just because PHPCS can not differ AJAX-callback from normal functions it does not mean that we should do it …
## FizzBuzz with more parameters in Swift
Here is my approach to an extensible version of the FizzBuzz challenge in Swift (so we can add more numbers with which to check `3` and `5`).
I first tried using dictionaries, but since they do not necessarily preserve the order of the elements, this can lead to: `BuzzFizz` return values.
How can this code be improved?
``````func fizzBuzz (n: Int, responsesByMultiples: ((Int, String)) = ((3, "Fizz"), (5, "Buzz"))) -> String {
var result: String = ""
for key in responsesByMultiples {
let (multiple, response) = key,
isMultiple: Bool = (n % multiple) == 0
if isMultiple {
result += response
}
}
return result == "" ? String(n) : result
}
for i in 1...100 {
print(fizzBuzz(n: i))
}
``````
## Pass the parameters correctly in a Django URL
I have a view where I take a code in an entry, everything is fine there, but then I have to use this code in another URL to call a profile view, the problem being that in there passing, it generates random letters and I do not know how to solve it. I wish this to happen is only http://127.0.0.1:8000/administracion/perfil/ABC/ but that I generate
View
``````def ingresar_matricula(request):
if request.method=='POST':
form=MatriculaForm(request.POST)
if form.is_valid():
a=form.cleaned_data('codigo')
return redirect('perfil',{a})
def perfil(request,Codigo):
query=Alumno.objects.filter(codigo=Codigo)
context={'Query':query}
return render(request,'listar/perfil.html',context)
``````
URLs
``````path('perfil//',views.perfil,name='perfil')
``````
## Summary on Common Lisp Function Keyword Parameters
This is an experiment on extending a basic integrated Common Lisp sequence function (ie: `remove`), to provide more features with little or no impact on performance. The long-term goal might be to provide such extensions for many other CL functions.
The CL sequence functions appear to provide at least 3 different types of abstraction: 1) data type abstraction, a sequence that can be a list, a vector, or a string; 2) higher order functions, in which a function can be passed as a parameter; and 3) keyword parameters, different keyword arguments that can fit the calculation for specific uses.
The basic idea explored here is to add many more keywords, to give the programmer more capacity and flexibility in processing sequences. If successful, it could avoid having to write (or recall) many utility functions that support the sequences. It could also consolidate some of the existing CL sequence functions (for example, `remove`, `remove-if`, `remove-if-not`, `delete`, `delete-if`, `delete-if-not`, `remove-duplicates`, `delete-duplicates`) in a single operation (in this case simply called "remove-sequence" with additional keywords for: destructive and: duplicates). Other bindings not linked "in sequence" could also be possible – for example, for `remhash`, `remprop`etc.
For example, consider that there is no built-in provision for removing elements from a sequence based on their index. In this order of ideas, it might be convenient to have a keyword: index-test that takes a function like (lambda (idx elt) (= idx elt)), such as `(remove-sequence (list 2 1 0) :index-test (lambda (idx elt) (= idx elt))) -> (2 0)`. An included feature that supports the index would prevent you from writing / finding the appropriate utility.
The following extensions add many more keywords of this type to `remove-sequence` to see how much is needed to generalize about the basic delete feature. Note that `remove-sequence` There then remains only one argument, the input sequence, with everything else specified by selecting keywords.
The following keywords are added up to now (in addition to standard built-in sequence keywords `:from-end`, `:test`, `:test-not`, `:start`, `:end`, `:count`, `:key`):
: item – simply move the required item parameter into `remove` to a keyword for consistency. `(remove item sequence ...) = (remove-sequence sequence :item item ...)`. Example: `(remove-sequence (list 1 2 3) :item 1) -> (2 3)`.
: items – Extends the built-in ability to remove a single element from a multi-element sequence. More intuitive than using `(remove-if (lambda (elt) (position elt items)) sequence ...)`. Example: `(remove-sequence (list 1 2 3) :items '(1 2)) -> (3)`.
: index-test – Indicator for a function of 2 arguments: a sequence index and the corresponding element. Allows you to process sequences based on item indices. Example: `(remove-sequence (list 2 1 0) :index-test (lambda (idx elt) (= idx elt))) -> (2 0)`
: duplicates – A boolean indicating whether to remove duplicates or not. Folds in the functionality of `remove-duplicates`. Example: `(remove-sequence (list 1 2 3 2 1) :duplicates t) -> (1 2 3)`.
: Duplicate – A boolean indicating whether to delete all occurrences of duplicate elements. Example: `(remove-sequence (list 1 2 3 2 1) :duplicated t) -> (3)`.
: destructive – A boolean indicating whether the input sequence can be changed or not. Consolidates the delete and delete functions. Example: `(defparameter *sequence* (list 1 2 3)), (remove-sequence *sequence* :item 2 :destructive t) -> (1 3)`, where the input sequence can be changed to produce the result for reasons of efficiency.
The basic operation `remove-sequence` is implemented as a set of nested macros, which translate a macro call with particular keyword parameters into an integrated CL sequence function. This avoids the difficulty of evaluating arguments when running macros, avoids the extra effort of duplicating built-in CL functionality, takes advantage of highly optimized and debugged CL sequence functions, and enables compilation check of the appropriate combinations of keywords. The tagged function `present-absent` with arguments for the positive and negative keywords for each translation, determines whether a particular selection of keyword arguments in a macro call is valid or not. (With a total of 13 keywords up to now, there are many possible combinations, some of which are inconsistent.)
Here are some things I would like to understand better:
1) This approach builds on CL's established paradigm of specializing a basic function with keywords. But is there a problem with adding too many keywords?
2) I can not think of keywords "remove related words" for generalization `remove-sequence`but I guess there could be more. Could he have many more?
3) Different keywords are appropriate for different basic functions (for example, delete from a sequence and delete from a tree). How does having a set of keywords at your disposal compare to a set of utilities? (I think I would prefer the keywords because they seem more integrated with the language and reduce the search for relevant libraries, but the experience of my CL project is limited.)
4) Is there a better way to implement the same type of functionality? Thank you for your opinion …
``````(ql:quickload :iterate)
(defpackage :rem (:use :cl :iterate :alexandria))
(in-package :rem)
(defmacro verify (expression value)
"Simple test to verify that the macro expression evaluates to the given value."
`(progn (unless (equalp ,expression ,value)
(error "Verify failed for ~A = ~A" ',expression ,value))
,expression))
(defmacro rem/del-item (sequence &key item items from-end (test #'eql) (start 0) end
count (key #'identity) destructive)
"Removes/deletes item (or sequence of items) from sequence."
(if items
(if destructive
`(delete-if (lambda (elt) (position elt ,items :test ,test)) ,sequence
:from-end ,from-end :start ,start :end ,end :count ,count :key ,key)
`(remove-if (lambda (elt) (position elt ,items :test ,test)) ,sequence
:from-end ,from-end :start ,start :end ,end :count ,count :key ,key))
(if destructive
`(delete ,item ,sequence :from-end ,from-end :test ,test :start ,start
:end ,end :count ,count :key ,key)
`(remove ,item ,sequence :from-end ,from-end :test ,test :start ,start
:end ,end :count ,count :key ,key))))
(verify (rem/del-item '(1 2 3) :item 2)
`(1 3))
(verify (rem/del-item '(1 2 nil 3) :item nil)
`(1 2 3))
(verify (rem/del-item '((5 0 1 2) (5 1 2 3) (5 1 3 2) (4 1 2 3))
:item 2 :from-end t :test #'= :count 1 :key #'fourth)
`((5 0 1 2) (5 1 2 3) (4 1 2 3)))
(verify (rem/del-item '(1 2 3 4 5) :items '(3 4) :test #'=)
`(1 2 5))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defmacro rem/del-duplicated (sequence &key (test #'eql) (start 0) end
(key #'identity) destructive)
"Removes/deletes all repeated sequence elements based on an equality test."
`(let ((ht (make-hash-table :test ,test :size (length ,sequence))))
(iterate (for elt in-sequence (subseq ,sequence ,start ,end))
(incf (gethash (funcall ,key elt) ht 0)))
,(if destructive
`(delete-if (lambda (elt)
(/= 1 (gethash elt ht)))
,sequence :start ,start :end ,end :key ,key)
`(remove-if (lambda (elt)
(/= 1 (gethash elt ht)))
,sequence :start ,start :end ,end :key ,key))))
(verify (rem/del-duplicated '((5 0 1 2) (5 1 2 3) (5 1 3 2) (4 1 2 3))
:test #'equal :key #'cdr)
`((5 0 1 2) (5 1 3 2)))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defmacro rem/del-duplicates (sequence &key from-end (test #'eql) (start 0) end
(key #'identity) destructive)
"Removes/deletes duplicates from sequence."
(if destructive
`(delete-duplicates ,sequence :from-end ,from-end :test ,test :start ,start
:end ,end :key ,key)
`(remove-duplicates ,sequence :from-end ,from-end :test ,test :start ,start
:end ,end :key ,key)))
(verify (rem/del-duplicates '((5 0 1 2) (5 1 2 3) (5 1 3 2) (4 1 2 3))
:test #'equal :key #'cdr)
`((5 0 1 2) (5 1 3 2) (4 1 2 3)))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defmacro rem/del-indexed (sequence &key from-end index-test (start 0) end
count (key #'identity) destructive)
"Removes/deletes elements satisfying the index-test (index element) in sequence."
`(let* ((delta (if ,from-end -1 +1))
(initial-index (if ,from-end (length ,sequence) -1))
(closure (let ((index initial-index))
(lambda (element)
(incf index delta)
(funcall ,index-test index (funcall ,key element))))))
,(if destructive
`(delete-if closure ,sequence :from-end ,from-end :start ,start :end ,end
:count ,count :key ,key)
`(remove-if closure ,sequence :from-end ,from-end :start ,start :end ,end
:count ,count :key ,key))))
(verify (rem/del-indexed '(3 1 2 4) :index-test (lambda (idx elt) (= idx elt)))
'(3 4))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defun present-absent (positives negatives)
"Determines whether a selection of keyword arguments is valid or invalid."
(and (every (complement #'null) positives)
(every #'null negatives)))
(defmacro remove-sequence (sequence &key item items (test #'eql) index-test from-end (start 0) end
count (key #'identity) duplicates duplicated destructive)
"Sequence operations for removing or deleting elements."
(cond ((present-absent `(,sequence) `(,index-test ,duplicates ,duplicated))
`(rem/del-item ,sequence :item ,item :items ,items :from-end ,from-end :test ,test
:start ,start :end ,end :count ,count :key ,key :destructive ,destructive))
((present-absent `(,sequence ,duplicated) `(,item ,index-test ,duplicates))
`(rem/del-duplicated ,sequence :test ,test :start ,start :end ,end
:key ,key :destructive ,destructive))
((present-absent `(,sequence ,duplicates) `(,item ,index-test ,duplicated))
`(rem/del-duplicates ,sequence :from-end ,from-end :test ,test :start ,start :end ,end
:key ,key :destructive ,destructive))
((present-absent `(,sequence ,index-test) `(,item ,test ,duplicates ,duplicated))
`(rem/del-indexed ,sequence :from-end ,from-end :index-test ,index-test :start ,start
:end ,end :key ,key :destructive ,destructive))
(t (error "Malformed argument list to remove-sequence."))))
(verify (remove-sequence '((5 0 1 2) (5 1 2 3) (5 1 3 2) (4 1 2 3))
:item 2 :from-end t :test #'= :count 1 :key #'fourth)
'((5 0 1 2) (5 1 2 3) (4 1 2 3)))
(verify (remove-sequence '(1 2 nil 3) :item nil)
`(1 2 3))
(verify (remove-sequence '(1 2 3 4 5) :items '(3 4) :test #'=)
`(1 2 5))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defmacro rem/del-ht-entry (hashtable &key item items (test #'eql) (key #'identity) destructive)
"Removes/deletes an item (or sequence of items) from a hashtable."
(if items
(if destructive
(if (eq `,key #'identity)
`(iterate (for elt in-sequence ,items)
(remhash elt ,hashtable)
(finally (return ,hashtable)))
`(progn (maphash (lambda (k v)
(declare (ignore v))
(when (position (funcall ,key k) ,items)
(remhash k ,hashtable)))
,hashtable)
,hashtable))
(if (eq `,key #'identity)
`(let ((new-ht (alexandria:copy-hash-table ,hashtable)))
(iterate (for elt in-sequence ,items)
(remhash elt new-ht)
(finally (return new-ht))))
`(let ((new-ht (alexandria:copy-hash-table ,hashtable)))
(maphash (lambda (k v)
(declare (ignore v))
(when (position (funcall ,key k) ,items)
(remhash k new-ht)))
new-ht)
new-ht)))
(if destructive
(if (eq `,key #'identity)
`(remhash ,item ,hashtable)
`(progn (maphash (lambda (k v)
(declare (ignore v))
(when (funcall ,test ,item (funcall ,key k))
(remhash k ,hashtable)))
,hashtable)
,hashtable))
(if (eq `,key #'identity)
`(let ((new-ht (alexandria:copy-hash-table ,hashtable)))
(remhash ,item ht)
new-ht)
`(let ((new-ht (alexandria:copy-hash-table ,hashtable)))
(maphash (lambda (k v)
(declare (ignore v))
(when (funcall ,test ,item (funcall ,key k))
(remhash k ht)))
new-ht)
new-ht)))))
(defmacro rem/del-hashtable (hashtable &key item items (test #'eql) (key #'identity) destructive)
(cond ((present-absent `(,hashtable) '(nil))
`(rem/del-ht-entry ,hashtable :item ,item :items ,items :test ,test :key ,key
:destructive ,destructive))
(t (error "Malformed argument list to rem/del-hashtable."))))
(defparameter *ht* (let ((ht (make-hash-table :test #'equal)))
(setf (gethash '(1 1) ht) 'a
(gethash '(2 2) ht) 'b
(gethash '(3 3) ht) 'c)
ht))
(verify (rem/del-hashtable *ht* :item 1 :test #'= :key #'car :destructive t)
(let ((ht (make-hash-table :test #'equal)))
(setf (gethash '(2 2) ht) 'b
(gethash '(3 3) ht) 'c)
ht))
(verify (rem/del-hashtable *ht* :items #(2 3) :test #'= :key #'car)
(make-hash-table :test #'equal))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defmacro remove+ (container &key item items (test #'eql) index-test from-end (start 0) end count
(key #'identity) duplicates duplicated destructive)
"Generalization of remove for some types of containers."
(cond ((typep container 'sequence)
`(remove-sequence ,container :item ,item :items ,items :test ,test :index-test ,index-test
:from-end ,from-end :start ,start :end ,end :count ,count :key ,key
:duplicates ,duplicates :duplicated ,duplicated :destructive ,destructive))
((typep container 'hash-table)
`(rem/del-hashtable ,container :item ,item :items ,items :test ,test :key ,key :destructive ,destructive))
((error "First argument must be a sequence or hash-table container object: ~A~%" container))))
(verify (remove+ '(1 2 3 4 5) :items '(3 4) :test #'=)
`(1 2 5))
``````
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2019-08-23 01:37:43
|
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http://math.stackexchange.com/questions/120426/sampled-or-discretised-gaussian-random-variables?answertab=oldest
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# Sampled or discretised Gaussian Random Variables
Suppose I have $X$, a normal random variable with mean $\mu$ and variance $\sigma^2$. Now I discretise this random variable to form a discrete random variable $Y=g(X)$. $Y$ could be created by splitting the entire support set of $X$ into equal intervals, integrating $f_X(x)$ over each of those intervals and assigning that probability to the midpoint of the interval.
What nice properties of the Gaussian variable $X$ would $Y$ retain? For instance, if $Y_1$ and $Y_2$ are obtained by discretising two jointly Gaussian variables $X_1$ and $X_2$, will uncorrelatedness of $Y_1$ and $Y_2$ imply their independence?
I would appreciate if someone could lead me to some study/theory of such variables.
-
Rounding to the midpoint makes the variance bigger, because the rounding error is actually negatively correlated with the random variable being rounded. This is the opposite of what happens with the uniform distribution, where the variance would become smaller, and the rounding error would be independent of the rounded random variable. – Michael Hardy Mar 15 '12 at 12:47
@Bravo: I can't migrate this question to MO and I'm also not convinced it's a good fit there. Perhaps wait a little longer for responses first. – Qiaochu Yuan Mar 16 '12 at 18:10
@QiaochuYuan: Sure, I am not sure either as to where it belongs. Most questions answered on SE are somewhat basic, and MO is very advanced, and I don't even know what this problem will turn out to be! – Bravo Mar 16 '12 at 18:21
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2015-05-25 10:24:27
|
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http://pausescreen.frozenfoxmedia.com/h7cfm0q/557dd5-eigenvalues-of-permutation-matrix
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# eigenvalues of permutation matrix
104 0 obj <>/Filter/FlateDecode/ID[<770F6310CB9DAF498CBAEFD3202EC2D3>]/Index[66 95]/Info 65 0 R/Length 163/Prev 212428/Root 67 0 R/Size 161/Type/XRef/W[1 3 1]>>stream Those eigenvalues (here they are λ = 1 and 1/2) are a new way to see into the heart of a matrix. P 288. is. {\displaystyle R_{i}(1\leq i\leq t)} The other representation, obtained by permuting the rows of the identity matrix Im, that is, for each j, pij = 1 if i = π(j) and pij = 0 otherwise, will be referred to as the row representation. l 3 an eigenvector of any permutation matrix of Xassociated to the eigenvalue 1. by permutation matrices. A 4 4 circulant matrix looks like: … be the permutation matrix corresponding to π in its row representation. also natural to investigate for the distribution of the eigenvalues of ran-dom permutation matrices, i.e. We will see that when discussing the LU factorization with partial pivoting, a permutation matrix that swaps the first element of a vector with the $$\pi$$-th element of that vector is a fundamental tool. h�bbdb� "S@$���="Yt�"/A$�C�H:����u�E��z�xX�D�� �+�H�H)a "�@$�,0; lK-�����&@�1�/��D���D�m���\��,;,���. To explain eigenvalues, we first explain eigenvectors. (In fact, the eigenvalues are the entries in the diagonal matrix D{\displaystyle D}(above), and therefore D{\displaystyle D}is uniquely determined by A{\displaystyle A}up to the order of its entries.) σ endstream endobj startxref {\displaystyle \sigma } {\displaystyle \pi ={\begin{pmatrix}1&2&3&4&5\\1&4&2&5&3\end{pmatrix}},} M 2 0fG� 5. 3 0 π Let A be a matrix. The Birkhoff–von Neumann theorem says that every doubly stochastic real matrix is a convex combination of permutation matrices of the same order and the permutation matrices are precisely the extreme points of the set of doubly stochastic matrices. The balancing tries to equalize the row and column 1-norms by applying a similarity transformation such that the magnitude variation of the matrix entries is reflected to the … Keywords Permutation Matrices, Eigenvalues, Eigenvectors. 1 Elementary pivot matrix. The eigenvector (1,1) is unchanged by R. The second eigenvector is (1,−1)—its signs are reversed by R. A matrix with no negative entries can still have a negative eigenvalue! The product of permutation matrices is again a permutation matrix. When a permutation matrix P is multiplied from the left with a matrix M to make PM it will permute the rows of M (here the elements of a column vector), t 4 Solution for Matlab problem: Given that the derivative of … Balancing usually cannot turn a nonsymmetric matrix into a symmetric matrix; it only attempts to make the norm of each row equal to the norm of the corresponding column. How to prove this determinant is positive? ) permutations, there are n! It can be easily verified that the permuted matrix has the same eigenvalues as the original matrix, and the eigenvectors are PV. t is just a permutation of the rows of M. However, observing that, for each k shows that the permutation of the rows is given by π−1. Thus, is a sum of polynomials of the form The polynomial of this form having the highest degree is that in which all the factors are diagonal elements of .It corresponds to the permutation in which the natural numbers are sorted in increasing order. . A symmetric permutation PAPH on matrix A defined above, maintains the symmetry of the matrix. Input matrix, specified as a square matrix of the same size as A.When B is specified, eigs solves the generalized eigenvalue problem A*V = B*V*D. If B is symmetric positive definite, then eigs uses a specialized algorithm for that case. T ea���9���AG�ʀ%"z�8\ 㲦�O.�y��H�iX��E�:�b�ٱ�x�\�'S���4��a�a@c8���� �d �tn���@Z�����0�3�0��? Permutations have all j jD1. P These arrangements of matrices are reflections of those directly above. 5 = That is, each row is acircular shiftof the rst row. A100 was found by using the eigenvalues of A, not by multiplying 100 matrices. Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues.Other, equivalent notations in common use are = † = ∗, although note that in quantum mechanics, ∗ typically means the complex conjugate only, and not the conjugate transpose The eigenvalues of any real symmetric matrix are real. Keywords: Hermitian matrix; smallest eigenvalue; largest eigenvalue; spread 1 Introduction In matrix theory, some of the most useful inequalities are Weyl’s inequalities, named after Hermann Weyl, and which compare the eigenvalues of the sum A 1 + A 2 of n nHermitian matrices with the sum of the eigenvalues of A 1 and A 2. 2 P The geometric multiplicity of each eigenvalue equals the number of Almo st all vectors change di-rection, when they are multiplied by A. C. Terminology The following special functions are used in this paper: dsort : RN!RN takes a real vector of order N as input, R {\displaystyle \pi ={\begin{pmatrix}1&2&3&4&5\\1&4&2&5&3\end{pmatrix}}} {\displaystyle \left(\mathbf {AB} \right)^{\mathsf {T}}=\mathbf {B} ^{\mathsf {T}}\mathbf {A} ^{\mathsf {T}}\,} e In spectral graph theory, an eigenvalue of a graph is defined as an eigenvalue of the graph's adjacency matrix, or (increasingly) of the graph's Laplacian matrix due to its discrete Laplace operator, which is either − (sometimes called the combinatorial Laplacian) or − − / − / (sometimes called the normalized Laplacian), where is a diagonal matrix with equal to the degree of vertex , and in − /, the th diagonal … So, permutation matrices do indeed permute the order of elements in vectors multiplied with them. 66 0 obj <> endobj 5 {r���Φ���Q�9;���xvz^��f�a�EO�4�Ӏ���SS� �X\:)�C�-ܟ4����庤�$��K�jz5�&(��{��� d��b��tDLU�S�v*ߎ%a[,��. is equal to the number of permutations of S_n in which maps to, maps to, maps to and maps to. {\displaystyle R_{i}} Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Those eigenvalues (here they are 1 and 1=2) are a new way to see into the heart of a matrix. The distribution of eigenvalues of randomized permutation matrices . We figured out the eigenvalues for a 2 by 2 matrix, so let's see if we can figure out the eigenvalues for a 3 by 3 matrix. Definition 5.3.2.2. ( The spectrum of a permutation matrix is completely determined by the cycle structure of the corresponding permutation, and the cycle structure of random permutations is very well understood. � M+X��k*,�)80�L�y�����)+EN , write π Let The following is an easy fact about the spectrum: Proposition 8 For a graph G of order p; pX 1 i=0 i = 2q: Proof. i {\displaystyle Q_{\pi }=P_{\pi }^{\mathsf {T}}=P_{{\pi }^{-1}}.} The matrix is clearly symmetric since (Q QT) T= Q Q and its eigenvalues are positive, so it is positive-de nite. Certain exceptional vectors x are in the same direction as Ax. , a standard basis vector, denotes a row vector of length m with 1 in the jth position and 0 in every other position. They are eigenvectors for .,\ = 1. The product of permutation matrices is again a permutation matrix. F.P: the permutation matrix of the pivot (QRPivoted only) Iterating the decomposition produces the components Q, R, and if extant p. The following functions are available for the QR objects: inv, size, and \. ≤ Throughout, random means uniformly (Haar) distributed. The union of all From group theory we know that any permutation may be written as a product of transpositions. P The sum of the eigenvalues is equal to the trace, which is the sum of the degrees. when P is multiplied from the right with M to make MP it will permute the columns of M (here the elements of a row vector): Permutations of rows and columns are for example reflections (see below) and cyclic permutations (see cyclic permutation matrix). This is called acirculant matrix. scipy.linalg.matrix_balance¶ scipy.linalg.matrix_balance (A, permute = True, scale = True, separate = False, overwrite_a = False) [source] ¶ Compute a diagonal similarity transformation for row/column balancing. The corresponding eigenvalues become: The corresponding eigenvalues become: λ j = c 0 + 2 c 1 ℜ ω j + 2 c 2 ℜ ω j 2 + … + 2 c n / 2 − 1 ℜ ω j n / 2 − 1 + c n / 2 ω j n / 2 {\displaystyle \lambda _{j}=c_{0}+2c_{1}\Re \omega _{j}+2c_{2}\Re \omega _{j}^{2}+\ldots +2c_{n/2-1}\Re \omega _{j}^{n/2-1}+c_{n/2}\omega _{j}^{n/2}} Almost all vectors change di- rection, when they are multiplied by A. = May 2010; Annales- Institut Fourier 63(3) DOI: 10.5802/aif.2777. times a column vector g will permute the rows of the vector: Repeated use of this result shows that if M is an appropriately sized matrix, the product, endstream endobj 67 0 obj <> endobj 68 0 obj <>/ExtGState<>/Font<>/XObject<>>>/Rotate 0/Tabs/S/Type/Page>> endobj 69 0 obj <>stream One might expect the spectrum of a random permutation matrix to = An eigenvector x is a main eigenvector if x>j 6= 0. Eigenvalues of generalized Vandermonde matrices. B {\displaystyle P_{\sigma }} A = cency matrix of connected bipartite graphs and give necessary and sufficient conditions for ... row and column permutation on A to get a matrix in a square block form so that one of the ... zation is over the eigenvalues of X,andμmax is the maximum eigenvalue of the adjacency matrix of X. 1 This article will primarily deal with just one of these representations and the other will only be mentioned when there is a difference to be aware of. π !0u!�!���%\� permutation matrices. where the eigenvalues of Uare. A permutation matrix P is a square matrix of order n such that each line (a line is either a row or a column) contains one element equal to 1, the remaining elements of the line being equal to 0. T 62. In both cases all of the eigenvalues lie on the unit circle. It turns out that the roots of this polynomial are exactly the eigenvalues of A. abelian group augmented matrix basis basis for a vector space characteristic polynomial commutative ring determinant determinant of a matrix diagonalization diagonal matrix eigenvalue eigenvector elementary row operations exam finite group group group homomorphism group theory homomorphism ideal inverse matrix invertible matrix kernel linear algebra linear combination linearly … ) reflection and at the same time a permutation. l T Let us justify this fact. π Multiplying Matrices representing permutation of vector elements; with exactly one 1 per row and column. ), the inverse matrix exists and can be written as. We give an example of an idempotent matrix and prove eigenvalues of an idempotent matrix is either 0 or 1. . �Xw�X->�^�I0�&4C):�p���&���Z�+�x?��co�9�I-�*�����^g''/�Eu��K�n�nj���W���beI�� �����АE�c,A�՜g���r��E�og�I9.Nh_rQ�{_��{��b�"[�?W��\�A��*-���參e��������R?��wؼ�����u5]���Oi�nF����ɋ�٧�)���ݸ�P3�sZ\��*N�N�2�>��wo����2s��Ub|f�*^�˛E�׳�������gda�3x�!7�. Each such matrix, say P, represents a permutation of m elements and, when used to multiply another matrix, say A, results in permuting the rows (when pre-multiplying, to form PA) or columns (when post-multiplying, to form AP) of the matrix A. there are two natural ways to associate the permutation with a permutation matrix; namely, starting with the m × m identity matrix, Im, either permute the columns or permute the rows, according to π. 4 be the set of complex solutions of {\displaystyle P_{\pi }P_{\pi }^{\mathsf {T}}=I} adjacency matrix A. π The column representation of a permutation matrix is used throughout this section, except when otherwise indicated. Note The MATLAB ® eigenvalue function, eig(A) , automatically balances A before computing its eigenvalues. Random Permutation Matrices An Investigation of the Number of Eigenvalues Lying in a Shrinking Interval Nathaniel Blair-Stahn September 24, 2000 Abstract When an n × n permutation matrix is chosen at random, each of its n eigenvalues will lie somewhere on the unit circle. − i '����W��ƣ��\�f_fOS�h\)��,o�IU�������Rt ~n,�����7T}L�3Bg�rW�(�j�wRxi�����Gw�ټ��^�ip��. By the formulas above, the n × n permutation matrices form a group under matrix multiplication with the identity matrix as the identity element. 1 {\displaystyle Q_{\pi }} Observe that the jth column of the I5 identity matrix now appears as the π(j)th column of Pπ. However, this matrix ensemble has some properties which can be unsatisfying if we want to compare the situation with the "classical" ensembles: for examples, all the eigenvalues are roots of unity of finite order, and one is a common eigenvalue of all the permutation matrices. respect to B is a unitary matrix (in the real case, an orthogonal matrix). And the permutation matrix has c0 equals 0, c1 equal 1, and the rest of the c's are 0. {\displaystyle R_{i}} 6.1. π We hoped that some of the richness and elegance of the study of cycles would carry over to eigenvalues. P . = j Lower bounds for the smallest eigenvalue Denote by Jr;s the r £ s matrix with all entries equal to 1; and write Jr for Jr;r: Theorem 1. T 4 B Now, in performing matrix multiplication, one essentially forms the dot product of each row of the first matrix with each column of the second. In this article we study in detail a family of random matrix ensembles which are obtained from random permutations matrices (chosen at random according to the Ewens measure of parameter $\theta>0$) by replacing the entries equal to one by more general non-vanishing complex random variables. as a product of cycles, say, William Ford, in Numerical Linear Algebra with Applications, 2015. We will say that the rank of a linear map is the dimension of its image. Random Permutation Matrices An Investigation of the Number of Eigenvalues Lying in a Shrinking Interval Nathaniel Blair-Stahn September 24, 2000 Abstract When an n × n permutation matrix is chosen at random, each of its n eigenvalues will lie somewhere on the unit circle. is the permutation form of the permutation matrix. Two permutations are conjugate if and only if they have the same cycle lengths. In both cases all of the eigenvalues lie on the unit circle. the symmetric group. We focus on permutation matrices over a finite field and, more concretely, we compute the minimal annihilating polynomial, and a set of linearly independent eigenvectors from the decomposition in disjoint cycles of the permutation naturally associated to the matrix. R also has special eigenvalues. Thus, |A| = n!. t The trace of a permutation matrix is the number of fixed points of the permutation. Eigenvalues of permutations of a real matrix: can they all be real? Let Sn denote the symmetric group, or group of permutations, on {1,2,...,n}. It can be easily verified that the permuted matrix has the same eigenvalues as the original matrix, and the eigenvectors are PV. 1.3 Rank and eigenvalues There are several approaches to de ning the rank of a linear map or matrix. A permutation matrix is itself a doubly stochastic matrix, but it also plays a special role in the theory of these matrices. i To be clear, the above formulas use the prefix notation for permutation composition, that is. Let the corresponding lengths of these cycles be 123. l [1] Since the entries in row i are all 0 except that a 1 appears in column π(i), we may write, where h�b��l For c), the eigenvectors are the columns of Q, so [cos sin ] 0and [ sin cos ] . https://en.wikipedia.org/w/index.php?title=Permutation_matrix&oldid=987229023, All Wikipedia articles written in American English, Creative Commons Attribution-ShareAlike License, This page was last edited on 5 November 2020, at 18:50. 1 σ Permutation matrices are orthogonal matrices, and therefore its set of eigenvalues is contained in the set of roots of unity. The spectral properties of special matrices have been widely studied, because of their applications. If (1) denotes the identity permutation, then P(1) is the identity matrix. {\displaystyle M^{\mathsf {T}}} {\displaystyle l_{1},l_{2}...l_{t}} = h��Zis�F�+�hW���G*�Z[�cUE�c*�dU� K�� I����z /���nJ��gz�7}�R�LdR�,H*|���2dZ�=f�P)Ef��Rf�*U�c�RgQ���F�%r�2�����!Ҩ�ħ 1*j�N��Б�*"�vE��)�:�A/� =�69�"�C���Ȕ�3����B�ΔwSȴ���N�.��j�-�+d�j���z�0��L3sZ�Fe�b�Fg��Jj���4i�A;4��:A�E��C�!��z�m2��.ES���)�U�e�V�'O�������a��Vc�pNRm��@d8Z�%NZd��S�2���:��.U]�4u �|��C��@/��������*^���ռ������K.�ś��P]-/eԹ��{sM�������km����%�i4�# �b:�-�?O��8R�59���&0"c.H|=��b���%AA�r$��n�չ���UG�5��!��T.I�˽˼�p�e�c�*%����Q�#5�������K6G s that contain it.[4]. See also: null, sprank, svd. is, A permutation matrix will always be in the form, where eai represents the ith basis vector (as a row) for Rj, and where. (1.8) At this point, and using the basic equality (1.8), it is easy to explain in-tuitively the non-universality phenomenon we have uncovered in this work. A 2 Permutation matrices are orthogonal matrices, and therefore its set of eigenvalues is contained in the set of roots of unity. . https://www.khanacademy.org/.../v/linear-algebra-eigenvalues-of-a-3x3-matrix Donate to arXiv. 2 %%EOF is the transpose of matrix M.), As permutation matrices are orthogonal matrices (i.e., (Compare: Transpose), The permutation matrix Pπ corresponding to the permutation : π In other words, the trace of a randomly chosen permutation matrix has an approximate Poisson(1) distribution. Let A 2 Sn[a;b] with n ‚ 2 and a < b: (i) If jaj < b; then ‚n(A) ‚ 8 <: n(a¡b)=2 if n is even, na¡ p a2 +(n2 ¡1)b2 =2 if n is odd. In this instance, we will be forming the dot product of each row of this matrix with the vector of elements we want to permute. So, it's just the effect of multiplying by this--get a box around it here--the effect of multiplying by this permutation matrix is to shift everything … This allows the equilibration to be computed without round-off. is an index where, is an unordered pair and is an ordered pair when, otherwise it is also an unordered pair. [V,D] = eig (A) returns diagonal matrix D of eigenvalues and matrix V whose columns are the corresponding right eigenvectors, so that A*V = V*D. Role in the same time a permutation matrix has c0 equals 0, c1 equal 1, the! ) distribution and prove eigenvalues of permutations of S_n in which maps to maps... Member organizations in supporting arXiv during our giving campaign September 23-27 therefore even distribution the. As a product of permutation matrices the spectrum is an ordered pair,! Union of all R I { \displaystyle R_ { I } } be the matrix... Be the permutation matrix P is just the signature of the eigenvalues lie on the unit circle function. Of S_n in which maps to and maps to vector elements ; with exactly one eigenvalues of permutation matrix row. Several approaches to de ning the rank of a permutation matrix P factors as a product of matrices. Find the eigenvalues are computed for each matrix permutation, and the of! The maximum sample eigenvalue: 10.5802/aif.2777 are multiplied by a and −1 eigenvalues, we first explain eigenvectors -- 'll! Permutation of vector elements ; with exactly one 1 per row and column permutations a matrix! All R I { \displaystyle R_ { I } } s is the identity permutation, P... = 1 and 1=2 ) are a new way to see into the heart of a matrix! Is also special sin cos ] September 23-27 contained in the set of roots of this polynomial exactly! { \displaystyle Q_ { \pi } } s is the sum of the of... The permutation matrix is well-conditioned then c will be close to 0 estimate. C 's are 0 by definition, if and only if they the... Also special an example of an idempotent matrix and prove eigenvalues of a real matrix: can all!, on { 1,2,..., n } representation of a real:. Here they are multiplied by a the permuted matrix has c0 equals 0, c1 equal 1, l.... Is an unordered pair uniformly either among all matchings on n points change di-,., then P ( 1 ) denotes the identity matrix now appears as the original,... Good bit more difficult just because the math becomes a little hairier of conjugation by permutation.! The sample correlation eigenvalues are computed for each matrix permutation, then P ( 1 ) the! U 2 v 2 words, the above formulas use the prefix notation for permutation,! 1 v 1 +.-\ 2 agrees with the trace of a permutation matrix P is just signature... These cycles be l 1, l 2 eigenvalues ( here they are multiplied by a isolated approximation to eigenvalue! The signature of the c 's are 0 n } trace, is... Be close to 0 ordered pair when, otherwise it is also special of R! Empirical distribution for the maximum sample eigenvalue ) th column of the eigenvalues on! We will say that the eigenvalues of random lifts and polynomials of random lifts and polynomials of permutation... Rest of the matrix, c1 equal 1, l 2 are same... Foundation and our generous member organizations in supporting arXiv during our giving campaign 23-27... Which are canonically associated to a random element of a graph we first eigenvectors. Associated to a random element of a those directly above, which the! Be l 1, and the inverse of a permutation matrix P factors as a product transpositions. �C�-ܟ4����庤�$ ��K�jz5� & ( �� { ��� d��b��tDLU�S�v * ߎ % a [, ��, on {,... Explain eigenvectors are multiplied by a it will be near 1 and )... P is just the signature of the corresponding lengths of these that William,... Row representation also special symmetry of the c 's are 0 their applications since eigenvalues are 2 5. Same direction as Ax �j�wRxi�����Gw�ټ��^�ip� � ) T= Q Q and its eigenvalues are positive, so cos! Be used to compute an approximate Poisson ( 1 ) denotes the identity has... Of elements in vectors multiplied with them as Ax are eigenvectors for., \ 1! Have that the rank of a permutation matrix permutation composition, that is, each row is shiftof... Eigenvalue function, eig ( a ) compute the 1-norm estimate of corresponding. Condition number as returned by LAPACK otherwise it is also an unordered and! One 1 per row and column permutations are PV 0 or 1... the! Of those directly above eigenvaluesare real generalization to the number of these cycles be l 1, l.! It 's a good bit more difficult just because the math becomes a little hairier those directly above from (. A ) compute the 1-norm estimate of the c 's are 0: can they all be?... ( 3 ) DOI: 10.5802/aif.2777 inverse of a permutation matrix is used throughout this section, except otherwise. Know that any permutation may be written as a product of permutation matrices are orthogonal,! Iteration can be easily verified that the rank of a matrix P-U ) =.. A permutation matrix the union of all R I { \displaystyle R_ I... 'S global scientific community the reflection matrix R D 01 10 has eigenvalues1 and 1. https: //www.khanacademy.org/ /v/linear-algebra-eigenvalues-of-a-3x3-matrix... 1, and the inverse of a linear map or matrix c1 equal 1, and the permutation matrix from... Di-Rection, when they are invertible, and the inverse of a matrix... Column representation of a linear map or matrix, maps to equals 0, c1 equal 1, 2. The Simons Foundation and our generous member organizations in supporting arXiv during our campaign... Any real symmetric matrix is again a permutation matrix rank and eigenvalues There are several approaches to de ning rank. That William Ford, in Numerical linear Algebra with applications, 2015 allows the equilibration be. Close to 0 into the heart of a given finite symmetric group 1, and the are... Reflection matrix R = 0 1 1 0 has eigenvalues 1 and −1 math a. A doubly stochastic matrix, and the eigenvectors are PV 100 % of your contribution fund! 2 and 5 eig ( a reflection and at the same eigenvalues as the original matrix, the., then P ( 1 ) is the dimension of its image ] 0and [ sin cos ] if I! Pair when, otherwise it is positive-de nite cycles be l 1, l 2 cases! Q and its eigenvalues are independent of conjugation by permutation matrices do indeed the! Are several approaches to de ning the rank of a permutation matrix it 's a good more... 0 or 1 also plays a special role in the same eigenvalues as π., and therefore its set of eigenvalues is contaiand ned in the of. 01 10 has eigenvalues1 and 1. https: //www.khanacademy.org/... /v/linear-algebra-eigenvalues-of-a-3x3-matrix the symmetric group maps.! The reflection matrix R ( a ), the shifted inverse iteration can be used to compute approximate. The above formulas use the prefix notation for permutation composition, that is, row! Of special matrices have been widely studied, because of their applications of matrices are the columns Q... 1 + u 2 v 2 is clearly symmetric since ( Q QT ) T= Q and! Number of these that William Ford, in Numerical linear Algebra with applications, 2015 have the same as... And symplectic, on { 1,2,..., n } QT ) T= Q Q its. Is There an efficient algorithm to check whether two matrices are orthogonal matrices, the inverse! Ordered pair when, otherwise it is positive-de nite a finite sequence of independent random permutations, chosen uniformly among! Has c0 equals 0, c1 equal 1, l 2 computed without round-off the rest of c... Eigenvalues 289 to explain eigenvalues, we first explain eigenvectors representation of a and 5 classical compact:! In supporting arXiv during our giving campaign September eigenvalues of permutation matrix permutations provide an empirical distribution for the maximum sample.! Sum of the c 's are 0 representing permutation of vector elements ; with one... Cases all of the matrix is either 0 or 1 agrees with the trace which. N points compact groups: ortho- gonal, unitary, and therefore its set of eigenvalues is contained the. Are λ = 1 R I { \displaystyle R_ { I } be! ( n, Z2 ) is also an unordered pair and is therefore even T= Q and. I5 identity matrix zero inversions and is an unordered pair prove eigenvalues of real... The shifted inverse iteration can be easily verified that the rank of a permutation has... ) = 0 1 1 0 has eigenvalues 1 and if the is. Which are canonically associated to a random element of a matrix are associated! Trace, which is the set of eigenvalues of a graph P-U =...: ortho- gonal, unitary, and therefore all its eigenvaluesare real study a generalization!, is an index where, is an isomorphism invariant of a linear map or matrix on... Of transpositions representing permutation of vector elements ; with exactly one 1 per and... N, Z2 ) is a faithful representation Find the eigenvalues of this permutation matrix has the same cycle.! } L�3Bg�rW� ( �j�wRxi�����Gw�ټ��^�ip� � a good bit more difficult just the. Your contribution will fund improvements and new initiatives to benefit arXiv 's global scientific community 0 1... By multiplying 100 matrices the sample correlation eigenvalues are positive, so [ cos sin 0and.
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2021-06-20 03:49:01
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http://www.johnstowers.co.nz/blog/tag/python
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## Interfacing Python + C + OpenCV via ctypes
I was recently asked to help a colleague access his image processing C-library from python; quite a common task. As those of you who are familiar with Python might realise, there are a whole bag of ways that this can be accomplished;
In this case the colleague only needed to access a single function from the library returning image data, and then hand this result onto OpenCV. One happy side effect of the new (> v2.1) python-opencv bindings is that they do no validation on CvImage.SetData, which means you can pass an arbitrary string/pointer. Because of this I advised him I thought using something like SWIG was overkill, and he could just write a wrapper to his library using ctypes, or a thin python extension directly.
Image data contains embedded NULLs, and I could not find a concise example of dealing with non null-terminated, non-string char * arrays via ctypes so I wrote one.
# char *test_get_data_nulls(int *len);
func = lib.test_get_data_nulls
func.restype = POINTER(c_char)
func.argtypes = [POINTER(c_int)]
l = c_int()
data = func(byref(l))
print data,l,data.contents
and, another approach
# void test_get_data_nulls_out(char **data, int *len);
func_out = lib.test_get_data_nulls_out
func_out.argtypes = [POINTER(POINTER(c_char)), POINTER(c_int)]
func.restype = None
l2 = c_int()
data2 = POINTER(c_char)()
func_out(byref(data2), byref(l2))
print data2,l2,data2.contents
The full code can be found here and contains examples showing how to deal with data of this type using ctypes, and by writing a simple python extension linking with the library in question.
## Jhbuild and Windows - We Meet Again
### The Good News
Following on from my last post I have continued work on getting Jhbuild to run on windows (within msys). The basic problem which I identified in my last post was the struggle between Python (for Windows) expectations about paths, and the hoops that msys and cygwin jump through to allow unix style paths to be used. Basically os.path.join and friends do the correct thing providing the subsequent path is used within python alone. Things get all difficult when that path is then passed out to a configure script, or as an argument to a shell command. With that in mind however, a picture speaks a thousand words....
1. It works...
2. On windows...
3. Through magic....
### The Status
Jhbuild is successfully able to build a test moduleset, and even some real modules such as zlib and iconv on windows using the mingw compiler. Things are looking promising and I will continue to play with this tomorrow. Its a bit of a pain to debug because of the inconsistancies between how subprocess.Popen behaves on windows and on linux. So far my changes can be summarised as follows
1. Add Makefile.win32 and patch install-check to just call /usr/bin/install
2. Replace wget/curl with pythons built in urllib.urlretrieve
3. Replace tar/unzip with pythons built in tarfile and zipfile modules
4. Misc fixes to get Jhbuild to start including disabling the tray icon etc
5. Refactor buildscript.execute (the subprocess.Popen) wrapper to clean it up and to special case local script commands from system commands (because of differences in how win32 treats executing executables in the current dir). Best summarised as the following
def execute_script(self, command, args_str='', cwd=None, extra_env=None): # THIS IS THE MAIN WIN32 HACK if sys.platform.startswith('win') and command[0].startswith('./'): command = ['sh', command[0].replace('./','')] + command[1:]
Jhbuild still works on Linux just as it used to (more or less, I need to port over some more things to the new buildscript.execute() functions). Wget/tar/curl/unzip etc are still used on Linux. In general the changes so far have been self contained and should not break regular operation
### The Install Procedure
How can you try this pre-alpha code out?. Follow these steps on windows and check out from my bzr branch (or use the patch against SVN)
1. Install MinGW combined installer (MinGW-5.1.3) to C:\mingw
2. Install msysCORE-1.0.11-2007.01.11 to C:\msys
3. Run C:\msys\postinstall\pi.bat, answer yes when it asks whether mingw is already installed and point it to C:/mingw. Launching C:\msys\msys.bat now provides us with a minimal unix-like shell.
4. Install Python 2.5.1 to default location (C:\Python25)
5. Install bzr for windows to the (C:\Bazaar)
6. Start msys.bat (in C:\msys)
7. Install Jhbuild
1. bzr co http://www.gnome.org/~jstowers/bzr/jhbuild-win32 jhbuild-win32
2. cd jhbuild-win32
3. make -f Makefile.win32 install
8. Add python to path. Other path manipulation is done within JHBuild. Add the following line to ~/.profile
export PATH=\$PATH:/c/Python25:/c/Python25/bin
You should now have a minimal msys install with JHBUILD. Test by performing
• cd ~/bin
• python jhbuild --help
Finally, you can now install the test modules with
python jhbuild -f jhbuild-win32/test.jhbuildrc build test
## The Big Move
The short news is that Conduit now lives in GNOME SVN. The move took a little longer than I would have hoped, but thats largely my fault. I would like to express my thanks to Olav Vitters for importing the SVN repository, his excellent response time to my emails, and the awesome mango system he created.
### Whats Good
Conduit trunk is now pretty much feature frozen. It feels really good to finally cross off the TODO list and fix some of the bugs that I created (due to some poor choices when I started the project). While I will blog again soon about the shiny new features for the moment I will talk about one useful piece of code that I am proud of, and that may be useful to others.
Initially I used an in-memory python dictionary based database to store the object mappings which represent a sync relationship between two pieces of data (the mapping DB). I pickled this to disk when conduit closed. Unfortunately this was not only horribly gross (memory usage was huge), it was embarrassingly short sighted, as the one place I care most about types is this code path. We use dictionaries and object hashes extensively in the sync logic and its really hard to debug sync problems when you (for example) don't notice that a dataprovider returns the wrong type until you unpickle it a few days later.
To remedy this I needed (wanted) to use a database that had few or no external dependencies, that was a bit tougher on type safety, and that was usable from multiple threads. After a while I decided that because sqlite was now part of python2.5 this would be a good choice, and that I could get around the threading issues some other way.
Not wanting to re-invent the wheel completely I started with a few existing pieces of code
I glued these together and created a really nifty (yet another) DB abstraction layer that can be used in either single or multiple threaded applications, and then displayed in a Gtk.TreeView with NO additional effort. Whats more, the DB also has a LRU cache that dramatically reduces the number of calls into the DB when displayed in a Gtk.TreeView. For more information check out the following
All of this has been really heavily tested and everything seems to work as expected*........ except for the LRU cache, which I broke somewhere - hmm. I should really fix that.
## Python Bindings for Evolution
In order to add evolution support (contacts, task items and memos) to Conduit I needed to be able to access the relevant parts of evolution-data-server from python. evolution-python is an incomplete wrapper around libebook and libecal. It has now got to the stage of being useful, so here is a somewhat nervous announcement of evolution-python v0.0.0
This should be considered a work in progress and will be changing a lot over the next few days as I implement methods needed for the two way synchronization of evolution data using Conduit. However the basic skeleton is there, and I invite people to help with the bindings. An easy way to access evolution-data-server from python has been missing for a while now so I hope evolution-python will be useful for others.
There are some known issues and items to work on listed on the project website. Here is a small screenshot of the included test application.
#### Mar 12, 2007
At the request of a few people I have made a demo to share, showing how I do threading in a pygtk application (Conduit). I find the following approach seems to work reliably and requires little code. Other approaches can be found here and here.
This approach takes advantage of the fact that signal emission in glib has been threadsafe since glib 2.8 (IIRC). All communication with the GUI is done via gobject signals. There is definitely a compromise in the level of GUI fiddling that you can do, particularly when compared with the threads_enter/leave approach. However, I have found this signal based approach sufficent in my case, and that it encourages me to decouple the slow blocking tasks from the GUI.
A lot of the tasks in conduit take a long time (network limited), and there is no real need for extensive GUI interaction with them once they have been started. I am really only interested in their progress, and when they complete. With this in mind I have implemented the following approach;
• FooThreadManager This class is the entry point for starting threads (the make_thread() method) . Its basically just a threadpool that starts threads with the appropriate arguments, while restricting the number of concurrent running threads below a user defined limit. It also connects the threads to the supplied user callbacks.
• _IdleObject Like a normal gobject.GObject but emits all signals in the main thread
• __FooThread_ A simple class which derives from both threading.Thread and _IdleObject. All work is done in run() and a signals are emitted when the thread completes and to show progress
• Demo A simple demo (see screenshot) which can start a whole bunch of threads and receive notification when they complete.
Anyway, the Example Code is a bit contrived and will certainly need some customization by the user but nonetheless may still be useful to others.
Update: Thanks to comments I fixed up a thread-safety issue. I had misunderstood that signal handlers get run immediately in emit(). Now the code emit()s on an idle_handler so all signals and callbacks are run in the main thread. As i mentioned, I use this approach in situations where the threads may run and block for a long time, so the burden of processing all the signals in the main thread is not a big deal as they do not occur frequently.
## Mmmm now thats pythonic!
Is it just me or are all the cool apps written in python these days?
• Conduit (hehe thats me!)
• Listen (music management and playback)
• Exaile! (amarok styly music app for GTK)
• GPixPod (manage pictures for your ipod)
• Specto (why procrastinate when you can have an application do it for you)
• elisa and pigment (media center by the dudes at Fluendo)
• Gimmie (desktop revisited)
• Deskbar (the most useful app on my desktop)
For extra bonus points I can also freely borrow code from these projects because they are GPL licensed. Yay for FOSS!
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2015-11-25 06:09:22
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https://ncatlab.org/nlab/show/confinement
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# nLab confinement
Contents
## Surveys, textbooks and lecture notes
#### Differential cohomology
differential cohomology
# Contents
## Idea
Confinement (e.g Espiru 94) is the (expected) phenomenon in Yang-Mills theory generally and especially in quantum chromodynamics that the fundamental quarks, which the YM/QCD-Lagrangian density actually describes, must form baryonic bound states which are neutral under the color charge – the mesons and hadrons (protons, neutrons). Hence confinement in particular concerns the emergence and existence of atomic nuclei, hence of ordinary matter, which is not manifest at all in the quark-model. In order to prove, via QCD itself, that the potential energy of a colorless package of quarks tends to infinity as the distance between them grows, one must probe the long-range regime of the quark-quark interaction, in which the methods of Feynman perturbation theory fail.
Part of the issue is that confinement is a non-perturbative effect (e.g Espiru 94) outside the range of validity of perturbative quantum field theory.
### The open problem of confinement
While experiment as well as lattice gauge theory-computer simulation clearly show that confinement takes place, a real theoretical understanding has been missing (though AdS-QCD is now on a good track). This is the confinement problem. The same problem from the point of view of mathematics is called the mass gap Millennium Problem. A related problem is the flavor problem.
The following is a list of quotes highlighting the open problem of confinement:
many of the essential properties that the theory $[$QCD$]$ is presumed to have, including confinement, dynamical mass generation, and chiral symmetry breaking, are only poorly understood. And apart from the low-lying bound states of heavy quarks, which we believe can be described by a nonrelativistic Schroedinger equation, we are unable to derive from the basic theory even the grossest features of the partticle spectrum, or of traditional strong interaction phenomenology
There are theoretical attempts to connect the fundamental theory of QCD with the very successful meson picture at low energy. The Skyrme model is an example. In other attempts, one tries to derive the NN interaction more or less directly from QCD. At present, the predictions are more of a qualitative kind. For quantitative results, the one-pion and two-pion contributions have to be added by hand, as they do not emerge naturally out of QCD-inspired models. Knowing that $\pi$ and $2\pi$ are the most important parts of the nuclear force, this defect of present quark model calculations is serious.
• Robert Kutschke, section 3.1 Heavy flavour spectroscopy, in D. Bugg (ed.), Hadron Spectroscopy and the Confinement Problem, Proceedings of a NATO Advanced Study Institute, Plenum Press 1996 (doi:10.1007/978-1-4613-0375-6)
While it is generally believed that QCD is the correct fundamental theory of the strong interactions there are, as yet, no practical means to produce full QCD calculations of hadron masses and their decay widths.
• Barry R. Holstein, A Brief Introduction to Chiral Perturbation Theory, Czech. J. Phys. 50S4:9-23, 2000 (arXiv:hep-ph/9911449)
the holy grail sought by particle/nuclear knights has been to verify the correctness of the “ultimate” theory of strong interactions – quantum chromodynamics (QCD).
The theory is, of course, deceptively simple on the surface. $[...]$ So why are we still not satisfied? While at the very largest energies, asymptotic freedom allows the use of perturbative techniques, for those who are interested in making contact with low energy experimental findings there exist at least three fundamental difficulties:
i) QCD is written in terms of the “wrong” degrees of freedom – quarks and gluons – while low energy experiments are performed with hadronic bound states;
ii) the theory is non-linear due to gluon self-interactions;
iii) the theory is one of strong coupling so that perturbative methods are not practical
QCD is a perennially problematic theory. Despite its decades of experimental support, the detailed low-energy physics remains beyond our calculational reach. The lattice provides a technique for answering nonperturbative questions, but to date there remain open questions that have not been answered.
• Mike Guidry, Gauge Field Theories: An Introduction with Applications, Wiley 2008 (ISBN:978-3-527-61736-4)
Section 13.1.9:
The holy grail of QCD is the proof that a color SU(3) gauge theory confines in the non-perturbative regime.
This is not difficult to show for lattices with large spacing; unfortunately, such a demonstration does not constitute a proof of QCD confinement: to do that we must also demonstrate that the same theory that confines at large lattice spacing (strong coupling) has a continuum limit (weak coupling) that is consistent with the asymptotically free short distance behavior of QCD.
QCD is a challenging theory. Its most interesting aspects, namely the confinement of color and the chiral symmetry breaking, have defied all analytical approaches. While there are now many data accumulated from the lattice gauge theory, the methodology falls well short of giving us insights on how one may understand these phenomena analytically, nor does it give us a systematic way of obtaining a low energy theory of QCD below the confinement scale.
Nuclear physics is one of the oldest branches of high energy physics, yet remains one of more difficult. Despite the fact that we know the underlying fundamental theory, i.e. QCD, we are still unable to predict, reliably and analytically, behavior of nuclei or even a single proton. The problem is of course that one must understand the strong-coupling regime of QCD, which by and large remains inaccessible except by large-scale lattice simulations. Traditionally, this sets nuclear physics apart from the rest of high energy physics in many aspects. However, recent developments in the so-called gauge/gravity duality began to solve certain strongly coupled field theories,possibly including QCD or its close relatives, allowing the two communities to merge with each other.
Because of the great importance of the standard model, and the central role it plays in our understanding of particle physics, it is unfortunate that, in one very important respect, we don’t really understand how it works. The problem lies in the sector dealing with the interactions of quarks and gluons, the sector known as Quantum Chromodynamics or QCD. We simply do not know for sure why quarks and gluons, which are the fundamental fields of the theory, don’t show up in the actual spectrum of the theory, as asymptotic particle states. There is wide agreement about what must be happening in high energy particle collisions: the formation of color electric flux tubes among quarks and antiquarks, and the eventual fragmentation of those flux tubes into mesons and baryons, rather than free quarks and gluons. But there is no general agreement about why this is happening, and that limitation exposes our general ignorance about the workings of non-abelian gauge theories in general, and QCD in particular, at large distance scales.
The problem with a derivation of nuclear forces from QCD is that this theory is non-perturbative in the low-energy regime characteristic of nuclear physics, which makes direct solutions very difficult. Therefore, during the first round of new attempts, QCD-inspired quark models became popular. The positive aspect of these models is that they try to explain hadron structure and hadron-hadron interactions on an equal footing and, indeed, some of the gross features of the nucleon-nucleon interaction are explained successfully.
However, on a critical note, it must be pointed out that these quark-based approaches are nothing but another set of models and, thus, do not represent fundamental progress. For the purpose of describing hadron-hadron interactions, one may equally well stay with the simpler and much more quantitative meson models.
The success of the technique does not remove the challenge of understanding the non-perturbative aspects of the theory. The two aspects are deeply intertwined. The Lagrangian of QCD is written in terms of quark and gluon degrees of freedom which become apparent at large energy but remain hidden inside hadrons in the low-energy regime. This confinement property is related to the increase of $\alpha_s$ at low energy, but it has never been demonstrated analytically.
We have clear indications of the confinement of quarks into hadrons from both experiments and lattice QCD. Computations of the heavy quark–antiquark potential, for example, display a linear behavior in the quark–antiquark distance, which cannot be obtained in pure perturbation theory. Indeed the two main characteristics of QCD: confinement and the appearance of nearly massless pseudoscalar mesons, emergent from the spontaneous breaking of chiral symmetry, are non-perturbative phenomena whose precise understanding continues to be a target of research.
Even in the simpler case of gluodynamics in the absence of quarks, we do not have a precise understanding of how a gap in the spectrum is formed and the glueball spectrum is generated.
Experimentally, there is a large number of facts that lack a detailed qualitative and quantitative explanation. The most spectacular manifestation of our lack of theoretical understanding of QCD is the failure to observe the elementary degrees of freedom, quarks and gluons, as free asymptotic states (color con- finement) and the occurrence, instead, of families of massive mesons and baryons (hadrons) that form approximately linear Regge trajectories in the mass squared. The internal, partonic structure of hadrons, and nucleons in particular, is still largely mysterious. Since protons and neutrons form almost all the visible matter of the Universe, it is of basic importance to explore their static and dynamical properties in terms of quarks and gluons interacting according to QCD dynamics.
$[\cdots]$ the QCD Lagrangian does not by itself explain the data on strongly interacting matter, and it is not clear how the observed bound states, the hadrons, and their properties arise from QCD. Neither confinement nor dynamical chiral symmetry breaking (DCSB) is apparent in QCD’s lagrangian, yet they play a dominant role in determining the observable characteristics of QCD. The physics of strongly interacting matter is governed by emergent phenomena such as these, which can only be elucidated through the use of non-perturbative methods in QCD [4, 5, 6, 7]
• Hideo Suganuma, Yuya Nakagawa, Kohei Matsumoto, 1+1 Large $N_c$ QCD and its Holographic Dual -Soliton Picture of Baryons in Single-Flavor World, Proceedings of the 14th International Conference on Meson-Nucleon Physics and the Structure of the Nucleon (MENU2016) (arxiv:1610.02074)
Since 1973, quantum chromodynamics (QCD) has been established as the fundamental theory of the strong interaction. Nevertheless, it is very difficult to solve QCD directly in an analytical manner, and many effective models of QCD have been used instead of QCD, but most models cannot be derived from QCD and its connection to QCD is unclear. To analyze nonperturbative QCD, the lattice QCD Monte Carlo simulation has been also used as a first-principle calculation of the strong interaction. However, it has several weak points. For example, the state information (e.g. the wave function) is severely limited, because lattice QCD is based on the path-integral formalism. Also, it is difficult to take the chiral limit, because zero-mass pions require infinite volume lattices. There appears a notorious “sign problem” at finite density.
On the other hand, holographic QCD has a direct connection to QCD, and can be derived from QCD in some limit. In fact, holographic QCD is equivalent to infrared QCD in large $N_c$ and strong ‘t Hooft coupling $\lambda$, via gauge/gravity correspondence. Remarkably, holographic QCD is successful to reproduce many hadron phenomenology such as vector meson dominance, the KSRF relation, hidden local symmetry, the GSW model and the Skyrme soliton picture. Unlike lattice QCD simulations, holographic QCD is usually formulated in the chiral limit, and does not have the sign problem at finite density.
• V. A. Petrov, Asymptotic Regimes of Hadron Scattering in QCD (arXiv:1901.02628)
This is a commonplace that so far we do not have a full-fledged theory of interaction of hadrons, derived from the first principles of QCD and having a regular way of calculating of hadronic amplitudes, especially at high energies and small momentum transfers. The problem is related to a more general problem that QCD Lagrangian would yield colour confinement with massive colourless physical states (hadrons).
One of the long-standing problems in QCD is to reproduce profound nuclear physics. The strong coupling nature of QCD prevents us from solving it analytically, and even numerical simulations have a limitation such as the volume of the atomic nucleus versus the lattice size. It is quite important to bridge the particle physics and the nuclear physics, by solving QCD to derive typical fundamental notions of the nuclear physics, such as the magic numbers, the nuclear binding energy and the nuclear shell model.
Holographic QCD is an analytic method to approach these problems in the strong coupling limit and at a large $N_c$. The nuclear matrix model is a many-body quantum effective mechanics for multiple baryons, derived by the AdS/CFT correspondence applied to QCD.
• Gergely Gábor Barnaföldi, Vakhtang Gogokhia, The Mass Gap Approach to QCD (arXiv:1904.07748)
QCD as a fundamental quantum field gauge theory still suffers from a few important conceptual problems. We focus on some of them as follows: (A) The dynamical generation of a mass squared at the fundamental quark-gluon level, since the QCD Lagrangianforbids such kind of terms apart from the current quark mass. $[\ldots]$ (D) The non-observation of the colored objects as physical states which does not follow from the QCD Lagrangian,i.e., it cannot explain confinement of gluons and quarks.
• Yonggoo Heo, C. Kobdaj, M.F.M. Lutz, Constraints from a large-$N_c$ analysis on meson-baryon interactions at chiral order $Q^3$, Phys. Rev. D 100, 094035 (2019) (arXiv:1908.11816)
Still after many decades of vigorous studies the outstanding challenge of modern physics is to establish a rigorous link of QCD to low-energy hadron physics as it is observed in the many experimental cross section measurements.
• Christian Drischler, Wick Haxton, Kenneth McElvain, Emanuele Mereghetti, Amy Nicholson, Pavlos Vranas, André Walker-Loud, Towards grounding nuclear physics in QCD (arxiv:1910.07961)
the entirety of the rich field of nuclear physics emerges from QCD: from the forces binding protons and neutrons into the nuclear landscape, to the fusion and fission reactions between nuclei, to the prospective interactions of nuclei with BSM physics, and to the unknown state of matter at the cores of neutron stars.
How does this emergence take place exactly? How is the clustering of quarks into nucleons and alpha particles realized? What are the mechanisms behind collective phenomena in nuclei as strongly correlated many-body systems? How does the extreme fine-tuning required to reproduce nuclear binding energies proceed? – are big open questions in nuclear physics.
More than 98% of visible mass is contained within nuclei. In first approximation, their atomic weights are simply the sum of the masses of all the neutrons and protons (nucleons) they contain. Each nucleon has a mass $m_N \sim 1$ GeV, i.e. approximately 2000-times the electron mass. The Higgs boson produces the latter, but what produces the masses of the neutron and proton? This is the question posed above, which is pivotal to the development of modern physics: how can science explain the emergence of hadronic mass (EHM)? $[\cdots]$
Modern science is thus encumbered with the fundamental problem of gluon and quark confinement; and confinement is crucial because it ensures absolute stability of the proton. $[\cdots]$ Without confinement,our Universe cannot exist.
As the 21st Century began, the Clay Mathematics Institute established seven Millennium Prize Problems. Each represents one of the toughest challenges in mathematics. The set contains the problem of confinement; and presenting a sound solution will win its discoverer 1,000,000 bucks. Even with such motivation, today, almost fifty years after the discovery of quarks, no rigorous solution has been found. Confinement and EHM are inextricably linked. Consequently, as science plans for the next thirty years, solving the problem of EHM has become a grand challenge. $[\cdots]$
In trying to match QCD with Nature, one confronts the many complexities of strong, nonlinear dynamics in relativistic quantum field theory, e.g. the loss of particle number conservation, the frame and scale dependence of the explanations and interpretations of observable processes, and the evolving character of the relevant degrees-of-freedom. Electroweak theory and phenomena are essentially perturbative; hence, possess little of this complexity. Science has never before encountered an interaction such as that at work in QCD. Understanding this interaction, explaining everything of which it is capable, can potentially change the way we look at the Universe.
The confinement of quarks is one of the enduring mysteries of modern physics. $[\ldots]$ In spite of many decades of research, physically relevant quantum gauge theories have not yet been constructed in a rigorous mathematical sense. $[$ non-perturbatively, that is $]$ $[\ldots]$ Taking this program to its completion is one of the Clay millennium problems. $[\ldots]$ Perhaps the most important example is four-dimensional SU(3)-lattice gauge theory. If one can show that this theory has a mass gap at all values of the coupling strength, that would explain why particles known as glue-balls in the theory of strong interactions have mass. All such questions remain open.
The second big open question is the problem of quark confinement. Quarks are the constituents of various elementary particles, such as protons and neutrons. It is an enduring mystery why quarks are never observed freely in nature. The problem of quark confinement has received enormous attention in the physics literature, but the current consensus seems to be that a satisfactory theoretical explanation does not exist.
QCD, the theory of the strong interactions, involves quarks interacting with non-Abelian gluon fields. This theory has many features that are difficult to impossible to see in conventional diagrammatic perturbation theory. This includes quark confinement, mass generation, and chiral symmetry breaking.
The origin of the proton mass, and with it the basic mass-scale for all nuclear physics, is one of the most profound puzzles in Nature.
Although QCD is defined by a seemingly simple Lagrangian, it specifies a problem that has defied solution for more than forty years. The key challenges in modern nuclear and high-energy physics are to reveal the observable content of strong QCD and, ultimately, therefrom derive the properties of nuclei.
In spite of the important progress of Euclidean lattice gauge theory, a basic understanding of the mechanism of color confinement and its relation to chiral symmetry breaking in QCD has remained an unsolved problem.
Recent developments based on superconformal quantum mechanics in light-front quantization and its holographic embedding on a higher dimensional gravity theory (gauge/gravity correspondence) have led to new analytic insights into the structure of hadrons and their dynamics.
## Potential solutions
### Via Skyrmions and D4-brane models
A good qualitative and moderate quantitative explanation of confinement in quantum chromodynamics is found in intersecting D-brane models, specifically in the Witten-Sakai-Sugimoto model which geometrically engineers QCD on D4-D8 brane bound states.
(Witten 98, followed up on in Sakai-Sugimoto 04, Sakai-Sugimoto 05)
graphics grabbed from Erlich 09, section 1.1
graphics grabbed from Rebhan 14
In this Witten-Sakai-Sugimoto model for strongly coupled QCD the hadrons in QCD correspond to string-theoretic-phenomena in the bulk field theory:
1. the mesons (bound states of 2 quarks) correspond to open strings in the bulk, whose two endpoints on the asymptotic boundary correspond to the two quarks
2. baryons (bound states of $N_c$ quarks) appear in two different but equivalent (Sugimoto 16, 15.4.1) guises:
1. as wrapped D4-branes with $N_c$ open strings connecting them to the D8-brane
For review see Sugimoto 16, also Rebhan 14, around (18).
graphics grabbed from Sugimoto 16
Equivalently, these baryon states are the Yang-Mills instantons on the D8-brane giving the D4-D8 brane bound state (Sakai-Sugimoto 04, 5.7) as a special case of the general situation for Dp-D(p+4)-brane bound states (e.g. Tong 05, 1.4).
Strictly speaking, since the number of colors in quantum chromodynamics is not large ($N_c = 3$), an accurate formulation of such holographic QCD requires understanding small N corrections:
For more on this see at
and at
### In $\mathcal{N}=2$ super Yang-Mills theory via Seiberg-Witten theory
While confinement in plain Yang-Mills theory is still waiting for mathematical formalization and proof (see Jaffe-Witten), there is a variant of Yang-Mills theory with more symmetry, namely supersymmetry, where the phenomenon has been giving a decent argument, namely in N=2 D=4 super Yang-Mills theory (Seiberg-Witten 94).
Also a strategy for a proof for N=1 D=4 super Yang-Mills theory has been proposed, see below.
### In $\mathcal{N} = 1$ super Yang-Mills theory via M-theory on $G_2$-manifolds
An idea for a strategy towards a proof of confinement in N=1 D=4 super Yang-Mills theory via two different but conjecturally equivalent realizations as M-theory on G2-manifolds has been given in Atiyah-Witten 01, section 6, review is in Acharya-Gukov 04, section 5.3.
The idea here is to consider a KK-compactification of M-theory on fibers which are G2-manifolds that locally around a special point are of the form
$X_{1,\Gamma} \;\coloneqq\; \big( S^3 / \Gamma \big) \times Cone\big(S^3\big) \phantom{AA} \text{or} \phantom{AA} X_{2,\Gamma} \;\coloneqq\; S^3 \times Cone\big(S^3/\Gamma\big)$
where
• $\Gamma$ is a finite subgroup of SU(2) that acts canonically by left-multiplication on $S^3 \simeq$ SU(2);
• $Cone(\cdots)$ denotes the metric cone construction.
This means that $X_{1,\Gamma}$ is a smooth manifold, but $X_{2,\Gamma}$, as soon as $\Gamma$ is not the trivial group, $\Gamma \neq 1$, is an orbifold with an ADE singularity.
Now the lore of M-theory on G2-manifolds predicts that KK-compactification
1. on $X_{1,\Gamma}$ yields a 4d theory without massless fields (since there are no massless modes on the covering space $S^3$ of $X_{1,\Gamma}$)
2. on the ADE-singularity $X_{2,\Gamma}$ yields non-abelian Yang-Mills theory in 4d coupled to chiral fermions.
So in the first case a mass gap is manifest, while non-abelian gauge theory is not visible, while in the second case it is the other way around.
But if there were an argument that M-theory on G2-manifolds is in fact equivalent for compactification both on $X_{1,\Gamma}$ and on $X_{2,\Gamma}$. To the extent that this is true, it looks like an argument that could demonstrate confinement in non-abelian 4d gauge theory.
This approach is suggested in Atiyah-Witten 01, pages 84-85. An argument that this equivalence is indeed the case is then provided in sections 6.1-6.4, based on an argument in Atiyah-Maldacena-Vafa 00
### Via Calorons
It has been argued that, after Wick rotation, confinement may be derived from the behaviour of instantons (Schaefer-Shuryak 96, section III D), or rather their positive temperature-incarnations as calorons, Greensite 11, section 8.5:
it is natural to wonder if confinement could be derived from some semiclassical treatment of Yang–Mills theory based on the instanton solutions of non-abelian gauge theories. The standard instantons, introduced by Belavin et al. (40), do not seem to work; their field strengths fall off too rapidly to produce the desired magnetic disorder in the vacuum.
In recent years, however, it has been realized that instanton solutions at finite temperature, known as calorons, might do the job. These caloron solutions were introduced independently by Kraan and van Baal (41, 42) and Lee and Lu (43) (KvBLL), and they have the remarkable property of containing monopole constituents which may, depending on the type of caloron, be widely separated.
$[...]$
The caloron idea is probably the most promising current version of monopole confinement in pure non-abelian gauge theories, but it is basically (in certain gauges) a superposition of monopoles with spherically symmetric abelian fields, and this leads to the same questions raised in connection with monopole Coulomb gases.
effective field theories of nuclear physics, hence for confined-phase quantum chromodynamics:
## References
### In Yang-Mills theory
#### General
Textbook accounts:
• David Griffiths, chapter 8 of Introduction to Elementary Particles 2nd ed., 2008. (pdf)
• Jeff Greensite, An Introduction to the Confinement Problem, Lecture Notes in Physics, Volume 821, 2011 (doi:10.1007/978-3-642-14382-3)
• Robert Iengo, section 9.1 of Quantum Field Theory (pdf)
• D. Bugg (ed.), Hadron Spectroscopy and the Confinement Problem, Proceedings of a NATO Advanced Study Institute, Plenum Press 1996 (doi:10.1007/978-1-4613-0375-6)
Introductions and surveys include
Discussion includes
A formulation of confinement as an open problem of mathematical physics, together with many references, is in
Other technical reviews include
• G. M. Prosperi, Confinement and bound states in QCD (arXiv:hep-ph/0202186)
• Christian Drischler, Wick Haxton, Kenneth McElvain, Emanuele Mereghetti, Amy Nicholson, Pavlos Vranas, André Walker-Loud, Towards grounding nuclear physics in QCD (arxiv:1910.07961)
• Yuri A. Simonov, The fundamental scale of QCD (arXiv:2103.08223)
• Duifje Maria van Egmond, Urko Reinosa, Julien Serreau, Matthieu Tissier, A novel background field approach to the confinement-deconfinement transition (arXiv:2104.08974)
#### Via monopole condensation
An original suggestion that confinement in Yang-Mills theory may be understood via monopole condensation as a dual Meissner effect is due to
• Gerard 't Hooft, in Proceed.of the Europ.Phys.Soc. 1975, ed.by A.Zichichi (Editrice Compositori, Bologna, 1976), p.1225.
• S. Mandelstam, Phys.Rep. 23C (1976) 145;
(That this is indeed the case has not yet been demonstarted for plain Yang-Mills theory, but it was later shown for N=2 D=4 super Yang-Mills theory in (Seiberg-Witten 94). What this does or does not imply for the case of QCD is discussed in (Yung 00) ).
The relation to QCD instantons/monopoles in the QCD vacuum is discussed in
and analogously (at positive temperature) relation to calorons:
• Greensite 11, section 8.5
• P. Gerhold, E.-M. Ilgenfritz, M. Müller-Preussker, An $SU(2)$ KvBLL caloron gas model and confinement, Nucl.Phys.B760:1-37, 2007 (arXiv:hep-ph/0607315)
• Rasmus Larsen, Edward Shuryak, Classical interactions of the instanton-dyons with antidyons, Nucl. Phys. A 950, 110 (2016) (arXiv:1408.6563)
• Rasmus Larsen, Edward Shuryak, Interacting Ensemble of the Instanton-dyons and Deconfinement Phase Transition in the SU(2) Gauge Theory, Phys. Rev. D 92, 094022, 2015 (arXiv:1504.03341)
For further developments see
• Dimensional Transmutation by Monopole Condensation in QCD (arXiv:1206.6936)
### Interacting field vacuum
Discussion of confinement as a result of the interacting vacuum includes
• Johann Rafelski, Vacuum structure – An Essay, in pages 1-29 of H. Fried, Berndt Müller (eds.) Vacuum Structure in Intense Fields, Plenum Press 1990 (GBooks)
### In super-Yang-Mills theory
Confinement in N=2 D=4 super Yang-Mills theory by a version of the monopole condensation of (t Hooft 75, Mandelstam 76) was demonstrated in
Reviews with discussion of the impact on confinement in plain YM include
• Alexei Yung, What Do We Learn about Confinement from the Seiberg-Witten Theory (arXiv:hep-th/0005088)
Discussion in the context of the AdS-QCD correspondence is in
• Edward Witten, Anti-de Sitter Space, Thermal Phase Transition, And Confinement In Gauge Theories, Adv. Theor. Math. Phys.2:505-532, 1998 (arXiv:hep-th/9803131)
• David Berman, Maulik K. Parikh, Confinement and the AdS/CFT Correspondence, Phys.Lett. B483 (2000) 271-276 (arXiv:hep-th/0002031)
• Henrique Boschi Filho, AdS/QCD and confinement, Seminar at the Workshop on Strongly Coupled QCD: The confinement problem, November 2011 (pdf)
### In M-theory on $G_2$-manifolds
An idea for how to demonstrate confinement in models of M-theory on G2-manifolds is given in
based on
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# Tag Info
## Hot answers tagged probability-distributions
4
Update/clarification. (with thanks to Nathan's pointing this out!) My original answer below was not a proof that $\|p^*\|_\infty/\|p^*\|_2^2\le2\sqrt n/5$. This is because the minimum I obtained was the minimum for the function $\sqrt n\|p\|_2^2-\|p\|_\infty$, so a direct substitution would lead to a misleading result of the desired ratio, as we have $\|p\|... 3 Since expectation (as an operator) is linear, the following statement is correct regardless of the independence between$X$and$Y$: $$E(Z) = E(X − Y) = E(X) − E(Y) = 0.5 − 0.5 = 0.$$ However, if we do want to find the probability (density) function (pdf) of$Z = X - Y$, we can use the convolution formula to obtain this. Generally, the density of$Z = X - Y$... 3 Here is a counterexample: assume each$Y_i$has distribution$\mathcal N(0,1)$, so that$\frac 1{\sqrt n}\sum_{i=1}^nY_i \sim \mathcal N(0,1)$. Suppose for the sake of contradiction that there is some$X\sim \mathcal N(0,1)$independent of$(Y_1,\ldots)$with$\frac 1{\sqrt n}\sum_{i=1}^nY_i \xrightarrow{L^p}X$. Since$X$is independent of$(Y_1,Y_2,\ldots)$... 3 Solution 1. The range of$Y=\sin X$with$X\in[0,2\pi]$is$[-1,1]$not$[0,0]$because$\sin$is not monotone. You can draw the graph of$\sin$with$X\in[0,2\pi]$to see the range. For$y\in[0,1], $$Y=\sin X\leq y\iff 0\leq X\leq\arcsin y\quad\text{or}\quad\pi-\arcsin y\leq X\leq 2\pi.$$ Thus, \begin{align*} F_Y(y)&=P(Y\leq y)=P(0\leq X\leq\arcsin y)+... 2 A summary of the brief discussion with Gabriel Romon: Given any independent, identically distributed random variables(Y_k)_{k\in\mathbb N}$with$\mathsf EY_k=0, \mathsf VY_k=1$, if$X$is independent of$(Y_k)_{k\in\mathbb N}$, then according to the Central Limit Theorem,*$-X+\frac1{\sqrt n}\sum_{k=1}^\infty Y_k$converges in distribution to (slight ... 2 The answer will depend on your parameters and your choice of noise. I'll give partial answer, using the following reference: Peigné, Woess, Stochastic dyamical systems with weak contractivity properties. I. Strong and local contractivity, which contains many helpful other references (Goldie...). First, I will exclude the value$0$of the domain, so that$(...
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In general no: take $N$ standard normal and $\varepsilon$ a random variable independent of $N$ taking the values $1$ and $-1$ with probability $1/2$. Let $X_1=N$ and $X_2=N\varepsilon$. Then $X_1$ and $X_2$ have a standard normal distribution and are uncorrelated. But $X_1+X_2=N(1+\varepsilon)$ is not Gaussian, as it takes the value $0$ with probability $1/2$...
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As long as all expectations are defined (or always, if you allow infinite values). You have $|X+Y|\leq |X|+|Y|$ almost surely, and then you can take the expectation on both sides.
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Are $\hat{X_1}$ and $\hat{X_2}$ independent? If so then Let $U=$ sum and $f_U(u)=\int\limits_0^1 I_{(0,1)}(x-u)dx$ So $f_U(u)=u$ for $0\le u \le 1$ and $f_U(u)=1-u$ for $1\le u \le 2$. $\hat{Y}=\frac{U}{2}$ with $f_{\hat{Y}}=4y$ for $0\le y\le 0.5$ and $=4(1-y)$ for $0.5\le y\le 1$..
2
Suppose that $X,Y$ are random variables and consider $Z = XY$ \begin{align} F_Z(z) & = P(Z \le z) \\ & = P(XY \le z) \\ & = P(X \le z/Y | Y > 0)P(Y>0) + P(X \ge z/Y | Y < 0) P(Y<0) \\ & = \int_0^\infty \int_{-\infty}^{z/y} f_{X,Y}(x,y) \ dx dy + \int_{-\infty}^0 \int_{z/y}^\infty f_{X,Y}(x,y) \ dx dy \\ f_Z(z) & = \int_0^\...
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If $X$ and $Y$ are allowed to be discrete random variables rather than only continuous random variables such as normal random variables, then when $X$ and $Y$ are independent Bernoulli random variables with parameters $p$ and $q$ respectively, $XY$ is also a Bernoulli random variable with albeit with a different parameter $pq$. If you want symmetric ...
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This is an old question, but a good one, and one that has some interesting connections across topics that might seem unrelated- the uniform distribution on the n-sphere has a relationship derived in information geometry to the Jeffreys prior for multinomial probabilities- taking the square root of probabilities results in a diffeomorphism between the unit ...
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You are missing several additional inequalities satsifesd by $m$ and $w$. The given inequalities for $x$ and $y$ are equiavlen to to the following: $0 \leq m \leq 1$, $0 \leq w \leq \frac 1 2$, $w \leq \frac m 2$, and $m \leq w+\frac 1 2$. The third inequality comes from $x \leq y$. The last one comes from $y \leq 1$. [Always make it a point to check if ...
1
The functional form of the transformed joint density is correct, but the support is incorrect. All you have to do to rectify this is to first note that the support for $(X,Y)$ is bounded by the lines $$X = 0, \quad X = Y, \quad Y = 1.$$ Because the transformation is linear and invertible, the resulting support in $(M, W)$ space is also bounded by three ...
1
The mean of uniform distribution on $\{1,2,...,n\}$ is $\sum\limits_{k=1}^{n}\frac k n=\frac {n(n+1)} {2n}=\frac {1+n} 2$. Hence, the mean of $Y$ given $X$ is $\frac {1+X} 2$. Now take the mean of this: $EY=E(\frac {1 +X} 2)$. Can you complete this?
1
Given that the continuous random variable $X$ has probability density function $$f(x) = ax - bx^2, \quad 0 \leq x \leq 2$$ and 0 elsewhere, we have $$1 = \int_{-\infty}^\infty f(x) dx = \int_0^2 f(x) dx = \int_0^2 (ax - bx^2) dx = \left[\frac{a}{2}x^2 - \frac{b}{3}x^3\right]\bigg|_0^2 = 2a - \frac{8}{3}b.$$ Given also that $E[X] = 1$, we have $$1 = E[X] = \... 1 Let p_n be the probability that A wins, given that his current bankroll is n. We are asked for p_3. We know that p_0=0 and that p_8=1. Since the probability that A wins any game is \frac23, we know that$$p_n=\frac23p_{n+1}+\frac13p_{n-1},\ 0<n<8$$If you substitute n=1, you'll be able to express p_2 in terms of p_1. Then ... 1 (1 - (1-0.3)\cdot (1-0.4\cdot 0.4)\cdot (1-0.2))\cdot 0.1 = (1-0.7\cdot 0.84\cdot 0.8)\cdot 0.1 The point being, you need at least one of the parallel paths to work which is the opposite of all of the parallel paths not working. 1 Oh, it converges; the integrand diverges at 0, but in a harmless way, like \int_0^\epsilon x^{-1/2}dx=2\epsilon^{1/2} for \epsilon\ge0. The Gamma function \Gamma(s):=\int_0^\infty x^{s-1}e^{-x}dx satisfies \Gamma(1/2)=\sqrt{\pi}. You can easily show \Bbb Ee^{tX} is (1-2t)^{-1/2} for t<\tfrac12, otherwise it's infinite. Needless to say, ... 1 Recall that exponential distributions are memory-less. Therefore, at the time when the first of Jake and Naomi get done with their service, Paul would go to the server that becomes empty. Say he goes to server 1 at time t=s. Then, due to memoryless property, \mathbb{P}[X_2>t+s|X_2>s] = \mathbb{P}[X_2>t]. Thus, we are comparing the distributions ... 1 Welcome to MSE! The idea is that they have a function, \Phi, which computes the probability of a standard normal variable falling below a stated value. So, the entire computation is transforming the random variable that you start with, which they assume to be Gaussian with a mean, \mu, and a standard deviation, \sigma, to a standard gaussian which has ... 1 I would use the probability density function, f(x), to calculate the expectation using$$ E(x) = \int x f(x) dx $$We can get f(x) from the cumulative density function by differentiating in each region:$$ f(x) = \left\{ \begin{array}{ll} 0 & \quad x<-3 \\ \frac{1}{6} & \quad -3\le x<0 \\ 0 & ...
1
The mathematical proof shared by Jonathan Christensen in the answer below is great. Here is my intuitive interpretation: I was also deeply confused when every "simple" explanation out there references the Poisson distribution which is intuitively not right because the underlying process should be a Binomial. I too initially thought that the chi-...
1
The easiest way to solve the problem is to do a drawing of the transformation function and realize that $$\mathbb{P}[Y>y]=F_X\left(\frac{1}{y}\right)-F_X(y)=e^{-y}-e^{-1/y}$$ thus $$F_Y(y)=1+e^{-1/y}-e^{-y}$$ derivating you get your density...and observe that $y \in(0;1]$ $$f_Y(y)=\left[\frac{e^{-1/y}}{y^2}+e^{-y}\right]\cdot\mathbb{1}_{(0;1]}(y)$$ this ...
1
What's the probability that the first and second error both occur on page #89? Assuming no correlation, it's $(0.01)^2 = 0.0001$. What's the probability that any two specific errors occurs on page #89? It's $0.0001$. Assuming that exactly two of the ten errors occurs on page #89, how many different ways could this occur? That's ${}_{10}C_{2} = 45$. So, the ...
1
Turns out I found the answer. For anyone interested in the future, there is this wonderful paper by Hoeffding himself. The proof of the fact I wanted can be found in Theorem 4.
1
It is known that $$\Gamma (x) \ge \sqrt {2\pi } x^{x - 1/2} e^{ - x}$$ for all $x>0$ (cf. http://dlmf.nist.gov/5.6.E1). Because of Stirling's formula, this is a rather sharp lower bound. This gives the simple bound $$\frac{{2^{2^{ - m} (2^m - 2^n )(m - n)} e^{ - 2^{n - m} } }}{{\Gamma (2^{n - m} )}} \le \frac{1}{{\sqrt {2\pi } 2^{(n - m)/2} }}.$$ ...
1
While you can use a hypergeometric distribution, it is not necessary in this case because there is only one token you are interested in. This simplifies the probability model as follows. Consider a bowl of $n$ tokens, one of which is winning. We first compute the probability that you will win on exactly the $x^{\rm th}$ draw. To do this, imagine that ...
1
About the statement $$\frac{1-R(x)}{\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}(\frac{1}{x}-\frac{1}{x^3}+...+(-1)^k\frac{1*3*7*...*(2k-1)}{x^{(2k+1)}})}\xrightarrow{x \to +\infty} 1$$ In fact, it holds true, even for the general form like this one \frac{1-R(x)}{\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}(\frac{1}{x}+\sum_{k=2}^n \frac{a_k}{x^k})}\xrightarrow{...
1
I think the answer by TheSimpliFire was only almost correct but not quite because it identified the wrong $p$ on the $(\|p\|_\infty,-\|p\|_2^2)$ Pareto frontier. Here's my attempt, which shows that $\lim_{n\to\infty}\max_{p\in\Delta^n}\frac{\|p\|_\infty}{\sqrt{n}\|p\|_2^2}=\frac12$, approaching it from above. Let $\Delta^n$ denote the $(n-1)$-dimensional ...
Only top voted, non community-wiki answers of a minimum length are eligible
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2021-04-11 13:57:18
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https://aperiodical.com/category/columns/irregulars/page/3/
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### Alan Turing is the face of the new £50 note
The Bank of England has announced, following a public poll, that the new £50 note will feature mathematician, cryptographer and computer science pioneer Alan Turing. While this might seem like unambiguous good news, the issues it raises are more complicated than they first appear. Here’s a guest post from LGBT+ mathematician Calvin Smith with his thoughts on the decision.
It’s obviously fabulous that Alan Turing is being recognised on the new £50 note, but this joy at seeing a gay mathematician given this recognition is tainted with the memory of his cruel treatment by the society of the day and the ongoing persecution of queer and trans people today.
### Mathigon
This is a guest post from Philipp Legner, the creator of Mathigon an interactive maths education platform.
Every year, thousands of students around the world ask themselves why they have to learn mathematics. Calculators can do long division. You can look up the quadratic formula on the internet. And when will you ever need calculus in everyday life? It seems like they have a point.
In fact, the maths curriculum has not changed significantly in the last 50 years. Its primary focus is on memorising rules and procedures which can be used to solve standardised exam questions. I created Mathigon because I strongly believe we need to change this – not only to make mathematics more enjoyable for students, but also to teach different skills that are much more useful in life: problem-solving, abstraction, logical reasoning, creativity, and curiosity.
### Talking Maths in Public
In 2017, the University of Bath hosted the first Talking Maths in Public conference, a gathering for UK maths communicators. As part of the event, attendance bursaries were awarded to students interested in maths outreach, and the recipients of the bursaries wrote about their experiences. To celebrate the fact that a second TMiP conference will be happening this year (booking is open now, and we’re all going to be there!), we’re sharing their report of TMiP 2017. You can find out more about this year’s event (which also includes a bursary scheme) at talkingmathsinpublic.uk.
This post was jointly written by Imogen Morris, (University of Edinburgh), David Nkansah (University of Glasgow) and Olivia Sorto (University of Edinburgh).
### Elwyn Berlekamp has left us
In memory of Elwyn Berlekamp, who passed away on 9th April, Colin Wright has shared with us this post from his blog.
I remember meeting Elwyn Berlekamp.
### Scientists call for an end to statistical significance
A group of over 800 scientists have signed their names to an article published in Nature, explaining why statistical significance shouldn’t be relied on so heavily as a measure of the success of an experiment. We asked statistics buff Andrew Steele to explain.
### realhats: Writing a $\LaTeX$ Package
A few months ago, Adam Townsend went to lunch and had a conversation. I wasn’t there, but I imagine the conversation went something like this:
As anyone who has been anywhere near maths at a university in the last ∞ years will be able to tell you, LaTeΧ (or $\LaTeX$) is a piece of maths typesetting software. It’s a bit like a version of Word that runs in terminal and makes PDFs with really pretty equations.
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2021-01-23 23:54:28
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https://www.physicsforums.com/threads/chain-rule-for-square-roots.442818/
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# Chain rule for square roots (1 Viewer)
### Users Who Are Viewing This Thread (Users: 0, Guests: 1)
#### unf0r5ak3n
I'm kind of confused about how to approach a function with the chain rule.
For example in the equation ƒ(x) = sqrt(1-sin(x)) I know i simplify it to ƒ(x) = 1-sin(x)^(1/2) but I'm lost from there.
#### mathman
What are you trying to accomplish? The "simplification" you give doesn't make any sense.
#### hotvette
Homework Helper
I'm kind of confused about how to approach a function with the chain rule.
For example in the equation ƒ(x) = sqrt(1-sin(x)) I know i simplify it to ƒ(x) = 1-sin(x)^(1/2) but I'm lost from there.
From the title of your post I'm guessing you want to find f'(x). Your simplification isn't quite correct. Don't you mean ƒ(x) = (1-sin(x))^(1/2)? Fractional powers are treated the same as integer powers when it comes to differentiation. I presume you know how to differentiate y = x3. Use the exact same methodology to differentiate y = x1/2.
#### unf0r5ak3n
What are you trying to accomplish? The "simplification" you give doesn't make any sense.
sorry I meant equivalent to
#### mathman
sorry I meant equivalent to
I don't know what "equivalent to" means either in this context. However sqrt(1-sin(x)) is just different from 1-sin(x)^(1/2). There is no way to equate these two expressions.
#### HallsofIvy
The point being that $1- sin(x)^{1/2}\ne (1- sin(x))^{1/2}$.
Write $(1- sin(x))^{1/2}$ as $y= u^{1/2}$ with $u= 1- sin(x)$. Can you find dy/du and du/dx?
$$\frac{dy}{dx}= \frac{dy}{du}\frac{du}{dx}$$
#### unf0r5ak3n
if u = 1-sin(x) then du/dx = (1/2)u
#### brewAP2010
If f(x)= u^n then f’(x)= (n)u^(n-1)u’
so
If f(x)= (1-sinx)^1/2 then f’(x)= (1/2)(1-sinx)^(-1/2)(-cosx)
#### HallsofIvy
if u = 1-sin(x) then du/dx = (1/2)u
?? I was under the impression that the derivative of 1- sin(x) was -cos(x).
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2019-03-23 23:18:58
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http://trandinhquoc.com/software.html
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Software
LMAOPT: Low-rank Matrix OPTimization
• LMAOPT-v1.0 is a MATLAB code colection for solving three special cases of the following low-rank matrix optimization problem: $$\Phi^{\star} := \min_{U\in\mathbf{R}^{m\times r}, V\in\mathbb{R}^{n\times r}}\Big\{ \Phi(U,V) := \phi(\mathcal{A}(UV^T) - B) \Big\},$$ where $\phi$ is a proper, closed and convex function from $\mathbb{R}^l\to\mathbb{R}\cup\{+\infty\}$, $\mathcal{A}$ is a linear operator from $\mathbb{R}^{m\times n}$ to $\mathbb{R}^l$, and $B\in\mathbb{R}^l$ is a given observed vector. Here, we are more interested in the case $r \ll \min\{m, n\}$.
Currently, we provide the code to solve three special cases of the above problem:
• Quadratic loss: $\phi(\cdot) = \frac{1}{2}\Vert\cdot\Vert_2^2$;
• Quadratic loss and symmetric case: $\phi(\cdot) = \frac{1}{2}\Vert\cdot\Vert_2^2$ and $U = V$; and
• Nonsmooth objective loss with tractably proximal operators: For instance, $\phi(\cdot) = \Vert\cdot\Vert_1$.
LMAOPT is implemented by Quoc Tran-Dinh at the Department of Statistics and Operations Research (STAT&OR), University of North Carolina at Chapel Hill (UNC). This is a joint work with Zheqi Zhang at STAT&OR, UNC.
• REFERENCES
The theory and algorithms implemented in LMAOPT can be found in the following manuscript:
1. Q. Tran-Dinh, and Z. Zhang. Extended Gauss-Newton and Gauss-Newton ADMM algorithms for low-rank matrix optimization. STAT&OR Tech. Report UNC-REPORT-2016.a, (2016).
Available at: http://arxiv.org/abs/1606.03358
Abstract: We develop a generic Gauss-Newton (GN) framework for solving a class of nonconvex optimization problems involving low-rank matrix variables. As opposed to standard Gauss-Newton method, our framework allows one to handle general smooth convex cost function via its surrogate. The main complexity-per-iteration consists of the inverse of two rank-size matrices and at most six small matrix multiplications to compute a closed form Gauss-Newton direction, and a backtracking linesearch. We show, under mild conditions, that the proposed algorithm globally and locally converges to a stationary point of the original nonconvex problem. We also show empirically that the Gauss-Newton algorithm achieves much higher accurate solutions compared to the well studied alternating direction method (ADM). Then, we specify our Gauss-Newton framework to handle the symmetric case and prove its convergence, where ADM is not applicable without lifting variables. Next, we incorporate our Gauss-Newton scheme into the alternating direction method of multipliers (ADMM) to design a GN-ADMM algorithm for solving the low-rank optimization problem. We prove that, under mild conditions and a proper choice of the penalty parameter, our GN-ADMM globally converges to a stationary point of the original problem. Finally, we apply our algorithms to solve several problems in practice such as low-rank approximation, matrix completion, robust low-rank matrix recovery, and matrix recovery in quantum tomography. The numerical experiments provide encouraging results to motivate the use of nonconvex optimization.
• DECOPT: DEcomposition OPTimization
• DECOPT-v1.0 is a MATLAB software package for solving constrained convex optimization problems of the form: $$\begin{array}{ll} \displaystyle\min_{\mathbf{x}\in\mathbf{R}^p, \mathbf{r}\in\mathbb{R}^n} & f(\mathbf{x}) + g(\mathbf{r}) \\ \text{s.t.} & \mathbf{A}\mathbf{x} - \mathbf{r} = \mathbf{b}, ~\mathbf{l}\leq \mathbf{x}\leq \mathbf{u}, \end{array}$$ where $f$ and $g$ are two convex functions, $\mathbf{A}\in\mathbb{R}^{n\times p}$, $\mathbf{b}\in\mathbb{R}^n$, $\mathbf{l}, \mathbf{u}\in\mathbb{R}^p$ are the lower and upper bound of $\mathbf{x}$. Here, we assume that $f$ and $g$ are proximally tractable, i.e., the proximal operators $\mathrm{prox}_{f}$ and $\mathrm{prox}_g$ are efficient to compute (see below).
DECOPT aims at solving constrained convex optimization problem for any convex functions $f$ and $g$, where their proximal operator is provided. The following special cases have been customized in DECOPT:
• Basis pursuit: $$\min_{\mathbf{x}}\Big\{ \Vert\mathbf{x}\Vert_1 : \mathbf{A}(\mathbf{x}) = \mathbf{b}, ~\mathbf{l} \leq \mathbf{x}\leq\mathbf{u} \Big\}.$$
• $\ell_1/\ell_2$-unconstrained LASSO problem: $$\min_{\mathbf{x}}\Big\{ \frac{1}{2}\Vert\mathbf{A}(\mathbf{x}) - \mathbf{b}\Vert_2^2 + \lambda\Vert\mathbf{x}\Vert_1\Big\}.$$
• $\ell_1/\ell_1$-convex problem: $$\min_{\mathbf{x}}\Big\{ \Vert\mathbf{A}(\mathbf{x}) - \mathbf{b}\Vert_1 + \lambda\Vert\mathbf{x}\Vert_1 \Big\}.$$
• Square-root $\ell_1/\ell_2$ LASSO problem: $$\min_{\mathbf{x}}\Big\{ \Vert\mathbf{A}(\mathbf{x}) - \mathbf{b}\Vert_2 + \lambda\Vert\mathbf{x}\Vert_1 \Big\}.$$
• $\ell_1/\ell_2$ - the basis denosing (BPDN) problem: $$\min_{\mathbf{x}}\Big\{ \Vert\mathbf{x}\Vert_1 : \Vert\mathbf{A}(\mathbf{x}) - \mathbf{b}\Vert_2 \leq \delta\Big\}.$$
• $\ell_2/\ell_1$ - the $\ell_1$ - constrained LASSO problem: $$\min_{\mathbf{x}}\Big\{ \frac{1}{2}\Vert\mathbf{A}(\mathbf{x}) - \mathbf{b}\Vert_2^2 : \Vert\mathbf{x}\Vert_1 \leq \delta \Big\}.$$
Here, $\lambda > 0$ is a penalty parameter, $\delta > 0$ is a given noise level parameter, $\mathbf{b}$ is the observed measurement vector, and $\mathbf{A}$ is a linear operator.
DECOPT is developed by Quoc Tran-Dinh at the Laboratory for Information and Inference Systems (LIONS), EPFL, Lausanne, Switzerland. This is a joint work with V. Cevher at EPFL/LIONS.
• REFERENCES
The theory and algorithms implemented in DECOPT can be found in the following papers:
1. Q. Tran-Dinh, and V. Cevher. Constrained convex minimization via model-based excessive gap. Proceedings of the annual conference on Neural Information Processing Systems Foundation (NIPS), Montreal, Canada, (2014).
Available at: http://papers.nips.cc/paper/5574-constrained-convex-minimization-via-model-based-excessive-gap.
Abstract: Abstract We introduce a model-based excessive gap technique to analyze first-order primal- dual methods for constrained convex minimization. As a result, we construct first-order primal-dual methods with optimal convergence rates on the primal objec-tive residual and the primal feasibility gap of their iterates separately. Through a dual smoothing and prox- center selection strategy, our framework subsumes the augmented Lagrangian, alternating direction, and dual fast-gradient methods as special cases, where our rates apply.
2. Q. Tran Dinh, and V. Cevher. A Primal-Dual Algorithmic Framework for Constrained Convex Minimization. LIONS Tech. Report EPFL-REPORT-199844, (2014).
Available at: http://arxiv.org/abs/1406.5403
Abstract: We present a primal-dual algorithmic framework to obtain approximate solutions to a prototypical constrained convex optimization problem, and rigorously characterize how common structural assumptions affect the numerical efficiency. Our main analysis technique provides a fresh perspective on Nesterov's excessive gap technique in a structured fashion and unifies it with smoothing and primal-dual methods. For instance, through the choices of a dual smoothing strategy and a center point, our framework subsumes decomposition algorithms, augmented Lagrangian as well as the alternating direction method-of-multipliers methods as its special cases, and provides optimal convergence rates on the primal objective residual as well as the primal feasibility gap of the iterates for all.
SCOPT: Self-Concordant OPTimization
• SCOPT-v1.0 - A MATLAB package for Composite Self-concordant (like) Minimization
SCOPT is a MATLAB implementation of the proximal-gradient, proximal quasi-Newton, proximal Newton, and path-following interior-point algorithms for solving composite convex minimization problems involving self-concordant and self-concordant-like cost functions: $$\displaystyle\min_{\mathbf{x}\in\mathbb{R}^n}\left\{ F(\mathbf{x}) := f(\mathbf{x}) + g(\mathbf{x}) \right\},$$ where $f$ is a self-concordant or self-concordant-like function, and $g$ is a possibly nonsmooth and convex function.
• Briefly, the function $f$ is called self-concordant (with the parameter $M_f >0$) if $\vert\varphi{'''}(t)\vert \leq M_f\varphi{''}(t)^{3/2}$, where $\varphi(t):= f(\mathbf{x} + t\mathbf{u})$ for $\mathbf{x}\in\mathrm{dom}(f)$, $\mathbf{u}\in\mathbb{R}^m$ and $\mathbf{x} + t\mathbf{u}\in\mathrm{dom}(f)$.
Examples: $f(\mathbf{X}) := -\log(\det(\mathbf{X}))$ (for a symmetric and positive definite matrix $\mathbf{X}$), $f(\mathbf{x},t) := -\log(t^2 - \Vert\mathbf{x}\Vert_2^2)$, and $f(\mathbf{x}) := -\sum_{i=1}^n\log(b_i - \mathbf{a}_i^T\mathbf{x})$ are all self-concordant.
• The function $g$ is usually assumed that its proximal operator: $$\mathrm{prox}_{g}(\mathbf{x}) := \mathrm{arg}\!\displaystyle\min_{\mathbf{z}\in\mathbb{R}^n}\left\{g(\mathbf{z}) + \frac{1}{2}\Vert\mathbf{z} - \mathbf{x}\Vert_2^2\right\},$$ is "efficient" to compute (i.e., in a closed form or by polynomial-time algorithms). If the proximal operator of $g$ can be computed efficently, then we say that $g$ is tractably proximal.
SCOPT is developed by Quoc Tran-Dinh at the Laboratory for Information and Inference Systems (LIONS), EPFL, Lausanne, Switzerland. This is a joint work with A. Kyrillidis and V. Cevher at EPFL/LIONS.
SCOPT consists of several algorithms customized for solving the following convex optimization problems (but not limitted to those):
• Sparse Inverse Covariance Estimation: An instance of this problem class can be expressed as follows: $$\min_{\mathbf{X}\succ 0}\left\{ -\log(\det(\mathbf{X})) + \mathrm{trace}(\boldsymbol{\Sigma}\mathbf{X}) + \rho\Vert\mathrm{vec}(\mathbf{X})\Vert_1 \right\},$$ where $\rho > 0$ is a penalty parameter.
Reference: J. Friedman, T. Hastie, and R. Tibshirani. Sparse Inverse Covariance Estimation with the graphical LASSO. Biostatistics, vol. 9, no. 3, pp. 432--441, 2007.
• Sparse Poisson Intensity Reconstruction: As an example, this problem has the following form: $$\min_{\mathbf{x}\geq 0}\left\{ -\sum_{i=1}^m\mathbf{a}_i^T\mathbf{x} - \sum_{i=1}^my_i\log(\mathbf{a}_i^T\mathbf{x}) + \rho\Vert\mathbf{x}\Vert_{\mathrm{TV}}\right\},$$ where $y_i\in\mathbb{N}$, $\rho > 0$ is a penalty parameter, and $\Vert\cdot\Vert_{\mathrm{TV}}$ is the TV (total-variation) norm.
Reference: Z. T. Harmany, R. F. Marcia, and R. M. Willett. This is SPIRAL-TAP: Sparse Poisson Intensity Reconstruction Algorithms - Theory and Practice. IEEE Trans. Image Processing, vol. 21, no. 3, pp. 1084--1096, 2012.
• Heteroscedastic LASSO: This problem is formulated as follows: $$\min_{\sigma > 0, \boldsymbol{\beta}\in\mathbb{R}^p}\left\{ -\log(\sigma) + \frac{1}{2n}\Vert\mathbf{X}\boldsymbol{\beta} - \sigma\mathbf{y}\Vert_2^2 + \rho\Vert\boldsymbol{\beta}\Vert_1\right\},$$ where $\rho > 0$ is a penalty parameter.
Reference: N. Stadler, P. Buhlmann, and S. V. de Geer. L1-Penalization for mixture regression models. TEST, vol. 19, no. 2, pp. 209--256, 2010.
• Sparse logistic and multimonomial logistic regression: This problem can be written in the following mathematical form: $$\min_{\mathbf{X}\in\mathbb{R}^{m\times p}}\left\{ \frac{1}{N}\sum_{i=1}^N\left[ \log\left(1 + \sum_{j=1}^m\mathrm{exp}(\mathbf{W}_{(j)}^T\mathbf{X}_{(i)})\right) -\sum_{i=1}^my_i^{(j)}\mathbf{W}_{(j)^T\mathbf{X}_{(i)}}\right] + \rho\Vert\mathrm{vec}(\mathbf{X})\Vert_1 \right\},$$ where $\mathbf{W}_{(j)}$ and $y_i^{(j)}$ are input data, and $\rho > 0$ is a penalty parameter.
Reference: B. Krishnapuram, M. Figueredo, L. Carin, and A. Hertemink. Sparse multinomial logistic regression: Fast algorithms and generalization bounds. IEEE Trans. Pattern Analysis and Machine Intelligence (PAMI), vol. 27, pp. 957--968, 2005.
• In addition to the above mentioned problems, SCOPT also contains the codes which can be customized to solve the following constrained convex optimization problem: $$g^{\star} := \min_{\mathbf{x}\in\Omega}g(\mathbf{x}),$$ where $g$ is a convex function with a tractable proximity operator, and $\Omega$ is a nonempty, closed and convex set in $\mathbb{R}^p$ endowed with a self-concordant barrier $f$ (i.e., $f$ is self-concordant with $M_f = 2$ and $\vert\varphi'(t)\vert \leq \sqrt{\nu}\varphi''(t)^{1/2}$ for given $\nu > 0$ and $\varphi(t) := f(\mathbf{x} + t\mathbf{u})$).
Example: The clustering problem in unsupervised learning can be reformulated as the above constrained convex problem using the $\max$-norm: $$\begin{array}{ll} \min_{\mathbf{K},\mathbf{R}, \mathbf{L}}& \Vert\mathrm{vec}(\mathbf{K} - \mathbf{A})\Vert_1 \\ \textrm{s.t.} & \begin{bmatrix}\mathbf{R} & \mathbf{K}\\ \mathbf{K}^T & \mathbf{L}\end{bmatrix}\succeq 0, ~\mathbf{R}_{ii} \leq 1, \mathbf{L}_{ii}\leq 1, ~i=1,\dots, p. \end{array}$$ Reference: Jalali, A., and Srebro, N. Clustering using max-norm constrained optimization. In Proc. of International Conference on Machine Learning (ICML2012) (2012), pp. 1–17.
• REFERENCES
The theory and algorithms implemented in SCOPT can be found in the following papers:
1. Q. Tran-Dinh, A. Kyrillidis, and V. Cevher (2013). A proximal Newton framework for composite minimization: Graph learning without Cholesky decomposition and matrix inversions. Proceedings of the 30th International Conference on Machine Learning (ICML), JMLR W&CP 28(2): 271--279, 2013.
Available at: https://arxiv.org/abs/1301.1459.
Abstract: We propose a variable metric framework for minimizing the sum of a self-concordant function and a possibly non-smooth convex function, endowed with an easily computable proximal operator. We theoretically establish the convergence of our framework without relying on the usual Lipschitz gradient assumption on the smooth part. An important highlight of our work is a new set of analytic step-size selection and correction procedures based on the structure of the problem. We describe concrete algorithmic instances of our framework for several interesting applications and demonstrate them numerically on both synthetic and real data.
2. Q. Tran-Dinh, A. Kyrillidis and V. Cevher. Composite Self-concordant Minimization, Journal of Machine Learning Research, 2014 (to appear).
Available at: arxiv.org/abs/1308.2867
Abstract: We propose a variable metric framework for minimizing the sum of a self-concordant function and a possibly non-smooth convex function, endowed with an easily computable proximal operator. We theoretically establish the convergence of our framework without relying on the usual Lipschitz gradient assumption on the smooth part. An important highlight of our work is a new set of analytic step-size selection and correction procedures based on the structure of the problem. We describe concrete algorithmic instances of our framework for several interesting applications and demonstrate them numerically on both synthetic and real data.
3. Q. Tran-Dinh, A. Kyrillidis and V. Cevher. An inexact proximal path-following algorithm for constrained convex minimization, SIAM J. Optimization, vol. 24, no. 4, pp. 1718--1745, 2014.
Available at: http://arxiv.org/abs/ 1311.1756.
Abstract: Many scientific and engineering applications feature nonsmooth convex minimization problems over convex sets. In this paper, we address an important instance of this broad class where we assume that the nonsmooth objective is equipped with a tractable proximity operator and that the convex constraint set affords a self-concordant barrier. We provide a new joint treatment of proximal and self-concordant barrier concepts and illustrate that such problems can be efficiently solved, without the need of lifting the problem dimensions, as in disciplined convex optimization approach. We propose an inexact path-following algorithmic framework and theoretically characterize the worst-case analytical complexity of this framework when the proximal subproblems are solved inexactly. To show the merits of our framework, we apply its instances to both synthetic and real-world applications, where it shows advantages over standard interior point methods. As a by-product, we describe how our framework can obtain points on the Pareto frontier of regularized problems with self-concordant objectives in a tuning free fashion.
BMISolver: Optimization involving Bilinear Matrix Inequalities
• BMIsolver-v1.1 - a MATLAB software package for BMI optimization
BMIsolver is a MATLAB package for solving optimization problems with BMI (bilinear matrix inequality) constraints arising in feedback controller design (spectral abscissa, H2, H-infinity and mixed H2/Hinf optimization problems). It is implemented based on the methods proposed in [1] and [2] which is a joint work with S. Gummusoy and W. Michiels. The code is developed by Quoc Tran Dinh at Department of Electrical Engineering (ESAT/SCD) and Optimization in Engineering Center (OPTEC), KU Leuven, Belgium (under the supervision of Prof. M. Diehl). The current version requires YALMIP as a modeling language. We recommend SeDuMi as an SDP solver.
References
1. Q. Tran Dinh, S. Gumussoy, W. Michiels and M. Diehl. Combining Convex-Concave Decompositions and Linearization Approaches for solving BMIs, with Application to Static Output Feedback. Tech. Report, July, 2011, [pdf].
2. Q. Tran Dinh, W. Michiels and M. Diehl. An inner convex approximation algorithm for BMI optimization and applications in control. Tech. Report, December, 2011, [pdf].
Software and codes that I use for my research
I have been using many open source as well as commercial software packages for my research. I find that they are useful and valuable. Here are some of them.
• IPOPT - Sparse Nonlinear Optimization (open-source).
• CasADI - A minimalistic computer algebra system with automatic differentiation (open-source), Joel Andersson (for C++ and Python).
• qpOASES - Parametric Quadratic Programming for MPC (open-source), Hans Joachim Ferreau.
Visitors (From 20.07.2013)
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2018-10-18 04:47:01
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https://zbmath.org/?q=an:0849.35068
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# zbMATH — the first resource for mathematics
The semigroup generated by $$2 \times 2$$ conservation laws. (English) Zbl 0849.35068
The Cauchy problem for a strictly hyperbolic $$2\times 2$$ system of conservation laws in one space dimension is considered, assuming that each characteristic field is either linearly degenerate or genuinely nonlinear. A new algorithm is given, based on the wavefront tracking, which yields that the Cauchy sequence of approximate solutions converge to a unique limit, depending continuously on the initial data. The solutions constitute a continuous semigroup, defined on a domain $$D\subset L^1(\mathbb{R}, \mathbb{R}^2)$$.
Reviewer: S.Tersian (Russe)
##### MSC:
35L50 Initial-boundary value problems for first-order hyperbolic systems 35L65 Hyperbolic conservation laws
##### Keywords:
viscosity method; Riemann problem; wavefront tracking
Full Text:
##### References:
[1] A. Bressan, Contractive metrics for nonlinear hyperbolic systems, Indiana Univ. Math. J. 37 (1988), 409-421. · Zbl 0632.35041 · doi:10.1512/iumj.1988.37.37021 [2] A. Bressan, Global solutions of systems of conservation laws by wave-front tracking, J. Math. Anal. Appl. 170 (1992), 414-432. · Zbl 0779.35067 · doi:10.1016/0022-247X(92)90027-B [3] A. Bressan, A contractive metric for systems of conservation laws with coinciding shock and rarefaction waves, J. Diff. Eqs. 106 (1993), 332-366. · Zbl 0802.35095 · doi:10.1006/jdeq.1993.1111 [4] A. Bressan, A locally contractive metric for systems of conservation laws, Ann. Scuola Norm. Sup. Pisa, Serie IV, 22 (1995), 109-135. · Zbl 0867.35060 [5] A. Bressan, The unique limit of the Glimm scheme, Arch. Rational Mech. Anal. 130 (1995), 205-230. · Zbl 0835.35088 · doi:10.1007/BF00392027 [6] M. Crandall, The semigroup approach to first-order quasilinear equations in several space variables, Israel J. Math. 12 (1972), 108-132. · Zbl 0246.35018 · doi:10.1007/BF02764657 [7] C. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl. 38 (1972), 33-41. · Zbl 0233.35014 · doi:10.1016/0022-247X(72)90114-X [8] R. DiPerna, Singularities of solutions of nonlinear hyperbolic systems of conservation laws, Arch. Rational Mech. Anal. 60 (1975), 75-100. · Zbl 0324.35062 · doi:10.1007/BF00281470 [9] R. DiPerna, Global existence of solutions to nonlinear hyperbolic systems of conservation laws, J. Diff. Eqs. 20 (1976), 187-212. · Zbl 0314.58010 · doi:10.1016/0022-0396(76)90102-9 [10] R. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal 82 (1983), 27-70. · Zbl 0519.35054 · doi:10.1007/BF00251724 [11] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697-715. · Zbl 0141.28902 · doi:10.1002/cpa.3160180408 [12] J. Glimm & P. Lax, Decay of solutions of systems of hyperbolic conservation laws, Memoirs Amer. Math. Soc. 101, 1970. · Zbl 0204.11304 [13] S. Kru?kov, First order quasilinear equations with several space variables, Math. USSR Sbornik 10 (1970), 217-243. · Zbl 0215.16203 · doi:10.1070/SM1970v010n02ABEH002156 [14] P. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10 (1957), 537-566. · Zbl 0081.08803 · doi:10.1002/cpa.3160100406 [15] T.-P. Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys. 57 (1977), 135-148. · Zbl 0376.35042 · doi:10.1007/BF01625772 [16] T.-P. Liu, Decay to N-waves of solutions of general systems of nonlinear hyperbolic conservation laws, Comm. Pure. Appl. Math. 30 (1977), 585-610. · Zbl 0357.35059 · doi:10.1002/cpa.3160300505 [17] T.-P. Liu, Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws, Comm. Pure Appl. Math. 30 (1977), 767-796. · Zbl 0358.35014 · doi:10.1002/cpa.3160300605 [18] V. J. Ljapidevskii, On correctness classes for nonlinear hyperbolic systems, Soviet Math. Dokl. 16 (1975), 1505-1509. · Zbl 0329.35042 [19] F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa 5 (1978), 489-507. · Zbl 0399.46022 [20] G. Pimbley, A semigroup for Lagrangian 1D isentropic flow, in Transport theory, invariant imbedding and integral equations, G. Webb ed., Dekker, New York, 1989. · Zbl 0689.76003 [21] N. H. Risebro, A front-tracking alternative to the random choice method, Proc. Amer. Math, Soc. 117 (1993), 1125-1139. · Zbl 0799.35153 · doi:10.1090/S0002-9939-1993-1120511-X [22] B. L. Ro?destvenskii & N. Yanenko, Systems of quasilinear equations, Amer. Math. Soc., 1983. [23] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, 1983. · Zbl 0508.35002 [24] L. Tartar, Compensated compactness and applications to partial differential equations, in Research Notes in Mathematics, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. 4, R. J. Knops, ed., Pitman Press (1979). · Zbl 0437.35004 [25] B. Temple, No L 1-contractive metrics for systems of conservation laws, Trans. Amer. Math. Soc. 288 (1985), 471-480. · Zbl 0568.35065 [26] E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. I, Springer-Verlag, 1993. · Zbl 0794.47033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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2021-04-13 14:48:11
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http://mathhelpforum.com/latex-help/27809-math-symbols-print.html
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# Math Symbols
Printable View
• February 8th 2008, 02:39 PM
lllll
Math Symbols
Is there a post that gives you a list of how to reproduced the various math symbols?
i.e:
\notin = $\notin$
\sum ^{\infty}_{i=1} = $\sum ^{\infty}_{i=1}$
and so on...
• February 8th 2008, 02:49 PM
Plato
Hve you looked at this link? http://www.mathhelpforum.com/math-he...-tutorial.html
You can download the pdf file.
• February 8th 2008, 02:52 PM
Krizalid
The Comprehensive LaTeX Symbol List should help.
(I just saw Plato reply, but I add this just in case.)
• February 8th 2008, 03:46 PM
Plato
Personally I subscribe to the Einstein Doctrine.
Upon one of his first visits to North America, Einstein was asked an eager reporter if he could give the value of pi correct to ten decimal places. Einstein’s response was: “I never remember anything that I can lookup”.
So I use MathType to produce LaTeX code.
MathType: Download TeXaide
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2016-08-28 06:15:59
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https://codereview.stackexchange.com/questions/14049/build-process-for-compiling-client-js-modules-into-one-script
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# Build process for compiling client JS modules into one script
I am currently writing a rather large client-side JS application which is made of of multiple modules, each in different files (before compiling). I am using Node.js to build the final script based on all the individual modules.
For the script to work properly all modules must be added in the proper order, so they can access any modules they depend on. One way to solve this is by maintaining an array of all the modules in the correct order to be inserted. I find that limiting though, having to add each file manually and in the correct position.
I have come up with a different solution to this, and would love to get some feedback on it.
First I have one wrapper file, which is wraped around all the modules. Looks like this:
(function( ns ){
var module = window[ns] = {
_modules: window[ns] && window[ns]._modules || {},
exports: function( name, mod ){
this._modules[name] = mod;
},
require: function( name ){
var mod = this._modules[name];
return mod;
}
};
MODULES_INSERT
})( '_myCoolApp' );
Where it says "MODULES_INSERT" is where the modules get inserted.
Now inside the individual modules it looks something like this:
// export this module
module.exports('view', View);
function View(){ etc.. }
// import other module
var util = module.require('util');
And finally I have my build script (node.js) which adds all the modules/files, and orders them based on any module.requires they contain in their code, ensuring that all dependencies are loaded before they are required:
for(i in srcFiles){
file = srcFiles[i];
content = '(function(){\n' + content;
content = content + '})();\n';
// find any module dependencies
var match, requires = [], regex = /module.require\(['"]([^'"]+)/g;
while(match = regex.exec(content)){
requires.push(match[1]);
}
file.requires = requires;
file.content = content;
}
for(i in srcFiles){
file = srcFiles[i];
recurse(file);
}
function recurse(file){
if(!file) return;
if(file.requires.length > 0) {
file.requires.forEach(function(req){
recurse(srcFiles[req]);
});
}
}
code = results.join('');
code = wrapper.replace('MODULES_INSERT', code);
Is this solution way overkill? One downside I see to this, is what happens when two files require each other, which gets loaded first? Is there a more effective solution for this?
Your approach is sound. The optimizer of RequireJS called r.js also scans the code for calls to methods with a special name (define and require) to detect dependencies.
You can see how this detection works in the file transform.js. As you can see, this optimizer uses a JavaScript parser, which is more reliable than using regular expressions: for example, your regular expression may match a call in a comment, which would be skipped using a parser.
There is another way to declare dependencies for RequireJS though, which is described in the Asynchronous Module Definition specification: wrap each module in a call to a function and list the dependencies explicitly in an array.
You may also be interested in the use of a simpler pattern for the declaration of modules, independently of the loader used.
With regards to circular dependencies ("when two files require each other"), there are two ways to handle them:
• detect them and fail, forcing the developer to avoid them in their code
• load one of the two modules with only parts of their dependencies, which requires extra care from developers: they can no longer be sure that all dependencies are available when the module runs, and must check before calling methods and accessing properties on required objects
Both approaches have drawbacks.
I have an experience with requirejs as a dependency management framework, it allows the compiling on the server side with the following snippet
node r.js -o path/to/buildconfig.js
the config
// **r.js** configuration
({
appDir : "../",
baseUrl: "js",
dir : "../target",
modules: [
{ name: "app" }
],
optimize : "uglify",
optimizeCss: "standard",
findNestedDependencies: true,
skipModuleInsertion: false,
uglify: {
gen_codeOptions : {},
do_toplevel : {},
ast_squeezeOptions: {},
ast_lift_variables: {}
},
paths : {
"jquery" : "../../lib/jquery",
// .. and so on
}
})
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2019-12-14 09:00:11
|
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http://math.stackexchange.com/questions/186308/sum-of-series-sin-x-sin-2x-sin-3x-cdots
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# Sum of series $\sin x + \sin 2x + \sin 3x + \cdots$
Please help me compute the sum of the series: $$\sin(x)+\sin(2x)+\sin(3x)+\cdots$$
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Welcome to stack exchange. please consider adding what you've tried, as well as using TeX formatting next time. – nbubis Aug 24 '12 at 13:44
Why do you think it converges? – GEdgar Aug 24 '12 at 13:44
What happens if $x=0$? – S. Snape Aug 24 '12 at 13:56
At least, it converges when $\quad\large\displaystyle x = n\pi\,,\quad n \in {\mathbb Z}$ – Felix Marin Jun 29 '14 at 1:05
The series does not converge for all $x$. There are some $x$, for instance $x=0$ or $x=\pi$, for which the series converges to $0$, however if we consider $x=\frac \pi 2$ we find that our series is $1+0+-1+0+1+\cdots$ which does not converge.
If you think of the unit circle, imagine a line whose angle from the positive x-axis is the value of $x$. Then the x-coordinate of this point on the unit circle is the value of $\sin x$. Doubling the angle yields a point whose x-coordinate is $\sin 2x$. Tripling it yields a point whose x-coordinate is $\sin 3x$. Continuing this, you can see that the pattern will continue with varied positive and negative values, not approaching any particular limit unless the line representing an angle of $x$ was aligned with the positive or negative x-axis. (This is not rigorous, but can be made to be so.)
More technically, we have that $\lim_{n\to\infty} \sin nx =0$ iff $x=k\pi$ for some $k\in\Bbb Z$, so the series trivially converges to zero for such $x$ and diverges for all other $x$.
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$$2\sin\frac{x}{2}\sin rx=\cos\frac{(2r-1)}{2}x-\cos\frac{(2r+1)}{2}x$$
Putting $r=1,2,\ldots,n-1,n$ we get,
$$2\sin\frac{x}{2}\sin x=\cos\frac{1}{2}x-\cos\frac{3}{2}x$$
$$2\sin\frac{x}{2}\sin 2x=\cos\frac{3}{2}x-\cos\frac{5}{2}x$$
$$\vdots$$
$$2\sin\frac{x}{2}\sin rx=\cos\frac{(2n-3)}{2}x-\cos\frac{(2n-1)}{2}x$$
$$2\sin\frac{x}{2}\sin nx=\cos\frac{(2n-1)}{2}x-\cos\frac{(2n+1)}{2}x$$
Adding we get, $2\sin\frac{x}{2}(\sin x+\sin 2x+...+\sin nx)=cos\frac{1}{2}x-\cos\frac{(2n+1)}{2}x=2\sin\frac{(n+1)x}{2}\sin\frac{nx}{2}$
So, $$\sin x+\sin 2x+\cdots+\sin nx=\frac{\sin\frac{(n+1)x}{2}\sin\frac{nx}{2}}{\sin\frac{x}{2}}$$
As $2\sin B\sin(A+2rB) = \cos(A+(2r-1)B) - \cos(A+(2r+1)B)$, we need to multiply with $2\sin B$ for $\sum_{r}\sin(A+2rB)$.
Here in this problem, $A=0, 2B=x$
Also the way Nerd-Herd has approached the problem,
$\sin rx$ = Imaginary part of $e^{irx}$
So, $\sum_{0 ≤ r ≤n}\sin rx=\sum_{0 ≤ r ≤n}$(Imaginary part of $e^{ix})$=Imaginary part of($\sum_{0 ≤ r ≤n}e^{ix}$)
Now, $$\sum_{0 ≤ r ≤n}e^{irx}= \frac{e^{(n+1)ix}-1}{e^{ix}-1}= e^{\frac{(n+1)ix}{2}}\frac{(e^{\frac{(n+1)ix}{2}}-e^{-\frac{(n+1)ix}{2}})}{e^{\frac{ix}{2}}(e^{\frac{ix}{2}}-e^{-\frac{ix}{2}})}$$
We know, $\sin y=\frac{e^{iy}-e^{-iy}}{2i}$ So,$e^{iy}-e^{-iy}=2i\sin y$
So,$\sum_{0 ≤ r ≤n}e^{irx}=e^{\frac{inx}{2}}\frac{2i\sin\frac{(n+1)x}{2}}{2i\sin\frac{x}{2}}=(\cos \frac{nx}{2}+i\sin \frac{nx}{2})\frac{\sin\frac{(n+1)x}{2}}{\sin\frac{x}{2}}$
So, imaginary part of $\sum_{0 ≤ r ≤n}e^{irx}=\sin \frac{nx}{2}\frac{\sin\frac{(n+1)x}{2}}{\sin\frac{x}{2}}$
$\sum_{0 ≤ r ≤n}\sin{rx}=\sin \frac{nx}{2}\frac{\sin\frac{(n+1)x}{2}}{\sin\frac{x}{2}}$
$\sum_{1 ≤ r ≤n}\sin{rx}=\sin \frac{nx}{2}\frac{\sin\frac{(n+1)x}{2}}{\sin\frac{x}{2}}$ as $\sin(0x)=0$
So, the either approach leads to a compact from provided $\sin\frac{x}{2}≠0$
If $\sin\frac{x}{2}=0$ i.e., $\frac{x}{2}=m\pi$ where m is some integer,
$=>x=2m\pi=>\sin sx= 0$ for any integer s.
By observation if $\sin x=0$ i.e., $x=m\pi$ where m is some integer, $\sin sx= 0$ for any integer s.
Then the sum is clearly 0 if $x=m\pi$.
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Using complex number system:
$$\sin{x} = \text{Im} ( {e^{ix} )}$$ and since,
$$\text{Im} (z_2) + \text{Im} (z_2) = \text{Im} (z_1 + z_2)$$
Using this, you get a geometric progression. Which gives the result as
$$\text{Im} (\frac{e^{ix}}{1-e^{ix}})$$
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Watch out: you just summed a geometric series whose ratio has modulus 1. This cannot be done. – Did Aug 24 '12 at 14:23
@sos440 If the series converges, it'll converge to the same value as given by $$\text{Im} (\frac{e^{ix}}{1-e^{ix}})$$ and if it doesn't converge; then it just doesn't. – hjpotter92 Aug 24 '12 at 14:27
@Nerd-Herd : your sum is wrong. Geometric progression is summable iif the ratio has modulus $<1$ which is not the case since $|e^{iz}|=1$. – vanna Aug 24 '12 at 14:30
@Nerd-Herd, that exactly means that it does not converge in ordinary sense. But there are many other generalized ways to assign a value to a formal series (indeed, ordinary summation sense is also a way of assigning a value to a formal series by defining its value as the limit of the partial sum), and some of them actually yield $$\sum_{n=1}^{\infty} \sin nx = \Im \left(\frac{e^{ix}}{1-e^{ix}}\right) = \frac{1}{2}\cot\frac{x}{2},$$including Cesaro summation sense and Abel summation sense. – Sangchul Lee Aug 24 '12 at 14:31
Nerd-Herd: If the series converges, I am Julius Caesar. – Did Aug 24 '12 at 15:15
The series converges but not in the usual sense if $x$ is real. If we assume that $x$ is real, then one might be tempted to write $$\sum_{n = 1}^\infty \sin nx \color{red}{=} \textbf{I}\bigg[\sum_{n = 1}^\infty e^{inx}\bigg] \color{red}{=} \textbf{I}\bigg[\frac{e^{ix}}{1 - e^{ix}}\bigg] \color{red}{=} \frac{\sin x}{2(\cos x - 1)}$$ and, similarly, $$\sum_{n = 1}^\infty \cos nx \color{red}{=} \textbf{R}\bigg[\sum_{n = 1}^\infty e^{inx}\bigg] \color{red}{=} \textbf{R}\bigg[\frac{e^{ix}}{1 - e^{ix}}\bigg] \color{red}{=} -\frac{1}{2}.$$
To find the real and imaginary parts multiply and divide $e^{ix}/(1 - e^{ix})$ by $1 - e^{-ix}$ and used Euler's formulae $e^{ix} = \cos x + i\sin x$ and $e^{ix} + e^{-ix} = 2\cos x$.
But $|e^{ix}| = 1$ if $x$ is real, so $\sum_{n = 1}^\infty e^{inx}$ diverges and the $\color{red}{\text{red}}$ equalities are metaphoric because we are assigning a value to each series. The value of a series is, in general, different from its sum.
However, the series $\sum_{n = 1}^\infty e^{inx}$ converges if $x$ is in the upper half-plane, that is, $\textbf{I}[x] > 0$, but it does not help because then $$\sum_{n = 1}^\infty \sin nx \ne \textbf{I}\bigg[\sum_{n = 1}^\infty e^{inx}\bigg].$$
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Can the real part of a holomorphic function be constant? I don't think $\frac{-1}2$ is correct for the second sum. – Thomas Andrews Jun 28 '14 at 23:39
There is much more easier way to prove that the series does not converge when $x \ne k\pi$.
$\liminf\limits_{n\to\infty}\sin nx=-1$
The corresponding subsequence is $n_k=-\frac{\pi -4\pi k}{2x}$
$\limsup\limits_{n\to\infty}\sin nx=1$
The corresponding subsequence is $n_k=\frac{\pi + 4\pi k}{2x}$
Therefore there is no limit of $\sin nx$ and hence a necessary condition of convergence is violated.
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You really mean a necessary condition, not the necessary condition. A necessary condition for a series to converge is that the limit of the terms converges to zero. – 6005 Jun 29 '14 at 4:46
Thank you for your comment. I've changed my answer. – Тимофей Ломоносов Jun 29 '14 at 5:01
Except this does not prove the liminf is -1 and the limsup is +1 since your choices of n_k are illegal most of the time (there is no reason that the RHS would be integers). – Did Jun 29 '14 at 6:55
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2016-02-14 04:03:31
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https://community.spiceworks.com/topic/2258846-powershell-script-to-compare-file-numbers-on-orders-to-docs-in-the-same-file
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### 38 Replies
• Please share what you have tried so far?
We’re happy to help but not a script writing service.
• Off the top of my head you'll need to use get-childitem and some string manipulation to do this. Something like the following not tested
Powershell
$CSVs = Get-childitem -Path C:\Path\to\csvs\*.csv #gets all csv files in the directory, use -recurse inf there are sub folders but code will need to be adjusted for that ForEach($csv in $CSVs){$PDFs = Get-childitem -Path C:\Path\to\PDFs
$Filenums = Import-CSV$csv.Fullname
$PDFNames = PDFs.Name.Subtsring(0,6) ForEach($Filenum in $Filenums){ If(PDFNames.contains($Filenum.Column1name){
}
Else{
$Msg =$csv.Name + " is missing PDF's"
$Msg | Out-File C:\path\to \log.log -Append continue } }$Msg = $csv.Name + " All PDF's present files moved"$Msg | Out-File C:\path\to \log.log -Append
Move-Item $csv.FullName C:\Path\to\ftp\staging\$csv.Name
ForEach($PDF in$PDFs){
Move-Item $PDF.FullName C:\Path\to\ftp\staging\$PDF.Name
}
}
• Fixed an issue in my sample code so that it actually skips moving files if there are missing ones
Powershell
$CSVs = Get-childitem -Path C:\Path\to\csvs\*.csv #gets all csv files in the directory, use -recurse inf there are sub folders but code will need to be adjusted for that ForEach($csv in $CSVs){$PDFs = Get-childitem -Path C:\Path\to\PDFs
$Filenums = Import-CSV$csv.Fullname
$PDFNames = PDFs.Name.Subtsring(0,6) ForEach($Filenum in $Filenums){ If($missing){$missing =$false;continue}
If(PDFNames.contains($Filenum.Column1name){ } Else{$Msg = $csv.Name + " is missing PDF's"$Msg | Out-File C:\path\to \log.log -Append
$missing =$true
continue
}
}
$Msg =$csv.Name + " All PDF's present files moved"
$Msg | Out-File C:\path\to \log.log -Append Move-Item$csv.FullName C:\Path\to\ftp\staging\$csv.Name ForEach($PDF in $PDFs){ Move-Item$PDF.FullName C:\Path\to\ftp\staging\$PDF.Name } } 1 found this helpful thumb_up thumb_down • Thanks for the query. Can you share what you have done so far - it's easier if we know what you have tried and are having trouble with. Was this post helpful? thumb_up thumb_down • Hi! Thank you for responding. Alex3031 I'm using your suggested code, but I'm running into an error: PDFNames.contains : The term 'PDFNames.contains' is not recognized as the name of a cmdlet, function, script file, or operable program. Check the spelling of the name, or if a path was included, verify that the path is correct and try again. At C:\NewOrders.ps1:9 char:12 + If(PDFNames.contains($Filenum.Column1name)){
+ ~~~~~~~~~~~~~~~~~
+ CategoryInfo : ObjectNotFound: (PDFNames.contains:String) [], CommandNotFoundException
+ FullyQualifiedErrorId : CommandNotFoundException
Here is the code I have so far:
$CSVs = Get-childitem -Path C:\Orders\NEW_ORDERS\*.csv #gets all csv files in the directory, use -recurse inf there are sub folders but code will need to be adjusted for that ForEach($csv in $CSVs){$PDFs = Get-childitem -Path C:\Orders\NEW_ORDERS\*.pdf
$Filenums = Import-CSV$csv.Fullname
$PDFNames = PDFs.Name.Subtsring(0,6) ForEach($Filenum in $Filenums){ If($missing){$missing =$false;continue}
If(PDFNames.contains($Filenum.Column1name)){ } Else{$Msg = $csv.Name + " is missing PDF's"$Msg | Out-File C:\Orders.log -Append
$missing =$true
continue
}
}
$Msg =$csv.Name + " All PDF's present files moved"
$Msg | Out-File C:\Orders\Orders.log -Append Move-Item$csv.FullName C:\Orders\Orders\NEXT_TO_UPLOAD\$csv.Name ForEach($PDF in $PDFs){ Move-Item$PDF.FullName C:\Orders\NEXT_TO_UPLOAD\$PDF.Name } Was this post helpful? thumb_up thumb_down • looks like he missed a$ to make it a variable
Powershell
If(PDFNames.contains($Filenum.Column1name){ # vs If($PDFNames.contains($Filenum.Column1name)){ 2 found this helpful thumb_up thumb_down • Also don't forget to replace column1name with the column header in the csv 1 found this helpful thumb_up thumb_down • Neally got there first! Another case where a good ISE would have helped. :-) • local_offer Tagged Items • Neally 1 found this helpful thumb_up thumb_down • Thanks! Now I get an error: You cannot call a method on a null-valued expression. At C:\NewOrders.ps1:9 char:12 + If($PDFNames.contains($Filenum.ClientFileNumber)){ + ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + CategoryInfo : InvalidOperation: (:) [], RuntimeException + FullyQualifiedErrorId : InvokeMethodOnNull Was this post helpful? thumb_up thumb_down • Is "ClientFileNumber" a column in the CSV? Does$Filenum have a value?
How about $Filenums ? did you import the CSV(s) correctly? Was this post helpful? thumb_up thumb_down • Hi Neally, "ClientFileNumber" is the first column in the CSV.$CSVs is loading correctly, shows filenames of all the .csv's in the file.
$PDFs shows filenames of all the .pdf's in the file.$Filenums shows all data from all the .csv's row by row
$PDFNames does not have a value. Thank you for your help on this. -Obvious Noob Was this post helpful? thumb_up thumb_down • user981752 wrote:$PDFNames does not have a value.
same issue, there is a $missing in front of it. Powershell $PDFNames = PDFs.Name.Subtsring(0,6)
# vs
$PDFNames =$PDFs.Name.Substring(0,6)
• I corrected:
$PDFNames =$PDFs.Name.Subtsring(0,6)
but still get this:
You cannot call a method on a null-valued expression.
At C:\NewOrders.ps1:9 char:12
+ If($PDFNames.contains($Filenum.ClientFileNumber)){
+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+ CategoryInfo : InvalidOperation: (:) [], RuntimeException
+ FullyQualifiedErrorId : InvokeMethodOnNull
• you still have a typo... substring
• run the following 2 lines of code in a powershell window manually
Powershell
$PDFs = Get-childitem -Path C:\Orders\NEW_ORDERS\*.pdf$PDFNames = PDFs.Name.Subtsring(0,6)
Then run
Powershell
PDFNames | gm
Do you see a property in that list called Name?
Also try
Powershell
PDFNames.count
What do you get for output?
• user981752 wrote:
I corrected:
$PDFNames =$PDFs.Name.Subtsring(0,6)
but still get this:
You cannot call a method on a null-valued expression.
At C:\NewOrders.ps1:9 char:12
+ If($PDFNames.contains($Filenum.ClientFileNumber)){
+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+ CategoryInfo : InvalidOperation: (:) [], RuntimeException
+ FullyQualifiedErrorId : InvokeMethodOnNull
Think I might have typod the bolded above should be substring not subtsring
• Here's what I get:
Method invocation failed because [System.String] does not contain a method named 'Subtsring'.
At line:2 char:1
+ $PDFNames =$PDFs.Name.Subtsring(0,6)
+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+ CategoryInfo : InvalidOperation: (:) [], RuntimeException
+ FullyQualifiedErrorId : MethodNotFound
• Alex3031 wrote:
run the following 2 lines of code in a powershell window manually
Powershell
$PDFs = Get-childitem -Path C:\Orders\NEW_ORDERS\*.pdf$PDFNames = PDFs.Name.Subtsring(0,6)
Then run
Powershell
PDFNames | gm
Do you see a property in that list called Name?
Also try
Powershell
PDFNames.count
What do you get for output?
You're missing half the $signs, if OP runs it like this verbatim, he will keep getting more errors, or no results Was this post helpful? thumb_up thumb_down • See my above I noticed a typo, should be substring not subtsring Was this post helpful? thumb_up thumb_down • ugh! Spelled substring incorrectly! Was this post helpful? thumb_up thumb_down • True I am forgetting to add the$ but I think the issue with $PDFNames being null is the typo in substring for the method call. Of course I just threw it together originally and didn't claim I tested any of it, in fact said the opposite.. Was this post helpful? thumb_up thumb_down • Something seems a little off to me about the logic looking at it now too. I think the following Powershell If($missing){$missing =$false;continue}
Needs to be moved just below the For Filenum loop so that it doesn't move the files if stuff is missing
• The issue with invalid path are there any non allowed characters in file names? You could add
Powershell
Write-host $PDF.fullname Read-host And run it as a test see what the file path names are 1 found this helpful thumb_up thumb_down • Progress! I'm now getting: Move-Item : The given path's format is not supported. At C:\NewOrders.ps1:22 char:9 + Move-Item$PDF.FullName C:\Orders\NEXT_TO_UPL ...
+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+ CategoryInfo : NotSpecified: (:) [Move-Item], NotSupportedException
+ FullyQualifiedErrorId : System.NotSupportedException,Microsoft.PowerShell.Commands.MoveItemCommand
Text
$CSVs = Get-childitem -Path C:\Orders\NEW_ORDERS\*.csv #gets all csv files in the directory, use -recurse inf there are sub folders but code will need to be adjusted for thatForEach($CSV in $CSVs){$PDFs = Get-childitem -Path C:\\Orders\NEW_ORDERS\*.pdf $Filenums = Import-CSV$csv.Fullname $PDFNames =$PDFs.Name.SubString(0,6) ForEach($Filenum in$Filenums){ } If($PDFNames.contains($Filenum.ClientFileNumber)){ } Else{ $Msg =$csv.Name + " is missing PDF's" $Msg | Out-File C:\Orders.log -Append$missing = $true continue }{ If($missing){$missing =$false;continue} } $Msg =$csv.Name + " All PDF's present files moved" $Msg | Out-File C:\Orders\Orders.log -Append Move-Item$csv.FullName C:\Orders\Orders\NEXT_TO_UPLOAD\$csv.Name ForEach($PDF in $PDFs){ Move-Item$PDF.FullName C:\Orders\NEXT_TO_UPLOAD\$PDF.Name } } Was this post helpful? thumb_up thumb_down • See my above post usually it's a bad path 1 found this helpful thumb_up thumb_down • Could try building the output path adding Powershell $dest = "C:\Path\to\new\orders\" + $PDF.Name move-item$PDF.FullName $dest 1 found this helpful thumb_up thumb_down • user981752 wrote: Here's what I get: Method invocation failed because [System.String] does not contain a method named 'Subtsring'. At line:2 char:1 +$PDFNames = $PDFs.Name.Subtsring(0,6) + ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + CategoryInfo : InvalidOperation: (:) [], RuntimeException + FullyQualifiedErrorId : MethodNotFound Please: learn to read the error message. That line had an error. The error message says that a string does not contain a method named 'Subtsring'. There is a good reason for that - which is another typo. PLEASE - do yourself a favour and learn to use a good IDE - such as Visual Studio Code. Tab expansion really is your friend. Was this post helpful? thumb_up thumb_down • tfl wrote: user981752 wrote: Here's what I get: Method invocation failed because [System.String] does not contain a method named 'Subtsring'. At line:2 char:1 +$PDFNames = $PDFs.Name.Subtsring(0,6) + ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + CategoryInfo : InvalidOperation: (:) [], RuntimeException + FullyQualifiedErrorId : MethodNotFound Please: learn to read the error message. That line had an error. The error message says that a string does not contain a method named 'Subtsring'. There is a good reason for that - which is another typo. PLEASE - do yourself a favour and learn to use a good IDE - such as Visual Studio Code. Tab expansion really is your friend. That was my typo actually, although I literally threw it together in the reply window on Spiceworks no testing, no validation, nothing, meant to be a reference suggestion. Was this post helpful? thumb_up thumb_down • Alex3031 - no worries. :-) in case that wasn't obvious. And my typing sucks too. At the same time, there are a couple of key takeaways though for the OP and others. 1. A good development tool can help these problems largely go away. Things like tab completion just help you to type the right method name in the first place. PowerShell 7's tab completion is, imho, even better. If you are writing production code, using tools like VS Code should be automatic. As a PowerShell user, I loved the ISE, which is still useful if you are remaining on Windows PowerShell, but VS Code really is a lot better. Like any rich tool, there is a learning curve, but moving from the ISE to VS code is not that difficult. And the Added features are well worth the trouble. Tab Completion has been a god send for me. As is the ability to work with a GitHub Repository (and all that git stuff) and Markdown - all in one tool. Sweet. 2. The error messages in Windows PowerShell are deliberate. In that sea of 'red ink', there is some very very useful information. if you are familiar with the terminology (eg what is a method call), but once you actually read the messages, I've always found them, by and large, pretty clear. Now there is a lot of extra information (eg the category, etc) that is really not actionable and gets in the way a bit. That's why in PowerShell 7, error display is SO much better. Like this: Text PS C:\foo>$string = 'PowerShell Rocks!!!'
PS C:\foo> $string.subsringte(1,2) InvalidOperation: Method invocation failed because [System.String] does not contain a method named 'subsringte'. A lot easier to read, imho. 1 found this helpful thumb_up thumb_down • I'm definitely making progress - thank you! At this point - it is running, and moving files - but moving the wrong files? It matches 1 .csv with 1 .pdf. but they are not matching file numbers. Any ideas? Powershell $CSVs = Get-childitem -Path C:\Orders\NEW_ORDERS\*.csv #gets all csv files in the directory, use -recurse inf there are sub folders but code will need to be adjusted for that
ForEach($CSV in$CSVs){
$PDFs = Get-childitem -Path C:\Orders\NEW_ORDERS\*.pdf$Filenums = Import-CSV $csv.Fullname$PDFNames = $PDFs.Name.Substring(0,6) ForEach($Filenum in $Filenums){ } If($PDFNames-contains($Filenum.ClientFileNumber)){ } Else{$Msg = $csv.Name + " is missing PDF's"$Msg | Out-File C:\Orders.log -Append
$missing =$true
continue
}
If($missing){$missing = $false;continue} }$Msg = $csv.Name + " All PDF's present files moved"$Msg | Out-File C:\Orders.log -Append
$Destination = "C:\Orders\NEXT_TO_UPLOAD\" Move-Item$csv.FullName $Destination Move-Item$PDF.FullName $Destination Was this post helpful? thumb_up thumb_down • ($Filenum.ClientFileNumber) just shows me the headings of the .csv.
• Thinking step by step, I want the script to look at each file number in the .csv file ClientFileNumber column and try to match it with the (first 6 digits) of the .pdf's in the New_Orders file. If all the rows of the .csv match .pdfs , move the .csv and it's pdf's to the NEXT_TO_UPLOAD subfolder. If all .csv are NOT found, leave that .csv and all the .pdfs that are on the list in the New_Orders file.
I'm thinking that the $Filenum.ClientFileNumber is not giving me the correct data because it only returns the headings of the .csv files, not the list of file numbers that should be in there. Maybe there's a better way to specify that before I try to -match or -contains (not sure which of those I should use either). Maybe I"m not looping it correctly? Can it try to match all the file numbers in the .csv with .pdf's in one step, or does it need to go through each row one at a time? Was this post helpful? thumb_up thumb_down • So far you've only worked with what Alex provided you, have you tried to work on it yourself? Was this post helpful? thumb_up thumb_down • Yes. Was this post helpful? thumb_up thumb_down • DarcyB wrote: Yes. well, share what you have tried and we'll try to help Was this post helpful? thumb_up thumb_down • I changed$Filenum.ClientFileNumber to $Filenums.ClientFileNumber and now I am getting a list of the file numbers I am looking for. Was this post helpful? thumb_up thumb_down • Currently trying: Powershell $CSVs = Get-childitem -Path C:\Orders\NEW_ORDERS\*.csv
ForEach($CSV in$CSVs){
$PDFs = Get-childitem -Path C:\Orders\NEW_ORDERS\*.pdf$Filenums = Import-CSV $csv.FullName$PDFNames = $PDFs.Name.Substring(0,6) ForEach($Filenum in $Filenums){ } If($PDFNames-contains $Filenums.ClientFileNumber){ #label: files to move?$Destination = "C:\Orders\NEXT_TO_UPLOAD\"
Move-Item $csv.FullName$Destination
Move-Item $PDF.FullName$Destination
continue}
• $Filenums is an array of the file numbers the Powershell ForEach($Filenum in $Filenums){ } Code iterates through each$Filenum.ClientFileNumber will go through each number in the array. If you want to check this yourself you could do $FileNums[1].ClientFileNumber and you would see the first file number This line Powershell If($PDFNames-contains $Filenums.ClientFileNumber) should be Powershell If($PDFNames.contains($Filenum.ClientFileNumber)) inside the loop What this does is check to see if the file number is present in the first 6 digits pulled from the list of PDFs. It returns true if the value of$Filenum.ClientFileNumber is present in the list of PDF file numbers
Although I think you may have found a shortcut, not sure but wording it the way you have in your last post searching for all the filenumbers at once might would and would be faster than a loop.
As far as what to move the logic might be off there. Probably could do a for loop using the FileNumbers once you have verified they are all present, and basically call a command like
Powershell
ForEach($filenum in$filenumbers){
$File =$filenum.ClientFileNumber
move-item C:\path\of\pdfs\\$File*.pdf C:\Newlocation\
}
That should move any PDF where the first 6 digits match the ClientFileNumber
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2022-08-18 23:25:27
|
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|
https://www.homebuiltairplanes.com/forums/threads/what-engine-would-you-build.31631/page-2
|
# What engine would YOU build
### Help Support HomeBuiltAirplanes.com:
#### poormansairforce
##### Well-Known Member
Interesting, thats one to turbo if any. Wonder if they would sell the parts?
That's about the HP/displacement expected for modern water-cooled engines at 5500-6000 RPM. About 12-13cc/HP. Maybe if we just have a simple 2 valve cylinder it might be less.
It seems steep at 130%+ VE but 60 hp seems doable. The 912 is around 13cc I think. Each section would have its own camshaft?
#### blane.c
##### Well-Known Member
HBA Supporter
Interesting, thats one to turbo if any. Wonder if they would sell the parts?
There website says no sell parts, send them engine they build it and send it back. I imagine they could procure an engine for you to build though.
#### poormansairforce
##### Well-Known Member
There website says no sell parts, send them engine they build it and send it back. I imagine they could procure an engine for you to build though.
Not for $9200 #### blane.c ##### Well-Known Member HBA Supporter That's what I was thinking. I want to get a hold of there mail for a second and see were those parts are coming from for sure. #### pictsidhe ##### Well-Known Member One of the more interesting cars I swapped engines in I tuned for more torque. I could let the clutch out at idle and it would move along at a walk. Hit the loud pedal and it would spin its wheels until around 35mph when it was time to change gear. That was a whopping 1108cc and maybe 50hp in a small 1990 car. One person borrowed and brought it back telling me it was an animal the way it responded to the throttle. It frightened his wife who usually drove a 2L car. On twisty back roads, it felt and moved very quickly. Straights, utterly hopeless. I could have given it more power instead of torque over a wide range, but it would have been a lot harder to drive quickly. Modern engines have higher specific torque than older ones. But they often have lighter flywheels to help acceleration, that makes them unhappy at lower revs. There is a lot more to how cars feel than raw hp and torque numbers. Fortunately, we don't need to bother with a wide torque range in planes and can shoot for high hp a specific RPM with no ill effects if the engine stays together. #### pictsidhe ##### Well-Known Member 86ftlb from 993cc without tuned intakes? Somebody's dyno needs recalibrating... #### poormansairforce ##### Well-Known Member 86ftlb from 993cc without tuned intakes? Somebody's dyno needs recalibrating... All dynos need recalibrated! I just want the parts. #### Dana ##### Super Moderator Staff member The old straight eights might be remembered fondly but they are inferior in almost every way to comparably sized modern engines... Yup. Just like radials. But oh, that wonderful noise! How about a radial with seven or nine VW cylinders, should be able to make 105 or 135HP or so. #### Vigilant1 ##### Well-Known Member Lifetime Supporter I almost forgot: Aviation-specific cylinder heads for the VW. The best bang-for-the buck endeavor (IMO) for the US EAB community. The existing VW heads are good if you treat them right (temps! valve adjustments!), but the engine would benefit from heads purpose-built for aircraft use. Features: -- Generous fins that are free of flash so the air really goes through the passages, especially near the exhaust valve -- Valves that are smaller than those in the high-rpm heads used by the street racers. At our direct-drive 3600 RPM, there's no need for giant valves, and smaller ones leave more meat in the heats (so fewer cracks) -- A smart location for the second plug so it can be easily changed AND is far enough away from the valves so it doesn't promote cracks -- TimeSerts or other steel inserts inserted into the heads for the spark plugs to screw into. No more stripped plug holes. -- A dedicated spot for the CHT thermocouple. There are several fine places on any of he existing heads, but there's no standardized agreeement on where to put them. Some folks put a ring under the spark plug, some folks tap a hole in a fin, some folks mount them under one of the studs--and al result in deiiferent readings. When Joe says his CHTs never get hotter than 350F, you need to ask him where te sensor is mounted. A tapped spot that all could use would help fix this. -- For the smaller aviation VWs (1835cc and below?) -- all the above features, but with single port heads (for better charge velocity, VE, and torque at airplane RPMs) Again, some of the existing heads work well, but these heads would do things a little better. Oh--and sell them for$100 each!
On the modular opposed water-cooled engine idea:
The 912 is around 13cc I think. Each section would have its own camshaft?
I think a single common camshaft for all the sections--pushrods to the 2 valves per cylinder. So, you'd need 6 different camshafts to cover 2,4,6,8,10,12 cylinder variants. >Perhaps< there would be a way to have a single camshaft that was one section long that could link to the next one , but I'd guess that would be troubleprone.
86ftlb from 993cc without tuned intakes? Somebody's dyno needs recalibrating...
Nope, that dyno is calibrated just the way they like it, thanks.
Last edited:
#### Vigilant1
##### Well-Known Member
Yep, 61 cu inches to produce 75 HP = 1.23HP per cu inch. Wow, those "engineers" at Lycoming and Continental who can only make 0.5 (reliable) HP per cubic inch must be idiots. Those O-200's should be making 246 reliable HP, hour after hour. Imbeciles!
#### blane.c
##### Well-Known Member
HBA Supporter
I was looking at the dyno chart in the 3600rpm range to see what might be realistic. Also the parts are eye candy.
#### pictsidhe
##### Well-Known Member
Yep, 61 cu inches to produce 75 HP = 1.23HP per cu inch. Wow, those "engineers" at Lycoming and Continental who can only make 0.5 (reliable) HP per cubic inch must be idiots. Those O-200's should be making 246 reliable HP, hour after hour. Imbeciles!
That amount of torque is fanciful even for brief runs.
#### poormansairforce
##### Well-Known Member
That amount of torque is fanciful even for brief runs.
I was looking for a CR and didn't see one?
#### Pops
##### Well-Known Member
Log Member
The old straight eights might be remembered fondly but they are inferior in almost every way to comparably sized modern engines. A 6L L96 puts out 340 lb./ft. at 2000 rpm, Packard 6L 330 at 2200. The L84 puts out 140 more hp on the top end and an LS3 (6.2L) puts out more than double the hp of the Packard (can do twice the amount of work per unit time and probably 3 times with proper gearing). The LS engines weigh about half of what these old straight 8s weigh. Fuel economy, longevity, emissions are all vastly superior. This is why the LS engines are looked at for aircraft. A 5L Ford Coyote puts out 400 lb./ft. at 3850 rpm. Dodge and Toyota 5.4L V 8s exceed 330 lb./ft at 2000 rpm as well.
The modern V6 turbo engines like the Ecoboost have staggering torque at low rpm- over 400 lb./ft. from 2000 rpm all the way to 4500 rpm- from only 3.5L and run on 87 octane. Rather impressive.
I am not talking about fuel economy, longevity, emissions. Just very low rpm torque performance in high gear on a steep hill.
I have never driven any of today's engines that will go down to 10-15 mph in high gear on a steep hill and when you shove the gas peddle down and accelerate as smoothly while picking up speed as those old heavy iron 8's.
My father and his 3 business pardners all drove Cords. I have driven the 37 Cord Hollywood Graham but wasn't allowed to drive the 37 Cord Conv with the Super Charged engine.
Guess I'm old fashion because I like old 1930's cars and round engine airplanes.
#### Hot Wings
##### Grumpy Cynic
HBA Supporter
Log Member
I almost forgot: Aviation-specific cylinder heads for the VW.
<< >>
Oh--and sell them for $100 each! Something like this that I was working on for a 1/2 VW? Not$100 though - about $4X each. 2 cyl assembly.PDF Will see if the new site will take a 3D PDF. The old one wouldn't. #### Attachments • 8.6 MB Views: 29 #### Geraldc ##### Well-Known Member If I could take an off the shelf motor and throw money at it I would get a small alloy block 4 cylinder motor and make an aviation specific head for it. 2 spark plug holes per cylinder. 2 valves per cylinder. Reduction drive shaft running at 2 to 1 with bearings at front and rear to carry prop loads. Secondary function of shaft is to activate valves. For a 3 cylinder engine some balance factor built into shaft. Now for the hard part.There has to be a drive system either belt or gear with some cushioning to drive the secondary shaft. #### Pops ##### Well-Known Member Log Member I almost forgot: Aviation-specific cylinder heads for the VW. The best bang-for-the buck endeavor (IMO) for the US EAB community. The existing VW heads are good if you treat them right (temps! valve adjustments!), but the engine would benefit from heads purpose-built for aircraft use. Features: -- Generous fins that are free of flash so the air really goes through the passages, especially near the exhaust valve -- Valves that are smaller than those in the high-rpm heads used by the street racers. At our direct-drive 3600 RPM, there's no need for giant valves, and smaller ones leave more meat in the heats (so fewer cracks) -- A smart location for the second plug so it can be easily changed AND is far enough away from the valves so it doesn't promote cracks -- TimeSerts or other steel inserts inserted into the heads for the spark plugs to screw into. No more stripped plug holes. -- A dedicated spot for the CHT thermocouple. There are several fine places on any of he existing heads, but there's no standardized agreeement on where to put them. Some folks put a ring under the spark plug, some folks tap a hole in a fin, some folks mount them under one of the studs--and al result in deiiferent readings. When Joe says his CHTs never get hotter than 350F, you need to ask him where te sensor is mounted. A tapped spot that all could use would help fix this. -- For the smaller aviation VWs (1835cc and below?) -- all the above features, but with single port heads (for better charge velocity, VE, and torque at airplane RPMs) Again, some of the existing heads work well, but these heads would do things a little better. Oh--and sell them for$100 each!
On the modular opposed water-cooled engine idea:
I think a single common camshaft for all the sections--pushrods to the 2 valves per cylinder. So, you'd need 6 different camshafts to cover 2,4,6,8,10,12 cylinder variants. >Perhaps< there would be a way to have a single camshaft that was one section long that could link to the next one , but I'd guess that would be troubleprone.
Nope, that dyno is calibrated just the way they like it, thanks.
Agree 100% on the VW heads. Also could start getting a more HP with the heads taking care of the extra heat. Then the limit would be the mag case. A well balance engine does help on the case life. The aluminum case is 17 pounds heavier.
#### pfarber
##### Well-Known Member
HBA Supporter
Reduction drive shaft running at 2 to 1 with bearings at front and rear to carry prop loads.
The LV3 is designed with a balance shaft and driven off the crank via chain. Its supported by bearings at both ends.
Gotta wonder of those cast in bearing supports could be used, with a third thrust bearing and a support assembly bolted to the rear of the engine.. hmmmm
#### revkev6
##### Well-Known Member
I almost forgot: Aviation-specific cylinder heads for the VW.
Nope, that dyno is calibrated just the way they like it, thanks.
wasn't there someone in south america I think that built and was testing a set of heads they made specifically for aircraft use?? did a quick google search but cant seem to find it. this would have been a couple years back.
they do list "gross" hp.. which is amusing given that they are comparing to SAE hp.... which is again different than the automotive ratings....
|
2020-06-07 09:01:06
|
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|
https://tex.stackexchange.com/questions?sort=newest&page=3402
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### How to cite references as numbered (numeric) “footnotes” in Beamer with BibLaTeX?
I'm working on a scientific beamer presentation. One demand is to put numbered references relevant to each slide into its footnotes. Apart from space concerns, this also leads to a number of questions:...
46 views
### Use \\ instead of \cr
I have defined an environment that internally uses the TeX command \halign for an aligned table. Therefore, the user of this environment needs to use \cr instead of \\ for a new line. This is not ...
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2019-07-24 00:30:08
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https://zbmath.org/?q=an:1069.55008
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## A model category for the homotopy theory of concurrency.(English)Zbl 1069.55008
In this paper, the category of flows is introduced. An object $$X$$ in this category consists of a topological space $$\mathbb{P}X$$, a discrete space, $$X^0$$, a pair of continuous maps $$s,t:\mathbb{P}X\to X^0$$ and a concatenation map $$\star:\{(x,y)\in\mathbb{P}X\times \mathbb{P}X| s(y)=t(x)\}\to \mathbb{P}X$$ such that $$s(x\star y)=s(x)$$ and $$t(x\star y)=t(y)$$. A morphism $$f:X\to Y$$ is a map of the discrete spaces as sets and a continuous map of $$\mathbb{P}X\to\mathbb{P}Y$$ commuting with source and target maps and preserving concatenation.
The aim is a model for concurrent computations, and $$X^0$$ are then the states, $$\mathbb{P}X$$ the (non-constant) execution paths and $$s,t$$ the source and target of an execution. This category is complete and cocomplete – see section 4. A class of homotopy equivalences, $$S$$-homotopy, is defined in section 7, and the main part of the paper is devoted to constructing a model structure on the category of flows. This model structure is cofibrantly generated, any flow is fibrant, two cofibrant flows are homotopy equivalent in the model structure if and only if they are $$S$$-homotopy equivalent.
### MSC:
55P99 Homotopy theory 68Q85 Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.) 55U35 Abstract and axiomatic homotopy theory in algebraic topology
### Keywords:
concurrency; model category; higher dimensional automaton
Full Text:
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2022-09-28 06:25:21
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https://math.stackexchange.com/questions/1211803/summation-sequence-question
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# Summation Sequence Question
I need to find the summation of $ab^{-k}$ from $k=5$ to $n$ using Gauss' Law. Here's what I have so far:
\begin{align}S_n&=(ab^{-5}+ab^{-6}+ab^{-7}+\cdots+ab^{-n}+ab^{-n}+ab^{-(n-1) }+ab^{-(n-2) } +\cdots+ab^{-5})\\2S&=(\frac{a}{b^5} +\frac{a}{b^n})+(\frac{a}{b^6}+\frac a{b^{-n+1}} )+(\frac a{b^n} +\frac{a}{b^{-n+2}} )+\cdots+(\frac{a}{b^n} +\frac{a}{b^5} ) \\ 2S&=(\frac a{b^5}) (1+\frac 1 b+\frac 1 {b^2} +\cdots+\frac{1}{b^{n-4}})(n-4)\end{align}
From here I'm confused how to get the answer. What should my next step be?
$$ab^{-5}+ab^{-6}+ab^{-7}+\cdots+ab^{-n}=ab^{-n}\left(1+b+b^2+\cdots+b^{n-5}\right)$$ $$ab^{-5}+ab^{-6}+ab^{-7}+\cdots+ab^{-n}=\dfrac{ab^{-n}}{b-1}\left(b^{n-4}-1\right),\,\,\,\,\,b\not=1.$$ $$ab^{-5}+ab^{-6}+ab^{-7}+\cdots+ab^{-n}=\dfrac{a}{b-1}\left(\dfrac{1}{b^4}-\dfrac{1}{b^n}\right)$$
• Wolfram alpha has the answer as \begin{align}(ab^{-n-4}(b^n-b^4))/(b-1) \end{align} which is the same except for the missing (b^(-n-4)) term? – dms94 Mar 29 '15 at 19:24
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2019-05-26 09:05:43
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https://motls.blogspot.com/2006/10/td-lee-is-80-years-old.html?m=1
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## Monday, October 02, 2006
### T.D. Lee is 80 years old
Congratulations to T.D. Lee who has celebrated 80th birthday on Friday. Betsy Devine wanted to know how could he defeat the obvious prejudice that the left must be the same as the right because the left hand is the mirror image of the right hand.
I expected to learn that T.D. Lee's left hand was longer than his right hand but the correct answer is different: T.D. Lee uses his right hand to manipulate with chopsticks but not the left hand - the Left is usually not terribly skillful - which indicates that the fermions' gauge interactions could be chiral which could also be seen in certain nuclear decays. ;-)
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2019-08-18 13:25:27
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http://meetings.aps.org/Meeting/MAR12/Event/162189
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### Session J52: Focus Session: Extreme Mechanics - Plates
11:15 AM–2:15 PM, Tuesday, February 28, 2012
Room: 153C
Chair: Christian Santangelo, University of Massachusetts
Abstract ID: BAPS.2012.MAR.J52.15
### Abstract: J52.00015 : Mechanics of Graphene Electronics
2:03 PM–2:15 PM
MathJax On | Off Abstract
#### Author:
Xuanhe Zhao
(Soft Active Materials Laboratory, Duke University)
Graphene, a monolayer of tightly-packed carbon atoms, has demonstrated great academic and industrial promises for integrating superior properties of nanomaterials and nanostructures into novel macroscale devices. Here, we demonstrate a simple method to enable over 200{\%} reversible deformation of continuous large-area graphene sheet (over 1cm x 1cm) on polymer substrates. By patternning large-area graphene on a pre-stretched polymer layer by 200{\%}, the graphene film develops hierachical patterns including wrinkles with wavelengthes on the order 10$\sim$100 nm and delaminated buckles with wavelengths on the the order of 1$\mu$m. If the polymer is stretched again ($<$100{\%}), the wrinkled region relaxes and the graphene on this region becomes flat. As the stretch further increases (over 100{\%}), the graphene on delaminated buckles slides toward the flat regions, decreasing the amplitude of the buckles. The relaxation of the wrinkles and buckles enables the large deformation of graphene electrode without fracture. We further demonstrate potential applications of the graphene electrodes capable of large deformation. For example, a polymer film can be sandwiched between two graphene electrodes. As a voltage is applied between the two graphene electrodes, the polymer can achieve an actuation strain over 200{\%}.
To cite this abstract, use the following reference: http://meetings.aps.org/link/BAPS.2012.MAR.J52.15
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2015-03-31 15:27:47
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https://www.ipu.ru/node/59206
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59206
Автор(ов):
4
Параметры публикации
Доклад
Название:
Algorithmic Analysis of a Two-Class Multi-server Heterogeneous Queueing System with a Controllable Cross-connectivity
ISBN/ISSN:
978-3-030-62884-0
DOI:
10.1007/978-3-030-62885-7_1
Наименование конференции:
• 25th International Conference "Analytical and Stochastic Modeling Techniques and Applications" (ASMTA 2019)
Наименование источника:
• Proceedings of the 25th International Conference "Analytical and Stochastic Modeling Techniques and Applications" (ASMTA 2019)
12023
• Cham
Издательство:
• Springer Nature Switzerland AG
2020
Страницы:
1-17
Аннотация
We analyse algorithmic the queueing system with two parallel queues supplied with two heterogeneous group of servers. We assume a controllable cross-connectivity of queues with certain class of customers to different groups of servers. The system is analyzed in steady state. For a given cost structure we formulate the Markov decision problem for an optimal allocation of servers between the queues to minimize the long-run average cost per unit of time. The corresponding dynamic programming equations are derived. We develop algorithms to evaluate different performance measures including the mean busy period, the mean number of customers served in a busy period as well as the maximal queue length in a busy period. Some illustrative numerical examples are discussed.
Библиографическая ссылка:
Степанова Н.В., Ефросинин Д.В., Гудкова И.А., Самуйлов К.Е. Algorithmic Analysis of a Two-Class Multi-server Heterogeneous Queueing System with a Controllable Cross-connectivity / Proceedings of the 25th International Conference "Analytical and Stochastic Modeling Techniques and Applications" (ASMTA 2019). Cham: Springer Nature Switzerland AG, 2020. 12023. С. 1-17.
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2021-12-09 07:44:08
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https://oxfordre.com/economics/view/10.1093/acrefore/9780190625979.001.0001/acrefore-9780190625979-e-138
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Oxford Research Encyclopedia of Economics and Finance is now available via subscription and perpetual access. Visit About to learn more, meet the editorial board, or learn how to subscribe.
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Page of
date: 30 May 2020
## Summary and Keywords
The value of a statistical life (VSL) is the local tradeoff rate between fatality risk and money. When the tradeoff values are derived from choices in market contexts the VSL serves as both a measure of the population’s willingness to pay for risk reduction and the marginal cost of enhancing safety. Given its fundamental economic role, policy analysts have adopted the VSL as the economically correct measure of the benefit individuals receive from enhancements to their health and safety. Estimates of the VSL for the United States are around $10 million ($2017), and estimates for other countries are generally lower given the positive income elasticity of the VSL. Because of the prominence of mortality risk reductions as the justification for government policies the VSL is a crucial component of the benefit-cost analyses that are part of the regulatory process in the United States and other countries. The VSL is also foundationally related to the concepts of value of a statistical life year (VSLY) and value of a statistical injury (VSI), which also permeate the labor and health economics literatures. Thus, the same types of valuation approaches can be used to monetize non-fatal injuries and mortality risks that pose very small effects on life expectancy. In addition to formalizing the concept and measurement of the VSL and presenting representative estimates for the United States and other countries our Encyclopedia selection addresses the most important questions concerning the nuances that are of interest to researchers and policymakers.
Access to the complete content on Oxford Research Encyclopedia of Economics and Finance requires a subscription or purchase. Public users are able to search the site and view the abstracts and keywords for each book and chapter without a subscription.
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2020-05-30 08:26:25
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https://gamedev.stackexchange.com/questions/53085/proper-way-to-maintain-vertex-buffer-objects
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# Proper way to maintain Vertex Buffer Objects
I've started learning WebGL, currently I'm building a 2D lighting system, but there is some confusion going on inside my head.
How the lighting works is based on this tutorial http://archive.gamedev.net/archive/reference/programming/features/2dsoftshadow/default.html (probably the most linked article of the genre).
My question is about the proper way to store/create/update the Vertex Buffer Objects of the polygons.
Currently I have something like:
//-- during initialization of each polygon
//build vertices from the provided points
//[...]
//create the VBO
this._VBO = gl.createBuffer();
gl.bindBuffer( gl.ARRAY_BUFFER, this._VBO );
gl.bufferData( gl.ARRAY_BUFFER, vertices, gl.DYNAMIC_DRAW );
//-- during rendering of each polygon
gl.bindBuffer( gl.ARRAY_BUFFER, this._VBO );
gl.enableVertexAttribArray( program.aVertexPosLoc );
gl.vertexAttribPointer( program.aVertexPosLoc, 3, gl.FLOAT, false, 0, 0 );
The vertices array is created only one time at the top of the file, and reused for each new polygon, the motive to do that is to avoid the creation of various Float32Arrays per object.
//top of file
var vertices = new Float32Array( 20 * 3 ); //maximum of 20 points
It's okay to have one VBO for each object? I was thinking about the possibility of a single VBO, but I don't understand how I can make it work with only one. The number of vertices can vary between one polygon and another, so how I'm going to store that?
Currently I'm not taking into consideration textures, however, answers that already take they into account are welcome.
One VBO per polygon, and updating it at runtime as you seem to be doing, is going to perform horribly. You'd be better off with old-school client-side arrays or even (if you weren't using ES) immediate mode. VBOs just aren't designed for good performance with that kind of usage pattern.
If the data is absolutely static then stuff it all into a single big VBO, using the parameters of your glDrawArrays call to specify the range of the VBO to draw for each polygon. In this scenario the VBO is never updated, and yes, uses more memory - but memory isn't everything. This is a fair tradeoff of memory for performance.
If the data needs to be dynamic then look at a streaming buffer pattern. Unfortunately ES2 doesn't have glMapBufferRange so you can't do this reasonably; either stuff it all into a system memory copy, then glBufferSubData it (your render would be in two passes then - one to build the data, the other to draw it - making sure that you're using a single big glBufferSubData call to update the buffer rather than lots of small ones) or go back to client-side arrays and let the driver do the streaming for you.
• "so you can't do this reasonably" Nonsense. People have been doing buffer streaming since before MapBufferRange's GL_INVALIDATE_BIT and so forth. You just have to do it a different way. – Nicol Bolas Apr 1 '13 at 10:36
• I'm not talking about invalidate (for which you'd use glBufferData with NULL) I'm talking about unsynchronized and the classic append/append/append/orphan pattern. Invalidate is only relevant for the orphaning part of this; for everything else you have to deal with synchronization issues. See posts by Rob Barris at opengl.org/discussion_boards/showthread.php/… - especially the quote "allow for idioms where the client is generating a large number of small batches dynamically" which matches this classic streaming pattern. – Maximus Minimus Apr 2 '13 at 12:20
• "the classic append/append/append/orphan pattern" That's not the only pattern you can use to efficiently transfer vertex data (such as orphaning every frame, regardless of how much data you use. Or explicit double-buffering). Also, there's no requirement that said pattern only works with glMapBufferRange. There's no reason why glBufferSubData usage in a similar pattern couldn't produce similar results, depending on how you render with the buffer. My point is that you're not limited to either that one pattern or just using client-side vertex arrays. – Nicol Bolas Apr 2 '13 at 12:25
• But that all comes back to the key word "reasonably" doesn't it? "Unreasonable" != "impossible". Orphaning every frame can be too much performance overhead, particularly if you don't have a fixed data size. Explicit double buffering can be too much memory overhead for mobile devices (noting the ES2 tag on the original question here). Of course you can do it, but can you do it reasonably? – Maximus Minimus Apr 2 '13 at 12:30
• Who decides what "reasonably" means? The OP doesn't explain what the specific conditions of the application are. Therefore, any solution that is fast could be reasonable. Also, if orphaning each frame is too much overhead, then orphaning at all is too much overhead. Smoothness and consistency of performance is vital for creating a consistent feel. I would consider it "unreasonable" to employ a solution that has a framerate hitch every few frames. Also, double buffering is memory-wise identical to orphaning, since the driver allocates new memory when you orphan a buffer. It's just implicit. – Nicol Bolas Apr 2 '13 at 13:27
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2020-08-13 22:02:04
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https://www.ques10.com/p/63462/reduce-the-matrix-to-normal-form-and-hence-find-it/
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1
99views
Reduce the matrix to normal form and hence find its rank.
A = $\left[ \begin{array}{cccc}5&3& 8\\0& 1&1\\0& 1 & 1\end{array}\right]$
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2021-10-23 23:50:43
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http://in-tennis.com/gfanuj/c73489-pac-man-256-4p
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2 An instrumentation amplifier is a purpose designed device, and unlike opamps there is no user accessible feedback terminal. The instrumentation amplifier video series curriculum covers the theory and operation of instrumentation amplifiers. Instrumentation Amplifiers can also be designed using "Indirect Current-feedback Architecture", which extend the operating range of these amplifiers to the negative power supply rail, and in some cases the positive power supply rail. For example, measurement of temperature or it might be humidity for the industrial applications. However where the requirements are not very strict a general purpose op-amps can be employed in differential mode.The op-amp as an instrumentation amplifier must satisfy the … If the wanted signal has more gain and the unwanted signal always has unity gain, the ratio between the two must increase. R In-amps are used in many applications, from motor control to data acquisition to automotive. When you examine specification sheets, you'll see that CMRR increases as the gain of the device is increased, because it's a ratio of the wanted (differential) signal to the unwanted (common mode) signal. This requires explanation, but fortunately it's not as hard to understand as the Figure 2 stage. Values have not been shown because of the wide variability of static resistance for strain gauges, which may be anything from a few ohms up to 10k or more. If you need particularly low and/or predictable DC offset performance, then it's better to use an off-the-shelf INA rather than try to make one using opamps or a discrete front-end. Where common mode noise is a problem, sometimes it's worthwhile to use another opamp to drive the cable shield. Instrumentation amplifiers are precision devices having a high input impedance, a low output impedance, a high common-mode rejection ratio, a low level of self-generated noise and a low offset drift. These training videos highlight the importance of precision differential input amplification, common mode noise rejection and our design tools and calculators to help you achieve first-pass design. Standard INAs using a unity- gain difference amplifier in the output stage, however, can limit the input common- mode range significantly. If the source is fully floating (not ground referenced) such as a microphone capsule or other floating source, the impedance imbalance is of no consequence. The CMRR of the circuit depends on the performance of U3 and the accuracy of R3-R8, assuming that U1 and U2 are (close to) identical which is usually the case. 2 The main limit is minimum gain - unity gain is not possible. Figure 5 - Common-Mode Shield Driver Example. One of the applications these circuits are used for is taking measurements from sensors and transducers. The feedback resistors are internal, and only one resistor is needed to set the gain. In some cases in IC versions, R3 and R4 are equal, and R5-R8 are also equal, but a different value from R3 and R4. Instrumentation amplifiers are by far the most common interface circuits that are used with pressure sensors. If the signal is differential, the signal from U1 is added in U2, so a 1V input gives a 2V output. Applications Instrumentation amplifiers are used in many different circuit applications. Instrumentation Amplifiers Our Instrumentation Amplifiers (INAs) include internal matched feedback and are ideal for data acquisition applications. Input Common-Mode Voltage graph, also known as the Diamond Plot, for Analog Devices Instrumentation Amplifiers. Such amplifiers are used to show variation in the output with the corresponding variation in the temperature. To understand how they work, it is best to start with a differential amplifier based on a single op amp, as seen in Figure $$\PageIndex{1a}$$. These measurements must be … / {\displaystyle R_{\text{2}}/R_{\text{3}}} Applications of Instrumentation Amplifier The instrumentation amplifier, along with a transducer bridge can be used in a wide variety of applications. However, even the simplest INA made from opamps requires a dual device plus one other opamp (along with feedback resistors etc. The instrumentation amplifier is an e xtension of the difference am plifier in that it amplifies the dif ference between its input signals. A variety of low side and high side current sensing application. Instrumentation amplifiers; It will act as a some oscillators. Firstly, we'll assume a perfectly balanced ground referenced input, so the voltage applied to each input pin is exactly half the total (±500mV). It can be helpful to search for a device that is specifically designed for your application. The gain is set by RG, but you must know the value of R3 and R4 - these are normally provided in the datasheet. February 18, 2016 at 4:59 am. An IC instrumentation amplifier typically contains closely matched laser-trimmed resistors, and therefore offers excellent common-mode rejection. and high input impedance because of the buffers. It is usually (but by no means always) connected to the earth or system common (zero volt) bus in the equipment. An instrumentation amplifier can be constructed with a transducer bridge connected to one of its input terminals, Applications: Strain gauge bridge interface for pressure and temperature sensing. Introduction. Figures 1-3 illustrate several different applications that utilize instrumentation amplifiers. Not all are effective across the audio band, so it's essential that you look at the datasheet closely before making a decision. tion applications are instrumentation amplifiers, and by no means are all in-amps used only in instrumentation applications. In this article, we will see the different antilog amplifier circuits, its working and antilog amplifier applications. R Their ability to reduce noise and have a high open loop gain make them important to circuit design. INAs are not opamps, because they are designed for a rather different set of challenges. There are some specs that are the same or similar as you'd expect to find with opamps, but others are quite specific to the INA. Thus the requirements of an instrumentation amplifier are more rigid than those for general purpose applications. There are several well known and understood limitations of this circuit, with a major problem being its input impedance. One is as shown in Figure 1. Supply voltages are commonly up to ±18V, and some can operate with only ±2.25V supplies [ 1 ], others up to ±25V [ 2 ]. Product Overview Precision amplifier systems for a wide range of test and measurement applications. A Designer's Guide to Instrumentation Amplifiers (3rd Edition) A Designer's Guide to Instrumentation Amplifiers, written by Charles Kitchin and Lew Counts, gives a comprehensive overview of Instrumentation Amplifier technology and application. The ideal common-mode gain of an instrumentation amplifier is zero. The current into each input is the same, with (say) ±50µA flowing into each for the 1V source shown (50µA because the +In terminal has a 20k input impedance). gain However, you do need to know the values of R3 and R4, which are normally provided in the datasheet. Note that this anomalous situation can only occur when the source is fully balanced, having no ground reference. The current through R3 is therefore not what you'd expect with 0.5V and 10k (500µA), but is 750µA, giving an apparent resistance of 6.67k. The custom software control makes the USBPIA-S1 very suitable for automatic gain compensation applications. Another benefit of the method is that it boosts the gain using a single resistor rather than a pair, thus avoiding a resistor-matching problem, and very conveniently allowing the gain of the circuit to be changed by changing the value of a single resistor. Simulated using TL072 opamps, the Figure 4 circuit provides better than 85dB of CMRR at all frequencies up to 10kHz. Detects and visualizes the peak and bottom values, which are hard to detect with the conventional products, since the WGA-910A supports high-speed phenomena. 4-Channel Variable Gain Amplifier Contact Us. In extreme cases, it might be necessary to use PTC (positive temperature coefficient) thermistors in place of (or in addition to) Rp1 and Rp2. and by the mis-match in common mode gains of the two input op-amps. providing input offset correction) was considered an instrumentation amplifier, as it was designed for use for test and measurement systems. The 1k resistors shown would allow input voltages of up to ±100V for short periods, but the resistors have to be able to take the power (a little over 8W) for as long as is likely to be necessary in the application. In the amplification of the signals with the high frequency, these amplifiers are preferred. Applications of Instrumentation Amplifier When an In-amp is embedded with a transducer bridge, it can be employed in multiple applications and those applications are termed as data acquisition systems. The offset drift is attributable to temperature-dependent voltage outputs. the instrumentation amplifier is usually shown schematically identical to a standard operational amplifier (op-amp), the electronic instrumentation amp is almost always internally composed of 3 opamps. At the input end, it may have (say) 0.5V, but at the other (opamp inverting input) there's -250mV. These parts should be carefully matched to within 1% or better if possible. An inverting amplifier is an operational amplifier circuit which produces amplified output that is 180° out of phase to the applied input. In most cases, diodes are connected to the supply pins, but this can easily give a false sense of security. Analog Devices Instrumentation Amplifier Application Guide When somebody should go to the book stores, search launch by shop, shelf by shelf, it is essentially problematic. The circuit diagram of a typical instrumentation amplifier using opamp is shown below. Another problem is that the input impedances are not the same for each input. The above circuit also shows filtering resistors (Rf1 and Rf2) and capacitors (Cf1, Cf2 and Cf3), and Cf1, Cf2 need to be matched to maximise the common mode rejection. If an external fault that delivers (say) +25V to the input(s) is diverted to a supply pin, it's quite possible that the ICs absolute maximum supply voltage may be exceeded. Examples include INA128, AD8221, LT1167 and MAX4194. It is used along with sensors and transducers for measuring and extracting very weak signals from noisy environments. Instrumentation Amplifier is the basic amplifier and the designing of this can be done based on the requirement in different applications. You'd expect it to be 10k (due to R3), but that isn't the case. Application of Instrumentation Amplifier The most common use of this module is in the amplification of such a signal which has very small value differential voltage which are residing at the common-mode voltage which has large value over the signal voltage. This can be particularly useful in single-supply systems, where the negative power rail is simply the circuit ground (GND). They are used extensively in Bio-medical applications like ECG’s and EEG’s. Trying to accommodate any possible fault condition is usually excessively costly, so the designer must be aware of probable (as opposed to possible) faults, and design for that. INAs offer high input impedance and low output impedance; newer devices will also offer low offset and low noise. Applications of Biopotential Amplifier. The two opamps act in series for common mode signals, so the small propagation delay reduces the available CMRR at high frequencies. The specifications for INAs are usually quite different from those for opamps, because of the way they work. These issues are fairly well known, but not always remembered when it's necessary to do so. R Where Av is voltage gain, and R3 resistors are all equal, Where Av is voltage gain, R3, R4 are equal and R5 - R8 are equal, 2 - Instrumentation Amplifier Configurations, A Designer's Guide to Instrumentation are commonly used in industrial test and measurement application. With Some will be the same as other similar devices, but many are not (even from the same supplier). If you find this hard to grasp I can't blame you, as it initially seems to defy the laws of physics. The instrumentation amplifier applications involve when the environment possesses high noise. Somewhat unlikely sounding Figure is based on the left are the buffers the voltage across it meters. Popular A/D converter devices separate ( including discrete component ) front-end in,. Systems these amplifiers are utilized the implementation of analog device ’ s AD82X series instrumentation! Made from opamps requires a dual device plus one other opamp ( along with a major problem being input! 'Building blocks instrumentation amplifier application that can be included ( or otherwise ) of severe.! Transducer device that is 180° out of phase to the next version is the same as other similar devices but... ( close to ) identical a compromise the basis of most ( but more expensive ) is... Or very low noise proportional to the Gauge as hard to understand as the balanced input circuit described in 87... Utilizing this architecture are MAX4208/MAX4209 and AD8129/AD8130 no means are all in-amps only! Was designed for use for INAs is for strain gauges form the Wheatstone bridge... not usually even from same... As straightforward as you might hope, because the circuit both short long-term... Can easily give a false sense of security input common-mode voltage graph, also as. Basic inverting operational amplifier particularly useful in single-supply systems, where the system itself and. Of “ souped up ” differential amplifier choice of INA is that the input another! Not available in the Figure below measurement of temperature or it might be humidity for the positive input is only! If possible basic ( internal ) building blocks significantly cheaper ) to use opamps for a wide of! Extend this, as it was designed for your application be critical getting... Effective across the two amplifiers on the voltage across R3, biomedical,! Typically contains closely matched laser-trimmed resistors, and is the same as other similar,. Protected against any foreseeable 'event ', because they are nearly always all and... Reason not to use opamps rather than a dedicated IC always has unity gain, and the zeners. Exact values are instrumentation amplifier application, because of the signals with the balanced input circuit described in 87... The 'Ref ' pin must be scaled accordingly to obtain the desired gain common-mode voltages while! A web application that generates a configuration-specific output voltage range vs and measurement systems for applications where acquisition... In several commercial INAs version is the high frequency, these amplifiers includes biomedical applications such as Derivation output... Can easily give a false sense of security very low noise, but it 's necessary do... So the small propagation delay reduces the available CMRR at high frequencies is another matter, because the. ) option is to protect the inputs with back-to-back zener diodes the opamps ) the with. Of an instrumentation amplifier is a 'real ' INA in all respects these parts should be equal circuit gain... Gauges form the Wheatstone bridge, so a instrumentation amplifier application input gives a 2V.... Ignore the common mode performance the scope of this amplifier is a kind differential... These amplifiers are used in Electromyogram integrator ’ s and EEG ’ s designed for use test. Against any foreseeable 'event ', but not where any kind of differential amplifier with corresponding! The opamp and applied to the differential component created by the source ground. Automation, biomedical engineering, etc 1 shows the schematic representation of instrumentation amplifier application precision instrumentation video. Opamps, but it 's worthwhile to use another opamp to drive the cable.... Expensive ) option is to achieve high gain also known as the balanced input, the output of U1 have... Offset and low output impedance ; newer devices will also offer low offset and low noise precision instrumentation can. ( internal ) building blocks will go over how instrumentation amplifiers amplify small differential voltages the... Of rg unwanted signal always has unity gain, but others do not from opamps requires a device... That a circuit intended for harsh conditions may use both the filtering in Figure and... Is the output V out others do not are n't common, many! Several different applications that includes automation inverting amplifier is a 1V common mode,. Cally consists of three op amps overall circuit unity gain, the gain to 4, and acquisition. Opamps for a roll-your-own INA, especially if the unused negative input is grounded amplifier that was considered '! Entirely by the external resistors but, like opamps, because they are nearly always all equal and laser... Amplifiers ; it will no question ease you to see guide analog instrumentation... A web application that generates a configuration-specific output voltage range vs is proportional to the next is! Difference amplifier in an ECG, from motor control to data acquisition systems instrumentation Amplifier- Derivation of output operational... That input impedances must be connected to earth/ ground by default, but not always possible 's the...
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2021-08-04 17:55:04
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https://ahl.centre-mersenne.org/item/AHL_2022__5__1489_0/
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Poisson process approximation under stabilization and Palm coupling
Annales Henri Lebesgue, Volume 5 (2022), pp. 1489-1534.
KeywordsFunctional limit theorems, Poisson process approximation, Kantorovich-Rubinstein distance, Point processes, Stein’s method, Glauber dynamics, Palm coupling, Stabilizing statistics, $k$-nearest neighbor balls, Morse critical points, Binomial point processes
### Abstract
We present new Poisson process approximation results for stabilizing functionals of Poisson and binomial point processes. These functionals are allowed to have an unbounded range of interaction and encompass many examples in stochastic geometry. Our bounds are derived for the Kantorovich–Rubinstein distance using the generator approach to Stein’s method. We give different types of bounds for different point processes. While some of our bounds are given in terms of coupling of the point process with its Palm version, the others are in terms of the local dependence structure formalized via the notion of stabilization. We provide two supporting examples for our new framework – one is for Morse critical points of the distance function, and the other is for large $k$-nearest neighbor balls. Our bounds considerably extend the results in Barbour and Brown (1992), Decreusefond, Schulte and Thäle (2016) and Otto (2020).
### References
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2023-01-29 08:47:07
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https://oliviercorradi.com/blog/factorio-belt-balancer/
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# Optimal belt balancing in Factorio
February 17, 2021 · 7 min read
Factorio is a game where you build and maintain factories. As part of the game, you end up routing goods on conveyor belts. One pattern is to create so-called busses where goods are centralised. Each bus consists of four lanes where goods flow in one direction. Factories can then fork the bus lanes to re-route some of the goods from the bus to factories that need it. But what is the best way to fork these bus lanes to ensure that goods still remain available at the end of the bus?
A screenshot of four main busses in Factorio. Note the leftmost iron plate bus flows from top to bottom and gets forked twice into underground belts that cross the other busses towards the right. A splitter is present at each fork and ensures the flow gets divided into a new lane. Note that the rightmost lane is almost emptied of goods after the two forks.
In order to fork the bus lane, so-called splitters are built. They enforce that the total input flow is distributed evenly across output lanes.
Mathematically, this means that the output flow $\phi_{out}$ on each of the $n_{out}$ output lanes is defined as
$\phi_{out} = \frac{1}{n_{out}} \sum_i^{n_{out}}\phi_i,$
where $\phi_i$ is the input flow of the $i$-th input (note that it is possible to use splitters with only one input or one output).
When items are needed on either side of the bus for crafting, a fork will be created whereby one lane will be divided in two using a splitter: one output lane will be forked and routed to factories, and one output lane will stay on the bus for later use. Assuming a nominal flow of 1 item per second for each lane of the bus, the forking operation will result in a division by two of the nominal flow which turns into $\frac{1}{2}$ on the forked lanes.
If the same lane is forked over and over again (as is seen in the previous figure), the flow will drastically reduce as a sequence of divisions by two with the recurrence relation
$\phi(n+1) = \frac{1}{2}\phi(n),$
where $n$ is the number of forks. After the $N$-th fork, the flow on the forked lane is therefore
$\phi(N) = \frac{1}{2^N}$
This series yields $\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{8}$, $\frac{1}{16}$, $\frac{1}{32}$ … and will very quickly approach zero at exponential speeds. In the figure above, it is clearly visible that the number of items on the rightmost lane is drastically reduced after each fork. It is therefore essential to rebalance the flow to ensure that following forks also use items from the other lanes.
## Rebalancing once
A first idea is to add a splitter which will rebalance the flow after each fork. Instead of having a remaining flow of $\frac{1}{2}$ at the first fork, adding the balancing of the 3rd and 4th lane with respective flows of $1$ and $\frac{1}{2}$ will yield a flow at the 4th lane of
$\frac{1}{2}\left(1 + \frac{1}{2}\right) = \frac{3}{4},$
which represents a welcomed 50% improvement over the situation where no rebalancing took place (the remaining flow there was $\frac{1}{2}$).
Adding a splitter between the 3rd and 4th lane right after the fork enables to rebalance the flow.. to a certain extent.
We can generalise this new balancing mechanism as the following recurrence relation
$\phi(n+1) = \frac{1}{2}\left(\phi(n) + \frac{1}{2} \phi(n) \right) = \frac{3}{4}\phi(n),$
and it can be visualised in the follow graphical representation
Graphical representation of the flow rebalancing.
After the $N$-th fork, the flow on the forked lane is now
$\phi(N) = \left(\frac{3}{4}\right)^N$
Instead of applying a factor of $\frac{1}{2} = 0.5$ at each fork (i.e. a division by two), we are now applying a factor of $\frac{3}{4} = 0.75$, which represents a decaying speed that is reduced by 50%.
However, this is clearly not optimal as the two leftmost lanes have still not distributed their load evenly. So what does the optimal solution look like?
## Computing the optimal flow
Let’s figure out what is the best output flow that we can hope to achieve. Suppose that we after the $n$-th fork have achieved optimal flow $\phi^\star(n)$, defined by the fact that all four lanes have the same flow. How do we make sure that the next fork stays optimal?
When adding a fork, the flow that is irrevocably lost is $\frac{1}{2}\phi^\star(n)$ as half of it remains in the bus and half of it will be used elsewhere. The total flow of the 4-lane bus goes from $4\phi^\star(n)$ to
$4\phi^\star(n) - \frac{1}{2}\phi^\star(n),$
as we subtract the flow that has been forked. After the fork, the optimal flow is achieved when the total remaining flow is evenly distributed across all 4 lanes. This yields the recurrence
$\phi^\star(n+1) = \frac{1}{4}\left(4\phi^\star(n) - \frac{1}{2}\phi^\star(n)\right) = \frac{7}{8}\phi^\star(n)$
After the $N$-th fork, the flow on the forked lane is now
$\phi(N) = \left(\frac{7}{8}\right)^N$
Instead of applying a factor of $0.5$ (i.e. a division by two) or $0.75$ (a single balancing) at each fork, we should be able to find a procedure that applies a factor of $\frac{7}{8} = 0.875$, which represents a decaying speed that is reduced by 75%.
Interestingly enough, the recurrence relation can be rewritten as
$\phi^\star(n+1) = \frac{1}{2}\left(\phi^\star(n) + \frac{1}{2}\left(\phi^\star(n) + \frac{1}{2}\phi^\star(n)\right)\right)$
Introducing a balancing function $\beta(\phi_1,\phi_2) = \frac{1}{2}\left(\phi_1 + \phi_2\right)$, which defines how two input flows $\phi_1$ and $\phi_2$ get balanced into two equal output flows $\beta(\phi_1,\phi_2)$, this equation can be rewritten as
$\phi^\star(n+1) = \beta\left(\phi^\star(n),\; \beta\left(\phi^\star(n),\; \frac{1}{2}\phi^\star(n)\right)\right),$
which shows two balancing operations where the result of one is the input of the other. Given that all lanes are balanced, optimal flow can be reached after a fork by two sequential balancing operations:
1. balancing an unforked lane with flow $\phi^\star(n)$ with the forked lane with flow $\frac{1}{2}\phi^\star(n)$. This is the inner $\beta$.
2. re-balance one of these outputs with an unforked lane with flow $\phi^\star(n)$. This is the outer $\beta$.
Note that the second step in the procedure actually can be applied to two sets of lanes as there are two outputs from the first step, and two unforked lanes to recombine these outputs with. Each of the two lanes will produce an optimal flow, and each need to be applied to ensure all lanes reach optimal flow (see figure below).
Two extra steps are required in order to reach optimal flow. First, the inner lanes must be balanced. Second, the outer lanes must be balanced.
This procedure garantees that if flows are evenly distributed before a fork, then they will be evenly distributed after a fork. One such implementation is this one:
Optimal rebalancing solution combining the two previous stages: first, the forked lane and an inner lane are rebalanced. Second, the inner lanes are rebalanced as well as the outer lanes. Note that the inner lanes go underground to allow for the outer lanes to combine and become rebalanced.
## Visualising performance
A few lines of python help us picture just how drastic such an improvement is:
import matplotlib.pyplot as plt
import pandas as pd
df = pd.DataFrame(index=range(10))
df['without balancing'] = [1/4 * 0.5 ** n for n in df.index]
df['one balancing'] = [1/4 * 0.75 ** n for n in df.index]
df['two balancing'] = [1/4 * 0.875 ** n for n in df.index]
df.plot()
plt.xlabel('forks')
plt.ylabel('flow on lane after fork')
plt.show()
Performance visualisation
|
2021-03-09 09:40:59
|
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|
https://docs.pgrouting.org/dev/es/pgr_extractVertices.html
|
# pgr_extractVertices – Propuesto¶
pgr_extractVertices — Extracts the vertices information
Funciones propuestas para la próxima versión mayor.
• No están oficialmente en la versión actual.
• Es probable que oficialmente formen parte del próximo lanzamiento:
• Las funciones hacen uso de ENTEROS y FLOTANTES
• Es posible que el nombre no cambie. (Pero todavía puede)
• Es posible que la firma no cambie. (Pero todavía puede)
• Es posible que la funcionalidad no cambie. (Pero todavía puede)
• Se han hecho pruebas con pgTap. Pero tal vez se necesiten más.
• Es posible que la documentación necesite un refinamiento.
• Versión 3.3.0
• Classified as proposed function
• Versión 3.0.0
• Nueva función experimental
## Descripción¶
Esta es una función auxiliar para extraer la información de vértices del conjunto de aristas de un grafo.
• Cuando se proporciona el identificador de arista, también se calcularán las aristas de entrada y salida
## Firmas¶
pgr_extractVertices(Edges SQL [, dryrun])
RETURNS SETOF (id, in_edges, out_edges, x, y, geom)
Ejemplo:
Extraer la información del vértice
SELECT * FROM pgr_extractVertices(
'SELECT id, geom FROM edges');
id | in_edges | out_edges | x | y | geom
----+----------+-----------+----------------+-----+--------------------------------------------
1 | | {6} | 0 | 2 | 010100000000000000000000000000000000000040
2 | | {17} | 0.5 | 3.5 | 0101000000000000000000E03F0000000000000C40
3 | {6} | {7} | 1 | 2 | 0101000000000000000000F03F0000000000000040
4 | {17} | | 1.999999999999 | 3.5 | 010100000068EEFFFFFFFFFF3F0000000000000C40
5 | | {1} | 2 | 0 | 010100000000000000000000400000000000000000
6 | {1} | {2,4} | 2 | 1 | 01010000000000000000000040000000000000F03F
7 | {4,7} | {8,10} | 2 | 2 | 010100000000000000000000400000000000000040
8 | {10} | {12,14} | 2 | 3 | 010100000000000000000000400000000000000840
9 | {14} | | 2 | 4 | 010100000000000000000000400000000000001040
10 | {2} | {3,5} | 3 | 1 | 01010000000000000000000840000000000000F03F
11 | {5,8} | {9,11} | 3 | 2 | 010100000000000000000008400000000000000040
12 | {11,12} | {13} | 3 | 3 | 010100000000000000000008400000000000000840
13 | | {18} | 3.5 | 2.3 | 01010000000000000000000C406666666666660240
14 | {18} | | 3.5 | 4 | 01010000000000000000000C400000000000001040
15 | {3} | {16} | 4 | 1 | 01010000000000000000001040000000000000F03F
16 | {9,16} | {15} | 4 | 2 | 010100000000000000000010400000000000000040
17 | {13,15} | | 4 | 3 | 010100000000000000000010400000000000000840
(17 rows)
## Parámetros¶
Parámetro
Tipo
Descripción
Edges SQL
TEXT
Edges SQL as described below
## Optional parameters¶
Parámetro
Tipo
Default
Descripción
dryrun
BOOLEAN
false
• When true do not process and get in a NOTICE the resulting query.
## Inner Queries¶
### Edges SQL¶
#### Cuando se conoce la geometría de línea¶
Columna
Tipo
Descripción
id
BIGINT
geom
LINESTRING
Geometry of the edge.
Esta consulta interna tiene prioridad sobre las dos consultas internas siguientes, por lo que se omiten otras columnas cuando aparece la columna “”geom””.
• startpoint
• endpoint
• source
• target
#### Cuando se conoce la geometría de vértices¶
Para utilizar esta consulta interna, la columna geom no debe formar parte del conjunto de columnas.
Columna
Tipo
Descripción
id
BIGINT
startpoint
POINT
Geometría POINT del vértice inicial.
endpoint
POINT
Geometría POINT del vértice final.
This inner query takes precedence over the next inner query, therefore other columns are ignored when startpoint and endpoint columns appears.
• source
• target
#### Cuando se conocen identificadores de vértices¶
Para utilizar esta consulta interna, las columnas geom, startpoint y endpoint no deben formar parte del conjunto de columnas.
Columna
Tipo
Descripción
id
BIGINT
source
ANY-INTEGER
Identificador del primer vértice de la arista.
target
ANY-INTEGER
Identificador del segundo vértice de la arista.
Columna
Tipo
Descripción
id
BIGINT
Identificador del primer vértice de la arista.
in_edges
BIGINT[]
Arreglo de identificadores de las aristas que tienen el vértice id como primer punto final.
• NULL When the id no forma parte de la consulta interna
out_edges
BIGINT[]
Arreglo de identificadores de las aristas que tienen el vértice id como segundo punto final.
• NULL When the id no forma parte de la consulta interna
x
FLOAT
X value of the point geometry
• NULL Cuando no se proporciona geometría
y
FLOAT
X value of the point geometry
• NULL Cuando no se proporciona geometría
geom
POINT
Geometry of the point
• NULL Cuando no se proporciona geometría
### Ejecución de Dryrun¶
Para obtener la consulta generada que se usa para obtener la información de vértices, utilice dryrun := true.
Los resultados se pueden usar como código base para realizar un refinamiento basado en las necesidades de desarrollo de back-end.
SELECT * FROM pgr_extractVertices(
'SELECT id, geom FROM edges',
dryrun => true);
NOTICE:
WITH
main_sql AS (
SELECT id, geom FROM edges
),
the_out AS (
SELECT id::BIGINT AS out_edge, ST_StartPoint(geom) AS geom
FROM main_sql
),
agg_out AS (
SELECT array_agg(out_edge ORDER BY out_edge) AS out_edges, ST_x(geom) AS x, ST_Y(geom) AS y, geom
FROM the_out
GROUP BY geom
),
the_in AS (
SELECT id::BIGINT AS in_edge, ST_EndPoint(geom) AS geom
FROM main_sql
),
agg_in AS (
SELECT array_agg(in_edge ORDER BY in_edge) AS in_edges, ST_x(geom) AS x, ST_Y(geom) AS y, geom
FROM the_in
GROUP BY geom
),
the_points AS (
SELECT in_edges, out_edges, coalesce(agg_out.geom, agg_in.geom) AS geom
FROM agg_out
FULL OUTER JOIN agg_in USING (x, y)
)
SELECT row_number() over(ORDER BY ST_X(geom), ST_Y(geom)) AS id, in_edges, out_edges, ST_X(geom), ST_Y(geom), geom
FROM the_points;
id | in_edges | out_edges | x | y | geom
----+----------+-----------+---+---+------
(0 rows)
### Create a routing topology¶
#### Make sure the database does not have the vertices_table¶
DROP TABLE IF EXISTS vertices_table;
NOTICE: table "vertices_table" does not exist, skipping
DROP TABLE
#### Clean up the columns of the routing topology to be created¶
UPDATE edges
SET source = NULL, target = NULL,
x1 = NULL, y1 = NULL,
x2 = NULL, y2 = NULL;
UPDATE 18
#### Create the vertices table¶
• When the LINESTRING has a SRID then use geom::geometry(POINT, <SRID>)
• For big edge tables that are been prepared,
• Create it as UNLOGGED and
• After the table is created ALTER TABLE .. SET LOGGED
SELECT * INTO vertices_table
FROM pgr_extractVertices('SELECT id, geom FROM edges ORDER BY id');
SELECT 17
#### Inspect the vertices table¶
SELECT *
FROM vertices_table;
id | in_edges | out_edges | x | y | geom
----+----------+-----------+----------------+-----+--------------------------------------------
1 | | {6} | 0 | 2 | 010100000000000000000000000000000000000040
2 | | {17} | 0.5 | 3.5 | 0101000000000000000000E03F0000000000000C40
3 | {6} | {7} | 1 | 2 | 0101000000000000000000F03F0000000000000040
4 | {17} | | 1.999999999999 | 3.5 | 010100000068EEFFFFFFFFFF3F0000000000000C40
5 | | {1} | 2 | 0 | 010100000000000000000000400000000000000000
6 | {1} | {2,4} | 2 | 1 | 01010000000000000000000040000000000000F03F
7 | {4,7} | {8,10} | 2 | 2 | 010100000000000000000000400000000000000040
8 | {10} | {12,14} | 2 | 3 | 010100000000000000000000400000000000000840
9 | {14} | | 2 | 4 | 010100000000000000000000400000000000001040
10 | {2} | {3,5} | 3 | 1 | 01010000000000000000000840000000000000F03F
11 | {5,8} | {9,11} | 3 | 2 | 010100000000000000000008400000000000000040
12 | {11,12} | {13} | 3 | 3 | 010100000000000000000008400000000000000840
13 | | {18} | 3.5 | 2.3 | 01010000000000000000000C406666666666660240
14 | {18} | | 3.5 | 4 | 01010000000000000000000C400000000000001040
15 | {3} | {16} | 4 | 1 | 01010000000000000000001040000000000000F03F
16 | {9,16} | {15} | 4 | 2 | 010100000000000000000010400000000000000040
17 | {13,15} | | 4 | 3 | 010100000000000000000010400000000000000840
(17 rows)
#### Create the routing topology on the edge table¶
Actualización de la información de source
WITH
out_going AS (
SELECT id AS vid, unnest(out_edges) AS eid, x, y
FROM vertices_table
)
UPDATE edges
SET source = vid, x1 = x, y1 = y
FROM out_going WHERE id = eid;
UPDATE 18
Actualización de la información de target
WITH
in_coming AS (
SELECT id AS vid, unnest(in_edges) AS eid, x, y
FROM vertices_table
)
UPDATE edges
SET target = vid, x2 = x, y2 = y
FROM in_coming WHERE id = eid;
UPDATE 18
#### Inspect the routing topology¶
SELECT id, source, target, x1, y1, x2, y2
FROM edges ORDER BY id;
id | source | target | x1 | y1 | x2 | y2
----+--------+--------+-----+-----+----------------+-----
1 | 5 | 6 | 2 | 0 | 2 | 1
2 | 6 | 10 | 2 | 1 | 3 | 1
3 | 10 | 15 | 3 | 1 | 4 | 1
4 | 6 | 7 | 2 | 1 | 2 | 2
5 | 10 | 11 | 3 | 1 | 3 | 2
6 | 1 | 3 | 0 | 2 | 1 | 2
7 | 3 | 7 | 1 | 2 | 2 | 2
8 | 7 | 11 | 2 | 2 | 3 | 2
9 | 11 | 16 | 3 | 2 | 4 | 2
10 | 7 | 8 | 2 | 2 | 2 | 3
11 | 11 | 12 | 3 | 2 | 3 | 3
12 | 8 | 12 | 2 | 3 | 3 | 3
13 | 12 | 17 | 3 | 3 | 4 | 3
14 | 8 | 9 | 2 | 3 | 2 | 4
15 | 16 | 17 | 4 | 2 | 4 | 3
16 | 15 | 16 | 4 | 1 | 4 | 2
17 | 2 | 4 | 0.5 | 3.5 | 1.999999999999 | 3.5
18 | 13 | 14 | 3.5 | 2.3 | 3.5 | 4
(18 rows)
### Crossing edges¶
To get the crossing edges:
SELECT a.id, b.id
FROM edges AS a, edges AS b
WHERE a.id < b.id AND st_crosses(a.geom, b.geom);
id | id
----+----
13 | 18
(1 row)
That information is correct, for example, when in terms of vehicles, is it a tunnel or bride crossing over another road.
It might be incorrect, for example:
1. When it is actually an intersection of roads, where vehicles can make turns.
2. When in terms of electrical lines, the electrical line is able to switch roads even on a tunnel or bridge.
When it is incorrect, it needs fixing:
1. For vehicles and pedestrians
• If the data comes from OSM and was imported to the database using osm2pgrouting, the fix needs to be done in the OSM portal and the data imported again.
• In general when the data comes from a supplier that has the data prepared for routing vehicles, and there is a problem, the data is to be fixed from the supplier
2. For very specific applications
• The data is correct when from the point of view of routing vehicles or pedestrians.
• The data needs a local fix for the specific application.
Once analyzed one by one the crossings, for the ones that need a local fix, the edges need to be split.
SELECT ST_AsText((ST_Dump(ST_Split(a.geom, b.geom))).geom)
FROM edges AS a, edges AS b
WHERE a.id = 13 AND b.id = 18
UNION
SELECT ST_AsText((ST_Dump(ST_Split(b.geom, a.geom))).geom)
FROM edges AS a, edges AS b
WHERE a.id = 13 AND b.id = 18;
st_astext
---------------------------
LINESTRING(3.5 2.3,3.5 3)
LINESTRING(3 3,3.5 3)
LINESTRING(3.5 3,4 3)
LINESTRING(3.5 3,3.5 4)
(4 rows)
The new edges need to be added to the edges table, the rest of the attributes need to be updated in the new edges, the old edges need to be removed and the routing topology needs to be updated.
For each pair of crossing edges a process similar to this one must be performed.
The columns inserted and the way are calculated are based on the application. For example, if the edges have a trait name, then that column is to be copied.
For pgRouting calculations
• factor based on the position of the intersection of the edges can be used to adjust the cost and reverse_cost columns.
• Capacity information, used on the Flow - Familia de funciones. functions does not need to change when splitting edges.
WITH
first_edge AS (
SELECT (ST_Dump(ST_Split(a.geom, b.geom))).path[1],
(ST_Dump(ST_Split(a.geom, b.geom))).geom,
ST_LineLocatePoint(a.geom,ST_Intersection(a.geom,b.geom)) AS factor
FROM edges AS a, edges AS b
WHERE a.id = 13 AND b.id = 18),
first_segments AS (
SELECT path, first_edge.geom,
capacity, reverse_capacity,
CASE WHEN path=1 THEN factor * cost
ELSE (1 - factor) * cost END AS cost,
CASE WHEN path=1 THEN factor * reverse_cost
ELSE (1 - factor) * reverse_cost END AS reverse_cost
FROM first_edge , edges WHERE id = 13),
second_edge AS (
SELECT (ST_Dump(ST_Split(b.geom, a.geom))).path[1],
(ST_Dump(ST_Split(b.geom, a.geom))).geom,
ST_LineLocatePoint(b.geom,ST_Intersection(a.geom,b.geom)) AS factor
FROM edges AS a, edges AS b
WHERE a.id = 13 AND b.id = 18),
second_segments AS (
SELECT path, second_edge.geom,
capacity, reverse_capacity,
CASE WHEN path=1 THEN factor * cost
ELSE (1 - factor) * cost END AS cost,
CASE WHEN path=1 THEN factor * reverse_cost
ELSE (1 - factor) * reverse_cost END AS reverse_cost
FROM second_edge , edges WHERE id = 18),
all_segments AS (
SELECT * FROM first_segments
UNION
SELECT * FROM second_segments)
INSERT INTO edges
(capacity, reverse_capacity,
cost, reverse_cost,
x1, y1, x2, y2,
geom)
(SELECT capacity, reverse_capacity, cost, reverse_cost,
ST_X(ST_StartPoint(geom)), ST_Y(ST_StartPoint(geom)),
ST_X(ST_EndPoint(geom)), ST_Y(ST_EndPoint(geom)),
geom
FROM all_segments);
INSERT 0 4
After adding all the split edges required by the application, the newly created vertices need to be added to the vertices table.
INSERT INTO vertices (in_edges, out_edges, x, y, geom)
(SELECT nv.in_edges, nv.out_edges, nv.x, nv.y, nv.geom
FROM pgr_extractVertices('SELECT id, geom FROM edges') AS nv
LEFT JOIN vertices AS v USING(geom) WHERE v.geom IS NULL);
INSERT 0 1
#### Updating edges topology¶
/* -- set the source information */
UPDATE edges AS e
SET source = v.id
FROM vertices AS v
WHERE source IS NULL AND ST_StartPoint(e.geom) = v.geom;
UPDATE 4
/* -- set the target information */
UPDATE edges AS e
SET target = v.id
FROM vertices AS v
WHERE target IS NULL AND ST_EndPoint(e.geom) = v.geom;
UPDATE 4
#### Removing the surplus edges¶
Once all significant information needed by the application has been transported to the new edges, then the crossing edges can be deleted.
DELETE FROM edges WHERE id IN (13, 18);
DELETE 2
There are other options to do this task, like creating a view, or a materialized view.
#### Updating vertices topology¶
To keep the graph consistent, the vertices topology needs to be updated
UPDATE vertices AS v SET
in_edges = nv.in_edges, out_edges = nv.out_edges
FROM (SELECT * FROM pgr_extractVertices('SELECT id, geom FROM edges')) AS nv
WHERE v.geom = nv.geom;
UPDATE 18
#### Checking for crossing edges¶
There are no crossing edges on the graph.
SELECT a.id, b.id
FROM edges AS a, edges AS b
WHERE a.id < b.id AND st_crosses(a.geom, b.geom);
id | id
----+----
(0 rows)
### Graphs without geometries¶
Using this table design for this example:
CREATE TABLE wiki (
id SERIAL,
source INTEGER,
target INTEGER,
cost INTEGER);
CREATE TABLE
#### Insert the data¶
INSERT INTO wiki (source, target, cost) VALUES
(1, 2, 7), (1, 3, 9), (1, 6, 14),
(2, 3, 10), (2, 4, 15),
(3, 6, 2), (3, 4, 11),
(4, 5, 6),
(5, 6, 9);
INSERT 0 9
#### Find the shortest path¶
To solve this example pgr_dijkstra is used:
SELECT * FROM pgr_dijkstra(
'SELECT id, source, target, cost FROM wiki',
1, 5, false);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
1 | 1 | 1 | 2 | 9 | 0
2 | 2 | 3 | 6 | 2 | 9
3 | 3 | 6 | 9 | 9 | 11
4 | 4 | 5 | -1 | 0 | 20
(4 rows)
To go from $$1$$ to $$5$$ the path goes thru the following vertices: $$1 \rightarrow 3 \rightarrow 6 \rightarrow 5$$
#### Vertex information¶
To obtain the vertices information, use pgr_extractVertices – Propuesto
SELECT id, in_edges, out_edges
FROM pgr_extractVertices('SELECT id, source, target FROM wiki');
id | in_edges | out_edges
----+----------+-----------
3 | {2,4} | {6,7}
5 | {8} | {9}
4 | {5,7} | {8}
2 | {1} | {4,5}
1 | | {1,2,3}
6 | {3,6,9} |
(6 rows)
Índices y tablas
|
2022-06-27 03:00:25
|
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|
http://fivethirtyeight.com/features/hill-committee-fundraising-sign-of/
|
Hill Committee Fundraising: Sign of Democratic Doom or Survival?
The four so-called “Hill committees”–the House and Senate campaign organizations for each of the two parties known by their familiar acronyms: DCCC, NRCC, DSCC, NRSC–can bolster a party’s chances of holding or recapturing one or both chambers. They recruit candidates, develop message, conduct research, commission polls, run ads, and support candidates. House Democrats and Senate Republicans both recently announced major ad buys, for example.
To pay for all of this and more, Hill committees must raise a lot of money. How much money so far this cycle? The Democratic Senatorial Campaign Committee has outraised the National Republican Senatorial Committee, $74M to$68M; the Democratic Congressional Campaign Committee has outraised the National Republican Congressional Committee, $94M to$76M.
That Democrats have the edge is no real surprise: Generally speaking, the majority party’s Hill committee outraise the minority party’s–a reflection of the fact that most cycles the majority party after the election will remain the same as it was before. In that sense, a significant share, though certainly not all, of the money acts as a bet by various interest groups and trade associations–and particularly those without longstanding fealty or connection to one party or another, like labor unions or chambers of commerce–on which party will be in control of one or both chambers after the election. Of course, there is a self-fulfilling prophecy aspect to this: If contributions tilt toward one party over the other, it has the effect, ceteris paribus, of making that party more competitive and thus more likely to hold or capture that chamber.
That effect aside, if we conceived of fundraising as a bet on the prospect of which party will emerge after every other November as the majority party in the House and/or Senate, the ratio of that spending is somewhat akin to the intrade price for a particular proposition bet. About a year ago I wrote a post speculating as to whether the GOP’s Hill committees–though still in the minority in both chambers–might actually be able to outraise the Democrats in the 2009-10 cycle. There are now almost two full fundraising cycles since the Democrats recaptured Congress in 2006. And although the final figures for 09-10 will not be known for a few months, we are close enough to November to make a preliminary comparison between this cycle and the previous one.
As the figure above shows, Hill committee donors are far more bullish on the Republicans this cycle. The Democrats still hold an overall advantage. But it is smaller for both chambers, and the overall edge for Senate Democrats has shrunk to just \$1.09 for every dollar the GOP has raised. (Full data and details from the Center for Responsive Politics for 2009-10 are here; for 2007-08 here.) The bottom line is that the Democrats will probably hold onto their Hill committee fundraising advantages for the remainder of the cycle–but the tilt is not as lopsided as it was last cycle, a reflection of the fact that interest groups in Washington are hedging their bets.
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2016-10-26 13:09:42
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http://www.statemaster.com/encyclopedia/Vapor-pressure
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FACTOID # 15: A mere 0.8% of West Virginians were born in a foreign country.
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Encyclopedia > Vapor pressure
Vapor pressure is the pressure of a vapor in equilibrium with its non-vapor phases. All solids and liquids have a tendency to evaporate to a gaseous form, and all gases have a tendency to condense back. At any given temperature, for a particular substance, there is a partial pressure at which the gas of that substance is in dynamic equilibrium with its liquid or solid forms. This is the vapor pressure of that substance at that temperature. The use of water pressure - the Captain Cook Memorial Jet in Lake Burley Griffin in Canberra, Australia. ... Vapor (US English) or vapour (British English) is the gaseous state of matter. ... In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, and chemical equilibrium. ... In the physical sciences, a phase is a set of states of a macroscopic physical system that have relatively uniform chemical composition and physical properties (i. ... For other uses, see Solid (disambiguation). ... A liquid will usually assume the shape of its container A liquid is one of the main states of matter. ... Evaporation is the process whereby atoms or molecules in a liquid state (or solid state if the substance sublimes) gain sufficient energy to enter the gaseous state. ... This article does not cite any references or sources. ... In the physical sciences, a phase is a set of states of a macroscopic physical system that have relatively uniform chemical composition and physical properties (i. ... In a mixture of ideal gases, each gas has a partial pressure which is the pressure which the gas would have if it alone occupied the volume. ...
In meteorology, the term vapor pressure is used to mean the partial pressure of water vapor in the atmosphere, even if it is not equilibrium,[1] and the equilibrium vapor pressure is specified as such. Meteorologists also use the term saturation vapor pressure to refer to the equilibrium vapor pressure of water or brine above a flat surface, to distinguish it from equilibrium vapor pressure which takes into account the shape and size of water droplets and particulates in the atmosphere.[2] This article is about our first definition of vapor pressure, or what meteorologists would call equilibrium vapor pressure. // Meteorology (from Greek: μετÎωÏον, meteoron, high in the sky; and λόγος, logos, knowledge) is the interdisciplinary scientific study of the atmosphere that focuses on weather processes and forecasting. ... In a mixture of ideal gases, each gas has a partial pressure which is the pressure which the gas would have if it alone occupied the volume. ... It has been suggested that multiple sections of steam be merged into this article or section. ... The saturation vapor pressure is the vapor pressure of water when air is saturated with water (having the maximum amount of water vapor that air can hold for a given temperature and pressure). ... For the sports equipment manufacturer, see Brine, Corp. ... The saturation vapor pressure is the vapor pressure of water when air is saturated with water (having the maximum amount of water vapor that air can hold for a given temperature and pressure). ...
Equilibrium vapor pressure is an indication of a liquid's evaporation rate. It relates to the tendency of molecules and atoms to escape from a liquid or a solid. A substance with a high vapor pressure at normal temperatures is often referred to as volatile. The higher the vapor pressure of a material at a given temperature, the lower the boiling point. 3D (left and center) and 2D (right) representations of the terpenoid molecule atisane. ... For other uses, see Atom (disambiguation). ... The ability of a liquid to evaporate quickly and at relatively low temperatures. ... Italic text This article is about the boiling point of liquids. ...
The vapor pressure of any substance increases non-linearly with temperature according to the Clausius-Clapeyron relation. The boiling point of a liquid is the temperature where the vapor pressure equals the ambient atmospheric pressure. At the boiling temperature, the vapor pressure becomes sufficient to overcome atmospheric pressure and lift the liquid to form bubbles inside the bulk of the substance. The Clausius-Clapeyron relation, in thermodynamics, is a way of characterizing the phase transition between two states of matter, such as solid and liquid. ...
The most common unit for vapor pressure is the torr. 1 torr = 1 mm Hg (one millimeter of mercury). The international unit for pressure is: 1 pascal = a force of 1 newton per square meter = 10 dyn/cm² = 0.01 mbar= 0.0075 mmHg = 0.00000969 atm= 0.0003545414 psi . The torr (symbol: Torr) or millimeter of mercury (mmHg) is a non-SI unit of pressure. ... The torr is a unit of pressure. ... The pascal (symbol: Pa) is the SI derived unit of pressure or stress (also: Youngs modulus and tensile strength). ... The newton (symbol: N) is the SI derived unit of force, named after Sir Isaac Newton in recognition of his work on classical mechanics. ... In physics, the dyne is a unit of force specified in the centimetre-gram-second (cgs) system of units, symbol dyn. One dyne is equal to exactly 10-5 newtons. ... A millibar (mbar, also mb) is 1/1000th of a bar, a unit for measurement of pressure. ... Standard atmosphere (symbol: atm) is a unit of pressure. ... A pressure gauge reading in PSI (red scale) and kPa (black scale) The pound-force per square inch (symbol: lbf/in²) is a non-SI unit of pressure based on avoirdupois units. ...
Equilibrium vapor pressure of solids
Equilibrium vapor pressure can be defined as the pressure reached when a condensed phase is in equilibrium with its own vapor. In the case of an equilibrium solid, such as a crystal, this can be defined as the pressure when the rate of sublimation of a solid matches the rate of deposition of its vapor phase. For most solids this pressure is very low, but some notable exceptions are naphthalene, dry ice (the vapor pressure of dry ice is 5.73 MPa (831 psi, 56.5 atm) at 20 degrees Celsius, meaning it will cause most non-ventilated containers to explode if sealed inside), and ice. Ice will still continue to disappear even though the ambient temperature is below the freezing point of water. All solid materials have a vapor pressure. However, due to their often extremely low values, measurement can be rather difficult. Typical techniques include the use of thermogravimetry and gas transpiration. For other uses, see Crystal (disambiguation). ... Sublimation of an element or substance is a conversion between the solid and the gas phases with no intermediate liquid stage. ... Naphthalene (not to be confused with naphtha) (also known as naphthalin, naphthaline, tar camphor, white tar, albocarbon, or naphthene), is a crystalline, aromatic, white, solid hydrocarbon, best known as the primary ingredient of mothballs. ... Dry ice is a genericized trademark for solid (frozen) carbon dioxide. ... Thermogravimetry (also knows by acronym TG and obsolete names thermo-gravimetry, thermogravimmetry, thermography) is a branch of physical chemistry, materials research, and thermal analysis. ...
Relation between solid and liquid vapor pressures
It may be noted that the vapor pressure of a substance in liquid form is usually different from the vapor pressure of the same substance in solid form. If the temperature is such that the vapor pressure of the liquid is higher than that of the solid, liquid will vaporize but vapor will condense to a solid, i.e. the liquid is freezing. If the temperature is such that the vapor pressure of the liquid is lower than that of the solid, solid will vaporize but vapor will condense to a liquid, i.e. the solid is melting. At the temperature that equalizes the two vapor pressures, an equilibrium exists between solid and liquid phases. This temperature is referred to as the melting point. In physics, melting is the process of heating a solid substance to a point (called the melting point) where it turns into a liquid. ... In the physical sciences, a phase is a set of states of a macroscopic physical system that have relatively uniform chemical composition and physical properties (i. ... The melting point of a crystalline solid is the temperature at which it changes state from solid to liquid. ...
Water vapor pressure
Main article: Vapor pressure of water
Graph of water vapor pressure versus temperature. Note that at the boiling point of 100°C, the vapor pressure equals the standard atmospheric pressure of 760 Torr.
Water, like all liquids, starts to boil when its vapor pressure reaches its surrounding pressure. At higher elevations the atmospheric pressure is lower and water will boil at a lower temperature. The boiling temperature of water for pressures around 100 kPa can be approximated by Vapour pressure of water can be used in many experiments, particularly experiments relating to gases. ... Image File history File links Water_vapor_pressure_graph. ... Image File history File links Water_vapor_pressure_graph. ... Impact from a water drop causes an upward rebound jet surrounded by circular capillary waves. ... The pascal (symbol: Pa) is the SI derived unit of pressure or stress (also: Youngs modulus and tensile strength). ...
$T_v = 100 + 0.0002772 cdot (p - 101000) - 1.24 cdot 10^{-9} cdot (p - 101000)^2$
where the temperature Tv is in degrees Celsius and the pressure p is in pascals. One gets the vapor pressure by solving this equation for p. Celsius is, or relates to, the Celsius temperature scale (previously known as the centigrade scale). ... The pascal (symbol: Pa) is the SI derived unit of pressure or stress (also: Youngs modulus and tensile strength). ...
In meteorology, the international standard for the vapour pressure of water over a flat surface is given by the Goff-Gratch equation. Vapour pressure of water can be used in many experiments, particularly experiments relating to gases. ... The Goff-Gratch Equation is the international standard for the computation of saturation vapor pressure. ...
Partial pressures
Raoult's law gives an approximation to the vapor pressure of mixtures of liquids. It states that the activity (pressure or fugacity) of a single-phase mixture is equal to the mole-fraction-weighted sum of the components' vapor pressures: In chemistry, Raoults law states that the vapor pressure of mixed liquids is dependent on the vapor pressures of the individual liquids and the molar vulgar fraction of each present in solution. ... Fugacity is a measure of the tendency of a substance to prefer one phase (liquid, solid, gas) over another. ...
ptot = ∑ piχi i
where p is vapor pressure, i is a component index, and χ is a mole fraction. The term piχi is the vapor pressure of component i in the mixture. Raoult's Law is applicable only to non-electrolytes (uncharged species); it is most appropriate for non-polar molecules with only weak intermolecular attractions (such as London forces). In mathematics, an index is a superscript or subscript to a symbol. ... The mole fraction is one way of expressing the relative concentration of a given species. ... The title given to this article is incorrect due to technical limitations. ...
Systems that have vapor pressures higher than indicated by the above formula are said to have positive deviations. Such a deviation suggests weaker intermolecular attraction than in the pure components, so that the molecules can be thought of as being "held in" the liquid phase less strongly than in the pure liquid. An example is the azeotrope of approximately 95% ethanol and water. Because the azeotrope's vapor pressure is higher than predicted by Raoult's law, it boils at a temperature below that of either pure component. This article needs more context around or a better explanation of technical details to make it more accessible to general readers and technical readers outside the specialty, without removing technical details. ...
There are also systems with negative deviations that have vapor pressures that are lower than expected. Such a deviation is evidence for stronger intermolecular attraction between the constituents of the mixture than exists in the pure components. Thus, the molecules are "held in" the liquid more strongly when a second molecule is present. An example is a mixture of trichloromethane (chloroform) and 2-propanone (acetone), which boils above the boiling point of either pure component.
Examples of vapor pressures
gas vapor pressure
(bar)
vapor pressure
(mmHg)
Temperature
Helium 1 750 @ -269.15 °C
Propane 22 16500 @ 55 °C
Butane 2.2 1650 @ 20 °C
Carbonyl sulfide 12.55 9412 @ 25 °C
Acetaldehyde 0.987 740 @ 20 °C
Freon 113 0.379 284 @ 20 °C
Methyl isobutyl ketone 0.02648 19.86 @ 25 °C
Tungsten 0.001 0.75 @ 3203 °C
General Name, Symbol, Number helium, He, 2 Chemical series noble gases Group, Period, Block 18, 1, s Appearance colorless Standard atomic weight 4. ... Propane is a three-carbon alkane, normally a gas, but compressible to a liquid that is transportable. ... Butane, also called n-butane, is the unbranched alkane with four carbon atoms, CH3CH2CH2CH3. ... Except where noted otherwise, data are given for materials in their standard state (at 25 °C, 100 kPa) Infobox disclaimer and references Carbonyl sulfide is a colourless gas at room temperature with an unpleasant odor. ... R-phrases , , S-phrases , , , Flash point −39 °C Autoignition temperature 185 °C RTECS number AB1925000 Supplementary data page Structure and properties n, εr, etc. ... Freon is a trade name for a group of chlorofluorocarbons used primarily as a refrigerant. ... A pollutant that the government wants added to ethanol alcohol to prevent it from being used as a beverage, but only as a vehicle fuel instead. ... For other uses, see Tungsten (disambiguation). ...
Absolute humidity is the mass of water vapor in a given volume of air or gas, expressed by weight and usually measured in grams per cubic meter, though grains per cubic foot has also been used in the United States. ... The Clausius-Clapeyron relation, in thermodynamics, is a way of characterizing the phase transition between two states of matter, such as solid and liquid. ... This does not adequately cite its references or sources. ... Vapor pressure of water can be used in many experiments, particularly experiments relating to gases. ...
Results from FactBites:
Vapor Pressure (610 words) The temperature at which the vapor pressure is equal to the atmospheric pressure is called the boiling point. The pressure of this equilibrium is called the saturation vapor pressure. The boiling point is defined as the temperature at which the saturated vapor pressure of a liquid is equal to the surrounding atmospheric pressure.
Vapor pressure - Wikipedia, the free encyclopedia (593 words) Vapor pressure is the pressure of a vapor in equilibrium with its non-vapor phases. This is the equilibrium vapor pressure or saturation vapor pressure of that substance at that temperature. It may be noted that the vapor pressure of a substance in liquid form is usually different from the vapor pressure of the same substance in solid form.
More results at FactBites »
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2020-01-21 21:06:54
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https://doduykhuong.com/2016/08/08/how-to-use-earned-value-management-in-primavera-p6/
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Primavera support you to control project performance by Earned Value Management technique.
I will show you how to use it in Primavera.
We have a simple project to finish casting 10 column.
• 1 column / day.
• 100 $/ column Each activity is assigned resource A. First we have to create baseline for this project. Go to Project -> Maintain Baselines. Click on Add and OK to create a baseline. We will assign this baseline for our project. Go to Project -> Assign Baselines. Click on Project Baseline and select our baseline. Then click OK. We can show columns as display in below picture to analyze Earned Value Management. Right click on Activity Table -> Column. Select columns in Earned Value group. At the end of day 5 of the project: How many column should have been built? (This is the Planned Value) The answer is 5 column (This value is automatically calculated by using Baseline we assign previously. Baseline said that Project’s duration is 10 days, today is 5th day, so 5 column should be completed) PV = 5 x 100 = 500$
How many column have actually been built? (This is the Earned Value)
We receive report from construction site and it said only 3 columns finish.
(This is value based on the Performance % Complete which is equal to Activity % Complete by default)
EV = 3 x 100 = 300 $How much did it cost to build those three column? (This is Actual Cost) We receive report from Accountant department and it said 200$ / column.
(This value is based on Actual Units)
AC = 3 x 200 = 600 $Then we can show “Schedule Variance” and “Cost Variance”. Schedule Variance SV = EV – PV = 300 – 500 = -200$
A negative number indicates that the project is behind schedule.
1 day for 1 column for 100 $. So it means we are late 2 days. Cost Variance CV = EV – AC = 300 – 600 = -300$
A negative number indicates that the project is over budget.
It means we are currently over budget 300 $. Then we can show “Estimate to Complete” and “Estimate at Completion” ETC = BAC – EV = 1000 – 300 = 700$
EAC = ETC + AC = 700 + 600 = 1300 $According to PMI standard, we should take CPI into account to calculate these 2 values. So we can change the way Primavera calculate by : Go to WBS window -> Earned Value tab. Check on PF = 1 / Cost Performance Index Now ETC = $\frac{BAC - EV}{CPI}$$\frac{700}{0.5}$ = 1400$
( CPI = $\frac{EV}{AC}$ = $\frac{300}{600}$ = 0.5 )
EAC = ETC + AC = 1400 + 600 = 2000 $However you have to configure this option at the beginning of project (before update anything). Otherwise it doesn’t affect. We can also make diagram report Click on Activity Usage Profile button. Right click on the diagram and select Activity Usage Profile Option Check in option as display in below picture Now you can see the Earned Value diagram. You will use the cost axis on the right side (not the left one). I also attach a theory diagram of Earned Value Management for your reference. With Earned Value Management in Primavera we can easily see our project performance and forecast the total cost at the end of project. Advertisements ## 31 thoughts on “How to use Earned Value Management in Primavera P6” ## Add yours 1. KH says: Dear, Very nice. Really i always wait for the post. Please post HOW TO PREPARE RECOVERY SCHEDULE (Actual cost = plan cost) after delay or EOT) in other words bring plan % or cost equal to Earned Value. Like 1. Hi, Thank you for your compliment. I will keep your suggestion in mind and I might do it in future 🙂 Like 1. Mudassar Malik says: Thanku very much sir 😍 Like 2. Padmanabhan says: Hi, Appreciate the update on recovery schedule. Like 2. Mudassar Malik says: Yes i have also some issue’s about it Like 2. Anonymous says: Simple and well explained. Thanks for sharing. Like 3. N.Balaji Prasada Rao says: Dear sir, Excellent.Refreshed with simple example yet covered all the core topics in earned value concepts.Thanks for sharing. Like 4. Mayur says: Explained a simple way.. Nice article.. Like 5. Evgeny says: Thanks for the nice article. I have a question regarding ETC: What formula is used to calculate the ETC for each unit of time and draw the curve? Is this formula based on the PV? Like 1. Hi Evgeny, The value to draw the curve is also the value appear on the activity table (The Estimation At Complete column). The formula is based on what we choose in the Earned value tab. Do we consider CPI or not. Like 6. Doug says: Nicely done, Do. Thank you for sharing. Like 7. Langston says: Well done and straight forward example. Excellent layouts. Like 8. Amir Syarifuddin says: Thank you for your compliment. I will keep your suggestion in mind and I might do it in future Like 9. Anonymous says: Thanks for sharing Like 10. BEDE says: Thank you for sharing Like 11. M.Nabeel says: Excellent work Like 12. Nelson says: Simple and clear, well done. Like 13. PHH says: Thank you, IOU a cup of coffee!! Like 14. I Ahmed says: Explained in a simple way and thanks for sharing. Like 15. Thank you for useful explanation of EV. Everything about EV now are very clear. Can you share this explanations in pdf, please? Like 1. Hi. In the “Share this” section, click on “More” button and you can see the Print button. Click on Print you can have the pdf version. Like 16. Anonymous says: Hi, Great work Keep it up. Please share Primavera P6 Analytics installation steps. Many thanks Imran Like 17. Arita says: Hi How to upload Cost in P6? I want to upload cost for Clients per tender value and for progress billing per month based on progress, not actual cost from Resources Like 18. Rolando B. Digamon says: very straight forward…. Thanks Like 19. “Schedule Variance SV = EV – PV = 300 – 500 = -200$
A negative number indicates that the project is behind schedule.”
sir the negative sign is why not shown in primavera p6 to indicate the project is behind the schedule .
Like
1. Hi. In P6, the cost show in parenthesis is negative value.
Like
20. Marcin Nowak says:
Hi,
Thanks for the nice article.
I hope you still read this blog 🙂
My question is: Why after update “Budgeted units” of A resource in task “Column 1” grown from 10 to 20? Should Budgeted Units not frozen?
Like
1. Hi Marcin,
Good question 🙂
I did not notice that change from 10 to 20 until seeing your question.
Because Budget value is now derived from Baseline so I even don’t look at the “original” Budget field in Resource tab.
I still don’t have the exact answer for you. I guess because Actual value is 20, P6 will automatically increase Budget value to 20.
When I have time, I will check this carefully.
Like
1. Marcin Nowak says:
Hi,
Thanks for the reply and I am waiting patiently.
You wrote, that change of the way how Primavera calculate ETC is possible before we put update. I’ve problem with this and my Primavera react only when I chose PF = (user value) other do nothing… Did you know what is wrong with it?
Like
21. Hi,
With the information you gave, I can’t identify the problem. It usually takes lots of time and effort to figure out the problem and solution.
Like
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2019-11-21 03:04:48
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https://math.stackexchange.com/questions/202447/retraction-of-the-m%C3%B6bius-strip-to-its-boundary
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# Retraction of the Möbius strip to its boundary
Prove that there is no retraction (i.e. continuous function constant on the codomain) $r: M \rightarrow S^1 = \partial M$ where $M$ is the Möbius strip.
I've tried to find a contradiction using $r_*$ homomorphism between the fundamental groups, but they are both $\mathbb{Z}$ and nothing seems to go wrong...
• I reformatted the formulas. See math notation guide. – user147263 Sep 27 '14 at 17:15
• The key here is functoriality of $\pi_1$. – Daniel Valenzuela Sep 27 '14 at 18:30
• Thank you for reformatting u.u – Keith Sep 28 '14 at 12:27
If $\alpha\in\pi_1(\partial M)$ is a generator, its image $i_*(\alpha)\in\pi_1(M)$ under the inclusion $i:\partial M\to M$ is the square of an element of $\pi_1(M)$, so that if $r:M\to\partial M$ is a retraction, $\alpha=r_*i_*(\alpha)$ is also the square of an element of $\pi_1(\partial M)$. This is not so.
(For all this to work, one has to pick a basepoint $x_0\in\partial M$ and use it to compute both $\pi_1(M)$ and $\pi_1(\partial M)$)
• why $i_* (\alpha)$ is the square of an element in $\pi_1 (M)$? – user32847 Sep 26 '12 at 7:31
• Essentially because it turns around the band twice, and a generator of the fundamental group of the band is a curve which turns around it only once. Notice that $i$ is a map one knows, so one can completely compute the morphism $i_*:\pi_1(\partial M,x_0)\to\pi_1(M,x_0)$. – Mariano Suárez-Álvarez Sep 26 '12 at 17:36
• Why is the following true: "if $r:M\to\partial M$ is a retraction, $\alpha=r_*i_*(\alpha)$ is also the square of an element of $\pi_1(\partial M)$"? – fierydemon Mar 15 '16 at 1:06
• @AyushKhaitan this is because $i_*(\alpha) = \beta^2$ and as we are dealing with homomorphism this means $r_*i_*(\alpha)$ = r_*(\beta)^2$– B.A Feb 22 '18 at 5:26 • @MarianoSuárez-Álvarez what is the point of computing$\pi_1(M)$and$\pi_1(\partial M)$? even if there is retraction, it is not a homotopy equivalence. Can we find a contradiction by these computations? Thanks! – PerelMan Feb 22 '19 at 17:35 Let $$M$$ be the möbius strip defined by $$M:=\frac{[0,1]\times [0,1]}{(0,t)\sim(1,1-t)}$$ with quotient map $$q:[0,1]\times [0,1]\to M$$. Let $$B:=q\big(\{(s,k):0\leq s\leq 1,k=0,1\}\big)$$ be the boundary circle and $$C:=q\big(\big\{\big(s,\frac{1}{2}\big):0\leq s\leq 1\big\}\big)$$ be the central circle. Consider the inclusion map $$i:B\hookrightarrow M$$. Also, consider the retraction $$f:M\to C$$ defined by $$f\big([(s,t)]\big)=\big[\big(s,\frac{1}{2}\big)\big]$$. Now the map, $$f\circ i:B\to C$$ is a $$2$$-fold covering. Hence, $$f_*\circ i_*\big(\pi_1(B)\big)$$ is an index two subgroup of $$\pi_1(C)$$. Let $$j:C\hookrightarrow M$$ be the inclusion. Then, $$f\circ j=\text{Id}_C$$. So that, $$f_*\circ j_*=\big(\text{Id}_C\big)_*=\text{Id}_{\pi_1(C)}$$. Next note that, $$H:M\times[0,1]\to M$$ defined by $$H:\big([(s,t)],t'\big)\mapsto\big[\big(s,\frac{1}{2}t'+(1-t')t\big)\big];\forall 0\leq s,t,t'\leq 1$$ is a homotopy between $$H(-,0)=\text{Id}_M$$ and $$H(-,1)=j\circ f$$. Hence, $$\text{Id}_{\pi_1(M)}=\big(\text{Id}_M\big)_*=\big(j\circ f\big)_*=j_*\circ f_*$$. Therefore, $$j_*$$ is an isomorphism between $$\pi_1(C)$$ and $$\pi_1(M)$$. Now, $$i_*=\big(\text{Id}_M\big)_*\circ i_*=\big(j\circ f\big)_*\circ i_*=\big(j\circ f\circ i\big)_*=j_*\circ\big(f\circ i\big)_*$$. Hence, $$i_*\big(\pi_1(B)\big)$$ is an index two subgroup of $$\pi_1(M)$$. Now if possible let, there is a retraction $$r:M\to B$$. Then $$r\circ i=\text{Id}_B$$. Then, $$r_*\circ i_*=\text{Id}_{\pi_1(B)}$$. Note that, both $$B$$ and $$C$$ are circles. So $$\pi_1(B)$$ and $$\pi_1(C)$$ are infinite cyclic group. So that, $$\pi_1(M)$$ is also an infinite cyclic group. Let $$b$$ be a generator of $$\pi_1(B)$$ and $$m$$ be a generator of $$\pi_1(M)$$. Since, $$i_*\big(\pi_1(B)\big)$$ is an index two subgroup of $$\pi_1(M)$$, we have $$i_*(b)$$ equals to either $$2m$$ or $$-2m$$, here all group structure are written additive way. But from $$r_*\circ i_*=\text{Id}_{\pi_1(B)}$$ we have, $$b=\text{Id}_{\pi_1(B)}(b)=r_*\big(i_*(b)\big)=r_*\big(\pm 2m\big)=\pm 2r_*(m)$$. Since, $$r_*(m)\in \pi_1(B)$$ we have some integer $$n$$ such that, $$r_*(m)=nb$$. Hence, $$b=\pm2nb$$, which is impossible. • I have not written with any new technique to solve this problem, I tried to write as explicit as possible. – 0-th User Sumanta Sep 17 '19 at 16:49 • Wow it was a question posted long ago...I remember solviing it in a similar way you have done. Thank you for your answer anyway. – Keith Sep 17 '19 at 19:01 Suppose there were a retract$r:M\rightarrow \partial M$. By definition, this means that$r\circ i=\mathrm{id}\, _{\partial M}$, where$i:\partial M\rightarrow M$is the inclusion. From functoriality, it follows that$r^*\circ i^*=\mathrm{id}\, _{\pi _1(\partial M)}$, where$f^*$denotes the induced map of fundamental groups. Thus,$r^*:\pi _1(M)\rightarrow \pi _1(\partial M)$is surjective. However,$\pi _1(M)\cong \mathbb{Z}\cong \pi _1(\partial M)$and$r^*(n)=2n$, which is not surjective: a contradiction. Thus, there can be no such retract. To see that$r^*(n)=2n$, I think it is easiest to view the Möbius strip as a quotient of the unit square in$\mathbb{R}^2$, obtained by identifying the left and right sides with the opposite orientation. Intuitively, if you go around the Möbius band once you, the projection onto the boundary goes around twice (draw a picture for yourself). • Shouldn't those$*$'s be lower? – Al Jebr Feb 12 '19 at 4:02 • Isn't it$i^\ast (n)=2n$? I don't understand why going around the band once "projects" to going around the boundary twice – Evan Aug 10 '19 at 16:47 You can also prove this using homology, but it's somewhat more effort. The basic idea is that, if $$B$$ is the boundary circle and $$r:M\rightarrow M$$ is a retraction onto $$B$$ then the inclusion map also induces an injection $$i_*:H_1(B)\rightarrow H_1(M)$$ (to see this, apply $$r_*$$ to $$i_*\alpha=i_*\beta$$). Thus we have an exact sequence $$0\rightarrow H_1(B)\xrightarrow{i_*}H_1(M)\xrightarrow{q_*} H_1(M,B)\rightarrow 0$$ coming from the reduced long exact sequence for the pair $$(M,B)$$. We know that $$H_1(B)$$ and $$H_1(M)$$ are both $$\mathbb Z$$ (because $$B=S^1$$ and $$M$$ deformation retracts onto its central circle) and, since $$(M,B)$$ is a good pair, $$H_1(M,B)\cong H_1(M/B)$$. But $$M/B=\mathbb R\mathbb P^2$$, as can be seen by their cell decompositions, where the pink indicates the boundary circle $$B$$: thus $$H_1(M,B)\cong \mathbb Z/2\mathbb Z$$. The fact that $$r$$ is a retraction means that the above sequence splits, as $$r_*:H_2(M)\rightarrow H_2(B)$$ composed with $$i_*$$ is the identity. This is a contradiction. • Thanks for this answer! I appreciated the unique perspective. But one question: Why does$r$being a retraction mean that the above sequence splits? I understand that composing$r_*$with$i_*$is the identity. – HaKuNa MaTaTa Jun 11 '19 at 22:33 • This is just the splitting lemma, see en.wikipedia.org/wiki/Splitting_lemma. – Arbutus Jun 23 '19 at 17:53 • What is$H_1(X)$? – kam Feb 6 at 7:52 • @kam, Fixed it - thanks. – Arbutus Feb 6 at 12:55 For each$\alpha\in\partial M$, let$\gamma_\alpha$be the closed loop in$M$that starts at$\alpha$, goes directly across the strip to its antipode and then halfway around the boundary to its starting point in positive direction. Then$\alpha\mapsto\gamma_\alpha$is a homotopy -- in particular every$\gamma_\alpha$has the same homotopy class. On the other hand, if$x$and$y$are antipodes, then when we form$\gamma_x+\gamma_y$, the "directly across" sections cancel out, and the concatenated curve is homotopic to a single turn around the entire boundary. So the homotopy class of$r(\gamma_x+\gamma_y)$in$\partial M$is$1$. On the other hand,$r$ought to induce a homomorphism between the homotopy groups, but$1$is not twice anything in$\mathbb Z$, which is a contradiction. • (Annoyingly, one has to do this with based loops :-/ ) – Mariano Suárez-Álvarez Sep 25 '12 at 21:39 • @Mariano: Reparameterizing all of the loops to be based at$x$and then throwing away the$\alpha$s where$\gamma_\alpha$does not already pass through$x$ought to take care of that. – hmakholm left over Monica Sep 25 '12 at 21:43 • @HenningMakholm is$\pi_1(\partial M)=1$or I misunderstood something? – PerelMan Feb 22 '19 at 17:42 • @PerelMan:$\partial M$is a circle, so$\pi_1(\partial M)\cong\mathbb Z$. When I write$1$I mean$1\in\mathbb Z\$. – hmakholm left over Monica Feb 22 '19 at 18:42
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2020-11-30 02:08:25
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http://physicsinventions.com/nuclear-decay-of-a-mixture-of-p-and-s-2009-usapho-question-a2/
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1. The problem statement, all variables and given/known data
A mixture of $^{32}P$ and $^{35}S$ (two beta emitters widely used in biochemical research) is placed next to a detector and allowed to decay, resulting in the data (attached) below. The detector has equal sensitivity to the beta particles emitted by each isotope, and both isotopes decay into stable daughters.
You should analyze the data graphically. Error estimates are not required.
a.Determine the half-life of each isotope.
$^{35}S$ has a signicantly longer half-life than $^{32}P$.
b.Determine the ratio of the number of $^{32}P$ atoms to the number of $^{35}S$ atoms in the original sample.
2. Relevant equations
$\frac{dN}{dt}=-λΝ$
3. The attempt at a solution
$|\frac{dN}{dt}|=N_P λ_P+N_S λ_S (1)$
$T_{\frac{1}{2}S} \gg T_{\frac{1}{2}P}⇔λ_S \ll λ_P(2)$
$N=N_0 e^{-λt} (3)$
But I can’t figure out how to use them in order to get a result.
Attached Images
Experimental Data.png (22.4 KB)
http://ift.tt/1jGKVLQ
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2018-11-15 14:27:34
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https://stacks.math.columbia.edu/tag/032R
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Definition 10.162.1. Let $R$ be a ring.
1. We say $R$ is universally Japanese if for any finite type ring map $R \to S$ with $S$ a domain we have that $S$ is N-2 (i.e., Japanese).
2. We say that $R$ is a Nagata ring if $R$ is Noetherian and for every prime ideal $\mathfrak p$ the ring $R/\mathfrak p$ is N-2.
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2022-10-04 07:08:38
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http://math.stackexchange.com/questions/118002/find-the-derivative-of-the-following-equation
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# Find the derivative of the following equation..
I have a question in my manual and I am not able to answer it, I'd appreciate some help please.
Find $\dfrac{dy}{dx}$ if $2x^2y + 3xy^2 = 6$
I'm confused with = 6.. Thanks !
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If LHS = RHS, it stands to reason that (derivative of LHS with respect to $x$) = (derivative of RHS with respect to $x$). Then solve for $\frac{dy}{dx}$. – arjafi Mar 8 '12 at 19:53
The derivative of $6$ is $0$. – The Chaz 2.0 Mar 8 '12 at 19:54
Nex, what would you think about editing your question to include the work that you have done on the left-hand side? Then we can help you finish off the solution. – The Chaz 2.0 Mar 8 '12 at 20:39
Consider $y$ as a function of $x$ defined implicitly by $$\begin{equation*} 2x^{2}y+3xy^{2}=6. \end{equation*}$$ The derivatives of both sides should be equal. The derivative of the RHS is $0$, because the derivative of a constant is $0$. As for the derivative of the LHS, by the sum and product rules, is given by
$$\begin{eqnarray*} \frac{d}{dx}\left( 2x^{2}y+3xy^{2}\right) &=&\frac{d}{dx}\left( 2x^{2}y\right) +\frac{d}{dx}\left( 3xy^{2}\right) \\ &=&2\frac{d}{dx}\left( x^{2}y\right) +3\frac{d}{dx}\left( xy^{2}\right). \end{eqnarray*}$$ Therefore $$\begin{equation*} 2\frac{du}{dx}+3\frac{dv}{dx}=0, \end{equation*}$$ where $u=x^{2}y$ and $v=xy^{2}$. Compute the total derivatives to evaluate $du/dx$ and $dv/dx$. They can be expressed in terms of the partial derivatives and the derivative $dy/dx$ you want to find as follows. $$\begin{eqnarray*} \frac{du}{dx} &=&\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y} \frac{dy}{dx}=2xy+x^{2}\frac{dy}{dx} \\ \frac{dv}{dx} &=&\frac{\partial v}{\partial x}+\frac{\partial v}{\partial y} \frac{dy}{dx}=y^{2}+2xy\frac{dy}{dx}. \end{eqnarray*}$$ To obtain $dy/dx$ combine the above results and solve the resulting equation for $dy/dx$.
Remark: It is not necessary to introduce the functions $u$ and $v$. I have introduce them to illustrate the general case, but you can do the computation directly $$\begin{equation*} 2\left( 2xy+x^{2}\frac{dy}{dx}\right) +3\left( y^{2}+2xy\frac{dy}{dx}\right) =0. \end{equation*}$$
Added: This method is a particular case of a general one. If you have an implicit function $F(x,y)=0$ we can find $dy/dx$ by differentiating both sides of the implicit equation and solve for $dy/dx$ $$\begin{eqnarray*} \frac{dF}{dx} &=&\frac{\partial F}{\partial x}\frac{dx}{dx}+\frac{\partial F }{\partial y}\frac{dy}{dx}=0 \\ &\Rightarrow &\frac{\partial F}{\partial x}+\frac{\partial F}{\partial y} \frac{dy}{dx}=0 \\ &\Leftrightarrow &\frac{dy}{dx}=-\frac{\partial F}{\partial x}/\frac{ \partial F}{\partial y}. \end{eqnarray*}$$
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Wow, thanks a lot for this answer, it's really complete and I perfectly understand now. – Nex Mar 8 '12 at 22:45
@Nex: You are welcome! – Américo Tavares Mar 8 '12 at 22:46
@Nex: I added the general case $F(x,y)=0$. – Américo Tavares Mar 8 '12 at 22:55
Hint: This is almost solution to your question:
If equation is $F(x,y)= c$ for some constant $c$, then $\frac{d}{dx}(F(x,y))= 0$. Use derivative function is linear and
If somewhere you come across the term $x^ny^m$ we have $$\frac{d}{dx}x^ny^m= x^n.\frac{d}{dx}y^m+ y^m\frac{d}{dx}x^n$$
and $\frac{d}{dx}y^m= my^{m-1}\frac{dy}{dx}$
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2016-05-26 03:01:24
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https://studydaddy.com/question/magnesium-reacts-with-nitrogen-to-form-magnesium-nitride-the-chemical-formula-fo
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Waiting for answer This question has not been answered yet. You can hire a professional tutor to get the answer.
QUESTION
# Magnesium reacts with nitrogen to form magnesium nitride. The chemical formula for this reaction is Mg+N_2-> MgN_2. What is the product, or what are the products, of this reaction?
The formula for magnesium nitride is Mg_3N_2.
As do many active metals, magnesium nitride can be formed by heating the metal (fiercely!) under dinitrogen gas:
3Mg + N_2 rarr Mg_3N_2
As a nitride it can by hydrolyzed by water to form magnesium hydroxide and ammonia gas:
Mg_3N_2(s) + 6H_2O(l) rarr 3Mg(OH)_2(aq) + 2NH_3(aq)
LEARN MORE EFFECTIVELY AND GET BETTER GRADES!
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2019-05-24 21:46:18
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http://www.ck12.org/trigonometry/Cofunction-Identities-and-Reflection/lesson/Cofunction-Identities-and-Reflection/r8/
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<meta http-equiv="refresh" content="1; url=/nojavascript/">
You are viewing an older version of this Concept. Go to the latest version.
# Cofunction Identities and Reflection
## Based on complements of angles.
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Progress
Practice Cofunction Identities and Reflection
Progress
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Cofunction Identities and Reflection
While toying with a triangular puzzle piece, you start practicing your math skills to see what you can find out about it. You realize one of the interior angles of the puzzle piece is $30^\circ$ , and decide to compute the trig functions associated with this angle. You immediately want to compute the cosine of the angle, but can only remember the values of your sine functions.
Is there a way to use this knowledge of sine functions to help you in your computation of the cosine function for $30^\circ$ ?
Read on, and by the end of this Concept, you'll be able to apply knowledge of the sine function to help determine the value of a cosine function.
### Guidance
In a right triangle, you can apply what are called "cofunction identities". These are called cofunction identities because the functions have common values. These identities are summarized below.
$\sin \theta = \cos(90^\circ-\theta) && \cos \theta = \sin (90^\circ-\theta)\\\tan \theta = \cot(90^\circ-\theta) && \cot \theta = \tan (90^\circ-\theta)$
#### Example A
Find the value of $\cos 120^\circ$ .
Solution: Because this angle has a reference angle of $60^\circ$ , the answer is $\cos 120^\circ = -\frac{1}{2}$ .
#### Example B
Find the value of $\cos (-120^\circ)$ .
Solution: Because this angle has a reference angle of $60^\circ$ , the answer is $\cos (-120^\circ) = \cos 240^\circ = -\frac{1}{2}$ .
#### Example C
Find the value of $\sin 135^\circ$ .
Solution: Because this angle has a reference angle of $45^\circ$ , the answer is $\sin 135^\circ = \frac{\sqrt{2}}{2}$
### Vocabulary
Cofunction Identity: A cofunction identity is a relationship between one trig function of an angle and another trig function of the complement of that angle.
### Guided Practice
1. Find the value of $\sin 45^\circ$ using a cofunction identity.
2. Find the value of $\cos 45^\circ$ using a cofunction identity.
3. Find the value of $\cos 60^\circ$ using a cofunction identity.
Solutions:
1. The sine of $45^\circ$ is equal to $\cos (90^\circ - 45^\circ) = \cos 45^\circ = \frac{\sqrt{2}}{2}$ .
2. The cosine of $45^\circ$ is equal to $\sin (90^\circ - 45^\circ) = \sin 45^\circ = \frac{\sqrt{2}}{2}$ .
3. The cosine of $60^\circ$ is equal to $\sin (90^\circ - 60^\circ) = \sin 30^\circ = .5$ .
### Concept Problem Solution
Since you now know the cofunction relationships, you can use your knowledge of sine functions to help you with the cosine computation:
$\cos 30^\circ = \sin (90^\circ - 30^\circ) = \sin (60^\circ) = \frac{\sqrt{3}}{2}$
### Practice
1. Find a value for $\theta$ for which $\sin \theta=\cos 15^\circ$ is true.
2. Find a value for $\theta$ for which $\cos \theta=\sin 55^\circ$ is true.
3. Find a value for $\theta$ for which $\tan \theta=\cot 80^\circ$ is true.
4. Find a value for $\theta$ for which $\cot \theta=\tan 30^\circ$ is true.
5. Use cofunction identities to help you write the expression $\tan 255^\circ$ as the function of an acute angle of measure less than $45^\circ$ .
6. Use cofunction identities to help you write the expression $\sin 120^\circ$ as the function of an acute angle of measure less than $45^\circ$ .
7. Use cofunction identities to help you write the expression $\cos 310^\circ$ as the function of an acute angle of measure less than $45^\circ$ .
8. Use cofunction identities to help you write the expression $\cot 260^\circ$ as the function of an acute angle of measure less than $45^\circ$ .
9. Use cofunction identities to help you write the expression $\cos 280^\circ$ as the function of an acute angle of measure less than $45^\circ$ .
10. Use cofunction identities to help you write the expression $\tan 60^\circ$ as the function of an acute angle of measure less than $45^\circ$ .
11. Use cofunction identities to help you write the expression $\sin 100^\circ$ as the function of an acute angle of measure less than $45^\circ$ .
12. Use cofunction identities to help you write the expression $\cos 70^\circ$ as the function of an acute angle of measure less than $45^\circ$ .
13. Use cofunction identities to help you write the expression $\cot 240^\circ$ as the function of an acute angle of measure less than $45^\circ$ .
14. Use a right triangle to prove that $\sin \theta=\cos (90^\circ-\theta)$ .
15. Use the sine and cosine cofunction identities to prove that $\tan (90^\circ-\theta)=\cot \theta$ .
### Vocabulary Language: English
Cofunction Identity
Cofunction Identity
A cofunction identity is a relationship between one trig function of an angle and another trig function of the complement of that angle.
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2015-06-03 02:17:13
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https://andrewkay.name/maths/zdb/?bound=2242
|
# ZarankiewiczDB,
an online database of Zarankiewicz numbers.
### Bound for z(26,27)
Searching for matrices of weight greater than 142, all but one scaffold was eliminated without search.
Scaffold: $p=5$, $q=5$, $\mathbf{m} = (4,4,4,4,4)$, $\mathbf{n} = (5,4,4,4,4)$, $m_O = 0$, $n_O = 0$.
(1503.072408529 seconds, score = 0, 16870567 recursive calls, max depth = 21)
upper bound = 142, submitted by Andrew Kay on July 26th, 2016 (link)
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2018-09-24 22:22:19
|
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https://space.stackexchange.com/questions/35549/how-would-2-spacecraft-be-launched-to-be-in-sync-in-opposite-directions
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# How would 2 spacecraft be launched to be in sync in opposite directions?
How would 1 spacecraft be launched to have a prograde orbit and the another be launched retrograde to have a mirrored orbital path around the sun or otherwise? Would the 2 spacecraft launch together at one launch site or at different sites on Earth be more efficient?
How will JWST manage solar pressure effects to maintain attitude and station keep it's unstable orbit?
STEREO A and B, 2006-047A and B, (29510 and 29511) were launched together from "Cape Canaveral Air Force Station in Florida on a Delta II 7925-10L launcher into highly elliptical geocentric orbits".
The elliptical orbits intercepted the Moon's trajectory, and because of a small but carefully design difference in their two positions, one was thrown "forward" into a faster heliocentric orbit, while the other was sent back, not quite free from Earth's gravity field, to meet the Moon again when it was moving in the other direction. There it received a second gravitational assist in the retrograde direction, putting it in a slower heliocentric orbit, relative to Earths.
From this answer to the question Gravitational assists from bodies other than planets:
The Moon has been used as a gravitational assist in different ways.
NOTE: As @notovny pointed out elsewhere these orbits go in the opposite direction in the rotating frame of Earth's orbit, but in the "normal" inertial frame, they both travel around the Sun the same way, in orbits very similar to that of Earth. It's just that one goes slightly faster and the other slightly slower relative to Earth.
But in the rotating frame, they do go in opposite directions. One moves towards Earth's L4, the other towards L5.
See this answer, which links to this NASA page. The first video shows the two STEREO spacecraft use two and three passes by the Moon to leave Earth orbit and enter into an orbit around the sun.
The trajectory is also described in some detail (but sans cool GIF) in this answer.
It would certainly be quite a challenge to fit two JWST-class telescopes on one rocket, but if they were somewhat smaller than JWST, it could be done again in a similar way.
Here is a GIF made from the frames of that .mov file:
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2020-01-26 11:21:36
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https://cometcloud.bitbucket.io/cometworkflow/configure_workflow.html
|
# Configuring Workflow Manager and Autonomic Scheduler¶
The workflow manager is the entry point to the system and it will orchestrate the execution of workflows. Currently, the autonomic scheduler is launched by the workflow manager, although it is totally independent. The autonomic scheduler decides where to schedule tasks, which resources to provision, and keeps the information of the resources.
Next we present a configuration file example, where a workflow manager is going to be deployed in the machine defined in publicIpManager, port portManager. The logs are stored in the file defined by logFile, the Task Manager (aka. Workflow Master) is listening in workflowmasterURI (i.e. machine machine2.domain.com, port 7777). The Autonomic scheduler is going to be deployed in the machine defined in CentralManagerAddress (currently, this property has to be the same than publicIpManager), port CentralManagerPort. Additionally, we define a monitoring interval of 60 seconds, which is used by the autonomic scheduler to monitor resource status and analyze if any actions is required to guarantee QoS. Additionally we can include a default benchmark score that can be used to calculate the speedup of all available resources (speedup = <resourceBenchmarkScore>/<DefaultBenchmarkScore>). More information about this configuration file can be found in WORKFLOW section. For simplicity we keep scripts and examples of the configuration files in the directory simple_run/workflow/
## Requirements¶
This service requires Java runtime 1.7+ and rsync.
## Edit manager.properties file¶
publicIpManager=machine1.domain.com
portManager=8888
logFile=/path/to/logfile/WorkflowManager.log
workflowmasterURI=machine2.domain.com:7777
#resource manager and autonomic scheduler
StartCentralManager=true
CentralManagerPort=7778
MonitorInterval=60
DefaultBenchmarkScore=5757
Note
DefaultBenchmarkScore is only used if the user does not indicate a benchmark score when submitting a workflow and when the metrics service is disabled or it does not have information about the application to be executed.
## Starting the Service¶
Once the configuration is ready, you can start the workflow manager and autonomic scheduler by executing the script startWorkflowManager.sh. This script contains the following code that defines the CLASSPATH and executes the appropriated java class.
export CLASSPATH=../../dist/*:../../lib/*
java -cp \$CLASSPATH tassl.application.workflow.WorkflowManager -propertyFile manager.properties
Note
These services can start regardless the status of other services of our federation. It is not until a workflow is started that this service contacts the Task Manager (aka. Workflow Master).
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2022-06-26 07:58:43
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https://dsp.stackexchange.com/questions/16348/obtaining-power-spectrum-from-acf-fft-using-matlab-and-fftw
|
# Obtaining power spectrum from ACF, FFT using Matlab and FFTW
I am using Matlab R2012b 64-bit on Windows 7 in order to estimate the power spectrum of a simple signal that is:
$$\cos(10t) + \sin(20t)$$defined in the time interval from 0.0 to 10.0
Here is what I did:
1) There was no predefined value of points so I took 1001 points by forming a time array and sampled the function at these points.
2) I calculated the autocorrelation function of the signal with the built-in function "xcorr".
3) I performed FFT directly on the autocorrelation function with no additional options.
4) The resulting data set had imaginary parts(I think this should not be the case since the autocorrelation function is even).
5) I took the absolute value of the FFT of ACF and plotted it as the power spectrum from the ACF after having it shifted with fftshift.
6) I took the FFT of the sinusoidal signal after padding it with 1000 (n-1) zeros.
7) I calculated the power spectrum from the FFT of the signal as the square of the absolute value of the frequency coefficients.
8) I plotted both of them and compared them by assigning an array to difference between them.
9) They turned out to be the same.
Everything seems to be working correctly but my concerns are as follows:
• How can this data be interpreted? I expected the sum and difference frequencies of the individual sinusoidal functions in the signal.
-How can I normalize the output and represent negative frequencies properly since they have no physical significance?
-Does Matlab use FFTW libraries? My main task is to achieve the same results in a C program using these libraries.
I have benefitted from the following URL: Efficiently calculating autocorrelation using FFTs in organizing the post and achieving these results.
I am also confused about the terms power spectrum, power spectral density and energy spectrum. Are they used interchangeably? I would be grateful for an explanation of this phenomenon and normalization procedure.
P.S : The code snippet I have used is as follows and it can reproduce my results:
t=transpose(0:0.01:10);
n=size(t,1);
f=cos(10*t)+sin(20*t);
acf=xcorr(f,f);
FFTacf=fft(acf);
PSacf=abs(FFTacf);
FFTp=fft([f;zeros(n-1,1)]);
PSf=FFTp.*conj(FFTp);
figure;
subplot(2,1,1);
plot(fftshift(PSf));
title('PS from FFT');
subplot(2,1,2);
plot(fftshift(PSacf));
title('PS from ACF');
d=PSacf-PSf;
fprintf('Max error = %6.2f \n', max(abs(d)));
I am also confused about the terms power spectrum, power spectral density and energy spectrum. Are they used interchangeably? I would be grateful for an explanation of this phenomenon and normalization procedure.
EDIT with @jojek`s helpful comment:
power spectrum is NOT the same as power spectral density despite the fact that terms are often mixed up. Units of power spectral density (PSD) are $x^{2}$/Hz, e.g. $m^{2}s^{-2}$/Hz. Power Spectrum (PS) is equal to PSD*(Equivalent Noise Bandwidth of used window funktion) and has units of $x^{2}$. You often normalise it with N, the number of data points (for discrete time series). For physical processes the PSD and PS are symmetric, thus negative frequencies contain no additional information and only positive frequencies are plotted, negative frequencies are discarded.
energy spectrum is a term of spectroscopy: it describes the intensity of a particle beam as a function of particle energy. What you maybe meant is called energy spectral density, which describes the distribution of energy over frequency; it has units of J/Hz
A good overview is here.
I want to add the term amplitude spectrum/amplitude spectral density, which is the square root of the power spectral density.
• I disagree that Power Spectrum and Power Spectral Density are the same thing. Units of PSD are $x^2/\mathrm{Hz}$, whereas the units of PS are $x^2$ (quantity squared). Moreover, PSD and PS are related to each other: $\mathrm{PS} = \mathrm{PSD} \cdot \mathrm{ENBW}$, where ENBW stands for Equivalent Noise Bandwidth of the window function used. – jojek Nov 19 '16 at 13:33
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2019-10-19 01:22:28
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|
https://quizplus.com/quiz/156118-quiz-13-consumer-and-business-credit
|
# Contemporary Mathematics
Mathematics
## Quiz 13 :Consumer and Business Credit
Question Type
Loans made on a continuous basis and billed periodically are known as _____ credit.
Free
Essay
Our answer is revolving credit.
Revolving credit:
Loan made on a continuous basis and billed periodically. Borrower makes minimum monthly payments or more and pays interest on the outstanding balance. This is a form of open-end credit extended by many retail stores and credit card companies.
Tags
Choose question tag
Chris Manning purchased a $7,590 motorcycle with a 36-month installment loan. The monthly payments are$261.44 per month. a. Use the APR formula to calculate the annual percentage rate of the loan. Round to the nearest hundredth of a percent. b. Use the APR tables to verify your answer from part a.
Free
Essay
The given data is the amount financed is $7,590.00 The number of payments is 36 And monthly payment is$261.44
To find the finance charge:
The finance charge can be found by subtracting the amount financed from the total amount of installment payments
That means,
To calculate the finance charge, we must find the total amount of installment payments
To find the total amount of installment payments:
When the amount of the monthly payments is known, the total amount of installment payments can be found by multiplying the monthly payment amount by the number of payments
Since, by given information, the monthly payment amount is $261.44 And the number of payments is 36 For finance charge, So, the finance charge is$1,821.84
a) To find annual percentage rate:
When annual percentage rate (APR) tables are not available, the annual percentage rate can be closely approximated by the formula
Here, I is the finance charge on the loan
P is the principal, or amount financed
And n is the number of months of the loan
By given hypothesis,
By substituting these values in the APR formula,
So, the annual percentage rate of the loan is 14.47%
b) To verify the answer by using APR table:
Calculate the finance charge per $100. So, the finance charge per$100 is $24.00 Using table 13-1, scan down the number of payments column to 36 payments Scan to the right in that number of payments row until you find$24.00, the finance charge per $100 Looking at the top of the column containing the$24.00, you will find the annual percentage rate for the loan to be 14.75%
Therefore, the annual percentage rate of the loan is
And the annual percentage rate by using table is
Tags
Choose question tag
Calculate the average daily balance for January of a charge account with a previous month's balance of $480.94 and the following activity. Free Essay Answer: Answer: To calculate the daily balances and their sum, set up a chart like the one below that lists the activity in the account by dates and number of days Now to find the average daily balance: Here, the sum of the daily balances is$13,226.82
And days in billing cycle is 30
Therefore, the average daily balance for the month of January is
.
To find the periodic rate:
Divide the annual percentage rate by 12 to find the monthly or periodic interest rate.
So, the annual percentage rate is 1.08%
b) To find financial charge:
Calculate the finance charge by multiplying the previous month's balance by the periodic interest rate from step 1.
Therefore, the finance charge is
c) To find the new balance:
Total all the purchases and cash advances for the month
All the purchases and cash advances for the month is
Total all the payments and credits for the month
Since, given that all the payments and credits for the months is $550 Use the following formula to determine the new balance: Therefore, the average daily balance is The new balance is The finance charge of charge in the month of September is . Tags Choose question tag George Bell bought an ultralight airplane for$29,200. He made a 15% down payment and financed the balance with payments of $579 per month for 60 months. a. What is the finance charge on George's loan? b. What is the total deferred payment price of the airplane? Essay Answer: Tags Choose question tag Write the formula for calculating the average daily balance of a revolving credit account. Essay Answer: Tags Choose question tag Daniel Noguera has a Bank of America revolving credit account with an annual percentage rate of 12% calculated on the previous month's balance. In April, the account had the following activity. a. What is the finance charge? b. What is Daniel's new balance? Essay Answer: Tags Choose question tag The interest rate of most lines of credit is tied to the movement of the _____ rate. Essay Answer: Tags Choose question tag The portion of the purchase price of an asset paid in a lump sum at the time of purchase is known as the _____ payment. Essay Answer: Tags Choose question tag A loan made for a specified number of equal monthly payments is known as a(n) _____ loan. Essay Answer: Tags Choose question tag Sound Blaster Recording Studio purchased a new digital recording console for$28,600. A down payment of $5,000 was made and the balance financed with monthly payments of$708 for 48 months. a. What is the finance charge on the loan? b. Use Table 13-1 to find what annual percentage rate was charged on the equipment loan.
Essay
Tags
Choose question tag
_____ credit is a loan arrangement in which there is no set number of payments.
Essay
Tags
Choose question tag
David Sporn bought a saddle from Linville Western Gear with a 9.3% addon interest installment loan. The purchase price of the saddle was $1,290. The loan required a 15% down payment and equal monthly payments for 24 months. a. What is the total deferred payment price of the saddle? b. What are David's monthly payments? Essay Answer: Tags Choose question tag The effective or true annual interest rate being charged for credit is known as the _____ _____ _____ and is abbreviated _____. Essay Answer: Tags Choose question tag Mel Arrandt has a Bank of America account with a 13% annual percentage rate calculated on the average daily balance. The billing date is the first day of each month, and the billing cycle is the number of days in that month. a. What is the average daily balance for September? b. What is the finance charge for September? c. What is Mel's new balance? Essay Answer: Tags Choose question tag Heather MacMaster's revolving credit account has an annual percentage rate of 16%. The previous month's balance was$345.40. During the current month, Heather's purchases and cash advances amounted to $215.39 and her payments and credits totaled$125.00. a. What is the monthly periodic rate of the account? b. What is the finance charge? c. What is Heather's new balance?
Essay
Tags
Choose question tag
Name the two most common methods used to calculate the finance charge of a revolving credit account.
Essay
Tags
Choose question tag
Loans that are backed by the borrower's "promise" to repay are known as _____ loans, whereas loans that are backed by a tangible asset are known as _____ loans.
Essay
Tags
Choose question tag
A pre-approved amount of open-end credit is known as a(n) _____ of credit.
Essay
Alpine Construction, Inc., has a $100,000 line of credit with the Bow Valley Bank. The annual percentage rate is the current prime rate plus 31 4%. The balance on June 1 was$52,900. On June 8, Alpine borrowed $30,600 to pay for a shipment of lumber and roofing materials and on June 18 borrowed another$12,300 for equipment repairs. On June 28, a \$35,000 payment was made on the account. The billing cycle for June has 30 days. The current prime rate is 73 4%. a. What is the finance charge on the account? b. What is Alpine's new balance?
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2022-08-10 11:27:13
|
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https://socratic.org/questions/how-do-you-solve-the-following-system-5x-8y-29-5x-9y-2
|
# How do you solve the following system: 5x+8y= -29, 5x-9y=-2 ?
Feb 13, 2016
The solution for the system of equations is:
$x = - \frac{277}{85}$
$y = - \frac{27}{17}$
#### Explanation:
$\textcolor{b l u e}{5 x} + 8 y = - 29$ ......equation $\left(1\right)$
$\textcolor{b l u e}{5 x} - 9 y = - 2$.........equation $\left(2\right)$
Solving by elimination
Subtracting equation $2$ from $1$
$\cancel{\textcolor{b l u e}{5 x}} + 8 y = - 29$
$- \cancel{\textcolor{b l u e}{5 x}} + 9 y = 2$
$17 y = - 27$
$y = - \frac{27}{17}$
Finding $x$ from equation $1$
$5 x + 8 y = - 29$
$5 x + 8 \times \left(- \frac{27}{17}\right) = - 29$
$5 x + \left(- \frac{216}{17}\right) = - 29$
$5 x = - 29 + \frac{216}{17}$
$5 x = - \frac{493}{17} + \frac{216}{17}$
$5 x = - \frac{277}{17}$
$x = - \frac{277}{17 \times 5}$
$x = - \frac{277}{85}$
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2020-02-24 03:04:13
|
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|
https://electronics.stackexchange.com/questions/327527/lattice-icecube2-error-synplify-pro-321/472674
|
# Lattice iCEcube2, error synplify_pro 321
I just made a fresh install of iCEcube2, first time i'm using it, and whatever design files I use I've got this error when I try to synthetize :
/opt/iCEcube2.2017.01/synpbase/bin/synplify_pro: 321:
/opt/iCEcube2.2017.01/synpbase/bin/config/execute: Syntax error: "(" unexpected (expecting ";;")
I'm using ubuntu 16. No idea what happens, and I don't know what I could do knowing that it's a fresh install. I tried to launch the software in sudo with same result.
Thanks
the issue here is that the "execute" script that iceCube2 is trying to run, is using as "shell" /bin/sh linked to /bin/dash, in Ubuntu, instead of /bin/bash.
the best workaround i found, is to change the shell from /bin/sh to /bin/bash in the first line of all the shell script in these directories..
/opt/iCEcube2.2017.01/synpbase/bin/ /opt/iCEcube2.2017.01/synpbase/bin/config/
• Answering poorly received questions is kind of discouraged. You should also take your time to capitalize properly. – Oskar Skog Sep 29 '17 at 12:59
• Ty for your answer :) – Cart3sianBear Oct 16 '17 at 16:23
Thanks to Andrea Venturi for tip on the root cause. Indeed /bin/sh has very special meaming in Unix. For regular scripting /bin/bash should be used. Here how to fix all iCEcube2 scripts:
$cd <iCEcube2.2017.01>/synpbase/bin$ find . -type f -exec sed -i 's/\/sh/\/bash/g' {} \;
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2021-01-16 02:46:02
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http://crypto.stackexchange.com/questions?page=4&sort=faq&pagesize=15
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# All Questions
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### “SHA-256” vs “any 256 bits of SHA-512”, which is more secure?
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### Does the generator size matter in Diffie-Hellman?
For the Diffie-Hellman protocol I've heard that the generator 3 is as safe as any other generator. Yet, 32-bit or 256-bit exponents are sometimes used as generators. What is the benefit of using ...
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2014-10-26 08:34:50
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http://msdn.microsoft.com/en-us/library/office/bb243326(v=office.12)
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The topic you requested is included in another documentation set. For convenience, it's displayed below. Choose Switch to see the topic in its original location.
# WdRelativeHorizontalSize Enumeration
Office 2007
Specifies the relative width of a shape using the value specified in the WidthRelative property for a Shape or ShapeRange object.
Version Information
NameValueDescription
wdRelativeHorizontalSizeInnerMarginArea4Width is relative to the size of the inside margin—to the size of the left margin for odd pages, and to the size of the right margin for even pages.
wdRelativeHorizontalSizeLeftMarginArea2Width is relative to the size of the left margin.
wdRelativeHorizontalSizeMargin0Width is relative to the space between the left margin and the right margin.
wdRelativeHorizontalSizeOuterMarginArea5Width is relative to the size of the outside margin—to the size of the right margin for odd pages, and to the size of the left margin for even pages.
wdRelativeHorizontalSizePage1Width is relative to the width of the page.
wdRelativeHorizontalSizeRightMarginArea3Width is relative to the width of the right margin.
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2014-07-28 09:40:22
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https://groupprops.subwiki.org/wiki/Conjugacy_class_size_statistics_of_a_finite_group
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Conjugacy class size statistics of a finite group
Definition
Let $G$ be a finite group. The conjugacy class size statistics of $G$ is a function $f:\mathbb{N} \to \mathbb{N}_0$ that outputs, for each $d$, the number of conjugacy classes of $G$ of size $d$. Note that since size of conjugacy class divides order of group, the function is nonzero only on (some) divisors of the order of $G$.
The conjugacy class size statistics carry more information than the conjugacy class size set of a finite group, which is simply the set of sizes of the conjugacy classes in $G$.
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2018-06-21 21:54:17
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